LMFDB - Embedded newform 1520.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.642074 q^{3} +1.00000 q^{5} -3.58774 q^{7} -2.58774 q^{9} +O(q^{10})$	$q+0.642074 q^{3} +1.00000 q^{5} -3.58774 q^{7} -2.58774 q^{9} +1.35793 q^{13} +0.642074 q^{15} +5.58774 q^{17} -1.00000 q^{19} -2.30359 q^{21} +4.87189 q^{23} +1.00000 q^{25} -3.58774 q^{27} +9.58774 q^{29} +7.17548 q^{31} -3.58774 q^{35} -0.945668 q^{37} +0.871889 q^{39} +10.4596 q^{41} +2.71585 q^{43} -2.58774 q^{45} +5.89134 q^{47} +5.87189 q^{49} +3.58774 q^{51} +9.81756 q^{53} -0.642074 q^{57} -10.1560 q^{59} +3.28415 q^{61} +9.28415 q^{63} +1.35793 q^{65} -10.3859 q^{67} +3.12811 q^{69} -14.3510 q^{71} -4.15604 q^{73} +0.642074 q^{75} -1.28415 q^{79} +5.45963 q^{81} +11.1755 q^{83} +5.58774 q^{85} +6.15604 q^{87} -6.45963 q^{89} -4.87189 q^{91} +4.60719 q^{93} -1.00000 q^{95} -13.4053 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9	$ 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9} + 5 q^{13} + q^{15} + 5 q^{17} - 3 q^{19} + 3 q^{21} + q^{23} + 3 q^{25} + q^{27} + 17 q^{29} - 2 q^{31} + q^{35} + 8 q^{37} - 11 q^{39} + 6 q^{41} + 10 q^{43} + 4 q^{45} - 4 q^{47} + 4 q^{49} - q^{51} + 5 q^{53} - q^{57} - 15 q^{59} + 8 q^{61} + 26 q^{63} + 5 q^{65} - 3 q^{67} + 23 q^{69} + 4 q^{71} + 3 q^{73} + q^{75} - 2 q^{79} - 9 q^{81} + 10 q^{83} + 5 q^{85} + 3 q^{87} + 6 q^{89} - q^{91} - 6 q^{93} - 3 q^{95} - 4 q^{97}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9 + 5 * q^13 + q^15 + 5 * q^17 - 3 * q^19 + 3 * q^21 + q^23 + 3 * q^25 + q^27 + 17 * q^29 - 2 * q^31 + q^35 + 8 * q^37 - 11 * q^39 + 6 * q^41 + 10 * q^43 + 4 * q^45 - 4 * q^47 + 4 * q^49 - q^51 + 5 * q^53 - q^57 - 15 * q^59 + 8 * q^61 + 26 * q^63 + 5 * q^65 - 3 * q^67 + 23 * q^69 + 4 * q^71 + 3 * q^73 + q^75 - 2 * q^79 - 9 * q^81 + 10 * q^83 + 5 * q^85 + 3 * q^87 + 6 * q^89 - q^91 - 6 * q^93 - 3 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.642074	0.370701	0.185351	−	0.982672i	$-0.440658\pi$
$3$	0.642074	0.370701	0.185351	+	0.982672i	$0.440658\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.58774	−1.35604	−0.678019	−	0.735044i	$-0.737161\pi$
$7$	−3.58774	−1.35604	−0.678019	+	0.735044i	$0.737161\pi$
$8$	0	0
$9$	−2.58774	−0.862580
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.35793	0.376621	0.188311	−	0.982110i	$-0.439699\pi$
$13$	1.35793	0.376621	0.188311	+	0.982110i	$0.439699\pi$
$14$	0	0
$15$	0.642074	0.165783
$16$	0	0
$17$	5.58774	1.35523	0.677613	−	0.735419i	$-0.263014\pi$
$17$	5.58774	1.35523	0.677613	+	0.735419i	$0.263014\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.30359	−0.502685
$22$	0	0
$23$	4.87189	1.01586	0.507930	−	0.861399i	$-0.330411\pi$
$23$	4.87189	1.01586	0.507930	+	0.861399i	$0.330411\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.58774	−0.690461
$28$	0	0
$29$	9.58774	1.78040	0.890199	−	0.455571i	$-0.150565\pi$
$29$	9.58774	1.78040	0.890199	+	0.455571i	$0.150565\pi$
$30$	0	0
$31$	7.17548	1.28875	0.644377	−	0.764708i	$-0.277117\pi$
$31$	7.17548	1.28875	0.644377	+	0.764708i	$0.277117\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.58774	−0.606439
$36$	0	0
$37$	−0.945668	−0.155467	−0.0777334	−	0.996974i	$-0.524768\pi$
$37$	−0.945668	−0.155467	−0.0777334	+	0.996974i	$0.524768\pi$
$38$	0	0
$39$	0.871889	0.139614
$40$	0	0
$41$	10.4596	1.63352	0.816760	−	0.576978i	$-0.195768\pi$
$41$	10.4596	1.63352	0.816760	+	0.576978i	$0.195768\pi$
$42$	0	0
$43$	2.71585	0.414164	0.207082	−	0.978324i	$-0.433603\pi$
$43$	2.71585	0.414164	0.207082	+	0.978324i	$0.433603\pi$
$44$	0	0
$45$	−2.58774	−0.385758
$46$	0	0
$47$	5.89134	0.859340	0.429670	−	0.902986i	$-0.358630\pi$
$47$	5.89134	0.859340	0.429670	+	0.902986i	$0.358630\pi$
$48$	0	0
$49$	5.87189	0.838841
$50$	0	0
$51$	3.58774	0.502384
$52$	0	0
$53$	9.81756	1.34855	0.674273	−	0.738483i	$-0.264457\pi$
$53$	9.81756	1.34855	0.674273	+	0.738483i	$0.264457\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.642074	−0.0850447
$58$	0	0
$59$	−10.1560	−1.32220	−0.661102	−	0.750296i	$-0.729911\pi$
$59$	−10.1560	−1.32220	−0.661102	+	0.750296i	$0.729911\pi$
$60$	0	0
$61$	3.28415	0.420492	0.210246	−	0.977649i	$-0.432574\pi$
$61$	3.28415	0.420492	0.210246	+	0.977649i	$0.432574\pi$
$62$	0	0
$63$	9.28415	1.16969
$64$	0	0
$65$	1.35793	0.168430
$66$	0	0
$67$	−10.3859	−1.26883	−0.634417	−	0.772991i	$-0.718760\pi$
$67$	−10.3859	−1.26883	−0.634417	+	0.772991i	$0.718760\pi$
$68$	0	0
$69$	3.12811	0.376580
$70$	0	0
$71$	−14.3510	−1.70315	−0.851573	−	0.524236i	$-0.824351\pi$
$71$	−14.3510	−1.70315	−0.851573	+	0.524236i	$0.824351\pi$
$72$	0	0
$73$	−4.15604	−0.486427	−0.243214	−	0.969973i	$-0.578202\pi$
$73$	−4.15604	−0.486427	−0.243214	+	0.969973i	$0.578202\pi$
$74$	0	0
$75$	0.642074	0.0741403
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.28415	−0.144478	−0.0722389	−	0.997387i	$-0.523014\pi$
$79$	−1.28415	−0.144478	−0.0722389	+	0.997387i	$0.523014\pi$
$80$	0	0
$81$	5.45963	0.606626
$82$	0	0
$83$	11.1755	1.22667	0.613334	−	0.789823i	$-0.289828\pi$
$83$	11.1755	1.22667	0.613334	+	0.789823i	$0.289828\pi$
$84$	0	0
$85$	5.58774	0.606076
$86$	0	0
$87$	6.15604	0.659996
$88$	0	0
$89$	−6.45963	−0.684719	−0.342360	−	0.939569i	$-0.611226\pi$
$89$	−6.45963	−0.684719	−0.342360	+	0.939569i	$0.611226\pi$
$90$	0	0
$91$	−4.87189	−0.510713
$92$	0	0
$93$	4.60719	0.477743
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−13.4053	−1.36110	−0.680551	−	0.732701i	$-0.738259\pi$
$97$	−13.4053	−1.36110	−0.680551	+	0.732701i	$0.738259\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.0668	−1.89722	−0.948610	−	0.316449i	$-0.897510\pi$
$101$	−19.0668	−1.89722	−0.948610	+	0.316449i	$0.897510\pi$
$102$	0	0
$103$	9.05433	0.892150	0.446075	−	0.894996i	$-0.352821\pi$
$103$	9.05433	0.892150	0.446075	+	0.894996i	$0.352821\pi$
$104$	0	0
$105$	−2.30359	−0.224808
$106$	0	0
$107$	5.24926	0.507465	0.253733	−	0.967274i	$-0.418342\pi$
$107$	5.24926	0.507465	0.253733	+	0.967274i	$0.418342\pi$
$108$	0	0
$109$	−6.04737	−0.579233	−0.289617	−	0.957143i	$-0.593528\pi$
$109$	−6.04737	−0.579233	−0.289617	+	0.957143i	$0.593528\pi$
$110$	0	0
$111$	−0.607188	−0.0576318
$112$	0	0
$113$	−6.22982	−0.586052	−0.293026	−	0.956105i	$-0.594662\pi$
$113$	−6.22982	−0.586052	−0.293026	+	0.956105i	$0.594662\pi$
$114$	0	0
$115$	4.87189	0.454306
$116$	0	0
$117$	−3.51396	−0.324866
$118$	0	0
$119$	−20.0474	−1.83774
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.71585	0.605548
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.1212	1.25305	0.626525	−	0.779402i	$-0.284477\pi$
$127$	14.1212	1.25305	0.626525	+	0.779402i	$0.284477\pi$
$128$	0	0
$129$	1.74378	0.153531
$130$	0	0
$131$	18.3510	1.60333	0.801666	−	0.597773i	$-0.203947\pi$
$131$	18.3510	1.60333	0.801666	+	0.597773i	$0.203947\pi$
$132$	0	0
$133$	3.58774	0.311097
$134$	0	0
$135$	−3.58774	−0.308784
$136$	0	0
$137$	3.01945	0.257969	0.128984	−	0.991647i	$-0.458828\pi$
$137$	3.01945	0.257969	0.128984	+	0.991647i	$0.458828\pi$
$138$	0	0
$139$	−11.7827	−0.999393	−0.499697	−	0.866201i	$-0.666555\pi$
$139$	−11.7827	−0.999393	−0.499697	+	0.866201i	$0.666555\pi$
$140$	0	0
$141$	3.78267	0.318558
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.58774	0.796219
$146$	0	0
$147$	3.77018	0.310960
$148$	0	0
$149$	20.4985	1.67930	0.839652	−	0.543124i	$-0.182759\pi$
$149$	20.4985	1.67930	0.839652	+	0.543124i	$0.182759\pi$
$150$	0	0
$151$	3.85244	0.313507	0.156754	−	0.987638i	$-0.449897\pi$
$151$	3.85244	0.313507	0.156754	+	0.987638i	$0.449897\pi$
$152$	0	0
$153$	−14.4596	−1.16899
$154$	0	0
$155$	7.17548	0.576349
$156$	0	0
$157$	3.89134	0.310562	0.155281	−	0.987870i	$-0.450372\pi$
$157$	3.89134	0.310562	0.155281	+	0.987870i	$0.450372\pi$
$158$	0	0
$159$	6.30359	0.499908
$160$	0	0
$161$	−17.4791	−1.37754
$162$	0	0
$163$	18.7159	1.46594	0.732969	−	0.680262i	$-0.238134\pi$
$163$	18.7159	1.46594	0.732969	+	0.680262i	$0.238134\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.09323	0.239361	0.119681	−	0.992812i	$-0.461813\pi$
$167$	3.09323	0.239361	0.119681	+	0.992812i	$0.461813\pi$
$168$	0	0
$169$	−11.1560	−0.858157
$170$	0	0
$171$	2.58774	0.197890
$172$	0	0
$173$	7.51396	0.571276	0.285638	−	0.958338i	$-0.407795\pi$
$173$	7.51396	0.571276	0.285638	+	0.958338i	$0.407795\pi$
$174$	0	0
$175$	−3.58774	−0.271208
$176$	0	0
$177$	−6.52092	−0.490143
$178$	0	0
$179$	4.45963	0.333328	0.166664	−	0.986014i	$-0.446700\pi$
$179$	4.45963	0.333328	0.166664	+	0.986014i	$0.446700\pi$
$180$	0	0
$181$	8.56829	0.636876	0.318438	−	0.947944i	$-0.396842\pi$
$181$	8.56829	0.636876	0.318438	+	0.947944i	$0.396842\pi$
$182$	0	0
$183$	2.10866	0.155877
$184$	0	0
$185$	−0.945668	−0.0695269
$186$	0	0
$187$	0	0
$188$	0	0
$189$	12.8719	0.936292
$190$	0	0
$191$	−7.73530	−0.559707	−0.279853	−	0.960043i	$-0.590286\pi$
$191$	−7.73530	−0.559707	−0.279853	+	0.960043i	$0.590286\pi$
$192$	0	0
$193$	6.08226	0.437810	0.218905	−	0.975746i	$-0.429751\pi$
$193$	6.08226	0.437810	0.218905	+	0.975746i	$0.429751\pi$
$194$	0	0
$195$	0.871889	0.0624372
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	6.91078	0.489892	0.244946	−	0.969537i	$-0.421230\pi$
$199$	6.91078	0.489892	0.244946	+	0.969537i	$0.421230\pi$
$200$	0	0
$201$	−6.66848	−0.470358
$202$	0	0
$203$	−34.3983	−2.41429
$204$	0	0
$205$	10.4596	0.730532
$206$	0	0
$207$	−12.6072	−0.876260
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.1949	0.839534	0.419767	−	0.907632i	$-0.362112\pi$
$211$	12.1949	0.839534	0.419767	+	0.907632i	$0.362112\pi$
$212$	0	0
$213$	−9.21438	−0.631359
$214$	0	0
$215$	2.71585	0.185220
$216$	0	0
$217$	−25.7438	−1.74760
$218$	0	0
$219$	−2.66848	−0.180319
$220$	0	0
$221$	7.58774	0.510407
$222$	0	0
$223$	1.87885	0.125817	0.0629085	−	0.998019i	$-0.479962\pi$
$223$	1.87885	0.125817	0.0629085	+	0.998019i	$0.479962\pi$
$224$	0	0
$225$	−2.58774	−0.172516
$226$	0	0
$227$	−7.35793	−0.488363	−0.244181	−	0.969730i	$-0.578519\pi$
$227$	−7.35793	−0.488363	−0.244181	+	0.969730i	$0.578519\pi$
$228$	0	0
$229$	13.0279	0.860909	0.430455	−	0.902612i	$-0.358353\pi$
$229$	13.0279	0.860909	0.430455	+	0.902612i	$0.358353\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.7019	−1.74930	−0.874651	−	0.484753i	$-0.838909\pi$
$233$	−26.7019	−1.74930	−0.874651	+	0.484753i	$0.838909\pi$
$234$	0	0
$235$	5.89134	0.384308
$236$	0	0
$237$	−0.824517	−0.0535581
$238$	0	0
$239$	9.47908	0.613151	0.306575	−	0.951846i	$-0.400817\pi$
$239$	9.47908	0.613151	0.306575	+	0.951846i	$0.400817\pi$
$240$	0	0
$241$	−24.2034	−1.55908	−0.779539	−	0.626353i	$-0.784547\pi$
$241$	−24.2034	−1.55908	−0.779539	+	0.626353i	$0.784547\pi$
$242$	0	0
$243$	14.2687	0.915338
$244$	0	0
$245$	5.87189	0.375141
$246$	0	0
$247$	−1.35793	−0.0864028
$248$	0	0
$249$	7.17548	0.454728
$250$	0	0
$251$	−9.43171	−0.595324	−0.297662	−	0.954671i	$-0.596207\pi$
$251$	−9.43171	−0.595324	−0.297662	+	0.954671i	$0.596207\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.58774	0.224673
$256$	0	0
$257$	25.1102	1.56633	0.783165	−	0.621814i	$-0.213604\pi$
$257$	25.1102	1.56633	0.783165	+	0.621814i	$0.213604\pi$
$258$	0	0
$259$	3.39281	0.210819
$260$	0	0
$261$	−24.8106	−1.53574
$262$	0	0
$263$	28.7019	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$263$	28.7019	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$264$	0	0
$265$	9.81756	0.603088
$266$	0	0
$267$	−4.14756	−0.253826
$268$	0	0
$269$	−0.350966	−0.0213988	−0.0106994	−	0.999943i	$-0.503406\pi$
$269$	−0.350966	−0.0213988	−0.0106994	+	0.999943i	$0.503406\pi$
$270$	0	0
$271$	−16.8719	−1.02489	−0.512447	−	0.858719i	$-0.671261\pi$
$271$	−16.8719	−1.02489	−0.512447	+	0.858719i	$0.671261\pi$
$272$	0	0
$273$	−3.12811	−0.189322
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.14756	−0.369371	−0.184685	−	0.982798i	$-0.559127\pi$
$277$	−6.14756	−0.369371	−0.184685	+	0.982798i	$0.559127\pi$
$278$	0	0
$279$	−18.5683	−1.11165
$280$	0	0
$281$	−17.0279	−1.01580	−0.507900	−	0.861416i	$-0.669578\pi$
$281$	−17.0279	−1.01580	−0.507900	+	0.861416i	$0.669578\pi$
$282$	0	0
$283$	−5.74378	−0.341432	−0.170716	−	0.985320i	$-0.554608\pi$
$283$	−5.74378	−0.341432	−0.170716	+	0.985320i	$0.554608\pi$
$284$	0	0
$285$	−0.642074	−0.0380332
$286$	0	0
$287$	−37.5264	−2.21512
$288$	0	0
$289$	14.2229	0.836639
$290$	0	0
$291$	−8.60719	−0.504562
$292$	0	0
$293$	−14.2772	−0.834082	−0.417041	−	0.908888i	$-0.636933\pi$
$293$	−14.2772	−0.834082	−0.417041	+	0.908888i	$0.636933\pi$
$294$	0	0
$295$	−10.1560	−0.591307
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.61567	0.382594
$300$	0	0
$301$	−9.74378	−0.561622
$302$	0	0
$303$	−12.2423	−0.703302
$304$	0	0
$305$	3.28415	0.188050
$306$	0	0
$307$	−8.83700	−0.504354	−0.252177	−	0.967681i	$-0.581147\pi$
$307$	−8.83700	−0.504354	−0.252177	+	0.967681i	$0.581147\pi$
$308$	0	0
$309$	5.81355	0.330721
$310$	0	0
$311$	−23.2229	−1.31685	−0.658424	−	0.752648i	$-0.728776\pi$
$311$	−23.2229	−1.31685	−0.658424	+	0.752648i	$0.728776\pi$
$312$	0	0
$313$	30.1421	1.70373	0.851867	−	0.523759i	$-0.175471\pi$
$313$	30.1421	1.70373	0.851867	+	0.523759i	$0.175471\pi$
$314$	0	0
$315$	9.28415	0.523102
$316$	0	0
$317$	5.96511	0.335034	0.167517	−	0.985869i	$-0.446425\pi$
$317$	5.96511	0.335034	0.167517	+	0.985869i	$0.446425\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	3.37041	0.188118
$322$	0	0
$323$	−5.58774	−0.310910
$324$	0	0
$325$	1.35793	0.0753242
$326$	0	0
$327$	−3.88286	−0.214723
$328$	0	0
$329$	−21.1366	−1.16530
$330$	0	0
$331$	2.15604	0.118506	0.0592532	−	0.998243i	$-0.481128\pi$
$331$	2.15604	0.118506	0.0592532	+	0.998243i	$0.481128\pi$
$332$	0	0
$333$	2.44714	0.134103
$334$	0	0
$335$	−10.3859	−0.567440
$336$	0	0
$337$	0.945668	0.0515138	0.0257569	−	0.999668i	$-0.491800\pi$
$337$	0.945668	0.0515138	0.0257569	+	0.999668i	$0.491800\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4.04737	0.218538
$344$	0	0
$345$	3.12811	0.168412
$346$	0	0
$347$	−10.2562	−0.550583	−0.275291	−	0.961361i	$-0.588774\pi$
$347$	−10.2562	−0.550583	−0.275291	+	0.961361i	$0.588774\pi$
$348$	0	0
$349$	−24.0558	−1.28768	−0.643840	−	0.765160i	$-0.722660\pi$
$349$	−24.0558	−1.28768	−0.643840	+	0.765160i	$0.722660\pi$
$350$	0	0
$351$	−4.87189	−0.260042
$352$	0	0
$353$	15.3315	0.816014	0.408007	−	0.912979i	$-0.366224\pi$
$353$	15.3315	0.816014	0.408007	+	0.912979i	$0.366224\pi$
$354$	0	0
$355$	−14.3510	−0.761670
$356$	0	0
$357$	−12.8719	−0.681253
$358$	0	0
$359$	−15.1281	−0.798431	−0.399216	−	0.916857i	$-0.630718\pi$
$359$	−15.1281	−0.798431	−0.399216	+	0.916857i	$0.630718\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.06281	−0.370701
$364$	0	0
$365$	−4.15604	−0.217537
$366$	0	0
$367$	33.6740	1.75777	0.878884	−	0.477035i	$-0.158288\pi$
$367$	33.6740	1.75777	0.878884	+	0.477035i	$0.158288\pi$
$368$	0	0
$369$	−27.0668	−1.40904
$370$	0	0
$371$	−35.2229	−1.82868
$372$	0	0
$373$	16.0738	0.832269	0.416134	−	0.909303i	$-0.363385\pi$
$373$	16.0738	0.832269	0.416134	+	0.909303i	$0.363385\pi$
$374$	0	0
$375$	0.642074	0.0331565
$376$	0	0
$377$	13.0194	0.670536
$378$	0	0
$379$	6.00848	0.308635	0.154317	−	0.988021i	$-0.450682\pi$
$379$	6.00848	0.308635	0.154317	+	0.988021i	$0.450682\pi$
$380$	0	0
$381$	9.06682	0.464507
$382$	0	0
$383$	15.8649	0.810660	0.405330	−	0.914170i	$-0.367157\pi$
$383$	15.8649	0.810660	0.405330	+	0.914170i	$0.367157\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.02792	−0.357249
$388$	0	0
$389$	15.7438	0.798241	0.399121	−	0.916898i	$-0.369315\pi$
$389$	15.7438	0.798241	0.399121	+	0.916898i	$0.369315\pi$
$390$	0	0
$391$	27.2229	1.37672
$392$	0	0
$393$	11.7827	0.594357
$394$	0	0
$395$	−1.28415	−0.0646125
$396$	0	0
$397$	−23.9861	−1.20383	−0.601913	−	0.798561i	$-0.705595\pi$
$397$	−23.9861	−1.20383	−0.601913	+	0.798561i	$0.705595\pi$
$398$	0	0
$399$	2.30359	0.115324
$400$	0	0
$401$	−24.5683	−1.22688	−0.613441	−	0.789741i	$-0.710215\pi$
$401$	−24.5683	−1.22688	−0.613441	+	0.789741i	$0.710215\pi$
$402$	0	0
$403$	9.74378	0.485372
$404$	0	0
$405$	5.45963	0.271291
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.9193	−0.737710	−0.368855	−	0.929487i	$-0.620250\pi$
$409$	−14.9193	−0.737710	−0.368855	+	0.929487i	$0.620250\pi$
$410$	0	0
$411$	1.93871	0.0956294
$412$	0	0
$413$	36.4372	1.79296
$414$	0	0
$415$	11.1755	0.548583
$416$	0	0
$417$	−7.56534	−0.370476
$418$	0	0
$419$	−14.0389	−0.685845	−0.342922	−	0.939364i	$-0.611417\pi$
$419$	−14.0389	−0.685845	−0.342922	+	0.939364i	$0.611417\pi$
$420$	0	0
$421$	39.9387	1.94649	0.973247	−	0.229762i	$-0.0737949\pi$
$421$	39.9387	1.94649	0.973247	+	0.229762i	$0.0737949\pi$
$422$	0	0
$423$	−15.2453	−0.741250
$424$	0	0
$425$	5.58774	0.271045
$426$	0	0
$427$	−11.7827	−0.570203
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.0279	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$431$	−27.0279	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$432$	0	0
$433$	−40.0683	−1.92556	−0.962781	−	0.270284i	$-0.912882\pi$
$433$	−40.0683	−1.92556	−0.962781	+	0.270284i	$0.912882\pi$
$434$	0	0
$435$	6.15604	0.295159
$436$	0	0
$437$	−4.87189	−0.233054
$438$	0	0
$439$	−6.42074	−0.306445	−0.153223	−	0.988192i	$-0.548965\pi$
$439$	−6.42074	−0.306445	−0.153223	+	0.988192i	$0.548965\pi$
$440$	0	0
$441$	−15.1949	−0.723568
$442$	0	0
$443$	35.2702	1.67574	0.837870	−	0.545871i	$-0.183801\pi$
$443$	35.2702	1.67574	0.837870	+	0.545871i	$0.183801\pi$
$444$	0	0
$445$	−6.45963	−0.306216
$446$	0	0
$447$	13.1616	0.622521
$448$	0	0
$449$	5.78267	0.272901	0.136451	−	0.990647i	$-0.456431\pi$
$449$	5.78267	0.272901	0.136451	+	0.990647i	$0.456431\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.47355	0.116218
$454$	0	0
$455$	−4.87189	−0.228398
$456$	0	0
$457$	−9.12811	−0.426995	−0.213498	−	0.976944i	$-0.568486\pi$
$457$	−9.12811	−0.426995	−0.213498	+	0.976944i	$0.568486\pi$
$458$	0	0
$459$	−20.0474	−0.935731
$460$	0	0
$461$	8.86341	0.412810	0.206405	−	0.978467i	$-0.433824\pi$
$461$	8.86341	0.412810	0.206405	+	0.978467i	$0.433824\pi$
$462$	0	0
$463$	21.1366	0.982301	0.491150	−	0.871075i	$-0.336577\pi$
$463$	21.1366	0.982301	0.491150	+	0.871075i	$0.336577\pi$
$464$	0	0
$465$	4.60719	0.213653
$466$	0	0
$467$	−7.78267	−0.360139	−0.180070	−	0.983654i	$-0.557632\pi$
$467$	−7.78267	−0.360139	−0.180070	+	0.983654i	$0.557632\pi$
$468$	0	0
$469$	37.2617	1.72059
$470$	0	0
$471$	2.49852	0.115126
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−25.4053	−1.16323
$478$	0	0
$479$	−40.4068	−1.84623	−0.923117	−	0.384519i	$-0.874367\pi$
$479$	−40.4068	−1.84623	−0.923117	+	0.384519i	$0.874367\pi$
$480$	0	0
$481$	−1.28415	−0.0585521
$482$	0	0
$483$	−11.2229	−0.510658
$484$	0	0
$485$	−13.4053	−0.608703
$486$	0	0
$487$	−5.82603	−0.264003	−0.132001	−	0.991250i	$-0.542140\pi$
$487$	−5.82603	−0.264003	−0.132001	+	0.991250i	$0.542140\pi$
$488$	0	0
$489$	12.0170	0.543426
$490$	0	0
$491$	−22.6630	−1.02277	−0.511384	−	0.859352i	$-0.670867\pi$
$491$	−22.6630	−1.02277	−0.511384	+	0.859352i	$0.670867\pi$
$492$	0	0
$493$	53.5738	2.41284
$494$	0	0
$495$	0	0
$496$	0	0
$497$	51.4876	2.30953
$498$	0	0
$499$	25.7438	1.15245	0.576225	−	0.817291i	$-0.304525\pi$
$499$	25.7438	1.15245	0.576225	+	0.817291i	$0.304525\pi$
$500$	0	0
$501$	1.98608	0.0887315
$502$	0	0
$503$	−25.4791	−1.13606	−0.568028	−	0.823009i	$-0.692293\pi$
$503$	−25.4791	−1.13606	−0.568028	+	0.823009i	$0.692293\pi$
$504$	0	0
$505$	−19.0668	−0.848462
$506$	0	0
$507$	−7.16300	−0.318120
$508$	0	0
$509$	−9.54037	−0.422869	−0.211435	−	0.977392i	$-0.567814\pi$
$509$	−9.54037	−0.422869	−0.211435	+	0.977392i	$0.567814\pi$
$510$	0	0
$511$	14.9108	0.659614
$512$	0	0
$513$	3.58774	0.158403
$514$	0	0
$515$	9.05433	0.398982
$516$	0	0
$517$	0	0
$518$	0	0
$519$	4.82452	0.211773
$520$	0	0
$521$	6.31207	0.276537	0.138268	−	0.990395i	$-0.455846\pi$
$521$	6.31207	0.276537	0.138268	+	0.990395i	$0.455846\pi$
$522$	0	0
$523$	−8.93719	−0.390796	−0.195398	−	0.980724i	$-0.562600\pi$
$523$	−8.93719	−0.390796	−0.195398	+	0.980724i	$0.562600\pi$
$524$	0	0
$525$	−2.30359	−0.100537
$526$	0	0
$527$	40.0947	1.74655
$528$	0	0
$529$	0.735300	0.0319696
$530$	0	0
$531$	26.2812	1.14051
$532$	0	0
$533$	14.2034	0.615218
$534$	0	0
$535$	5.24926	0.226945
$536$	0	0
$537$	2.86341	0.123565
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.71585	0.374724	0.187362	−	0.982291i	$-0.440006\pi$
$541$	8.71585	0.374724	0.187362	+	0.982291i	$0.440006\pi$
$542$	0	0
$543$	5.50148	0.236091
$544$	0	0
$545$	−6.04737	−0.259041
$546$	0	0
$547$	26.9457	1.15211	0.576057	−	0.817410i	$-0.304591\pi$
$547$	26.9457	1.15211	0.576057	+	0.817410i	$0.304591\pi$
$548$	0	0
$549$	−8.49852	−0.362708
$550$	0	0
$551$	−9.58774	−0.408452
$552$	0	0
$553$	4.60719	0.195918
$554$	0	0
$555$	−0.607188	−0.0257737
$556$	0	0
$557$	−23.3619	−0.989877	−0.494938	−	0.868928i	$-0.664809\pi$
$557$	−23.3619	−0.989877	−0.494938	+	0.868928i	$0.664809\pi$
$558$	0	0
$559$	3.68793	0.155983
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.6476	−1.33379	−0.666894	−	0.745153i	$-0.732376\pi$
$563$	−31.6476	−1.33379	−0.666894	+	0.745153i	$0.732376\pi$
$564$	0	0
$565$	−6.22982	−0.262090
$566$	0	0
$567$	−19.5877	−0.822608
$568$	0	0
$569$	−20.8804	−0.875350	−0.437675	−	0.899133i	$-0.644198\pi$
$569$	−20.8804	−0.875350	−0.437675	+	0.899133i	$0.644198\pi$
$570$	0	0
$571$	−42.6630	−1.78539	−0.892696	−	0.450659i	$-0.851189\pi$
$571$	−42.6630	−1.78539	−0.892696	+	0.450659i	$0.851189\pi$
$572$	0	0
$573$	−4.96663	−0.207484
$574$	0	0
$575$	4.87189	0.203172
$576$	0	0
$577$	−8.59871	−0.357969	−0.178985	−	0.983852i	$-0.557281\pi$
$577$	−8.59871	−0.357969	−0.178985	+	0.983852i	$0.557281\pi$
$578$	0	0
$579$	3.90526	0.162297
$580$	0	0
$581$	−40.0947	−1.66341
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−3.51396	−0.145284
$586$	0	0
$587$	−29.7438	−1.22766	−0.613829	−	0.789439i	$-0.710371\pi$
$587$	−29.7438	−1.22766	−0.613829	+	0.789439i	$0.710371\pi$
$588$	0	0
$589$	−7.17548	−0.295661
$590$	0	0
$591$	−1.28415	−0.0528228
$592$	0	0
$593$	4.56829	0.187597	0.0937987	−	0.995591i	$-0.470099\pi$
$593$	4.56829	0.187597	0.0937987	+	0.995591i	$0.470099\pi$
$594$	0	0
$595$	−20.0474	−0.821862
$596$	0	0
$597$	4.43723	0.181604
$598$	0	0
$599$	−14.7159	−0.601273	−0.300637	−	0.953739i	$-0.597199\pi$
$599$	−14.7159	−0.601273	−0.300637	+	0.953739i	$0.597199\pi$
$600$	0	0
$601$	37.3230	1.52244	0.761219	−	0.648495i	$-0.224601\pi$
$601$	37.3230	1.52244	0.761219	+	0.648495i	$0.224601\pi$
$602$	0	0
$603$	26.8759	1.09447
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	−23.0404	−0.935181	−0.467591	−	0.883945i	$-0.654878\pi$
$607$	−23.0404	−0.935181	−0.467591	+	0.883945i	$0.654878\pi$
$608$	0	0
$609$	−22.0863	−0.894981
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	32.4985	1.31260	0.656302	−	0.754499i	$-0.272120\pi$
$613$	32.4985	1.31260	0.656302	+	0.754499i	$0.272120\pi$
$614$	0	0
$615$	6.71585	0.270809
$616$	0	0
$617$	−32.2982	−1.30027	−0.650137	−	0.759817i	$-0.725289\pi$
$617$	−32.2982	−1.30027	−0.650137	+	0.759817i	$0.725289\pi$
$618$	0	0
$619$	34.5683	1.38942	0.694709	−	0.719291i	$-0.255533\pi$
$619$	34.5683	1.38942	0.694709	+	0.719291i	$0.255533\pi$
$620$	0	0
$621$	−17.4791	−0.701411
$622$	0	0
$623$	23.1755	0.928506
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.28415	−0.210693
$630$	0	0
$631$	19.2702	0.767136	0.383568	−	0.923513i	$-0.374695\pi$
$631$	19.2702	0.767136	0.383568	+	0.923513i	$0.374695\pi$
$632$	0	0
$633$	7.83004	0.311216
$634$	0	0
$635$	14.1212	0.560381
$636$	0	0
$637$	7.97359	0.315925
$638$	0	0
$639$	37.1366	1.46910
$640$	0	0
$641$	−3.06682	−0.121132	−0.0605660	−	0.998164i	$-0.519291\pi$
$641$	−3.06682	−0.121132	−0.0605660	+	0.998164i	$0.519291\pi$
$642$	0	0
$643$	25.5264	1.00666	0.503332	−	0.864093i	$-0.332107\pi$
$643$	25.5264	1.00666	0.503332	+	0.864093i	$0.332107\pi$
$644$	0	0
$645$	1.74378	0.0686612
$646$	0	0
$647$	−25.4791	−1.00169	−0.500843	−	0.865538i	$-0.666977\pi$
$647$	−25.4791	−1.00169	−0.500843	+	0.865538i	$0.666977\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−16.5294	−0.647838
$652$	0	0
$653$	−36.2034	−1.41675	−0.708374	−	0.705837i	$-0.750571\pi$
$653$	−36.2034	−1.41675	−0.708374	+	0.705837i	$0.750571\pi$
$654$	0	0
$655$	18.3510	0.717032
$656$	0	0
$657$	10.7547	0.419583
$658$	0	0
$659$	42.6406	1.66104	0.830522	−	0.556986i	$-0.188042\pi$
$659$	42.6406	1.66104	0.830522	+	0.556986i	$0.188042\pi$
$660$	0	0
$661$	24.4681	0.951699	0.475850	−	0.879527i	$-0.342141\pi$
$661$	24.4681	0.951699	0.475850	+	0.879527i	$0.342141\pi$
$662$	0	0
$663$	4.87189	0.189208
$664$	0	0
$665$	3.58774	0.139127
$666$	0	0
$667$	46.7104	1.80863
$668$	0	0
$669$	1.20636	0.0466406
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−26.7672	−1.03180	−0.515901	−	0.856649i	$-0.672543\pi$
$673$	−26.7672	−1.03180	−0.515901	+	0.856649i	$0.672543\pi$
$674$	0	0
$675$	−3.58774	−0.138092
$676$	0	0
$677$	−5.65304	−0.217264	−0.108632	−	0.994082i	$-0.534647\pi$
$677$	−5.65304	−0.217264	−0.108632	+	0.994082i	$0.534647\pi$
$678$	0	0
$679$	48.0947	1.84571
$680$	0	0
$681$	−4.72433	−0.181037
$682$	0	0
$683$	11.1102	0.425119	0.212560	−	0.977148i	$-0.431820\pi$
$683$	11.1102	0.425119	0.212560	+	0.977148i	$0.431820\pi$
$684$	0	0
$685$	3.01945	0.115367
$686$	0	0
$687$	8.36489	0.319140
$688$	0	0
$689$	13.3315	0.507890
$690$	0	0
$691$	5.96111	0.226771	0.113386	−	0.993551i	$-0.463830\pi$
$691$	5.96111	0.226771	0.113386	+	0.993551i	$0.463830\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.7827	−0.446942
$696$	0	0
$697$	58.4457	2.21379
$698$	0	0
$699$	−17.1446	−0.648469
$700$	0	0
$701$	−25.8524	−0.976433	−0.488217	−	0.872722i	$-0.662352\pi$
$701$	−25.8524	−0.976433	−0.488217	+	0.872722i	$0.662352\pi$
$702$	0	0
$703$	0.945668	0.0356665
$704$	0	0
$705$	3.78267	0.142464
$706$	0	0
$707$	68.4068	2.57270
$708$	0	0
$709$	−23.4317	−0.879996	−0.439998	−	0.897999i	$-0.645021\pi$
$709$	−23.4317	−0.879996	−0.439998	+	0.897999i	$0.645021\pi$
$710$	0	0
$711$	3.32304	0.124624
$712$	0	0
$713$	34.9582	1.30919
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.08627	0.227296
$718$	0	0
$719$	6.20885	0.231551	0.115776	−	0.993275i	$-0.463065\pi$
$719$	6.20885	0.231551	0.115776	+	0.993275i	$0.463065\pi$
$720$	0	0
$721$	−32.4846	−1.20979
$722$	0	0
$723$	−15.5404	−0.577953
$724$	0	0
$725$	9.58774	0.356080
$726$	0	0
$727$	−29.7214	−1.10230	−0.551152	−	0.834405i	$-0.685812\pi$
$727$	−29.7214	−1.10230	−0.551152	+	0.834405i	$0.685812\pi$
$728$	0	0
$729$	−7.21733	−0.267308
$730$	0	0
$731$	15.1755	0.561286
$732$	0	0
$733$	−7.52645	−0.277996	−0.138998	−	0.990293i	$-0.544388\pi$
$733$	−7.52645	−0.277996	−0.138998	+	0.990293i	$0.544388\pi$
$734$	0	0
$735$	3.77018	0.139065
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−22.9582	−0.844529	−0.422265	−	0.906473i	$-0.638765\pi$
$739$	−22.9582	−0.844529	−0.422265	+	0.906473i	$0.638765\pi$
$740$	0	0
$741$	−0.871889	−0.0320296
$742$	0	0
$743$	−0.524931	−0.0192579	−0.00962893	−	0.999954i	$-0.503065\pi$
$743$	−0.524931	−0.0192579	−0.00962893	+	0.999954i	$0.503065\pi$
$744$	0	0
$745$	20.4985	0.751008
$746$	0	0
$747$	−28.9193	−1.05810
$748$	0	0
$749$	−18.8330	−0.688143
$750$	0	0
$751$	−21.3619	−0.779508	−0.389754	−	0.920919i	$-0.627440\pi$
$751$	−21.3619	−0.779508	−0.389754	+	0.920919i	$0.627440\pi$
$752$	0	0
$753$	−6.05585	−0.220687
$754$	0	0
$755$	3.85244	0.140205
$756$	0	0
$757$	50.4068	1.83207	0.916033	−	0.401102i	$-0.131373\pi$
$757$	50.4068	1.83207	0.916033	+	0.401102i	$0.131373\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.2338	−0.733476	−0.366738	−	0.930324i	$-0.619525\pi$
$761$	−20.2338	−0.733476	−0.366738	+	0.930324i	$0.619525\pi$
$762$	0	0
$763$	21.6964	0.785463
$764$	0	0
$765$	−14.4596	−0.522789
$766$	0	0
$767$	−13.7911	−0.497970
$768$	0	0
$769$	−27.1142	−0.977763	−0.488881	−	0.872350i	$-0.662595\pi$
$769$	−27.1142	−0.977763	−0.488881	+	0.872350i	$0.662595\pi$
$770$	0	0
$771$	16.1226	0.580641
$772$	0	0
$773$	−12.1685	−0.437671	−0.218836	−	0.975762i	$-0.570226\pi$
$773$	−12.1685	−0.437671	−0.218836	+	0.975762i	$0.570226\pi$
$774$	0	0
$775$	7.17548	0.257751
$776$	0	0
$777$	2.17843	0.0781509
$778$	0	0
$779$	−10.4596	−0.374755
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−34.3983	−1.22930
$784$	0	0
$785$	3.89134	0.138888
$786$	0	0
$787$	36.1296	1.28788	0.643941	−	0.765075i	$-0.277298\pi$
$787$	36.1296	1.28788	0.643941	+	0.765075i	$0.277298\pi$
$788$	0	0
$789$	18.4288	0.656081
$790$	0	0
$791$	22.3510	0.794709
$792$	0	0
$793$	4.45963	0.158366
$794$	0	0
$795$	6.30359	0.223565
$796$	0	0
$797$	42.4417	1.50336	0.751681	−	0.659527i	$-0.229243\pi$
$797$	42.4417	1.50336	0.751681	+	0.659527i	$0.229243\pi$
$798$	0	0
$799$	32.9193	1.16460
$800$	0	0
$801$	16.7159	0.590626
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−17.4791	−0.616057
$806$	0	0
$807$	−0.225346	−0.00793255
$808$	0	0
$809$	−37.3006	−1.31142	−0.655710	−	0.755013i	$-0.727631\pi$
$809$	−37.3006	−1.31142	−0.655710	+	0.755013i	$0.727631\pi$
$810$	0	0
$811$	−27.4402	−0.963555	−0.481778	−	0.876294i	$-0.660009\pi$
$811$	−27.4402	−0.963555	−0.481778	+	0.876294i	$0.660009\pi$
$812$	0	0
$813$	−10.8330	−0.379930
$814$	0	0
$815$	18.7159	0.655588
$816$	0	0
$817$	−2.71585	−0.0950157
$818$	0	0
$819$	12.6072	0.440531
$820$	0	0
$821$	47.0140	1.64080	0.820400	−	0.571790i	$-0.193751\pi$
$821$	47.0140	1.64080	0.820400	+	0.571790i	$0.193751\pi$
$822$	0	0
$823$	−35.9776	−1.25410	−0.627050	−	0.778979i	$-0.715738\pi$
$823$	−35.9776	−1.25410	−0.627050	+	0.778979i	$0.715738\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.6032	−0.507802	−0.253901	−	0.967230i	$-0.581714\pi$
$827$	−14.6032	−0.507802	−0.253901	+	0.967230i	$0.581714\pi$
$828$	0	0
$829$	−6.96663	−0.241961	−0.120981	−	0.992655i	$-0.538604\pi$
$829$	−6.96663	−0.241961	−0.120981	+	0.992655i	$0.538604\pi$
$830$	0	0
$831$	−3.94719	−0.136926
$832$	0	0
$833$	32.8106	1.13682
$834$	0	0
$835$	3.09323	0.107046
$836$	0	0
$837$	−25.7438	−0.889835
$838$	0	0
$839$	0.459630	0.0158682	0.00793410	−	0.999969i	$-0.497474\pi$
$839$	0.459630	0.0158682	0.00793410	+	0.999969i	$0.497474\pi$
$840$	0	0
$841$	62.9248	2.16982
$842$	0	0
$843$	−10.9332	−0.376559
$844$	0	0
$845$	−11.1560	−0.383779
$846$	0	0
$847$	39.4652	1.35604
$848$	0	0
$849$	−3.68793	−0.126569
$850$	0	0
$851$	−4.60719	−0.157932
$852$	0	0
$853$	−5.48755	−0.187890	−0.0939451	−	0.995577i	$-0.529948\pi$
$853$	−5.48755	−0.187890	−0.0939451	+	0.995577i	$0.529948\pi$
$854$	0	0
$855$	2.58774	0.0884989
$856$	0	0
$857$	52.6586	1.79878	0.899391	−	0.437145i	$-0.144010\pi$
$857$	52.6586	1.79878	0.899391	+	0.437145i	$0.144010\pi$
$858$	0	0
$859$	4.29512	0.146547	0.0732737	−	0.997312i	$-0.476655\pi$
$859$	4.29512	0.146547	0.0732737	+	0.997312i	$0.476655\pi$
$860$	0	0
$861$	−24.0947	−0.821147
$862$	0	0
$863$	14.6506	0.498711	0.249355	−	0.968412i	$-0.419781\pi$
$863$	14.6506	0.498711	0.249355	+	0.968412i	$0.419781\pi$
$864$	0	0
$865$	7.51396	0.255482
$866$	0	0
$867$	9.13212	0.310143
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−14.1032	−0.477869
$872$	0	0
$873$	34.6894	1.17406
$874$	0	0
$875$	−3.58774	−0.121288
$876$	0	0
$877$	−19.7089	−0.665522	−0.332761	−	0.943011i	$-0.607980\pi$
$877$	−19.7089	−0.665522	−0.332761	+	0.943011i	$0.607980\pi$
$878$	0	0
$879$	−9.16701	−0.309195
$880$	0	0
$881$	1.32304	0.0445744	0.0222872	−	0.999752i	$-0.492905\pi$
$881$	1.32304	0.0445744	0.0222872	+	0.999752i	$0.492905\pi$
$882$	0	0
$883$	25.0668	0.843566	0.421783	−	0.906697i	$-0.361404\pi$
$883$	25.0668	0.843566	0.421783	+	0.906697i	$0.361404\pi$
$884$	0	0
$885$	−6.52092	−0.219198
$886$	0	0
$887$	36.5669	1.22779	0.613897	−	0.789386i	$-0.289601\pi$
$887$	36.5669	1.22779	0.613897	+	0.789386i	$0.289601\pi$
$888$	0	0
$889$	−50.6630	−1.69918
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.89134	−0.197146
$894$	0	0
$895$	4.45963	0.149069
$896$	0	0
$897$	4.24774	0.141828
$898$	0	0
$899$	68.7967	2.29450
$900$	0	0
$901$	54.8580	1.82758
$902$	0	0
$903$	−6.25622	−0.208194
$904$	0	0
$905$	8.56829	0.284820
$906$	0	0
$907$	40.7368	1.35264	0.676322	−	0.736606i	$-0.263573\pi$
$907$	40.7368	1.35264	0.676322	+	0.736606i	$0.263573\pi$
$908$	0	0
$909$	49.3400	1.63650
$910$	0	0
$911$	58.2982	1.93150	0.965752	−	0.259467i	$-0.0835469\pi$
$911$	58.2982	1.93150	0.965752	+	0.259467i	$0.0835469\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.10866	0.0697103
$916$	0	0
$917$	−65.8385	−2.17418
$918$	0	0
$919$	14.5209	0.479001	0.239501	−	0.970896i	$-0.423016\pi$
$919$	14.5209	0.479001	0.239501	+	0.970896i	$0.423016\pi$
$920$	0	0
$921$	−5.67401	−0.186965
$922$	0	0
$923$	−19.4876	−0.641441
$924$	0	0
$925$	−0.945668	−0.0310934
$926$	0	0
$927$	−23.4303	−0.769551
$928$	0	0
$929$	13.7523	0.451197	0.225598	−	0.974220i	$-0.427566\pi$
$929$	13.7523	0.451197	0.225598	+	0.974220i	$0.427566\pi$
$930$	0	0
$931$	−5.87189	−0.192443
$932$	0	0
$933$	−14.9108	−0.488157
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.4512	−0.406761	−0.203381	−	0.979100i	$-0.565193\pi$
$937$	−12.4512	−0.406761	−0.203381	+	0.979100i	$0.565193\pi$
$938$	0	0
$939$	19.3535	0.631576
$940$	0	0
$941$	−34.3594	−1.12009	−0.560043	−	0.828464i	$-0.689215\pi$
$941$	−34.3594	−1.12009	−0.560043	+	0.828464i	$0.689215\pi$
$942$	0	0
$943$	50.9582	1.65943
$944$	0	0
$945$	12.8719	0.418723
$946$	0	0
$947$	−23.4876	−0.763243	−0.381621	−	0.924319i	$-0.624634\pi$
$947$	−23.4876	−0.763243	−0.381621	+	0.924319i	$0.624634\pi$
$948$	0	0
$949$	−5.64359	−0.183199
$950$	0	0
$951$	3.83004	0.124198
$952$	0	0
$953$	48.5808	1.57369	0.786843	−	0.617153i	$-0.211714\pi$
$953$	48.5808	1.57369	0.786843	+	0.617153i	$0.211714\pi$
$954$	0	0
$955$	−7.73530	−0.250308
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.8330	−0.349816
$960$	0	0
$961$	20.4876	0.660889
$962$	0	0
$963$	−13.5837	−0.437730
$964$	0	0
$965$	6.08226	0.195795
$966$	0	0
$967$	−53.3261	−1.71485	−0.857425	−	0.514608i	$-0.827937\pi$
$967$	−53.3261	−1.71485	−0.857425	+	0.514608i	$0.827937\pi$
$968$	0	0
$969$	−3.58774	−0.115255
$970$	0	0
$971$	33.1366	1.06340	0.531702	−	0.846932i	$-0.321553\pi$
$971$	33.1366	1.06340	0.531702	+	0.846932i	$0.321553\pi$
$972$	0	0
$973$	42.2732	1.35522
$974$	0	0
$975$	0.871889	0.0279228
$976$	0	0
$977$	5.62263	0.179884	0.0899419	−	0.995947i	$-0.471332\pi$
$977$	5.62263	0.179884	0.0899419	+	0.995947i	$0.471332\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	15.6490	0.499635
$982$	0	0
$983$	−21.3664	−0.681482	−0.340741	−	0.940157i	$-0.610678\pi$
$983$	−21.3664	−0.681482	−0.340741	+	0.940157i	$0.610678\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	−13.5712	−0.431978
$988$	0	0
$989$	13.2313	0.420732
$990$	0	0
$991$	−41.1446	−1.30700	−0.653501	−	0.756926i	$-0.726700\pi$
$991$	−41.1446	−1.30700	−0.653501	+	0.756926i	$0.726700\pi$
$992$	0	0
$993$	1.38433	0.0439305
$994$	0	0
$995$	6.91078	0.219087
$996$	0	0
$997$	−48.5155	−1.53650	−0.768250	−	0.640150i	$-0.778872\pi$
$997$	−48.5155	−1.53650	−0.768250	+	0.640150i	$0.778872\pi$
$998$	0	0
$999$	3.39281	0.107344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.s.1.2		3
4.3	odd	2		760.2.a.j.1.2	✓	3
5.4	even	2		7600.2.a.bq.1.2		3
8.3	odd	2		6080.2.a.bv.1.2		3
8.5	even	2		6080.2.a.bq.1.2		3
12.11	even	2		6840.2.a.bg.1.3		3
20.3	even	4		3800.2.d.l.3649.3		6
20.7	even	4		3800.2.d.l.3649.4		6
20.19	odd	2		3800.2.a.x.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.j.1.2	✓	3	4.3	odd	2
1520.2.a.s.1.2		3	1.1	even	1	trivial
3800.2.a.x.1.2		3	20.19	odd	2
3800.2.d.l.3649.3		6	20.3	even	4
3800.2.d.l.3649.4		6	20.7	even	4
6080.2.a.bq.1.2		3	8.5	even	2
6080.2.a.bv.1.2		3	8.3	odd	2
6840.2.a.bg.1.3		3	12.11	even	2
7600.2.a.bq.1.2		3	5.4	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q+0.642074 q^{3} +1.00000 q^{5} -3.58774 q^{7} -2.58774 q^{9} +O(q^{10})\)	\(q+0.642074 q^{3} +1.00000 q^{5} -3.58774 q^{7} -2.58774 q^{9} +1.35793 q^{13} +0.642074 q^{15} +5.58774 q^{17} -1.00000 q^{19} -2.30359 q^{21} +4.87189 q^{23} +1.00000 q^{25} -3.58774 q^{27} +9.58774 q^{29} +7.17548 q^{31} -3.58774 q^{35} -0.945668 q^{37} +0.871889 q^{39} +10.4596 q^{41} +2.71585 q^{43} -2.58774 q^{45} +5.89134 q^{47} +5.87189 q^{49} +3.58774 q^{51} +9.81756 q^{53} -0.642074 q^{57} -10.1560 q^{59} +3.28415 q^{61} +9.28415 q^{63} +1.35793 q^{65} -10.3859 q^{67} +3.12811 q^{69} -14.3510 q^{71} -4.15604 q^{73} +0.642074 q^{75} -1.28415 q^{79} +5.45963 q^{81} +11.1755 q^{83} +5.58774 q^{85} +6.15604 q^{87} -6.45963 q^{89} -4.87189 q^{91} +4.60719 q^{93} -1.00000 q^{95} -13.4053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9	\( 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9} + 5 q^{13} + q^{15} + 5 q^{17} - 3 q^{19} + 3 q^{21} + q^{23} + 3 q^{25} + q^{27} + 17 q^{29} - 2 q^{31} + q^{35} + 8 q^{37} - 11 q^{39} + 6 q^{41} + 10 q^{43} + 4 q^{45} - 4 q^{47} + 4 q^{49} - q^{51} + 5 q^{53} - q^{57} - 15 q^{59} + 8 q^{61} + 26 q^{63} + 5 q^{65} - 3 q^{67} + 23 q^{69} + 4 q^{71} + 3 q^{73} + q^{75} - 2 q^{79} - 9 q^{81} + 10 q^{83} + 5 q^{85} + 3 q^{87} + 6 q^{89} - q^{91} - 6 q^{93} - 3 q^{95} - 4 q^{97}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9 + 5 * q^13 + q^15 + 5 * q^17 - 3 * q^19 + 3 * q^21 + q^23 + 3 * q^25 + q^27 + 17 * q^29 - 2 * q^31 + q^35 + 8 * q^37 - 11 * q^39 + 6 * q^41 + 10 * q^43 + 4 * q^45 - 4 * q^47 + 4 * q^49 - q^51 + 5 * q^53 - q^57 - 15 * q^59 + 8 * q^61 + 26 * q^63 + 5 * q^65 - 3 * q^67 + 23 * q^69 + 4 * q^71 + 3 * q^73 + q^75 - 2 * q^79 - 9 * q^81 + 10 * q^83 + 5 * q^85 + 3 * q^87 + 6 * q^89 - q^91 - 6 * q^93 - 3 * q^95 - 4 * q^97

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