LMFDB - Embedded newform 324.8.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,8,Mod(1,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("324.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	324.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:	$101.212748257$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 1289x^{5} + 4994x^{4} + 496633x^{3} - 2291461x^{2} - 56851263x + 373225328 $ x^7 - 3x^6 - 1289x^5 + 4994x^4 + 496633x^3 - 2291461x^2 - 56851263x + 373225328
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{15} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-17.3440$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+114.920 q^{5} -1474.10 q^{7} +O(q^{10})$	$q+114.920 q^{5} -1474.10 q^{7} -352.372 q^{11} -112.314 q^{13} +32247.7 q^{17} +17076.5 q^{19} +36946.8 q^{23} -64918.5 q^{25} +183778. q^{29} +79382.6 q^{31} -169403. q^{35} -510721. q^{37} -444591. q^{41} -71944.9 q^{43} +753882. q^{47} +1.34944e6 q^{49} -2.03509e6 q^{53} -40494.5 q^{55} -2.17034e6 q^{59} -2.03715e6 q^{61} -12907.0 q^{65} +1.72117e6 q^{67} +3.11124e6 q^{71} -730423. q^{73} +519433. q^{77} +1.08536e6 q^{79} +5.65476e6 q^{83} +3.70589e6 q^{85} -9.10792e6 q^{89} +165562. q^{91} +1.96242e6 q^{95} +5.45479e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 321 q^{5} + 83 q^{7}+O(q^{10}) $ 7 * q - 321 * q^5 + 83 * q^7	$ 7 q - 321 q^{5} + 83 q^{7} + 111 q^{11} + 1847 q^{13} - 48 q^{17} + 10124 q^{19} - 19119 q^{23} + 73378 q^{25} - 6045 q^{29} + 153089 q^{31} + 13713 q^{35} + 69674 q^{37} - 446631 q^{41} + 384347 q^{43} - 298413 q^{47} + 351876 q^{49} - 454038 q^{53} - 1263483 q^{55} - 2619543 q^{59} + 146231 q^{61} - 2535735 q^{65} - 1637419 q^{67} - 4353492 q^{71} - 2132260 q^{73} - 9785451 q^{77} - 2402185 q^{79} - 12936357 q^{83} + 1015002 q^{85} - 19684830 q^{89} - 492203 q^{91} - 22685196 q^{95} + 2853257 q^{97}+O(q^{100}) $ 7 * q - 321 * q^5 + 83 * q^7 + 111 * q^11 + 1847 * q^13 - 48 * q^17 + 10124 * q^19 - 19119 * q^23 + 73378 * q^25 - 6045 * q^29 + 153089 * q^31 + 13713 * q^35 + 69674 * q^37 - 446631 * q^41 + 384347 * q^43 - 298413 * q^47 + 351876 * q^49 - 454038 * q^53 - 1263483 * q^55 - 2619543 * q^59 + 146231 * q^61 - 2535735 * q^65 - 1637419 * q^67 - 4353492 * q^71 - 2132260 * q^73 - 9785451 * q^77 - 2402185 * q^79 - 12936357 * q^83 + 1015002 * q^85 - 19684830 * q^89 - 492203 * q^91 - 22685196 * q^95 + 2853257 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	114.920	0.411149	0.205574	−	0.978641i	$-0.434094\pi$
$5$	114.920	0.411149	0.205574	+	0.978641i	$0.434094\pi$
$6$	0	0
$7$	−1474.10	−1.62437	−0.812185	−	0.583400i	$-0.801722\pi$
$7$	−1474.10	−1.62437	−0.812185	+	0.583400i	$0.801722\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−352.372	−0.0798228	−0.0399114	−	0.999203i	$-0.512708\pi$
$11$	−352.372	−0.0798228	−0.0399114	+	0.999203i	$0.512708\pi$
$12$	0	0
$13$	−112.314	−0.0141785	−0.00708926	−	0.999975i	$-0.502257\pi$
$13$	−112.314	−0.0141785	−0.00708926	+	0.999975i	$0.502257\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	32247.7	1.59194	0.795972	−	0.605334i	$-0.206960\pi$
$17$	32247.7	1.59194	0.795972	+	0.605334i	$0.206960\pi$
$18$	0	0
$19$	17076.5	0.571163	0.285582	−	0.958354i	$-0.407813\pi$
$19$	17076.5	0.571163	0.285582	+	0.958354i	$0.407813\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	36946.8	0.633183	0.316591	−	0.948562i	$-0.397462\pi$
$23$	36946.8	0.633183	0.316591	+	0.948562i	$0.397462\pi$
$24$	0	0
$25$	−64918.5	−0.830957
$26$	0	0
$27$	0	0
$28$	0	0
$29$	183778.	1.39927	0.699634	−	0.714501i	$-0.253346\pi$
$29$	183778.	1.39927	0.699634	+	0.714501i	$0.253346\pi$
$30$	0	0
$31$	79382.6	0.478585	0.239293	−	0.970948i	$-0.423085\pi$
$31$	79382.6	0.478585	0.239293	+	0.970948i	$0.423085\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−169403.	−0.667858
$36$	0	0
$37$	−510721.	−1.65759	−0.828796	−	0.559551i	$-0.810973\pi$
$37$	−510721.	−1.65759	−0.828796	+	0.559551i	$0.810973\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−444591.	−1.00744	−0.503718	−	0.863868i	$-0.668035\pi$
$41$	−444591.	−1.00744	−0.503718	+	0.863868i	$0.668035\pi$
$42$	0	0
$43$	−71944.9	−0.137994	−0.0689971	−	0.997617i	$-0.521980\pi$
$43$	−71944.9	−0.137994	−0.0689971	+	0.997617i	$0.521980\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	753882.	1.05916	0.529579	−	0.848261i	$-0.322350\pi$
$47$	753882.	1.05916	0.529579	+	0.848261i	$0.322350\pi$
$48$	0	0
$49$	1.34944e6	1.63858
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.03509e6	−1.87767	−0.938833	−	0.344373i	$-0.888091\pi$
$53$	−2.03509e6	−1.87767	−0.938833	+	0.344373i	$0.888091\pi$
$54$	0	0
$55$	−40494.5	−0.0328191
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.17034e6	−1.37577	−0.687884	−	0.725821i	$-0.741460\pi$
$59$	−2.17034e6	−1.37577	−0.687884	+	0.725821i	$0.741460\pi$
$60$	0	0
$61$	−2.03715e6	−1.14913	−0.574563	−	0.818460i	$-0.694828\pi$
$61$	−2.03715e6	−1.14913	−0.574563	+	0.818460i	$0.694828\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12907.0	−0.00582948
$66$	0	0
$67$	1.72117e6	0.699138	0.349569	−	0.936911i	$-0.386328\pi$
$67$	1.72117e6	0.699138	0.349569	+	0.936911i	$0.386328\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.11124e6	1.03164	0.515821	−	0.856696i	$-0.327487\pi$
$71$	3.11124e6	1.03164	0.515821	+	0.856696i	$0.327487\pi$
$72$	0	0
$73$	−730423.	−0.219758	−0.109879	−	0.993945i	$-0.535046\pi$
$73$	−730423.	−0.219758	−0.109879	+	0.993945i	$0.535046\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	519433.	0.129662
$78$	0	0
$79$	1.08536e6	0.247673	0.123837	−	0.992303i	$-0.460480\pi$
$79$	1.08536e6	0.247673	0.123837	+	0.992303i	$0.460480\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.65476e6	1.08553	0.542764	−	0.839885i	$-0.317378\pi$
$83$	5.65476e6	1.08553	0.542764	+	0.839885i	$0.317378\pi$
$84$	0	0
$85$	3.70589e6	0.654526
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.10792e6	−1.36948	−0.684738	−	0.728790i	$-0.740083\pi$
$89$	−9.10792e6	−1.36948	−0.684738	+	0.728790i	$0.740083\pi$
$90$	0	0
$91$	165562.	0.0230312
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.96242e6	0.234833
$96$	0	0
$97$	5.45479e6	0.606844	0.303422	−	0.952856i	$-0.401871\pi$
$97$	5.45479e6	0.606844	0.303422	+	0.952856i	$0.401871\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.42605e7	−1.37724	−0.688620	−	0.725123i	$-0.741783\pi$
$101$	−1.42605e7	−1.37724	−0.688620	+	0.725123i	$0.741783\pi$
$102$	0	0
$103$	−1.70734e7	−1.53953	−0.769767	−	0.638326i	$-0.779627\pi$
$103$	−1.70734e7	−1.53953	−0.769767	+	0.638326i	$0.779627\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.58251e6	0.361626	0.180813	−	0.983517i	$-0.442127\pi$
$107$	4.58251e6	0.361626	0.180813	+	0.983517i	$0.442127\pi$
$108$	0	0
$109$	−1.13806e6	−0.0841731	−0.0420865	−	0.999114i	$-0.513401\pi$
$109$	−1.13806e6	−0.0841731	−0.0420865	+	0.999114i	$0.513401\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.39293e7	−0.908143	−0.454072	−	0.890965i	$-0.650029\pi$
$113$	−1.39293e7	−0.908143	−0.454072	+	0.890965i	$0.650029\pi$
$114$	0	0
$115$	4.24591e6	0.260332
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.75365e7	−2.58591
$120$	0	0
$121$	−1.93630e7	−0.993628
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.64385e7	−0.752796
$126$	0	0
$127$	2.65498e7	1.15013	0.575066	−	0.818107i	$-0.304976\pi$
$127$	2.65498e7	1.15013	0.575066	+	0.818107i	$0.304976\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.04200e7	−0.404964	−0.202482	−	0.979286i	$-0.564901\pi$
$131$	−1.04200e7	−0.404964	−0.202482	+	0.979286i	$0.564901\pi$
$132$	0	0
$133$	−2.51725e7	−0.927780
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.27119e7	−1.41914	−0.709572	−	0.704633i	$-0.751112\pi$
$137$	−4.27119e7	−1.41914	−0.709572	+	0.704633i	$0.751112\pi$
$138$	0	0
$139$	2.60656e6	0.0823220	0.0411610	−	0.999153i	$-0.486894\pi$
$139$	2.60656e6	0.0823220	0.0411610	+	0.999153i	$0.486894\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	39576.2	0.00113177
$144$	0	0
$145$	2.11197e7	0.575308
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.39490e7	−0.345454	−0.172727	−	0.984970i	$-0.555258\pi$
$149$	−1.39490e7	−0.345454	−0.172727	+	0.984970i	$0.555258\pi$
$150$	0	0
$151$	7.50274e7	1.77338	0.886688	−	0.462368i	$-0.153000\pi$
$151$	7.50274e7	1.77338	0.886688	+	0.462368i	$0.153000\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.12261e6	0.196770
$156$	0	0
$157$	−2.58820e7	−0.533764	−0.266882	−	0.963729i	$-0.585993\pi$
$157$	−2.58820e7	−0.533764	−0.266882	+	0.963729i	$0.585993\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.44634e7	−1.02852
$162$	0	0
$163$	−1.08241e8	−1.95766	−0.978830	−	0.204677i	$-0.934386\pi$
$163$	−1.08241e8	−1.95766	−0.978830	+	0.204677i	$0.934386\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.10911e7	−0.516569	−0.258285	−	0.966069i	$-0.583157\pi$
$167$	−3.10911e7	−0.516569	−0.258285	+	0.966069i	$0.583157\pi$
$168$	0	0
$169$	−6.27359e7	−0.999799
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.23248e7	1.35568	0.677840	−	0.735209i	$-0.262916\pi$
$173$	9.23248e7	1.35568	0.677840	+	0.735209i	$0.262916\pi$
$174$	0	0
$175$	9.56966e7	1.34978
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.21336e7	−1.07037	−0.535187	−	0.844734i	$-0.679759\pi$
$179$	−8.21336e7	−1.07037	−0.535187	+	0.844734i	$0.679759\pi$
$180$	0	0
$181$	7.30657e7	0.915880	0.457940	−	0.888983i	$-0.348587\pi$
$181$	7.30657e7	0.915880	0.457940	+	0.888983i	$0.348587\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.86918e7	−0.681517
$186$	0	0
$187$	−1.13632e7	−0.127073
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.17476e8	−1.21992	−0.609962	−	0.792430i	$-0.708815\pi$
$191$	−1.17476e8	−1.21992	−0.609962	+	0.792430i	$0.708815\pi$
$192$	0	0
$193$	1.81917e8	1.82147	0.910735	−	0.412992i	$-0.135516\pi$
$193$	1.81917e8	1.82147	0.910735	+	0.412992i	$0.135516\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.42583e7	−0.878390	−0.439195	−	0.898392i	$-0.644736\pi$
$197$	−9.42583e7	−0.878390	−0.439195	+	0.898392i	$0.644736\pi$
$198$	0	0
$199$	−2.47261e7	−0.222418	−0.111209	−	0.993797i	$-0.535472\pi$
$199$	−2.47261e7	−0.222418	−0.111209	+	0.993797i	$0.535472\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.70908e8	−2.27293
$204$	0	0
$205$	−5.10922e7	−0.414206
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.01727e6	−0.0455918
$210$	0	0
$211$	−1.96521e8	−1.44019	−0.720095	−	0.693875i	$-0.755902\pi$
$211$	−1.96521e8	−1.44019	−0.720095	+	0.693875i	$0.755902\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.26788e6	−0.0567361
$216$	0	0
$217$	−1.17018e8	−0.777399
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.62186e6	−0.0225714
$222$	0	0
$223$	−9.85237e7	−0.594941	−0.297470	−	0.954731i	$-0.596143\pi$
$223$	−9.85237e7	−0.594941	−0.297470	+	0.954731i	$0.596143\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.52419e8	−0.864867	−0.432433	−	0.901666i	$-0.642345\pi$
$227$	−1.52419e8	−0.864867	−0.432433	+	0.901666i	$0.642345\pi$
$228$	0	0
$229$	1.14220e8	0.628520	0.314260	−	0.949337i	$-0.398244\pi$
$229$	1.14220e8	0.628520	0.314260	+	0.949337i	$0.398244\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.09636e8	−1.60364	−0.801818	−	0.597569i	$-0.796134\pi$
$233$	−3.09636e8	−1.60364	−0.801818	+	0.597569i	$0.796134\pi$
$234$	0	0
$235$	8.66358e7	0.435472
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.88074e7	−0.136493	−0.0682467	−	0.997668i	$-0.521740\pi$
$239$	−2.88074e7	−0.136493	−0.0682467	+	0.997668i	$0.521740\pi$
$240$	0	0
$241$	−1.38963e8	−0.639499	−0.319749	−	0.947502i	$-0.603599\pi$
$241$	−1.38963e8	−0.639499	−0.319749	+	0.947502i	$0.603599\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.55077e8	0.673699
$246$	0	0
$247$	−1.91792e6	−0.00809824
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.19284e8	1.27444	0.637220	−	0.770682i	$-0.280084\pi$
$251$	3.19284e8	1.27444	0.637220	+	0.770682i	$0.280084\pi$
$252$	0	0
$253$	−1.30190e7	−0.0505424
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.70270e8	1.36067	0.680334	−	0.732902i	$-0.261835\pi$
$257$	3.70270e8	1.36067	0.680334	+	0.732902i	$0.261835\pi$
$258$	0	0
$259$	7.52855e8	2.69254
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.79428e7	0.0608200	0.0304100	−	0.999538i	$-0.490319\pi$
$263$	1.79428e7	0.0608200	0.0304100	+	0.999538i	$0.490319\pi$
$264$	0	0
$265$	−2.33872e8	−0.772000
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.71720e7	0.304374	0.152187	−	0.988352i	$-0.451368\pi$
$269$	9.71720e7	0.304374	0.152187	+	0.988352i	$0.451368\pi$
$270$	0	0
$271$	−1.54239e8	−0.470763	−0.235382	−	0.971903i	$-0.575634\pi$
$271$	−1.54239e8	−0.470763	−0.235382	+	0.971903i	$0.575634\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.28755e7	0.0663293
$276$	0	0
$277$	−1.68510e8	−0.476373	−0.238187	−	0.971219i	$-0.576553\pi$
$277$	−1.68510e8	−0.476373	−0.238187	+	0.971219i	$0.576553\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.92051e8	−1.05407	−0.527036	−	0.849843i	$-0.676697\pi$
$281$	−3.92051e8	−1.05407	−0.527036	+	0.849843i	$0.676697\pi$
$282$	0	0
$283$	−5.41260e8	−1.41956	−0.709780	−	0.704424i	$-0.751206\pi$
$283$	−5.41260e8	−1.41956	−0.709780	+	0.704424i	$0.751206\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.55374e8	1.63645
$288$	0	0
$289$	6.29576e8	1.53428
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.83589e8	1.12316	0.561578	−	0.827424i	$-0.310194\pi$
$293$	4.83589e8	1.12316	0.561578	+	0.827424i	$0.310194\pi$
$294$	0	0
$295$	−2.49414e8	−0.565645
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.14963e6	−0.00897759
$300$	0	0
$301$	1.06054e8	0.224153
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.34108e8	−0.472462
$306$	0	0
$307$	3.21712e8	0.634575	0.317287	−	0.948329i	$-0.397228\pi$
$307$	3.21712e8	0.634575	0.317287	+	0.948329i	$0.397228\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.59142e8	0.865536	0.432768	−	0.901505i	$-0.357537\pi$
$311$	4.59142e8	0.865536	0.432768	+	0.901505i	$0.357537\pi$
$312$	0	0
$313$	2.02200e6	0.00372714	0.00186357	−	0.999998i	$-0.499407\pi$
$313$	2.02200e6	0.00372714	0.00186357	+	0.999998i	$0.499407\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.51130e8	−0.266467	−0.133234	−	0.991085i	$-0.542536\pi$
$317$	−1.51130e8	−0.266467	−0.133234	+	0.991085i	$0.542536\pi$
$318$	0	0
$319$	−6.47583e7	−0.111694
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.50677e8	0.909259
$324$	0	0
$325$	7.29123e6	0.0117817
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.11130e9	−1.72047
$330$	0	0
$331$	3.12554e8	0.473726	0.236863	−	0.971543i	$-0.423881\pi$
$331$	3.12554e8	0.473726	0.236863	+	0.971543i	$0.423881\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.97797e8	0.287450
$336$	0	0
$337$	3.77782e8	0.537696	0.268848	−	0.963183i	$-0.413357\pi$
$337$	3.77782e8	0.537696	0.268848	+	0.963183i	$0.413357\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.79722e7	−0.0382020
$342$	0	0
$343$	−7.75226e8	−1.03729
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.36944e8	1.20382	0.601908	−	0.798565i	$-0.294407\pi$
$347$	9.36944e8	1.20382	0.601908	+	0.798565i	$0.294407\pi$
$348$	0	0
$349$	4.81813e8	0.606721	0.303361	−	0.952876i	$-0.401891\pi$
$349$	4.81813e8	0.606721	0.303361	+	0.952876i	$0.401891\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.86966e8	0.831234	0.415617	−	0.909540i	$-0.363566\pi$
$353$	6.86966e8	0.831234	0.415617	+	0.909540i	$0.363566\pi$
$354$	0	0
$355$	3.57542e8	0.424158
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.80471e8	−0.319932	−0.159966	−	0.987123i	$-0.551138\pi$
$359$	−2.80471e8	−0.319932	−0.159966	+	0.987123i	$0.551138\pi$
$360$	0	0
$361$	−6.02266e8	−0.673773
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.39399e7	−0.0903532
$366$	0	0
$367$	−1.59141e9	−1.68055	−0.840275	−	0.542160i	$-0.817607\pi$
$367$	−1.59141e9	−1.68055	−0.840275	+	0.542160i	$0.817607\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.99994e9	3.05002
$372$	0	0
$373$	4.32539e8	0.431564	0.215782	−	0.976442i	$-0.430770\pi$
$373$	4.32539e8	0.431564	0.215782	+	0.976442i	$0.430770\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.06408e7	−0.0198396
$378$	0	0
$379$	−2.72458e8	−0.257077	−0.128538	−	0.991705i	$-0.541029\pi$
$379$	−2.72458e8	−0.257077	−0.128538	+	0.991705i	$0.541029\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.71823e9	1.56274	0.781368	−	0.624070i	$-0.214522\pi$
$383$	1.71823e9	1.56274	0.781368	+	0.624070i	$0.214522\pi$
$384$	0	0
$385$	5.96930e7	0.0533103
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.56490e9	1.34792	0.673958	−	0.738770i	$-0.264593\pi$
$389$	1.56490e9	1.34792	0.673958	+	0.738770i	$0.264593\pi$
$390$	0	0
$391$	1.19145e9	1.00799
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.24729e8	0.101830
$396$	0	0
$397$	−8.70691e8	−0.698389	−0.349194	−	0.937050i	$-0.613545\pi$
$397$	−8.70691e8	−0.698389	−0.349194	+	0.937050i	$0.613545\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.90460e8	−0.302393	−0.151196	−	0.988504i	$-0.548313\pi$
$401$	−3.90460e8	−0.302393	−0.151196	+	0.988504i	$0.548313\pi$
$402$	0	0
$403$	−8.91574e6	−0.00678563
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.79964e8	0.132314
$408$	0	0
$409$	−1.57719e9	−1.13986	−0.569929	−	0.821694i	$-0.693029\pi$
$409$	−1.57719e9	−1.13986	−0.569929	+	0.821694i	$0.693029\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.19930e9	2.23475
$414$	0	0
$415$	6.49843e8	0.446313
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.66413e8	−0.442583	−0.221291	−	0.975208i	$-0.571027\pi$
$419$	−6.66413e8	−0.442583	−0.221291	+	0.975208i	$0.571027\pi$
$420$	0	0
$421$	−7.79981e8	−0.509444	−0.254722	−	0.967014i	$-0.581984\pi$
$421$	−7.79981e8	−0.509444	−0.254722	+	0.967014i	$0.581984\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.09347e9	−1.32284
$426$	0	0
$427$	3.00296e9	1.86661
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.18039e8	0.251505	0.125752	−	0.992062i	$-0.459866\pi$
$431$	4.18039e8	0.251505	0.125752	+	0.992062i	$0.459866\pi$
$432$	0	0
$433$	2.06750e9	1.22388	0.611938	−	0.790906i	$-0.290390\pi$
$433$	2.06750e9	1.22388	0.611938	+	0.790906i	$0.290390\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.30920e8	0.361650
$438$	0	0
$439$	4.84061e8	0.273070	0.136535	−	0.990635i	$-0.456403\pi$
$439$	4.84061e8	0.273070	0.136535	+	0.990635i	$0.456403\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.44900e8	0.243136	0.121568	−	0.992583i	$-0.461208\pi$
$443$	4.44900e8	0.243136	0.121568	+	0.992583i	$0.461208\pi$
$444$	0	0
$445$	−1.04668e9	−0.563058
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.21561e9	−1.15513	−0.577564	−	0.816345i	$-0.695997\pi$
$449$	−2.21561e9	−1.15513	−0.577564	+	0.816345i	$0.695997\pi$
$450$	0	0
$451$	1.56662e8	0.0804164
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.90263e7	0.00946923
$456$	0	0
$457$	7.31664e8	0.358596	0.179298	−	0.983795i	$-0.442617\pi$
$457$	7.31664e8	0.358596	0.179298	+	0.983795i	$0.442617\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.96678e8	−0.331191	−0.165596	−	0.986194i	$-0.552955\pi$
$461$	−6.96678e8	−0.331191	−0.165596	+	0.986194i	$0.552955\pi$
$462$	0	0
$463$	−3.18278e9	−1.49030	−0.745149	−	0.666898i	$-0.767622\pi$
$463$	−3.18278e9	−1.49030	−0.745149	+	0.666898i	$0.767622\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.15917e9	−0.981019	−0.490509	−	0.871436i	$-0.663189\pi$
$467$	−2.15917e9	−0.981019	−0.490509	+	0.871436i	$0.663189\pi$
$468$	0	0
$469$	−2.53719e9	−1.13566
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.53514e7	0.0110151
$474$	0	0
$475$	−1.10858e9	−0.474612
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.27171e9	−0.528705	−0.264352	−	0.964426i	$-0.585158\pi$
$479$	−1.27171e9	−0.528705	−0.264352	+	0.964426i	$0.585158\pi$
$480$	0	0
$481$	5.73609e7	0.0235022
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.26862e8	0.249503
$486$	0	0
$487$	3.06244e9	1.20148	0.600740	−	0.799444i	$-0.294873\pi$
$487$	3.06244e9	1.20148	0.600740	+	0.799444i	$0.294873\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.40156e8	0.244062	0.122031	−	0.992526i	$-0.461059\pi$
$491$	6.40156e8	0.244062	0.122031	+	0.992526i	$0.461059\pi$
$492$	0	0
$493$	5.92643e9	2.22756
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.58629e9	−1.67577
$498$	0	0
$499$	−2.56246e9	−0.923219	−0.461610	−	0.887083i	$-0.652728\pi$
$499$	−2.56246e9	−0.923219	−0.461610	+	0.887083i	$0.652728\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.38081e8	0.293628	0.146814	−	0.989164i	$-0.453098\pi$
$503$	8.38081e8	0.293628	0.146814	+	0.989164i	$0.453098\pi$
$504$	0	0
$505$	−1.63881e9	−0.566250
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.69994e9	−0.907489	−0.453744	−	0.891132i	$-0.649912\pi$
$509$	−2.69994e9	−0.907489	−0.453744	+	0.891132i	$0.649912\pi$
$510$	0	0
$511$	1.07672e9	0.356968
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.96206e9	−0.632977
$516$	0	0
$517$	−2.65647e8	−0.0845450
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.04292e8	0.125246	0.0626228	−	0.998037i	$-0.480053\pi$
$521$	4.04292e8	0.125246	0.0626228	+	0.998037i	$0.480053\pi$
$522$	0	0
$523$	2.81239e9	0.859645	0.429822	−	0.902913i	$-0.358576\pi$
$523$	2.81239e9	0.859645	0.429822	+	0.902913i	$0.358576\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.55991e9	0.761880
$528$	0	0
$529$	−2.03976e9	−0.599080
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.99337e7	0.0142839
$534$	0	0
$535$	5.26620e8	0.148682
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.75505e8	−0.130796
$540$	0	0
$541$	−4.60374e8	−0.125003	−0.0625015	−	0.998045i	$-0.519908\pi$
$541$	−4.60374e8	−0.125003	−0.0625015	+	0.998045i	$0.519908\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.30786e8	−0.0346077
$546$	0	0
$547$	−5.42090e9	−1.41617	−0.708086	−	0.706126i	$-0.750441\pi$
$547$	−5.42090e9	−1.41617	−0.708086	+	0.706126i	$0.750441\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.13828e9	0.799211
$552$	0	0
$553$	−1.59993e9	−0.402313
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.20349e9	0.540280	0.270140	−	0.962821i	$-0.412930\pi$
$557$	2.20349e9	0.540280	0.270140	+	0.962821i	$0.412930\pi$
$558$	0	0
$559$	8.08040e6	0.00195655
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.40427e9	0.331644	0.165822	−	0.986156i	$-0.446972\pi$
$563$	1.40427e9	0.331644	0.165822	+	0.986156i	$0.446972\pi$
$564$	0	0
$565$	−1.60075e9	−0.373382
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.23706e9	−1.19178	−0.595888	−	0.803067i	$-0.703200\pi$
$569$	−5.23706e9	−1.19178	−0.595888	+	0.803067i	$0.703200\pi$
$570$	0	0
$571$	−1.12562e8	−0.0253027	−0.0126513	−	0.999920i	$-0.504027\pi$
$571$	−1.12562e8	−0.0253027	−0.0126513	+	0.999920i	$0.504027\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.39853e9	−0.526147
$576$	0	0
$577$	1.62505e9	0.352169	0.176085	−	0.984375i	$-0.443657\pi$
$577$	1.62505e9	0.352169	0.176085	+	0.984375i	$0.443657\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.33571e9	−1.76330
$582$	0	0
$583$	7.17109e8	0.149881
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.22611e9	−0.454270	−0.227135	−	0.973863i	$-0.572936\pi$
$587$	−2.22611e9	−0.454270	−0.227135	+	0.973863i	$0.572936\pi$
$588$	0	0
$589$	1.35557e9	0.273350
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.49410e9	−1.67273	−0.836365	−	0.548173i	$-0.815324\pi$
$593$	−8.49410e9	−1.67273	−0.836365	+	0.548173i	$0.815324\pi$
$594$	0	0
$595$	−5.46287e9	−1.06319
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.06535e9	−1.53331	−0.766653	−	0.642061i	$-0.778080\pi$
$599$	−8.06535e9	−1.53331	−0.766653	+	0.642061i	$0.778080\pi$
$600$	0	0
$601$	3.23711e9	0.608271	0.304135	−	0.952629i	$-0.401632\pi$
$601$	3.23711e9	0.608271	0.304135	+	0.952629i	$0.401632\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.22519e9	−0.408529
$606$	0	0
$607$	6.89785e9	1.25185	0.625926	−	0.779882i	$-0.284721\pi$
$607$	6.89785e9	1.25185	0.625926	+	0.779882i	$0.284721\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.46712e7	−0.0150173
$612$	0	0
$613$	−5.89630e8	−0.103387	−0.0516937	−	0.998663i	$-0.516462\pi$
$613$	−5.89630e8	−0.103387	−0.0516937	+	0.998663i	$0.516462\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.20626e9	−0.549542	−0.274771	−	0.961510i	$-0.588602\pi$
$617$	−3.20626e9	−0.549542	−0.274771	+	0.961510i	$0.588602\pi$
$618$	0	0
$619$	−3.11954e8	−0.0528655	−0.0264328	−	0.999651i	$-0.508415\pi$
$619$	−3.11954e8	−0.0528655	−0.0264328	+	0.999651i	$0.508415\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.34260e10	2.22453
$624$	0	0
$625$	3.18265e9	0.521446
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.64696e10	−2.63879
$630$	0	0
$631$	−6.52609e9	−1.03407	−0.517035	−	0.855964i	$-0.672964\pi$
$631$	−6.52609e9	−1.03407	−0.517035	+	0.855964i	$0.672964\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.05109e9	0.472875
$636$	0	0
$637$	−1.51560e8	−0.0232326
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.14242e9	−0.471261	−0.235630	−	0.971843i	$-0.575715\pi$
$641$	−3.14242e9	−0.471261	−0.235630	+	0.971843i	$0.575715\pi$
$642$	0	0
$643$	8.13467e9	1.20671	0.603353	−	0.797474i	$-0.293831\pi$
$643$	8.13467e9	1.20671	0.603353	+	0.797474i	$0.293831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.51431e9	−0.364968	−0.182484	−	0.983209i	$-0.558414\pi$
$647$	−2.51431e9	−0.364968	−0.182484	+	0.983209i	$0.558414\pi$
$648$	0	0
$649$	7.64766e8	0.109818
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.11575e10	−1.56809	−0.784045	−	0.620703i	$-0.786847\pi$
$653$	−1.11575e10	−1.56809	−0.784045	+	0.620703i	$0.786847\pi$
$654$	0	0
$655$	−1.19746e9	−0.166501
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.71729e9	−0.233746	−0.116873	−	0.993147i	$-0.537287\pi$
$659$	−1.71729e9	−0.233746	−0.116873	+	0.993147i	$0.537287\pi$
$660$	0	0
$661$	5.07302e9	0.683221	0.341610	−	0.939842i	$-0.389028\pi$
$661$	5.07302e9	0.683221	0.341610	+	0.939842i	$0.389028\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.89281e9	−0.381456
$666$	0	0
$667$	6.79001e9	0.885993
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.17833e8	0.0917265
$672$	0	0
$673$	−1.03813e10	−1.31280	−0.656402	−	0.754411i	$-0.727923\pi$
$673$	−1.03813e10	−1.31280	−0.656402	+	0.754411i	$0.727923\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.13301e9	−0.511925	−0.255962	−	0.966687i	$-0.582392\pi$
$677$	−4.13301e9	−0.511925	−0.255962	+	0.966687i	$0.582392\pi$
$678$	0	0
$679$	−8.04093e9	−0.985739
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.46699e10	1.76179	0.880894	−	0.473314i	$-0.156943\pi$
$683$	1.46699e10	1.76179	0.880894	+	0.473314i	$0.156943\pi$
$684$	0	0
$685$	−4.90843e9	−0.583479
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.28568e8	0.0266225
$690$	0	0
$691$	9.96753e9	1.14925	0.574625	−	0.818417i	$-0.305148\pi$
$691$	9.96753e9	1.14925	0.574625	+	0.818417i	$0.305148\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.99545e8	0.0338466
$696$	0	0
$697$	−1.43370e10	−1.60378
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.24856e9	0.685121	0.342560	−	0.939496i	$-0.388706\pi$
$701$	6.24856e9	0.685121	0.342560	+	0.939496i	$0.388706\pi$
$702$	0	0
$703$	−8.72130e9	−0.946755
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.10214e10	2.23715
$708$	0	0
$709$	3.02300e9	0.318549	0.159275	−	0.987234i	$-0.449084\pi$
$709$	3.02300e9	0.318549	0.159275	+	0.987234i	$0.449084\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.93293e9	0.303032
$714$	0	0
$715$	4.54808e6	0.000465325 0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.74337e9	0.877259	0.438629	−	0.898668i	$-0.355464\pi$
$719$	8.74337e9	0.877259	0.438629	+	0.898668i	$0.355464\pi$
$720$	0	0
$721$	2.51679e10	2.50077
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.19306e10	−1.16273
$726$	0	0
$727$	−2.54321e9	−0.245477	−0.122739	−	0.992439i	$-0.539168\pi$
$727$	−2.54321e9	−0.245477	−0.122739	+	0.992439i	$0.539168\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.32006e9	−0.219679
$732$	0	0
$733$	1.17359e10	1.10066	0.550328	−	0.834949i	$-0.314503\pi$
$733$	1.17359e10	1.10066	0.550328	+	0.834949i	$0.314503\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.06494e8	−0.0558072
$738$	0	0
$739$	1.14571e10	1.04429	0.522143	−	0.852858i	$-0.325133\pi$
$739$	1.14571e10	1.04429	0.522143	+	0.852858i	$0.325133\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.16806e10	1.04473	0.522366	−	0.852722i	$-0.325050\pi$
$743$	1.16806e10	1.04473	0.522366	+	0.852722i	$0.325050\pi$
$744$	0	0
$745$	−1.60301e9	−0.142033
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.75509e9	−0.587415
$750$	0	0
$751$	−1.64854e10	−1.42024	−0.710118	−	0.704083i	$-0.751358\pi$
$751$	−1.64854e10	−1.42024	−0.710118	+	0.704083i	$0.751358\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.62212e9	0.729121
$756$	0	0
$757$	−1.38249e10	−1.15831	−0.579157	−	0.815216i	$-0.696618\pi$
$757$	−1.38249e10	−1.15831	−0.579157	+	0.815216i	$0.696618\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.37944e10	1.95717	0.978584	−	0.205848i	$-0.0659952\pi$
$761$	2.37944e10	1.95717	0.978584	+	0.205848i	$0.0659952\pi$
$762$	0	0
$763$	1.67762e9	0.136728
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.43758e8	0.0195063
$768$	0	0
$769$	−8.28185e7	−0.00656727	−0.00328364	−	0.999995i	$-0.501045\pi$
$769$	−8.28185e7	−0.00656727	−0.00328364	+	0.999995i	$0.501045\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.08224e9	−0.473626	−0.236813	−	0.971555i	$-0.576103\pi$
$773$	−6.08224e9	−0.473626	−0.236813	+	0.971555i	$0.576103\pi$
$774$	0	0
$775$	−5.15340e9	−0.397683
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.59204e9	−0.575410
$780$	0	0
$781$	−1.09631e9	−0.0823486
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.97435e9	−0.219456
$786$	0	0
$787$	−2.06191e10	−1.50785	−0.753924	−	0.656962i	$-0.771841\pi$
$787$	−2.06191e10	−1.50785	−0.753924	+	0.656962i	$0.771841\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.05332e10	1.47516
$792$	0	0
$793$	2.28799e8	0.0162929
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.25105e9	−0.297435	−0.148718	−	0.988880i	$-0.547515\pi$
$797$	−4.25105e9	−0.297435	−0.148718	+	0.988880i	$0.547515\pi$
$798$	0	0
$799$	2.43110e10	1.68612
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.57381e8	0.0175417
$804$	0	0
$805$	−6.25891e9	−0.422876
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.46466e10	−0.972561	−0.486281	−	0.873803i	$-0.661647\pi$
$809$	−1.46466e10	−0.972561	−0.486281	+	0.873803i	$0.661647\pi$
$810$	0	0
$811$	−2.59479e10	−1.70816	−0.854080	−	0.520142i	$-0.825879\pi$
$811$	−2.59479e10	−1.70816	−0.854080	+	0.520142i	$0.825879\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.24391e10	−0.804889
$816$	0	0
$817$	−1.22856e9	−0.0788171
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.27079e10	0.801442	0.400721	−	0.916200i	$-0.368760\pi$
$821$	1.27079e10	0.801442	0.400721	+	0.916200i	$0.368760\pi$
$822$	0	0
$823$	−2.90967e9	−0.181947	−0.0909735	−	0.995853i	$-0.528998\pi$
$823$	−2.90967e9	−0.181947	−0.0909735	+	0.995853i	$0.528998\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.67366e8	0.0102896	0.00514480	−	0.999987i	$-0.498362\pi$
$827$	1.67366e8	0.0102896	0.00514480	+	0.999987i	$0.498362\pi$
$828$	0	0
$829$	1.53623e10	0.936514	0.468257	−	0.883592i	$-0.344882\pi$
$829$	1.53623e10	0.936514	0.468257	+	0.883592i	$0.344882\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.35163e10	2.60852
$834$	0	0
$835$	−3.57298e9	−0.212387
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.37095e9	−0.547793	−0.273897	−	0.961759i	$-0.588313\pi$
$839$	−9.37095e9	−0.547793	−0.273897	+	0.961759i	$0.588313\pi$
$840$	0	0
$841$	1.65246e10	0.957953
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.20958e9	−0.411066
$846$	0	0
$847$	2.85431e10	1.61402
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.88695e10	−1.04956
$852$	0	0
$853$	−9.77437e9	−0.539221	−0.269611	−	0.962969i	$-0.586895\pi$
$853$	−9.77437e9	−0.539221	−0.269611	+	0.962969i	$0.586895\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.58079e10	0.857908	0.428954	−	0.903326i	$-0.358882\pi$
$857$	1.58079e10	0.857908	0.428954	+	0.903326i	$0.358882\pi$
$858$	0	0
$859$	2.95850e10	1.59256	0.796281	−	0.604927i	$-0.206798\pi$
$859$	2.95850e10	1.59256	0.796281	+	0.604927i	$0.206798\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.57761e10	1.36514	0.682572	−	0.730818i	$-0.260861\pi$
$863$	2.57761e10	1.36514	0.682572	+	0.730818i	$0.260861\pi$
$864$	0	0
$865$	1.06099e10	0.557386
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.82450e8	−0.0197700
$870$	0	0
$871$	−1.93311e8	−0.00991274
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.42321e10	1.22282
$876$	0	0
$877$	1.06134e10	0.531320	0.265660	−	0.964067i	$-0.414410\pi$
$877$	1.06134e10	0.531320	0.265660	+	0.964067i	$0.414410\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.38830e10	0.684018	0.342009	−	0.939697i	$-0.388893\pi$
$881$	1.38830e10	0.684018	0.342009	+	0.939697i	$0.388893\pi$
$882$	0	0
$883$	1.01691e10	0.497073	0.248537	−	0.968622i	$-0.420050\pi$
$883$	1.01691e10	0.497073	0.248537	+	0.968622i	$0.420050\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.76067e10	0.847119	0.423559	−	0.905868i	$-0.360780\pi$
$887$	1.76067e10	0.847119	0.423559	+	0.905868i	$0.360780\pi$
$888$	0	0
$889$	−3.91371e10	−1.86824
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.28736e10	0.604952
$894$	0	0
$895$	−9.43876e9	−0.440083
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.45888e10	0.669669
$900$	0	0
$901$	−6.56270e10	−2.98914
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.39668e9	0.376563
$906$	0	0
$907$	7.70535e9	0.342900	0.171450	−	0.985193i	$-0.445155\pi$
$907$	7.70535e9	0.342900	0.171450	+	0.985193i	$0.445155\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.89399e10	1.26819	0.634093	−	0.773256i	$-0.281373\pi$
$911$	2.89399e10	1.26819	0.634093	+	0.773256i	$0.281373\pi$
$912$	0	0
$913$	−1.99258e9	−0.0866499
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.53601e10	0.657812
$918$	0	0
$919$	−5.25163e9	−0.223198	−0.111599	−	0.993753i	$-0.535597\pi$
$919$	−5.25163e9	−0.223198	−0.111599	+	0.993753i	$0.535597\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.49434e8	−0.0146272
$924$	0	0
$925$	3.31552e10	1.37739
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.40568e7	−0.000575215 0	−0.000287607	−	1.00000i	$-0.500092\pi$
$929$	−1.40568e7	−0.000575215 0	−0.000287607		1.00000i	$0.500092\pi$
$930$	0	0
$931$	2.30436e10	0.935895
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.30585e9	−0.0522461
$936$	0	0
$937$	2.39142e10	0.949656	0.474828	−	0.880079i	$-0.342510\pi$
$937$	2.39142e10	0.949656	0.474828	+	0.880079i	$0.342510\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.93413e10	−1.93040	−0.965199	−	0.261518i	$-0.915777\pi$
$941$	−4.93413e10	−1.93040	−0.965199	+	0.261518i	$0.915777\pi$
$942$	0	0
$943$	−1.64262e10	−0.637891
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.80272e10	1.45502	0.727510	−	0.686097i	$-0.240677\pi$
$947$	3.80272e10	1.45502	0.727510	+	0.686097i	$0.240677\pi$
$948$	0	0
$949$	8.20364e7	0.00311584
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.55641e10	−0.956765	−0.478383	−	0.878151i	$-0.658777\pi$
$953$	−2.55641e10	−0.956765	−0.478383	+	0.878151i	$0.658777\pi$
$954$	0	0
$955$	−1.35003e10	−0.501570
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.29617e10	2.30521
$960$	0	0
$961$	−2.12110e10	−0.770956
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.09058e10	0.748895
$966$	0	0
$967$	4.34305e10	1.54455	0.772276	−	0.635287i	$-0.219118\pi$
$967$	4.34305e10	1.54455	0.772276	+	0.635287i	$0.219118\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.53632e10	−1.59015	−0.795073	−	0.606514i	$-0.792567\pi$
$971$	−4.53632e10	−1.59015	−0.795073	+	0.606514i	$0.792567\pi$
$972$	0	0
$973$	−3.84234e9	−0.133721
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.58177e10	−0.542640	−0.271320	−	0.962489i	$-0.587460\pi$
$977$	−1.58177e10	−0.542640	−0.271320	+	0.962489i	$0.587460\pi$
$978$	0	0
$979$	3.20938e9	0.109315
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.35559e10	−1.12676	−0.563381	−	0.826198i	$-0.690499\pi$
$983$	−3.35559e10	−1.12676	−0.563381	+	0.826198i	$0.690499\pi$
$984$	0	0
$985$	−1.08321e10	−0.361149
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.65813e9	−0.0873755
$990$	0	0
$991$	−2.25908e10	−0.737351	−0.368675	−	0.929558i	$-0.620189\pi$
$991$	−2.25908e10	−0.737351	−0.368675	+	0.929558i	$0.620189\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.84151e9	−0.0914467
$996$	0	0
$997$	2.76455e10	0.883469	0.441735	−	0.897146i	$-0.354363\pi$
$997$	2.76455e10	0.883469	0.441735	+	0.897146i	$0.354363\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.8.a.c.1.4		7
3.2	odd	2		324.8.a.d.1.4		7
9.2	odd	6		108.8.e.a.37.4		14
9.4	even	3		36.8.e.a.25.6	yes	14
9.5	odd	6		108.8.e.a.73.4		14
9.7	even	3		36.8.e.a.13.6	✓	14
36.7	odd	6		144.8.i.d.49.2		14
36.11	even	6		432.8.i.d.145.4		14
36.23	even	6		432.8.i.d.289.4		14
36.31	odd	6		144.8.i.d.97.2		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.8.e.a.13.6	✓	14	9.7	even	3
36.8.e.a.25.6	yes	14	9.4	even	3
108.8.e.a.37.4		14	9.2	odd	6
108.8.e.a.73.4		14	9.5	odd	6
144.8.i.d.49.2		14	36.7	odd	6
144.8.i.d.97.2		14	36.31	odd	6
324.8.a.c.1.4		7	1.1	even	1	trivial
324.8.a.d.1.4		7	3.2	odd	2
432.8.i.d.145.4		14	36.11	even	6
432.8.i.d.289.4		14	36.23	even	6

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q+114.920 q^{5} -1474.10 q^{7} +O(q^{10})\)	\(q+114.920 q^{5} -1474.10 q^{7} -352.372 q^{11} -112.314 q^{13} +32247.7 q^{17} +17076.5 q^{19} +36946.8 q^{23} -64918.5 q^{25} +183778. q^{29} +79382.6 q^{31} -169403. q^{35} -510721. q^{37} -444591. q^{41} -71944.9 q^{43} +753882. q^{47} +1.34944e6 q^{49} -2.03509e6 q^{53} -40494.5 q^{55} -2.17034e6 q^{59} -2.03715e6 q^{61} -12907.0 q^{65} +1.72117e6 q^{67} +3.11124e6 q^{71} -730423. q^{73} +519433. q^{77} +1.08536e6 q^{79} +5.65476e6 q^{83} +3.70589e6 q^{85} -9.10792e6 q^{89} +165562. q^{91} +1.96242e6 q^{95} +5.45479e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 321 q^{5} + 83 q^{7}+O(q^{10}) \) 7 * q - 321 * q^5 + 83 * q^7	\( 7 q - 321 q^{5} + 83 q^{7} + 111 q^{11} + 1847 q^{13} - 48 q^{17} + 10124 q^{19} - 19119 q^{23} + 73378 q^{25} - 6045 q^{29} + 153089 q^{31} + 13713 q^{35} + 69674 q^{37} - 446631 q^{41} + 384347 q^{43} - 298413 q^{47} + 351876 q^{49} - 454038 q^{53} - 1263483 q^{55} - 2619543 q^{59} + 146231 q^{61} - 2535735 q^{65} - 1637419 q^{67} - 4353492 q^{71} - 2132260 q^{73} - 9785451 q^{77} - 2402185 q^{79} - 12936357 q^{83} + 1015002 q^{85} - 19684830 q^{89} - 492203 q^{91} - 22685196 q^{95} + 2853257 q^{97}+O(q^{100}) \) 7 * q - 321 * q^5 + 83 * q^7 + 111 * q^11 + 1847 * q^13 - 48 * q^17 + 10124 * q^19 - 19119 * q^23 + 73378 * q^25 - 6045 * q^29 + 153089 * q^31 + 13713 * q^35 + 69674 * q^37 - 446631 * q^41 + 384347 * q^43 - 298413 * q^47 + 351876 * q^49 - 454038 * q^53 - 1263483 * q^55 - 2619543 * q^59 + 146231 * q^61 - 2535735 * q^65 - 1637419 * q^67 - 4353492 * q^71 - 2132260 * q^73 - 9785451 * q^77 - 2402185 * q^79 - 12936357 * q^83 + 1015002 * q^85 - 19684830 * q^89 - 492203 * q^91 - 22685196 * q^95 + 2853257 * q^97

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