LMFDB - Embedded newform 1035.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,2,Mod(1,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1035.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1035.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:	$8.26451660920$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1035.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.00000 q^{5} +1.82843 q^{7} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +1.00000 q^{5} +1.82843 q^{7} -2.82843 q^{8} +1.41421 q^{10} +2.58579 q^{11} +3.41421 q^{13} +2.58579 q^{14} -4.00000 q^{16} +6.07107 q^{17} -6.24264 q^{19} +3.65685 q^{22} +1.00000 q^{23} +1.00000 q^{25} +4.82843 q^{26} +3.58579 q^{29} -4.17157 q^{31} +8.58579 q^{34} +1.82843 q^{35} +3.00000 q^{37} -8.82843 q^{38} -2.82843 q^{40} +10.4142 q^{41} +6.00000 q^{43} +1.41421 q^{46} +8.58579 q^{47} -3.65685 q^{49} +1.41421 q^{50} -7.24264 q^{53} +2.58579 q^{55} -5.17157 q^{56} +5.07107 q^{58} +6.89949 q^{59} +5.41421 q^{61} -5.89949 q^{62} +8.00000 q^{64} +3.41421 q^{65} -11.4853 q^{67} +2.58579 q^{70} -8.89949 q^{71} -10.2426 q^{73} +4.24264 q^{74} +4.72792 q^{77} -15.3137 q^{79} -4.00000 q^{80} +14.7279 q^{82} -14.0711 q^{83} +6.07107 q^{85} +8.48528 q^{86} -7.31371 q^{88} -10.1421 q^{89} +6.24264 q^{91} +12.1421 q^{94} -6.24264 q^{95} -1.17157 q^{97} -5.17157 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{5} - 2 q^{7} + 8 q^{11} + 4 q^{13} + 8 q^{14} - 8 q^{16} - 2 q^{17} - 4 q^{19} - 4 q^{22} + 2 q^{23} + 2 q^{25} + 4 q^{26} + 10 q^{29} - 14 q^{31} + 20 q^{34} - 2 q^{35} + 6 q^{37} - 12 q^{38} + 18 q^{41} + 12 q^{43} + 20 q^{47} + 4 q^{49} - 6 q^{53} + 8 q^{55} - 16 q^{56} - 4 q^{58} - 6 q^{59} + 8 q^{61} + 8 q^{62} + 16 q^{64} + 4 q^{65} - 6 q^{67} + 8 q^{70} + 2 q^{71} - 12 q^{73} - 16 q^{77} - 8 q^{79} - 8 q^{80} + 4 q^{82} - 14 q^{83} - 2 q^{85} + 8 q^{88} + 8 q^{89} + 4 q^{91} - 4 q^{94} - 4 q^{95} - 8 q^{97} - 16 q^{98}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 + 8 * q^11 + 4 * q^13 + 8 * q^14 - 8 * q^16 - 2 * q^17 - 4 * q^19 - 4 * q^22 + 2 * q^23 + 2 * q^25 + 4 * q^26 + 10 * q^29 - 14 * q^31 + 20 * q^34 - 2 * q^35 + 6 * q^37 - 12 * q^38 + 18 * q^41 + 12 * q^43 + 20 * q^47 + 4 * q^49 - 6 * q^53 + 8 * q^55 - 16 * q^56 - 4 * q^58 - 6 * q^59 + 8 * q^61 + 8 * q^62 + 16 * q^64 + 4 * q^65 - 6 * q^67 + 8 * q^70 + 2 * q^71 - 12 * q^73 - 16 * q^77 - 8 * q^79 - 8 * q^80 + 4 * q^82 - 14 * q^83 - 2 * q^85 + 8 * q^88 + 8 * q^89 + 4 * q^91 - 4 * q^94 - 4 * q^95 - 8 * q^97 - 16 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.82843	0.691080	0.345540	−	0.938404i	$-0.387696\pi$
$7$	1.82843	0.691080	0.345540	+	0.938404i	$0.387696\pi$
$8$	−2.82843	−1.00000
$9$	0	0
$10$	1.41421	0.447214
$11$	2.58579	0.779644	0.389822	−	0.920890i	$-0.372537\pi$
$11$	2.58579	0.779644	0.389822	+	0.920890i	$0.372537\pi$
$12$	0	0
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	2.58579	0.691080
$15$	0	0
$16$	−4.00000	−1.00000
$17$	6.07107	1.47245	0.736225	−	0.676737i	$-0.236606\pi$
$17$	6.07107	1.47245	0.736225	+	0.676737i	$0.236606\pi$
$18$	0	0
$19$	−6.24264	−1.43216	−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264	−1.43216	−0.716080	+	0.698018i	$0.754065\pi$
$20$	0	0
$21$	0	0
$22$	3.65685	0.779644
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	4.82843	0.946932
$27$	0	0
$28$	0	0
$29$	3.58579	0.665864	0.332932	−	0.942951i	$-0.391962\pi$
$29$	3.58579	0.665864	0.332932	+	0.942951i	$0.391962\pi$
$30$	0	0
$31$	−4.17157	−0.749237	−0.374618	−	0.927179i	$-0.622226\pi$
$31$	−4.17157	−0.749237	−0.374618	+	0.927179i	$0.622226\pi$
$32$	0	0
$33$	0	0
$34$	8.58579	1.47245
$35$	1.82843	0.309061
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−8.82843	−1.43216
$39$	0	0
$40$	−2.82843	−0.447214
$41$	10.4142	1.62643	0.813213	−	0.581966i	$-0.197716\pi$
$41$	10.4142	1.62643	0.813213	+	0.581966i	$0.197716\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	0	0
$46$	1.41421	0.208514
$47$	8.58579	1.25237	0.626183	−	0.779676i	$-0.284616\pi$
$47$	8.58579	1.25237	0.626183	+	0.779676i	$0.284616\pi$
$48$	0	0
$49$	−3.65685	−0.522408
$50$	1.41421	0.200000
$51$	0	0
$52$	0	0
$53$	−7.24264	−0.994853	−0.497427	−	0.867506i	$-0.665722\pi$
$53$	−7.24264	−0.994853	−0.497427	+	0.867506i	$0.665722\pi$
$54$	0	0
$55$	2.58579	0.348667
$56$	−5.17157	−0.691080
$57$	0	0
$58$	5.07107	0.665864
$59$	6.89949	0.898238	0.449119	−	0.893472i	$-0.351738\pi$
$59$	6.89949	0.898238	0.449119	+	0.893472i	$0.351738\pi$
$60$	0	0
$61$	5.41421	0.693219	0.346610	−	0.938010i	$-0.387333\pi$
$61$	5.41421	0.693219	0.346610	+	0.938010i	$0.387333\pi$
$62$	−5.89949	−0.749237
$63$	0	0
$64$	8.00000	1.00000
$65$	3.41421	0.423481
$66$	0	0
$67$	−11.4853	−1.40315	−0.701575	−	0.712595i	$-0.747520\pi$
$67$	−11.4853	−1.40315	−0.701575	+	0.712595i	$0.747520\pi$
$68$	0	0
$69$	0	0
$70$	2.58579	0.309061
$71$	−8.89949	−1.05618	−0.528088	−	0.849190i	$-0.677091\pi$
$71$	−8.89949	−1.05618	−0.528088	+	0.849190i	$0.677091\pi$
$72$	0	0
$73$	−10.2426	−1.19881	−0.599405	−	0.800446i	$-0.704596\pi$
$73$	−10.2426	−1.19881	−0.599405	+	0.800446i	$0.704596\pi$
$74$	4.24264	0.493197
$75$	0	0
$76$	0	0
$77$	4.72792	0.538797
$78$	0	0
$79$	−15.3137	−1.72293	−0.861463	−	0.507820i	$-0.830452\pi$
$79$	−15.3137	−1.72293	−0.861463	+	0.507820i	$0.830452\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	14.7279	1.62643
$83$	−14.0711	−1.54450	−0.772250	−	0.635319i	$-0.780869\pi$
$83$	−14.0711	−1.54450	−0.772250	+	0.635319i	$0.780869\pi$
$84$	0	0
$85$	6.07107	0.658500
$86$	8.48528	0.914991
$87$	0	0
$88$	−7.31371	−0.779644
$89$	−10.1421	−1.07506	−0.537532	−	0.843243i	$-0.680643\pi$
$89$	−10.1421	−1.07506	−0.537532	+	0.843243i	$0.680643\pi$
$90$	0	0
$91$	6.24264	0.654407
$92$	0	0
$93$	0	0
$94$	12.1421	1.25237
$95$	−6.24264	−0.640481
$96$	0	0
$97$	−1.17157	−0.118955	−0.0594776	−	0.998230i	$-0.518943\pi$
$97$	−1.17157	−0.118955	−0.0594776	+	0.998230i	$0.518943\pi$
$98$	−5.17157	−0.522408
$99$	0	0
$100$	0	0
$101$	−12.5563	−1.24940	−0.624702	−	0.780863i	$-0.714779\pi$
$101$	−12.5563	−1.24940	−0.624702	+	0.780863i	$0.714779\pi$
$102$	0	0
$103$	0.828427	0.0816274	0.0408137	−	0.999167i	$-0.487005\pi$
$103$	0.828427	0.0816274	0.0408137	+	0.999167i	$0.487005\pi$
$104$	−9.65685	−0.946932
$105$	0	0
$106$	−10.2426	−0.994853
$107$	4.41421	0.426738	0.213369	−	0.976972i	$-0.431556\pi$
$107$	4.41421	0.426738	0.213369	+	0.976972i	$0.431556\pi$
$108$	0	0
$109$	14.2426	1.36420	0.682099	−	0.731260i	$-0.261067\pi$
$109$	14.2426	1.36420	0.682099	+	0.731260i	$0.261067\pi$
$110$	3.65685	0.348667
$111$	0	0
$112$	−7.31371	−0.691080
$113$	10.4142	0.979687	0.489843	−	0.871810i	$-0.337054\pi$
$113$	10.4142	0.979687	0.489843	+	0.871810i	$0.337054\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	9.75736	0.898238
$119$	11.1005	1.01758
$120$	0	0
$121$	−4.31371	−0.392155
$122$	7.65685	0.693219
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.242641	−0.0215309	−0.0107654	−	0.999942i	$-0.503427\pi$
$127$	−0.242641	−0.0215309	−0.0107654	+	0.999942i	$0.503427\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	4.82843	0.423481
$131$	13.6569	1.19320	0.596602	−	0.802537i	$-0.296517\pi$
$131$	13.6569	1.19320	0.596602	+	0.802537i	$0.296517\pi$
$132$	0	0
$133$	−11.4142	−0.989738
$134$	−16.2426	−1.40315
$135$	0	0
$136$	−17.1716	−1.47245
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	0	0
$142$	−12.5858	−1.05618
$143$	8.82843	0.738270
$144$	0	0
$145$	3.58579	0.297783
$146$	−14.4853	−1.19881
$147$	0	0
$148$	0	0
$149$	−9.07107	−0.743131	−0.371565	−	0.928407i	$-0.621179\pi$
$149$	−9.07107	−0.743131	−0.371565	+	0.928407i	$0.621179\pi$
$150$	0	0
$151$	−9.31371	−0.757939	−0.378969	−	0.925409i	$-0.623721\pi$
$151$	−9.31371	−0.757939	−0.378969	+	0.925409i	$0.623721\pi$
$152$	17.6569	1.43216
$153$	0	0
$154$	6.68629	0.538797
$155$	−4.17157	−0.335069
$156$	0	0
$157$	−1.48528	−0.118538	−0.0592692	−	0.998242i	$-0.518877\pi$
$157$	−1.48528	−0.118538	−0.0592692	+	0.998242i	$0.518877\pi$
$158$	−21.6569	−1.72293
$159$	0	0
$160$	0	0
$161$	1.82843	0.144100
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	0	0
$166$	−19.8995	−1.54450
$167$	6.72792	0.520622	0.260311	−	0.965525i	$-0.416175\pi$
$167$	6.72792	0.520622	0.260311	+	0.965525i	$0.416175\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	8.58579	0.658500
$171$	0	0
$172$	0	0
$173$	−13.6569	−1.03831	−0.519156	−	0.854680i	$-0.673754\pi$
$173$	−13.6569	−1.03831	−0.519156	+	0.854680i	$0.673754\pi$
$174$	0	0
$175$	1.82843	0.138216
$176$	−10.3431	−0.779644
$177$	0	0
$178$	−14.3431	−1.07506
$179$	7.65685	0.572300	0.286150	−	0.958185i	$-0.407624\pi$
$179$	7.65685	0.572300	0.286150	+	0.958185i	$0.407624\pi$
$180$	0	0
$181$	−6.82843	−0.507553	−0.253776	−	0.967263i	$-0.581673\pi$
$181$	−6.82843	−0.507553	−0.253776	+	0.967263i	$0.581673\pi$
$182$	8.82843	0.654407
$183$	0	0
$184$	−2.82843	−0.208514
$185$	3.00000	0.220564
$186$	0	0
$187$	15.6985	1.14799
$188$	0	0
$189$	0	0
$190$	−8.82843	−0.640481
$191$	−10.2426	−0.741131	−0.370566	−	0.928806i	$-0.620836\pi$
$191$	−10.2426	−0.741131	−0.370566	+	0.928806i	$0.620836\pi$
$192$	0	0
$193$	−8.48528	−0.610784	−0.305392	−	0.952227i	$-0.598787\pi$
$193$	−8.48528	−0.610784	−0.305392	+	0.952227i	$0.598787\pi$
$194$	−1.65685	−0.118955
$195$	0	0
$196$	0	0
$197$	−16.8284	−1.19898	−0.599488	−	0.800384i	$-0.704629\pi$
$197$	−16.8284	−1.19898	−0.599488	+	0.800384i	$0.704629\pi$
$198$	0	0
$199$	10.4853	0.743282	0.371641	−	0.928377i	$-0.378795\pi$
$199$	10.4853	0.743282	0.371641	+	0.928377i	$0.378795\pi$
$200$	−2.82843	−0.200000
$201$	0	0
$202$	−17.7574	−1.24940
$203$	6.55635	0.460166
$204$	0	0
$205$	10.4142	0.727360
$206$	1.17157	0.0816274
$207$	0	0
$208$	−13.6569	−0.946932
$209$	−16.1421	−1.11657
$210$	0	0
$211$	0.656854	0.0452197	0.0226099	−	0.999744i	$-0.492802\pi$
$211$	0.656854	0.0452197	0.0226099	+	0.999744i	$0.492802\pi$
$212$	0	0
$213$	0	0
$214$	6.24264	0.426738
$215$	6.00000	0.409197
$216$	0	0
$217$	−7.62742	−0.517783
$218$	20.1421	1.36420
$219$	0	0
$220$	0	0
$221$	20.7279	1.39431
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	0	0
$226$	14.7279	0.979687
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	23.7990	1.57268	0.786341	−	0.617793i	$-0.211973\pi$
$229$	23.7990	1.57268	0.786341	+	0.617793i	$0.211973\pi$
$230$	1.41421	0.0932505
$231$	0	0
$232$	−10.1421	−0.665864
$233$	−9.65685	−0.632642	−0.316321	−	0.948652i	$-0.602448\pi$
$233$	−9.65685	−0.632642	−0.316321	+	0.948652i	$0.602448\pi$
$234$	0	0
$235$	8.58579	0.560075
$236$	0	0
$237$	0	0
$238$	15.6985	1.01758
$239$	1.58579	0.102576	0.0512880	−	0.998684i	$-0.483667\pi$
$239$	1.58579	0.102576	0.0512880	+	0.998684i	$0.483667\pi$
$240$	0	0
$241$	23.4142	1.50824	0.754121	−	0.656735i	$-0.228063\pi$
$241$	23.4142	1.50824	0.754121	+	0.656735i	$0.228063\pi$
$242$	−6.10051	−0.392155
$243$	0	0
$244$	0	0
$245$	−3.65685	−0.233628
$246$	0	0
$247$	−21.3137	−1.35616
$248$	11.7990	0.749237
$249$	0	0
$250$	1.41421	0.0894427
$251$	20.6274	1.30199	0.650996	−	0.759082i	$-0.274352\pi$
$251$	20.6274	1.30199	0.650996	+	0.759082i	$0.274352\pi$
$252$	0	0
$253$	2.58579	0.162567
$254$	−0.343146	−0.0215309
$255$	0	0
$256$	0	0
$257$	−30.0416	−1.87395	−0.936973	−	0.349403i	$-0.886385\pi$
$257$	−30.0416	−1.87395	−0.936973	+	0.349403i	$0.886385\pi$
$258$	0	0
$259$	5.48528	0.340839
$260$	0	0
$261$	0	0
$262$	19.3137	1.19320
$263$	−3.72792	−0.229874	−0.114937	−	0.993373i	$-0.536667\pi$
$263$	−3.72792	−0.229874	−0.114937	+	0.993373i	$0.536667\pi$
$264$	0	0
$265$	−7.24264	−0.444912
$266$	−16.1421	−0.989738
$267$	0	0
$268$	0	0
$269$	26.5563	1.61917	0.809585	−	0.587003i	$-0.199692\pi$
$269$	26.5563	1.61917	0.809585	+	0.587003i	$0.199692\pi$
$270$	0	0
$271$	−6.51472	−0.395741	−0.197870	−	0.980228i	$-0.563403\pi$
$271$	−6.51472	−0.395741	−0.197870	+	0.980228i	$0.563403\pi$
$272$	−24.2843	−1.47245
$273$	0	0
$274$	−4.00000	−0.241649
$275$	2.58579	0.155929
$276$	0	0
$277$	25.3137	1.52095	0.760477	−	0.649365i	$-0.224965\pi$
$277$	25.3137	1.52095	0.760477	+	0.649365i	$0.224965\pi$
$278$	−15.5563	−0.933008
$279$	0	0
$280$	−5.17157	−0.309061
$281$	−17.4142	−1.03884	−0.519422	−	0.854518i	$-0.673853\pi$
$281$	−17.4142	−1.03884	−0.519422	+	0.854518i	$0.673853\pi$
$282$	0	0
$283$	−28.6569	−1.70347	−0.851737	−	0.523970i	$-0.824450\pi$
$283$	−28.6569	−1.70347	−0.851737	+	0.523970i	$0.824450\pi$
$284$	0	0
$285$	0	0
$286$	12.4853	0.738270
$287$	19.0416	1.12399
$288$	0	0
$289$	19.8579	1.16811
$290$	5.07107	0.297783
$291$	0	0
$292$	0	0
$293$	3.10051	0.181133	0.0905667	−	0.995890i	$-0.471132\pi$
$293$	3.10051	0.181133	0.0905667	+	0.995890i	$0.471132\pi$
$294$	0	0
$295$	6.89949	0.401704
$296$	−8.48528	−0.493197
$297$	0	0
$298$	−12.8284	−0.743131
$299$	3.41421	0.197449
$300$	0	0
$301$	10.9706	0.632333
$302$	−13.1716	−0.757939
$303$	0	0
$304$	24.9706	1.43216
$305$	5.41421	0.310017
$306$	0	0
$307$	27.2132	1.55314	0.776570	−	0.630031i	$-0.216958\pi$
$307$	27.2132	1.55314	0.776570	+	0.630031i	$0.216958\pi$
$308$	0	0
$309$	0	0
$310$	−5.89949	−0.335069
$311$	−17.6569	−1.00123	−0.500614	−	0.865671i	$-0.666892\pi$
$311$	−17.6569	−1.00123	−0.500614	+	0.865671i	$0.666892\pi$
$312$	0	0
$313$	25.9706	1.46794	0.733971	−	0.679180i	$-0.237665\pi$
$313$	25.9706	1.46794	0.733971	+	0.679180i	$0.237665\pi$
$314$	−2.10051	−0.118538
$315$	0	0
$316$	0	0
$317$	29.5563	1.66005	0.830025	−	0.557726i	$-0.188326\pi$
$317$	29.5563	1.66005	0.830025	+	0.557726i	$0.188326\pi$
$318$	0	0
$319$	9.27208	0.519137
$320$	8.00000	0.447214
$321$	0	0
$322$	2.58579	0.144100
$323$	−37.8995	−2.10878
$324$	0	0
$325$	3.41421	0.189386
$326$	−2.82843	−0.156652
$327$	0	0
$328$	−29.4558	−1.62643
$329$	15.6985	0.865485
$330$	0	0
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	0	0
$333$	0	0
$334$	9.51472	0.520622
$335$	−11.4853	−0.627508
$336$	0	0
$337$	30.9706	1.68707	0.843537	−	0.537071i	$-0.180469\pi$
$337$	30.9706	1.68707	0.843537	+	0.537071i	$0.180469\pi$
$338$	−1.89949	−0.103319
$339$	0	0
$340$	0	0
$341$	−10.7868	−0.584138
$342$	0	0
$343$	−19.4853	−1.05211
$344$	−16.9706	−0.914991
$345$	0	0
$346$	−19.3137	−1.03831
$347$	−13.1716	−0.707087	−0.353544	−	0.935418i	$-0.615023\pi$
$347$	−13.1716	−0.707087	−0.353544	+	0.935418i	$0.615023\pi$
$348$	0	0
$349$	−32.1127	−1.71895	−0.859477	−	0.511175i	$-0.829210\pi$
$349$	−32.1127	−1.71895	−0.859477	+	0.511175i	$0.829210\pi$
$350$	2.58579	0.138216
$351$	0	0
$352$	0	0
$353$	34.5269	1.83768	0.918841	−	0.394628i	$-0.129126\pi$
$353$	34.5269	1.83768	0.918841	+	0.394628i	$0.129126\pi$
$354$	0	0
$355$	−8.89949	−0.472336
$356$	0	0
$357$	0	0
$358$	10.8284	0.572300
$359$	10.9289	0.576807	0.288403	−	0.957509i	$-0.406876\pi$
$359$	10.9289	0.576807	0.288403	+	0.957509i	$0.406876\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	−9.65685	−0.507553
$363$	0	0
$364$	0	0
$365$	−10.2426	−0.536124
$366$	0	0
$367$	−13.9706	−0.729257	−0.364629	−	0.931153i	$-0.618804\pi$
$367$	−13.9706	−0.729257	−0.364629	+	0.931153i	$0.618804\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	4.24264	0.220564
$371$	−13.2426	−0.687524
$372$	0	0
$373$	−12.9706	−0.671590	−0.335795	−	0.941935i	$-0.609005\pi$
$373$	−12.9706	−0.671590	−0.335795	+	0.941935i	$0.609005\pi$
$374$	22.2010	1.14799
$375$	0	0
$376$	−24.2843	−1.25237
$377$	12.2426	0.630528
$378$	0	0
$379$	28.9706	1.48812	0.744059	−	0.668114i	$-0.232898\pi$
$379$	28.9706	1.48812	0.744059	+	0.668114i	$0.232898\pi$
$380$	0	0
$381$	0	0
$382$	−14.4853	−0.741131
$383$	−14.4142	−0.736532	−0.368266	−	0.929720i	$-0.620048\pi$
$383$	−14.4142	−0.736532	−0.368266	+	0.929720i	$0.620048\pi$
$384$	0	0
$385$	4.72792	0.240957
$386$	−12.0000	−0.610784
$387$	0	0
$388$	0	0
$389$	−14.4853	−0.734433	−0.367216	−	0.930136i	$-0.619689\pi$
$389$	−14.4853	−0.734433	−0.367216	+	0.930136i	$0.619689\pi$
$390$	0	0
$391$	6.07107	0.307027
$392$	10.3431	0.522408
$393$	0	0
$394$	−23.7990	−1.19898
$395$	−15.3137	−0.770516
$396$	0	0
$397$	−15.3137	−0.768573	−0.384286	−	0.923214i	$-0.625553\pi$
$397$	−15.3137	−0.768573	−0.384286	+	0.923214i	$0.625553\pi$
$398$	14.8284	0.743282
$399$	0	0
$400$	−4.00000	−0.200000
$401$	32.1421	1.60510	0.802551	−	0.596584i	$-0.203476\pi$
$401$	32.1421	1.60510	0.802551	+	0.596584i	$0.203476\pi$
$402$	0	0
$403$	−14.2426	−0.709476
$404$	0	0
$405$	0	0
$406$	9.27208	0.460166
$407$	7.75736	0.384518
$408$	0	0
$409$	1.48528	0.0734424	0.0367212	−	0.999326i	$-0.488309\pi$
$409$	1.48528	0.0734424	0.0367212	+	0.999326i	$0.488309\pi$
$410$	14.7279	0.727360
$411$	0	0
$412$	0	0
$413$	12.6152	0.620755
$414$	0	0
$415$	−14.0711	−0.690722
$416$	0	0
$417$	0	0
$418$	−22.8284	−1.11657
$419$	24.0416	1.17451	0.587255	−	0.809402i	$-0.300208\pi$
$419$	24.0416	1.17451	0.587255	+	0.809402i	$0.300208\pi$
$420$	0	0
$421$	−1.75736	−0.0856485	−0.0428242	−	0.999083i	$-0.513636\pi$
$421$	−1.75736	−0.0856485	−0.0428242	+	0.999083i	$0.513636\pi$
$422$	0.928932	0.0452197
$423$	0	0
$424$	20.4853	0.994853
$425$	6.07107	0.294490
$426$	0	0
$427$	9.89949	0.479070
$428$	0	0
$429$	0	0
$430$	8.48528	0.409197
$431$	27.7990	1.33903	0.669515	−	0.742798i	$-0.266502\pi$
$431$	27.7990	1.33903	0.669515	+	0.742798i	$0.266502\pi$
$432$	0	0
$433$	−1.82843	−0.0878686	−0.0439343	−	0.999034i	$-0.513989\pi$
$433$	−1.82843	−0.0878686	−0.0439343	+	0.999034i	$0.513989\pi$
$434$	−10.7868	−0.517783
$435$	0	0
$436$	0	0
$437$	−6.24264	−0.298626
$438$	0	0
$439$	−18.4853	−0.882254	−0.441127	−	0.897445i	$-0.645421\pi$
$439$	−18.4853	−0.882254	−0.441127	+	0.897445i	$0.645421\pi$
$440$	−7.31371	−0.348667
$441$	0	0
$442$	29.3137	1.39431
$443$	−30.5269	−1.45038	−0.725189	−	0.688550i	$-0.758247\pi$
$443$	−30.5269	−1.45038	−0.725189	+	0.688550i	$0.758247\pi$
$444$	0	0
$445$	−10.1421	−0.480783
$446$	−19.7990	−0.937509
$447$	0	0
$448$	14.6274	0.691080
$449$	13.2426	0.624959	0.312479	−	0.949925i	$-0.398840\pi$
$449$	13.2426	0.624959	0.312479	+	0.949925i	$0.398840\pi$
$450$	0	0
$451$	26.9289	1.26803
$452$	0	0
$453$	0	0
$454$	−24.9706	−1.17193
$455$	6.24264	0.292660
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	33.6569	1.57268
$459$	0	0
$460$	0	0
$461$	9.51472	0.443145	0.221572	−	0.975144i	$-0.428881\pi$
$461$	9.51472	0.443145	0.221572	+	0.975144i	$0.428881\pi$
$462$	0	0
$463$	17.0711	0.793360	0.396680	−	0.917957i	$-0.370162\pi$
$463$	17.0711	0.793360	0.396680	+	0.917957i	$0.370162\pi$
$464$	−14.3431	−0.665864
$465$	0	0
$466$	−13.6569	−0.632642
$467$	−3.58579	−0.165930	−0.0829652	−	0.996552i	$-0.526439\pi$
$467$	−3.58579	−0.165930	−0.0829652	+	0.996552i	$0.526439\pi$
$468$	0	0
$469$	−21.0000	−0.969690
$470$	12.1421	0.560075
$471$	0	0
$472$	−19.5147	−0.898238
$473$	15.5147	0.713368
$474$	0	0
$475$	−6.24264	−0.286432
$476$	0	0
$477$	0	0
$478$	2.24264	0.102576
$479$	18.7279	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$479$	18.7279	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$480$	0	0
$481$	10.2426	0.467024
$482$	33.1127	1.50824
$483$	0	0
$484$	0	0
$485$	−1.17157	−0.0531984
$486$	0	0
$487$	10.3848	0.470579	0.235290	−	0.971925i	$-0.424396\pi$
$487$	10.3848	0.470579	0.235290	+	0.971925i	$0.424396\pi$
$488$	−15.3137	−0.693219
$489$	0	0
$490$	−5.17157	−0.233628
$491$	−2.07107	−0.0934660	−0.0467330	−	0.998907i	$-0.514881\pi$
$491$	−2.07107	−0.0934660	−0.0467330	+	0.998907i	$0.514881\pi$
$492$	0	0
$493$	21.7696	0.980451
$494$	−30.1421	−1.35616
$495$	0	0
$496$	16.6863	0.749237
$497$	−16.2721	−0.729902
$498$	0	0
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	0	0
$502$	29.1716	1.30199
$503$	14.2721	0.636361	0.318180	−	0.948030i	$-0.396928\pi$
$503$	14.2721	0.636361	0.318180	+	0.948030i	$0.396928\pi$
$504$	0	0
$505$	−12.5563	−0.558750
$506$	3.65685	0.162567
$507$	0	0
$508$	0	0
$509$	31.7990	1.40947	0.704733	−	0.709473i	$-0.251067\pi$
$509$	31.7990	1.40947	0.704733	+	0.709473i	$0.251067\pi$
$510$	0	0
$511$	−18.7279	−0.828474
$512$	−22.6274	−1.00000
$513$	0	0
$514$	−42.4853	−1.87395
$515$	0.828427	0.0365049
$516$	0	0
$517$	22.2010	0.976399
$518$	7.75736	0.340839
$519$	0	0
$520$	−9.65685	−0.423481
$521$	0.443651	0.0194367	0.00971835	−	0.999953i	$-0.496907\pi$
$521$	0.443651	0.0194367	0.00971835	+	0.999953i	$0.496907\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	0	0
$525$	0	0
$526$	−5.27208	−0.229874
$527$	−25.3259	−1.10321
$528$	0	0
$529$	1.00000	0.0434783
$530$	−10.2426	−0.444912
$531$	0	0
$532$	0	0
$533$	35.5563	1.54012
$534$	0	0
$535$	4.41421	0.190843
$536$	32.4853	1.40315
$537$	0	0
$538$	37.5563	1.61917
$539$	−9.45584	−0.407292
$540$	0	0
$541$	34.1421	1.46789	0.733943	−	0.679212i	$-0.237678\pi$
$541$	34.1421	1.46789	0.733943	+	0.679212i	$0.237678\pi$
$542$	−9.21320	−0.395741
$543$	0	0
$544$	0	0
$545$	14.2426	0.610088
$546$	0	0
$547$	−20.9706	−0.896637	−0.448318	−	0.893874i	$-0.647977\pi$
$547$	−20.9706	−0.896637	−0.448318	+	0.893874i	$0.647977\pi$
$548$	0	0
$549$	0	0
$550$	3.65685	0.155929
$551$	−22.3848	−0.953624
$552$	0	0
$553$	−28.0000	−1.19068
$554$	35.7990	1.52095
$555$	0	0
$556$	0	0
$557$	−12.8995	−0.546569	−0.273285	−	0.961933i	$-0.588110\pi$
$557$	−12.8995	−0.546569	−0.273285	+	0.961933i	$0.588110\pi$
$558$	0	0
$559$	20.4853	0.866435
$560$	−7.31371	−0.309061
$561$	0	0
$562$	−24.6274	−1.03884
$563$	−1.10051	−0.0463808	−0.0231904	−	0.999731i	$-0.507382\pi$
$563$	−1.10051	−0.0463808	−0.0231904	+	0.999731i	$0.507382\pi$
$564$	0	0
$565$	10.4142	0.438129
$566$	−40.5269	−1.70347
$567$	0	0
$568$	25.1716	1.05618
$569$	16.1421	0.676714	0.338357	−	0.941018i	$-0.390129\pi$
$569$	16.1421	0.676714	0.338357	+	0.941018i	$0.390129\pi$
$570$	0	0
$571$	3.27208	0.136932	0.0684661	−	0.997653i	$-0.478190\pi$
$571$	3.27208	0.136932	0.0684661	+	0.997653i	$0.478190\pi$
$572$	0	0
$573$	0	0
$574$	26.9289	1.12399
$575$	1.00000	0.0417029
$576$	0	0
$577$	−36.9706	−1.53910	−0.769552	−	0.638584i	$-0.779521\pi$
$577$	−36.9706	−1.53910	−0.769552	+	0.638584i	$0.779521\pi$
$578$	28.0833	1.16811
$579$	0	0
$580$	0	0
$581$	−25.7279	−1.06737
$582$	0	0
$583$	−18.7279	−0.775631
$584$	28.9706	1.19881
$585$	0	0
$586$	4.38478	0.181133
$587$	−40.6274	−1.67687	−0.838436	−	0.544999i	$-0.816530\pi$
$587$	−40.6274	−1.67687	−0.838436	+	0.544999i	$0.816530\pi$
$588$	0	0
$589$	26.0416	1.07303
$590$	9.75736	0.401704
$591$	0	0
$592$	−12.0000	−0.493197
$593$	2.44365	0.100349	0.0501744	−	0.998740i	$-0.484022\pi$
$593$	2.44365	0.100349	0.0501744	+	0.998740i	$0.484022\pi$
$594$	0	0
$595$	11.1005	0.455076
$596$	0	0
$597$	0	0
$598$	4.82843	0.197449
$599$	−33.1716	−1.35535	−0.677677	−	0.735360i	$-0.737013\pi$
$599$	−33.1716	−1.35535	−0.677677	+	0.735360i	$0.737013\pi$
$600$	0	0
$601$	−18.3137	−0.747032	−0.373516	−	0.927624i	$-0.621848\pi$
$601$	−18.3137	−0.747032	−0.373516	+	0.927624i	$0.621848\pi$
$602$	15.5147	0.632333
$603$	0	0
$604$	0	0
$605$	−4.31371	−0.175377
$606$	0	0
$607$	−38.3848	−1.55799	−0.778995	−	0.627030i	$-0.784270\pi$
$607$	−38.3848	−1.55799	−0.778995	+	0.627030i	$0.784270\pi$
$608$	0	0
$609$	0	0
$610$	7.65685	0.310017
$611$	29.3137	1.18591
$612$	0	0
$613$	−32.6274	−1.31781	−0.658904	−	0.752227i	$-0.728980\pi$
$613$	−32.6274	−1.31781	−0.658904	+	0.752227i	$0.728980\pi$
$614$	38.4853	1.55314
$615$	0	0
$616$	−13.3726	−0.538797
$617$	−4.89949	−0.197246	−0.0986231	−	0.995125i	$-0.531444\pi$
$617$	−4.89949	−0.197246	−0.0986231	+	0.995125i	$0.531444\pi$
$618$	0	0
$619$	−48.2843	−1.94071	−0.970354	−	0.241687i	$-0.922299\pi$
$619$	−48.2843	−1.94071	−0.970354	+	0.241687i	$0.922299\pi$
$620$	0	0
$621$	0	0
$622$	−24.9706	−1.00123
$623$	−18.5442	−0.742956
$624$	0	0
$625$	1.00000	0.0400000
$626$	36.7279	1.46794
$627$	0	0
$628$	0	0
$629$	18.2132	0.726208
$630$	0	0
$631$	33.2132	1.32220	0.661098	−	0.750299i	$-0.270091\pi$
$631$	33.2132	1.32220	0.661098	+	0.750299i	$0.270091\pi$
$632$	43.3137	1.72293
$633$	0	0
$634$	41.7990	1.66005
$635$	−0.242641	−0.00962890
$636$	0	0
$637$	−12.4853	−0.494685
$638$	13.1127	0.519137
$639$	0	0
$640$	11.3137	0.447214
$641$	−4.92893	−0.194681	−0.0973406	−	0.995251i	$-0.531034\pi$
$641$	−4.92893	−0.194681	−0.0973406	+	0.995251i	$0.531034\pi$
$642$	0	0
$643$	33.0000	1.30139	0.650696	−	0.759338i	$-0.274477\pi$
$643$	33.0000	1.30139	0.650696	+	0.759338i	$0.274477\pi$
$644$	0	0
$645$	0	0
$646$	−53.5980	−2.10878
$647$	−21.2132	−0.833977	−0.416989	−	0.908912i	$-0.636914\pi$
$647$	−21.2132	−0.833977	−0.416989	+	0.908912i	$0.636914\pi$
$648$	0	0
$649$	17.8406	0.700306
$650$	4.82843	0.189386
$651$	0	0
$652$	0	0
$653$	2.58579	0.101190	0.0505948	−	0.998719i	$-0.483888\pi$
$653$	2.58579	0.101190	0.0505948	+	0.998719i	$0.483888\pi$
$654$	0	0
$655$	13.6569	0.533617
$656$	−41.6569	−1.62643
$657$	0	0
$658$	22.2010	0.865485
$659$	−2.38478	−0.0928977	−0.0464488	−	0.998921i	$-0.514790\pi$
$659$	−2.38478	−0.0928977	−0.0464488	+	0.998921i	$0.514790\pi$
$660$	0	0
$661$	11.4558	0.445581	0.222790	−	0.974866i	$-0.428483\pi$
$661$	11.4558	0.445581	0.222790	+	0.974866i	$0.428483\pi$
$662$	1.41421	0.0549650
$663$	0	0
$664$	39.7990	1.54450
$665$	−11.4142	−0.442624
$666$	0	0
$667$	3.58579	0.138842
$668$	0	0
$669$	0	0
$670$	−16.2426	−0.627508
$671$	14.0000	0.540464
$672$	0	0
$673$	41.6985	1.60736	0.803679	−	0.595063i	$-0.202873\pi$
$673$	41.6985	1.60736	0.803679	+	0.595063i	$0.202873\pi$
$674$	43.7990	1.68707
$675$	0	0
$676$	0	0
$677$	19.2426	0.739555	0.369777	−	0.929120i	$-0.379434\pi$
$677$	19.2426	0.739555	0.369777	+	0.929120i	$0.379434\pi$
$678$	0	0
$679$	−2.14214	−0.0822076
$680$	−17.1716	−0.658500
$681$	0	0
$682$	−15.2548	−0.584138
$683$	32.1838	1.23148	0.615739	−	0.787950i	$-0.288858\pi$
$683$	32.1838	1.23148	0.615739	+	0.787950i	$0.288858\pi$
$684$	0	0
$685$	−2.82843	−0.108069
$686$	−27.5563	−1.05211
$687$	0	0
$688$	−24.0000	−0.914991
$689$	−24.7279	−0.942059
$690$	0	0
$691$	−15.6569	−0.595615	−0.297807	−	0.954626i	$-0.596255\pi$
$691$	−15.6569	−0.595615	−0.297807	+	0.954626i	$0.596255\pi$
$692$	0	0
$693$	0	0
$694$	−18.6274	−0.707087
$695$	−11.0000	−0.417254
$696$	0	0
$697$	63.2254	2.39483
$698$	−45.4142	−1.71895
$699$	0	0
$700$	0	0
$701$	36.3848	1.37423	0.687117	−	0.726547i	$-0.258876\pi$
$701$	36.3848	1.37423	0.687117	+	0.726547i	$0.258876\pi$
$702$	0	0
$703$	−18.7279	−0.706337
$704$	20.6863	0.779644
$705$	0	0
$706$	48.8284	1.83768
$707$	−22.9584	−0.863438
$708$	0	0
$709$	16.7279	0.628230	0.314115	−	0.949385i	$-0.398292\pi$
$709$	16.7279	0.628230	0.314115	+	0.949385i	$0.398292\pi$
$710$	−12.5858	−0.472336
$711$	0	0
$712$	28.6863	1.07506
$713$	−4.17157	−0.156227
$714$	0	0
$715$	8.82843	0.330164
$716$	0	0
$717$	0	0
$718$	15.4558	0.576807
$719$	−36.2132	−1.35052	−0.675262	−	0.737578i	$-0.735970\pi$
$719$	−36.2132	−1.35052	−0.675262	+	0.737578i	$0.735970\pi$
$720$	0	0
$721$	1.51472	0.0564111
$722$	28.2426	1.05108
$723$	0	0
$724$	0	0
$725$	3.58579	0.133173
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	−17.6569	−0.654407
$729$	0	0
$730$	−14.4853	−0.536124
$731$	36.4264	1.34728
$732$	0	0
$733$	−17.4853	−0.645834	−0.322917	−	0.946427i	$-0.604663\pi$
$733$	−17.4853	−0.645834	−0.322917	+	0.946427i	$0.604663\pi$
$734$	−19.7574	−0.729257
$735$	0	0
$736$	0	0
$737$	−29.6985	−1.09396
$738$	0	0
$739$	−53.6274	−1.97272	−0.986358	−	0.164613i	$-0.947362\pi$
$739$	−53.6274	−1.97272	−0.986358	+	0.164613i	$0.947362\pi$
$740$	0	0
$741$	0	0
$742$	−18.7279	−0.687524
$743$	25.6569	0.941259	0.470629	−	0.882331i	$-0.344027\pi$
$743$	25.6569	0.941259	0.470629	+	0.882331i	$0.344027\pi$
$744$	0	0
$745$	−9.07107	−0.332338
$746$	−18.3431	−0.671590
$747$	0	0
$748$	0	0
$749$	8.07107	0.294910
$750$	0	0
$751$	−9.27208	−0.338343	−0.169171	−	0.985587i	$-0.554109\pi$
$751$	−9.27208	−0.338343	−0.169171	+	0.985587i	$0.554109\pi$
$752$	−34.3431	−1.25237
$753$	0	0
$754$	17.3137	0.630528
$755$	−9.31371	−0.338961
$756$	0	0
$757$	27.0000	0.981332	0.490666	−	0.871348i	$-0.336754\pi$
$757$	27.0000	0.981332	0.490666	+	0.871348i	$0.336754\pi$
$758$	40.9706	1.48812
$759$	0	0
$760$	17.6569	0.640481
$761$	19.9289	0.722423	0.361212	−	0.932484i	$-0.382363\pi$
$761$	19.9289	0.722423	0.361212	+	0.932484i	$0.382363\pi$
$762$	0	0
$763$	26.0416	0.942770
$764$	0	0
$765$	0	0
$766$	−20.3848	−0.736532
$767$	23.5563	0.850570
$768$	0	0
$769$	20.0416	0.722720	0.361360	−	0.932426i	$-0.382313\pi$
$769$	20.0416	0.722720	0.361360	+	0.932426i	$0.382313\pi$
$770$	6.68629	0.240957
$771$	0	0
$772$	0	0
$773$	3.17157	0.114074	0.0570368	−	0.998372i	$-0.481835\pi$
$773$	3.17157	0.114074	0.0570368	+	0.998372i	$0.481835\pi$
$774$	0	0
$775$	−4.17157	−0.149847
$776$	3.31371	0.118955
$777$	0	0
$778$	−20.4853	−0.734433
$779$	−65.0122	−2.32930
$780$	0	0
$781$	−23.0122	−0.823441
$782$	8.58579	0.307027
$783$	0	0
$784$	14.6274	0.522408
$785$	−1.48528	−0.0530120
$786$	0	0
$787$	3.62742	0.129303	0.0646517	−	0.997908i	$-0.479406\pi$
$787$	3.62742	0.129303	0.0646517	+	0.997908i	$0.479406\pi$
$788$	0	0
$789$	0	0
$790$	−21.6569	−0.770516
$791$	19.0416	0.677042
$792$	0	0
$793$	18.4853	0.656432
$794$	−21.6569	−0.768573
$795$	0	0
$796$	0	0
$797$	−16.7574	−0.593576	−0.296788	−	0.954943i	$-0.595915\pi$
$797$	−16.7574	−0.593576	−0.296788	+	0.954943i	$0.595915\pi$
$798$	0	0
$799$	52.1249	1.84405
$800$	0	0
$801$	0	0
$802$	45.4558	1.60510
$803$	−26.4853	−0.934645
$804$	0	0
$805$	1.82843	0.0644436
$806$	−20.1421	−0.709476
$807$	0	0
$808$	35.5147	1.24940
$809$	5.87006	0.206380	0.103190	−	0.994662i	$-0.467095\pi$
$809$	5.87006	0.206380	0.103190	+	0.994662i	$0.467095\pi$
$810$	0	0
$811$	39.2843	1.37946	0.689729	−	0.724068i	$-0.257730\pi$
$811$	39.2843	1.37946	0.689729	+	0.724068i	$0.257730\pi$
$812$	0	0
$813$	0	0
$814$	10.9706	0.384518
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	−37.4558	−1.31041
$818$	2.10051	0.0734424
$819$	0	0
$820$	0	0
$821$	43.7990	1.52860	0.764298	−	0.644864i	$-0.223086\pi$
$821$	43.7990	1.52860	0.764298	+	0.644864i	$0.223086\pi$
$822$	0	0
$823$	45.6569	1.59150	0.795749	−	0.605627i	$-0.207077\pi$
$823$	45.6569	1.59150	0.795749	+	0.605627i	$0.207077\pi$
$824$	−2.34315	−0.0816274
$825$	0	0
$826$	17.8406	0.620755
$827$	50.0122	1.73909	0.869547	−	0.493850i	$-0.164411\pi$
$827$	50.0122	1.73909	0.869547	+	0.493850i	$0.164411\pi$
$828$	0	0
$829$	23.4853	0.815678	0.407839	−	0.913054i	$-0.366283\pi$
$829$	23.4853	0.815678	0.407839	+	0.913054i	$0.366283\pi$
$830$	−19.8995	−0.690722
$831$	0	0
$832$	27.3137	0.946932
$833$	−22.2010	−0.769219
$834$	0	0
$835$	6.72792	0.232829
$836$	0	0
$837$	0	0
$838$	34.0000	1.17451
$839$	4.62742	0.159756	0.0798781	−	0.996805i	$-0.474547\pi$
$839$	4.62742	0.159756	0.0798781	+	0.996805i	$0.474547\pi$
$840$	0	0
$841$	−16.1421	−0.556625
$842$	−2.48528	−0.0856485
$843$	0	0
$844$	0	0
$845$	−1.34315	−0.0462056
$846$	0	0
$847$	−7.88730	−0.271011
$848$	28.9706	0.994853
$849$	0	0
$850$	8.58579	0.294490
$851$	3.00000	0.102839
$852$	0	0
$853$	−31.4558	−1.07703	−0.538514	−	0.842617i	$-0.681014\pi$
$853$	−31.4558	−1.07703	−0.538514	+	0.842617i	$0.681014\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	−12.4853	−0.426738
$857$	−21.5980	−0.737773	−0.368886	−	0.929474i	$-0.620261\pi$
$857$	−21.5980	−0.737773	−0.368886	+	0.929474i	$0.620261\pi$
$858$	0	0
$859$	−57.0000	−1.94481	−0.972407	−	0.233289i	$-0.925051\pi$
$859$	−57.0000	−1.94481	−0.972407	+	0.233289i	$0.925051\pi$
$860$	0	0
$861$	0	0
$862$	39.3137	1.33903
$863$	−11.1716	−0.380285	−0.190142	−	0.981757i	$-0.560895\pi$
$863$	−11.1716	−0.380285	−0.190142	+	0.981757i	$0.560895\pi$
$864$	0	0
$865$	−13.6569	−0.464347
$866$	−2.58579	−0.0878686
$867$	0	0
$868$	0	0
$869$	−39.5980	−1.34327
$870$	0	0
$871$	−39.2132	−1.32869
$872$	−40.2843	−1.36420
$873$	0	0
$874$	−8.82843	−0.298626
$875$	1.82843	0.0618121
$876$	0	0
$877$	−1.51472	−0.0511484	−0.0255742	−	0.999673i	$-0.508141\pi$
$877$	−1.51472	−0.0511484	−0.0255742	+	0.999673i	$0.508141\pi$
$878$	−26.1421	−0.882254
$879$	0	0
$880$	−10.3431	−0.348667
$881$	−19.4142	−0.654081	−0.327041	−	0.945010i	$-0.606051\pi$
$881$	−19.4142	−0.654081	−0.327041	+	0.945010i	$0.606051\pi$
$882$	0	0
$883$	−1.89949	−0.0639231	−0.0319615	−	0.999489i	$-0.510175\pi$
$883$	−1.89949	−0.0639231	−0.0319615	+	0.999489i	$0.510175\pi$
$884$	0	0
$885$	0	0
$886$	−43.1716	−1.45038
$887$	−21.7990	−0.731938	−0.365969	−	0.930627i	$-0.619262\pi$
$887$	−21.7990	−0.731938	−0.365969	+	0.930627i	$0.619262\pi$
$888$	0	0
$889$	−0.443651	−0.0148796
$890$	−14.3431	−0.480783
$891$	0	0
$892$	0	0
$893$	−53.5980	−1.79359
$894$	0	0
$895$	7.65685	0.255940
$896$	20.6863	0.691080
$897$	0	0
$898$	18.7279	0.624959
$899$	−14.9584	−0.498890
$900$	0	0
$901$	−43.9706	−1.46487
$902$	38.0833	1.26803
$903$	0	0
$904$	−29.4558	−0.979687
$905$	−6.82843	−0.226985
$906$	0	0
$907$	31.1421	1.03406	0.517029	−	0.855968i	$-0.327038\pi$
$907$	31.1421	1.03406	0.517029	+	0.855968i	$0.327038\pi$
$908$	0	0
$909$	0	0
$910$	8.82843	0.292660
$911$	5.65685	0.187420	0.0937100	−	0.995600i	$-0.470127\pi$
$911$	5.65685	0.187420	0.0937100	+	0.995600i	$0.470127\pi$
$912$	0	0
$913$	−36.3848	−1.20416
$914$	−43.8406	−1.45012
$915$	0	0
$916$	0	0
$917$	24.9706	0.824601
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−2.82843	−0.0932505
$921$	0	0
$922$	13.4558	0.443145
$923$	−30.3848	−1.00013
$924$	0	0
$925$	3.00000	0.0986394
$926$	24.1421	0.793360
$927$	0	0
$928$	0	0
$929$	−41.8701	−1.37371	−0.686856	−	0.726794i	$-0.741010\pi$
$929$	−41.8701	−1.37371	−0.686856	+	0.726794i	$0.741010\pi$
$930$	0	0
$931$	22.8284	0.748171
$932$	0	0
$933$	0	0
$934$	−5.07107	−0.165930
$935$	15.6985	0.513395
$936$	0	0
$937$	−59.7990	−1.95355	−0.976774	−	0.214272i	$-0.931262\pi$
$937$	−59.7990	−1.95355	−0.976774	+	0.214272i	$0.931262\pi$
$938$	−29.6985	−0.969690
$939$	0	0
$940$	0	0
$941$	7.41421	0.241696	0.120848	−	0.992671i	$-0.461439\pi$
$941$	7.41421	0.241696	0.120848	+	0.992671i	$0.461439\pi$
$942$	0	0
$943$	10.4142	0.339133
$944$	−27.5980	−0.898238
$945$	0	0
$946$	21.9411	0.713368
$947$	3.79899	0.123451	0.0617253	−	0.998093i	$-0.480340\pi$
$947$	3.79899	0.123451	0.0617253	+	0.998093i	$0.480340\pi$
$948$	0	0
$949$	−34.9706	−1.13519
$950$	−8.82843	−0.286432
$951$	0	0
$952$	−31.3970	−1.01758
$953$	−7.37258	−0.238821	−0.119411	−	0.992845i	$-0.538101\pi$
$953$	−7.37258	−0.238821	−0.119411	+	0.992845i	$0.538101\pi$
$954$	0	0
$955$	−10.2426	−0.331444
$956$	0	0
$957$	0	0
$958$	26.4853	0.855701
$959$	−5.17157	−0.166999
$960$	0	0
$961$	−13.5980	−0.438645
$962$	14.4853	0.467024
$963$	0	0
$964$	0	0
$965$	−8.48528	−0.273151
$966$	0	0
$967$	0.443651	0.0142668	0.00713342	−	0.999975i	$-0.497729\pi$
$967$	0.443651	0.0142668	0.00713342	+	0.999975i	$0.497729\pi$
$968$	12.2010	0.392155
$969$	0	0
$970$	−1.65685	−0.0531984
$971$	−30.4264	−0.976430	−0.488215	−	0.872723i	$-0.662352\pi$
$971$	−30.4264	−0.976430	−0.488215	+	0.872723i	$0.662352\pi$
$972$	0	0
$973$	−20.1127	−0.644784
$974$	14.6863	0.470579
$975$	0	0
$976$	−21.6569	−0.693219
$977$	15.5858	0.498633	0.249317	−	0.968422i	$-0.419794\pi$
$977$	15.5858	0.498633	0.249317	+	0.968422i	$0.419794\pi$
$978$	0	0
$979$	−26.2254	−0.838167
$980$	0	0
$981$	0	0
$982$	−2.92893	−0.0934660
$983$	−24.0122	−0.765870	−0.382935	−	0.923775i	$-0.625087\pi$
$983$	−24.0122	−0.765870	−0.382935	+	0.923775i	$0.625087\pi$
$984$	0	0
$985$	−16.8284	−0.536198
$986$	30.7868	0.980451
$987$	0	0
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	1.34315	0.0426664	0.0213332	−	0.999772i	$-0.493209\pi$
$991$	1.34315	0.0426664	0.0213332	+	0.999772i	$0.493209\pi$
$992$	0	0
$993$	0	0
$994$	−23.0122	−0.729902
$995$	10.4853	0.332406
$996$	0	0
$997$	−34.1421	−1.08129	−0.540646	−	0.841250i	$-0.681820\pi$
$997$	−34.1421	−1.08129	−0.540646	+	0.841250i	$0.681820\pi$
$998$	−18.3848	−0.581960
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.2.a.j.1.2		2
3.2	odd	2		345.2.a.h.1.1	✓	2
5.4	even	2		5175.2.a.bj.1.1		2
12.11	even	2		5520.2.a.bm.1.1		2
15.2	even	4		1725.2.b.s.1174.2		4
15.8	even	4		1725.2.b.s.1174.3		4
15.14	odd	2		1725.2.a.z.1.2		2
69.68	even	2		7935.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.h.1.1	✓	2	3.2	odd	2
1035.2.a.j.1.2		2	1.1	even	1	trivial
1725.2.a.z.1.2		2	15.14	odd	2
1725.2.b.s.1174.2		4	15.2	even	4
1725.2.b.s.1174.3		4	15.8	even	4
5175.2.a.bj.1.1		2	5.4	even	2
5520.2.a.bm.1.1		2	12.11	even	2
7935.2.a.q.1.1		2	69.68	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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1035.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.00000 q^{5} +1.82843 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +1.00000 q^{5} +1.82843 q^{7} -2.82843 q^{8} +1.41421 q^{10} +2.58579 q^{11} +3.41421 q^{13} +2.58579 q^{14} -4.00000 q^{16} +6.07107 q^{17} -6.24264 q^{19} +3.65685 q^{22} +1.00000 q^{23} +1.00000 q^{25} +4.82843 q^{26} +3.58579 q^{29} -4.17157 q^{31} +8.58579 q^{34} +1.82843 q^{35} +3.00000 q^{37} -8.82843 q^{38} -2.82843 q^{40} +10.4142 q^{41} +6.00000 q^{43} +1.41421 q^{46} +8.58579 q^{47} -3.65685 q^{49} +1.41421 q^{50} -7.24264 q^{53} +2.58579 q^{55} -5.17157 q^{56} +5.07107 q^{58} +6.89949 q^{59} +5.41421 q^{61} -5.89949 q^{62} +8.00000 q^{64} +3.41421 q^{65} -11.4853 q^{67} +2.58579 q^{70} -8.89949 q^{71} -10.2426 q^{73} +4.24264 q^{74} +4.72792 q^{77} -15.3137 q^{79} -4.00000 q^{80} +14.7279 q^{82} -14.0711 q^{83} +6.07107 q^{85} +8.48528 q^{86} -7.31371 q^{88} -10.1421 q^{89} +6.24264 q^{91} +12.1421 q^{94} -6.24264 q^{95} -1.17157 q^{97} -5.17157 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{5} - 2 q^{7} + 8 q^{11} + 4 q^{13} + 8 q^{14} - 8 q^{16} - 2 q^{17} - 4 q^{19} - 4 q^{22} + 2 q^{23} + 2 q^{25} + 4 q^{26} + 10 q^{29} - 14 q^{31} + 20 q^{34} - 2 q^{35} + 6 q^{37} - 12 q^{38} + 18 q^{41} + 12 q^{43} + 20 q^{47} + 4 q^{49} - 6 q^{53} + 8 q^{55} - 16 q^{56} - 4 q^{58} - 6 q^{59} + 8 q^{61} + 8 q^{62} + 16 q^{64} + 4 q^{65} - 6 q^{67} + 8 q^{70} + 2 q^{71} - 12 q^{73} - 16 q^{77} - 8 q^{79} - 8 q^{80} + 4 q^{82} - 14 q^{83} - 2 q^{85} + 8 q^{88} + 8 q^{89} + 4 q^{91} - 4 q^{94} - 4 q^{95} - 8 q^{97} - 16 q^{98}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 + 8 * q^11 + 4 * q^13 + 8 * q^14 - 8 * q^16 - 2 * q^17 - 4 * q^19 - 4 * q^22 + 2 * q^23 + 2 * q^25 + 4 * q^26 + 10 * q^29 - 14 * q^31 + 20 * q^34 - 2 * q^35 + 6 * q^37 - 12 * q^38 + 18 * q^41 + 12 * q^43 + 20 * q^47 + 4 * q^49 - 6 * q^53 + 8 * q^55 - 16 * q^56 - 4 * q^58 - 6 * q^59 + 8 * q^61 + 8 * q^62 + 16 * q^64 + 4 * q^65 - 6 * q^67 + 8 * q^70 + 2 * q^71 - 12 * q^73 - 16 * q^77 - 8 * q^79 - 8 * q^80 + 4 * q^82 - 14 * q^83 - 2 * q^85 + 8 * q^88 + 8 * q^89 + 4 * q^91 - 4 * q^94 - 4 * q^95 - 8 * q^97 - 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more