LMFDB - Embedded newform 156.8.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [156,8,Mod(1,156)] 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("156.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 156 = 2^{2} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	156.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-81] 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:	yes
Analytic conductor:	$48.7320639755$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11078x - 379248 $ x^3 - x^2 - 11078*x - 379248
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-40.2803$ of defining polynomial
Character	$\chi$	$=$	156.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} +480.358 q^{5} -800.424 q^{7} +729.000 q^{9} +7629.09 q^{11} +2197.00 q^{13} -12969.7 q^{15} -21796.8 q^{17} +33672.2 q^{19} +21611.5 q^{21} -26192.8 q^{23} +152618. q^{25} -19683.0 q^{27} -113083. q^{29} +88423.6 q^{31} -205985. q^{33} -384490. q^{35} -379766. q^{37} -59319.0 q^{39} -399092. q^{41} +849618. q^{43} +350181. q^{45} +939767. q^{47} -182864. q^{49} +588513. q^{51} +1.14570e6 q^{53} +3.66469e6 q^{55} -909150. q^{57} +731851. q^{59} +632912. q^{61} -583509. q^{63} +1.05535e6 q^{65} +1.77055e6 q^{67} +707205. q^{69} +5.46845e6 q^{71} +984549. q^{73} -4.12070e6 q^{75} -6.10651e6 q^{77} +6.31489e6 q^{79} +531441. q^{81} -3.30152e6 q^{83} -1.04703e7 q^{85} +3.05325e6 q^{87} +7.70391e6 q^{89} -1.75853e6 q^{91} -2.38744e6 q^{93} +1.61747e7 q^{95} -1.39202e6 q^{97} +5.56160e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 81 q^{3} + 524 q^{5} + 722 q^{7} + 2187 q^{9} + 6752 q^{11} + 6591 q^{13} - 14148 q^{15} - 15418 q^{17} + 1826 q^{19} - 19494 q^{21} + 16536 q^{23} + 10513 q^{25} - 59049 q^{27} - 151194 q^{29} + 46338 q^{31}+ \cdots + 4922208 q^{99}+O(q^{100}) $ 3 * q - 81 * q^3 + 524 * q^5 + 722 * q^7 + 2187 * q^9 + 6752 * q^11 + 6591 * q^13 - 14148 * q^15 - 15418 * q^17 + 1826 * q^19 - 19494 * q^21 + 16536 * q^23 + 10513 * q^25 - 59049 * q^27 - 151194 * q^29 + 46338 * q^31 - 182304 * q^33 - 223864 * q^35 - 112378 * q^37 - 177957 * q^39 + 324944 * q^41 + 1282940 * q^43 + 381996 * q^45 + 917072 * q^47 + 559359 * q^49 + 416286 * q^51 + 2688314 * q^53 + 4049992 * q^55 - 49302 * q^57 + 3176620 * q^59 + 2165222 * q^61 + 526338 * q^63 + 1151228 * q^65 + 6699990 * q^67 - 446472 * q^69 + 7734556 * q^71 + 5784690 * q^73 - 283851 * q^75 - 2868232 * q^77 + 8426640 * q^79 + 1594323 * q^81 + 4126284 * q^83 - 8292584 * q^85 + 4082238 * q^87 + 5207756 * q^89 + 1586234 * q^91 - 1251126 * q^93 + 18113000 * q^95 - 7634614 * q^97 + 4922208 * 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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	480.358	1.71858	0.859290	−	0.511489i	$-0.170906\pi$
$5$	480.358	1.71858	0.859290	+	0.511489i	$0.170906\pi$
$6$	0	0
$7$	−800.424	−0.882017	−0.441009	−	0.897503i	$-0.645379\pi$
$7$	−800.424	−0.882017	−0.441009	+	0.897503i	$0.645379\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	7629.09	1.72822	0.864108	−	0.503306i	$-0.167883\pi$
$11$	7629.09	1.72822	0.864108	+	0.503306i	$0.167883\pi$
$12$	0	0
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	0	0
$15$	−12969.7	−0.992222
$16$	0	0
$17$	−21796.8	−1.07602	−0.538011	−	0.842938i	$-0.680824\pi$
$17$	−21796.8	−1.07602	−0.538011	+	0.842938i	$0.680824\pi$
$18$	0	0
$19$	33672.2	1.12625	0.563124	−	0.826372i	$-0.309599\pi$
$19$	33672.2	1.12625	0.563124	+	0.826372i	$0.309599\pi$
$20$	0	0
$21$	21611.5	0.509233
$22$	0	0
$23$	−26192.8	−0.448884	−0.224442	−	0.974487i	$-0.572056\pi$
$23$	−26192.8	−0.448884	−0.224442	+	0.974487i	$0.572056\pi$
$24$	0	0
$25$	152618.	1.95352
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−113083.	−0.861004	−0.430502	−	0.902590i	$-0.641664\pi$
$29$	−113083.	−0.861004	−0.430502	+	0.902590i	$0.641664\pi$
$30$	0	0
$31$	88423.6	0.533092	0.266546	−	0.963822i	$-0.414118\pi$
$31$	88423.6	0.533092	0.266546	+	0.963822i	$0.414118\pi$
$32$	0	0
$33$	−205985.	−0.997786
$34$	0	0
$35$	−384490.	−1.51582
$36$	0	0
$37$	−379766.	−1.23257	−0.616283	−	0.787525i	$-0.711362\pi$
$37$	−379766.	−1.23257	−0.616283	+	0.787525i	$0.711362\pi$
$38$	0	0
$39$	−59319.0	−0.160128
$40$	0	0
$41$	−399092.	−0.904336	−0.452168	−	0.891933i	$-0.649349\pi$
$41$	−399092.	−0.904336	−0.452168	+	0.891933i	$0.649349\pi$
$42$	0	0
$43$	849618.	1.62961	0.814806	−	0.579733i	$-0.196843\pi$
$43$	849618.	1.62961	0.814806	+	0.579733i	$0.196843\pi$
$44$	0	0
$45$	350181.	0.572860
$46$	0	0
$47$	939767.	1.32032	0.660158	−	0.751127i	$-0.270489\pi$
$47$	939767.	1.32032	0.660158	+	0.751127i	$0.270489\pi$
$48$	0	0
$49$	−182864.	−0.222046
$50$	0	0
$51$	588513.	0.621242
$52$	0	0
$53$	1.14570e6	1.05707	0.528536	−	0.848911i	$-0.322741\pi$
$53$	1.14570e6	1.05707	0.528536	+	0.848911i	$0.322741\pi$
$54$	0	0
$55$	3.66469e6	2.97008
$56$	0	0
$57$	−909150.	−0.650240
$58$	0	0
$59$	731851.	0.463917	0.231959	−	0.972726i	$-0.425487\pi$
$59$	731851.	0.463917	0.231959	+	0.972726i	$0.425487\pi$
$60$	0	0
$61$	632912.	0.357017	0.178508	−	0.983938i	$-0.442873\pi$
$61$	632912.	0.357017	0.178508	+	0.983938i	$0.442873\pi$
$62$	0	0
$63$	−583509.	−0.294006
$64$	0	0
$65$	1.05535e6	0.476648
$66$	0	0
$67$	1.77055e6	0.719196	0.359598	−	0.933107i	$-0.382914\pi$
$67$	1.77055e6	0.719196	0.359598	+	0.933107i	$0.382914\pi$
$68$	0	0
$69$	707205.	0.259163
$70$	0	0
$71$	5.46845e6	1.81326	0.906629	−	0.421928i	$-0.138646\pi$
$71$	5.46845e6	1.81326	0.906629	+	0.421928i	$0.138646\pi$
$72$	0	0
$73$	984549.	0.296215	0.148108	−	0.988971i	$-0.452682\pi$
$73$	984549.	0.296215	0.148108	+	0.988971i	$0.452682\pi$
$74$	0	0
$75$	−4.12070e6	−1.12786
$76$	0	0
$77$	−6.10651e6	−1.52432
$78$	0	0
$79$	6.31489e6	1.44102	0.720511	−	0.693443i	$-0.243907\pi$
$79$	6.31489e6	1.44102	0.720511	+	0.693443i	$0.243907\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−3.30152e6	−0.633783	−0.316892	−	0.948462i	$-0.602639\pi$
$83$	−3.30152e6	−0.633783	−0.316892	+	0.948462i	$0.602639\pi$
$84$	0	0
$85$	−1.04703e7	−1.84923
$86$	0	0
$87$	3.05325e6	0.497101
$88$	0	0
$89$	7.70391e6	1.15837	0.579184	−	0.815197i	$-0.303371\pi$
$89$	7.70391e6	1.15837	0.579184	+	0.815197i	$0.303371\pi$
$90$	0	0
$91$	−1.75853e6	−0.244628
$92$	0	0
$93$	−2.38744e6	−0.307781
$94$	0	0
$95$	1.61747e7	1.93555
$96$	0	0
$97$	−1.39202e6	−0.154862	−0.0774308	−	0.996998i	$-0.524672\pi$
$97$	−1.39202e6	−0.154862	−0.0774308	+	0.996998i	$0.524672\pi$
$98$	0	0
$99$	5.56160e6	0.576072
$100$	0	0
$101$	1.56117e7	1.50773	0.753866	−	0.657028i	$-0.228187\pi$
$101$	1.56117e7	1.50773	0.753866	+	0.657028i	$0.228187\pi$
$102$	0	0
$103$	−4.93242e6	−0.444764	−0.222382	−	0.974960i	$-0.571383\pi$
$103$	−4.93242e6	−0.444764	−0.222382	+	0.974960i	$0.571383\pi$
$104$	0	0
$105$	1.03812e7	0.875157
$106$	0	0
$107$	−1.36458e7	−1.07685	−0.538425	−	0.842673i	$-0.680981\pi$
$107$	−1.36458e7	−1.07685	−0.538425	+	0.842673i	$0.680981\pi$
$108$	0	0
$109$	−3.07725e6	−0.227599	−0.113799	−	0.993504i	$-0.536302\pi$
$109$	−3.07725e6	−0.227599	−0.113799	+	0.993504i	$0.536302\pi$
$110$	0	0
$111$	1.02537e7	0.711622
$112$	0	0
$113$	1.04413e7	0.680739	0.340369	−	0.940292i	$-0.389448\pi$
$113$	1.04413e7	0.680739	0.340369	+	0.940292i	$0.389448\pi$
$114$	0	0
$115$	−1.25819e7	−0.771443
$116$	0	0
$117$	1.60161e6	0.0924500
$118$	0	0
$119$	1.74467e7	0.949070
$120$	0	0
$121$	3.87158e7	1.98673
$122$	0	0
$123$	1.07755e7	0.522118
$124$	0	0
$125$	3.57835e7	1.63869
$126$	0	0
$127$	−2.96218e7	−1.28321	−0.641606	−	0.767035i	$-0.721731\pi$
$127$	−2.96218e7	−1.28321	−0.641606	+	0.767035i	$0.721731\pi$
$128$	0	0
$129$	−2.29397e7	−0.940857
$130$	0	0
$131$	−4.74229e7	−1.84306	−0.921529	−	0.388310i	$-0.873059\pi$
$131$	−4.74229e7	−1.84306	−0.921529	+	0.388310i	$0.873059\pi$
$132$	0	0
$133$	−2.69521e7	−0.993371
$134$	0	0
$135$	−9.45488e6	−0.330741
$136$	0	0
$137$	−2.73115e7	−0.907452	−0.453726	−	0.891141i	$-0.649905\pi$
$137$	−2.73115e7	−0.907452	−0.453726	+	0.891141i	$0.649905\pi$
$138$	0	0
$139$	2.91049e7	0.919209	0.459605	−	0.888124i	$-0.347991\pi$
$139$	2.91049e7	0.919209	0.459605	+	0.888124i	$0.347991\pi$
$140$	0	0
$141$	−2.53737e7	−0.762284
$142$	0	0
$143$	1.67611e7	0.479321
$144$	0	0
$145$	−5.43204e7	−1.47970
$146$	0	0
$147$	4.93733e6	0.128198
$148$	0	0
$149$	−6.81525e7	−1.68783	−0.843917	−	0.536474i	$-0.819756\pi$
$149$	−6.81525e7	−1.68783	−0.843917	+	0.536474i	$0.819756\pi$
$150$	0	0
$151$	5.30712e7	1.25441	0.627205	−	0.778854i	$-0.284199\pi$
$151$	5.30712e7	1.25441	0.627205	+	0.778854i	$0.284199\pi$
$152$	0	0
$153$	−1.58899e7	−0.358674
$154$	0	0
$155$	4.24749e7	0.916161
$156$	0	0
$157$	3.34739e7	0.690331	0.345166	−	0.938542i	$-0.387823\pi$
$157$	3.34739e7	0.690331	0.345166	+	0.938542i	$0.387823\pi$
$158$	0	0
$159$	−3.09338e7	−0.610300
$160$	0	0
$161$	2.09653e7	0.395924
$162$	0	0
$163$	−4.33468e7	−0.783973	−0.391986	−	0.919971i	$-0.628212\pi$
$163$	−4.33468e7	−0.783973	−0.391986	+	0.919971i	$0.628212\pi$
$164$	0	0
$165$	−9.89466e7	−1.71478
$166$	0	0
$167$	2.64651e7	0.439710	0.219855	−	0.975533i	$-0.429442\pi$
$167$	2.64651e7	0.439710	0.219855	+	0.975533i	$0.429442\pi$
$168$	0	0
$169$	4.82681e6	0.0769231
$170$	0	0
$171$	2.45471e7	0.375416
$172$	0	0
$173$	6.24515e7	0.917025	0.458513	−	0.888688i	$-0.348382\pi$
$173$	6.24515e7	0.917025	0.458513	+	0.888688i	$0.348382\pi$
$174$	0	0
$175$	−1.22160e8	−1.72303
$176$	0	0
$177$	−1.97600e7	−0.267843
$178$	0	0
$179$	4.26317e7	0.555581	0.277790	−	0.960642i	$-0.410398\pi$
$179$	4.26317e7	0.555581	0.277790	+	0.960642i	$0.410398\pi$
$180$	0	0
$181$	1.43389e8	1.79738	0.898689	−	0.438587i	$-0.144521\pi$
$181$	1.43389e8	1.79738	0.898689	+	0.438587i	$0.144521\pi$
$182$	0	0
$183$	−1.70886e7	−0.206124
$184$	0	0
$185$	−1.82423e8	−2.11826
$186$	0	0
$187$	−1.66290e8	−1.85960
$188$	0	0
$189$	1.57547e7	0.169744
$190$	0	0
$191$	1.29924e7	0.134918	0.0674592	−	0.997722i	$-0.478511\pi$
$191$	1.29924e7	0.134918	0.0674592	+	0.997722i	$0.478511\pi$
$192$	0	0
$193$	7.26618e7	0.727538	0.363769	−	0.931489i	$-0.381490\pi$
$193$	7.26618e7	0.727538	0.363769	+	0.931489i	$0.381490\pi$
$194$	0	0
$195$	−2.84943e7	−0.275193
$196$	0	0
$197$	−1.48236e8	−1.38141	−0.690705	−	0.723137i	$-0.742700\pi$
$197$	−1.48236e8	−1.38141	−0.690705	+	0.723137i	$0.742700\pi$
$198$	0	0
$199$	−6.62122e7	−0.595597	−0.297799	−	0.954629i	$-0.596252\pi$
$199$	−6.62122e7	−0.595597	−0.297799	+	0.954629i	$0.596252\pi$
$200$	0	0
$201$	−4.78049e7	−0.415228
$202$	0	0
$203$	9.05146e7	0.759421
$204$	0	0
$205$	−1.91707e8	−1.55417
$206$	0	0
$207$	−1.90945e7	−0.149628
$208$	0	0
$209$	2.56888e8	1.94640
$210$	0	0
$211$	−1.76246e8	−1.29160	−0.645802	−	0.763505i	$-0.723477\pi$
$211$	−1.76246e8	−1.29160	−0.645802	+	0.763505i	$0.723477\pi$
$212$	0	0
$213$	−1.47648e8	−1.04689
$214$	0	0
$215$	4.08121e8	2.80062
$216$	0	0
$217$	−7.07763e7	−0.470196
$218$	0	0
$219$	−2.65828e7	−0.171020
$220$	0	0
$221$	−4.78875e7	−0.298435
$222$	0	0
$223$	−1.08567e8	−0.655587	−0.327793	−	0.944749i	$-0.606305\pi$
$223$	−1.08567e8	−0.655587	−0.327793	+	0.944749i	$0.606305\pi$
$224$	0	0
$225$	1.11259e8	0.651172
$226$	0	0
$227$	−1.84826e8	−1.04875	−0.524377	−	0.851486i	$-0.675702\pi$
$227$	−1.84826e8	−1.04875	−0.524377	+	0.851486i	$0.675702\pi$
$228$	0	0
$229$	2.69737e7	0.148428	0.0742142	−	0.997242i	$-0.476355\pi$
$229$	2.69737e7	0.148428	0.0742142	+	0.997242i	$0.476355\pi$
$230$	0	0
$231$	1.64876e8	0.880065
$232$	0	0
$233$	2.14981e8	1.11341	0.556705	−	0.830710i	$-0.312065\pi$
$233$	2.14981e8	1.11341	0.556705	+	0.830710i	$0.312065\pi$
$234$	0	0
$235$	4.51424e8	2.26907
$236$	0	0
$237$	−1.70502e8	−0.831975
$238$	0	0
$239$	−1.06247e8	−0.503413	−0.251706	−	0.967804i	$-0.580992\pi$
$239$	−1.06247e8	−0.503413	−0.251706	+	0.967804i	$0.580992\pi$
$240$	0	0
$241$	−4.26426e8	−1.96238	−0.981192	−	0.193035i	$-0.938167\pi$
$241$	−4.26426e8	−1.96238	−0.981192	+	0.193035i	$0.938167\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−8.78402e7	−0.381603
$246$	0	0
$247$	7.39779e7	0.312365
$248$	0	0
$249$	8.91411e7	0.365915
$250$	0	0
$251$	4.84674e8	1.93460	0.967300	−	0.253634i	$-0.0816260\pi$
$251$	4.84674e8	1.93460	0.967300	+	0.253634i	$0.0816260\pi$
$252$	0	0
$253$	−1.99827e8	−0.775769
$254$	0	0
$255$	2.82697e8	1.06765
$256$	0	0
$257$	2.21477e8	0.813883	0.406942	−	0.913454i	$-0.366595\pi$
$257$	2.21477e8	0.813883	0.406942	+	0.913454i	$0.366595\pi$
$258$	0	0
$259$	3.03974e8	1.08714
$260$	0	0
$261$	−8.24377e7	−0.287001
$262$	0	0
$263$	−6.77193e7	−0.229545	−0.114772	−	0.993392i	$-0.536614\pi$
$263$	−6.77193e7	−0.229545	−0.114772	+	0.993392i	$0.536614\pi$
$264$	0	0
$265$	5.50344e8	1.81666
$266$	0	0
$267$	−2.08006e8	−0.668784
$268$	0	0
$269$	−2.68963e8	−0.842481	−0.421240	−	0.906949i	$-0.638405\pi$
$269$	−2.68963e8	−0.842481	−0.421240	+	0.906949i	$0.638405\pi$
$270$	0	0
$271$	−5.32945e8	−1.62663	−0.813316	−	0.581822i	$-0.802340\pi$
$271$	−5.32945e8	−1.62663	−0.813316	+	0.581822i	$0.802340\pi$
$272$	0	0
$273$	4.74804e7	0.141236
$274$	0	0
$275$	1.16434e9	3.37610
$276$	0	0
$277$	−3.93629e7	−0.111278	−0.0556388	−	0.998451i	$-0.517720\pi$
$277$	−3.93629e7	−0.111278	−0.0556388	+	0.998451i	$0.517720\pi$
$278$	0	0
$279$	6.44608e7	0.177697
$280$	0	0
$281$	5.25113e8	1.41182	0.705912	−	0.708299i	$-0.250537\pi$
$281$	5.25113e8	1.41182	0.705912	+	0.708299i	$0.250537\pi$
$282$	0	0
$283$	2.40271e8	0.630158	0.315079	−	0.949065i	$-0.397969\pi$
$283$	2.40271e8	0.630158	0.315079	+	0.949065i	$0.397969\pi$
$284$	0	0
$285$	−4.36717e8	−1.11749
$286$	0	0
$287$	3.19443e8	0.797640
$288$	0	0
$289$	6.47612e7	0.157824
$290$	0	0
$291$	3.75845e7	0.0894094
$292$	0	0
$293$	4.00851e8	0.930993	0.465497	−	0.885050i	$-0.345876\pi$
$293$	4.00851e8	0.930993	0.465497	+	0.885050i	$0.345876\pi$
$294$	0	0
$295$	3.51550e8	0.797279
$296$	0	0
$297$	−1.50163e8	−0.332595
$298$	0	0
$299$	−5.75456e7	−0.124498
$300$	0	0
$301$	−6.80055e8	−1.43735
$302$	0	0
$303$	−4.21515e8	−0.870490
$304$	0	0
$305$	3.04024e8	0.613562
$306$	0	0
$307$	−4.37804e8	−0.863565	−0.431782	−	0.901978i	$-0.642115\pi$
$307$	−4.37804e8	−0.863565	−0.431782	+	0.901978i	$0.642115\pi$
$308$	0	0
$309$	1.33175e8	0.256785
$310$	0	0
$311$	−7.39551e7	−0.139414	−0.0697070	−	0.997568i	$-0.522206\pi$
$311$	−7.39551e7	−0.139414	−0.0697070	+	0.997568i	$0.522206\pi$
$312$	0	0
$313$	−1.22974e8	−0.226677	−0.113338	−	0.993556i	$-0.536154\pi$
$313$	−1.22974e8	−0.226677	−0.113338	+	0.993556i	$0.536154\pi$
$314$	0	0
$315$	−2.80293e8	−0.505272
$316$	0	0
$317$	−3.43640e8	−0.605893	−0.302946	−	0.953008i	$-0.597970\pi$
$317$	−3.43640e8	−0.605893	−0.302946	+	0.953008i	$0.597970\pi$
$318$	0	0
$319$	−8.62722e8	−1.48800
$320$	0	0
$321$	3.68436e8	0.621720
$322$	0	0
$323$	−7.33947e8	−1.21187
$324$	0	0
$325$	3.35303e8	0.541808
$326$	0	0
$327$	8.30858e7	0.131404
$328$	0	0
$329$	−7.52212e8	−1.16454
$330$	0	0
$331$	1.24712e8	0.189020	0.0945102	−	0.995524i	$-0.469871\pi$
$331$	1.24712e8	0.189020	0.0945102	+	0.995524i	$0.469871\pi$
$332$	0	0
$333$	−2.76849e8	−0.410855
$334$	0	0
$335$	8.50499e8	1.23600
$336$	0	0
$337$	1.97813e8	0.281547	0.140773	−	0.990042i	$-0.455041\pi$
$337$	1.97813e8	0.281547	0.140773	+	0.990042i	$0.455041\pi$
$338$	0	0
$339$	−2.81915e8	−0.393025
$340$	0	0
$341$	6.74591e8	0.921298
$342$	0	0
$343$	8.05553e8	1.07787
$344$	0	0
$345$	3.39711e8	0.445393
$346$	0	0
$347$	6.43590e8	0.826906	0.413453	−	0.910526i	$-0.364323\pi$
$347$	6.43590e8	0.826906	0.413453	+	0.910526i	$0.364323\pi$
$348$	0	0
$349$	5.79925e8	0.730270	0.365135	−	0.930955i	$-0.381023\pi$
$349$	5.79925e8	0.730270	0.365135	+	0.930955i	$0.381023\pi$
$350$	0	0
$351$	−4.32436e7	−0.0533761
$352$	0	0
$353$	−1.41627e9	−1.71369	−0.856847	−	0.515571i	$-0.827580\pi$
$353$	−1.41627e9	−1.71369	−0.856847	+	0.515571i	$0.827580\pi$
$354$	0	0
$355$	2.62681e9	3.11623
$356$	0	0
$357$	−4.71060e8	−0.547946
$358$	0	0
$359$	1.38549e8	0.158042	0.0790210	−	0.996873i	$-0.474821\pi$
$359$	1.38549e8	0.158042	0.0790210	+	0.996873i	$0.474821\pi$
$360$	0	0
$361$	2.39948e8	0.268437
$362$	0	0
$363$	−1.04533e9	−1.14704
$364$	0	0
$365$	4.72936e8	0.509069
$366$	0	0
$367$	1.29015e9	1.36241	0.681207	−	0.732090i	$-0.261455\pi$
$367$	1.29015e9	1.36241	0.681207	+	0.732090i	$0.261455\pi$
$368$	0	0
$369$	−2.90938e8	−0.301445
$370$	0	0
$371$	−9.17044e8	−0.932355
$372$	0	0
$373$	−1.64979e9	−1.64607	−0.823034	−	0.567993i	$-0.807720\pi$
$373$	−1.64979e9	−1.64607	−0.823034	+	0.567993i	$0.807720\pi$
$374$	0	0
$375$	−9.66155e8	−0.946100
$376$	0	0
$377$	−2.48444e8	−0.238800
$378$	0	0
$379$	−3.88541e7	−0.0366606	−0.0183303	−	0.999832i	$-0.505835\pi$
$379$	−3.88541e7	−0.0366606	−0.0183303	+	0.999832i	$0.505835\pi$
$380$	0	0
$381$	7.99788e8	0.740863
$382$	0	0
$383$	−1.13507e9	−1.03235	−0.516177	−	0.856482i	$-0.672645\pi$
$383$	−1.13507e9	−1.03235	−0.516177	+	0.856482i	$0.672645\pi$
$384$	0	0
$385$	−2.93331e9	−2.61966
$386$	0	0
$387$	6.19372e8	0.543204
$388$	0	0
$389$	−1.56751e9	−1.35017	−0.675083	−	0.737742i	$-0.735892\pi$
$389$	−1.56751e9	−1.35017	−0.675083	+	0.737742i	$0.735892\pi$
$390$	0	0
$391$	5.70919e8	0.483009
$392$	0	0
$393$	1.28042e9	1.06409
$394$	0	0
$395$	3.03340e9	2.47651
$396$	0	0
$397$	−6.92611e8	−0.555550	−0.277775	−	0.960646i	$-0.589597\pi$
$397$	−6.92611e8	−0.555550	−0.277775	+	0.960646i	$0.589597\pi$
$398$	0	0
$399$	7.27706e8	0.573523
$400$	0	0
$401$	1.78673e9	1.38374	0.691870	−	0.722022i	$-0.256787\pi$
$401$	1.78673e9	1.38374	0.691870	+	0.722022i	$0.256787\pi$
$402$	0	0
$403$	1.94267e8	0.147853
$404$	0	0
$405$	2.55282e8	0.190953
$406$	0	0
$407$	−2.89727e9	−2.13014
$408$	0	0
$409$	−3.45334e8	−0.249579	−0.124790	−	0.992183i	$-0.539826\pi$
$409$	−3.45334e8	−0.249579	−0.124790	+	0.992183i	$0.539826\pi$
$410$	0	0
$411$	7.37411e8	0.523918
$412$	0	0
$413$	−5.85791e8	−0.409183
$414$	0	0
$415$	−1.58591e9	−1.08921
$416$	0	0
$417$	−7.85833e8	−0.530706
$418$	0	0
$419$	−9.49769e8	−0.630767	−0.315384	−	0.948964i	$-0.602133\pi$
$419$	−9.49769e8	−0.630767	−0.315384	+	0.948964i	$0.602133\pi$
$420$	0	0
$421$	−2.49548e9	−1.62992	−0.814960	−	0.579517i	$-0.803241\pi$
$421$	−2.49548e9	−1.62992	−0.814960	+	0.579517i	$0.803241\pi$
$422$	0	0
$423$	6.85090e8	0.440105
$424$	0	0
$425$	−3.32659e9	−2.10203
$426$	0	0
$427$	−5.06598e8	−0.314895
$428$	0	0
$429$	−4.52550e8	−0.276736
$430$	0	0
$431$	9.09506e8	0.547186	0.273593	−	0.961846i	$-0.411788\pi$
$431$	9.09506e8	0.547186	0.273593	+	0.961846i	$0.411788\pi$
$432$	0	0
$433$	−3.17167e9	−1.87750	−0.938752	−	0.344594i	$-0.888017\pi$
$433$	−3.17167e9	−1.87750	−0.938752	+	0.344594i	$0.888017\pi$
$434$	0	0
$435$	1.46665e9	0.854308
$436$	0	0
$437$	−8.81970e8	−0.505555
$438$	0	0
$439$	−1.47389e9	−0.831458	−0.415729	−	0.909489i	$-0.636474\pi$
$439$	−1.47389e9	−0.831458	−0.415729	+	0.909489i	$0.636474\pi$
$440$	0	0
$441$	−1.33308e8	−0.0740152
$442$	0	0
$443$	−2.63296e9	−1.43890	−0.719450	−	0.694545i	$-0.755606\pi$
$443$	−2.63296e9	−1.43890	−0.719450	+	0.694545i	$0.755606\pi$
$444$	0	0
$445$	3.70063e9	1.99075
$446$	0	0
$447$	1.84012e9	0.974471
$448$	0	0
$449$	1.72268e9	0.898137	0.449069	−	0.893497i	$-0.351756\pi$
$449$	1.72268e9	0.898137	0.449069	+	0.893497i	$0.351756\pi$
$450$	0	0
$451$	−3.04471e9	−1.56289
$452$	0	0
$453$	−1.43292e9	−0.724234
$454$	0	0
$455$	−8.44724e8	−0.420412
$456$	0	0
$457$	−2.85445e9	−1.39899	−0.699497	−	0.714635i	$-0.746593\pi$
$457$	−2.85445e9	−1.39899	−0.699497	+	0.714635i	$0.746593\pi$
$458$	0	0
$459$	4.29026e8	0.207081
$460$	0	0
$461$	−4.16130e8	−0.197822	−0.0989112	−	0.995096i	$-0.531536\pi$
$461$	−4.16130e8	−0.197822	−0.0989112	+	0.995096i	$0.531536\pi$
$462$	0	0
$463$	1.20291e9	0.563246	0.281623	−	0.959525i	$-0.409127\pi$
$463$	1.20291e9	0.563246	0.281623	+	0.959525i	$0.409127\pi$
$464$	0	0
$465$	−1.14682e9	−0.528946
$466$	0	0
$467$	1.78997e9	0.813276	0.406638	−	0.913589i	$-0.366701\pi$
$467$	1.78997e9	0.813276	0.406638	+	0.913589i	$0.366701\pi$
$468$	0	0
$469$	−1.41719e9	−0.634343
$470$	0	0
$471$	−9.03795e8	−0.398563
$472$	0	0
$473$	6.48181e9	2.81632
$474$	0	0
$475$	5.13901e9	2.20015
$476$	0	0
$477$	8.35213e8	0.352357
$478$	0	0
$479$	−1.09621e9	−0.455742	−0.227871	−	0.973691i	$-0.573176\pi$
$479$	−1.09621e9	−0.455742	−0.227871	+	0.973691i	$0.573176\pi$
$480$	0	0
$481$	−8.34346e8	−0.341852
$482$	0	0
$483$	−5.66064e8	−0.228587
$484$	0	0
$485$	−6.68666e8	−0.266142
$486$	0	0
$487$	4.23704e9	1.66231	0.831154	−	0.556042i	$-0.187681\pi$
$487$	4.23704e9	1.66231	0.831154	+	0.556042i	$0.187681\pi$
$488$	0	0
$489$	1.17036e9	0.452627
$490$	0	0
$491$	−3.77577e9	−1.43953	−0.719763	−	0.694219i	$-0.755750\pi$
$491$	−3.77577e9	−1.43953	−0.719763	+	0.694219i	$0.755750\pi$
$492$	0	0
$493$	2.46485e9	0.926460
$494$	0	0
$495$	2.67156e9	0.990026
$496$	0	0
$497$	−4.37708e9	−1.59933
$498$	0	0
$499$	4.09554e9	1.47557	0.737784	−	0.675037i	$-0.235872\pi$
$499$	4.09554e9	1.47557	0.737784	+	0.675037i	$0.235872\pi$
$500$	0	0
$501$	−7.14559e8	−0.253867
$502$	0	0
$503$	3.00181e9	1.05171	0.525854	−	0.850575i	$-0.323746\pi$
$503$	3.00181e9	1.05171	0.525854	+	0.850575i	$0.323746\pi$
$504$	0	0
$505$	7.49918e9	2.59116
$506$	0	0
$507$	−1.30324e8	−0.0444116
$508$	0	0
$509$	2.35466e9	0.791435	0.395718	−	0.918372i	$-0.370496\pi$
$509$	2.35466e9	0.791435	0.395718	+	0.918372i	$0.370496\pi$
$510$	0	0
$511$	−7.88057e8	−0.261267
$512$	0	0
$513$	−6.62771e8	−0.216747
$514$	0	0
$515$	−2.36933e9	−0.764363
$516$	0	0
$517$	7.16956e9	2.28179
$518$	0	0
$519$	−1.68619e9	−0.529445
$520$	0	0
$521$	−3.51002e9	−1.08737	−0.543685	−	0.839289i	$-0.682971\pi$
$521$	−3.51002e9	−1.08737	−0.543685	+	0.839289i	$0.682971\pi$
$522$	0	0
$523$	2.45850e9	0.751474	0.375737	−	0.926726i	$-0.377390\pi$
$523$	2.45850e9	0.751474	0.375737	+	0.926726i	$0.377390\pi$
$524$	0	0
$525$	3.29831e9	0.994795
$526$	0	0
$527$	−1.92735e9	−0.573619
$528$	0	0
$529$	−2.71876e9	−0.798503
$530$	0	0
$531$	5.33519e8	0.154639
$532$	0	0
$533$	−8.76805e8	−0.250818
$534$	0	0
$535$	−6.55486e9	−1.85065
$536$	0	0
$537$	−1.15106e9	−0.320765
$538$	0	0
$539$	−1.39509e9	−0.383743
$540$	0	0
$541$	3.51448e9	0.954267	0.477134	−	0.878831i	$-0.341676\pi$
$541$	3.51448e9	0.954267	0.477134	+	0.878831i	$0.341676\pi$
$542$	0	0
$543$	−3.87149e9	−1.03772
$544$	0	0
$545$	−1.47818e9	−0.391147
$546$	0	0
$547$	3.24109e9	0.846712	0.423356	−	0.905963i	$-0.360852\pi$
$547$	3.24109e9	0.846712	0.423356	+	0.905963i	$0.360852\pi$
$548$	0	0
$549$	4.61393e8	0.119006
$550$	0	0
$551$	−3.80777e9	−0.969705
$552$	0	0
$553$	−5.05459e9	−1.27101
$554$	0	0
$555$	4.92543e9	1.22298
$556$	0	0
$557$	3.67574e9	0.901263	0.450632	−	0.892710i	$-0.351199\pi$
$557$	3.67574e9	0.901263	0.450632	+	0.892710i	$0.351199\pi$
$558$	0	0
$559$	1.86661e9	0.451973
$560$	0	0
$561$	4.48982e9	1.07364
$562$	0	0
$563$	7.07829e8	0.167166	0.0835831	−	0.996501i	$-0.473364\pi$
$563$	7.07829e8	0.167166	0.0835831	+	0.996501i	$0.473364\pi$
$564$	0	0
$565$	5.01556e9	1.16990
$566$	0	0
$567$	−4.25378e8	−0.0980019
$568$	0	0
$569$	−1.20435e9	−0.274069	−0.137034	−	0.990566i	$-0.543757\pi$
$569$	−1.20435e9	−0.274069	−0.137034	+	0.990566i	$0.543757\pi$
$570$	0	0
$571$	−8.05515e9	−1.81070	−0.905351	−	0.424663i	$-0.860392\pi$
$571$	−8.05515e9	−1.81070	−0.905351	+	0.424663i	$0.860392\pi$
$572$	0	0
$573$	−3.50794e8	−0.0778952
$574$	0	0
$575$	−3.99750e9	−0.876903
$576$	0	0
$577$	−1.02226e9	−0.221538	−0.110769	−	0.993846i	$-0.535331\pi$
$577$	−1.02226e9	−0.221538	−0.110769	+	0.993846i	$0.535331\pi$
$578$	0	0
$579$	−1.96187e9	−0.420044
$580$	0	0
$581$	2.64262e9	0.559008
$582$	0	0
$583$	8.74062e9	1.82685
$584$	0	0
$585$	7.69347e8	0.158883
$586$	0	0
$587$	−6.83523e9	−1.39483	−0.697413	−	0.716670i	$-0.745665\pi$
$587$	−6.83523e9	−1.39483	−0.697413	+	0.716670i	$0.745665\pi$
$588$	0	0
$589$	2.97742e9	0.600394
$590$	0	0
$591$	4.00238e9	0.797558
$592$	0	0
$593$	−9.50693e8	−0.187219	−0.0936093	−	0.995609i	$-0.529840\pi$
$593$	−9.50693e8	−0.187219	−0.0936093	+	0.995609i	$0.529840\pi$
$594$	0	0
$595$	8.38064e9	1.63105
$596$	0	0
$597$	1.78773e9	0.343868
$598$	0	0
$599$	3.01345e9	0.572889	0.286444	−	0.958097i	$-0.407527\pi$
$599$	3.01345e9	0.572889	0.286444	+	0.958097i	$0.407527\pi$
$600$	0	0
$601$	4.27628e9	0.803537	0.401768	−	0.915741i	$-0.368396\pi$
$601$	4.27628e9	0.803537	0.401768	+	0.915741i	$0.368396\pi$
$602$	0	0
$603$	1.29073e9	0.239732
$604$	0	0
$605$	1.85974e10	3.41436
$606$	0	0
$607$	−6.64232e9	−1.20548	−0.602739	−	0.797938i	$-0.705924\pi$
$607$	−6.64232e9	−1.20548	−0.602739	+	0.797938i	$0.705924\pi$
$608$	0	0
$609$	−2.44389e9	−0.438452
$610$	0	0
$611$	2.06467e9	0.366190
$612$	0	0
$613$	−4.75171e9	−0.833178	−0.416589	−	0.909095i	$-0.636775\pi$
$613$	−4.75171e9	−0.833178	−0.416589	+	0.909095i	$0.636775\pi$
$614$	0	0
$615$	5.17609e9	0.897302
$616$	0	0
$617$	−6.33786e9	−1.08629	−0.543143	−	0.839640i	$-0.682766\pi$
$617$	−6.33786e9	−1.08629	−0.543143	+	0.839640i	$0.682766\pi$
$618$	0	0
$619$	1.06126e10	1.79848	0.899241	−	0.437454i	$-0.144120\pi$
$619$	1.06126e10	1.79848	0.899241	+	0.437454i	$0.144120\pi$
$620$	0	0
$621$	5.15553e8	0.0863878
$622$	0	0
$623$	−6.16640e9	−1.02170
$624$	0	0
$625$	5.26556e9	0.862710
$626$	0	0
$627$	−6.93599e9	−1.12376
$628$	0	0
$629$	8.27768e9	1.32627
$630$	0	0
$631$	−8.73323e9	−1.38380	−0.691898	−	0.721995i	$-0.743225\pi$
$631$	−8.73323e9	−1.38380	−0.691898	+	0.721995i	$0.743225\pi$
$632$	0	0
$633$	4.75863e9	0.745708
$634$	0	0
$635$	−1.42291e10	−2.20530
$636$	0	0
$637$	−4.01753e8	−0.0615844
$638$	0	0
$639$	3.98650e9	0.604420
$640$	0	0
$641$	−6.68290e9	−1.00222	−0.501109	−	0.865384i	$-0.667074\pi$
$641$	−6.68290e9	−1.00222	−0.501109	+	0.865384i	$0.667074\pi$
$642$	0	0
$643$	−6.08421e9	−0.902539	−0.451269	−	0.892388i	$-0.649029\pi$
$643$	−6.08421e9	−0.902539	−0.451269	+	0.892388i	$0.649029\pi$
$644$	0	0
$645$	−1.10193e10	−1.61694
$646$	0	0
$647$	−3.35194e9	−0.486554	−0.243277	−	0.969957i	$-0.578222\pi$
$647$	−3.35194e9	−0.486554	−0.243277	+	0.969957i	$0.578222\pi$
$648$	0	0
$649$	5.58335e9	0.801749
$650$	0	0
$651$	1.91096e9	0.271468
$652$	0	0
$653$	5.17471e9	0.727261	0.363630	−	0.931543i	$-0.381537\pi$
$653$	5.17471e9	0.727261	0.363630	+	0.931543i	$0.381537\pi$
$654$	0	0
$655$	−2.27800e10	−3.16744
$656$	0	0
$657$	7.17736e8	0.0987384
$658$	0	0
$659$	9.38741e9	1.27775	0.638876	−	0.769309i	$-0.279400\pi$
$659$	9.38741e9	1.27775	0.638876	+	0.769309i	$0.279400\pi$
$660$	0	0
$661$	−1.30856e10	−1.76233	−0.881165	−	0.472809i	$-0.843240\pi$
$661$	−1.30856e10	−1.76233	−0.881165	+	0.472809i	$0.843240\pi$
$662$	0	0
$663$	1.29296e9	0.172301
$664$	0	0
$665$	−1.29466e10	−1.70719
$666$	0	0
$667$	2.96197e9	0.386491
$668$	0	0
$669$	2.93131e9	0.378503
$670$	0	0
$671$	4.82854e9	0.617003
$672$	0	0
$673$	4.10451e9	0.519049	0.259524	−	0.965737i	$-0.416434\pi$
$673$	4.10451e9	0.519049	0.259524	+	0.965737i	$0.416434\pi$
$674$	0	0
$675$	−3.00399e9	−0.375954
$676$	0	0
$677$	1.12936e10	1.39885	0.699424	−	0.714707i	$-0.253440\pi$
$677$	1.12936e10	1.39885	0.699424	+	0.714707i	$0.253440\pi$
$678$	0	0
$679$	1.11420e9	0.136591
$680$	0	0
$681$	4.99031e9	0.605498
$682$	0	0
$683$	−6.54596e9	−0.786142	−0.393071	−	0.919508i	$-0.628587\pi$
$683$	−6.54596e9	−0.786142	−0.393071	+	0.919508i	$0.628587\pi$
$684$	0	0
$685$	−1.31193e10	−1.55953
$686$	0	0
$687$	−7.28291e8	−0.0856952
$688$	0	0
$689$	2.51710e9	0.293179
$690$	0	0
$691$	−1.47731e10	−1.70333	−0.851664	−	0.524089i	$-0.824406\pi$
$691$	−1.47731e10	−1.70333	−0.851664	+	0.524089i	$0.824406\pi$
$692$	0	0
$693$	−4.45164e9	−0.508106
$694$	0	0
$695$	1.39808e10	1.57973
$696$	0	0
$697$	8.69893e9	0.973085
$698$	0	0
$699$	−5.80450e9	−0.642827
$700$	0	0
$701$	−1.02539e9	−0.112429	−0.0562143	−	0.998419i	$-0.517903\pi$
$701$	−1.02539e9	−0.112429	−0.0562143	+	0.998419i	$0.517903\pi$
$702$	0	0
$703$	−1.27876e10	−1.38818
$704$	0	0
$705$	−1.21885e10	−1.31005
$706$	0	0
$707$	−1.24959e10	−1.32985
$708$	0	0
$709$	1.66619e10	1.75575	0.877877	−	0.478887i	$-0.158959\pi$
$709$	1.66619e10	1.75575	0.877877	+	0.478887i	$0.158959\pi$
$710$	0	0
$711$	4.60355e9	0.480341
$712$	0	0
$713$	−2.31606e9	−0.239296
$714$	0	0
$715$	8.05132e9	0.823751
$716$	0	0
$717$	2.86867e9	0.290645
$718$	0	0
$719$	5.70558e9	0.572465	0.286232	−	0.958160i	$-0.407597\pi$
$719$	5.70558e9	0.572465	0.286232	+	0.958160i	$0.407597\pi$
$720$	0	0
$721$	3.94803e9	0.392290
$722$	0	0
$723$	1.15135e10	1.13298
$724$	0	0
$725$	−1.72586e10	−1.68199
$726$	0	0
$727$	−5.74555e9	−0.554576	−0.277288	−	0.960787i	$-0.589436\pi$
$727$	−5.74555e9	−0.554576	−0.277288	+	0.960787i	$0.589436\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.85190e10	−1.75350
$732$	0	0
$733$	−1.26217e10	−1.18373	−0.591867	−	0.806036i	$-0.701609\pi$
$733$	−1.26217e10	−1.18373	−0.591867	+	0.806036i	$0.701609\pi$
$734$	0	0
$735$	2.37169e9	0.220319
$736$	0	0
$737$	1.35077e10	1.24293
$738$	0	0
$739$	1.52125e9	0.138658	0.0693289	−	0.997594i	$-0.477914\pi$
$739$	1.52125e9	0.138658	0.0693289	+	0.997594i	$0.477914\pi$
$740$	0	0
$741$	−1.99740e9	−0.180344
$742$	0	0
$743$	1.67640e10	1.49940	0.749701	−	0.661777i	$-0.230197\pi$
$743$	1.67640e10	1.49940	0.749701	+	0.661777i	$0.230197\pi$
$744$	0	0
$745$	−3.27376e10	−2.90068
$746$	0	0
$747$	−2.40681e9	−0.211261
$748$	0	0
$749$	1.09224e10	0.949801
$750$	0	0
$751$	1.41040e10	1.21507	0.607536	−	0.794292i	$-0.292158\pi$
$751$	1.41040e10	1.21507	0.607536	+	0.794292i	$0.292158\pi$
$752$	0	0
$753$	−1.30862e10	−1.11694
$754$	0	0
$755$	2.54931e10	2.15580
$756$	0	0
$757$	−1.24712e10	−1.04489	−0.522446	−	0.852672i	$-0.674980\pi$
$757$	−1.24712e10	−1.04489	−0.522446	+	0.852672i	$0.674980\pi$
$758$	0	0
$759$	5.39533e9	0.447890
$760$	0	0
$761$	−2.36111e9	−0.194209	−0.0971045	−	0.995274i	$-0.530958\pi$
$761$	−2.36111e9	−0.194209	−0.0971045	+	0.995274i	$0.530958\pi$
$762$	0	0
$763$	2.46311e9	0.200746
$764$	0	0
$765$	−7.63281e9	−0.616410
$766$	0	0
$767$	1.60788e9	0.128667
$768$	0	0
$769$	1.72321e9	0.136646	0.0683228	−	0.997663i	$-0.478235\pi$
$769$	1.72321e9	0.136646	0.0683228	+	0.997663i	$0.478235\pi$
$770$	0	0
$771$	−5.97987e9	−0.469896
$772$	0	0
$773$	−1.63809e10	−1.27558	−0.637792	−	0.770209i	$-0.720152\pi$
$773$	−1.63809e10	−1.27558	−0.637792	+	0.770209i	$0.720152\pi$
$774$	0	0
$775$	1.34951e10	1.04140
$776$	0	0
$777$	−8.20729e9	−0.627663
$778$	0	0
$779$	−1.34383e10	−1.01851
$780$	0	0
$781$	4.17193e10	3.13370
$782$	0	0
$783$	2.22582e9	0.165700
$784$	0	0
$785$	1.60794e10	1.18639
$786$	0	0
$787$	−6.69903e9	−0.489892	−0.244946	−	0.969537i	$-0.578770\pi$
$787$	−6.69903e9	−0.489892	−0.244946	+	0.969537i	$0.578770\pi$
$788$	0	0
$789$	1.82842e9	0.132528
$790$	0	0
$791$	−8.35748e9	−0.600423
$792$	0	0
$793$	1.39051e9	0.0990187
$794$	0	0
$795$	−1.48593e10	−1.04885
$796$	0	0
$797$	−8.74251e9	−0.611691	−0.305845	−	0.952081i	$-0.598939\pi$
$797$	−8.74251e9	−0.611691	−0.305845	+	0.952081i	$0.598939\pi$
$798$	0	0
$799$	−2.04839e10	−1.42069
$800$	0	0
$801$	5.61615e9	0.386122
$802$	0	0
$803$	7.51121e9	0.511924
$804$	0	0
$805$	1.00709e10	0.680426
$806$	0	0
$807$	7.26201e9	0.486407
$808$	0	0
$809$	−2.57182e10	−1.70773	−0.853867	−	0.520492i	$-0.825749\pi$
$809$	−2.57182e10	−1.70773	−0.853867	+	0.520492i	$0.825749\pi$
$810$	0	0
$811$	−1.61196e9	−0.106116	−0.0530579	−	0.998591i	$-0.516897\pi$
$811$	−1.61196e9	−0.106116	−0.0530579	+	0.998591i	$0.516897\pi$
$812$	0	0
$813$	1.43895e10	0.939137
$814$	0	0
$815$	−2.08220e10	−1.34732
$816$	0	0
$817$	2.86086e10	1.83535
$818$	0	0
$819$	−1.28197e9	−0.0815425
$820$	0	0
$821$	2.02299e10	1.27583	0.637914	−	0.770108i	$-0.279798\pi$
$821$	2.02299e10	1.27583	0.637914	+	0.770108i	$0.279798\pi$
$822$	0	0
$823$	−3.78840e9	−0.236895	−0.118448	−	0.992960i	$-0.537792\pi$
$823$	−3.78840e9	−0.236895	−0.118448	+	0.992960i	$0.537792\pi$
$824$	0	0
$825$	−3.14372e10	−1.94919
$826$	0	0
$827$	−7.14220e9	−0.439099	−0.219550	−	0.975601i	$-0.570459\pi$
$827$	−7.14220e9	−0.439099	−0.219550	+	0.975601i	$0.570459\pi$
$828$	0	0
$829$	−2.06083e10	−1.25632	−0.628161	−	0.778083i	$-0.716192\pi$
$829$	−2.06083e10	−1.25632	−0.628161	+	0.778083i	$0.716192\pi$
$830$	0	0
$831$	1.06280e9	0.0642462
$832$	0	0
$833$	3.98585e9	0.238926
$834$	0	0
$835$	1.27127e10	0.755677
$836$	0	0
$837$	−1.74044e9	−0.102594
$838$	0	0
$839$	2.32247e9	0.135764	0.0678819	−	0.997693i	$-0.478376\pi$
$839$	2.32247e9	0.135764	0.0678819	+	0.997693i	$0.478376\pi$
$840$	0	0
$841$	−4.46205e9	−0.258672
$842$	0	0
$843$	−1.41781e10	−0.815117
$844$	0	0
$845$	2.31859e9	0.132198
$846$	0	0
$847$	−3.09891e10	−1.75233
$848$	0	0
$849$	−6.48733e9	−0.363822
$850$	0	0
$851$	9.94713e9	0.553279
$852$	0	0
$853$	−3.07680e10	−1.69738	−0.848688	−	0.528894i	$-0.822607\pi$
$853$	−3.07680e10	−1.69738	−0.848688	+	0.528894i	$0.822607\pi$
$854$	0	0
$855$	1.17914e10	0.645183
$856$	0	0
$857$	1.85601e10	1.00727	0.503637	−	0.863915i	$-0.331995\pi$
$857$	1.85601e10	1.00727	0.503637	+	0.863915i	$0.331995\pi$
$858$	0	0
$859$	4.84997e9	0.261074	0.130537	−	0.991443i	$-0.458330\pi$
$859$	4.84997e9	0.261074	0.130537	+	0.991443i	$0.458330\pi$
$860$	0	0
$861$	−8.62496e9	−0.460517
$862$	0	0
$863$	1.93700e10	1.02587	0.512933	−	0.858429i	$-0.328559\pi$
$863$	1.93700e10	1.02587	0.512933	+	0.858429i	$0.328559\pi$
$864$	0	0
$865$	2.99990e10	1.57598
$866$	0	0
$867$	−1.74855e9	−0.0911196
$868$	0	0
$869$	4.81768e10	2.49040
$870$	0	0
$871$	3.88991e9	0.199469
$872$	0	0
$873$	−1.01478e9	−0.0516205
$874$	0	0
$875$	−2.86420e10	−1.44536
$876$	0	0
$877$	2.54274e10	1.27293	0.636464	−	0.771307i	$-0.280396\pi$
$877$	2.54274e10	1.27293	0.636464	+	0.771307i	$0.280396\pi$
$878$	0	0
$879$	−1.08230e10	−0.537509
$880$	0	0
$881$	2.55758e10	1.26013	0.630063	−	0.776544i	$-0.283029\pi$
$881$	2.55758e10	1.26013	0.630063	+	0.776544i	$0.283029\pi$
$882$	0	0
$883$	1.41560e10	0.691955	0.345977	−	0.938243i	$-0.387547\pi$
$883$	1.41560e10	0.691955	0.345977	+	0.938243i	$0.387547\pi$
$884$	0	0
$885$	−9.49185e9	−0.460309
$886$	0	0
$887$	3.99228e9	0.192083	0.0960413	−	0.995377i	$-0.469382\pi$
$887$	3.99228e9	0.192083	0.0960413	+	0.995377i	$0.469382\pi$
$888$	0	0
$889$	2.37100e10	1.13181
$890$	0	0
$891$	4.05441e9	0.192024
$892$	0	0
$893$	3.16441e10	1.48700
$894$	0	0
$895$	2.04785e10	0.954810
$896$	0	0
$897$	1.55373e9	0.0718790
$898$	0	0
$899$	−9.99922e9	−0.458994
$900$	0	0
$901$	−2.49725e10	−1.13743
$902$	0	0
$903$	1.83615e10	0.829852
$904$	0	0
$905$	6.88778e10	3.08894
$906$	0	0
$907$	8.00293e9	0.356142	0.178071	−	0.984018i	$-0.443014\pi$
$907$	8.00293e9	0.356142	0.178071	+	0.984018i	$0.443014\pi$
$908$	0	0
$909$	1.13809e10	0.502577
$910$	0	0
$911$	9.08756e9	0.398229	0.199114	−	0.979976i	$-0.436193\pi$
$911$	9.08756e9	0.398229	0.199114	+	0.979976i	$0.436193\pi$
$912$	0	0
$913$	−2.51876e10	−1.09531
$914$	0	0
$915$	−8.20865e9	−0.354240
$916$	0	0
$917$	3.79585e10	1.62561
$918$	0	0
$919$	−1.91516e10	−0.813955	−0.406977	−	0.913438i	$-0.633417\pi$
$919$	−1.91516e10	−0.813955	−0.406977	+	0.913438i	$0.633417\pi$
$920$	0	0
$921$	1.18207e10	0.498579
$922$	0	0
$923$	1.20142e10	0.502908
$924$	0	0
$925$	−5.79593e10	−2.40784
$926$	0	0
$927$	−3.59574e9	−0.148255
$928$	0	0
$929$	4.20832e10	1.72208	0.861040	−	0.508537i	$-0.169813\pi$
$929$	4.20832e10	1.72208	0.861040	+	0.508537i	$0.169813\pi$
$930$	0	0
$931$	−6.15745e9	−0.250079
$932$	0	0
$933$	1.99679e9	0.0804907
$934$	0	0
$935$	−7.98785e10	−3.19587
$936$	0	0
$937$	−1.76568e10	−0.701170	−0.350585	−	0.936531i	$-0.614017\pi$
$937$	−1.76568e10	−0.701170	−0.350585	+	0.936531i	$0.614017\pi$
$938$	0	0
$939$	3.32029e9	0.130872
$940$	0	0
$941$	3.42121e10	1.33849	0.669246	−	0.743041i	$-0.266617\pi$
$941$	3.42121e10	1.33849	0.669246	+	0.743041i	$0.266617\pi$
$942$	0	0
$943$	1.04533e10	0.405942
$944$	0	0
$945$	7.56791e9	0.291719
$946$	0	0
$947$	−1.52599e10	−0.583883	−0.291942	−	0.956436i	$-0.594301\pi$
$947$	−1.52599e10	−0.583883	−0.291942	+	0.956436i	$0.594301\pi$
$948$	0	0
$949$	2.16305e9	0.0821553
$950$	0	0
$951$	9.27827e9	0.349812
$952$	0	0
$953$	6.96741e9	0.260763	0.130382	−	0.991464i	$-0.458380\pi$
$953$	6.96741e9	0.260763	0.130382	+	0.991464i	$0.458380\pi$
$954$	0	0
$955$	6.24098e9	0.231868
$956$	0	0
$957$	2.32935e10	0.859098
$958$	0	0
$959$	2.18608e10	0.800388
$960$	0	0
$961$	−1.96939e10	−0.715813
$962$	0	0
$963$	−9.94778e9	−0.358950
$964$	0	0
$965$	3.49036e10	1.25033
$966$	0	0
$967$	3.60920e10	1.28357	0.641783	−	0.766886i	$-0.278195\pi$
$967$	3.60920e10	1.28357	0.641783	+	0.766886i	$0.278195\pi$
$968$	0	0
$969$	1.98166e10	0.699673
$970$	0	0
$971$	−5.48838e8	−0.0192388	−0.00961938	−	0.999954i	$-0.503062\pi$
$971$	−5.48838e8	−0.0192388	−0.00961938	+	0.999954i	$0.503062\pi$
$972$	0	0
$973$	−2.32963e10	−0.810758
$974$	0	0
$975$	−9.05317e9	−0.312813
$976$	0	0
$977$	−4.65666e10	−1.59751	−0.798755	−	0.601657i	$-0.794508\pi$
$977$	−4.65666e10	−1.59751	−0.798755	+	0.601657i	$0.794508\pi$
$978$	0	0
$979$	5.87738e10	2.00191
$980$	0	0
$981$	−2.24332e9	−0.0758663
$982$	0	0
$983$	3.55800e10	1.19473	0.597363	−	0.801971i	$-0.296215\pi$
$983$	3.55800e10	1.19473	0.597363	+	0.801971i	$0.296215\pi$
$984$	0	0
$985$	−7.12064e10	−2.37406
$986$	0	0
$987$	2.03097e10	0.672348
$988$	0	0
$989$	−2.22539e10	−0.731507
$990$	0	0
$991$	2.33963e10	0.763642	0.381821	−	0.924236i	$-0.375297\pi$
$991$	2.33963e10	0.763642	0.381821	+	0.924236i	$0.375297\pi$
$992$	0	0
$993$	−3.36721e9	−0.109131
$994$	0	0
$995$	−3.18056e10	−1.02358
$996$	0	0
$997$	−3.75614e10	−1.20035	−0.600176	−	0.799868i	$-0.704903\pi$
$997$	−3.75614e10	−1.20035	−0.600176	+	0.799868i	$0.704903\pi$
$998$	0	0
$999$	7.47493e9	0.237207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	156.8.a.c.1.3	✓	3
3.2	odd	2		468.8.a.d.1.1		3
4.3	odd	2		624.8.a.m.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.8.a.c.1.3	✓	3	1.1	even	1	trivial
468.8.a.d.1.1		3	3.2	odd	2
624.8.a.m.1.3		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	156.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +480.358 q^{5} -800.424 q^{7} +729.000 q^{9} +7629.09 q^{11} +2197.00 q^{13} -12969.7 q^{15} -21796.8 q^{17} +33672.2 q^{19} +21611.5 q^{21} -26192.8 q^{23} +152618. q^{25} -19683.0 q^{27} -113083. q^{29} +88423.6 q^{31} -205985. q^{33} -384490. q^{35} -379766. q^{37} -59319.0 q^{39} -399092. q^{41} +849618. q^{43} +350181. q^{45} +939767. q^{47} -182864. q^{49} +588513. q^{51} +1.14570e6 q^{53} +3.66469e6 q^{55} -909150. q^{57} +731851. q^{59} +632912. q^{61} -583509. q^{63} +1.05535e6 q^{65} +1.77055e6 q^{67} +707205. q^{69} +5.46845e6 q^{71} +984549. q^{73} -4.12070e6 q^{75} -6.10651e6 q^{77} +6.31489e6 q^{79} +531441. q^{81} -3.30152e6 q^{83} -1.04703e7 q^{85} +3.05325e6 q^{87} +7.70391e6 q^{89} -1.75853e6 q^{91} -2.38744e6 q^{93} +1.61747e7 q^{95} -1.39202e6 q^{97} +5.56160e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 81 q^{3} + 524 q^{5} + 722 q^{7} + 2187 q^{9} + 6752 q^{11} + 6591 q^{13} - 14148 q^{15} - 15418 q^{17} + 1826 q^{19} - 19494 q^{21} + 16536 q^{23} + 10513 q^{25} - 59049 q^{27} - 151194 q^{29} + 46338 q^{31}+ \cdots + 4922208 q^{99}+O(q^{100}) \) 3 * q - 81 * q^3 + 524 * q^5 + 722 * q^7 + 2187 * q^9 + 6752 * q^11 + 6591 * q^13 - 14148 * q^15 - 15418 * q^17 + 1826 * q^19 - 19494 * q^21 + 16536 * q^23 + 10513 * q^25 - 59049 * q^27 - 151194 * q^29 + 46338 * q^31 - 182304 * q^33 - 223864 * q^35 - 112378 * q^37 - 177957 * q^39 + 324944 * q^41 + 1282940 * q^43 + 381996 * q^45 + 917072 * q^47 + 559359 * q^49 + 416286 * q^51 + 2688314 * q^53 + 4049992 * q^55 - 49302 * q^57 + 3176620 * q^59 + 2165222 * q^61 + 526338 * q^63 + 1151228 * q^65 + 6699990 * q^67 - 446472 * q^69 + 7734556 * q^71 + 5784690 * q^73 - 283851 * q^75 - 2868232 * q^77 + 8426640 * q^79 + 1594323 * q^81 + 4126284 * q^83 - 8292584 * q^85 + 4082238 * q^87 + 5207756 * q^89 + 1586234 * q^91 - 1251126 * q^93 + 18113000 * q^95 - 7634614 * q^97 + 4922208 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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