LMFDB - Embedded newform 6240.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} -2.56155 q^{11} +1.00000 q^{13} -1.00000 q^{15} +0.561553 q^{17} +1.12311 q^{19} -2.56155 q^{21} +6.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.876894 q^{29} +6.24621 q^{31} -2.56155 q^{33} +2.56155 q^{35} -5.68466 q^{37} +1.00000 q^{39} -9.68466 q^{41} -9.12311 q^{43} -1.00000 q^{45} +8.00000 q^{47} -0.438447 q^{49} +0.561553 q^{51} +0.561553 q^{53} +2.56155 q^{55} +1.12311 q^{57} -4.00000 q^{59} +9.68466 q^{61} -2.56155 q^{63} -1.00000 q^{65} +6.24621 q^{67} +6.56155 q^{69} +3.68466 q^{71} -16.2462 q^{73} +1.00000 q^{75} +6.56155 q^{77} -5.43845 q^{79} +1.00000 q^{81} -4.00000 q^{83} -0.561553 q^{85} -0.876894 q^{87} -4.56155 q^{89} -2.56155 q^{91} +6.24621 q^{93} -1.12311 q^{95} +5.68466 q^{97} -2.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 2 q^{15} - 3 q^{17} - 6 q^{19} - q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - q^{33} + q^{35} + q^{37} + 2 q^{39} - 7 q^{41} - 10 q^{43} - 2 q^{45} + 16 q^{47} - 5 q^{49} - 3 q^{51} - 3 q^{53} + q^{55} - 6 q^{57} - 8 q^{59} + 7 q^{61} - q^{63} - 2 q^{65} - 4 q^{67} + 9 q^{69} - 5 q^{71} - 16 q^{73} + 2 q^{75} + 9 q^{77} - 15 q^{79} + 2 q^{81} - 8 q^{83} + 3 q^{85} - 10 q^{87} - 5 q^{89} - q^{91} - 4 q^{93} + 6 q^{95} - q^{97} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - q^11 + 2 * q^13 - 2 * q^15 - 3 * q^17 - 6 * q^19 - q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - q^33 + q^35 + q^37 + 2 * q^39 - 7 * q^41 - 10 * q^43 - 2 * q^45 + 16 * q^47 - 5 * q^49 - 3 * q^51 - 3 * q^53 + q^55 - 6 * q^57 - 8 * q^59 + 7 * q^61 - q^63 - 2 * q^65 - 4 * q^67 + 9 * q^69 - 5 * q^71 - 16 * q^73 + 2 * q^75 + 9 * q^77 - 15 * q^79 + 2 * q^81 - 8 * q^83 + 3 * q^85 - 10 * q^87 - 5 * q^89 - q^91 - 4 * q^93 + 6 * q^95 - q^97 - 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.56155	−0.968176	−0.484088	−	0.875019i	$-0.660849\pi$
$7$	−2.56155	−0.968176	−0.484088	+	0.875019i	$0.660849\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.561553	0.136197	0.0680983	−	0.997679i	$-0.478307\pi$
$17$	0.561553	0.136197	0.0680983	+	0.997679i	$0.478307\pi$
$18$	0	0
$19$	1.12311	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$20$	0	0
$21$	−2.56155	−0.558977
$22$	0	0
$23$	6.56155	1.36818	0.684089	−	0.729398i	$-0.260200\pi$
$23$	6.56155	1.36818	0.684089	+	0.729398i	$0.260200\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.876894	−0.162835	−0.0814176	−	0.996680i	$-0.525945\pi$
$29$	−0.876894	−0.162835	−0.0814176	+	0.996680i	$0.525945\pi$
$30$	0	0
$31$	6.24621	1.12185	0.560926	−	0.827866i	$-0.310445\pi$
$31$	6.24621	1.12185	0.560926	+	0.827866i	$0.310445\pi$
$32$	0	0
$33$	−2.56155	−0.445909
$34$	0	0
$35$	2.56155	0.432981
$36$	0	0
$37$	−5.68466	−0.934552	−0.467276	−	0.884111i	$-0.654765\pi$
$37$	−5.68466	−0.934552	−0.467276	+	0.884111i	$0.654765\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−9.68466	−1.51249	−0.756245	−	0.654289i	$-0.772968\pi$
$41$	−9.68466	−1.51249	−0.756245	+	0.654289i	$0.772968\pi$
$42$	0	0
$43$	−9.12311	−1.39126	−0.695630	−	0.718400i	$-0.744875\pi$
$43$	−9.12311	−1.39126	−0.695630	+	0.718400i	$0.744875\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	0.561553	0.0786331
$52$	0	0
$53$	0.561553	0.0771352	0.0385676	−	0.999256i	$-0.487721\pi$
$53$	0.561553	0.0771352	0.0385676	+	0.999256i	$0.487721\pi$
$54$	0	0
$55$	2.56155	0.345400
$56$	0	0
$57$	1.12311	0.148759
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	9.68466	1.23999	0.619997	−	0.784604i	$-0.287134\pi$
$61$	9.68466	1.23999	0.619997	+	0.784604i	$0.287134\pi$
$62$	0	0
$63$	−2.56155	−0.322725
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	0	0
$69$	6.56155	0.789918
$70$	0	0
$71$	3.68466	0.437289	0.218644	−	0.975805i	$-0.429837\pi$
$71$	3.68466	0.437289	0.218644	+	0.975805i	$0.429837\pi$
$72$	0	0
$73$	−16.2462	−1.90148	−0.950738	−	0.309997i	$-0.899672\pi$
$73$	−16.2462	−1.90148	−0.950738	+	0.309997i	$0.899672\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	6.56155	0.747758
$78$	0	0
$79$	−5.43845	−0.611873	−0.305937	−	0.952052i	$-0.598970\pi$
$79$	−5.43845	−0.611873	−0.305937	+	0.952052i	$0.598970\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−0.561553	−0.0609090
$86$	0	0
$87$	−0.876894	−0.0940129
$88$	0	0
$89$	−4.56155	−0.483524	−0.241762	−	0.970336i	$-0.577725\pi$
$89$	−4.56155	−0.483524	−0.241762	+	0.970336i	$0.577725\pi$
$90$	0	0
$91$	−2.56155	−0.268524
$92$	0	0
$93$	6.24621	0.647702
$94$	0	0
$95$	−1.12311	−0.115228
$96$	0	0
$97$	5.68466	0.577190	0.288595	−	0.957451i	$-0.406812\pi$
$97$	5.68466	0.577190	0.288595	+	0.957451i	$0.406812\pi$
$98$	0	0
$99$	−2.56155	−0.257446
$100$	0	0
$101$	−3.12311	−0.310761	−0.155380	−	0.987855i	$-0.549660\pi$
$101$	−3.12311	−0.310761	−0.155380	+	0.987855i	$0.549660\pi$
$102$	0	0
$103$	−16.4924	−1.62505	−0.812523	−	0.582929i	$-0.801907\pi$
$103$	−16.4924	−1.62505	−0.812523	+	0.582929i	$0.801907\pi$
$104$	0	0
$105$	2.56155	0.249982
$106$	0	0
$107$	2.56155	0.247635	0.123817	−	0.992305i	$-0.460486\pi$
$107$	2.56155	0.247635	0.123817	+	0.992305i	$0.460486\pi$
$108$	0	0
$109$	−4.87689	−0.467122	−0.233561	−	0.972342i	$-0.575038\pi$
$109$	−4.87689	−0.467122	−0.233561	+	0.972342i	$0.575038\pi$
$110$	0	0
$111$	−5.68466	−0.539564
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−6.56155	−0.611868
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−1.43845	−0.131862
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	−9.68466	−0.873236
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−9.12311	−0.803245
$130$	0	0
$131$	−16.4924	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$131$	−16.4924	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$132$	0	0
$133$	−2.87689	−0.249458
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−0.315342	−0.0267469	−0.0133735	−	0.999911i	$-0.504257\pi$
$139$	−0.315342	−0.0267469	−0.0133735	+	0.999911i	$0.504257\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−2.56155	−0.214208
$144$	0	0
$145$	0.876894	0.0728221
$146$	0	0
$147$	−0.438447	−0.0361625
$148$	0	0
$149$	−17.0540	−1.39712	−0.698558	−	0.715553i	$-0.746175\pi$
$149$	−17.0540	−1.39712	−0.698558	+	0.715553i	$0.746175\pi$
$150$	0	0
$151$	−16.4924	−1.34213	−0.671067	−	0.741397i	$-0.734164\pi$
$151$	−16.4924	−1.34213	−0.671067	+	0.741397i	$0.734164\pi$
$152$	0	0
$153$	0.561553	0.0453989
$154$	0	0
$155$	−6.24621	−0.501708
$156$	0	0
$157$	−20.2462	−1.61582	−0.807912	−	0.589303i	$-0.799402\pi$
$157$	−20.2462	−1.61582	−0.807912	+	0.589303i	$0.799402\pi$
$158$	0	0
$159$	0.561553	0.0445340
$160$	0	0
$161$	−16.8078	−1.32464
$162$	0	0
$163$	−18.5616	−1.45385	−0.726927	−	0.686715i	$-0.759052\pi$
$163$	−18.5616	−1.45385	−0.726927	+	0.686715i	$0.759052\pi$
$164$	0	0
$165$	2.56155	0.199417
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.12311	0.0858860
$172$	0	0
$173$	4.24621	0.322833	0.161417	−	0.986886i	$-0.448394\pi$
$173$	4.24621	0.322833	0.161417	+	0.986886i	$0.448394\pi$
$174$	0	0
$175$	−2.56155	−0.193635
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	1.12311	0.0839449	0.0419724	−	0.999119i	$-0.486636\pi$
$179$	1.12311	0.0839449	0.0419724	+	0.999119i	$0.486636\pi$
$180$	0	0
$181$	−0.561553	−0.0417399	−0.0208699	−	0.999782i	$-0.506644\pi$
$181$	−0.561553	−0.0417399	−0.0208699	+	0.999782i	$0.506644\pi$
$182$	0	0
$183$	9.68466	0.715911
$184$	0	0
$185$	5.68466	0.417944
$186$	0	0
$187$	−1.43845	−0.105190
$188$	0	0
$189$	−2.56155	−0.186326
$190$	0	0
$191$	23.3693	1.69094	0.845472	−	0.534019i	$-0.179319\pi$
$191$	23.3693	1.69094	0.845472	+	0.534019i	$0.179319\pi$
$192$	0	0
$193$	−17.6847	−1.27297	−0.636485	−	0.771289i	$-0.719612\pi$
$193$	−17.6847	−1.27297	−0.636485	+	0.771289i	$0.719612\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−18.4924	−1.31753	−0.658765	−	0.752349i	$-0.728921\pi$
$197$	−18.4924	−1.31753	−0.658765	+	0.752349i	$0.728921\pi$
$198$	0	0
$199$	−6.24621	−0.442782	−0.221391	−	0.975185i	$-0.571060\pi$
$199$	−6.24621	−0.442782	−0.221391	+	0.975185i	$0.571060\pi$
$200$	0	0
$201$	6.24621	0.440574
$202$	0	0
$203$	2.24621	0.157653
$204$	0	0
$205$	9.68466	0.676406
$206$	0	0
$207$	6.56155	0.456059
$208$	0	0
$209$	−2.87689	−0.198999
$210$	0	0
$211$	14.2462	0.980750	0.490375	−	0.871512i	$-0.336860\pi$
$211$	14.2462	0.980750	0.490375	+	0.871512i	$0.336860\pi$
$212$	0	0
$213$	3.68466	0.252469
$214$	0	0
$215$	9.12311	0.622191
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	−16.2462	−1.09782
$220$	0	0
$221$	0.561553	0.0377741
$222$	0	0
$223$	−6.24621	−0.418277	−0.209139	−	0.977886i	$-0.567066\pi$
$223$	−6.24621	−0.418277	−0.209139	+	0.977886i	$0.567066\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−1.12311	−0.0745431	−0.0372716	−	0.999305i	$-0.511867\pi$
$227$	−1.12311	−0.0745431	−0.0372716	+	0.999305i	$0.511867\pi$
$228$	0	0
$229$	23.6155	1.56056	0.780279	−	0.625432i	$-0.215077\pi$
$229$	23.6155	1.56056	0.780279	+	0.625432i	$0.215077\pi$
$230$	0	0
$231$	6.56155	0.431718
$232$	0	0
$233$	−2.31534	−0.151683	−0.0758415	−	0.997120i	$-0.524164\pi$
$233$	−2.31534	−0.151683	−0.0758415	+	0.997120i	$0.524164\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	−5.43845	−0.353265
$238$	0	0
$239$	−0.807764	−0.0522499	−0.0261250	−	0.999659i	$-0.508317\pi$
$239$	−0.807764	−0.0522499	−0.0261250	+	0.999659i	$0.508317\pi$
$240$	0	0
$241$	9.36932	0.603531	0.301765	−	0.953382i	$-0.402424\pi$
$241$	9.36932	0.603531	0.301765	+	0.953382i	$0.402424\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.438447	0.0280114
$246$	0	0
$247$	1.12311	0.0714615
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−16.8078	−1.05670
$254$	0	0
$255$	−0.561553	−0.0351658
$256$	0	0
$257$	−3.75379	−0.234155	−0.117077	−	0.993123i	$-0.537353\pi$
$257$	−3.75379	−0.234155	−0.117077	+	0.993123i	$0.537353\pi$
$258$	0	0
$259$	14.5616	0.904811
$260$	0	0
$261$	−0.876894	−0.0542784
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−0.561553	−0.0344959
$266$	0	0
$267$	−4.56155	−0.279162
$268$	0	0
$269$	−5.36932	−0.327373	−0.163687	−	0.986512i	$-0.552339\pi$
$269$	−5.36932	−0.327373	−0.163687	+	0.986512i	$0.552339\pi$
$270$	0	0
$271$	−19.3693	−1.17660	−0.588301	−	0.808642i	$-0.700203\pi$
$271$	−19.3693	−1.17660	−0.588301	+	0.808642i	$0.700203\pi$
$272$	0	0
$273$	−2.56155	−0.155032
$274$	0	0
$275$	−2.56155	−0.154467
$276$	0	0
$277$	−12.8769	−0.773698	−0.386849	−	0.922143i	$-0.626436\pi$
$277$	−12.8769	−0.773698	−0.386849	+	0.922143i	$0.626436\pi$
$278$	0	0
$279$	6.24621	0.373951
$280$	0	0
$281$	−8.24621	−0.491928	−0.245964	−	0.969279i	$-0.579104\pi$
$281$	−8.24621	−0.491928	−0.245964	+	0.969279i	$0.579104\pi$
$282$	0	0
$283$	6.24621	0.371299	0.185649	−	0.982616i	$-0.440561\pi$
$283$	6.24621	0.371299	0.185649	+	0.982616i	$0.440561\pi$
$284$	0	0
$285$	−1.12311	−0.0665270
$286$	0	0
$287$	24.8078	1.46436
$288$	0	0
$289$	−16.6847	−0.981450
$290$	0	0
$291$	5.68466	0.333241
$292$	0	0
$293$	−16.8769	−0.985959	−0.492979	−	0.870041i	$-0.664092\pi$
$293$	−16.8769	−0.985959	−0.492979	+	0.870041i	$0.664092\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−2.56155	−0.148636
$298$	0	0
$299$	6.56155	0.379464
$300$	0	0
$301$	23.3693	1.34699
$302$	0	0
$303$	−3.12311	−0.179418
$304$	0	0
$305$	−9.68466	−0.554542
$306$	0	0
$307$	−10.5616	−0.602780	−0.301390	−	0.953501i	$-0.597451\pi$
$307$	−10.5616	−0.602780	−0.301390	+	0.953501i	$0.597451\pi$
$308$	0	0
$309$	−16.4924	−0.938221
$310$	0	0
$311$	5.12311	0.290505	0.145252	−	0.989395i	$-0.453601\pi$
$311$	5.12311	0.290505	0.145252	+	0.989395i	$0.453601\pi$
$312$	0	0
$313$	4.24621	0.240010	0.120005	−	0.992773i	$-0.461709\pi$
$313$	4.24621	0.240010	0.120005	+	0.992773i	$0.461709\pi$
$314$	0	0
$315$	2.56155	0.144327
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	2.24621	0.125764
$320$	0	0
$321$	2.56155	0.142972
$322$	0	0
$323$	0.630683	0.0350921
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−4.87689	−0.269693
$328$	0	0
$329$	−20.4924	−1.12978
$330$	0	0
$331$	−14.8769	−0.817708	−0.408854	−	0.912600i	$-0.634071\pi$
$331$	−14.8769	−0.817708	−0.408854	+	0.912600i	$0.634071\pi$
$332$	0	0
$333$	−5.68466	−0.311517
$334$	0	0
$335$	−6.24621	−0.341267
$336$	0	0
$337$	7.75379	0.422376	0.211188	−	0.977445i	$-0.432267\pi$
$337$	7.75379	0.422376	0.211188	+	0.977445i	$0.432267\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	19.0540	1.02882
$344$	0	0
$345$	−6.56155	−0.353262
$346$	0	0
$347$	−15.6847	−0.841997	−0.420998	−	0.907061i	$-0.638320\pi$
$347$	−15.6847	−0.841997	−0.420998	+	0.907061i	$0.638320\pi$
$348$	0	0
$349$	−15.7538	−0.843281	−0.421640	−	0.906763i	$-0.638546\pi$
$349$	−15.7538	−0.843281	−0.421640	+	0.906763i	$0.638546\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	7.12311	0.379125	0.189562	−	0.981869i	$-0.439293\pi$
$353$	7.12311	0.379125	0.189562	+	0.981869i	$0.439293\pi$
$354$	0	0
$355$	−3.68466	−0.195561
$356$	0	0
$357$	−1.43845	−0.0761307
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	−4.43845	−0.232958
$364$	0	0
$365$	16.2462	0.850366
$366$	0	0
$367$	27.3693	1.42867	0.714333	−	0.699806i	$-0.246730\pi$
$367$	27.3693	1.42867	0.714333	+	0.699806i	$0.246730\pi$
$368$	0	0
$369$	−9.68466	−0.504163
$370$	0	0
$371$	−1.43845	−0.0746805
$372$	0	0
$373$	−32.7386	−1.69514	−0.847571	−	0.530682i	$-0.821936\pi$
$373$	−32.7386	−1.69514	−0.847571	+	0.530682i	$0.821936\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−0.876894	−0.0451624
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	−13.1231	−0.670559	−0.335280	−	0.942119i	$-0.608831\pi$
$383$	−13.1231	−0.670559	−0.335280	+	0.942119i	$0.608831\pi$
$384$	0	0
$385$	−6.56155	−0.334408
$386$	0	0
$387$	−9.12311	−0.463754
$388$	0	0
$389$	29.8617	1.51405	0.757025	−	0.653386i	$-0.226652\pi$
$389$	29.8617	1.51405	0.757025	+	0.653386i	$0.226652\pi$
$390$	0	0
$391$	3.68466	0.186341
$392$	0	0
$393$	−16.4924	−0.831933
$394$	0	0
$395$	5.43845	0.273638
$396$	0	0
$397$	−13.6847	−0.686813	−0.343407	−	0.939187i	$-0.611581\pi$
$397$	−13.6847	−0.686813	−0.343407	+	0.939187i	$0.611581\pi$
$398$	0	0
$399$	−2.87689	−0.144025
$400$	0	0
$401$	32.7386	1.63489	0.817445	−	0.576007i	$-0.195390\pi$
$401$	32.7386	1.63489	0.817445	+	0.576007i	$0.195390\pi$
$402$	0	0
$403$	6.24621	0.311146
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	14.5616	0.721789
$408$	0	0
$409$	21.8617	1.08099	0.540497	−	0.841346i	$-0.318236\pi$
$409$	21.8617	1.08099	0.540497	+	0.841346i	$0.318236\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	10.2462	0.504183
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−0.315342	−0.0154423
$418$	0	0
$419$	25.1231	1.22734	0.613672	−	0.789561i	$-0.289692\pi$
$419$	25.1231	1.22734	0.613672	+	0.789561i	$0.289692\pi$
$420$	0	0
$421$	−12.8769	−0.627581	−0.313791	−	0.949492i	$-0.601599\pi$
$421$	−12.8769	−0.627581	−0.313791	+	0.949492i	$0.601599\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	0.561553	0.0272393
$426$	0	0
$427$	−24.8078	−1.20053
$428$	0	0
$429$	−2.56155	−0.123673
$430$	0	0
$431$	32.9848	1.58882	0.794412	−	0.607379i	$-0.207779\pi$
$431$	32.9848	1.58882	0.794412	+	0.607379i	$0.207779\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	0.876894	0.0420439
$436$	0	0
$437$	7.36932	0.352522
$438$	0	0
$439$	−18.5616	−0.885895	−0.442947	−	0.896548i	$-0.646067\pi$
$439$	−18.5616	−0.885895	−0.442947	+	0.896548i	$0.646067\pi$
$440$	0	0
$441$	−0.438447	−0.0208784
$442$	0	0
$443$	20.8078	0.988607	0.494303	−	0.869289i	$-0.335423\pi$
$443$	20.8078	0.988607	0.494303	+	0.869289i	$0.335423\pi$
$444$	0	0
$445$	4.56155	0.216238
$446$	0	0
$447$	−17.0540	−0.806625
$448$	0	0
$449$	−23.4384	−1.10613	−0.553064	−	0.833139i	$-0.686542\pi$
$449$	−23.4384	−1.10613	−0.553064	+	0.833139i	$0.686542\pi$
$450$	0	0
$451$	24.8078	1.16815
$452$	0	0
$453$	−16.4924	−0.774882
$454$	0	0
$455$	2.56155	0.120087
$456$	0	0
$457$	28.4233	1.32959	0.664793	−	0.747028i	$-0.268520\pi$
$457$	28.4233	1.32959	0.664793	+	0.747028i	$0.268520\pi$
$458$	0	0
$459$	0.561553	0.0262110
$460$	0	0
$461$	8.56155	0.398751	0.199376	−	0.979923i	$-0.436109\pi$
$461$	8.56155	0.398751	0.199376	+	0.979923i	$0.436109\pi$
$462$	0	0
$463$	1.93087	0.0897351	0.0448676	−	0.998993i	$-0.485713\pi$
$463$	1.93087	0.0897351	0.0448676	+	0.998993i	$0.485713\pi$
$464$	0	0
$465$	−6.24621	−0.289661
$466$	0	0
$467$	−7.68466	−0.355604	−0.177802	−	0.984066i	$-0.556899\pi$
$467$	−7.68466	−0.355604	−0.177802	+	0.984066i	$0.556899\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−20.2462	−0.932896
$472$	0	0
$473$	23.3693	1.07452
$474$	0	0
$475$	1.12311	0.0515316
$476$	0	0
$477$	0.561553	0.0257117
$478$	0	0
$479$	−26.4233	−1.20731	−0.603656	−	0.797245i	$-0.706290\pi$
$479$	−26.4233	−1.20731	−0.603656	+	0.797245i	$0.706290\pi$
$480$	0	0
$481$	−5.68466	−0.259198
$482$	0	0
$483$	−16.8078	−0.764780
$484$	0	0
$485$	−5.68466	−0.258127
$486$	0	0
$487$	16.3153	0.739319	0.369659	−	0.929167i	$-0.379474\pi$
$487$	16.3153	0.739319	0.369659	+	0.929167i	$0.379474\pi$
$488$	0	0
$489$	−18.5616	−0.839382
$490$	0	0
$491$	39.8617	1.79894	0.899468	−	0.436988i	$-0.143955\pi$
$491$	39.8617	1.79894	0.899468	+	0.436988i	$0.143955\pi$
$492$	0	0
$493$	−0.492423	−0.0221776
$494$	0	0
$495$	2.56155	0.115133
$496$	0	0
$497$	−9.43845	−0.423372
$498$	0	0
$499$	−5.61553	−0.251386	−0.125693	−	0.992069i	$-0.540115\pi$
$499$	−5.61553	−0.251386	−0.125693	+	0.992069i	$0.540115\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	3.12311	0.138976
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−1.05398	−0.0467166	−0.0233583	−	0.999727i	$-0.507436\pi$
$509$	−1.05398	−0.0467166	−0.0233583	+	0.999727i	$0.507436\pi$
$510$	0	0
$511$	41.6155	1.84096
$512$	0	0
$513$	1.12311	0.0495863
$514$	0	0
$515$	16.4924	0.726743
$516$	0	0
$517$	−20.4924	−0.901256
$518$	0	0
$519$	4.24621	0.186388
$520$	0	0
$521$	−2.49242	−0.109195	−0.0545975	−	0.998508i	$-0.517388\pi$
$521$	−2.49242	−0.109195	−0.0545975	+	0.998508i	$0.517388\pi$
$522$	0	0
$523$	−29.6155	−1.29500	−0.647498	−	0.762067i	$-0.724185\pi$
$523$	−29.6155	−1.29500	−0.647498	+	0.762067i	$0.724185\pi$
$524$	0	0
$525$	−2.56155	−0.111795
$526$	0	0
$527$	3.50758	0.152792
$528$	0	0
$529$	20.0540	0.871912
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−9.68466	−0.419489
$534$	0	0
$535$	−2.56155	−0.110746
$536$	0	0
$537$	1.12311	0.0484656
$538$	0	0
$539$	1.12311	0.0483756
$540$	0	0
$541$	24.2462	1.04243	0.521213	−	0.853427i	$-0.325480\pi$
$541$	24.2462	1.04243	0.521213	+	0.853427i	$0.325480\pi$
$542$	0	0
$543$	−0.561553	−0.0240985
$544$	0	0
$545$	4.87689	0.208903
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	9.68466	0.413331
$550$	0	0
$551$	−0.984845	−0.0419558
$552$	0	0
$553$	13.9309	0.592401
$554$	0	0
$555$	5.68466	0.241300
$556$	0	0
$557$	−6.63068	−0.280951	−0.140476	−	0.990084i	$-0.544863\pi$
$557$	−6.63068	−0.280951	−0.140476	+	0.990084i	$0.544863\pi$
$558$	0	0
$559$	−9.12311	−0.385866
$560$	0	0
$561$	−1.43845	−0.0607313
$562$	0	0
$563$	−1.93087	−0.0813765	−0.0406882	−	0.999172i	$-0.512955\pi$
$563$	−1.93087	−0.0813765	−0.0406882	+	0.999172i	$0.512955\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	−2.56155	−0.107575
$568$	0	0
$569$	−3.75379	−0.157367	−0.0786835	−	0.996900i	$-0.525072\pi$
$569$	−3.75379	−0.157367	−0.0786835	+	0.996900i	$0.525072\pi$
$570$	0	0
$571$	−39.6847	−1.66075	−0.830376	−	0.557204i	$-0.811874\pi$
$571$	−39.6847	−1.66075	−0.830376	+	0.557204i	$0.811874\pi$
$572$	0	0
$573$	23.3693	0.976267
$574$	0	0
$575$	6.56155	0.273636
$576$	0	0
$577$	−19.3002	−0.803477	−0.401739	−	0.915754i	$-0.631594\pi$
$577$	−19.3002	−0.803477	−0.401739	+	0.915754i	$0.631594\pi$
$578$	0	0
$579$	−17.6847	−0.734949
$580$	0	0
$581$	10.2462	0.425084
$582$	0	0
$583$	−1.43845	−0.0595744
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	39.8617	1.64527	0.822635	−	0.568570i	$-0.192503\pi$
$587$	39.8617	1.64527	0.822635	+	0.568570i	$0.192503\pi$
$588$	0	0
$589$	7.01515	0.289054
$590$	0	0
$591$	−18.4924	−0.760677
$592$	0	0
$593$	−33.8617	−1.39053	−0.695267	−	0.718751i	$-0.744714\pi$
$593$	−33.8617	−1.39053	−0.695267	+	0.718751i	$0.744714\pi$
$594$	0	0
$595$	1.43845	0.0589706
$596$	0	0
$597$	−6.24621	−0.255640
$598$	0	0
$599$	36.4924	1.49104	0.745520	−	0.666483i	$-0.232201\pi$
$599$	36.4924	1.49104	0.745520	+	0.666483i	$0.232201\pi$
$600$	0	0
$601$	12.0691	0.492310	0.246155	−	0.969230i	$-0.420833\pi$
$601$	12.0691	0.492310	0.246155	+	0.969230i	$0.420833\pi$
$602$	0	0
$603$	6.24621	0.254365
$604$	0	0
$605$	4.43845	0.180449
$606$	0	0
$607$	−42.1080	−1.70911	−0.854554	−	0.519362i	$-0.826170\pi$
$607$	−42.1080	−1.70911	−0.854554	+	0.519362i	$0.826170\pi$
$608$	0	0
$609$	2.24621	0.0910211
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−14.3153	−0.578191	−0.289096	−	0.957300i	$-0.593355\pi$
$613$	−14.3153	−0.578191	−0.289096	+	0.957300i	$0.593355\pi$
$614$	0	0
$615$	9.68466	0.390523
$616$	0	0
$617$	−40.2462	−1.62025	−0.810126	−	0.586256i	$-0.800601\pi$
$617$	−40.2462	−1.62025	−0.810126	+	0.586256i	$0.800601\pi$
$618$	0	0
$619$	−36.9848	−1.48655	−0.743273	−	0.668988i	$-0.766728\pi$
$619$	−36.9848	−1.48655	−0.743273	+	0.668988i	$0.766728\pi$
$620$	0	0
$621$	6.56155	0.263306
$622$	0	0
$623$	11.6847	0.468136
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.87689	−0.114892
$628$	0	0
$629$	−3.19224	−0.127283
$630$	0	0
$631$	30.2462	1.20408	0.602041	−	0.798465i	$-0.294354\pi$
$631$	30.2462	1.20408	0.602041	+	0.798465i	$0.294354\pi$
$632$	0	0
$633$	14.2462	0.566236
$634$	0	0
$635$	−12.0000	−0.476205
$636$	0	0
$637$	−0.438447	−0.0173719
$638$	0	0
$639$	3.68466	0.145763
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	−8.31534	−0.327925	−0.163963	−	0.986467i	$-0.552428\pi$
$643$	−8.31534	−0.327925	−0.163963	+	0.986467i	$0.552428\pi$
$644$	0	0
$645$	9.12311	0.359222
$646$	0	0
$647$	25.4384	1.00009	0.500044	−	0.866000i	$-0.333317\pi$
$647$	25.4384	1.00009	0.500044	+	0.866000i	$0.333317\pi$
$648$	0	0
$649$	10.2462	0.402199
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	5.50758	0.215528	0.107764	−	0.994176i	$-0.465631\pi$
$653$	5.50758	0.215528	0.107764	+	0.994176i	$0.465631\pi$
$654$	0	0
$655$	16.4924	0.644412
$656$	0	0
$657$	−16.2462	−0.633825
$658$	0	0
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	−42.9848	−1.67192	−0.835958	−	0.548793i	$-0.815088\pi$
$661$	−42.9848	−1.67192	−0.835958	+	0.548793i	$0.815088\pi$
$662$	0	0
$663$	0.561553	0.0218089
$664$	0	0
$665$	2.87689	0.111561
$666$	0	0
$667$	−5.75379	−0.222788
$668$	0	0
$669$	−6.24621	−0.241492
$670$	0	0
$671$	−24.8078	−0.957693
$672$	0	0
$673$	15.1231	0.582953	0.291476	−	0.956578i	$-0.405854\pi$
$673$	15.1231	0.582953	0.291476	+	0.956578i	$0.405854\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−6.17708	−0.237405	−0.118702	−	0.992930i	$-0.537873\pi$
$677$	−6.17708	−0.237405	−0.118702	+	0.992930i	$0.537873\pi$
$678$	0	0
$679$	−14.5616	−0.558821
$680$	0	0
$681$	−1.12311	−0.0430375
$682$	0	0
$683$	−13.6155	−0.520984	−0.260492	−	0.965476i	$-0.583885\pi$
$683$	−13.6155	−0.520984	−0.260492	+	0.965476i	$0.583885\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	23.6155	0.900989
$688$	0	0
$689$	0.561553	0.0213935
$690$	0	0
$691$	27.3693	1.04118	0.520589	−	0.853807i	$-0.325712\pi$
$691$	27.3693	1.04118	0.520589	+	0.853807i	$0.325712\pi$
$692$	0	0
$693$	6.56155	0.249253
$694$	0	0
$695$	0.315342	0.0119616
$696$	0	0
$697$	−5.43845	−0.205996
$698$	0	0
$699$	−2.31534	−0.0875743
$700$	0	0
$701$	45.8617	1.73217	0.866087	−	0.499893i	$-0.166627\pi$
$701$	45.8617	1.73217	0.866087	+	0.499893i	$0.166627\pi$
$702$	0	0
$703$	−6.38447	−0.240795
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	8.00000	0.300871
$708$	0	0
$709$	−23.1231	−0.868406	−0.434203	−	0.900815i	$-0.642970\pi$
$709$	−23.1231	−0.868406	−0.434203	+	0.900815i	$0.642970\pi$
$710$	0	0
$711$	−5.43845	−0.203958
$712$	0	0
$713$	40.9848	1.53489
$714$	0	0
$715$	2.56155	0.0957966
$716$	0	0
$717$	−0.807764	−0.0301665
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	42.2462	1.57333
$722$	0	0
$723$	9.36932	0.348449
$724$	0	0
$725$	−0.876894	−0.0325670
$726$	0	0
$727$	30.8769	1.14516	0.572580	−	0.819849i	$-0.305942\pi$
$727$	30.8769	1.14516	0.572580	+	0.819849i	$0.305942\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.12311	−0.189485
$732$	0	0
$733$	−37.6847	−1.39191	−0.695957	−	0.718083i	$-0.745020\pi$
$733$	−37.6847	−1.39191	−0.695957	+	0.718083i	$0.745020\pi$
$734$	0	0
$735$	0.438447	0.0161724
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−12.6307	−0.464627	−0.232314	−	0.972641i	$-0.574630\pi$
$739$	−12.6307	−0.464627	−0.232314	+	0.972641i	$0.574630\pi$
$740$	0	0
$741$	1.12311	0.0412583
$742$	0	0
$743$	6.38447	0.234224	0.117112	−	0.993119i	$-0.462636\pi$
$743$	6.38447	0.234224	0.117112	+	0.993119i	$0.462636\pi$
$744$	0	0
$745$	17.0540	0.624809
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−6.56155	−0.239754
$750$	0	0
$751$	13.4384	0.490376	0.245188	−	0.969476i	$-0.421150\pi$
$751$	13.4384	0.490376	0.245188	+	0.969476i	$0.421150\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	16.4924	0.600221
$756$	0	0
$757$	−28.8769	−1.04955	−0.524774	−	0.851241i	$-0.675850\pi$
$757$	−28.8769	−1.04955	−0.524774	+	0.851241i	$0.675850\pi$
$758$	0	0
$759$	−16.8078	−0.610083
$760$	0	0
$761$	40.7386	1.47677	0.738387	−	0.674377i	$-0.235588\pi$
$761$	40.7386	1.47677	0.738387	+	0.674377i	$0.235588\pi$
$762$	0	0
$763$	12.4924	0.452256
$764$	0	0
$765$	−0.561553	−0.0203030
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−22.9848	−0.828855	−0.414427	−	0.910082i	$-0.636018\pi$
$769$	−22.9848	−0.828855	−0.414427	+	0.910082i	$0.636018\pi$
$770$	0	0
$771$	−3.75379	−0.135189
$772$	0	0
$773$	−0.876894	−0.0315397	−0.0157698	−	0.999876i	$-0.505020\pi$
$773$	−0.876894	−0.0315397	−0.0157698	+	0.999876i	$0.505020\pi$
$774$	0	0
$775$	6.24621	0.224371
$776$	0	0
$777$	14.5616	0.522393
$778$	0	0
$779$	−10.8769	−0.389705
$780$	0	0
$781$	−9.43845	−0.337734
$782$	0	0
$783$	−0.876894	−0.0313376
$784$	0	0
$785$	20.2462	0.722618
$786$	0	0
$787$	−1.75379	−0.0625158	−0.0312579	−	0.999511i	$-0.509951\pi$
$787$	−1.75379	−0.0625158	−0.0312579	+	0.999511i	$0.509951\pi$
$788$	0	0
$789$	8.00000	0.284808
$790$	0	0
$791$	−5.12311	−0.182157
$792$	0	0
$793$	9.68466	0.343912
$794$	0	0
$795$	−0.561553	−0.0199162
$796$	0	0
$797$	−28.5616	−1.01170	−0.505851	−	0.862621i	$-0.668821\pi$
$797$	−28.5616	−1.01170	−0.505851	+	0.862621i	$0.668821\pi$
$798$	0	0
$799$	4.49242	0.158930
$800$	0	0
$801$	−4.56155	−0.161175
$802$	0	0
$803$	41.6155	1.46858
$804$	0	0
$805$	16.8078	0.592396
$806$	0	0
$807$	−5.36932	−0.189009
$808$	0	0
$809$	−9.50758	−0.334269	−0.167134	−	0.985934i	$-0.553451\pi$
$809$	−9.50758	−0.334269	−0.167134	+	0.985934i	$0.553451\pi$
$810$	0	0
$811$	−31.8617	−1.11882	−0.559408	−	0.828892i	$-0.688972\pi$
$811$	−31.8617	−1.11882	−0.559408	+	0.828892i	$0.688972\pi$
$812$	0	0
$813$	−19.3693	−0.679312
$814$	0	0
$815$	18.5616	0.650183
$816$	0	0
$817$	−10.2462	−0.358470
$818$	0	0
$819$	−2.56155	−0.0895079
$820$	0	0
$821$	30.3153	1.05801	0.529006	−	0.848618i	$-0.322565\pi$
$821$	30.3153	1.05801	0.529006	+	0.848618i	$0.322565\pi$
$822$	0	0
$823$	−28.9848	−1.01035	−0.505174	−	0.863017i	$-0.668572\pi$
$823$	−28.9848	−1.01035	−0.505174	+	0.863017i	$0.668572\pi$
$824$	0	0
$825$	−2.56155	−0.0891818
$826$	0	0
$827$	−34.7386	−1.20798	−0.603990	−	0.796992i	$-0.706423\pi$
$827$	−34.7386	−1.20798	−0.603990	+	0.796992i	$0.706423\pi$
$828$	0	0
$829$	33.2311	1.15416	0.577081	−	0.816687i	$-0.304192\pi$
$829$	33.2311	1.15416	0.577081	+	0.816687i	$0.304192\pi$
$830$	0	0
$831$	−12.8769	−0.446695
$832$	0	0
$833$	−0.246211	−0.00853071
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	6.24621	0.215901
$838$	0	0
$839$	13.9309	0.480947	0.240474	−	0.970656i	$-0.422697\pi$
$839$	13.9309	0.480947	0.240474	+	0.970656i	$0.422697\pi$
$840$	0	0
$841$	−28.2311	−0.973485
$842$	0	0
$843$	−8.24621	−0.284015
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	11.3693	0.390654
$848$	0	0
$849$	6.24621	0.214369
$850$	0	0
$851$	−37.3002	−1.27863
$852$	0	0
$853$	32.4233	1.11015	0.555076	−	0.831800i	$-0.312689\pi$
$853$	32.4233	1.11015	0.555076	+	0.831800i	$0.312689\pi$
$854$	0	0
$855$	−1.12311	−0.0384094
$856$	0	0
$857$	27.4384	0.937280	0.468640	−	0.883389i	$-0.344744\pi$
$857$	27.4384	0.937280	0.468640	+	0.883389i	$0.344744\pi$
$858$	0	0
$859$	22.4233	0.765073	0.382536	−	0.923940i	$-0.375051\pi$
$859$	22.4233	0.765073	0.382536	+	0.923940i	$0.375051\pi$
$860$	0	0
$861$	24.8078	0.845446
$862$	0	0
$863$	33.6155	1.14429	0.572143	−	0.820154i	$-0.306112\pi$
$863$	33.6155	1.14429	0.572143	+	0.820154i	$0.306112\pi$
$864$	0	0
$865$	−4.24621	−0.144376
$866$	0	0
$867$	−16.6847	−0.566641
$868$	0	0
$869$	13.9309	0.472572
$870$	0	0
$871$	6.24621	0.211645
$872$	0	0
$873$	5.68466	0.192397
$874$	0	0
$875$	2.56155	0.0865963
$876$	0	0
$877$	25.2311	0.851992	0.425996	−	0.904725i	$-0.359924\pi$
$877$	25.2311	0.851992	0.425996	+	0.904725i	$0.359924\pi$
$878$	0	0
$879$	−16.8769	−0.569244
$880$	0	0
$881$	20.8769	0.703360	0.351680	−	0.936120i	$-0.385610\pi$
$881$	20.8769	0.703360	0.351680	+	0.936120i	$0.385610\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	0.177081	0.00594580	0.00297290	−	0.999996i	$-0.499054\pi$
$887$	0.177081	0.00594580	0.00297290	+	0.999996i	$0.499054\pi$
$888$	0	0
$889$	−30.7386	−1.03094
$890$	0	0
$891$	−2.56155	−0.0858152
$892$	0	0
$893$	8.98485	0.300666
$894$	0	0
$895$	−1.12311	−0.0375413
$896$	0	0
$897$	6.56155	0.219084
$898$	0	0
$899$	−5.47727	−0.182677
$900$	0	0
$901$	0.315342	0.0105056
$902$	0	0
$903$	23.3693	0.777682
$904$	0	0
$905$	0.561553	0.0186666
$906$	0	0
$907$	−29.6155	−0.983367	−0.491684	−	0.870774i	$-0.663618\pi$
$907$	−29.6155	−0.983367	−0.491684	+	0.870774i	$0.663618\pi$
$908$	0	0
$909$	−3.12311	−0.103587
$910$	0	0
$911$	−23.3693	−0.774260	−0.387130	−	0.922025i	$-0.626534\pi$
$911$	−23.3693	−0.774260	−0.387130	+	0.922025i	$0.626534\pi$
$912$	0	0
$913$	10.2462	0.339100
$914$	0	0
$915$	−9.68466	−0.320165
$916$	0	0
$917$	42.2462	1.39509
$918$	0	0
$919$	−28.1771	−0.929476	−0.464738	−	0.885448i	$-0.653852\pi$
$919$	−28.1771	−0.929476	−0.464738	+	0.885448i	$0.653852\pi$
$920$	0	0
$921$	−10.5616	−0.348015
$922$	0	0
$923$	3.68466	0.121282
$924$	0	0
$925$	−5.68466	−0.186910
$926$	0	0
$927$	−16.4924	−0.541682
$928$	0	0
$929$	−19.3002	−0.633219	−0.316609	−	0.948556i	$-0.602544\pi$
$929$	−19.3002	−0.633219	−0.316609	+	0.948556i	$0.602544\pi$
$930$	0	0
$931$	−0.492423	−0.0161385
$932$	0	0
$933$	5.12311	0.167723
$934$	0	0
$935$	1.43845	0.0470423
$936$	0	0
$937$	−29.3693	−0.959454	−0.479727	−	0.877418i	$-0.659264\pi$
$937$	−29.3693	−0.959454	−0.479727	+	0.877418i	$0.659264\pi$
$938$	0	0
$939$	4.24621	0.138570
$940$	0	0
$941$	−14.8078	−0.482719	−0.241360	−	0.970436i	$-0.577593\pi$
$941$	−14.8078	−0.482719	−0.241360	+	0.970436i	$0.577593\pi$
$942$	0	0
$943$	−63.5464	−2.06936
$944$	0	0
$945$	2.56155	0.0833273
$946$	0	0
$947$	18.7386	0.608924	0.304462	−	0.952525i	$-0.401523\pi$
$947$	18.7386	0.608924	0.304462	+	0.952525i	$0.401523\pi$
$948$	0	0
$949$	−16.2462	−0.527374
$950$	0	0
$951$	−22.0000	−0.713399
$952$	0	0
$953$	14.3153	0.463719	0.231860	−	0.972749i	$-0.425519\pi$
$953$	14.3153	0.463719	0.231860	+	0.972749i	$0.425519\pi$
$954$	0	0
$955$	−23.3693	−0.756213
$956$	0	0
$957$	2.24621	0.0726097
$958$	0	0
$959$	15.3693	0.496301
$960$	0	0
$961$	8.01515	0.258553
$962$	0	0
$963$	2.56155	0.0825449
$964$	0	0
$965$	17.6847	0.569289
$966$	0	0
$967$	17.7538	0.570923	0.285462	−	0.958390i	$-0.407853\pi$
$967$	17.7538	0.570923	0.285462	+	0.958390i	$0.407853\pi$
$968$	0	0
$969$	0.630683	0.0202605
$970$	0	0
$971$	−44.3542	−1.42339	−0.711696	−	0.702487i	$-0.752073\pi$
$971$	−44.3542	−1.42339	−0.711696	+	0.702487i	$0.752073\pi$
$972$	0	0
$973$	0.807764	0.0258957
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	28.2462	0.903676	0.451838	−	0.892100i	$-0.350768\pi$
$977$	28.2462	0.903676	0.451838	+	0.892100i	$0.350768\pi$
$978$	0	0
$979$	11.6847	0.373443
$980$	0	0
$981$	−4.87689	−0.155707
$982$	0	0
$983$	19.5076	0.622195	0.311098	−	0.950378i	$-0.399303\pi$
$983$	19.5076	0.622195	0.311098	+	0.950378i	$0.399303\pi$
$984$	0	0
$985$	18.4924	0.589218
$986$	0	0
$987$	−20.4924	−0.652281
$988$	0	0
$989$	−59.8617	−1.90349
$990$	0	0
$991$	−33.3002	−1.05782	−0.528908	−	0.848679i	$-0.677398\pi$
$991$	−33.3002	−1.05782	−0.528908	+	0.848679i	$0.677398\pi$
$992$	0	0
$993$	−14.8769	−0.472104
$994$	0	0
$995$	6.24621	0.198018
$996$	0	0
$997$	−0.384472	−0.0121763	−0.00608817	−	0.999981i	$-0.501938\pi$
$997$	−0.384472	−0.0121763	−0.00608817	+	0.999981i	$0.501938\pi$
$998$	0	0
$999$	−5.68466	−0.179855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bq.1.1	yes	2
4.3	odd	2		6240.2.a.bi.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bi.1.2	✓	2	4.3	odd	2
6240.2.a.bq.1.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} -2.56155 q^{11} +1.00000 q^{13} -1.00000 q^{15} +0.561553 q^{17} +1.12311 q^{19} -2.56155 q^{21} +6.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.876894 q^{29} +6.24621 q^{31} -2.56155 q^{33} +2.56155 q^{35} -5.68466 q^{37} +1.00000 q^{39} -9.68466 q^{41} -9.12311 q^{43} -1.00000 q^{45} +8.00000 q^{47} -0.438447 q^{49} +0.561553 q^{51} +0.561553 q^{53} +2.56155 q^{55} +1.12311 q^{57} -4.00000 q^{59} +9.68466 q^{61} -2.56155 q^{63} -1.00000 q^{65} +6.24621 q^{67} +6.56155 q^{69} +3.68466 q^{71} -16.2462 q^{73} +1.00000 q^{75} +6.56155 q^{77} -5.43845 q^{79} +1.00000 q^{81} -4.00000 q^{83} -0.561553 q^{85} -0.876894 q^{87} -4.56155 q^{89} -2.56155 q^{91} +6.24621 q^{93} -1.12311 q^{95} +5.68466 q^{97} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 2 q^{15} - 3 q^{17} - 6 q^{19} - q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - q^{33} + q^{35} + q^{37} + 2 q^{39} - 7 q^{41} - 10 q^{43} - 2 q^{45} + 16 q^{47} - 5 q^{49} - 3 q^{51} - 3 q^{53} + q^{55} - 6 q^{57} - 8 q^{59} + 7 q^{61} - q^{63} - 2 q^{65} - 4 q^{67} + 9 q^{69} - 5 q^{71} - 16 q^{73} + 2 q^{75} + 9 q^{77} - 15 q^{79} + 2 q^{81} - 8 q^{83} + 3 q^{85} - 10 q^{87} - 5 q^{89} - q^{91} - 4 q^{93} + 6 q^{95} - q^{97} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - q^11 + 2 * q^13 - 2 * q^15 - 3 * q^17 - 6 * q^19 - q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 - 4 * q^31 - q^33 + q^35 + q^37 + 2 * q^39 - 7 * q^41 - 10 * q^43 - 2 * q^45 + 16 * q^47 - 5 * q^49 - 3 * q^51 - 3 * q^53 + q^55 - 6 * q^57 - 8 * q^59 + 7 * q^61 - q^63 - 2 * q^65 - 4 * q^67 + 9 * q^69 - 5 * q^71 - 16 * q^73 + 2 * q^75 + 9 * q^77 - 15 * q^79 + 2 * q^81 - 8 * q^83 + 3 * q^85 - 10 * q^87 - 5 * q^89 - q^91 - 4 * q^93 + 6 * q^95 - q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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