LMFDB - Embedded newform 336.7.bh.f.241.2

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} - 2x^{7} + 193x^{6} + 306x^{5} + 29845x^{4} + 16988x^{3} + 1125468x^{2} + 214128x + 35378704 $ x^8 - 2x^7 + 193x^6 + 306x^5 + 29845x^4 + 16988x^3 + 1125468x^2 + 214128*x + 35378704
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 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			241.2
Root			$-3.27778 - 5.67728i$ of defining polynomial
Character	$\chi$	$=$	336.241
Dual form			336.7.bh.f.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.5000 + 7.79423i) q^{3} +(63.2658 - 36.5265i) q^{5} +(271.352 + 209.802i) q^{7} +(121.500 + 210.444i) q^{9} +O(q^{10})$	\(q+(13.5000 + 7.79423i) q^{3} +(63.2658 - 36.5265i) q^{5} +(271.352 + 209.802i) q^{7} +(121.500 + 210.444i) q^{9} +(-1073.02 + 1858.52i) q^{11} -2550.94i q^{13} +1138.79 q^{15} +(2870.93 + 1657.53i) q^{17} +(-3813.46 + 2201.70i) q^{19} +(2028.01 + 4947.31i) q^{21} +(8548.49 + 14806.4i) q^{23} +(-5144.12 + 8909.88i) q^{25} +3788.00i q^{27} -33248.2 q^{29} +(-10149.7 - 5859.92i) q^{31} +(-28971.4 + 16726.7i) q^{33} +(24830.7 + 3361.73i) q^{35} +(-24322.8 - 42128.4i) q^{37} +(19882.6 - 34437.7i) q^{39} +71461.4i q^{41} -86836.9 q^{43} +(15373.6 + 8875.95i) q^{45} +(143252. - 82706.7i) q^{47} +(29615.3 + 113861. i) q^{49} +(25838.4 + 44753.4i) q^{51} +(42419.4 - 73472.5i) q^{53} +156774. i q^{55} -68642.2 q^{57} +(18534.0 + 10700.6i) q^{59} +(235369. - 135891. i) q^{61} +(-11182.3 + 82595.5i) q^{63} +(-93177.0 - 161387. i) q^{65} +(-239484. + 414798. i) q^{67} +266516. i q^{69} -337334. q^{71} +(340100. + 196357. i) q^{73} +(-138891. + 80188.9i) q^{75} +(-681087. + 279193. i) q^{77} +(5038.26 + 8726.52i) q^{79} +(-29524.5 + 51137.9i) q^{81} +77744.9i q^{83} +242176. q^{85} +(-448851. - 259144. i) q^{87} +(478066. - 276012. i) q^{89} +(535192. - 692203. i) q^{91} +(-91347.1 - 158218. i) q^{93} +(-160841. + 278585. i) q^{95} +1.06023e6i q^{97} -521486. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 108 q^{3} + 462 q^{5} - 580 q^{7} + 972 q^{9}+O(q^{10}) $ 8 * q + 108 * q^3 + 462 * q^5 - 580 * q^7 + 972 * q^9	$ 8 q + 108 q^{3} + 462 q^{5} - 580 q^{7} + 972 q^{9} - 1806 q^{11} + 8316 q^{15} + 9564 q^{17} - 23022 q^{19} - 1350 q^{21} - 2400 q^{23} + 32762 q^{25} - 2484 q^{29} - 148416 q^{31} - 48762 q^{33} - 11412 q^{35} - 84046 q^{37} + 75222 q^{39} - 92972 q^{43} + 112266 q^{45} + 323124 q^{47} - 8644 q^{49} + 86076 q^{51} + 358086 q^{53} - 414396 q^{57} - 719382 q^{59} + 421536 q^{61} + 104490 q^{63} - 322740 q^{65} - 267010 q^{67} + 464664 q^{71} + 1944486 q^{73} + 884574 q^{75} - 1713498 q^{77} + 685904 q^{79} - 236196 q^{81} - 3876168 q^{85} - 33534 q^{87} + 4130604 q^{89} + 484266 q^{91} - 1335744 q^{93} + 2105232 q^{95} - 877716 q^{99}+O(q^{100}) $ 8 * q + 108 * q^3 + 462 * q^5 - 580 * q^7 + 972 * q^9 - 1806 * q^11 + 8316 * q^15 + 9564 * q^17 - 23022 * q^19 - 1350 * q^21 - 2400 * q^23 + 32762 * q^25 - 2484 * q^29 - 148416 * q^31 - 48762 * q^33 - 11412 * q^35 - 84046 * q^37 + 75222 * q^39 - 92972 * q^43 + 112266 * q^45 + 323124 * q^47 - 8644 * q^49 + 86076 * q^51 + 358086 * q^53 - 414396 * q^57 - 719382 * q^59 + 421536 * q^61 + 104490 * q^63 - 322740 * q^65 - 267010 * q^67 + 464664 * q^71 + 1944486 * q^73 + 884574 * q^75 - 1713498 * q^77 + 685904 * q^79 - 236196 * q^81 - 3876168 * q^85 - 33534 * q^87 + 4130604 * q^89 + 484266 * q^91 - 1335744 * q^93 + 2105232 * q^95 - 877716 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	63.2658	−	36.5265i	0.506127	−	0.292212i	−0.225113	−	0.974333i	$-0.572275\pi$
$5$	63.2658	−	36.5265i	0.506127	−	0.292212i	0.731240	+	0.682120i	$0.238942\pi$
$6$		0			0
$7$	271.352	+	209.802i	0.791115	+	0.611667i
$7$	271.352	+	209.802i	0.791115	+	0.611667i
$8$		0			0
$9$	121.500	+	210.444i	0.166667	+	0.288675i
$10$		0			0
$11$	−1073.02	+	1858.52i	−0.806173	+	1.39633i	0.109323	+	0.994006i	$0.465132\pi$
$11$	−1073.02	+	1858.52i	−0.806173	+	1.39633i	−0.915496	+	0.402327i	$0.868202\pi$
$12$		0			0
$13$		−	2550.94i		−	1.16110i	−0.814224	−	0.580550i	$-0.802837\pi$
$13$		−	2550.94i		−	1.16110i	0.814224	−	0.580550i	$-0.197163\pi$
$14$		0			0
$15$	1138.79			0.337418
$16$		0			0
$17$	2870.93	+	1657.53i	0.584355	+	0.337377i	0.762862	−	0.646561i	$-0.223794\pi$
$17$	2870.93	+	1657.53i	0.584355	+	0.337377i	−0.178508	+	0.983939i	$0.557127\pi$
$18$		0			0
$19$	−3813.46	+	2201.70i	−0.555978	+	0.320994i	−0.751530	−	0.659699i	$-0.770684\pi$
$19$	−3813.46	+	2201.70i	−0.555978	+	0.320994i	0.195551	+	0.980693i	$0.437350\pi$
$20$		0			0
$21$	2028.01	+	4947.31i	0.218984	+	0.534209i
$22$		0			0
$23$	8548.49	+	14806.4i	0.702597	+	1.21693i	0.967552	+	0.252673i	$0.0813095\pi$
$23$	8548.49	+	14806.4i	0.702597	+	1.21693i	−0.264955	+	0.964261i	$0.585357\pi$
$24$		0			0
$25$	−5144.12	+	8909.88i	−0.329224	+	0.570232i
$26$		0			0
$27$			3788.00i			0.192450i
$28$		0			0
$29$	−33248.2			−1.36325			−0.681623	−	0.731703i	$-0.738726\pi$
$29$	−33248.2			−1.36325			−0.681623	+	0.731703i	$0.738726\pi$
$30$		0			0
$31$	−10149.7	−	5859.92i	−0.340696	−	0.196701i	0.319884	−	0.947457i	$-0.396356\pi$
$31$	−10149.7	−	5859.92i	−0.340696	−	0.196701i	−0.660580	+	0.750756i	$0.729689\pi$
$32$		0			0
$33$	−28971.4	+	16726.7i	−0.806173	+	0.465444i
$34$		0			0
$35$	24830.7	+	3361.73i	0.579141	+	0.0784076i
$36$		0			0
$37$	−24322.8	−	42128.4i	−0.480185	−	0.831706i	0.519556	−	0.854436i	$-0.326097\pi$
$37$	−24322.8	−	42128.4i	−0.480185	−	0.831706i	−0.999742	+	0.0227307i	$0.992764\pi$
$38$		0			0
$39$	19882.6	−	34437.7i	0.335181	−	0.580550i
$40$		0			0
$41$			71461.4i			1.03686i	0.855120	+	0.518430i	$0.173483\pi$
$41$			71461.4i			1.03686i	−0.855120	+	0.518430i	$0.826517\pi$
$42$		0			0
$43$	−86836.9			−1.09219			−0.546096	−	0.837723i	$-0.683887\pi$
$43$	−86836.9			−1.09219			−0.546096	+	0.837723i	$0.683887\pi$
$44$		0			0
$45$	15373.6	+	8875.95i	0.168709	+	0.0974041i
$46$		0			0
$47$	143252.	−	82706.7i	1.37977	−	0.796612i	0.387641	−	0.921810i	$-0.373290\pi$
$47$	143252.	−	82706.7i	1.37977	−	0.796612i	0.992132	+	0.125198i	$0.0399566\pi$
$48$		0			0
$49$	29615.3	+	113861.i	0.251726	+	0.967799i
$50$		0			0
$51$	25838.4	+	44753.4i	0.194785	+	0.337377i
$52$		0			0
$53$	42419.4	−	73472.5i	0.284929	−	0.493512i	−0.687663	−	0.726030i	$-0.741363\pi$
$53$	42419.4	−	73472.5i	0.284929	−	0.493512i	0.972592	+	0.232519i	$0.0746966\pi$
$54$		0			0
$55$			156774.i			0.942295i
$56$		0			0
$57$	−68642.2			−0.370652
$58$		0			0
$59$	18534.0	+	10700.6i	0.0902428	+	0.0521017i	0.544442	−	0.838798i	$-0.316741\pi$
$59$	18534.0	+	10700.6i	0.0902428	+	0.0521017i	−0.454199	+	0.890900i	$0.650075\pi$
$60$		0			0
$61$	235369.	−	135891.i	1.03696	−	0.598687i	0.117986	−	0.993015i	$-0.462356\pi$
$61$	235369.	−	135891.i	1.03696	−	0.598687i	0.918970	+	0.394328i	$0.129023\pi$
$62$		0			0
$63$	−11182.3	+	82595.5i	−0.0447207	+	0.330320i
$64$		0			0
$65$	−93177.0	−	161387.i	−0.339288	−	0.587664i
$66$		0			0
$67$	−239484.	+	414798.i	−0.796253	+	1.37915i	0.125787	+	0.992057i	$0.459854\pi$
$67$	−239484.	+	414798.i	−0.796253	+	1.37915i	−0.922040	+	0.387094i	$0.873479\pi$
$68$		0			0
$69$			266516.i			0.811289i
$70$		0			0
$71$	−337334.			−0.942507			−0.471254	−	0.881998i	$-0.656198\pi$
$71$	−337334.			−0.942507			−0.471254	+	0.881998i	$0.656198\pi$
$72$		0			0
$73$	340100.	+	196357.i	0.874254	+	0.504751i	0.868760	−	0.495234i	$-0.164918\pi$
$73$	340100.	+	196357.i	0.874254	+	0.504751i	0.00549449	+	0.999985i	$0.498251\pi$
$74$		0			0
$75$	−138891.	+	80188.9i	−0.329224	+	0.190077i
$76$		0			0
$77$	−681087.	+	279193.i	−1.49187	+	0.611550i
$78$		0			0
$79$	5038.26	+	8726.52i	0.0102188	+	0.0176994i	0.871090	−	0.491124i	$-0.163414\pi$
$79$	5038.26	+	8726.52i	0.0102188	+	0.0176994i	−0.860871	+	0.508824i	$0.830081\pi$
$80$		0			0
$81$	−29524.5	+	51137.9i	−0.0555556	+	0.0962250i
$82$		0			0
$83$			77744.9i			0.135968i	0.997686	+	0.0679841i	$0.0216567\pi$
$83$			77744.9i			0.135968i	−0.997686	+	0.0679841i	$0.978343\pi$
$84$		0			0
$85$	242176.			0.394343
$86$		0			0
$87$	−448851.	−	259144.i	−0.681623	−	0.393535i
$88$		0			0
$89$	478066.	−	276012.i	0.678138	−	0.391523i	−0.121015	−	0.992651i	$-0.538615\pi$
$89$	478066.	−	276012.i	0.678138	−	0.391523i	0.799153	+	0.601128i	$0.205282\pi$
$90$		0			0
$91$	535192.	−	692203.i	0.710208	−	0.918564i
$92$		0			0
$93$	−91347.1	−	158218.i	−0.113565	−	0.196701i
$94$		0			0
$95$	−160841.	+	278585.i	−0.187597	+	0.324928i
$96$		0			0
$97$			1.06023e6i			1.16168i	0.814020	+	0.580838i	$0.197275\pi$
$97$			1.06023e6i			1.16168i	−0.814020	+	0.580838i	$0.802725\pi$
$98$		0			0
$99$	−521486.			−0.537449
$100$		0			0
$101$	−763243.	−	440658.i	−0.740796	−	0.427699i	0.0815629	−	0.996668i	$-0.474009\pi$
$101$	−763243.	−	440658.i	−0.740796	−	0.427699i	−0.822359	+	0.568970i	$0.807342\pi$
$102$		0			0
$103$	82998.1	−	47918.9i	0.0759550	−	0.0438526i	−0.461542	−	0.887119i	$-0.652703\pi$
$103$	82998.1	−	47918.9i	0.0759550	−	0.0438526i	0.537496	+	0.843266i	$0.319370\pi$
$104$		0			0
$105$	309012.	+	238919.i	0.266936	+	0.206387i
$106$		0			0
$107$	463383.	+	802602.i	0.378258	+	0.655162i	0.990809	−	0.135269i	$-0.0431898\pi$
$107$	463383.	+	802602.i	0.378258	+	0.655162i	−0.612551	+	0.790431i	$0.709856\pi$
$108$		0			0
$109$	−1.17618e6	+	2.03720e6i	−0.908224	+	1.57309i	−0.0916936	+	0.995787i	$0.529228\pi$
$109$	−1.17618e6	+	2.03720e6i	−0.908224	+	1.57309i	−0.816530	+	0.577303i	$0.804105\pi$
$110$		0			0
$111$		−	758311.i		−	0.554470i
$112$		0			0
$113$	−658872.			−0.456631			−0.228316	−	0.973587i	$-0.573322\pi$
$113$	−658872.			−0.456631			−0.228316	+	0.973587i	$0.573322\pi$
$114$		0			0
$115$	1.08166e6	+	624494.i	0.711206	+	0.410615i
$116$		0			0
$117$	536830.	−	309939.i	0.335181	−	0.193517i
$118$		0			0
$119$	431281.	+	1.05210e6i	0.255929	+	0.624335i
$120$		0			0
$121$	−1.41695e6	−	2.45423e6i	−0.799831	−	1.38535i
$122$		0			0
$123$	−556986.	+	964729.i	−0.299316	+	0.518430i
$124$		0			0
$125$			1.89304e6i			0.969238i
$126$		0			0
$127$	−3.10460e6			−1.51564			−0.757818	−	0.652465i	$-0.773735\pi$
$127$	−3.10460e6			−1.51564			−0.757818	+	0.652465i	$0.773735\pi$
$128$		0			0
$129$	−1.17230e6	−	676826.i	−0.546096	−	0.315288i
$130$		0			0
$131$	−3.03561e6	+	1.75261e6i	−1.35031	+	0.779600i	−0.988292	−	0.152572i	$-0.951244\pi$
$131$	−3.03561e6	+	1.75261e6i	−1.35031	+	0.779600i	−0.362015	+	0.932172i	$0.617911\pi$
$132$		0			0
$133$	−1.49671e6	−	202634.i	−0.636185	−	0.0861305i
$134$		0			0
$135$	138362.	+	239651.i	0.0562363	+	0.0974041i
$136$		0			0
$137$	−142894.	+	247500.i	−0.0555717	+	0.0962530i	−0.892473	−	0.451101i	$-0.851031\pi$
$137$	−142894.	+	247500.i	−0.0555717	+	0.0962530i	0.836901	+	0.547354i	$0.184365\pi$
$138$		0			0
$139$		−	2.54890e6i		−	0.949093i	−0.880230	−	0.474547i	$-0.842612\pi$
$139$		−	2.54890e6i		−	0.949093i	0.880230	−	0.474547i	$-0.157388\pi$
$140$		0			0
$141$	2.57854e6			0.919849
$142$		0			0
$143$	4.74097e6	+	2.73720e6i	1.62128	+	0.936048i
$144$		0			0
$145$	−2.10348e6	+	1.21444e6i	−0.689975	+	0.398358i
$146$		0			0
$147$	−487648.	+	1.76795e6i	−0.153516	+	0.556566i
$148$		0			0
$149$	405472.	+	702298.i	0.122575	+	0.212306i	0.920782	−	0.390077i	$-0.127551\pi$
$149$	405472.	+	702298.i	0.122575	+	0.212306i	−0.798207	+	0.602383i	$0.794218\pi$
$150$		0			0
$151$	−1.10258e6	+	1.90972e6i	−0.320241	+	0.554674i	−0.980538	−	0.196331i	$-0.937097\pi$
$151$	−1.10258e6	+	1.90972e6i	−0.320241	+	0.554674i	0.660296	+	0.751005i	$0.270431\pi$
$152$		0			0
$153$			805562.i			0.224918i
$154$		0			0
$155$	−856171.			−0.229914
$156$		0			0
$157$	3.89107e6	+	2.24651e6i	1.00547	+	0.580510i	0.909863	−	0.414909i	$-0.136187\pi$
$157$	3.89107e6	+	2.24651e6i	1.00547	+	0.580510i	0.0956099	+	0.995419i	$0.469520\pi$
$158$		0			0
$159$	1.14532e6	−	661253.i	0.284929	−	0.164504i
$160$		0			0
$161$	−786762.	+	5.81125e6i	−0.188524	+	1.39249i
$162$		0			0
$163$	3.67950e6	+	6.37308e6i	0.849622	+	1.47159i	0.881546	+	0.472098i	$0.156503\pi$
$163$	3.67950e6	+	6.37308e6i	0.849622	+	1.47159i	−0.0319242	+	0.999490i	$0.510164\pi$
$164$		0			0
$165$	−1.22194e6	+	2.11645e6i	−0.272017	+	0.471148i
$166$		0			0
$167$		−	2.46735e6i		−	0.529763i	−0.964281	−	0.264881i	$-0.914667\pi$
$167$		−	2.46735e6i		−	0.529763i	0.964281	−	0.264881i	$-0.0853328\pi$
$168$		0			0
$169$	−1.68048e6			−0.348155
$170$		0			0
$171$	−926670.	−	535013.i	−0.185326	−	0.106998i
$172$		0			0
$173$	607596.	−	350796.i	0.117348	−	0.0677511i	−0.440177	−	0.897911i	$-0.645084\pi$
$173$	607596.	−	350796.i	0.117348	−	0.0677511i	0.557525	+	0.830160i	$0.311751\pi$
$174$		0			0
$175$	−3.26518e6	+	1.33847e6i	−0.609246	+	0.249744i
$176$		0			0
$177$	166806.	+	288916.i	0.0300809	+	0.0521017i
$178$		0			0
$179$	1.12866e6	−	1.95490e6i	0.196791	−	0.340852i	−0.750695	−	0.660649i	$-0.770281\pi$
$179$	1.12866e6	−	1.95490e6i	0.196791	−	0.340852i	0.947486	+	0.319797i	$0.103615\pi$
$180$		0			0
$181$			2.39583e6i			0.404036i	0.979382	+	0.202018i	$0.0647499\pi$
$181$			2.39583e6i			0.404036i	−0.979382	+	0.202018i	$0.935250\pi$
$182$		0			0
$183$	4.23665e6			0.691304
$184$		0			0
$185$	−3.07761e6	−	1.77686e6i	−0.486069	−	0.280632i
$186$		0			0
$187$	−6.16112e6	+	3.55712e6i	−0.942182	+	0.543969i
$188$		0			0
$189$	−794729.	+	1.02788e6i	−0.117715	+	0.152250i
$190$		0			0
$191$	648809.	+	1.12377e6i	0.0931144	+	0.161279i	0.908820	−	0.417188i	$-0.136984\pi$
$191$	648809.	+	1.12377e6i	0.0931144	+	0.161279i	−0.815706	+	0.578467i	$0.803651\pi$
$192$		0			0
$193$	−1.49829e6	+	2.59511e6i	−0.208412	+	0.360980i	−0.951214	−	0.308530i	$-0.900163\pi$
$193$	−1.49829e6	+	2.59511e6i	−0.208412	+	0.360980i	0.742802	+	0.669511i	$0.233496\pi$
$194$		0			0
$195$		−	2.90497e6i		−	0.391776i
$196$		0			0
$197$	6.72245e6			0.879283			0.439641	−	0.898173i	$-0.355106\pi$
$197$	6.72245e6			0.879283			0.439641	+	0.898173i	$0.355106\pi$
$198$		0			0
$199$	−9.71212e6	−	5.60729e6i	−1.23241	−	0.711531i	−0.264877	−	0.964282i	$-0.585331\pi$
$199$	−9.71212e6	−	5.60729e6i	−1.23241	−	0.711531i	−0.967531	+	0.252751i	$0.918665\pi$
$200$		0			0
$201$	−6.46606e6	+	3.73318e6i	−0.796253	+	0.459717i
$202$		0			0
$203$	−9.02199e6	−	6.97554e6i	−1.07848	−	0.833853i
$204$		0			0
$205$	2.61024e6	+	4.52106e6i	0.302983	+	0.524782i
$206$		0			0
$207$	−2.07728e6	+	3.59796e6i	−0.234199	+	0.405644i
$208$		0			0
$209$		−	9.44984e6i		−	1.03511i
$210$		0			0
$211$	273141.			0.0290763			0.0145381	−	0.999894i	$-0.495372\pi$
$211$	273141.			0.0290763			0.0145381	+	0.999894i	$0.495372\pi$
$212$		0			0
$213$	−4.55400e6	−	2.62926e6i	−0.471254	−	0.272078i
$214$		0			0
$215$	−5.49381e6	+	3.17185e6i	−0.552787	+	0.319152i
$216$		0			0
$217$	−1.52472e6	−	3.71953e6i	−0.149214	−	0.364006i
$218$		0			0
$219$	3.06090e6	+	5.30163e6i	0.291418	+	0.504751i
$220$		0			0
$221$	4.22827e6	−	7.32358e6i	0.391729	−	0.678495i
$222$		0			0
$223$			2.12335e7i			1.91472i	0.288890	+	0.957362i	$0.406714\pi$
$223$			2.12335e7i			1.91472i	−0.288890	+	0.957362i	$0.593286\pi$
$224$		0			0
$225$	−2.50004e6			−0.219483
$226$		0			0
$227$	4.68100e6	+	2.70258e6i	0.400186	+	0.231047i	0.686564	−	0.727069i	$-0.259118\pi$
$227$	4.68100e6	+	2.70258e6i	0.400186	+	0.231047i	−0.286378	+	0.958117i	$0.592451\pi$
$228$		0			0
$229$	2.00874e7	−	1.15975e7i	1.67270	−	0.965732i	0.706578	−	0.707635i	$-0.250238\pi$
$229$	2.00874e7	−	1.15975e7i	1.67270	−	0.965732i	0.966119	−	0.258097i	$-0.0830953\pi$
$230$		0			0
$231$	−1.13708e7	−	1.53944e6i	−0.922473	−	0.124890i
$232$		0			0
$233$	3.77717e6	+	6.54226e6i	0.298607	+	0.517202i	0.975817	−	0.218587i	$-0.0701448\pi$
$233$	3.77717e6	+	6.54226e6i	0.298607	+	0.517202i	−0.677211	+	0.735789i	$0.736812\pi$
$234$		0			0
$235$	6.04198e6	−	1.04650e7i	0.465560	−	0.806374i
$236$		0			0
$237$			157077.i			0.0117996i
$238$		0			0
$239$	2.64637e7			1.93846			0.969230	−	0.246155i	$-0.0791673\pi$
$239$	2.64637e7			1.93846			0.969230	+	0.246155i	$0.0791673\pi$
$240$		0			0
$241$	−5.18988e6	−	2.99638e6i	−0.370771	−	0.214065i	0.303024	−	0.952983i	$-0.402004\pi$
$241$	−5.18988e6	−	2.99638e6i	−0.370771	−	0.214065i	−0.673795	+	0.738918i	$0.735337\pi$
$242$		0			0
$243$	−797162.	+	460241.i	−0.0555556	+	0.0320750i
$244$		0			0
$245$	6.03257e6	+	6.12174e6i	0.410208	+	0.416271i
$246$		0			0
$247$	5.61640e6	+	9.72789e6i	0.372707	+	0.645547i
$248$		0			0
$249$	−605961.	+	1.04956e6i	−0.0392506	+	0.0679841i
$250$		0			0
$251$		−	2.47242e7i		−	1.56351i	−0.623583	−	0.781757i	$-0.714324\pi$
$251$		−	2.47242e7i		−	1.56351i	0.623583	−	0.781757i	$-0.285676\pi$
$252$		0			0
$253$	−3.66907e7			−2.26566
$254$		0			0
$255$	3.26938e6	+	1.88758e6i	0.197172	+	0.113837i
$256$		0			0
$257$	1.30814e7	−	7.55255e6i	0.770646	−	0.444932i	−0.0624592	−	0.998048i	$-0.519894\pi$
$257$	1.30814e7	−	7.55255e6i	0.770646	−	0.444932i	0.833105	+	0.553115i	$0.186561\pi$
$258$		0			0
$259$	2.23856e6	−	1.65346e7i	0.128845	−	0.951689i
$260$		0			0
$261$	−4.03966e6	−	6.99689e6i	−0.227208	−	0.393535i
$262$		0			0
$263$	7.01596e6	−	1.21520e7i	0.385674	−	0.668007i	−0.606189	−	0.795321i	$-0.707302\pi$
$263$	7.01596e6	−	1.21520e7i	0.385674	−	0.668007i	0.991862	+	0.127314i	$0.0406357\pi$
$264$		0			0
$265$		−	6.19773e6i		−	0.333039i
$266$		0			0
$267$	8.60519e6			0.452092
$268$		0			0
$269$	2.14076e7	+	1.23597e7i	1.09979	+	0.634965i	0.936166	−	0.351558i	$-0.114348\pi$
$269$	2.14076e7	+	1.23597e7i	1.09979	+	0.634965i	0.163625	+	0.986523i	$0.447681\pi$
$270$		0			0
$271$	2.96220e7	−	1.71022e7i	1.48835	−	0.859301i	0.488441	−	0.872597i	$-0.337566\pi$
$271$	2.96220e7	−	1.71022e7i	1.48835	−	0.859301i	0.999912	+	0.0132962i	$0.00423244\pi$
$272$		0			0
$273$	1.26203e7	−	5.17334e6i	0.620270	−	0.254263i
$274$		0			0
$275$	−1.10395e7	−	1.91209e7i	−0.530823	−	0.919412i
$276$		0			0
$277$	−1.91114e7	+	3.31020e7i	−0.899196	+	1.55745i	−0.0706716	+	0.997500i	$0.522514\pi$
$277$	−1.91114e7	+	3.31020e7i	−0.899196	+	1.55745i	−0.828524	+	0.559953i	$0.810819\pi$
$278$		0			0
$279$		−	2.84792e6i		−	0.131134i
$280$		0			0
$281$	−4.38890e6			−0.197805			−0.0989025	−	0.995097i	$-0.531533\pi$
$281$	−4.38890e6			−0.197805			−0.0989025	+	0.995097i	$0.531533\pi$
$282$		0			0
$283$	−1.92918e7	−	1.11381e7i	−0.851166	−	0.491421i	0.00987835	−	0.999951i	$-0.496856\pi$
$283$	−1.92918e7	−	1.11381e7i	−0.851166	−	0.491421i	−0.861044	+	0.508531i	$0.830189\pi$
$284$		0			0
$285$	−4.34271e6	+	2.50726e6i	−0.187597	+	0.108309i
$286$		0			0
$287$	−1.49927e7	+	1.93912e7i	−0.634213	+	0.820275i
$288$		0			0
$289$	−6.57394e6	−	1.13864e7i	−0.272353	−	0.471730i
$290$		0			0
$291$	−8.26367e6	+	1.43131e7i	−0.335347	+	0.580838i
$292$		0			0
$293$		−	4.03269e7i		−	1.60322i	−0.597849	−	0.801608i	$-0.703978\pi$
$293$		−	4.03269e7i		−	1.60322i	0.597849	−	0.801608i	$-0.296022\pi$
$294$		0			0
$295$	1.56342e6			0.0608991
$296$		0			0
$297$	−7.04006e6	−	4.06458e6i	−0.268724	−	0.155148i
$298$		0			0
$299$	3.77703e7	−	2.18067e7i	1.41298	−	0.815786i
$300$		0			0
$301$	−2.35634e7	−	1.82185e7i	−0.864049	−	0.668058i
$302$		0			0
$303$	−6.86918e6	−	1.18978e7i	−0.246932	−	0.427699i
$304$		0			0
$305$	9.92722e6	−	1.71945e7i	0.349887	−	0.606023i
$306$		0			0
$307$			1.99914e7i			0.690921i	0.938433	+	0.345461i	$0.112277\pi$
$307$			1.99914e7i			0.690921i	−0.938433	+	0.345461i	$0.887723\pi$
$308$		0			0
$309$	1.49396e6			0.0506366
$310$		0			0
$311$	1.04940e7	+	6.05872e6i	0.348868	+	0.201419i	0.664186	−	0.747567i	$-0.268778\pi$
$311$	1.04940e7	+	6.05872e6i	0.348868	+	0.201419i	−0.315319	+	0.948986i	$0.602111\pi$
$312$		0			0
$313$	4.91209e7	−	2.83600e7i	1.60189	−	0.924853i	0.610785	−	0.791797i	$-0.290854\pi$
$313$	4.91209e7	−	2.83600e7i	1.60189	−	0.924853i	0.991108	−	0.133057i	$-0.0424792\pi$
$314$		0			0
$315$	2.30947e6	+	5.63392e6i	0.0738892	+	0.180252i
$316$		0			0
$317$	−1.24114e7	−	2.14971e7i	−0.389621	−	0.674843i	0.602778	−	0.797909i	$-0.294061\pi$
$317$	−1.24114e7	−	2.14971e7i	−0.389621	−	0.674843i	−0.992398	+	0.123066i	$0.960727\pi$
$318$		0			0
$319$	3.56759e7	−	6.17925e7i	1.09901	−	1.90355i
$320$		0			0
$321$			1.44468e7i			0.436775i
$322$		0			0
$323$	−1.45976e7			−0.433185
$324$		0			0
$325$	2.27286e7	+	1.31223e7i	0.662097	+	0.382262i
$326$		0			0
$327$	−3.17568e7	+	1.83348e7i	−0.908224	+	0.524363i
$328$		0			0
$329$	5.62239e7	+	7.61192e6i	1.57882	+	0.213750i
$330$		0			0
$331$	−4.53145e6	−	7.84871e6i	−0.124955	−	0.216428i	0.796760	−	0.604295i	$-0.206545\pi$
$331$	−4.53145e6	−	7.84871e6i	−0.124955	−	0.216428i	−0.921715	+	0.387867i	$0.873212\pi$
$332$		0			0
$333$	5.91045e6	−	1.02372e7i	0.160062	−	0.277235i
$334$		0			0
$335$			3.49900e7i			0.930700i
$336$		0			0
$337$	3.31353e7			0.865767			0.432884	−	0.901450i	$-0.357496\pi$
$337$	3.31353e7			0.865767			0.432884	+	0.901450i	$0.357496\pi$
$338$		0			0
$339$	−8.89477e6	−	5.13540e6i	−0.228316	−	0.131818i
$340$		0			0
$341$	2.17815e7	−	1.25756e7i	0.549320	−	0.317150i
$342$		0			0
$343$	−1.58520e7	+	3.71097e7i	−0.392827	+	0.919613i
$344$		0			0
$345$	9.73490e6	+	1.68613e7i	0.237069	+	0.410615i
$346$		0			0
$347$	1.35157e7	−	2.34098e7i	0.323481	−	0.560286i	−0.657722	−	0.753260i	$-0.728480\pi$
$347$	1.35157e7	−	2.34098e7i	0.323481	−	0.560286i	0.981204	+	0.192974i	$0.0618133\pi$
$348$		0			0
$349$		−	1.26526e7i		−	0.297649i	−0.988864	−	0.148824i	$-0.952451\pi$
$349$		−	1.26526e7i		−	0.297649i	0.988864	−	0.148824i	$-0.0475489\pi$
$350$		0			0
$351$	9.66294e6			0.223454
$352$		0			0
$353$	−6.20503e7	−	3.58248e7i	−1.41065	−	0.814441i	−0.415203	−	0.909729i	$-0.636289\pi$
$353$	−6.20503e7	−	3.58248e7i	−1.41065	−	0.814441i	−0.995450	+	0.0952883i	$0.969623\pi$
$354$		0			0
$355$	−2.13417e7	+	1.23216e7i	−0.477028	+	0.275412i
$356$		0			0
$357$	−2.37804e6	+	1.75649e7i	−0.0522654	+	0.386048i
$358$		0			0
$359$	5.75025e6	+	9.95972e6i	0.124281	+	0.215260i	0.921451	−	0.388493i	$-0.127004\pi$
$359$	5.75025e6	+	9.95972e6i	0.124281	+	0.215260i	−0.797171	+	0.603754i	$0.793671\pi$
$360$		0			0
$361$	−1.38280e7	+	2.39508e7i	−0.293925	+	0.509094i
$362$		0			0
$363$		−	4.41761e7i		−	0.923565i
$364$		0			0
$365$	2.86889e7			0.589978
$366$		0			0
$367$	−9.40666e6	−	5.43094e6i	−0.190299	−	0.109869i	0.401823	−	0.915717i	$-0.368377\pi$
$367$	−9.40666e6	−	5.43094e6i	−0.190299	−	0.109869i	−0.592123	+	0.805848i	$0.701710\pi$
$368$		0			0
$369$	−1.50386e7	+	8.68256e6i	−0.299316	+	0.172810i
$370$		0			0
$371$	2.69253e7	−	1.10373e7i	0.527277	−	0.216143i
$372$		0			0
$373$	2.63819e6	+	4.56949e6i	0.0508370	+	0.0880523i	0.890324	−	0.455327i	$-0.150478\pi$
$373$	2.63819e6	+	4.56949e6i	0.0508370	+	0.0880523i	−0.839487	+	0.543380i	$0.817144\pi$
$374$		0			0
$375$	−1.47548e7	+	2.55561e7i	−0.279795	+	0.484619i
$376$		0			0
$377$			8.48142e7i			1.58287i
$378$		0			0
$379$	−7.77449e7			−1.42809			−0.714043	−	0.700102i	$-0.753138\pi$
$379$	−7.77449e7			−1.42809			−0.714043	+	0.700102i	$0.753138\pi$
$380$		0			0
$381$	−4.19122e7	−	2.41980e7i	−0.757818	−	0.437527i
$382$		0			0
$383$	5.70488e7	−	3.29371e7i	1.01543	−	0.586259i	0.102653	−	0.994717i	$-0.467267\pi$
$383$	5.70488e7	−	3.29371e7i	1.01543	−	0.586259i	0.912777	+	0.408459i	$0.133934\pi$
$384$		0			0
$385$	−3.28916e7	+	4.25411e7i	−0.576371	+	0.745464i
$386$		0			0
$387$	−1.05507e7	−	1.82743e7i	−0.182032	−	0.315288i
$388$		0			0
$389$	1.99115e7	−	3.44878e7i	0.338264	−	0.585890i	−0.645842	−	0.763471i	$-0.723494\pi$
$389$	1.99115e7	−	3.44878e7i	0.338264	−	0.585890i	0.984106	+	0.177581i	$0.0568271\pi$
$390$		0			0
$391$			5.66777e7i			0.948161i
$392$		0			0
$393$	−5.46410e7			−0.900205
$394$		0			0
$395$	637499.	+	368060.i	0.0103440	+	0.00597211i
$396$		0			0
$397$	1.44625e7	−	8.34992e6i	0.231138	−	0.133448i	−0.379959	−	0.925003i	$-0.624062\pi$
$397$	1.44625e7	−	8.34992e6i	0.231138	−	0.133448i	0.611097	+	0.791556i	$0.290729\pi$
$398$		0			0
$399$	−1.86262e7	−	1.44013e7i	−0.293229	−	0.226716i
$400$		0			0
$401$	−1.77044e7	−	3.06649e7i	−0.274567	−	0.475564i	0.695459	−	0.718566i	$-0.255201\pi$
$401$	−1.77044e7	−	3.06649e7i	−0.274567	−	0.475564i	−0.970026	+	0.243002i	$0.921868\pi$
$402$		0			0
$403$	−1.49483e7	+	2.58912e7i	−0.228390	+	0.395583i
$404$		0			0
$405$			4.31371e6i			0.0649361i
$406$		0			0
$407$	1.04395e8			1.54845
$408$		0			0
$409$	3.81349e7	+	2.20172e7i	0.557381	+	0.321804i	0.752094	−	0.659056i	$-0.229044\pi$
$409$	3.81349e7	+	2.20172e7i	0.557381	+	0.321804i	−0.194713	+	0.980860i	$0.562377\pi$
$410$		0			0
$411$	−3.85815e6	+	2.22750e6i	−0.0555717	+	0.0320843i
$412$		0			0
$413$	2.78424e6	+	6.79210e6i	0.0395235	+	0.0964171i
$414$		0			0
$415$	2.83975e6	+	4.91859e6i	0.0397316	+	0.0688172i
$416$		0			0
$417$	1.98667e7	−	3.44102e7i	0.273980	−	0.474547i
$418$		0			0
$419$		−	1.45445e8i		−	1.97723i	−0.150456	−	0.988617i	$-0.548074\pi$
$419$		−	1.45445e8i		−	1.97723i	0.150456	−	0.988617i	$-0.451926\pi$
$420$		0			0
$421$	1.72505e7			0.231182			0.115591	−	0.993297i	$-0.463124\pi$
$421$	1.72505e7			0.231182			0.115591	+	0.993297i	$0.463124\pi$
$422$		0			0
$423$	3.48103e7	+	2.00977e7i	0.459924	+	0.265537i
$424$		0			0
$425$	−2.95369e7	+	1.70531e7i	−0.384767	+	0.222145i
$426$		0			0
$427$	9.23781e7	+	1.25067e7i	1.18655	+	0.160642i
$428$		0			0
$429$	4.26687e7	+	7.39044e7i	0.540428	+	0.936048i
$430$		0			0
$431$	4.44477e6	−	7.69857e6i	0.0555159	−	0.0961564i	−0.836932	−	0.547307i	$-0.815653\pi$
$431$	4.44477e6	−	7.69857e6i	0.0555159	−	0.0961564i	0.892448	+	0.451151i	$0.148986\pi$
$432$		0			0
$433$		−	1.64906e7i		−	0.203130i	−0.994829	−	0.101565i	$-0.967615\pi$
$433$		−	1.64906e7i		−	0.203130i	0.994829	−	0.101565i	$-0.0323850\pi$
$434$		0			0
$435$	−3.78626e7			−0.459984
$436$		0			0
$437$	−6.51986e7	−	3.76424e7i	−0.781257	−	0.451059i
$438$		0			0
$439$	−7.87931e7	+	4.54912e7i	−0.931310	+	0.537692i	−0.887226	−	0.461336i	$-0.847370\pi$
$439$	−7.87931e7	+	4.54912e7i	−0.931310	+	0.537692i	−0.0440846	+	0.999028i	$0.514037\pi$
$440$		0			0
$441$	−2.03630e7	+	2.00664e7i	−0.237425	+	0.233967i
$442$		0			0
$443$	8.56112e7	+	1.48283e8i	0.984735	+	1.70561i	0.643108	+	0.765776i	$0.277645\pi$
$443$	8.56112e7	+	1.48283e8i	0.984735	+	1.70561i	0.341627	+	0.939835i	$0.389022\pi$
$444$		0			0
$445$	2.01635e7	−	3.49242e7i	0.228816	−	0.396320i
$446$		0			0
$447$			1.26414e7i			0.141537i
$448$		0			0
$449$	1.66699e8			1.84159			0.920796	−	0.390045i	$-0.127540\pi$
$449$	1.66699e8			1.84159			0.920796	+	0.390045i	$0.127540\pi$
$450$		0			0
$451$	−1.32812e8	−	7.66793e7i	−1.44780	−	0.835888i
$452$		0			0
$453$	−2.97695e7	+	1.71874e7i	−0.320241	+	0.184891i
$454$		0			0
$455$	8.57556e6	−	6.33415e7i	0.0910391	−	0.672441i
$456$		0			0
$457$	−5.01139e7	−	8.67997e7i	−0.525060	−	0.909431i	−0.999574	−	0.0291831i	$-0.990709\pi$
$457$	−5.01139e7	−	8.67997e7i	−0.525060	−	0.909431i	0.474514	−	0.880248i	$-0.342624\pi$
$458$		0			0
$459$	−6.27873e6	+	1.08751e7i	−0.0649283	+	0.112459i
$460$		0			0
$461$		−	7.26582e7i		−	0.741621i	−0.928709	−	0.370810i	$-0.879080\pi$
$461$		−	7.26582e7i		−	0.741621i	0.928709	−	0.370810i	$-0.120920\pi$
$462$		0			0
$463$	4.61814e7			0.465291			0.232645	−	0.972562i	$-0.425262\pi$
$463$	4.61814e7			0.465291			0.232645	+	0.972562i	$0.425262\pi$
$464$		0			0
$465$	−1.15583e7	−	6.67319e6i	−0.114957	−	0.0663704i
$466$		0			0
$467$	8.53270e7	−	4.92635e7i	0.837791	−	0.483699i	−0.0187219	−	0.999825i	$-0.505960\pi$
$467$	8.53270e7	−	4.92635e7i	0.837791	−	0.483699i	0.856513	+	0.516126i	$0.172626\pi$
$468$		0			0
$469$	−1.52010e8	+	6.23123e7i	−1.47351	+	0.604025i
$470$		0			0
$471$	3.50197e7	+	6.06558e7i	0.335158	+	0.580510i
$472$		0			0
$473$	9.31774e7	−	1.61388e8i	0.880495	−	1.52506i
$474$		0			0
$475$		−	4.53033e7i		−	0.422716i
$476$		0			0
$477$	2.06158e7			0.189953
$478$		0			0
$479$	1.06047e7	+	6.12260e6i	0.0964917	+	0.0557095i	0.547469	−	0.836826i	$-0.315591\pi$
$479$	1.06047e7	+	6.12260e6i	0.0964917	+	0.0557095i	−0.450978	+	0.892535i	$0.648925\pi$
$480$		0			0
$481$	−1.07467e8	+	6.20460e7i	−0.965694	+	0.557544i
$482$		0			0
$483$	−5.59155e7	+	7.23197e7i	−0.496239	+	0.641823i
$484$		0			0
$485$	3.87265e7	+	6.70763e7i	0.339456	+	0.587955i
$486$		0			0
$487$	−6.09340e7	+	1.05541e8i	−0.527561	+	0.913763i	0.471923	+	0.881640i	$0.343560\pi$
$487$	−6.09340e7	+	1.05541e8i	−0.527561	+	0.913763i	−0.999484	+	0.0321229i	$0.989773\pi$
$488$		0			0
$489$			1.14715e8i			0.981059i
$490$		0			0
$491$	−1.25121e8			−1.05703			−0.528513	−	0.848925i	$-0.677250\pi$
$491$	−1.25121e8			−1.05703			−0.528513	+	0.848925i	$0.677250\pi$
$492$		0			0
$493$	−9.54534e7	−	5.51101e7i	−0.796619	−	0.459928i
$494$		0			0
$495$	−3.29923e7	+	1.90481e7i	−0.272017	+	0.157049i
$496$		0			0
$497$	−9.15363e7	−	7.07732e7i	−0.745632	−	0.576501i
$498$		0			0
$499$	−1.01359e8	−	1.75559e8i	−0.815758	−	1.41293i	−0.908782	−	0.417270i	$-0.862987\pi$
$499$	−1.01359e8	−	1.75559e8i	−0.815758	−	1.41293i	0.0930245	−	0.995664i	$-0.470347\pi$
$500$		0			0
$501$	1.92311e7	−	3.33092e7i	0.152929	−	0.264881i
$502$		0			0
$503$			1.91474e8i			1.50455i	0.658851	+	0.752274i	$0.271043\pi$
$503$			1.91474e8i			1.50455i	−0.658851	+	0.752274i	$0.728957\pi$
$504$		0			0
$505$	−6.43829e7			−0.499915
$506$		0			0
$507$	−2.26865e7	−	1.30980e7i	−0.174078	−	0.100504i
$508$		0			0
$509$	−8.96152e7	+	5.17394e7i	−0.679561	+	0.392345i	−0.799690	−	0.600414i	$-0.795003\pi$
$509$	−8.96152e7	+	5.17394e7i	−0.679561	+	0.392345i	0.120129	+	0.992758i	$0.461669\pi$
$510$		0			0
$511$	5.10909e7	+	1.24635e8i	0.382896	+	0.934069i
$512$		0			0
$513$	−8.34003e6	−	1.44454e7i	−0.0617754	−	0.106998i
$514$		0			0
$515$	3.50063e6	−	6.06326e6i	0.0256286	−	0.0443900i
$516$		0			0
$517$			3.54983e8i			2.56883i
$518$		0			0
$519$	1.09367e7			0.0782322
$520$		0			0
$521$	1.72264e8	+	9.94565e7i	1.21809	+	0.703267i	0.964510	−	0.264047i	$-0.0850573\pi$
$521$	1.72264e8	+	9.94565e7i	1.21809	+	0.703267i	0.253584	+	0.967313i	$0.418391\pi$
$522$		0			0
$523$	−1.02914e8	+	5.94177e7i	−0.719401	+	0.415346i	−0.814532	−	0.580118i	$-0.803006\pi$
$523$	−1.02914e8	+	5.94177e7i	−0.719401	+	0.415346i	0.0951311	+	0.995465i	$0.469673\pi$
$524$		0			0
$525$	−5.45123e7	−	7.38020e6i	−0.376718	−	0.0510023i
$526$		0			0
$527$	−1.94260e7	−	3.36469e7i	−0.132725	−	0.229886i
$528$		0			0
$529$	−7.21356e7	+	1.24942e8i	−0.487284	+	0.844001i
$530$		0			0
$531$			5.20049e6i			0.0347345i
$532$		0			0
$533$	1.82294e8			1.20390
$534$		0			0
$535$	5.86326e7	+	3.38515e7i	0.382893	+	0.221063i
$536$		0			0
$537$	3.04739e7	−	1.75941e7i	0.196791	−	0.113617i
$538$		0			0
$539$	−2.43390e8	−	6.71336e7i	−1.55430	−	0.428720i
$540$		0			0
$541$	−1.70729e6	−	2.95711e6i	−0.0107824	−	0.0186757i	0.860584	−	0.509309i	$-0.170099\pi$
$541$	−1.70729e6	−	2.95711e6i	−0.0107824	−	0.0186757i	−0.871366	+	0.490633i	$0.836766\pi$
$542$		0			0
$543$	−1.86736e7	+	3.23437e7i	−0.116635	+	0.202018i
$544$		0			0
$545$			1.71847e8i			1.06158i
$546$		0			0
$547$	1.24520e8			0.760813			0.380406	−	0.924819i	$-0.375784\pi$
$547$	1.24520e8			0.760813			0.380406	+	0.924819i	$0.375784\pi$
$548$		0			0
$549$	5.71947e7	+	3.30214e7i	0.345652	+	0.199562i
$550$		0			0
$551$	1.26791e8	−	7.32026e7i	0.757936	−	0.437594i
$552$		0			0
$553$	−463697.	+	3.42500e6i	−0.00274194	+	0.0202528i
$554$		0			0
$555$	−2.76985e7	−	4.79752e7i	−0.162023	−	0.280632i
$556$		0			0
$557$	−1.83383e7	+	3.17629e7i	−0.106119	+	0.183804i	−0.914195	−	0.405275i	$-0.867176\pi$
$557$	−1.83383e7	+	3.17629e7i	−0.106119	+	0.183804i	0.808076	+	0.589078i	$0.200509\pi$
$558$		0			0
$559$			2.21515e8i			1.26814i
$560$		0			0
$561$	−1.10900e8			−0.628121
$562$		0			0
$563$	4.08896e7	+	2.36076e7i	0.229133	+	0.132290i	0.610172	−	0.792269i	$-0.291100\pi$
$563$	4.08896e7	+	2.36076e7i	0.229133	+	0.132290i	−0.381039	+	0.924559i	$0.624434\pi$
$564$		0			0
$565$	−4.16841e7	+	2.40663e7i	−0.231113	+	0.133433i
$566$		0			0
$567$	−1.87404e7	+	7.68211e6i	−0.102809	+	0.0421436i
$568$		0			0
$569$	7.48737e7	+	1.29685e8i	0.406437	+	0.703969i	0.994488	−	0.104855i	$-0.0334379\pi$
$569$	7.48737e7	+	1.29685e8i	0.406437	+	0.703969i	−0.588051	+	0.808824i	$0.700105\pi$
$570$		0			0
$571$	4.86751e7	−	8.43077e7i	0.261456	−	0.452855i	−0.705173	−	0.709035i	$-0.749131\pi$
$571$	4.86751e7	−	8.43077e7i	0.261456	−	0.452855i	0.966629	+	0.256180i	$0.0824641\pi$
$572$		0			0
$573$			2.02279e7i			0.107519i
$574$		0			0
$575$	−1.75898e8			−0.925246
$576$		0			0
$577$	1.19272e8	+	6.88620e7i	0.620887	+	0.358469i	0.777214	−	0.629236i	$-0.216632\pi$
$577$	1.19272e8	+	6.88620e7i	0.620887	+	0.358469i	−0.156327	+	0.987705i	$0.549965\pi$
$578$		0			0
$579$	−4.04537e7	+	2.33560e7i	−0.208412	+	0.120327i
$580$		0			0
$581$	−1.63110e7	+	2.10963e7i	−0.0831673	+	0.107567i
$582$		0			0
$583$	9.10334e7	+	1.57674e8i	0.459404	+	0.795712i
$584$		0			0
$585$	2.26420e7	−	3.92171e7i	0.113096	−	0.195888i
$586$		0			0
$587$			2.35474e8i			1.16420i	0.813116	+	0.582101i	$0.197769\pi$
$587$			2.35474e8i			1.16420i	−0.813116	+	0.582101i	$0.802231\pi$
$588$		0			0
$589$	5.16071e7			0.252560
$590$		0			0
$591$	9.07530e7	+	5.23963e7i	0.439641	+	0.253827i
$592$		0			0
$593$	−1.56277e7	+	9.02267e6i	−0.0749431	+	0.0432684i	−0.537003	−	0.843580i	$-0.680444\pi$
$593$	−1.56277e7	+	9.02267e6i	−0.0749431	+	0.0432684i	0.462060	+	0.886849i	$0.347110\pi$
$594$		0			0
$595$	6.57151e7	+	5.08090e7i	0.311971	+	0.241207i
$596$		0			0
$597$	−8.74090e7	−	1.51397e8i	−0.410803	−	0.711531i
$598$		0			0
$599$	−4.75156e7	+	8.22994e7i	−0.221083	+	0.382927i	−0.955137	−	0.296164i	$-0.904293\pi$
$599$	−4.75156e7	+	8.22994e7i	−0.221083	+	0.382927i	0.734054	+	0.679091i	$0.237626\pi$
$600$		0			0
$601$		−	3.02566e8i		−	1.39379i	−0.717175	−	0.696894i	$-0.754565\pi$
$601$		−	3.02566e8i		−	1.39379i	0.717175	−	0.696894i	$-0.245435\pi$
$602$		0			0
$603$	−1.16389e8			−0.530836
$604$		0			0
$605$	−1.79289e8	−	1.03512e8i	−0.809631	−	0.467441i
$606$		0			0
$607$	5.14464e7	−	2.97026e7i	0.230032	−	0.132809i	−0.380555	−	0.924758i	$-0.624267\pi$
$607$	5.14464e7	−	2.97026e7i	0.230032	−	0.132809i	0.610587	+	0.791949i	$0.290934\pi$
$608$		0			0
$609$	−6.74279e7	−	1.64489e8i	−0.298530	−	0.728258i
$610$		0			0
$611$	−2.10980e8	−	3.65427e8i	−0.924947	−	1.60206i
$612$		0			0
$613$	9.34358e7	−	1.61836e8i	0.405632	−	0.702575i	−0.588763	−	0.808306i	$-0.700385\pi$
$613$	9.34358e7	−	1.61836e8i	0.405632	−	0.702575i	0.994395	+	0.105731i	$0.0337182\pi$
$614$		0			0
$615$			8.13792e7i			0.349855i
$616$		0			0
$617$	7.25028e7			0.308674			0.154337	−	0.988018i	$-0.450676\pi$
$617$	7.25028e7			0.308674			0.154337	+	0.988018i	$0.450676\pi$
$618$		0			0
$619$	−1.69087e8	−	9.76227e7i	−0.712918	−	0.411603i	0.0992227	−	0.995065i	$-0.468364\pi$
$619$	−1.69087e8	−	9.76227e7i	−0.712918	−	0.411603i	−0.812140	+	0.583462i	$0.801698\pi$
$620$		0			0
$621$	−5.60867e7	+	3.23817e7i	−0.234199	+	0.135215i
$622$		0			0
$623$	1.87632e8	+	2.54028e7i	0.775967	+	0.105055i
$624$		0			0
$625$	−1.12306e7	−	1.94520e7i	−0.0460005	−	0.0796752i
$626$		0			0
$627$	7.36542e7	−	1.27573e8i	0.298810	−	0.517554i
$628$		0			0
$629$		−	1.61264e8i		−	0.648015i
$630$		0			0
$631$	7.65603e7			0.304730			0.152365	−	0.988324i	$-0.451311\pi$
$631$	7.65603e7			0.304730			0.152365	+	0.988324i	$0.451311\pi$
$632$		0			0
$633$	3.68740e6	+	2.12892e6i	0.0145381	+	0.00839360i
$634$		0			0
$635$	−1.96415e8	+	1.13401e8i	−0.767104	+	0.442888i
$636$		0			0
$637$	2.90451e8	−	7.55468e7i	1.12371	−	0.292279i
$638$		0			0
$639$	−4.09860e7	−	7.09899e7i	−0.157085	−	0.272078i
$640$		0			0
$641$	−1.89649e8	+	3.28481e8i	−0.720071	+	1.24720i	0.240899	+	0.970550i	$0.422558\pi$
$641$	−1.89649e8	+	3.28481e8i	−0.720071	+	1.24720i	−0.960971	+	0.276650i	$0.910776\pi$
$642$		0			0
$643$		−	2.99610e7i		−	0.112700i	−0.998411	−	0.0563500i	$-0.982054\pi$
$643$		−	2.99610e7i		−	0.112700i	0.998411	−	0.0563500i	$-0.0179463\pi$
$644$		0			0
$645$	−9.88885e7			−0.368525
$646$		0			0
$647$	3.22339e7	+	1.86103e7i	0.119015	+	0.0687132i	0.558326	−	0.829622i	$-0.311444\pi$
$647$	3.22339e7	+	1.86103e7i	0.119015	+	0.0687132i	−0.439311	+	0.898335i	$0.644777\pi$
$648$		0			0
$649$	−3.97745e7	+	2.29638e7i	−0.145503	+	0.0840060i
$650$		0			0
$651$	8.40714e6	−	6.20976e7i	0.0304723	−	0.225077i
$652$		0			0
$653$	9.92694e7	+	1.71940e8i	0.356513	+	0.617499i	0.987376	−	0.158396i	$-0.0506321\pi$
$653$	9.92694e7	+	1.71940e8i	0.356513	+	0.617499i	−0.630862	+	0.775895i	$0.717299\pi$
$654$		0			0
$655$	−1.28034e8	+	2.21761e8i	−0.455618	+	0.789153i
$656$		0			0
$657$			9.54293e7i			0.336501i
$658$		0			0
$659$	5.59686e8			1.95564			0.977818	−	0.209454i	$-0.0671687\pi$
$659$	5.59686e8			1.95564			0.977818	+	0.209454i	$0.0671687\pi$
$660$		0			0
$661$	−1.31767e8	−	7.60755e7i	−0.456248	−	0.263415i	0.254217	−	0.967147i	$-0.418182\pi$
$661$	−1.31767e8	−	7.60755e7i	−0.456248	−	0.263415i	−0.710465	+	0.703732i	$0.751516\pi$
$662$		0			0
$663$	1.14163e8	−	6.59122e7i	0.391729	−	0.226165i
$664$		0			0
$665$	−1.02092e8	+	4.18499e7i	−0.347158	+	0.142308i
$666$		0			0
$667$	−2.84222e8	−	4.92287e8i	−0.957813	−	1.65898i
$668$		0			0
$669$	−1.65499e8	+	2.86652e8i	−0.552733	+	0.957362i
$670$		0			0
$671$			5.83251e8i			1.93058i
$672$		0			0
$673$	2.07554e8			0.680903			0.340451	−	0.940262i	$-0.389420\pi$
$673$	2.07554e8			0.680903			0.340451	+	0.940262i	$0.389420\pi$
$674$		0			0
$675$	−3.37506e7	−	1.94859e7i	−0.109741	−	0.0633592i
$676$		0			0
$677$	1.72463e8	−	9.95717e7i	0.555815	−	0.320900i	−0.195649	−	0.980674i	$-0.562681\pi$
$677$	1.72463e8	−	9.95717e7i	0.555815	−	0.320900i	0.751464	+	0.659774i	$0.229348\pi$
$678$		0			0
$679$	−2.22438e8	+	2.87696e8i	−0.710559	+	0.919019i
$680$		0			0
$681$	4.21290e7	+	7.29696e7i	0.133395	+	0.231047i
$682$		0			0
$683$	2.58231e8	−	4.47269e8i	0.810486	−	1.40380i	−0.102037	−	0.994781i	$-0.532536\pi$
$683$	2.58231e8	−	4.47269e8i	0.810486	−	1.40380i	0.912524	−	0.409023i	$-0.134130\pi$
$684$		0			0
$685$			2.08778e7i			0.0649550i
$686$		0			0
$687$	3.61573e8			1.11513
$688$		0			0
$689$	−1.87424e8	−	1.08209e8i	−0.573017	−	0.330831i
$690$		0			0
$691$	5.07493e8	−	2.93001e8i	1.53814	−	0.888046i	0.539192	−	0.842183i	$-0.318730\pi$
$691$	5.07493e8	−	2.93001e8i	1.53814	−	0.888046i	0.998948	−	0.0458631i	$-0.0146038\pi$
$692$		0			0
$693$	−1.41507e8	−	1.09409e8i	−0.425184	−	0.328740i
$694$		0			0
$695$	−9.31026e7	−	1.61259e8i	−0.277337	−	0.480362i
$696$		0			0
$697$	−1.18450e8	+	2.05161e8i	−0.349813	+	0.605894i
$698$		0			0
$699$			1.17761e8i			0.344801i
$700$		0			0
$701$	8.34906e7			0.242373			0.121186	−	0.992630i	$-0.461330\pi$
$701$	8.34906e7			0.242373			0.121186	+	0.992630i	$0.461330\pi$
$702$		0			0
$703$	1.85508e8	+	1.07103e8i	0.533945	+	0.308274i
$704$		0			0
$705$	1.63133e8	−	9.41851e7i	0.465560	−	0.268791i
$706$		0			0
$707$	−1.14657e8	−	2.79703e8i	−0.324445	−	0.791479i
$708$		0			0
$709$	−7.43808e7	−	1.28831e8i	−0.208700	−	0.361479i	0.742605	−	0.669729i	$-0.233590\pi$
$709$	−7.43808e7	−	1.28831e8i	−0.208700	−	0.361479i	−0.951305	+	0.308251i	$0.900256\pi$
$710$		0			0
$711$	−1.22430e6	+	2.12054e6i	−0.00340626	+	0.00589981i
$712$		0			0
$713$		−	2.00374e8i		−	0.552806i
$714$		0			0
$715$	3.99922e8			1.09410
$716$		0			0
$717$	3.57260e8	+	2.06264e8i	0.969230	+	0.559585i
$718$		0			0
$719$	3.24690e8	−	1.87460e8i	0.873540	−	0.504338i	0.00501672	−	0.999987i	$-0.498403\pi$
$719$	3.24690e8	−	1.87460e8i	0.873540	−	0.504338i	0.868523	+	0.495649i	$0.165070\pi$
$720$		0			0
$721$	3.25752e7	+	4.41023e6i	0.0869123	+	0.0117667i
$722$		0			0
$723$	−4.67089e7	−	8.09022e7i	−0.123590	−	0.214065i
$724$		0			0
$725$	1.71033e8	−	2.96238e8i	0.448813	−	0.777367i
$726$		0			0
$727$		−	5.14759e8i		−	1.33968i	−0.742506	−	0.669840i	$-0.766363\pi$
$727$		−	5.14759e8i		−	1.33968i	0.742506	−	0.669840i	$-0.233637\pi$
$728$		0			0
$729$	−1.43489e7			−0.0370370
$730$		0			0
$731$	−2.49303e8	−	1.43935e8i	−0.638227	−	0.368481i
$732$		0			0
$733$	2.31672e8	−	1.33756e8i	0.588249	−	0.339626i	−0.176156	−	0.984362i	$-0.556366\pi$
$733$	2.31672e8	−	1.33756e8i	0.588249	−	0.339626i	0.764405	+	0.644737i	$0.223033\pi$
$734$		0			0
$735$	3.37255e7	+	1.29663e8i	0.0849368	+	0.326552i
$736$		0			0
$737$	−5.13940e8	−	8.90170e8i	−1.28384	−	2.22367i
$738$		0			0
$739$	−2.45455e8	+	4.25141e8i	−0.608189	+	1.05341i	0.383349	+	0.923603i	$0.374770\pi$
$739$	−2.45455e8	+	4.25141e8i	−0.608189	+	1.05341i	−0.991539	+	0.129811i	$0.958563\pi$
$740$		0			0
$741$			1.75102e8i			0.430365i
$742$		0			0
$743$	−8.00568e7			−0.195178			−0.0975892	−	0.995227i	$-0.531113\pi$
$743$	−8.00568e7			−0.195178			−0.0975892	+	0.995227i	$0.531113\pi$
$744$		0			0
$745$	5.13050e7	+	2.96210e7i	0.124077	+	0.0716358i
$746$		0			0
$747$	−1.63610e7	+	9.44600e6i	−0.0392506	+	0.0226614i
$748$		0			0
$749$	−4.26475e7	+	3.15007e8i	−0.101496	+	0.749677i
$750$		0			0
$751$	−2.18469e8	−	3.78400e8i	−0.515788	−	0.893370i	−0.999832	−	0.0183270i	$-0.994166\pi$
$751$	−2.18469e8	−	3.78400e8i	−0.515788	−	0.893370i	0.484044	−	0.875043i	$-0.339167\pi$
$752$		0			0
$753$	1.92706e8	−	3.33777e8i	0.451348	−	0.781757i
$754$		0			0
$755$			1.61093e8i			0.374314i
$756$		0			0
$757$	−1.56135e8			−0.359925			−0.179962	−	0.983673i	$-0.557598\pi$
$757$	−1.56135e8			−0.359925			−0.179962	+	0.983673i	$0.557598\pi$
$758$		0			0
$759$	−4.95325e8	−	2.85976e8i	−1.13283	−	0.654039i
$760$		0			0
$761$	−3.13313e8	+	1.80891e8i	−0.710926	+	0.410453i	−0.811404	−	0.584486i	$-0.801296\pi$
$761$	−3.13313e8	+	1.80891e8i	−0.710926	+	0.410453i	0.100478	+	0.994939i	$0.467963\pi$
$762$		0			0
$763$	−7.46566e8	+	3.06034e8i	−1.68072	+	0.688964i
$764$		0			0
$765$	2.94244e7	+	5.09645e7i	0.0657239	+	0.113837i
$766$		0			0
$767$	2.72966e7	−	4.72791e7i	0.0604954	−	0.104781i
$768$		0			0
$769$			1.92526e8i			0.423361i	0.977339	+	0.211680i	$0.0678936\pi$
$769$			1.92526e8i			0.423361i	−0.977339	+	0.211680i	$0.932106\pi$
$770$		0			0
$771$	2.35465e8			0.513764
$772$		0			0
$773$	3.54462e8	+	2.04648e8i	0.767416	+	0.443068i	0.831952	−	0.554848i	$-0.187223\pi$
$773$	3.54462e8	+	2.04648e8i	0.767416	+	0.443068i	−0.0645363	+	0.997915i	$0.520557\pi$
$774$		0			0
$775$	1.04422e8	−	6.02883e7i	0.224331	−	0.129517i
$776$		0			0
$777$	1.59095e8	−	2.05770e8i	0.339151	−	0.438650i
$778$		0			0
$779$	−1.57337e8	−	2.72515e8i	−0.332826	−	0.576471i
$780$		0			0
$781$	3.61965e8	−	6.26941e8i	0.759824	−	1.31605i
$782$		0			0
$783$		−	1.25944e8i		−	0.262357i
$784$		0			0
$785$	3.28229e8			0.678529
$786$		0			0
$787$	−2.32826e7	−	1.34422e7i	−0.0477648	−	0.0275770i	0.475927	−	0.879485i	$-0.342112\pi$
$787$	−2.32826e7	−	1.34422e7i	−0.0477648	−	0.0275770i	−0.523692	+	0.851908i	$0.675446\pi$
$788$		0			0
$789$	1.89431e8	−	1.09368e8i	0.385674	−	0.222669i
$790$		0			0
$791$	−1.78787e8	−	1.38233e8i	−0.361248	−	0.279306i
$792$		0			0
$793$	−3.46648e8	−	6.00413e8i	−0.695136	−	1.20401i
$794$		0			0
$795$	4.83066e7	−	8.36694e7i	0.0961401	−	0.166520i
$796$		0			0
$797$		−	3.01544e8i		−	0.595630i	−0.954624	−	0.297815i	$-0.903742\pi$
$797$		−	3.01544e8i		−	0.595630i	0.954624	−	0.297815i	$-0.0962578\pi$
$798$		0			0
$799$	5.48357e8			1.07504
$800$		0			0
$801$	1.16170e8	+	6.70708e7i	0.226046	+	0.130508i
$802$		0			0
$803$	−7.29865e8	+	4.21388e8i	−1.40960	+	0.813833i
$804$		0			0
$805$	1.62490e8	+	3.96391e8i	0.311486	+	0.759865i
$806$		0			0
$807$	1.92668e8	+	3.33711e8i	0.366597	+	0.634965i
$808$		0			0
$809$	1.91561e8	−	3.31793e8i	0.361794	−	0.626646i	−0.626462	−	0.779452i	$-0.715498\pi$
$809$	1.91561e8	−	3.31793e8i	0.361794	−	0.626646i	0.988256	+	0.152806i	$0.0488310\pi$
$810$		0			0
$811$		−	2.36880e8i		−	0.444085i	−0.975037	−	0.222042i	$-0.928728\pi$
$811$		−	2.36880e8i		−	0.444085i	0.975037	−	0.222042i	$-0.0712723\pi$
$812$		0			0
$813$	5.33195e8			0.992235
$814$		0			0
$815$	4.65573e8	+	2.68799e8i	0.860033	+	0.496540i
$816$		0			0
$817$	3.31149e8	−	1.91189e8i	0.607235	−	0.350587i
$818$		0			0
$819$	2.10696e8	+	2.85253e7i	0.383535	+	0.0519252i
$820$		0			0
$821$	−3.06951e8	−	5.31655e8i	−0.554676	−	0.960727i	−0.997929	−	0.0643305i	$-0.979509\pi$
$821$	−3.06951e8	−	5.31655e8i	−0.554676	−	0.960727i	0.443252	−	0.896397i	$-0.353825\pi$
$822$		0			0
$823$	3.29240e8	−	5.70261e8i	0.590628	−	1.02300i	−0.403520	−	0.914971i	$-0.632214\pi$
$823$	3.29240e8	−	5.70261e8i	0.590628	−	1.02300i	0.994148	−	0.108026i	$-0.0344531\pi$
$824$		0			0
$825$		−	3.44176e8i		−	0.612941i
$826$		0			0
$827$	6.88166e7			0.121668			0.0608341	−	0.998148i	$-0.480624\pi$
$827$	6.88166e7			0.121668			0.0608341	+	0.998148i	$0.480624\pi$
$828$		0			0
$829$	−4.17661e8	−	2.41137e8i	−0.733096	−	0.423253i	0.0864579	−	0.996256i	$-0.472445\pi$
$829$	−4.17661e8	−	2.41137e8i	−0.733096	−	0.423253i	−0.819554	+	0.573003i	$0.805779\pi$
$830$		0			0
$831$	−5.16009e8	+	2.97918e8i	−0.899196	+	0.519151i
$832$		0			0
$833$	−1.03704e8	+	3.75974e8i	−0.179416	+	0.650464i
$834$		0			0
$835$	−9.01238e7	−	1.56099e8i	−0.154803	−	0.268127i
$836$		0			0
$837$	2.21973e7	−	3.84469e7i	0.0378551	−	0.0655670i
$838$		0			0
$839$			2.98764e8i			0.505874i	0.967483	+	0.252937i	$0.0813965\pi$
$839$			2.98764e8i			0.505874i	−0.967483	+	0.252937i	$0.918603\pi$
$840$		0			0
$841$	5.10621e8			0.858441
$842$		0			0
$843$	−5.92502e7	−	3.42081e7i	−0.0989025	−	0.0571014i
$844$		0			0
$845$	−1.06317e8	+	6.13821e7i	−0.176211	+	0.101735i
$846$		0			0
$847$	1.30409e8	−	9.63239e8i	0.214614	−	1.58520i
$848$		0			0
$849$	−1.73626e8	−	3.00730e8i	−0.283722	−	0.491421i
$850$		0			0
$851$	4.15847e8	−	7.20268e8i	0.674753	−	1.16871i
$852$		0			0
$853$			1.21454e9i			1.95688i	0.206537	+	0.978439i	$0.433781\pi$
$853$			1.21454e9i			1.95688i	−0.206537	+	0.978439i	$0.566219\pi$
$854$		0			0
$855$	−7.81687e7			−0.125065
$856$		0			0
$857$	2.95805e8	+	1.70783e8i	0.469962	+	0.271333i	0.716224	−	0.697871i	$-0.245869\pi$
$857$	2.95805e8	+	1.70783e8i	0.469962	+	0.271333i	−0.246262	+	0.969203i	$0.579202\pi$
$858$		0			0
$859$	4.92186e7	−	2.84164e7i	0.0776515	−	0.0448321i	−0.460672	−	0.887571i	$-0.652391\pi$
$859$	4.92186e7	−	2.84164e7i	0.0776515	−	0.0448321i	0.538323	+	0.842739i	$0.319058\pi$
$860$		0			0
$861$	−3.53542e8	+	1.44925e8i	−0.553900	+	0.227056i
$862$		0			0
$863$	2.73857e8	+	4.74335e8i	0.426081	+	0.737993i	0.996521	−	0.0833459i	$-0.0265606\pi$
$863$	2.73857e8	+	4.74335e8i	0.426081	+	0.737993i	−0.570440	+	0.821339i	$0.693227\pi$
$864$		0			0
$865$	2.56267e7	−	4.43868e7i	0.0395954	−	0.0685812i
$866$		0			0
$867$		−	2.04955e8i		−	0.314486i
$868$		0			0
$869$	−2.16245e7			−0.0329524
$870$		0			0
$871$	1.05812e9	+	6.10908e8i	1.60133	+	0.924530i
$872$		0			0
$873$	−2.23119e8	+	1.28818e8i	−0.335347	+	0.193613i
$874$		0			0
$875$	−3.97164e8	+	5.13682e8i	−0.592851	+	0.766779i
$876$		0			0
$877$	2.77263e8	+	4.80234e8i	0.411049	+	0.711958i	0.995005	−	0.0998278i	$-0.0318292\pi$
$877$	2.77263e8	+	4.80234e8i	0.411049	+	0.711958i	−0.583956	+	0.811785i	$0.698496\pi$
$878$		0			0
$879$	3.14317e8	−	5.44414e8i	0.462809	−	0.801608i
$880$		0			0
$881$			1.02196e9i			1.49454i	0.664521	+	0.747270i	$0.268636\pi$
$881$			1.02196e9i			1.49454i	−0.664521	+	0.747270i	$0.731364\pi$
$882$		0			0
$883$	9.10455e8			1.32244			0.661220	−	0.750192i	$-0.270039\pi$
$883$	9.10455e8			1.32244			0.661220	+	0.750192i	$0.270039\pi$
$884$		0			0
$885$	2.11062e7	+	1.21857e7i	0.0304495	+	0.0175801i
$886$		0			0
$887$	5.43780e7	−	3.13951e7i	0.0779206	−	0.0449875i	−0.460533	−	0.887642i	$-0.652342\pi$
$887$	5.43780e7	−	3.13951e7i	0.0779206	−	0.0449875i	0.538454	+	0.842655i	$0.319009\pi$
$888$		0			0
$889$	−8.42442e8	−	6.51352e8i	−1.19904	−	0.927066i
$890$		0			0
$891$	−6.33606e7	−	1.09744e8i	−0.0895748	−	0.155148i
$892$		0			0
$893$	−3.64191e8	+	6.30797e8i	−0.511416	+	0.885798i
$894$		0			0
$895$		−	1.64905e8i		−	0.230019i
$896$		0			0
$897$	6.79865e8			0.941988
$898$		0			0
$899$	3.37459e8	+	1.94832e8i	0.464453	+	0.268152i
$900$		0			0
$901$	2.43566e8	−	1.40623e8i	0.332999	−	0.192257i
$902$		0			0
$903$	−1.76106e8	−	4.29609e8i	−0.239173	−	0.583458i
$904$		0			0
$905$	8.75114e7	+	1.51574e8i	0.118064	+	0.204493i
$906$		0			0
$907$	−2.52448e8	+	4.37253e8i	−0.338338	+	0.586018i	−0.984120	−	0.177503i	$-0.943198\pi$
$907$	−2.52448e8	+	4.37253e8i	−0.338338	+	0.586018i	0.645782	+	0.763521i	$0.276531\pi$
$908$		0			0
$909$		−	2.14160e8i		−	0.285132i
$910$		0			0
$911$	−1.07163e9			−1.41739			−0.708694	−	0.705516i	$-0.750715\pi$
$911$	−1.07163e9			−1.41739			−0.708694	+	0.705516i	$0.750715\pi$
$912$		0			0
$913$	−1.44490e8	−	8.34215e7i	−0.189857	−	0.109614i
$914$		0			0
$915$	2.68035e8	−	1.54750e8i	0.349887	−	0.202008i
$916$		0			0
$917$	−1.19142e9	−	1.61302e8i	−1.54510	−	0.209185i
$918$		0			0
$919$	2.08740e8	+	3.61547e8i	0.268942	+	0.465821i	0.968589	−	0.248668i	$-0.0799928\pi$
$919$	2.08740e8	+	3.61547e8i	0.268942	+	0.465821i	−0.699647	+	0.714489i	$0.746659\pi$
$920$		0			0
$921$	−1.55818e8	+	2.69884e8i	−0.199452	+	0.345461i
$922$		0			0
$923$			8.60517e8i			1.09435i
$924$		0			0
$925$	5.00479e8			0.632354
$926$		0			0
$927$	2.01685e7	+	1.16443e7i	0.0253183	+	0.0146175i
$928$		0			0
$929$	−8.29914e8	+	4.79151e8i	−1.03511	+	0.597620i	−0.918444	−	0.395551i	$-0.870554\pi$
$929$	−8.29914e8	+	4.79151e8i	−1.03511	+	0.597620i	−0.116664	+	0.993171i	$0.537220\pi$
$930$		0			0
$931$	−3.63623e8	−	3.68998e8i	−0.450612	−	0.457272i
$932$		0			0
$933$	9.44462e7	+	1.63586e8i	0.116289	+	0.201419i
$934$		0			0
$935$	−2.59859e8	+	4.50089e8i	−0.317909	+	0.550634i
$936$		0			0
$937$			1.53993e9i			1.87190i	0.352130	+	0.935951i	$0.385457\pi$
$937$			1.53993e9i			1.87190i	−0.352130	+	0.935951i	$0.614543\pi$
$938$		0			0
$939$	8.84177e8			1.06793
$940$		0			0
$941$	−8.92799e8	−	5.15458e8i	−1.07148	−	0.618620i	−0.142896	−	0.989738i	$-0.545641\pi$
$941$	−8.92799e8	−	5.15458e8i	−1.07148	−	0.618620i	−0.928586	+	0.371117i	$0.878975\pi$
$942$		0			0
$943$	−1.05809e9	+	6.10887e8i	−1.26179	+	0.728494i
$944$		0			0
$945$	−1.27342e7	+	9.40585e7i	−0.0150895	+	0.111456i
$946$		0			0
$947$	1.43760e8	+	2.48999e8i	0.169273	+	0.293189i	0.938164	−	0.346190i	$-0.112525\pi$
$947$	1.43760e8	+	2.48999e8i	0.169273	+	0.293189i	−0.768892	+	0.639379i	$0.779191\pi$
$948$		0			0
$949$	5.00894e8	−	8.67573e8i	0.586067	−	1.01510i
$950$		0			0
$951$		−	3.86948e8i		−	0.449895i
$952$		0			0
$953$	4.50715e8			0.520743			0.260371	−	0.965509i	$-0.416155\pi$
$953$	4.50715e8			0.520743			0.260371	+	0.965509i	$0.416155\pi$
$954$		0			0
$955$	8.20949e7	+	4.73975e7i	0.0942554	+	0.0544184i
$956$		0			0
$957$	9.63249e8	−	5.56132e8i	1.09901	−	0.634515i
$958$		0			0
$959$	−9.07008e7	+	3.71803e7i	−0.102838	+	0.0421558i
$960$		0			0
$961$	−3.75075e8	−	6.49648e8i	−0.422617	−	0.731995i
$962$		0			0
$963$	−1.12602e8	+	1.95032e8i	−0.126086	+	0.218387i
$964$		0			0
$965$			2.18909e8i			0.243602i
$966$		0			0
$967$	−8.35037e8			−0.923478			−0.461739	−	0.887016i	$-0.652774\pi$
$967$	−8.35037e8			−0.923478			−0.461739	+	0.887016i	$0.652774\pi$
$968$		0			0
$969$	−1.97067e8	−	1.13777e8i	−0.216592	−	0.125050i
$970$		0			0
$971$	−6.46725e7	+	3.73387e7i	−0.0706419	+	0.0407851i	−0.534905	−	0.844912i	$-0.679653\pi$
$971$	−6.46725e7	+	3.73387e7i	−0.0706419	+	0.0407851i	0.464263	+	0.885697i	$0.346319\pi$
$972$		0			0
$973$	5.34765e8	−	6.91651e8i	0.580530	−	0.750842i
$974$		0			0
$975$	2.04557e8	+	3.54303e8i	0.220699	+	0.382262i
$976$		0			0
$977$	−2.79211e8	+	4.83607e8i	−0.299398	+	0.518572i	−0.975998	−	0.217778i	$-0.930119\pi$
$977$	−2.79211e8	+	4.83607e8i	−0.299398	+	0.518572i	0.676601	+	0.736350i	$0.263452\pi$
$978$		0			0
$979$			1.18466e9i			1.26254i
$980$		0			0
$981$	−5.71622e8			−0.605483
$982$		0			0
$983$	−1.49108e9	−	8.60876e8i	−1.56979	−	0.906317i	−0.996193	−	0.0871772i	$-0.972215\pi$
$983$	−1.49108e9	−	8.60876e8i	−1.56979	−	0.906317i	−0.573594	−	0.819140i	$-0.694451\pi$
$984$		0			0
$985$	4.25301e8	−	2.45548e8i	0.445029	−	0.256937i
$986$		0			0
$987$	6.99693e8	+	5.40982e8i	0.727706	+	0.562641i
$988$		0			0
$989$	−7.42324e8	−	1.28574e9i	−0.767370	−	1.32912i
$990$		0			0
$991$	3.95954e8	−	6.85813e8i	0.406841	−	0.704668i	−0.587693	−	0.809084i	$-0.699964\pi$
$991$	3.95954e8	−	6.85813e8i	0.406841	−	0.704668i	0.994534	+	0.104415i	$0.0332971\pi$
$992$		0			0
$993$		−	1.41277e8i		−	0.144286i
$994$		0			0
$995$	−8.19260e8			−0.831673
$996$		0			0
$997$	−1.41568e9	−	8.17343e8i	−1.42850	−	0.824743i	−0.431494	−	0.902116i	$-0.642014\pi$
$997$	−1.41568e9	−	8.17343e8i	−1.42850	−	0.824743i	−0.997002	+	0.0773726i	$0.975347\pi$
$998$		0			0
$999$	1.59582e8	−	9.21348e7i	0.160062	−	0.0924117i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.7.bh.f.241.2		8
4.3	odd	2		42.7.g.a.31.3	yes	8
7.5	odd	6	inner	336.7.bh.f.145.2		8
12.11	even	2		126.7.n.a.73.2		8
28.3	even	6		294.7.c.b.97.2		8
28.11	odd	6		294.7.c.b.97.3		8
28.19	even	6		42.7.g.a.19.3	✓	8
28.23	odd	6		294.7.g.d.19.4		8
28.27	even	2		294.7.g.d.31.4		8
84.47	odd	6		126.7.n.a.19.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.7.g.a.19.3	✓	8	28.19	even	6
42.7.g.a.31.3	yes	8	4.3	odd	2
126.7.n.a.19.2		8	84.47	odd	6
126.7.n.a.73.2		8	12.11	even	2
294.7.c.b.97.2		8	28.3	even	6
294.7.c.b.97.3		8	28.11	odd	6
294.7.g.d.19.4		8	28.23	odd	6
294.7.g.d.31.4		8	28.27	even	2
336.7.bh.f.145.2		8	7.5	odd	6	inner
336.7.bh.f.241.2		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(13.5000 + 7.79423i) q^{3} +(63.2658 - 36.5265i) q^{5} +(271.352 + 209.802i) q^{7} +(121.500 + 210.444i) q^{9} +O(q^{10})\)	\(q+(13.5000 + 7.79423i) q^{3} +(63.2658 - 36.5265i) q^{5} +(271.352 + 209.802i) q^{7} +(121.500 + 210.444i) q^{9} +(-1073.02 + 1858.52i) q^{11} -2550.94i q^{13} +1138.79 q^{15} +(2870.93 + 1657.53i) q^{17} +(-3813.46 + 2201.70i) q^{19} +(2028.01 + 4947.31i) q^{21} +(8548.49 + 14806.4i) q^{23} +(-5144.12 + 8909.88i) q^{25} +3788.00i q^{27} -33248.2 q^{29} +(-10149.7 - 5859.92i) q^{31} +(-28971.4 + 16726.7i) q^{33} +(24830.7 + 3361.73i) q^{35} +(-24322.8 - 42128.4i) q^{37} +(19882.6 - 34437.7i) q^{39} +71461.4i q^{41} -86836.9 q^{43} +(15373.6 + 8875.95i) q^{45} +(143252. - 82706.7i) q^{47} +(29615.3 + 113861. i) q^{49} +(25838.4 + 44753.4i) q^{51} +(42419.4 - 73472.5i) q^{53} +156774. i q^{55} -68642.2 q^{57} +(18534.0 + 10700.6i) q^{59} +(235369. - 135891. i) q^{61} +(-11182.3 + 82595.5i) q^{63} +(-93177.0 - 161387. i) q^{65} +(-239484. + 414798. i) q^{67} +266516. i q^{69} -337334. q^{71} +(340100. + 196357. i) q^{73} +(-138891. + 80188.9i) q^{75} +(-681087. + 279193. i) q^{77} +(5038.26 + 8726.52i) q^{79} +(-29524.5 + 51137.9i) q^{81} +77744.9i q^{83} +242176. q^{85} +(-448851. - 259144. i) q^{87} +(478066. - 276012. i) q^{89} +(535192. - 692203. i) q^{91} +(-91347.1 - 158218. i) q^{93} +(-160841. + 278585. i) q^{95} +1.06023e6i q^{97} -521486. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 108 q^{3} + 462 q^{5} - 580 q^{7} + 972 q^{9}+O(q^{10}) \) 8 * q + 108 * q^3 + 462 * q^5 - 580 * q^7 + 972 * q^9	\( 8 q + 108 q^{3} + 462 q^{5} - 580 q^{7} + 972 q^{9} - 1806 q^{11} + 8316 q^{15} + 9564 q^{17} - 23022 q^{19} - 1350 q^{21} - 2400 q^{23} + 32762 q^{25} - 2484 q^{29} - 148416 q^{31} - 48762 q^{33} - 11412 q^{35} - 84046 q^{37} + 75222 q^{39} - 92972 q^{43} + 112266 q^{45} + 323124 q^{47} - 8644 q^{49} + 86076 q^{51} + 358086 q^{53} - 414396 q^{57} - 719382 q^{59} + 421536 q^{61} + 104490 q^{63} - 322740 q^{65} - 267010 q^{67} + 464664 q^{71} + 1944486 q^{73} + 884574 q^{75} - 1713498 q^{77} + 685904 q^{79} - 236196 q^{81} - 3876168 q^{85} - 33534 q^{87} + 4130604 q^{89} + 484266 q^{91} - 1335744 q^{93} + 2105232 q^{95} - 877716 q^{99}+O(q^{100}) \) 8 * q + 108 * q^3 + 462 * q^5 - 580 * q^7 + 972 * q^9 - 1806 * q^11 + 8316 * q^15 + 9564 * q^17 - 23022 * q^19 - 1350 * q^21 - 2400 * q^23 + 32762 * q^25 - 2484 * q^29 - 148416 * q^31 - 48762 * q^33 - 11412 * q^35 - 84046 * q^37 + 75222 * q^39 - 92972 * q^43 + 112266 * q^45 + 323124 * q^47 - 8644 * q^49 + 86076 * q^51 + 358086 * q^53 - 414396 * q^57 - 719382 * q^59 + 421536 * q^61 + 104490 * q^63 - 322740 * q^65 - 267010 * q^67 + 464664 * q^71 + 1944486 * q^73 + 884574 * q^75 - 1713498 * q^77 + 685904 * q^79 - 236196 * q^81 - 3876168 * q^85 - 33534 * q^87 + 4130604 * q^89 + 484266 * q^91 - 1335744 * q^93 + 2105232 * q^95 - 877716 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more