LMFDB - Embedded newform 520.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,6,Mod(1,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("520.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	520.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$83.3995863027$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 1392x^{6} - 960x^{5} + 541704x^{4} + 955392x^{3} - 49992640x^{2} - 201007872x + 5544720 $ x^8 - 1392x^6 - 960x^5 + 541704x^4 + 955392x^3 - 49992640x^2 - 201007872x + 5544720
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-27.7866$ of defining polynomial
Character	$\chi$	$=$	520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.7866 q^{3} -25.0000 q^{5} +138.753 q^{7} +644.243 q^{9} -132.506 q^{11} +169.000 q^{13} +744.666 q^{15} +1951.61 q^{17} +1920.72 q^{19} -4133.00 q^{21} +1713.94 q^{23} +625.000 q^{25} -11951.7 q^{27} +1118.88 q^{29} -2283.69 q^{31} +3946.89 q^{33} -3468.83 q^{35} -3956.62 q^{37} -5033.94 q^{39} +7403.63 q^{41} -1767.31 q^{43} -16106.1 q^{45} -15917.2 q^{47} +2445.51 q^{49} -58131.7 q^{51} +1515.57 q^{53} +3312.64 q^{55} -57211.7 q^{57} -43798.7 q^{59} +19517.9 q^{61} +89390.9 q^{63} -4225.00 q^{65} -9820.52 q^{67} -51052.4 q^{69} +17669.6 q^{71} -23941.5 q^{73} -18616.6 q^{75} -18385.6 q^{77} +106118. q^{79} +199449. q^{81} +95434.8 q^{83} -48790.1 q^{85} -33327.7 q^{87} +108550. q^{89} +23449.3 q^{91} +68023.5 q^{93} -48017.9 q^{95} +74315.7 q^{97} -85365.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{3} - 200 q^{5} + 184 q^{7} + 872 q^{9} + 128 q^{11} + 1352 q^{13} + 400 q^{15} + 1656 q^{17} - 2112 q^{19} - 1352 q^{21} - 1144 q^{23} + 5000 q^{25} - 6112 q^{27} - 6672 q^{29} - 2984 q^{31}+ \cdots + 312944 q^{99}+O(q^{100}) $ 8 * q - 16 * q^3 - 200 * q^5 + 184 * q^7 + 872 * q^9 + 128 * q^11 + 1352 * q^13 + 400 * q^15 + 1656 * q^17 - 2112 * q^19 - 1352 * q^21 - 1144 * q^23 + 5000 * q^25 - 6112 * q^27 - 6672 * q^29 - 2984 * q^31 - 10840 * q^33 - 4600 * q^35 - 13432 * q^37 - 2704 * q^39 - 10136 * q^41 - 3632 * q^43 - 21800 * q^45 + 25256 * q^47 - 1368 * q^49 - 20736 * q^51 + 26744 * q^53 - 3200 * q^55 + 22536 * q^57 + 55344 * q^59 + 22784 * q^61 + 112696 * q^63 - 33800 * q^65 + 16984 * q^67 + 33104 * q^69 + 118856 * q^71 + 120744 * q^73 - 10000 * q^75 - 24200 * q^77 + 154720 * q^79 + 171944 * q^81 + 179048 * q^83 - 41400 * q^85 + 190992 * q^87 + 254336 * q^89 + 31096 * q^91 + 247672 * q^93 + 52800 * q^95 + 134320 * q^97 + 312944 * q^99

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
$3$	−29.7866	−1.91081	−0.955406	−	0.295294i	$-0.904582\pi$
$3$	−29.7866	−1.91081	−0.955406	+	0.295294i	$0.904582\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	138.753	1.07028	0.535141	−	0.844763i	$-0.320258\pi$
$7$	138.753	1.07028	0.535141	+	0.844763i	$0.320258\pi$
$8$	0	0
$9$	644.243	2.65121
$10$	0	0
$11$	−132.506	−0.330181	−0.165091	−	0.986278i	$-0.552792\pi$
$11$	−132.506	−0.330181	−0.165091	+	0.986278i	$0.552792\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	744.666	0.854541
$16$	0	0
$17$	1951.61	1.63783	0.818917	−	0.573912i	$-0.194575\pi$
$17$	1951.61	1.63783	0.818917	+	0.573912i	$0.194575\pi$
$18$	0	0
$19$	1920.72	1.22062	0.610308	−	0.792164i	$-0.291046\pi$
$19$	1920.72	1.22062	0.610308	+	0.792164i	$0.291046\pi$
$20$	0	0
$21$	−4133.00	−2.04511
$22$	0	0
$23$	1713.94	0.675577	0.337789	−	0.941222i	$-0.390321\pi$
$23$	1713.94	0.675577	0.337789	+	0.941222i	$0.390321\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−11951.7	−3.15514
$28$	0	0
$29$	1118.88	0.247053	0.123526	−	0.992341i	$-0.460580\pi$
$29$	1118.88	0.247053	0.123526	+	0.992341i	$0.460580\pi$
$30$	0	0
$31$	−2283.69	−0.426809	−0.213404	−	0.976964i	$-0.568455\pi$
$31$	−2283.69	−0.426809	−0.213404	+	0.976964i	$0.568455\pi$
$32$	0	0
$33$	3946.89	0.630915
$34$	0	0
$35$	−3468.83	−0.478645
$36$	0	0
$37$	−3956.62	−0.475138	−0.237569	−	0.971371i	$-0.576351\pi$
$37$	−3956.62	−0.475138	−0.237569	+	0.971371i	$0.576351\pi$
$38$	0	0
$39$	−5033.94	−0.529964
$40$	0	0
$41$	7403.63	0.687836	0.343918	−	0.939000i	$-0.388246\pi$
$41$	7403.63	0.687836	0.343918	+	0.939000i	$0.388246\pi$
$42$	0	0
$43$	−1767.31	−0.145761	−0.0728807	−	0.997341i	$-0.523219\pi$
$43$	−1767.31	−0.145761	−0.0728807	+	0.997341i	$0.523219\pi$
$44$	0	0
$45$	−16106.1	−1.18566
$46$	0	0
$47$	−15917.2	−1.05105	−0.525523	−	0.850780i	$-0.676130\pi$
$47$	−15917.2	−1.05105	−0.525523	+	0.850780i	$0.676130\pi$
$48$	0	0
$49$	2445.51	0.145505
$50$	0	0
$51$	−58131.7	−3.12959
$52$	0	0
$53$	1515.57	0.0741115	0.0370558	−	0.999313i	$-0.488202\pi$
$53$	1515.57	0.0741115	0.0370558	+	0.999313i	$0.488202\pi$
$54$	0	0
$55$	3312.64	0.147662
$56$	0	0
$57$	−57211.7	−2.33237
$58$	0	0
$59$	−43798.7	−1.63807	−0.819033	−	0.573746i	$-0.805490\pi$
$59$	−43798.7	−1.63807	−0.819033	+	0.573746i	$0.805490\pi$
$60$	0	0
$61$	19517.9	0.671597	0.335799	−	0.941934i	$-0.390994\pi$
$61$	19517.9	0.671597	0.335799	+	0.941934i	$0.390994\pi$
$62$	0	0
$63$	89390.9	2.83754
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	−9820.52	−0.267268	−0.133634	−	0.991031i	$-0.542665\pi$
$67$	−9820.52	−0.267268	−0.133634	+	0.991031i	$0.542665\pi$
$68$	0	0
$69$	−51052.4	−1.29090
$70$	0	0
$71$	17669.6	0.415987	0.207994	−	0.978130i	$-0.433307\pi$
$71$	17669.6	0.415987	0.207994	+	0.978130i	$0.433307\pi$
$72$	0	0
$73$	−23941.5	−0.525829	−0.262915	−	0.964819i	$-0.584684\pi$
$73$	−23941.5	−0.525829	−0.262915	+	0.964819i	$0.584684\pi$
$74$	0	0
$75$	−18616.6	−0.382163
$76$	0	0
$77$	−18385.6	−0.353387
$78$	0	0
$79$	106118.	1.91303	0.956517	−	0.291676i	$-0.0942129\pi$
$79$	106118.	1.91303	0.956517	+	0.291676i	$0.0942129\pi$
$80$	0	0
$81$	199449.	3.37768
$82$	0	0
$83$	95434.8	1.52059	0.760294	−	0.649579i	$-0.225055\pi$
$83$	95434.8	1.52059	0.760294	+	0.649579i	$0.225055\pi$
$84$	0	0
$85$	−48790.1	−0.732461
$86$	0	0
$87$	−33327.7	−0.472072
$88$	0	0
$89$	108550.	1.45263	0.726315	−	0.687362i	$-0.241231\pi$
$89$	108550.	1.45263	0.726315	+	0.687362i	$0.241231\pi$
$90$	0	0
$91$	23449.3	0.296843
$92$	0	0
$93$	68023.5	0.815552
$94$	0	0
$95$	−48017.9	−0.545876
$96$	0	0
$97$	74315.7	0.801958	0.400979	−	0.916087i	$-0.368670\pi$
$97$	74315.7	0.801958	0.400979	+	0.916087i	$0.368670\pi$
$98$	0	0
$99$	−85365.8	−0.875378
$100$	0	0
$101$	145059.	1.41495	0.707473	−	0.706740i	$-0.249835\pi$
$101$	145059.	1.41495	0.707473	+	0.706740i	$0.249835\pi$
$102$	0	0
$103$	−180232.	−1.67394	−0.836968	−	0.547252i	$-0.815674\pi$
$103$	−180232.	−1.67394	−0.836968	+	0.547252i	$0.815674\pi$
$104$	0	0
$105$	103325.	0.914601
$106$	0	0
$107$	−98023.6	−0.827697	−0.413848	−	0.910346i	$-0.635816\pi$
$107$	−98023.6	−0.827697	−0.413848	+	0.910346i	$0.635816\pi$
$108$	0	0
$109$	83119.8	0.670098	0.335049	−	0.942201i	$-0.391247\pi$
$109$	83119.8	0.670098	0.335049	+	0.942201i	$0.391247\pi$
$110$	0	0
$111$	117854.	0.907900
$112$	0	0
$113$	121689.	0.896510	0.448255	−	0.893906i	$-0.352046\pi$
$113$	121689.	0.896510	0.448255	+	0.893906i	$0.352046\pi$
$114$	0	0
$115$	−42848.4	−0.302127
$116$	0	0
$117$	108877.	0.735312
$118$	0	0
$119$	270792.	1.75294
$120$	0	0
$121$	−143493.	−0.890980
$122$	0	0
$123$	−220529.	−1.31433
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−98878.5	−0.543992	−0.271996	−	0.962298i	$-0.587684\pi$
$127$	−98878.5	−0.543992	−0.271996	+	0.962298i	$0.587684\pi$
$128$	0	0
$129$	52642.3	0.278523
$130$	0	0
$131$	−273826.	−1.39411	−0.697053	−	0.717019i	$-0.745506\pi$
$131$	−273826.	−1.39411	−0.697053	+	0.717019i	$0.745506\pi$
$132$	0	0
$133$	266506.	1.30641
$134$	0	0
$135$	298792.	1.41102
$136$	0	0
$137$	−57626.1	−0.262312	−0.131156	−	0.991362i	$-0.541869\pi$
$137$	−57626.1	−0.262312	−0.131156	+	0.991362i	$0.541869\pi$
$138$	0	0
$139$	90435.9	0.397012	0.198506	−	0.980100i	$-0.436391\pi$
$139$	90435.9	0.397012	0.198506	+	0.980100i	$0.436391\pi$
$140$	0	0
$141$	474119.	2.00835
$142$	0	0
$143$	−22393.4	−0.0915758
$144$	0	0
$145$	−27972.1	−0.110485
$146$	0	0
$147$	−72843.4	−0.278033
$148$	0	0
$149$	276038.	1.01860	0.509299	−	0.860590i	$-0.329905\pi$
$149$	276038.	1.01860	0.509299	+	0.860590i	$0.329905\pi$
$150$	0	0
$151$	359786.	1.28411	0.642055	−	0.766659i	$-0.278082\pi$
$151$	359786.	1.28411	0.642055	+	0.766659i	$0.278082\pi$
$152$	0	0
$153$	1.25731e6	4.34223
$154$	0	0
$155$	57092.3	0.190875
$156$	0	0
$157$	−466305.	−1.50981	−0.754903	−	0.655836i	$-0.772316\pi$
$157$	−466305.	−1.50981	−0.754903	+	0.655836i	$0.772316\pi$
$158$	0	0
$159$	−45143.7	−0.141613
$160$	0	0
$161$	237815.	0.723059
$162$	0	0
$163$	−524255.	−1.54552	−0.772758	−	0.634700i	$-0.781124\pi$
$163$	−524255.	−1.54552	−0.772758	+	0.634700i	$0.781124\pi$
$164$	0	0
$165$	−98672.3	−0.282154
$166$	0	0
$167$	−39887.6	−0.110674	−0.0553372	−	0.998468i	$-0.517623\pi$
$167$	−39887.6	−0.110674	−0.0553372	+	0.998468i	$0.517623\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	1.23741e6	3.23611
$172$	0	0
$173$	598657.	1.52077	0.760383	−	0.649474i	$-0.225011\pi$
$173$	598657.	1.52077	0.760383	+	0.649474i	$0.225011\pi$
$174$	0	0
$175$	86720.9	0.214057
$176$	0	0
$177$	1.30462e6	3.13004
$178$	0	0
$179$	−672025.	−1.56766	−0.783832	−	0.620973i	$-0.786737\pi$
$179$	−672025.	−1.56766	−0.783832	+	0.620973i	$0.786737\pi$
$180$	0	0
$181$	142064.	0.322319	0.161160	−	0.986928i	$-0.448477\pi$
$181$	142064.	0.322319	0.161160	+	0.986928i	$0.448477\pi$
$182$	0	0
$183$	−581373.	−1.28330
$184$	0	0
$185$	98915.4	0.212488
$186$	0	0
$187$	−258599.	−0.540782
$188$	0	0
$189$	−1.65833e6	−3.37690
$190$	0	0
$191$	−93947.8	−0.186339	−0.0931694	−	0.995650i	$-0.529700\pi$
$191$	−93947.8	−0.186339	−0.0931694	+	0.995650i	$0.529700\pi$
$192$	0	0
$193$	−29673.1	−0.0573416	−0.0286708	−	0.999589i	$-0.509127\pi$
$193$	−29673.1	−0.0573416	−0.0286708	+	0.999589i	$0.509127\pi$
$194$	0	0
$195$	125848.	0.237007
$196$	0	0
$197$	−462903.	−0.849815	−0.424907	−	0.905237i	$-0.639693\pi$
$197$	−462903.	−0.849815	−0.424907	+	0.905237i	$0.639693\pi$
$198$	0	0
$199$	−807986.	−1.44634	−0.723171	−	0.690669i	$-0.757316\pi$
$199$	−807986.	−1.44634	−0.723171	+	0.690669i	$0.757316\pi$
$200$	0	0
$201$	292520.	0.510700
$202$	0	0
$203$	155249.	0.264416
$204$	0	0
$205$	−185091.	−0.307610
$206$	0	0
$207$	1.10419e6	1.79109
$208$	0	0
$209$	−254506.	−0.403025
$210$	0	0
$211$	466889.	0.721950	0.360975	−	0.932575i	$-0.382444\pi$
$211$	466889.	0.721950	0.360975	+	0.932575i	$0.382444\pi$
$212$	0	0
$213$	−526316.	−0.794873
$214$	0	0
$215$	44182.8	0.0651865
$216$	0	0
$217$	−316870.	−0.456806
$218$	0	0
$219$	713137.	1.00476
$220$	0	0
$221$	329821.	0.454253
$222$	0	0
$223$	6930.21	0.00933220	0.00466610	−	0.999989i	$-0.498515\pi$
$223$	6930.21	0.00933220	0.00466610	+	0.999989i	$0.498515\pi$
$224$	0	0
$225$	402652.	0.530241
$226$	0	0
$227$	673243.	0.867175	0.433588	−	0.901111i	$-0.357247\pi$
$227$	673243.	0.867175	0.433588	+	0.901111i	$0.357247\pi$
$228$	0	0
$229$	−753445.	−0.949430	−0.474715	−	0.880140i	$-0.657449\pi$
$229$	−753445.	−0.949430	−0.474715	+	0.880140i	$0.657449\pi$
$230$	0	0
$231$	547645.	0.675257
$232$	0	0
$233$	49965.8	0.0602953	0.0301476	−	0.999545i	$-0.490402\pi$
$233$	49965.8	0.0602953	0.0301476	+	0.999545i	$0.490402\pi$
$234$	0	0
$235$	397929.	0.470042
$236$	0	0
$237$	−3.16091e6	−3.65545
$238$	0	0
$239$	1.74531e6	1.97642	0.988209	−	0.153108i	$-0.0489283\pi$
$239$	1.74531e6	1.97642	0.988209	+	0.153108i	$0.0489283\pi$
$240$	0	0
$241$	−601566.	−0.667176	−0.333588	−	0.942719i	$-0.608259\pi$
$241$	−601566.	−0.667176	−0.333588	+	0.942719i	$0.608259\pi$
$242$	0	0
$243$	−3.03665e6	−3.29898
$244$	0	0
$245$	−61137.6	−0.0650719
$246$	0	0
$247$	324601.	0.338538
$248$	0	0
$249$	−2.84268e6	−2.90556
$250$	0	0
$251$	560000.	0.561052	0.280526	−	0.959846i	$-0.409491\pi$
$251$	560000.	0.561052	0.280526	+	0.959846i	$0.409491\pi$
$252$	0	0
$253$	−227106.	−0.223063
$254$	0	0
$255$	1.45329e6	1.39960
$256$	0	0
$257$	1.70370e6	1.60902	0.804510	−	0.593940i	$-0.202428\pi$
$257$	1.70370e6	1.60902	0.804510	+	0.593940i	$0.202428\pi$
$258$	0	0
$259$	−548994.	−0.508532
$260$	0	0
$261$	720832.	0.654988
$262$	0	0
$263$	−2.06742e6	−1.84306	−0.921529	−	0.388310i	$-0.873059\pi$
$263$	−2.06742e6	−1.84306	−0.921529	+	0.388310i	$0.873059\pi$
$264$	0	0
$265$	−37889.2	−0.0331437
$266$	0	0
$267$	−3.23334e6	−2.77570
$268$	0	0
$269$	1.59362e6	1.34278	0.671388	−	0.741106i	$-0.265698\pi$
$269$	1.59362e6	1.34278	0.671388	+	0.741106i	$0.265698\pi$
$270$	0	0
$271$	1.43967e6	1.19080	0.595402	−	0.803428i	$-0.296993\pi$
$271$	1.43967e6	1.19080	0.595402	+	0.803428i	$0.296993\pi$
$272$	0	0
$273$	−698476.	−0.567211
$274$	0	0
$275$	−82816.0	−0.0660363
$276$	0	0
$277$	240544.	0.188363	0.0941815	−	0.995555i	$-0.469977\pi$
$277$	240544.	0.188363	0.0941815	+	0.995555i	$0.469977\pi$
$278$	0	0
$279$	−1.47125e6	−1.13156
$280$	0	0
$281$	1.66707e6	1.25947	0.629736	−	0.776809i	$-0.283163\pi$
$281$	1.66707e6	1.25947	0.629736	+	0.776809i	$0.283163\pi$
$282$	0	0
$283$	1.48417e6	1.10159	0.550793	−	0.834642i	$-0.314326\pi$
$283$	1.48417e6	1.10159	0.550793	+	0.834642i	$0.314326\pi$
$284$	0	0
$285$	1.43029e6	1.04307
$286$	0	0
$287$	1.02728e6	0.736179
$288$	0	0
$289$	2.38891e6	1.68250
$290$	0	0
$291$	−2.21361e6	−1.53239
$292$	0	0
$293$	189974.	0.129278	0.0646390	−	0.997909i	$-0.479410\pi$
$293$	189974.	0.129278	0.0646390	+	0.997909i	$0.479410\pi$
$294$	0	0
$295$	1.09497e6	0.732566
$296$	0	0
$297$	1.58366e6	1.04177
$298$	0	0
$299$	289655.	0.187371
$300$	0	0
$301$	−245221.	−0.156006
$302$	0	0
$303$	−4.32081e6	−2.70370
$304$	0	0
$305$	−487948.	−0.300347
$306$	0	0
$307$	1.73353e6	1.04975	0.524873	−	0.851180i	$-0.324113\pi$
$307$	1.73353e6	1.04975	0.524873	+	0.851180i	$0.324113\pi$
$308$	0	0
$309$	5.36850e6	3.19858
$310$	0	0
$311$	538447.	0.315676	0.157838	−	0.987465i	$-0.449548\pi$
$311$	538447.	0.315676	0.157838	+	0.987465i	$0.449548\pi$
$312$	0	0
$313$	−2.18751e6	−1.26208	−0.631042	−	0.775748i	$-0.717373\pi$
$313$	−2.18751e6	−1.26208	−0.631042	+	0.775748i	$0.717373\pi$
$314$	0	0
$315$	−2.23477e6	−1.26899
$316$	0	0
$317$	−3.01629e6	−1.68587	−0.842936	−	0.538014i	$-0.819175\pi$
$317$	−3.01629e6	−1.68587	−0.842936	+	0.538014i	$0.819175\pi$
$318$	0	0
$319$	−148258.	−0.0815722
$320$	0	0
$321$	2.91979e6	1.58157
$322$	0	0
$323$	3.74848e6	1.99917
$324$	0	0
$325$	105625.	0.0554700
$326$	0	0
$327$	−2.47586e6	−1.28043
$328$	0	0
$329$	−2.20856e6	−1.12492
$330$	0	0
$331$	1.62436e6	0.814913	0.407457	−	0.913225i	$-0.366416\pi$
$331$	1.62436e6	0.814913	0.407457	+	0.913225i	$0.366416\pi$
$332$	0	0
$333$	−2.54902e6	−1.25969
$334$	0	0
$335$	245513.	0.119526
$336$	0	0
$337$	−3.00154e6	−1.43969	−0.719846	−	0.694134i	$-0.755788\pi$
$337$	−3.00154e6	−1.43969	−0.719846	+	0.694134i	$0.755788\pi$
$338$	0	0
$339$	−3.62470e6	−1.71306
$340$	0	0
$341$	302602.	0.140924
$342$	0	0
$343$	−1.99271e6	−0.914551
$344$	0	0
$345$	1.27631e6	0.577309
$346$	0	0
$347$	−458177.	−0.204272	−0.102136	−	0.994770i	$-0.532568\pi$
$347$	−458177.	−0.204272	−0.102136	+	0.994770i	$0.532568\pi$
$348$	0	0
$349$	1.17764e6	0.517548	0.258774	−	0.965938i	$-0.416682\pi$
$349$	1.17764e6	0.517548	0.258774	+	0.965938i	$0.416682\pi$
$350$	0	0
$351$	−2.01983e6	−0.875079
$352$	0	0
$353$	3.59518e6	1.53562	0.767811	−	0.640677i	$-0.221346\pi$
$353$	3.59518e6	1.53562	0.767811	+	0.640677i	$0.221346\pi$
$354$	0	0
$355$	−441739.	−0.186035
$356$	0	0
$357$	−8.06598e6	−3.34955
$358$	0	0
$359$	−1.64641e6	−0.674219	−0.337110	−	0.941465i	$-0.609449\pi$
$359$	−1.64641e6	−0.674219	−0.337110	+	0.941465i	$0.609449\pi$
$360$	0	0
$361$	1.21305e6	0.489905
$362$	0	0
$363$	4.27418e6	1.70250
$364$	0	0
$365$	598538.	0.235158
$366$	0	0
$367$	−1.12688e6	−0.436730	−0.218365	−	0.975867i	$-0.570072\pi$
$367$	−1.12688e6	−0.436730	−0.218365	+	0.975867i	$0.570072\pi$
$368$	0	0
$369$	4.76973e6	1.82359
$370$	0	0
$371$	210290.	0.0793203
$372$	0	0
$373$	3.41831e6	1.27215	0.636076	−	0.771626i	$-0.280556\pi$
$373$	3.41831e6	1.27215	0.636076	+	0.771626i	$0.280556\pi$
$374$	0	0
$375$	465416.	0.170908
$376$	0	0
$377$	189091.	0.0685201
$378$	0	0
$379$	505431.	0.180744	0.0903720	−	0.995908i	$-0.471194\pi$
$379$	505431.	0.180744	0.0903720	+	0.995908i	$0.471194\pi$
$380$	0	0
$381$	2.94526e6	1.03947
$382$	0	0
$383$	5.36338e6	1.86828	0.934139	−	0.356908i	$-0.116169\pi$
$383$	5.36338e6	1.86828	0.934139	+	0.356908i	$0.116169\pi$
$384$	0	0
$385$	459640.	0.158040
$386$	0	0
$387$	−1.13858e6	−0.386443
$388$	0	0
$389$	−812527.	−0.272248	−0.136124	−	0.990692i	$-0.543464\pi$
$389$	−812527.	−0.272248	−0.136124	+	0.990692i	$0.543464\pi$
$390$	0	0
$391$	3.34493e6	1.10648
$392$	0	0
$393$	8.15634e6	2.66388
$394$	0	0
$395$	−2.65296e6	−0.855535
$396$	0	0
$397$	−2.00210e6	−0.637542	−0.318771	−	0.947832i	$-0.603270\pi$
$397$	−2.00210e6	−0.637542	−0.318771	+	0.947832i	$0.603270\pi$
$398$	0	0
$399$	−7.93831e6	−2.49630
$400$	0	0
$401$	−1.85573e6	−0.576307	−0.288153	−	0.957584i	$-0.593041\pi$
$401$	−1.85573e6	−0.576307	−0.288153	+	0.957584i	$0.593041\pi$
$402$	0	0
$403$	−385944.	−0.118375
$404$	0	0
$405$	−4.98622e6	−1.51055
$406$	0	0
$407$	524274.	0.156882
$408$	0	0
$409$	4.58839e6	1.35629	0.678144	−	0.734929i	$-0.262785\pi$
$409$	4.58839e6	1.35629	0.678144	+	0.734929i	$0.262785\pi$
$410$	0	0
$411$	1.71649e6	0.501229
$412$	0	0
$413$	−6.07722e6	−1.75319
$414$	0	0
$415$	−2.38587e6	−0.680028
$416$	0	0
$417$	−2.69378e6	−0.758616
$418$	0	0
$419$	6.66322e6	1.85417	0.927085	−	0.374852i	$-0.122307\pi$
$419$	6.66322e6	1.85417	0.927085	+	0.374852i	$0.122307\pi$
$420$	0	0
$421$	6.30703e6	1.73428	0.867141	−	0.498062i	$-0.165955\pi$
$421$	6.30703e6	1.73428	0.867141	+	0.498062i	$0.165955\pi$
$422$	0	0
$423$	−1.02545e7	−2.78654
$424$	0	0
$425$	1.21975e6	0.327567
$426$	0	0
$427$	2.70818e6	0.718799
$428$	0	0
$429$	667025.	0.174984
$430$	0	0
$431$	−1.42774e6	−0.370217	−0.185109	−	0.982718i	$-0.559264\pi$
$431$	−1.42774e6	−0.370217	−0.185109	+	0.982718i	$0.559264\pi$
$432$	0	0
$433$	−2.98454e6	−0.764993	−0.382496	−	0.923957i	$-0.624936\pi$
$433$	−2.98454e6	−0.764993	−0.382496	+	0.923957i	$0.624936\pi$
$434$	0	0
$435$	833194.	0.211117
$436$	0	0
$437$	3.29199e6	0.824621
$438$	0	0
$439$	−1.50983e6	−0.373910	−0.186955	−	0.982368i	$-0.559862\pi$
$439$	−1.50983e6	−0.373910	−0.186955	+	0.982368i	$0.559862\pi$
$440$	0	0
$441$	1.57550e6	0.385764
$442$	0	0
$443$	−7.11802e6	−1.72326	−0.861629	−	0.507539i	$-0.830555\pi$
$443$	−7.11802e6	−1.72326	−0.861629	+	0.507539i	$0.830555\pi$
$444$	0	0
$445$	−2.71375e6	−0.649636
$446$	0	0
$447$	−8.22223e6	−1.94635
$448$	0	0
$449$	6.99203e6	1.63677	0.818384	−	0.574671i	$-0.194870\pi$
$449$	6.99203e6	1.63677	0.818384	+	0.574671i	$0.194870\pi$
$450$	0	0
$451$	−981022.	−0.227111
$452$	0	0
$453$	−1.07168e7	−2.45369
$454$	0	0
$455$	−586233.	−0.132752
$456$	0	0
$457$	−4.03690e6	−0.904185	−0.452092	−	0.891971i	$-0.649322\pi$
$457$	−4.03690e6	−0.904185	−0.452092	+	0.891971i	$0.649322\pi$
$458$	0	0
$459$	−2.33249e7	−5.16760
$460$	0	0
$461$	−7.20483e6	−1.57896	−0.789481	−	0.613775i	$-0.789650\pi$
$461$	−7.20483e6	−1.57896	−0.789481	+	0.613775i	$0.789650\pi$
$462$	0	0
$463$	−4.43724e6	−0.961968	−0.480984	−	0.876729i	$-0.659721\pi$
$463$	−4.43724e6	−0.961968	−0.480984	+	0.876729i	$0.659721\pi$
$464$	0	0
$465$	−1.70059e6	−0.364726
$466$	0	0
$467$	8.13996e6	1.72715	0.863575	−	0.504220i	$-0.168220\pi$
$467$	8.13996e6	1.72715	0.863575	+	0.504220i	$0.168220\pi$
$468$	0	0
$469$	−1.36263e6	−0.286053
$470$	0	0
$471$	1.38897e7	2.88496
$472$	0	0
$473$	234179.	0.0481277
$474$	0	0
$475$	1.20045e6	0.244123
$476$	0	0
$477$	976394.	0.196485
$478$	0	0
$479$	−7.40297e6	−1.47424	−0.737118	−	0.675764i	$-0.763814\pi$
$479$	−7.40297e6	−1.47424	−0.737118	+	0.675764i	$0.763814\pi$
$480$	0	0
$481$	−668668.	−0.131780
$482$	0	0
$483$	−7.08369e6	−1.38163
$484$	0	0
$485$	−1.85789e6	−0.358646
$486$	0	0
$487$	1.85372e6	0.354179	0.177089	−	0.984195i	$-0.443332\pi$
$487$	1.85372e6	0.354179	0.177089	+	0.984195i	$0.443332\pi$
$488$	0	0
$489$	1.56158e7	2.95319
$490$	0	0
$491$	−4.16070e6	−0.778866	−0.389433	−	0.921055i	$-0.627329\pi$
$491$	−4.16070e6	−0.778866	−0.389433	+	0.921055i	$0.627329\pi$
$492$	0	0
$493$	2.18362e6	0.404631
$494$	0	0
$495$	2.13414e6	0.391481
$496$	0	0
$497$	2.45171e6	0.445224
$498$	0	0
$499$	832530.	0.149675	0.0748374	−	0.997196i	$-0.476156\pi$
$499$	832530.	0.149675	0.0748374	+	0.997196i	$0.476156\pi$
$500$	0	0
$501$	1.18812e6	0.211478
$502$	0	0
$503$	1.03297e7	1.82040	0.910199	−	0.414172i	$-0.135929\pi$
$503$	1.03297e7	1.82040	0.910199	+	0.414172i	$0.135929\pi$
$504$	0	0
$505$	−3.62647e6	−0.632783
$506$	0	0
$507$	−850736.	−0.146986
$508$	0	0
$509$	−1.00971e7	−1.72743	−0.863716	−	0.503979i	$-0.831869\pi$
$509$	−1.00971e7	−1.72743	−0.863716	+	0.503979i	$0.831869\pi$
$510$	0	0
$511$	−3.32197e6	−0.562786
$512$	0	0
$513$	−2.29558e7	−3.85122
$514$	0	0
$515$	4.50580e6	0.748607
$516$	0	0
$517$	2.10911e6	0.347035
$518$	0	0
$519$	−1.78320e7	−2.90590
$520$	0	0
$521$	−6.85650e6	−1.10665	−0.553323	−	0.832967i	$-0.686640\pi$
$521$	−6.85650e6	−1.10665	−0.553323	+	0.832967i	$0.686640\pi$
$522$	0	0
$523$	5.96539e6	0.953641	0.476820	−	0.879001i	$-0.341789\pi$
$523$	5.96539e6	0.953641	0.476820	+	0.879001i	$0.341789\pi$
$524$	0	0
$525$	−2.58312e6	−0.409022
$526$	0	0
$527$	−4.45687e6	−0.699042
$528$	0	0
$529$	−3.49877e6	−0.543595
$530$	0	0
$531$	−2.82170e7	−4.34285
$532$	0	0
$533$	1.25121e6	0.190771
$534$	0	0
$535$	2.45059e6	0.370157
$536$	0	0
$537$	2.00174e7	2.99551
$538$	0	0
$539$	−324043.	−0.0480431
$540$	0	0
$541$	604353.	0.0887764	0.0443882	−	0.999014i	$-0.485866\pi$
$541$	604353.	0.0887764	0.0443882	+	0.999014i	$0.485866\pi$
$542$	0	0
$543$	−4.23159e6	−0.615892
$544$	0	0
$545$	−2.07800e6	−0.299677
$546$	0	0
$547$	1.09035e7	1.55810	0.779051	−	0.626960i	$-0.215701\pi$
$547$	1.09035e7	1.55810	0.779051	+	0.626960i	$0.215701\pi$
$548$	0	0
$549$	1.25743e7	1.78054
$550$	0	0
$551$	2.14906e6	0.301557
$552$	0	0
$553$	1.47243e7	2.04749
$554$	0	0
$555$	−2.94636e6	−0.406025
$556$	0	0
$557$	5.46209e6	0.745969	0.372985	−	0.927838i	$-0.378334\pi$
$557$	5.46209e6	0.745969	0.372985	+	0.927838i	$0.378334\pi$
$558$	0	0
$559$	−298676.	−0.0404269
$560$	0	0
$561$	7.70278e6	1.03333
$562$	0	0
$563$	502323.	0.0667901	0.0333950	−	0.999442i	$-0.489368\pi$
$563$	502323.	0.0667901	0.0333950	+	0.999442i	$0.489368\pi$
$564$	0	0
$565$	−3.04223e6	−0.400932
$566$	0	0
$567$	2.76742e7	3.61508
$568$	0	0
$569$	7.80010e6	1.01000	0.504998	−	0.863121i	$-0.331493\pi$
$569$	7.80010e6	1.01000	0.504998	+	0.863121i	$0.331493\pi$
$570$	0	0
$571$	−4.37874e6	−0.562029	−0.281014	−	0.959704i	$-0.590671\pi$
$571$	−4.37874e6	−0.562029	−0.281014	+	0.959704i	$0.590671\pi$
$572$	0	0
$573$	2.79839e6	0.356058
$574$	0	0
$575$	1.07121e6	0.135115
$576$	0	0
$577$	−7.24603e6	−0.906067	−0.453034	−	0.891493i	$-0.649658\pi$
$577$	−7.24603e6	−0.906067	−0.453034	+	0.891493i	$0.649658\pi$
$578$	0	0
$579$	883862.	0.109569
$580$	0	0
$581$	1.32419e7	1.62746
$582$	0	0
$583$	−200821.	−0.0244702
$584$	0	0
$585$	−2.72193e6	−0.328842
$586$	0	0
$587$	4.36722e6	0.523131	0.261565	−	0.965186i	$-0.415761\pi$
$587$	4.36722e6	0.523131	0.261565	+	0.965186i	$0.415761\pi$
$588$	0	0
$589$	−4.38633e6	−0.520970
$590$	0	0
$591$	1.37883e7	1.62384
$592$	0	0
$593$	4.11389e6	0.480414	0.240207	−	0.970722i	$-0.422785\pi$
$593$	4.11389e6	0.480414	0.240207	+	0.970722i	$0.422785\pi$
$594$	0	0
$595$	−6.76980e6	−0.783941
$596$	0	0
$597$	2.40672e7	2.76369
$598$	0	0
$599$	1.32500e7	1.50886	0.754428	−	0.656382i	$-0.227914\pi$
$599$	1.32500e7	1.50886	0.754428	+	0.656382i	$0.227914\pi$
$600$	0	0
$601$	3.41114e6	0.385224	0.192612	−	0.981275i	$-0.438304\pi$
$601$	3.41114e6	0.385224	0.192612	+	0.981275i	$0.438304\pi$
$602$	0	0
$603$	−6.32680e6	−0.708583
$604$	0	0
$605$	3.58733e6	0.398459
$606$	0	0
$607$	1.53270e7	1.68844	0.844222	−	0.535994i	$-0.180063\pi$
$607$	1.53270e7	1.68844	0.844222	+	0.535994i	$0.180063\pi$
$608$	0	0
$609$	−4.62434e6	−0.505250
$610$	0	0
$611$	−2.69000e6	−0.291507
$612$	0	0
$613$	248888.	0.0267518	0.0133759	−	0.999911i	$-0.495742\pi$
$613$	248888.	0.0267518	0.0133759	+	0.999911i	$0.495742\pi$
$614$	0	0
$615$	5.51323e6	0.587784
$616$	0	0
$617$	−3.31170e6	−0.350218	−0.175109	−	0.984549i	$-0.556028\pi$
$617$	−3.31170e6	−0.350218	−0.175109	+	0.984549i	$0.556028\pi$
$618$	0	0
$619$	1.53763e7	1.61296	0.806482	−	0.591259i	$-0.201369\pi$
$619$	1.53763e7	1.61296	0.806482	+	0.591259i	$0.201369\pi$
$620$	0	0
$621$	−2.04844e7	−2.13154
$622$	0	0
$623$	1.50617e7	1.55472
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	7.58087e6	0.770105
$628$	0	0
$629$	−7.72175e6	−0.778197
$630$	0	0
$631$	4.70542e6	0.470462	0.235231	−	0.971939i	$-0.424415\pi$
$631$	4.70542e6	0.470462	0.235231	+	0.971939i	$0.424415\pi$
$632$	0	0
$633$	−1.39070e7	−1.37951
$634$	0	0
$635$	2.47196e6	0.243281
$636$	0	0
$637$	413290.	0.0403559
$638$	0	0
$639$	1.13835e7	1.10287
$640$	0	0
$641$	8.41272e6	0.808708	0.404354	−	0.914603i	$-0.367496\pi$
$641$	8.41272e6	0.808708	0.404354	+	0.914603i	$0.367496\pi$
$642$	0	0
$643$	−1.17891e7	−1.12448	−0.562241	−	0.826974i	$-0.690060\pi$
$643$	−1.17891e7	−1.12448	−0.562241	+	0.826974i	$0.690060\pi$
$644$	0	0
$645$	−1.31606e6	−0.124559
$646$	0	0
$647$	3.30240e6	0.310148	0.155074	−	0.987903i	$-0.450438\pi$
$647$	3.30240e6	0.310148	0.155074	+	0.987903i	$0.450438\pi$
$648$	0	0
$649$	5.80358e6	0.540859
$650$	0	0
$651$	9.43849e6	0.872871
$652$	0	0
$653$	1.31169e7	1.20379	0.601893	−	0.798577i	$-0.294413\pi$
$653$	1.31169e7	1.20379	0.601893	+	0.798577i	$0.294413\pi$
$654$	0	0
$655$	6.84564e6	0.623464
$656$	0	0
$657$	−1.54242e7	−1.39408
$658$	0	0
$659$	1.20047e7	1.07680	0.538402	−	0.842688i	$-0.319028\pi$
$659$	1.20047e7	1.07680	0.538402	+	0.842688i	$0.319028\pi$
$660$	0	0
$661$	−1.44664e7	−1.28783	−0.643914	−	0.765098i	$-0.722690\pi$
$661$	−1.44664e7	−1.28783	−0.643914	+	0.765098i	$0.722690\pi$
$662$	0	0
$663$	−9.82426e6	−0.867993
$664$	0	0
$665$	−6.66265e6	−0.584242
$666$	0	0
$667$	1.91769e6	0.166903
$668$	0	0
$669$	−206428.	−0.0178321
$670$	0	0
$671$	−2.58623e6	−0.221749
$672$	0	0
$673$	1.19827e7	1.01980	0.509902	−	0.860233i	$-0.329682\pi$
$673$	1.19827e7	1.01980	0.509902	+	0.860233i	$0.329682\pi$
$674$	0	0
$675$	−7.46979e6	−0.631029
$676$	0	0
$677$	−3.19621e6	−0.268018	−0.134009	−	0.990980i	$-0.542785\pi$
$677$	−3.19621e6	−0.268018	−0.134009	+	0.990980i	$0.542785\pi$
$678$	0	0
$679$	1.03116e7	0.858321
$680$	0	0
$681$	−2.00536e7	−1.65701
$682$	0	0
$683$	−2.39098e7	−1.96121	−0.980606	−	0.195991i	$-0.937208\pi$
$683$	−2.39098e7	−1.96121	−0.980606	+	0.195991i	$0.937208\pi$
$684$	0	0
$685$	1.44065e6	0.117309
$686$	0	0
$687$	2.24426e7	1.81418
$688$	0	0
$689$	256131.	0.0205548
$690$	0	0
$691$	5.92812e6	0.472304	0.236152	−	0.971716i	$-0.424114\pi$
$691$	5.92812e6	0.472304	0.236152	+	0.971716i	$0.424114\pi$
$692$	0	0
$693$	−1.18448e7	−0.936902
$694$	0	0
$695$	−2.26090e6	−0.177549
$696$	0	0
$697$	1.44490e7	1.12656
$698$	0	0
$699$	−1.48831e6	−0.115213
$700$	0	0
$701$	7.00857e6	0.538684	0.269342	−	0.963045i	$-0.413194\pi$
$701$	7.00857e6	0.538684	0.269342	+	0.963045i	$0.413194\pi$
$702$	0	0
$703$	−7.59954e6	−0.579961
$704$	0	0
$705$	−1.18530e7	−0.898162
$706$	0	0
$707$	2.01274e7	1.51439
$708$	0	0
$709$	−3.08805e6	−0.230712	−0.115356	−	0.993324i	$-0.536801\pi$
$709$	−3.08805e6	−0.230712	−0.115356	+	0.993324i	$0.536801\pi$
$710$	0	0
$711$	6.83660e7	5.07185
$712$	0	0
$713$	−3.91410e6	−0.288342
$714$	0	0
$715$	559836.	0.0409539
$716$	0	0
$717$	−5.19870e7	−3.77657
$718$	0	0
$719$	1.69455e7	1.22245	0.611227	−	0.791455i	$-0.290676\pi$
$719$	1.69455e7	1.22245	0.611227	+	0.791455i	$0.290676\pi$
$720$	0	0
$721$	−2.50078e7	−1.79158
$722$	0	0
$723$	1.79186e7	1.27485
$724$	0	0
$725$	699302.	0.0494105
$726$	0	0
$727$	−3.86513e6	−0.271224	−0.135612	−	0.990762i	$-0.543300\pi$
$727$	−3.86513e6	−0.271224	−0.135612	+	0.990762i	$0.543300\pi$
$728$	0	0
$729$	4.19855e7	2.92604
$730$	0	0
$731$	−3.44910e6	−0.238733
$732$	0	0
$733$	495013.	0.0340296	0.0170148	−	0.999855i	$-0.494584\pi$
$733$	495013.	0.0340296	0.0170148	+	0.999855i	$0.494584\pi$
$734$	0	0
$735$	1.82108e6	0.124340
$736$	0	0
$737$	1.30127e6	0.0882470
$738$	0	0
$739$	2.25289e7	1.51750	0.758750	−	0.651382i	$-0.225810\pi$
$739$	2.25289e7	1.51750	0.758750	+	0.651382i	$0.225810\pi$
$740$	0	0
$741$	−9.66877e6	−0.646883
$742$	0	0
$743$	−1.12215e7	−0.745727	−0.372863	−	0.927886i	$-0.621624\pi$
$743$	−1.12215e7	−0.745727	−0.372863	+	0.927886i	$0.621624\pi$
$744$	0	0
$745$	−6.90095e6	−0.455531
$746$	0	0
$747$	6.14832e7	4.03139
$748$	0	0
$749$	−1.36011e7	−0.885870
$750$	0	0
$751$	−1.00330e7	−0.649129	−0.324565	−	0.945863i	$-0.605218\pi$
$751$	−1.00330e7	−0.649129	−0.324565	+	0.945863i	$0.605218\pi$
$752$	0	0
$753$	−1.66805e7	−1.07207
$754$	0	0
$755$	−8.99466e6	−0.574272
$756$	0	0
$757$	2.55119e6	0.161809	0.0809046	−	0.996722i	$-0.474219\pi$
$757$	2.55119e6	0.161809	0.0809046	+	0.996722i	$0.474219\pi$
$758$	0	0
$759$	6.76472e6	0.426232
$760$	0	0
$761$	1.50208e7	0.940225	0.470112	−	0.882607i	$-0.344213\pi$
$761$	1.50208e7	0.940225	0.470112	+	0.882607i	$0.344213\pi$
$762$	0	0
$763$	1.15332e7	0.717194
$764$	0	0
$765$	−3.14327e7	−1.94191
$766$	0	0
$767$	−7.40199e6	−0.454318
$768$	0	0
$769$	−2.02354e7	−1.23394	−0.616971	−	0.786986i	$-0.711641\pi$
$769$	−2.02354e7	−1.23394	−0.616971	+	0.786986i	$0.711641\pi$
$770$	0	0
$771$	−5.07476e7	−3.07453
$772$	0	0
$773$	−1.01400e7	−0.610366	−0.305183	−	0.952294i	$-0.598718\pi$
$773$	−1.01400e7	−0.610366	−0.305183	+	0.952294i	$0.598718\pi$
$774$	0	0
$775$	−1.42731e6	−0.0853618
$776$	0	0
$777$	1.63527e7	0.971709
$778$	0	0
$779$	1.42203e7	0.839584
$780$	0	0
$781$	−2.34132e6	−0.137351
$782$	0	0
$783$	−1.33725e7	−0.779487
$784$	0	0
$785$	1.16576e7	0.675206
$786$	0	0
$787$	3.71438e6	0.213771	0.106886	−	0.994271i	$-0.465912\pi$
$787$	3.71438e6	0.213771	0.106886	+	0.994271i	$0.465912\pi$
$788$	0	0
$789$	6.15814e7	3.52174
$790$	0	0
$791$	1.68848e7	0.959519
$792$	0	0
$793$	3.29853e6	0.186268
$794$	0	0
$795$	1.12859e6	0.0633314
$796$	0	0
$797$	−900746.	−0.0502292	−0.0251146	−	0.999685i	$-0.507995\pi$
$797$	−900746.	−0.0502292	−0.0251146	+	0.999685i	$0.507995\pi$
$798$	0	0
$799$	−3.10640e7	−1.72144
$800$	0	0
$801$	6.99326e7	3.85122
$802$	0	0
$803$	3.17239e6	0.173619
$804$	0	0
$805$	−5.94536e6	−0.323362
$806$	0	0
$807$	−4.74685e7	−2.56579
$808$	0	0
$809$	−3.25046e7	−1.74612	−0.873060	−	0.487613i	$-0.837868\pi$
$809$	−3.25046e7	−1.74612	−0.873060	+	0.487613i	$0.837868\pi$
$810$	0	0
$811$	2.11218e7	1.12766	0.563831	−	0.825890i	$-0.309327\pi$
$811$	2.11218e7	1.12766	0.563831	+	0.825890i	$0.309327\pi$
$812$	0	0
$813$	−4.28830e7	−2.27540
$814$	0	0
$815$	1.31064e7	0.691176
$816$	0	0
$817$	−3.39451e6	−0.177919
$818$	0	0
$819$	1.51071e7	0.786992
$820$	0	0
$821$	1.96246e7	1.01611	0.508057	−	0.861324i	$-0.330364\pi$
$821$	1.96246e7	1.01611	0.508057	+	0.861324i	$0.330364\pi$
$822$	0	0
$823$	2.44244e6	0.125697	0.0628483	−	0.998023i	$-0.479982\pi$
$823$	2.44244e6	0.125697	0.0628483	+	0.998023i	$0.479982\pi$
$824$	0	0
$825$	2.46681e6	0.126183
$826$	0	0
$827$	−2.31313e7	−1.17608	−0.588039	−	0.808833i	$-0.700100\pi$
$827$	−2.31313e7	−1.17608	−0.588039	+	0.808833i	$0.700100\pi$
$828$	0	0
$829$	1.29962e7	0.656797	0.328399	−	0.944539i	$-0.393491\pi$
$829$	1.29962e7	0.656797	0.328399	+	0.944539i	$0.393491\pi$
$830$	0	0
$831$	−7.16500e6	−0.359926
$832$	0	0
$833$	4.77266e6	0.238313
$834$	0	0
$835$	997191.	0.0494951
$836$	0	0
$837$	2.72939e7	1.34664
$838$	0	0
$839$	4.93644e6	0.242108	0.121054	−	0.992646i	$-0.461373\pi$
$839$	4.93644e6	0.242108	0.121054	+	0.992646i	$0.461373\pi$
$840$	0	0
$841$	−1.92592e7	−0.938965
$842$	0	0
$843$	−4.96564e7	−2.40661
$844$	0	0
$845$	−714025.	−0.0344010
$846$	0	0
$847$	−1.99102e7	−0.953601
$848$	0	0
$849$	−4.42085e7	−2.10492
$850$	0	0
$851$	−6.78139e6	−0.320992
$852$	0	0
$853$	2.40642e7	1.13240	0.566199	−	0.824268i	$-0.308413\pi$
$853$	2.40642e7	1.13240	0.566199	+	0.824268i	$0.308413\pi$
$854$	0	0
$855$	−3.09352e7	−1.44723
$856$	0	0
$857$	2.27231e7	1.05686	0.528428	−	0.848978i	$-0.322782\pi$
$857$	2.27231e7	1.05686	0.528428	+	0.848978i	$0.322782\pi$
$858$	0	0
$859$	−3.16681e7	−1.46433	−0.732165	−	0.681127i	$-0.761490\pi$
$859$	−3.16681e7	−1.46433	−0.732165	+	0.681127i	$0.761490\pi$
$860$	0	0
$861$	−3.05992e7	−1.40670
$862$	0	0
$863$	4.62656e6	0.211461	0.105731	−	0.994395i	$-0.466282\pi$
$863$	4.62656e6	0.211461	0.105731	+	0.994395i	$0.466282\pi$
$864$	0	0
$865$	−1.49664e7	−0.680108
$866$	0	0
$867$	−7.11575e7	−3.21494
$868$	0	0
$869$	−1.40613e7	−0.631648
$870$	0	0
$871$	−1.65967e6	−0.0741269
$872$	0	0
$873$	4.78774e7	2.12615
$874$	0	0
$875$	−2.16802e6	−0.0957290
$876$	0	0
$877$	2.85149e7	1.25191	0.625954	−	0.779860i	$-0.284710\pi$
$877$	2.85149e7	1.25191	0.625954	+	0.779860i	$0.284710\pi$
$878$	0	0
$879$	−5.65868e6	−0.247026
$880$	0	0
$881$	−2.08083e7	−0.903225	−0.451612	−	0.892214i	$-0.649151\pi$
$881$	−2.08083e7	−0.903225	−0.451612	+	0.892214i	$0.649151\pi$
$882$	0	0
$883$	−1.09855e7	−0.474155	−0.237077	−	0.971491i	$-0.576189\pi$
$883$	−1.09855e7	−0.474155	−0.237077	+	0.971491i	$0.576189\pi$
$884$	0	0
$885$	−3.26154e7	−1.39980
$886$	0	0
$887$	4.22561e7	1.80335	0.901675	−	0.432415i	$-0.142338\pi$
$887$	4.22561e7	1.80335	0.901675	+	0.432415i	$0.142338\pi$
$888$	0	0
$889$	−1.37197e7	−0.582225
$890$	0	0
$891$	−2.64281e7	−1.11525
$892$	0	0
$893$	−3.05724e7	−1.28292
$894$	0	0
$895$	1.68006e7	0.701080
$896$	0	0
$897$	−8.62785e6	−0.358032
$898$	0	0
$899$	−2.55518e6	−0.105444
$900$	0	0
$901$	2.95779e6	0.121382
$902$	0	0
$903$	7.30430e6	0.298098
$904$	0	0
$905$	−3.55159e6	−0.144146
$906$	0	0
$907$	2.47455e7	0.998797	0.499399	−	0.866372i	$-0.333554\pi$
$907$	2.47455e7	0.998797	0.499399	+	0.866372i	$0.333554\pi$
$908$	0	0
$909$	9.34530e7	3.75131
$910$	0	0
$911$	−564871.	−0.0225504	−0.0112752	−	0.999936i	$-0.503589\pi$
$911$	−564871.	−0.0225504	−0.0112752	+	0.999936i	$0.503589\pi$
$912$	0	0
$913$	−1.26456e7	−0.502070
$914$	0	0
$915$	1.45343e7	0.573908
$916$	0	0
$917$	−3.79943e7	−1.49209
$918$	0	0
$919$	−2.85775e7	−1.11618	−0.558092	−	0.829779i	$-0.688466\pi$
$919$	−2.85775e7	−1.11618	−0.558092	+	0.829779i	$0.688466\pi$
$920$	0	0
$921$	−5.16359e7	−2.00587
$922$	0	0
$923$	2.98616e6	0.115374
$924$	0	0
$925$	−2.47289e6	−0.0950276
$926$	0	0
$927$	−1.16113e8	−4.43795
$928$	0	0
$929$	4.50673e6	0.171326	0.0856628	−	0.996324i	$-0.472699\pi$
$929$	4.50673e6	0.171326	0.0856628	+	0.996324i	$0.472699\pi$
$930$	0	0
$931$	4.69712e6	0.177606
$932$	0	0
$933$	−1.60385e7	−0.603198
$934$	0	0
$935$	6.46497e6	0.241845
$936$	0	0
$937$	−2.23744e7	−0.832534	−0.416267	−	0.909242i	$-0.636662\pi$
$937$	−2.23744e7	−0.832534	−0.416267	+	0.909242i	$0.636662\pi$
$938$	0	0
$939$	6.51584e7	2.41161
$940$	0	0
$941$	5.11297e7	1.88234	0.941172	−	0.337929i	$-0.109726\pi$
$941$	5.11297e7	1.88234	0.941172	+	0.337929i	$0.109726\pi$
$942$	0	0
$943$	1.26893e7	0.464686
$944$	0	0
$945$	4.14584e7	1.51019
$946$	0	0
$947$	1.66708e7	0.604062	0.302031	−	0.953298i	$-0.402335\pi$
$947$	1.66708e7	0.604062	0.302031	+	0.953298i	$0.402335\pi$
$948$	0	0
$949$	−4.04612e6	−0.145839
$950$	0	0
$951$	8.98450e7	3.22139
$952$	0	0
$953$	−3.05426e7	−1.08937	−0.544684	−	0.838642i	$-0.683350\pi$
$953$	−3.05426e7	−1.08937	−0.544684	+	0.838642i	$0.683350\pi$
$954$	0	0
$955$	2.34870e6	0.0833332
$956$	0	0
$957$	4.41611e6	0.155869
$958$	0	0
$959$	−7.99582e6	−0.280748
$960$	0	0
$961$	−2.34139e7	−0.817834
$962$	0	0
$963$	−6.31510e7	−2.19439
$964$	0	0
$965$	741828.	0.0256439
$966$	0	0
$967$	1.03889e7	0.357274	0.178637	−	0.983915i	$-0.442831\pi$
$967$	1.03889e7	0.357274	0.178637	+	0.983915i	$0.442831\pi$
$968$	0	0
$969$	−1.11655e8	−3.82003
$970$	0	0
$971$	4.18374e7	1.42402	0.712010	−	0.702169i	$-0.247785\pi$
$971$	4.18374e7	1.42402	0.712010	+	0.702169i	$0.247785\pi$
$972$	0	0
$973$	1.25483e7	0.424915
$974$	0	0
$975$	−3.14621e6	−0.105993
$976$	0	0
$977$	−1.44074e7	−0.482890	−0.241445	−	0.970414i	$-0.577621\pi$
$977$	−1.44074e7	−0.482890	−0.241445	+	0.970414i	$0.577621\pi$
$978$	0	0
$979$	−1.43835e7	−0.479631
$980$	0	0
$981$	5.35493e7	1.77657
$982$	0	0
$983$	2.11505e7	0.698130	0.349065	−	0.937098i	$-0.386499\pi$
$983$	2.11505e7	0.698130	0.349065	+	0.937098i	$0.386499\pi$
$984$	0	0
$985$	1.15726e7	0.380049
$986$	0	0
$987$	6.57856e7	2.14950
$988$	0	0
$989$	−3.02906e6	−0.0984731
$990$	0	0
$991$	−5.90549e7	−1.91017	−0.955084	−	0.296335i	$-0.904236\pi$
$991$	−5.90549e7	−1.91017	−0.955084	+	0.296335i	$0.904236\pi$
$992$	0	0
$993$	−4.83841e7	−1.55715
$994$	0	0
$995$	2.01997e7	0.646824
$996$	0	0
$997$	−1.01289e7	−0.322718	−0.161359	−	0.986896i	$-0.551588\pi$
$997$	−1.01289e7	−0.322718	−0.161359	+	0.986896i	$0.551588\pi$
$998$	0	0
$999$	4.72882e7	1.49913

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.6.a.e.1.1	✓	8
4.3	odd	2		1040.6.a.z.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.6.a.e.1.1	✓	8	1.1	even	1	trivial
1040.6.a.z.1.8		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	520.a (trivial)

\(f(q)\)	\(=\)	\(q-29.7866 q^{3} -25.0000 q^{5} +138.753 q^{7} +644.243 q^{9} -132.506 q^{11} +169.000 q^{13} +744.666 q^{15} +1951.61 q^{17} +1920.72 q^{19} -4133.00 q^{21} +1713.94 q^{23} +625.000 q^{25} -11951.7 q^{27} +1118.88 q^{29} -2283.69 q^{31} +3946.89 q^{33} -3468.83 q^{35} -3956.62 q^{37} -5033.94 q^{39} +7403.63 q^{41} -1767.31 q^{43} -16106.1 q^{45} -15917.2 q^{47} +2445.51 q^{49} -58131.7 q^{51} +1515.57 q^{53} +3312.64 q^{55} -57211.7 q^{57} -43798.7 q^{59} +19517.9 q^{61} +89390.9 q^{63} -4225.00 q^{65} -9820.52 q^{67} -51052.4 q^{69} +17669.6 q^{71} -23941.5 q^{73} -18616.6 q^{75} -18385.6 q^{77} +106118. q^{79} +199449. q^{81} +95434.8 q^{83} -48790.1 q^{85} -33327.7 q^{87} +108550. q^{89} +23449.3 q^{91} +68023.5 q^{93} -48017.9 q^{95} +74315.7 q^{97} -85365.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{3} - 200 q^{5} + 184 q^{7} + 872 q^{9} + 128 q^{11} + 1352 q^{13} + 400 q^{15} + 1656 q^{17} - 2112 q^{19} - 1352 q^{21} - 1144 q^{23} + 5000 q^{25} - 6112 q^{27} - 6672 q^{29} - 2984 q^{31}+ \cdots + 312944 q^{99}+O(q^{100}) \) 8 * q - 16 * q^3 - 200 * q^5 + 184 * q^7 + 872 * q^9 + 128 * q^11 + 1352 * q^13 + 400 * q^15 + 1656 * q^17 - 2112 * q^19 - 1352 * q^21 - 1144 * q^23 + 5000 * q^25 - 6112 * q^27 - 6672 * q^29 - 2984 * q^31 - 10840 * q^33 - 4600 * q^35 - 13432 * q^37 - 2704 * q^39 - 10136 * q^41 - 3632 * q^43 - 21800 * q^45 + 25256 * q^47 - 1368 * q^49 - 20736 * q^51 + 26744 * q^53 - 3200 * q^55 + 22536 * q^57 + 55344 * q^59 + 22784 * q^61 + 112696 * q^63 - 33800 * q^65 + 16984 * q^67 + 33104 * q^69 + 118856 * q^71 + 120744 * q^73 - 10000 * q^75 - 24200 * q^77 + 154720 * q^79 + 171944 * q^81 + 179048 * q^83 - 41400 * q^85 + 190992 * q^87 + 254336 * q^89 + 31096 * q^91 + 247672 * q^93 + 52800 * q^95 + 134320 * q^97 + 312944 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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