LMFDB - Embedded newform 7938.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +1.00000 q^{8} -2.64575 q^{10} -2.00000 q^{11} +2.64575 q^{13} +1.00000 q^{16} +7.93725 q^{17} -5.29150 q^{19} -2.64575 q^{20} -2.00000 q^{22} -6.00000 q^{23} +2.00000 q^{25} +2.64575 q^{26} -5.00000 q^{29} +5.29150 q^{31} +1.00000 q^{32} +7.93725 q^{34} +3.00000 q^{37} -5.29150 q^{38} -2.64575 q^{40} -8.00000 q^{43} -2.00000 q^{44} -6.00000 q^{46} +5.29150 q^{47} +2.00000 q^{50} +2.64575 q^{52} -2.00000 q^{53} +5.29150 q^{55} -5.00000 q^{58} +5.29150 q^{59} -2.64575 q^{61} +5.29150 q^{62} +1.00000 q^{64} -7.00000 q^{65} -2.00000 q^{67} +7.93725 q^{68} -8.00000 q^{71} +13.2288 q^{73} +3.00000 q^{74} -5.29150 q^{76} +10.0000 q^{79} -2.64575 q^{80} -15.8745 q^{83} -21.0000 q^{85} -8.00000 q^{86} -2.00000 q^{88} -13.2288 q^{89} -6.00000 q^{92} +5.29150 q^{94} +14.0000 q^{95} -10.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 4 q^{11} + 2 q^{16} - 4 q^{22} - 12 q^{23} + 4 q^{25} - 10 q^{29} + 2 q^{32} + 6 q^{37} - 16 q^{43} - 4 q^{44} - 12 q^{46} + 4 q^{50} - 4 q^{53} - 10 q^{58} + 2 q^{64}+ \cdots + 28 q^{95}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 4 * q^11 + 2 * q^16 - 4 * q^22 - 12 * q^23 + 4 * q^25 - 10 * q^29 + 2 * q^32 + 6 * q^37 - 16 * q^43 - 4 * q^44 - 12 * q^46 + 4 * q^50 - 4 * q^53 - 10 * q^58 + 2 * q^64 - 14 * q^65 - 4 * q^67 - 16 * q^71 + 6 * q^74 + 20 * q^79 - 42 * q^85 - 16 * q^86 - 4 * q^88 - 12 * q^92 + 28 * q^95

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.64575	−1.18322	−0.591608	−	0.806226i	$-0.701507\pi$
$5$	−2.64575	−1.18322	−0.591608	+	0.806226i	$0.701507\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.64575	−0.836660
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.64575	0.733799	0.366900	−	0.930261i	$-0.380419\pi$
$13$	2.64575	0.733799	0.366900	+	0.930261i	$0.380419\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.93725	1.92507	0.962533	−	0.271163i	$-0.0874083\pi$
$17$	7.93725	1.92507	0.962533	+	0.271163i	$0.0874083\pi$
$18$	0	0
$19$	−5.29150	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$20$	−2.64575	−0.591608
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	2.64575	0.518875
$27$	0	0
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	5.29150	0.950382	0.475191	−	0.879883i	$-0.342379\pi$
$31$	5.29150	0.950382	0.475191	+	0.879883i	$0.342379\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.93725	1.36123
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−5.29150	−0.858395
$39$	0	0
$40$	−2.64575	−0.418330
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−6.00000	−0.884652
$47$	5.29150	0.771845	0.385922	−	0.922531i	$-0.373883\pi$
$47$	5.29150	0.771845	0.385922	+	0.922531i	$0.373883\pi$
$48$	0	0
$49$	0	0
$50$	2.00000	0.282843
$51$	0	0
$52$	2.64575	0.366900
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	5.29150	0.713506
$56$	0	0
$57$	0	0
$58$	−5.00000	−0.656532
$59$	5.29150	0.688895	0.344447	−	0.938806i	$-0.388066\pi$
$59$	5.29150	0.688895	0.344447	+	0.938806i	$0.388066\pi$
$60$	0	0
$61$	−2.64575	−0.338754	−0.169377	−	0.985551i	$-0.554176\pi$
$61$	−2.64575	−0.338754	−0.169377	+	0.985551i	$0.554176\pi$
$62$	5.29150	0.672022
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.00000	−0.868243
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	7.93725	0.962533
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	13.2288	1.54831	0.774154	−	0.632997i	$-0.218175\pi$
$73$	13.2288	1.54831	0.774154	+	0.632997i	$0.218175\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	−5.29150	−0.606977
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−2.64575	−0.295804
$81$	0	0
$82$	0	0
$83$	−15.8745	−1.74245	−0.871227	−	0.490881i	$-0.836675\pi$
$83$	−15.8745	−1.74245	−0.871227	+	0.490881i	$0.836675\pi$
$84$	0	0
$85$	−21.0000	−2.27777
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−13.2288	−1.40225	−0.701123	−	0.713041i	$-0.747317\pi$
$89$	−13.2288	−1.40225	−0.701123	+	0.713041i	$0.747317\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	5.29150	0.545777
$95$	14.0000	1.43637
$96$	0	0
$97$	−10.5830	−1.07454	−0.537271	−	0.843410i	$-0.680545\pi$
$97$	−10.5830	−1.07454	−0.537271	+	0.843410i	$0.680545\pi$
$98$	0	0
$99$	0	0
$100$	2.00000	0.200000
$101$	−10.5830	−1.05305	−0.526524	−	0.850160i	$-0.676505\pi$
$101$	−10.5830	−1.05305	−0.526524	+	0.850160i	$0.676505\pi$
$102$	0	0
$103$	10.5830	1.04277	0.521387	−	0.853320i	$-0.325415\pi$
$103$	10.5830	1.04277	0.521387	+	0.853320i	$0.325415\pi$
$104$	2.64575	0.259437
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	5.29150	0.504525
$111$	0	0
$112$	0	0
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	0	0
$115$	15.8745	1.48031
$116$	−5.00000	−0.464238
$117$	0	0
$118$	5.29150	0.487122
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−2.64575	−0.239535
$123$	0	0
$124$	5.29150	0.475191
$125$	7.93725	0.709930
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−7.00000	−0.613941
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	7.93725	0.680614
$137$	−23.0000	−1.96502	−0.982511	−	0.186203i	$-0.940382\pi$
$137$	−23.0000	−1.96502	−0.982511	+	0.186203i	$0.940382\pi$
$138$	0	0
$139$	15.8745	1.34646	0.673229	−	0.739434i	$-0.264907\pi$
$139$	15.8745	1.34646	0.673229	+	0.739434i	$0.264907\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	−5.29150	−0.442498
$144$	0	0
$145$	13.2288	1.09859
$146$	13.2288	1.09482
$147$	0	0
$148$	3.00000	0.246598
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−5.29150	−0.429198
$153$	0	0
$154$	0	0
$155$	−14.0000	−1.12451
$156$	0	0
$157$	−13.2288	−1.05577	−0.527885	−	0.849316i	$-0.677015\pi$
$157$	−13.2288	−1.05577	−0.527885	+	0.849316i	$0.677015\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	−2.64575	−0.209165
$161$	0	0
$162$	0	0
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	0	0
$165$	0	0
$166$	−15.8745	−1.23210
$167$	−5.29150	−0.409469	−0.204734	−	0.978818i	$-0.565633\pi$
$167$	−5.29150	−0.409469	−0.204734	+	0.978818i	$0.565633\pi$
$168$	0	0
$169$	−6.00000	−0.461538
$170$	−21.0000	−1.61063
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−13.2288	−1.00576	−0.502882	−	0.864355i	$-0.667727\pi$
$173$	−13.2288	−1.00576	−0.502882	+	0.864355i	$0.667727\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−13.2288	−0.991537
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−10.5830	−0.786629	−0.393314	−	0.919404i	$-0.628672\pi$
$181$	−10.5830	−0.786629	−0.393314	+	0.919404i	$0.628672\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−7.93725	−0.583559
$186$	0	0
$187$	−15.8745	−1.16086
$188$	5.29150	0.385922
$189$	0	0
$190$	14.0000	1.01567
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	−10.5830	−0.759815
$195$	0	0
$196$	0	0
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	−26.4575	−1.87552	−0.937762	−	0.347279i	$-0.887106\pi$
$199$	−26.4575	−1.87552	−0.937762	+	0.347279i	$0.887106\pi$
$200$	2.00000	0.141421
$201$	0	0
$202$	−10.5830	−0.744618
$203$	0	0
$204$	0	0
$205$	0	0
$206$	10.5830	0.737353
$207$	0	0
$208$	2.64575	0.183450
$209$	10.5830	0.732042
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−4.00000	−0.273434
$215$	21.1660	1.44351
$216$	0	0
$217$	0	0
$218$	−17.0000	−1.15139
$219$	0	0
$220$	5.29150	0.356753
$221$	21.0000	1.41261
$222$	0	0
$223$	21.1660	1.41738	0.708690	−	0.705520i	$-0.249286\pi$
$223$	21.1660	1.41738	0.708690	+	0.705520i	$0.249286\pi$
$224$	0	0
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	10.5830	0.702419	0.351209	−	0.936297i	$-0.385771\pi$
$227$	10.5830	0.702419	0.351209	+	0.936297i	$0.385771\pi$
$228$	0	0
$229$	−13.2288	−0.874181	−0.437090	−	0.899418i	$-0.643991\pi$
$229$	−13.2288	−0.874181	−0.437090	+	0.899418i	$0.643991\pi$
$230$	15.8745	1.04673
$231$	0	0
$232$	−5.00000	−0.328266
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	0	0
$235$	−14.0000	−0.913259
$236$	5.29150	0.344447
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−18.5203	−1.19299	−0.596497	−	0.802615i	$-0.703441\pi$
$241$	−18.5203	−1.19299	−0.596497	+	0.802615i	$0.703441\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−2.64575	−0.169377
$245$	0	0
$246$	0	0
$247$	−14.0000	−0.890799
$248$	5.29150	0.336011
$249$	0	0
$250$	7.93725	0.501996
$251$	−26.4575	−1.66998	−0.834992	−	0.550263i	$-0.814528\pi$
$251$	−26.4575	−1.66998	−0.834992	+	0.550263i	$0.814528\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.2288	−0.825187	−0.412594	−	0.910915i	$-0.635377\pi$
$257$	−13.2288	−0.825187	−0.412594	+	0.910915i	$0.635377\pi$
$258$	0	0
$259$	0	0
$260$	−7.00000	−0.434122
$261$	0	0
$262$	0	0
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	0	0
$265$	5.29150	0.325054
$266$	0	0
$267$	0	0
$268$	−2.00000	−0.122169
$269$	−2.64575	−0.161314	−0.0806572	−	0.996742i	$-0.525702\pi$
$269$	−2.64575	−0.161314	−0.0806572	+	0.996742i	$0.525702\pi$
$270$	0	0
$271$	−5.29150	−0.321436	−0.160718	−	0.987000i	$-0.551381\pi$
$271$	−5.29150	−0.321436	−0.160718	+	0.987000i	$0.551381\pi$
$272$	7.93725	0.481267
$273$	0	0
$274$	−23.0000	−1.38948
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	15.8745	0.952090
$279$	0	0
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	0	0
$283$	−10.5830	−0.629094	−0.314547	−	0.949242i	$-0.601853\pi$
$283$	−10.5830	−0.629094	−0.314547	+	0.949242i	$0.601853\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−5.29150	−0.312893
$287$	0	0
$288$	0	0
$289$	46.0000	2.70588
$290$	13.2288	0.776819
$291$	0	0
$292$	13.2288	0.774154
$293$	−7.93725	−0.463699	−0.231850	−	0.972752i	$-0.574478\pi$
$293$	−7.93725	−0.463699	−0.231850	+	0.972752i	$0.574478\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	3.00000	0.174371
$297$	0	0
$298$	−11.0000	−0.637213
$299$	−15.8745	−0.918046
$300$	0	0
$301$	0	0
$302$	−2.00000	−0.115087
$303$	0	0
$304$	−5.29150	−0.303488
$305$	7.00000	0.400819
$306$	0	0
$307$	−10.5830	−0.604004	−0.302002	−	0.953307i	$-0.597655\pi$
$307$	−10.5830	−0.604004	−0.302002	+	0.953307i	$0.597655\pi$
$308$	0	0
$309$	0	0
$310$	−14.0000	−0.795147
$311$	−5.29150	−0.300054	−0.150027	−	0.988682i	$-0.547936\pi$
$311$	−5.29150	−0.300054	−0.150027	+	0.988682i	$0.547936\pi$
$312$	0	0
$313$	7.93725	0.448640	0.224320	−	0.974516i	$-0.427984\pi$
$313$	7.93725	0.448640	0.224320	+	0.974516i	$0.427984\pi$
$314$	−13.2288	−0.746542
$315$	0	0
$316$	10.0000	0.562544
$317$	−27.0000	−1.51647	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000	−1.51647	−0.758236	+	0.651981i	$0.773938\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	−2.64575	−0.147902
$321$	0	0
$322$	0	0
$323$	−42.0000	−2.33694
$324$	0	0
$325$	5.29150	0.293520
$326$	24.0000	1.32924
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−15.8745	−0.871227
$333$	0	0
$334$	−5.29150	−0.289538
$335$	5.29150	0.289106
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−6.00000	−0.326357
$339$	0	0
$340$	−21.0000	−1.13888
$341$	−10.5830	−0.573102
$342$	0	0
$343$	0	0
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−13.2288	−0.711182
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−31.7490	−1.69949	−0.849743	−	0.527197i	$-0.823243\pi$
$349$	−31.7490	−1.69949	−0.849743	+	0.527197i	$0.823243\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	31.7490	1.68983	0.844915	−	0.534901i	$-0.179651\pi$
$353$	31.7490	1.68983	0.844915	+	0.534901i	$0.179651\pi$
$354$	0	0
$355$	21.1660	1.12338
$356$	−13.2288	−0.701123
$357$	0	0
$358$	18.0000	0.951330
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	−10.5830	−0.556230
$363$	0	0
$364$	0	0
$365$	−35.0000	−1.83198
$366$	0	0
$367$	5.29150	0.276214	0.138107	−	0.990417i	$-0.455898\pi$
$367$	5.29150	0.276214	0.138107	+	0.990417i	$0.455898\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	−7.93725	−0.412638
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−15.8745	−0.820851
$375$	0	0
$376$	5.29150	0.272888
$377$	−13.2288	−0.681316
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	14.0000	0.718185
$381$	0	0
$382$	−18.0000	−0.920960
$383$	31.7490	1.62230	0.811149	−	0.584839i	$-0.198842\pi$
$383$	31.7490	1.62230	0.811149	+	0.584839i	$0.198842\pi$
$384$	0	0
$385$	0	0
$386$	11.0000	0.559885
$387$	0	0
$388$	−10.5830	−0.537271
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−47.6235	−2.40843
$392$	0	0
$393$	0	0
$394$	−15.0000	−0.755689
$395$	−26.4575	−1.33122
$396$	0	0
$397$	−29.1033	−1.46065	−0.730325	−	0.683099i	$-0.760632\pi$
$397$	−29.1033	−1.46065	−0.730325	+	0.683099i	$0.760632\pi$
$398$	−26.4575	−1.32620
$399$	0	0
$400$	2.00000	0.100000
$401$	−39.0000	−1.94757	−0.973784	−	0.227477i	$-0.926952\pi$
$401$	−39.0000	−1.94757	−0.973784	+	0.227477i	$0.926952\pi$
$402$	0	0
$403$	14.0000	0.697390
$404$	−10.5830	−0.526524
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	7.93725	0.392472	0.196236	−	0.980557i	$-0.437128\pi$
$409$	7.93725	0.392472	0.196236	+	0.980557i	$0.437128\pi$
$410$	0	0
$411$	0	0
$412$	10.5830	0.521387
$413$	0	0
$414$	0	0
$415$	42.0000	2.06170
$416$	2.64575	0.129719
$417$	0	0
$418$	10.5830	0.517632
$419$	−10.5830	−0.517014	−0.258507	−	0.966009i	$-0.583230\pi$
$419$	−10.5830	−0.517014	−0.258507	+	0.966009i	$0.583230\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	−22.0000	−1.07094
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	15.8745	0.770027
$426$	0	0
$427$	0	0
$428$	−4.00000	−0.193347
$429$	0	0
$430$	21.1660	1.02072
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	18.5203	0.890027	0.445013	−	0.895524i	$-0.353199\pi$
$433$	18.5203	0.890027	0.445013	+	0.895524i	$0.353199\pi$
$434$	0	0
$435$	0	0
$436$	−17.0000	−0.814152
$437$	31.7490	1.51876
$438$	0	0
$439$	31.7490	1.51530	0.757649	−	0.652662i	$-0.226348\pi$
$439$	31.7490	1.51530	0.757649	+	0.652662i	$0.226348\pi$
$440$	5.29150	0.252262
$441$	0	0
$442$	21.0000	0.998868
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	35.0000	1.65916
$446$	21.1660	1.00224
$447$	0	0
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	−1.00000	−0.0470360
$453$	0	0
$454$	10.5830	0.496685
$455$	0	0
$456$	0	0
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	−13.2288	−0.618139
$459$	0	0
$460$	15.8745	0.740153
$461$	31.7490	1.47870	0.739350	−	0.673322i	$-0.235133\pi$
$461$	31.7490	1.47870	0.739350	+	0.673322i	$0.235133\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	1.00000	0.0463241
$467$	−5.29150	−0.244862	−0.122431	−	0.992477i	$-0.539069\pi$
$467$	−5.29150	−0.244862	−0.122431	+	0.992477i	$0.539069\pi$
$468$	0	0
$469$	0	0
$470$	−14.0000	−0.645772
$471$	0	0
$472$	5.29150	0.243561
$473$	16.0000	0.735681
$474$	0	0
$475$	−10.5830	−0.485582
$476$	0	0
$477$	0	0
$478$	6.00000	0.274434
$479$	−15.8745	−0.725325	−0.362662	−	0.931921i	$-0.618132\pi$
$479$	−15.8745	−0.725325	−0.362662	+	0.931921i	$0.618132\pi$
$480$	0	0
$481$	7.93725	0.361908
$482$	−18.5203	−0.843575
$483$	0	0
$484$	−7.00000	−0.318182
$485$	28.0000	1.27141
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−2.64575	−0.119768
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−39.6863	−1.78738
$494$	−14.0000	−0.629890
$495$	0	0
$496$	5.29150	0.237595
$497$	0	0
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	7.93725	0.354965
$501$	0	0
$502$	−26.4575	−1.18086
$503$	26.4575	1.17968	0.589841	−	0.807519i	$-0.299190\pi$
$503$	26.4575	1.17968	0.589841	+	0.807519i	$0.299190\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	12.0000	0.533465
$507$	0	0
$508$	20.0000	0.887357
$509$	10.5830	0.469083	0.234542	−	0.972106i	$-0.424641\pi$
$509$	10.5830	0.469083	0.234542	+	0.972106i	$0.424641\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−13.2288	−0.583495
$515$	−28.0000	−1.23383
$516$	0	0
$517$	−10.5830	−0.465440
$518$	0	0
$519$	0	0
$520$	−7.00000	−0.306970
$521$	10.5830	0.463650	0.231825	−	0.972758i	$-0.425530\pi$
$521$	10.5830	0.463650	0.231825	+	0.972758i	$0.425530\pi$
$522$	0	0
$523$	42.3320	1.85105	0.925525	−	0.378686i	$-0.123624\pi$
$523$	42.3320	1.85105	0.925525	+	0.378686i	$0.123624\pi$
$524$	0	0
$525$	0	0
$526$	−2.00000	−0.0872041
$527$	42.0000	1.82955
$528$	0	0
$529$	13.0000	0.565217
$530$	5.29150	0.229848
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	10.5830	0.457543
$536$	−2.00000	−0.0863868
$537$	0	0
$538$	−2.64575	−0.114066
$539$	0	0
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	−5.29150	−0.227289
$543$	0	0
$544$	7.93725	0.340307
$545$	44.9778	1.92664
$546$	0	0
$547$	34.0000	1.45374	0.726868	−	0.686778i	$-0.240975\pi$
$547$	34.0000	1.45374	0.726868	+	0.686778i	$0.240975\pi$
$548$	−23.0000	−0.982511
$549$	0	0
$550$	−4.00000	−0.170561
$551$	26.4575	1.12713
$552$	0	0
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	15.8745	0.673229
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	0	0
$559$	−21.1660	−0.895227
$560$	0	0
$561$	0	0
$562$	−15.0000	−0.632737
$563$	−21.1660	−0.892041	−0.446020	−	0.895023i	$-0.647159\pi$
$563$	−21.1660	−0.892041	−0.446020	+	0.895023i	$0.647159\pi$
$564$	0	0
$565$	2.64575	0.111308
$566$	−10.5830	−0.444837
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−5.29150	−0.221249
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	13.2288	0.550720	0.275360	−	0.961341i	$-0.411203\pi$
$577$	13.2288	0.550720	0.275360	+	0.961341i	$0.411203\pi$
$578$	46.0000	1.91335
$579$	0	0
$580$	13.2288	0.549294
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	13.2288	0.547410
$585$	0	0
$586$	−7.93725	−0.327885
$587$	5.29150	0.218404	0.109202	−	0.994020i	$-0.465171\pi$
$587$	5.29150	0.218404	0.109202	+	0.994020i	$0.465171\pi$
$588$	0	0
$589$	−28.0000	−1.15372
$590$	−14.0000	−0.576371
$591$	0	0
$592$	3.00000	0.123299
$593$	23.8118	0.977832	0.488916	−	0.872331i	$-0.337392\pi$
$593$	23.8118	0.977832	0.488916	+	0.872331i	$0.337392\pi$
$594$	0	0
$595$	0	0
$596$	−11.0000	−0.450578
$597$	0	0
$598$	−15.8745	−0.649157
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	18.5203	0.755457	0.377729	−	0.925916i	$-0.376705\pi$
$601$	18.5203	0.755457	0.377729	+	0.925916i	$0.376705\pi$
$602$	0	0
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	18.5203	0.752956
$606$	0	0
$607$	−47.6235	−1.93298	−0.966490	−	0.256706i	$-0.917363\pi$
$607$	−47.6235	−1.93298	−0.966490	+	0.256706i	$0.917363\pi$
$608$	−5.29150	−0.214599
$609$	0	0
$610$	7.00000	0.283422
$611$	14.0000	0.566379
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−10.5830	−0.427095
$615$	0	0
$616$	0	0
$617$	−19.0000	−0.764911	−0.382456	−	0.923974i	$-0.624922\pi$
$617$	−19.0000	−0.764911	−0.382456	+	0.923974i	$0.624922\pi$
$618$	0	0
$619$	21.1660	0.850734	0.425367	−	0.905021i	$-0.360145\pi$
$619$	21.1660	0.850734	0.425367	+	0.905021i	$0.360145\pi$
$620$	−14.0000	−0.562254
$621$	0	0
$622$	−5.29150	−0.212170
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	7.93725	0.317236
$627$	0	0
$628$	−13.2288	−0.527885
$629$	23.8118	0.949437
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−27.0000	−1.07231
$635$	−52.9150	−2.09987
$636$	0	0
$637$	0	0
$638$	10.0000	0.395904
$639$	0	0
$640$	−2.64575	−0.104583
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	0	0
$643$	47.6235	1.87809	0.939044	−	0.343796i	$-0.111713\pi$
$643$	47.6235	1.87809	0.939044	+	0.343796i	$0.111713\pi$
$644$	0	0
$645$	0	0
$646$	−42.0000	−1.65247
$647$	31.7490	1.24818	0.624091	−	0.781351i	$-0.285469\pi$
$647$	31.7490	1.24818	0.624091	+	0.781351i	$0.285469\pi$
$648$	0	0
$649$	−10.5830	−0.415419
$650$	5.29150	0.207550
$651$	0	0
$652$	24.0000	0.939913
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	34.3948	1.33780	0.668901	−	0.743352i	$-0.266765\pi$
$661$	34.3948	1.33780	0.668901	+	0.743352i	$0.266765\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	−15.8745	−0.616050
$665$	0	0
$666$	0	0
$667$	30.0000	1.16160
$668$	−5.29150	−0.204734
$669$	0	0
$670$	5.29150	0.204429
$671$	5.29150	0.204276
$672$	0	0
$673$	−43.0000	−1.65753	−0.828764	−	0.559598i	$-0.810955\pi$
$673$	−43.0000	−1.65753	−0.828764	+	0.559598i	$0.810955\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	−6.00000	−0.230769
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	−21.0000	−0.805313
$681$	0	0
$682$	−10.5830	−0.405244
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	60.8523	2.32505
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−5.29150	−0.201590
$690$	0	0
$691$	−42.3320	−1.61039	−0.805193	−	0.593013i	$-0.797938\pi$
$691$	−42.3320	−1.61039	−0.805193	+	0.593013i	$0.797938\pi$
$692$	−13.2288	−0.502882
$693$	0	0
$694$	12.0000	0.455514
$695$	−42.0000	−1.59315
$696$	0	0
$697$	0	0
$698$	−31.7490	−1.20172
$699$	0	0
$700$	0	0
$701$	−1.00000	−0.0377695	−0.0188847	−	0.999822i	$-0.506012\pi$
$701$	−1.00000	−0.0377695	−0.0188847	+	0.999822i	$0.506012\pi$
$702$	0	0
$703$	−15.8745	−0.598718
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	31.7490	1.19489
$707$	0	0
$708$	0	0
$709$	31.0000	1.16423	0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000	1.16423	0.582115	+	0.813107i	$0.302225\pi$
$710$	21.1660	0.794346
$711$	0	0
$712$	−13.2288	−0.495769
$713$	−31.7490	−1.18901
$714$	0	0
$715$	14.0000	0.523570
$716$	18.0000	0.672692
$717$	0	0
$718$	−4.00000	−0.149279
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	9.00000	0.334945
$723$	0	0
$724$	−10.5830	−0.393314
$725$	−10.0000	−0.371391
$726$	0	0
$727$	42.3320	1.57001	0.785004	−	0.619491i	$-0.212661\pi$
$727$	42.3320	1.57001	0.785004	+	0.619491i	$0.212661\pi$
$728$	0	0
$729$	0	0
$730$	−35.0000	−1.29541
$731$	−63.4980	−2.34856
$732$	0	0
$733$	−10.5830	−0.390892	−0.195446	−	0.980714i	$-0.562615\pi$
$733$	−10.5830	−0.390892	−0.195446	+	0.980714i	$0.562615\pi$
$734$	5.29150	0.195313
$735$	0	0
$736$	−6.00000	−0.221163
$737$	4.00000	0.147342
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	−7.93725	−0.291779
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	29.1033	1.06626
$746$	10.0000	0.366126
$747$	0	0
$748$	−15.8745	−0.580429
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	5.29150	0.192961
$753$	0	0
$754$	−13.2288	−0.481763
$755$	5.29150	0.192577
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	6.00000	0.217930
$759$	0	0
$760$	14.0000	0.507833
$761$	−2.64575	−0.0959084	−0.0479542	−	0.998850i	$-0.515270\pi$
$761$	−2.64575	−0.0959084	−0.0479542	+	0.998850i	$0.515270\pi$
$762$	0	0
$763$	0	0
$764$	−18.0000	−0.651217
$765$	0	0
$766$	31.7490	1.14714
$767$	14.0000	0.505511
$768$	0	0
$769$	−13.2288	−0.477041	−0.238521	−	0.971137i	$-0.576662\pi$
$769$	−13.2288	−0.477041	−0.238521	+	0.971137i	$0.576662\pi$
$770$	0	0
$771$	0	0
$772$	11.0000	0.395899
$773$	7.93725	0.285483	0.142742	−	0.989760i	$-0.454408\pi$
$773$	7.93725	0.285483	0.142742	+	0.989760i	$0.454408\pi$
$774$	0	0
$775$	10.5830	0.380153
$776$	−10.5830	−0.379908
$777$	0	0
$778$	18.0000	0.645331
$779$	0	0
$780$	0	0
$781$	16.0000	0.572525
$782$	−47.6235	−1.70301
$783$	0	0
$784$	0	0
$785$	35.0000	1.24920
$786$	0	0
$787$	−42.3320	−1.50897	−0.754487	−	0.656315i	$-0.772114\pi$
$787$	−42.3320	−1.50897	−0.754487	+	0.656315i	$0.772114\pi$
$788$	−15.0000	−0.534353
$789$	0	0
$790$	−26.4575	−0.941316
$791$	0	0
$792$	0	0
$793$	−7.00000	−0.248577
$794$	−29.1033	−1.03284
$795$	0	0
$796$	−26.4575	−0.937762
$797$	7.93725	0.281152	0.140576	−	0.990070i	$-0.455105\pi$
$797$	7.93725	0.281152	0.140576	+	0.990070i	$0.455105\pi$
$798$	0	0
$799$	42.0000	1.48585
$800$	2.00000	0.0707107
$801$	0	0
$802$	−39.0000	−1.37714
$803$	−26.4575	−0.933665
$804$	0	0
$805$	0	0
$806$	14.0000	0.493129
$807$	0	0
$808$	−10.5830	−0.372309
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	−37.0405	−1.30067	−0.650334	−	0.759648i	$-0.725371\pi$
$811$	−37.0405	−1.30067	−0.650334	+	0.759648i	$0.725371\pi$
$812$	0	0
$813$	0	0
$814$	−6.00000	−0.210300
$815$	−63.4980	−2.22424
$816$	0	0
$817$	42.3320	1.48101
$818$	7.93725	0.277520
$819$	0	0
$820$	0	0
$821$	−11.0000	−0.383903	−0.191951	−	0.981404i	$-0.561482\pi$
$821$	−11.0000	−0.383903	−0.191951	+	0.981404i	$0.561482\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	10.5830	0.368676
$825$	0	0
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	21.1660	0.735126	0.367563	−	0.929999i	$-0.380192\pi$
$829$	21.1660	0.735126	0.367563	+	0.929999i	$0.380192\pi$
$830$	42.0000	1.45784
$831$	0	0
$832$	2.64575	0.0917249
$833$	0	0
$834$	0	0
$835$	14.0000	0.484490
$836$	10.5830	0.366021
$837$	0	0
$838$	−10.5830	−0.365584
$839$	−31.7490	−1.09610	−0.548049	−	0.836446i	$-0.684629\pi$
$839$	−31.7490	−1.09610	−0.548049	+	0.836446i	$0.684629\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−15.0000	−0.516934
$843$	0	0
$844$	−22.0000	−0.757271
$845$	15.8745	0.546100
$846$	0	0
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	15.8745	0.544491
$851$	−18.0000	−0.617032
$852$	0	0
$853$	42.3320	1.44942	0.724710	−	0.689054i	$-0.241974\pi$
$853$	42.3320	1.44942	0.724710	+	0.689054i	$0.241974\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	−23.8118	−0.813394	−0.406697	−	0.913563i	$-0.633320\pi$
$857$	−23.8118	−0.813394	−0.406697	+	0.913563i	$0.633320\pi$
$858$	0	0
$859$	−31.7490	−1.08326	−0.541631	−	0.840616i	$-0.682193\pi$
$859$	−31.7490	−1.08326	−0.541631	+	0.840616i	$0.682193\pi$
$860$	21.1660	0.721755
$861$	0	0
$862$	32.0000	1.08992
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	0	0
$865$	35.0000	1.19004
$866$	18.5203	0.629344
$867$	0	0
$868$	0	0
$869$	−20.0000	−0.678454
$870$	0	0
$871$	−5.29150	−0.179296
$872$	−17.0000	−0.575693
$873$	0	0
$874$	31.7490	1.07393
$875$	0	0
$876$	0	0
$877$	3.00000	0.101303	0.0506514	−	0.998716i	$-0.483870\pi$
$877$	3.00000	0.101303	0.0506514	+	0.998716i	$0.483870\pi$
$878$	31.7490	1.07148
$879$	0	0
$880$	5.29150	0.178377
$881$	21.1660	0.713101	0.356551	−	0.934276i	$-0.383953\pi$
$881$	21.1660	0.713101	0.356551	+	0.934276i	$0.383953\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	21.0000	0.706306
$885$	0	0
$886$	−6.00000	−0.201574
$887$	47.6235	1.59904	0.799521	−	0.600639i	$-0.205087\pi$
$887$	47.6235	1.59904	0.799521	+	0.600639i	$0.205087\pi$
$888$	0	0
$889$	0	0
$890$	35.0000	1.17320
$891$	0	0
$892$	21.1660	0.708690
$893$	−28.0000	−0.936984
$894$	0	0
$895$	−47.6235	−1.59188
$896$	0	0
$897$	0	0
$898$	6.00000	0.200223
$899$	−26.4575	−0.882407
$900$	0	0
$901$	−15.8745	−0.528857
$902$	0	0
$903$	0	0
$904$	−1.00000	−0.0332595
$905$	28.0000	0.930751
$906$	0	0
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	10.5830	0.351209
$909$	0	0
$910$	0	0
$911$	44.0000	1.45779	0.728893	−	0.684628i	$-0.240035\pi$
$911$	44.0000	1.45779	0.728893	+	0.684628i	$0.240035\pi$
$912$	0	0
$913$	31.7490	1.05074
$914$	−25.0000	−0.826927
$915$	0	0
$916$	−13.2288	−0.437090
$917$	0	0
$918$	0	0
$919$	−18.0000	−0.593765	−0.296883	−	0.954914i	$-0.595947\pi$
$919$	−18.0000	−0.593765	−0.296883	+	0.954914i	$0.595947\pi$
$920$	15.8745	0.523367
$921$	0	0
$922$	31.7490	1.04560
$923$	−21.1660	−0.696688
$924$	0	0
$925$	6.00000	0.197279
$926$	−22.0000	−0.722965
$927$	0	0
$928$	−5.00000	−0.164133
$929$	23.8118	0.781239	0.390619	−	0.920552i	$-0.372261\pi$
$929$	23.8118	0.781239	0.390619	+	0.920552i	$0.372261\pi$
$930$	0	0
$931$	0	0
$932$	1.00000	0.0327561
$933$	0	0
$934$	−5.29150	−0.173143
$935$	42.0000	1.37355
$936$	0	0
$937$	−7.93725	−0.259299	−0.129649	−	0.991560i	$-0.541385\pi$
$937$	−7.93725	−0.259299	−0.129649	+	0.991560i	$0.541385\pi$
$938$	0	0
$939$	0	0
$940$	−14.0000	−0.456630
$941$	55.5608	1.81123	0.905615	−	0.424101i	$-0.139410\pi$
$941$	55.5608	1.81123	0.905615	+	0.424101i	$0.139410\pi$
$942$	0	0
$943$	0	0
$944$	5.29150	0.172224
$945$	0	0
$946$	16.0000	0.520205
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	0	0
$949$	35.0000	1.13615
$950$	−10.5830	−0.343358
$951$	0	0
$952$	0	0
$953$	51.0000	1.65205	0.826026	−	0.563632i	$-0.190596\pi$
$953$	51.0000	1.65205	0.826026	+	0.563632i	$0.190596\pi$
$954$	0	0
$955$	47.6235	1.54106
$956$	6.00000	0.194054
$957$	0	0
$958$	−15.8745	−0.512882
$959$	0	0
$960$	0	0
$961$	−3.00000	−0.0967742
$962$	7.93725	0.255907
$963$	0	0
$964$	−18.5203	−0.596497
$965$	−29.1033	−0.936867
$966$	0	0
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	28.0000	0.899026
$971$	−58.2065	−1.86794	−0.933968	−	0.357356i	$-0.883678\pi$
$971$	−58.2065	−1.86794	−0.933968	+	0.357356i	$0.883678\pi$
$972$	0	0
$973$	0	0
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−2.64575	−0.0846884
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	26.4575	0.845586
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	−10.5830	−0.337545	−0.168773	−	0.985655i	$-0.553980\pi$
$983$	−10.5830	−0.337545	−0.168773	+	0.985655i	$0.553980\pi$
$984$	0	0
$985$	39.6863	1.26451
$986$	−39.6863	−1.26387
$987$	0	0
$988$	−14.0000	−0.445399
$989$	48.0000	1.52631
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	5.29150	0.168005
$993$	0	0
$994$	0	0
$995$	70.0000	2.21915
$996$	0	0
$997$	18.5203	0.586542	0.293271	−	0.956029i	$-0.405256\pi$
$997$	18.5203	0.586542	0.293271	+	0.956029i	$0.405256\pi$
$998$	−6.00000	−0.189927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bo.1.1	yes	2
3.2	odd	2		7938.2.a.bl.1.2	yes	2
7.6	odd	2	inner	7938.2.a.bo.1.2	yes	2
21.20	even	2		7938.2.a.bl.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7938.2.a.bl.1.1	✓	2	21.20	even	2
7938.2.a.bl.1.2	yes	2	3.2	odd	2
7938.2.a.bo.1.1	yes	2	1.1	even	1	trivial
7938.2.a.bo.1.2	yes	2	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +1.00000 q^{8} -2.64575 q^{10} -2.00000 q^{11} +2.64575 q^{13} +1.00000 q^{16} +7.93725 q^{17} -5.29150 q^{19} -2.64575 q^{20} -2.00000 q^{22} -6.00000 q^{23} +2.00000 q^{25} +2.64575 q^{26} -5.00000 q^{29} +5.29150 q^{31} +1.00000 q^{32} +7.93725 q^{34} +3.00000 q^{37} -5.29150 q^{38} -2.64575 q^{40} -8.00000 q^{43} -2.00000 q^{44} -6.00000 q^{46} +5.29150 q^{47} +2.00000 q^{50} +2.64575 q^{52} -2.00000 q^{53} +5.29150 q^{55} -5.00000 q^{58} +5.29150 q^{59} -2.64575 q^{61} +5.29150 q^{62} +1.00000 q^{64} -7.00000 q^{65} -2.00000 q^{67} +7.93725 q^{68} -8.00000 q^{71} +13.2288 q^{73} +3.00000 q^{74} -5.29150 q^{76} +10.0000 q^{79} -2.64575 q^{80} -15.8745 q^{83} -21.0000 q^{85} -8.00000 q^{86} -2.00000 q^{88} -13.2288 q^{89} -6.00000 q^{92} +5.29150 q^{94} +14.0000 q^{95} -10.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 4 q^{11} + 2 q^{16} - 4 q^{22} - 12 q^{23} + 4 q^{25} - 10 q^{29} + 2 q^{32} + 6 q^{37} - 16 q^{43} - 4 q^{44} - 12 q^{46} + 4 q^{50} - 4 q^{53} - 10 q^{58} + 2 q^{64}+ \cdots + 28 q^{95}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 4 * q^11 + 2 * q^16 - 4 * q^22 - 12 * q^23 + 4 * q^25 - 10 * q^29 + 2 * q^32 + 6 * q^37 - 16 * q^43 - 4 * q^44 - 12 * q^46 + 4 * q^50 - 4 * q^53 - 10 * q^58 + 2 * q^64 - 14 * q^65 - 4 * q^67 - 16 * q^71 + 6 * q^74 + 20 * q^79 - 42 * q^85 - 16 * q^86 - 4 * q^88 - 12 * q^92 + 28 * q^95

Introduction

L-functions

Modular forms

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