LMFDB - Embedded newform 624.6.c.d.337.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,6,Mod(337,624)] 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("624.337"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,54,0,0,0,0,0,486,0,0,0,530] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$100.079503563$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 $ x^6 - 2x^5 + 2x^4 - 2946x^3 + 131769x^2 - 1332936*x + 6741792
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$9.96927 - 9.96927i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.6.c.d.337.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -86.8538i q^{5} -98.7774i q^{7} +81.0000 q^{9} -610.754i q^{11} +(592.276 - 143.186i) q^{13} -781.684i q^{15} -1148.60 q^{17} -2267.14i q^{19} -888.996i q^{21} -433.224 q^{23} -4418.59 q^{25} +729.000 q^{27} +7669.59 q^{29} -7367.13i q^{31} -5496.79i q^{33} -8579.19 q^{35} +10575.0i q^{37} +(5330.48 - 1288.67i) q^{39} -3699.17i q^{41} +6061.57 q^{43} -7035.16i q^{45} -8740.69i q^{47} +7050.03 q^{49} -10337.4 q^{51} +34784.8 q^{53} -53046.3 q^{55} -20404.3i q^{57} +11949.6i q^{59} -45400.2 q^{61} -8000.97i q^{63} +(-12436.2 - 51441.4i) q^{65} +45698.1i q^{67} -3899.01 q^{69} +23826.9i q^{71} +37759.1i q^{73} -39767.3 q^{75} -60328.7 q^{77} +35307.0 q^{79} +6561.00 q^{81} -31484.4i q^{83} +99760.7i q^{85} +69026.3 q^{87} +59051.5i q^{89} +(-14143.5 - 58503.5i) q^{91} -66304.2i q^{93} -196910. q^{95} +4965.78i q^{97} -49471.1i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 54 q^{3} + 486 q^{9} + 530 q^{13} - 836 q^{17} + 416 q^{23} + 718 q^{25} + 4374 q^{27} + 18788 q^{29} - 6112 q^{35} + 4770 q^{39} + 24200 q^{43} - 3038 q^{49} - 7524 q^{51} - 42396 q^{53} - 124656 q^{55}+ \cdots - 567168 q^{95}+O(q^{100}) $ 6 * q + 54 * q^3 + 486 * q^9 + 530 * q^13 - 836 * q^17 + 416 * q^23 + 718 * q^25 + 4374 * q^27 + 18788 * q^29 - 6112 * q^35 + 4770 * q^39 + 24200 * q^43 - 3038 * q^49 - 7524 * q^51 - 42396 * q^53 - 124656 * q^55 - 3196 * q^61 - 17168 * q^65 + 3744 * q^69 + 6462 * q^75 - 114024 * q^77 + 169328 * q^79 + 39366 * q^81 + 169092 * q^87 - 236152 * q^91 - 567168 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	9.00000			0.577350
$3$	9.00000			0.577350
$4$		0			0
$5$		−	86.8538i		−	1.55369i	−0.629693	−	0.776844i	$-0.716819\pi$
$5$		−	86.8538i		−	1.55369i	0.629693	−	0.776844i	$-0.283181\pi$
$6$		0			0
$7$		−	98.7774i		−	0.761925i	−0.924590	−	0.380963i	$-0.875593\pi$
$7$		−	98.7774i		−	0.761925i	0.924590	−	0.380963i	$-0.124407\pi$
$8$		0			0
$9$	81.0000			0.333333
$10$		0			0
$11$		−	610.754i		−	1.52190i	−0.648813	−	0.760948i	$-0.724734\pi$
$11$		−	610.754i		−	1.52190i	0.648813	−	0.760948i	$-0.275266\pi$
$12$		0			0
$13$	592.276	−	143.186i	0.971999	−	0.234986i
$13$	592.276	−	143.186i	0.971999	−	0.234986i
$14$		0			0
$15$		−	781.684i		−	0.897023i
$16$		0			0
$17$	−1148.60			−0.963936			−0.481968	−	0.876189i	$-0.660078\pi$
$17$	−1148.60			−0.963936			−0.481968	+	0.876189i	$0.660078\pi$
$18$		0			0
$19$		−	2267.14i		−	1.44077i	−0.693574	−	0.720386i	$-0.743965\pi$
$19$		−	2267.14i		−	1.44077i	0.693574	−	0.720386i	$-0.256035\pi$
$20$		0			0
$21$		−	888.996i		−	0.439898i
$22$		0			0
$23$	−433.224			−0.170763			−0.0853813	−	0.996348i	$-0.527211\pi$
$23$	−433.224			−0.170763			−0.0853813	+	0.996348i	$0.527211\pi$
$24$		0			0
$25$	−4418.59			−1.41395
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	7669.59			1.69347			0.846734	−	0.532016i	$-0.178566\pi$
$29$	7669.59			1.69347			0.846734	+	0.532016i	$0.178566\pi$
$30$		0			0
$31$		−	7367.13i		−	1.37687i	−0.725296	−	0.688437i	$-0.758297\pi$
$31$		−	7367.13i		−	1.37687i	0.725296	−	0.688437i	$-0.241703\pi$
$32$		0			0
$33$		−	5496.79i		−	0.878667i
$34$		0			0
$35$	−8579.19			−1.18379
$36$		0			0
$37$			10575.0i			1.26991i	0.772547	+	0.634957i	$0.218982\pi$
$37$			10575.0i			1.26991i	−0.772547	+	0.634957i	$0.781018\pi$
$38$		0			0
$39$	5330.48	−	1288.67i	0.561184	−	0.135669i
$40$		0			0
$41$		−	3699.17i		−	0.343672i	−0.985126	−	0.171836i	$-0.945030\pi$
$41$		−	3699.17i		−	0.343672i	0.985126	−	0.171836i	$-0.0549700\pi$
$42$		0			0
$43$	6061.57			0.499936			0.249968	−	0.968254i	$-0.419580\pi$
$43$	6061.57			0.499936			0.249968	+	0.968254i	$0.419580\pi$
$44$		0			0
$45$		−	7035.16i		−	0.517896i
$46$		0			0
$47$		−	8740.69i		−	0.577166i	−0.957455	−	0.288583i	$-0.906816\pi$
$47$		−	8740.69i		−	0.577166i	0.957455	−	0.288583i	$-0.0931841\pi$
$48$		0			0
$49$	7050.03			0.419470
$50$		0			0
$51$	−10337.4			−0.556529
$52$		0			0
$53$	34784.8			1.70098			0.850491	−	0.525990i	$-0.176305\pi$
$53$	34784.8			1.70098			0.850491	+	0.525990i	$0.176305\pi$
$54$		0			0
$55$	−53046.3			−2.36455
$56$		0			0
$57$		−	20404.3i		−	0.831830i
$58$		0			0
$59$			11949.6i			0.446915i	0.974714	+	0.223457i	$0.0717344\pi$
$59$			11949.6i			0.446915i	−0.974714	+	0.223457i	$0.928266\pi$
$60$		0			0
$61$	−45400.2			−1.56219			−0.781094	−	0.624414i	$-0.785338\pi$
$61$	−45400.2			−1.56219			−0.781094	+	0.624414i	$0.785338\pi$
$62$		0			0
$63$		−	8000.97i		−	0.253975i
$64$		0			0
$65$	−12436.2	−	51441.4i	−0.365095	−	1.51018i
$66$		0			0
$67$			45698.1i			1.24369i	0.783141	+	0.621844i	$0.213616\pi$
$67$			45698.1i			1.24369i	−0.783141	+	0.621844i	$0.786384\pi$
$68$		0			0
$69$	−3899.01			−0.0985898
$70$		0			0
$71$			23826.9i			0.560946i	0.959862	+	0.280473i	$0.0904913\pi$
$71$			23826.9i			0.560946i	−0.959862	+	0.280473i	$0.909509\pi$
$72$		0			0
$73$			37759.1i			0.829305i	0.909980	+	0.414653i	$0.136097\pi$
$73$			37759.1i			0.829305i	−0.909980	+	0.414653i	$0.863903\pi$
$74$		0			0
$75$	−39767.3			−0.816343
$76$		0			0
$77$	−60328.7			−1.15957
$78$		0			0
$79$	35307.0			0.636492			0.318246	−	0.948008i	$-0.396906\pi$
$79$	35307.0			0.636492			0.318246	+	0.948008i	$0.396906\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$		−	31484.4i		−	0.501649i	−0.968033	−	0.250824i	$-0.919298\pi$
$83$		−	31484.4i		−	0.501649i	0.968033	−	0.250824i	$-0.0807017\pi$
$84$		0			0
$85$			99760.7i			1.49766i
$86$		0			0
$87$	69026.3			0.977724
$88$		0			0
$89$			59051.5i			0.790235i	0.918631	+	0.395117i	$0.129296\pi$
$89$			59051.5i			0.790235i	−0.918631	+	0.395117i	$0.870704\pi$
$90$		0			0
$91$	−14143.5	−	58503.5i	−0.179042	−	0.740590i
$92$		0			0
$93$		−	66304.2i		−	0.794938i
$94$		0			0
$95$	−196910.			−2.23851
$96$		0			0
$97$			4965.78i			0.0535868i	0.999641	+	0.0267934i	$0.00852963\pi$
$97$			4965.78i			0.0535868i	−0.999641	+	0.0267934i	$0.991470\pi$
$98$		0			0
$99$		−	49471.1i		−	0.507298i
$100$		0			0
$101$	−125605.			−1.22519			−0.612596	−	0.790396i	$-0.709875\pi$
$101$	−125605.			−1.22519			−0.612596	+	0.790396i	$0.709875\pi$
$102$		0			0
$103$	126352.			1.17351			0.586756	−	0.809764i	$-0.300405\pi$
$103$	126352.			1.17351			0.586756	+	0.809764i	$0.300405\pi$
$104$		0			0
$105$	−77212.7			−0.683464
$106$		0			0
$107$	−33067.8			−0.279219			−0.139610	−	0.990207i	$-0.544585\pi$
$107$	−33067.8			−0.279219			−0.139610	+	0.990207i	$0.544585\pi$
$108$		0			0
$109$			147666.i			1.19046i	0.803555	+	0.595230i	$0.202939\pi$
$109$			147666.i			1.19046i	−0.803555	+	0.595230i	$0.797061\pi$
$110$		0			0
$111$			95174.7i			0.733186i
$112$		0			0
$113$	118111.			0.870152			0.435076	−	0.900394i	$-0.356722\pi$
$113$	118111.			0.870152			0.435076	+	0.900394i	$0.356722\pi$
$114$		0			0
$115$			37627.1i			0.265312i
$116$		0			0
$117$	47974.4	−	11598.1i	0.324000	−	0.0783286i
$118$		0			0
$119$			113456.i			0.734447i
$120$		0			0
$121$	−211970.			−1.31616
$122$		0			0
$123$		−	33292.5i		−	0.198419i
$124$		0			0
$125$			112353.i			0.643147i
$126$		0			0
$127$	−322666.			−1.77518			−0.887592	−	0.460631i	$-0.847623\pi$
$127$	−322666.			−1.77518			−0.887592	+	0.460631i	$0.847623\pi$
$128$		0			0
$129$	54554.2			0.288638
$130$		0			0
$131$	357874.			1.82202			0.911008	−	0.412390i	$-0.135306\pi$
$131$	357874.			1.82202			0.911008	+	0.412390i	$0.135306\pi$
$132$		0			0
$133$	−223942.			−1.09776
$134$		0			0
$135$		−	63316.4i		−	0.299008i
$136$		0			0
$137$		−	347064.i		−	1.57982i	−0.613221	−	0.789912i	$-0.710126\pi$
$137$		−	347064.i		−	1.57982i	0.613221	−	0.789912i	$-0.289874\pi$
$138$		0			0
$139$	103030.			0.452299			0.226149	−	0.974093i	$-0.427386\pi$
$139$	103030.			0.452299			0.226149	+	0.974093i	$0.427386\pi$
$140$		0			0
$141$		−	78666.2i		−	0.333227i
$142$		0			0
$143$	−87451.4	−	361735.i	−0.357624	−	1.47928i
$144$		0			0
$145$		−	666133.i		−	2.63112i
$146$		0			0
$147$	63450.3			0.242181
$148$		0			0
$149$			90695.3i			0.334672i	0.985900	+	0.167336i	$0.0535164\pi$
$149$			90695.3i			0.334672i	−0.985900	+	0.167336i	$0.946484\pi$
$150$		0			0
$151$			104288.i			0.372215i	0.982529	+	0.186107i	$0.0595873\pi$
$151$			104288.i			0.372215i	−0.982529	+	0.186107i	$0.940413\pi$
$152$		0			0
$153$	−93036.9			−0.321312
$154$		0			0
$155$	−639863.			−2.13923
$156$		0			0
$157$	−314419.			−1.01803			−0.509014	−	0.860758i	$-0.669990\pi$
$157$	−314419.			−1.01803			−0.509014	+	0.860758i	$0.669990\pi$
$158$		0			0
$159$	313063.			0.982062
$160$		0			0
$161$			42792.7i			0.130108i
$162$		0			0
$163$		−	140732.i		−	0.414881i	−0.978248	−	0.207441i	$-0.933487\pi$
$163$		−	140732.i		−	0.414881i	0.978248	−	0.207441i	$-0.0665134\pi$
$164$		0			0
$165$	−477417.			−1.36517
$166$		0			0
$167$		−	108944.i		−	0.302281i	−0.988512	−	0.151141i	$-0.951705\pi$
$167$		−	108944.i		−	0.302281i	0.988512	−	0.151141i	$-0.0482946\pi$
$168$		0			0
$169$	330289.	−	169611.i	0.889563	−	0.456812i
$170$		0			0
$171$		−	183639.i		−	0.480257i
$172$		0			0
$173$	401188.			1.01914			0.509569	−	0.860430i	$-0.329805\pi$
$173$	401188.			1.01914			0.509569	+	0.860430i	$0.329805\pi$
$174$		0			0
$175$			436456.i			1.07732i
$176$		0			0
$177$			107547.i			0.258026i
$178$		0			0
$179$	209535.			0.488792			0.244396	−	0.969676i	$-0.421410\pi$
$179$	209535.			0.488792			0.244396	+	0.969676i	$0.421410\pi$
$180$		0			0
$181$	236277.			0.536075			0.268038	−	0.963408i	$-0.413625\pi$
$181$	236277.			0.536075			0.268038	+	0.963408i	$0.413625\pi$
$182$		0			0
$183$	−408602.			−0.901929
$184$		0			0
$185$	918476.			1.97305
$186$		0			0
$187$			701515.i			1.46701i
$188$		0			0
$189$		−	72008.7i		−	0.146633i
$190$		0			0
$191$	−509719.			−1.01099			−0.505496	−	0.862829i	$-0.668690\pi$
$191$	−509719.			−1.01099			−0.505496	+	0.862829i	$0.668690\pi$
$192$		0			0
$193$		−	58146.8i		−	0.112365i	−0.998421	−	0.0561827i	$-0.982107\pi$
$193$		−	58146.8i		−	0.112365i	0.998421	−	0.0561827i	$-0.0178929\pi$
$194$		0			0
$195$	−111926.	−	462973.i	−0.210788	−	0.871905i
$196$		0			0
$197$			91830.0i			0.168585i	0.996441	+	0.0842925i	$0.0268630\pi$
$197$			91830.0i			0.168585i	−0.996441	+	0.0842925i	$0.973137\pi$
$198$		0			0
$199$	−562291.			−1.00653			−0.503267	−	0.864131i	$-0.667869\pi$
$199$	−562291.			−1.00653			−0.503267	+	0.864131i	$0.667869\pi$
$200$		0			0
$201$			411283.i			0.718043i
$202$		0			0
$203$		−	757582.i		−	1.29030i
$204$		0			0
$205$	−321287.			−0.533960
$206$		0			0
$207$	−35091.1			−0.0569208
$208$		0			0
$209$	−1.38467e6			−2.19270
$210$		0			0
$211$	−650662.			−1.00612			−0.503059	−	0.864252i	$-0.667792\pi$
$211$	−650662.			−1.00612			−0.503059	+	0.864252i	$0.667792\pi$
$212$		0			0
$213$			214442.i			0.323862i
$214$		0			0
$215$		−	526471.i		−	0.776745i
$216$		0			0
$217$	−727706.			−1.04907
$218$		0			0
$219$			339832.i			0.478800i
$220$		0			0
$221$	−680291.	+	164464.i	−0.936944	+	0.226511i
$222$		0			0
$223$		−	302795.i		−	0.407743i	−0.978998	−	0.203871i	$-0.934648\pi$
$223$		−	302795.i		−	0.407743i	0.978998	−	0.203871i	$-0.0653525\pi$
$224$		0			0
$225$	−357906.			−0.471316
$226$		0			0
$227$		−	723191.i		−	0.931512i	−0.884913	−	0.465756i	$-0.845782\pi$
$227$		−	723191.i		−	0.931512i	0.884913	−	0.465756i	$-0.154218\pi$
$228$		0			0
$229$			1.39378e6i			1.75633i	0.478356	+	0.878166i	$0.341233\pi$
$229$			1.39378e6i			1.75633i	−0.478356	+	0.878166i	$0.658767\pi$
$230$		0			0
$231$	−542958.			−0.669478
$232$		0			0
$233$	−319385.			−0.385411			−0.192706	−	0.981257i	$-0.561726\pi$
$233$	−319385.			−0.385411			−0.192706	+	0.981257i	$0.561726\pi$
$234$		0			0
$235$	−759162.			−0.896736
$236$		0			0
$237$	317763.			0.367479
$238$		0			0
$239$			1.52265e6i			1.72427i	0.506682	+	0.862133i	$0.330872\pi$
$239$			1.52265e6i			1.72427i	−0.506682	+	0.862133i	$0.669128\pi$
$240$		0			0
$241$			607839.i			0.674134i	0.941481	+	0.337067i	$0.109435\pi$
$241$			607839.i			0.674134i	−0.941481	+	0.337067i	$0.890565\pi$
$242$		0			0
$243$	59049.0			0.0641500
$244$		0			0
$245$		−	612323.i		−	0.651726i
$246$		0			0
$247$	−324623.	−	1.34277e6i	−0.338561	−	1.40043i
$248$		0			0
$249$		−	283359.i		−	0.289627i
$250$		0			0
$251$	1.05049e6			1.05247			0.526234	−	0.850340i	$-0.323604\pi$
$251$	1.05049e6			1.05247			0.526234	+	0.850340i	$0.323604\pi$
$252$		0			0
$253$			264593.i			0.259883i
$254$		0			0
$255$			897846.i			0.864672i
$256$		0			0
$257$	236240.			0.223111			0.111555	−	0.993758i	$-0.464417\pi$
$257$	236240.			0.223111			0.111555	+	0.993758i	$0.464417\pi$
$258$		0			0
$259$	1.04457e6			0.967580
$260$		0			0
$261$	621237.			0.564489
$262$		0			0
$263$	−928964.			−0.828151			−0.414076	−	0.910242i	$-0.635895\pi$
$263$	−928964.			−0.828151			−0.414076	+	0.910242i	$0.635895\pi$
$264$		0			0
$265$		−	3.02119e6i		−	2.64280i
$266$		0			0
$267$			531464.i			0.456242i
$268$		0			0
$269$	827650.			0.697374			0.348687	−	0.937239i	$-0.386628\pi$
$269$	827650.			0.697374			0.348687	+	0.937239i	$0.386628\pi$
$270$		0			0
$271$		−	660781.i		−	0.546555i	−0.961935	−	0.273278i	$-0.911892\pi$
$271$		−	660781.i		−	0.546555i	0.961935	−	0.273278i	$-0.0881078\pi$
$272$		0			0
$273$	−127292.	−	526531.i	−0.103370	−	0.427580i
$274$		0			0
$275$			2.69867e6i			2.15188i
$276$		0			0
$277$	−1.38275e6			−1.08279			−0.541395	−	0.840769i	$-0.682104\pi$
$277$	−1.38275e6			−1.08279			−0.541395	+	0.840769i	$0.682104\pi$
$278$		0			0
$279$		−	596738.i		−	0.458958i
$280$		0			0
$281$			1.44893e6i			1.09467i	0.836915	+	0.547333i	$0.184357\pi$
$281$			1.44893e6i			1.09467i	−0.836915	+	0.547333i	$0.815643\pi$
$282$		0			0
$283$	−1.22044e6			−0.905841			−0.452921	−	0.891551i	$-0.649618\pi$
$283$	−1.22044e6			−0.905841			−0.452921	+	0.891551i	$0.649618\pi$
$284$		0			0
$285$	−1.77219e6			−1.29240
$286$		0			0
$287$	−365394.			−0.261853
$288$		0			0
$289$	−100565.			−0.0708279
$290$		0			0
$291$			44692.0i			0.0309384i
$292$		0			0
$293$		−	2.19465e6i		−	1.49347i	−0.665123	−	0.746734i	$-0.731621\pi$
$293$		−	2.19465e6i		−	1.49347i	0.665123	−	0.746734i	$-0.268379\pi$
$294$		0			0
$295$	1.03787e6			0.694366
$296$		0			0
$297$		−	445240.i		−	0.292889i
$298$		0			0
$299$	−256588.	+	62031.5i	−0.165981	+	0.0401268i
$300$		0			0
$301$		−	598746.i		−	0.380914i
$302$		0			0
$303$	−1.13045e6			−0.707365
$304$		0			0
$305$			3.94318e6i			2.42715i
$306$		0			0
$307$			83619.1i			0.0506360i	0.999679	+	0.0253180i	$0.00805983\pi$
$307$			83619.1i			0.0506360i	−0.999679	+	0.0253180i	$0.991940\pi$
$308$		0			0
$309$	1.13716e6			0.677528
$310$		0			0
$311$	−491767.			−0.288309			−0.144155	−	0.989555i	$-0.546046\pi$
$311$	−491767.			−0.288309			−0.144155	+	0.989555i	$0.546046\pi$
$312$		0			0
$313$	896969.			0.517508			0.258754	−	0.965943i	$-0.416688\pi$
$313$	896969.			0.517508			0.258754	+	0.965943i	$0.416688\pi$
$314$		0			0
$315$	−694915.			−0.394598
$316$		0			0
$317$		−	3.15449e6i		−	1.76312i	−0.472074	−	0.881559i	$-0.656494\pi$
$317$		−	3.15449e6i		−	1.76312i	0.472074	−	0.881559i	$-0.343506\pi$
$318$		0			0
$319$		−	4.68423e6i		−	2.57728i
$320$		0			0
$321$	−297610.			−0.161207
$322$		0			0
$323$			2.60405e6i			1.38881i
$324$		0			0
$325$	−2.61702e6	+	632679.i	−1.37436	+	0.332258i
$326$		0			0
$327$			1.32900e6i			0.687312i
$328$		0			0
$329$	−863382.			−0.439757
$330$		0			0
$331$			597196.i			0.299604i	0.988716	+	0.149802i	$0.0478636\pi$
$331$			597196.i			0.299604i	−0.988716	+	0.149802i	$0.952136\pi$
$332$		0			0
$333$			856572.i			0.423305i
$334$		0			0
$335$	3.96906e6			1.93230
$336$		0			0
$337$	3.61808e6			1.73541			0.867707	−	0.497077i	$-0.165593\pi$
$337$	3.61808e6			1.73541			0.867707	+	0.497077i	$0.165593\pi$
$338$		0			0
$339$	1.06300e6			0.502382
$340$		0			0
$341$	−4.49951e6			−2.09546
$342$		0			0
$343$		−	2.35653e6i		−	1.08153i
$344$		0			0
$345$			338644.i			0.153178i
$346$		0			0
$347$	1.62893e6			0.726236			0.363118	−	0.931743i	$-0.381712\pi$
$347$	1.62893e6			0.726236			0.363118	+	0.931743i	$0.381712\pi$
$348$		0			0
$349$		−	2.60140e6i		−	1.14326i	−0.820513	−	0.571628i	$-0.806312\pi$
$349$		−	2.60140e6i		−	1.14326i	0.820513	−	0.571628i	$-0.193688\pi$
$350$		0			0
$351$	431769.	−	104383.i	0.187061	−	0.0452231i
$352$		0			0
$353$			1.30441e6i			0.557155i	0.960414	+	0.278578i	$0.0898630\pi$
$353$			1.30441e6i			0.557155i	−0.960414	+	0.278578i	$0.910137\pi$
$354$		0			0
$355$	2.06945e6			0.871535
$356$		0			0
$357$			1.02110e6i			0.424033i
$358$		0			0
$359$		−	213674.i		−	0.0875013i	−0.999042	−	0.0437507i	$-0.986069\pi$
$359$		−	213674.i		−	0.0875013i	0.999042	−	0.0437507i	$-0.0139307\pi$
$360$		0			0
$361$	−2.66384e6			−1.07582
$362$		0			0
$363$	−1.90773e6			−0.759888
$364$		0			0
$365$	3.27952e6			1.28848
$366$		0			0
$367$	−1.15670e6			−0.448285			−0.224142	−	0.974556i	$-0.571958\pi$
$367$	−1.15670e6			−0.448285			−0.224142	+	0.974556i	$0.571958\pi$
$368$		0			0
$369$		−	299633.i		−	0.114557i
$370$		0			0
$371$		−	3.43595e6i		−	1.29602i
$372$		0			0
$373$	1.97983e6			0.736810			0.368405	−	0.929665i	$-0.379904\pi$
$373$	1.97983e6			0.736810			0.368405	+	0.929665i	$0.379904\pi$
$374$		0			0
$375$			1.01118e6i			0.371321i
$376$		0			0
$377$	4.54251e6	−	1.09818e6i	1.64605	−	0.397941i
$378$		0			0
$379$			3.86155e6i			1.38090i	0.723379	+	0.690451i	$0.242588\pi$
$379$			3.86155e6i			1.38090i	−0.723379	+	0.690451i	$0.757412\pi$
$380$		0			0
$381$	−2.90399e6			−1.02490
$382$		0			0
$383$			1.55518e6i			0.541731i	0.962617	+	0.270866i	$0.0873099\pi$
$383$			1.55518e6i			0.541731i	−0.962617	+	0.270866i	$0.912690\pi$
$384$		0			0
$385$			5.23978e6i			1.80161i
$386$		0			0
$387$	490988.			0.166645
$388$		0			0
$389$	2.85249e6			0.955763			0.477881	−	0.878424i	$-0.341405\pi$
$389$	2.85249e6			0.955763			0.477881	+	0.878424i	$0.341405\pi$
$390$		0			0
$391$	497603.			0.164604
$392$		0			0
$393$	3.22087e6			1.05194
$394$		0			0
$395$		−	3.06655e6i		−	0.988911i
$396$		0			0
$397$		−	1.25222e6i		−	0.398752i	−0.979923	−	0.199376i	$-0.936109\pi$
$397$		−	1.25222e6i		−	0.398752i	0.979923	−	0.199376i	$-0.0638915\pi$
$398$		0			0
$399$	−2.01548e6			−0.633792
$400$		0			0
$401$			4.20050e6i			1.30449i	0.758009	+	0.652244i	$0.226172\pi$
$401$			4.20050e6i			1.30449i	−0.758009	+	0.652244i	$0.773828\pi$
$402$		0			0
$403$	−1.05487e6	−	4.36337e6i	−0.323546	−	1.33832i
$404$		0			0
$405$		−	569848.i		−	0.172632i
$406$		0			0
$407$	6.45870e6			1.93268
$408$		0			0
$409$		−	650694.i		−	0.192340i	−0.995365	−	0.0961698i	$-0.969341\pi$
$409$		−	650694.i		−	0.192340i	0.995365	−	0.0961698i	$-0.0306592\pi$
$410$		0			0
$411$		−	3.12358e6i		−	0.912111i
$412$		0			0
$413$	1.18035e6			0.340516
$414$		0			0
$415$	−2.73454e6			−0.779406
$416$		0			0
$417$	927267.			0.261135
$418$		0			0
$419$	6.82010e6			1.89782			0.948912	−	0.315541i	$-0.102186\pi$
$419$	6.82010e6			1.89782			0.948912	+	0.315541i	$0.102186\pi$
$420$		0			0
$421$		−	3.11659e6i		−	0.856988i	−0.903545	−	0.428494i	$-0.859044\pi$
$421$		−	3.11659e6i		−	0.856988i	0.903545	−	0.428494i	$-0.140956\pi$
$422$		0			0
$423$		−	707996.i		−	0.192389i
$424$		0			0
$425$	5.07521e6			1.36296
$426$		0			0
$427$			4.48451e6i			1.19027i
$428$		0			0
$429$	−787062.	−	3.25562e6i	−0.206474	−	0.854063i
$430$		0			0
$431$			4.88213e6i			1.26595i	0.774173	+	0.632974i	$0.218166\pi$
$431$			4.88213e6i			1.26595i	−0.774173	+	0.632974i	$0.781834\pi$
$432$		0			0
$433$	3.64275e6			0.933705			0.466852	−	0.884335i	$-0.345388\pi$
$433$	3.64275e6			0.933705			0.466852	+	0.884335i	$0.345388\pi$
$434$		0			0
$435$		−	5.99520e6i		−	1.51908i
$436$		0			0
$437$			982180.i			0.246030i
$438$		0			0
$439$	1.70122e6			0.421308			0.210654	−	0.977561i	$-0.432441\pi$
$439$	1.70122e6			0.421308			0.210654	+	0.977561i	$0.432441\pi$
$440$		0			0
$441$	571053.			0.139823
$442$		0			0
$443$	5.85057e6			1.41641			0.708205	−	0.706007i	$-0.249505\pi$
$443$	5.85057e6			1.41641			0.708205	+	0.706007i	$0.249505\pi$
$444$		0			0
$445$	5.12885e6			1.22778
$446$		0			0
$447$			816258.i			0.193223i
$448$		0			0
$449$		−	1.56399e6i		−	0.366115i	−0.983102	−	0.183058i	$-0.941401\pi$
$449$		−	1.56399e6i		−	0.366115i	0.983102	−	0.183058i	$-0.0585995\pi$
$450$		0			0
$451$	−2.25928e6			−0.523033
$452$		0			0
$453$			938596.i			0.214898i
$454$		0			0
$455$	−5.08125e6	+	1.22842e6i	−1.15065	+	0.278175i
$456$		0			0
$457$			336948.i			0.0754697i	0.999288	+	0.0377349i	$0.0120142\pi$
$457$			336948.i			0.0754697i	−0.999288	+	0.0377349i	$0.987986\pi$
$458$		0			0
$459$	−837332.			−0.185510
$460$		0			0
$461$			3.89174e6i			0.852887i	0.904514	+	0.426444i	$0.140234\pi$
$461$			3.89174e6i			0.852887i	−0.904514	+	0.426444i	$0.859766\pi$
$462$		0			0
$463$			6.34214e6i			1.37494i	0.726213	+	0.687470i	$0.241279\pi$
$463$			6.34214e6i			1.37494i	−0.726213	+	0.687470i	$0.758721\pi$
$464$		0			0
$465$	−5.75877e6			−1.23509
$466$		0			0
$467$	−2.48691e6			−0.527677			−0.263839	−	0.964567i	$-0.584989\pi$
$467$	−2.48691e6			−0.527677			−0.263839	+	0.964567i	$0.584989\pi$
$468$		0			0
$469$	4.51394e6			0.947597
$470$		0			0
$471$	−2.82977e6			−0.587759
$472$		0			0
$473$		−	3.70213e6i		−	0.760850i
$474$		0			0
$475$			1.00176e7i			2.03718i
$476$		0			0
$477$	2.81757e6			0.566994
$478$		0			0
$479$			5.31079e6i			1.05760i	0.848747	+	0.528798i	$0.177357\pi$
$479$			5.31079e6i			1.05760i	−0.848747	+	0.528798i	$0.822643\pi$
$480$		0			0
$481$	1.51419e6	+	6.26330e6i	0.298412	+	1.23436i
$482$		0			0
$483$			385134.i			0.0751180i
$484$		0			0
$485$	431297.			0.0832572
$486$		0			0
$487$			7.73506e6i			1.47789i	0.673767	+	0.738944i	$0.264675\pi$
$487$			7.73506e6i			1.47789i	−0.673767	+	0.738944i	$0.735325\pi$
$488$		0			0
$489$		−	1.26659e6i		−	0.239532i
$490$		0			0
$491$	322490.			0.0603689			0.0301844	−	0.999544i	$-0.490391\pi$
$491$	322490.			0.0603689			0.0301844	+	0.999544i	$0.490391\pi$
$492$		0			0
$493$	−8.80932e6			−1.63239
$494$		0			0
$495$	−4.29675e6			−0.788184
$496$		0			0
$497$	2.35355e6			0.427399
$498$		0			0
$499$		−	414653.i		−	0.0745476i	−0.999305	−	0.0372738i	$-0.988133\pi$
$499$		−	414653.i		−	0.0745476i	0.999305	−	0.0372738i	$-0.0118674\pi$
$500$		0			0
$501$		−	980493.i		−	0.174522i
$502$		0			0
$503$	1.11227e7			1.96016			0.980080	−	0.198604i	$-0.0636408\pi$
$503$	1.11227e7			1.96016			0.980080	+	0.198604i	$0.0636408\pi$
$504$		0			0
$505$			1.09093e7i			1.90357i
$506$		0			0
$507$	2.97260e6	−	1.52650e6i	0.513590	−	0.263741i
$508$		0			0
$509$		−	754143.i		−	0.129021i	−0.997917	−	0.0645103i	$-0.979451\pi$
$509$		−	754143.i		−	0.129021i	0.997917	−	0.0645103i	$-0.0205485\pi$
$510$		0			0
$511$	3.72974e6			0.631869
$512$		0			0
$513$		−	1.65275e6i		−	0.277277i
$514$		0			0
$515$		−	1.09741e7i		−	1.82327i
$516$		0			0
$517$	−5.33841e6			−0.878386
$518$		0			0
$519$	3.61069e6			0.588399
$520$		0			0
$521$	6.34188e6			1.02358			0.511792	−	0.859109i	$-0.328982\pi$
$521$	6.34188e6			1.02358			0.511792	+	0.859109i	$0.328982\pi$
$522$		0			0
$523$	−7.64810e6			−1.22264			−0.611321	−	0.791382i	$-0.709362\pi$
$523$	−7.64810e6			−1.22264			−0.611321	+	0.791382i	$0.709362\pi$
$524$		0			0
$525$			3.92811e6i			0.621993i
$526$		0			0
$527$			8.46192e6i			1.32722i
$528$		0			0
$529$	−6.24866e6			−0.970840
$530$		0			0
$531$			967921.i			0.148972i
$532$		0			0
$533$	−529669.	−	2.19093e6i	−0.0807582	−	0.334049i
$534$		0			0
$535$			2.87206e6i			0.433820i
$536$		0			0
$537$	1.88581e6			0.282204
$538$		0			0
$539$		−	4.30584e6i		−	0.638390i
$540$		0			0
$541$			5.40473e6i			0.793927i	0.917834	+	0.396964i	$0.129936\pi$
$541$			5.40473e6i			0.793927i	−0.917834	+	0.396964i	$0.870064\pi$
$542$		0			0
$543$	2.12650e6			0.309503
$544$		0			0
$545$	1.28254e7			1.84960
$546$		0			0
$547$	−6.06535e6			−0.866737			−0.433368	−	0.901217i	$-0.642675\pi$
$547$	−6.06535e6			−0.866737			−0.433368	+	0.901217i	$0.642675\pi$
$548$		0			0
$549$	−3.67742e6			−0.520729
$550$		0			0
$551$		−	1.73881e7i		−	2.43990i
$552$		0			0
$553$		−	3.48753e6i		−	0.484960i
$554$		0			0
$555$	8.26629e6			1.13914
$556$		0			0
$557$			6.81169e6i			0.930287i	0.885235	+	0.465143i	$0.153997\pi$
$557$			6.81169e6i			0.930287i	−0.885235	+	0.465143i	$0.846003\pi$
$558$		0			0
$559$	3.59012e6	−	867932.i	0.485937	−	0.117478i
$560$		0			0
$561$			6.31363e6i			0.846978i
$562$		0			0
$563$	−4.93741e6			−0.656491			−0.328245	−	0.944593i	$-0.606457\pi$
$563$	−4.93741e6			−0.656491			−0.328245	+	0.944593i	$0.606457\pi$
$564$		0			0
$565$		−	1.02584e7i		−	1.35194i
$566$		0			0
$567$		−	648078.i		−	0.0846583i
$568$		0			0
$569$	9.18371e6			1.18915			0.594576	−	0.804039i	$-0.297320\pi$
$569$	9.18371e6			1.18915			0.594576	+	0.804039i	$0.297320\pi$
$570$		0			0
$571$	1.11741e7			1.43424			0.717120	−	0.696949i	$-0.245460\pi$
$571$	1.11741e7			1.43424			0.717120	+	0.696949i	$0.245460\pi$
$572$		0			0
$573$	−4.58748e6			−0.583697
$574$		0			0
$575$	1.91424e6			0.241449
$576$		0			0
$577$		−	8.62732e6i		−	1.07879i	−0.842053	−	0.539394i	$-0.818653\pi$
$577$		−	8.62732e6i		−	1.07879i	0.842053	−	0.539394i	$-0.181347\pi$
$578$		0			0
$579$		−	523322.i		−	0.0648742i
$580$		0			0
$581$	−3.10994e6			−0.382219
$582$		0			0
$583$		−	2.12450e7i		−	2.58872i
$584$		0			0
$585$	−1.00734e6	−	4.16676e6i	−0.121698	−	0.503394i
$586$		0			0
$587$		−	5.86783e6i		−	0.702882i	−0.936210	−	0.351441i	$-0.885692\pi$
$587$		−	5.86783e6i		−	0.702882i	0.936210	−	0.351441i	$-0.114308\pi$
$588$		0			0
$589$	−1.67023e7			−1.98376
$590$		0			0
$591$			826470.i			0.0973326i
$592$		0			0
$593$		−	1.28187e7i		−	1.49694i	−0.663166	−	0.748472i	$-0.730788\pi$
$593$		−	1.28187e7i		−	1.49694i	0.663166	−	0.748472i	$-0.269212\pi$
$594$		0			0
$595$	9.85410e6			1.14110
$596$		0			0
$597$	−5.06062e6			−0.581123
$598$		0			0
$599$	8.82262e6			1.00469			0.502343	−	0.864668i	$-0.332471\pi$
$599$	8.82262e6			1.00469			0.502343	+	0.864668i	$0.332471\pi$
$600$		0			0
$601$	8.16061e6			0.921587			0.460793	−	0.887507i	$-0.347565\pi$
$601$	8.16061e6			0.921587			0.460793	+	0.887507i	$0.347565\pi$
$602$		0			0
$603$			3.70155e6i			0.414563i
$604$		0			0
$605$			1.84104e7i			2.04491i
$606$		0			0
$607$	−8.03055e6			−0.884654			−0.442327	−	0.896854i	$-0.645847\pi$
$607$	−8.03055e6			−0.884654			−0.442327	+	0.896854i	$0.645847\pi$
$608$		0			0
$609$		−	6.81823e6i		−	0.744953i
$610$		0			0
$611$	−1.25154e6	−	5.17690e6i	−0.135626	−	0.561005i
$612$		0			0
$613$		−	3.99383e6i		−	0.429278i	−0.976693	−	0.214639i	$-0.931143\pi$
$613$		−	3.99383e6i		−	0.429278i	0.976693	−	0.214639i	$-0.0688575\pi$
$614$		0			0
$615$	−2.89158e6			−0.308282
$616$		0			0
$617$		−	1.20933e7i		−	1.27888i	−0.768840	−	0.639442i	$-0.779166\pi$
$617$		−	1.20933e7i		−	1.27888i	0.768840	−	0.639442i	$-0.220834\pi$
$618$		0			0
$619$			1.31953e7i			1.38418i	0.721813	+	0.692088i	$0.243309\pi$
$619$			1.31953e7i			1.38418i	−0.721813	+	0.692088i	$0.756691\pi$
$620$		0			0
$621$	−315820.			−0.0328633
$622$		0			0
$623$	5.83295e6			0.602100
$624$		0			0
$625$	−4.04979e6			−0.414699
$626$		0			0
$627$	−1.24620e7			−1.26596
$628$		0			0
$629$		−	1.21464e7i		−	1.22412i
$630$		0			0
$631$		−	1.16207e7i		−	1.16188i	−0.813947	−	0.580939i	$-0.802686\pi$
$631$		−	1.16207e7i		−	1.16188i	0.813947	−	0.580939i	$-0.197314\pi$
$632$		0			0
$633$	−5.85595e6			−0.580883
$634$		0			0
$635$			2.80247e7i			2.75808i
$636$		0			0
$637$	4.17557e6	−	1.00947e6i	0.407724	−	0.0985696i
$638$		0			0
$639$			1.92998e6i			0.186982i
$640$		0			0
$641$	−1.15298e7			−1.10835			−0.554173	−	0.832402i	$-0.686965\pi$
$641$	−1.15298e7			−1.10835			−0.554173	+	0.832402i	$0.686965\pi$
$642$		0			0
$643$			2.90732e6i			0.277310i	0.990341	+	0.138655i	$0.0442778\pi$
$643$			2.90732e6i			0.277310i	−0.990341	+	0.138655i	$0.955722\pi$
$644$		0			0
$645$		−	4.73824e6i		−	0.448454i
$646$		0			0
$647$	4.58122e6			0.430249			0.215125	−	0.976587i	$-0.430984\pi$
$647$	4.58122e6			0.430249			0.215125	+	0.976587i	$0.430984\pi$
$648$		0			0
$649$	7.29829e6			0.680157
$650$		0			0
$651$	−6.54935e6			−0.605684
$652$		0			0
$653$	1.95066e7			1.79019			0.895096	−	0.445874i	$-0.147107\pi$
$653$	1.95066e7			1.79019			0.895096	+	0.445874i	$0.147107\pi$
$654$		0			0
$655$		−	3.10827e7i		−	2.83084i
$656$		0			0
$657$			3.05849e6i			0.276435i
$658$		0			0
$659$	−8.41225e6			−0.754568			−0.377284	−	0.926098i	$-0.623142\pi$
$659$	−8.41225e6			−0.754568			−0.377284	+	0.926098i	$0.623142\pi$
$660$		0			0
$661$			1.64620e7i			1.46548i	0.680510	+	0.732739i	$0.261758\pi$
$661$			1.64620e7i			1.46548i	−0.680510	+	0.732739i	$0.738242\pi$
$662$		0			0
$663$	−6.12262e6	+	1.48018e6i	−0.540945	+	0.130776i
$664$		0			0
$665$			1.94503e7i			1.70558i
$666$		0			0
$667$	−3.32265e6			−0.289181
$668$		0			0
$669$		−	2.72515e6i		−	0.235410i
$670$		0			0
$671$			2.77284e7i			2.37748i
$672$		0			0
$673$	9.88194e6			0.841016			0.420508	−	0.907289i	$-0.361852\pi$
$673$	9.88194e6			0.841016			0.420508	+	0.907289i	$0.361852\pi$
$674$		0			0
$675$	−3.22115e6			−0.272114
$676$		0			0
$677$	−731933.			−0.0613761			−0.0306881	−	0.999529i	$-0.509770\pi$
$677$	−731933.			−0.0613761			−0.0306881	+	0.999529i	$0.509770\pi$
$678$		0			0
$679$	490506.			0.0408291
$680$		0			0
$681$		−	6.50872e6i		−	0.537809i
$682$		0			0
$683$			6.46151e6i			0.530008i	0.964247	+	0.265004i	$0.0853733\pi$
$683$			6.46151e6i			0.530008i	−0.964247	+	0.265004i	$0.914627\pi$
$684$		0			0
$685$	−3.01439e7			−2.45455
$686$		0			0
$687$			1.25440e7i			1.01402i
$688$		0			0
$689$	2.06022e7	−	4.98069e6i	1.65335	−	0.399707i
$690$		0			0
$691$		−	1.16211e7i		−	0.925877i	−0.886390	−	0.462939i	$-0.846795\pi$
$691$		−	1.16211e7i		−	0.925877i	0.886390	−	0.462939i	$-0.153205\pi$
$692$		0			0
$693$	−4.88662e6			−0.386523
$694$		0			0
$695$		−	8.94852e6i		−	0.702731i
$696$		0			0
$697$			4.24888e6i			0.331278i
$698$		0			0
$699$	−2.87446e6			−0.222517
$700$		0			0
$701$	−1.39904e6			−0.107531			−0.0537655	−	0.998554i	$-0.517122\pi$
$701$	−1.39904e6			−0.107531			−0.0537655	+	0.998554i	$0.517122\pi$
$702$		0			0
$703$	2.39750e7			1.82966
$704$		0			0
$705$	−6.83246e6			−0.517731
$706$		0			0
$707$			1.24069e7i			0.933504i
$708$		0			0
$709$		−	1.84802e7i		−	1.38067i	−0.723489	−	0.690336i	$-0.757463\pi$
$709$		−	1.84802e7i		−	1.38067i	0.723489	−	0.690336i	$-0.242537\pi$
$710$		0			0
$711$	2.85987e6			0.212164
$712$		0			0
$713$			3.19162e6i			0.235118i
$714$		0			0
$715$	−3.14181e7	+	7.59549e6i	−2.29834	+	0.555636i
$716$		0			0
$717$			1.37038e7i			0.995505i
$718$		0			0
$719$	−7.61846e6			−0.549598			−0.274799	−	0.961502i	$-0.588611\pi$
$719$	−7.61846e6			−0.549598			−0.274799	+	0.961502i	$0.588611\pi$
$720$		0			0
$721$		−	1.24807e7i		−	0.894128i
$722$		0			0
$723$			5.47055e6i			0.389211i
$724$		0			0
$725$	−3.38888e7			−2.39448
$726$		0			0
$727$	−2.55002e7			−1.78940			−0.894699	−	0.446670i	$-0.852610\pi$
$727$	−2.55002e7			−1.78940			−0.894699	+	0.446670i	$0.852610\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−6.96235e6			−0.481906
$732$		0			0
$733$			1.27470e7i			0.876293i	0.898903	+	0.438147i	$0.144365\pi$
$733$			1.27470e7i			0.876293i	−0.898903	+	0.438147i	$0.855635\pi$
$734$		0			0
$735$		−	5.51090e6i		−	0.376274i
$736$		0			0
$737$	2.79103e7			1.89276
$738$		0			0
$739$		−	1.54752e7i		−	1.04238i	−0.853441	−	0.521190i	$-0.825488\pi$
$739$		−	1.54752e7i		−	1.04238i	0.853441	−	0.521190i	$-0.174512\pi$
$740$		0			0
$741$	−2.92161e6	−	1.20850e7i	−0.195468	−	0.808537i
$742$		0			0
$743$		−	3.52596e6i		−	0.234318i	−0.993113	−	0.117159i	$-0.962621\pi$
$743$		−	3.52596e6i		−	0.234318i	0.993113	−	0.117159i	$-0.0373787\pi$
$744$		0			0
$745$	7.87723e6			0.519976
$746$		0			0
$747$		−	2.55023e6i		−	0.167216i
$748$		0			0
$749$			3.26635e6i			0.212744i
$750$		0			0
$751$	−8.20700e6			−0.530988			−0.265494	−	0.964113i	$-0.585535\pi$
$751$	−8.20700e6			−0.530988			−0.265494	+	0.964113i	$0.585535\pi$
$752$		0			0
$753$	9.45443e6			0.607642
$754$		0			0
$755$	9.05785e6			0.578306
$756$		0			0
$757$	−1.27038e7			−0.805736			−0.402868	−	0.915258i	$-0.631987\pi$
$757$	−1.27038e7			−0.805736			−0.402868	+	0.915258i	$0.631987\pi$
$758$		0			0
$759$			2.38134e6i			0.150043i
$760$		0			0
$761$			1.41876e7i			0.888073i	0.896009	+	0.444037i	$0.146454\pi$
$761$			1.41876e7i			0.888073i	−0.896009	+	0.444037i	$0.853546\pi$
$762$		0			0
$763$	1.45861e7			0.907041
$764$		0			0
$765$			8.08061e6i			0.499219i
$766$		0			0
$767$	1.71102e6	+	7.07748e6i	0.105019	+	0.434401i
$768$		0			0
$769$		−	1.10700e7i		−	0.675045i	−0.941318	−	0.337522i	$-0.890411\pi$
$769$		−	1.10700e7i		−	0.675045i	0.941318	−	0.337522i	$-0.109589\pi$
$770$		0			0
$771$	2.12616e6			0.128813
$772$		0			0
$773$			5.17639e6i			0.311586i	0.987790	+	0.155793i	$0.0497933\pi$
$773$			5.17639e6i			0.311586i	−0.987790	+	0.155793i	$0.950207\pi$
$774$		0			0
$775$			3.25523e7i			1.94683i
$776$		0			0
$777$	9.40110e6			0.558633
$778$		0			0
$779$	−8.38655e6			−0.495153
$780$		0			0
$781$	1.45524e7			0.853701
$782$		0			0
$783$	5.59113e6			0.325908
$784$		0			0
$785$			2.73085e7i			1.58170i
$786$		0			0
$787$		−	1.43044e7i		−	0.823252i	−0.911353	−	0.411626i	$-0.864961\pi$
$787$		−	1.43044e7i		−	0.823252i	0.911353	−	0.411626i	$-0.135039\pi$
$788$		0			0
$789$	−8.36068e6			−0.478133
$790$		0			0
$791$		−	1.16667e7i		−	0.662990i
$792$		0			0
$793$	−2.68894e7	+	6.50067e6i	−1.51844	+	0.367092i
$794$		0			0
$795$		−	2.71907e7i		−	1.52582i
$796$		0			0
$797$	−1.27192e7			−0.709275			−0.354638	−	0.935004i	$-0.615396\pi$
$797$	−1.27192e7			−0.709275			−0.354638	+	0.935004i	$0.615396\pi$
$798$		0			0
$799$			1.00396e7i			0.556351i
$800$		0			0
$801$			4.78317e6i			0.263412i
$802$		0			0
$803$	2.30615e7			1.26212
$804$		0			0
$805$	3.71671e6			0.202148
$806$		0			0
$807$	7.44885e6			0.402629
$808$		0			0
$809$	5.39790e6			0.289970			0.144985	−	0.989434i	$-0.453687\pi$
$809$	5.39790e6			0.289970			0.144985	+	0.989434i	$0.453687\pi$
$810$		0			0
$811$		−	1.46019e7i		−	0.779576i	−0.920905	−	0.389788i	$-0.872548\pi$
$811$		−	1.46019e7i		−	0.779576i	0.920905	−	0.389788i	$-0.127452\pi$
$812$		0			0
$813$		−	5.94703e6i		−	0.315554i
$814$		0			0
$815$	−1.22231e7			−0.644597
$816$		0			0
$817$		−	1.37425e7i		−	0.720293i
$818$		0			0
$819$	−1.14563e6	−	4.73878e6i	−0.0596806	−	0.246863i
$820$		0			0
$821$			2.73956e7i			1.41848i	0.704967	+	0.709240i	$0.250962\pi$
$821$			2.73956e7i			1.41848i	−0.704967	+	0.709240i	$0.749038\pi$
$822$		0			0
$823$	−2.28175e7			−1.17427			−0.587136	−	0.809488i	$-0.699745\pi$
$823$	−2.28175e7			−1.17427			−0.587136	+	0.809488i	$0.699745\pi$
$824$		0			0
$825$			2.42880e7i			1.24239i
$826$		0			0
$827$		−	1.17556e7i		−	0.597698i	−0.954300	−	0.298849i	$-0.903397\pi$
$827$		−	1.17556e7i		−	0.597698i	0.954300	−	0.298849i	$-0.0966028\pi$
$828$		0			0
$829$	−2.04885e7			−1.03544			−0.517718	−	0.855551i	$-0.673219\pi$
$829$	−2.04885e7			−1.03544			−0.517718	+	0.855551i	$0.673219\pi$
$830$		0			0
$831$	−1.24447e7			−0.625149
$832$		0			0
$833$	−8.09770e6			−0.404342
$834$		0			0
$835$	−9.46218e6			−0.469651
$836$		0			0
$837$		−	5.37064e6i		−	0.264979i
$838$		0			0
$839$		−	2.38084e7i		−	1.16769i	−0.811867	−	0.583843i	$-0.801549\pi$
$839$		−	2.38084e7i		−	1.16769i	0.811867	−	0.583843i	$-0.198451\pi$
$840$		0			0
$841$	3.83114e7			1.86783
$842$		0			0
$843$			1.30404e7i			0.632006i
$844$		0			0
$845$	−1.47314e7	−	2.86868e7i	−0.709744	−	1.38210i
$846$		0			0
$847$			2.09378e7i			1.00282i
$848$		0			0
$849$	−1.09840e7			−0.522988
$850$		0			0
$851$		−	4.58133e6i		−	0.216854i
$852$		0			0
$853$			9.87851e6i			0.464857i	0.972614	+	0.232428i	$0.0746671\pi$
$853$			9.87851e6i			0.464857i	−0.972614	+	0.232428i	$0.925333\pi$
$854$		0			0
$855$	−1.59497e7			−0.746170
$856$		0			0
$857$	−1.97784e6			−0.0919898			−0.0459949	−	0.998942i	$-0.514646\pi$
$857$	−1.97784e6			−0.0919898			−0.0459949	+	0.998942i	$0.514646\pi$
$858$		0			0
$859$	5.87858e6			0.271825			0.135913	−	0.990721i	$-0.456603\pi$
$859$	5.87858e6			0.271825			0.135913	+	0.990721i	$0.456603\pi$
$860$		0			0
$861$	−3.28855e6			−0.151181
$862$		0			0
$863$			1.96178e7i			0.896651i	0.893870	+	0.448325i	$0.147979\pi$
$863$			1.96178e7i			0.896651i	−0.893870	+	0.448325i	$0.852021\pi$
$864$		0			0
$865$		−	3.48447e7i		−	1.58342i
$866$		0			0
$867$	−905089.			−0.0408925
$868$		0			0
$869$		−	2.15639e7i		−	0.968675i
$870$		0			0
$871$	6.54333e6	+	2.70659e7i	0.292249	+	1.20886i
$872$		0			0
$873$			402228.i			0.0178623i
$874$		0			0
$875$	1.10979e7			0.490030
$876$		0			0
$877$		−	1.90004e7i		−	0.834189i	−0.908863	−	0.417095i	$-0.863048\pi$
$877$		−	1.90004e7i		−	0.834189i	0.908863	−	0.417095i	$-0.136952\pi$
$878$		0			0
$879$		−	1.97518e7i		−	0.862254i
$880$		0			0
$881$	6.81302e6			0.295733			0.147866	−	0.989007i	$-0.452759\pi$
$881$	6.81302e6			0.295733			0.147866	+	0.989007i	$0.452759\pi$
$882$		0			0
$883$	−2.84750e7			−1.22903			−0.614513	−	0.788906i	$-0.710648\pi$
$883$	−2.84750e7			−1.22903			−0.614513	+	0.788906i	$0.710648\pi$
$884$		0			0
$885$	9.34085e6			0.400893
$886$		0			0
$887$	−3.43992e7			−1.46804			−0.734022	−	0.679126i	$-0.762359\pi$
$887$	−3.43992e7			−1.46804			−0.734022	+	0.679126i	$0.762359\pi$
$888$		0			0
$889$			3.18721e7i			1.35256i
$890$		0			0
$891$		−	4.00716e6i		−	0.169099i
$892$		0			0
$893$	−1.98164e7			−0.831564
$894$		0			0
$895$		−	1.81989e7i		−	0.759430i
$896$		0			0
$897$	−2.30929e6	+	558284.i	−0.0958292	+	0.0231672i
$898$		0			0
$899$		−	5.65029e7i		−	2.33169i
$900$		0			0
$901$	−3.99540e7			−1.63964
$902$		0			0
$903$		−	5.38872e6i		−	0.219921i
$904$		0			0
$905$		−	2.05216e7i		−	0.832894i
$906$		0			0
$907$	−2.13493e7			−0.861719			−0.430859	−	0.902419i	$-0.641789\pi$
$907$	−2.13493e7			−0.861719			−0.430859	+	0.902419i	$0.641789\pi$
$908$		0			0
$909$	−1.01740e7			−0.408397
$910$		0			0
$911$	7.55882e6			0.301758			0.150879	−	0.988552i	$-0.451790\pi$
$911$	7.55882e6			0.301758			0.150879	+	0.988552i	$0.451790\pi$
$912$		0			0
$913$	−1.92292e7			−0.763457
$914$		0			0
$915$			3.54886e7i			1.40132i
$916$		0			0
$917$		−	3.53499e7i		−	1.38824i
$918$		0			0
$919$	4.56027e7			1.78115			0.890577	−	0.454832i	$-0.150301\pi$
$919$	4.56027e7			1.78115			0.890577	+	0.454832i	$0.150301\pi$
$920$		0			0
$921$			752572.i			0.0292347i
$922$		0			0
$923$	3.41167e6	+	1.41121e7i	0.131814	+	0.545239i
$924$		0			0
$925$		−	4.67264e7i		−	1.79559i
$926$		0			0
$927$	1.02345e7			0.391171
$928$		0			0
$929$		−	4.74157e7i		−	1.80253i	−0.433267	−	0.901266i	$-0.642639\pi$
$929$		−	4.74157e7i		−	1.80253i	0.433267	−	0.901266i	$-0.357361\pi$
$930$		0			0
$931$		−	1.59834e7i		−	0.604360i
$932$		0			0
$933$	−4.42590e6			−0.166455
$934$		0			0
$935$	6.09292e7			2.27928
$936$		0			0
$937$	4.02496e7			1.49766			0.748829	−	0.662764i	$-0.230617\pi$
$937$	4.02496e7			1.49766			0.748829	+	0.662764i	$0.230617\pi$
$938$		0			0
$939$	8.07272e6			0.298783
$940$		0			0
$941$			8.70389e6i			0.320434i	0.987082	+	0.160217i	$0.0512195\pi$
$941$			8.70389e6i			0.320434i	−0.987082	+	0.160217i	$0.948781\pi$
$942$		0			0
$943$			1.60257e6i			0.0586864i
$944$		0			0
$945$	−6.25423e6			−0.227821
$946$		0			0
$947$			9.78178e6i			0.354440i	0.984171	+	0.177220i	$0.0567105\pi$
$947$			9.78178e6i			0.354440i	−0.984171	+	0.177220i	$0.943290\pi$
$948$		0			0
$949$	5.40657e6	+	2.23638e7i	0.194875	+	0.806084i
$950$		0			0
$951$		−	2.83904e7i		−	1.01794i
$952$		0			0
$953$	−1.18962e7			−0.424303			−0.212151	−	0.977237i	$-0.568047\pi$
$953$	−1.18962e7			−0.424303			−0.212151	+	0.977237i	$0.568047\pi$
$954$		0			0
$955$			4.42711e7i			1.57077i
$956$		0			0
$957$		−	4.21581e7i		−	1.48799i
$958$		0			0
$959$	−3.42821e7			−1.20371
$960$		0			0
$961$	−2.56455e7			−0.895781
$962$		0			0
$963$	−2.67849e6			−0.0930731
$964$		0			0
$965$	−5.05028e6			−0.174581
$966$		0			0
$967$		−	7.21381e6i		−	0.248084i	−0.992277	−	0.124042i	$-0.960414\pi$
$967$		−	7.21381e6i		−	0.248084i	0.992277	−	0.124042i	$-0.0395857\pi$
$968$		0			0
$969$			2.34365e7i			0.801830i
$970$		0			0
$971$	−9.17081e6			−0.312147			−0.156074	−	0.987745i	$-0.549884\pi$
$971$	−9.17081e6			−0.312147			−0.156074	+	0.987745i	$0.549884\pi$
$972$		0			0
$973$		−	1.01770e7i		−	0.344618i
$974$		0			0
$975$	−2.35532e7	+	5.69412e6i	−0.793485	+	0.191829i
$976$		0			0
$977$			2.18053e7i			0.730845i	0.930842	+	0.365423i	$0.119076\pi$
$977$			2.18053e7i			0.730845i	−0.930842	+	0.365423i	$0.880924\pi$
$978$		0			0
$979$	3.60660e7			1.20265
$980$		0			0
$981$			1.19610e7i			0.396820i
$982$		0			0
$983$			1.96569e7i			0.648831i	0.945915	+	0.324415i	$0.105168\pi$
$983$			1.96569e7i			0.648831i	−0.945915	+	0.324415i	$0.894832\pi$
$984$		0			0
$985$	7.97579e6			0.261929
$986$		0			0
$987$	−7.77044e6			−0.253894
$988$		0			0
$989$	−2.62602e6			−0.0853703
$990$		0			0
$991$	−2.15229e7			−0.696173			−0.348087	−	0.937462i	$-0.613168\pi$
$991$	−2.15229e7			−0.696173			−0.348087	+	0.937462i	$0.613168\pi$
$992$		0			0
$993$			5.37477e6i			0.172976i
$994$		0			0
$995$			4.88372e7i			1.56384i
$996$		0			0
$997$	−1.16375e6			−0.0370783			−0.0185392	−	0.999828i	$-0.505902\pi$
$997$	−1.16375e6			−0.0370783			−0.0185392	+	0.999828i	$0.505902\pi$
$998$		0			0
$999$			7.70915e6i			0.244395i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.c.d.337.1		6
4.3	odd	2		78.6.b.a.25.4	yes	6
12.11	even	2		234.6.b.c.181.3		6
13.12	even	2	inner	624.6.c.d.337.6		6
52.31	even	4		1014.6.a.q.1.1		3
52.47	even	4		1014.6.a.o.1.3		3
52.51	odd	2		78.6.b.a.25.3	✓	6
156.155	even	2		234.6.b.c.181.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.b.a.25.3	✓	6	52.51	odd	2
78.6.b.a.25.4	yes	6	4.3	odd	2
234.6.b.c.181.3		6	12.11	even	2
234.6.b.c.181.4		6	156.155	even	2
624.6.c.d.337.1		6	1.1	even	1	trivial
624.6.c.d.337.6		6	13.12	even	2	inner
1014.6.a.o.1.3		3	52.47	even	4
1014.6.a.q.1.1		3	52.31	even	4

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -86.8538i q^{5} -98.7774i q^{7} +81.0000 q^{9} -610.754i q^{11} +(592.276 - 143.186i) q^{13} -781.684i q^{15} -1148.60 q^{17} -2267.14i q^{19} -888.996i q^{21} -433.224 q^{23} -4418.59 q^{25} +729.000 q^{27} +7669.59 q^{29} -7367.13i q^{31} -5496.79i q^{33} -8579.19 q^{35} +10575.0i q^{37} +(5330.48 - 1288.67i) q^{39} -3699.17i q^{41} +6061.57 q^{43} -7035.16i q^{45} -8740.69i q^{47} +7050.03 q^{49} -10337.4 q^{51} +34784.8 q^{53} -53046.3 q^{55} -20404.3i q^{57} +11949.6i q^{59} -45400.2 q^{61} -8000.97i q^{63} +(-12436.2 - 51441.4i) q^{65} +45698.1i q^{67} -3899.01 q^{69} +23826.9i q^{71} +37759.1i q^{73} -39767.3 q^{75} -60328.7 q^{77} +35307.0 q^{79} +6561.00 q^{81} -31484.4i q^{83} +99760.7i q^{85} +69026.3 q^{87} +59051.5i q^{89} +(-14143.5 - 58503.5i) q^{91} -66304.2i q^{93} -196910. q^{95} +4965.78i q^{97} -49471.1i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 54 q^{3} + 486 q^{9} + 530 q^{13} - 836 q^{17} + 416 q^{23} + 718 q^{25} + 4374 q^{27} + 18788 q^{29} - 6112 q^{35} + 4770 q^{39} + 24200 q^{43} - 3038 q^{49} - 7524 q^{51} - 42396 q^{53} - 124656 q^{55}+ \cdots - 567168 q^{95}+O(q^{100}) \) 6 * q + 54 * q^3 + 486 * q^9 + 530 * q^13 - 836 * q^17 + 416 * q^23 + 718 * q^25 + 4374 * q^27 + 18788 * q^29 - 6112 * q^35 + 4770 * q^39 + 24200 * q^43 - 3038 * q^49 - 7524 * q^51 - 42396 * q^53 - 124656 * q^55 - 3196 * q^61 - 17168 * q^65 + 3744 * q^69 + 6462 * q^75 - 114024 * q^77 + 169328 * q^79 + 39366 * q^81 + 169092 * q^87 - 236152 * q^91 - 567168 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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