LMFDB - Embedded newform 2960.2.a.y.1.4

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:	$5$
Coefficient field:	5.5.6397264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - 2x^{2} + 14x - 4 $ x^5 - 10x^3 - 2x^2 + 14*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.925120$ 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)

$f(q)$	$=$	$q+0.925120 q^{3} -1.00000 q^{5} -0.763238 q^{7} -2.14415 q^{9} +O(q^{10})$	$q+0.925120 q^{3} -1.00000 q^{5} -0.763238 q^{7} -2.14415 q^{9} +6.44732 q^{11} -2.67063 q^{13} -0.925120 q^{15} +5.30316 q^{17} -4.88966 q^{19} -0.706087 q^{21} -0.149760 q^{23} +1.00000 q^{25} -4.75896 q^{27} +5.21771 q^{29} -0.912998 q^{31} +5.96454 q^{33} +0.763238 q^{35} +1.00000 q^{37} -2.47065 q^{39} +10.7711 q^{41} +8.12355 q^{43} +2.14415 q^{45} -3.51008 q^{47} -6.41747 q^{49} +4.90606 q^{51} +10.5971 q^{53} -6.44732 q^{55} -4.52352 q^{57} -4.45447 q^{59} -8.26210 q^{61} +1.63650 q^{63} +2.67063 q^{65} +6.88966 q^{67} -0.138546 q^{69} -2.03546 q^{71} +9.81478 q^{73} +0.925120 q^{75} -4.92084 q^{77} +1.83384 q^{79} +2.02985 q^{81} -1.03942 q^{83} -5.30316 q^{85} +4.82700 q^{87} +16.9301 q^{89} +2.03833 q^{91} -0.844633 q^{93} +4.88966 q^{95} -5.18225 q^{97} -13.8240 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * 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
$3$	0.925120	0.534118	0.267059	−	0.963680i	$-0.413948\pi$
$3$	0.925120	0.534118	0.267059	+	0.963680i	$0.413948\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.763238	−0.288477	−0.144239	−	0.989543i	$-0.546073\pi$
$7$	−0.763238	−0.288477	−0.144239	+	0.989543i	$0.546073\pi$
$8$	0	0
$9$	−2.14415	−0.714718
$10$	0	0
$11$	6.44732	1.94394	0.971970	−	0.235106i	$-0.0755438\pi$
$11$	6.44732	1.94394	0.971970	+	0.235106i	$0.0755438\pi$
$12$	0	0
$13$	−2.67063	−0.740699	−0.370350	−	0.928892i	$-0.620762\pi$
$13$	−2.67063	−0.740699	−0.370350	+	0.928892i	$0.620762\pi$
$14$	0	0
$15$	−0.925120	−0.238865
$16$	0	0
$17$	5.30316	1.28621	0.643103	−	0.765780i	$-0.277647\pi$
$17$	5.30316	1.28621	0.643103	+	0.765780i	$0.277647\pi$
$18$	0	0
$19$	−4.88966	−1.12177	−0.560883	−	0.827895i	$-0.689538\pi$
$19$	−4.88966	−1.12177	−0.560883	+	0.827895i	$0.689538\pi$
$20$	0	0
$21$	−0.706087	−0.154081
$22$	0	0
$23$	−0.149760	−0.0312271	−0.0156136	−	0.999878i	$-0.504970\pi$
$23$	−0.149760	−0.0312271	−0.0156136	+	0.999878i	$0.504970\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.75896	−0.915862
$28$	0	0
$29$	5.21771	0.968904	0.484452	−	0.874818i	$-0.339019\pi$
$29$	5.21771	0.968904	0.484452	+	0.874818i	$0.339019\pi$
$30$	0	0
$31$	−0.912998	−0.163979	−0.0819897	−	0.996633i	$-0.526127\pi$
$31$	−0.912998	−0.163979	−0.0819897	+	0.996633i	$0.526127\pi$
$32$	0	0
$33$	5.96454	1.03829
$34$	0	0
$35$	0.763238	0.129011
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−2.47065	−0.395621
$40$	0	0
$41$	10.7711	1.68216	0.841080	−	0.540911i	$-0.181920\pi$
$41$	10.7711	1.68216	0.841080	+	0.540911i	$0.181920\pi$
$42$	0	0
$43$	8.12355	1.23883	0.619415	−	0.785064i	$-0.287370\pi$
$43$	8.12355	1.23883	0.619415	+	0.785064i	$0.287370\pi$
$44$	0	0
$45$	2.14415	0.319631
$46$	0	0
$47$	−3.51008	−0.511997	−0.255999	−	0.966677i	$-0.582404\pi$
$47$	−3.51008	−0.511997	−0.255999	+	0.966677i	$0.582404\pi$
$48$	0	0
$49$	−6.41747	−0.916781
$50$	0	0
$51$	4.90606	0.686986
$52$	0	0
$53$	10.5971	1.45562	0.727810	−	0.685779i	$-0.240538\pi$
$53$	10.5971	1.45562	0.727810	+	0.685779i	$0.240538\pi$
$54$	0	0
$55$	−6.44732	−0.869356
$56$	0	0
$57$	−4.52352	−0.599156
$58$	0	0
$59$	−4.45447	−0.579922	−0.289961	−	0.957038i	$-0.593642\pi$
$59$	−4.45447	−0.579922	−0.289961	+	0.957038i	$0.593642\pi$
$60$	0	0
$61$	−8.26210	−1.05785	−0.528927	−	0.848667i	$-0.677405\pi$
$61$	−8.26210	−1.05785	−0.528927	+	0.848667i	$0.677405\pi$
$62$	0	0
$63$	1.63650	0.206180
$64$	0	0
$65$	2.67063	0.331251
$66$	0	0
$67$	6.88966	0.841706	0.420853	−	0.907129i	$-0.361731\pi$
$67$	6.88966	0.841706	0.420853	+	0.907129i	$0.361731\pi$
$68$	0	0
$69$	−0.138546	−0.0166790
$70$	0	0
$71$	−2.03546	−0.241564	−0.120782	−	0.992679i	$-0.538540\pi$
$71$	−2.03546	−0.241564	−0.120782	+	0.992679i	$0.538540\pi$
$72$	0	0
$73$	9.81478	1.14873	0.574367	−	0.818598i	$-0.305248\pi$
$73$	9.81478	1.14873	0.574367	+	0.818598i	$0.305248\pi$
$74$	0	0
$75$	0.925120	0.106824
$76$	0	0
$77$	−4.92084	−0.560782
$78$	0	0
$79$	1.83384	0.206323	0.103161	−	0.994665i	$-0.467104\pi$
$79$	1.83384	0.206323	0.103161	+	0.994665i	$0.467104\pi$
$80$	0	0
$81$	2.02985	0.225539
$82$	0	0
$83$	−1.03942	−0.114091	−0.0570457	−	0.998372i	$-0.518168\pi$
$83$	−1.03942	−0.114091	−0.0570457	+	0.998372i	$0.518168\pi$
$84$	0	0
$85$	−5.30316	−0.575209
$86$	0	0
$87$	4.82700	0.517509
$88$	0	0
$89$	16.9301	1.79459	0.897293	−	0.441435i	$-0.145531\pi$
$89$	16.9301	1.79459	0.897293	+	0.441435i	$0.145531\pi$
$90$	0	0
$91$	2.03833	0.213675
$92$	0	0
$93$	−0.844633	−0.0875844
$94$	0	0
$95$	4.88966	0.501669
$96$	0	0
$97$	−5.18225	−0.526178	−0.263089	−	0.964772i	$-0.584741\pi$
$97$	−5.18225	−0.526178	−0.263089	+	0.964772i	$0.584741\pi$
$98$	0	0
$99$	−13.8240	−1.38937
$100$	0	0
$101$	8.84463	0.880074	0.440037	−	0.897980i	$-0.354965\pi$
$101$	8.84463	0.880074	0.440037	+	0.897980i	$0.354965\pi$
$102$	0	0
$103$	−5.30580	−0.522796	−0.261398	−	0.965231i	$-0.584184\pi$
$103$	−5.30580	−0.522796	−0.261398	+	0.965231i	$0.584184\pi$
$104$	0	0
$105$	0.706087	0.0689071
$106$	0	0
$107$	11.3249	1.09482	0.547408	−	0.836866i	$-0.315615\pi$
$107$	11.3249	1.09482	0.547408	+	0.836866i	$0.315615\pi$
$108$	0	0
$109$	7.24460	0.693907	0.346954	−	0.937882i	$-0.387216\pi$
$109$	7.24460	0.693907	0.346954	+	0.937882i	$0.387216\pi$
$110$	0	0
$111$	0.925120	0.0878085
$112$	0	0
$113$	14.3207	1.34717	0.673587	−	0.739108i	$-0.264753\pi$
$113$	14.3207	1.34717	0.673587	+	0.739108i	$0.264753\pi$
$114$	0	0
$115$	0.149760	0.0139652
$116$	0	0
$117$	5.72624	0.529391
$118$	0	0
$119$	−4.04758	−0.371041
$120$	0	0
$121$	30.5679	2.77890
$122$	0	0
$123$	9.96454	0.898473
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.8310	1.04983	0.524914	−	0.851155i	$-0.324097\pi$
$127$	11.8310	1.04983	0.524914	+	0.851155i	$0.324097\pi$
$128$	0	0
$129$	7.51526	0.661682
$130$	0	0
$131$	−0.127834	−0.0111689	−0.00558444	−	0.999984i	$-0.501778\pi$
$131$	−0.127834	−0.0111689	−0.00558444	+	0.999984i	$0.501778\pi$
$132$	0	0
$133$	3.73198	0.323604
$134$	0	0
$135$	4.75896	0.409586
$136$	0	0
$137$	5.81478	0.496790	0.248395	−	0.968659i	$-0.420097\pi$
$137$	5.81478	0.496790	0.248395	+	0.968659i	$0.420097\pi$
$138$	0	0
$139$	−21.1922	−1.79750	−0.898749	−	0.438463i	$-0.855523\pi$
$139$	−21.1922	−1.79750	−0.898749	+	0.438463i	$0.855523\pi$
$140$	0	0
$141$	−3.24724	−0.273467
$142$	0	0
$143$	−17.2184	−1.43987
$144$	0	0
$145$	−5.21771	−0.433307
$146$	0	0
$147$	−5.93693	−0.489670
$148$	0	0
$149$	−2.08546	−0.170847	−0.0854237	−	0.996345i	$-0.527224\pi$
$149$	−2.08546	−0.170847	−0.0854237	+	0.996345i	$0.527224\pi$
$150$	0	0
$151$	−4.13855	−0.336790	−0.168395	−	0.985720i	$-0.553858\pi$
$151$	−4.13855	−0.336790	−0.168395	+	0.985720i	$0.553858\pi$
$152$	0	0
$153$	−11.3708	−0.919274
$154$	0	0
$155$	0.912998	0.0733338
$156$	0	0
$157$	−4.42307	−0.353000	−0.176500	−	0.984301i	$-0.556478\pi$
$157$	−4.42307	−0.353000	−0.176500	+	0.984301i	$0.556478\pi$
$158$	0	0
$159$	9.80357	0.777474
$160$	0	0
$161$	0.114303	0.00900830
$162$	0	0
$163$	13.9795	1.09496	0.547481	−	0.836818i	$-0.315587\pi$
$163$	13.9795	1.09496	0.547481	+	0.836818i	$0.315587\pi$
$164$	0	0
$165$	−5.96454	−0.464339
$166$	0	0
$167$	7.01750	0.543030	0.271515	−	0.962434i	$-0.412475\pi$
$167$	7.01750	0.543030	0.271515	+	0.962434i	$0.412475\pi$
$168$	0	0
$169$	−5.86774	−0.451364
$170$	0	0
$171$	10.4842	0.801746
$172$	0	0
$173$	−0.823027	−0.0625736	−0.0312868	−	0.999510i	$-0.509961\pi$
$173$	−0.823027	−0.0625736	−0.0312868	+	0.999510i	$0.509961\pi$
$174$	0	0
$175$	−0.763238	−0.0576954
$176$	0	0
$177$	−4.12092	−0.309747
$178$	0	0
$179$	−1.54575	−0.115535	−0.0577673	−	0.998330i	$-0.518398\pi$
$179$	−1.54575	−0.115535	−0.0577673	+	0.998330i	$0.518398\pi$
$180$	0	0
$181$	8.20285	0.609713	0.304856	−	0.952398i	$-0.401392\pi$
$181$	8.20285	0.609713	0.304856	+	0.952398i	$0.401392\pi$
$182$	0	0
$183$	−7.64343	−0.565019
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	34.1912	2.50031
$188$	0	0
$189$	3.63222	0.264205
$190$	0	0
$191$	22.2642	1.61098	0.805491	−	0.592609i	$-0.201902\pi$
$191$	22.2642	1.61098	0.805491	+	0.592609i	$0.201902\pi$
$192$	0	0
$193$	−23.7893	−1.71239	−0.856195	−	0.516654i	$-0.827178\pi$
$193$	−23.7893	−1.71239	−0.856195	+	0.516654i	$0.827178\pi$
$194$	0	0
$195$	2.47065	0.176927
$196$	0	0
$197$	−3.20845	−0.228593	−0.114296	−	0.993447i	$-0.536461\pi$
$197$	−3.20845	−0.228593	−0.114296	+	0.993447i	$0.536461\pi$
$198$	0	0
$199$	−15.2403	−1.08036	−0.540179	−	0.841550i	$-0.681643\pi$
$199$	−15.2403	−1.08036	−0.540179	+	0.841550i	$0.681643\pi$
$200$	0	0
$201$	6.37377	0.449571
$202$	0	0
$203$	−3.98235	−0.279506
$204$	0	0
$205$	−10.7711	−0.752285
$206$	0	0
$207$	0.321108	0.0223186
$208$	0	0
$209$	−31.5252	−2.18064
$210$	0	0
$211$	−5.93834	−0.408812	−0.204406	−	0.978886i	$-0.565526\pi$
$211$	−5.93834	−0.408812	−0.204406	+	0.978886i	$0.565526\pi$
$212$	0	0
$213$	−1.88304	−0.129024
$214$	0	0
$215$	−8.12355	−0.554022
$216$	0	0
$217$	0.696835	0.0473043
$218$	0	0
$219$	9.07985	0.613560
$220$	0	0
$221$	−14.1628	−0.952692
$222$	0	0
$223$	19.6010	1.31258	0.656291	−	0.754507i	$-0.272124\pi$
$223$	19.6010	1.31258	0.656291	+	0.754507i	$0.272124\pi$
$224$	0	0
$225$	−2.14415	−0.142944
$226$	0	0
$227$	−2.09666	−0.139160	−0.0695800	−	0.997576i	$-0.522166\pi$
$227$	−2.09666	−0.139160	−0.0695800	+	0.997576i	$0.522166\pi$
$228$	0	0
$229$	25.9036	1.71176	0.855878	−	0.517178i	$-0.173017\pi$
$229$	25.9036	1.71176	0.855878	+	0.517178i	$0.173017\pi$
$230$	0	0
$231$	−4.55237	−0.299524
$232$	0	0
$233$	4.00266	0.262223	0.131111	−	0.991368i	$-0.458145\pi$
$233$	4.00266	0.262223	0.131111	+	0.991368i	$0.458145\pi$
$234$	0	0
$235$	3.51008	0.228972
$236$	0	0
$237$	1.69652	0.110201
$238$	0	0
$239$	−24.2402	−1.56797	−0.783983	−	0.620782i	$-0.786815\pi$
$239$	−24.2402	−1.56797	−0.783983	+	0.620782i	$0.786815\pi$
$240$	0	0
$241$	−21.7439	−1.40065	−0.700323	−	0.713826i	$-0.746961\pi$
$241$	−21.7439	−1.40065	−0.700323	+	0.713826i	$0.746961\pi$
$242$	0	0
$243$	16.1547	1.03633
$244$	0	0
$245$	6.41747	0.409997
$246$	0	0
$247$	13.0585	0.830891
$248$	0	0
$249$	−0.961591	−0.0609383
$250$	0	0
$251$	1.95957	0.123687	0.0618435	−	0.998086i	$-0.480302\pi$
$251$	1.95957	0.123687	0.0618435	+	0.998086i	$0.480302\pi$
$252$	0	0
$253$	−0.965550	−0.0607036
$254$	0	0
$255$	−4.90606	−0.307230
$256$	0	0
$257$	−8.23522	−0.513699	−0.256849	−	0.966451i	$-0.582684\pi$
$257$	−8.23522	−0.513699	−0.256849	+	0.966451i	$0.582684\pi$
$258$	0	0
$259$	−0.763238	−0.0474253
$260$	0	0
$261$	−11.1876	−0.692492
$262$	0	0
$263$	−27.5352	−1.69789	−0.848947	−	0.528477i	$-0.822763\pi$
$263$	−27.5352	−1.69789	−0.848947	+	0.528477i	$0.822763\pi$
$264$	0	0
$265$	−10.5971	−0.650973
$266$	0	0
$267$	15.6624	0.958521
$268$	0	0
$269$	21.7093	1.32364	0.661819	−	0.749663i	$-0.269785\pi$
$269$	21.7093	1.32364	0.661819	+	0.749663i	$0.269785\pi$
$270$	0	0
$271$	−26.2359	−1.59372	−0.796859	−	0.604166i	$-0.793506\pi$
$271$	−26.2359	−1.59372	−0.796859	+	0.604166i	$0.793506\pi$
$272$	0	0
$273$	1.88570	0.114128
$274$	0	0
$275$	6.44732	0.388788
$276$	0	0
$277$	25.7007	1.54421	0.772103	−	0.635497i	$-0.219205\pi$
$277$	25.7007	1.54421	0.772103	+	0.635497i	$0.219205\pi$
$278$	0	0
$279$	1.95761	0.117199
$280$	0	0
$281$	−0.867736	−0.0517648	−0.0258824	−	0.999665i	$-0.508240\pi$
$281$	−0.867736	−0.0517648	−0.0258824	+	0.999665i	$0.508240\pi$
$282$	0	0
$283$	−17.6481	−1.04907	−0.524534	−	0.851389i	$-0.675761\pi$
$283$	−17.6481	−1.04907	−0.524534	+	0.851389i	$0.675761\pi$
$284$	0	0
$285$	4.52352	0.267950
$286$	0	0
$287$	−8.22090	−0.485265
$288$	0	0
$289$	11.1236	0.654327
$290$	0	0
$291$	−4.79420	−0.281041
$292$	0	0
$293$	−16.5092	−0.964478	−0.482239	−	0.876040i	$-0.660176\pi$
$293$	−16.5092	−0.964478	−0.482239	+	0.876040i	$0.660176\pi$
$294$	0	0
$295$	4.45447	0.259349
$296$	0	0
$297$	−30.6825	−1.78038
$298$	0	0
$299$	0.399953	0.0231299
$300$	0	0
$301$	−6.20021	−0.357374
$302$	0	0
$303$	8.18235	0.470064
$304$	0	0
$305$	8.26210	0.473086
$306$	0	0
$307$	23.4962	1.34100	0.670500	−	0.741910i	$-0.266080\pi$
$307$	23.4962	1.34100	0.670500	+	0.741910i	$0.266080\pi$
$308$	0	0
$309$	−4.90850	−0.279235
$310$	0	0
$311$	8.41970	0.477438	0.238719	−	0.971089i	$-0.423273\pi$
$311$	8.41970	0.477438	0.238719	+	0.971089i	$0.423273\pi$
$312$	0	0
$313$	−13.1041	−0.740687	−0.370344	−	0.928895i	$-0.620760\pi$
$313$	−13.1041	−0.740687	−0.370344	+	0.928895i	$0.620760\pi$
$314$	0	0
$315$	−1.63650	−0.0922063
$316$	0	0
$317$	−16.7653	−0.941635	−0.470818	−	0.882231i	$-0.656041\pi$
$317$	−16.7653	−0.941635	−0.470818	+	0.882231i	$0.656041\pi$
$318$	0	0
$319$	33.6402	1.88349
$320$	0	0
$321$	10.4769	0.584761
$322$	0	0
$323$	−25.9307	−1.44282
$324$	0	0
$325$	−2.67063	−0.148140
$326$	0	0
$327$	6.70213	0.370629
$328$	0	0
$329$	2.67902	0.147699
$330$	0	0
$331$	14.2690	0.784297	0.392149	−	0.919902i	$-0.371732\pi$
$331$	14.2690	0.784297	0.392149	+	0.919902i	$0.371732\pi$
$332$	0	0
$333$	−2.14415	−0.117499
$334$	0	0
$335$	−6.88966	−0.376422
$336$	0	0
$337$	10.1570	0.553290	0.276645	−	0.960972i	$-0.410777\pi$
$337$	10.1570	0.553290	0.276645	+	0.960972i	$0.410777\pi$
$338$	0	0
$339$	13.2483	0.719551
$340$	0	0
$341$	−5.88639	−0.318766
$342$	0	0
$343$	10.2407	0.552947
$344$	0	0
$345$	0.138546	0.00745906
$346$	0	0
$347$	5.98570	0.321329	0.160665	−	0.987009i	$-0.448636\pi$
$347$	5.98570	0.321329	0.160665	+	0.987009i	$0.448636\pi$
$348$	0	0
$349$	−7.43246	−0.397850	−0.198925	−	0.980015i	$-0.563745\pi$
$349$	−7.43246	−0.397850	−0.198925	+	0.980015i	$0.563745\pi$
$350$	0	0
$351$	12.7094	0.678379
$352$	0	0
$353$	15.9383	0.848312	0.424156	−	0.905589i	$-0.360571\pi$
$353$	15.9383	0.848312	0.424156	+	0.905589i	$0.360571\pi$
$354$	0	0
$355$	2.03546	0.108031
$356$	0	0
$357$	−3.74450	−0.198180
$358$	0	0
$359$	−8.73759	−0.461152	−0.230576	−	0.973054i	$-0.574061\pi$
$359$	−8.73759	−0.461152	−0.230576	+	0.973054i	$0.574061\pi$
$360$	0	0
$361$	4.90880	0.258358
$362$	0	0
$363$	28.2790	1.48426
$364$	0	0
$365$	−9.81478	−0.513729
$366$	0	0
$367$	−36.6585	−1.91356	−0.956778	−	0.290819i	$-0.906072\pi$
$367$	−36.6585	−1.91356	−0.956778	+	0.290819i	$0.906072\pi$
$368$	0	0
$369$	−23.0948	−1.20227
$370$	0	0
$371$	−8.08810	−0.419913
$372$	0	0
$373$	14.1385	0.732066	0.366033	−	0.930602i	$-0.380716\pi$
$373$	14.1385	0.732066	0.366033	+	0.930602i	$0.380716\pi$
$374$	0	0
$375$	−0.925120	−0.0477730
$376$	0	0
$377$	−13.9346	−0.717666
$378$	0	0
$379$	1.84450	0.0947456	0.0473728	−	0.998877i	$-0.484915\pi$
$379$	1.84450	0.0947456	0.0473728	+	0.998877i	$0.484915\pi$
$380$	0	0
$381$	10.9451	0.560733
$382$	0	0
$383$	−15.1041	−0.771783	−0.385892	−	0.922544i	$-0.626106\pi$
$383$	−15.1041	−0.771783	−0.385892	+	0.922544i	$0.626106\pi$
$384$	0	0
$385$	4.92084	0.250789
$386$	0	0
$387$	−17.4181	−0.885414
$388$	0	0
$389$	−22.5262	−1.14212	−0.571061	−	0.820908i	$-0.693468\pi$
$389$	−22.5262	−1.14212	−0.571061	+	0.820908i	$0.693468\pi$
$390$	0	0
$391$	−0.794201	−0.0401645
$392$	0	0
$393$	−0.118262	−0.00596551
$394$	0	0
$395$	−1.83384	−0.0922704
$396$	0	0
$397$	4.93009	0.247434	0.123717	−	0.992318i	$-0.460518\pi$
$397$	4.93009	0.247434	0.123717	+	0.992318i	$0.460518\pi$
$398$	0	0
$399$	3.45253	0.172843
$400$	0	0
$401$	−13.2106	−0.659708	−0.329854	−	0.944032i	$-0.607000\pi$
$401$	−13.2106	−0.659708	−0.329854	+	0.944032i	$0.607000\pi$
$402$	0	0
$403$	2.43828	0.121459
$404$	0	0
$405$	−2.02985	−0.100864
$406$	0	0
$407$	6.44732	0.319582
$408$	0	0
$409$	9.31575	0.460634	0.230317	−	0.973116i	$-0.426024\pi$
$409$	9.31575	0.460634	0.230317	+	0.973116i	$0.426024\pi$
$410$	0	0
$411$	5.37937	0.265345
$412$	0	0
$413$	3.39982	0.167294
$414$	0	0
$415$	1.03942	0.0510232
$416$	0	0
$417$	−19.6053	−0.960077
$418$	0	0
$419$	37.9278	1.85290	0.926448	−	0.376424i	$-0.122846\pi$
$419$	37.9278	1.85290	0.926448	+	0.376424i	$0.122846\pi$
$420$	0	0
$421$	32.8534	1.60118	0.800589	−	0.599213i	$-0.204520\pi$
$421$	32.8534	1.60118	0.800589	+	0.599213i	$0.204520\pi$
$422$	0	0
$423$	7.52614	0.365933
$424$	0	0
$425$	5.30316	0.257241
$426$	0	0
$427$	6.30595	0.305166
$428$	0	0
$429$	−15.9291	−0.769063
$430$	0	0
$431$	−14.2266	−0.685269	−0.342635	−	0.939469i	$-0.611319\pi$
$431$	−14.2266	−0.685269	−0.342635	+	0.939469i	$0.611319\pi$
$432$	0	0
$433$	4.07885	0.196017	0.0980084	−	0.995186i	$-0.468753\pi$
$433$	4.07885	0.196017	0.0980084	+	0.995186i	$0.468753\pi$
$434$	0	0
$435$	−4.82700	−0.231437
$436$	0	0
$437$	0.732275	0.0350295
$438$	0	0
$439$	−23.1333	−1.10409	−0.552045	−	0.833814i	$-0.686152\pi$
$439$	−23.1333	−1.10409	−0.552045	+	0.833814i	$0.686152\pi$
$440$	0	0
$441$	13.7600	0.655240
$442$	0	0
$443$	22.2393	1.05662	0.528310	−	0.849052i	$-0.322826\pi$
$443$	22.2393	1.05662	0.528310	+	0.849052i	$0.322826\pi$
$444$	0	0
$445$	−16.9301	−0.802563
$446$	0	0
$447$	−1.92930	−0.0912528
$448$	0	0
$449$	8.07885	0.381264	0.190632	−	0.981662i	$-0.438946\pi$
$449$	8.07885	0.381264	0.190632	+	0.981662i	$0.438946\pi$
$450$	0	0
$451$	69.4446	3.27002
$452$	0	0
$453$	−3.82865	−0.179886
$454$	0	0
$455$	−2.03833	−0.0955583
$456$	0	0
$457$	−1.48177	−0.0693142	−0.0346571	−	0.999399i	$-0.511034\pi$
$457$	−1.48177	−0.0693142	−0.0346571	+	0.999399i	$0.511034\pi$
$458$	0	0
$459$	−25.2375	−1.17799
$460$	0	0
$461$	−39.4069	−1.83536	−0.917682	−	0.397316i	$-0.869942\pi$
$461$	−39.4069	−1.83536	−0.917682	+	0.397316i	$0.869942\pi$
$462$	0	0
$463$	−36.7687	−1.70878	−0.854392	−	0.519629i	$-0.826070\pi$
$463$	−36.7687	−1.70878	−0.854392	+	0.519629i	$0.826070\pi$
$464$	0	0
$465$	0.844633	0.0391689
$466$	0	0
$467$	9.80288	0.453623	0.226811	−	0.973939i	$-0.427170\pi$
$467$	9.80288	0.453623	0.226811	+	0.973939i	$0.427170\pi$
$468$	0	0
$469$	−5.25846	−0.242813
$470$	0	0
$471$	−4.09187	−0.188544
$472$	0	0
$473$	52.3751	2.40821
$474$	0	0
$475$	−4.88966	−0.224353
$476$	0	0
$477$	−22.7218	−1.04036
$478$	0	0
$479$	−40.3721	−1.84465	−0.922325	−	0.386416i	$-0.873713\pi$
$479$	−40.3721	−1.84465	−0.922325	+	0.386416i	$0.873713\pi$
$480$	0	0
$481$	−2.67063	−0.121770
$482$	0	0
$483$	0.105744	0.00481150
$484$	0	0
$485$	5.18225	0.235314
$486$	0	0
$487$	28.4425	1.28885	0.644427	−	0.764666i	$-0.277096\pi$
$487$	28.4425	1.28885	0.644427	+	0.764666i	$0.277096\pi$
$488$	0	0
$489$	12.9327	0.584839
$490$	0	0
$491$	−21.2761	−0.960176	−0.480088	−	0.877220i	$-0.659395\pi$
$491$	−21.2761	−0.960176	−0.480088	+	0.877220i	$0.659395\pi$
$492$	0	0
$493$	27.6703	1.24621
$494$	0	0
$495$	13.8240	0.621344
$496$	0	0
$497$	1.55354	0.0696858
$498$	0	0
$499$	15.8222	0.708297	0.354149	−	0.935189i	$-0.384771\pi$
$499$	15.8222	0.708297	0.354149	+	0.935189i	$0.384771\pi$
$500$	0	0
$501$	6.49203	0.290042
$502$	0	0
$503$	−9.47252	−0.422359	−0.211179	−	0.977447i	$-0.567730\pi$
$503$	−9.47252	−0.422359	−0.211179	+	0.977447i	$0.567730\pi$
$504$	0	0
$505$	−8.84463	−0.393581
$506$	0	0
$507$	−5.42836	−0.241082
$508$	0	0
$509$	−13.8991	−0.616067	−0.308033	−	0.951376i	$-0.599671\pi$
$509$	−13.8991	−0.616067	−0.308033	+	0.951376i	$0.599671\pi$
$510$	0	0
$511$	−7.49102	−0.331383
$512$	0	0
$513$	23.2697	1.02738
$514$	0	0
$515$	5.30580	0.233802
$516$	0	0
$517$	−22.6306	−0.995291
$518$	0	0
$519$	−0.761399	−0.0334217
$520$	0	0
$521$	−21.0655	−0.922898	−0.461449	−	0.887167i	$-0.652670\pi$
$521$	−21.0655	−0.922898	−0.461449	+	0.887167i	$0.652670\pi$
$522$	0	0
$523$	24.2202	1.05908	0.529538	−	0.848286i	$-0.322365\pi$
$523$	24.2202	1.05908	0.529538	+	0.848286i	$0.322365\pi$
$524$	0	0
$525$	−0.706087	−0.0308162
$526$	0	0
$527$	−4.84178	−0.210911
$528$	0	0
$529$	−22.9776	−0.999025
$530$	0	0
$531$	9.55106	0.414481
$532$	0	0
$533$	−28.7656	−1.24598
$534$	0	0
$535$	−11.3249	−0.489616
$536$	0	0
$537$	−1.43000	−0.0617092
$538$	0	0
$539$	−41.3754	−1.78217
$540$	0	0
$541$	36.9977	1.59066	0.795328	−	0.606179i	$-0.207299\pi$
$541$	36.9977	1.59066	0.795328	+	0.606179i	$0.207299\pi$
$542$	0	0
$543$	7.58862	0.325659
$544$	0	0
$545$	−7.24460	−0.310325
$546$	0	0
$547$	−18.3652	−0.785239	−0.392619	−	0.919701i	$-0.628431\pi$
$547$	−18.3652	−0.785239	−0.392619	+	0.919701i	$0.628431\pi$
$548$	0	0
$549$	17.7152	0.756067
$550$	0	0
$551$	−25.5128	−1.08688
$552$	0	0
$553$	−1.39966	−0.0595194
$554$	0	0
$555$	−0.925120	−0.0392692
$556$	0	0
$557$	−11.4948	−0.487051	−0.243525	−	0.969895i	$-0.578304\pi$
$557$	−11.4948	−0.487051	−0.243525	+	0.969895i	$0.578304\pi$
$558$	0	0
$559$	−21.6950	−0.917601
$560$	0	0
$561$	31.6310	1.33546
$562$	0	0
$563$	−9.78904	−0.412559	−0.206279	−	0.978493i	$-0.566136\pi$
$563$	−9.78904	−0.412559	−0.206279	+	0.978493i	$0.566136\pi$
$564$	0	0
$565$	−14.3207	−0.602475
$566$	0	0
$567$	−1.54926	−0.0650628
$568$	0	0
$569$	−10.5575	−0.442592	−0.221296	−	0.975207i	$-0.571029\pi$
$569$	−10.5575	−0.442592	−0.221296	+	0.975207i	$0.571029\pi$
$570$	0	0
$571$	−8.64556	−0.361806	−0.180903	−	0.983501i	$-0.557902\pi$
$571$	−8.64556	−0.361806	−0.180903	+	0.983501i	$0.557902\pi$
$572$	0	0
$573$	20.5971	0.860455
$574$	0	0
$575$	−0.149760	−0.00624542
$576$	0	0
$577$	−4.16430	−0.173362	−0.0866811	−	0.996236i	$-0.527626\pi$
$577$	−4.16430	−0.173362	−0.0866811	+	0.996236i	$0.527626\pi$
$578$	0	0
$579$	−22.0079	−0.914618
$580$	0	0
$581$	0.793327	0.0329128
$582$	0	0
$583$	68.3227	2.82964
$584$	0	0
$585$	−5.72624	−0.236751
$586$	0	0
$587$	−31.3122	−1.29239	−0.646197	−	0.763171i	$-0.723642\pi$
$587$	−31.3122	−1.29239	−0.646197	+	0.763171i	$0.723642\pi$
$588$	0	0
$589$	4.46425	0.183946
$590$	0	0
$591$	−2.96820	−0.122096
$592$	0	0
$593$	27.6422	1.13513	0.567564	−	0.823329i	$-0.307886\pi$
$593$	27.6422	1.13513	0.567564	+	0.823329i	$0.307886\pi$
$594$	0	0
$595$	4.04758	0.165935
$596$	0	0
$597$	−14.0991	−0.577039
$598$	0	0
$599$	−20.1328	−0.822604	−0.411302	−	0.911499i	$-0.634926\pi$
$599$	−20.1328	−0.822604	−0.411302	+	0.911499i	$0.634926\pi$
$600$	0	0
$601$	6.88733	0.280940	0.140470	−	0.990085i	$-0.455139\pi$
$601$	6.88733	0.280940	0.140470	+	0.990085i	$0.455139\pi$
$602$	0	0
$603$	−14.7725	−0.601582
$604$	0	0
$605$	−30.5679	−1.24276
$606$	0	0
$607$	35.0574	1.42294	0.711468	−	0.702718i	$-0.248030\pi$
$607$	35.0574	1.42294	0.711468	+	0.702718i	$0.248030\pi$
$608$	0	0
$609$	−3.68415	−0.149289
$610$	0	0
$611$	9.37411	0.379236
$612$	0	0
$613$	22.4218	0.905608	0.452804	−	0.891610i	$-0.350424\pi$
$613$	22.4218	0.905608	0.452804	+	0.891610i	$0.350424\pi$
$614$	0	0
$615$	−9.96454	−0.401809
$616$	0	0
$617$	3.94076	0.158649	0.0793246	−	0.996849i	$-0.474724\pi$
$617$	3.94076	0.158649	0.0793246	+	0.996849i	$0.474724\pi$
$618$	0	0
$619$	0.426159	0.0171288	0.00856439	−	0.999963i	$-0.497274\pi$
$619$	0.426159	0.0171288	0.00856439	+	0.999963i	$0.497274\pi$
$620$	0	0
$621$	0.712701	0.0285997
$622$	0	0
$623$	−12.9217	−0.517697
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−29.1646	−1.16472
$628$	0	0
$629$	5.30316	0.211451
$630$	0	0
$631$	−12.7364	−0.507028	−0.253514	−	0.967332i	$-0.581586\pi$
$631$	−12.7364	−0.507028	−0.253514	+	0.967332i	$0.581586\pi$
$632$	0	0
$633$	−5.49367	−0.218354
$634$	0	0
$635$	−11.8310	−0.469498
$636$	0	0
$637$	17.1387	0.679059
$638$	0	0
$639$	4.36433	0.172650
$640$	0	0
$641$	0.0295361	0.00116660	0.000583302	−	1.00000i	$-0.499814\pi$
$641$	0.0295361	0.00116660	0.000583302		1.00000i	$0.499814\pi$
$642$	0	0
$643$	−28.6321	−1.12914	−0.564569	−	0.825386i	$-0.690958\pi$
$643$	−28.6321	−1.12914	−0.564569	+	0.825386i	$0.690958\pi$
$644$	0	0
$645$	−7.51526	−0.295913
$646$	0	0
$647$	12.3744	0.486489	0.243245	−	0.969965i	$-0.421788\pi$
$647$	12.3744	0.486489	0.243245	+	0.969965i	$0.421788\pi$
$648$	0	0
$649$	−28.7194	−1.12733
$650$	0	0
$651$	0.644656	0.0252661
$652$	0	0
$653$	19.4413	0.760795	0.380398	−	0.924823i	$-0.375787\pi$
$653$	19.4413	0.760795	0.380398	+	0.924823i	$0.375787\pi$
$654$	0	0
$655$	0.127834	0.00499488
$656$	0	0
$657$	−21.0444	−0.821020
$658$	0	0
$659$	−47.4358	−1.84784	−0.923918	−	0.382591i	$-0.875032\pi$
$659$	−47.4358	−1.84784	−0.923918	+	0.382591i	$0.875032\pi$
$660$	0	0
$661$	−37.0577	−1.44138	−0.720688	−	0.693260i	$-0.756174\pi$
$661$	−37.0577	−1.44138	−0.720688	+	0.693260i	$0.756174\pi$
$662$	0	0
$663$	−13.1023	−0.508850
$664$	0	0
$665$	−3.73198	−0.144720
$666$	0	0
$667$	−0.781403	−0.0302560
$668$	0	0
$669$	18.1333	0.701075
$670$	0	0
$671$	−53.2684	−2.05640
$672$	0	0
$673$	−41.0843	−1.58368	−0.791842	−	0.610726i	$-0.790878\pi$
$673$	−41.0843	−1.58368	−0.791842	+	0.610726i	$0.790878\pi$
$674$	0	0
$675$	−4.75896	−0.183172
$676$	0	0
$677$	−1.26815	−0.0487391	−0.0243696	−	0.999703i	$-0.507758\pi$
$677$	−1.26815	−0.0487391	−0.0243696	+	0.999703i	$0.507758\pi$
$678$	0	0
$679$	3.95529	0.151790
$680$	0	0
$681$	−1.93966	−0.0743279
$682$	0	0
$683$	−2.43429	−0.0931454	−0.0465727	−	0.998915i	$-0.514830\pi$
$683$	−2.43429	−0.0931454	−0.0465727	+	0.998915i	$0.514830\pi$
$684$	0	0
$685$	−5.81478	−0.222171
$686$	0	0
$687$	23.9639	0.914280
$688$	0	0
$689$	−28.3009	−1.07818
$690$	0	0
$691$	12.8823	0.490066	0.245033	−	0.969515i	$-0.421201\pi$
$691$	12.8823	0.490066	0.245033	+	0.969515i	$0.421201\pi$
$692$	0	0
$693$	10.5510	0.400801
$694$	0	0
$695$	21.1922	0.803866
$696$	0	0
$697$	57.1208	2.16361
$698$	0	0
$699$	3.70294	0.140058
$700$	0	0
$701$	23.9659	0.905181	0.452590	−	0.891719i	$-0.350500\pi$
$701$	23.9659	0.905181	0.452590	+	0.891719i	$0.350500\pi$
$702$	0	0
$703$	−4.88966	−0.184417
$704$	0	0
$705$	3.24724	0.122298
$706$	0	0
$707$	−6.75056	−0.253881
$708$	0	0
$709$	−50.1842	−1.88471	−0.942353	−	0.334620i	$-0.891392\pi$
$709$	−50.1842	−1.88471	−0.942353	+	0.334620i	$0.891392\pi$
$710$	0	0
$711$	−3.93203	−0.147463
$712$	0	0
$713$	0.136731	0.00512060
$714$	0	0
$715$	17.2184	0.643932
$716$	0	0
$717$	−22.4251	−0.837480
$718$	0	0
$719$	20.2628	0.755675	0.377837	−	0.925872i	$-0.376668\pi$
$719$	20.2628	0.755675	0.377837	+	0.925872i	$0.376668\pi$
$720$	0	0
$721$	4.04959	0.150815
$722$	0	0
$723$	−20.1157	−0.748110
$724$	0	0
$725$	5.21771	0.193781
$726$	0	0
$727$	−41.8924	−1.55370	−0.776851	−	0.629685i	$-0.783184\pi$
$727$	−41.8924	−1.55370	−0.776851	+	0.629685i	$0.783184\pi$
$728$	0	0
$729$	8.85552	0.327982
$730$	0	0
$731$	43.0805	1.59339
$732$	0	0
$733$	28.9524	1.06938	0.534690	−	0.845048i	$-0.320428\pi$
$733$	28.9524	1.06938	0.534690	+	0.845048i	$0.320428\pi$
$734$	0	0
$735$	5.93693	0.218987
$736$	0	0
$737$	44.4198	1.63623
$738$	0	0
$739$	0.191814	0.00705599	0.00352800	−	0.999994i	$-0.498877\pi$
$739$	0.191814	0.00705599	0.00352800	+	0.999994i	$0.498877\pi$
$740$	0	0
$741$	12.0807	0.443794
$742$	0	0
$743$	43.5661	1.59828	0.799142	−	0.601142i	$-0.205287\pi$
$743$	43.5661	1.59828	0.799142	+	0.601142i	$0.205287\pi$
$744$	0	0
$745$	2.08546	0.0764053
$746$	0	0
$747$	2.22868	0.0815432
$748$	0	0
$749$	−8.64357	−0.315829
$750$	0	0
$751$	−35.1963	−1.28433	−0.642166	−	0.766565i	$-0.721964\pi$
$751$	−35.1963	−1.28433	−0.642166	+	0.766565i	$0.721964\pi$
$752$	0	0
$753$	1.81284	0.0660635
$754$	0	0
$755$	4.13855	0.150617
$756$	0	0
$757$	−25.9437	−0.942939	−0.471469	−	0.881882i	$-0.656276\pi$
$757$	−25.9437	−0.942939	−0.471469	+	0.881882i	$0.656276\pi$
$758$	0	0
$759$	−0.893249	−0.0324229
$760$	0	0
$761$	−36.3048	−1.31605	−0.658025	−	0.752996i	$-0.728608\pi$
$761$	−36.3048	−1.31605	−0.658025	+	0.752996i	$0.728608\pi$
$762$	0	0
$763$	−5.52936	−0.200176
$764$	0	0
$765$	11.3708	0.411112
$766$	0	0
$767$	11.8962	0.429548
$768$	0	0
$769$	23.8359	0.859546	0.429773	−	0.902937i	$-0.358594\pi$
$769$	23.8359	0.859546	0.429773	+	0.902937i	$0.358594\pi$
$770$	0	0
$771$	−7.61857	−0.274376
$772$	0	0
$773$	53.4424	1.92219	0.961095	−	0.276220i	$-0.0890818\pi$
$773$	53.4424	1.92219	0.961095	+	0.276220i	$0.0890818\pi$
$774$	0	0
$775$	−0.912998	−0.0327959
$776$	0	0
$777$	−0.706087	−0.0253307
$778$	0	0
$779$	−52.6670	−1.88699
$780$	0	0
$781$	−13.1232	−0.469586
$782$	0	0
$783$	−24.8308	−0.887382
$784$	0	0
$785$	4.42307	0.157866
$786$	0	0
$787$	35.0591	1.24972	0.624860	−	0.780737i	$-0.285156\pi$
$787$	35.0591	1.24972	0.624860	+	0.780737i	$0.285156\pi$
$788$	0	0
$789$	−25.4734	−0.906877
$790$	0	0
$791$	−10.9301	−0.388629
$792$	0	0
$793$	22.0650	0.783552
$794$	0	0
$795$	−9.80357	−0.347697
$796$	0	0
$797$	23.3477	0.827018	0.413509	−	0.910500i	$-0.364303\pi$
$797$	23.3477	0.827018	0.413509	+	0.910500i	$0.364303\pi$
$798$	0	0
$799$	−18.6145	−0.658534
$800$	0	0
$801$	−36.3007	−1.28262
$802$	0	0
$803$	63.2790	2.23307
$804$	0	0
$805$	−0.114303	−0.00402863
$806$	0	0
$807$	20.0837	0.706980
$808$	0	0
$809$	29.9233	1.05205	0.526024	−	0.850469i	$-0.323682\pi$
$809$	29.9233	1.05205	0.526024	+	0.850469i	$0.323682\pi$
$810$	0	0
$811$	−3.99426	−0.140257	−0.0701287	−	0.997538i	$-0.522341\pi$
$811$	−3.99426	−0.140257	−0.0701287	+	0.997538i	$0.522341\pi$
$812$	0	0
$813$	−24.2714	−0.851233
$814$	0	0
$815$	−13.9795	−0.489682
$816$	0	0
$817$	−39.7214	−1.38968
$818$	0	0
$819$	−4.37049	−0.152717
$820$	0	0
$821$	41.8542	1.46072	0.730362	−	0.683061i	$-0.239352\pi$
$821$	41.8542	1.46072	0.730362	+	0.683061i	$0.239352\pi$
$822$	0	0
$823$	48.8580	1.70308	0.851541	−	0.524287i	$-0.175668\pi$
$823$	48.8580	1.70308	0.851541	+	0.524287i	$0.175668\pi$
$824$	0	0
$825$	5.96454	0.207659
$826$	0	0
$827$	−36.9524	−1.28496	−0.642480	−	0.766302i	$-0.722094\pi$
$827$	−36.9524	−1.28496	−0.642480	+	0.766302i	$0.722094\pi$
$828$	0	0
$829$	2.04955	0.0711840	0.0355920	−	0.999366i	$-0.488668\pi$
$829$	2.04955	0.0711840	0.0355920	+	0.999366i	$0.488668\pi$
$830$	0	0
$831$	23.7763	0.824789
$832$	0	0
$833$	−34.0329	−1.17917
$834$	0	0
$835$	−7.01750	−0.242850
$836$	0	0
$837$	4.34492	0.150182
$838$	0	0
$839$	−51.2252	−1.76849	−0.884246	−	0.467022i	$-0.845327\pi$
$839$	−51.2252	−1.76849	−0.884246	+	0.467022i	$0.845327\pi$
$840$	0	0
$841$	−1.77555	−0.0612259
$842$	0	0
$843$	−0.802760	−0.0276485
$844$	0	0
$845$	5.86774	0.201856
$846$	0	0
$847$	−23.3306	−0.801649
$848$	0	0
$849$	−16.3266	−0.560327
$850$	0	0
$851$	−0.149760	−0.00513370
$852$	0	0
$853$	−56.0152	−1.91792	−0.958962	−	0.283534i	$-0.908493\pi$
$853$	−56.0152	−1.91792	−0.958962	+	0.283534i	$0.908493\pi$
$854$	0	0
$855$	−10.4842	−0.358552
$856$	0	0
$857$	53.9011	1.84123	0.920614	−	0.390474i	$-0.127689\pi$
$857$	53.9011	1.84123	0.920614	+	0.390474i	$0.127689\pi$
$858$	0	0
$859$	−38.9051	−1.32742	−0.663712	−	0.747988i	$-0.731020\pi$
$859$	−38.9051	−1.32742	−0.663712	+	0.747988i	$0.731020\pi$
$860$	0	0
$861$	−7.60532	−0.259189
$862$	0	0
$863$	−33.0560	−1.12524	−0.562619	−	0.826716i	$-0.690206\pi$
$863$	−33.0560	−1.12524	−0.562619	+	0.826716i	$0.690206\pi$
$864$	0	0
$865$	0.823027	0.0279838
$866$	0	0
$867$	10.2906	0.349488
$868$	0	0
$869$	11.8233	0.401079
$870$	0	0
$871$	−18.3997	−0.623451
$872$	0	0
$873$	11.1115	0.376068
$874$	0	0
$875$	0.763238	0.0258022
$876$	0	0
$877$	13.6062	0.459448	0.229724	−	0.973256i	$-0.426218\pi$
$877$	13.6062	0.459448	0.229724	+	0.973256i	$0.426218\pi$
$878$	0	0
$879$	−15.2730	−0.515145
$880$	0	0
$881$	−31.8714	−1.07377	−0.536887	−	0.843654i	$-0.680400\pi$
$881$	−31.8714	−1.07377	−0.536887	+	0.843654i	$0.680400\pi$
$882$	0	0
$883$	24.5744	0.826994	0.413497	−	0.910506i	$-0.364307\pi$
$883$	24.5744	0.826994	0.413497	+	0.910506i	$0.364307\pi$
$884$	0	0
$885$	4.12092	0.138523
$886$	0	0
$887$	35.2709	1.18428	0.592141	−	0.805834i	$-0.298283\pi$
$887$	35.2709	1.18428	0.592141	+	0.805834i	$0.298283\pi$
$888$	0	0
$889$	−9.02985	−0.302851
$890$	0	0
$891$	13.0871	0.438434
$892$	0	0
$893$	17.1631	0.574341
$894$	0	0
$895$	1.54575	0.0516687
$896$	0	0
$897$	0.370005	0.0123541
$898$	0	0
$899$	−4.76376	−0.158880
$900$	0	0
$901$	56.1980	1.87223
$902$	0	0
$903$	−5.73594	−0.190880
$904$	0	0
$905$	−8.20285	−0.272672
$906$	0	0
$907$	11.1346	0.369719	0.184859	−	0.982765i	$-0.440817\pi$
$907$	11.1346	0.369719	0.184859	+	0.982765i	$0.440817\pi$
$908$	0	0
$909$	−18.9642	−0.629004
$910$	0	0
$911$	−1.81469	−0.0601235	−0.0300618	−	0.999548i	$-0.509570\pi$
$911$	−1.81469	−0.0601235	−0.0300618	+	0.999548i	$0.509570\pi$
$912$	0	0
$913$	−6.70149	−0.221787
$914$	0	0
$915$	7.64343	0.252684
$916$	0	0
$917$	0.0975676	0.00322197
$918$	0	0
$919$	22.9790	0.758009	0.379004	−	0.925395i	$-0.376267\pi$
$919$	22.9790	0.758009	0.379004	+	0.925395i	$0.376267\pi$
$920$	0	0
$921$	21.7368	0.716252
$922$	0	0
$923$	5.43595	0.178927
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	11.3765	0.373652
$928$	0	0
$929$	−47.1891	−1.54822	−0.774112	−	0.633049i	$-0.781803\pi$
$929$	−47.1891	−1.54822	−0.774112	+	0.633049i	$0.781803\pi$
$930$	0	0
$931$	31.3792	1.02841
$932$	0	0
$933$	7.78924	0.255008
$934$	0	0
$935$	−34.1912	−1.11817
$936$	0	0
$937$	23.4516	0.766132	0.383066	−	0.923721i	$-0.374868\pi$
$937$	23.4516	0.766132	0.383066	+	0.923721i	$0.374868\pi$
$938$	0	0
$939$	−12.1229	−0.395615
$940$	0	0
$941$	39.9562	1.30253	0.651267	−	0.758848i	$-0.274238\pi$
$941$	39.9562	1.30253	0.651267	+	0.758848i	$0.274238\pi$
$942$	0	0
$943$	−1.61308	−0.0525290
$944$	0	0
$945$	−3.63222	−0.118156
$946$	0	0
$947$	19.9868	0.649485	0.324742	−	0.945803i	$-0.394722\pi$
$947$	19.9868	0.649485	0.324742	+	0.945803i	$0.394722\pi$
$948$	0	0
$949$	−26.2117	−0.850866
$950$	0	0
$951$	−15.5100	−0.502945
$952$	0	0
$953$	−56.0082	−1.81428	−0.907142	−	0.420825i	$-0.861741\pi$
$953$	−56.0082	−1.81428	−0.907142	+	0.420825i	$0.861741\pi$
$954$	0	0
$955$	−22.2642	−0.720453
$956$	0	0
$957$	31.1212	1.00601
$958$	0	0
$959$	−4.43807	−0.143313
$960$	0	0
$961$	−30.1664	−0.973111
$962$	0	0
$963$	−24.2822	−0.782484
$964$	0	0
$965$	23.7893	0.765804
$966$	0	0
$967$	−59.2145	−1.90421	−0.952105	−	0.305773i	$-0.901085\pi$
$967$	−59.2145	−1.90421	−0.952105	+	0.305773i	$0.901085\pi$
$968$	0	0
$969$	−23.9890	−0.770638
$970$	0	0
$971$	15.5034	0.497526	0.248763	−	0.968564i	$-0.419976\pi$
$971$	15.5034	0.497526	0.248763	+	0.968564i	$0.419976\pi$
$972$	0	0
$973$	16.1747	0.518537
$974$	0	0
$975$	−2.47065	−0.0791242
$976$	0	0
$977$	−2.51823	−0.0805654	−0.0402827	−	0.999188i	$-0.512826\pi$
$977$	−2.51823	−0.0805654	−0.0402827	+	0.999188i	$0.512826\pi$
$978$	0	0
$979$	109.154	3.48857
$980$	0	0
$981$	−15.5335	−0.495948
$982$	0	0
$983$	36.5856	1.16690	0.583449	−	0.812150i	$-0.301703\pi$
$983$	36.5856	1.16690	0.583449	+	0.812150i	$0.301703\pi$
$984$	0	0
$985$	3.20845	0.102230
$986$	0	0
$987$	2.47842	0.0788890
$988$	0	0
$989$	−1.21658	−0.0386851
$990$	0	0
$991$	−32.6644	−1.03762	−0.518810	−	0.854890i	$-0.673625\pi$
$991$	−32.6644	−1.03762	−0.518810	+	0.854890i	$0.673625\pi$
$992$	0	0
$993$	13.2006	0.418907
$994$	0	0
$995$	15.2403	0.483151
$996$	0	0
$997$	2.24180	0.0709985	0.0354992	−	0.999370i	$-0.488698\pi$
$997$	2.24180	0.0709985	0.0354992	+	0.999370i	$0.488698\pi$
$998$	0	0
$999$	−4.75896	−0.150567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.y.1.4		5
4.3	odd	2		1480.2.a.i.1.2	✓	5
20.19	odd	2		7400.2.a.p.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.i.1.2	✓	5	4.3	odd	2
2960.2.a.y.1.4		5	1.1	even	1	trivial
7400.2.a.p.1.4		5	20.19	odd	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)

\(f(q)\)	\(=\)	\(q+0.925120 q^{3} -1.00000 q^{5} -0.763238 q^{7} -2.14415 q^{9} +O(q^{10})\)	\(q+0.925120 q^{3} -1.00000 q^{5} -0.763238 q^{7} -2.14415 q^{9} +6.44732 q^{11} -2.67063 q^{13} -0.925120 q^{15} +5.30316 q^{17} -4.88966 q^{19} -0.706087 q^{21} -0.149760 q^{23} +1.00000 q^{25} -4.75896 q^{27} +5.21771 q^{29} -0.912998 q^{31} +5.96454 q^{33} +0.763238 q^{35} +1.00000 q^{37} -2.47065 q^{39} +10.7711 q^{41} +8.12355 q^{43} +2.14415 q^{45} -3.51008 q^{47} -6.41747 q^{49} +4.90606 q^{51} +10.5971 q^{53} -6.44732 q^{55} -4.52352 q^{57} -4.45447 q^{59} -8.26210 q^{61} +1.63650 q^{63} +2.67063 q^{65} +6.88966 q^{67} -0.138546 q^{69} -2.03546 q^{71} +9.81478 q^{73} +0.925120 q^{75} -4.92084 q^{77} +1.83384 q^{79} +2.02985 q^{81} -1.03942 q^{83} -5.30316 q^{85} +4.82700 q^{87} +16.9301 q^{89} +2.03833 q^{91} -0.844633 q^{93} +4.88966 q^{95} -5.18225 q^{97} -13.8240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * q^99

Introduction

L-functions

Modular forms

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Database

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