LMFDB - Embedded newform 9522.2.a.bq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9522,2,Mod(1,9522)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9522, 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("9522.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.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:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.91899$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.47889 q^{5} +3.20362 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.47889 q^{5} +3.20362 q^{7} -1.00000 q^{8} +1.47889 q^{10} +0.0552927 q^{11} -0.805738 q^{13} -3.20362 q^{14} +1.00000 q^{16} +2.05954 q^{17} +3.67657 q^{19} -1.47889 q^{20} -0.0552927 q^{22} -2.81288 q^{25} +0.805738 q^{26} +3.20362 q^{28} -7.34575 q^{29} -7.95546 q^{31} -1.00000 q^{32} -2.05954 q^{34} -4.73780 q^{35} +3.08816 q^{37} -3.67657 q^{38} +1.47889 q^{40} +1.45973 q^{41} +12.5322 q^{43} +0.0552927 q^{44} +1.27459 q^{47} +3.26315 q^{49} +2.81288 q^{50} -0.805738 q^{52} -11.0449 q^{53} -0.0817718 q^{55} -3.20362 q^{56} +7.34575 q^{58} +5.36077 q^{59} -12.2015 q^{61} +7.95546 q^{62} +1.00000 q^{64} +1.19160 q^{65} -4.52116 q^{67} +2.05954 q^{68} +4.73780 q^{70} +14.6029 q^{71} -9.86160 q^{73} -3.08816 q^{74} +3.67657 q^{76} +0.177136 q^{77} -3.61983 q^{79} -1.47889 q^{80} -1.45973 q^{82} -10.0343 q^{83} -3.04583 q^{85} -12.5322 q^{86} -0.0552927 q^{88} -8.25248 q^{89} -2.58128 q^{91} -1.27459 q^{94} -5.43725 q^{95} -7.01618 q^{97} -3.26315 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8} + 7 q^{10} - 13 q^{11} - 4 q^{13} - 7 q^{14} + 5 q^{16} - 9 q^{17} + 11 q^{19} - 7 q^{20} + 13 q^{22} - 2 q^{25} + 4 q^{26} + 7 q^{28} + 7 q^{29} - 8 q^{31} - 5 q^{32} + 9 q^{34} - q^{35} + 12 q^{37} - 11 q^{38} + 7 q^{40} + 10 q^{41} + 4 q^{43} - 13 q^{44} + 24 q^{47} - 12 q^{49} + 2 q^{50} - 4 q^{52} - 9 q^{53} + 16 q^{55} - 7 q^{56} - 7 q^{58} + 14 q^{59} + 5 q^{61} + 8 q^{62} + 5 q^{64} - 12 q^{65} + 13 q^{67} - 9 q^{68} + q^{70} + 19 q^{71} + 4 q^{73} - 12 q^{74} + 11 q^{76} - 5 q^{77} + 4 q^{79} - 7 q^{80} - 10 q^{82} - 24 q^{83} + 17 q^{85} - 4 q^{86} + 13 q^{88} - 4 q^{89} - 21 q^{91} - 24 q^{94} + 11 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8 + 7 * q^10 - 13 * q^11 - 4 * q^13 - 7 * q^14 + 5 * q^16 - 9 * q^17 + 11 * q^19 - 7 * q^20 + 13 * q^22 - 2 * q^25 + 4 * q^26 + 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 + 9 * q^34 - q^35 + 12 * q^37 - 11 * q^38 + 7 * q^40 + 10 * q^41 + 4 * q^43 - 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 - 9 * q^53 + 16 * q^55 - 7 * q^56 - 7 * q^58 + 14 * q^59 + 5 * q^61 + 8 * q^62 + 5 * q^64 - 12 * q^65 + 13 * q^67 - 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 - 12 * q^74 + 11 * q^76 - 5 * q^77 + 4 * q^79 - 7 * q^80 - 10 * q^82 - 24 * q^83 + 17 * q^85 - 4 * q^86 + 13 * q^88 - 4 * q^89 - 21 * q^91 - 24 * q^94 + 11 * q^95 - 9 * q^97 + 12 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.47889	−0.661380	−0.330690	−	0.943739i	$-0.607282\pi$
$5$	−1.47889	−0.661380	−0.330690	+	0.943739i	$0.607282\pi$
$6$	0	0
$7$	3.20362	1.21085	0.605426	−	0.795901i	$-0.293003\pi$
$7$	3.20362	1.21085	0.605426	+	0.795901i	$0.293003\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.47889	0.467667
$11$	0.0552927	0.0166714	0.00833568	−	0.999965i	$-0.497347\pi$
$11$	0.0552927	0.0166714	0.00833568	+	0.999965i	$0.497347\pi$
$12$	0	0
$13$	−0.805738	−0.223472	−0.111736	−	0.993738i	$-0.535641\pi$
$13$	−0.805738	−0.223472	−0.111736	+	0.993738i	$0.535641\pi$
$14$	−3.20362	−0.856202
$15$	0	0
$16$	1.00000	0.250000
$17$	2.05954	0.499511	0.249756	−	0.968309i	$-0.419650\pi$
$17$	2.05954	0.499511	0.249756	+	0.968309i	$0.419650\pi$
$18$	0	0
$19$	3.67657	0.843464	0.421732	−	0.906721i	$-0.361422\pi$
$19$	3.67657	0.843464	0.421732	+	0.906721i	$0.361422\pi$
$20$	−1.47889	−0.330690
$21$	0	0
$22$	−0.0552927	−0.0117884
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.81288	−0.562576
$26$	0.805738	0.158018
$27$	0	0
$28$	3.20362	0.605426
$29$	−7.34575	−1.36407	−0.682036	−	0.731319i	$-0.738905\pi$
$29$	−7.34575	−1.36407	−0.682036	+	0.731319i	$0.738905\pi$
$30$	0	0
$31$	−7.95546	−1.42884	−0.714422	−	0.699715i	$-0.753310\pi$
$31$	−7.95546	−1.42884	−0.714422	+	0.699715i	$0.753310\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.05954	−0.353208
$35$	−4.73780	−0.800834
$36$	0	0
$37$	3.08816	0.507690	0.253845	−	0.967245i	$-0.418305\pi$
$37$	3.08816	0.507690	0.253845	+	0.967245i	$0.418305\pi$
$38$	−3.67657	−0.596419
$39$	0	0
$40$	1.47889	0.233833
$41$	1.45973	0.227972	0.113986	−	0.993482i	$-0.463638\pi$
$41$	1.45973	0.227972	0.113986	+	0.993482i	$0.463638\pi$
$42$	0	0
$43$	12.5322	1.91115	0.955574	−	0.294750i	$-0.0952365\pi$
$43$	12.5322	1.91115	0.955574	+	0.294750i	$0.0952365\pi$
$44$	0.0552927	0.00833568
$45$	0	0
$46$	0	0
$47$	1.27459	0.185918	0.0929591	−	0.995670i	$-0.470367\pi$
$47$	1.27459	0.185918	0.0929591	+	0.995670i	$0.470367\pi$
$48$	0	0
$49$	3.26315	0.466165
$50$	2.81288	0.397801
$51$	0	0
$52$	−0.805738	−0.111736
$53$	−11.0449	−1.51713	−0.758564	−	0.651599i	$-0.774099\pi$
$53$	−11.0449	−1.51713	−0.758564	+	0.651599i	$0.774099\pi$
$54$	0	0
$55$	−0.0817718	−0.0110261
$56$	−3.20362	−0.428101
$57$	0	0
$58$	7.34575	0.964544
$59$	5.36077	0.697913	0.348956	−	0.937139i	$-0.386536\pi$
$59$	5.36077	0.697913	0.348956	+	0.937139i	$0.386536\pi$
$60$	0	0
$61$	−12.2015	−1.56224	−0.781120	−	0.624380i	$-0.785352\pi$
$61$	−12.2015	−1.56224	−0.781120	+	0.624380i	$0.785352\pi$
$62$	7.95546	1.01034
$63$	0	0
$64$	1.00000	0.125000
$65$	1.19160	0.147800
$66$	0	0
$67$	−4.52116	−0.552348	−0.276174	−	0.961108i	$-0.589067\pi$
$67$	−4.52116	−0.552348	−0.276174	+	0.961108i	$0.589067\pi$
$68$	2.05954	0.249756
$69$	0	0
$70$	4.73780	0.566275
$71$	14.6029	1.73304	0.866522	−	0.499138i	$-0.166350\pi$
$71$	14.6029	1.73304	0.866522	+	0.499138i	$0.166350\pi$
$72$	0	0
$73$	−9.86160	−1.15421	−0.577106	−	0.816669i	$-0.695818\pi$
$73$	−9.86160	−1.15421	−0.577106	+	0.816669i	$0.695818\pi$
$74$	−3.08816	−0.358991
$75$	0	0
$76$	3.67657	0.421732
$77$	0.177136	0.0201866
$78$	0	0
$79$	−3.61983	−0.407262	−0.203631	−	0.979048i	$-0.565274\pi$
$79$	−3.61983	−0.407262	−0.203631	+	0.979048i	$0.565274\pi$
$80$	−1.47889	−0.165345
$81$	0	0
$82$	−1.45973	−0.161201
$83$	−10.0343	−1.10140	−0.550702	−	0.834702i	$-0.685640\pi$
$83$	−10.0343	−1.10140	−0.550702	+	0.834702i	$0.685640\pi$
$84$	0	0
$85$	−3.04583	−0.330367
$86$	−12.5322	−1.35139
$87$	0	0
$88$	−0.0552927	−0.00589422
$89$	−8.25248	−0.874762	−0.437381	−	0.899276i	$-0.644094\pi$
$89$	−8.25248	−0.874762	−0.437381	+	0.899276i	$0.644094\pi$
$90$	0	0
$91$	−2.58128	−0.270591
$92$	0	0
$93$	0	0
$94$	−1.27459	−0.131464
$95$	−5.43725	−0.557850
$96$	0	0
$97$	−7.01618	−0.712386	−0.356193	−	0.934412i	$-0.615925\pi$
$97$	−7.01618	−0.712386	−0.356193	+	0.934412i	$0.615925\pi$
$98$	−3.26315	−0.329628
$99$	0	0
$100$	−2.81288	−0.281288
$101$	−15.3507	−1.52745	−0.763725	−	0.645542i	$-0.776632\pi$
$101$	−15.3507	−1.52745	−0.763725	+	0.645542i	$0.776632\pi$
$102$	0	0
$103$	−15.3219	−1.50971	−0.754854	−	0.655893i	$-0.772292\pi$
$103$	−15.3219	−1.50971	−0.754854	+	0.655893i	$0.772292\pi$
$104$	0.805738	0.0790091
$105$	0	0
$106$	11.0449	1.07277
$107$	−9.67542	−0.935358	−0.467679	−	0.883898i	$-0.654910\pi$
$107$	−9.67542	−0.935358	−0.467679	+	0.883898i	$0.654910\pi$
$108$	0	0
$109$	13.4002	1.28350	0.641752	−	0.766912i	$-0.278208\pi$
$109$	13.4002	1.28350	0.641752	+	0.766912i	$0.278208\pi$
$110$	0.0817718	0.00779664
$111$	0	0
$112$	3.20362	0.302713
$113$	8.22890	0.774110	0.387055	−	0.922057i	$-0.373492\pi$
$113$	8.22890	0.774110	0.387055	+	0.922057i	$0.373492\pi$
$114$	0	0
$115$	0	0
$116$	−7.34575	−0.682036
$117$	0	0
$118$	−5.36077	−0.493499
$119$	6.59797	0.604835
$120$	0	0
$121$	−10.9969	−0.999722
$122$	12.2015	1.10467
$123$	0	0
$124$	−7.95546	−0.714422
$125$	11.5544	1.03346
$126$	0	0
$127$	19.3524	1.71725	0.858624	−	0.512606i	$-0.171320\pi$
$127$	19.3524	1.71725	0.858624	+	0.512606i	$0.171320\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.19160	−0.104510
$131$	20.9028	1.82628	0.913141	−	0.407643i	$-0.133649\pi$
$131$	20.9028	1.82628	0.913141	+	0.407643i	$0.133649\pi$
$132$	0	0
$133$	11.7783	1.02131
$134$	4.52116	0.390569
$135$	0	0
$136$	−2.05954	−0.176604
$137$	5.89909	0.503993	0.251997	−	0.967728i	$-0.418913\pi$
$137$	5.89909	0.503993	0.251997	+	0.967728i	$0.418913\pi$
$138$	0	0
$139$	−4.04356	−0.342971	−0.171485	−	0.985187i	$-0.554857\pi$
$139$	−4.04356	−0.342971	−0.171485	+	0.985187i	$0.554857\pi$
$140$	−4.73780	−0.400417
$141$	0	0
$142$	−14.6029	−1.22545
$143$	−0.0445514	−0.00372558
$144$	0	0
$145$	10.8636	0.902170
$146$	9.86160	0.816152
$147$	0	0
$148$	3.08816	0.253845
$149$	−8.21355	−0.672880	−0.336440	−	0.941705i	$-0.609223\pi$
$149$	−8.21355	−0.672880	−0.336440	+	0.941705i	$0.609223\pi$
$150$	0	0
$151$	−12.8490	−1.04564	−0.522820	−	0.852443i	$-0.675120\pi$
$151$	−12.8490	−1.04564	−0.522820	+	0.852443i	$0.675120\pi$
$152$	−3.67657	−0.298209
$153$	0	0
$154$	−0.177136	−0.0142741
$155$	11.7653	0.945009
$156$	0	0
$157$	−3.91879	−0.312754	−0.156377	−	0.987697i	$-0.549981\pi$
$157$	−3.91879	−0.312754	−0.156377	+	0.987697i	$0.549981\pi$
$158$	3.61983	0.287978
$159$	0	0
$160$	1.47889	0.116917
$161$	0	0
$162$	0	0
$163$	15.7247	1.23165	0.615825	−	0.787883i	$-0.288823\pi$
$163$	15.7247	1.23165	0.615825	+	0.787883i	$0.288823\pi$
$164$	1.45973	0.113986
$165$	0	0
$166$	10.0343	0.778810
$167$	4.01965	0.311050	0.155525	−	0.987832i	$-0.450293\pi$
$167$	4.01965	0.311050	0.155525	+	0.987832i	$0.450293\pi$
$168$	0	0
$169$	−12.3508	−0.950060
$170$	3.04583	0.233605
$171$	0	0
$172$	12.5322	0.955574
$173$	−5.25447	−0.399490	−0.199745	−	0.979848i	$-0.564011\pi$
$173$	−5.25447	−0.399490	−0.199745	+	0.979848i	$0.564011\pi$
$174$	0	0
$175$	−9.01139	−0.681197
$176$	0.0552927	0.00416784
$177$	0	0
$178$	8.25248	0.618550
$179$	2.11290	0.157926	0.0789629	−	0.996878i	$-0.474839\pi$
$179$	2.11290	0.157926	0.0789629	+	0.996878i	$0.474839\pi$
$180$	0	0
$181$	0.230880	0.0171612	0.00858058	−	0.999963i	$-0.497269\pi$
$181$	0.230880	0.0171612	0.00858058	+	0.999963i	$0.497269\pi$
$182$	2.58128	0.191337
$183$	0	0
$184$	0	0
$185$	−4.56705	−0.335776
$186$	0	0
$187$	0.113877	0.00832753
$188$	1.27459	0.0929591
$189$	0	0
$190$	5.43725	0.394460
$191$	4.40318	0.318603	0.159302	−	0.987230i	$-0.449076\pi$
$191$	4.40318	0.318603	0.159302	+	0.987230i	$0.449076\pi$
$192$	0	0
$193$	21.4973	1.54741	0.773704	−	0.633547i	$-0.218402\pi$
$193$	21.4973	1.54741	0.773704	+	0.633547i	$0.218402\pi$
$194$	7.01618	0.503733
$195$	0	0
$196$	3.26315	0.233082
$197$	−8.89209	−0.633535	−0.316768	−	0.948503i	$-0.602598\pi$
$197$	−8.89209	−0.633535	−0.316768	+	0.948503i	$0.602598\pi$
$198$	0	0
$199$	−18.1550	−1.28698	−0.643488	−	0.765456i	$-0.722513\pi$
$199$	−18.1550	−1.28698	−0.643488	+	0.765456i	$0.722513\pi$
$200$	2.81288	0.198901
$201$	0	0
$202$	15.3507	1.08007
$203$	−23.5330	−1.65169
$204$	0	0
$205$	−2.15879	−0.150776
$206$	15.3219	1.06752
$207$	0	0
$208$	−0.805738	−0.0558679
$209$	0.203288	0.0140617
$210$	0	0
$211$	17.2040	1.18437	0.592187	−	0.805801i	$-0.298265\pi$
$211$	17.2040	1.18437	0.592187	+	0.805801i	$0.298265\pi$
$212$	−11.0449	−0.758564
$213$	0	0
$214$	9.67542	0.661398
$215$	−18.5338	−1.26400
$216$	0	0
$217$	−25.4862	−1.73012
$218$	−13.4002	−0.907575
$219$	0	0
$220$	−0.0817718	−0.00551306
$221$	−1.65945	−0.111627
$222$	0	0
$223$	−9.64578	−0.645929	−0.322964	−	0.946411i	$-0.604679\pi$
$223$	−9.64578	−0.645929	−0.322964	+	0.946411i	$0.604679\pi$
$224$	−3.20362	−0.214051
$225$	0	0
$226$	−8.22890	−0.547378
$227$	−13.2879	−0.881951	−0.440975	−	0.897519i	$-0.645367\pi$
$227$	−13.2879	−0.881951	−0.440975	+	0.897519i	$0.645367\pi$
$228$	0	0
$229$	−10.3343	−0.682912	−0.341456	−	0.939898i	$-0.610920\pi$
$229$	−10.3343	−0.682912	−0.341456	+	0.939898i	$0.610920\pi$
$230$	0	0
$231$	0	0
$232$	7.34575	0.482272
$233$	0.221377	0.0145029	0.00725144	−	0.999974i	$-0.497692\pi$
$233$	0.221377	0.0145029	0.00725144	+	0.999974i	$0.497692\pi$
$234$	0	0
$235$	−1.88498	−0.122963
$236$	5.36077	0.348956
$237$	0	0
$238$	−6.59797	−0.427683
$239$	−10.0933	−0.652880	−0.326440	−	0.945218i	$-0.605849\pi$
$239$	−10.0933	−0.652880	−0.326440	+	0.945218i	$0.605849\pi$
$240$	0	0
$241$	2.49660	0.160820	0.0804099	−	0.996762i	$-0.474377\pi$
$241$	2.49660	0.160820	0.0804099	+	0.996762i	$0.474377\pi$
$242$	10.9969	0.706910
$243$	0	0
$244$	−12.2015	−0.781120
$245$	−4.82585	−0.308312
$246$	0	0
$247$	−2.96236	−0.188490
$248$	7.95546	0.505172
$249$	0	0
$250$	−11.5544	−0.730765
$251$	14.3756	0.907379	0.453690	−	0.891160i	$-0.350107\pi$
$251$	14.3756	0.907379	0.453690	+	0.891160i	$0.350107\pi$
$252$	0	0
$253$	0	0
$254$	−19.3524	−1.21428
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.1087	1.06721	0.533606	−	0.845733i	$-0.320836\pi$
$257$	17.1087	1.06721	0.533606	+	0.845733i	$0.320836\pi$
$258$	0	0
$259$	9.89326	0.614738
$260$	1.19160	0.0738999
$261$	0	0
$262$	−20.9028	−1.29138
$263$	−9.82969	−0.606125	−0.303062	−	0.952971i	$-0.598009\pi$
$263$	−9.82969	−0.606125	−0.303062	+	0.952971i	$0.598009\pi$
$264$	0	0
$265$	16.3341	1.00340
$266$	−11.7783	−0.722176
$267$	0	0
$268$	−4.52116	−0.276174
$269$	−27.0424	−1.64881	−0.824404	−	0.566002i	$-0.808489\pi$
$269$	−27.0424	−1.64881	−0.824404	+	0.566002i	$0.808489\pi$
$270$	0	0
$271$	14.0124	0.851195	0.425597	−	0.904913i	$-0.360064\pi$
$271$	14.0124	0.851195	0.425597	+	0.904913i	$0.360064\pi$
$272$	2.05954	0.124878
$273$	0	0
$274$	−5.89909	−0.356377
$275$	−0.155532	−0.00937891
$276$	0	0
$277$	−6.65528	−0.399877	−0.199938	−	0.979808i	$-0.564074\pi$
$277$	−6.65528	−0.399877	−0.199938	+	0.979808i	$0.564074\pi$
$278$	4.04356	0.242517
$279$	0	0
$280$	4.73780	0.283138
$281$	−14.7244	−0.878384	−0.439192	−	0.898393i	$-0.644735\pi$
$281$	−14.7244	−0.878384	−0.439192	+	0.898393i	$0.644735\pi$
$282$	0	0
$283$	13.6907	0.813827	0.406914	−	0.913467i	$-0.366605\pi$
$283$	13.6907	0.813827	0.406914	+	0.913467i	$0.366605\pi$
$284$	14.6029	0.866522
$285$	0	0
$286$	0.0445514	0.00263438
$287$	4.67642	0.276041
$288$	0	0
$289$	−12.7583	−0.750489
$290$	−10.8636	−0.637931
$291$	0	0
$292$	−9.86160	−0.577106
$293$	29.6195	1.73039	0.865194	−	0.501437i	$-0.167195\pi$
$293$	29.6195	1.73039	0.865194	+	0.501437i	$0.167195\pi$
$294$	0	0
$295$	−7.92800	−0.461586
$296$	−3.08816	−0.179495
$297$	0	0
$298$	8.21355	0.475798
$299$	0	0
$300$	0	0
$301$	40.1485	2.31412
$302$	12.8490	0.739380
$303$	0	0
$304$	3.67657	0.210866
$305$	18.0447	1.03324
$306$	0	0
$307$	9.12670	0.520888	0.260444	−	0.965489i	$-0.416131\pi$
$307$	9.12670	0.520888	0.260444	+	0.965489i	$0.416131\pi$
$308$	0.177136	0.0100933
$309$	0	0
$310$	−11.7653	−0.668222
$311$	−22.9754	−1.30281	−0.651407	−	0.758728i	$-0.725821\pi$
$311$	−22.9754	−1.30281	−0.651407	+	0.758728i	$0.725821\pi$
$312$	0	0
$313$	−19.7421	−1.11589	−0.557946	−	0.829877i	$-0.688410\pi$
$313$	−19.7421	−1.11589	−0.557946	+	0.829877i	$0.688410\pi$
$314$	3.91879	0.221150
$315$	0	0
$316$	−3.61983	−0.203631
$317$	20.8318	1.17003	0.585016	−	0.811022i	$-0.301088\pi$
$317$	20.8318	1.17003	0.585016	+	0.811022i	$0.301088\pi$
$318$	0	0
$319$	−0.406166	−0.0227409
$320$	−1.47889	−0.0826725
$321$	0	0
$322$	0	0
$323$	7.57204	0.421320
$324$	0	0
$325$	2.26645	0.125720
$326$	−15.7247	−0.870909
$327$	0	0
$328$	−1.45973	−0.0806003
$329$	4.08330	0.225120
$330$	0	0
$331$	−15.5468	−0.854527	−0.427264	−	0.904127i	$-0.640522\pi$
$331$	−15.5468	−0.854527	−0.427264	+	0.904127i	$0.640522\pi$
$332$	−10.0343	−0.550702
$333$	0	0
$334$	−4.01965	−0.219945
$335$	6.68631	0.365312
$336$	0	0
$337$	−7.07769	−0.385546	−0.192773	−	0.981243i	$-0.561748\pi$
$337$	−7.07769	−0.385546	−0.192773	+	0.981243i	$0.561748\pi$
$338$	12.3508	0.671794
$339$	0	0
$340$	−3.04583	−0.165183
$341$	−0.439879	−0.0238208
$342$	0	0
$343$	−11.9714	−0.646396
$344$	−12.5322	−0.675693
$345$	0	0
$346$	5.25447	0.282482
$347$	25.4158	1.36439	0.682195	−	0.731170i	$-0.261025\pi$
$347$	25.4158	1.36439	0.682195	+	0.731170i	$0.261025\pi$
$348$	0	0
$349$	−23.1316	−1.23821	−0.619104	−	0.785309i	$-0.712504\pi$
$349$	−23.1316	−1.23821	−0.619104	+	0.785309i	$0.712504\pi$
$350$	9.01139	0.481679
$351$	0	0
$352$	−0.0552927	−0.00294711
$353$	−24.0678	−1.28100	−0.640500	−	0.767958i	$-0.721273\pi$
$353$	−24.0678	−1.28100	−0.640500	+	0.767958i	$0.721273\pi$
$354$	0	0
$355$	−21.5961	−1.14620
$356$	−8.25248	−0.437381
$357$	0	0
$358$	−2.11290	−0.111670
$359$	19.1724	1.01188	0.505940	−	0.862569i	$-0.331146\pi$
$359$	19.1724	1.01188	0.505940	+	0.862569i	$0.331146\pi$
$360$	0	0
$361$	−5.48281	−0.288569
$362$	−0.230880	−0.0121348
$363$	0	0
$364$	−2.58128	−0.135296
$365$	14.5842	0.763374
$366$	0	0
$367$	26.2686	1.37121	0.685606	−	0.727973i	$-0.259537\pi$
$367$	26.2686	1.37121	0.685606	+	0.727973i	$0.259537\pi$
$368$	0	0
$369$	0	0
$370$	4.56705	0.237429
$371$	−35.3835	−1.83702
$372$	0	0
$373$	−20.4049	−1.05653	−0.528263	−	0.849081i	$-0.677156\pi$
$373$	−20.4049	−1.05653	−0.528263	+	0.849081i	$0.677156\pi$
$374$	−0.113877	−0.00588846
$375$	0	0
$376$	−1.27459	−0.0657320
$377$	5.91875	0.304831
$378$	0	0
$379$	−3.99029	−0.204968	−0.102484	−	0.994735i	$-0.532679\pi$
$379$	−3.99029	−0.204968	−0.102484	+	0.994735i	$0.532679\pi$
$380$	−5.43725	−0.278925
$381$	0	0
$382$	−4.40318	−0.225286
$383$	6.32220	0.323049	0.161525	−	0.986869i	$-0.448359\pi$
$383$	6.32220	0.323049	0.161525	+	0.986869i	$0.448359\pi$
$384$	0	0
$385$	−0.261966	−0.0133510
$386$	−21.4973	−1.09418
$387$	0	0
$388$	−7.01618	−0.356193
$389$	−25.4165	−1.28867	−0.644335	−	0.764743i	$-0.722866\pi$
$389$	−25.4165	−1.28867	−0.644335	+	0.764743i	$0.722866\pi$
$390$	0	0
$391$	0	0
$392$	−3.26315	−0.164814
$393$	0	0
$394$	8.89209	0.447977
$395$	5.35333	0.269355
$396$	0	0
$397$	−3.43509	−0.172402	−0.0862011	−	0.996278i	$-0.527473\pi$
$397$	−3.43509	−0.172402	−0.0862011	+	0.996278i	$0.527473\pi$
$398$	18.1550	0.910029
$399$	0	0
$400$	−2.81288	−0.140644
$401$	−8.35498	−0.417228	−0.208614	−	0.977998i	$-0.566895\pi$
$401$	−8.35498	−0.417228	−0.208614	+	0.977998i	$0.566895\pi$
$402$	0	0
$403$	6.41002	0.319306
$404$	−15.3507	−0.763725
$405$	0	0
$406$	23.5330	1.16792
$407$	0.170752	0.00846388
$408$	0	0
$409$	−4.18657	−0.207013	−0.103506	−	0.994629i	$-0.533006\pi$
$409$	−4.18657	−0.207013	−0.103506	+	0.994629i	$0.533006\pi$
$410$	2.15879	0.106615
$411$	0	0
$412$	−15.3219	−0.754854
$413$	17.1738	0.845070
$414$	0	0
$415$	14.8396	0.728447
$416$	0.805738	0.0395046
$417$	0	0
$418$	−0.203288	−0.00994312
$419$	8.98508	0.438950	0.219475	−	0.975618i	$-0.429566\pi$
$419$	8.98508	0.438950	0.219475	+	0.975618i	$0.429566\pi$
$420$	0	0
$421$	6.75886	0.329407	0.164703	−	0.986343i	$-0.447333\pi$
$421$	6.75886	0.329407	0.164703	+	0.986343i	$0.447333\pi$
$422$	−17.2040	−0.837478
$423$	0	0
$424$	11.0449	0.536386
$425$	−5.79323	−0.281013
$426$	0	0
$427$	−39.0889	−1.89164
$428$	−9.67542	−0.467679
$429$	0	0
$430$	18.5338	0.893780
$431$	−22.6737	−1.09216	−0.546078	−	0.837734i	$-0.683880\pi$
$431$	−22.6737	−1.09216	−0.546078	+	0.837734i	$0.683880\pi$
$432$	0	0
$433$	−29.9541	−1.43950	−0.719752	−	0.694231i	$-0.755744\pi$
$433$	−29.9541	−1.43950	−0.719752	+	0.694231i	$0.755744\pi$
$434$	25.4862	1.22338
$435$	0	0
$436$	13.4002	0.641752
$437$	0	0
$438$	0	0
$439$	−1.50141	−0.0716585	−0.0358292	−	0.999358i	$-0.511407\pi$
$439$	−1.50141	−0.0716585	−0.0358292	+	0.999358i	$0.511407\pi$
$440$	0.0817718	0.00389832
$441$	0	0
$442$	1.65945	0.0789319
$443$	5.89537	0.280097	0.140049	−	0.990145i	$-0.455274\pi$
$443$	5.89537	0.280097	0.140049	+	0.990145i	$0.455274\pi$
$444$	0	0
$445$	12.2045	0.578550
$446$	9.64578	0.456741
$447$	0	0
$448$	3.20362	0.151357
$449$	36.8587	1.73947	0.869734	−	0.493520i	$-0.164290\pi$
$449$	36.8587	1.73947	0.869734	+	0.493520i	$0.164290\pi$
$450$	0	0
$451$	0.0807125	0.00380061
$452$	8.22890	0.387055
$453$	0	0
$454$	13.2879	0.623633
$455$	3.81743	0.178964
$456$	0	0
$457$	5.79767	0.271203	0.135602	−	0.990763i	$-0.456703\pi$
$457$	5.79767	0.271203	0.135602	+	0.990763i	$0.456703\pi$
$458$	10.3343	0.482892
$459$	0	0
$460$	0	0
$461$	19.1277	0.890864	0.445432	−	0.895316i	$-0.353050\pi$
$461$	19.1277	0.890864	0.445432	+	0.895316i	$0.353050\pi$
$462$	0	0
$463$	−20.3367	−0.945126	−0.472563	−	0.881297i	$-0.656671\pi$
$463$	−20.3367	−0.945126	−0.472563	+	0.881297i	$0.656671\pi$
$464$	−7.34575	−0.341018
$465$	0	0
$466$	−0.221377	−0.0102551
$467$	27.1467	1.25620	0.628101	−	0.778132i	$-0.283833\pi$
$467$	27.1467	1.25620	0.628101	+	0.778132i	$0.283833\pi$
$468$	0	0
$469$	−14.4841	−0.668812
$470$	1.88498	0.0869477
$471$	0	0
$472$	−5.36077	−0.246749
$473$	0.692941	0.0318615
$474$	0	0
$475$	−10.3418	−0.474512
$476$	6.59797	0.302417
$477$	0	0
$478$	10.0933	0.461656
$479$	−3.73359	−0.170592	−0.0852961	−	0.996356i	$-0.527184\pi$
$479$	−3.73359	−0.170592	−0.0852961	+	0.996356i	$0.527184\pi$
$480$	0	0
$481$	−2.48825	−0.113454
$482$	−2.49660	−0.113717
$483$	0	0
$484$	−10.9969	−0.499861
$485$	10.3762	0.471158
$486$	0	0
$487$	23.0573	1.04483	0.522413	−	0.852693i	$-0.325032\pi$
$487$	23.0573	1.04483	0.522413	+	0.852693i	$0.325032\pi$
$488$	12.2015	0.552336
$489$	0	0
$490$	4.82585	0.218010
$491$	1.12229	0.0506483	0.0253241	−	0.999679i	$-0.491938\pi$
$491$	1.12229	0.0506483	0.0253241	+	0.999679i	$0.491938\pi$
$492$	0	0
$493$	−15.1288	−0.681369
$494$	2.96236	0.133283
$495$	0	0
$496$	−7.95546	−0.357211
$497$	46.7821	2.09846
$498$	0	0
$499$	39.6576	1.77532	0.887659	−	0.460502i	$-0.152331\pi$
$499$	39.6576	1.77532	0.887659	+	0.460502i	$0.152331\pi$
$500$	11.5544	0.516729
$501$	0	0
$502$	−14.3756	−0.641614
$503$	−34.0385	−1.51770	−0.758850	−	0.651266i	$-0.774238\pi$
$503$	−34.0385	−1.51770	−0.758850	+	0.651266i	$0.774238\pi$
$504$	0	0
$505$	22.7020	1.01023
$506$	0	0
$507$	0	0
$508$	19.3524	0.858624
$509$	−0.940632	−0.0416928	−0.0208464	−	0.999783i	$-0.506636\pi$
$509$	−0.940632	−0.0416928	−0.0208464	+	0.999783i	$0.506636\pi$
$510$	0	0
$511$	−31.5928	−1.39758
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−17.1087	−0.754633
$515$	22.6594	0.998491
$516$	0	0
$517$	0.0704755	0.00309951
$518$	−9.89326	−0.434685
$519$	0	0
$520$	−1.19160	−0.0522551
$521$	−40.4064	−1.77024	−0.885119	−	0.465366i	$-0.845923\pi$
$521$	−40.4064	−1.77024	−0.885119	+	0.465366i	$0.845923\pi$
$522$	0	0
$523$	14.0908	0.616146	0.308073	−	0.951363i	$-0.400316\pi$
$523$	14.0908	0.616146	0.308073	+	0.951363i	$0.400316\pi$
$524$	20.9028	0.913141
$525$	0	0
$526$	9.82969	0.428595
$527$	−16.3846	−0.713723
$528$	0	0
$529$	0	0
$530$	−16.3341	−0.709510
$531$	0	0
$532$	11.7783	0.510655
$533$	−1.17616	−0.0509453
$534$	0	0
$535$	14.3089	0.618628
$536$	4.52116	0.195285
$537$	0	0
$538$	27.0424	1.16588
$539$	0.180428	0.00777160
$540$	0	0
$541$	−32.0267	−1.37694	−0.688468	−	0.725267i	$-0.741716\pi$
$541$	−32.0267	−1.37694	−0.688468	+	0.725267i	$0.741716\pi$
$542$	−14.0124	−0.601885
$543$	0	0
$544$	−2.05954	−0.0883019
$545$	−19.8174	−0.848885
$546$	0	0
$547$	−34.2269	−1.46344	−0.731718	−	0.681607i	$-0.761281\pi$
$547$	−34.2269	−1.46344	−0.731718	+	0.681607i	$0.761281\pi$
$548$	5.89909	0.251997
$549$	0	0
$550$	0.155532	0.00663189
$551$	−27.0072	−1.15054
$552$	0	0
$553$	−11.5965	−0.493135
$554$	6.65528	0.282756
$555$	0	0
$556$	−4.04356	−0.171485
$557$	−11.3076	−0.479117	−0.239559	−	0.970882i	$-0.577003\pi$
$557$	−11.3076	−0.479117	−0.239559	+	0.970882i	$0.577003\pi$
$558$	0	0
$559$	−10.0977	−0.427087
$560$	−4.73780	−0.200209
$561$	0	0
$562$	14.7244	0.621111
$563$	17.2976	0.729006	0.364503	−	0.931202i	$-0.381239\pi$
$563$	17.2976	0.729006	0.364503	+	0.931202i	$0.381239\pi$
$564$	0	0
$565$	−12.1697	−0.511981
$566$	−13.6907	−0.575463
$567$	0	0
$568$	−14.6029	−0.612724
$569$	29.0600	1.21826	0.609129	−	0.793071i	$-0.291519\pi$
$569$	29.0600	1.21826	0.609129	+	0.793071i	$0.291519\pi$
$570$	0	0
$571$	−3.14635	−0.131671	−0.0658354	−	0.997830i	$-0.520971\pi$
$571$	−3.14635	−0.131671	−0.0658354	+	0.997830i	$0.520971\pi$
$572$	−0.0445514	−0.00186279
$573$	0	0
$574$	−4.67642	−0.195190
$575$	0	0
$576$	0	0
$577$	10.8526	0.451799	0.225900	−	0.974151i	$-0.427468\pi$
$577$	10.8526	0.451799	0.225900	+	0.974151i	$0.427468\pi$
$578$	12.7583	0.530676
$579$	0	0
$580$	10.8636	0.451085
$581$	−32.1459	−1.33364
$582$	0	0
$583$	−0.610700	−0.0252926
$584$	9.86160	0.408076
$585$	0	0
$586$	−29.6195	−1.22357
$587$	32.2820	1.33242	0.666210	−	0.745764i	$-0.267915\pi$
$587$	32.2820	1.33242	0.666210	+	0.745764i	$0.267915\pi$
$588$	0	0
$589$	−29.2488	−1.20518
$590$	7.92800	0.326390
$591$	0	0
$592$	3.08816	0.126922
$593$	−35.1225	−1.44231	−0.721154	−	0.692774i	$-0.756388\pi$
$593$	−35.1225	−1.44231	−0.721154	+	0.692774i	$0.756388\pi$
$594$	0	0
$595$	−9.75768	−0.400026
$596$	−8.21355	−0.336440
$597$	0	0
$598$	0	0
$599$	33.0831	1.35174	0.675870	−	0.737021i	$-0.263768\pi$
$599$	33.0831	1.35174	0.675870	+	0.737021i	$0.263768\pi$
$600$	0	0
$601$	−40.9247	−1.66935	−0.834677	−	0.550740i	$-0.814345\pi$
$601$	−40.9247	−1.66935	−0.834677	+	0.550740i	$0.814345\pi$
$602$	−40.1485	−1.63633
$603$	0	0
$604$	−12.8490	−0.522820
$605$	16.2633	0.661197
$606$	0	0
$607$	0.867250	0.0352006	0.0176003	−	0.999845i	$-0.494397\pi$
$607$	0.867250	0.0352006	0.0176003	+	0.999845i	$0.494397\pi$
$608$	−3.67657	−0.149105
$609$	0	0
$610$	−18.0447	−0.730608
$611$	−1.02699	−0.0415474
$612$	0	0
$613$	−13.5982	−0.549227	−0.274614	−	0.961555i	$-0.588550\pi$
$613$	−13.5982	−0.549227	−0.274614	+	0.961555i	$0.588550\pi$
$614$	−9.12670	−0.368324
$615$	0	0
$616$	−0.177136	−0.00713703
$617$	−42.6638	−1.71758	−0.858790	−	0.512328i	$-0.828783\pi$
$617$	−42.6638	−1.71758	−0.858790	+	0.512328i	$0.828783\pi$
$618$	0	0
$619$	31.5251	1.26710	0.633551	−	0.773701i	$-0.281597\pi$
$619$	31.5251	1.26710	0.633551	+	0.773701i	$0.281597\pi$
$620$	11.7653	0.472505
$621$	0	0
$622$	22.9754	0.921229
$623$	−26.4378	−1.05921
$624$	0	0
$625$	−3.02331	−0.120932
$626$	19.7421	0.789054
$627$	0	0
$628$	−3.91879	−0.156377
$629$	6.36017	0.253597
$630$	0	0
$631$	48.7324	1.94001	0.970003	−	0.243093i	$-0.0781622\pi$
$631$	48.7324	1.94001	0.970003	+	0.243093i	$0.0781622\pi$
$632$	3.61983	0.143989
$633$	0	0
$634$	−20.8318	−0.827337
$635$	−28.6201	−1.13575
$636$	0	0
$637$	−2.62925	−0.104175
$638$	0.406166	0.0160803
$639$	0	0
$640$	1.47889	0.0584583
$641$	30.1834	1.19217	0.596086	−	0.802920i	$-0.296722\pi$
$641$	30.1834	1.19217	0.596086	+	0.802920i	$0.296722\pi$
$642$	0	0
$643$	−34.4723	−1.35946	−0.679728	−	0.733464i	$-0.737902\pi$
$643$	−34.4723	−1.35946	−0.679728	+	0.733464i	$0.737902\pi$
$644$	0	0
$645$	0	0
$646$	−7.57204	−0.297918
$647$	−13.9668	−0.549092	−0.274546	−	0.961574i	$-0.588528\pi$
$647$	−13.9668	−0.549092	−0.274546	+	0.961574i	$0.588528\pi$
$648$	0	0
$649$	0.296411	0.0116352
$650$	−2.26645	−0.0888973
$651$	0	0
$652$	15.7247	0.615825
$653$	−36.9272	−1.44507	−0.722537	−	0.691332i	$-0.757024\pi$
$653$	−36.9272	−1.44507	−0.722537	+	0.691332i	$0.757024\pi$
$654$	0	0
$655$	−30.9129	−1.20787
$656$	1.45973	0.0569930
$657$	0	0
$658$	−4.08330	−0.159184
$659$	−11.0352	−0.429872	−0.214936	−	0.976628i	$-0.568954\pi$
$659$	−11.0352	−0.429872	−0.214936	+	0.976628i	$0.568954\pi$
$660$	0	0
$661$	−33.9751	−1.32148	−0.660738	−	0.750616i	$-0.729757\pi$
$661$	−33.9751	−1.32148	−0.660738	+	0.750616i	$0.729757\pi$
$662$	15.5468	0.604242
$663$	0	0
$664$	10.0343	0.389405
$665$	−17.4189	−0.675475
$666$	0	0
$667$	0	0
$668$	4.01965	0.155525
$669$	0	0
$670$	−6.68631	−0.258315
$671$	−0.674653	−0.0260447
$672$	0	0
$673$	−31.3928	−1.21010	−0.605052	−	0.796186i	$-0.706848\pi$
$673$	−31.3928	−1.21010	−0.605052	+	0.796186i	$0.706848\pi$
$674$	7.07769	0.272622
$675$	0	0
$676$	−12.3508	−0.475030
$677$	−39.3382	−1.51189	−0.755944	−	0.654636i	$-0.772822\pi$
$677$	−39.3382	−1.51189	−0.755944	+	0.654636i	$0.772822\pi$
$678$	0	0
$679$	−22.4772	−0.862594
$680$	3.04583	0.116802
$681$	0	0
$682$	0.439879	0.0168438
$683$	−31.6224	−1.21000	−0.604998	−	0.796227i	$-0.706826\pi$
$683$	−31.6224	−1.21000	−0.604998	+	0.796227i	$0.706826\pi$
$684$	0	0
$685$	−8.72412	−0.333331
$686$	11.9714	0.457071
$687$	0	0
$688$	12.5322	0.477787
$689$	8.89927	0.339035
$690$	0	0
$691$	22.2620	0.846886	0.423443	−	0.905923i	$-0.360821\pi$
$691$	22.2620	0.846886	0.423443	+	0.905923i	$0.360821\pi$
$692$	−5.25447	−0.199745
$693$	0	0
$694$	−25.4158	−0.964770
$695$	5.97999	0.226834
$696$	0	0
$697$	3.00638	0.113875
$698$	23.1316	0.875545
$699$	0	0
$700$	−9.01139	−0.340598
$701$	13.8462	0.522964	0.261482	−	0.965208i	$-0.415789\pi$
$701$	13.8462	0.522964	0.261482	+	0.965208i	$0.415789\pi$
$702$	0	0
$703$	11.3538	0.428218
$704$	0.0552927	0.00208392
$705$	0	0
$706$	24.0678	0.905804
$707$	−49.1777	−1.84952
$708$	0	0
$709$	−49.1026	−1.84409	−0.922043	−	0.387087i	$-0.873481\pi$
$709$	−49.1026	−1.84409	−0.922043	+	0.387087i	$0.873481\pi$
$710$	21.5961	0.810487
$711$	0	0
$712$	8.25248	0.309275
$713$	0	0
$714$	0	0
$715$	0.0658867	0.00246402
$716$	2.11290	0.0789629
$717$	0	0
$718$	−19.1724	−0.715507
$719$	−36.5928	−1.36468	−0.682341	−	0.731034i	$-0.739038\pi$
$719$	−36.5928	−1.36468	−0.682341	+	0.731034i	$0.739038\pi$
$720$	0	0
$721$	−49.0853	−1.82803
$722$	5.48281	0.204049
$723$	0	0
$724$	0.230880	0.00858058
$725$	20.6627	0.767394
$726$	0	0
$727$	−18.6107	−0.690232	−0.345116	−	0.938560i	$-0.612160\pi$
$727$	−18.6107	−0.690232	−0.345116	+	0.938560i	$0.612160\pi$
$728$	2.58128	0.0956684
$729$	0	0
$730$	−14.5842	−0.539787
$731$	25.8106	0.954640
$732$	0	0
$733$	22.6156	0.835327	0.417664	−	0.908602i	$-0.362849\pi$
$733$	22.6156	0.835327	0.417664	+	0.908602i	$0.362849\pi$
$734$	−26.2686	−0.969593
$735$	0	0
$736$	0	0
$737$	−0.249987	−0.00920840
$738$	0	0
$739$	−17.8869	−0.657982	−0.328991	−	0.944333i	$-0.606709\pi$
$739$	−17.8869	−0.657982	−0.328991	+	0.944333i	$0.606709\pi$
$740$	−4.56705	−0.167888
$741$	0	0
$742$	35.3835	1.29897
$743$	−22.4379	−0.823168	−0.411584	−	0.911372i	$-0.635024\pi$
$743$	−22.4379	−0.823168	−0.411584	+	0.911372i	$0.635024\pi$
$744$	0	0
$745$	12.1469	0.445030
$746$	20.4049	0.747076
$747$	0	0
$748$	0.113877	0.00416377
$749$	−30.9963	−1.13258
$750$	0	0
$751$	−46.9846	−1.71449	−0.857246	−	0.514907i	$-0.827826\pi$
$751$	−46.9846	−1.71449	−0.857246	+	0.514907i	$0.827826\pi$
$752$	1.27459	0.0464795
$753$	0	0
$754$	−5.91875	−0.215548
$755$	19.0023	0.691566
$756$	0	0
$757$	−28.7306	−1.04423	−0.522115	−	0.852875i	$-0.674857\pi$
$757$	−28.7306	−1.04423	−0.522115	+	0.852875i	$0.674857\pi$
$758$	3.99029	0.144934
$759$	0	0
$760$	5.43725	0.197230
$761$	−15.6869	−0.568651	−0.284326	−	0.958728i	$-0.591770\pi$
$761$	−15.6869	−0.568651	−0.284326	+	0.958728i	$0.591770\pi$
$762$	0	0
$763$	42.9290	1.55413
$764$	4.40318	0.159302
$765$	0	0
$766$	−6.32220	−0.228430
$767$	−4.31938	−0.155964
$768$	0	0
$769$	19.8140	0.714509	0.357255	−	0.934007i	$-0.383713\pi$
$769$	19.8140	0.714509	0.357255	+	0.934007i	$0.383713\pi$
$770$	0.261966	0.00944058
$771$	0	0
$772$	21.4973	0.773704
$773$	19.5152	0.701915	0.350957	−	0.936391i	$-0.385856\pi$
$773$	19.5152	0.701915	0.350957	+	0.936391i	$0.385856\pi$
$774$	0	0
$775$	22.3778	0.803833
$776$	7.01618	0.251866
$777$	0	0
$778$	25.4165	0.911227
$779$	5.36682	0.192286
$780$	0	0
$781$	0.807433	0.0288922
$782$	0	0
$783$	0	0
$784$	3.26315	0.116541
$785$	5.79547	0.206849
$786$	0	0
$787$	−20.0112	−0.713321	−0.356661	−	0.934234i	$-0.616085\pi$
$787$	−20.0112	−0.713321	−0.356661	+	0.934234i	$0.616085\pi$
$788$	−8.89209	−0.316768
$789$	0	0
$790$	−5.35333	−0.190463
$791$	26.3622	0.937333
$792$	0	0
$793$	9.83121	0.349116
$794$	3.43509	0.121907
$795$	0	0
$796$	−18.1550	−0.643488
$797$	−19.7315	−0.698927	−0.349464	−	0.936950i	$-0.613636\pi$
$797$	−19.7315	−0.698927	−0.349464	+	0.936950i	$0.613636\pi$
$798$	0	0
$799$	2.62507	0.0928682
$800$	2.81288	0.0994503
$801$	0	0
$802$	8.35498	0.295024
$803$	−0.545274	−0.0192423
$804$	0	0
$805$	0	0
$806$	−6.41002	−0.225783
$807$	0	0
$808$	15.3507	0.540035
$809$	−27.7606	−0.976009	−0.488005	−	0.872841i	$-0.662275\pi$
$809$	−27.7606	−0.976009	−0.488005	+	0.872841i	$0.662275\pi$
$810$	0	0
$811$	18.4982	0.649560	0.324780	−	0.945790i	$-0.394710\pi$
$811$	18.4982	0.649560	0.324780	+	0.945790i	$0.394710\pi$
$812$	−23.5330	−0.825845
$813$	0	0
$814$	−0.170752	−0.00598487
$815$	−23.2551	−0.814590
$816$	0	0
$817$	46.0757	1.61198
$818$	4.18657	0.146380
$819$	0	0
$820$	−2.15879	−0.0753881
$821$	3.89872	0.136066	0.0680332	−	0.997683i	$-0.478328\pi$
$821$	3.89872	0.136066	0.0680332	+	0.997683i	$0.478328\pi$
$822$	0	0
$823$	−13.6654	−0.476345	−0.238173	−	0.971223i	$-0.576548\pi$
$823$	−13.6654	−0.476345	−0.238173	+	0.971223i	$0.576548\pi$
$824$	15.3219	0.533762
$825$	0	0
$826$	−17.1738	−0.597554
$827$	1.97288	0.0686037	0.0343019	−	0.999412i	$-0.489079\pi$
$827$	1.97288	0.0686037	0.0343019	+	0.999412i	$0.489079\pi$
$828$	0	0
$829$	6.37133	0.221285	0.110643	−	0.993860i	$-0.464709\pi$
$829$	6.37133	0.221285	0.110643	+	0.993860i	$0.464709\pi$
$830$	−14.8396	−0.515090
$831$	0	0
$832$	−0.805738	−0.0279339
$833$	6.72059	0.232855
$834$	0	0
$835$	−5.94462	−0.205722
$836$	0.203288	0.00703085
$837$	0	0
$838$	−8.98508	−0.310384
$839$	−0.347463	−0.0119957	−0.00599787	−	0.999982i	$-0.501909\pi$
$839$	−0.347463	−0.0119957	−0.00599787	+	0.999982i	$0.501909\pi$
$840$	0	0
$841$	24.9600	0.860691
$842$	−6.75886	−0.232926
$843$	0	0
$844$	17.2040	0.592187
$845$	18.2655	0.628351
$846$	0	0
$847$	−35.2300	−1.21052
$848$	−11.0449	−0.379282
$849$	0	0
$850$	5.79323	0.198706
$851$	0	0
$852$	0	0
$853$	−22.0786	−0.755958	−0.377979	−	0.925814i	$-0.623381\pi$
$853$	−22.0786	−0.755958	−0.377979	+	0.925814i	$0.623381\pi$
$854$	39.0889	1.33759
$855$	0	0
$856$	9.67542	0.330699
$857$	19.3638	0.661454	0.330727	−	0.943726i	$-0.392706\pi$
$857$	19.3638	0.661454	0.330727	+	0.943726i	$0.392706\pi$
$858$	0	0
$859$	−10.1662	−0.346866	−0.173433	−	0.984846i	$-0.555486\pi$
$859$	−10.1662	−0.346866	−0.173433	+	0.984846i	$0.555486\pi$
$860$	−18.5338	−0.631998
$861$	0	0
$862$	22.6737	0.772271
$863$	11.7222	0.399027	0.199514	−	0.979895i	$-0.436064\pi$
$863$	11.7222	0.399027	0.199514	+	0.979895i	$0.436064\pi$
$864$	0	0
$865$	7.77079	0.264215
$866$	29.9541	1.01788
$867$	0	0
$868$	−25.4862	−0.865060
$869$	−0.200150	−0.00678962
$870$	0	0
$871$	3.64287	0.123434
$872$	−13.4002	−0.453787
$873$	0	0
$874$	0	0
$875$	37.0159	1.25136
$876$	0	0
$877$	−30.3909	−1.02623	−0.513114	−	0.858321i	$-0.671508\pi$
$877$	−30.3909	−1.02623	−0.513114	+	0.858321i	$0.671508\pi$
$878$	1.50141	0.0506702
$879$	0	0
$880$	−0.0817718	−0.00275653
$881$	−33.3403	−1.12326	−0.561632	−	0.827387i	$-0.689826\pi$
$881$	−33.3403	−1.12326	−0.561632	+	0.827387i	$0.689826\pi$
$882$	0	0
$883$	−5.16297	−0.173748	−0.0868740	−	0.996219i	$-0.527688\pi$
$883$	−5.16297	−0.173748	−0.0868740	+	0.996219i	$0.527688\pi$
$884$	−1.65945	−0.0558133
$885$	0	0
$886$	−5.89537	−0.198059
$887$	13.6271	0.457555	0.228777	−	0.973479i	$-0.426527\pi$
$887$	13.6271	0.457555	0.228777	+	0.973479i	$0.426527\pi$
$888$	0	0
$889$	61.9977	2.07933
$890$	−12.2045	−0.409097
$891$	0	0
$892$	−9.64578	−0.322964
$893$	4.68613	0.156815
$894$	0	0
$895$	−3.12476	−0.104449
$896$	−3.20362	−0.107025
$897$	0	0
$898$	−36.8587	−1.22999
$899$	58.4388	1.94904
$900$	0	0
$901$	−22.7473	−0.757822
$902$	−0.0807125	−0.00268743
$903$	0	0
$904$	−8.22890	−0.273689
$905$	−0.341446	−0.0113500
$906$	0	0
$907$	22.2146	0.737623	0.368811	−	0.929504i	$-0.379765\pi$
$907$	22.2146	0.737623	0.368811	+	0.929504i	$0.379765\pi$
$908$	−13.2879	−0.440975
$909$	0	0
$910$	−3.81743	−0.126546
$911$	−18.1822	−0.602402	−0.301201	−	0.953561i	$-0.597388\pi$
$911$	−18.1822	−0.602402	−0.301201	+	0.953561i	$0.597388\pi$
$912$	0	0
$913$	−0.554821	−0.0183619
$914$	−5.79767	−0.191770
$915$	0	0
$916$	−10.3343	−0.341456
$917$	66.9644	2.21136
$918$	0	0
$919$	30.2025	0.996289	0.498145	−	0.867094i	$-0.334015\pi$
$919$	30.2025	0.996289	0.498145	+	0.867094i	$0.334015\pi$
$920$	0	0
$921$	0	0
$922$	−19.1277	−0.629936
$923$	−11.7661	−0.387286
$924$	0	0
$925$	−8.68661	−0.285614
$926$	20.3367	0.668305
$927$	0	0
$928$	7.34575	0.241136
$929$	25.8080	0.846733	0.423366	−	0.905959i	$-0.360848\pi$
$929$	25.8080	0.846733	0.423366	+	0.905959i	$0.360848\pi$
$930$	0	0
$931$	11.9972	0.393193
$932$	0.221377	0.00725144
$933$	0	0
$934$	−27.1467	−0.888268
$935$	−0.168412	−0.00550767
$936$	0	0
$937$	15.1514	0.494976	0.247488	−	0.968891i	$-0.420395\pi$
$937$	15.1514	0.494976	0.247488	+	0.968891i	$0.420395\pi$
$938$	14.4841	0.472922
$939$	0	0
$940$	−1.88498	−0.0614813
$941$	25.1968	0.821391	0.410695	−	0.911773i	$-0.365286\pi$
$941$	25.1968	0.821391	0.410695	+	0.911773i	$0.365286\pi$
$942$	0	0
$943$	0	0
$944$	5.36077	0.174478
$945$	0	0
$946$	−0.692941	−0.0225295
$947$	0.612556	0.0199054	0.00995271	−	0.999950i	$-0.496832\pi$
$947$	0.612556	0.0199054	0.00995271	+	0.999950i	$0.496832\pi$
$948$	0	0
$949$	7.94587	0.257934
$950$	10.3418	0.335531
$951$	0	0
$952$	−6.59797	−0.213841
$953$	34.2779	1.11037	0.555185	−	0.831727i	$-0.312647\pi$
$953$	34.2779	1.11037	0.555185	+	0.831727i	$0.312647\pi$
$954$	0	0
$955$	−6.51183	−0.210718
$956$	−10.0933	−0.326440
$957$	0	0
$958$	3.73359	0.120627
$959$	18.8984	0.610262
$960$	0	0
$961$	32.2894	1.04159
$962$	2.48825	0.0802242
$963$	0	0
$964$	2.49660	0.0804099
$965$	−31.7921	−1.02343
$966$	0	0
$967$	32.9004	1.05801	0.529003	−	0.848620i	$-0.322566\pi$
$967$	32.9004	1.05801	0.529003	+	0.848620i	$0.322566\pi$
$968$	10.9969	0.353455
$969$	0	0
$970$	−10.3762	−0.333159
$971$	−15.3464	−0.492488	−0.246244	−	0.969208i	$-0.579197\pi$
$971$	−15.3464	−0.492488	−0.246244	+	0.969208i	$0.579197\pi$
$972$	0	0
$973$	−12.9540	−0.415287
$974$	−23.0573	−0.738804
$975$	0	0
$976$	−12.2015	−0.390560
$977$	−7.59382	−0.242948	−0.121474	−	0.992595i	$-0.538762\pi$
$977$	−7.59382	−0.242948	−0.121474	+	0.992595i	$0.538762\pi$
$978$	0	0
$979$	−0.456302	−0.0145835
$980$	−4.82585	−0.154156
$981$	0	0
$982$	−1.12229	−0.0358137
$983$	2.79957	0.0892924	0.0446462	−	0.999003i	$-0.485784\pi$
$983$	2.79957	0.0892924	0.0446462	+	0.999003i	$0.485784\pi$
$984$	0	0
$985$	13.1504	0.419008
$986$	15.1288	0.481801
$987$	0	0
$988$	−2.96236	−0.0942451
$989$	0	0
$990$	0	0
$991$	0.506808	0.0160993	0.00804964	−	0.999968i	$-0.497438\pi$
$991$	0.506808	0.0160993	0.00804964	+	0.999968i	$0.497438\pi$
$992$	7.95546	0.252586
$993$	0	0
$994$	−46.7821	−1.48384
$995$	26.8493	0.851181
$996$	0	0
$997$	12.1136	0.383640	0.191820	−	0.981430i	$-0.438561\pi$
$997$	12.1136	0.383640	0.191820	+	0.981430i	$0.438561\pi$
$998$	−39.6576	−1.25534
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bq.1.3		5
3.2	odd	2		3174.2.a.bd.1.3		5
23.17	odd	22		414.2.i.d.289.1		10
23.19	odd	22		414.2.i.d.361.1		10
23.22	odd	2		9522.2.a.bt.1.3		5
69.17	even	22		138.2.e.a.13.1	✓	10
69.65	even	22		138.2.e.a.85.1	yes	10
69.68	even	2		3174.2.a.bc.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.a.13.1	✓	10	69.17	even	22
138.2.e.a.85.1	yes	10	69.65	even	22
414.2.i.d.289.1		10	23.17	odd	22
414.2.i.d.361.1		10	23.19	odd	22
3174.2.a.bc.1.3		5	69.68	even	2
3174.2.a.bd.1.3		5	3.2	odd	2
9522.2.a.bq.1.3		5	1.1	even	1	trivial
9522.2.a.bt.1.3		5	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.47889 q^{5} +3.20362 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.47889 q^{5} +3.20362 q^{7} -1.00000 q^{8} +1.47889 q^{10} +0.0552927 q^{11} -0.805738 q^{13} -3.20362 q^{14} +1.00000 q^{16} +2.05954 q^{17} +3.67657 q^{19} -1.47889 q^{20} -0.0552927 q^{22} -2.81288 q^{25} +0.805738 q^{26} +3.20362 q^{28} -7.34575 q^{29} -7.95546 q^{31} -1.00000 q^{32} -2.05954 q^{34} -4.73780 q^{35} +3.08816 q^{37} -3.67657 q^{38} +1.47889 q^{40} +1.45973 q^{41} +12.5322 q^{43} +0.0552927 q^{44} +1.27459 q^{47} +3.26315 q^{49} +2.81288 q^{50} -0.805738 q^{52} -11.0449 q^{53} -0.0817718 q^{55} -3.20362 q^{56} +7.34575 q^{58} +5.36077 q^{59} -12.2015 q^{61} +7.95546 q^{62} +1.00000 q^{64} +1.19160 q^{65} -4.52116 q^{67} +2.05954 q^{68} +4.73780 q^{70} +14.6029 q^{71} -9.86160 q^{73} -3.08816 q^{74} +3.67657 q^{76} +0.177136 q^{77} -3.61983 q^{79} -1.47889 q^{80} -1.45973 q^{82} -10.0343 q^{83} -3.04583 q^{85} -12.5322 q^{86} -0.0552927 q^{88} -8.25248 q^{89} -2.58128 q^{91} -1.27459 q^{94} -5.43725 q^{95} -7.01618 q^{97} -3.26315 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8} + 7 q^{10} - 13 q^{11} - 4 q^{13} - 7 q^{14} + 5 q^{16} - 9 q^{17} + 11 q^{19} - 7 q^{20} + 13 q^{22} - 2 q^{25} + 4 q^{26} + 7 q^{28} + 7 q^{29} - 8 q^{31} - 5 q^{32} + 9 q^{34} - q^{35} + 12 q^{37} - 11 q^{38} + 7 q^{40} + 10 q^{41} + 4 q^{43} - 13 q^{44} + 24 q^{47} - 12 q^{49} + 2 q^{50} - 4 q^{52} - 9 q^{53} + 16 q^{55} - 7 q^{56} - 7 q^{58} + 14 q^{59} + 5 q^{61} + 8 q^{62} + 5 q^{64} - 12 q^{65} + 13 q^{67} - 9 q^{68} + q^{70} + 19 q^{71} + 4 q^{73} - 12 q^{74} + 11 q^{76} - 5 q^{77} + 4 q^{79} - 7 q^{80} - 10 q^{82} - 24 q^{83} + 17 q^{85} - 4 q^{86} + 13 q^{88} - 4 q^{89} - 21 q^{91} - 24 q^{94} + 11 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8 + 7 * q^10 - 13 * q^11 - 4 * q^13 - 7 * q^14 + 5 * q^16 - 9 * q^17 + 11 * q^19 - 7 * q^20 + 13 * q^22 - 2 * q^25 + 4 * q^26 + 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 + 9 * q^34 - q^35 + 12 * q^37 - 11 * q^38 + 7 * q^40 + 10 * q^41 + 4 * q^43 - 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 - 9 * q^53 + 16 * q^55 - 7 * q^56 - 7 * q^58 + 14 * q^59 + 5 * q^61 + 8 * q^62 + 5 * q^64 - 12 * q^65 + 13 * q^67 - 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 - 12 * q^74 + 11 * q^76 - 5 * q^77 + 4 * q^79 - 7 * q^80 - 10 * q^82 - 24 * q^83 + 17 * q^85 - 4 * q^86 + 13 * q^88 - 4 * q^89 - 21 * q^91 - 24 * q^94 + 11 * q^95 - 9 * q^97 + 12 * q^98

Introduction

L-functions

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