LMFDB - Embedded newform 312.6.q.d.217.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [312,6,Mod(217,312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("312.217"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	312.q (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,90] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$50.0397517816$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 45625 x^{18} - 258130 x^{17} + 874261585 x^{16} + 8836032648 x^{15} - 9129028858830 x^{14} + \cdots + 26\!\cdots\!43 $ x^20 - 45625x^18 - 258130x^17 + 874261585x^16 + 8836032648x^15 - 9129028858830x^14 - 118714904615180x^13 + 56637774639083265x^12 + 793081367997391880x^11 - 215324381403097127519x^10 - 2781362820580322566070x^9 + 504016461228571684639795x^8 + 5073527973108012320509800x^7 - 707056493606854944779576550x^6 - 4366052589943248282014274900x^5 + 549723668065017075497674094355x^4 + 1316164761437113609151985766800x^3 - 199931629936443700846273893579105x^2 - 100383689835189661319905240398330x + 26657550825189106770810531324990243
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{44}\cdot 3\cdot 13 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.8
Root			$50.0867 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	312.217
Dual form			312.6.q.d.289.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.50000 - 7.79423i) q^{3} +49.5867 q^{5} +(-115.331 - 199.759i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(202.563 - 350.850i) q^{11} +(599.553 + 108.761i) q^{13} +(223.140 - 386.490i) q^{15} +(116.089 + 201.071i) q^{17} +(-174.248 - 301.807i) q^{19} -2075.96 q^{21} +(2021.35 - 3501.09i) q^{23} -666.156 q^{25} -729.000 q^{27} +(-3717.32 + 6438.59i) q^{29} -7157.59 q^{31} +(-1823.07 - 3157.65i) q^{33} +(-5718.89 - 9905.41i) q^{35} +(-49.9573 + 86.5286i) q^{37} +(3545.70 - 4183.63i) q^{39} +(7005.95 - 12134.7i) q^{41} +(7341.52 + 12715.9i) q^{43} +(-2008.26 - 3478.41i) q^{45} +12657.0 q^{47} +(-18199.0 + 31521.6i) q^{49} +2089.60 q^{51} +14624.1 q^{53} +(10044.4 - 17397.5i) q^{55} -3136.47 q^{57} +(-18042.8 - 31251.0i) q^{59} +(-21691.6 - 37570.9i) q^{61} +(-9341.82 + 16180.5i) q^{63} +(29729.9 + 5393.09i) q^{65} +(-14136.4 + 24485.0i) q^{67} +(-18192.2 - 31509.8i) q^{69} +(-7539.63 - 13059.0i) q^{71} +25512.0 q^{73} +(-2997.70 + 5192.17i) q^{75} -93447.3 q^{77} -67991.9 q^{79} +(-3280.50 + 5681.99i) q^{81} -120312. q^{83} +(5756.46 + 9970.47i) q^{85} +(33455.9 + 57947.3i) q^{87} +(11401.5 - 19748.0i) q^{89} +(-47421.2 - 132310. i) q^{91} +(-32209.2 + 55787.9i) q^{93} +(-8640.41 - 14965.6i) q^{95} +(-73278.9 - 126923. i) q^{97} -32815.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 90 q^{3} - 10 q^{5} - 9 q^{7} - 810 q^{9} + 72 q^{11} + 182 q^{13} - 45 q^{15} + 937 q^{17} + 566 q^{19} - 162 q^{21} - 324 q^{23} + 28770 q^{25} - 14580 q^{27} - 5599 q^{29} - 5342 q^{31} - 648 q^{33}+ \cdots - 11664 q^{99}+O(q^{100}) $ 20 * q + 90 * q^3 - 10 * q^5 - 9 * q^7 - 810 * q^9 + 72 * q^11 + 182 * q^13 - 45 * q^15 + 937 * q^17 + 566 * q^19 - 162 * q^21 - 324 * q^23 + 28770 * q^25 - 14580 * q^27 - 5599 * q^29 - 5342 * q^31 - 648 * q^33 - 132 * q^35 + 25035 * q^37 - 4257 * q^39 - 555 * q^41 + 23875 * q^43 + 405 * q^45 - 23432 * q^47 - 37161 * q^49 + 16866 * q^51 - 50606 * q^53 + 30558 * q^55 + 10188 * q^57 + 5290 * q^59 - 101912 * q^61 - 729 * q^63 - 57477 * q^65 + 39267 * q^67 + 2916 * q^69 + 41652 * q^71 + 49824 * q^73 + 129465 * q^75 + 100772 * q^77 - 169090 * q^79 - 65610 * q^81 + 37548 * q^83 + 185295 * q^85 + 50391 * q^87 - 40010 * q^89 + 404103 * q^91 - 24039 * q^93 - 34138 * q^95 + 111143 * q^97 - 11664 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$4$		0			0
$5$	49.5867			0.887034			0.443517	−	0.896266i	$-0.353731\pi$
$5$	49.5867			0.887034			0.443517	+	0.896266i	$0.353731\pi$
$6$		0			0
$7$	−115.331	−	199.759i	−0.889613	−	1.54086i	−0.840333	−	0.542071i	$-0.817640\pi$
$7$	−115.331	−	199.759i	−0.889613	−	1.54086i	−0.0492806	−	0.998785i	$-0.515693\pi$
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	202.563	−	350.850i	0.504753	−	0.874257i	−0.495232	−	0.868761i	$-0.664917\pi$
$11$	202.563	−	350.850i	0.504753	−	0.874257i	0.999985	−	0.00549658i	$-0.00174962\pi$
$12$		0			0
$13$	599.553	+	108.761i	0.983942	+	0.178490i
$13$	599.553	+	108.761i	0.983942	+	0.178490i
$14$		0			0
$15$	223.140	−	386.490i	0.256065	−	0.443517i
$16$		0			0
$17$	116.089	+	201.071i	0.0974243	+	0.168744i	0.910618	−	0.413250i	$-0.135606\pi$
$17$	116.089	+	201.071i	0.0974243	+	0.168744i	−0.813194	+	0.581993i	$0.802273\pi$
$18$		0			0
$19$	−174.248	−	301.807i	−0.110735	−	0.191799i	0.805332	−	0.592824i	$-0.201987\pi$
$19$	−174.248	−	301.807i	−0.110735	−	0.191799i	−0.916067	+	0.401026i	$0.868654\pi$
$20$		0			0
$21$	−2075.96			−1.02724
$22$		0			0
$23$	2021.35	−	3501.09i	0.796751	−	1.38001i	−0.124970	−	0.992161i	$-0.539883\pi$
$23$	2021.35	−	3501.09i	0.796751	−	1.38001i	0.921721	−	0.387853i	$-0.126783\pi$
$24$		0			0
$25$	−666.156			−0.213170
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−3717.32	+	6438.59i	−0.820796	+	1.42166i	0.0842937	+	0.996441i	$0.473137\pi$
$29$	−3717.32	+	6438.59i	−0.820796	+	1.42166i	−0.905090	+	0.425220i	$0.860197\pi$
$30$		0			0
$31$	−7157.59			−1.33771			−0.668856	−	0.743392i	$-0.733216\pi$
$31$	−7157.59			−1.33771			−0.668856	+	0.743392i	$0.733216\pi$
$32$		0			0
$33$	−1823.07	−	3157.65i	−0.291419	−	0.504753i
$34$		0			0
$35$	−5718.89	−	9905.41i	−0.789118	−	1.36679i
$36$		0			0
$37$	−49.9573	+	86.5286i	−0.00599922	+	0.0103910i	−0.869009	−	0.494796i	$-0.835243\pi$
$37$	−49.9573	+	86.5286i	−0.00599922	+	0.0103910i	0.863010	+	0.505186i	$0.168576\pi$
$38$		0			0
$39$	3545.70	−	4183.63i	0.373285	−	0.440445i
$40$		0			0
$41$	7005.95	−	12134.7i	0.650890	−	1.12737i	−0.332017	−	0.943273i	$-0.607729\pi$
$41$	7005.95	−	12134.7i	0.650890	−	1.12737i	0.982907	−	0.184101i	$-0.0589374\pi$
$42$		0			0
$43$	7341.52	+	12715.9i	0.605501	+	1.04876i	0.991972	+	0.126457i	$0.0403604\pi$
$43$	7341.52	+	12715.9i	0.605501	+	1.04876i	−0.386471	+	0.922301i	$0.626306\pi$
$44$		0			0
$45$	−2008.26	−	3478.41i	−0.147839	−	0.256065i
$46$		0			0
$47$	12657.0			0.835766			0.417883	−	0.908501i	$-0.362772\pi$
$47$	12657.0			0.835766			0.417883	+	0.908501i	$0.362772\pi$
$48$		0			0
$49$	−18199.0	+	31521.6i	−1.08282	+	1.87551i
$50$		0			0
$51$	2089.60			0.112496
$52$		0			0
$53$	14624.1			0.715123			0.357561	−	0.933890i	$-0.383608\pi$
$53$	14624.1			0.715123			0.357561	+	0.933890i	$0.383608\pi$
$54$		0			0
$55$	10044.4	−	17397.5i	0.447733	−	0.775496i
$56$		0			0
$57$	−3136.47			−0.127866
$58$		0			0
$59$	−18042.8	−	31251.0i	−0.674797	−	1.16878i	−0.976528	−	0.215389i	$-0.930898\pi$
$59$	−18042.8	−	31251.0i	−0.674797	−	1.16878i	0.301732	−	0.953393i	$-0.402435\pi$
$60$		0			0
$61$	−21691.6	−	37570.9i	−0.746391	−	1.29279i	−0.949542	−	0.313640i	$-0.898451\pi$
$61$	−21691.6	−	37570.9i	−0.746391	−	1.29279i	0.203150	−	0.979148i	$-0.434882\pi$
$62$		0			0
$63$	−9341.82	+	16180.5i	−0.296538	+	0.513619i
$64$		0			0
$65$	29729.9	+	5393.09i	0.872790	+	0.158327i
$66$		0			0
$67$	−14136.4	+	24485.0i	−0.384727	+	0.666366i	−0.991731	−	0.128332i	$-0.959038\pi$
$67$	−14136.4	+	24485.0i	−0.384727	+	0.666366i	0.607005	+	0.794698i	$0.292371\pi$
$68$		0			0
$69$	−18192.2	−	31509.8i	−0.460005	−	0.796751i
$70$		0			0
$71$	−7539.63	−	13059.0i	−0.177502	−	0.307443i	0.763522	−	0.645782i	$-0.223468\pi$
$71$	−7539.63	−	13059.0i	−0.177502	−	0.307443i	−0.941024	+	0.338339i	$0.890135\pi$
$72$		0			0
$73$	25512.0			0.560321			0.280160	−	0.959953i	$-0.409612\pi$
$73$	25512.0			0.560321			0.280160	+	0.959953i	$0.409612\pi$
$74$		0			0
$75$	−2997.70	+	5192.17i	−0.0615369	+	0.106585i
$76$		0			0
$77$	−93447.3			−1.79614
$78$		0			0
$79$	−67991.9			−1.22572			−0.612858	−	0.790193i	$-0.709980\pi$
$79$	−67991.9			−1.22572			−0.612858	+	0.790193i	$0.709980\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−120312.			−1.91696			−0.958479	−	0.285164i	$-0.907952\pi$
$83$	−120312.			−1.91696			−0.958479	+	0.285164i	$0.907952\pi$
$84$		0			0
$85$	5756.46	+	9970.47i	0.0864187	+	0.149682i
$86$		0			0
$87$	33455.9	+	57947.3i	0.473887	+	0.820796i
$88$		0			0
$89$	11401.5	−	19748.0i	0.152577	−	0.264271i	−0.779597	−	0.626281i	$-0.784576\pi$
$89$	11401.5	−	19748.0i	0.152577	−	0.264271i	0.932174	+	0.362010i	$0.117910\pi$
$90$		0			0
$91$	−47421.2	−	132310.i	−0.600300	−	1.67490i
$92$		0			0
$93$	−32209.2	+	55787.9i	−0.386164	+	0.668856i
$94$		0			0
$95$	−8640.41	−	14965.6i	−0.0982258	−	0.170132i
$96$		0			0
$97$	−73278.9	−	126923.i	−0.790769	−	1.36965i	−0.925491	−	0.378769i	$-0.876347\pi$
$97$	−73278.9	−	126923.i	−0.790769	−	1.36965i	0.134722	−	0.990883i	$-0.456986\pi$
$98$		0			0
$99$	−32815.2			−0.336502
$100$		0			0
$101$	2743.48	−	4751.84i	0.0267607	−	0.0463509i	−0.852335	−	0.522996i	$-0.824814\pi$
$101$	2743.48	−	4751.84i	0.0267607	−	0.0463509i	0.879096	+	0.476645i	$0.158147\pi$
$102$		0			0
$103$	117634.			1.09255			0.546274	−	0.837607i	$-0.316046\pi$
$103$	117634.			1.09255			0.546274	+	0.837607i	$0.316046\pi$
$104$		0			0
$105$	−102940.			−0.911195
$106$		0			0
$107$	−61292.0	+	106161.i	−0.517540	+	0.896406i	0.482252	+	0.876033i	$0.339819\pi$
$107$	−61292.0	+	106161.i	−0.517540	+	0.896406i	−0.999792	+	0.0203737i	$0.993514\pi$
$108$		0			0
$109$	−89811.5			−0.724045			−0.362023	−	0.932169i	$-0.617914\pi$
$109$	−89811.5			−0.724045			−0.362023	+	0.932169i	$0.617914\pi$
$110$		0			0
$111$	449.616	+	778.758i	0.00346365	+	0.00599922i
$112$		0			0
$113$	113200.	+	196069.i	0.833972	+	1.44448i	0.894865	+	0.446337i	$0.147272\pi$
$113$	113200.	+	196069.i	0.833972	+	1.44448i	−0.0608934	+	0.998144i	$0.519395\pi$
$114$		0			0
$115$	100232.	−	173608.i	0.706746	−	1.22412i
$116$		0			0
$117$	−16652.5	−	46462.3i	−0.112465	−	0.313788i
$118$		0			0
$119$	26777.3	−	46379.6i	0.173340	−	0.300234i
$120$		0			0
$121$	−1538.10	−	2664.08i	−0.00955042	−	0.0165418i
$122$		0			0
$123$	−63053.6	−	109212.i	−0.375792	−	0.650890i
$124$		0			0
$125$	−187991.			−1.07612
$126$		0			0
$127$	179657.	−	311176.i	0.988406	−	1.71197i	0.362713	−	0.931901i	$-0.381850\pi$
$127$	179657.	−	311176.i	0.988406	−	1.71197i	0.625693	−	0.780069i	$-0.284816\pi$
$128$		0			0
$129$	132147.			0.699172
$130$		0			0
$131$	115715.			0.589130			0.294565	−	0.955632i	$-0.404825\pi$
$131$	115715.			0.589130			0.294565	+	0.955632i	$0.404825\pi$
$132$		0			0
$133$	−40192.5	+	69615.5i	−0.197023	+	0.341253i
$134$		0			0
$135$	−36148.7			−0.170710
$136$		0			0
$137$	−144019.	−	249448.i	−0.655569	−	1.13548i	−0.981751	−	0.190172i	$-0.939096\pi$
$137$	−144019.	−	249448.i	−0.655569	−	1.13548i	0.326182	−	0.945307i	$-0.394238\pi$
$138$		0			0
$139$	−30707.0	−	53186.1i	−0.134803	−	0.233486i	0.790719	−	0.612179i	$-0.209707\pi$
$139$	−30707.0	−	53186.1i	−0.134803	−	0.233486i	−0.925522	+	0.378693i	$0.876374\pi$
$140$		0			0
$141$	56956.3	−	98651.2i	0.241265	−	0.417883i
$142$		0			0
$143$	159606.	−	188322.i	0.652693	−	0.770125i
$144$		0			0
$145$	−184330.	+	319269.i	−0.728075	+	1.26106i
$146$		0			0
$147$	163791.	+	283695.i	0.625169	+	1.08282i
$148$		0			0
$149$	−33587.8	−	58175.7i	−0.123941	−	0.214672i	0.797377	−	0.603481i	$-0.206220\pi$
$149$	−33587.8	−	58175.7i	−0.123941	−	0.214672i	−0.921319	+	0.388809i	$0.872887\pi$
$150$		0			0
$151$	265159.			0.946376			0.473188	−	0.880962i	$-0.343103\pi$
$151$	265159.			0.946376			0.473188	+	0.880962i	$0.343103\pi$
$152$		0			0
$153$	9403.18	−	16286.8i	0.0324748	−	0.0562480i
$154$		0			0
$155$	−354921.			−1.18660
$156$		0			0
$157$	−64807.3			−0.209834			−0.104917	−	0.994481i	$-0.533458\pi$
$157$	−64807.3			−0.209834			−0.104917	+	0.994481i	$0.533458\pi$
$158$		0			0
$159$	65808.6	−	113984.i	0.206438	−	0.357561i
$160$		0			0
$161$	−932500.			−2.83520
$162$		0			0
$163$	166052.	+	287610.i	0.489524	+	0.847881i	0.999927	−	0.0120545i	$-0.00383717\pi$
$163$	166052.	+	287610.i	0.489524	+	0.847881i	−0.510403	+	0.859935i	$0.670504\pi$
$164$		0			0
$165$	−90400.0	−	156577.i	−0.258499	−	0.447733i
$166$		0			0
$167$	56447.8	−	97770.5i	0.156623	−	0.271279i	−0.777026	−	0.629469i	$-0.783272\pi$
$167$	56447.8	−	97770.5i	0.156623	−	0.271279i	0.933649	+	0.358190i	$0.116606\pi$
$168$		0			0
$169$	347635.	+	130416.i	0.936283	+	0.351248i
$170$		0			0
$171$	−14114.1	+	24446.4i	−0.0369117	+	0.0639329i
$172$		0			0
$173$	84494.6	+	146349.i	0.214642	+	0.371770i	0.953162	−	0.302461i	$-0.0978083\pi$
$173$	84494.6	+	146349.i	0.214642	+	0.371770i	−0.738520	+	0.674232i	$0.764475\pi$
$174$		0			0
$175$	76828.5	+	133071.i	0.189639	+	0.328464i
$176$		0			0
$177$	−324770.			−0.779188
$178$		0			0
$179$	−46275.6	+	80151.6i	−0.107949	+	0.186973i	−0.914939	−	0.403592i	$-0.867762\pi$
$179$	−46275.6	+	80151.6i	−0.107949	+	0.186973i	0.806990	+	0.590565i	$0.201095\pi$
$180$		0			0
$181$	−405706.			−0.920482			−0.460241	−	0.887794i	$-0.652237\pi$
$181$	−405706.			−0.920482			−0.460241	+	0.887794i	$0.652237\pi$
$182$		0			0
$183$	−390448.			−0.861859
$184$		0			0
$185$	−2477.22	+	4290.67i	−0.00532152	+	0.00921714i
$186$		0			0
$187$	94061.1			0.196701
$188$		0			0
$189$	84076.4	+	145625.i	0.171206	+	0.296538i
$190$		0			0
$191$	116450.	+	201697.i	0.230969	+	0.400051i	0.958094	−	0.286455i	$-0.0924769\pi$
$191$	116450.	+	201697.i	0.230969	+	0.400051i	−0.727124	+	0.686506i	$0.759144\pi$
$192$		0			0
$193$	350194.	−	606553.i	0.676730	−	1.17213i	−0.299231	−	0.954181i	$-0.596730\pi$
$193$	350194.	−	606553.i	0.676730	−	1.17213i	0.975960	−	0.217949i	$-0.0699367\pi$
$194$		0			0
$195$	175819.	−	207453.i	0.331116	−	0.390690i
$196$		0			0
$197$	−254820.	+	441361.i	−0.467808	+	0.810267i	−0.999323	−	0.0367819i	$-0.988289\pi$
$197$	−254820.	+	441361.i	−0.467808	+	0.810267i	0.531516	+	0.847048i	$0.321623\pi$
$198$		0			0
$199$	493479.	+	854731.i	0.883357	+	1.53002i	0.847586	+	0.530658i	$0.178055\pi$
$199$	493479.	+	854731.i	0.883357	+	1.53002i	0.0357707	+	0.999360i	$0.488611\pi$
$200$		0			0
$201$	127228.	+	220365.i	0.222122	+	0.384727i
$202$		0			0
$203$	1.71489e6			2.92077
$204$		0			0
$205$	347402.	−	601719.i	0.577362	−	1.00002i
$206$		0			0
$207$	−327459.			−0.531168
$208$		0			0
$209$	−141185.			−0.223575
$210$		0			0
$211$	350051.	−	606305.i	0.541283	−	0.937530i	−0.457547	−	0.889185i	$-0.651272\pi$
$211$	350051.	−	606305.i	0.541283	−	0.937530i	0.998831	−	0.0483451i	$-0.0153947\pi$
$212$		0			0
$213$	−135713.			−0.204962
$214$		0			0
$215$	364042.	+	630539.i	0.537100	+	0.930284i
$216$		0			0
$217$	825493.	+	1.42980e6i	1.19005	+	2.06122i
$218$		0			0
$219$	114804.	−	198846.i	0.161751	−	0.280160i
$220$		0			0
$221$	47732.6	+	133179.i	0.0657408	+	0.183423i
$222$		0			0
$223$	286906.	−	496935.i	0.386346	−	0.669171i	−0.605609	−	0.795763i	$-0.707070\pi$
$223$	286906.	−	496935.i	0.386346	−	0.669171i	0.991955	+	0.126591i	$0.0404036\pi$
$224$		0			0
$225$	26979.3	+	46729.5i	0.0355283	+	0.0615369i
$226$		0			0
$227$	−296780.	−	514038.i	−0.382270	−	0.662110i	0.609117	−	0.793081i	$-0.291524\pi$
$227$	−296780.	−	514038.i	−0.382270	−	0.662110i	−0.991386	+	0.130970i	$0.958191\pi$
$228$		0			0
$229$	777766.			0.980077			0.490038	−	0.871701i	$-0.336983\pi$
$229$	777766.			0.980077			0.490038	+	0.871701i	$0.336983\pi$
$230$		0			0
$231$	−420513.	+	728350.i	−0.518501	+	0.898069i
$232$		0			0
$233$	292792.			0.353321			0.176660	−	0.984272i	$-0.443471\pi$
$233$	292792.			0.353321			0.176660	+	0.984272i	$0.443471\pi$
$234$		0			0
$235$	627617.			0.741353
$236$		0			0
$237$	−305964.	+	529945.i	−0.353834	+	0.612858i
$238$		0			0
$239$	897569.			1.01642			0.508210	−	0.861233i	$-0.330307\pi$
$239$	897569.			1.01642			0.508210	+	0.861233i	$0.330307\pi$
$240$		0			0
$241$	320248.	+	554686.i	0.355176	+	0.615183i	0.987148	−	0.159808i	$-0.0510876\pi$
$241$	320248.	+	554686.i	0.355176	+	0.615183i	−0.631972	+	0.774991i	$0.717754\pi$
$242$		0			0
$243$	29524.5	+	51137.9i	0.0320750	+	0.0555556i
$244$		0			0
$245$	−902430.	+	1.56306e6i	−0.960502	+	1.66364i
$246$		0			0
$247$	−71646.5	−	199901.i	−0.0747227	−	0.208484i
$248$		0			0
$249$	−541402.	+	937737.i	−0.553378	+	0.958479i
$250$		0			0
$251$	−136563.	−	236534.i	−0.136820	−	0.236978i	0.789471	−	0.613787i	$-0.210355\pi$
$251$	−136563.	−	236534.i	−0.136820	−	0.236978i	−0.926291	+	0.376809i	$0.877021\pi$
$252$		0			0
$253$	−818903.	−	1.41838e6i	−0.804325	−	1.39313i
$254$		0			0
$255$	103616.			0.0997877
$256$		0			0
$257$	29930.6	−	51841.3i	0.0282672	−	0.0489602i	−0.851546	−	0.524280i	$-0.824334\pi$
$257$	29930.6	−	51841.3i	0.0282672	−	0.0489602i	0.879813	+	0.475320i	$0.157668\pi$
$258$		0			0
$259$	23046.5			0.0213480
$260$		0			0
$261$	602206.			0.547198
$262$		0			0
$263$	170710.	−	295679.i	0.152185	−	0.263591i	−0.779846	−	0.625972i	$-0.784703\pi$
$263$	170710.	−	295679.i	0.152185	−	0.263591i	0.932030	+	0.362380i	$0.118036\pi$
$264$		0			0
$265$	725163.			0.634338
$266$		0			0
$267$	−102614.	−	177732.i	−0.0880903	−	0.152577i
$268$		0			0
$269$	978681.	+	1.69513e6i	0.824633	+	1.42831i	0.902200	+	0.431319i	$0.141952\pi$
$269$	978681.	+	1.69513e6i	0.824633	+	1.42831i	−0.0775667	+	0.996987i	$0.524715\pi$
$270$		0			0
$271$	−413425.	+	716073.i	−0.341959	+	0.592290i	−0.984796	−	0.173713i	$-0.944424\pi$
$271$	−413425.	+	716073.i	−0.341959	+	0.592290i	0.642838	+	0.766002i	$0.277757\pi$
$272$		0			0
$273$	−1.24465e6	−	225783.i	−1.01074	−	0.183352i
$274$		0			0
$275$	−134939.	+	233721.i	−0.107598	+	0.186365i
$276$		0			0
$277$	226361.	+	392069.i	0.177256	+	0.307017i	0.940940	−	0.338574i	$-0.109944\pi$
$277$	226361.	+	392069.i	0.177256	+	0.307017i	−0.763683	+	0.645591i	$0.776611\pi$
$278$		0			0
$279$	289882.	+	502091.i	0.222952	+	0.386164i
$280$		0			0
$281$	889133.			0.671740			0.335870	−	0.941908i	$-0.390970\pi$
$281$	889133.			0.671740			0.335870	+	0.941908i	$0.390970\pi$
$282$		0			0
$283$	993652.	−	1.72106e6i	0.737510	−	1.27741i	−0.216103	−	0.976371i	$-0.569335\pi$
$283$	993652.	−	1.72106e6i	0.737510	−	1.27741i	0.953613	−	0.301035i	$-0.0973320\pi$
$284$		0			0
$285$	−155527.			−0.113421
$286$		0			0
$287$	−3.23202e6			−2.31616
$288$		0			0
$289$	682975.	−	1.18295e6i	0.481017	−	0.833146i
$290$		0			0
$291$	−1.31902e6			−0.913102
$292$		0			0
$293$	950996.	+	1.64717e6i	0.647157	+	1.12091i	0.983799	+	0.179276i	$0.0573755\pi$
$293$	950996.	+	1.64717e6i	0.647157	+	1.12091i	−0.336642	+	0.941633i	$0.609291\pi$
$294$		0			0
$295$	−894681.	−	1.54963e6i	−0.598568	−	1.03675i
$296$		0			0
$297$	−147668.	+	255769.i	−0.0971397	+	0.168251i
$298$		0			0
$299$	1.59269e6	−	1.87924e6i	1.03028	−	1.21564i
$300$		0			0
$301$	1.69341e6	−	2.93307e6i	1.07732	−	1.86598i
$302$		0			0
$303$	−24691.3	−	42766.6i	−0.0154503	−	0.0267607i
$304$		0			0
$305$	−1.07561e6	−	1.86302e6i	−0.662075	−	1.14675i
$306$		0			0
$307$	1.24800e6			0.755733			0.377866	−	0.925860i	$-0.376658\pi$
$307$	1.24800e6			0.755733			0.377866	+	0.925860i	$0.376658\pi$
$308$		0			0
$309$	529353.	−	916867.i	0.315391	−	0.546274i
$310$		0			0
$311$	−418523.			−0.245368			−0.122684	−	0.992446i	$-0.539150\pi$
$311$	−418523.			−0.245368			−0.122684	+	0.992446i	$0.539150\pi$
$312$		0			0
$313$	−1.44250e6			−0.832252			−0.416126	−	0.909307i	$-0.636612\pi$
$313$	−1.44250e6			−0.832252			−0.416126	+	0.909307i	$0.636612\pi$
$314$		0			0
$315$	−463230.	+	802338.i	−0.263039	+	0.455597i
$316$		0			0
$317$	−796659.			−0.445271			−0.222636	−	0.974902i	$-0.571466\pi$
$317$	−796659.			−0.445271			−0.222636	+	0.974902i	$0.571466\pi$
$318$		0			0
$319$	1.50599e6	+	2.60844e6i	0.828598	+	1.43517i
$320$		0			0
$321$	551628.	+	955447.i	0.298802	+	0.517540i
$322$		0			0
$323$	40456.5	−	70072.8i	0.0215766	−	0.0373717i
$324$		0			0
$325$	−399396.	−	72451.6i	−0.209747	−	0.0380487i
$326$		0			0
$327$	−404152.	+	700011.i	−0.209014	+	0.362023i
$328$		0			0
$329$	−1.45974e6	−	2.52835e6i	−0.743509	−	1.28779i
$330$		0			0
$331$	1.25187e6	+	2.16831e6i	0.628044	+	1.08780i	0.987944	+	0.154813i	$0.0494775\pi$
$331$	1.25187e6	+	2.16831e6i	0.628044	+	1.08780i	−0.359900	+	0.932991i	$0.617189\pi$
$332$		0			0
$333$	8093.09			0.00399948
$334$		0			0
$335$	−700978.	+	1.21413e6i	−0.341266	+	0.591090i
$336$		0			0
$337$	−317710.			−0.152390			−0.0761950	−	0.997093i	$-0.524277\pi$
$337$	−317710.			−0.152390			−0.0761950	+	0.997093i	$0.524277\pi$
$338$		0			0
$339$	2.03760e6			0.962987
$340$		0			0
$341$	−1.44986e6	+	2.51124e6i	−0.675214	+	1.16950i
$342$		0			0
$343$	4.51892e6			2.07395
$344$		0			0
$345$	−902091.	−	1.56247e6i	−0.408040	−	0.706746i
$346$		0			0
$347$	−827156.	−	1.43268e6i	−0.368777	−	0.638740i	0.620598	−	0.784129i	$-0.286890\pi$
$347$	−827156.	−	1.43268e6i	−0.368777	−	0.638740i	−0.989375	+	0.145389i	$0.953557\pi$
$348$		0			0
$349$	629445.	−	1.09023e6i	0.276627	−	0.479131i	−0.693918	−	0.720054i	$-0.744117\pi$
$349$	629445.	−	1.09023e6i	0.276627	−	0.479131i	0.970544	+	0.240923i	$0.0774502\pi$
$350$		0			0
$351$	−437074.	−	79286.6i	−0.189360	−	0.0343504i
$352$		0			0
$353$	−522756.	+	905439.i	−0.223286	+	0.386743i	−0.955804	−	0.294005i	$-0.905012\pi$
$353$	−522756.	+	905439.i	−0.223286	+	0.386743i	0.732518	+	0.680748i	$0.238345\pi$
$354$		0			0
$355$	−373866.	−	647554.i	−0.157451	−	0.272713i
$356$		0			0
$357$	−240995.	−	417416.i	−0.100078	−	0.173340i
$358$		0			0
$359$	−4.06398e6			−1.66424			−0.832120	−	0.554596i	$-0.812873\pi$
$359$	−4.06398e6			−1.66424			−0.832120	+	0.554596i	$0.812873\pi$
$360$		0			0
$361$	1.17732e6	−	2.03919e6i	0.475476	−	0.823548i
$362$		0			0
$363$	−27685.9			−0.0110279
$364$		0			0
$365$	1.26506e6			0.497024
$366$		0			0
$367$	200743.	−	347698.i	0.0777994	−	0.134752i	−0.824501	−	0.565861i	$-0.808544\pi$
$367$	200743.	−	347698.i	0.0777994	−	0.134752i	0.902300	+	0.431108i	$0.141877\pi$
$368$		0			0
$369$	−1.13496e6			−0.433927
$370$		0			0
$371$	−1.68662e6	−	2.92131e6i	−0.636183	−	1.10190i
$372$		0			0
$373$	1.74375e6	+	3.02026e6i	0.648950	+	1.12401i	0.983374	+	0.181592i	$0.0581249\pi$
$373$	1.74375e6	+	3.02026e6i	0.648950	+	1.12401i	−0.334424	+	0.942423i	$0.608542\pi$
$374$		0			0
$375$	−845960.	+	1.46525e6i	−0.310650	+	0.538062i
$376$		0			0
$377$	−2.92900e6	+	3.45598e6i	−1.06137	+	1.25233i
$378$		0			0
$379$	−782352.	+	1.35507e6i	−0.279772	+	0.484579i	−0.971328	−	0.237744i	$-0.923592\pi$
$379$	−782352.	+	1.35507e6i	−0.279772	+	0.484579i	0.691556	+	0.722323i	$0.256926\pi$
$380$		0			0
$381$	−1.61692e6	−	2.80058e6i	−0.570657	−	0.988406i
$382$		0			0
$383$	562258.	+	973860.i	0.195857	+	0.339234i	0.947181	−	0.320699i	$-0.103918\pi$
$383$	562258.	+	973860.i	0.195857	+	0.339234i	−0.751324	+	0.659933i	$0.770585\pi$
$384$		0			0
$385$	−4.63375e6			−1.59324
$386$		0			0
$387$	594663.	−	1.02999e6i	0.201834	−	0.349586i
$388$		0			0
$389$	5.41179e6			1.81329			0.906644	−	0.421897i	$-0.138635\pi$
$389$	5.41179e6			1.81329			0.906644	+	0.421897i	$0.138635\pi$
$390$		0			0
$391$	938625.			0.310492
$392$		0			0
$393$	520717.	−	901908.i	0.170067	−	0.294565i
$394$		0			0
$395$	−3.37150e6			−1.08725
$396$		0			0
$397$	539068.	+	933693.i	0.171659	+	0.297323i	0.939000	−	0.343917i	$-0.111754\pi$
$397$	539068.	+	933693.i	0.171659	+	0.297323i	−0.767341	+	0.641239i	$0.778421\pi$
$398$		0			0
$399$	361733.	+	626540.i	0.113751	+	0.197023i
$400$		0			0
$401$	1.54097e6	−	2.66903e6i	0.478556	−	0.828883i	−0.521142	−	0.853470i	$-0.674494\pi$
$401$	1.54097e6	−	2.66903e6i	0.478556	−	0.828883i	0.999698	+	0.0245873i	$0.00782718\pi$
$402$		0			0
$403$	−4.29136e6	−	778465.i	−1.31623	−	0.238768i
$404$		0			0
$405$	−162669.	+	281751.i	−0.0492797	+	0.0853549i
$406$		0			0
$407$	20239.0	+	35055.0i	0.00605625	+	0.0104897i
$408$		0			0
$409$	−661329.	−	1.14546e6i	−0.195483	−	0.338587i	0.751576	−	0.659647i	$-0.229294\pi$
$409$	−661329.	−	1.14546e6i	−0.195483	−	0.338587i	−0.947059	+	0.321060i	$0.895961\pi$
$410$		0			0
$411$	−2.59234e6			−0.756986
$412$		0			0
$413$	−4.16178e6	+	7.20842e6i	−1.20062	+	2.07953i
$414$		0			0
$415$	−5.96586e6			−1.70041
$416$		0			0
$417$	−552727.			−0.155658
$418$		0			0
$419$	1.10039e6	−	1.90593e6i	0.306205	−	0.530362i	−0.671324	−	0.741164i	$-0.734274\pi$
$419$	1.10039e6	−	1.90593e6i	0.306205	−	0.530362i	0.977529	+	0.210802i	$0.0676075\pi$
$420$		0			0
$421$	5.39913e6			1.48463			0.742316	−	0.670051i	$-0.233728\pi$
$421$	5.39913e6			1.48463			0.742316	+	0.670051i	$0.233728\pi$
$422$		0			0
$423$	−512607.	−	887861.i	−0.139294	−	0.241265i
$424$		0			0
$425$	−77333.1	−	133945.i	−0.0207679	−	0.0359711i
$426$		0			0
$427$	−5.00343e6	+	8.66619e6i	−1.32800	+	2.30016i
$428$		0			0
$429$	−749598.	−	2.09146e6i	−0.196646	−	0.548663i
$430$		0			0
$431$	2.58738e6	−	4.48148e6i	0.670915	−	1.16206i	−0.306730	−	0.951797i	$-0.599235\pi$
$431$	2.58738e6	−	4.48148e6i	0.670915	−	1.16206i	0.977645	−	0.210262i	$-0.0674319\pi$
$432$		0			0
$433$	−3.81020e6	−	6.59946e6i	−0.976625	−	1.69156i	−0.674464	−	0.738308i	$-0.735625\pi$
$433$	−3.81020e6	−	6.59946e6i	−0.976625	−	1.69156i	−0.302161	−	0.953257i	$-0.597708\pi$
$434$		0			0
$435$	1.65897e6	+	2.87342e6i	0.420354	+	0.728075i
$436$		0			0
$437$	−1.40887e6			−0.352913
$438$		0			0
$439$	2.70953e6	−	4.69304e6i	0.671015	−	1.16223i	−0.306602	−	0.951838i	$-0.599192\pi$
$439$	2.70953e6	−	4.69304e6i	0.671015	−	1.16223i	0.977617	−	0.210394i	$-0.0674747\pi$
$440$		0			0
$441$	2.94824e6			0.721883
$442$		0			0
$443$	−4.22896e6			−1.02382			−0.511911	−	0.859039i	$-0.671062\pi$
$443$	−4.22896e6			−1.02382			−0.511911	+	0.859039i	$0.671062\pi$
$444$		0			0
$445$	565365.	−	979241.i	0.135341	−	0.234417i
$446$		0			0
$447$	−604580.			−0.143115
$448$		0			0
$449$	−2.84596e6	−	4.92935e6i	−0.666214	−	1.15392i	−0.978955	−	0.204078i	$-0.934581\pi$
$449$	−2.84596e6	−	4.92935e6i	−0.666214	−	1.15392i	0.312741	−	0.949838i	$-0.398753\pi$
$450$		0			0
$451$	−2.83830e6	−	4.91607e6i	−0.657077	−	1.13809i
$452$		0			0
$453$	1.19321e6	−	2.06671e6i	0.273195	−	0.473188i
$454$		0			0
$455$	−2.35146e6	−	6.56081e6i	−0.532487	−	1.48569i
$456$		0			0
$457$	1.53862e6	−	2.66496e6i	0.344619	−	0.596898i	−0.640665	−	0.767820i	$-0.721341\pi$
$457$	1.53862e6	−	2.66496e6i	0.344619	−	0.596898i	0.985285	+	0.170922i	$0.0546746\pi$
$458$		0			0
$459$	−84628.6	−	146581.i	−0.0187493	−	0.0324748i
$460$		0			0
$461$	2.48159e6	+	4.29824e6i	0.543848	+	0.941972i	0.998678	+	0.0513940i	$0.0163664\pi$
$461$	2.48159e6	+	4.29824e6i	0.543848	+	0.941972i	−0.454831	+	0.890578i	$0.650300\pi$
$462$		0			0
$463$	8.20115e6			1.77796			0.888981	−	0.457945i	$-0.151414\pi$
$463$	8.20115e6			1.77796			0.888981	+	0.457945i	$0.151414\pi$
$464$		0			0
$465$	−1.59715e6	+	2.76634e6i	−0.342541	+	0.593298i
$466$		0			0
$467$	3.27493e6			0.694880			0.347440	−	0.937702i	$-0.387051\pi$
$467$	3.27493e6			0.694880			0.347440	+	0.937702i	$0.387051\pi$
$468$		0			0
$469$	6.52147e6			1.36903
$470$		0			0
$471$	−291633.	+	505123.i	−0.0605737	+	0.104917i
$472$		0			0
$473$	5.94848e6			1.22251
$474$		0			0
$475$	116077.	+	201051.i	0.0236054	+	0.0408857i
$476$		0			0
$477$	−592278.	−	1.02585e6i	−0.119187	−	0.206438i
$478$		0			0
$479$	−2.24686e6	+	3.89167e6i	−0.447442	+	0.774993i	−0.998219	−	0.0596599i	$-0.980998\pi$
$479$	−2.24686e6	+	3.89167e6i	−0.447442	+	0.774993i	0.550776	+	0.834653i	$0.314332\pi$
$480$		0			0
$481$	−39363.0	+	46445.1i	−0.00775757	+	0.00915329i
$482$		0			0
$483$	−4.19625e6	+	7.26812e6i	−0.818453	+	1.41760i
$484$		0			0
$485$	−3.63366e6	−	6.29369e6i	−0.701440	−	1.21493i
$486$		0			0
$487$	−2.86547e6	−	4.96313e6i	−0.547486	−	0.948273i	−0.998446	−	0.0557291i	$-0.982252\pi$
$487$	−2.86547e6	−	4.96313e6i	−0.547486	−	0.948273i	0.450960	−	0.892544i	$-0.351082\pi$
$488$		0			0
$489$	2.98893e6			0.565254
$490$		0			0
$491$	−3.24813e6	+	5.62593e6i	−0.608037	+	1.05315i	0.383526	+	0.923530i	$0.374709\pi$
$491$	−3.24813e6	+	5.62593e6i	−0.608037	+	1.05315i	−0.991564	+	0.129621i	$0.958624\pi$
$492$		0			0
$493$	−1.72616e6			−0.319862
$494$		0			0
$495$	−1.62720e6			−0.298489
$496$		0			0
$497$	−1.73911e6	+	3.01222e6i	−0.315817	+	0.547011i
$498$		0			0
$499$	−2.15828e6			−0.388022			−0.194011	−	0.980999i	$-0.562150\pi$
$499$	−2.15828e6			−0.388022			−0.194011	+	0.980999i	$0.562150\pi$
$500$		0			0
$501$	−508031.	−	879935.i	−0.0904265	−	0.156623i
$502$		0			0
$503$	443631.	+	768392.i	0.0781812	+	0.135414i	0.902465	−	0.430763i	$-0.141755\pi$
$503$	443631.	+	768392.i	0.0781812	+	0.135414i	−0.824284	+	0.566176i	$0.808422\pi$
$504$		0			0
$505$	136040.	−	235628.i	0.0237377	−	0.0411149i
$506$		0			0
$507$	2.58085e6	−	2.12268e6i	0.445905	−	0.366745i
$508$		0			0
$509$	2.76523e6	−	4.78952e6i	0.473082	−	0.819403i	−0.526443	−	0.850210i	$-0.676475\pi$
$509$	2.76523e6	−	4.78952e6i	0.473082	−	0.819403i	0.999525	+	0.0308078i	$0.00980799\pi$
$510$		0			0
$511$	−2.94232e6	−	5.09625e6i	−0.498469	−	0.863374i
$512$		0			0
$513$	127027.	+	220017.i	0.0213110	+	0.0369117i
$514$		0			0
$515$	5.83309e6			0.969127
$516$		0			0
$517$	2.56383e6	−	4.44069e6i	0.421855	−	0.730674i
$518$		0			0
$519$	1.52090e6			0.247847
$520$		0			0
$521$	−2.12885e6			−0.343598			−0.171799	−	0.985132i	$-0.554958\pi$
$521$	−2.12885e6			−0.343598			−0.171799	+	0.985132i	$0.554958\pi$
$522$		0			0
$523$	−4.74981e6	+	8.22692e6i	−0.759316	+	1.31517i	0.183885	+	0.982948i	$0.441133\pi$
$523$	−4.74981e6	+	8.22692e6i	−0.759316	+	1.31517i	−0.943200	+	0.332225i	$0.892201\pi$
$524$		0			0
$525$	1.38291e6			0.218976
$526$		0			0
$527$	−830915.	−	1.43919e6i	−0.130326	−	0.225731i
$528$		0			0
$529$	−4.95357e6	−	8.57984e6i	−0.769625	−	1.33303i
$530$		0			0
$531$	−1.46146e6	+	2.53133e6i	−0.224932	+	0.389594i
$532$		0			0
$533$	5.52022e6	−	6.51341e6i	0.841663	−	0.993094i
$534$		0			0
$535$	−3.03927e6	+	5.26417e6i	−0.459076	+	0.795143i
$536$		0			0
$537$	416480.	+	721365.i	0.0623245	+	0.107949i
$538$		0			0
$539$	7.37290e6	+	1.27702e7i	1.09312	+	1.89333i
$540$		0			0
$541$	−1.66231e6			−0.244185			−0.122092	−	0.992519i	$-0.538960\pi$
$541$	−1.66231e6			−0.244185			−0.122092	+	0.992519i	$0.538960\pi$
$542$		0			0
$543$	−1.82568e6	+	3.16217e6i	−0.265720	+	0.460241i
$544$		0			0
$545$	−4.45346e6			−0.642253
$546$		0			0
$547$	−6.76157e6			−0.966228			−0.483114	−	0.875558i	$-0.660494\pi$
$547$	−6.76157e6			−0.966228			−0.483114	+	0.875558i	$0.660494\pi$
$548$		0			0
$549$	−1.75702e6	+	3.04324e6i	−0.248797	+	0.430929i
$550$		0			0
$551$	2.59095e6			0.363564
$552$		0			0
$553$	7.84158e6	+	1.35820e7i	1.09041	+	1.88865i
$554$		0			0
$555$	22295.0	+	38616.0i	0.00307238	+	0.00532152i
$556$		0			0
$557$	461179.	−	798786.i	0.0629842	−	0.109092i	−0.832814	−	0.553553i	$-0.813272\pi$
$557$	461179.	−	798786.i	0.0629842	−	0.109092i	0.895798	+	0.444461i	$0.146605\pi$
$558$		0			0
$559$	3.01864e6	+	8.42232e6i	0.408585	+	1.13999i
$560$		0			0
$561$	423275.	−	733134.i	0.0567826	−	0.0983504i
$562$		0			0
$563$	−4.19758e6	−	7.27043e6i	−0.558121	−	0.966694i	−0.997653	−	0.0684669i	$-0.978189\pi$
$563$	−4.19758e6	−	7.27043e6i	−0.558121	−	0.966694i	0.439533	−	0.898227i	$-0.355144\pi$
$564$		0			0
$565$	5.61323e6	+	9.72240e6i	0.739762	+	1.28130i
$566$		0			0
$567$	1.51337e6			0.197692
$568$		0			0
$569$	293870.	−	508998.i	0.0380518	−	0.0659076i	−0.846372	−	0.532592i	$-0.821218\pi$
$569$	293870.	−	508998.i	0.0380518	−	0.0659076i	0.884424	+	0.466684i	$0.154551\pi$
$570$		0			0
$571$	−2.40735e6			−0.308993			−0.154497	−	0.987993i	$-0.549376\pi$
$571$	−2.40735e6			−0.308993			−0.154497	+	0.987993i	$0.549376\pi$
$572$		0			0
$573$	2.09609e6			0.266700
$574$		0			0
$575$	−1.34654e6	+	2.33227e6i	−0.169843	+	0.294177i
$576$		0			0
$577$	4.85220e6			0.606735			0.303368	−	0.952874i	$-0.401889\pi$
$577$	4.85220e6			0.606735			0.303368	+	0.952874i	$0.401889\pi$
$578$		0			0
$579$	−3.15174e6	−	5.45898e6i	−0.390710	−	0.676730i
$580$		0			0
$581$	1.38757e7	+	2.40334e7i	1.70535	+	2.95376i
$582$		0			0
$583$	2.96231e6	−	5.13087e6i	0.360960	−	0.625201i
$584$		0			0
$585$	−825745.	−	2.30391e6i	−0.0997600	−	0.278341i
$586$		0			0
$587$	6.16410e6	−	1.06765e7i	0.738370	−	1.27890i	−0.214858	−	0.976645i	$-0.568929\pi$
$587$	6.16410e6	−	1.06765e7i	0.738370	−	1.27890i	0.953229	−	0.302250i	$-0.0977377\pi$
$588$		0			0
$589$	1.24720e6	+	2.16021e6i	0.148132	+	0.256571i
$590$		0			0
$591$	2.29338e6	+	3.97224e6i	0.270089	+	0.467808i
$592$		0			0
$593$	3.25898e6			0.380579			0.190290	−	0.981728i	$-0.439057\pi$
$593$	3.25898e6			0.380579			0.190290	+	0.981728i	$0.439057\pi$
$594$		0			0
$595$	1.32780e6	−	2.29981e6i	0.153759	−	0.266318i
$596$		0			0
$597$	8.88262e6			1.02001
$598$		0			0
$599$	9.24697e6			1.05301			0.526505	−	0.850172i	$-0.323502\pi$
$599$	9.24697e6			1.05301			0.526505	+	0.850172i	$0.323502\pi$
$600$		0			0
$601$	−413758.	+	716651.i	−0.0467262	+	0.0809322i	−0.888443	−	0.458988i	$-0.848212\pi$
$601$	−413758.	+	716651.i	−0.0467262	+	0.0809322i	0.841716	+	0.539920i	$0.181546\pi$
$602$		0			0
$603$	2.29010e6			0.256484
$604$		0			0
$605$	−76269.6	−	132103.i	−0.00847155	−	0.0146732i
$606$		0			0
$607$	3.24829e6	+	5.62620e6i	0.357835	+	0.619788i	0.987599	−	0.156998i	$-0.0501818\pi$
$607$	3.24829e6	+	5.62620e6i	0.357835	+	0.619788i	−0.629764	+	0.776786i	$0.716848\pi$
$608$		0			0
$609$	7.71701e6	−	1.33663e7i	0.843152	−	1.46038i
$610$		0			0
$611$	7.58852e6	+	1.37658e6i	0.822345	+	0.149176i
$612$		0			0
$613$	717877.	−	1.24340e6i	0.0771611	−	0.133647i	−0.824863	−	0.565333i	$-0.808748\pi$
$613$	717877.	−	1.24340e6i	0.0771611	−	0.133647i	0.902024	+	0.431686i	$0.142081\pi$
$614$		0			0
$615$	−3.12662e6	−	5.41547e6i	−0.333340	−	0.577362i
$616$		0			0
$617$	82138.6	+	142268.i	0.00868629	+	0.0150451i	0.870336	−	0.492459i	$-0.163902\pi$
$617$	82138.6	+	142268.i	0.00868629	+	0.0150451i	−0.861650	+	0.507504i	$0.830568\pi$
$618$		0			0
$619$	−1.09551e7			−1.14918			−0.574591	−	0.818441i	$-0.694839\pi$
$619$	−1.09551e7			−1.14918			−0.574591	+	0.818441i	$0.694839\pi$
$620$		0			0
$621$	−1.47357e6	+	2.55229e6i	−0.153335	+	0.265584i
$622$		0			0
$623$	−5.25981e6			−0.542938
$624$		0			0
$625$	−7.24012e6			−0.741389
$626$		0			0
$627$	−635334.	+	1.10043e6i	−0.0645406	+	0.111788i
$628$		0			0
$629$	−23197.9			−0.00233788
$630$		0			0
$631$	1.27240e6	+	2.20386e6i	0.127219	+	0.220349i	0.922598	−	0.385763i	$-0.126062\pi$
$631$	1.27240e6	+	2.20386e6i	0.127219	+	0.220349i	−0.795379	+	0.606112i	$0.792728\pi$
$632$		0			0
$633$	−3.15046e6	−	5.45675e6i	−0.312510	−	0.541283i
$634$		0			0
$635$	8.90862e6	−	1.54302e7i	0.876750	−	1.51858i
$636$		0			0
$637$	−1.43396e7	+	1.69196e7i	−1.40020	+	1.65212i
$638$		0			0
$639$	−610710.	+	1.05778e6i	−0.0591675	+	0.102481i
$640$		0			0
$641$	5.65311e6	+	9.79147e6i	0.543428	+	0.941245i	0.998704	+	0.0508948i	$0.0162073\pi$
$641$	5.65311e6	+	9.79147e6i	0.543428	+	0.941245i	−0.455276	+	0.890350i	$0.650459\pi$
$642$		0			0
$643$	3.04907e6	+	5.28115e6i	0.290831	+	0.503733i	0.974006	−	0.226521i	$-0.0727350\pi$
$643$	3.04907e6	+	5.28115e6i	0.290831	+	0.503733i	−0.683176	+	0.730254i	$0.739402\pi$
$644$		0			0
$645$	6.55275e6			0.620190
$646$		0			0
$647$	9.27040e6	−	1.60568e7i	0.870638	−	1.50799i	0.00930040	−	0.999957i	$-0.497040\pi$
$647$	9.27040e6	−	1.60568e7i	0.870638	−	1.50799i	0.861338	−	0.508033i	$-0.169627\pi$
$648$		0			0
$649$	−1.46192e7			−1.36242
$650$		0			0
$651$	1.48589e7			1.37415
$652$		0			0
$653$	−5.21107e6	+	9.02583e6i	−0.478238	+	0.828332i	−0.999689	−	0.0249494i	$-0.992058\pi$
$653$	−5.21107e6	+	9.02583e6i	−0.478238	+	0.828332i	0.521451	+	0.853281i	$0.325391\pi$
$654$		0			0
$655$	5.73792e6			0.522578
$656$		0			0
$657$	−1.03323e6	−	1.78962e6i	−0.0933868	−	0.161751i
$658$		0			0
$659$	−1.52031e6	−	2.63325e6i	−0.136370	−	0.236199i	0.789750	−	0.613429i	$-0.210210\pi$
$659$	−1.52031e6	−	2.63325e6i	−0.136370	−	0.236199i	−0.926120	+	0.377229i	$0.876877\pi$
$660$		0			0
$661$	4.20245e6	−	7.27886e6i	0.374110	−	0.647978i	−0.616083	−	0.787681i	$-0.711282\pi$
$661$	4.20245e6	−	7.27886e6i	0.374110	−	0.647978i	0.990193	+	0.139703i	$0.0446149\pi$
$662$		0			0
$663$	1.25282e6	+	227266.i	0.110689	+	0.0200794i
$664$		0			0
$665$	−1.99302e6	+	3.45201e6i	−0.174766	+	0.302704i
$666$		0			0
$667$	1.50281e7	+	2.60294e7i	1.30794	+	2.26542i
$668$		0			0
$669$	−2.58215e6	−	4.47241e6i	−0.223057	−	0.386346i
$670$		0			0
$671$	−1.75757e7			−1.50697
$672$		0			0
$673$	−7.76794e6	+	1.34545e7i	−0.661101	+	1.14506i	0.319225	+	0.947679i	$0.396577\pi$
$673$	−7.76794e6	+	1.34545e7i	−0.661101	+	1.14506i	−0.980327	+	0.197382i	$0.936756\pi$
$674$		0			0
$675$	485628.			0.0410246
$676$		0			0
$677$	7.89626e6			0.662140			0.331070	−	0.943606i	$-0.392590\pi$
$677$	7.89626e6			0.662140			0.331070	+	0.943606i	$0.392590\pi$
$678$		0			0
$679$	−1.69027e7	+	2.92763e7i	−1.40696	+	2.43692i
$680$		0			0
$681$	−5.34204e6			−0.441407
$682$		0			0
$683$	−4.92619e6	−	8.53241e6i	−0.404073	−	0.699875i	0.590140	−	0.807301i	$-0.299072\pi$
$683$	−4.92619e6	−	8.53241e6i	−0.404073	−	0.699875i	−0.994213	+	0.107426i	$0.965739\pi$
$684$		0			0
$685$	−7.14143e6	−	1.23693e7i	−0.581512	−	1.00721i
$686$		0			0
$687$	3.49995e6	−	6.06208e6i	0.282924	−	0.490038i
$688$		0			0
$689$	8.76795e6	+	1.59053e6i	0.703639	+	0.127642i
$690$		0			0
$691$	7.54612e6	−	1.30703e7i	0.601213	−	1.04133i	−0.391425	−	0.920210i	$-0.628018\pi$
$691$	7.54612e6	−	1.30703e7i	0.601213	−	1.04133i	0.992638	−	0.121121i	$-0.0386490\pi$
$692$		0			0
$693$	3.78462e6	+	6.55515e6i	0.299356	+	0.518501i
$694$		0			0
$695$	−1.52266e6	−	2.63733e6i	−0.119575	−	0.207110i
$696$		0			0
$697$	3.25325e6			0.253650
$698$		0			0
$699$	1.31756e6	−	2.28209e6i	0.101995	−	0.176660i
$700$		0			0
$701$	−1.18511e7			−0.910887			−0.455443	−	0.890265i	$-0.650519\pi$
$701$	−1.18511e7			−0.910887			−0.455443	+	0.890265i	$0.650519\pi$
$702$		0			0
$703$	34820.0			0.00265730
$704$		0			0
$705$	2.82428e6	−	4.89179e6i	0.214010	−	0.370677i
$706$		0			0
$707$	−1.26563e6			−0.0952268
$708$		0			0
$709$	6.48050e6	+	1.12246e7i	0.484165	+	0.838598i	0.999835	−	0.0181897i	$-0.00579029\pi$
$709$	6.48050e6	+	1.12246e7i	0.484165	+	0.838598i	−0.515670	+	0.856787i	$0.672457\pi$
$710$		0			0
$711$	2.75367e6	+	4.76950e6i	0.204286	+	0.353834i
$712$		0			0
$713$	−1.44680e7	+	2.50594e7i	−1.06582	+	1.84606i
$714$		0			0
$715$	7.91434e6	−	9.33828e6i	0.578961	−	0.683127i
$716$		0			0
$717$	4.03906e6	−	6.99586e6i	0.293415	−	0.508210i
$718$		0			0
$719$	3.85829e6	+	6.68276e6i	0.278338	+	0.482096i	0.970972	−	0.239194i	$-0.0768831\pi$
$719$	3.85829e6	+	6.68276e6i	0.278338	+	0.482096i	−0.692634	+	0.721290i	$0.743550\pi$
$720$		0			0
$721$	−1.35669e7	−	2.34985e7i	−0.971945	−	1.68346i
$722$		0			0
$723$	5.76446e6			0.410122
$724$		0			0
$725$	2.47632e6	−	4.28911e6i	0.174969	−	0.303055i
$726$		0			0
$727$	−2.55539e7			−1.79317			−0.896583	−	0.442876i	$-0.853958\pi$
$727$	−2.55539e7			−1.79317			−0.896583	+	0.442876i	$0.853958\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−1.70453e6	+	2.95234e6i	−0.117981	+	0.204349i
$732$		0			0
$733$	9.02994e6			0.620762			0.310381	−	0.950612i	$-0.399543\pi$
$733$	9.02994e6			0.620762			0.310381	+	0.950612i	$0.399543\pi$
$734$		0			0
$735$	8.12187e6	+	1.40675e7i	0.554546	+	0.960502i
$736$		0			0
$737$	5.72703e6	+	9.91951e6i	0.388383	+	0.672700i
$738$		0			0
$739$	−253779.	+	439558.i	−0.0170940	+	0.0296077i	−0.874446	−	0.485123i	$-0.838775\pi$
$739$	−253779.	+	439558.i	−0.0170940	+	0.0296077i	0.857352	+	0.514731i	$0.172108\pi$
$740$		0			0
$741$	−1.88048e6	−	341125.i	−0.125812	−	0.0228228i
$742$		0			0
$743$	−7.97646e6	+	1.38156e7i	−0.530076	+	0.918119i	0.469308	+	0.883035i	$0.344503\pi$
$743$	−7.97646e6	+	1.38156e7i	−0.530076	+	0.918119i	−0.999384	+	0.0350845i	$0.988830\pi$
$744$		0			0
$745$	−1.66551e6	−	2.88474e6i	−0.109940	−	0.190422i
$746$		0			0
$747$	4.87262e6	+	8.43963e6i	0.319493	+	0.553378i
$748$		0			0
$749$	2.82755e7			1.84164
$750$		0			0
$751$	1.21597e7	−	2.10612e7i	0.786723	−	1.36265i	−0.141241	−	0.989975i	$-0.545109\pi$
$751$	1.21597e7	−	2.10612e7i	0.786723	−	1.36265i	0.927964	−	0.372670i	$-0.121558\pi$
$752$		0			0
$753$	−2.45813e6			−0.157986
$754$		0			0
$755$	1.31484e7			0.839468
$756$		0			0
$757$	6.16267e6	−	1.06741e7i	0.390867	−	0.677002i	−0.601697	−	0.798724i	$-0.705508\pi$
$757$	6.16267e6	−	1.06741e7i	0.390867	−	0.677002i	0.992564	+	0.121723i	$0.0388418\pi$
$758$		0			0
$759$	−1.47403e7			−0.928754
$760$		0			0
$761$	−1.27189e7	−	2.20298e7i	−0.796137	−	1.37895i	−0.922115	−	0.386916i	$-0.873540\pi$
$761$	−1.27189e7	−	2.20298e7i	−0.796137	−	1.37895i	0.125978	−	0.992033i	$-0.459793\pi$
$762$		0			0
$763$	1.03581e7	+	1.79407e7i	0.644120	+	1.11565i
$764$		0			0
$765$	466273.	−	807608.i	0.0288062	−	0.0498939i
$766$		0			0
$767$	−7.41871e6	−	2.06990e7i	−0.455345	−	1.27046i
$768$		0			0
$769$	1.03862e7	−	1.79894e7i	0.633344	−	1.09698i	−0.353520	−	0.935427i	$-0.615015\pi$
$769$	1.03862e7	−	1.79894e7i	0.633344	−	1.09698i	0.986863	−	0.161556i	$-0.0516514\pi$
$770$		0			0
$771$	−269375.	−	466572.i	−0.0163201	−	0.0282672i
$772$		0			0
$773$	−2.00461e6	−	3.47209e6i	−0.120665	−	0.208998i	0.799365	−	0.600846i	$-0.205169\pi$
$773$	−2.00461e6	−	3.47209e6i	−0.120665	−	0.208998i	−0.920030	+	0.391848i	$0.871836\pi$
$774$		0			0
$775$	4.76807e6			0.285160
$776$		0			0
$777$	103709.	−	179630.i	0.00616262	−	0.0106740i
$778$		0			0
$779$	−4.88311e6			−0.288305
$780$		0			0
$781$	−6.10900e6			−0.358379
$782$		0			0
$783$	2.70993e6	−	4.69373e6i	0.157962	−	0.273599i
$784$		0			0
$785$	−3.21358e6			−0.186130
$786$		0			0
$787$	9.56737e6	+	1.65712e7i	0.550625	+	0.953710i	0.998230	+	0.0594783i	$0.0189437\pi$
$787$	9.56737e6	+	1.65712e7i	0.550625	+	0.953710i	−0.447605	+	0.894231i	$0.647723\pi$
$788$		0			0
$789$	−1.53639e6	−	2.66111e6i	−0.0878638	−	0.152185i
$790$		0			0
$791$	2.61110e7	−	4.52256e7i	1.48382	−	2.57006i
$792$		0			0
$793$	−8.91902e6	−	2.48850e7i	−0.503656	−	1.40525i
$794$		0			0
$795$	3.26323e6	−	5.65209e6i	0.183118	−	0.317169i
$796$		0			0
$797$	−2.75865e6	−	4.77813e6i	−0.153834	−	0.266448i	0.778800	−	0.627272i	$-0.215829\pi$
$797$	−2.75865e6	−	4.77813e6i	−0.153834	−	0.266448i	−0.932634	+	0.360825i	$0.882495\pi$
$798$		0			0
$799$	1.46933e6	+	2.54495e6i	0.0814239	+	0.141030i
$800$		0			0
$801$	−1.84705e6			−0.101718
$802$		0			0
$803$	5.16778e6	−	8.95086e6i	0.282823	−	0.489865i
$804$		0			0
$805$	−4.62396e7			−2.51492
$806$		0			0
$807$	1.76163e7			0.952204
$808$		0			0
$809$	−6.81953e6	+	1.18118e7i	−0.366339	+	0.634518i	−0.988990	−	0.147982i	$-0.952722\pi$
$809$	−6.81953e6	+	1.18118e7i	−0.366339	+	0.634518i	0.622651	+	0.782500i	$0.286056\pi$
$810$		0			0
$811$	−2.45217e7			−1.30917			−0.654587	−	0.755986i	$-0.727158\pi$
$811$	−2.45217e7			−1.30917			−0.654587	+	0.755986i	$0.727158\pi$
$812$		0			0
$813$	3.72083e6	+	6.44466e6i	0.197430	+	0.341959i
$814$		0			0
$815$	8.23396e6	+	1.42616e7i	0.434225	+	0.752099i
$816$		0			0
$817$	2.55850e6	−	4.43145e6i	0.134100	−	0.232268i
$818$		0			0
$819$	−7.36072e6	+	8.68505e6i	−0.383452	+	0.452442i
$820$		0			0
$821$	−945954.	+	1.63844e6i	−0.0489793	+	0.0848346i	−0.889476	−	0.456983i	$-0.848930\pi$
$821$	−945954.	+	1.63844e6i	−0.0489793	+	0.0848346i	0.840496	+	0.541817i	$0.182263\pi$
$822$		0			0
$823$	−1.33092e7	−	2.30522e7i	−0.684938	−	1.18635i	−0.973456	−	0.228873i	$-0.926496\pi$
$823$	−1.33092e7	−	2.30522e7i	−0.684938	−	1.18635i	0.288518	−	0.957474i	$-0.406837\pi$
$824$		0			0
$825$	1.21445e6	+	2.10348e6i	0.0621218	+	0.107598i
$826$		0			0
$827$	−2.86078e7			−1.45452			−0.727261	−	0.686362i	$-0.759207\pi$
$827$	−2.86078e7			−1.45452			−0.727261	+	0.686362i	$0.759207\pi$
$828$		0			0
$829$	−1.58973e7	+	2.75349e7i	−0.803410	+	1.39155i	0.113950	+	0.993487i	$0.463650\pi$
$829$	−1.58973e7	+	2.75349e7i	−0.803410	+	1.39155i	−0.917359	+	0.398060i	$0.869684\pi$
$830$		0			0
$831$	4.07450e6			0.204678
$832$		0			0
$833$	−8.45080e6			−0.421974
$834$		0			0
$835$	2.79906e6	−	4.84812e6i	0.138930	−	0.240634i
$836$		0			0
$837$	5.21788e6			0.257443
$838$		0			0
$839$	588888.	+	1.01998e6i	0.0288820	+	0.0500251i	0.880105	−	0.474779i	$-0.157472\pi$
$839$	588888.	+	1.01998e6i	0.0288820	+	0.0500251i	−0.851223	+	0.524804i	$0.824139\pi$
$840$		0			0
$841$	−1.73814e7	−	3.01055e7i	−0.847413	−	1.46776i
$842$		0			0
$843$	4.00110e6	−	6.93011e6i	0.193915	−	0.335870i
$844$		0			0
$845$	1.72381e7	+	6.46689e6i	0.830515	+	0.311569i
$846$		0			0
$847$	−354783.	+	614502.i	−0.0169924	+	0.0294316i
$848$		0			0
$849$	−8.94286e6	−	1.54895e7i	−0.425802	−	0.737510i
$850$		0			0
$851$	201963.	+	349810.i	0.00955977	+	0.0165580i
$852$		0			0
$853$	2.05782e7			0.968356			0.484178	−	0.874969i	$-0.339119\pi$
$853$	2.05782e7			0.968356			0.484178	+	0.874969i	$0.339119\pi$
$854$		0			0
$855$	−699873.	+	1.21222e6i	−0.0327419	+	0.0567107i
$856$		0			0
$857$	−3.33776e7			−1.55240			−0.776198	−	0.630489i	$-0.782855\pi$
$857$	−3.33776e7			−1.55240			−0.776198	+	0.630489i	$0.782855\pi$
$858$		0			0
$859$	−2.39037e7			−1.10531			−0.552654	−	0.833411i	$-0.686385\pi$
$859$	−2.39037e7			−1.10531			−0.552654	+	0.833411i	$0.686385\pi$
$860$		0			0
$861$	−1.45441e7	+	2.51911e7i	−0.668618	+	1.15808i
$862$		0			0
$863$	1.90582e7			0.871075			0.435538	−	0.900171i	$-0.356558\pi$
$863$	1.90582e7			0.871075			0.435538	+	0.900171i	$0.356558\pi$
$864$		0			0
$865$	4.18981e6	+	7.25697e6i	0.190394	+	0.329773i
$866$		0			0
$867$	−6.14678e6	−	1.06465e7i	−0.277715	−	0.481017i
$868$		0			0
$869$	−1.37727e7	+	2.38549e7i	−0.618683	+	1.07159i
$870$		0			0
$871$	−1.11385e7	+	1.31426e7i	−0.497488	+	0.586995i
$872$		0			0
$873$	−5.93559e6	+	1.02808e7i	−0.263590	+	0.456551i
$874$		0			0
$875$	2.16812e7	+	3.75530e7i	0.957334	+	1.65815i
$876$		0			0
$877$	1.30726e7	+	2.26425e7i	0.573937	+	0.994088i	0.996156	+	0.0875945i	$0.0279180\pi$
$877$	1.30726e7	+	2.26425e7i	0.573937	+	0.994088i	−0.422219	+	0.906494i	$0.638749\pi$
$878$		0			0
$879$	1.71179e7			0.747273
$880$		0			0
$881$	−2.22594e6	+	3.85545e6i	−0.0966217	+	0.167354i	−0.910284	−	0.413984i	$-0.864137\pi$
$881$	−2.22594e6	+	3.85545e6i	−0.0966217	+	0.167354i	0.813663	+	0.581337i	$0.197470\pi$
$882$		0			0
$883$	−3.06418e7			−1.32255			−0.661276	−	0.750143i	$-0.729985\pi$
$883$	−3.06418e7			−1.32255			−0.661276	+	0.750143i	$0.729985\pi$
$884$		0			0
$885$	−1.61043e7			−0.691167
$886$		0			0
$887$	1.56704e7	−	2.71420e7i	0.668762	−	1.15833i	−0.309489	−	0.950903i	$-0.600158\pi$
$887$	1.56704e7	−	2.71420e7i	0.668762	−	1.15833i	0.978251	−	0.207426i	$-0.0665088\pi$
$888$		0			0
$889$	−8.28803e7			−3.51720
$890$		0			0
$891$	1.32902e6	+	2.30192e6i	0.0560836	+	0.0971397i
$892$		0			0
$893$	−2.20546e6	−	3.81996e6i	−0.0925486	−	0.160299i
$894$		0			0
$895$	−2.29465e6	+	3.97446e6i	−0.0957546	+	0.165852i
$896$		0			0
$897$	−7.48016e6	−	2.08704e7i	−0.310406	−	0.866063i
$898$		0			0
$899$	2.66071e7	−	4.60848e7i	1.09799	−	1.90177i
$900$		0			0
$901$	1.69770e6	+	2.94050e6i	0.0696703	+	0.120673i
$902$		0			0
$903$	−1.52407e7	−	2.63977e7i	−0.621993	−	1.07732i
$904$		0			0
$905$	−2.01176e7			−0.816499
$906$		0			0
$907$	2.21987e7	−	3.84493e7i	0.896004	−	1.55192i	0.0634475	−	0.997985i	$-0.479790\pi$
$907$	2.21987e7	−	3.84493e7i	0.896004	−	1.55192i	0.832557	−	0.553940i	$-0.186876\pi$
$908$		0			0
$909$	−444443.			−0.0178405
$910$		0			0
$911$	−2.39276e7			−0.955219			−0.477610	−	0.878572i	$-0.658497\pi$
$911$	−2.39276e7			−0.955219			−0.477610	+	0.878572i	$0.658497\pi$
$912$		0			0
$913$	−2.43707e7	+	4.22113e7i	−0.967589	+	1.67591i
$914$		0			0
$915$	−1.93611e7			−0.764498
$916$		0			0
$917$	−1.33455e7	−	2.31151e7i	−0.524098	−	0.907764i
$918$		0			0
$919$	1.80912e7	+	3.13349e7i	0.706609	+	1.22388i	0.966108	+	0.258139i	$0.0831093\pi$
$919$	1.80912e7	+	3.13349e7i	0.706609	+	1.22388i	−0.259499	+	0.965743i	$0.583557\pi$
$920$		0			0
$921$	5.61600e6	−	9.72719e6i	0.218161	−	0.377866i
$922$		0			0
$923$	−3.10010e6	−	8.64960e6i	−0.119777	−	0.334189i
$924$		0			0
$925$	33279.4	−	57641.6i	0.00127885	−	0.00221504i
$926$		0			0
$927$	−4.76418e6	−	8.25180e6i	−0.182091	−	0.315391i
$928$		0			0
$929$	9.93627e6	+	1.72101e7i	0.377732	+	0.654252i	0.990732	−	0.135832i	$-0.0433706\pi$
$929$	9.93627e6	+	1.72101e7i	0.377732	+	0.654252i	−0.613000	+	0.790083i	$0.710037\pi$
$930$		0			0
$931$	1.26846e7			0.479626
$932$		0			0
$933$	−1.88335e6	+	3.26207e6i	−0.0708317	+	0.122684i
$934$		0			0
$935$	4.66418e6			0.174480
$936$		0			0
$937$	4.15734e7			1.54692			0.773458	−	0.633847i	$-0.218525\pi$
$937$	4.15734e7			1.54692			0.773458	+	0.633847i	$0.218525\pi$
$938$		0			0
$939$	−6.49124e6	+	1.12432e7i	−0.240250	+	0.416126i
$940$		0			0
$941$	−6.96322e6			−0.256352			−0.128176	−	0.991751i	$-0.540912\pi$
$941$	−6.96322e6			−0.256352			−0.128176	+	0.991751i	$0.540912\pi$
$942$		0			0
$943$	−2.83230e7	−	4.90569e7i	−1.03720	−	1.79647i
$944$		0			0
$945$	4.16907e6	+	7.22105e6i	0.151866	+	0.263039i
$946$		0			0
$947$	1.63283e7	−	2.82815e7i	0.591653	−	1.02477i	−0.402357	−	0.915483i	$-0.631809\pi$
$947$	1.63283e7	−	2.82815e7i	0.591653	−	1.02477i	0.994010	−	0.109290i	$-0.0348577\pi$
$948$		0			0
$949$	1.52958e7	+	2.77470e6i	0.551323	+	0.100012i
$950$		0			0
$951$	−3.58497e6	+	6.20934e6i	−0.128539	+	0.222636i
$952$		0			0
$953$	−4.89665e6	−	8.48124e6i	−0.174649	−	0.302501i	0.765391	−	0.643566i	$-0.222546\pi$
$953$	−4.89665e6	−	8.48124e6i	−0.174649	−	0.302501i	−0.940040	+	0.341065i	$0.889212\pi$
$954$		0			0
$955$	5.77435e6	+	1.00015e7i	0.204878	+	0.354859i
$956$		0			0
$957$	2.71077e7			0.956783
$958$		0			0
$959$	−3.32197e7	+	5.75383e7i	−1.16641	+	2.02027i
$960$		0			0
$961$	2.26019e7			0.789473
$962$		0			0
$963$	9.92930e6			0.345027
$964$		0			0
$965$	1.73650e7	−	3.00770e7i	0.600282	−	1.03972i
$966$		0			0
$967$	9.49538e6			0.326547			0.163274	−	0.986581i	$-0.447795\pi$
$967$	9.49538e6			0.326547			0.163274	+	0.986581i	$0.447795\pi$
$968$		0			0
$969$	−364109.	−	630655.i	−0.0124572	−	0.0215766i
$970$		0			0
$971$	466610.	+	808192.i	0.0158820	+	0.0275085i	0.873857	−	0.486183i	$-0.161611\pi$
$971$	466610.	+	808192.i	0.0158820	+	0.0275085i	−0.857975	+	0.513691i	$0.828278\pi$
$972$		0			0
$973$	−7.08295e6	+	1.22680e7i	−0.239846	+	0.415425i
$974$		0			0
$975$	−2.36199e6	+	2.78695e6i	−0.0795730	+	0.0938897i
$976$		0			0
$977$	2.67300e7	−	4.62977e7i	0.895906	−	1.55175i	0.0632259	−	0.997999i	$-0.479861\pi$
$977$	2.67300e7	−	4.62977e7i	0.895906	−	1.55175i	0.832680	−	0.553755i	$-0.186806\pi$
$978$		0			0
$979$	−4.61906e6	−	8.00045e6i	−0.154027	−	0.266783i
$980$		0			0
$981$	3.63736e6	+	6.30010e6i	0.120674	+	0.209014i
$982$		0			0
$983$	6.86792e6			0.226695			0.113347	−	0.993555i	$-0.463843\pi$
$983$	6.86792e6			0.226695			0.113347	+	0.993555i	$0.463843\pi$
$984$		0			0
$985$	−1.26357e7	+	2.18856e7i	−0.414961	+	0.718734i
$986$		0			0
$987$	−2.62753e7			−0.858530
$988$		0			0
$989$	5.93592e7			1.92973
$990$		0			0
$991$	−247836.	+	429265.i	−0.00801642	+	0.0138849i	−0.870006	−	0.493042i	$-0.835885\pi$
$991$	−247836.	+	429265.i	−0.00801642	+	0.0138849i	0.861989	+	0.506926i	$0.169218\pi$
$992$		0			0
$993$	2.25337e7			0.725203
$994$		0			0
$995$	2.44700e7	+	4.23833e7i	0.783568	+	1.35718i
$996$		0			0
$997$	2.34073e7	+	4.05426e7i	0.745785	+	1.29174i	0.949827	+	0.312775i	$0.101259\pi$
$997$	2.34073e7	+	4.05426e7i	0.745785	+	1.29174i	−0.204042	+	0.978962i	$0.565408\pi$
$998$		0			0
$999$	36418.9	−	63079.4i	0.00115455	−	0.00199974i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.6.q.d.217.8	✓	20
13.3	even	3	inner	312.6.q.d.289.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.6.q.d.217.8	✓	20	1.1	even	1	trivial
312.6.q.d.289.8	yes	20	13.3	even	3	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	312.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 - 7.79423i) q^{3} +49.5867 q^{5} +(-115.331 - 199.759i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(202.563 - 350.850i) q^{11} +(599.553 + 108.761i) q^{13} +(223.140 - 386.490i) q^{15} +(116.089 + 201.071i) q^{17} +(-174.248 - 301.807i) q^{19} -2075.96 q^{21} +(2021.35 - 3501.09i) q^{23} -666.156 q^{25} -729.000 q^{27} +(-3717.32 + 6438.59i) q^{29} -7157.59 q^{31} +(-1823.07 - 3157.65i) q^{33} +(-5718.89 - 9905.41i) q^{35} +(-49.9573 + 86.5286i) q^{37} +(3545.70 - 4183.63i) q^{39} +(7005.95 - 12134.7i) q^{41} +(7341.52 + 12715.9i) q^{43} +(-2008.26 - 3478.41i) q^{45} +12657.0 q^{47} +(-18199.0 + 31521.6i) q^{49} +2089.60 q^{51} +14624.1 q^{53} +(10044.4 - 17397.5i) q^{55} -3136.47 q^{57} +(-18042.8 - 31251.0i) q^{59} +(-21691.6 - 37570.9i) q^{61} +(-9341.82 + 16180.5i) q^{63} +(29729.9 + 5393.09i) q^{65} +(-14136.4 + 24485.0i) q^{67} +(-18192.2 - 31509.8i) q^{69} +(-7539.63 - 13059.0i) q^{71} +25512.0 q^{73} +(-2997.70 + 5192.17i) q^{75} -93447.3 q^{77} -67991.9 q^{79} +(-3280.50 + 5681.99i) q^{81} -120312. q^{83} +(5756.46 + 9970.47i) q^{85} +(33455.9 + 57947.3i) q^{87} +(11401.5 - 19748.0i) q^{89} +(-47421.2 - 132310. i) q^{91} +(-32209.2 + 55787.9i) q^{93} +(-8640.41 - 14965.6i) q^{95} +(-73278.9 - 126923. i) q^{97} -32815.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 90 q^{3} - 10 q^{5} - 9 q^{7} - 810 q^{9} + 72 q^{11} + 182 q^{13} - 45 q^{15} + 937 q^{17} + 566 q^{19} - 162 q^{21} - 324 q^{23} + 28770 q^{25} - 14580 q^{27} - 5599 q^{29} - 5342 q^{31} - 648 q^{33}+ \cdots - 11664 q^{99}+O(q^{100}) \) 20 * q + 90 * q^3 - 10 * q^5 - 9 * q^7 - 810 * q^9 + 72 * q^11 + 182 * q^13 - 45 * q^15 + 937 * q^17 + 566 * q^19 - 162 * q^21 - 324 * q^23 + 28770 * q^25 - 14580 * q^27 - 5599 * q^29 - 5342 * q^31 - 648 * q^33 - 132 * q^35 + 25035 * q^37 - 4257 * q^39 - 555 * q^41 + 23875 * q^43 + 405 * q^45 - 23432 * q^47 - 37161 * q^49 + 16866 * q^51 - 50606 * q^53 + 30558 * q^55 + 10188 * q^57 + 5290 * q^59 - 101912 * q^61 - 729 * q^63 - 57477 * q^65 + 39267 * q^67 + 2916 * q^69 + 41652 * q^71 + 49824 * q^73 + 129465 * q^75 + 100772 * q^77 - 169090 * q^79 - 65610 * q^81 + 37548 * q^83 + 185295 * q^85 + 50391 * q^87 - 40010 * q^89 + 404103 * q^91 - 24039 * q^93 - 34138 * q^95 + 111143 * q^97 - 11664 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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