LMFDB - Embedded newform 336.7.bh.e.145.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,7,Mod(145,336)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("336.145");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	336.bh (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$77.2981720963$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 82x^{6} - 165x^{5} + 5606x^{4} - 7807x^{3} + 102447x^{2} + 132594x + 1162084 $ x^8 - x^7 + 82x^6 - 165x^5 + 5606x^4 - 7807x^3 + 102447x^2 + 132594x + 1162084
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.4
Root			$3.42459 + 5.93157i$ of defining polynomial
Character	$\chi$	$=$	336.145
Dual form			336.7.bh.e.241.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.5000 - 7.79423i) q^{3} +(201.438 + 116.300i) q^{5} +(-184.358 - 289.242i) q^{7} +(121.500 - 210.444i) q^{9} +O(q^{10})$	\(q+(13.5000 - 7.79423i) q^{3} +(201.438 + 116.300i) q^{5} +(-184.358 - 289.242i) q^{7} +(121.500 - 210.444i) q^{9} +(460.090 + 796.899i) q^{11} -536.864i q^{13} +3625.88 q^{15} +(2610.93 - 1507.42i) q^{17} +(-7374.63 - 4257.75i) q^{19} +(-4743.25 - 2467.85i) q^{21} +(10535.7 - 18248.4i) q^{23} +(19239.0 + 33322.9i) q^{25} -3788.00i q^{27} -3910.02 q^{29} +(19930.8 - 11507.1i) q^{31} +(12422.4 + 7172.09i) q^{33} +(-3497.70 - 79705.3i) q^{35} +(22097.3 - 38273.6i) q^{37} +(-4184.44 - 7247.66i) q^{39} -56637.1i q^{41} +120736. q^{43} +(48949.4 - 28261.0i) q^{45} +(-131321. - 75818.3i) q^{47} +(-49673.4 + 106648. i) q^{49} +(23498.4 - 40700.4i) q^{51} +(115683. + 200370. i) q^{53} +214034. i q^{55} -132743. q^{57} +(7463.43 - 4309.02i) q^{59} +(68573.4 + 39590.9i) q^{61} +(-83268.9 + 3654.08i) q^{63} +(62437.4 - 108145. i) q^{65} +(156755. + 271508. i) q^{67} -328471. i q^{69} +212805. q^{71} +(-198776. + 114764. i) q^{73} +(519453. + 299906. i) q^{75} +(145676. - 279992. i) q^{77} +(-379148. + 656704. i) q^{79} +(-29524.5 - 51137.9i) q^{81} -161117. i q^{83} +701254. q^{85} +(-52785.2 + 30475.6i) q^{87} +(-486510. - 280886. i) q^{89} +(-155284. + 98975.1i) q^{91} +(179377. - 310691. i) q^{93} +(-990354. - 1.71534e6i) q^{95} -199575. i q^{97} +223604. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 108 q^{3} - 42 q^{5} + 92 q^{7} + 972 q^{9}+O(q^{10}) $ 8 * q + 108 * q^3 - 42 * q^5 + 92 * q^7 + 972 * q^9	$ 8 q + 108 q^{3} - 42 q^{5} + 92 q^{7} + 972 q^{9} - 126 q^{11} - 756 q^{15} - 2532 q^{17} - 19998 q^{19} - 5886 q^{21} + 15648 q^{23} + 42698 q^{25} + 129468 q^{29} + 42096 q^{31} - 3402 q^{33} - 73524 q^{35} + 9866 q^{37} - 29106 q^{39} + 71764 q^{43} - 10206 q^{45} - 86988 q^{47} + 314588 q^{49} - 22788 q^{51} + 391710 q^{53} - 359964 q^{57} + 553434 q^{59} - 1009104 q^{61} - 181278 q^{63} + 589452 q^{65} - 229762 q^{67} - 208488 q^{71} - 1249290 q^{73} + 1152846 q^{75} + 1009566 q^{77} - 693808 q^{79} - 236196 q^{81} - 1302600 q^{85} + 1747818 q^{87} - 1414692 q^{89} + 766410 q^{91} + 378864 q^{93} - 3047568 q^{95} - 61236 q^{99}+O(q^{100}) $ 8 * q + 108 * q^3 - 42 * q^5 + 92 * q^7 + 972 * q^9 - 126 * q^11 - 756 * q^15 - 2532 * q^17 - 19998 * q^19 - 5886 * q^21 + 15648 * q^23 + 42698 * q^25 + 129468 * q^29 + 42096 * q^31 - 3402 * q^33 - 73524 * q^35 + 9866 * q^37 - 29106 * q^39 + 71764 * q^43 - 10206 * q^45 - 86988 * q^47 + 314588 * q^49 - 22788 * q^51 + 391710 * q^53 - 359964 * q^57 + 553434 * q^59 - 1009104 * q^61 - 181278 * q^63 + 589452 * q^65 - 229762 * q^67 - 208488 * q^71 - 1249290 * q^73 + 1152846 * q^75 + 1009566 * q^77 - 693808 * q^79 - 236196 * q^81 - 1302600 * q^85 + 1747818 * q^87 - 1414692 * q^89 + 766410 * q^91 + 378864 * q^93 - 3047568 * q^95 - 61236 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	13.5000	−	7.79423i	0.500000	−	0.288675i
$3$	13.5000	−	7.79423i	0.500000	−	0.288675i
$4$		0			0
$5$	201.438	+	116.300i	1.61150	+	0.930402i	0.989022	+	0.147768i	$0.0472088\pi$
$5$	201.438	+	116.300i	1.61150	+	0.930402i	0.622482	+	0.782634i	$0.286124\pi$
$6$		0			0
$7$	−184.358	−	289.242i	−0.537486	−	0.843272i
$7$	−184.358	−	289.242i	−0.537486	−	0.843272i
$8$		0			0
$9$	121.500	−	210.444i	0.166667	−	0.288675i
$10$		0			0
$11$	460.090	+	796.899i	0.345672	+	0.598722i	0.985476	−	0.169817i	$-0.0543175\pi$
$11$	460.090	+	796.899i	0.345672	+	0.598722i	−0.639803	+	0.768539i	$0.720984\pi$
$12$		0			0
$13$		−	536.864i		−	0.244362i	−0.992508	−	0.122181i	$-0.961011\pi$
$13$		−	536.864i		−	0.244362i	0.992508	−	0.122181i	$-0.0389889\pi$
$14$		0			0
$15$	3625.88			1.07434
$16$		0			0
$17$	2610.93	−	1507.42i	0.531433	−	0.306823i	−0.210167	−	0.977666i	$-0.567401\pi$
$17$	2610.93	−	1507.42i	0.531433	−	0.306823i	0.741600	+	0.670843i	$0.234067\pi$
$18$		0			0
$19$	−7374.63	−	4257.75i	−1.07518	−	0.620753i	−0.145585	−	0.989346i	$-0.546506\pi$
$19$	−7374.63	−	4257.75i	−1.07518	−	0.620753i	−0.929591	+	0.368592i	$0.879840\pi$
$20$		0			0
$21$	−4743.25	−	2467.85i	−0.512175	−	0.266477i
$22$		0			0
$23$	10535.7	−	18248.4i	0.865924	−	1.49982i	−0.000202610	−	1.00000i	$-0.500064\pi$
$23$	10535.7	−	18248.4i	0.865924	−	1.49982i	0.866127	−	0.499825i	$-0.166602\pi$
$24$		0			0
$25$	19239.0	+	33322.9i	1.23130	+	2.13267i
$26$		0			0
$27$		−	3788.00i		−	0.192450i
$28$		0			0
$29$	−3910.02			−0.160319			−0.0801594	−	0.996782i	$-0.525543\pi$
$29$	−3910.02			−0.160319			−0.0801594	+	0.996782i	$0.525543\pi$
$30$		0			0
$31$	19930.8	−	11507.1i	0.669021	−	0.386260i	−0.126684	−	0.991943i	$-0.540434\pi$
$31$	19930.8	−	11507.1i	0.669021	−	0.386260i	0.795706	+	0.605683i	$0.207100\pi$
$32$		0			0
$33$	12422.4	+	7172.09i	0.345672	+	0.199574i
$34$		0			0
$35$	−3497.70	−	79705.3i	−0.0815790	−	1.85902i
$36$		0			0
$37$	22097.3	−	38273.6i	0.436249	−	0.755605i	−0.561148	−	0.827715i	$-0.689640\pi$
$37$	22097.3	−	38273.6i	0.436249	−	0.755605i	0.997397	+	0.0721108i	$0.0229735\pi$
$38$		0			0
$39$	−4184.44	−	7247.66i	−0.0705413	−	0.122181i
$40$		0			0
$41$		−	56637.1i		−	0.821768i	−0.911688	−	0.410884i	$-0.865220\pi$
$41$		−	56637.1i		−	0.821768i	0.911688	−	0.410884i	$-0.134780\pi$
$42$		0			0
$43$	120736.			1.51856			0.759280	−	0.650764i	$-0.225551\pi$
$43$	120736.			1.51856			0.759280	+	0.650764i	$0.225551\pi$
$44$		0			0
$45$	48949.4	−	28261.0i	0.537168	−	0.310134i
$46$		0			0
$47$	−131321.	−	75818.3i	−1.26486	−	0.730265i	−0.290845	−	0.956770i	$-0.593937\pi$
$47$	−131321.	−	75818.3i	−1.26486	−	0.730265i	−0.974010	+	0.226505i	$0.927270\pi$
$48$		0			0
$49$	−49673.4	+	106648.i	−0.422217	+	0.906495i
$50$		0			0
$51$	23498.4	−	40700.4i	0.177144	−	0.306823i
$52$		0			0
$53$	115683.	+	200370.i	0.777041	+	1.34587i	0.933640	+	0.358212i	$0.116613\pi$
$53$	115683.	+	200370.i	0.777041	+	1.34587i	−0.156600	+	0.987662i	$0.550053\pi$
$54$		0			0
$55$			214034.i			1.28646i
$56$		0			0
$57$	−132743.			−0.716784
$58$		0			0
$59$	7463.43	−	4309.02i	0.0363398	−	0.0209808i	−0.481720	−	0.876325i	$-0.659988\pi$
$59$	7463.43	−	4309.02i	0.0363398	−	0.0209808i	0.518060	+	0.855344i	$0.326654\pi$
$60$		0			0
$61$	68573.4	+	39590.9i	0.302111	+	0.174424i	0.643391	−	0.765538i	$-0.277527\pi$
$61$	68573.4	+	39590.9i	0.302111	+	0.174424i	−0.341280	+	0.939962i	$0.610860\pi$
$62$		0			0
$63$	−83268.9	+	3654.08i	−0.333013	+	0.0146136i
$64$		0			0
$65$	62437.4	−	108145.i	0.227355	−	0.393791i
$66$		0			0
$67$	156755.	+	271508.i	0.521192	+	0.902730i	0.999696	+	0.0246453i	$0.00784565\pi$
$67$	156755.	+	271508.i	0.521192	+	0.902730i	−0.478505	+	0.878085i	$0.658821\pi$
$68$		0			0
$69$		−	328471.i		−	0.999883i
$70$		0			0
$71$	212805.			0.594576			0.297288	−	0.954788i	$-0.403918\pi$
$71$	212805.			0.594576			0.297288	+	0.954788i	$0.403918\pi$
$72$		0			0
$73$	−198776.	+	114764.i	−0.510971	+	0.295009i	−0.733233	−	0.679978i	$-0.761989\pi$
$73$	−198776.	+	114764.i	−0.510971	+	0.295009i	0.222262	+	0.974987i	$0.428656\pi$
$74$		0			0
$75$	519453.	+	299906.i	1.23130	+	0.710889i
$76$		0			0
$77$	145676.	−	279992.i	0.319092	−	0.613301i
$78$		0			0
$79$	−379148.	+	656704.i	−0.769003	+	1.33195i	0.169101	+	0.985599i	$0.445914\pi$
$79$	−379148.	+	656704.i	−0.769003	+	1.33195i	−0.938104	+	0.346353i	$0.887420\pi$
$80$		0			0
$81$	−29524.5	−	51137.9i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	161117.i		−	0.281779i	−0.990025	−	0.140889i	$-0.955004\pi$
$83$		−	161117.i		−	0.281779i	0.990025	−	0.140889i	$-0.0449962\pi$
$84$		0			0
$85$	701254.			1.14187
$86$		0			0
$87$	−52785.2	+	30475.6i	−0.0801594	+	0.0462801i
$88$		0			0
$89$	−486510.	−	280886.i	−0.690115	−	0.398438i	0.113540	−	0.993533i	$-0.463781\pi$
$89$	−486510.	−	280886.i	−0.690115	−	0.398438i	−0.803655	+	0.595095i	$0.797114\pi$
$90$		0			0
$91$	−155284.	+	98975.1i	−0.206064	+	0.131341i
$92$		0			0
$93$	179377.	−	310691.i	0.223007	−	0.386260i
$94$		0			0
$95$	−990354.	−	1.71534e6i	−1.15510	−	2.00069i
$96$		0			0
$97$		−	199575.i		−	0.218670i	−0.994005	−	0.109335i	$-0.965128\pi$
$97$		−	199575.i		−	0.218670i	0.994005	−	0.109335i	$-0.0348722\pi$
$98$		0			0
$99$	223604.			0.230448
$100$		0			0
$101$	958861.	−	553599.i	0.930661	−	0.537318i	0.0436406	−	0.999047i	$-0.486104\pi$
$101$	958861.	−	553599.i	0.930661	−	0.537318i	0.887021	+	0.461730i	$0.152771\pi$
$102$		0			0
$103$	1.19877e6	+	692111.i	1.09705	+	0.633380i	0.935443	−	0.353476i	$-0.115001\pi$
$103$	1.19877e6	+	692111.i	1.09705	+	0.633380i	0.161602	+	0.986856i	$0.448334\pi$
$104$		0			0
$105$	−668460.	−	1.04876e6i	−0.577441	−	0.905958i
$106$		0			0
$107$	786950.	−	1.36304e6i	0.642386	−	1.11264i	−0.342513	−	0.939513i	$-0.611278\pi$
$107$	786950.	−	1.36304e6i	0.642386	−	1.11264i	0.984899	−	0.173131i	$-0.0553885\pi$
$108$		0			0
$109$	−1.05699e6	−	1.83076e6i	−0.816191	−	1.41368i	−0.908469	−	0.417952i	$-0.862748\pi$
$109$	−1.05699e6	−	1.83076e6i	−0.816191	−	1.41368i	0.0922779	−	0.995733i	$-0.470585\pi$
$110$		0			0
$111$		−	688926.i		−	0.503736i
$112$		0			0
$113$	2.61150e6			1.80990			0.904951	−	0.425516i	$-0.139907\pi$
$113$	2.61150e6			1.80990			0.904951	+	0.425516i	$0.139907\pi$
$114$		0			0
$115$	4.24458e6	−	2.45061e6i	2.79088	−	1.61132i
$116$		0			0
$117$	−112980.	−	65229.0i	−0.0705413	−	0.0407271i
$118$		0			0
$119$	−917356.	−	477287.i	−0.544373	−	0.283230i
$120$		0			0
$121$	462415.	−	800926.i	0.261021	−	0.452102i
$122$		0			0
$123$	−441443.	−	764601.i	−0.237224	−	0.410884i
$124$		0			0
$125$			5.31562e6i			2.72160i
$126$		0			0
$127$	−2.99941e6			−1.46428			−0.732141	−	0.681154i	$-0.761479\pi$
$127$	−2.99941e6			−1.46428			−0.732141	+	0.681154i	$0.761479\pi$
$128$		0			0
$129$	1.62994e6	−	941046.i	0.759280	−	0.438371i
$130$		0			0
$131$	36359.2	+	20992.0i	0.0161734	+	0.00933769i	0.508065	−	0.861319i	$-0.330361\pi$
$131$	36359.2	+	20992.0i	0.0161734	+	0.00933769i	−0.491892	+	0.870656i	$0.663694\pi$
$132$		0			0
$133$	128051.	+	2.91801e6i	0.0544286	+	1.24031i
$134$		0			0
$135$	440545.	−	763046.i	0.179056	−	0.310134i
$136$		0			0
$137$	1.40463e6	+	2.43290e6i	0.546263	+	0.946155i	0.998526	+	0.0542704i	$0.0172833\pi$
$137$	1.40463e6	+	2.43290e6i	0.546263	+	0.946155i	−0.452264	+	0.891884i	$0.649383\pi$
$138$		0			0
$139$		−	224325.i		−	0.0835281i	−0.999128	−	0.0417640i	$-0.986702\pi$
$139$		−	224325.i		−	0.0835281i	0.999128	−	0.0417640i	$-0.0132978\pi$
$140$		0			0
$141$	−2.36378e6			−0.843237
$142$		0			0
$143$	427827.	−	247006.i	0.146305	−	0.0844693i
$144$		0			0
$145$	−787626.	−	454736.i	−0.258354	−	0.149161i
$146$		0			0
$147$	160650.	+	1.82692e6i	0.0505743	+	0.575131i
$148$		0			0
$149$	2.03342e6	−	3.52199e6i	0.614707	−	1.06470i	−0.375729	−	0.926730i	$-0.622608\pi$
$149$	2.03342e6	−	3.52199e6i	0.614707	−	1.06470i	0.990436	−	0.137974i	$-0.0440591\pi$
$150$		0			0
$151$	−562185.	−	973733.i	−0.163286	−	0.282819i	0.772759	−	0.634699i	$-0.218876\pi$
$151$	−562185.	−	973733.i	−0.163286	−	0.282819i	−0.936045	+	0.351880i	$0.885543\pi$
$152$		0			0
$153$		−	732607.i		−	0.204549i
$154$		0			0
$155$	5.35310e6			1.43751
$156$		0			0
$157$	5.12354e6	−	2.95807e6i	1.32395	−	0.764382i	0.339592	−	0.940573i	$-0.389711\pi$
$157$	5.12354e6	−	2.95807e6i	1.32395	−	0.764382i	0.984356	+	0.176191i	$0.0563778\pi$
$158$		0			0
$159$	3.12345e6	+	1.80333e6i	0.777041	+	0.448625i
$160$		0			0
$161$	−7.22054e6	+	316858.i	−1.73018	+	0.0759255i
$162$		0			0
$163$	−2.01094e6	+	3.48305e6i	−0.464340	+	0.804260i	−0.999171	−	0.0406983i	$-0.987042\pi$
$163$	−2.01094e6	+	3.48305e6i	−0.464340	+	0.804260i	0.534832	+	0.844959i	$0.320375\pi$
$164$		0			0
$165$	1.66823e6	+	2.88946e6i	0.371368	+	0.643229i
$166$		0			0
$167$			5.97767e6i			1.28346i	0.766930	+	0.641730i	$0.221783\pi$
$167$			5.97767e6i			1.28346i	−0.766930	+	0.641730i	$0.778217\pi$
$168$		0			0
$169$	4.53859e6			0.940287
$170$		0			0
$171$	−1.79204e6	+	1.03463e6i	−0.358392	+	0.206918i
$172$		0			0
$173$	1.81172e6	+	1.04600e6i	0.349907	+	0.202019i	0.664644	−	0.747160i	$-0.268583\pi$
$173$	1.81172e6	+	1.04600e6i	0.349907	+	0.202019i	−0.314737	+	0.949179i	$0.601916\pi$
$174$		0			0
$175$	6.09154e6	−	1.17081e7i	1.13661	−	2.18460i
$176$		0			0
$177$	67170.9	−	116343.i	0.0121133	−	0.0209808i
$178$		0			0
$179$	−2.05976e6	−	3.56761e6i	−0.359135	−	0.622040i	0.628682	−	0.777663i	$-0.283595\pi$
$179$	−2.05976e6	−	3.56761e6i	−0.359135	−	0.622040i	−0.987817	+	0.155623i	$0.950262\pi$
$180$		0			0
$181$		−	651520.i		−	0.109873i	−0.998490	−	0.0549366i	$-0.982504\pi$
$181$		−	651520.i		−	0.109873i	0.998490	−	0.0549366i	$-0.0174957\pi$
$182$		0			0
$183$	1.23432e6			0.201407
$184$		0			0
$185$	8.90247e6	−	5.13984e6i	1.40603	−	0.811773i
$186$		0			0
$187$	2.40253e6	+	1.38710e6i	0.367403	+	0.212120i
$188$		0			0
$189$	−1.09565e6	+	698347.i	−0.162288	+	0.103439i
$190$		0			0
$191$	−3.48252e6	+	6.03189e6i	−0.499796	+	0.865673i	−1.00000		0.000235235i	$-0.999925\pi$
$191$	−3.48252e6	+	6.03189e6i	−0.499796	+	0.865673i	0.500204	+	0.865908i	$0.333258\pi$
$192$		0			0
$193$	−6.65162e6	−	1.15209e7i	−0.925242	−	1.60257i	−0.791172	−	0.611594i	$-0.790529\pi$
$193$	−6.65162e6	−	1.15209e7i	−0.925242	−	1.60257i	−0.134070	−	0.990972i	$-0.542805\pi$
$194$		0			0
$195$		−	1.94661e6i		−	0.262527i
$196$		0			0
$197$	−5.32260e6			−0.696186			−0.348093	−	0.937460i	$-0.613171\pi$
$197$	−5.32260e6			−0.696186			−0.348093	+	0.937460i	$0.613171\pi$
$198$		0			0
$199$	4.54482e6	−	2.62395e6i	0.576710	−	0.332963i	−0.183115	−	0.983091i	$-0.558618\pi$
$199$	4.54482e6	−	2.62395e6i	0.576710	−	0.332963i	0.759825	+	0.650128i	$0.225285\pi$
$200$		0			0
$201$	4.23239e6	+	2.44357e6i	0.521192	+	0.300910i
$202$		0			0
$203$	720842.	+	1.13094e6i	0.0861692	+	0.135192i
$204$		0			0
$205$	6.58691e6	−	1.14089e7i	0.764575	−	1.32428i
$206$		0			0
$207$	−2.56017e6	−	4.43435e6i	−0.288641	−	0.499942i
$208$		0			0
$209$		−	7.83579e6i		−	0.858309i
$210$		0			0
$211$	−522144.			−0.0555831			−0.0277916	−	0.999614i	$-0.508847\pi$
$211$	−522144.			−0.0555831			−0.0277916	+	0.999614i	$0.508847\pi$
$212$		0			0
$213$	2.87287e6	−	1.65865e6i	0.297288	−	0.171639i
$214$		0			0
$215$	2.43209e7	+	1.40417e7i	2.44717	+	1.41287i
$216$		0			0
$217$	−7.00273e6	−	3.64342e6i	−0.685312	−	0.356558i
$218$		0			0
$219$	−1.78899e6	+	3.09862e6i	−0.170324	+	0.295009i
$220$		0			0
$221$	−809280.	−	1.40171e6i	−0.0749760	−	0.129862i
$222$		0			0
$223$			1.02146e6i			0.0921096i	0.998939	+	0.0460548i	$0.0146649\pi$
$223$			1.02146e6i			0.0921096i	−0.998939	+	0.0460548i	$0.985335\pi$
$224$		0			0
$225$	9.35016e6			0.820864
$226$		0			0
$227$	−9.65010e6	+	5.57149e6i	−0.825001	+	0.476314i	−0.852138	−	0.523317i	$-0.824694\pi$
$227$	−9.65010e6	+	5.57149e6i	−0.825001	+	0.476314i	0.0271371	+	0.999632i	$0.491361\pi$
$228$		0			0
$229$	4.09488e6	+	2.36418e6i	0.340984	+	0.196867i	0.660707	−	0.750644i	$-0.270256\pi$
$229$	4.09488e6	+	2.36418e6i	0.340984	+	0.196867i	−0.319723	+	0.947511i	$0.603590\pi$
$230$		0			0
$231$	−215699.	−	4.91533e6i	−0.0174989	−	0.398764i
$232$		0			0
$233$	480558.	−	832352.i	0.0379908	−	0.0658020i	−0.846405	−	0.532540i	$-0.821238\pi$
$233$	480558.	−	832352.i	0.0379908	−	0.0658020i	0.884396	+	0.466738i	$0.154571\pi$
$234$		0			0
$235$	−1.76354e7	−	3.05454e7i	−1.35888	−	2.35365i
$236$		0			0
$237$			1.18207e7i			0.887968i
$238$		0			0
$239$	−1.35988e7			−0.996108			−0.498054	−	0.867146i	$-0.665952\pi$
$239$	−1.35988e7			−0.996108			−0.498054	+	0.867146i	$0.665952\pi$
$240$		0			0
$241$	−1.11183e7	+	6.41918e6i	−0.794308	+	0.458594i	−0.841477	−	0.540293i	$-0.818313\pi$
$241$	−1.11183e7	+	6.41918e6i	−0.794308	+	0.458594i	0.0471692	+	0.998887i	$0.484980\pi$
$242$		0			0
$243$	−797162.	−	460241.i	−0.0555556	−	0.0320750i
$244$		0			0
$245$	−2.24093e7	+	1.57060e7i	−1.52381	+	1.06799i
$246$		0			0
$247$	−2.28583e6	+	3.95918e6i	−0.151689	+	0.262733i
$248$		0			0
$249$	−1.25579e6	−	2.17509e6i	−0.0813425	−	0.140889i
$250$		0			0
$251$		−	1.03894e7i		−	0.657007i	−0.944503	−	0.328504i	$-0.893456\pi$
$251$		−	1.03894e7i		−	0.657007i	0.944503	−	0.328504i	$-0.106544\pi$
$252$		0			0
$253$	1.93895e7			1.19730
$254$		0			0
$255$	9.46693e6	−	5.46573e6i	0.570937	−	0.329631i
$256$		0			0
$257$	−1.65816e7	−	9.57337e6i	−0.976846	−	0.563982i	−0.0755293	−	0.997144i	$-0.524065\pi$
$257$	−1.65816e7	−	9.57337e6i	−0.976846	−	0.563982i	−0.901316	+	0.433162i	$0.857398\pi$
$258$		0			0
$259$	−1.51442e7	+	664571.i	−0.871658	+	0.0382509i
$260$		0			0
$261$	−475067.	+	822840.i	−0.0267198	+	0.0462801i
$262$		0			0
$263$	2.00250e6	+	3.46843e6i	0.110079	+	0.190663i	0.915802	−	0.401630i	$-0.131556\pi$
$263$	2.00250e6	+	3.46843e6i	0.110079	+	0.190663i	−0.805723	+	0.592293i	$0.798223\pi$
$264$		0			0
$265$			5.38161e7i			2.89184i
$266$		0			0
$267$	−8.75717e6			−0.460077
$268$		0			0
$269$	−2.73993e7	+	1.58190e7i	−1.40761	+	0.812685i	−0.995158	−	0.0982929i	$-0.968662\pi$
$269$	−2.73993e7	+	1.58190e7i	−1.40761	+	0.812685i	−0.412455	+	0.910978i	$0.635328\pi$
$270$		0			0
$271$	3.16572e7	+	1.82773e7i	1.59061	+	0.918341i	0.993202	+	0.116407i	$0.0371378\pi$
$271$	3.16572e7	+	1.82773e7i	1.59061	+	0.918341i	0.597413	+	0.801934i	$0.296196\pi$
$272$		0			0
$273$	−1.32490e6	+	2.54648e6i	−0.0651170	+	0.125156i
$274$		0			0
$275$	−1.77033e7	+	3.06631e7i	−0.851250	+	1.47441i
$276$		0			0
$277$	1.95609e7	+	3.38804e7i	0.920341	+	1.59408i	0.798888	+	0.601479i	$0.205422\pi$
$277$	1.95609e7	+	3.38804e7i	0.920341	+	1.59408i	0.121452	+	0.992597i	$0.461245\pi$
$278$		0			0
$279$		−	5.59243e6i		−	0.257506i
$280$		0			0
$281$	−480023.			−0.0216343			−0.0108172	−	0.999941i	$-0.503443\pi$
$281$	−480023.			−0.0216343			−0.0108172	+	0.999941i	$0.503443\pi$
$282$		0			0
$283$	−3.30947e7	+	1.91072e7i	−1.46015	+	0.843020i	−0.999018	−	0.0443090i	$-0.985891\pi$
$283$	−3.30947e7	+	1.91072e7i	−1.46015	+	0.843020i	−0.461136	+	0.887329i	$0.652558\pi$
$284$		0			0
$285$	−2.67396e7	−	1.54381e7i	−1.15510	−	0.666897i
$286$		0			0
$287$	−1.63819e7	+	1.04415e7i	−0.692975	+	0.441689i
$288$		0			0
$289$	−7.52415e6	+	1.30322e7i	−0.311719	+	0.539914i
$290$		0			0
$291$	−1.55553e6	−	2.69426e6i	−0.0631247	−	0.109335i
$292$		0			0
$293$			1.03603e7i			0.411878i	0.978565	+	0.205939i	$0.0660248\pi$
$293$			1.03603e7i			0.411878i	−0.978565	+	0.205939i	$0.933975\pi$
$294$		0			0
$295$	2.00456e6			0.0780823
$296$		0			0
$297$	3.01865e6	−	1.74282e6i	0.115224	−	0.0665247i
$298$		0			0
$299$	−9.79689e6	−	5.65624e6i	−0.366501	−	0.211599i
$300$		0			0
$301$	−2.22587e7	−	3.49220e7i	−0.816206	−	1.28056i
$302$		0			0
$303$	8.62975e6	−	1.49472e7i	0.310220	−	0.537318i
$304$		0			0
$305$	9.20886e6	+	1.59502e7i	0.324569	+	0.562169i
$306$		0			0
$307$			8.06102e6i			0.278596i	0.990251	+	0.139298i	$0.0444846\pi$
$307$			8.06102e6i			0.278596i	−0.990251	+	0.139298i	$0.955515\pi$
$308$		0			0
$309$	2.15779e7			0.731364
$310$		0			0
$311$	2.21674e7	−	1.27983e7i	0.736942	−	0.425474i	−0.0840143	−	0.996465i	$-0.526774\pi$
$311$	2.21674e7	−	1.27983e7i	0.736942	−	0.425474i	0.820956	+	0.570991i	$0.193441\pi$
$312$		0			0
$313$	−3.03923e7	−	1.75470e7i	−0.991129	−	0.572229i	−0.0855178	−	0.996337i	$-0.527254\pi$
$313$	−3.03923e7	−	1.75470e7i	−0.991129	−	0.572229i	−0.905612	+	0.424108i	$0.860588\pi$
$314$		0			0
$315$	−1.71985e7	−	8.94812e6i	−0.550248	−	0.286286i
$316$		0			0
$317$	1.04219e7	−	1.80513e7i	0.327167	−	0.566669i	−0.654782	−	0.755818i	$-0.727239\pi$
$317$	1.04219e7	−	1.80513e7i	0.327167	−	0.566669i	0.981948	+	0.189149i	$0.0605728\pi$
$318$		0			0
$319$	−1.79896e6	−	3.11589e6i	−0.0554178	−	0.0959864i
$320$		0			0
$321$		−	2.45347e7i		−	0.741763i
$322$		0			0
$323$	−2.56729e7			−0.761845
$324$		0			0
$325$	1.78899e7	−	1.03287e7i	0.521144	−	0.300882i
$326$		0			0
$327$	−2.85388e7	−	1.64769e7i	−0.816191	−	0.471228i
$328$		0			0
$329$	2.28022e6	+	5.19613e7i	0.0640307	+	1.45912i
$330$		0			0
$331$	−1.10153e7	+	1.90791e7i	−0.303748	+	0.526107i	−0.976982	−	0.213323i	$-0.931571\pi$
$331$	−1.10153e7	+	1.90791e7i	−0.303748	+	0.526107i	0.673234	+	0.739430i	$0.264905\pi$
$332$		0			0
$333$	−5.36964e6	−	9.30049e6i	−0.145416	−	0.251868i
$334$		0			0
$335$			7.29227e7i			1.93967i
$336$		0			0
$337$	−5.41316e7			−1.41436			−0.707181	−	0.707032i	$-0.750034\pi$
$337$	−5.41316e7			−1.41436			−0.707181	+	0.707032i	$0.750034\pi$
$338$		0			0
$339$	3.52553e7	−	2.03546e7i	0.904951	−	0.522474i
$340$		0			0
$341$	1.83399e7	+	1.05886e7i	0.462524	+	0.267039i
$342$		0			0
$343$	4.00049e7	−	5.29379e6i	0.991358	−	0.131185i
$344$		0			0
$345$	3.82012e7	−	6.61664e7i	0.930293	−	1.61132i
$346$		0			0
$347$	4.14201e6	+	7.17416e6i	0.0991339	+	0.171705i	0.911326	−	0.411685i	$-0.135059\pi$
$347$	4.14201e6	+	7.17416e6i	0.0991339	+	0.171705i	−0.812192	+	0.583390i	$0.801726\pi$
$348$		0			0
$349$		−	7.43895e7i		−	1.74999i	−0.484133	−	0.874994i	$-0.660865\pi$
$349$		−	7.43895e7i		−	1.74999i	0.484133	−	0.874994i	$-0.339135\pi$
$350$		0			0
$351$	−2.03364e6			−0.0470276
$352$		0			0
$353$	3.81691e6	−	2.20369e6i	0.0867736	−	0.0500988i	−0.455985	−	0.889987i	$-0.650713\pi$
$353$	3.81691e6	−	2.20369e6i	0.0867736	−	0.0500988i	0.542759	+	0.839889i	$0.317380\pi$
$354$		0			0
$355$	4.28670e7	+	2.47493e7i	0.958161	+	0.553194i
$356$		0			0
$357$	−1.61044e7	+	706707.i	−0.353948	+	0.0155323i
$358$		0			0
$359$	−1.12174e7	+	1.94292e7i	−0.242443	+	0.419924i	−0.961410	−	0.275121i	$-0.911282\pi$
$359$	−1.12174e7	+	1.94292e7i	−0.242443	+	0.419924i	0.718966	+	0.695045i	$0.244616\pi$
$360$		0			0
$361$	1.27339e7	+	2.20557e7i	0.270669	+	0.468813i
$362$		0			0
$363$		−	1.44167e7i		−	0.301401i
$364$		0			0
$365$	−5.33882e7			−1.09791
$366$		0			0
$367$	−5.41637e7	+	3.12714e7i	−1.09575	+	0.632630i	−0.935101	−	0.354382i	$-0.884691\pi$
$367$	−5.41637e7	+	3.12714e7i	−1.09575	+	0.632630i	−0.160646	+	0.987012i	$0.551358\pi$
$368$		0			0
$369$	−1.19189e7	−	6.88141e6i	−0.237224	−	0.136961i
$370$		0			0
$371$	3.66282e7	−	7.04003e7i	0.717289	−	1.37865i
$372$		0			0
$373$	4.97129e6	−	8.61053e6i	0.0957950	−	0.165922i	−0.814145	−	0.580661i	$-0.802794\pi$
$373$	4.97129e6	−	8.61053e6i	0.0957950	−	0.165922i	0.909940	+	0.414740i	$0.136127\pi$
$374$		0			0
$375$	4.14312e7	+	7.17609e7i	0.785658	+	1.36080i
$376$		0			0
$377$			2.09915e6i			0.0391759i
$378$		0			0
$379$	1.56872e7			0.288157			0.144078	−	0.989566i	$-0.453978\pi$
$379$	1.56872e7			0.288157			0.144078	+	0.989566i	$0.453978\pi$
$380$		0			0
$381$	−4.04920e7	+	2.33781e7i	−0.732141	+	0.422702i
$382$		0			0
$383$	−5.32230e7	−	3.07283e7i	−0.947333	−	0.546943i	−0.0550816	−	0.998482i	$-0.517542\pi$
$383$	−5.32230e7	−	3.07283e7i	−0.947333	−	0.546943i	−0.892251	+	0.451539i	$0.850875\pi$
$384$		0			0
$385$	6.19078e7	−	3.94589e7i	1.08483	−	0.691453i
$386$		0			0
$387$	1.46695e7	−	2.54082e7i	0.253093	−	0.438371i
$388$		0			0
$389$	4.45518e7	+	7.71659e7i	0.756861	+	1.31092i	0.944444	+	0.328673i	$0.106601\pi$
$389$	4.45518e7	+	7.71659e7i	0.756861	+	1.31092i	−0.187583	+	0.982249i	$0.560065\pi$
$390$		0			0
$391$		−	6.35269e7i		−	1.06274i
$392$		0			0
$393$	654465.			0.0107822
$394$		0			0
$395$	−1.52750e8	+	8.81901e7i	−2.47850	+	1.43096i
$396$		0			0
$397$	5.73411e7	+	3.31059e7i	0.916421	+	0.529096i	0.882491	−	0.470329i	$-0.155865\pi$
$397$	5.73411e7	+	3.31059e7i	0.916421	+	0.529096i	0.0339293	+	0.999424i	$0.489198\pi$
$398$		0			0
$399$	2.44723e7	+	3.83950e7i	0.385262	+	0.604444i
$400$		0			0
$401$	−3.38142e7	+	5.85679e7i	−0.524404	+	0.908294i	0.475193	+	0.879882i	$0.342378\pi$
$401$	−3.38142e7	+	5.85679e7i	−0.524404	+	0.908294i	−0.999596	+	0.0284121i	$0.990955\pi$
$402$		0			0
$403$	−6.17773e6	−	1.07001e7i	−0.0943873	−	0.163484i
$404$		0			0
$405$		−	1.37348e7i		−	0.206756i
$406$		0			0
$407$	4.06670e7			0.603196
$408$		0			0
$409$	−4.54171e7	+	2.62216e7i	−0.663818	+	0.383256i	−0.793730	−	0.608270i	$-0.791864\pi$
$409$	−4.54171e7	+	2.62216e7i	−0.663818	+	0.383256i	0.129912	+	0.991526i	$0.458530\pi$
$410$		0			0
$411$	3.79251e7	+	2.18961e7i	0.546263	+	0.315385i
$412$		0			0
$413$	−2.62229e6	−	1.36434e6i	−0.0372247	−	0.0193675i
$414$		0			0
$415$	1.87380e7	−	3.24552e7i	0.262168	−	0.454088i
$416$		0			0
$417$	−1.74844e6	−	3.02838e6i	−0.0241125	−	0.0417640i
$418$		0			0
$419$		−	8.47286e6i		−	0.115183i	−0.998340	−	0.0575915i	$-0.981658\pi$
$419$		−	8.47286e6i		−	0.115183i	0.998340	−	0.0575915i	$-0.0183421\pi$
$420$		0			0
$421$	7.34469e6			0.0984299			0.0492149	−	0.998788i	$-0.484328\pi$
$421$	7.34469e6			0.0984299			0.0492149	+	0.998788i	$0.484328\pi$
$422$		0			0
$423$	−3.19110e7	+	1.84238e7i	−0.421618	+	0.243422i
$424$		0			0
$425$	1.00463e8	+	5.80026e7i	1.30870	+	0.755580i
$426$		0			0
$427$	−1.19069e6	−	2.71332e7i	−0.0152937	−	0.348512i
$428$		0			0
$429$	3.85044e6	−	6.66916e6i	0.0487684	−	0.0844693i
$430$		0			0
$431$	2.67489e7	+	4.63304e7i	0.334098	+	0.578675i	0.983311	−	0.181932i	$-0.0582350\pi$
$431$	2.67489e7	+	4.63304e7i	0.334098	+	0.578675i	−0.649213	+	0.760607i	$0.724902\pi$
$432$		0			0
$433$		−	8.43895e7i		−	1.03950i	−0.854318	−	0.519751i	$-0.826025\pi$
$433$		−	8.43895e7i		−	1.03950i	0.854318	−	0.519751i	$-0.173975\pi$
$434$		0			0
$435$	−1.41773e7			−0.172236
$436$		0			0
$437$	−1.55394e8	+	8.97167e7i	−1.86204	+	1.07505i
$438$		0			0
$439$	6.39114e6	+	3.68993e6i	0.0755414	+	0.0436138i	0.537295	−	0.843394i	$-0.319446\pi$
$439$	6.39114e6	+	3.68993e6i	0.0755414	+	0.0436138i	−0.461754	+	0.887008i	$0.652780\pi$
$440$		0			0
$441$	1.64082e7	+	2.34112e7i	0.191313	+	0.272966i
$442$		0			0
$443$	3.86184e7	−	6.68890e7i	0.444204	−	0.769384i	−0.553792	−	0.832655i	$-0.686820\pi$
$443$	3.86184e7	−	6.68890e7i	0.444204	−	0.769384i	0.997996	+	0.0632707i	$0.0201531\pi$
$444$		0			0
$445$	−6.53343e7	−	1.13162e8i	−0.741415	−	1.28417i
$446$		0			0
$447$		−	6.33958e7i		−	0.709803i
$448$		0			0
$449$	1.13928e7			0.125861			0.0629305	−	0.998018i	$-0.479955\pi$
$449$	1.13928e7			0.125861			0.0629305	+	0.998018i	$0.479955\pi$
$450$		0			0
$451$	4.51341e7	−	2.60582e7i	0.492011	−	0.284063i
$452$		0			0
$453$	−1.51790e7	−	8.76360e6i	−0.163286	−	0.0942731i
$454$		0			0
$455$	−4.27909e7	+	1.87779e6i	−0.454273	+	0.0199348i
$456$		0			0
$457$	−7.68556e7	+	1.33118e8i	−0.805243	+	1.39472i	0.110884	+	0.993833i	$0.464632\pi$
$457$	−7.68556e7	+	1.33118e8i	−0.805243	+	1.39472i	−0.916127	+	0.400888i	$0.868702\pi$
$458$		0			0
$459$	−5.71010e6	−	9.89019e6i	−0.0590481	−	0.102274i
$460$		0			0
$461$			8.01923e7i			0.818521i	0.912418	+	0.409261i	$0.134213\pi$
$461$			8.01923e7i			0.818521i	−0.912418	+	0.409261i	$0.865787\pi$
$462$		0			0
$463$	−3.37769e7			−0.340312			−0.170156	−	0.985417i	$-0.554427\pi$
$463$	−3.37769e7			−0.340312			−0.170156	+	0.985417i	$0.554427\pi$
$464$		0			0
$465$	7.22668e7	−	4.17233e7i	0.718754	−	0.414973i
$466$		0			0
$467$	−1.47999e8	−	8.54475e7i	−1.45315	−	0.838975i	−0.454488	−	0.890753i	$-0.650178\pi$
$467$	−1.47999e8	−	8.54475e7i	−1.45315	−	0.838975i	−0.998659	+	0.0517781i	$0.983511\pi$
$468$		0			0
$469$	4.96326e7	−	9.53949e7i	0.481114	−	0.924712i
$470$		0			0
$471$	4.61118e7	−	7.98680e7i	0.441316	−	0.764382i
$472$		0			0
$473$	5.55495e7	+	9.62146e7i	0.524925	+	0.909196i
$474$		0			0
$475$		−	3.27659e8i		−	3.05732i
$476$		0			0
$477$	5.62222e7			0.518027
$478$		0			0
$479$	−2.19520e7	+	1.26740e7i	−0.199741	+	0.115321i	−0.596535	−	0.802587i	$-0.703456\pi$
$479$	−2.19520e7	+	1.26740e7i	−0.199741	+	0.115321i	0.396794	+	0.917908i	$0.370123\pi$
$480$		0			0
$481$	−2.05477e7	−	1.18632e7i	−0.184641	−	0.106603i
$482$		0			0
$483$	−9.50076e7	+	6.05561e7i	−0.843174	+	0.537424i
$484$		0			0
$485$	2.32106e7	−	4.02019e7i	0.203451	−	0.352388i
$486$		0			0
$487$	8.99675e7	+	1.55828e8i	0.778930	+	1.34915i	0.932559	+	0.361018i	$0.117571\pi$
$487$	8.99675e7	+	1.55828e8i	0.778930	+	1.34915i	−0.153628	+	0.988129i	$0.549096\pi$
$488$		0			0
$489$			6.26949e7i			0.536174i
$490$		0			0
$491$	1.07343e8			0.906838			0.453419	−	0.891298i	$-0.350204\pi$
$491$	1.07343e8			0.906838			0.453419	+	0.891298i	$0.350204\pi$
$492$		0			0
$493$	−1.02088e7	+	5.89404e6i	−0.0851987	+	0.0491895i
$494$		0			0
$495$	4.50423e7	+	2.60052e7i	0.371368	+	0.214410i
$496$		0			0
$497$	−3.92323e7	−	6.15523e7i	−0.319576	−	0.501389i
$498$		0			0
$499$	−1.37396e7	+	2.37977e7i	−0.110579	+	0.191529i	−0.916004	−	0.401169i	$-0.868604\pi$
$499$	−1.37396e7	+	2.37977e7i	−0.110579	+	0.191529i	0.805425	+	0.592698i	$0.201937\pi$
$500$		0			0
$501$	4.65913e7	+	8.06985e7i	0.370503	+	0.641730i
$502$		0			0
$503$			1.17139e8i			0.920442i	0.887805	+	0.460221i	$0.152230\pi$
$503$			1.17139e8i			0.920442i	−0.887805	+	0.460221i	$0.847770\pi$
$504$		0			0
$505$	2.57535e8			1.99969
$506$		0			0
$507$	6.12709e7	−	3.53748e7i	0.470144	−	0.271437i
$508$		0			0
$509$	−8.18289e7	−	4.72439e7i	−0.620517	−	0.358255i	0.156554	−	0.987669i	$-0.449962\pi$
$509$	−8.18289e7	−	4.72439e7i	−0.620517	−	0.358255i	−0.777070	+	0.629414i	$0.783295\pi$
$510$		0			0
$511$	6.98405e7	+	3.63370e7i	0.523413	+	0.272324i
$512$		0			0
$513$	−1.61283e7	+	2.79351e7i	−0.119464	+	0.206918i
$514$		0			0
$515$	1.60985e8	+	2.78835e8i	1.17860	+	2.04139i
$516$		0			0
$517$		−	1.39533e8i		−	1.00973i
$518$		0			0
$519$	3.26110e7			0.233272
$520$		0			0
$521$	1.43465e8	−	8.28297e7i	1.01446	−	0.585697i	0.101964	−	0.994788i	$-0.467488\pi$
$521$	1.43465e8	−	8.28297e7i	1.01446	−	0.585697i	0.912494	+	0.409091i	$0.134154\pi$
$522$		0			0
$523$	−3.74398e7	−	2.16159e7i	−0.261715	−	0.151101i	0.363402	−	0.931633i	$-0.381615\pi$
$523$	−3.74398e7	−	2.16159e7i	−0.261715	−	0.151101i	−0.625117	+	0.780531i	$0.714949\pi$
$524$		0			0
$525$	−9.01961e6	−	2.05538e8i	−0.0623318	−	1.42041i
$526$		0			0
$527$	3.46920e7	−	6.00883e7i	0.237027	−	0.410542i
$528$		0			0
$529$	−1.47984e8	−	2.56316e8i	−0.999649	−	1.73144i
$530$		0			0
$531$		−	2.09418e6i		−	0.0139872i
$532$		0			0
$533$	−3.04064e7			−0.200809
$534$		0			0
$535$	3.17043e8	−	1.83045e8i	2.07041	−	1.19535i
$536$		0			0
$537$	−5.56135e7	−	3.21085e7i	−0.359135	−	0.207347i
$538$		0			0
$539$	−1.07842e8	+	9.48312e6i	−0.688687	+	0.0605599i
$540$		0			0
$541$	2.35853e7	−	4.08510e7i	0.148953	−	0.257995i	−0.781887	−	0.623420i	$-0.785743\pi$
$541$	2.35853e7	−	4.08510e7i	0.148953	−	0.257995i	0.930841	+	0.365425i	$0.119076\pi$
$542$		0			0
$543$	−5.07809e6	−	8.79552e6i	−0.0317177	−	0.0549366i
$544$		0			0
$545$		−	4.91714e8i		−	3.03754i
$546$		0			0
$547$	−2.08291e8			−1.27265			−0.636323	−	0.771423i	$-0.719546\pi$
$547$	−2.08291e8			−1.27265			−0.636323	+	0.771423i	$0.719546\pi$
$548$		0			0
$549$	1.66633e7	−	9.62059e6i	0.100704	−	0.0581413i
$550$		0			0
$551$	2.88349e7	+	1.66479e7i	0.172371	+	0.0995184i
$552$		0			0
$553$	2.59846e8	−	1.14028e7i	1.53653	−	0.0674273i
$554$		0			0
$555$	8.01222e7	−	1.38776e8i	0.468677	−	0.811773i
$556$		0			0
$557$	3.52394e6	+	6.10364e6i	0.0203921	+	0.0353202i	0.876041	−	0.482236i	$-0.160175\pi$
$557$	3.52394e6	+	6.10364e6i	0.0203921	+	0.0353202i	−0.855649	+	0.517556i	$0.826842\pi$
$558$		0			0
$559$		−	6.48189e7i		−	0.371079i
$560$		0			0
$561$	4.32455e7			0.244936
$562$		0			0
$563$	−1.60089e8	+	9.24276e7i	−0.897092	+	0.517936i	−0.876256	−	0.481847i	$-0.839966\pi$
$563$	−1.60089e8	+	9.24276e7i	−0.897092	+	0.517936i	−0.0208364	+	0.999783i	$0.506633\pi$
$564$		0			0
$565$	5.26056e8	+	3.03718e8i	2.91666	+	1.68394i
$566$		0			0
$567$	−9.34819e6	+	1.79674e7i	−0.0512836	+	0.0985681i
$568$		0			0
$569$	−1.29070e8	+	2.23556e8i	−0.700629	+	1.21353i	0.267616	+	0.963526i	$0.413764\pi$
$569$	−1.29070e8	+	2.23556e8i	−0.700629	+	1.21353i	−0.968246	+	0.250000i	$0.919569\pi$
$570$		0			0
$571$	−8.79110e7	−	1.52266e8i	−0.472209	−	0.817891i	0.527285	−	0.849689i	$-0.323210\pi$
$571$	−8.79110e7	−	1.52266e8i	−0.472209	−	0.817891i	−0.999494	+	0.0317978i	$0.989877\pi$
$572$		0			0
$573$			1.08574e8i			0.577115i
$574$		0			0
$575$	8.10785e8			4.26484
$576$		0			0
$577$	−1.14796e8	+	6.62773e7i	−0.597582	+	0.345014i	−0.768090	−	0.640342i	$-0.778793\pi$
$577$	−1.14796e8	+	6.62773e7i	−0.597582	+	0.345014i	0.170508	+	0.985356i	$0.445459\pi$
$578$		0			0
$579$	−1.79594e8	−	1.03688e8i	−0.925242	−	0.534189i
$580$		0			0
$581$	−4.66020e7	+	2.97033e7i	−0.237616	+	0.151452i
$582$		0			0
$583$	−1.06450e8	+	1.84376e8i	−0.537203	+	0.930463i
$584$		0			0
$585$	−1.51723e7	−	2.62792e7i	−0.0757851	−	0.131264i
$586$		0			0
$587$			2.64040e8i			1.30543i	0.757602	+	0.652717i	$0.226371\pi$
$587$			2.64040e8i			1.30543i	−0.757602	+	0.652717i	$0.773629\pi$
$588$		0			0
$589$	−1.95977e8			−0.959088
$590$		0			0
$591$	−7.18551e7	+	4.14855e7i	−0.348093	+	0.200971i
$592$		0			0
$593$	5.83488e7	+	3.36877e7i	0.279813	+	0.161550i	0.633339	−	0.773875i	$-0.281684\pi$
$593$	5.83488e7	+	3.36877e7i	0.279813	+	0.161550i	−0.353526	+	0.935425i	$0.615017\pi$
$594$		0			0
$595$	−1.29282e8	−	2.02832e8i	−0.613742	−	0.962912i
$596$		0			0
$597$	4.09034e7	−	7.08467e7i	0.192237	−	0.332963i
$598$		0			0
$599$	−3.28119e7	−	5.68319e7i	−0.152669	−	0.264430i	0.779539	−	0.626354i	$-0.215454\pi$
$599$	−3.28119e7	−	5.68319e7i	−0.152669	−	0.264430i	−0.932208	+	0.361923i	$0.882120\pi$
$600$		0			0
$601$		−	2.08034e8i		−	0.958320i	−0.877728	−	0.479160i	$-0.840941\pi$
$601$		−	2.08034e8i		−	0.958320i	0.877728	−	0.479160i	$-0.159059\pi$
$602$		0			0
$603$	7.61830e7			0.347461
$604$		0			0
$605$	1.86296e8	−	1.07558e8i	0.841273	−	0.485709i
$606$		0			0
$607$	3.48090e8	+	2.00970e8i	1.55642	+	0.898598i	0.997595	+	0.0693104i	$0.0220799\pi$
$607$	3.48090e8	+	2.00970e8i	1.55642	+	0.898598i	0.558822	+	0.829288i	$0.311253\pi$
$608$		0			0
$609$	1.85462e7	+	9.64932e6i	0.0821113	+	0.0427213i
$610$		0			0
$611$	−4.07041e7	+	7.05016e7i	−0.178449	+	0.309083i
$612$		0			0
$613$	−5.14774e7	−	8.91614e7i	−0.223478	−	0.387075i	0.732384	−	0.680892i	$-0.238408\pi$
$613$	−5.14774e7	−	8.91614e7i	−0.223478	−	0.387075i	−0.955862	+	0.293817i	$0.905074\pi$
$614$		0			0
$615$		−	2.05360e8i		−	0.882855i
$616$		0			0
$617$	−3.54994e8			−1.51135			−0.755676	−	0.654945i	$-0.772692\pi$
$617$	−3.54994e8			−1.51135			−0.755676	+	0.654945i	$0.772692\pi$
$618$		0			0
$619$	−8.87611e7	+	5.12462e7i	−0.374240	+	0.216068i	−0.675309	−	0.737535i	$-0.735990\pi$
$619$	−8.87611e7	+	5.12462e7i	−0.374240	+	0.216068i	0.301069	+	0.953602i	$0.402657\pi$
$620$		0			0
$621$	−6.91247e7	−	3.99092e7i	−0.288641	−	0.166647i
$622$		0			0
$623$	8.44759e6	+	1.92503e8i	0.0349356	+	0.796110i
$624$		0			0
$625$	−3.17599e8	+	5.50097e8i	−1.30089	+	2.25320i
$626$		0			0
$627$	−6.10739e7	−	1.05783e8i	−0.247772	−	0.429155i
$628$		0			0
$629$		−	1.33240e8i		−	0.535404i
$630$		0			0
$631$	5.10491e7			0.203189			0.101595	−	0.994826i	$-0.467606\pi$
$631$	5.10491e7			0.203189			0.101595	+	0.994826i	$0.467606\pi$
$632$		0			0
$633$	−7.04894e6	+	4.06971e6i	−0.0277916	+	0.0160455i
$634$		0			0
$635$	−6.04195e8	−	3.48832e8i	−2.35969	−	1.36237i
$636$		0			0
$637$	5.72556e7	+	2.66678e7i	0.221513	+	0.103174i
$638$		0			0
$639$	2.58558e7	−	4.47836e7i	0.0990959	−	0.171639i
$640$		0			0
$641$	1.19118e8	+	2.06319e8i	0.452276	+	0.783365i	0.998527	−	0.0542563i	$-0.0172788\pi$
$641$	1.19118e8	+	2.06319e8i	0.452276	+	0.783365i	−0.546251	+	0.837622i	$0.683945\pi$
$642$		0			0
$643$			2.38667e8i			0.897758i	0.893593	+	0.448879i	$0.148177\pi$
$643$			2.38667e8i			0.897758i	−0.893593	+	0.448879i	$0.851823\pi$
$644$		0			0
$645$	4.37775e8			1.63144
$646$		0			0
$647$	2.59300e8	−	1.49707e8i	0.957391	−	0.552750i	0.0620217	−	0.998075i	$-0.480245\pi$
$647$	2.59300e8	−	1.49707e8i	0.957391	−	0.552750i	0.895369	+	0.445325i	$0.146912\pi$
$648$		0			0
$649$	6.86770e6	+	3.96507e6i	0.0251233	+	0.0145050i
$650$		0			0
$651$	−1.22935e8	+	5.39473e6i	−0.445585	+	0.0195536i
$652$		0			0
$653$	1.90019e8	−	3.29122e8i	0.682429	−	1.18200i	−0.291809	−	0.956477i	$-0.594257\pi$
$653$	1.90019e8	−	3.29122e8i	0.682429	−	1.18200i	0.974237	−	0.225525i	$-0.0724096\pi$
$654$		0			0
$655$	4.88275e6	+	8.45716e6i	0.0173756	+	0.0300954i
$656$		0			0
$657$			5.57751e7i			0.196673i
$658$		0			0
$659$	−6.92456e7			−0.241956			−0.120978	−	0.992655i	$-0.538603\pi$
$659$	−6.92456e7			−0.241956			−0.120978	+	0.992655i	$0.538603\pi$
$660$		0			0
$661$	−4.39149e8	+	2.53543e8i	−1.52057	+	0.877904i	−0.520869	+	0.853637i	$0.674392\pi$
$661$	−4.39149e8	+	2.53543e8i	−1.52057	+	0.877904i	−0.999706	+	0.0242670i	$0.992275\pi$
$662$		0			0
$663$	−2.18506e7	−	1.26154e7i	−0.0749760	−	0.0432874i
$664$		0			0
$665$	−3.13571e8	+	6.02690e8i	−1.06628	+	2.04941i
$666$		0			0
$667$	−4.11948e7	+	7.13514e7i	−0.138824	+	0.240450i
$668$		0			0
$669$	7.96146e6	+	1.37896e7i	0.0265897	+	0.0460548i
$670$		0			0
$671$			7.28615e7i			0.241174i
$672$		0			0
$673$	3.94455e8			1.29405			0.647027	−	0.762467i	$-0.276012\pi$
$673$	3.94455e8			1.29405			0.647027	+	0.762467i	$0.276012\pi$
$674$		0			0
$675$	1.26227e8	−	7.28773e7i	0.410432	−	0.236963i
$676$		0			0
$677$	3.66631e8	+	2.11675e8i	1.18158	+	0.682186i	0.956380	−	0.292126i	$-0.0943627\pi$
$677$	3.66631e8	+	2.11675e8i	1.18158	+	0.682186i	0.225201	+	0.974312i	$0.427696\pi$
$678$		0			0
$679$	−5.77254e7	+	3.67931e7i	−0.184399	+	0.117532i
$680$		0			0
$681$	−8.68509e7	+	1.50430e8i	−0.275000	+	0.476314i
$682$		0			0
$683$	−9.89199e7	−	1.71334e8i	−0.310471	−	0.537752i	0.667993	−	0.744168i	$-0.267154\pi$
$683$	−9.89199e7	−	1.71334e8i	−0.310471	−	0.537752i	−0.978464	+	0.206415i	$0.933820\pi$
$684$		0			0
$685$			6.53437e8i			2.03298i
$686$		0			0
$687$	7.37078e7			0.227323
$688$		0			0
$689$	1.07571e8	−	6.21063e7i	0.328881	−	0.189879i
$690$		0			0
$691$	−9.88719e6	−	5.70837e6i	−0.0299667	−	0.0173013i	0.484942	−	0.874546i	$-0.338841\pi$
$691$	−9.88719e6	−	5.70837e6i	−0.0299667	−	0.0173013i	−0.514909	+	0.857245i	$0.672174\pi$
$692$		0			0
$693$	−4.12231e7	−	6.46757e7i	−0.123863	−	0.194331i
$694$		0			0
$695$	2.60890e7	−	4.51875e7i	0.0777147	−	0.134606i
$696$		0			0
$697$	−8.53760e7	−	1.47875e8i	−0.252137	−	0.436715i
$698$		0			0
$699$		−	1.49823e7i		−	0.0438680i
$700$		0			0
$701$	−2.88915e8			−0.838717			−0.419359	−	0.907821i	$-0.637745\pi$
$701$	−2.88915e8			−0.838717			−0.419359	+	0.907821i	$0.637745\pi$
$702$		0			0
$703$	−3.25919e8	+	1.88169e8i	−0.938088	+	0.541605i
$704$		0			0
$705$	−4.76155e8	−	2.74908e8i	−1.35888	−	0.784549i
$706$		0			0
$707$	−3.36898e8	−	1.75283e8i	−0.953323	−	0.496000i
$708$		0			0
$709$	−1.87067e8	+	3.24009e8i	−0.524877	+	0.909113i	0.474703	+	0.880146i	$0.342555\pi$
$709$	−1.87067e8	+	3.24009e8i	−0.524877	+	0.909113i	−0.999580	+	0.0289676i	$0.990778\pi$
$710$		0			0
$711$	9.21331e7	+	1.59579e8i	0.256334	+	0.443984i
$712$		0			0
$713$		−	4.84940e8i		−	1.33789i
$714$		0			0
$715$	1.14907e8			0.314362
$716$		0			0
$717$	−1.83584e8	+	1.05992e8i	−0.498054	+	0.287552i
$718$		0			0
$719$	3.32566e7	+	1.92007e7i	0.0894728	+	0.0516572i	0.544069	−	0.839041i	$-0.316883\pi$
$719$	3.32566e7	+	1.92007e7i	0.0894728	+	0.0516572i	−0.454596	+	0.890698i	$0.650216\pi$
$720$		0			0
$721$	−2.08151e7	−	4.74332e8i	−0.0555357	−	1.26554i
$722$		0			0
$723$	−1.00065e8	+	1.73318e8i	−0.264769	+	0.458594i
$724$		0			0
$725$	−7.52248e7	−	1.30293e8i	−0.197400	−	0.341907i
$726$		0			0
$727$		−	3.25563e8i		−	0.847290i	−0.905828	−	0.423645i	$-0.860750\pi$
$727$		−	3.25563e8i		−	0.847290i	0.905828	−	0.423645i	$-0.139250\pi$
$728$		0			0
$729$	−1.43489e7			−0.0370370
$730$		0			0
$731$	3.15234e8	−	1.82000e8i	0.807013	−	0.465929i
$732$		0			0
$733$	−1.37433e8	−	7.93472e7i	−0.348964	−	0.201474i	0.315265	−	0.949004i	$-0.397907\pi$
$733$	−1.37433e8	−	7.93472e7i	−0.348964	−	0.201474i	−0.664229	+	0.747529i	$0.731240\pi$
$734$		0			0
$735$	−1.80110e8	+	3.86694e8i	−0.453602	+	0.973880i
$736$		0			0
$737$	−1.44243e8	+	2.49836e8i	−0.360323	+	0.624098i
$738$		0			0
$739$	−1.17829e8	−	2.04087e8i	−0.291958	−	0.505686i	0.682315	−	0.731059i	$-0.260973\pi$
$739$	−1.17829e8	−	2.04087e8i	−0.291958	−	0.505686i	−0.974273	+	0.225373i	$0.927640\pi$
$740$		0			0
$741$			7.12652e7i			0.175155i
$742$		0			0
$743$	−1.12156e8			−0.273436			−0.136718	−	0.990610i	$-0.543655\pi$
$743$	−1.12156e8			−0.273436			−0.136718	+	0.990610i	$0.543655\pi$
$744$		0			0
$745$	8.19216e8	−	4.72975e8i	1.98121	−	1.14385i
$746$		0			0
$747$	−3.39062e7	−	1.95758e7i	−0.0813425	−	0.0469631i
$748$		0			0
$749$	−5.39329e8	+	2.36673e7i	−1.28354	+	0.0563253i
$750$		0			0
$751$	−1.19826e8	+	2.07544e8i	−0.282898	+	0.489994i	−0.972097	−	0.234578i	$-0.924629\pi$
$751$	−1.19826e8	+	2.07544e8i	−0.282898	+	0.489994i	0.689199	+	0.724572i	$0.257962\pi$
$752$		0			0
$753$	−8.09775e7	−	1.40257e8i	−0.189662	−	0.328504i
$754$		0			0
$755$		−	2.61529e8i		−	0.607686i
$756$		0			0
$757$	2.05564e7			0.0473871			0.0236936	−	0.999719i	$-0.492457\pi$
$757$	2.05564e7			0.0473871			0.0236936	+	0.999719i	$0.492457\pi$
$758$		0			0
$759$	2.61758e8	−	1.51126e8i	0.598652	−	0.345632i
$760$		0			0
$761$	−4.34822e8	−	2.51044e8i	−0.986637	−	0.569635i	−0.0823696	−	0.996602i	$-0.526249\pi$
$761$	−4.34822e8	−	2.51044e8i	−0.986637	−	0.569635i	−0.904267	+	0.426967i	$0.859582\pi$
$762$		0			0
$763$	−3.34670e8	+	6.43242e8i	−0.753430	+	1.44811i
$764$		0			0
$765$	8.52023e7	−	1.47575e8i	0.190312	−	0.329631i
$766$		0			0
$767$	−2.31336e6	−	4.00685e6i	−0.00512692	−	0.00888008i
$768$		0			0
$769$			1.32826e8i			0.292081i	0.989279	+	0.146041i	$0.0466530\pi$
$769$			1.32826e8i			0.292081i	−0.989279	+	0.146041i	$0.953347\pi$
$770$		0			0
$771$	−2.98468e8			−0.651230
$772$		0			0
$773$	6.75312e7	−	3.89892e7i	0.146206	−	0.0844122i	−0.425112	−	0.905141i	$-0.639765\pi$
$773$	6.75312e7	−	3.89892e7i	0.146206	−	0.0844122i	0.571319	+	0.820728i	$0.306432\pi$
$774$		0			0
$775$	7.66898e8	+	4.42769e8i	1.64753	+	0.951200i
$776$		0			0
$777$	−1.99266e8	+	1.27009e8i	−0.424787	+	0.270751i
$778$		0			0
$779$	−2.41146e8	+	4.17678e8i	−0.510115	+	0.883546i
$780$		0			0
$781$	9.79095e7	+	1.69584e8i	0.205528	+	0.355986i
$782$		0			0
$783$			1.48111e7i			0.0308534i
$784$		0			0
$785$	1.37610e9			2.84473
$786$		0			0
$787$	1.28535e8	−	7.42095e7i	0.263691	−	0.152242i	−0.362326	−	0.932051i	$-0.618017\pi$
$787$	1.28535e8	−	7.42095e7i	0.263691	−	0.152242i	0.626017	+	0.779809i	$0.284684\pi$
$788$		0			0
$789$	5.40675e7	+	3.12159e7i	0.110079	+	0.0635542i
$790$		0			0
$791$	−4.81451e8	−	7.55357e8i	−0.972798	−	1.52624i
$792$		0			0
$793$	2.12549e7	−	3.68146e7i	0.0426226	−	0.0738245i
$794$		0			0
$795$	4.19455e8	+	7.26517e8i	0.834802	+	1.44592i
$796$		0			0
$797$			3.50628e7i			0.0692583i	0.999400	+	0.0346292i	$0.0110250\pi$
$797$			3.50628e7i			0.0692583i	−0.999400	+	0.0346292i	$0.988975\pi$
$798$		0			0
$799$	−4.57160e8			−0.896248
$800$		0			0
$801$	−1.18222e8	+	6.82554e7i	−0.230038	+	0.132813i
$802$		0			0
$803$	−1.82910e8	−	1.05603e8i	−0.353257	−	0.203953i
$804$		0			0
$805$	−1.49134e9	−	7.75923e8i	−2.85884	−	1.48741i
$806$		0			0
$807$	−2.46594e8	+	4.27113e8i	−0.469204	+	0.812685i
$808$		0			0
$809$	−1.88273e8	−	3.26099e8i	−0.355585	−	0.615892i	0.631633	−	0.775268i	$-0.282385\pi$
$809$	−1.88273e8	−	3.26099e8i	−0.355585	−	0.615892i	−0.987218	+	0.159376i	$0.949052\pi$
$810$		0			0
$811$		−	5.58329e8i		−	1.04671i	−0.852114	−	0.523356i	$-0.824680\pi$
$811$		−	5.58329e8i		−	1.04671i	0.852114	−	0.523356i	$-0.175320\pi$
$812$		0			0
$813$	5.69830e8			1.06041
$814$		0			0
$815$	−8.10159e8	+	4.67745e8i	−1.49657	+	0.864046i
$816$		0			0
$817$	−8.90385e8	−	5.14064e8i	−1.63272	−	0.942652i
$818$		0			0
$819$	1.96175e6	+	4.47041e7i	0.00357101	+	0.0813758i
$820$		0			0
$821$	−1.85684e8	+	3.21615e8i	−0.335541	+	0.581174i	−0.983589	−	0.180426i	$-0.942253\pi$
$821$	−1.85684e8	+	3.21615e8i	−0.335541	+	0.581174i	0.648047	+	0.761600i	$0.275586\pi$
$822$		0			0
$823$	−4.07943e7	−	7.06579e7i	−0.0731813	−	0.126754i	0.827113	−	0.562036i	$-0.189982\pi$
$823$	−4.07943e7	−	7.06579e7i	−0.0731813	−	0.126754i	−0.900294	+	0.435282i	$0.856649\pi$
$824$		0			0
$825$			5.51936e8i			0.982939i
$826$		0			0
$827$	9.21411e8			1.62906			0.814529	−	0.580123i	$-0.196995\pi$
$827$	9.21411e8			1.62906			0.814529	+	0.580123i	$0.196995\pi$
$828$		0			0
$829$	−4.35666e7	+	2.51532e7i	−0.0764698	+	0.0441499i	−0.537747	−	0.843106i	$-0.680725\pi$
$829$	−4.35666e7	+	2.51532e7i	−0.0764698	+	0.0441499i	0.461278	+	0.887256i	$0.347391\pi$
$830$		0			0
$831$	5.28143e8	+	3.04924e8i	0.920341	+	0.531359i
$832$		0			0
$833$	3.10701e7	+	3.53330e8i	0.0537536	+	0.611287i
$834$		0			0
$835$	−6.95205e8	+	1.20413e9i	−1.19413	+	2.06830i
$836$		0			0
$837$	−4.35887e7	−	7.54978e7i	−0.0743357	−	0.128753i
$838$		0			0
$839$			1.18275e8i			0.200266i	0.994974	+	0.100133i	$0.0319269\pi$
$839$			1.18275e8i			0.200266i	−0.994974	+	0.100133i	$0.968073\pi$
$840$		0			0
$841$	−5.79535e8			−0.974298
$842$		0			0
$843$	−6.48031e6	+	3.74141e6i	−0.0108172	+	0.00624529i
$844$		0			0
$845$	9.14244e8	+	5.27839e8i	1.51528	+	0.874845i
$846$		0			0
$847$	−3.16912e8	+	1.39070e7i	−0.521540	+	0.0228867i
$848$		0			0
$849$	−2.97852e8	+	5.15895e8i	−0.486718	+	0.843020i
$850$		0			0
$851$	−4.65621e8	−	8.06479e8i	−0.755516	−	1.30859i
$852$		0			0
$853$		−	5.72140e8i		−	0.921840i	−0.887442	−	0.460920i	$-0.847520\pi$
$853$		−	5.72140e8i		−	0.921840i	0.887442	−	0.460920i	$-0.152480\pi$
$854$		0			0
$855$	−4.81312e8			−0.770067
$856$		0			0
$857$	3.38731e8	−	1.95567e8i	0.538162	−	0.310708i	−0.206172	−	0.978516i	$-0.566101\pi$
$857$	3.38731e8	−	1.95567e8i	0.538162	−	0.310708i	0.744334	+	0.667808i	$0.232767\pi$
$858$		0			0
$859$	6.62106e8	+	3.82267e8i	1.04460	+	0.603097i	0.921131	−	0.389252i	$-0.127266\pi$
$859$	6.62106e8	+	3.82267e8i	1.04460	+	0.603097i	0.123464	+	0.992349i	$0.460600\pi$
$860$		0			0
$861$	−1.39772e8	+	2.68644e8i	−0.218983	+	0.420889i
$862$		0			0
$863$	−6.28637e8	+	1.08883e9i	−0.978065	+	1.69406i	−0.308639	+	0.951179i	$0.599874\pi$
$863$	−6.28637e8	+	1.08883e9i	−0.978065	+	1.69406i	−0.669426	+	0.742879i	$0.733460\pi$
$864$		0			0
$865$	2.43300e8	+	4.21407e8i	0.375918	+	0.651109i
$866$		0			0
$867$			2.34580e8i			0.359943i
$868$		0			0
$869$	−6.97770e8			−1.06329
$870$		0			0
$871$	1.45763e8	−	8.41562e7i	0.220593	−	0.127360i
$872$		0			0
$873$	−4.19993e7	−	2.42483e7i	−0.0631247	−	0.0364451i
$874$		0			0
$875$	1.53750e9	−	9.79977e8i	2.29505	−	1.46282i
$876$		0			0
$877$	−3.91957e8	+	6.78889e8i	−0.581084	+	1.00647i	0.414267	+	0.910155i	$0.364038\pi$
$877$	−3.91957e8	+	6.78889e8i	−0.581084	+	1.00647i	−0.995351	+	0.0963122i	$0.969295\pi$
$878$		0			0
$879$	8.07504e7	+	1.39864e8i	0.118899	+	0.205939i
$880$		0			0
$881$			3.67078e7i			0.0536823i	0.999640	+	0.0268412i	$0.00854483\pi$
$881$			3.67078e7i			0.0536823i	−0.999640	+	0.0268412i	$0.991455\pi$
$882$		0			0
$883$	7.33429e8			1.06531			0.532655	−	0.846332i	$-0.321194\pi$
$883$	7.33429e8			1.06531			0.532655	+	0.846332i	$0.321194\pi$
$884$		0			0
$885$	2.70615e7	−	1.56240e7i	0.0390412	−	0.0225404i
$886$		0			0
$887$	−4.97098e8	−	2.87000e8i	−0.712313	−	0.411254i	0.0996037	−	0.995027i	$-0.468242\pi$
$887$	−4.97098e8	−	2.87000e8i	−0.712313	−	0.411254i	−0.811917	+	0.583773i	$0.801576\pi$
$888$		0			0
$889$	5.52965e8	+	8.67556e8i	0.787031	+	1.23479i
$890$		0			0
$891$	2.71679e7	−	4.70561e7i	0.0384080	−	0.0665247i
$892$		0			0
$893$	6.45630e8	+	1.11826e9i	0.906628	+	1.57033i
$894$		0			0
$895$		−	9.58203e8i		−	1.33656i
$896$		0			0
$897$	−1.76344e8			−0.244334
$898$		0			0
$899$	−7.79298e7	+	4.49928e7i	−0.107257	+	0.0619247i
$900$		0			0
$901$	6.04083e8	+	3.48767e8i	0.825890	+	0.476828i
$902$		0			0
$903$	−5.72682e8	−	2.97958e8i	−0.777769	−	0.404662i
$904$		0			0
$905$	7.57719e7	−	1.31241e8i	0.102226	−	0.177061i
$906$		0			0
$907$	−5.53301e8	−	9.58345e8i	−0.741548	−	1.28440i	−0.951790	−	0.306750i	$-0.900759\pi$
$907$	−5.53301e8	−	9.58345e8i	−0.741548	−	1.28440i	0.210242	−	0.977649i	$-0.432575\pi$
$908$		0			0
$909$		−	2.69049e8i		−	0.358212i
$910$		0			0
$911$	7.63025e8			1.00921			0.504607	−	0.863349i	$-0.331637\pi$
$911$	7.63025e8			1.00921			0.504607	+	0.863349i	$0.331637\pi$
$912$		0			0
$913$	1.28394e8	−	7.41285e7i	0.168707	−	0.0974031i
$914$		0			0
$915$	2.48639e8	+	1.43552e8i	0.324569	+	0.187390i
$916$		0			0
$917$	−631328.	−	1.43867e7i	−0.000818743	−	0.0186574i
$918$		0			0
$919$	−9.28030e7	+	1.60740e8i	−0.119568	+	0.207098i	−0.919597	−	0.392864i	$-0.871484\pi$
$919$	−9.28030e7	+	1.60740e8i	−0.119568	+	0.207098i	0.800028	+	0.599962i	$0.204818\pi$
$920$		0			0
$921$	6.28294e7	+	1.08824e8i	0.0804237	+	0.139298i
$922$		0			0
$923$		−	1.14247e8i		−	0.145292i
$924$		0			0
$925$	1.70052e9			2.14860
$926$		0			0
$927$	2.91301e8	−	1.68183e8i	0.365682	−	0.211127i
$928$		0			0
$929$	1.21183e7	+	6.99652e6i	0.0151146	+	0.00872639i	0.507538	−	0.861629i	$-0.330556\pi$
$929$	1.21183e7	+	6.99652e6i	0.0151146	+	0.00872639i	−0.492424	+	0.870356i	$0.663889\pi$
$930$		0			0
$931$	8.20404e8	−	5.74995e8i	1.01667	−	0.712549i
$932$		0			0
$933$	1.99507e8	−	3.45555e8i	0.245647	−	0.425474i
$934$		0			0
$935$	3.22640e8	+	5.58829e8i	0.394715	+	0.683666i
$936$		0			0
$937$		−	5.41893e8i		−	0.658711i	−0.944206	−	0.329355i	$-0.893169\pi$
$937$		−	5.41893e8i		−	0.658711i	0.944206	−	0.329355i	$-0.106831\pi$
$938$		0			0
$939$	−5.47061e8			−0.660753
$940$		0			0
$941$	−1.09435e9	+	6.31821e8i	−1.31337	+	0.758273i	−0.982652	−	0.185458i	$-0.940623\pi$
$941$	−1.09435e9	+	6.31821e8i	−1.31337	+	0.758273i	−0.330715	+	0.943731i	$0.607290\pi$
$942$		0			0
$943$	−1.03353e9	−	5.96711e8i	−1.23251	−	0.711589i
$944$		0			0
$945$	−3.01923e8	+	1.32493e7i	−0.357768	+	0.0156999i
$946$		0			0
$947$	1.41001e8	−	2.44221e8i	0.166024	−	0.287562i	−0.770994	−	0.636842i	$-0.780240\pi$
$947$	1.41001e8	−	2.44221e8i	0.166024	−	0.287562i	0.937019	+	0.349280i	$0.113574\pi$
$948$		0			0
$949$	6.16125e7	+	1.06716e8i	0.0720892	+	0.124862i
$950$		0			0
$951$		−	3.24923e8i		−	0.377780i
$952$		0			0
$953$	4.16665e8			0.481403			0.240701	−	0.970599i	$-0.422623\pi$
$953$	4.16665e8			0.481403			0.240701	+	0.970599i	$0.422623\pi$
$954$		0			0
$955$	−1.40302e9	+	8.10035e8i	−1.61085	+	0.930023i
$956$		0			0
$957$	−4.85719e7	−	2.80430e7i	−0.0554178	−	0.0319955i
$958$		0			0
$959$	4.44742e8	−	8.54804e8i	0.504257	−	0.969194i
$960$		0			0
$961$	−1.78927e8	+	3.09911e8i	−0.201607	+	0.349194i
$962$		0			0
$963$	−1.91229e8	−	3.31218e8i	−0.214129	−	0.370881i
$964$		0			0
$965$		−	3.09434e9i		−	3.44339i
$966$		0			0
$967$	3.24846e8			0.359251			0.179626	−	0.983735i	$-0.442511\pi$
$967$	3.24846e8			0.359251			0.179626	+	0.983735i	$0.442511\pi$
$968$		0			0
$969$	−3.46584e8	+	2.00100e8i	−0.380923	+	0.219926i
$970$		0			0
$971$	1.37705e9	+	7.95041e8i	1.50416	+	0.868425i	0.999988	+	0.00481836i	$0.00153374\pi$
$971$	1.37705e9	+	7.95041e8i	1.50416	+	0.868425i	0.504167	+	0.863606i	$0.331800\pi$
$972$		0			0
$973$	−6.48842e7	+	4.13560e7i	−0.0704369	+	0.0448952i
$974$		0			0
$975$	1.61009e8	−	2.78876e8i	0.173715	−	0.300882i
$976$		0			0
$977$	3.51811e8	+	6.09355e8i	0.377247	+	0.653411i	0.990661	−	0.136351i	$-0.0435374\pi$
$977$	3.51811e8	+	6.09355e8i	0.377247	+	0.653411i	−0.613413	+	0.789762i	$0.710204\pi$
$978$		0			0
$979$		−	5.16932e8i		−	0.550916i
$980$		0			0
$981$	−5.13698e8			−0.544128
$982$		0			0
$983$	−1.23385e8	+	7.12362e7i	−0.129897	+	0.0749963i	−0.563541	−	0.826088i	$-0.690561\pi$
$983$	−1.23385e8	+	7.12362e7i	−0.129897	+	0.0749963i	0.433643	+	0.901085i	$0.357228\pi$
$984$		0			0
$985$	−1.07217e9	−	6.19020e8i	−1.12191	−	0.647733i
$986$		0			0
$987$	4.35781e8	+	6.83705e8i	0.453228	+	0.711078i
$988$		0			0
$989$	1.27204e9	−	2.20324e9i	1.31496	−	2.27757i
$990$		0			0
$991$	2.14982e8	+	3.72360e8i	0.220893	+	0.382598i	0.955079	−	0.296350i	$-0.0957695\pi$
$991$	2.14982e8	+	3.72360e8i	0.220893	+	0.382598i	−0.734186	+	0.678948i	$0.762436\pi$
$992$		0			0
$993$			3.43424e8i			0.350738i
$994$		0			0
$995$	1.22067e9			1.23916
$996$		0			0
$997$	−5.64708e8	+	3.26034e8i	−0.569821	+	0.328986i	−0.757078	−	0.653325i	$-0.773374\pi$
$997$	−5.64708e8	+	3.26034e8i	−0.569821	+	0.328986i	0.187257	+	0.982311i	$0.440040\pi$
$998$		0			0
$999$	−1.44980e8	−	8.37044e7i	−0.145416	−	0.0839561i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.7.bh.e.145.4		8
4.3	odd	2		84.7.m.a.61.4	✓	8
7.3	odd	6	inner	336.7.bh.e.241.4		8
12.11	even	2		252.7.z.d.145.1		8
28.3	even	6		84.7.m.a.73.4	yes	8
28.11	odd	6		588.7.m.c.325.1		8
28.19	even	6		588.7.d.b.97.8		8
28.23	odd	6		588.7.d.b.97.1		8
28.27	even	2		588.7.m.c.313.1		8
84.59	odd	6		252.7.z.d.73.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.7.m.a.61.4	✓	8	4.3	odd	2
84.7.m.a.73.4	yes	8	28.3	even	6
252.7.z.d.73.1		8	84.59	odd	6
252.7.z.d.145.1		8	12.11	even	2
336.7.bh.e.145.4		8	1.1	even	1	trivial
336.7.bh.e.241.4		8	7.3	odd	6	inner
588.7.d.b.97.1		8	28.23	odd	6
588.7.d.b.97.8		8	28.19	even	6
588.7.m.c.313.1		8	28.27	even	2
588.7.m.c.325.1		8	28.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	336.bh (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(13.5000 - 7.79423i) q^{3} +(201.438 + 116.300i) q^{5} +(-184.358 - 289.242i) q^{7} +(121.500 - 210.444i) q^{9} +O(q^{10})\)	\(q+(13.5000 - 7.79423i) q^{3} +(201.438 + 116.300i) q^{5} +(-184.358 - 289.242i) q^{7} +(121.500 - 210.444i) q^{9} +(460.090 + 796.899i) q^{11} -536.864i q^{13} +3625.88 q^{15} +(2610.93 - 1507.42i) q^{17} +(-7374.63 - 4257.75i) q^{19} +(-4743.25 - 2467.85i) q^{21} +(10535.7 - 18248.4i) q^{23} +(19239.0 + 33322.9i) q^{25} -3788.00i q^{27} -3910.02 q^{29} +(19930.8 - 11507.1i) q^{31} +(12422.4 + 7172.09i) q^{33} +(-3497.70 - 79705.3i) q^{35} +(22097.3 - 38273.6i) q^{37} +(-4184.44 - 7247.66i) q^{39} -56637.1i q^{41} +120736. q^{43} +(48949.4 - 28261.0i) q^{45} +(-131321. - 75818.3i) q^{47} +(-49673.4 + 106648. i) q^{49} +(23498.4 - 40700.4i) q^{51} +(115683. + 200370. i) q^{53} +214034. i q^{55} -132743. q^{57} +(7463.43 - 4309.02i) q^{59} +(68573.4 + 39590.9i) q^{61} +(-83268.9 + 3654.08i) q^{63} +(62437.4 - 108145. i) q^{65} +(156755. + 271508. i) q^{67} -328471. i q^{69} +212805. q^{71} +(-198776. + 114764. i) q^{73} +(519453. + 299906. i) q^{75} +(145676. - 279992. i) q^{77} +(-379148. + 656704. i) q^{79} +(-29524.5 - 51137.9i) q^{81} -161117. i q^{83} +701254. q^{85} +(-52785.2 + 30475.6i) q^{87} +(-486510. - 280886. i) q^{89} +(-155284. + 98975.1i) q^{91} +(179377. - 310691. i) q^{93} +(-990354. - 1.71534e6i) q^{95} -199575. i q^{97} +223604. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 108 q^{3} - 42 q^{5} + 92 q^{7} + 972 q^{9}+O(q^{10}) \) 8 * q + 108 * q^3 - 42 * q^5 + 92 * q^7 + 972 * q^9	\( 8 q + 108 q^{3} - 42 q^{5} + 92 q^{7} + 972 q^{9} - 126 q^{11} - 756 q^{15} - 2532 q^{17} - 19998 q^{19} - 5886 q^{21} + 15648 q^{23} + 42698 q^{25} + 129468 q^{29} + 42096 q^{31} - 3402 q^{33} - 73524 q^{35} + 9866 q^{37} - 29106 q^{39} + 71764 q^{43} - 10206 q^{45} - 86988 q^{47} + 314588 q^{49} - 22788 q^{51} + 391710 q^{53} - 359964 q^{57} + 553434 q^{59} - 1009104 q^{61} - 181278 q^{63} + 589452 q^{65} - 229762 q^{67} - 208488 q^{71} - 1249290 q^{73} + 1152846 q^{75} + 1009566 q^{77} - 693808 q^{79} - 236196 q^{81} - 1302600 q^{85} + 1747818 q^{87} - 1414692 q^{89} + 766410 q^{91} + 378864 q^{93} - 3047568 q^{95} - 61236 q^{99}+O(q^{100}) \) 8 * q + 108 * q^3 - 42 * q^5 + 92 * q^7 + 972 * q^9 - 126 * q^11 - 756 * q^15 - 2532 * q^17 - 19998 * q^19 - 5886 * q^21 + 15648 * q^23 + 42698 * q^25 + 129468 * q^29 + 42096 * q^31 - 3402 * q^33 - 73524 * q^35 + 9866 * q^37 - 29106 * q^39 + 71764 * q^43 - 10206 * q^45 - 86988 * q^47 + 314588 * q^49 - 22788 * q^51 + 391710 * q^53 - 359964 * q^57 + 553434 * q^59 - 1009104 * q^61 - 181278 * q^63 + 589452 * q^65 - 229762 * q^67 - 208488 * q^71 - 1249290 * q^73 + 1152846 * q^75 + 1009566 * q^77 - 693808 * q^79 - 236196 * q^81 - 1302600 * q^85 + 1747818 * q^87 - 1414692 * q^89 + 766410 * q^91 + 378864 * q^93 - 3047568 * q^95 - 61236 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more