LMFDB - Embedded newform 2240.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.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:	$17.8864900528$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})$	$q+2.56155 q^{3} +1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +1.43845 q^{11} -0.561553 q^{13} +2.56155 q^{15} +5.68466 q^{17} +4.00000 q^{19} +2.56155 q^{21} +1.00000 q^{25} +1.43845 q^{27} -4.56155 q^{29} -5.12311 q^{31} +3.68466 q^{33} +1.00000 q^{35} -0.876894 q^{37} -1.43845 q^{39} -8.24621 q^{41} +3.56155 q^{45} -6.56155 q^{47} +1.00000 q^{49} +14.5616 q^{51} +7.12311 q^{53} +1.43845 q^{55} +10.2462 q^{57} +14.2462 q^{59} +8.24621 q^{61} +3.56155 q^{63} -0.561553 q^{65} -2.24621 q^{67} +2.24621 q^{71} -8.24621 q^{73} +2.56155 q^{75} +1.43845 q^{77} +1.43845 q^{79} -7.00000 q^{81} -6.24621 q^{83} +5.68466 q^{85} -11.6847 q^{87} -16.2462 q^{89} -0.561553 q^{91} -13.1231 q^{93} +4.00000 q^{95} +5.68466 q^{97} +5.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9	$ 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} + 7 q^{11} + 3 q^{13} + q^{15} - q^{17} + 8 q^{19} + q^{21} + 2 q^{25} + 7 q^{27} - 5 q^{29} - 2 q^{31} - 5 q^{33} + 2 q^{35} - 10 q^{37} - 7 q^{39} + 3 q^{45} - 9 q^{47} + 2 q^{49} + 25 q^{51} + 6 q^{53} + 7 q^{55} + 4 q^{57} + 12 q^{59} + 3 q^{63} + 3 q^{65} + 12 q^{67} - 12 q^{71} + q^{75} + 7 q^{77} + 7 q^{79} - 14 q^{81} + 4 q^{83} - q^{85} - 11 q^{87} - 16 q^{89} + 3 q^{91} - 18 q^{93} + 8 q^{95} - q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 + 7 * q^11 + 3 * q^13 + q^15 - q^17 + 8 * q^19 + q^21 + 2 * q^25 + 7 * q^27 - 5 * q^29 - 2 * q^31 - 5 * q^33 + 2 * q^35 - 10 * q^37 - 7 * q^39 + 3 * q^45 - 9 * q^47 + 2 * q^49 + 25 * q^51 + 6 * q^53 + 7 * q^55 + 4 * q^57 + 12 * q^59 + 3 * q^63 + 3 * q^65 + 12 * q^67 - 12 * q^71 + q^75 + 7 * q^77 + 7 * q^79 - 14 * q^81 + 4 * q^83 - q^85 - 11 * q^87 - 16 * q^89 + 3 * q^91 - 18 * q^93 + 8 * q^95 - q^97 + 2 * 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
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	1.43845	0.433708	0.216854	−	0.976204i	$-0.430420\pi$
$11$	1.43845	0.433708	0.216854	+	0.976204i	$0.430420\pi$
$12$	0	0
$13$	−0.561553	−0.155747	−0.0778734	−	0.996963i	$-0.524813\pi$
$13$	−0.561553	−0.155747	−0.0778734	+	0.996963i	$0.524813\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	5.68466	1.37873	0.689366	−	0.724413i	$-0.257889\pi$
$17$	5.68466	1.37873	0.689366	+	0.724413i	$0.257889\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	−4.56155	−0.847059	−0.423530	−	0.905882i	$-0.639209\pi$
$29$	−4.56155	−0.847059	−0.423530	+	0.905882i	$0.639209\pi$
$30$	0	0
$31$	−5.12311	−0.920137	−0.460068	−	0.887883i	$-0.652175\pi$
$31$	−5.12311	−0.920137	−0.460068	+	0.887883i	$0.652175\pi$
$32$	0	0
$33$	3.68466	0.641417
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−0.876894	−0.144161	−0.0720803	−	0.997399i	$-0.522964\pi$
$37$	−0.876894	−0.144161	−0.0720803	+	0.997399i	$0.522964\pi$
$38$	0	0
$39$	−1.43845	−0.230336
$40$	0	0
$41$	−8.24621	−1.28784	−0.643921	−	0.765092i	$-0.722693\pi$
$41$	−8.24621	−1.28784	−0.643921	+	0.765092i	$0.722693\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	−6.56155	−0.957101	−0.478550	−	0.878060i	$-0.658838\pi$
$47$	−6.56155	−0.957101	−0.478550	+	0.878060i	$0.658838\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	14.5616	2.03903
$52$	0	0
$53$	7.12311	0.978434	0.489217	−	0.872162i	$-0.337283\pi$
$53$	7.12311	0.978434	0.489217	+	0.872162i	$0.337283\pi$
$54$	0	0
$55$	1.43845	0.193960
$56$	0	0
$57$	10.2462	1.35714
$58$	0	0
$59$	14.2462	1.85470	0.927349	−	0.374197i	$-0.122082\pi$
$59$	14.2462	1.85470	0.927349	+	0.374197i	$0.122082\pi$
$60$	0	0
$61$	8.24621	1.05582	0.527910	−	0.849301i	$-0.322976\pi$
$61$	8.24621	1.05582	0.527910	+	0.849301i	$0.322976\pi$
$62$	0	0
$63$	3.56155	0.448713
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	−2.24621	−0.274418	−0.137209	−	0.990542i	$-0.543813\pi$
$67$	−2.24621	−0.274418	−0.137209	+	0.990542i	$0.543813\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.24621	0.266576	0.133288	−	0.991077i	$-0.457446\pi$
$71$	2.24621	0.266576	0.133288	+	0.991077i	$0.457446\pi$
$72$	0	0
$73$	−8.24621	−0.965146	−0.482573	−	0.875856i	$-0.660298\pi$
$73$	−8.24621	−0.965146	−0.482573	+	0.875856i	$0.660298\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	1.43845	0.163926
$78$	0	0
$79$	1.43845	0.161838	0.0809190	−	0.996721i	$-0.474214\pi$
$79$	1.43845	0.161838	0.0809190	+	0.996721i	$0.474214\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−6.24621	−0.685611	−0.342805	−	0.939406i	$-0.611377\pi$
$83$	−6.24621	−0.685611	−0.342805	+	0.939406i	$0.611377\pi$
$84$	0	0
$85$	5.68466	0.616588
$86$	0	0
$87$	−11.6847	−1.25273
$88$	0	0
$89$	−16.2462	−1.72209	−0.861047	−	0.508525i	$-0.830191\pi$
$89$	−16.2462	−1.72209	−0.861047	+	0.508525i	$0.830191\pi$
$90$	0	0
$91$	−0.561553	−0.0588667
$92$	0	0
$93$	−13.1231	−1.36080
$94$	0	0
$95$	4.00000	0.410391
$96$	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
$99$	5.12311	0.514891
$100$	0	0
$101$	19.1231	1.90282	0.951410	−	0.307927i	$-0.0996352\pi$
$101$	19.1231	1.90282	0.951410	+	0.307927i	$0.0996352\pi$
$102$	0	0
$103$	9.43845	0.929998	0.464999	−	0.885311i	$-0.346055\pi$
$103$	9.43845	0.929998	0.464999	+	0.885311i	$0.346055\pi$
$104$	0	0
$105$	2.56155	0.249982
$106$	0	0
$107$	15.3693	1.48581	0.742904	−	0.669398i	$-0.233448\pi$
$107$	15.3693	1.48581	0.742904	+	0.669398i	$0.233448\pi$
$108$	0	0
$109$	−11.9309	−1.14277	−0.571385	−	0.820682i	$-0.693594\pi$
$109$	−11.9309	−1.14277	−0.571385	+	0.820682i	$0.693594\pi$
$110$	0	0
$111$	−2.24621	−0.213201
$112$	0	0
$113$	−5.36932	−0.505103	−0.252551	−	0.967583i	$-0.581270\pi$
$113$	−5.36932	−0.505103	−0.252551	+	0.967583i	$0.581270\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	5.68466	0.521112
$120$	0	0
$121$	−8.93087	−0.811897
$122$	0	0
$123$	−21.1231	−1.90461
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.4924	1.81841	0.909204	−	0.416350i	$-0.136691\pi$
$127$	20.4924	1.81841	0.909204	+	0.416350i	$0.136691\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	1.43845	0.123802
$136$	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
$139$	−21.6155	−1.83341	−0.916703	−	0.399570i	$-0.869159\pi$
$139$	−21.6155	−1.83341	−0.916703	+	0.399570i	$0.869159\pi$
$140$	0	0
$141$	−16.8078	−1.41547
$142$	0	0
$143$	−0.807764	−0.0675486
$144$	0	0
$145$	−4.56155	−0.378816
$146$	0	0
$147$	2.56155	0.211273
$148$	0	0
$149$	−24.2462	−1.98633	−0.993164	−	0.116731i	$-0.962758\pi$
$149$	−24.2462	−1.98633	−0.993164	+	0.116731i	$0.962758\pi$
$150$	0	0
$151$	9.43845	0.768090	0.384045	−	0.923314i	$-0.374531\pi$
$151$	9.43845	0.768090	0.384045	+	0.923314i	$0.374531\pi$
$152$	0	0
$153$	20.2462	1.63681
$154$	0	0
$155$	−5.12311	−0.411498
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	18.2462	1.44702
$160$	0	0
$161$	0	0
$162$	0	0
$163$	7.36932	0.577209	0.288605	−	0.957448i	$-0.406809\pi$
$163$	7.36932	0.577209	0.288605	+	0.957448i	$0.406809\pi$
$164$	0	0
$165$	3.68466	0.286850
$166$	0	0
$167$	−22.5616	−1.74587	−0.872933	−	0.487841i	$-0.837785\pi$
$167$	−22.5616	−1.74587	−0.872933	+	0.487841i	$0.837785\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	14.2462	1.08944
$172$	0	0
$173$	−23.9309	−1.81943	−0.909715	−	0.415233i	$-0.863700\pi$
$173$	−23.9309	−1.81943	−0.909715	+	0.415233i	$0.863700\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	36.4924	2.74294
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−15.1231	−1.12409	−0.562046	−	0.827106i	$-0.689986\pi$
$181$	−15.1231	−1.12409	−0.562046	+	0.827106i	$0.689986\pi$
$182$	0	0
$183$	21.1231	1.56146
$184$	0	0
$185$	−0.876894	−0.0644706
$186$	0	0
$187$	8.17708	0.597967
$188$	0	0
$189$	1.43845	0.104632
$190$	0	0
$191$	8.80776	0.637307	0.318654	−	0.947871i	$-0.396769\pi$
$191$	8.80776	0.637307	0.318654	+	0.947871i	$0.396769\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−1.43845	−0.103009
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−18.2462	−1.29344	−0.646720	−	0.762728i	$-0.723860\pi$
$199$	−18.2462	−1.29344	−0.646720	+	0.762728i	$0.723860\pi$
$200$	0	0
$201$	−5.75379	−0.405841
$202$	0	0
$203$	−4.56155	−0.320158
$204$	0	0
$205$	−8.24621	−0.575940
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.75379	0.397998
$210$	0	0
$211$	22.5616	1.55320	0.776601	−	0.629993i	$-0.216942\pi$
$211$	22.5616	1.55320	0.776601	+	0.629993i	$0.216942\pi$
$212$	0	0
$213$	5.75379	0.394243
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.12311	−0.347779
$218$	0	0
$219$	−21.1231	−1.42737
$220$	0	0
$221$	−3.19224	−0.214733
$222$	0	0
$223$	17.4384	1.16776	0.583882	−	0.811838i	$-0.301533\pi$
$223$	17.4384	1.16776	0.583882	+	0.811838i	$0.301533\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	−9.93087	−0.659135	−0.329567	−	0.944132i	$-0.606903\pi$
$227$	−9.93087	−0.659135	−0.329567	+	0.944132i	$0.606903\pi$
$228$	0	0
$229$	3.75379	0.248057	0.124029	−	0.992279i	$-0.460419\pi$
$229$	3.75379	0.248057	0.124029	+	0.992279i	$0.460419\pi$
$230$	0	0
$231$	3.68466	0.242433
$232$	0	0
$233$	−3.12311	−0.204601	−0.102301	−	0.994754i	$-0.532620\pi$
$233$	−3.12311	−0.204601	−0.102301	+	0.994754i	$0.532620\pi$
$234$	0	0
$235$	−6.56155	−0.428029
$236$	0	0
$237$	3.68466	0.239344
$238$	0	0
$239$	−11.0540	−0.715022	−0.357511	−	0.933909i	$-0.616375\pi$
$239$	−11.0540	−0.715022	−0.357511	+	0.933909i	$0.616375\pi$
$240$	0	0
$241$	1.36932	0.0882055	0.0441027	−	0.999027i	$-0.485957\pi$
$241$	1.36932	0.0882055	0.0441027	+	0.999027i	$0.485957\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.24621	−0.142923
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−11.3693	−0.717625	−0.358812	−	0.933410i	$-0.616818\pi$
$251$	−11.3693	−0.717625	−0.358812	+	0.933410i	$0.616818\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	14.5616	0.911880
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−0.876894	−0.0544876
$260$	0	0
$261$	−16.2462	−1.00562
$262$	0	0
$263$	−5.12311	−0.315904	−0.157952	−	0.987447i	$-0.550489\pi$
$263$	−5.12311	−0.315904	−0.157952	+	0.987447i	$0.550489\pi$
$264$	0	0
$265$	7.12311	0.437569
$266$	0	0
$267$	−41.6155	−2.54683
$268$	0	0
$269$	−4.24621	−0.258896	−0.129448	−	0.991586i	$-0.541321\pi$
$269$	−4.24621	−0.258896	−0.129448	+	0.991586i	$0.541321\pi$
$270$	0	0
$271$	13.1231	0.797172	0.398586	−	0.917131i	$-0.369501\pi$
$271$	13.1231	0.797172	0.398586	+	0.917131i	$0.369501\pi$
$272$	0	0
$273$	−1.43845	−0.0870588
$274$	0	0
$275$	1.43845	0.0867416
$276$	0	0
$277$	4.24621	0.255130	0.127565	−	0.991830i	$-0.459284\pi$
$277$	4.24621	0.255130	0.127565	+	0.991830i	$0.459284\pi$
$278$	0	0
$279$	−18.2462	−1.09237
$280$	0	0
$281$	6.31534	0.376742	0.188371	−	0.982098i	$-0.439679\pi$
$281$	6.31534	0.376742	0.188371	+	0.982098i	$0.439679\pi$
$282$	0	0
$283$	−18.5616	−1.10337	−0.551685	−	0.834053i	$-0.686015\pi$
$283$	−18.5616	−1.10337	−0.551685	+	0.834053i	$0.686015\pi$
$284$	0	0
$285$	10.2462	0.606933
$286$	0	0
$287$	−8.24621	−0.486758
$288$	0	0
$289$	15.3153	0.900902
$290$	0	0
$291$	14.5616	0.853613
$292$	0	0
$293$	6.80776	0.397714	0.198857	−	0.980029i	$-0.436277\pi$
$293$	6.80776	0.397714	0.198857	+	0.980029i	$0.436277\pi$
$294$	0	0
$295$	14.2462	0.829446
$296$	0	0
$297$	2.06913	0.120063
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	48.9848	2.81411
$304$	0	0
$305$	8.24621	0.472177
$306$	0	0
$307$	−8.31534	−0.474582	−0.237291	−	0.971439i	$-0.576259\pi$
$307$	−8.31534	−0.474582	−0.237291	+	0.971439i	$0.576259\pi$
$308$	0	0
$309$	24.1771	1.37539
$310$	0	0
$311$	−10.2462	−0.581009	−0.290505	−	0.956874i	$-0.593823\pi$
$311$	−10.2462	−0.581009	−0.290505	+	0.956874i	$0.593823\pi$
$312$	0	0
$313$	−25.6847	−1.45178	−0.725891	−	0.687809i	$-0.758572\pi$
$313$	−25.6847	−1.45178	−0.725891	+	0.687809i	$0.758572\pi$
$314$	0	0
$315$	3.56155	0.200671
$316$	0	0
$317$	16.7386	0.940135	0.470068	−	0.882630i	$-0.344230\pi$
$317$	16.7386	0.940135	0.470068	+	0.882630i	$0.344230\pi$
$318$	0	0
$319$	−6.56155	−0.367376
$320$	0	0
$321$	39.3693	2.19738
$322$	0	0
$323$	22.7386	1.26521
$324$	0	0
$325$	−0.561553	−0.0311493
$326$	0	0
$327$	−30.5616	−1.69006
$328$	0	0
$329$	−6.56155	−0.361750
$330$	0	0
$331$	18.2462	1.00290	0.501451	−	0.865186i	$-0.332800\pi$
$331$	18.2462	1.00290	0.501451	+	0.865186i	$0.332800\pi$
$332$	0	0
$333$	−3.12311	−0.171145
$334$	0	0
$335$	−2.24621	−0.122724
$336$	0	0
$337$	−8.24621	−0.449200	−0.224600	−	0.974451i	$-0.572108\pi$
$337$	−8.24621	−0.449200	−0.224600	+	0.974451i	$0.572108\pi$
$338$	0	0
$339$	−13.7538	−0.747003
$340$	0	0
$341$	−7.36932	−0.399071
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.4924	−1.10009	−0.550045	−	0.835135i	$-0.685390\pi$
$347$	−20.4924	−1.10009	−0.550045	+	0.835135i	$0.685390\pi$
$348$	0	0
$349$	19.1231	1.02364	0.511818	−	0.859094i	$-0.328972\pi$
$349$	19.1231	1.02364	0.511818	+	0.859094i	$0.328972\pi$
$350$	0	0
$351$	−0.807764	−0.0431153
$352$	0	0
$353$	−33.0540	−1.75929	−0.879643	−	0.475635i	$-0.842218\pi$
$353$	−33.0540	−1.75929	−0.879643	+	0.475635i	$0.842218\pi$
$354$	0	0
$355$	2.24621	0.119217
$356$	0	0
$357$	14.5616	0.770679
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−22.8769	−1.20073
$364$	0	0
$365$	−8.24621	−0.431626
$366$	0	0
$367$	32.1771	1.67963	0.839815	−	0.542873i	$-0.182663\pi$
$367$	32.1771	1.67963	0.839815	+	0.542873i	$0.182663\pi$
$368$	0	0
$369$	−29.3693	−1.52891
$370$	0	0
$371$	7.12311	0.369813
$372$	0	0
$373$	−1.50758	−0.0780594	−0.0390297	−	0.999238i	$-0.512427\pi$
$373$	−1.50758	−0.0780594	−0.0390297	+	0.999238i	$0.512427\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	2.56155	0.131927
$378$	0	0
$379$	28.4924	1.46356	0.731779	−	0.681542i	$-0.238690\pi$
$379$	28.4924	1.46356	0.731779	+	0.681542i	$0.238690\pi$
$380$	0	0
$381$	52.4924	2.68927
$382$	0	0
$383$	−5.75379	−0.294005	−0.147002	−	0.989136i	$-0.546963\pi$
$383$	−5.75379	−0.294005	−0.147002	+	0.989136i	$0.546963\pi$
$384$	0	0
$385$	1.43845	0.0733101
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.19224	−0.263257	−0.131628	−	0.991299i	$-0.542021\pi$
$389$	−5.19224	−0.263257	−0.131628	+	0.991299i	$0.542021\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	51.2311	2.58426
$394$	0	0
$395$	1.43845	0.0723761
$396$	0	0
$397$	22.8078	1.14469	0.572344	−	0.820013i	$-0.306034\pi$
$397$	22.8078	1.14469	0.572344	+	0.820013i	$0.306034\pi$
$398$	0	0
$399$	10.2462	0.512952
$400$	0	0
$401$	26.1771	1.30722	0.653611	−	0.756831i	$-0.273253\pi$
$401$	26.1771	1.30722	0.653611	+	0.756831i	$0.273253\pi$
$402$	0	0
$403$	2.87689	0.143308
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	−1.26137	−0.0625236
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	−15.3693	−0.758112
$412$	0	0
$413$	14.2462	0.701010
$414$	0	0
$415$	−6.24621	−0.306614
$416$	0	0
$417$	−55.3693	−2.71145
$418$	0	0
$419$	−24.4924	−1.19653	−0.598267	−	0.801297i	$-0.704144\pi$
$419$	−24.4924	−1.19653	−0.598267	+	0.801297i	$0.704144\pi$
$420$	0	0
$421$	−17.6847	−0.861898	−0.430949	−	0.902376i	$-0.641821\pi$
$421$	−17.6847	−0.861898	−0.430949	+	0.902376i	$0.641821\pi$
$422$	0	0
$423$	−23.3693	−1.13626
$424$	0	0
$425$	5.68466	0.275746
$426$	0	0
$427$	8.24621	0.399062
$428$	0	0
$429$	−2.06913	−0.0998986
$430$	0	0
$431$	−29.9309	−1.44172	−0.720860	−	0.693081i	$-0.756253\pi$
$431$	−29.9309	−1.44172	−0.720860	+	0.693081i	$0.756253\pi$
$432$	0	0
$433$	7.75379	0.372623	0.186312	−	0.982491i	$-0.440347\pi$
$433$	7.75379	0.372623	0.186312	+	0.982491i	$0.440347\pi$
$434$	0	0
$435$	−11.6847	−0.560236
$436$	0	0
$437$	0	0
$438$	0	0
$439$	17.6155	0.840743	0.420372	−	0.907352i	$-0.361900\pi$
$439$	17.6155	0.840743	0.420372	+	0.907352i	$0.361900\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	−3.50758	−0.166650	−0.0833250	−	0.996522i	$-0.526554\pi$
$443$	−3.50758	−0.166650	−0.0833250	+	0.996522i	$0.526554\pi$
$444$	0	0
$445$	−16.2462	−0.770144
$446$	0	0
$447$	−62.1080	−2.93761
$448$	0	0
$449$	23.3002	1.09960	0.549802	−	0.835295i	$-0.314703\pi$
$449$	23.3002	1.09960	0.549802	+	0.835295i	$0.314703\pi$
$450$	0	0
$451$	−11.8617	−0.558547
$452$	0	0
$453$	24.1771	1.13594
$454$	0	0
$455$	−0.561553	−0.0263260
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	8.17708	0.381673
$460$	0	0
$461$	−1.36932	−0.0637754	−0.0318877	−	0.999491i	$-0.510152\pi$
$461$	−1.36932	−0.0637754	−0.0318877	+	0.999491i	$0.510152\pi$
$462$	0	0
$463$	−25.6155	−1.19045	−0.595227	−	0.803557i	$-0.702938\pi$
$463$	−25.6155	−1.19045	−0.595227	+	0.803557i	$0.702938\pi$
$464$	0	0
$465$	−13.1231	−0.608569
$466$	0	0
$467$	17.9309	0.829742	0.414871	−	0.909880i	$-0.363827\pi$
$467$	17.9309	0.829742	0.414871	+	0.909880i	$0.363827\pi$
$468$	0	0
$469$	−2.24621	−0.103720
$470$	0	0
$471$	15.3693	0.708181
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	25.3693	1.16158
$478$	0	0
$479$	22.7386	1.03895	0.519477	−	0.854484i	$-0.326127\pi$
$479$	22.7386	1.03895	0.519477	+	0.854484i	$0.326127\pi$
$480$	0	0
$481$	0.492423	0.0224525
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.68466	0.258127
$486$	0	0
$487$	−13.1231	−0.594665	−0.297332	−	0.954774i	$-0.596097\pi$
$487$	−13.1231	−0.594665	−0.297332	+	0.954774i	$0.596097\pi$
$488$	0	0
$489$	18.8769	0.853643
$490$	0	0
$491$	27.0540	1.22093	0.610464	−	0.792044i	$-0.290983\pi$
$491$	27.0540	1.22093	0.610464	+	0.792044i	$0.290983\pi$
$492$	0	0
$493$	−25.9309	−1.16787
$494$	0	0
$495$	5.12311	0.230266
$496$	0	0
$497$	2.24621	0.100756
$498$	0	0
$499$	−31.5464	−1.41221	−0.706105	−	0.708107i	$-0.749550\pi$
$499$	−31.5464	−1.41221	−0.706105	+	0.708107i	$0.749550\pi$
$500$	0	0
$501$	−57.7926	−2.58198
$502$	0	0
$503$	3.05398	0.136170	0.0680850	−	0.997680i	$-0.478311\pi$
$503$	3.05398	0.136170	0.0680850	+	0.997680i	$0.478311\pi$
$504$	0	0
$505$	19.1231	0.850967
$506$	0	0
$507$	−32.4924	−1.44304
$508$	0	0
$509$	−5.50758	−0.244119	−0.122060	−	0.992523i	$-0.538950\pi$
$509$	−5.50758	−0.244119	−0.122060	+	0.992523i	$0.538950\pi$
$510$	0	0
$511$	−8.24621	−0.364791
$512$	0	0
$513$	5.75379	0.254036
$514$	0	0
$515$	9.43845	0.415908
$516$	0	0
$517$	−9.43845	−0.415102
$518$	0	0
$519$	−61.3002	−2.69078
$520$	0	0
$521$	−6.63068	−0.290496	−0.145248	−	0.989395i	$-0.546398\pi$
$521$	−6.63068	−0.290496	−0.145248	+	0.989395i	$0.546398\pi$
$522$	0	0
$523$	−16.4924	−0.721163	−0.360582	−	0.932728i	$-0.617422\pi$
$523$	−16.4924	−0.721163	−0.360582	+	0.932728i	$0.617422\pi$
$524$	0	0
$525$	2.56155	0.111795
$526$	0	0
$527$	−29.1231	−1.26862
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	50.7386	2.20187
$532$	0	0
$533$	4.63068	0.200577
$534$	0	0
$535$	15.3693	0.664474
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.43845	0.0619583
$540$	0	0
$541$	3.43845	0.147830	0.0739152	−	0.997265i	$-0.476451\pi$
$541$	3.43845	0.147830	0.0739152	+	0.997265i	$0.476451\pi$
$542$	0	0
$543$	−38.7386	−1.66243
$544$	0	0
$545$	−11.9309	−0.511062
$546$	0	0
$547$	−13.1231	−0.561103	−0.280552	−	0.959839i	$-0.590517\pi$
$547$	−13.1231	−0.561103	−0.280552	+	0.959839i	$0.590517\pi$
$548$	0	0
$549$	29.3693	1.25345
$550$	0	0
$551$	−18.2462	−0.777315
$552$	0	0
$553$	1.43845	0.0611690
$554$	0	0
$555$	−2.24621	−0.0953464
$556$	0	0
$557$	−31.6155	−1.33959	−0.669796	−	0.742545i	$-0.733619\pi$
$557$	−31.6155	−1.33959	−0.669796	+	0.742545i	$0.733619\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	20.9460	0.884342
$562$	0	0
$563$	−28.9848	−1.22157	−0.610783	−	0.791798i	$-0.709145\pi$
$563$	−28.9848	−1.22157	−0.610783	+	0.791798i	$0.709145\pi$
$564$	0	0
$565$	−5.36932	−0.225889
$566$	0	0
$567$	−7.00000	−0.293972
$568$	0	0
$569$	8.73863	0.366343	0.183171	−	0.983081i	$-0.441364\pi$
$569$	8.73863	0.366343	0.183171	+	0.983081i	$0.441364\pi$
$570$	0	0
$571$	−2.24621	−0.0940010	−0.0470005	−	0.998895i	$-0.514966\pi$
$571$	−2.24621	−0.0940010	−0.0470005	+	0.998895i	$0.514966\pi$
$572$	0	0
$573$	22.5616	0.942522
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.5616	0.689466	0.344733	−	0.938701i	$-0.387969\pi$
$577$	16.5616	0.689466	0.344733	+	0.938701i	$0.387969\pi$
$578$	0	0
$579$	−35.8617	−1.49036
$580$	0	0
$581$	−6.24621	−0.259137
$582$	0	0
$583$	10.2462	0.424355
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	16.4924	0.680715	0.340358	−	0.940296i	$-0.389452\pi$
$587$	16.4924	0.680715	0.340358	+	0.940296i	$0.389452\pi$
$588$	0	0
$589$	−20.4924	−0.844376
$590$	0	0
$591$	25.6155	1.05368
$592$	0	0
$593$	−19.9309	−0.818463	−0.409231	−	0.912431i	$-0.634203\pi$
$593$	−19.9309	−0.818463	−0.409231	+	0.912431i	$0.634203\pi$
$594$	0	0
$595$	5.68466	0.233048
$596$	0	0
$597$	−46.7386	−1.91288
$598$	0	0
$599$	−17.4384	−0.712516	−0.356258	−	0.934388i	$-0.615948\pi$
$599$	−17.4384	−0.712516	−0.356258	+	0.934388i	$0.615948\pi$
$600$	0	0
$601$	15.7538	0.642610	0.321305	−	0.946976i	$-0.395878\pi$
$601$	15.7538	0.642610	0.321305	+	0.946976i	$0.395878\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	−8.93087	−0.363091
$606$	0	0
$607$	−21.9309	−0.890147	−0.445073	−	0.895494i	$-0.646822\pi$
$607$	−21.9309	−0.890147	−0.445073	+	0.895494i	$0.646822\pi$
$608$	0	0
$609$	−11.6847	−0.473486
$610$	0	0
$611$	3.68466	0.149065
$612$	0	0
$613$	−3.75379	−0.151614	−0.0758070	−	0.997123i	$-0.524153\pi$
$613$	−3.75379	−0.151614	−0.0758070	+	0.997123i	$0.524153\pi$
$614$	0	0
$615$	−21.1231	−0.851766
$616$	0	0
$617$	−20.7386	−0.834906	−0.417453	−	0.908699i	$-0.637077\pi$
$617$	−20.7386	−0.834906	−0.417453	+	0.908699i	$0.637077\pi$
$618$	0	0
$619$	34.7386	1.39626	0.698132	−	0.715969i	$-0.254015\pi$
$619$	34.7386	1.39626	0.698132	+	0.715969i	$0.254015\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.2462	−0.650891
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	14.7386	0.588604
$628$	0	0
$629$	−4.98485	−0.198759
$630$	0	0
$631$	14.5616	0.579686	0.289843	−	0.957074i	$-0.406397\pi$
$631$	14.5616	0.579686	0.289843	+	0.957074i	$0.406397\pi$
$632$	0	0
$633$	57.7926	2.29705
$634$	0	0
$635$	20.4924	0.813217
$636$	0	0
$637$	−0.561553	−0.0222495
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−8.24621	−0.325706	−0.162853	−	0.986650i	$-0.552070\pi$
$641$	−8.24621	−0.325706	−0.162853	+	0.986650i	$0.552070\pi$
$642$	0	0
$643$	34.5616	1.36297	0.681487	−	0.731830i	$-0.261333\pi$
$643$	34.5616	1.36297	0.681487	+	0.731830i	$0.261333\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.7538	−0.855230	−0.427615	−	0.903961i	$-0.640646\pi$
$647$	−21.7538	−0.855230	−0.427615	+	0.903961i	$0.640646\pi$
$648$	0	0
$649$	20.4924	0.804398
$650$	0	0
$651$	−13.1231	−0.514335
$652$	0	0
$653$	−39.6155	−1.55028	−0.775138	−	0.631792i	$-0.782320\pi$
$653$	−39.6155	−1.55028	−0.775138	+	0.631792i	$0.782320\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	0	0
$657$	−29.3693	−1.14581
$658$	0	0
$659$	36.6695	1.42844	0.714221	−	0.699921i	$-0.246781\pi$
$659$	36.6695	1.42844	0.714221	+	0.699921i	$0.246781\pi$
$660$	0	0
$661$	24.8769	0.967599	0.483800	−	0.875179i	$-0.339256\pi$
$661$	24.8769	0.967599	0.483800	+	0.875179i	$0.339256\pi$
$662$	0	0
$663$	−8.17708	−0.317572
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	0	0
$668$	0	0
$669$	44.6695	1.72702
$670$	0	0
$671$	11.8617	0.457917
$672$	0	0
$673$	−18.4924	−0.712831	−0.356415	−	0.934328i	$-0.616001\pi$
$673$	−18.4924	−0.712831	−0.356415	+	0.934328i	$0.616001\pi$
$674$	0	0
$675$	1.43845	0.0553659
$676$	0	0
$677$	−39.3002	−1.51043	−0.755214	−	0.655478i	$-0.772467\pi$
$677$	−39.3002	−1.51043	−0.755214	+	0.655478i	$0.772467\pi$
$678$	0	0
$679$	5.68466	0.218157
$680$	0	0
$681$	−25.4384	−0.974803
$682$	0	0
$683$	−10.8769	−0.416193	−0.208096	−	0.978108i	$-0.566727\pi$
$683$	−10.8769	−0.416193	−0.208096	+	0.978108i	$0.566727\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	9.61553	0.366855
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	14.2462	0.541951	0.270976	−	0.962586i	$-0.412654\pi$
$691$	14.2462	0.541951	0.270976	+	0.962586i	$0.412654\pi$
$692$	0	0
$693$	5.12311	0.194611
$694$	0	0
$695$	−21.6155	−0.819924
$696$	0	0
$697$	−46.8769	−1.77559
$698$	0	0
$699$	−8.00000	−0.302588
$700$	0	0
$701$	20.4233	0.771377	0.385689	−	0.922629i	$-0.373964\pi$
$701$	20.4233	0.771377	0.385689	+	0.922629i	$0.373964\pi$
$702$	0	0
$703$	−3.50758	−0.132291
$704$	0	0
$705$	−16.8078	−0.633017
$706$	0	0
$707$	19.1231	0.719198
$708$	0	0
$709$	−28.5616	−1.07265	−0.536326	−	0.844011i	$-0.680188\pi$
$709$	−28.5616	−1.07265	−0.536326	+	0.844011i	$0.680188\pi$
$710$	0	0
$711$	5.12311	0.192131
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−0.807764	−0.0302087
$716$	0	0
$717$	−28.3153	−1.05746
$718$	0	0
$719$	44.4924	1.65929	0.829644	−	0.558293i	$-0.188544\pi$
$719$	44.4924	1.65929	0.829644	+	0.558293i	$0.188544\pi$
$720$	0	0
$721$	9.43845	0.351506
$722$	0	0
$723$	3.50758	0.130448
$724$	0	0
$725$	−4.56155	−0.169412
$726$	0	0
$727$	48.0000	1.78022	0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000	1.78022	0.890111	+	0.455744i	$0.150627\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−2.80776	−0.103707	−0.0518536	−	0.998655i	$-0.516513\pi$
$733$	−2.80776	−0.103707	−0.0518536	+	0.998655i	$0.516513\pi$
$734$	0	0
$735$	2.56155	0.0944843
$736$	0	0
$737$	−3.23106	−0.119017
$738$	0	0
$739$	26.4233	0.971997	0.485998	−	0.873960i	$-0.338456\pi$
$739$	26.4233	0.971997	0.485998	+	0.873960i	$0.338456\pi$
$740$	0	0
$741$	−5.75379	−0.211371
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	−24.2462	−0.888312
$746$	0	0
$747$	−22.2462	−0.813946
$748$	0	0
$749$	15.3693	0.561583
$750$	0	0
$751$	51.6847	1.88600	0.943000	−	0.332793i	$-0.107991\pi$
$751$	51.6847	1.88600	0.943000	+	0.332793i	$0.107991\pi$
$752$	0	0
$753$	−29.1231	−1.06130
$754$	0	0
$755$	9.43845	0.343500
$756$	0	0
$757$	−9.86174	−0.358431	−0.179216	−	0.983810i	$-0.557356\pi$
$757$	−9.86174	−0.358431	−0.179216	+	0.983810i	$0.557356\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25.8617	−0.937487	−0.468744	−	0.883334i	$-0.655293\pi$
$761$	−25.8617	−0.937487	−0.468744	+	0.883334i	$0.655293\pi$
$762$	0	0
$763$	−11.9309	−0.431926
$764$	0	0
$765$	20.2462	0.732003
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	40.7386	1.46907	0.734536	−	0.678569i	$-0.237400\pi$
$769$	40.7386	1.46907	0.734536	+	0.678569i	$0.237400\pi$
$770$	0	0
$771$	−15.3693	−0.553512
$772$	0	0
$773$	25.6847	0.923813	0.461906	−	0.886929i	$-0.347166\pi$
$773$	25.6847	0.923813	0.461906	+	0.886929i	$0.347166\pi$
$774$	0	0
$775$	−5.12311	−0.184027
$776$	0	0
$777$	−2.24621	−0.0805824
$778$	0	0
$779$	−32.9848	−1.18180
$780$	0	0
$781$	3.23106	0.115616
$782$	0	0
$783$	−6.56155	−0.234491
$784$	0	0
$785$	6.00000	0.214149
$786$	0	0
$787$	−41.9309	−1.49467	−0.747337	−	0.664445i	$-0.768668\pi$
$787$	−41.9309	−1.49467	−0.747337	+	0.664445i	$0.768668\pi$
$788$	0	0
$789$	−13.1231	−0.467195
$790$	0	0
$791$	−5.36932	−0.190911
$792$	0	0
$793$	−4.63068	−0.164440
$794$	0	0
$795$	18.2462	0.647126
$796$	0	0
$797$	12.5616	0.444953	0.222477	−	0.974938i	$-0.428586\pi$
$797$	12.5616	0.444953	0.222477	+	0.974938i	$0.428586\pi$
$798$	0	0
$799$	−37.3002	−1.31959
$800$	0	0
$801$	−57.8617	−2.04444
$802$	0	0
$803$	−11.8617	−0.418592
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.8769	−0.382885
$808$	0	0
$809$	23.9309	0.841365	0.420682	−	0.907208i	$-0.361791\pi$
$809$	23.9309	0.841365	0.420682	+	0.907208i	$0.361791\pi$
$810$	0	0
$811$	2.38447	0.0837301	0.0418651	−	0.999123i	$-0.486670\pi$
$811$	2.38447	0.0837301	0.0418651	+	0.999123i	$0.486670\pi$
$812$	0	0
$813$	33.6155	1.17895
$814$	0	0
$815$	7.36932	0.258136
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	7.30019	0.254778	0.127389	−	0.991853i	$-0.459340\pi$
$821$	7.30019	0.254778	0.127389	+	0.991853i	$0.459340\pi$
$822$	0	0
$823$	21.1231	0.736305	0.368153	−	0.929765i	$-0.379990\pi$
$823$	21.1231	0.736305	0.368153	+	0.929765i	$0.379990\pi$
$824$	0	0
$825$	3.68466	0.128283
$826$	0	0
$827$	−35.8617	−1.24703	−0.623517	−	0.781809i	$-0.714297\pi$
$827$	−35.8617	−1.24703	−0.623517	+	0.781809i	$0.714297\pi$
$828$	0	0
$829$	8.24621	0.286403	0.143201	−	0.989694i	$-0.454260\pi$
$829$	8.24621	0.286403	0.143201	+	0.989694i	$0.454260\pi$
$830$	0	0
$831$	10.8769	0.377315
$832$	0	0
$833$	5.68466	0.196962
$834$	0	0
$835$	−22.5616	−0.780775
$836$	0	0
$837$	−7.36932	−0.254721
$838$	0	0
$839$	28.4924	0.983668	0.491834	−	0.870689i	$-0.336327\pi$
$839$	28.4924	0.983668	0.491834	+	0.870689i	$0.336327\pi$
$840$	0	0
$841$	−8.19224	−0.282491
$842$	0	0
$843$	16.1771	0.557168
$844$	0	0
$845$	−12.6847	−0.436366
$846$	0	0
$847$	−8.93087	−0.306868
$848$	0	0
$849$	−47.5464	−1.63179
$850$	0	0
$851$	0	0
$852$	0	0
$853$	24.2462	0.830174	0.415087	−	0.909782i	$-0.363751\pi$
$853$	24.2462	0.830174	0.415087	+	0.909782i	$0.363751\pi$
$854$	0	0
$855$	14.2462	0.487210
$856$	0	0
$857$	36.2462	1.23815	0.619073	−	0.785333i	$-0.287508\pi$
$857$	36.2462	1.23815	0.619073	+	0.785333i	$0.287508\pi$
$858$	0	0
$859$	48.4924	1.65454	0.827270	−	0.561804i	$-0.189893\pi$
$859$	48.4924	1.65454	0.827270	+	0.561804i	$0.189893\pi$
$860$	0	0
$861$	−21.1231	−0.719874
$862$	0	0
$863$	−16.6307	−0.566115	−0.283058	−	0.959103i	$-0.591349\pi$
$863$	−16.6307	−0.566115	−0.283058	+	0.959103i	$0.591349\pi$
$864$	0	0
$865$	−23.9309	−0.813674
$866$	0	0
$867$	39.2311	1.33236
$868$	0	0
$869$	2.06913	0.0701904
$870$	0	0
$871$	1.26137	0.0427398
$872$	0	0
$873$	20.2462	0.685230
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−48.6004	−1.64112	−0.820559	−	0.571562i	$-0.806338\pi$
$877$	−48.6004	−1.64112	−0.820559	+	0.571562i	$0.806338\pi$
$878$	0	0
$879$	17.4384	0.588184
$880$	0	0
$881$	9.36932	0.315660	0.157830	−	0.987466i	$-0.449550\pi$
$881$	9.36932	0.315660	0.157830	+	0.987466i	$0.449550\pi$
$882$	0	0
$883$	51.8617	1.74529	0.872643	−	0.488358i	$-0.162404\pi$
$883$	51.8617	1.74529	0.872643	+	0.488358i	$0.162404\pi$
$884$	0	0
$885$	36.4924	1.22668
$886$	0	0
$887$	−20.4924	−0.688068	−0.344034	−	0.938957i	$-0.611794\pi$
$887$	−20.4924	−0.688068	−0.344034	+	0.938957i	$0.611794\pi$
$888$	0	0
$889$	20.4924	0.687294
$890$	0	0
$891$	−10.0691	−0.337329
$892$	0	0
$893$	−26.2462	−0.878296
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.3693	0.779410
$900$	0	0
$901$	40.4924	1.34900
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.1231	−0.502709
$906$	0	0
$907$	−44.4924	−1.47735	−0.738673	−	0.674064i	$-0.764547\pi$
$907$	−44.4924	−1.47735	−0.738673	+	0.674064i	$0.764547\pi$
$908$	0	0
$909$	68.1080	2.25900
$910$	0	0
$911$	−8.98485	−0.297681	−0.148841	−	0.988861i	$-0.547554\pi$
$911$	−8.98485	−0.297681	−0.148841	+	0.988861i	$0.547554\pi$
$912$	0	0
$913$	−8.98485	−0.297355
$914$	0	0
$915$	21.1231	0.698308
$916$	0	0
$917$	20.0000	0.660458
$918$	0	0
$919$	−49.7926	−1.64251	−0.821253	−	0.570564i	$-0.806725\pi$
$919$	−49.7926	−1.64251	−0.821253	+	0.570564i	$0.806725\pi$
$920$	0	0
$921$	−21.3002	−0.701865
$922$	0	0
$923$	−1.26137	−0.0415184
$924$	0	0
$925$	−0.876894	−0.0288321
$926$	0	0
$927$	33.6155	1.10408
$928$	0	0
$929$	58.3542	1.91454	0.957269	−	0.289199i	$-0.0933890\pi$
$929$	58.3542	1.91454	0.957269	+	0.289199i	$0.0933890\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	−26.2462	−0.859262
$934$	0	0
$935$	8.17708	0.267419
$936$	0	0
$937$	−41.0540	−1.34117	−0.670587	−	0.741830i	$-0.733958\pi$
$937$	−41.0540	−1.34117	−0.670587	+	0.741830i	$0.733958\pi$
$938$	0	0
$939$	−65.7926	−2.14706
$940$	0	0
$941$	35.7538	1.16554	0.582770	−	0.812637i	$-0.301969\pi$
$941$	35.7538	1.16554	0.582770	+	0.812637i	$0.301969\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.43845	0.0467927
$946$	0	0
$947$	−23.3693	−0.759401	−0.379700	−	0.925110i	$-0.623973\pi$
$947$	−23.3693	−0.759401	−0.379700	+	0.925110i	$0.623973\pi$
$948$	0	0
$949$	4.63068	0.150318
$950$	0	0
$951$	42.8769	1.39038
$952$	0	0
$953$	−16.2462	−0.526266	−0.263133	−	0.964760i	$-0.584756\pi$
$953$	−16.2462	−0.526266	−0.263133	+	0.964760i	$0.584756\pi$
$954$	0	0
$955$	8.80776	0.285013
$956$	0	0
$957$	−16.8078	−0.543318
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−4.75379	−0.153348
$962$	0	0
$963$	54.7386	1.76393
$964$	0	0
$965$	−14.0000	−0.450676
$966$	0	0
$967$	55.3693	1.78056	0.890279	−	0.455416i	$-0.150510\pi$
$967$	55.3693	1.78056	0.890279	+	0.455416i	$0.150510\pi$
$968$	0	0
$969$	58.2462	1.87114
$970$	0	0
$971$	−35.3693	−1.13506	−0.567528	−	0.823354i	$-0.692100\pi$
$971$	−35.3693	−1.13506	−0.567528	+	0.823354i	$0.692100\pi$
$972$	0	0
$973$	−21.6155	−0.692962
$974$	0	0
$975$	−1.43845	−0.0460672
$976$	0	0
$977$	40.1080	1.28317	0.641584	−	0.767053i	$-0.278278\pi$
$977$	40.1080	1.28317	0.641584	+	0.767053i	$0.278278\pi$
$978$	0	0
$979$	−23.3693	−0.746887
$980$	0	0
$981$	−42.4924	−1.35668
$982$	0	0
$983$	−28.6695	−0.914415	−0.457208	−	0.889360i	$-0.651150\pi$
$983$	−28.6695	−0.914415	−0.457208	+	0.889360i	$0.651150\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	−16.8078	−0.534997
$988$	0	0
$989$	0	0
$990$	0	0
$991$	28.4924	0.905092	0.452546	−	0.891741i	$-0.350516\pi$
$991$	28.4924	0.905092	0.452546	+	0.891741i	$0.350516\pi$
$992$	0	0
$993$	46.7386	1.48321
$994$	0	0
$995$	−18.2462	−0.578444
$996$	0	0
$997$	−25.1922	−0.797846	−0.398923	−	0.916984i	$-0.630616\pi$
$997$	−25.1922	−0.797846	−0.398923	+	0.916984i	$0.630616\pi$
$998$	0	0
$999$	−1.26137	−0.0399079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.a.bl.1.2		2
4.3	odd	2		2240.2.a.bf.1.1		2
8.3	odd	2		1120.2.a.s.1.2	yes	2
8.5	even	2		1120.2.a.q.1.1	✓	2
40.19	odd	2		5600.2.a.bb.1.1		2
40.29	even	2		5600.2.a.bg.1.2		2
56.13	odd	2		7840.2.a.bi.1.2		2
56.27	even	2		7840.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.a.q.1.1	✓	2	8.5	even	2
1120.2.a.s.1.2	yes	2	8.3	odd	2
2240.2.a.bf.1.1		2	4.3	odd	2
2240.2.a.bl.1.2		2	1.1	even	1	trivial
5600.2.a.bb.1.1		2	40.19	odd	2
5600.2.a.bg.1.2		2	40.29	even	2
7840.2.a.bd.1.1		2	56.27	even	2
7840.2.a.bi.1.2		2	56.13	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q+2.56155 q^{3} +1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +1.43845 q^{11} -0.561553 q^{13} +2.56155 q^{15} +5.68466 q^{17} +4.00000 q^{19} +2.56155 q^{21} +1.00000 q^{25} +1.43845 q^{27} -4.56155 q^{29} -5.12311 q^{31} +3.68466 q^{33} +1.00000 q^{35} -0.876894 q^{37} -1.43845 q^{39} -8.24621 q^{41} +3.56155 q^{45} -6.56155 q^{47} +1.00000 q^{49} +14.5616 q^{51} +7.12311 q^{53} +1.43845 q^{55} +10.2462 q^{57} +14.2462 q^{59} +8.24621 q^{61} +3.56155 q^{63} -0.561553 q^{65} -2.24621 q^{67} +2.24621 q^{71} -8.24621 q^{73} +2.56155 q^{75} +1.43845 q^{77} +1.43845 q^{79} -7.00000 q^{81} -6.24621 q^{83} +5.68466 q^{85} -11.6847 q^{87} -16.2462 q^{89} -0.561553 q^{91} -13.1231 q^{93} +4.00000 q^{95} +5.68466 q^{97} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9	\( 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} + 7 q^{11} + 3 q^{13} + q^{15} - q^{17} + 8 q^{19} + q^{21} + 2 q^{25} + 7 q^{27} - 5 q^{29} - 2 q^{31} - 5 q^{33} + 2 q^{35} - 10 q^{37} - 7 q^{39} + 3 q^{45} - 9 q^{47} + 2 q^{49} + 25 q^{51} + 6 q^{53} + 7 q^{55} + 4 q^{57} + 12 q^{59} + 3 q^{63} + 3 q^{65} + 12 q^{67} - 12 q^{71} + q^{75} + 7 q^{77} + 7 q^{79} - 14 q^{81} + 4 q^{83} - q^{85} - 11 q^{87} - 16 q^{89} + 3 q^{91} - 18 q^{93} + 8 q^{95} - q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 + 7 * q^11 + 3 * q^13 + q^15 - q^17 + 8 * q^19 + q^21 + 2 * q^25 + 7 * q^27 - 5 * q^29 - 2 * q^31 - 5 * q^33 + 2 * q^35 - 10 * q^37 - 7 * q^39 + 3 * q^45 - 9 * q^47 + 2 * q^49 + 25 * q^51 + 6 * q^53 + 7 * q^55 + 4 * q^57 + 12 * q^59 + 3 * q^63 + 3 * q^65 + 12 * q^67 - 12 * q^71 + q^75 + 7 * q^77 + 7 * q^79 - 14 * q^81 + 4 * q^83 - q^85 - 11 * q^87 - 16 * q^89 + 3 * q^91 - 18 * q^93 + 8 * q^95 - q^97 + 2 * q^99

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