LMFDB - Embedded newform 6640.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6640,2,Mod(1,6640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6640, 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("6640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6640 = 2^{4} \cdot 5 \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6640.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:	$53.0206669421$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.887108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 6x - 9 $ x^5 - 2x^4 - 7x^3 + 13x^2 + 6x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 830)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.90136$ of defining polynomial
Character	$\chi$	$=$	6640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90136 q^{3} +1.00000 q^{5} -0.676454 q^{7} +0.615178 q^{9} +O(q^{10})$	$q-1.90136 q^{3} +1.00000 q^{5} -0.676454 q^{7} +0.615178 q^{9} -3.66496 q^{11} -4.18150 q^{13} -1.90136 q^{15} +1.93328 q^{17} +3.24278 q^{19} +1.28618 q^{21} -2.41614 q^{23} +1.00000 q^{25} +4.53441 q^{27} -0.713816 q^{29} +6.46164 q^{31} +6.96842 q^{33} -0.676454 q^{35} +11.1001 q^{37} +7.95055 q^{39} -1.26227 q^{41} +11.5560 q^{43} +0.615178 q^{45} -5.69628 q^{47} -6.54241 q^{49} -3.67586 q^{51} -6.76477 q^{53} -3.66496 q^{55} -6.16570 q^{57} -12.3803 q^{59} +10.8459 q^{61} -0.416140 q^{63} -4.18150 q^{65} +13.1206 q^{67} +4.59396 q^{69} +13.6665 q^{71} +2.40210 q^{73} -1.90136 q^{75} +2.47918 q^{77} -4.25254 q^{79} -10.4671 q^{81} -1.00000 q^{83} +1.93328 q^{85} +1.35722 q^{87} -6.64709 q^{89} +2.82859 q^{91} -12.2859 q^{93} +3.24278 q^{95} +0.225504 q^{97} -2.25460 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 3 * q^9	$ 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 3 q^{9} - q^{11} + 4 q^{13} - 2 q^{15} - q^{17} - 7 q^{19} - q^{21} - 8 q^{23} + 5 q^{25} + q^{27} - 11 q^{29} - 10 q^{31} - 5 q^{33} - 5 q^{35} + 8 q^{37} + 14 q^{39} - 3 q^{41} + 2 q^{43} + 3 q^{45} - 7 q^{47} - 12 q^{49} - 19 q^{51} - 12 q^{53} - q^{55} - 6 q^{57} + 3 q^{59} + 11 q^{61} + 2 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 4 q^{71} - 3 q^{73} - 2 q^{75} - q^{77} + 12 q^{79} - 19 q^{81} - 5 q^{83} - q^{85} - 9 q^{87} - 30 q^{89} - 14 q^{91} - 10 q^{93} - 7 q^{95} - 7 q^{97} + 36 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 3 * q^9 - q^11 + 4 * q^13 - 2 * q^15 - q^17 - 7 * q^19 - q^21 - 8 * q^23 + 5 * q^25 + q^27 - 11 * q^29 - 10 * q^31 - 5 * q^33 - 5 * q^35 + 8 * q^37 + 14 * q^39 - 3 * q^41 + 2 * q^43 + 3 * q^45 - 7 * q^47 - 12 * q^49 - 19 * q^51 - 12 * q^53 - q^55 - 6 * q^57 + 3 * q^59 + 11 * q^61 + 2 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 4 * q^71 - 3 * q^73 - 2 * q^75 - q^77 + 12 * q^79 - 19 * q^81 - 5 * q^83 - q^85 - 9 * q^87 - 30 * q^89 - 14 * q^91 - 10 * q^93 - 7 * q^95 - 7 * q^97 + 36 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.90136	−1.09775	−0.548876	−	0.835904i	$-0.684944\pi$
$3$	−1.90136	−1.09775	−0.548876	+	0.835904i	$0.684944\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.676454	−0.255676	−0.127838	−	0.991795i	$-0.540804\pi$
$7$	−0.676454	−0.255676	−0.127838	+	0.991795i	$0.540804\pi$
$8$	0	0
$9$	0.615178	0.205059
$10$	0	0
$11$	−3.66496	−1.10503	−0.552514	−	0.833504i	$-0.686331\pi$
$11$	−3.66496	−1.10503	−0.552514	+	0.833504i	$0.686331\pi$
$12$	0	0
$13$	−4.18150	−1.15974	−0.579870	−	0.814709i	$-0.696897\pi$
$13$	−4.18150	−1.15974	−0.579870	+	0.814709i	$0.696897\pi$
$14$	0	0
$15$	−1.90136	−0.490930
$16$	0	0
$17$	1.93328	0.468888	0.234444	−	0.972130i	$-0.424673\pi$
$17$	1.93328	0.468888	0.234444	+	0.972130i	$0.424673\pi$
$18$	0	0
$19$	3.24278	0.743944	0.371972	−	0.928244i	$-0.378682\pi$
$19$	3.24278	0.743944	0.371972	+	0.928244i	$0.378682\pi$
$20$	0	0
$21$	1.28618	0.280668
$22$	0	0
$23$	−2.41614	−0.503800	−0.251900	−	0.967753i	$-0.581055\pi$
$23$	−2.41614	−0.503800	−0.251900	+	0.967753i	$0.581055\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.53441	0.872648
$28$	0	0
$29$	−0.713816	−0.132552	−0.0662761	−	0.997801i	$-0.521112\pi$
$29$	−0.713816	−0.132552	−0.0662761	+	0.997801i	$0.521112\pi$
$30$	0	0
$31$	6.46164	1.16055	0.580273	−	0.814422i	$-0.302946\pi$
$31$	6.46164	1.16055	0.580273	+	0.814422i	$0.302946\pi$
$32$	0	0
$33$	6.96842	1.21305
$34$	0	0
$35$	−0.676454	−0.114342
$36$	0	0
$37$	11.1001	1.82485	0.912426	−	0.409242i	$-0.134207\pi$
$37$	11.1001	1.82485	0.912426	+	0.409242i	$0.134207\pi$
$38$	0	0
$39$	7.95055	1.27311
$40$	0	0
$41$	−1.26227	−0.197133	−0.0985667	−	0.995130i	$-0.531426\pi$
$41$	−1.26227	−0.197133	−0.0985667	+	0.995130i	$0.531426\pi$
$42$	0	0
$43$	11.5560	1.76227	0.881137	−	0.472862i	$-0.156779\pi$
$43$	11.5560	1.76227	0.881137	+	0.472862i	$0.156779\pi$
$44$	0	0
$45$	0.615178	0.0917053
$46$	0	0
$47$	−5.69628	−0.830888	−0.415444	−	0.909619i	$-0.636374\pi$
$47$	−5.69628	−0.830888	−0.415444	+	0.909619i	$0.636374\pi$
$48$	0	0
$49$	−6.54241	−0.934630
$50$	0	0
$51$	−3.67586	−0.514723
$52$	0	0
$53$	−6.76477	−0.929212	−0.464606	−	0.885517i	$-0.653804\pi$
$53$	−6.76477	−0.929212	−0.464606	+	0.885517i	$0.653804\pi$
$54$	0	0
$55$	−3.66496	−0.494183
$56$	0	0
$57$	−6.16570	−0.816666
$58$	0	0
$59$	−12.3803	−1.61177	−0.805887	−	0.592069i	$-0.798311\pi$
$59$	−12.3803	−1.61177	−0.805887	+	0.592069i	$0.798311\pi$
$60$	0	0
$61$	10.8459	1.38867	0.694336	−	0.719651i	$-0.255698\pi$
$61$	10.8459	1.38867	0.694336	+	0.719651i	$0.255698\pi$
$62$	0	0
$63$	−0.416140	−0.0524287
$64$	0	0
$65$	−4.18150	−0.518652
$66$	0	0
$67$	13.1206	1.60293	0.801466	−	0.598041i	$-0.204054\pi$
$67$	13.1206	1.60293	0.801466	+	0.598041i	$0.204054\pi$
$68$	0	0
$69$	4.59396	0.553047
$70$	0	0
$71$	13.6665	1.62191	0.810955	−	0.585108i	$-0.198948\pi$
$71$	13.6665	1.62191	0.810955	+	0.585108i	$0.198948\pi$
$72$	0	0
$73$	2.40210	0.281144	0.140572	−	0.990070i	$-0.455106\pi$
$73$	2.40210	0.281144	0.140572	+	0.990070i	$0.455106\pi$
$74$	0	0
$75$	−1.90136	−0.219550
$76$	0	0
$77$	2.47918	0.282529
$78$	0	0
$79$	−4.25254	−0.478448	−0.239224	−	0.970964i	$-0.576893\pi$
$79$	−4.25254	−0.478448	−0.239224	+	0.970964i	$0.576893\pi$
$80$	0	0
$81$	−10.4671	−1.16301
$82$	0	0
$83$	−1.00000	−0.109764
$83$	−1.00000	−0.109764
$84$	0	0
$85$	1.93328	0.209693
$86$	0	0
$87$	1.35722	0.145510
$88$	0	0
$89$	−6.64709	−0.704590	−0.352295	−	0.935889i	$-0.614599\pi$
$89$	−6.64709	−0.704590	−0.352295	+	0.935889i	$0.614599\pi$
$90$	0	0
$91$	2.82859	0.296517
$92$	0	0
$93$	−12.2859	−1.27399
$94$	0	0
$95$	3.24278	0.332702
$96$	0	0
$97$	0.225504	0.0228965	0.0114482	−	0.999934i	$-0.496356\pi$
$97$	0.225504	0.0228965	0.0114482	+	0.999934i	$0.496356\pi$
$98$	0	0
$99$	−2.25460	−0.226596
$100$	0	0
$101$	1.99827	0.198835	0.0994177	−	0.995046i	$-0.468302\pi$
$101$	1.99827	0.198835	0.0994177	+	0.995046i	$0.468302\pi$
$102$	0	0
$103$	−8.00033	−0.788296	−0.394148	−	0.919047i	$-0.628960\pi$
$103$	−8.00033	−0.788296	−0.394148	+	0.919047i	$0.628960\pi$
$104$	0	0
$105$	1.28618	0.125519
$106$	0	0
$107$	−9.67855	−0.935661	−0.467830	−	0.883818i	$-0.654964\pi$
$107$	−9.67855	−0.935661	−0.467830	+	0.883818i	$0.654964\pi$
$108$	0	0
$109$	−7.62760	−0.730592	−0.365296	−	0.930891i	$-0.619032\pi$
$109$	−7.62760	−0.730592	−0.365296	+	0.930891i	$0.619032\pi$
$110$	0	0
$111$	−21.1054	−2.00323
$112$	0	0
$113$	−13.0659	−1.22914	−0.614569	−	0.788863i	$-0.710670\pi$
$113$	−13.0659	−1.22914	−0.614569	+	0.788863i	$0.710670\pi$
$114$	0	0
$115$	−2.41614	−0.225306
$116$	0	0
$117$	−2.57237	−0.237816
$118$	0	0
$119$	−1.30777	−0.119883
$120$	0	0
$121$	2.43195	0.221086
$122$	0	0
$123$	2.40003	0.216404
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.6118	−1.20785	−0.603927	−	0.797040i	$-0.706398\pi$
$127$	−13.6118	−1.20785	−0.603927	+	0.797040i	$0.706398\pi$
$128$	0	0
$129$	−21.9721	−1.93454
$130$	0	0
$131$	−3.64650	−0.318596	−0.159298	−	0.987231i	$-0.550923\pi$
$131$	−3.64650	−0.318596	−0.159298	+	0.987231i	$0.550923\pi$
$132$	0	0
$133$	−2.19359	−0.190208
$134$	0	0
$135$	4.53441	0.390260
$136$	0	0
$137$	−7.63943	−0.652680	−0.326340	−	0.945252i	$-0.605815\pi$
$137$	−7.63943	−0.652680	−0.326340	+	0.945252i	$0.605815\pi$
$138$	0	0
$139$	−13.3906	−1.13578	−0.567889	−	0.823105i	$-0.692240\pi$
$139$	−13.3906	−1.13578	−0.567889	+	0.823105i	$0.692240\pi$
$140$	0	0
$141$	10.8307	0.912109
$142$	0	0
$143$	15.3250	1.28154
$144$	0	0
$145$	−0.713816	−0.0592792
$146$	0	0
$147$	12.4395	1.02599
$148$	0	0
$149$	−21.5526	−1.76566	−0.882829	−	0.469694i	$-0.844364\pi$
$149$	−21.5526	−1.76566	−0.882829	+	0.469694i	$0.844364\pi$
$150$	0	0
$151$	−18.2521	−1.48533	−0.742667	−	0.669661i	$-0.766439\pi$
$151$	−18.2521	−1.48533	−0.742667	+	0.669661i	$0.766439\pi$
$152$	0	0
$153$	1.18931	0.0961499
$154$	0	0
$155$	6.46164	0.519012
$156$	0	0
$157$	7.61634	0.607850	0.303925	−	0.952696i	$-0.401703\pi$
$157$	7.61634	0.607850	0.303925	+	0.952696i	$0.401703\pi$
$158$	0	0
$159$	12.8623	1.02004
$160$	0	0
$161$	1.63441	0.128809
$162$	0	0
$163$	19.1298	1.49836	0.749182	−	0.662364i	$-0.230447\pi$
$163$	19.1298	1.49836	0.749182	+	0.662364i	$0.230447\pi$
$164$	0	0
$165$	6.96842	0.542491
$166$	0	0
$167$	1.28131	0.0991505	0.0495752	−	0.998770i	$-0.484213\pi$
$167$	1.28131	0.0991505	0.0495752	+	0.998770i	$0.484213\pi$
$168$	0	0
$169$	4.48496	0.344997
$170$	0	0
$171$	1.99489	0.152553
$172$	0	0
$173$	1.64571	0.125121	0.0625604	−	0.998041i	$-0.480073\pi$
$173$	1.64571	0.125121	0.0625604	+	0.998041i	$0.480073\pi$
$174$	0	0
$175$	−0.676454	−0.0511351
$176$	0	0
$177$	23.5394	1.76933
$178$	0	0
$179$	4.69850	0.351182	0.175591	−	0.984463i	$-0.443816\pi$
$179$	4.69850	0.351182	0.175591	+	0.984463i	$0.443816\pi$
$180$	0	0
$181$	8.38260	0.623074	0.311537	−	0.950234i	$-0.399156\pi$
$181$	8.38260	0.623074	0.311537	+	0.950234i	$0.399156\pi$
$182$	0	0
$183$	−20.6219	−1.52442
$184$	0	0
$185$	11.1001	0.816098
$186$	0	0
$187$	−7.08538	−0.518135
$188$	0	0
$189$	−3.06732	−0.223115
$190$	0	0
$191$	1.46961	0.106337	0.0531686	−	0.998586i	$-0.483068\pi$
$191$	1.46961	0.106337	0.0531686	+	0.998586i	$0.483068\pi$
$192$	0	0
$193$	−3.04517	−0.219196	−0.109598	−	0.993976i	$-0.534956\pi$
$193$	−3.04517	−0.219196	−0.109598	+	0.993976i	$0.534956\pi$
$194$	0	0
$195$	7.95055	0.569351
$196$	0	0
$197$	26.7931	1.90893	0.954463	−	0.298328i	$-0.0964289\pi$
$197$	26.7931	1.90893	0.954463	+	0.298328i	$0.0964289\pi$
$198$	0	0
$199$	16.1059	1.14172	0.570859	−	0.821048i	$-0.306610\pi$
$199$	16.1059	1.14172	0.570859	+	0.821048i	$0.306610\pi$
$200$	0	0
$201$	−24.9469	−1.75962
$202$	0	0
$203$	0.482864	0.0338904
$204$	0	0
$205$	−1.26227	−0.0881608
$206$	0	0
$207$	−1.48636	−0.103309
$208$	0	0
$209$	−11.8847	−0.822079
$210$	0	0
$211$	12.4698	0.858455	0.429228	−	0.903196i	$-0.358786\pi$
$211$	12.4698	0.858455	0.429228	+	0.903196i	$0.358786\pi$
$212$	0	0
$213$	−25.9849	−1.78046
$214$	0	0
$215$	11.5560	0.788113
$216$	0	0
$217$	−4.37100	−0.296723
$218$	0	0
$219$	−4.56725	−0.308626
$220$	0	0
$221$	−8.08400	−0.543789
$222$	0	0
$223$	−11.8562	−0.793952	−0.396976	−	0.917829i	$-0.629940\pi$
$223$	−11.8562	−0.793952	−0.396976	+	0.917829i	$0.629940\pi$
$224$	0	0
$225$	0.615178	0.0410119
$226$	0	0
$227$	19.1281	1.26958	0.634788	−	0.772686i	$-0.281087\pi$
$227$	19.1281	1.26958	0.634788	+	0.772686i	$0.281087\pi$
$228$	0	0
$229$	−21.0266	−1.38948	−0.694738	−	0.719263i	$-0.744480\pi$
$229$	−21.0266	−1.38948	−0.694738	+	0.719263i	$0.744480\pi$
$230$	0	0
$231$	−4.71382	−0.310146
$232$	0	0
$233$	26.9296	1.76422	0.882110	−	0.471044i	$-0.156123\pi$
$233$	26.9296	1.76422	0.882110	+	0.471044i	$0.156123\pi$
$234$	0	0
$235$	−5.69628	−0.371584
$236$	0	0
$237$	8.08562	0.525217
$238$	0	0
$239$	2.85483	0.184664	0.0923319	−	0.995728i	$-0.470568\pi$
$239$	2.85483	0.184664	0.0923319	+	0.995728i	$0.470568\pi$
$240$	0	0
$241$	14.5844	0.939465	0.469733	−	0.882809i	$-0.344350\pi$
$241$	14.5844	0.939465	0.469733	+	0.882809i	$0.344350\pi$
$242$	0	0
$243$	6.29850	0.404049
$244$	0	0
$245$	−6.54241	−0.417979
$246$	0	0
$247$	−13.5597	−0.862782
$248$	0	0
$249$	1.90136	0.120494
$250$	0	0
$251$	−10.6811	−0.674183	−0.337092	−	0.941472i	$-0.609443\pi$
$251$	−10.6811	−0.674183	−0.337092	+	0.941472i	$0.609443\pi$
$252$	0	0
$253$	8.85506	0.556713
$254$	0	0
$255$	−3.67586	−0.230191
$256$	0	0
$257$	6.67532	0.416395	0.208198	−	0.978087i	$-0.433240\pi$
$257$	6.67532	0.416395	0.208198	+	0.978087i	$0.433240\pi$
$258$	0	0
$259$	−7.50873	−0.466570
$260$	0	0
$261$	−0.439124	−0.0271811
$262$	0	0
$263$	−22.1289	−1.36453	−0.682263	−	0.731107i	$-0.739004\pi$
$263$	−22.1289	−1.36453	−0.682263	+	0.731107i	$0.739004\pi$
$264$	0	0
$265$	−6.76477	−0.415556
$266$	0	0
$267$	12.6385	0.773465
$268$	0	0
$269$	−25.9955	−1.58497	−0.792486	−	0.609889i	$-0.791214\pi$
$269$	−25.9955	−1.58497	−0.792486	+	0.609889i	$0.791214\pi$
$270$	0	0
$271$	−18.8894	−1.14745	−0.573724	−	0.819049i	$-0.694502\pi$
$271$	−18.8894	−1.14745	−0.573724	+	0.819049i	$0.694502\pi$
$272$	0	0
$273$	−5.37818	−0.325502
$274$	0	0
$275$	−3.66496	−0.221006
$276$	0	0
$277$	−10.6147	−0.637773	−0.318886	−	0.947793i	$-0.603309\pi$
$277$	−10.6147	−0.637773	−0.318886	+	0.947793i	$0.603309\pi$
$278$	0	0
$279$	3.97506	0.237981
$280$	0	0
$281$	−21.3410	−1.27310	−0.636550	−	0.771235i	$-0.719639\pi$
$281$	−21.3410	−1.27310	−0.636550	+	0.771235i	$0.719639\pi$
$282$	0	0
$283$	−0.954875	−0.0567614	−0.0283807	−	0.999597i	$-0.509035\pi$
$283$	−0.954875	−0.0567614	−0.0283807	+	0.999597i	$0.509035\pi$
$284$	0	0
$285$	−6.16570	−0.365224
$286$	0	0
$287$	0.853868	0.0504022
$288$	0	0
$289$	−13.2624	−0.780144
$290$	0	0
$291$	−0.428765	−0.0251347
$292$	0	0
$293$	−0.503318	−0.0294042	−0.0147021	−	0.999892i	$-0.504680\pi$
$293$	−0.503318	−0.0294042	−0.0147021	+	0.999892i	$0.504680\pi$
$294$	0	0
$295$	−12.3803	−0.720808
$296$	0	0
$297$	−16.6184	−0.964300
$298$	0	0
$299$	10.1031	0.584277
$300$	0	0
$301$	−7.81710	−0.450570
$302$	0	0
$303$	−3.79944	−0.218272
$304$	0	0
$305$	10.8459	0.621033
$306$	0	0
$307$	−24.8490	−1.41821	−0.709103	−	0.705105i	$-0.750900\pi$
$307$	−24.8490	−1.41821	−0.709103	+	0.705105i	$0.750900\pi$
$308$	0	0
$309$	15.2115	0.865354
$310$	0	0
$311$	8.40037	0.476341	0.238171	−	0.971223i	$-0.423452\pi$
$311$	8.40037	0.476341	0.238171	+	0.971223i	$0.423452\pi$
$312$	0	0
$313$	−9.18012	−0.518891	−0.259445	−	0.965758i	$-0.583540\pi$
$313$	−9.18012	−0.518891	−0.259445	+	0.965758i	$0.583540\pi$
$314$	0	0
$315$	−0.416140	−0.0234468
$316$	0	0
$317$	13.7169	0.770417	0.385209	−	0.922830i	$-0.374130\pi$
$317$	13.7169	0.770417	0.385209	+	0.922830i	$0.374130\pi$
$318$	0	0
$319$	2.61611	0.146474
$320$	0	0
$321$	18.4024	1.02712
$322$	0	0
$323$	6.26919	0.348827
$324$	0	0
$325$	−4.18150	−0.231948
$326$	0	0
$327$	14.5028	0.802008
$328$	0	0
$329$	3.85327	0.212438
$330$	0	0
$331$	−19.0525	−1.04722	−0.523609	−	0.851958i	$-0.675415\pi$
$331$	−19.0525	−1.04722	−0.523609	+	0.851958i	$0.675415\pi$
$332$	0	0
$333$	6.82856	0.374203
$334$	0	0
$335$	13.1206	0.716853
$336$	0	0
$337$	−0.933610	−0.0508570	−0.0254285	−	0.999677i	$-0.508095\pi$
$337$	−0.933610	−0.0508570	−0.0254285	+	0.999677i	$0.508095\pi$
$338$	0	0
$339$	24.8431	1.34929
$340$	0	0
$341$	−23.6817	−1.28243
$342$	0	0
$343$	9.16082	0.494638
$344$	0	0
$345$	4.59396	0.247330
$346$	0	0
$347$	−8.09029	−0.434310	−0.217155	−	0.976137i	$-0.569678\pi$
$347$	−8.09029	−0.434310	−0.217155	+	0.976137i	$0.569678\pi$
$348$	0	0
$349$	21.1017	1.12955	0.564773	−	0.825246i	$-0.308964\pi$
$349$	21.1017	1.12955	0.564773	+	0.825246i	$0.308964\pi$
$350$	0	0
$351$	−18.9606	−1.01204
$352$	0	0
$353$	17.0214	0.905955	0.452978	−	0.891522i	$-0.350362\pi$
$353$	17.0214	0.905955	0.452978	+	0.891522i	$0.350362\pi$
$354$	0	0
$355$	13.6665	0.725341
$356$	0	0
$357$	2.48655	0.131602
$358$	0	0
$359$	10.4739	0.552793	0.276396	−	0.961044i	$-0.410860\pi$
$359$	10.4739	0.552793	0.276396	+	0.961044i	$0.410860\pi$
$360$	0	0
$361$	−8.48439	−0.446547
$362$	0	0
$363$	−4.62401	−0.242698
$364$	0	0
$365$	2.40210	0.125731
$366$	0	0
$367$	−35.6683	−1.86187	−0.930934	−	0.365188i	$-0.881005\pi$
$367$	−35.6683	−1.86187	−0.930934	+	0.365188i	$0.881005\pi$
$368$	0	0
$369$	−0.776521	−0.0404241
$370$	0	0
$371$	4.57605	0.237577
$372$	0	0
$373$	15.3023	0.792324	0.396162	−	0.918181i	$-0.370342\pi$
$373$	15.3023	0.792324	0.396162	+	0.918181i	$0.370342\pi$
$374$	0	0
$375$	−1.90136	−0.0981859
$376$	0	0
$377$	2.98482	0.153726
$378$	0	0
$379$	−9.90264	−0.508664	−0.254332	−	0.967117i	$-0.581856\pi$
$379$	−9.90264	−0.508664	−0.254332	+	0.967117i	$0.581856\pi$
$380$	0	0
$381$	25.8810	1.32592
$382$	0	0
$383$	−10.1907	−0.520720	−0.260360	−	0.965512i	$-0.583841\pi$
$383$	−10.1907	−0.520720	−0.260360	+	0.965512i	$0.583841\pi$
$384$	0	0
$385$	2.47918	0.126351
$386$	0	0
$387$	7.10900	0.361371
$388$	0	0
$389$	20.8935	1.05934	0.529670	−	0.848204i	$-0.322316\pi$
$389$	20.8935	1.05934	0.529670	+	0.848204i	$0.322316\pi$
$390$	0	0
$391$	−4.67106	−0.236226
$392$	0	0
$393$	6.93331	0.349739
$394$	0	0
$395$	−4.25254	−0.213969
$396$	0	0
$397$	19.7012	0.988774	0.494387	−	0.869242i	$-0.335393\pi$
$397$	19.7012	0.988774	0.494387	+	0.869242i	$0.335393\pi$
$398$	0	0
$399$	4.17081	0.208802
$400$	0	0
$401$	−8.55599	−0.427266	−0.213633	−	0.976914i	$-0.568530\pi$
$401$	−8.55599	−0.427266	−0.213633	+	0.976914i	$0.568530\pi$
$402$	0	0
$403$	−27.0194	−1.34593
$404$	0	0
$405$	−10.4671	−0.520114
$406$	0	0
$407$	−40.6816	−2.01651
$408$	0	0
$409$	19.7809	0.978103	0.489051	−	0.872255i	$-0.337343\pi$
$409$	19.7809	0.978103	0.489051	+	0.872255i	$0.337343\pi$
$410$	0	0
$411$	14.5253	0.716481
$412$	0	0
$413$	8.37469	0.412092
$414$	0	0
$415$	−1.00000	−0.0490881
$416$	0	0
$417$	25.4604	1.24680
$418$	0	0
$419$	6.17741	0.301786	0.150893	−	0.988550i	$-0.451785\pi$
$419$	6.17741	0.301786	0.150893	+	0.988550i	$0.451785\pi$
$420$	0	0
$421$	−20.8153	−1.01447	−0.507237	−	0.861807i	$-0.669333\pi$
$421$	−20.8153	−1.01447	−0.507237	+	0.861807i	$0.669333\pi$
$422$	0	0
$423$	−3.50423	−0.170381
$424$	0	0
$425$	1.93328	0.0937777
$426$	0	0
$427$	−7.33673	−0.355049
$428$	0	0
$429$	−29.1385	−1.40682
$430$	0	0
$431$	−26.8450	−1.29308	−0.646540	−	0.762880i	$-0.723785\pi$
$431$	−26.8450	−1.29308	−0.646540	+	0.762880i	$0.723785\pi$
$432$	0	0
$433$	14.7687	0.709740	0.354870	−	0.934916i	$-0.384525\pi$
$433$	14.7687	0.709740	0.354870	+	0.934916i	$0.384525\pi$
$434$	0	0
$435$	1.35722	0.0650738
$436$	0	0
$437$	−7.83500	−0.374799
$438$	0	0
$439$	−7.60197	−0.362822	−0.181411	−	0.983407i	$-0.558066\pi$
$439$	−7.60197	−0.362822	−0.181411	+	0.983407i	$0.558066\pi$
$440$	0	0
$441$	−4.02475	−0.191655
$442$	0	0
$443$	−27.9734	−1.32906	−0.664528	−	0.747263i	$-0.731368\pi$
$443$	−27.9734	−1.32906	−0.664528	+	0.747263i	$0.731368\pi$
$444$	0	0
$445$	−6.64709	−0.315102
$446$	0	0
$447$	40.9793	1.93826
$448$	0	0
$449$	−20.2209	−0.954285	−0.477142	−	0.878826i	$-0.658327\pi$
$449$	−20.2209	−0.954285	−0.477142	+	0.878826i	$0.658327\pi$
$450$	0	0
$451$	4.62617	0.217838
$452$	0	0
$453$	34.7038	1.63053
$454$	0	0
$455$	2.82859	0.132607
$456$	0	0
$457$	−3.40867	−0.159451	−0.0797254	−	0.996817i	$-0.525404\pi$
$457$	−3.40867	−0.159451	−0.0797254	+	0.996817i	$0.525404\pi$
$458$	0	0
$459$	8.76627	0.409174
$460$	0	0
$461$	−18.4764	−0.860533	−0.430266	−	0.902702i	$-0.641580\pi$
$461$	−18.4764	−0.860533	−0.430266	+	0.902702i	$0.641580\pi$
$462$	0	0
$463$	−13.7825	−0.640526	−0.320263	−	0.947329i	$-0.603771\pi$
$463$	−13.7825	−0.640526	−0.320263	+	0.947329i	$0.603771\pi$
$464$	0	0
$465$	−12.2859	−0.569746
$466$	0	0
$467$	−32.6364	−1.51023	−0.755116	−	0.655591i	$-0.772419\pi$
$467$	−32.6364	−1.51023	−0.755116	+	0.655591i	$0.772419\pi$
$468$	0	0
$469$	−8.87546	−0.409830
$470$	0	0
$471$	−14.4814	−0.667269
$472$	0	0
$473$	−42.3523	−1.94736
$474$	0	0
$475$	3.24278	0.148789
$476$	0	0
$477$	−4.16154	−0.190544
$478$	0	0
$479$	−27.4332	−1.25346	−0.626728	−	0.779238i	$-0.715606\pi$
$479$	−27.4332	−1.25346	−0.626728	+	0.779238i	$0.715606\pi$
$480$	0	0
$481$	−46.4153	−2.11635
$482$	0	0
$483$	−3.10760	−0.141401
$484$	0	0
$485$	0.225504	0.0102396
$486$	0	0
$487$	−14.2221	−0.644466	−0.322233	−	0.946660i	$-0.604433\pi$
$487$	−14.2221	−0.644466	−0.322233	+	0.946660i	$0.604433\pi$
$488$	0	0
$489$	−36.3727	−1.64483
$490$	0	0
$491$	5.55428	0.250661	0.125331	−	0.992115i	$-0.460001\pi$
$491$	5.55428	0.250661	0.125331	+	0.992115i	$0.460001\pi$
$492$	0	0
$493$	−1.38000	−0.0621522
$494$	0	0
$495$	−2.25460	−0.101337
$496$	0	0
$497$	−9.24473	−0.414683
$498$	0	0
$499$	−28.5067	−1.27614	−0.638069	−	0.769980i	$-0.720266\pi$
$499$	−28.5067	−1.27614	−0.638069	+	0.769980i	$0.720266\pi$
$500$	0	0
$501$	−2.43623	−0.108843
$502$	0	0
$503$	−1.72926	−0.0771037	−0.0385518	−	0.999257i	$-0.512274\pi$
$503$	−1.72926	−0.0771037	−0.0385518	+	0.999257i	$0.512274\pi$
$504$	0	0
$505$	1.99827	0.0889219
$506$	0	0
$507$	−8.52753	−0.378721
$508$	0	0
$509$	−19.4684	−0.862922	−0.431461	−	0.902132i	$-0.642002\pi$
$509$	−19.4684	−0.862922	−0.431461	+	0.902132i	$0.642002\pi$
$510$	0	0
$511$	−1.62491	−0.0718817
$512$	0	0
$513$	14.7041	0.649201
$514$	0	0
$515$	−8.00033	−0.352537
$516$	0	0
$517$	20.8766	0.918154
$518$	0	0
$519$	−3.12908	−0.137352
$520$	0	0
$521$	17.1975	0.753437	0.376719	−	0.926328i	$-0.377052\pi$
$521$	17.1975	0.753437	0.376719	+	0.926328i	$0.377052\pi$
$522$	0	0
$523$	−40.3732	−1.76540	−0.882699	−	0.469938i	$-0.844276\pi$
$523$	−40.3732	−1.76540	−0.882699	+	0.469938i	$0.844276\pi$
$524$	0	0
$525$	1.28618	0.0561337
$526$	0	0
$527$	12.4921	0.544166
$528$	0	0
$529$	−17.1623	−0.746186
$530$	0	0
$531$	−7.61608	−0.330510
$532$	0	0
$533$	5.27818	0.228624
$534$	0	0
$535$	−9.67855	−0.418440
$536$	0	0
$537$	−8.93355	−0.385511
$538$	0	0
$539$	23.9777	1.03279
$540$	0	0
$541$	−5.70138	−0.245122	−0.122561	−	0.992461i	$-0.539111\pi$
$541$	−5.70138	−0.245122	−0.122561	+	0.992461i	$0.539111\pi$
$542$	0	0
$543$	−15.9384	−0.683981
$544$	0	0
$545$	−7.62760	−0.326730
$546$	0	0
$547$	15.6918	0.670933	0.335467	−	0.942052i	$-0.391106\pi$
$547$	15.6918	0.670933	0.335467	+	0.942052i	$0.391106\pi$
$548$	0	0
$549$	6.67214	0.284760
$550$	0	0
$551$	−2.31475	−0.0986115
$552$	0	0
$553$	2.87665	0.122328
$554$	0	0
$555$	−21.1054	−0.895874
$556$	0	0
$557$	−17.3131	−0.733580	−0.366790	−	0.930304i	$-0.619543\pi$
$557$	−17.3131	−0.733580	−0.366790	+	0.930304i	$0.619543\pi$
$558$	0	0
$559$	−48.3214	−2.04378
$560$	0	0
$561$	13.4719	0.568783
$562$	0	0
$563$	−20.2198	−0.852163	−0.426082	−	0.904685i	$-0.640106\pi$
$563$	−20.2198	−0.852163	−0.426082	+	0.904685i	$0.640106\pi$
$564$	0	0
$565$	−13.0659	−0.549688
$566$	0	0
$567$	7.08051	0.297353
$568$	0	0
$569$	24.1625	1.01294	0.506472	−	0.862256i	$-0.330949\pi$
$569$	24.1625	1.01294	0.506472	+	0.862256i	$0.330949\pi$
$570$	0	0
$571$	5.04750	0.211231	0.105616	−	0.994407i	$-0.466319\pi$
$571$	5.04750	0.211231	0.105616	+	0.994407i	$0.466319\pi$
$572$	0	0
$573$	−2.79426	−0.116732
$574$	0	0
$575$	−2.41614	−0.100760
$576$	0	0
$577$	29.1851	1.21499	0.607495	−	0.794324i	$-0.292175\pi$
$577$	29.1851	1.21499	0.607495	+	0.794324i	$0.292175\pi$
$578$	0	0
$579$	5.78997	0.240623
$580$	0	0
$581$	0.676454	0.0280640
$582$	0	0
$583$	24.7926	1.02681
$584$	0	0
$585$	−2.57237	−0.106354
$586$	0	0
$587$	32.3034	1.33330	0.666651	−	0.745370i	$-0.267727\pi$
$587$	32.3034	1.33330	0.666651	+	0.745370i	$0.267727\pi$
$588$	0	0
$589$	20.9537	0.863381
$590$	0	0
$591$	−50.9433	−2.09553
$592$	0	0
$593$	−40.4996	−1.66312	−0.831560	−	0.555436i	$-0.812552\pi$
$593$	−40.4996	−1.66312	−0.831560	+	0.555436i	$0.812552\pi$
$594$	0	0
$595$	−1.30777	−0.0536134
$596$	0	0
$597$	−30.6232	−1.25332
$598$	0	0
$599$	10.3844	0.424294	0.212147	−	0.977238i	$-0.431954\pi$
$599$	10.3844	0.424294	0.212147	+	0.977238i	$0.431954\pi$
$600$	0	0
$601$	12.1056	0.493797	0.246899	−	0.969041i	$-0.420589\pi$
$601$	12.1056	0.493797	0.246899	+	0.969041i	$0.420589\pi$
$602$	0	0
$603$	8.07148	0.328696
$604$	0	0
$605$	2.43195	0.0988727
$606$	0	0
$607$	35.1071	1.42495	0.712476	−	0.701696i	$-0.247574\pi$
$607$	35.1071	1.42495	0.712476	+	0.701696i	$0.247574\pi$
$608$	0	0
$609$	−0.918099	−0.0372032
$610$	0	0
$611$	23.8190	0.963614
$612$	0	0
$613$	16.5252	0.667447	0.333723	−	0.942671i	$-0.391695\pi$
$613$	16.5252	0.667447	0.333723	+	0.942671i	$0.391695\pi$
$614$	0	0
$615$	2.40003	0.0967786
$616$	0	0
$617$	15.7815	0.635340	0.317670	−	0.948201i	$-0.397100\pi$
$617$	15.7815	0.635340	0.317670	+	0.948201i	$0.397100\pi$
$618$	0	0
$619$	−29.1384	−1.17117	−0.585585	−	0.810611i	$-0.699135\pi$
$619$	−29.1384	−1.17117	−0.585585	+	0.810611i	$0.699135\pi$
$620$	0	0
$621$	−10.9558	−0.439640
$622$	0	0
$623$	4.49645	0.180147
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	22.5970	0.902439
$628$	0	0
$629$	21.4596	0.855652
$630$	0	0
$631$	29.0781	1.15758	0.578791	−	0.815476i	$-0.303525\pi$
$631$	29.0781	1.15758	0.578791	+	0.815476i	$0.303525\pi$
$632$	0	0
$633$	−23.7096	−0.942371
$634$	0	0
$635$	−13.6118	−0.540169
$636$	0	0
$637$	27.3571	1.08393
$638$	0	0
$639$	8.40731	0.332588
$640$	0	0
$641$	−46.4146	−1.83327	−0.916633	−	0.399730i	$-0.869104\pi$
$641$	−46.4146	−1.83327	−0.916633	+	0.399730i	$0.869104\pi$
$642$	0	0
$643$	3.86264	0.152328	0.0761638	−	0.997095i	$-0.475733\pi$
$643$	3.86264	0.152328	0.0761638	+	0.997095i	$0.475733\pi$
$644$	0	0
$645$	−21.9721	−0.865152
$646$	0	0
$647$	−13.1160	−0.515644	−0.257822	−	0.966192i	$-0.583005\pi$
$647$	−13.1160	−0.515644	−0.257822	+	0.966192i	$0.583005\pi$
$648$	0	0
$649$	45.3732	1.78106
$650$	0	0
$651$	8.31086	0.325728
$652$	0	0
$653$	5.17260	0.202419	0.101210	−	0.994865i	$-0.467729\pi$
$653$	5.17260	0.202419	0.101210	+	0.994865i	$0.467729\pi$
$654$	0	0
$655$	−3.64650	−0.142480
$656$	0	0
$657$	1.47772	0.0576512
$658$	0	0
$659$	27.1920	1.05925	0.529626	−	0.848232i	$-0.322332\pi$
$659$	27.1920	1.05925	0.529626	+	0.848232i	$0.322332\pi$
$660$	0	0
$661$	16.2686	0.632777	0.316388	−	0.948630i	$-0.397530\pi$
$661$	16.2686	0.632777	0.316388	+	0.948630i	$0.397530\pi$
$662$	0	0
$663$	15.3706	0.596945
$664$	0	0
$665$	−2.19359	−0.0850638
$666$	0	0
$667$	1.72468	0.0667798
$668$	0	0
$669$	22.5430	0.871562
$670$	0	0
$671$	−39.7497	−1.53452
$672$	0	0
$673$	−7.76224	−0.299212	−0.149606	−	0.988746i	$-0.547801\pi$
$673$	−7.76224	−0.299212	−0.149606	+	0.988746i	$0.547801\pi$
$674$	0	0
$675$	4.53441	0.174530
$676$	0	0
$677$	−22.2329	−0.854478	−0.427239	−	0.904139i	$-0.640514\pi$
$677$	−22.2329	−0.854478	−0.427239	+	0.904139i	$0.640514\pi$
$678$	0	0
$679$	−0.152543	−0.00585407
$680$	0	0
$681$	−36.3695	−1.39368
$682$	0	0
$683$	−12.6116	−0.482571	−0.241285	−	0.970454i	$-0.577569\pi$
$683$	−12.6116	−0.482571	−0.241285	+	0.970454i	$0.577569\pi$
$684$	0	0
$685$	−7.63943	−0.291887
$686$	0	0
$687$	39.9791	1.52530
$688$	0	0
$689$	28.2869	1.07764
$690$	0	0
$691$	4.93392	0.187695	0.0938475	−	0.995587i	$-0.470083\pi$
$691$	4.93392	0.187695	0.0938475	+	0.995587i	$0.470083\pi$
$692$	0	0
$693$	1.52514	0.0579351
$694$	0	0
$695$	−13.3906	−0.507936
$696$	0	0
$697$	−2.44032	−0.0924336
$698$	0	0
$699$	−51.2030	−1.93667
$700$	0	0
$701$	−41.1971	−1.55599	−0.777997	−	0.628268i	$-0.783764\pi$
$701$	−41.1971	−1.55599	−0.777997	+	0.628268i	$0.783764\pi$
$702$	0	0
$703$	35.9953	1.35759
$704$	0	0
$705$	10.8307	0.407907
$706$	0	0
$707$	−1.35174	−0.0508373
$708$	0	0
$709$	4.73433	0.177802	0.0889008	−	0.996040i	$-0.471665\pi$
$709$	4.73433	0.177802	0.0889008	+	0.996040i	$0.471665\pi$
$710$	0	0
$711$	−2.61607	−0.0981103
$712$	0	0
$713$	−15.6122	−0.584683
$714$	0	0
$715$	15.3250	0.573124
$716$	0	0
$717$	−5.42807	−0.202715
$718$	0	0
$719$	41.1298	1.53388	0.766941	−	0.641717i	$-0.221778\pi$
$719$	41.1298	1.53388	0.766941	+	0.641717i	$0.221778\pi$
$720$	0	0
$721$	5.41186	0.201548
$722$	0	0
$723$	−27.7303	−1.03130
$724$	0	0
$725$	−0.713816	−0.0265105
$726$	0	0
$727$	38.8338	1.44027	0.720133	−	0.693836i	$-0.244081\pi$
$727$	38.8338	1.44027	0.720133	+	0.693836i	$0.244081\pi$
$728$	0	0
$729$	19.4255	0.719465
$730$	0	0
$731$	22.3409	0.826309
$732$	0	0
$733$	−11.4181	−0.421737	−0.210868	−	0.977514i	$-0.567629\pi$
$733$	−11.4181	−0.421737	−0.210868	+	0.977514i	$0.567629\pi$
$734$	0	0
$735$	12.4395	0.458838
$736$	0	0
$737$	−48.0864	−1.77128
$738$	0	0
$739$	0.0168706	0.000620594 0	0.000310297	−	1.00000i	$-0.499901\pi$
$739$	0.0168706	0.000620594 0	0.000310297		1.00000i	$0.499901\pi$
$740$	0	0
$741$	25.7819	0.947121
$742$	0	0
$743$	−22.2784	−0.817316	−0.408658	−	0.912688i	$-0.634003\pi$
$743$	−22.2784	−0.817316	−0.408658	+	0.912688i	$0.634003\pi$
$744$	0	0
$745$	−21.5526	−0.789627
$746$	0	0
$747$	−0.615178	−0.0225082
$748$	0	0
$749$	6.54709	0.239226
$750$	0	0
$751$	−25.8159	−0.942034	−0.471017	−	0.882124i	$-0.656113\pi$
$751$	−25.8159	−0.942034	−0.471017	+	0.882124i	$0.656113\pi$
$752$	0	0
$753$	20.3086	0.740086
$754$	0	0
$755$	−18.2521	−0.664261
$756$	0	0
$757$	−13.4878	−0.490223	−0.245111	−	0.969495i	$-0.578825\pi$
$757$	−13.4878	−0.490223	−0.245111	+	0.969495i	$0.578825\pi$
$758$	0	0
$759$	−16.8367	−0.611133
$760$	0	0
$761$	−39.8508	−1.44459	−0.722295	−	0.691585i	$-0.756913\pi$
$761$	−39.8508	−1.44459	−0.722295	+	0.691585i	$0.756913\pi$
$762$	0	0
$763$	5.15972	0.186794
$764$	0	0
$765$	1.18931	0.0429996
$766$	0	0
$767$	51.7682	1.86924
$768$	0	0
$769$	−17.8357	−0.643171	−0.321585	−	0.946881i	$-0.604216\pi$
$769$	−17.8357	−0.643171	−0.321585	+	0.946881i	$0.604216\pi$
$770$	0	0
$771$	−12.6922	−0.457098
$772$	0	0
$773$	−4.63147	−0.166582	−0.0832912	−	0.996525i	$-0.526543\pi$
$773$	−4.63147	−0.166582	−0.0832912	+	0.996525i	$0.526543\pi$
$774$	0	0
$775$	6.46164	0.232109
$776$	0	0
$777$	14.2768	0.512178
$778$	0	0
$779$	−4.09326	−0.146656
$780$	0	0
$781$	−50.0871	−1.79226
$782$	0	0
$783$	−3.23673	−0.115671
$784$	0	0
$785$	7.61634	0.271839
$786$	0	0
$787$	22.1862	0.790852	0.395426	−	0.918498i	$-0.370597\pi$
$787$	22.1862	0.790852	0.395426	+	0.918498i	$0.370597\pi$
$788$	0	0
$789$	42.0750	1.49791
$790$	0	0
$791$	8.83850	0.314261
$792$	0	0
$793$	−45.3520	−1.61050
$794$	0	0
$795$	12.8623	0.456178
$796$	0	0
$797$	44.0369	1.55987	0.779934	−	0.625862i	$-0.215253\pi$
$797$	44.0369	1.55987	0.779934	+	0.625862i	$0.215253\pi$
$798$	0	0
$799$	−11.0125	−0.389594
$800$	0	0
$801$	−4.08914	−0.144483
$802$	0	0
$803$	−8.80359	−0.310672
$804$	0	0
$805$	1.63441	0.0576053
$806$	0	0
$807$	49.4268	1.73991
$808$	0	0
$809$	−36.9398	−1.29873	−0.649367	−	0.760475i	$-0.724966\pi$
$809$	−36.9398	−1.29873	−0.649367	+	0.760475i	$0.724966\pi$
$810$	0	0
$811$	3.99794	0.140387	0.0701933	−	0.997533i	$-0.477638\pi$
$811$	3.99794	0.140387	0.0701933	+	0.997533i	$0.477638\pi$
$812$	0	0
$813$	35.9156	1.25961
$814$	0	0
$815$	19.1298	0.670089
$816$	0	0
$817$	37.4735	1.31103
$818$	0	0
$819$	1.74009	0.0608036
$820$	0	0
$821$	56.5450	1.97343	0.986717	−	0.162447i	$-0.0519385\pi$
$821$	56.5450	1.97343	0.986717	+	0.162447i	$0.0519385\pi$
$822$	0	0
$823$	45.4241	1.58338	0.791692	−	0.610921i	$-0.209201\pi$
$823$	45.4241	1.58338	0.791692	+	0.610921i	$0.209201\pi$
$824$	0	0
$825$	6.96842	0.242609
$826$	0	0
$827$	49.9030	1.73530	0.867648	−	0.497179i	$-0.165631\pi$
$827$	49.9030	1.73530	0.867648	+	0.497179i	$0.165631\pi$
$828$	0	0
$829$	−0.655282	−0.0227589	−0.0113794	−	0.999935i	$-0.503622\pi$
$829$	−0.655282	−0.0227589	−0.0113794	+	0.999935i	$0.503622\pi$
$830$	0	0
$831$	20.1823	0.700116
$832$	0	0
$833$	−12.6483	−0.438237
$834$	0	0
$835$	1.28131	0.0443414
$836$	0	0
$837$	29.2997	1.01275
$838$	0	0
$839$	1.17572	0.0405904	0.0202952	−	0.999794i	$-0.493539\pi$
$839$	1.17572	0.0405904	0.0202952	+	0.999794i	$0.493539\pi$
$840$	0	0
$841$	−28.4905	−0.982430
$842$	0	0
$843$	40.5771	1.39755
$844$	0	0
$845$	4.48496	0.154287
$846$	0	0
$847$	−1.64510	−0.0565263
$848$	0	0
$849$	1.81556	0.0623100
$850$	0	0
$851$	−26.8195	−0.919360
$852$	0	0
$853$	8.01797	0.274530	0.137265	−	0.990534i	$-0.456169\pi$
$853$	8.01797	0.274530	0.137265	+	0.990534i	$0.456169\pi$
$854$	0	0
$855$	1.99489	0.0682237
$856$	0	0
$857$	−54.7273	−1.86945	−0.934725	−	0.355373i	$-0.884354\pi$
$857$	−54.7273	−1.86945	−0.934725	+	0.355373i	$0.884354\pi$
$858$	0	0
$859$	−3.50130	−0.119463	−0.0597314	−	0.998214i	$-0.519024\pi$
$859$	−3.50130	−0.119463	−0.0597314	+	0.998214i	$0.519024\pi$
$860$	0	0
$861$	−1.62351	−0.0553291
$862$	0	0
$863$	10.0421	0.341837	0.170919	−	0.985285i	$-0.445326\pi$
$863$	10.0421	0.341837	0.170919	+	0.985285i	$0.445326\pi$
$864$	0	0
$865$	1.64571	0.0559557
$866$	0	0
$867$	25.2167	0.856404
$868$	0	0
$869$	15.5854	0.528698
$870$	0	0
$871$	−54.8636	−1.85898
$872$	0	0
$873$	0.138725	0.00469514
$874$	0	0
$875$	−0.676454	−0.0228683
$876$	0	0
$877$	0.842542	0.0284506	0.0142253	−	0.999899i	$-0.495472\pi$
$877$	0.842542	0.0284506	0.0142253	+	0.999899i	$0.495472\pi$
$878$	0	0
$879$	0.956990	0.0322785
$880$	0	0
$881$	−43.2206	−1.45614	−0.728069	−	0.685504i	$-0.759582\pi$
$881$	−43.2206	−1.45614	−0.728069	+	0.685504i	$0.759582\pi$
$882$	0	0
$883$	53.3290	1.79466	0.897331	−	0.441357i	$-0.145503\pi$
$883$	53.3290	1.79466	0.897331	+	0.441357i	$0.145503\pi$
$884$	0	0
$885$	23.5394	0.791268
$886$	0	0
$887$	29.9794	1.00661	0.503306	−	0.864108i	$-0.332117\pi$
$887$	29.9794	1.00661	0.503306	+	0.864108i	$0.332117\pi$
$888$	0	0
$889$	9.20778	0.308819
$890$	0	0
$891$	38.3615	1.28516
$892$	0	0
$893$	−18.4718	−0.618134
$894$	0	0
$895$	4.69850	0.157053
$896$	0	0
$897$	−19.2096	−0.641391
$898$	0	0
$899$	−4.61242	−0.153833
$900$	0	0
$901$	−13.0782	−0.435697
$902$	0	0
$903$	14.8631	0.494614
$904$	0	0
$905$	8.38260	0.278647
$906$	0	0
$907$	29.3692	0.975189	0.487595	−	0.873070i	$-0.337874\pi$
$907$	29.3692	0.975189	0.487595	+	0.873070i	$0.337874\pi$
$908$	0	0
$909$	1.22929	0.0407730
$910$	0	0
$911$	40.8914	1.35479	0.677397	−	0.735618i	$-0.263108\pi$
$911$	40.8914	1.35479	0.677397	+	0.735618i	$0.263108\pi$
$912$	0	0
$913$	3.66496	0.121293
$914$	0	0
$915$	−20.6219	−0.681740
$916$	0	0
$917$	2.46669	0.0814572
$918$	0	0
$919$	−24.1702	−0.797303	−0.398651	−	0.917103i	$-0.630522\pi$
$919$	−24.1702	−0.797303	−0.398651	+	0.917103i	$0.630522\pi$
$920$	0	0
$921$	47.2469	1.55684
$922$	0	0
$923$	−57.1463	−1.88099
$924$	0	0
$925$	11.1001	0.364970
$926$	0	0
$927$	−4.92163	−0.161648
$928$	0	0
$929$	−49.7408	−1.63194	−0.815971	−	0.578092i	$-0.803797\pi$
$929$	−49.7408	−1.63194	−0.815971	+	0.578092i	$0.803797\pi$
$930$	0	0
$931$	−21.2156	−0.695313
$932$	0	0
$933$	−15.9721	−0.522904
$934$	0	0
$935$	−7.08538	−0.231717
$936$	0	0
$937$	50.8620	1.66159	0.830794	−	0.556580i	$-0.187887\pi$
$937$	50.8620	1.66159	0.830794	+	0.556580i	$0.187887\pi$
$938$	0	0
$939$	17.4547	0.569613
$940$	0	0
$941$	−10.2878	−0.335374	−0.167687	−	0.985840i	$-0.553630\pi$
$941$	−10.2878	−0.335374	−0.167687	+	0.985840i	$0.553630\pi$
$942$	0	0
$943$	3.04982	0.0993158
$944$	0	0
$945$	−3.06732	−0.0997799
$946$	0	0
$947$	39.6451	1.28829	0.644146	−	0.764902i	$-0.277213\pi$
$947$	39.6451	1.28829	0.644146	+	0.764902i	$0.277213\pi$
$948$	0	0
$949$	−10.0444	−0.326054
$950$	0	0
$951$	−26.0808	−0.845727
$952$	0	0
$953$	54.0991	1.75244	0.876222	−	0.481908i	$-0.160056\pi$
$953$	54.0991	1.75244	0.876222	+	0.481908i	$0.160056\pi$
$954$	0	0
$955$	1.46961	0.0475554
$956$	0	0
$957$	−4.97417	−0.160792
$958$	0	0
$959$	5.16772	0.166874
$960$	0	0
$961$	10.7528	0.346865
$962$	0	0
$963$	−5.95403	−0.191866
$964$	0	0
$965$	−3.04517	−0.0980274
$966$	0	0
$967$	−16.0995	−0.517724	−0.258862	−	0.965914i	$-0.583347\pi$
$967$	−16.0995	−0.517724	−0.258862	+	0.965914i	$0.583347\pi$
$968$	0	0
$969$	−11.9200	−0.382925
$970$	0	0
$971$	9.76754	0.313455	0.156728	−	0.987642i	$-0.449905\pi$
$971$	9.76754	0.313455	0.156728	+	0.987642i	$0.449905\pi$
$972$	0	0
$973$	9.05815	0.290391
$974$	0	0
$975$	7.95055	0.254621
$976$	0	0
$977$	−15.5385	−0.497119	−0.248560	−	0.968617i	$-0.579957\pi$
$977$	−15.5385	−0.497119	−0.248560	+	0.968617i	$0.579957\pi$
$978$	0	0
$979$	24.3613	0.778592
$980$	0	0
$981$	−4.69233	−0.149815
$982$	0	0
$983$	−55.0941	−1.75723	−0.878614	−	0.477533i	$-0.841531\pi$
$983$	−55.0941	−1.75723	−0.878614	+	0.477533i	$0.841531\pi$
$984$	0	0
$985$	26.7931	0.853698
$986$	0	0
$987$	−7.32646	−0.233204
$988$	0	0
$989$	−27.9209	−0.887833
$990$	0	0
$991$	31.3337	0.995347	0.497673	−	0.867364i	$-0.334188\pi$
$991$	31.3337	0.995347	0.497673	+	0.867364i	$0.334188\pi$
$992$	0	0
$993$	36.2257	1.14959
$994$	0	0
$995$	16.1059	0.510592
$996$	0	0
$997$	−19.4580	−0.616241	−0.308120	−	0.951347i	$-0.599700\pi$
$997$	−19.4580	−0.616241	−0.308120	+	0.951347i	$0.599700\pi$
$998$	0	0
$999$	50.3326	1.59245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6640.2.a.z.1.2		5
4.3	odd	2		830.2.a.k.1.4	✓	5
12.11	even	2		7470.2.a.bv.1.3		5
20.19	odd	2		4150.2.a.bd.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
830.2.a.k.1.4	✓	5	4.3	odd	2
4150.2.a.bd.1.2		5	20.19	odd	2
6640.2.a.z.1.2		5	1.1	even	1	trivial
7470.2.a.bv.1.3		5	12.11	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:	\( N \)	\(=\)	\( 6640 = 2^{4} \cdot 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6640.a (trivial)

\(f(q)\)	\(=\)	\(q-1.90136 q^{3} +1.00000 q^{5} -0.676454 q^{7} +0.615178 q^{9} +O(q^{10})\)	\(q-1.90136 q^{3} +1.00000 q^{5} -0.676454 q^{7} +0.615178 q^{9} -3.66496 q^{11} -4.18150 q^{13} -1.90136 q^{15} +1.93328 q^{17} +3.24278 q^{19} +1.28618 q^{21} -2.41614 q^{23} +1.00000 q^{25} +4.53441 q^{27} -0.713816 q^{29} +6.46164 q^{31} +6.96842 q^{33} -0.676454 q^{35} +11.1001 q^{37} +7.95055 q^{39} -1.26227 q^{41} +11.5560 q^{43} +0.615178 q^{45} -5.69628 q^{47} -6.54241 q^{49} -3.67586 q^{51} -6.76477 q^{53} -3.66496 q^{55} -6.16570 q^{57} -12.3803 q^{59} +10.8459 q^{61} -0.416140 q^{63} -4.18150 q^{65} +13.1206 q^{67} +4.59396 q^{69} +13.6665 q^{71} +2.40210 q^{73} -1.90136 q^{75} +2.47918 q^{77} -4.25254 q^{79} -10.4671 q^{81} -1.00000 q^{83} +1.93328 q^{85} +1.35722 q^{87} -6.64709 q^{89} +2.82859 q^{91} -12.2859 q^{93} +3.24278 q^{95} +0.225504 q^{97} -2.25460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 3 * q^9	\( 5 q - 2 q^{3} + 5 q^{5} - 5 q^{7} + 3 q^{9} - q^{11} + 4 q^{13} - 2 q^{15} - q^{17} - 7 q^{19} - q^{21} - 8 q^{23} + 5 q^{25} + q^{27} - 11 q^{29} - 10 q^{31} - 5 q^{33} - 5 q^{35} + 8 q^{37} + 14 q^{39} - 3 q^{41} + 2 q^{43} + 3 q^{45} - 7 q^{47} - 12 q^{49} - 19 q^{51} - 12 q^{53} - q^{55} - 6 q^{57} + 3 q^{59} + 11 q^{61} + 2 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 4 q^{71} - 3 q^{73} - 2 q^{75} - q^{77} + 12 q^{79} - 19 q^{81} - 5 q^{83} - q^{85} - 9 q^{87} - 30 q^{89} - 14 q^{91} - 10 q^{93} - 7 q^{95} - 7 q^{97} + 36 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 5 * q^5 - 5 * q^7 + 3 * q^9 - q^11 + 4 * q^13 - 2 * q^15 - q^17 - 7 * q^19 - q^21 - 8 * q^23 + 5 * q^25 + q^27 - 11 * q^29 - 10 * q^31 - 5 * q^33 - 5 * q^35 + 8 * q^37 + 14 * q^39 - 3 * q^41 + 2 * q^43 + 3 * q^45 - 7 * q^47 - 12 * q^49 - 19 * q^51 - 12 * q^53 - q^55 - 6 * q^57 + 3 * q^59 + 11 * q^61 + 2 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 4 * q^71 - 3 * q^73 - 2 * q^75 - q^77 + 12 * q^79 - 19 * q^81 - 5 * q^83 - q^85 - 9 * q^87 - 30 * q^89 - 14 * q^91 - 10 * q^93 - 7 * q^95 - 7 * q^97 + 36 * q^99

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