LMFDB - Embedded newform 3680.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.876604.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 8x^{2} + 18x - 16 $ x^5 - x^4 - 9x^3 + 8x^2 + 18*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.60527$ of defining polynomial
Character	$\chi$	$=$	3680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.352411 q^{3} +1.00000 q^{5} -3.95190 q^{7} -2.87581 q^{9} +O(q^{10})$	$q+0.352411 q^{3} +1.00000 q^{5} -3.95190 q^{7} -2.87581 q^{9} -5.63905 q^{11} +2.53457 q^{13} +0.352411 q^{15} +1.26403 q^{17} +6.83349 q^{19} -1.39270 q^{21} +1.00000 q^{23} +1.00000 q^{25} -2.07070 q^{27} +7.69701 q^{29} -9.04028 q^{31} -1.98726 q^{33} -3.95190 q^{35} -2.90308 q^{37} +0.893211 q^{39} +0.188663 q^{41} +2.97692 q^{43} -2.87581 q^{45} -4.82698 q^{47} +8.61755 q^{49} +0.445459 q^{51} +0.328956 q^{53} -5.63905 q^{55} +2.40820 q^{57} -0.966221 q^{59} +1.31009 q^{61} +11.3649 q^{63} +2.53457 q^{65} +3.94987 q^{67} +0.352411 q^{69} +7.69365 q^{71} +0.617173 q^{73} +0.352411 q^{75} +22.2850 q^{77} +15.1828 q^{79} +7.89768 q^{81} -1.30673 q^{83} +1.26403 q^{85} +2.71251 q^{87} -17.3242 q^{89} -10.0164 q^{91} -3.18590 q^{93} +6.83349 q^{95} +8.26476 q^{97} +16.2168 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9	$ 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9} - 3 q^{11} + 7 q^{13} - q^{15} + 9 q^{17} - q^{19} + 20 q^{21} + 5 q^{23} + 5 q^{25} - 4 q^{27} - 10 q^{29} - 21 q^{31} + 7 q^{33} - q^{35} + 8 q^{37} + 24 q^{39} - 13 q^{41} + 6 q^{43} + 6 q^{45} + 24 q^{49} + 17 q^{51} - 6 q^{53} - 3 q^{55} + 26 q^{57} - 18 q^{59} - 11 q^{61} - 4 q^{63} + 7 q^{65} - 38 q^{67} - q^{69} + 21 q^{71} - 12 q^{73} - q^{75} + 46 q^{77} + 18 q^{79} + 9 q^{81} - 20 q^{83} + 9 q^{85} + 6 q^{87} - 16 q^{89} + 3 q^{91} + 22 q^{93} - q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100}) $ 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9 - 3 * q^11 + 7 * q^13 - q^15 + 9 * q^17 - q^19 + 20 * q^21 + 5 * q^23 + 5 * q^25 - 4 * q^27 - 10 * q^29 - 21 * q^31 + 7 * q^33 - q^35 + 8 * q^37 + 24 * q^39 - 13 * q^41 + 6 * q^43 + 6 * q^45 + 24 * q^49 + 17 * q^51 - 6 * q^53 - 3 * q^55 + 26 * q^57 - 18 * q^59 - 11 * q^61 - 4 * q^63 + 7 * q^65 - 38 * q^67 - q^69 + 21 * q^71 - 12 * q^73 - q^75 + 46 * q^77 + 18 * q^79 + 9 * q^81 - 20 * q^83 + 9 * q^85 + 6 * q^87 - 16 * q^89 + 3 * q^91 + 22 * q^93 - q^95 + 29 * q^97 - 29 * q^99

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
$3$	0.352411	0.203465	0.101732	−	0.994812i	$-0.467561\pi$
$3$	0.352411	0.203465	0.101732	+	0.994812i	$0.467561\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.95190	−1.49368	−0.746840	−	0.665004i	$-0.768430\pi$
$7$	−3.95190	−1.49368	−0.746840	+	0.665004i	$0.768430\pi$
$8$	0	0
$9$	−2.87581	−0.958602
$10$	0	0
$11$	−5.63905	−1.70024	−0.850118	−	0.526592i	$-0.823470\pi$
$11$	−5.63905	−1.70024	−0.850118	+	0.526592i	$0.823470\pi$
$12$	0	0
$13$	2.53457	0.702963	0.351481	−	0.936195i	$-0.385678\pi$
$13$	2.53457	0.702963	0.351481	+	0.936195i	$0.385678\pi$
$14$	0	0
$15$	0.352411	0.0909922
$16$	0	0
$17$	1.26403	0.306573	0.153286	−	0.988182i	$-0.451014\pi$
$17$	1.26403	0.306573	0.153286	+	0.988182i	$0.451014\pi$
$18$	0	0
$19$	6.83349	1.56771	0.783855	−	0.620944i	$-0.213251\pi$
$19$	6.83349	1.56771	0.783855	+	0.620944i	$0.213251\pi$
$20$	0	0
$21$	−1.39270	−0.303911
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.07070	−0.398506
$28$	0	0
$29$	7.69701	1.42930	0.714650	−	0.699483i	$-0.246586\pi$
$29$	7.69701	1.42930	0.714650	+	0.699483i	$0.246586\pi$
$30$	0	0
$31$	−9.04028	−1.62368	−0.811842	−	0.583878i	$-0.801535\pi$
$31$	−9.04028	−1.62368	−0.811842	+	0.583878i	$0.801535\pi$
$32$	0	0
$33$	−1.98726	−0.345938
$34$	0	0
$35$	−3.95190	−0.667994
$36$	0	0
$37$	−2.90308	−0.477263	−0.238632	−	0.971110i	$-0.576699\pi$
$37$	−2.90308	−0.477263	−0.238632	+	0.971110i	$0.576699\pi$
$38$	0	0
$39$	0.893211	0.143028
$40$	0	0
$41$	0.188663	0.0294643	0.0147321	−	0.999891i	$-0.495310\pi$
$41$	0.188663	0.0294643	0.0147321	+	0.999891i	$0.495310\pi$
$42$	0	0
$43$	2.97692	0.453976	0.226988	−	0.973898i	$-0.427112\pi$
$43$	2.97692	0.453976	0.226988	+	0.973898i	$0.427112\pi$
$44$	0	0
$45$	−2.87581	−0.428700
$46$	0	0
$47$	−4.82698	−0.704088	−0.352044	−	0.935984i	$-0.614513\pi$
$47$	−4.82698	−0.704088	−0.352044	+	0.935984i	$0.614513\pi$
$48$	0	0
$49$	8.61755	1.23108
$50$	0	0
$51$	0.445459	0.0623767
$52$	0	0
$53$	0.328956	0.0451856	0.0225928	−	0.999745i	$-0.492808\pi$
$53$	0.328956	0.0451856	0.0225928	+	0.999745i	$0.492808\pi$
$54$	0	0
$55$	−5.63905	−0.760369
$56$	0	0
$57$	2.40820	0.318974
$58$	0	0
$59$	−0.966221	−0.125791	−0.0628957	−	0.998020i	$-0.520034\pi$
$59$	−0.966221	−0.125791	−0.0628957	+	0.998020i	$0.520034\pi$
$60$	0	0
$61$	1.31009	0.167740	0.0838700	−	0.996477i	$-0.473272\pi$
$61$	1.31009	0.167740	0.0838700	+	0.996477i	$0.473272\pi$
$62$	0	0
$63$	11.3649	1.43184
$64$	0	0
$65$	2.53457	0.314375
$66$	0	0
$67$	3.94987	0.482553	0.241277	−	0.970456i	$-0.422434\pi$
$67$	3.94987	0.482553	0.241277	+	0.970456i	$0.422434\pi$
$68$	0	0
$69$	0.352411	0.0424253
$70$	0	0
$71$	7.69365	0.913068	0.456534	−	0.889706i	$-0.349091\pi$
$71$	7.69365	0.913068	0.456534	+	0.889706i	$0.349091\pi$
$72$	0	0
$73$	0.617173	0.0722346	0.0361173	−	0.999348i	$-0.488501\pi$
$73$	0.617173	0.0722346	0.0361173	+	0.999348i	$0.488501\pi$
$74$	0	0
$75$	0.352411	0.0406929
$76$	0	0
$77$	22.2850	2.53961
$78$	0	0
$79$	15.1828	1.70819	0.854097	−	0.520114i	$-0.174111\pi$
$79$	15.1828	1.70819	0.854097	+	0.520114i	$0.174111\pi$
$80$	0	0
$81$	7.89768	0.877520
$82$	0	0
$83$	−1.30673	−0.143432	−0.0717161	−	0.997425i	$-0.522848\pi$
$83$	−1.30673	−0.143432	−0.0717161	+	0.997425i	$0.522848\pi$
$84$	0	0
$85$	1.26403	0.137103
$86$	0	0
$87$	2.71251	0.290812
$88$	0	0
$89$	−17.3242	−1.83636	−0.918178	−	0.396167i	$-0.870340\pi$
$89$	−17.3242	−1.83636	−0.918178	+	0.396167i	$0.870340\pi$
$90$	0	0
$91$	−10.0164	−1.05000
$92$	0	0
$93$	−3.18590	−0.330362
$94$	0	0
$95$	6.83349	0.701101
$96$	0	0
$97$	8.26476	0.839159	0.419580	−	0.907718i	$-0.362177\pi$
$97$	8.26476	0.839159	0.419580	+	0.907718i	$0.362177\pi$
$98$	0	0
$99$	16.2168	1.62985
$100$	0	0
$101$	17.8545	1.77659	0.888296	−	0.459271i	$-0.151889\pi$
$101$	17.8545	1.77659	0.888296	+	0.459271i	$0.151889\pi$
$102$	0	0
$103$	18.0624	1.77974	0.889872	−	0.456210i	$-0.150793\pi$
$103$	18.0624	1.77974	0.889872	+	0.456210i	$0.150793\pi$
$104$	0	0
$105$	−1.39270	−0.135913
$106$	0	0
$107$	12.7747	1.23498	0.617488	−	0.786580i	$-0.288150\pi$
$107$	12.7747	1.23498	0.617488	+	0.786580i	$0.288150\pi$
$108$	0	0
$109$	14.7596	1.41372	0.706859	−	0.707355i	$-0.250112\pi$
$109$	14.7596	1.41372	0.706859	+	0.707355i	$0.250112\pi$
$110$	0	0
$111$	−1.02308	−0.0971062
$112$	0	0
$113$	−9.11666	−0.857623	−0.428812	−	0.903394i	$-0.641068\pi$
$113$	−9.11666	−0.857623	−0.428812	+	0.903394i	$0.641068\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−7.28893	−0.673862
$118$	0	0
$119$	−4.99533	−0.457921
$120$	0	0
$121$	20.7989	1.89081
$122$	0	0
$123$	0.0664871	0.00599494
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.87377	−0.343742	−0.171871	−	0.985120i	$-0.554981\pi$
$127$	−3.87377	−0.343742	−0.171871	+	0.985120i	$0.554981\pi$
$128$	0	0
$129$	1.04910	0.0923682
$130$	0	0
$131$	−2.55488	−0.223221	−0.111610	−	0.993752i	$-0.535601\pi$
$131$	−2.55488	−0.223221	−0.111610	+	0.993752i	$0.535601\pi$
$132$	0	0
$133$	−27.0053	−2.34166
$134$	0	0
$135$	−2.07070	−0.178217
$136$	0	0
$137$	3.37778	0.288583	0.144292	−	0.989535i	$-0.453910\pi$
$137$	3.37778	0.288583	0.144292	+	0.989535i	$0.453910\pi$
$138$	0	0
$139$	−1.41965	−0.120413	−0.0602064	−	0.998186i	$-0.519176\pi$
$139$	−1.41965	−0.120413	−0.0602064	+	0.998186i	$0.519176\pi$
$140$	0	0
$141$	−1.70108	−0.143257
$142$	0	0
$143$	−14.2926	−1.19520
$144$	0	0
$145$	7.69701	0.639202
$146$	0	0
$147$	3.03692	0.250481
$148$	0	0
$149$	13.3961	1.09745	0.548724	−	0.836004i	$-0.315114\pi$
$149$	13.3961	1.09745	0.548724	+	0.836004i	$0.315114\pi$
$150$	0	0
$151$	13.6090	1.10749	0.553743	−	0.832688i	$-0.313199\pi$
$151$	13.6090	1.10749	0.553743	+	0.832688i	$0.313199\pi$
$152$	0	0
$153$	−3.63511	−0.293881
$154$	0	0
$155$	−9.04028	−0.726133
$156$	0	0
$157$	19.8029	1.58045	0.790223	−	0.612819i	$-0.209965\pi$
$157$	19.8029	1.58045	0.790223	+	0.612819i	$0.209965\pi$
$158$	0	0
$159$	0.115928	0.00919367
$160$	0	0
$161$	−3.95190	−0.311454
$162$	0	0
$163$	−20.4606	−1.60260	−0.801300	−	0.598263i	$-0.795858\pi$
$163$	−20.4606	−1.60260	−0.801300	+	0.598263i	$0.795858\pi$
$164$	0	0
$165$	−1.98726	−0.154708
$166$	0	0
$167$	20.1227	1.55714	0.778569	−	0.627559i	$-0.215946\pi$
$167$	20.1227	1.55714	0.778569	+	0.627559i	$0.215946\pi$
$168$	0	0
$169$	−6.57596	−0.505843
$170$	0	0
$171$	−19.6518	−1.50281
$172$	0	0
$173$	24.2937	1.84702	0.923509	−	0.383576i	$-0.125308\pi$
$173$	24.2937	1.84702	0.923509	+	0.383576i	$0.125308\pi$
$174$	0	0
$175$	−3.95190	−0.298736
$176$	0	0
$177$	−0.340507	−0.0255941
$178$	0	0
$179$	−15.3672	−1.14860	−0.574299	−	0.818645i	$-0.694725\pi$
$179$	−15.3672	−1.14860	−0.574299	+	0.818645i	$0.694725\pi$
$180$	0	0
$181$	−14.0440	−1.04388	−0.521942	−	0.852981i	$-0.674792\pi$
$181$	−14.0440	−1.04388	−0.521942	+	0.852981i	$0.674792\pi$
$182$	0	0
$183$	0.461691	0.0341292
$184$	0	0
$185$	−2.90308	−0.213439
$186$	0	0
$187$	−7.12793	−0.521246
$188$	0	0
$189$	8.18321	0.595241
$190$	0	0
$191$	−2.51098	−0.181688	−0.0908441	−	0.995865i	$-0.528956\pi$
$191$	−2.51098	−0.181688	−0.0908441	+	0.995865i	$0.528956\pi$
$192$	0	0
$193$	18.6016	1.33897	0.669484	−	0.742826i	$-0.266515\pi$
$193$	18.6016	1.33897	0.669484	+	0.742826i	$0.266515\pi$
$194$	0	0
$195$	0.893211	0.0639641
$196$	0	0
$197$	−17.8033	−1.26843	−0.634216	−	0.773156i	$-0.718677\pi$
$197$	−17.8033	−1.26843	−0.634216	+	0.773156i	$0.718677\pi$
$198$	0	0
$199$	−9.14740	−0.648442	−0.324221	−	0.945981i	$-0.605102\pi$
$199$	−9.14740	−0.648442	−0.324221	+	0.945981i	$0.605102\pi$
$200$	0	0
$201$	1.39198	0.0981826
$202$	0	0
$203$	−30.4179	−2.13491
$204$	0	0
$205$	0.188663	0.0131768
$206$	0	0
$207$	−2.87581	−0.199882
$208$	0	0
$209$	−38.5344	−2.66548
$210$	0	0
$211$	−18.6161	−1.28159	−0.640793	−	0.767714i	$-0.721394\pi$
$211$	−18.6161	−1.28159	−0.640793	+	0.767714i	$0.721394\pi$
$212$	0	0
$213$	2.71133	0.185777
$214$	0	0
$215$	2.97692	0.203024
$216$	0	0
$217$	35.7263	2.42526
$218$	0	0
$219$	0.217499	0.0146972
$220$	0	0
$221$	3.20377	0.215509
$222$	0	0
$223$	−15.7475	−1.05453	−0.527267	−	0.849700i	$-0.676783\pi$
$223$	−15.7475	−1.05453	−0.527267	+	0.849700i	$0.676783\pi$
$224$	0	0
$225$	−2.87581	−0.191720
$226$	0	0
$227$	1.28041	0.0849835	0.0424918	−	0.999097i	$-0.486470\pi$
$227$	1.28041	0.0849835	0.0424918	+	0.999097i	$0.486470\pi$
$228$	0	0
$229$	−0.257424	−0.0170110	−0.00850552	−	0.999964i	$-0.502707\pi$
$229$	−0.257424	−0.0170110	−0.00850552	+	0.999964i	$0.502707\pi$
$230$	0	0
$231$	7.85348	0.516721
$232$	0	0
$233$	12.2036	0.799483	0.399741	−	0.916628i	$-0.369100\pi$
$233$	12.2036	0.799483	0.399741	+	0.916628i	$0.369100\pi$
$234$	0	0
$235$	−4.82698	−0.314878
$236$	0	0
$237$	5.35057	0.347557
$238$	0	0
$239$	7.31120	0.472922	0.236461	−	0.971641i	$-0.424012\pi$
$239$	7.31120	0.472922	0.236461	+	0.971641i	$0.424012\pi$
$240$	0	0
$241$	−11.7293	−0.755548	−0.377774	−	0.925898i	$-0.623310\pi$
$241$	−11.7293	−0.755548	−0.377774	+	0.925898i	$0.623310\pi$
$242$	0	0
$243$	8.99533	0.577051
$244$	0	0
$245$	8.61755	0.550555
$246$	0	0
$247$	17.3199	1.10204
$248$	0	0
$249$	−0.460506	−0.0291834
$250$	0	0
$251$	6.21099	0.392034	0.196017	−	0.980600i	$-0.437199\pi$
$251$	6.21099	0.392034	0.196017	+	0.980600i	$0.437199\pi$
$252$	0	0
$253$	−5.63905	−0.354524
$254$	0	0
$255$	0.445459	0.0278957
$256$	0	0
$257$	27.0382	1.68660	0.843300	−	0.537443i	$-0.180610\pi$
$257$	27.0382	1.68660	0.843300	+	0.537443i	$0.180610\pi$
$258$	0	0
$259$	11.4727	0.712878
$260$	0	0
$261$	−22.1351	−1.37013
$262$	0	0
$263$	22.3100	1.37569	0.687846	−	0.725857i	$-0.258556\pi$
$263$	22.3100	1.37569	0.687846	+	0.725857i	$0.258556\pi$
$264$	0	0
$265$	0.328956	0.0202076
$266$	0	0
$267$	−6.10523	−0.373634
$268$	0	0
$269$	−14.2720	−0.870178	−0.435089	−	0.900387i	$-0.643283\pi$
$269$	−14.2720	−0.870178	−0.435089	+	0.900387i	$0.643283\pi$
$270$	0	0
$271$	18.0785	1.09819	0.549096	−	0.835759i	$-0.314972\pi$
$271$	18.0785	1.09819	0.549096	+	0.835759i	$0.314972\pi$
$272$	0	0
$273$	−3.52988	−0.213638
$274$	0	0
$275$	−5.63905	−0.340047
$276$	0	0
$277$	4.97476	0.298905	0.149452	−	0.988769i	$-0.452249\pi$
$277$	4.97476	0.298905	0.149452	+	0.988769i	$0.452249\pi$
$278$	0	0
$279$	25.9981	1.55647
$280$	0	0
$281$	−5.15895	−0.307757	−0.153878	−	0.988090i	$-0.549176\pi$
$281$	−5.15895	−0.307757	−0.153878	+	0.988090i	$0.549176\pi$
$282$	0	0
$283$	−11.8800	−0.706193	−0.353096	−	0.935587i	$-0.614871\pi$
$283$	−11.8800	−0.706193	−0.353096	+	0.935587i	$0.614871\pi$
$284$	0	0
$285$	2.40820	0.142649
$286$	0	0
$287$	−0.745580	−0.0440102
$288$	0	0
$289$	−15.4022	−0.906013
$290$	0	0
$291$	2.91259	0.170739
$292$	0	0
$293$	−24.0075	−1.40253	−0.701266	−	0.712900i	$-0.747381\pi$
$293$	−24.0075	−1.40253	−0.701266	+	0.712900i	$0.747381\pi$
$294$	0	0
$295$	−0.966221	−0.0562556
$296$	0	0
$297$	11.6768	0.677555
$298$	0	0
$299$	2.53457	0.146578
$300$	0	0
$301$	−11.7645	−0.678095
$302$	0	0
$303$	6.29214	0.361474
$304$	0	0
$305$	1.31009	0.0750156
$306$	0	0
$307$	−20.8389	−1.18934	−0.594669	−	0.803971i	$-0.702717\pi$
$307$	−20.8389	−1.18934	−0.594669	+	0.803971i	$0.702717\pi$
$308$	0	0
$309$	6.36540	0.362115
$310$	0	0
$311$	−23.6092	−1.33875	−0.669376	−	0.742924i	$-0.733438\pi$
$311$	−23.6092	−1.33875	−0.669376	+	0.742924i	$0.733438\pi$
$312$	0	0
$313$	15.7933	0.892692	0.446346	−	0.894861i	$-0.352725\pi$
$313$	15.7933	0.892692	0.446346	+	0.894861i	$0.352725\pi$
$314$	0	0
$315$	11.3649	0.640340
$316$	0	0
$317$	−32.3058	−1.81447	−0.907236	−	0.420621i	$-0.861812\pi$
$317$	−32.3058	−1.81447	−0.907236	+	0.420621i	$0.861812\pi$
$318$	0	0
$319$	−43.4038	−2.43015
$320$	0	0
$321$	4.50194	0.251274
$322$	0	0
$323$	8.63774	0.480617
$324$	0	0
$325$	2.53457	0.140593
$326$	0	0
$327$	5.20146	0.287642
$328$	0	0
$329$	19.0758	1.05168
$330$	0	0
$331$	14.4140	0.792265	0.396132	−	0.918193i	$-0.370352\pi$
$331$	14.4140	0.792265	0.396132	+	0.918193i	$0.370352\pi$
$332$	0	0
$333$	8.34869	0.457506
$334$	0	0
$335$	3.94987	0.215804
$336$	0	0
$337$	9.34929	0.509288	0.254644	−	0.967035i	$-0.418042\pi$
$337$	9.34929	0.509288	0.254644	+	0.967035i	$0.418042\pi$
$338$	0	0
$339$	−3.21281	−0.174496
$340$	0	0
$341$	50.9786	2.76065
$342$	0	0
$343$	−6.39240	−0.345157
$344$	0	0
$345$	0.352411	0.0189732
$346$	0	0
$347$	−1.13655	−0.0610133	−0.0305067	−	0.999535i	$-0.509712\pi$
$347$	−1.13655	−0.0610133	−0.0305067	+	0.999535i	$0.509712\pi$
$348$	0	0
$349$	−8.75535	−0.468663	−0.234332	−	0.972157i	$-0.575290\pi$
$349$	−8.75535	−0.468663	−0.234332	+	0.972157i	$0.575290\pi$
$350$	0	0
$351$	−5.24833	−0.280135
$352$	0	0
$353$	22.1257	1.17763	0.588815	−	0.808268i	$-0.299595\pi$
$353$	22.1257	1.17763	0.588815	+	0.808268i	$0.299595\pi$
$354$	0	0
$355$	7.69365	0.408336
$356$	0	0
$357$	−1.76041	−0.0931708
$358$	0	0
$359$	−8.55850	−0.451700	−0.225850	−	0.974162i	$-0.572516\pi$
$359$	−8.55850	−0.451700	−0.225850	+	0.974162i	$0.572516\pi$
$360$	0	0
$361$	27.6965	1.45771
$362$	0	0
$363$	7.32975	0.384712
$364$	0	0
$365$	0.617173	0.0323043
$366$	0	0
$367$	12.4672	0.650784	0.325392	−	0.945579i	$-0.394504\pi$
$367$	12.4672	0.650784	0.325392	+	0.945579i	$0.394504\pi$
$368$	0	0
$369$	−0.542559	−0.0282445
$370$	0	0
$371$	−1.30000	−0.0674928
$372$	0	0
$373$	17.6684	0.914837	0.457418	−	0.889252i	$-0.348774\pi$
$373$	17.6684	0.914837	0.457418	+	0.889252i	$0.348774\pi$
$374$	0	0
$375$	0.352411	0.0181984
$376$	0	0
$377$	19.5086	1.00474
$378$	0	0
$379$	−6.67680	−0.342964	−0.171482	−	0.985187i	$-0.554856\pi$
$379$	−6.67680	−0.342964	−0.171482	+	0.985187i	$0.554856\pi$
$380$	0	0
$381$	−1.36516	−0.0699393
$382$	0	0
$383$	4.27134	0.218256	0.109128	−	0.994028i	$-0.465194\pi$
$383$	4.27134	0.218256	0.109128	+	0.994028i	$0.465194\pi$
$384$	0	0
$385$	22.2850	1.13575
$386$	0	0
$387$	−8.56105	−0.435183
$388$	0	0
$389$	−14.4378	−0.732023	−0.366011	−	0.930610i	$-0.619277\pi$
$389$	−14.4378	−0.732023	−0.366011	+	0.930610i	$0.619277\pi$
$390$	0	0
$391$	1.26403	0.0639248
$392$	0	0
$393$	−0.900369	−0.0454176
$394$	0	0
$395$	15.1828	0.763927
$396$	0	0
$397$	9.49861	0.476722	0.238361	−	0.971177i	$-0.423390\pi$
$397$	9.49861	0.476722	0.238361	+	0.971177i	$0.423390\pi$
$398$	0	0
$399$	−9.51697	−0.476444
$400$	0	0
$401$	−31.6884	−1.58244	−0.791221	−	0.611530i	$-0.790554\pi$
$401$	−31.6884	−1.58244	−0.791221	+	0.611530i	$0.790554\pi$
$402$	0	0
$403$	−22.9132	−1.14139
$404$	0	0
$405$	7.89768	0.392439
$406$	0	0
$407$	16.3706	0.811461
$408$	0	0
$409$	−26.2552	−1.29824	−0.649119	−	0.760687i	$-0.724862\pi$
$409$	−26.2552	−1.29824	−0.649119	+	0.760687i	$0.724862\pi$
$410$	0	0
$411$	1.19037	0.0587165
$412$	0	0
$413$	3.81841	0.187892
$414$	0	0
$415$	−1.30673	−0.0641448
$416$	0	0
$417$	−0.500299	−0.0244998
$418$	0	0
$419$	−25.3855	−1.24016	−0.620082	−	0.784537i	$-0.712901\pi$
$419$	−25.3855	−1.24016	−0.620082	+	0.784537i	$0.712901\pi$
$420$	0	0
$421$	16.2678	0.792846	0.396423	−	0.918068i	$-0.370251\pi$
$421$	16.2678	0.792846	0.396423	+	0.918068i	$0.370251\pi$
$422$	0	0
$423$	13.8815	0.674940
$424$	0	0
$425$	1.26403	0.0613145
$426$	0	0
$427$	−5.17736	−0.250550
$428$	0	0
$429$	−5.03686	−0.243182
$430$	0	0
$431$	22.2534	1.07191	0.535953	−	0.844248i	$-0.319952\pi$
$431$	22.2534	1.07191	0.535953	+	0.844248i	$0.319952\pi$
$432$	0	0
$433$	6.72589	0.323226	0.161613	−	0.986854i	$-0.448330\pi$
$433$	6.72589	0.323226	0.161613	+	0.986854i	$0.448330\pi$
$434$	0	0
$435$	2.71251	0.130055
$436$	0	0
$437$	6.83349	0.326890
$438$	0	0
$439$	11.3391	0.541188	0.270594	−	0.962694i	$-0.412780\pi$
$439$	11.3391	0.541188	0.270594	+	0.962694i	$0.412780\pi$
$440$	0	0
$441$	−24.7824	−1.18011
$442$	0	0
$443$	−6.39714	−0.303937	−0.151969	−	0.988385i	$-0.548561\pi$
$443$	−6.39714	−0.303937	−0.151969	+	0.988385i	$0.548561\pi$
$444$	0	0
$445$	−17.3242	−0.821244
$446$	0	0
$447$	4.72092	0.223292
$448$	0	0
$449$	8.84945	0.417631	0.208816	−	0.977955i	$-0.433039\pi$
$449$	8.84945	0.417631	0.208816	+	0.977955i	$0.433039\pi$
$450$	0	0
$451$	−1.06388	−0.0500962
$452$	0	0
$453$	4.79597	0.225334
$454$	0	0
$455$	−10.0164	−0.469575
$456$	0	0
$457$	28.8347	1.34883	0.674415	−	0.738353i	$-0.264396\pi$
$457$	28.8347	1.34883	0.674415	+	0.738353i	$0.264396\pi$
$458$	0	0
$459$	−2.61743	−0.122171
$460$	0	0
$461$	−28.5500	−1.32970	−0.664852	−	0.746975i	$-0.731505\pi$
$461$	−28.5500	−1.32970	−0.664852	+	0.746975i	$0.731505\pi$
$462$	0	0
$463$	−27.4923	−1.27768	−0.638838	−	0.769341i	$-0.720585\pi$
$463$	−27.4923	−1.27768	−0.638838	+	0.769341i	$0.720585\pi$
$464$	0	0
$465$	−3.18590	−0.147742
$466$	0	0
$467$	34.0356	1.57498	0.787490	−	0.616327i	$-0.211380\pi$
$467$	34.0356	1.57498	0.787490	+	0.616327i	$0.211380\pi$
$468$	0	0
$469$	−15.6095	−0.720780
$470$	0	0
$471$	6.97878	0.321565
$472$	0	0
$473$	−16.7870	−0.771867
$474$	0	0
$475$	6.83349	0.313542
$476$	0	0
$477$	−0.946014	−0.0433150
$478$	0	0
$479$	7.53712	0.344380	0.172190	−	0.985064i	$-0.444916\pi$
$479$	7.53712	0.344380	0.172190	+	0.985064i	$0.444916\pi$
$480$	0	0
$481$	−7.35805	−0.335498
$482$	0	0
$483$	−1.39270	−0.0633698
$484$	0	0
$485$	8.26476	0.375283
$486$	0	0
$487$	−24.6238	−1.11581	−0.557906	−	0.829904i	$-0.688395\pi$
$487$	−24.6238	−1.11581	−0.557906	+	0.829904i	$0.688395\pi$
$488$	0	0
$489$	−7.21056	−0.326073
$490$	0	0
$491$	−14.0638	−0.634692	−0.317346	−	0.948310i	$-0.602792\pi$
$491$	−14.0638	−0.634692	−0.317346	+	0.948310i	$0.602792\pi$
$492$	0	0
$493$	9.72927	0.438184
$494$	0	0
$495$	16.2168	0.728891
$496$	0	0
$497$	−30.4046	−1.36383
$498$	0	0
$499$	26.5470	1.18841	0.594204	−	0.804315i	$-0.297467\pi$
$499$	26.5470	1.18841	0.594204	+	0.804315i	$0.297467\pi$
$500$	0	0
$501$	7.09145	0.316823
$502$	0	0
$503$	36.3219	1.61952	0.809758	−	0.586764i	$-0.199599\pi$
$503$	36.3219	1.61952	0.809758	+	0.586764i	$0.199599\pi$
$504$	0	0
$505$	17.8545	0.794516
$506$	0	0
$507$	−2.31744	−0.102921
$508$	0	0
$509$	1.72157	0.0763074	0.0381537	−	0.999272i	$-0.487852\pi$
$509$	1.72157	0.0763074	0.0381537	+	0.999272i	$0.487852\pi$
$510$	0	0
$511$	−2.43901	−0.107895
$512$	0	0
$513$	−14.1501	−0.624742
$514$	0	0
$515$	18.0624	0.795926
$516$	0	0
$517$	27.2196	1.19712
$518$	0	0
$519$	8.56139	0.375803
$520$	0	0
$521$	−45.2230	−1.98126	−0.990629	−	0.136582i	$-0.956388\pi$
$521$	−45.2230	−1.98126	−0.990629	+	0.136582i	$0.956388\pi$
$522$	0	0
$523$	−27.2150	−1.19003	−0.595013	−	0.803716i	$-0.702853\pi$
$523$	−27.2150	−1.19003	−0.595013	+	0.803716i	$0.702853\pi$
$524$	0	0
$525$	−1.39270	−0.0607822
$526$	0	0
$527$	−11.4272	−0.497777
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.77867	0.120584
$532$	0	0
$533$	0.478180	0.0207123
$534$	0	0
$535$	12.7747	0.552298
$536$	0	0
$537$	−5.41558	−0.233699
$538$	0	0
$539$	−48.5948	−2.09313
$540$	0	0
$541$	−5.81174	−0.249866	−0.124933	−	0.992165i	$-0.539872\pi$
$541$	−5.81174	−0.249866	−0.124933	+	0.992165i	$0.539872\pi$
$542$	0	0
$543$	−4.94927	−0.212394
$544$	0	0
$545$	14.7596	0.632234
$546$	0	0
$547$	−25.4942	−1.09005	−0.545027	−	0.838419i	$-0.683481\pi$
$547$	−25.4942	−1.09005	−0.545027	+	0.838419i	$0.683481\pi$
$548$	0	0
$549$	−3.76757	−0.160796
$550$	0	0
$551$	52.5974	2.24073
$552$	0	0
$553$	−60.0008	−2.55149
$554$	0	0
$555$	−1.02308	−0.0434272
$556$	0	0
$557$	−0.522188	−0.0221258	−0.0110629	−	0.999939i	$-0.503522\pi$
$557$	−0.522188	−0.0221258	−0.0110629	+	0.999939i	$0.503522\pi$
$558$	0	0
$559$	7.54521	0.319129
$560$	0	0
$561$	−2.51196	−0.106055
$562$	0	0
$563$	22.8837	0.964432	0.482216	−	0.876052i	$-0.339832\pi$
$563$	22.8837	0.964432	0.482216	+	0.876052i	$0.339832\pi$
$564$	0	0
$565$	−9.11666	−0.383541
$566$	0	0
$567$	−31.2109	−1.31073
$568$	0	0
$569$	−2.19530	−0.0920319	−0.0460159	−	0.998941i	$-0.514652\pi$
$569$	−2.19530	−0.0920319	−0.0460159	+	0.998941i	$0.514652\pi$
$570$	0	0
$571$	−18.5189	−0.774991	−0.387495	−	0.921872i	$-0.626660\pi$
$571$	−18.5189	−0.774991	−0.387495	+	0.921872i	$0.626660\pi$
$572$	0	0
$573$	−0.884898	−0.0369671
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−30.5975	−1.27379	−0.636895	−	0.770951i	$-0.719781\pi$
$577$	−30.5975	−1.27379	−0.636895	+	0.770951i	$0.719781\pi$
$578$	0	0
$579$	6.55540	0.272433
$580$	0	0
$581$	5.16407	0.214242
$582$	0	0
$583$	−1.85500	−0.0768262
$584$	0	0
$585$	−7.28893	−0.301360
$586$	0	0
$587$	14.0369	0.579365	0.289683	−	0.957123i	$-0.406450\pi$
$587$	14.0369	0.579365	0.289683	+	0.957123i	$0.406450\pi$
$588$	0	0
$589$	−61.7767	−2.54546
$590$	0	0
$591$	−6.27408	−0.258081
$592$	0	0
$593$	−20.1236	−0.826376	−0.413188	−	0.910646i	$-0.635585\pi$
$593$	−20.1236	−0.826376	−0.413188	+	0.910646i	$0.635585\pi$
$594$	0	0
$595$	−4.99533	−0.204789
$596$	0	0
$597$	−3.22365	−0.131935
$598$	0	0
$599$	−35.1489	−1.43615	−0.718073	−	0.695968i	$-0.754975\pi$
$599$	−35.1489	−1.43615	−0.718073	+	0.695968i	$0.754975\pi$
$600$	0	0
$601$	−13.1732	−0.537346	−0.268673	−	0.963231i	$-0.586585\pi$
$601$	−13.1732	−0.537346	−0.268673	+	0.963231i	$0.586585\pi$
$602$	0	0
$603$	−11.3591	−0.462577
$604$	0	0
$605$	20.7989	0.845594
$606$	0	0
$607$	33.4291	1.35685	0.678423	−	0.734672i	$-0.262664\pi$
$607$	33.4291	1.35685	0.678423	+	0.734672i	$0.262664\pi$
$608$	0	0
$609$	−10.7196	−0.434380
$610$	0	0
$611$	−12.2343	−0.494947
$612$	0	0
$613$	−18.1013	−0.731103	−0.365551	−	0.930791i	$-0.619120\pi$
$613$	−18.1013	−0.731103	−0.365551	+	0.930791i	$0.619120\pi$
$614$	0	0
$615$	0.0664871	0.00268102
$616$	0	0
$617$	4.09327	0.164789	0.0823944	−	0.996600i	$-0.473743\pi$
$617$	4.09327	0.164789	0.0823944	+	0.996600i	$0.473743\pi$
$618$	0	0
$619$	29.9861	1.20525	0.602623	−	0.798026i	$-0.294122\pi$
$619$	29.9861	1.20525	0.602623	+	0.798026i	$0.294122\pi$
$620$	0	0
$621$	−2.07070	−0.0830943
$622$	0	0
$623$	68.4634	2.74293
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−13.5799	−0.542331
$628$	0	0
$629$	−3.66958	−0.146316
$630$	0	0
$631$	37.5829	1.49615	0.748077	−	0.663612i	$-0.230978\pi$
$631$	37.5829	1.49615	0.748077	+	0.663612i	$0.230978\pi$
$632$	0	0
$633$	−6.56053	−0.260758
$634$	0	0
$635$	−3.87377	−0.153726
$636$	0	0
$637$	21.8418	0.865403
$638$	0	0
$639$	−22.1254	−0.875269
$640$	0	0
$641$	28.9195	1.14225	0.571126	−	0.820863i	$-0.306507\pi$
$641$	28.9195	1.14225	0.571126	+	0.820863i	$0.306507\pi$
$642$	0	0
$643$	4.05771	0.160020	0.0800102	−	0.996794i	$-0.474505\pi$
$643$	4.05771	0.160020	0.0800102	+	0.996794i	$0.474505\pi$
$644$	0	0
$645$	1.04910	0.0413083
$646$	0	0
$647$	−2.81376	−0.110621	−0.0553103	−	0.998469i	$-0.517615\pi$
$647$	−2.81376	−0.110621	−0.0553103	+	0.998469i	$0.517615\pi$
$648$	0	0
$649$	5.44857	0.213875
$650$	0	0
$651$	12.5904	0.493455
$652$	0	0
$653$	23.7672	0.930083	0.465041	−	0.885289i	$-0.346039\pi$
$653$	23.7672	0.930083	0.465041	+	0.885289i	$0.346039\pi$
$654$	0	0
$655$	−2.55488	−0.0998275
$656$	0	0
$657$	−1.77487	−0.0692443
$658$	0	0
$659$	22.2518	0.866807	0.433403	−	0.901200i	$-0.357313\pi$
$659$	22.2518	0.866807	0.433403	+	0.901200i	$0.357313\pi$
$660$	0	0
$661$	45.0505	1.75226	0.876131	−	0.482074i	$-0.160116\pi$
$661$	45.0505	1.75226	0.876131	+	0.482074i	$0.160116\pi$
$662$	0	0
$663$	1.12905	0.0438485
$664$	0	0
$665$	−27.0053	−1.04722
$666$	0	0
$667$	7.69701	0.298029
$668$	0	0
$669$	−5.54961	−0.214560
$670$	0	0
$671$	−7.38767	−0.285198
$672$	0	0
$673$	−38.8944	−1.49927	−0.749635	−	0.661851i	$-0.769771\pi$
$673$	−38.8944	−1.49927	−0.749635	+	0.661851i	$0.769771\pi$
$674$	0	0
$675$	−2.07070	−0.0797013
$676$	0	0
$677$	30.7421	1.18151	0.590757	−	0.806849i	$-0.298829\pi$
$677$	30.7421	1.18151	0.590757	+	0.806849i	$0.298829\pi$
$678$	0	0
$679$	−32.6616	−1.25344
$680$	0	0
$681$	0.451229	0.0172911
$682$	0	0
$683$	50.7299	1.94113	0.970564	−	0.240844i	$-0.0774243\pi$
$683$	50.7299	1.94113	0.970564	+	0.240844i	$0.0774243\pi$
$684$	0	0
$685$	3.37778	0.129058
$686$	0	0
$687$	−0.0907190	−0.00346114
$688$	0	0
$689$	0.833762	0.0317638
$690$	0	0
$691$	11.3810	0.432954	0.216477	−	0.976288i	$-0.430543\pi$
$691$	11.3810	0.432954	0.216477	+	0.976288i	$0.430543\pi$
$692$	0	0
$693$	−64.0873	−2.43447
$694$	0	0
$695$	−1.41965	−0.0538503
$696$	0	0
$697$	0.238476	0.00903294
$698$	0	0
$699$	4.30068	0.162667
$700$	0	0
$701$	−1.38246	−0.0522149	−0.0261074	−	0.999659i	$-0.508311\pi$
$701$	−1.38246	−0.0522149	−0.0261074	+	0.999659i	$0.508311\pi$
$702$	0	0
$703$	−19.8382	−0.748210
$704$	0	0
$705$	−1.70108	−0.0640665
$706$	0	0
$707$	−70.5594	−2.65366
$708$	0	0
$709$	−35.4080	−1.32978	−0.664888	−	0.746944i	$-0.731521\pi$
$709$	−35.4080	−1.32978	−0.664888	+	0.746944i	$0.731521\pi$
$710$	0	0
$711$	−43.6627	−1.63748
$712$	0	0
$713$	−9.04028	−0.338561
$714$	0	0
$715$	−14.2926	−0.534511
$716$	0	0
$717$	2.57655	0.0962230
$718$	0	0
$719$	−10.0867	−0.376169	−0.188085	−	0.982153i	$-0.560228\pi$
$719$	−10.0867	−0.376169	−0.188085	+	0.982153i	$0.560228\pi$
$720$	0	0
$721$	−71.3810	−2.65837
$722$	0	0
$723$	−4.13352	−0.153727
$724$	0	0
$725$	7.69701	0.285860
$726$	0	0
$727$	13.4646	0.499374	0.249687	−	0.968327i	$-0.419672\pi$
$727$	13.4646	0.499374	0.249687	+	0.968327i	$0.419672\pi$
$728$	0	0
$729$	−20.5230	−0.760111
$730$	0	0
$731$	3.76292	0.139177
$732$	0	0
$733$	19.2923	0.712577	0.356289	−	0.934376i	$-0.384042\pi$
$733$	19.2923	0.712577	0.356289	+	0.934376i	$0.384042\pi$
$734$	0	0
$735$	3.03692	0.112019
$736$	0	0
$737$	−22.2735	−0.820455
$738$	0	0
$739$	41.9597	1.54351	0.771756	−	0.635918i	$-0.219379\pi$
$739$	41.9597	1.54351	0.771756	+	0.635918i	$0.219379\pi$
$740$	0	0
$741$	6.10374	0.224227
$742$	0	0
$743$	−26.2397	−0.962640	−0.481320	−	0.876545i	$-0.659843\pi$
$743$	−26.2397	−0.962640	−0.481320	+	0.876545i	$0.659843\pi$
$744$	0	0
$745$	13.3961	0.490794
$746$	0	0
$747$	3.75790	0.137494
$748$	0	0
$749$	−50.4844	−1.84466
$750$	0	0
$751$	43.8563	1.60034	0.800170	−	0.599773i	$-0.204742\pi$
$751$	43.8563	1.60034	0.800170	+	0.599773i	$0.204742\pi$
$752$	0	0
$753$	2.18882	0.0797652
$754$	0	0
$755$	13.6090	0.495283
$756$	0	0
$757$	26.2435	0.953835	0.476918	−	0.878948i	$-0.341754\pi$
$757$	26.2435	0.953835	0.476918	+	0.878948i	$0.341754\pi$
$758$	0	0
$759$	−1.98726	−0.0721331
$760$	0	0
$761$	−45.7738	−1.65930	−0.829650	−	0.558283i	$-0.811460\pi$
$761$	−45.7738	−1.65930	−0.829650	+	0.558283i	$0.811460\pi$
$762$	0	0
$763$	−58.3287	−2.11164
$764$	0	0
$765$	−3.63511	−0.131428
$766$	0	0
$767$	−2.44895	−0.0884266
$768$	0	0
$769$	26.7930	0.966181	0.483091	−	0.875570i	$-0.339514\pi$
$769$	26.7930	0.966181	0.483091	+	0.875570i	$0.339514\pi$
$770$	0	0
$771$	9.52858	0.343164
$772$	0	0
$773$	−40.8103	−1.46784	−0.733922	−	0.679234i	$-0.762312\pi$
$773$	−40.8103	−1.46784	−0.733922	+	0.679234i	$0.762312\pi$
$774$	0	0
$775$	−9.04028	−0.324737
$776$	0	0
$777$	4.04311	0.145046
$778$	0	0
$779$	1.28923	0.0461914
$780$	0	0
$781$	−43.3849	−1.55243
$782$	0	0
$783$	−15.9382	−0.569585
$784$	0	0
$785$	19.8029	0.706797
$786$	0	0
$787$	−7.34419	−0.261792	−0.130896	−	0.991396i	$-0.541785\pi$
$787$	−7.34419	−0.261792	−0.130896	+	0.991396i	$0.541785\pi$
$788$	0	0
$789$	7.86228	0.279905
$790$	0	0
$791$	36.0282	1.28101
$792$	0	0
$793$	3.32052	0.117915
$794$	0	0
$795$	0.115928	0.00411154
$796$	0	0
$797$	−3.62790	−0.128507	−0.0642535	−	0.997934i	$-0.520467\pi$
$797$	−3.62790	−0.128507	−0.0642535	+	0.997934i	$0.520467\pi$
$798$	0	0
$799$	−6.10146	−0.215854
$800$	0	0
$801$	49.8209	1.76034
$802$	0	0
$803$	−3.48027	−0.122816
$804$	0	0
$805$	−3.95190	−0.139286
$806$	0	0
$807$	−5.02961	−0.177051
$808$	0	0
$809$	8.82493	0.310268	0.155134	−	0.987893i	$-0.450419\pi$
$809$	8.82493	0.310268	0.155134	+	0.987893i	$0.450419\pi$
$810$	0	0
$811$	17.9422	0.630035	0.315018	−	0.949086i	$-0.397990\pi$
$811$	17.9422	0.630035	0.315018	+	0.949086i	$0.397990\pi$
$812$	0	0
$813$	6.37108	0.223443
$814$	0	0
$815$	−20.4606	−0.716705
$816$	0	0
$817$	20.3428	0.711703
$818$	0	0
$819$	28.8052	1.00653
$820$	0	0
$821$	−38.7051	−1.35082	−0.675409	−	0.737444i	$-0.736033\pi$
$821$	−38.7051	−1.35082	−0.675409	+	0.737444i	$0.736033\pi$
$822$	0	0
$823$	25.7406	0.897259	0.448630	−	0.893718i	$-0.351912\pi$
$823$	25.7406	0.897259	0.448630	+	0.893718i	$0.351912\pi$
$824$	0	0
$825$	−1.98726	−0.0691876
$826$	0	0
$827$	−54.3061	−1.88841	−0.944205	−	0.329359i	$-0.893167\pi$
$827$	−54.3061	−1.88841	−0.944205	+	0.329359i	$0.893167\pi$
$828$	0	0
$829$	44.6143	1.54952	0.774759	−	0.632257i	$-0.217871\pi$
$829$	44.6143	1.54952	0.774759	+	0.632257i	$0.217871\pi$
$830$	0	0
$831$	1.75316	0.0608165
$832$	0	0
$833$	10.8929	0.377415
$834$	0	0
$835$	20.1227	0.696373
$836$	0	0
$837$	18.7197	0.647048
$838$	0	0
$839$	26.1843	0.903984	0.451992	−	0.892022i	$-0.350714\pi$
$839$	26.1843	0.903984	0.451992	+	0.892022i	$0.350714\pi$
$840$	0	0
$841$	30.2440	1.04290
$842$	0	0
$843$	−1.81807	−0.0626177
$844$	0	0
$845$	−6.57596	−0.226220
$846$	0	0
$847$	−82.1951	−2.82426
$848$	0	0
$849$	−4.18665	−0.143685
$850$	0	0
$851$	−2.90308	−0.0995163
$852$	0	0
$853$	8.97182	0.307189	0.153595	−	0.988134i	$-0.450915\pi$
$853$	8.97182	0.307189	0.153595	+	0.988134i	$0.450915\pi$
$854$	0	0
$855$	−19.6518	−0.672077
$856$	0	0
$857$	7.25709	0.247898	0.123949	−	0.992289i	$-0.460444\pi$
$857$	7.25709	0.247898	0.123949	+	0.992289i	$0.460444\pi$
$858$	0	0
$859$	34.9010	1.19081	0.595403	−	0.803427i	$-0.296993\pi$
$859$	34.9010	1.19081	0.595403	+	0.803427i	$0.296993\pi$
$860$	0	0
$861$	−0.262751	−0.00895452
$862$	0	0
$863$	−15.7663	−0.536690	−0.268345	−	0.963323i	$-0.586477\pi$
$863$	−15.7663	−0.536690	−0.268345	+	0.963323i	$0.586477\pi$
$864$	0	0
$865$	24.2937	0.826012
$866$	0	0
$867$	−5.42792	−0.184342
$868$	0	0
$869$	−85.6163	−2.90433
$870$	0	0
$871$	10.0112	0.339217
$872$	0	0
$873$	−23.7679	−0.804420
$874$	0	0
$875$	−3.95190	−0.133599
$876$	0	0
$877$	−3.07478	−0.103828	−0.0519139	−	0.998652i	$-0.516532\pi$
$877$	−3.07478	−0.103828	−0.0519139	+	0.998652i	$0.516532\pi$
$878$	0	0
$879$	−8.46051	−0.285366
$880$	0	0
$881$	8.27798	0.278892	0.139446	−	0.990230i	$-0.455468\pi$
$881$	8.27798	0.278892	0.139446	+	0.990230i	$0.455468\pi$
$882$	0	0
$883$	−11.5521	−0.388758	−0.194379	−	0.980927i	$-0.562269\pi$
$883$	−11.5521	−0.388758	−0.194379	+	0.980927i	$0.562269\pi$
$884$	0	0
$885$	−0.340507	−0.0114460
$886$	0	0
$887$	44.6832	1.50031	0.750157	−	0.661260i	$-0.229978\pi$
$887$	44.6832	1.50031	0.750157	+	0.661260i	$0.229978\pi$
$888$	0	0
$889$	15.3088	0.513440
$890$	0	0
$891$	−44.5354	−1.49199
$892$	0	0
$893$	−32.9851	−1.10380
$894$	0	0
$895$	−15.3672	−0.513669
$896$	0	0
$897$	0.893211	0.0298234
$898$	0	0
$899$	−69.5832	−2.32073
$900$	0	0
$901$	0.415811	0.0138527
$902$	0	0
$903$	−4.14595	−0.137968
$904$	0	0
$905$	−14.0440	−0.466839
$906$	0	0
$907$	5.55427	0.184426	0.0922132	−	0.995739i	$-0.470606\pi$
$907$	5.55427	0.184426	0.0922132	+	0.995739i	$0.470606\pi$
$908$	0	0
$909$	−51.3462	−1.70304
$910$	0	0
$911$	−13.1325	−0.435100	−0.217550	−	0.976049i	$-0.569807\pi$
$911$	−13.1325	−0.435100	−0.217550	+	0.976049i	$0.569807\pi$
$912$	0	0
$913$	7.36871	0.243869
$914$	0	0
$915$	0.461691	0.0152630
$916$	0	0
$917$	10.0966	0.333421
$918$	0	0
$919$	12.5417	0.413712	0.206856	−	0.978371i	$-0.433677\pi$
$919$	12.5417	0.413712	0.206856	+	0.978371i	$0.433677\pi$
$920$	0	0
$921$	−7.34386	−0.241988
$922$	0	0
$923$	19.5001	0.641853
$924$	0	0
$925$	−2.90308	−0.0954527
$926$	0	0
$927$	−51.9441	−1.70607
$928$	0	0
$929$	−8.24051	−0.270362	−0.135181	−	0.990821i	$-0.543162\pi$
$929$	−8.24051	−0.270362	−0.135181	+	0.990821i	$0.543162\pi$
$930$	0	0
$931$	58.8879	1.92997
$932$	0	0
$933$	−8.32013	−0.272389
$934$	0	0
$935$	−7.12793	−0.233108
$936$	0	0
$937$	11.4798	0.375030	0.187515	−	0.982262i	$-0.439957\pi$
$937$	11.4798	0.375030	0.187515	+	0.982262i	$0.439957\pi$
$938$	0	0
$939$	5.56575	0.181631
$940$	0	0
$941$	−28.3769	−0.925061	−0.462531	−	0.886603i	$-0.653058\pi$
$941$	−28.3769	−0.925061	−0.462531	+	0.886603i	$0.653058\pi$
$942$	0	0
$943$	0.188663	0.00614373
$944$	0	0
$945$	8.18321	0.266200
$946$	0	0
$947$	7.88882	0.256352	0.128176	−	0.991751i	$-0.459088\pi$
$947$	7.88882	0.256352	0.128176	+	0.991751i	$0.459088\pi$
$948$	0	0
$949$	1.56427	0.0507783
$950$	0	0
$951$	−11.3849	−0.369181
$952$	0	0
$953$	−9.20245	−0.298096	−0.149048	−	0.988830i	$-0.547621\pi$
$953$	−9.20245	−0.298096	−0.149048	+	0.988830i	$0.547621\pi$
$954$	0	0
$955$	−2.51098	−0.0812534
$956$	0	0
$957$	−15.2960	−0.494449
$958$	0	0
$959$	−13.3487	−0.431051
$960$	0	0
$961$	50.7267	1.63635
$962$	0	0
$963$	−36.7375	−1.18385
$964$	0	0
$965$	18.6016	0.598805
$966$	0	0
$967$	19.6278	0.631186	0.315593	−	0.948895i	$-0.397797\pi$
$967$	19.6278	0.631186	0.315593	+	0.948895i	$0.397797\pi$
$968$	0	0
$969$	3.04404	0.0977886
$970$	0	0
$971$	−7.09502	−0.227690	−0.113845	−	0.993499i	$-0.536317\pi$
$971$	−7.09502	−0.227690	−0.113845	+	0.993499i	$0.536317\pi$
$972$	0	0
$973$	5.61031	0.179858
$974$	0	0
$975$	0.893211	0.0286056
$976$	0	0
$977$	40.5255	1.29652	0.648262	−	0.761417i	$-0.275496\pi$
$977$	40.5255	1.29652	0.648262	+	0.761417i	$0.275496\pi$
$978$	0	0
$979$	97.6917	3.12224
$980$	0	0
$981$	−42.4459	−1.35519
$982$	0	0
$983$	48.8845	1.55917	0.779587	−	0.626294i	$-0.215429\pi$
$983$	48.8845	1.55917	0.779587	+	0.626294i	$0.215429\pi$
$984$	0	0
$985$	−17.8033	−0.567260
$986$	0	0
$987$	6.72251	0.213980
$988$	0	0
$989$	2.97692	0.0946606
$990$	0	0
$991$	−44.9031	−1.42639	−0.713197	−	0.700963i	$-0.752754\pi$
$991$	−44.9031	−1.42639	−0.713197	+	0.700963i	$0.752754\pi$
$992$	0	0
$993$	5.07965	0.161198
$994$	0	0
$995$	−9.14740	−0.289992
$996$	0	0
$997$	55.6027	1.76096	0.880478	−	0.474087i	$-0.157222\pi$
$997$	55.6027	1.76096	0.880478	+	0.474087i	$0.157222\pi$
$998$	0	0
$999$	6.01141	0.190192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.x.1.3	✓	5
4.3	odd	2		3680.2.a.ba.1.3	yes	5
8.3	odd	2		7360.2.a.cl.1.3		5
8.5	even	2		7360.2.a.cq.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.x.1.3	✓	5	1.1	even	1	trivial
3680.2.a.ba.1.3	yes	5	4.3	odd	2
7360.2.a.cl.1.3		5	8.3	odd	2
7360.2.a.cq.1.3		5	8.5	even	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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.a (trivial)

\(f(q)\)	\(=\)	\(q+0.352411 q^{3} +1.00000 q^{5} -3.95190 q^{7} -2.87581 q^{9} +O(q^{10})\)	\(q+0.352411 q^{3} +1.00000 q^{5} -3.95190 q^{7} -2.87581 q^{9} -5.63905 q^{11} +2.53457 q^{13} +0.352411 q^{15} +1.26403 q^{17} +6.83349 q^{19} -1.39270 q^{21} +1.00000 q^{23} +1.00000 q^{25} -2.07070 q^{27} +7.69701 q^{29} -9.04028 q^{31} -1.98726 q^{33} -3.95190 q^{35} -2.90308 q^{37} +0.893211 q^{39} +0.188663 q^{41} +2.97692 q^{43} -2.87581 q^{45} -4.82698 q^{47} +8.61755 q^{49} +0.445459 q^{51} +0.328956 q^{53} -5.63905 q^{55} +2.40820 q^{57} -0.966221 q^{59} +1.31009 q^{61} +11.3649 q^{63} +2.53457 q^{65} +3.94987 q^{67} +0.352411 q^{69} +7.69365 q^{71} +0.617173 q^{73} +0.352411 q^{75} +22.2850 q^{77} +15.1828 q^{79} +7.89768 q^{81} -1.30673 q^{83} +1.26403 q^{85} +2.71251 q^{87} -17.3242 q^{89} -10.0164 q^{91} -3.18590 q^{93} +6.83349 q^{95} +8.26476 q^{97} +16.2168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9	\( 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9} - 3 q^{11} + 7 q^{13} - q^{15} + 9 q^{17} - q^{19} + 20 q^{21} + 5 q^{23} + 5 q^{25} - 4 q^{27} - 10 q^{29} - 21 q^{31} + 7 q^{33} - q^{35} + 8 q^{37} + 24 q^{39} - 13 q^{41} + 6 q^{43} + 6 q^{45} + 24 q^{49} + 17 q^{51} - 6 q^{53} - 3 q^{55} + 26 q^{57} - 18 q^{59} - 11 q^{61} - 4 q^{63} + 7 q^{65} - 38 q^{67} - q^{69} + 21 q^{71} - 12 q^{73} - q^{75} + 46 q^{77} + 18 q^{79} + 9 q^{81} - 20 q^{83} + 9 q^{85} + 6 q^{87} - 16 q^{89} + 3 q^{91} + 22 q^{93} - q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100}) \) 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9 - 3 * q^11 + 7 * q^13 - q^15 + 9 * q^17 - q^19 + 20 * q^21 + 5 * q^23 + 5 * q^25 - 4 * q^27 - 10 * q^29 - 21 * q^31 + 7 * q^33 - q^35 + 8 * q^37 + 24 * q^39 - 13 * q^41 + 6 * q^43 + 6 * q^45 + 24 * q^49 + 17 * q^51 - 6 * q^53 - 3 * q^55 + 26 * q^57 - 18 * q^59 - 11 * q^61 - 4 * q^63 + 7 * q^65 - 38 * q^67 - q^69 + 21 * q^71 - 12 * q^73 - q^75 + 46 * q^77 + 18 * q^79 + 9 * q^81 - 20 * q^83 + 9 * q^85 + 6 * q^87 - 16 * q^89 + 3 * q^91 + 22 * q^93 - q^95 + 29 * q^97 - 29 * q^99

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