LMFDB - Embedded newform 2960.2.a.w.1.3

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.973904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 $ x^5 - 2x^4 - 8x^3 + 6x^2 + 19x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.383115$ 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)

$f(q)$	$=$	$q-1.38311 q^{3} -1.00000 q^{5} +2.62521 q^{7} -1.08699 q^{9} +O(q^{10})$	$q-1.38311 q^{3} -1.00000 q^{5} +2.62521 q^{7} -1.08699 q^{9} +1.64446 q^{11} +2.44254 q^{13} +1.38311 q^{15} -0.578749 q^{17} -5.20156 q^{19} -3.63096 q^{21} -8.22913 q^{23} +1.00000 q^{25} +5.65278 q^{27} +0.766229 q^{29} -4.21452 q^{31} -2.27447 q^{33} -2.62521 q^{35} +1.00000 q^{37} -3.37831 q^{39} -1.64446 q^{41} +1.91893 q^{43} +1.08699 q^{45} +9.56543 q^{47} -0.108279 q^{49} +0.800477 q^{51} +7.74217 q^{53} -1.64446 q^{55} +7.19435 q^{57} +13.0359 q^{59} -3.86379 q^{61} -2.85359 q^{63} -2.44254 q^{65} -11.4566 q^{67} +11.3818 q^{69} +2.54690 q^{71} -9.79732 q^{73} -1.38311 q^{75} +4.31704 q^{77} +1.81364 q^{79} -4.55746 q^{81} -10.9822 q^{83} +0.578749 q^{85} -1.05978 q^{87} -8.85915 q^{89} +6.41217 q^{91} +5.82917 q^{93} +5.20156 q^{95} -10.5605 q^{97} -1.78751 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 - 5 * q^5 - 11 * q^7 + 6 * q^9	$ 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41} - 10 q^{43} - 6 q^{45} - 7 q^{47} + 22 q^{49} + 2 q^{51} - q^{53} - 5 q^{55} - 8 q^{57} + 30 q^{59} - 14 q^{61} - 30 q^{63} - 4 q^{65} - 24 q^{67} + 8 q^{69} + 7 q^{71} + 5 q^{73} - 3 q^{75} - 17 q^{77} - 28 q^{79} - 31 q^{81} - 27 q^{83} - 36 q^{87} + 6 q^{89} + 14 q^{91} - 22 q^{93} - 4 q^{95} - 26 q^{97} + 10 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 5 * q^5 - 11 * q^7 + 6 * q^9 + 5 * q^11 + 4 * q^13 + 3 * q^15 + 4 * q^19 + 3 * q^21 - 4 * q^23 + 5 * q^25 - 3 * q^27 - 4 * q^29 - 8 * q^31 + 5 * q^33 + 11 * q^35 + 5 * q^37 - 2 * q^39 - 5 * q^41 - 10 * q^43 - 6 * q^45 - 7 * q^47 + 22 * q^49 + 2 * q^51 - q^53 - 5 * q^55 - 8 * q^57 + 30 * q^59 - 14 * q^61 - 30 * q^63 - 4 * q^65 - 24 * q^67 + 8 * q^69 + 7 * q^71 + 5 * q^73 - 3 * q^75 - 17 * q^77 - 28 * q^79 - 31 * q^81 - 27 * q^83 - 36 * q^87 + 6 * q^89 + 14 * q^91 - 22 * q^93 - 4 * q^95 - 26 * q^97 + 10 * 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
$3$	−1.38311	−0.798542	−0.399271	−	0.916833i	$-0.630737\pi$
$3$	−1.38311	−0.798542	−0.399271	+	0.916833i	$0.630737\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.62521	0.992236	0.496118	−	0.868255i	$-0.334758\pi$
$7$	2.62521	0.992236	0.496118	+	0.868255i	$0.334758\pi$
$8$	0	0
$9$	−1.08699	−0.362331
$10$	0	0
$11$	1.64446	0.495823	0.247911	−	0.968783i	$-0.420256\pi$
$11$	1.64446	0.495823	0.247911	+	0.968783i	$0.420256\pi$
$12$	0	0
$13$	2.44254	0.677438	0.338719	−	0.940888i	$-0.390006\pi$
$13$	2.44254	0.677438	0.338719	+	0.940888i	$0.390006\pi$
$14$	0	0
$15$	1.38311	0.357119
$16$	0	0
$17$	−0.578749	−0.140367	−0.0701837	−	0.997534i	$-0.522359\pi$
$17$	−0.578749	−0.140367	−0.0701837	+	0.997534i	$0.522359\pi$
$18$	0	0
$19$	−5.20156	−1.19332	−0.596660	−	0.802494i	$-0.703506\pi$
$19$	−5.20156	−1.19332	−0.596660	+	0.802494i	$0.703506\pi$
$20$	0	0
$21$	−3.63096	−0.792341
$22$	0	0
$23$	−8.22913	−1.71589	−0.857946	−	0.513739i	$-0.828260\pi$
$23$	−8.22913	−1.71589	−0.857946	+	0.513739i	$0.828260\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65278	1.08788
$28$	0	0
$29$	0.766229	0.142285	0.0711426	−	0.997466i	$-0.477335\pi$
$29$	0.766229	0.142285	0.0711426	+	0.997466i	$0.477335\pi$
$30$	0	0
$31$	−4.21452	−0.756950	−0.378475	−	0.925611i	$-0.623551\pi$
$31$	−4.21452	−0.756950	−0.378475	+	0.925611i	$0.623551\pi$
$32$	0	0
$33$	−2.27447	−0.395935
$34$	0	0
$35$	−2.62521	−0.443741
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−3.37831	−0.540962
$40$	0	0
$41$	−1.64446	−0.256821	−0.128411	−	0.991721i	$-0.540988\pi$
$41$	−1.64446	−0.256821	−0.128411	+	0.991721i	$0.540988\pi$
$42$	0	0
$43$	1.91893	0.292634	0.146317	−	0.989238i	$-0.453258\pi$
$43$	1.91893	0.292634	0.146317	+	0.989238i	$0.453258\pi$
$44$	0	0
$45$	1.08699	0.162039
$46$	0	0
$47$	9.56543	1.39526	0.697630	−	0.716458i	$-0.254238\pi$
$47$	9.56543	1.39526	0.697630	+	0.716458i	$0.254238\pi$
$48$	0	0
$49$	−0.108279	−0.0154684
$50$	0	0
$51$	0.800477	0.112089
$52$	0	0
$53$	7.74217	1.06347	0.531735	−	0.846911i	$-0.321540\pi$
$53$	7.74217	1.06347	0.531735	+	0.846911i	$0.321540\pi$
$54$	0	0
$55$	−1.64446	−0.221739
$56$	0	0
$57$	7.19435	0.952915
$58$	0	0
$59$	13.0359	1.69713	0.848565	−	0.529092i	$-0.177467\pi$
$59$	13.0359	1.69713	0.848565	+	0.529092i	$0.177467\pi$
$60$	0	0
$61$	−3.86379	−0.494707	−0.247354	−	0.968925i	$-0.579561\pi$
$61$	−3.86379	−0.494707	−0.247354	+	0.968925i	$0.579561\pi$
$62$	0	0
$63$	−2.85359	−0.359518
$64$	0	0
$65$	−2.44254	−0.302959
$66$	0	0
$67$	−11.4566	−1.39965	−0.699824	−	0.714315i	$-0.746738\pi$
$67$	−11.4566	−1.39965	−0.699824	+	0.714315i	$0.746738\pi$
$68$	0	0
$69$	11.3818	1.37021
$70$	0	0
$71$	2.54690	0.302261	0.151131	−	0.988514i	$-0.451709\pi$
$71$	2.54690	0.302261	0.151131	+	0.988514i	$0.451709\pi$
$72$	0	0
$73$	−9.79732	−1.14669	−0.573345	−	0.819314i	$-0.694354\pi$
$73$	−9.79732	−1.14669	−0.573345	+	0.819314i	$0.694354\pi$
$74$	0	0
$75$	−1.38311	−0.159708
$76$	0	0
$77$	4.31704	0.491973
$78$	0	0
$79$	1.81364	0.204050	0.102025	−	0.994782i	$-0.467468\pi$
$79$	1.81364	0.204050	0.102025	+	0.994782i	$0.467468\pi$
$80$	0	0
$81$	−4.55746	−0.506385
$82$	0	0
$83$	−10.9822	−1.20546	−0.602728	−	0.797947i	$-0.705920\pi$
$83$	−10.9822	−1.20546	−0.602728	+	0.797947i	$0.705920\pi$
$84$	0	0
$85$	0.578749	0.0627742
$86$	0	0
$87$	−1.05978	−0.113621
$88$	0	0
$89$	−8.85915	−0.939068	−0.469534	−	0.882914i	$-0.655578\pi$
$89$	−8.85915	−0.939068	−0.469534	+	0.882914i	$0.655578\pi$
$90$	0	0
$91$	6.41217	0.672178
$92$	0	0
$93$	5.82917	0.604456
$94$	0	0
$95$	5.20156	0.533669
$96$	0	0
$97$	−10.5605	−1.07225	−0.536126	−	0.844138i	$-0.680113\pi$
$97$	−10.5605	−1.07225	−0.536126	+	0.844138i	$0.680113\pi$
$98$	0	0
$99$	−1.78751	−0.179652
$100$	0	0
$101$	0.908368	0.0903860	0.0451930	−	0.998978i	$-0.485610\pi$
$101$	0.908368	0.0903860	0.0451930	+	0.998978i	$0.485610\pi$
$102$	0	0
$103$	−18.9017	−1.86244	−0.931221	−	0.364455i	$-0.881255\pi$
$103$	−18.9017	−1.86244	−0.931221	+	0.364455i	$0.881255\pi$
$104$	0	0
$105$	3.63096	0.354346
$106$	0	0
$107$	−9.67742	−0.935552	−0.467776	−	0.883847i	$-0.654945\pi$
$107$	−9.67742	−0.935552	−0.467776	+	0.883847i	$0.654945\pi$
$108$	0	0
$109$	10.9954	1.05316	0.526582	−	0.850124i	$-0.323473\pi$
$109$	10.9954	1.05316	0.526582	+	0.850124i	$0.323473\pi$
$110$	0	0
$111$	−1.38311	−0.131279
$112$	0	0
$113$	−2.26855	−0.213407	−0.106704	−	0.994291i	$-0.534030\pi$
$113$	−2.26855	−0.213407	−0.106704	+	0.994291i	$0.534030\pi$
$114$	0	0
$115$	8.22913	0.767371
$116$	0	0
$117$	−2.65502	−0.245457
$118$	0	0
$119$	−1.51934	−0.139278
$120$	0	0
$121$	−8.29576	−0.754160
$122$	0	0
$123$	2.27447	0.205082
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.5147	−1.82038	−0.910192	−	0.414186i	$-0.864066\pi$
$127$	−20.5147	−1.82038	−0.910192	+	0.414186i	$0.864066\pi$
$128$	0	0
$129$	−2.65410	−0.233681
$130$	0	0
$131$	12.9339	1.13004	0.565021	−	0.825076i	$-0.308868\pi$
$131$	12.9339	1.13004	0.565021	+	0.825076i	$0.308868\pi$
$132$	0	0
$133$	−13.6552	−1.18405
$134$	0	0
$135$	−5.65278	−0.486514
$136$	0	0
$137$	13.2882	1.13529	0.567643	−	0.823275i	$-0.307855\pi$
$137$	13.2882	1.13529	0.567643	+	0.823275i	$0.307855\pi$
$138$	0	0
$139$	−12.0833	−1.02489	−0.512446	−	0.858719i	$-0.671261\pi$
$139$	−12.0833	−1.02489	−0.512446	+	0.858719i	$0.671261\pi$
$140$	0	0
$141$	−13.2301	−1.11417
$142$	0	0
$143$	4.01665	0.335889
$144$	0	0
$145$	−0.766229	−0.0636319
$146$	0	0
$147$	0.149762	0.0123522
$148$	0	0
$149$	−20.2139	−1.65599	−0.827995	−	0.560736i	$-0.810518\pi$
$149$	−20.2139	−1.65599	−0.827995	+	0.560736i	$0.810518\pi$
$150$	0	0
$151$	−15.8330	−1.28847	−0.644237	−	0.764826i	$-0.722825\pi$
$151$	−15.8330	−1.28847	−0.644237	+	0.764826i	$0.722825\pi$
$152$	0	0
$153$	0.629097	0.0508595
$154$	0	0
$155$	4.21452	0.338519
$156$	0	0
$157$	17.0862	1.36363	0.681815	−	0.731525i	$-0.261191\pi$
$157$	17.0862	1.36363	0.681815	+	0.731525i	$0.261191\pi$
$158$	0	0
$159$	−10.7083	−0.849225
$160$	0	0
$161$	−21.6032	−1.70257
$162$	0	0
$163$	−6.82137	−0.534291	−0.267146	−	0.963656i	$-0.586080\pi$
$163$	−6.82137	−0.534291	−0.267146	+	0.963656i	$0.586080\pi$
$164$	0	0
$165$	2.27447	0.177068
$166$	0	0
$167$	4.90916	0.379882	0.189941	−	0.981796i	$-0.439170\pi$
$167$	4.90916	0.379882	0.189941	+	0.981796i	$0.439170\pi$
$168$	0	0
$169$	−7.03402	−0.541078
$170$	0	0
$171$	5.65406	0.432377
$172$	0	0
$173$	−21.2573	−1.61616	−0.808080	−	0.589073i	$-0.799493\pi$
$173$	−21.2573	−1.61616	−0.808080	+	0.589073i	$0.799493\pi$
$174$	0	0
$175$	2.62521	0.198447
$176$	0	0
$177$	−18.0301	−1.35523
$178$	0	0
$179$	7.66910	0.573215	0.286608	−	0.958048i	$-0.407472\pi$
$179$	7.66910	0.573215	0.286608	+	0.958048i	$0.407472\pi$
$180$	0	0
$181$	−10.3972	−0.772817	−0.386409	−	0.922328i	$-0.626285\pi$
$181$	−10.3972	−0.772817	−0.386409	+	0.922328i	$0.626285\pi$
$182$	0	0
$183$	5.34406	0.395044
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−0.951729	−0.0695973
$188$	0	0
$189$	14.8397	1.07943
$190$	0	0
$191$	14.0187	1.01436	0.507178	−	0.861841i	$-0.330689\pi$
$191$	14.0187	1.01436	0.507178	+	0.861841i	$0.330689\pi$
$192$	0	0
$193$	−16.5915	−1.19428	−0.597142	−	0.802136i	$-0.703697\pi$
$193$	−16.5915	−1.19428	−0.597142	+	0.802136i	$0.703697\pi$
$194$	0	0
$195$	3.37831	0.241926
$196$	0	0
$197$	−20.3482	−1.44975	−0.724873	−	0.688882i	$-0.758102\pi$
$197$	−20.3482	−1.44975	−0.724873	+	0.688882i	$0.758102\pi$
$198$	0	0
$199$	11.5118	0.816047	0.408024	−	0.912971i	$-0.366218\pi$
$199$	11.5118	0.816047	0.408024	+	0.912971i	$0.366218\pi$
$200$	0	0
$201$	15.8458	1.11768
$202$	0	0
$203$	2.01151	0.141180
$204$	0	0
$205$	1.64446	0.114854
$206$	0	0
$207$	8.94501	0.621721
$208$	0	0
$209$	−8.55374	−0.591675
$210$	0	0
$211$	1.51976	0.104624	0.0523122	−	0.998631i	$-0.483341\pi$
$211$	1.51976	0.104624	0.0523122	+	0.998631i	$0.483341\pi$
$212$	0	0
$213$	−3.52266	−0.241368
$214$	0	0
$215$	−1.91893	−0.130870
$216$	0	0
$217$	−11.0640	−0.751073
$218$	0	0
$219$	13.5508	0.915679
$220$	0	0
$221$	−1.41362	−0.0950901
$222$	0	0
$223$	−6.89820	−0.461938	−0.230969	−	0.972961i	$-0.574190\pi$
$223$	−6.89820	−0.461938	−0.230969	+	0.972961i	$0.574190\pi$
$224$	0	0
$225$	−1.08699	−0.0724662
$226$	0	0
$227$	26.8277	1.78062	0.890308	−	0.455359i	$-0.150489\pi$
$227$	26.8277	1.78062	0.890308	+	0.455359i	$0.150489\pi$
$228$	0	0
$229$	26.0120	1.71892	0.859462	−	0.511200i	$-0.170799\pi$
$229$	26.0120	1.71892	0.859462	+	0.511200i	$0.170799\pi$
$230$	0	0
$231$	−5.97097	−0.392861
$232$	0	0
$233$	11.3141	0.741212	0.370606	−	0.928790i	$-0.379150\pi$
$233$	11.3141	0.741212	0.370606	+	0.928790i	$0.379150\pi$
$234$	0	0
$235$	−9.56543	−0.623980
$236$	0	0
$237$	−2.50847	−0.162943
$238$	0	0
$239$	−21.2819	−1.37661	−0.688306	−	0.725420i	$-0.741645\pi$
$239$	−21.2819	−1.37661	−0.688306	+	0.725420i	$0.741645\pi$
$240$	0	0
$241$	21.1975	1.36545	0.682726	−	0.730675i	$-0.260794\pi$
$241$	21.1975	1.36545	0.682726	+	0.730675i	$0.260794\pi$
$242$	0	0
$243$	−10.6548	−0.683509
$244$	0	0
$245$	0.108279	0.00691770
$246$	0	0
$247$	−12.7050	−0.808400
$248$	0	0
$249$	15.1897	0.962607
$250$	0	0
$251$	6.06998	0.383134	0.191567	−	0.981480i	$-0.438643\pi$
$251$	6.06998	0.383134	0.191567	+	0.981480i	$0.438643\pi$
$252$	0	0
$253$	−13.5325	−0.850778
$254$	0	0
$255$	−0.800477	−0.0501278
$256$	0	0
$257$	11.2639	0.702623	0.351312	−	0.936259i	$-0.385736\pi$
$257$	11.2639	0.702623	0.351312	+	0.936259i	$0.385736\pi$
$258$	0	0
$259$	2.62521	0.163123
$260$	0	0
$261$	−0.832887	−0.0515544
$262$	0	0
$263$	−27.5622	−1.69956	−0.849780	−	0.527137i	$-0.823265\pi$
$263$	−27.5622	−1.69956	−0.849780	+	0.527137i	$0.823265\pi$
$264$	0	0
$265$	−7.74217	−0.475598
$266$	0	0
$267$	12.2532	0.749885
$268$	0	0
$269$	−22.9078	−1.39672	−0.698358	−	0.715749i	$-0.746085\pi$
$269$	−22.9078	−1.39672	−0.698358	+	0.715749i	$0.746085\pi$
$270$	0	0
$271$	−5.89472	−0.358079	−0.179039	−	0.983842i	$-0.557299\pi$
$271$	−5.89472	−0.358079	−0.179039	+	0.983842i	$0.557299\pi$
$272$	0	0
$273$	−8.86876	−0.536762
$274$	0	0
$275$	1.64446	0.0991645
$276$	0	0
$277$	20.8630	1.25354	0.626769	−	0.779205i	$-0.284377\pi$
$277$	20.8630	1.25354	0.626769	+	0.779205i	$0.284377\pi$
$278$	0	0
$279$	4.58116	0.274267
$280$	0	0
$281$	−0.481953	−0.0287509	−0.0143754	−	0.999897i	$-0.504576\pi$
$281$	−0.481953	−0.0287509	−0.0143754	+	0.999897i	$0.504576\pi$
$282$	0	0
$283$	−0.890931	−0.0529603	−0.0264802	−	0.999649i	$-0.508430\pi$
$283$	−0.890931	−0.0529603	−0.0264802	+	0.999649i	$0.508430\pi$
$284$	0	0
$285$	−7.19435	−0.426157
$286$	0	0
$287$	−4.31704	−0.254827
$288$	0	0
$289$	−16.6650	−0.980297
$290$	0	0
$291$	14.6063	0.856238
$292$	0	0
$293$	−16.5945	−0.969460	−0.484730	−	0.874664i	$-0.661082\pi$
$293$	−16.5945	−0.969460	−0.484730	+	0.874664i	$0.661082\pi$
$294$	0	0
$295$	−13.0359	−0.758979
$296$	0	0
$297$	9.29576	0.539395
$298$	0	0
$299$	−20.1000	−1.16241
$300$	0	0
$301$	5.03760	0.290362
$302$	0	0
$303$	−1.25638	−0.0721770
$304$	0	0
$305$	3.86379	0.221240
$306$	0	0
$307$	−24.7733	−1.41389	−0.706943	−	0.707271i	$-0.749926\pi$
$307$	−24.7733	−1.41389	−0.706943	+	0.707271i	$0.749926\pi$
$308$	0	0
$309$	26.1432	1.48724
$310$	0	0
$311$	−27.1396	−1.53895	−0.769473	−	0.638680i	$-0.779481\pi$
$311$	−27.1396	−1.53895	−0.769473	+	0.638680i	$0.779481\pi$
$312$	0	0
$313$	−8.71332	−0.492506	−0.246253	−	0.969206i	$-0.579199\pi$
$313$	−8.71332	−0.492506	−0.246253	+	0.969206i	$0.579199\pi$
$314$	0	0
$315$	2.85359	0.160781
$316$	0	0
$317$	7.50140	0.421321	0.210660	−	0.977559i	$-0.432439\pi$
$317$	7.50140	0.421321	0.210660	+	0.977559i	$0.432439\pi$
$318$	0	0
$319$	1.26003	0.0705482
$320$	0	0
$321$	13.3850	0.747077
$322$	0	0
$323$	3.01040	0.167503
$324$	0	0
$325$	2.44254	0.135488
$326$	0	0
$327$	−15.2078	−0.840996
$328$	0	0
$329$	25.1112	1.38443
$330$	0	0
$331$	10.4401	0.573841	0.286921	−	0.957954i	$-0.407368\pi$
$331$	10.4401	0.573841	0.286921	+	0.957954i	$0.407368\pi$
$332$	0	0
$333$	−1.08699	−0.0595669
$334$	0	0
$335$	11.4566	0.625942
$336$	0	0
$337$	−11.4990	−0.626389	−0.313194	−	0.949689i	$-0.601399\pi$
$337$	−11.4990	−0.626389	−0.313194	+	0.949689i	$0.601399\pi$
$338$	0	0
$339$	3.13766	0.170414
$340$	0	0
$341$	−6.93060	−0.375313
$342$	0	0
$343$	−18.6607	−1.00758
$344$	0	0
$345$	−11.3818	−0.612777
$346$	0	0
$347$	−19.6061	−1.05251	−0.526257	−	0.850326i	$-0.676405\pi$
$347$	−19.6061	−1.05251	−0.526257	+	0.850326i	$0.676405\pi$
$348$	0	0
$349$	−21.9395	−1.17439	−0.587196	−	0.809445i	$-0.699768\pi$
$349$	−21.9395	−1.17439	−0.587196	+	0.809445i	$0.699768\pi$
$350$	0	0
$351$	13.8071	0.736970
$352$	0	0
$353$	2.11029	0.112319	0.0561597	−	0.998422i	$-0.482114\pi$
$353$	2.11029	0.112319	0.0561597	+	0.998422i	$0.482114\pi$
$354$	0	0
$355$	−2.54690	−0.135175
$356$	0	0
$357$	2.10142	0.111219
$358$	0	0
$359$	6.97594	0.368176	0.184088	−	0.982910i	$-0.441067\pi$
$359$	6.97594	0.368176	0.184088	+	0.982910i	$0.441067\pi$
$360$	0	0
$361$	8.05622	0.424012
$362$	0	0
$363$	11.4740	0.602228
$364$	0	0
$365$	9.79732	0.512815
$366$	0	0
$367$	−9.57842	−0.499989	−0.249995	−	0.968247i	$-0.580429\pi$
$367$	−9.57842	−0.499989	−0.249995	+	0.968247i	$0.580429\pi$
$368$	0	0
$369$	1.78751	0.0930543
$370$	0	0
$371$	20.3248	1.05521
$372$	0	0
$373$	5.34611	0.276811	0.138405	−	0.990376i	$-0.455802\pi$
$373$	5.34611	0.276811	0.138405	+	0.990376i	$0.455802\pi$
$374$	0	0
$375$	1.38311	0.0714237
$376$	0	0
$377$	1.87154	0.0963894
$378$	0	0
$379$	30.8519	1.58476	0.792378	−	0.610031i	$-0.208843\pi$
$379$	30.8519	1.58476	0.792378	+	0.610031i	$0.208843\pi$
$380$	0	0
$381$	28.3742	1.45365
$382$	0	0
$383$	9.29595	0.475001	0.237500	−	0.971387i	$-0.423672\pi$
$383$	9.29595	0.475001	0.237500	+	0.971387i	$0.423672\pi$
$384$	0	0
$385$	−4.31704	−0.220017
$386$	0	0
$387$	−2.08587	−0.106031
$388$	0	0
$389$	19.0668	0.966726	0.483363	−	0.875420i	$-0.339415\pi$
$389$	19.0668	0.966726	0.483363	+	0.875420i	$0.339415\pi$
$390$	0	0
$391$	4.76261	0.240855
$392$	0	0
$393$	−17.8891	−0.902386
$394$	0	0
$395$	−1.81364	−0.0912540
$396$	0	0
$397$	−8.23564	−0.413335	−0.206667	−	0.978411i	$-0.566262\pi$
$397$	−8.23564	−0.413335	−0.206667	+	0.978411i	$0.566262\pi$
$398$	0	0
$399$	18.8867	0.945517
$400$	0	0
$401$	11.2408	0.561339	0.280669	−	0.959804i	$-0.409444\pi$
$401$	11.2408	0.561339	0.280669	+	0.959804i	$0.409444\pi$
$402$	0	0
$403$	−10.2941	−0.512787
$404$	0	0
$405$	4.55746	0.226462
$406$	0	0
$407$	1.64446	0.0815127
$408$	0	0
$409$	22.0826	1.09191	0.545957	−	0.837813i	$-0.316166\pi$
$409$	22.0826	1.09191	0.545957	+	0.837813i	$0.316166\pi$
$410$	0	0
$411$	−18.3791	−0.906574
$412$	0	0
$413$	34.2219	1.68395
$414$	0	0
$415$	10.9822	0.539097
$416$	0	0
$417$	16.7126	0.818419
$418$	0	0
$419$	15.6870	0.766361	0.383181	−	0.923673i	$-0.374829\pi$
$419$	15.6870	0.766361	0.383181	+	0.923673i	$0.374829\pi$
$420$	0	0
$421$	2.52196	0.122913	0.0614564	−	0.998110i	$-0.480425\pi$
$421$	2.52196	0.122913	0.0614564	+	0.998110i	$0.480425\pi$
$422$	0	0
$423$	−10.3976	−0.505546
$424$	0	0
$425$	−0.578749	−0.0280735
$426$	0	0
$427$	−10.1432	−0.490866
$428$	0	0
$429$	−5.55548	−0.268221
$430$	0	0
$431$	−24.9936	−1.20390	−0.601951	−	0.798533i	$-0.705610\pi$
$431$	−24.9936	−1.20390	−0.601951	+	0.798533i	$0.705610\pi$
$432$	0	0
$433$	−18.0649	−0.868146	−0.434073	−	0.900878i	$-0.642924\pi$
$433$	−18.0649	−0.868146	−0.434073	+	0.900878i	$0.642924\pi$
$434$	0	0
$435$	1.05978	0.0508127
$436$	0	0
$437$	42.8043	2.04761
$438$	0	0
$439$	11.2220	0.535595	0.267798	−	0.963475i	$-0.413704\pi$
$439$	11.2220	0.535595	0.267798	+	0.963475i	$0.413704\pi$
$440$	0	0
$441$	0.117699	0.00560470
$442$	0	0
$443$	38.4723	1.82788	0.913938	−	0.405854i	$-0.133026\pi$
$443$	38.4723	1.82788	0.913938	+	0.405854i	$0.133026\pi$
$444$	0	0
$445$	8.85915	0.419964
$446$	0	0
$447$	27.9582	1.32238
$448$	0	0
$449$	39.8327	1.87982	0.939910	−	0.341423i	$-0.110909\pi$
$449$	39.8327	1.87982	0.939910	+	0.341423i	$0.110909\pi$
$450$	0	0
$451$	−2.70424	−0.127338
$452$	0	0
$453$	21.8989	1.02890
$454$	0	0
$455$	−6.41217	−0.300607
$456$	0	0
$457$	−11.8205	−0.552939	−0.276470	−	0.961023i	$-0.589165\pi$
$457$	−11.8205	−0.552939	−0.276470	+	0.961023i	$0.589165\pi$
$458$	0	0
$459$	−3.27154	−0.152703
$460$	0	0
$461$	12.7422	0.593464	0.296732	−	0.954961i	$-0.404103\pi$
$461$	12.7422	0.593464	0.296732	+	0.954961i	$0.404103\pi$
$462$	0	0
$463$	0.694775	0.0322889	0.0161445	−	0.999870i	$-0.494861\pi$
$463$	0.694775	0.0322889	0.0161445	+	0.999870i	$0.494861\pi$
$464$	0	0
$465$	−5.82917	−0.270321
$466$	0	0
$467$	−2.28132	−0.105567	−0.0527834	−	0.998606i	$-0.516809\pi$
$467$	−2.28132	−0.105567	−0.0527834	+	0.998606i	$0.516809\pi$
$468$	0	0
$469$	−30.0760	−1.38878
$470$	0	0
$471$	−23.6322	−1.08892
$472$	0	0
$473$	3.15560	0.145095
$474$	0	0
$475$	−5.20156	−0.238664
$476$	0	0
$477$	−8.41569	−0.385328
$478$	0	0
$479$	25.7331	1.17578	0.587888	−	0.808942i	$-0.299959\pi$
$479$	25.7331	1.17578	0.587888	+	0.808942i	$0.299959\pi$
$480$	0	0
$481$	2.44254	0.111370
$482$	0	0
$483$	29.8797	1.35957
$484$	0	0
$485$	10.5605	0.479526
$486$	0	0
$487$	18.7007	0.847412	0.423706	−	0.905800i	$-0.360729\pi$
$487$	18.7007	0.847412	0.423706	+	0.905800i	$0.360729\pi$
$488$	0	0
$489$	9.43474	0.426654
$490$	0	0
$491$	−18.5675	−0.837939	−0.418970	−	0.908000i	$-0.637609\pi$
$491$	−18.5675	−0.837939	−0.418970	+	0.908000i	$0.637609\pi$
$492$	0	0
$493$	−0.443455	−0.0199722
$494$	0	0
$495$	1.78751	0.0803428
$496$	0	0
$497$	6.68615	0.299915
$498$	0	0
$499$	9.35537	0.418804	0.209402	−	0.977830i	$-0.432848\pi$
$499$	9.35537	0.418804	0.209402	+	0.977830i	$0.432848\pi$
$500$	0	0
$501$	−6.78993	−0.303352
$502$	0	0
$503$	25.6072	1.14177	0.570885	−	0.821030i	$-0.306600\pi$
$503$	25.6072	1.14177	0.570885	+	0.821030i	$0.306600\pi$
$504$	0	0
$505$	−0.908368	−0.0404218
$506$	0	0
$507$	9.72885	0.432074
$508$	0	0
$509$	−32.5502	−1.44276	−0.721382	−	0.692537i	$-0.756493\pi$
$509$	−32.5502	−1.44276	−0.721382	+	0.692537i	$0.756493\pi$
$510$	0	0
$511$	−25.7200	−1.13779
$512$	0	0
$513$	−29.4033	−1.29819
$514$	0	0
$515$	18.9017	0.832909
$516$	0	0
$517$	15.7299	0.691802
$518$	0	0
$519$	29.4012	1.29057
$520$	0	0
$521$	3.09643	0.135657	0.0678285	−	0.997697i	$-0.478393\pi$
$521$	3.09643	0.135657	0.0678285	+	0.997697i	$0.478393\pi$
$522$	0	0
$523$	−24.9299	−1.09011	−0.545054	−	0.838401i	$-0.683491\pi$
$523$	−24.9299	−1.09011	−0.545054	+	0.838401i	$0.683491\pi$
$524$	0	0
$525$	−3.63096	−0.158468
$526$	0	0
$527$	2.43915	0.106251
$528$	0	0
$529$	44.7186	1.94429
$530$	0	0
$531$	−14.1699	−0.614923
$532$	0	0
$533$	−4.01665	−0.173980
$534$	0	0
$535$	9.67742	0.418392
$536$	0	0
$537$	−10.6072	−0.457736
$538$	0	0
$539$	−0.178060	−0.00766960
$540$	0	0
$541$	−37.4914	−1.61188	−0.805940	−	0.591997i	$-0.798340\pi$
$541$	−37.4914	−1.61188	−0.805940	+	0.591997i	$0.798340\pi$
$542$	0	0
$543$	14.3805	0.617127
$544$	0	0
$545$	−10.9954	−0.470990
$546$	0	0
$547$	8.33942	0.356568	0.178284	−	0.983979i	$-0.442945\pi$
$547$	8.33942	0.356568	0.178284	+	0.983979i	$0.442945\pi$
$548$	0	0
$549$	4.19991	0.179248
$550$	0	0
$551$	−3.98559	−0.169792
$552$	0	0
$553$	4.76118	0.202466
$554$	0	0
$555$	1.38311	0.0587100
$556$	0	0
$557$	−28.8556	−1.22265	−0.611327	−	0.791378i	$-0.709364\pi$
$557$	−28.8556	−1.22265	−0.611327	+	0.791378i	$0.709364\pi$
$558$	0	0
$559$	4.68706	0.198241
$560$	0	0
$561$	1.31635	0.0555764
$562$	0	0
$563$	−16.2728	−0.685815	−0.342908	−	0.939369i	$-0.611412\pi$
$563$	−16.2728	−0.685815	−0.342908	+	0.939369i	$0.611412\pi$
$564$	0	0
$565$	2.26855	0.0954386
$566$	0	0
$567$	−11.9643	−0.502453
$568$	0	0
$569$	35.5680	1.49109	0.745544	−	0.666457i	$-0.232190\pi$
$569$	35.5680	1.49109	0.745544	+	0.666457i	$0.232190\pi$
$570$	0	0
$571$	−11.0308	−0.461623	−0.230812	−	0.972998i	$-0.574138\pi$
$571$	−11.0308	−0.461623	−0.230812	+	0.972998i	$0.574138\pi$
$572$	0	0
$573$	−19.3894	−0.810006
$574$	0	0
$575$	−8.22913	−0.343179
$576$	0	0
$577$	−33.6759	−1.40195	−0.700973	−	0.713188i	$-0.747251\pi$
$577$	−33.6759	−1.40195	−0.700973	+	0.713188i	$0.747251\pi$
$578$	0	0
$579$	22.9480	0.953685
$580$	0	0
$581$	−28.8306	−1.19610
$582$	0	0
$583$	12.7317	0.527292
$584$	0	0
$585$	2.65502	0.109772
$586$	0	0
$587$	−4.39536	−0.181416	−0.0907080	−	0.995878i	$-0.528913\pi$
$587$	−4.39536	−0.181416	−0.0907080	+	0.995878i	$0.528913\pi$
$588$	0	0
$589$	21.9221	0.903284
$590$	0	0
$591$	28.1438	1.15768
$592$	0	0
$593$	−0.696484	−0.0286012	−0.0143006	−	0.999898i	$-0.504552\pi$
$593$	−0.696484	−0.0286012	−0.0143006	+	0.999898i	$0.504552\pi$
$594$	0	0
$595$	1.51934	0.0622868
$596$	0	0
$597$	−15.9221	−0.651648
$598$	0	0
$599$	−31.3890	−1.28252	−0.641260	−	0.767324i	$-0.721588\pi$
$599$	−31.3890	−1.28252	−0.641260	+	0.767324i	$0.721588\pi$
$600$	0	0
$601$	41.4460	1.69062	0.845309	−	0.534278i	$-0.179417\pi$
$601$	41.4460	1.69062	0.845309	+	0.534278i	$0.179417\pi$
$602$	0	0
$603$	12.4533	0.507136
$604$	0	0
$605$	8.29576	0.337271
$606$	0	0
$607$	−11.3837	−0.462051	−0.231026	−	0.972948i	$-0.574208\pi$
$607$	−11.3837	−0.462051	−0.231026	+	0.972948i	$0.574208\pi$
$608$	0	0
$609$	−2.78215	−0.112739
$610$	0	0
$611$	23.3639	0.945202
$612$	0	0
$613$	−14.4059	−0.581848	−0.290924	−	0.956746i	$-0.593963\pi$
$613$	−14.4059	−0.581848	−0.290924	+	0.956746i	$0.593963\pi$
$614$	0	0
$615$	−2.27447	−0.0917156
$616$	0	0
$617$	−22.8874	−0.921413	−0.460707	−	0.887552i	$-0.652404\pi$
$617$	−22.8874	−0.921413	−0.460707	+	0.887552i	$0.652404\pi$
$618$	0	0
$619$	40.5987	1.63180	0.815901	−	0.578192i	$-0.196242\pi$
$619$	40.5987	1.63180	0.815901	+	0.578192i	$0.196242\pi$
$620$	0	0
$621$	−46.5175	−1.86668
$622$	0	0
$623$	−23.2571	−0.931777
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	11.8308	0.472477
$628$	0	0
$629$	−0.578749	−0.0230763
$630$	0	0
$631$	31.2970	1.24592	0.622958	−	0.782256i	$-0.285931\pi$
$631$	31.2970	1.24592	0.622958	+	0.782256i	$0.285931\pi$
$632$	0	0
$633$	−2.10200	−0.0835469
$634$	0	0
$635$	20.5147	0.814101
$636$	0	0
$637$	−0.264476	−0.0104789
$638$	0	0
$639$	−2.76846	−0.109519
$640$	0	0
$641$	29.7434	1.17479	0.587397	−	0.809299i	$-0.300153\pi$
$641$	29.7434	1.17479	0.587397	+	0.809299i	$0.300153\pi$
$642$	0	0
$643$	29.2023	1.15163	0.575814	−	0.817581i	$-0.304685\pi$
$643$	29.2023	1.15163	0.575814	+	0.817581i	$0.304685\pi$
$644$	0	0
$645$	2.65410	0.104505
$646$	0	0
$647$	−0.300586	−0.0118173	−0.00590863	−	0.999983i	$-0.501881\pi$
$647$	−0.300586	−0.0118173	−0.00590863	+	0.999983i	$0.501881\pi$
$648$	0	0
$649$	21.4370	0.841475
$650$	0	0
$651$	15.3028	0.599763
$652$	0	0
$653$	29.4692	1.15322	0.576610	−	0.817020i	$-0.304375\pi$
$653$	29.4692	1.15322	0.576610	+	0.817020i	$0.304375\pi$
$654$	0	0
$655$	−12.9339	−0.505370
$656$	0	0
$657$	10.6496	0.415481
$658$	0	0
$659$	1.80662	0.0703757	0.0351879	−	0.999381i	$-0.488797\pi$
$659$	1.80662	0.0703757	0.0351879	+	0.999381i	$0.488797\pi$
$660$	0	0
$661$	50.0378	1.94624	0.973122	−	0.230291i	$-0.0739679\pi$
$661$	50.0378	1.94624	0.973122	+	0.230291i	$0.0739679\pi$
$662$	0	0
$663$	1.95519	0.0759334
$664$	0	0
$665$	13.6552	0.529525
$666$	0	0
$667$	−6.30540	−0.244146
$668$	0	0
$669$	9.54101	0.368877
$670$	0	0
$671$	−6.35383	−0.245287
$672$	0	0
$673$	16.4032	0.632299	0.316149	−	0.948709i	$-0.397610\pi$
$673$	16.4032	0.632299	0.316149	+	0.948709i	$0.397610\pi$
$674$	0	0
$675$	5.65278	0.217576
$676$	0	0
$677$	−36.3118	−1.39558	−0.697789	−	0.716304i	$-0.745833\pi$
$677$	−36.3118	−1.39558	−0.697789	+	0.716304i	$0.745833\pi$
$678$	0	0
$679$	−27.7234	−1.06393
$680$	0	0
$681$	−37.1058	−1.42190
$682$	0	0
$683$	−25.4014	−0.971959	−0.485980	−	0.873970i	$-0.661537\pi$
$683$	−25.4014	−0.971959	−0.485980	+	0.873970i	$0.661537\pi$
$684$	0	0
$685$	−13.2882	−0.507716
$686$	0	0
$687$	−35.9776	−1.37263
$688$	0	0
$689$	18.9105	0.720434
$690$	0	0
$691$	4.47905	0.170391	0.0851956	−	0.996364i	$-0.472848\pi$
$691$	4.47905	0.170391	0.0851956	+	0.996364i	$0.472848\pi$
$692$	0	0
$693$	−4.69260	−0.178257
$694$	0	0
$695$	12.0833	0.458346
$696$	0	0
$697$	0.951729	0.0360493
$698$	0	0
$699$	−15.6487	−0.591889
$700$	0	0
$701$	−20.9639	−0.791795	−0.395898	−	0.918295i	$-0.629567\pi$
$701$	−20.9639	−0.791795	−0.395898	+	0.918295i	$0.629567\pi$
$702$	0	0
$703$	−5.20156	−0.196181
$704$	0	0
$705$	13.2301	0.498274
$706$	0	0
$707$	2.38466	0.0896842
$708$	0	0
$709$	−39.5543	−1.48549	−0.742746	−	0.669573i	$-0.766477\pi$
$709$	−39.5543	−1.48549	−0.742746	+	0.669573i	$0.766477\pi$
$710$	0	0
$711$	−1.97141	−0.0739337
$712$	0	0
$713$	34.6819	1.29885
$714$	0	0
$715$	−4.01665	−0.150214
$716$	0	0
$717$	29.4353	1.09928
$718$	0	0
$719$	44.0927	1.64438	0.822190	−	0.569213i	$-0.192752\pi$
$719$	44.0927	1.64438	0.822190	+	0.569213i	$0.192752\pi$
$720$	0	0
$721$	−49.6210	−1.84798
$722$	0	0
$723$	−29.3186	−1.09037
$724$	0	0
$725$	0.766229	0.0284570
$726$	0	0
$727$	34.7329	1.28817	0.644086	−	0.764953i	$-0.277238\pi$
$727$	34.7329	1.28817	0.644086	+	0.764953i	$0.277238\pi$
$728$	0	0
$729$	28.4093	1.05220
$730$	0	0
$731$	−1.11058	−0.0410763
$732$	0	0
$733$	19.7348	0.728921	0.364461	−	0.931219i	$-0.381253\pi$
$733$	19.7348	0.728921	0.364461	+	0.931219i	$0.381253\pi$
$734$	0	0
$735$	−0.149762	−0.00552407
$736$	0	0
$737$	−18.8399	−0.693977
$738$	0	0
$739$	−26.1822	−0.963128	−0.481564	−	0.876411i	$-0.659931\pi$
$739$	−26.1822	−0.963128	−0.481564	+	0.876411i	$0.659931\pi$
$740$	0	0
$741$	17.5725	0.645541
$742$	0	0
$743$	−38.3253	−1.40602	−0.703010	−	0.711180i	$-0.748161\pi$
$743$	−38.3253	−1.40602	−0.703010	+	0.711180i	$0.748161\pi$
$744$	0	0
$745$	20.2139	0.740581
$746$	0	0
$747$	11.9376	0.436774
$748$	0	0
$749$	−25.4053	−0.928288
$750$	0	0
$751$	−24.2638	−0.885399	−0.442699	−	0.896670i	$-0.645979\pi$
$751$	−24.2638	−0.885399	−0.442699	+	0.896670i	$0.645979\pi$
$752$	0	0
$753$	−8.39549	−0.305948
$754$	0	0
$755$	15.8330	0.576224
$756$	0	0
$757$	−9.26589	−0.336775	−0.168387	−	0.985721i	$-0.553856\pi$
$757$	−9.26589	−0.336775	−0.168387	+	0.985721i	$0.553856\pi$
$758$	0	0
$759$	18.7169	0.679382
$760$	0	0
$761$	15.8829	0.575753	0.287877	−	0.957667i	$-0.407051\pi$
$761$	15.8829	0.575753	0.287877	+	0.957667i	$0.407051\pi$
$762$	0	0
$763$	28.8651	1.04499
$764$	0	0
$765$	−0.629097	−0.0227451
$766$	0	0
$767$	31.8406	1.14970
$768$	0	0
$769$	−16.2832	−0.587188	−0.293594	−	0.955930i	$-0.594851\pi$
$769$	−16.2832	−0.587188	−0.293594	+	0.955930i	$0.594851\pi$
$770$	0	0
$771$	−15.5793	−0.561074
$772$	0	0
$773$	3.98058	0.143172	0.0715858	−	0.997434i	$-0.477194\pi$
$773$	3.98058	0.143172	0.0715858	+	0.997434i	$0.477194\pi$
$774$	0	0
$775$	−4.21452	−0.151390
$776$	0	0
$777$	−3.63096	−0.130260
$778$	0	0
$779$	8.55374	0.306470
$780$	0	0
$781$	4.18827	0.149868
$782$	0	0
$783$	4.33133	0.154789
$784$	0	0
$785$	−17.0862	−0.609834
$786$	0	0
$787$	5.01650	0.178819	0.0894094	−	0.995995i	$-0.471502\pi$
$787$	5.01650	0.178819	0.0894094	+	0.995995i	$0.471502\pi$
$788$	0	0
$789$	38.1217	1.35717
$790$	0	0
$791$	−5.95541	−0.211750
$792$	0	0
$793$	−9.43744	−0.335133
$794$	0	0
$795$	10.7083	0.379785
$796$	0	0
$797$	2.99628	0.106134	0.0530668	−	0.998591i	$-0.483100\pi$
$797$	2.99628	0.106134	0.0530668	+	0.998591i	$0.483100\pi$
$798$	0	0
$799$	−5.53599	−0.195849
$800$	0	0
$801$	9.62984	0.340254
$802$	0	0
$803$	−16.1113	−0.568555
$804$	0	0
$805$	21.6032	0.761412
$806$	0	0
$807$	31.6842	1.11534
$808$	0	0
$809$	17.7054	0.622488	0.311244	−	0.950330i	$-0.399254\pi$
$809$	17.7054	0.622488	0.311244	+	0.950330i	$0.399254\pi$
$810$	0	0
$811$	25.3743	0.891011	0.445505	−	0.895279i	$-0.353024\pi$
$811$	25.3743	0.891011	0.445505	+	0.895279i	$0.353024\pi$
$812$	0	0
$813$	8.15307	0.285941
$814$	0	0
$815$	6.82137	0.238942
$816$	0	0
$817$	−9.98144	−0.349206
$818$	0	0
$819$	−6.96998	−0.243551
$820$	0	0
$821$	15.2958	0.533826	0.266913	−	0.963721i	$-0.413996\pi$
$821$	15.2958	0.533826	0.266913	+	0.963721i	$0.413996\pi$
$822$	0	0
$823$	−44.7798	−1.56092	−0.780462	−	0.625203i	$-0.785016\pi$
$823$	−44.7798	−1.56092	−0.780462	+	0.625203i	$0.785016\pi$
$824$	0	0
$825$	−2.27447	−0.0791870
$826$	0	0
$827$	13.6099	0.473263	0.236631	−	0.971600i	$-0.423957\pi$
$827$	13.6099	0.473263	0.236631	+	0.971600i	$0.423957\pi$
$828$	0	0
$829$	15.3300	0.532434	0.266217	−	0.963913i	$-0.414226\pi$
$829$	15.3300	0.532434	0.266217	+	0.963913i	$0.414226\pi$
$830$	0	0
$831$	−28.8560	−1.00100
$832$	0	0
$833$	0.0626665	0.00217126
$834$	0	0
$835$	−4.90916	−0.169888
$836$	0	0
$837$	−23.8238	−0.823470
$838$	0	0
$839$	−23.5040	−0.811447	−0.405723	−	0.913996i	$-0.632980\pi$
$839$	−23.5040	−0.811447	−0.405723	+	0.913996i	$0.632980\pi$
$840$	0	0
$841$	−28.4129	−0.979755
$842$	0	0
$843$	0.666596	0.0229588
$844$	0	0
$845$	7.03402	0.241978
$846$	0	0
$847$	−21.7781	−0.748304
$848$	0	0
$849$	1.23226	0.0422910
$850$	0	0
$851$	−8.22913	−0.282091
$852$	0	0
$853$	−16.6766	−0.570996	−0.285498	−	0.958379i	$-0.592159\pi$
$853$	−16.6766	−0.570996	−0.285498	+	0.958379i	$0.592159\pi$
$854$	0	0
$855$	−5.65406	−0.193365
$856$	0	0
$857$	−48.2788	−1.64917	−0.824585	−	0.565737i	$-0.808592\pi$
$857$	−48.2788	−1.64917	−0.824585	+	0.565737i	$0.808592\pi$
$858$	0	0
$859$	30.5872	1.04362	0.521812	−	0.853061i	$-0.325256\pi$
$859$	30.5872	1.04362	0.521812	+	0.853061i	$0.325256\pi$
$860$	0	0
$861$	5.97097	0.203490
$862$	0	0
$863$	30.2802	1.03075	0.515374	−	0.856965i	$-0.327653\pi$
$863$	30.2802	1.03075	0.515374	+	0.856965i	$0.327653\pi$
$864$	0	0
$865$	21.2573	0.722769
$866$	0	0
$867$	23.0497	0.782808
$868$	0	0
$869$	2.98245	0.101173
$870$	0	0
$871$	−27.9832	−0.948174
$872$	0	0
$873$	11.4792	0.388510
$874$	0	0
$875$	−2.62521	−0.0887483
$876$	0	0
$877$	21.4151	0.723137	0.361569	−	0.932345i	$-0.382241\pi$
$877$	21.4151	0.723137	0.361569	+	0.932345i	$0.382241\pi$
$878$	0	0
$879$	22.9521	0.774154
$880$	0	0
$881$	1.54785	0.0521484	0.0260742	−	0.999660i	$-0.491699\pi$
$881$	1.54785	0.0521484	0.0260742	+	0.999660i	$0.491699\pi$
$882$	0	0
$883$	−10.5279	−0.354291	−0.177146	−	0.984185i	$-0.556686\pi$
$883$	−10.5279	−0.354291	−0.177146	+	0.984185i	$0.556686\pi$
$884$	0	0
$885$	18.0301	0.606077
$886$	0	0
$887$	−17.9508	−0.602730	−0.301365	−	0.953509i	$-0.597442\pi$
$887$	−17.9508	−0.602730	−0.301365	+	0.953509i	$0.597442\pi$
$888$	0	0
$889$	−53.8553	−1.80625
$890$	0	0
$891$	−7.49456	−0.251077
$892$	0	0
$893$	−49.7551	−1.66499
$894$	0	0
$895$	−7.66910	−0.256350
$896$	0	0
$897$	27.8005	0.928233
$898$	0	0
$899$	−3.22929	−0.107703
$900$	0	0
$901$	−4.48078	−0.149276
$902$	0	0
$903$	−6.96757	−0.231866
$904$	0	0
$905$	10.3972	0.345614
$906$	0	0
$907$	20.8340	0.691782	0.345891	−	0.938275i	$-0.387577\pi$
$907$	20.8340	0.691782	0.345891	+	0.938275i	$0.387577\pi$
$908$	0	0
$909$	−0.987390	−0.0327497
$910$	0	0
$911$	−29.5904	−0.980373	−0.490186	−	0.871618i	$-0.663071\pi$
$911$	−29.5904	−0.980373	−0.490186	+	0.871618i	$0.663071\pi$
$912$	0	0
$913$	−18.0598	−0.597693
$914$	0	0
$915$	−5.34406	−0.176669
$916$	0	0
$917$	33.9543	1.12127
$918$	0	0
$919$	−22.2046	−0.732462	−0.366231	−	0.930524i	$-0.619352\pi$
$919$	−22.2046	−0.732462	−0.366231	+	0.930524i	$0.619352\pi$
$920$	0	0
$921$	34.2643	1.12905
$922$	0	0
$923$	6.22090	0.204763
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	20.5460	0.674821
$928$	0	0
$929$	−12.6544	−0.415178	−0.207589	−	0.978216i	$-0.566562\pi$
$929$	−12.6544	−0.415178	−0.207589	+	0.978216i	$0.566562\pi$
$930$	0	0
$931$	0.563220	0.0184588
$932$	0	0
$933$	37.5372	1.22891
$934$	0	0
$935$	0.951729	0.0311249
$936$	0	0
$937$	59.1488	1.93231	0.966154	−	0.257968i	$-0.0830528\pi$
$937$	59.1488	1.93231	0.966154	+	0.257968i	$0.0830528\pi$
$938$	0	0
$939$	12.0515	0.393287
$940$	0	0
$941$	20.7402	0.676110	0.338055	−	0.941126i	$-0.390231\pi$
$941$	20.7402	0.676110	0.338055	+	0.941126i	$0.390231\pi$
$942$	0	0
$943$	13.5325	0.440677
$944$	0	0
$945$	−14.8397	−0.482736
$946$	0	0
$947$	48.0626	1.56182	0.780912	−	0.624641i	$-0.214755\pi$
$947$	48.0626	1.56182	0.780912	+	0.624641i	$0.214755\pi$
$948$	0	0
$949$	−23.9303	−0.776810
$950$	0	0
$951$	−10.3753	−0.336442
$952$	0	0
$953$	0.993315	0.0321766	0.0160883	−	0.999871i	$-0.494879\pi$
$953$	0.993315	0.0321766	0.0160883	+	0.999871i	$0.494879\pi$
$954$	0	0
$955$	−14.0187	−0.453634
$956$	0	0
$957$	−1.74277	−0.0563357
$958$	0	0
$959$	34.8843	1.12647
$960$	0	0
$961$	−13.2378	−0.427026
$962$	0	0
$963$	10.5193	0.338980
$964$	0	0
$965$	16.5915	0.534100
$966$	0	0
$967$	31.4163	1.01028	0.505140	−	0.863037i	$-0.331441\pi$
$967$	31.4163	1.01028	0.505140	+	0.863037i	$0.331441\pi$
$968$	0	0
$969$	−4.16373	−0.133758
$970$	0	0
$971$	36.6023	1.17462	0.587312	−	0.809361i	$-0.300186\pi$
$971$	36.6023	1.17462	0.587312	+	0.809361i	$0.300186\pi$
$972$	0	0
$973$	−31.7212	−1.01693
$974$	0	0
$975$	−3.37831	−0.108192
$976$	0	0
$977$	4.13108	0.132165	0.0660825	−	0.997814i	$-0.478950\pi$
$977$	4.13108	0.132165	0.0660825	+	0.997814i	$0.478950\pi$
$978$	0	0
$979$	−14.5685	−0.465611
$980$	0	0
$981$	−11.9519	−0.381594
$982$	0	0
$983$	−19.1236	−0.609947	−0.304974	−	0.952361i	$-0.598648\pi$
$983$	−19.1236	−0.609947	−0.304974	+	0.952361i	$0.598648\pi$
$984$	0	0
$985$	20.3482	0.648346
$986$	0	0
$987$	−34.7317	−1.10552
$988$	0	0
$989$	−15.7911	−0.502129
$990$	0	0
$991$	−3.96555	−0.125970	−0.0629850	−	0.998014i	$-0.520062\pi$
$991$	−3.96555	−0.125970	−0.0629850	+	0.998014i	$0.520062\pi$
$992$	0	0
$993$	−14.4399	−0.458236
$994$	0	0
$995$	−11.5118	−0.364947
$996$	0	0
$997$	29.1697	0.923813	0.461906	−	0.886929i	$-0.347166\pi$
$997$	29.1697	0.923813	0.461906	+	0.886929i	$0.347166\pi$
$998$	0	0
$999$	5.65278	0.178846

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.w.1.3		5
4.3	odd	2		185.2.a.e.1.4	✓	5
12.11	even	2		1665.2.a.p.1.2		5
20.3	even	4		925.2.b.f.149.3		10
20.7	even	4		925.2.b.f.149.8		10
20.19	odd	2		925.2.a.f.1.2		5
28.27	even	2		9065.2.a.k.1.4		5
60.59	even	2		8325.2.a.ch.1.4		5
148.147	odd	2		6845.2.a.f.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.e.1.4	✓	5	4.3	odd	2
925.2.a.f.1.2		5	20.19	odd	2
925.2.b.f.149.3		10	20.3	even	4
925.2.b.f.149.8		10	20.7	even	4
1665.2.a.p.1.2		5	12.11	even	2
2960.2.a.w.1.3		5	1.1	even	1	trivial
6845.2.a.f.1.2		5	148.147	odd	2
8325.2.a.ch.1.4		5	60.59	even	2
9065.2.a.k.1.4		5	28.27	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)

\(f(q)\)	\(=\)	\(q-1.38311 q^{3} -1.00000 q^{5} +2.62521 q^{7} -1.08699 q^{9} +O(q^{10})\)	\(q-1.38311 q^{3} -1.00000 q^{5} +2.62521 q^{7} -1.08699 q^{9} +1.64446 q^{11} +2.44254 q^{13} +1.38311 q^{15} -0.578749 q^{17} -5.20156 q^{19} -3.63096 q^{21} -8.22913 q^{23} +1.00000 q^{25} +5.65278 q^{27} +0.766229 q^{29} -4.21452 q^{31} -2.27447 q^{33} -2.62521 q^{35} +1.00000 q^{37} -3.37831 q^{39} -1.64446 q^{41} +1.91893 q^{43} +1.08699 q^{45} +9.56543 q^{47} -0.108279 q^{49} +0.800477 q^{51} +7.74217 q^{53} -1.64446 q^{55} +7.19435 q^{57} +13.0359 q^{59} -3.86379 q^{61} -2.85359 q^{63} -2.44254 q^{65} -11.4566 q^{67} +11.3818 q^{69} +2.54690 q^{71} -9.79732 q^{73} -1.38311 q^{75} +4.31704 q^{77} +1.81364 q^{79} -4.55746 q^{81} -10.9822 q^{83} +0.578749 q^{85} -1.05978 q^{87} -8.85915 q^{89} +6.41217 q^{91} +5.82917 q^{93} +5.20156 q^{95} -10.5605 q^{97} -1.78751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 - 5 * q^5 - 11 * q^7 + 6 * q^9	\( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41} - 10 q^{43} - 6 q^{45} - 7 q^{47} + 22 q^{49} + 2 q^{51} - q^{53} - 5 q^{55} - 8 q^{57} + 30 q^{59} - 14 q^{61} - 30 q^{63} - 4 q^{65} - 24 q^{67} + 8 q^{69} + 7 q^{71} + 5 q^{73} - 3 q^{75} - 17 q^{77} - 28 q^{79} - 31 q^{81} - 27 q^{83} - 36 q^{87} + 6 q^{89} + 14 q^{91} - 22 q^{93} - 4 q^{95} - 26 q^{97} + 10 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 5 * q^5 - 11 * q^7 + 6 * q^9 + 5 * q^11 + 4 * q^13 + 3 * q^15 + 4 * q^19 + 3 * q^21 - 4 * q^23 + 5 * q^25 - 3 * q^27 - 4 * q^29 - 8 * q^31 + 5 * q^33 + 11 * q^35 + 5 * q^37 - 2 * q^39 - 5 * q^41 - 10 * q^43 - 6 * q^45 - 7 * q^47 + 22 * q^49 + 2 * q^51 - q^53 - 5 * q^55 - 8 * q^57 + 30 * q^59 - 14 * q^61 - 30 * q^63 - 4 * q^65 - 24 * q^67 + 8 * q^69 + 7 * q^71 + 5 * q^73 - 3 * q^75 - 17 * q^77 - 28 * q^79 - 31 * q^81 - 27 * q^83 - 36 * q^87 + 6 * q^89 + 14 * q^91 - 22 * q^93 - 4 * q^95 - 26 * q^97 + 10 * q^99

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