LMFDB - Embedded newform 2960.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, 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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.286164.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} - 3x + 12 $ x^4 - x^3 - 11x^2 - 3x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 740)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.51951$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51951 q^{3} +1.00000 q^{5} -0.519509 q^{7} +3.34793 q^{9} +O(q^{10})$	$q-2.51951 q^{3} +1.00000 q^{5} -0.519509 q^{7} +3.34793 q^{9} -6.16241 q^{11} -6.51034 q^{13} -2.51951 q^{15} -2.51034 q^{17} -6.68192 q^{19} +1.30891 q^{21} +3.03902 q^{23} +1.00000 q^{25} -0.876607 q^{27} -3.03902 q^{29} +4.33876 q^{31} +15.5263 q^{33} -0.519509 q^{35} +1.00000 q^{37} +16.4029 q^{39} +12.1624 q^{41} +2.34316 q^{43} +3.34793 q^{45} -9.54019 q^{47} -6.73011 q^{49} +6.32482 q^{51} -9.20143 q^{53} -6.16241 q^{55} +16.8352 q^{57} +9.37778 q^{59} +2.00000 q^{61} -1.73928 q^{63} -6.51034 q^{65} -0.357097 q^{67} -7.65684 q^{69} +2.87661 q^{71} -10.2404 q^{73} -2.51951 q^{75} +3.20143 q^{77} -0.681922 q^{79} -7.83516 q^{81} -6.17635 q^{83} -2.51034 q^{85} +7.65684 q^{87} -11.0207 q^{89} +3.38218 q^{91} -10.9315 q^{93} -6.68192 q^{95} +6.34316 q^{97} -20.6313 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9	$ 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9} - 5 q^{11} - 6 q^{13} - 3 q^{15} + 10 q^{17} + 19 q^{21} - 2 q^{23} + 4 q^{25} - 9 q^{27} + 2 q^{29} + 4 q^{31} - 11 q^{33} + 5 q^{35} + 4 q^{37} - 2 q^{39} + 29 q^{41} - 4 q^{43} + 13 q^{45} + 9 q^{47} + q^{49} - 14 q^{51} - 3 q^{53} - 5 q^{55} + 8 q^{57} + 10 q^{59} + 8 q^{61} + 8 q^{63} - 6 q^{65} - 14 q^{67} - 44 q^{69} + 17 q^{71} + 7 q^{73} - 3 q^{75} - 21 q^{77} + 24 q^{79} + 28 q^{81} - 31 q^{83} + 10 q^{85} + 44 q^{87} - 4 q^{89} - 14 q^{91} + 18 q^{93} + 12 q^{97} + 22 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9 - 5 * q^11 - 6 * q^13 - 3 * q^15 + 10 * q^17 + 19 * q^21 - 2 * q^23 + 4 * q^25 - 9 * q^27 + 2 * q^29 + 4 * q^31 - 11 * q^33 + 5 * q^35 + 4 * q^37 - 2 * q^39 + 29 * q^41 - 4 * q^43 + 13 * q^45 + 9 * q^47 + q^49 - 14 * q^51 - 3 * q^53 - 5 * q^55 + 8 * q^57 + 10 * q^59 + 8 * q^61 + 8 * q^63 - 6 * q^65 - 14 * q^67 - 44 * q^69 + 17 * q^71 + 7 * q^73 - 3 * q^75 - 21 * q^77 + 24 * q^79 + 28 * q^81 - 31 * q^83 + 10 * q^85 + 44 * q^87 - 4 * q^89 - 14 * q^91 + 18 * q^93 + 12 * q^97 + 22 * 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
$3$	−2.51951	−1.45464	−0.727320	−	0.686299i	$-0.759234\pi$
$3$	−2.51951	−1.45464	−0.727320	+	0.686299i	$0.759234\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.519509	−0.196356	−0.0981780	−	0.995169i	$-0.531301\pi$
$7$	−0.519509	−0.196356	−0.0981780	+	0.995169i	$0.531301\pi$
$8$	0	0
$9$	3.34793	1.11598
$10$	0	0
$11$	−6.16241	−1.85804	−0.929019	−	0.370033i	$-0.879346\pi$
$11$	−6.16241	−1.85804	−0.929019	+	0.370033i	$0.879346\pi$
$12$	0	0
$13$	−6.51034	−1.80564	−0.902822	−	0.430015i	$-0.858508\pi$
$13$	−6.51034	−1.80564	−0.902822	+	0.430015i	$0.858508\pi$
$14$	0	0
$15$	−2.51951	−0.650535
$16$	0	0
$17$	−2.51034	−0.608847	−0.304423	−	0.952537i	$-0.598464\pi$
$17$	−2.51034	−0.608847	−0.304423	+	0.952537i	$0.598464\pi$
$18$	0	0
$19$	−6.68192	−1.53294	−0.766469	−	0.642281i	$-0.777988\pi$
$19$	−6.68192	−1.53294	−0.766469	+	0.642281i	$0.777988\pi$
$20$	0	0
$21$	1.30891	0.285627
$22$	0	0
$23$	3.03902	0.633679	0.316840	−	0.948479i	$-0.397378\pi$
$23$	3.03902	0.633679	0.316840	+	0.948479i	$0.397378\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.876607	−0.168703
$28$	0	0
$29$	−3.03902	−0.564332	−0.282166	−	0.959366i	$-0.591053\pi$
$29$	−3.03902	−0.564332	−0.282166	+	0.959366i	$0.591053\pi$
$30$	0	0
$31$	4.33876	0.779264	0.389632	−	0.920971i	$-0.372602\pi$
$31$	4.33876	0.779264	0.389632	+	0.920971i	$0.372602\pi$
$32$	0	0
$33$	15.5263	2.70277
$34$	0	0
$35$	−0.519509	−0.0878131
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	16.4029	2.62656
$40$	0	0
$41$	12.1624	1.89945	0.949725	−	0.313086i	$-0.101363\pi$
$41$	12.1624	1.89945	0.949725	+	0.313086i	$0.101363\pi$
$42$	0	0
$43$	2.34316	0.357329	0.178665	−	0.983910i	$-0.442822\pi$
$43$	2.34316	0.357329	0.178665	+	0.983910i	$0.442822\pi$
$44$	0	0
$45$	3.34793	0.499080
$46$	0	0
$47$	−9.54019	−1.39158	−0.695790	−	0.718245i	$-0.744945\pi$
$47$	−9.54019	−1.39158	−0.695790	+	0.718245i	$0.744945\pi$
$48$	0	0
$49$	−6.73011	−0.961444
$50$	0	0
$51$	6.32482	0.885653
$52$	0	0
$53$	−9.20143	−1.26391	−0.631957	−	0.775004i	$-0.717748\pi$
$53$	−9.20143	−1.26391	−0.631957	+	0.775004i	$0.717748\pi$
$54$	0	0
$55$	−6.16241	−0.830939
$56$	0	0
$57$	16.8352	2.22987
$58$	0	0
$59$	9.37778	1.22088	0.610441	−	0.792062i	$-0.290992\pi$
$59$	9.37778	1.22088	0.610441	+	0.792062i	$0.290992\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−1.73928	−0.219129
$64$	0	0
$65$	−6.51034	−0.807508
$66$	0	0
$67$	−0.357097	−0.0436264	−0.0218132	−	0.999762i	$-0.506944\pi$
$67$	−0.357097	−0.0436264	−0.0218132	+	0.999762i	$0.506944\pi$
$68$	0	0
$69$	−7.65684	−0.921775
$70$	0	0
$71$	2.87661	0.341390	0.170695	−	0.985324i	$-0.445399\pi$
$71$	2.87661	0.341390	0.170695	+	0.985324i	$0.445399\pi$
$72$	0	0
$73$	−10.2404	−1.19855	−0.599277	−	0.800542i	$-0.704545\pi$
$73$	−10.2404	−1.19855	−0.599277	+	0.800542i	$0.704545\pi$
$74$	0	0
$75$	−2.51951	−0.290928
$76$	0	0
$77$	3.20143	0.364837
$78$	0	0
$79$	−0.681922	−0.0767222	−0.0383611	−	0.999264i	$-0.512214\pi$
$79$	−0.681922	−0.0767222	−0.0383611	+	0.999264i	$0.512214\pi$
$80$	0	0
$81$	−7.83516	−0.870574
$82$	0	0
$83$	−6.17635	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$83$	−6.17635	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$84$	0	0
$85$	−2.51034	−0.272285
$86$	0	0
$87$	7.65684	0.820899
$88$	0	0
$89$	−11.0207	−1.16819	−0.584095	−	0.811685i	$-0.698550\pi$
$89$	−11.0207	−1.16819	−0.584095	+	0.811685i	$0.698550\pi$
$90$	0	0
$91$	3.38218	0.354549
$92$	0	0
$93$	−10.9315	−1.13355
$94$	0	0
$95$	−6.68192	−0.685551
$96$	0	0
$97$	6.34316	0.644051	0.322025	−	0.946731i	$-0.395636\pi$
$97$	6.34316	0.644051	0.322025	+	0.946731i	$0.395636\pi$
$98$	0	0
$99$	−20.6313	−2.07352
$100$	0	0
$101$	19.4466	1.93501	0.967507	−	0.252845i	$-0.0813664\pi$
$101$	19.4466	1.93501	0.967507	+	0.252845i	$0.0813664\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	1.30891	0.127736
$106$	0	0
$107$	15.0251	1.45253	0.726265	−	0.687415i	$-0.241255\pi$
$107$	15.0251	1.45253	0.726265	+	0.687415i	$0.241255\pi$
$108$	0	0
$109$	15.7070	1.50446	0.752229	−	0.658902i	$-0.228979\pi$
$109$	15.7070	1.50446	0.752229	+	0.658902i	$0.228979\pi$
$110$	0	0
$111$	−2.51951	−0.239141
$112$	0	0
$113$	−2.18552	−0.205596	−0.102798	−	0.994702i	$-0.532780\pi$
$113$	−2.18552	−0.205596	−0.102798	+	0.994702i	$0.532780\pi$
$114$	0	0
$115$	3.03902	0.283390
$116$	0	0
$117$	−21.7961	−2.01505
$118$	0	0
$119$	1.30414	0.119551
$120$	0	0
$121$	26.9753	2.45230
$122$	0	0
$123$	−30.6433	−2.76301
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.17635	−0.193119	−0.0965597	−	0.995327i	$-0.530784\pi$
$127$	−2.17635	−0.193119	−0.0965597	+	0.995327i	$0.530784\pi$
$128$	0	0
$129$	−5.90362	−0.519785
$130$	0	0
$131$	12.7416	1.11324	0.556620	−	0.830767i	$-0.312098\pi$
$131$	12.7416	1.11324	0.556620	+	0.830767i	$0.312098\pi$
$132$	0	0
$133$	3.47132	0.301002
$134$	0	0
$135$	−0.876607	−0.0754463
$136$	0	0
$137$	−4.26513	−0.364394	−0.182197	−	0.983262i	$-0.558321\pi$
$137$	−4.26513	−0.364394	−0.182197	+	0.983262i	$0.558321\pi$
$138$	0	0
$139$	11.6290	0.986356	0.493178	−	0.869928i	$-0.335835\pi$
$139$	11.6290	0.986356	0.493178	+	0.869928i	$0.335835\pi$
$140$	0	0
$141$	24.0366	2.02425
$142$	0	0
$143$	40.1194	3.35495
$144$	0	0
$145$	−3.03902	−0.252377
$146$	0	0
$147$	16.9566	1.39855
$148$	0	0
$149$	−17.3591	−1.42211	−0.711056	−	0.703136i	$-0.751783\pi$
$149$	−17.3591	−1.42211	−0.711056	+	0.703136i	$0.751783\pi$
$150$	0	0
$151$	−0.343164	−0.0279263	−0.0139631	−	0.999903i	$-0.504445\pi$
$151$	−0.343164	−0.0279263	−0.0139631	+	0.999903i	$0.504445\pi$
$152$	0	0
$153$	−8.40444	−0.679458
$154$	0	0
$155$	4.33876	0.348497
$156$	0	0
$157$	−4.16241	−0.332197	−0.166098	−	0.986109i	$-0.553117\pi$
$157$	−4.16241	−0.332197	−0.166098	+	0.986109i	$0.553117\pi$
$158$	0	0
$159$	23.1831	1.83854
$160$	0	0
$161$	−1.57880	−0.124427
$162$	0	0
$163$	−19.3455	−1.51526	−0.757628	−	0.652686i	$-0.773642\pi$
$163$	−19.3455	−1.51526	−0.757628	+	0.652686i	$0.773642\pi$
$164$	0	0
$165$	15.5263	1.20872
$166$	0	0
$167$	0.979321	0.0757821	0.0378911	−	0.999282i	$-0.487936\pi$
$167$	0.979321	0.0757821	0.0378911	+	0.999282i	$0.487936\pi$
$168$	0	0
$169$	29.3845	2.26035
$170$	0	0
$171$	−22.3706	−1.71072
$172$	0	0
$173$	−22.9363	−1.74381	−0.871907	−	0.489671i	$-0.837117\pi$
$173$	−22.9363	−1.74381	−0.871907	+	0.489671i	$0.837117\pi$
$174$	0	0
$175$	−0.519509	−0.0392712
$176$	0	0
$177$	−23.6274	−1.77594
$178$	0	0
$179$	16.7600	1.25270	0.626349	−	0.779543i	$-0.284548\pi$
$179$	16.7600	1.25270	0.626349	+	0.779543i	$0.284548\pi$
$180$	0	0
$181$	−11.4466	−0.850822	−0.425411	−	0.905000i	$-0.639871\pi$
$181$	−11.4466	−0.850822	−0.425411	+	0.905000i	$0.639871\pi$
$182$	0	0
$183$	−5.03902	−0.372495
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	15.4697	1.13126
$188$	0	0
$189$	0.455405	0.0331259
$190$	0	0
$191$	10.4351	0.755060	0.377530	−	0.925997i	$-0.376774\pi$
$191$	10.4351	0.755060	0.377530	+	0.925997i	$0.376774\pi$
$192$	0	0
$193$	20.4029	1.46863	0.734315	−	0.678809i	$-0.237503\pi$
$193$	20.4029	1.46863	0.734315	+	0.678809i	$0.237503\pi$
$194$	0	0
$195$	16.4029	1.17463
$196$	0	0
$197$	−11.1051	−0.791202	−0.395601	−	0.918422i	$-0.629464\pi$
$197$	−11.1051	−0.791202	−0.395601	+	0.918422i	$0.629464\pi$
$198$	0	0
$199$	0.622223	0.0441083	0.0220541	−	0.999757i	$-0.492979\pi$
$199$	0.622223	0.0441083	0.0220541	+	0.999757i	$0.492979\pi$
$200$	0	0
$201$	0.899710	0.0634606
$202$	0	0
$203$	1.57880	0.110810
$204$	0	0
$205$	12.1624	0.849460
$206$	0	0
$207$	10.1744	0.707171
$208$	0	0
$209$	41.1768	2.84826
$210$	0	0
$211$	18.3001	1.25983	0.629917	−	0.776662i	$-0.283089\pi$
$211$	18.3001	1.25983	0.629917	+	0.776662i	$0.283089\pi$
$212$	0	0
$213$	−7.24764	−0.496600
$214$	0	0
$215$	2.34316	0.159802
$216$	0	0
$217$	−2.25403	−0.153013
$218$	0	0
$219$	25.8009	1.74346
$220$	0	0
$221$	16.3432	1.09936
$222$	0	0
$223$	0.784636	0.0525431	0.0262715	−	0.999655i	$-0.491637\pi$
$223$	0.784636	0.0525431	0.0262715	+	0.999655i	$0.491637\pi$
$224$	0	0
$225$	3.34793	0.223195
$226$	0	0
$227$	−6.32482	−0.419793	−0.209897	−	0.977724i	$-0.567313\pi$
$227$	−6.32482	−0.419793	−0.209897	+	0.977724i	$0.567313\pi$
$228$	0	0
$229$	2.08437	0.137739	0.0688697	−	0.997626i	$-0.478061\pi$
$229$	2.08437	0.137739	0.0688697	+	0.997626i	$0.478061\pi$
$230$	0	0
$231$	−8.06604	−0.530706
$232$	0	0
$233$	26.4626	1.73362	0.866810	−	0.498639i	$-0.166167\pi$
$233$	26.4626	1.73362	0.866810	+	0.498639i	$0.166167\pi$
$234$	0	0
$235$	−9.54019	−0.622333
$236$	0	0
$237$	1.71811	0.111603
$238$	0	0
$239$	−18.6636	−1.20725	−0.603623	−	0.797270i	$-0.706277\pi$
$239$	−18.6636	−1.20725	−0.603623	+	0.797270i	$0.706277\pi$
$240$	0	0
$241$	−0.0596981	−0.00384549	−0.00192274	−	0.999998i	$-0.500612\pi$
$241$	−0.0596981	−0.00384549	−0.00192274	+	0.999998i	$0.500612\pi$
$242$	0	0
$243$	22.3706	1.43507
$244$	0	0
$245$	−6.73011	−0.429971
$246$	0	0
$247$	43.5016	2.76794
$248$	0	0
$249$	15.5614	0.986161
$250$	0	0
$251$	−3.05295	−0.192701	−0.0963503	−	0.995347i	$-0.530717\pi$
$251$	−3.05295	−0.192701	−0.0963503	+	0.995347i	$0.530717\pi$
$252$	0	0
$253$	−18.7277	−1.17740
$254$	0	0
$255$	6.32482	0.396076
$256$	0	0
$257$	−20.9132	−1.30453	−0.652265	−	0.757991i	$-0.726181\pi$
$257$	−20.9132	−1.30453	−0.652265	+	0.757991i	$0.726181\pi$
$258$	0	0
$259$	−0.519509	−0.0322807
$260$	0	0
$261$	−10.1744	−0.629780
$262$	0	0
$263$	7.91122	0.487827	0.243913	−	0.969797i	$-0.421569\pi$
$263$	7.91122	0.487827	0.243913	+	0.969797i	$0.421569\pi$
$264$	0	0
$265$	−9.20143	−0.565239
$266$	0	0
$267$	27.7667	1.69929
$268$	0	0
$269$	−0.606717	−0.0369922	−0.0184961	−	0.999829i	$-0.505888\pi$
$269$	−0.606717	−0.0369922	−0.0184961	+	0.999829i	$0.505888\pi$
$270$	0	0
$271$	−20.5931	−1.25094	−0.625472	−	0.780247i	$-0.715094\pi$
$271$	−20.5931	−1.25094	−0.625472	+	0.780247i	$0.715094\pi$
$272$	0	0
$273$	−8.52144	−0.515741
$274$	0	0
$275$	−6.16241	−0.371607
$276$	0	0
$277$	21.2842	1.27885	0.639423	−	0.768855i	$-0.279173\pi$
$277$	21.2842	1.27885	0.639423	+	0.768855i	$0.279173\pi$
$278$	0	0
$279$	14.5258	0.869640
$280$	0	0
$281$	−10.3432	−0.617021	−0.308511	−	0.951221i	$-0.599831\pi$
$281$	−10.3432	−0.617021	−0.308511	+	0.951221i	$0.599831\pi$
$282$	0	0
$283$	−1.67518	−0.0995789	−0.0497894	−	0.998760i	$-0.515855\pi$
$283$	−1.67518	−0.0995789	−0.0497894	+	0.998760i	$0.515855\pi$
$284$	0	0
$285$	16.8352	0.997229
$286$	0	0
$287$	−6.31849	−0.372969
$288$	0	0
$289$	−10.6982	−0.629306
$290$	0	0
$291$	−15.9817	−0.936862
$292$	0	0
$293$	9.96332	0.582063	0.291032	−	0.956713i	$-0.406001\pi$
$293$	9.96332	0.582063	0.291032	+	0.956713i	$0.406001\pi$
$294$	0	0
$295$	9.37778	0.545995
$296$	0	0
$297$	5.40201	0.313456
$298$	0	0
$299$	−19.7850	−1.14420
$300$	0	0
$301$	−1.21730	−0.0701637
$302$	0	0
$303$	−48.9960	−2.81475
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−26.2265	−1.49683	−0.748413	−	0.663233i	$-0.769184\pi$
$307$	−26.2265	−1.49683	−0.748413	+	0.663233i	$0.769184\pi$
$308$	0	0
$309$	5.03902	0.286660
$310$	0	0
$311$	−1.74881	−0.0991658	−0.0495829	−	0.998770i	$-0.515789\pi$
$311$	−1.74881	−0.0991658	−0.0495829	+	0.998770i	$0.515789\pi$
$312$	0	0
$313$	−21.7762	−1.23087	−0.615433	−	0.788189i	$-0.711019\pi$
$313$	−21.7762	−1.23087	−0.615433	+	0.788189i	$0.711019\pi$
$314$	0	0
$315$	−1.73928	−0.0979973
$316$	0	0
$317$	4.42120	0.248319	0.124160	−	0.992262i	$-0.460376\pi$
$317$	4.42120	0.248319	0.124160	+	0.992262i	$0.460376\pi$
$318$	0	0
$319$	18.7277	1.04855
$320$	0	0
$321$	−37.8558	−2.11291
$322$	0	0
$323$	16.7739	0.933324
$324$	0	0
$325$	−6.51034	−0.361129
$326$	0	0
$327$	−39.5740	−2.18844
$328$	0	0
$329$	4.95622	0.273245
$330$	0	0
$331$	−32.4168	−1.78179	−0.890894	−	0.454211i	$-0.849921\pi$
$331$	−32.4168	−1.78179	−0.890894	+	0.454211i	$0.849921\pi$
$332$	0	0
$333$	3.34793	0.183465
$334$	0	0
$335$	−0.357097	−0.0195103
$336$	0	0
$337$	29.8789	1.62761	0.813805	−	0.581138i	$-0.197392\pi$
$337$	29.8789	1.62761	0.813805	+	0.581138i	$0.197392\pi$
$338$	0	0
$339$	5.50643	0.299068
$340$	0	0
$341$	−26.7372	−1.44790
$342$	0	0
$343$	7.13292	0.385142
$344$	0	0
$345$	−7.65684	−0.412230
$346$	0	0
$347$	−2.88294	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$347$	−2.88294	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$348$	0	0
$349$	31.5494	1.68880	0.844399	−	0.535714i	$-0.179958\pi$
$349$	31.5494	1.68880	0.844399	+	0.535714i	$0.179958\pi$
$350$	0	0
$351$	5.70701	0.304617
$352$	0	0
$353$	−2.68633	−0.142979	−0.0714894	−	0.997441i	$-0.522775\pi$
$353$	−2.68633	−0.142979	−0.0714894	+	0.997441i	$0.522775\pi$
$354$	0	0
$355$	2.87661	0.152674
$356$	0	0
$357$	−3.28581	−0.173903
$358$	0	0
$359$	−21.9570	−1.15885	−0.579423	−	0.815027i	$-0.696722\pi$
$359$	−21.9570	−1.15885	−0.579423	+	0.815027i	$0.696722\pi$
$360$	0	0
$361$	25.6481	1.34990
$362$	0	0
$363$	−67.9646	−3.56722
$364$	0	0
$365$	−10.2404	−0.536010
$366$	0	0
$367$	25.3499	1.32325	0.661627	−	0.749833i	$-0.269866\pi$
$367$	25.3499	1.32325	0.661627	+	0.749833i	$0.269866\pi$
$368$	0	0
$369$	40.7189	2.11974
$370$	0	0
$371$	4.78023	0.248177
$372$	0	0
$373$	23.6322	1.22363	0.611813	−	0.791002i	$-0.290440\pi$
$373$	23.6322	1.22363	0.611813	+	0.791002i	$0.290440\pi$
$374$	0	0
$375$	−2.51951	−0.130107
$376$	0	0
$377$	19.7850	1.01898
$378$	0	0
$379$	18.8216	0.966800	0.483400	−	0.875400i	$-0.339402\pi$
$379$	18.8216	0.966800	0.483400	+	0.875400i	$0.339402\pi$
$380$	0	0
$381$	5.48332	0.280919
$382$	0	0
$383$	−11.0485	−0.564554	−0.282277	−	0.959333i	$-0.591090\pi$
$383$	−11.0485	−0.564554	−0.282277	+	0.959333i	$0.591090\pi$
$384$	0	0
$385$	3.20143	0.163160
$386$	0	0
$387$	7.84474	0.398771
$388$	0	0
$389$	29.4235	1.49183	0.745916	−	0.666040i	$-0.232012\pi$
$389$	29.4235	1.49183	0.745916	+	0.666040i	$0.232012\pi$
$390$	0	0
$391$	−7.62897	−0.385814
$392$	0	0
$393$	−32.1026	−1.61936
$394$	0	0
$395$	−0.681922	−0.0343112
$396$	0	0
$397$	17.2014	0.863315	0.431658	−	0.902038i	$-0.357929\pi$
$397$	17.2014	0.863315	0.431658	+	0.902038i	$0.357929\pi$
$398$	0	0
$399$	−8.74603	−0.437849
$400$	0	0
$401$	18.6496	0.931319	0.465660	−	0.884964i	$-0.345817\pi$
$401$	18.6496	0.931319	0.465660	+	0.884964i	$0.345817\pi$
$402$	0	0
$403$	−28.2468	−1.40707
$404$	0	0
$405$	−7.83516	−0.389332
$406$	0	0
$407$	−6.16241	−0.305459
$408$	0	0
$409$	11.0390	0.545844	0.272922	−	0.962036i	$-0.412010\pi$
$409$	11.0390	0.545844	0.272922	+	0.962036i	$0.412010\pi$
$410$	0	0
$411$	10.7460	0.530062
$412$	0	0
$413$	−4.87184	−0.239728
$414$	0	0
$415$	−6.17635	−0.303185
$416$	0	0
$417$	−29.2993	−1.43479
$418$	0	0
$419$	9.95698	0.486430	0.243215	−	0.969972i	$-0.421798\pi$
$419$	9.95698	0.486430	0.243215	+	0.969972i	$0.421798\pi$
$420$	0	0
$421$	22.8153	1.11195	0.555974	−	0.831200i	$-0.312346\pi$
$421$	22.8153	1.11195	0.555974	+	0.831200i	$0.312346\pi$
$422$	0	0
$423$	−31.9399	−1.55297
$424$	0	0
$425$	−2.51034	−0.121769
$426$	0	0
$427$	−1.03902	−0.0502816
$428$	0	0
$429$	−101.081	−4.88025
$430$	0	0
$431$	−8.67239	−0.417735	−0.208867	−	0.977944i	$-0.566978\pi$
$431$	−8.67239	−0.417735	−0.208867	+	0.977944i	$0.566978\pi$
$432$	0	0
$433$	−5.38852	−0.258956	−0.129478	−	0.991582i	$-0.541330\pi$
$433$	−5.38852	−0.258956	−0.129478	+	0.991582i	$0.541330\pi$
$434$	0	0
$435$	7.65684	0.367117
$436$	0	0
$437$	−20.3065	−0.971391
$438$	0	0
$439$	−0.603884	−0.0288218	−0.0144109	−	0.999896i	$-0.504587\pi$
$439$	−0.603884	−0.0288218	−0.0144109	+	0.999896i	$0.504587\pi$
$440$	0	0
$441$	−22.5319	−1.07295
$442$	0	0
$443$	−27.2655	−1.29542	−0.647712	−	0.761885i	$-0.724274\pi$
$443$	−27.2655	−1.29542	−0.647712	+	0.761885i	$0.724274\pi$
$444$	0	0
$445$	−11.0207	−0.522430
$446$	0	0
$447$	43.7364	2.06866
$448$	0	0
$449$	9.28581	0.438224	0.219112	−	0.975700i	$-0.429684\pi$
$449$	9.28581	0.438224	0.219112	+	0.975700i	$0.429684\pi$
$450$	0	0
$451$	−74.9498	−3.52925
$452$	0	0
$453$	0.864604	0.0406227
$454$	0	0
$455$	3.38218	0.158559
$456$	0	0
$457$	12.2651	0.573738	0.286869	−	0.957970i	$-0.407385\pi$
$457$	12.2651	0.573738	0.286869	+	0.957970i	$0.407385\pi$
$458$	0	0
$459$	2.20058	0.102714
$460$	0	0
$461$	1.81291	0.0844357	0.0422178	−	0.999108i	$-0.486558\pi$
$461$	1.81291	0.0844357	0.0422178	+	0.999108i	$0.486558\pi$
$462$	0	0
$463$	23.7349	1.10305	0.551527	−	0.834157i	$-0.314046\pi$
$463$	23.7349	1.10305	0.551527	+	0.834157i	$0.314046\pi$
$464$	0	0
$465$	−10.9315	−0.506938
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	0.185515	0.00856630
$470$	0	0
$471$	10.4872	0.483226
$472$	0	0
$473$	−14.4395	−0.663931
$474$	0	0
$475$	−6.68192	−0.306588
$476$	0	0
$477$	−30.8057	−1.41050
$478$	0	0
$479$	9.62456	0.439758	0.219879	−	0.975527i	$-0.429434\pi$
$479$	9.62456	0.439758	0.219879	+	0.975527i	$0.429434\pi$
$480$	0	0
$481$	−6.51034	−0.296846
$482$	0	0
$483$	3.97780	0.180996
$484$	0	0
$485$	6.34316	0.288028
$486$	0	0
$487$	31.7165	1.43721	0.718607	−	0.695417i	$-0.244780\pi$
$487$	31.7165	1.43721	0.718607	+	0.695417i	$0.244780\pi$
$488$	0	0
$489$	48.7412	2.20415
$490$	0	0
$491$	5.02068	0.226580	0.113290	−	0.993562i	$-0.463861\pi$
$491$	5.02068	0.226580	0.113290	+	0.993562i	$0.463861\pi$
$492$	0	0
$493$	7.62897	0.343591
$494$	0	0
$495$	−20.6313	−0.927308
$496$	0	0
$497$	−1.49442	−0.0670341
$498$	0	0
$499$	−34.8794	−1.56141	−0.780707	−	0.624897i	$-0.785141\pi$
$499$	−34.8794	−1.56141	−0.780707	+	0.624897i	$0.785141\pi$
$500$	0	0
$501$	−2.46741	−0.110236
$502$	0	0
$503$	−20.2158	−0.901377	−0.450688	−	0.892681i	$-0.648821\pi$
$503$	−20.2158	−0.901377	−0.450688	+	0.892681i	$0.648821\pi$
$504$	0	0
$505$	19.4466	0.865364
$506$	0	0
$507$	−74.0346	−3.28799
$508$	0	0
$509$	4.18075	0.185309	0.0926543	−	0.995698i	$-0.470465\pi$
$509$	4.18075	0.185309	0.0926543	+	0.995698i	$0.470465\pi$
$510$	0	0
$511$	5.32001	0.235343
$512$	0	0
$513$	5.85742	0.258611
$514$	0	0
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	58.7906	2.58561
$518$	0	0
$519$	57.7882	2.53662
$520$	0	0
$521$	−6.26036	−0.274271	−0.137136	−	0.990552i	$-0.543790\pi$
$521$	−6.26036	−0.274271	−0.137136	+	0.990552i	$0.543790\pi$
$522$	0	0
$523$	−30.3432	−1.32681	−0.663407	−	0.748259i	$-0.730890\pi$
$523$	−30.3432	−1.32681	−0.663407	+	0.748259i	$0.730890\pi$
$524$	0	0
$525$	1.30891	0.0571255
$526$	0	0
$527$	−10.8918	−0.474452
$528$	0	0
$529$	−13.7644	−0.598451
$530$	0	0
$531$	31.3961	1.36248
$532$	0	0
$533$	−79.1814	−3.42973
$534$	0	0
$535$	15.0251	0.649591
$536$	0	0
$537$	−42.2269	−1.82222
$538$	0	0
$539$	41.4737	1.78640
$540$	0	0
$541$	−16.0502	−0.690051	−0.345025	−	0.938593i	$-0.612130\pi$
$541$	−16.0502	−0.690051	−0.345025	+	0.938593i	$0.612130\pi$
$542$	0	0
$543$	28.8399	1.23764
$544$	0	0
$545$	15.7070	0.672814
$546$	0	0
$547$	−21.0804	−0.901332	−0.450666	−	0.892693i	$-0.648813\pi$
$547$	−21.0804	−0.901332	−0.450666	+	0.892693i	$0.648813\pi$
$548$	0	0
$549$	6.69586	0.285772
$550$	0	0
$551$	20.3065	0.865085
$552$	0	0
$553$	0.354265	0.0150649
$554$	0	0
$555$	−2.51951	−0.106947
$556$	0	0
$557$	0.959408	0.0406514	0.0203257	−	0.999793i	$-0.493530\pi$
$557$	0.959408	0.0406514	0.0203257	+	0.999793i	$0.493530\pi$
$558$	0	0
$559$	−15.2548	−0.645209
$560$	0	0
$561$	−38.9762	−1.64558
$562$	0	0
$563$	41.3957	1.74462	0.872310	−	0.488954i	$-0.162621\pi$
$563$	41.3957	1.74462	0.872310	+	0.488954i	$0.162621\pi$
$564$	0	0
$565$	−2.18552	−0.0919453
$566$	0	0
$567$	4.07044	0.170942
$568$	0	0
$569$	−26.4626	−1.10937	−0.554684	−	0.832061i	$-0.687161\pi$
$569$	−26.4626	−1.10937	−0.554684	+	0.832061i	$0.687161\pi$
$570$	0	0
$571$	−17.4482	−0.730185	−0.365093	−	0.930971i	$-0.618963\pi$
$571$	−17.4482	−0.730185	−0.365093	+	0.930971i	$0.618963\pi$
$572$	0	0
$573$	−26.2914	−1.09834
$574$	0	0
$575$	3.03902	0.126736
$576$	0	0
$577$	30.2954	1.26121	0.630607	−	0.776103i	$-0.282806\pi$
$577$	30.2954	1.26121	0.630607	+	0.776103i	$0.282806\pi$
$578$	0	0
$579$	−51.4052	−2.13633
$580$	0	0
$581$	3.20867	0.133118
$582$	0	0
$583$	56.7030	2.34840
$584$	0	0
$585$	−21.7961	−0.901160
$586$	0	0
$587$	−21.2858	−0.878559	−0.439280	−	0.898350i	$-0.644766\pi$
$587$	−21.2858	−0.878559	−0.439280	+	0.898350i	$0.644766\pi$
$588$	0	0
$589$	−28.9912	−1.19456
$590$	0	0
$591$	27.9793	1.15091
$592$	0	0
$593$	−17.5629	−0.721223	−0.360612	−	0.932716i	$-0.617432\pi$
$593$	−17.5629	−0.721223	−0.360612	+	0.932716i	$0.617432\pi$
$594$	0	0
$595$	1.30414	0.0534647
$596$	0	0
$597$	−1.56770	−0.0641616
$598$	0	0
$599$	12.8901	0.526675	0.263338	−	0.964704i	$-0.415177\pi$
$599$	12.8901	0.526675	0.263338	+	0.964704i	$0.415177\pi$
$600$	0	0
$601$	−13.8424	−0.564641	−0.282321	−	0.959320i	$-0.591104\pi$
$601$	−13.8424	−0.564641	−0.282321	+	0.959320i	$0.591104\pi$
$602$	0	0
$603$	−1.19554	−0.0486860
$604$	0	0
$605$	26.9753	1.09670
$606$	0	0
$607$	−24.7460	−1.00441	−0.502205	−	0.864749i	$-0.667478\pi$
$607$	−24.7460	−1.00441	−0.502205	+	0.864749i	$0.667478\pi$
$608$	0	0
$609$	−3.97780	−0.161189
$610$	0	0
$611$	62.1099	2.51270
$612$	0	0
$613$	−11.9753	−0.483679	−0.241839	−	0.970316i	$-0.577751\pi$
$613$	−11.9753	−0.483679	−0.241839	+	0.970316i	$0.577751\pi$
$614$	0	0
$615$	−30.6433	−1.23566
$616$	0	0
$617$	−25.5167	−1.02726	−0.513632	−	0.858010i	$-0.671700\pi$
$617$	−25.5167	−1.02726	−0.513632	+	0.858010i	$0.671700\pi$
$618$	0	0
$619$	24.0660	0.967296	0.483648	−	0.875263i	$-0.339312\pi$
$619$	24.0660	0.967296	0.483648	+	0.875263i	$0.339312\pi$
$620$	0	0
$621$	−2.66402	−0.106904
$622$	0	0
$623$	5.72535	0.229381
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−103.745	−4.14318
$628$	0	0
$629$	−2.51034	−0.100094
$630$	0	0
$631$	24.5545	0.977500	0.488750	−	0.872424i	$-0.337453\pi$
$631$	24.5545	0.977500	0.488750	+	0.872424i	$0.337453\pi$
$632$	0	0
$633$	−46.1074	−1.83260
$634$	0	0
$635$	−2.17635	−0.0863656
$636$	0	0
$637$	43.8153	1.73603
$638$	0	0
$639$	9.63067	0.380983
$640$	0	0
$641$	−5.20300	−0.205506	−0.102753	−	0.994707i	$-0.532765\pi$
$641$	−5.20300	−0.205506	−0.102753	+	0.994707i	$0.532765\pi$
$642$	0	0
$643$	0.0875650	0.00345323	0.00172661	−	0.999999i	$-0.499450\pi$
$643$	0.0875650	0.00345323	0.00172661	+	0.999999i	$0.499450\pi$
$644$	0	0
$645$	−5.90362	−0.232455
$646$	0	0
$647$	5.81606	0.228653	0.114326	−	0.993443i	$-0.463529\pi$
$647$	5.81606	0.228653	0.114326	+	0.993443i	$0.463529\pi$
$648$	0	0
$649$	−57.7897	−2.26845
$650$	0	0
$651$	5.67904	0.222579
$652$	0	0
$653$	12.5716	0.491965	0.245983	−	0.969274i	$-0.420889\pi$
$653$	12.5716	0.491965	0.245983	+	0.969274i	$0.420889\pi$
$654$	0	0
$655$	12.7416	0.497856
$656$	0	0
$657$	−34.2843	−1.33756
$658$	0	0
$659$	−21.7730	−0.848157	−0.424079	−	0.905625i	$-0.639402\pi$
$659$	−21.7730	−0.848157	−0.424079	+	0.905625i	$0.639402\pi$
$660$	0	0
$661$	−9.29462	−0.361519	−0.180759	−	0.983527i	$-0.557856\pi$
$661$	−9.29462	−0.361519	−0.180759	+	0.983527i	$0.557856\pi$
$662$	0	0
$663$	−41.1768	−1.59917
$664$	0	0
$665$	3.47132	0.134612
$666$	0	0
$667$	−9.23563	−0.357605
$668$	0	0
$669$	−1.97690	−0.0764312
$670$	0	0
$671$	−12.3248	−0.475795
$672$	0	0
$673$	34.2221	1.31917	0.659583	−	0.751632i	$-0.270733\pi$
$673$	34.2221	1.31917	0.659583	+	0.751632i	$0.270733\pi$
$674$	0	0
$675$	−0.876607	−0.0337406
$676$	0	0
$677$	15.4761	0.594794	0.297397	−	0.954754i	$-0.403881\pi$
$677$	15.4761	0.594794	0.297397	+	0.954754i	$0.403881\pi$
$678$	0	0
$679$	−3.29533	−0.126463
$680$	0	0
$681$	15.9355	0.610648
$682$	0	0
$683$	−12.3750	−0.473516	−0.236758	−	0.971569i	$-0.576085\pi$
$683$	−12.3750	−0.473516	−0.236758	+	0.971569i	$0.576085\pi$
$684$	0	0
$685$	−4.26513	−0.162962
$686$	0	0
$687$	−5.25160	−0.200361
$688$	0	0
$689$	59.9044	2.28218
$690$	0	0
$691$	−1.69819	−0.0646024	−0.0323012	−	0.999478i	$-0.510284\pi$
$691$	−1.69819	−0.0646024	−0.0323012	+	0.999478i	$0.510284\pi$
$692$	0	0
$693$	10.7182	0.407149
$694$	0	0
$695$	11.6290	0.441112
$696$	0	0
$697$	−30.5318	−1.15647
$698$	0	0
$699$	−66.6727	−2.52179
$700$	0	0
$701$	33.3869	1.26100	0.630502	−	0.776187i	$-0.282849\pi$
$701$	33.3869	1.26100	0.630502	+	0.776187i	$0.282849\pi$
$702$	0	0
$703$	−6.68192	−0.252013
$704$	0	0
$705$	24.0366	0.905271
$706$	0	0
$707$	−10.1027	−0.379952
$708$	0	0
$709$	−19.9403	−0.748874	−0.374437	−	0.927252i	$-0.622164\pi$
$709$	−19.9403	−0.748874	−0.374437	+	0.927252i	$0.622164\pi$
$710$	0	0
$711$	−2.28302	−0.0856201
$712$	0	0
$713$	13.1856	0.493803
$714$	0	0
$715$	40.1194	1.50038
$716$	0	0
$717$	47.0231	1.75611
$718$	0	0
$719$	−43.8694	−1.63605	−0.818027	−	0.575180i	$-0.804932\pi$
$719$	−43.8694	−1.63605	−0.818027	+	0.575180i	$0.804932\pi$
$720$	0	0
$721$	1.03902	0.0386951
$722$	0	0
$723$	0.150410	0.00559380
$724$	0	0
$725$	−3.03902	−0.112866
$726$	0	0
$727$	41.4052	1.53563	0.767817	−	0.640669i	$-0.221343\pi$
$727$	41.4052	1.53563	0.767817	+	0.640669i	$0.221343\pi$
$728$	0	0
$729$	−32.8574	−1.21694
$730$	0	0
$731$	−5.88214	−0.217559
$732$	0	0
$733$	25.6919	0.948950	0.474475	−	0.880269i	$-0.342638\pi$
$733$	25.6919	0.948950	0.474475	+	0.880269i	$0.342638\pi$
$734$	0	0
$735$	16.9566	0.625453
$736$	0	0
$737$	2.20058	0.0810594
$738$	0	0
$739$	46.4195	1.70757	0.853785	−	0.520625i	$-0.174301\pi$
$739$	46.4195	1.70757	0.853785	+	0.520625i	$0.174301\pi$
$740$	0	0
$741$	−109.603	−4.02635
$742$	0	0
$743$	−9.86501	−0.361912	−0.180956	−	0.983491i	$-0.557919\pi$
$743$	−9.86501	−0.361912	−0.180956	+	0.983491i	$0.557919\pi$
$744$	0	0
$745$	−17.3591	−0.635987
$746$	0	0
$747$	−20.6780	−0.756567
$748$	0	0
$749$	−7.80567	−0.285213
$750$	0	0
$751$	−37.5860	−1.37153	−0.685765	−	0.727823i	$-0.740532\pi$
$751$	−37.5860	−1.37153	−0.685765	+	0.727823i	$0.740532\pi$
$752$	0	0
$753$	7.69194	0.280310
$754$	0	0
$755$	−0.343164	−0.0124890
$756$	0	0
$757$	47.1672	1.71432	0.857161	−	0.515049i	$-0.172226\pi$
$757$	47.1672	1.71432	0.857161	+	0.515049i	$0.172226\pi$
$758$	0	0
$759$	47.1846	1.71269
$760$	0	0
$761$	−30.0167	−1.08810	−0.544052	−	0.839052i	$-0.683111\pi$
$761$	−30.0167	−1.08810	−0.544052	+	0.839052i	$0.683111\pi$
$762$	0	0
$763$	−8.15994	−0.295410
$764$	0	0
$765$	−8.40444	−0.303863
$766$	0	0
$767$	−61.0525	−2.20448
$768$	0	0
$769$	−32.3662	−1.16715	−0.583577	−	0.812058i	$-0.698347\pi$
$769$	−32.3662	−1.16715	−0.583577	+	0.812058i	$0.698347\pi$
$770$	0	0
$771$	52.6910	1.89762
$772$	0	0
$773$	−40.6338	−1.46150	−0.730748	−	0.682648i	$-0.760828\pi$
$773$	−40.6338	−1.46150	−0.730748	+	0.682648i	$0.760828\pi$
$774$	0	0
$775$	4.33876	0.155853
$776$	0	0
$777$	1.30891	0.0469568
$778$	0	0
$779$	−81.2683	−2.91174
$780$	0	0
$781$	−17.7268	−0.634316
$782$	0	0
$783$	2.66402	0.0952044
$784$	0	0
$785$	−4.16241	−0.148563
$786$	0	0
$787$	13.7089	0.488671	0.244335	−	0.969691i	$-0.421430\pi$
$787$	13.7089	0.488671	0.244335	+	0.969691i	$0.421430\pi$
$788$	0	0
$789$	−19.9324	−0.709612
$790$	0	0
$791$	1.13540	0.0403700
$792$	0	0
$793$	−13.0207	−0.462378
$794$	0	0
$795$	23.1831	0.822220
$796$	0	0
$797$	18.3321	0.649355	0.324677	−	0.945825i	$-0.394744\pi$
$797$	18.3321	0.649355	0.324677	+	0.945825i	$0.394744\pi$
$798$	0	0
$799$	23.9491	0.847259
$800$	0	0
$801$	−36.8964	−1.30367
$802$	0	0
$803$	63.1059	2.22696
$804$	0	0
$805$	−1.57880	−0.0556453
$806$	0	0
$807$	1.52863	0.0538103
$808$	0	0
$809$	18.3248	0.644267	0.322133	−	0.946694i	$-0.395600\pi$
$809$	18.3248	0.644267	0.322133	+	0.946694i	$0.395600\pi$
$810$	0	0
$811$	9.45775	0.332106	0.166053	−	0.986117i	$-0.446898\pi$
$811$	9.45775	0.332106	0.166053	+	0.986117i	$0.446898\pi$
$812$	0	0
$813$	51.8846	1.81967
$814$	0	0
$815$	−19.3455	−0.677643
$816$	0	0
$817$	−15.6568	−0.547763
$818$	0	0
$819$	11.3233	0.395668
$820$	0	0
$821$	−32.4689	−1.13317	−0.566586	−	0.824003i	$-0.691736\pi$
$821$	−32.4689	−1.13317	−0.566586	+	0.824003i	$0.691736\pi$
$822$	0	0
$823$	−32.0370	−1.11674	−0.558369	−	0.829593i	$-0.688573\pi$
$823$	−32.0370	−1.11674	−0.558369	+	0.829593i	$0.688573\pi$
$824$	0	0
$825$	15.5263	0.540555
$826$	0	0
$827$	−2.20543	−0.0766903	−0.0383451	−	0.999265i	$-0.512209\pi$
$827$	−2.20543	−0.0766903	−0.0383451	+	0.999265i	$0.512209\pi$
$828$	0	0
$829$	26.9745	0.936862	0.468431	−	0.883500i	$-0.344819\pi$
$829$	26.9745	0.936862	0.468431	+	0.883500i	$0.344819\pi$
$830$	0	0
$831$	−53.6258	−1.86026
$832$	0	0
$833$	16.8949	0.585372
$834$	0	0
$835$	0.979321	0.0338908
$836$	0	0
$837$	−3.80338	−0.131464
$838$	0	0
$839$	20.8424	0.719560	0.359780	−	0.933037i	$-0.382852\pi$
$839$	20.8424	0.719560	0.359780	+	0.933037i	$0.382852\pi$
$840$	0	0
$841$	−19.7644	−0.681530
$842$	0	0
$843$	26.0597	0.897544
$844$	0	0
$845$	29.3845	1.01086
$846$	0	0
$847$	−14.0139	−0.481524
$848$	0	0
$849$	4.22062	0.144851
$850$	0	0
$851$	3.03902	0.104176
$852$	0	0
$853$	−48.7093	−1.66778	−0.833888	−	0.551933i	$-0.813890\pi$
$853$	−48.7093	−1.66778	−0.833888	+	0.551933i	$0.813890\pi$
$854$	0	0
$855$	−22.3706	−0.765058
$856$	0	0
$857$	−28.4212	−0.970850	−0.485425	−	0.874278i	$-0.661335\pi$
$857$	−28.4212	−0.970850	−0.485425	+	0.874278i	$0.661335\pi$
$858$	0	0
$859$	−16.2926	−0.555895	−0.277947	−	0.960596i	$-0.589654\pi$
$859$	−16.2926	−0.555895	−0.277947	+	0.960596i	$0.589654\pi$
$860$	0	0
$861$	15.9195	0.542535
$862$	0	0
$863$	−28.1333	−0.957670	−0.478835	−	0.877905i	$-0.658941\pi$
$863$	−28.1333	−0.957670	−0.478835	+	0.877905i	$0.658941\pi$
$864$	0	0
$865$	−22.9363	−0.779858
$866$	0	0
$867$	26.9542	0.915413
$868$	0	0
$869$	4.20228	0.142553
$870$	0	0
$871$	2.32482	0.0787737
$872$	0	0
$873$	21.2365	0.718745
$874$	0	0
$875$	−0.519509	−0.0175626
$876$	0	0
$877$	−25.0899	−0.847226	−0.423613	−	0.905843i	$-0.639238\pi$
$877$	−25.0899	−0.847226	−0.423613	+	0.905843i	$0.639238\pi$
$878$	0	0
$879$	−25.1027	−0.846692
$880$	0	0
$881$	29.2197	0.984436	0.492218	−	0.870472i	$-0.336186\pi$
$881$	29.2197	0.984436	0.492218	+	0.870472i	$0.336186\pi$
$882$	0	0
$883$	−26.9745	−0.907763	−0.453882	−	0.891062i	$-0.649961\pi$
$883$	−26.9745	−0.907763	−0.453882	+	0.891062i	$0.649961\pi$
$884$	0	0
$885$	−23.6274	−0.794226
$886$	0	0
$887$	31.2839	1.05041	0.525205	−	0.850976i	$-0.323989\pi$
$887$	31.2839	1.05041	0.525205	+	0.850976i	$0.323989\pi$
$888$	0	0
$889$	1.13063	0.0379202
$890$	0	0
$891$	48.2835	1.61756
$892$	0	0
$893$	63.7468	2.13321
$894$	0	0
$895$	16.7600	0.560224
$896$	0	0
$897$	49.8486	1.66440
$898$	0	0
$899$	−13.1856	−0.439763
$900$	0	0
$901$	23.0987	0.769530
$902$	0	0
$903$	3.06699	0.102063
$904$	0	0
$905$	−11.4466	−0.380499
$906$	0	0
$907$	−17.4697	−0.580073	−0.290037	−	0.957016i	$-0.593668\pi$
$907$	−17.4697	−0.580073	−0.290037	+	0.957016i	$0.593668\pi$
$908$	0	0
$909$	65.1060	2.15943
$910$	0	0
$911$	37.6659	1.24793	0.623964	−	0.781453i	$-0.285521\pi$
$911$	37.6659	1.24793	0.623964	+	0.781453i	$0.285521\pi$
$912$	0	0
$913$	38.0612	1.25964
$914$	0	0
$915$	−5.03902	−0.166585
$916$	0	0
$917$	−6.61939	−0.218592
$918$	0	0
$919$	43.6119	1.43862	0.719312	−	0.694687i	$-0.244457\pi$
$919$	43.6119	1.43862	0.719312	+	0.694687i	$0.244457\pi$
$920$	0	0
$921$	66.0780	2.17734
$922$	0	0
$923$	−18.7277	−0.616429
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−6.69586	−0.219921
$928$	0	0
$929$	22.8367	0.749249	0.374625	−	0.927177i	$-0.377772\pi$
$929$	22.8367	0.749249	0.374625	+	0.927177i	$0.377772\pi$
$930$	0	0
$931$	44.9701	1.47383
$932$	0	0
$933$	4.40614	0.144250
$934$	0	0
$935$	15.4697	0.505915
$936$	0	0
$937$	−14.2309	−0.464904	−0.232452	−	0.972608i	$-0.574675\pi$
$937$	−14.2309	−0.464904	−0.232452	+	0.972608i	$0.574675\pi$
$938$	0	0
$939$	54.8654	1.79047
$940$	0	0
$941$	16.4140	0.535080	0.267540	−	0.963547i	$-0.413789\pi$
$941$	16.4140	0.535080	0.267540	+	0.963547i	$0.413789\pi$
$942$	0	0
$943$	36.9618	1.20364
$944$	0	0
$945$	0.455405	0.0148143
$946$	0	0
$947$	33.9506	1.10325	0.551624	−	0.834093i	$-0.314008\pi$
$947$	33.9506	1.10325	0.551624	+	0.834093i	$0.314008\pi$
$948$	0	0
$949$	66.6688	2.16416
$950$	0	0
$951$	−11.1393	−0.361215
$952$	0	0
$953$	−41.5358	−1.34548	−0.672738	−	0.739881i	$-0.734882\pi$
$953$	−41.5358	−1.34548	−0.672738	+	0.739881i	$0.734882\pi$
$954$	0	0
$955$	10.4351	0.337673
$956$	0	0
$957$	−47.1846	−1.52526
$958$	0	0
$959$	2.21577	0.0715510
$960$	0	0
$961$	−12.1752	−0.392748
$962$	0	0
$963$	50.3029	1.62099
$964$	0	0
$965$	20.4029	0.656791
$966$	0	0
$967$	16.2341	0.522054	0.261027	−	0.965332i	$-0.415939\pi$
$967$	16.2341	0.522054	0.261027	+	0.965332i	$0.415939\pi$
$968$	0	0
$969$	−42.2620	−1.35765
$970$	0	0
$971$	−30.7277	−0.986098	−0.493049	−	0.870001i	$-0.664118\pi$
$971$	−30.7277	−0.986098	−0.493049	+	0.870001i	$0.664118\pi$
$972$	0	0
$973$	−6.04136	−0.193677
$974$	0	0
$975$	16.4029	0.525312
$976$	0	0
$977$	−20.8654	−0.667544	−0.333772	−	0.942654i	$-0.608322\pi$
$977$	−20.8654	−0.667544	−0.333772	+	0.942654i	$0.608322\pi$
$978$	0	0
$979$	67.9140	2.17054
$980$	0	0
$981$	52.5859	1.67894
$982$	0	0
$983$	20.2360	0.645430	0.322715	−	0.946496i	$-0.395405\pi$
$983$	20.2360	0.645430	0.322715	+	0.946496i	$0.395405\pi$
$984$	0	0
$985$	−11.1051	−0.353836
$986$	0	0
$987$	−12.4872	−0.397473
$988$	0	0
$989$	7.12092	0.226432
$990$	0	0
$991$	9.33642	0.296581	0.148291	−	0.988944i	$-0.452623\pi$
$991$	9.33642	0.296581	0.148291	+	0.988944i	$0.452623\pi$
$992$	0	0
$993$	81.6744	2.59186
$994$	0	0
$995$	0.622223	0.0197258
$996$	0	0
$997$	−13.9506	−0.441821	−0.220911	−	0.975294i	$-0.570903\pi$
$997$	−13.9506	−0.441821	−0.220911	+	0.975294i	$0.570903\pi$
$998$	0	0
$999$	−0.876607	−0.0277346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.v.1.2		4
4.3	odd	2		740.2.a.f.1.3	✓	4
12.11	even	2		6660.2.a.r.1.3		4
20.3	even	4		3700.2.d.i.149.6		8
20.7	even	4		3700.2.d.i.149.3		8
20.19	odd	2		3700.2.a.j.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.f.1.3	✓	4	4.3	odd	2
2960.2.a.v.1.2		4	1.1	even	1	trivial
3700.2.a.j.1.2		4	20.19	odd	2
3700.2.d.i.149.3		8	20.7	even	4
3700.2.d.i.149.6		8	20.3	even	4
6660.2.a.r.1.3		4	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:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51951 q^{3} +1.00000 q^{5} -0.519509 q^{7} +3.34793 q^{9} +O(q^{10})\)	\(q-2.51951 q^{3} +1.00000 q^{5} -0.519509 q^{7} +3.34793 q^{9} -6.16241 q^{11} -6.51034 q^{13} -2.51951 q^{15} -2.51034 q^{17} -6.68192 q^{19} +1.30891 q^{21} +3.03902 q^{23} +1.00000 q^{25} -0.876607 q^{27} -3.03902 q^{29} +4.33876 q^{31} +15.5263 q^{33} -0.519509 q^{35} +1.00000 q^{37} +16.4029 q^{39} +12.1624 q^{41} +2.34316 q^{43} +3.34793 q^{45} -9.54019 q^{47} -6.73011 q^{49} +6.32482 q^{51} -9.20143 q^{53} -6.16241 q^{55} +16.8352 q^{57} +9.37778 q^{59} +2.00000 q^{61} -1.73928 q^{63} -6.51034 q^{65} -0.357097 q^{67} -7.65684 q^{69} +2.87661 q^{71} -10.2404 q^{73} -2.51951 q^{75} +3.20143 q^{77} -0.681922 q^{79} -7.83516 q^{81} -6.17635 q^{83} -2.51034 q^{85} +7.65684 q^{87} -11.0207 q^{89} +3.38218 q^{91} -10.9315 q^{93} -6.68192 q^{95} +6.34316 q^{97} -20.6313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9	\( 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9} - 5 q^{11} - 6 q^{13} - 3 q^{15} + 10 q^{17} + 19 q^{21} - 2 q^{23} + 4 q^{25} - 9 q^{27} + 2 q^{29} + 4 q^{31} - 11 q^{33} + 5 q^{35} + 4 q^{37} - 2 q^{39} + 29 q^{41} - 4 q^{43} + 13 q^{45} + 9 q^{47} + q^{49} - 14 q^{51} - 3 q^{53} - 5 q^{55} + 8 q^{57} + 10 q^{59} + 8 q^{61} + 8 q^{63} - 6 q^{65} - 14 q^{67} - 44 q^{69} + 17 q^{71} + 7 q^{73} - 3 q^{75} - 21 q^{77} + 24 q^{79} + 28 q^{81} - 31 q^{83} + 10 q^{85} + 44 q^{87} - 4 q^{89} - 14 q^{91} + 18 q^{93} + 12 q^{97} + 22 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9 - 5 * q^11 - 6 * q^13 - 3 * q^15 + 10 * q^17 + 19 * q^21 - 2 * q^23 + 4 * q^25 - 9 * q^27 + 2 * q^29 + 4 * q^31 - 11 * q^33 + 5 * q^35 + 4 * q^37 - 2 * q^39 + 29 * q^41 - 4 * q^43 + 13 * q^45 + 9 * q^47 + q^49 - 14 * q^51 - 3 * q^53 - 5 * q^55 + 8 * q^57 + 10 * q^59 + 8 * q^61 + 8 * q^63 - 6 * q^65 - 14 * q^67 - 44 * q^69 + 17 * q^71 + 7 * q^73 - 3 * q^75 - 21 * q^77 + 24 * q^79 + 28 * q^81 - 31 * q^83 + 10 * q^85 + 44 * q^87 - 4 * q^89 - 14 * q^91 + 18 * q^93 + 12 * q^97 + 22 * q^99

Introduction

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