LMFDB - Embedded newform 2960.2.a.u.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:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.91729$ 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)

$f(q)$	$=$	$q-0.406728 q^{3} -1.00000 q^{5} -2.91729 q^{7} -2.83457 q^{9} -6.51056 q^{11} -0.813457 q^{13} +0.406728 q^{15} -2.51056 q^{17} -0.406728 q^{19} +1.18654 q^{21} -5.02112 q^{23} +1.00000 q^{25} +2.37309 q^{27} -5.32401 q^{29} +8.75186 q^{31} +2.64803 q^{33} +2.91729 q^{35} -1.00000 q^{37} +0.330856 q^{39} +6.34513 q^{41} +7.32401 q^{43} +2.83457 q^{45} +5.42784 q^{47} +1.51056 q^{49} +1.02112 q^{51} -2.34513 q^{53} +6.51056 q^{55} +0.165428 q^{57} +1.42784 q^{59} -1.32401 q^{61} +8.26926 q^{63} +0.813457 q^{65} +5.42784 q^{67} +2.04223 q^{69} -14.6480 q^{71} -11.0211 q^{73} -0.406728 q^{75} +18.9932 q^{77} +1.75870 q^{79} +7.53851 q^{81} +7.05476 q^{83} +2.51056 q^{85} +2.16543 q^{87} +6.00000 q^{89} +2.37309 q^{91} -3.55963 q^{93} +0.406728 q^{95} +2.34513 q^{97} +18.4546 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + q^{7} + 11 q^{9} - 11 q^{11} + q^{17} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 5 q^{29} - 3 q^{31} - 14 q^{33} - q^{35} - 3 q^{37} + 40 q^{39} - 9 q^{41} + 11 q^{43} - 11 q^{45}+ \cdots - 19 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + q^7 + 11 * q^9 - 11 * q^11 + q^17 + 6 * q^21 + 2 * q^23 + 3 * q^25 + 12 * q^27 - 5 * q^29 - 3 * q^31 - 14 * q^33 - q^35 - 3 * q^37 + 40 * q^39 - 9 * q^41 + 11 * q^43 - 11 * q^45 - 2 * q^47 - 4 * q^49 - 14 * q^51 + 21 * q^53 + 11 * q^55 + 20 * q^57 - 14 * q^59 + 7 * q^61 + 37 * q^63 - 2 * q^67 - 28 * q^69 - 22 * q^71 - 16 * q^73 + 7 * q^77 + 26 * q^79 + 47 * q^81 - 2 * q^83 - q^85 + 26 * q^87 + 18 * q^89 + 12 * q^91 - 18 * q^93 - 21 * q^97 - 19 * q^99

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
$3$	−0.406728	−0.234825	−0.117412	−	0.993083i	$-0.537460\pi$
$3$	−0.406728	−0.234825	−0.117412	+	0.993083i	$0.537460\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.91729	−1.10263	−0.551315	−	0.834297i	$-0.685874\pi$
$7$	−2.91729	−1.10263	−0.551315	+	0.834297i	$0.685874\pi$
$8$	0	0
$9$	−2.83457	−0.944857
$10$	0	0
$11$	−6.51056	−1.96301	−0.981503	−	0.191444i	$-0.938683\pi$
$11$	−6.51056	−1.96301	−0.981503	+	0.191444i	$0.938683\pi$
$12$	0	0
$13$	−0.813457	−0.225612	−0.112806	−	0.993617i	$-0.535984\pi$
$13$	−0.813457	−0.225612	−0.112806	+	0.993617i	$0.535984\pi$
$14$	0	0
$15$	0.406728	0.105017
$16$	0	0
$17$	−2.51056	−0.608900	−0.304450	−	0.952528i	$-0.598473\pi$
$17$	−2.51056	−0.608900	−0.304450	+	0.952528i	$0.598473\pi$
$18$	0	0
$19$	−0.406728	−0.0933099	−0.0466550	−	0.998911i	$-0.514856\pi$
$19$	−0.406728	−0.0933099	−0.0466550	+	0.998911i	$0.514856\pi$
$20$	0	0
$21$	1.18654	0.258925
$22$	0	0
$23$	−5.02112	−1.04697	−0.523487	−	0.852033i	$-0.675369\pi$
$23$	−5.02112	−1.04697	−0.523487	+	0.852033i	$0.675369\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.37309	0.456701
$28$	0	0
$29$	−5.32401	−0.988645	−0.494322	−	0.869279i	$-0.664584\pi$
$29$	−5.32401	−0.988645	−0.494322	+	0.869279i	$0.664584\pi$
$30$	0	0
$31$	8.75186	1.57188	0.785940	−	0.618303i	$-0.212179\pi$
$31$	8.75186	1.57188	0.785940	+	0.618303i	$0.212179\pi$
$32$	0	0
$33$	2.64803	0.460963
$34$	0	0
$35$	2.91729	0.493111
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	0.330856	0.0529794
$40$	0	0
$41$	6.34513	0.990943	0.495471	−	0.868624i	$-0.334995\pi$
$41$	6.34513	0.990943	0.495471	+	0.868624i	$0.334995\pi$
$42$	0	0
$43$	7.32401	1.11690	0.558451	−	0.829538i	$-0.311396\pi$
$43$	7.32401	1.11690	0.558451	+	0.829538i	$0.311396\pi$
$44$	0	0
$45$	2.83457	0.422553
$46$	0	0
$47$	5.42784	0.791732	0.395866	−	0.918308i	$-0.370444\pi$
$47$	5.42784	0.791732	0.395866	+	0.918308i	$0.370444\pi$
$48$	0	0
$49$	1.51056	0.215794
$50$	0	0
$51$	1.02112	0.142985
$52$	0	0
$53$	−2.34513	−0.322128	−0.161064	−	0.986944i	$-0.551493\pi$
$53$	−2.34513	−0.322128	−0.161064	+	0.986944i	$0.551493\pi$
$54$	0	0
$55$	6.51056	0.877883
$56$	0	0
$57$	0.165428	0.0219115
$58$	0	0
$59$	1.42784	0.185889	0.0929447	−	0.995671i	$-0.470372\pi$
$59$	1.42784	0.185889	0.0929447	+	0.995671i	$0.470372\pi$
$60$	0	0
$61$	−1.32401	−0.169523	−0.0847613	−	0.996401i	$-0.527013\pi$
$61$	−1.32401	−0.169523	−0.0847613	+	0.996401i	$0.527013\pi$
$62$	0	0
$63$	8.26926	1.04183
$64$	0	0
$65$	0.813457	0.100897
$66$	0	0
$67$	5.42784	0.663117	0.331558	−	0.943435i	$-0.392426\pi$
$67$	5.42784	0.663117	0.331558	+	0.943435i	$0.392426\pi$
$68$	0	0
$69$	2.04223	0.245856
$70$	0	0
$71$	−14.6480	−1.73840	−0.869201	−	0.494460i	$-0.835366\pi$
$71$	−14.6480	−1.73840	−0.869201	+	0.494460i	$0.835366\pi$
$72$	0	0
$73$	−11.0211	−1.28992	−0.644962	−	0.764215i	$-0.723127\pi$
$73$	−11.0211	−1.28992	−0.644962	+	0.764215i	$0.723127\pi$
$74$	0	0
$75$	−0.406728	−0.0469650
$76$	0	0
$77$	18.9932	2.16447
$78$	0	0
$79$	1.75870	0.197869	0.0989346	−	0.995094i	$-0.468457\pi$
$79$	1.75870	0.197869	0.0989346	+	0.995094i	$0.468457\pi$
$80$	0	0
$81$	7.53851	0.837613
$82$	0	0
$83$	7.05476	0.774360	0.387180	−	0.922004i	$-0.373449\pi$
$83$	7.05476	0.774360	0.387180	+	0.922004i	$0.373449\pi$
$84$	0	0
$85$	2.51056	0.272308
$86$	0	0
$87$	2.16543	0.232158
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	2.37309	0.248767
$92$	0	0
$93$	−3.55963	−0.369116
$94$	0	0
$95$	0.406728	0.0417295
$96$	0	0
$97$	2.34513	0.238112	0.119056	−	0.992888i	$-0.462013\pi$
$97$	2.34513	0.238112	0.119056	+	0.992888i	$0.462013\pi$
$98$	0	0
$99$	18.4546	1.85476
$100$	0	0
$101$	−8.20766	−0.816693	−0.408346	−	0.912827i	$-0.633894\pi$
$101$	−8.20766	−0.816693	−0.408346	+	0.912827i	$0.633894\pi$
$102$	0	0
$103$	8.81346	0.868416	0.434208	−	0.900813i	$-0.357028\pi$
$103$	8.81346	0.868416	0.434208	+	0.900813i	$0.357028\pi$
$104$	0	0
$105$	−1.18654	−0.115795
$106$	0	0
$107$	−15.2624	−1.47547	−0.737737	−	0.675089i	$-0.764105\pi$
$107$	−15.2624	−1.47547	−0.737737	+	0.675089i	$0.764105\pi$
$108$	0	0
$109$	4.30290	0.412143	0.206072	−	0.978537i	$-0.433932\pi$
$109$	4.30290	0.412143	0.206072	+	0.978537i	$0.433932\pi$
$110$	0	0
$111$	0.406728	0.0386050
$112$	0	0
$113$	−9.15859	−0.861567	−0.430784	−	0.902455i	$-0.641763\pi$
$113$	−9.15859	−0.861567	−0.430784	+	0.902455i	$0.641763\pi$
$114$	0	0
$115$	5.02112	0.468221
$116$	0	0
$117$	2.30580	0.213171
$118$	0	0
$119$	7.32401	0.671391
$120$	0	0
$121$	31.3874	2.85340
$122$	0	0
$123$	−2.58074	−0.232698
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.61439	0.764403	0.382202	−	0.924079i	$-0.375166\pi$
$127$	8.61439	0.764403	0.382202	+	0.924079i	$0.375166\pi$
$128$	0	0
$129$	−2.97888	−0.262276
$130$	0	0
$131$	−13.4278	−1.17320	−0.586598	−	0.809878i	$-0.699533\pi$
$131$	−13.4278	−1.17320	−0.586598	+	0.809878i	$0.699533\pi$
$132$	0	0
$133$	1.18654	0.102886
$134$	0	0
$135$	−2.37309	−0.204243
$136$	0	0
$137$	−14.6903	−1.25507	−0.627537	−	0.778587i	$-0.715937\pi$
$137$	−14.6903	−1.25507	−0.627537	+	0.778587i	$0.715937\pi$
$138$	0	0
$139$	−17.3662	−1.47299	−0.736493	−	0.676445i	$-0.763519\pi$
$139$	−17.3662	−1.47299	−0.736493	+	0.676445i	$0.763519\pi$
$140$	0	0
$141$	−2.20766	−0.185918
$142$	0	0
$143$	5.29606	0.442879
$144$	0	0
$145$	5.32401	0.442135
$146$	0	0
$147$	−0.614387	−0.0506738
$148$	0	0
$149$	−0.207658	−0.0170120	−0.00850601	−	0.999964i	$-0.502708\pi$
$149$	−0.207658	−0.0170120	−0.00850601	+	0.999964i	$0.502708\pi$
$150$	0	0
$151$	−0.813457	−0.0661982	−0.0330991	−	0.999452i	$-0.510538\pi$
$151$	−0.813457	−0.0661982	−0.0330991	+	0.999452i	$0.510538\pi$
$152$	0	0
$153$	7.11636	0.575323
$154$	0	0
$155$	−8.75186	−0.702966
$156$	0	0
$157$	5.32401	0.424903	0.212451	−	0.977172i	$-0.431855\pi$
$157$	5.32401	0.424903	0.212451	+	0.977172i	$0.431855\pi$
$158$	0	0
$159$	0.953831	0.0756437
$160$	0	0
$161$	14.6480	1.15443
$162$	0	0
$163$	−15.2008	−1.19062	−0.595310	−	0.803496i	$-0.702971\pi$
$163$	−15.2008	−1.19062	−0.595310	+	0.803496i	$0.702971\pi$
$164$	0	0
$165$	−2.64803	−0.206149
$166$	0	0
$167$	12.4826	0.965933	0.482966	−	0.875639i	$-0.339559\pi$
$167$	12.4826	0.965933	0.482966	+	0.875639i	$0.339559\pi$
$168$	0	0
$169$	−12.3383	−0.949099
$170$	0	0
$171$	1.15290	0.0881645
$172$	0	0
$173$	23.0354	1.75135	0.875674	−	0.482903i	$-0.160417\pi$
$173$	23.0354	1.75135	0.875674	+	0.482903i	$0.160417\pi$
$174$	0	0
$175$	−2.91729	−0.220526
$176$	0	0
$177$	−0.580745	−0.0436514
$178$	0	0
$179$	−24.2835	−1.81504	−0.907518	−	0.420013i	$-0.862026\pi$
$179$	−24.2835	−1.81504	−0.907518	+	0.420013i	$0.862026\pi$
$180$	0	0
$181$	2.16543	0.160955	0.0804775	−	0.996756i	$-0.474355\pi$
$181$	2.16543	0.160955	0.0804775	+	0.996756i	$0.474355\pi$
$182$	0	0
$183$	0.538514	0.0398081
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	16.3451	1.19527
$188$	0	0
$189$	−6.92297	−0.503572
$190$	0	0
$191$	19.3326	1.39886	0.699429	−	0.714702i	$-0.253438\pi$
$191$	19.3326	1.39886	0.699429	+	0.714702i	$0.253438\pi$
$192$	0	0
$193$	−20.0422	−1.44267	−0.721336	−	0.692586i	$-0.756471\pi$
$193$	−20.0422	−1.44267	−0.721336	+	0.692586i	$0.756471\pi$
$194$	0	0
$195$	−0.330856	−0.0236931
$196$	0	0
$197$	15.6269	1.11337	0.556686	−	0.830723i	$-0.312073\pi$
$197$	15.6269	1.11337	0.556686	+	0.830723i	$0.312073\pi$
$198$	0	0
$199$	−20.2835	−1.43786	−0.718931	−	0.695082i	$-0.755368\pi$
$199$	−20.2835	−1.43786	−0.718931	+	0.695082i	$0.755368\pi$
$200$	0	0
$201$	−2.20766	−0.155716
$202$	0	0
$203$	15.5317	1.09011
$204$	0	0
$205$	−6.34513	−0.443163
$206$	0	0
$207$	14.2327	0.989242
$208$	0	0
$209$	2.64803	0.183168
$210$	0	0
$211$	−18.7182	−1.28862	−0.644308	−	0.764766i	$-0.722854\pi$
$211$	−18.7182	−1.28862	−0.644308	+	0.764766i	$0.722854\pi$
$212$	0	0
$213$	5.95777	0.408220
$214$	0	0
$215$	−7.32401	−0.499494
$216$	0	0
$217$	−25.5317	−1.73320
$218$	0	0
$219$	4.48260	0.302906
$220$	0	0
$221$	2.04223	0.137375
$222$	0	0
$223$	15.7307	1.05341	0.526704	−	0.850049i	$-0.323428\pi$
$223$	15.7307	1.05341	0.526704	+	0.850049i	$0.323428\pi$
$224$	0	0
$225$	−2.83457	−0.188971
$226$	0	0
$227$	−12.8836	−0.855117	−0.427559	−	0.903988i	$-0.640626\pi$
$227$	−12.8836	−0.855117	−0.427559	+	0.903988i	$0.640626\pi$
$228$	0	0
$229$	5.25383	0.347183	0.173591	−	0.984818i	$-0.444463\pi$
$229$	5.25383	0.347183	0.173591	+	0.984818i	$0.444463\pi$
$230$	0	0
$231$	−7.72506	−0.508271
$232$	0	0
$233$	28.3172	1.85512	0.927560	−	0.373675i	$-0.121902\pi$
$233$	28.3172	1.85512	0.927560	+	0.373675i	$0.121902\pi$
$234$	0	0
$235$	−5.42784	−0.354073
$236$	0	0
$237$	−0.715313	−0.0464646
$238$	0	0
$239$	−14.3788	−0.930085	−0.465043	−	0.885288i	$-0.653961\pi$
$239$	−14.3788	−0.930085	−0.465043	+	0.885288i	$0.653961\pi$
$240$	0	0
$241$	18.2749	1.17719	0.588596	−	0.808427i	$-0.299681\pi$
$241$	18.2749	1.17719	0.588596	+	0.808427i	$0.299681\pi$
$242$	0	0
$243$	−10.1854	−0.653393
$244$	0	0
$245$	−1.51056	−0.0965060
$246$	0	0
$247$	0.330856	0.0210519
$248$	0	0
$249$	−2.86937	−0.181839
$250$	0	0
$251$	1.22019	0.0770174	0.0385087	−	0.999258i	$-0.487739\pi$
$251$	1.22019	0.0770174	0.0385087	+	0.999258i	$0.487739\pi$
$252$	0	0
$253$	32.6903	2.05522
$254$	0	0
$255$	−1.02112	−0.0639447
$256$	0	0
$257$	27.1306	1.69236	0.846181	−	0.532895i	$-0.178896\pi$
$257$	27.1306	1.69236	0.846181	+	0.532895i	$0.178896\pi$
$258$	0	0
$259$	2.91729	0.181271
$260$	0	0
$261$	15.0913	0.934128
$262$	0	0
$263$	20.2133	1.24641	0.623204	−	0.782059i	$-0.285831\pi$
$263$	20.2133	1.24641	0.623204	+	0.782059i	$0.285831\pi$
$264$	0	0
$265$	2.34513	0.144060
$266$	0	0
$267$	−2.44037	−0.149348
$268$	0	0
$269$	−0.207658	−0.0126611	−0.00633057	−	0.999980i	$-0.502015\pi$
$269$	−0.207658	−0.0126611	−0.00633057	+	0.999980i	$0.502015\pi$
$270$	0	0
$271$	30.6480	1.86174	0.930868	−	0.365357i	$-0.119053\pi$
$271$	30.6480	1.86174	0.930868	+	0.365357i	$0.119053\pi$
$272$	0	0
$273$	−0.965202	−0.0584167
$274$	0	0
$275$	−6.51056	−0.392601
$276$	0	0
$277$	−16.8135	−1.01022	−0.505111	−	0.863054i	$-0.668549\pi$
$277$	−16.8135	−1.01022	−0.505111	+	0.863054i	$0.668549\pi$
$278$	0	0
$279$	−24.8078	−1.48520
$280$	0	0
$281$	−14.2749	−0.851572	−0.425786	−	0.904824i	$-0.640002\pi$
$281$	−14.2749	−0.851572	−0.425786	+	0.904824i	$0.640002\pi$
$282$	0	0
$283$	13.2288	0.786369	0.393184	−	0.919460i	$-0.371373\pi$
$283$	13.2288	0.786369	0.393184	+	0.919460i	$0.371373\pi$
$284$	0	0
$285$	−0.165428	−0.00979911
$286$	0	0
$287$	−18.5106	−1.09264
$288$	0	0
$289$	−10.6971	−0.629241
$290$	0	0
$291$	−0.953831	−0.0559146
$292$	0	0
$293$	2.34513	0.137004	0.0685020	−	0.997651i	$-0.478178\pi$
$293$	2.34513	0.137004	0.0685020	+	0.997651i	$0.478178\pi$
$294$	0	0
$295$	−1.42784	−0.0831323
$296$	0	0
$297$	−15.4501	−0.896507
$298$	0	0
$299$	4.08446	0.236210
$300$	0	0
$301$	−21.3662	−1.23153
$302$	0	0
$303$	3.33829	0.191780
$304$	0	0
$305$	1.32401	0.0758128
$306$	0	0
$307$	23.2624	1.32766	0.663828	−	0.747885i	$-0.268931\pi$
$307$	23.2624	1.32766	0.663828	+	0.747885i	$0.268931\pi$
$308$	0	0
$309$	−3.58468	−0.203926
$310$	0	0
$311$	31.3999	1.78052	0.890262	−	0.455449i	$-0.150521\pi$
$311$	31.3999	1.78052	0.890262	+	0.455449i	$0.150521\pi$
$312$	0	0
$313$	−8.04223	−0.454574	−0.227287	−	0.973828i	$-0.572986\pi$
$313$	−8.04223	−0.454574	−0.227287	+	0.973828i	$0.572986\pi$
$314$	0	0
$315$	−8.26926	−0.465920
$316$	0	0
$317$	−12.3029	−0.691000	−0.345500	−	0.938419i	$-0.612291\pi$
$317$	−12.3029	−0.691000	−0.345500	+	0.938419i	$0.612291\pi$
$318$	0	0
$319$	34.6623	1.94072
$320$	0	0
$321$	6.20766	0.346478
$322$	0	0
$323$	1.02112	0.0568164
$324$	0	0
$325$	−0.813457	−0.0451225
$326$	0	0
$327$	−1.75011	−0.0967814
$328$	0	0
$329$	−15.8346	−0.872988
$330$	0	0
$331$	6.77981	0.372652	0.186326	−	0.982488i	$-0.440342\pi$
$331$	6.77981	0.372652	0.186326	+	0.982488i	$0.440342\pi$
$332$	0	0
$333$	2.83457	0.155334
$334$	0	0
$335$	−5.42784	−0.296555
$336$	0	0
$337$	10.6903	0.582336	0.291168	−	0.956672i	$-0.405956\pi$
$337$	10.6903	0.582336	0.291168	+	0.956672i	$0.405956\pi$
$338$	0	0
$339$	3.72506	0.202317
$340$	0	0
$341$	−56.9795	−3.08561
$342$	0	0
$343$	16.0143	0.864689
$344$	0	0
$345$	−2.04223	−0.109950
$346$	0	0
$347$	−5.62691	−0.302069	−0.151034	−	0.988529i	$-0.548260\pi$
$347$	−5.62691	−0.302069	−0.151034	+	0.988529i	$0.548260\pi$
$348$	0	0
$349$	15.1306	0.809924	0.404962	−	0.914334i	$-0.367285\pi$
$349$	15.1306	0.809924	0.404962	+	0.914334i	$0.367285\pi$
$350$	0	0
$351$	−1.93040	−0.103037
$352$	0	0
$353$	35.7816	1.90446	0.952230	−	0.305381i	$-0.0987839\pi$
$353$	35.7816	1.90446	0.952230	+	0.305381i	$0.0987839\pi$
$354$	0	0
$355$	14.6480	0.777437
$356$	0	0
$357$	−2.97888	−0.157659
$358$	0	0
$359$	−4.48260	−0.236583	−0.118291	−	0.992979i	$-0.537742\pi$
$359$	−4.48260	−0.236583	−0.118291	+	0.992979i	$0.537742\pi$
$360$	0	0
$361$	−18.8346	−0.991293
$362$	0	0
$363$	−12.7661	−0.670048
$364$	0	0
$365$	11.0211	0.576872
$366$	0	0
$367$	21.5653	1.12570	0.562850	−	0.826559i	$-0.309705\pi$
$367$	21.5653	1.12570	0.562850	+	0.826559i	$0.309705\pi$
$368$	0	0
$369$	−17.9857	−0.936300
$370$	0	0
$371$	6.84141	0.355188
$372$	0	0
$373$	16.9230	0.876238	0.438119	−	0.898917i	$-0.355645\pi$
$373$	16.9230	0.876238	0.438119	+	0.898917i	$0.355645\pi$
$374$	0	0
$375$	0.406728	0.0210034
$376$	0	0
$377$	4.33086	0.223050
$378$	0	0
$379$	−31.2710	−1.60628	−0.803142	−	0.595788i	$-0.796840\pi$
$379$	−31.2710	−1.60628	−0.803142	+	0.595788i	$0.796840\pi$
$380$	0	0
$381$	−3.50372	−0.179501
$382$	0	0
$383$	−1.62691	−0.0831314	−0.0415657	−	0.999136i	$-0.513235\pi$
$383$	−1.62691	−0.0831314	−0.0415657	+	0.999136i	$0.513235\pi$
$384$	0	0
$385$	−18.9932	−0.967981
$386$	0	0
$387$	−20.7604	−1.05531
$388$	0	0
$389$	−31.6412	−1.60427	−0.802136	−	0.597141i	$-0.796303\pi$
$389$	−31.6412	−1.60427	−0.802136	+	0.597141i	$0.796303\pi$
$390$	0	0
$391$	12.6058	0.637503
$392$	0	0
$393$	5.46149	0.275496
$394$	0	0
$395$	−1.75870	−0.0884898
$396$	0	0
$397$	−39.2961	−1.97221	−0.986106	−	0.166116i	$-0.946878\pi$
$397$	−39.2961	−1.97221	−0.986106	+	0.166116i	$0.946878\pi$
$398$	0	0
$399$	−0.482601	−0.0241603
$400$	0	0
$401$	−9.66914	−0.482854	−0.241427	−	0.970419i	$-0.577615\pi$
$401$	−9.66914	−0.482854	−0.241427	+	0.970419i	$0.577615\pi$
$402$	0	0
$403$	−7.11926	−0.354636
$404$	0	0
$405$	−7.53851	−0.374592
$406$	0	0
$407$	6.51056	0.322716
$408$	0	0
$409$	−26.2749	−1.29921	−0.649606	−	0.760271i	$-0.725066\pi$
$409$	−26.2749	−1.29921	−0.649606	+	0.760271i	$0.725066\pi$
$410$	0	0
$411$	5.97495	0.294722
$412$	0	0
$413$	−4.16543	−0.204967
$414$	0	0
$415$	−7.05476	−0.346304
$416$	0	0
$417$	7.06335	0.345894
$418$	0	0
$419$	30.1940	1.47507	0.737536	−	0.675308i	$-0.235989\pi$
$419$	30.1940	1.47507	0.737536	+	0.675308i	$0.235989\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	−15.3856	−0.748074
$424$	0	0
$425$	−2.51056	−0.121780
$426$	0	0
$427$	3.86253	0.186921
$428$	0	0
$429$	−2.15406	−0.103999
$430$	0	0
$431$	−25.5653	−1.23144	−0.615719	−	0.787966i	$-0.711134\pi$
$431$	−25.5653	−1.23144	−0.615719	+	0.787966i	$0.711134\pi$
$432$	0	0
$433$	32.0422	1.53985	0.769926	−	0.638134i	$-0.220293\pi$
$433$	32.0422	1.53985	0.769926	+	0.638134i	$0.220293\pi$
$434$	0	0
$435$	−2.16543	−0.103824
$436$	0	0
$437$	2.04223	0.0976931
$438$	0	0
$439$	−3.93840	−0.187970	−0.0939848	−	0.995574i	$-0.529961\pi$
$439$	−3.93840	−0.187970	−0.0939848	+	0.995574i	$0.529961\pi$
$440$	0	0
$441$	−4.28178	−0.203894
$442$	0	0
$443$	0.0758724	0.00360481	0.00180240	−	0.999998i	$-0.499426\pi$
$443$	0.0758724	0.00360481	0.00180240	+	0.999998i	$0.499426\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	0.0844605	0.00399485
$448$	0	0
$449$	19.2961	0.910637	0.455319	−	0.890329i	$-0.349525\pi$
$449$	19.2961	0.910637	0.455319	+	0.890329i	$0.349525\pi$
$450$	0	0
$451$	−41.3103	−1.94523
$452$	0	0
$453$	0.330856	0.0155450
$454$	0	0
$455$	−2.37309	−0.111252
$456$	0	0
$457$	3.00684	0.140654	0.0703271	−	0.997524i	$-0.477596\pi$
$457$	3.00684	0.140654	0.0703271	+	0.997524i	$0.477596\pi$
$458$	0	0
$459$	−5.95777	−0.278085
$460$	0	0
$461$	−0.633755	−0.0295169	−0.0147585	−	0.999891i	$-0.504698\pi$
$461$	−0.633755	−0.0295169	−0.0147585	+	0.999891i	$0.504698\pi$
$462$	0	0
$463$	−18.0422	−0.838494	−0.419247	−	0.907872i	$-0.637706\pi$
$463$	−18.0422	−0.838494	−0.419247	+	0.907872i	$0.637706\pi$
$464$	0	0
$465$	3.55963	0.165074
$466$	0	0
$467$	23.4758	1.08633	0.543164	−	0.839626i	$-0.317226\pi$
$467$	23.4758	1.08633	0.543164	+	0.839626i	$0.317226\pi$
$468$	0	0
$469$	−15.8346	−0.731173
$470$	0	0
$471$	−2.16543	−0.0997777
$472$	0	0
$473$	−47.6834	−2.19249
$474$	0	0
$475$	−0.406728	−0.0186620
$476$	0	0
$477$	6.64744	0.304365
$478$	0	0
$479$	8.19907	0.374625	0.187313	−	0.982300i	$-0.440022\pi$
$479$	8.19907	0.374625	0.187313	+	0.982300i	$0.440022\pi$
$480$	0	0
$481$	0.813457	0.0370904
$482$	0	0
$483$	−5.95777	−0.271088
$484$	0	0
$485$	−2.34513	−0.106487
$486$	0	0
$487$	−0.953831	−0.0432222	−0.0216111	−	0.999766i	$-0.506880\pi$
$487$	−0.953831	−0.0432222	−0.0216111	+	0.999766i	$0.506880\pi$
$488$	0	0
$489$	6.18260	0.279587
$490$	0	0
$491$	−19.3383	−0.872725	−0.436362	−	0.899771i	$-0.643733\pi$
$491$	−19.3383	−0.872725	−0.436362	+	0.899771i	$0.643733\pi$
$492$	0	0
$493$	13.3662	0.601985
$494$	0	0
$495$	−18.4546	−0.829475
$496$	0	0
$497$	42.7325	1.91681
$498$	0	0
$499$	5.63550	0.252280	0.126140	−	0.992012i	$-0.459741\pi$
$499$	5.63550	0.252280	0.126140	+	0.992012i	$0.459741\pi$
$500$	0	0
$501$	−5.07703	−0.226825
$502$	0	0
$503$	33.6269	1.49935	0.749675	−	0.661806i	$-0.230210\pi$
$503$	33.6269	1.49935	0.749675	+	0.661806i	$0.230210\pi$
$504$	0	0
$505$	8.20766	0.365236
$506$	0	0
$507$	5.01833	0.222872
$508$	0	0
$509$	−18.6903	−0.828431	−0.414216	−	0.910179i	$-0.635944\pi$
$509$	−18.6903	−0.828431	−0.414216	+	0.910179i	$0.635944\pi$
$510$	0	0
$511$	32.1517	1.42231
$512$	0	0
$513$	−0.965202	−0.0426147
$514$	0	0
$515$	−8.81346	−0.388367
$516$	0	0
$517$	−35.3383	−1.55418
$518$	0	0
$519$	−9.36915	−0.411260
$520$	0	0
$521$	25.1586	1.10222	0.551109	−	0.834433i	$-0.314205\pi$
$521$	25.1586	1.10222	0.551109	+	0.834433i	$0.314205\pi$
$522$	0	0
$523$	7.25383	0.317188	0.158594	−	0.987344i	$-0.449304\pi$
$523$	7.25383	0.317188	0.158594	+	0.987344i	$0.449304\pi$
$524$	0	0
$525$	1.18654	0.0517850
$526$	0	0
$527$	−21.9720	−0.957117
$528$	0	0
$529$	2.21160	0.0961564
$530$	0	0
$531$	−4.04733	−0.175639
$532$	0	0
$533$	−5.16149	−0.223569
$534$	0	0
$535$	15.2624	0.659852
$536$	0	0
$537$	9.87680	0.426215
$538$	0	0
$539$	−9.83457	−0.423605
$540$	0	0
$541$	16.0422	0.689709	0.344855	−	0.938656i	$-0.387928\pi$
$541$	16.0422	0.689709	0.344855	+	0.938656i	$0.387928\pi$
$542$	0	0
$543$	−0.880741	−0.0377962
$544$	0	0
$545$	−4.30290	−0.184316
$546$	0	0
$547$	−16.0143	−0.684721	−0.342360	−	0.939569i	$-0.611226\pi$
$547$	−16.0143	−0.684721	−0.342360	+	0.939569i	$0.611226\pi$
$548$	0	0
$549$	3.75301	0.160175
$550$	0	0
$551$	2.16543	0.0922503
$552$	0	0
$553$	−5.13063	−0.218177
$554$	0	0
$555$	−0.406728	−0.0172647
$556$	0	0
$557$	−10.8557	−0.459970	−0.229985	−	0.973194i	$-0.573868\pi$
$557$	−10.8557	−0.459970	−0.229985	+	0.973194i	$0.573868\pi$
$558$	0	0
$559$	−5.95777	−0.251987
$560$	0	0
$561$	−6.64803	−0.280680
$562$	0	0
$563$	−31.0776	−1.30977	−0.654883	−	0.755731i	$-0.727282\pi$
$563$	−31.0776	−1.30977	−0.654883	+	0.755731i	$0.727282\pi$
$564$	0	0
$565$	9.15859	0.385305
$566$	0	0
$567$	−21.9920	−0.923577
$568$	0	0
$569$	30.0845	1.26121	0.630603	−	0.776105i	$-0.282808\pi$
$569$	30.0845	1.26121	0.630603	+	0.776105i	$0.282808\pi$
$570$	0	0
$571$	−5.55673	−0.232542	−0.116271	−	0.993218i	$-0.537094\pi$
$571$	−5.55673	−0.232542	−0.116271	+	0.993218i	$0.537094\pi$
$572$	0	0
$573$	−7.86312	−0.328487
$574$	0	0
$575$	−5.02112	−0.209395
$576$	0	0
$577$	−22.2499	−0.926275	−0.463137	−	0.886286i	$-0.653276\pi$
$577$	−22.2499	−0.926275	−0.463137	+	0.886286i	$0.653276\pi$
$578$	0	0
$579$	8.15174	0.338775
$580$	0	0
$581$	−20.5807	−0.853833
$582$	0	0
$583$	15.2681	0.632340
$584$	0	0
$585$	−2.30580	−0.0953332
$586$	0	0
$587$	−45.3103	−1.87016	−0.935079	−	0.354440i	$-0.884672\pi$
$587$	−45.3103	−1.87016	−0.935079	+	0.354440i	$0.884672\pi$
$588$	0	0
$589$	−3.55963	−0.146672
$590$	0	0
$591$	−6.35591	−0.261447
$592$	0	0
$593$	−0.232712	−0.00955635	−0.00477817	−	0.999989i	$-0.501521\pi$
$593$	−0.232712	−0.00955635	−0.00477817	+	0.999989i	$0.501521\pi$
$594$	0	0
$595$	−7.32401	−0.300255
$596$	0	0
$597$	8.24989	0.337645
$598$	0	0
$599$	−43.4056	−1.77350	−0.886752	−	0.462246i	$-0.847044\pi$
$599$	−43.4056	−1.77350	−0.886752	+	0.462246i	$0.847044\pi$
$600$	0	0
$601$	35.5317	1.44937	0.724684	−	0.689082i	$-0.241986\pi$
$601$	35.5317	1.44937	0.724684	+	0.689082i	$0.241986\pi$
$602$	0	0
$603$	−15.3856	−0.626551
$604$	0	0
$605$	−31.3874	−1.27608
$606$	0	0
$607$	−26.7325	−1.08504	−0.542519	−	0.840043i	$-0.682529\pi$
$607$	−26.7325	−1.08504	−0.542519	+	0.840043i	$0.682529\pi$
$608$	0	0
$609$	−6.31717	−0.255985
$610$	0	0
$611$	−4.41532	−0.178625
$612$	0	0
$613$	48.0565	1.94098	0.970492	−	0.241134i	$-0.0775192\pi$
$613$	48.0565	1.94098	0.970492	+	0.241134i	$0.0775192\pi$
$614$	0	0
$615$	2.58074	0.104066
$616$	0	0
$617$	12.9789	0.522510	0.261255	−	0.965270i	$-0.415864\pi$
$617$	12.9789	0.522510	0.261255	+	0.965270i	$0.415864\pi$
$618$	0	0
$619$	−25.3662	−1.01956	−0.509778	−	0.860306i	$-0.670272\pi$
$619$	−25.3662	−1.01956	−0.509778	+	0.860306i	$0.670272\pi$
$620$	0	0
$621$	−11.9155	−0.478154
$622$	0	0
$623$	−17.5037	−0.701272
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.07703	−0.0430124
$628$	0	0
$629$	2.51056	0.100102
$630$	0	0
$631$	31.6075	1.25828	0.629138	−	0.777293i	$-0.283408\pi$
$631$	31.6075	1.25828	0.629138	+	0.777293i	$0.283408\pi$
$632$	0	0
$633$	7.61323	0.302599
$634$	0	0
$635$	−8.61439	−0.341852
$636$	0	0
$637$	−1.22877	−0.0486858
$638$	0	0
$639$	41.5209	1.64254
$640$	0	0
$641$	−23.8625	−0.942513	−0.471257	−	0.881996i	$-0.656199\pi$
$641$	−23.8625	−0.942513	−0.471257	+	0.881996i	$0.656199\pi$
$642$	0	0
$643$	−32.8277	−1.29460	−0.647300	−	0.762236i	$-0.724102\pi$
$643$	−32.8277	−1.29460	−0.647300	+	0.762236i	$0.724102\pi$
$644$	0	0
$645$	2.97888	0.117293
$646$	0	0
$647$	13.9578	0.548737	0.274368	−	0.961625i	$-0.411531\pi$
$647$	13.9578	0.548737	0.274368	+	0.961625i	$0.411531\pi$
$648$	0	0
$649$	−9.29606	−0.364902
$650$	0	0
$651$	10.3845	0.406999
$652$	0	0
$653$	28.5921	1.11890	0.559448	−	0.828865i	$-0.311013\pi$
$653$	28.5921	1.11890	0.559448	+	0.828865i	$0.311013\pi$
$654$	0	0
$655$	13.4278	0.524669
$656$	0	0
$657$	31.2401	1.21879
$658$	0	0
$659$	5.62691	0.219193	0.109597	−	0.993976i	$-0.465044\pi$
$659$	5.62691	0.219193	0.109597	+	0.993976i	$0.465044\pi$
$660$	0	0
$661$	13.3240	0.518244	0.259122	−	0.965845i	$-0.416567\pi$
$661$	13.3240	0.518244	0.259122	+	0.965845i	$0.416567\pi$
$662$	0	0
$663$	−0.830633	−0.0322591
$664$	0	0
$665$	−1.18654	−0.0460122
$666$	0	0
$667$	26.7325	1.03509
$668$	0	0
$669$	−6.39814	−0.247366
$670$	0	0
$671$	8.62007	0.332774
$672$	0	0
$673$	29.6691	1.14366	0.571831	−	0.820372i	$-0.306233\pi$
$673$	29.6691	1.14366	0.571831	+	0.820372i	$0.306233\pi$
$674$	0	0
$675$	2.37309	0.0913401
$676$	0	0
$677$	−1.25383	−0.0481885	−0.0240943	−	0.999710i	$-0.507670\pi$
$677$	−1.25383	−0.0481885	−0.0240943	+	0.999710i	$0.507670\pi$
$678$	0	0
$679$	−6.84141	−0.262549
$680$	0	0
$681$	5.24014	0.200803
$682$	0	0
$683$	4.76045	0.182153	0.0910767	−	0.995844i	$-0.470969\pi$
$683$	4.76045	0.182153	0.0910767	+	0.995844i	$0.470969\pi$
$684$	0	0
$685$	14.6903	0.561286
$686$	0	0
$687$	−2.13688	−0.0815271
$688$	0	0
$689$	1.90766	0.0726761
$690$	0	0
$691$	20.2219	0.769279	0.384639	−	0.923067i	$-0.374326\pi$
$691$	20.2219	0.769279	0.384639	+	0.923067i	$0.374326\pi$
$692$	0	0
$693$	−53.8375	−2.04512
$694$	0	0
$695$	17.3662	0.658739
$696$	0	0
$697$	−15.9298	−0.603385
$698$	0	0
$699$	−11.5174	−0.435628
$700$	0	0
$701$	−22.0845	−0.834119	−0.417059	−	0.908879i	$-0.636939\pi$
$701$	−22.0845	−0.834119	−0.417059	+	0.908879i	$0.636939\pi$
$702$	0	0
$703$	0.406728	0.0153401
$704$	0	0
$705$	2.20766	0.0831452
$706$	0	0
$707$	23.9441	0.900510
$708$	0	0
$709$	5.59896	0.210273	0.105137	−	0.994458i	$-0.466472\pi$
$709$	5.59896	0.210273	0.105137	+	0.994458i	$0.466472\pi$
$710$	0	0
$711$	−4.98516	−0.186958
$712$	0	0
$713$	−43.9441	−1.64572
$714$	0	0
$715$	−5.29606	−0.198061
$716$	0	0
$717$	5.84826	0.218407
$718$	0	0
$719$	−40.2921	−1.50264	−0.751321	−	0.659937i	$-0.770583\pi$
$719$	−40.2921	−1.50264	−0.751321	+	0.659937i	$0.770583\pi$
$720$	0	0
$721$	−25.7114	−0.957542
$722$	0	0
$723$	−7.43294	−0.276434
$724$	0	0
$725$	−5.32401	−0.197729
$726$	0	0
$727$	−11.2538	−0.417381	−0.208691	−	0.977982i	$-0.566920\pi$
$727$	−11.2538	−0.417381	−0.208691	+	0.977982i	$0.566920\pi$
$728$	0	0
$729$	−18.4729	−0.684180
$730$	0	0
$731$	−18.3874	−0.680081
$732$	0	0
$733$	−42.2892	−1.56199	−0.780994	−	0.624539i	$-0.785287\pi$
$733$	−42.2892	−1.56199	−0.780994	+	0.624539i	$0.785287\pi$
$734$	0	0
$735$	0.614387	0.0226620
$736$	0	0
$737$	−35.3383	−1.30170
$738$	0	0
$739$	29.3103	1.07820	0.539099	−	0.842242i	$-0.318765\pi$
$739$	29.3103	1.07820	0.539099	+	0.842242i	$0.318765\pi$
$740$	0	0
$741$	−0.134569	−0.00494350
$742$	0	0
$743$	4.39245	0.161144	0.0805718	−	0.996749i	$-0.474325\pi$
$743$	4.39245	0.161144	0.0805718	+	0.996749i	$0.474325\pi$
$744$	0	0
$745$	0.207658	0.00760801
$746$	0	0
$747$	−19.9972	−0.731660
$748$	0	0
$749$	44.5248	1.62690
$750$	0	0
$751$	−2.58074	−0.0941727	−0.0470864	−	0.998891i	$-0.514994\pi$
$751$	−2.58074	−0.0941727	−0.0470864	+	0.998891i	$0.514994\pi$
$752$	0	0
$753$	−0.496284	−0.0180856
$754$	0	0
$755$	0.813457	0.0296047
$756$	0	0
$757$	27.3805	0.995162	0.497581	−	0.867418i	$-0.334222\pi$
$757$	27.3805	0.995162	0.497581	+	0.867418i	$0.334222\pi$
$758$	0	0
$759$	−13.2961	−0.482616
$760$	0	0
$761$	−42.9509	−1.55697	−0.778485	−	0.627663i	$-0.784011\pi$
$761$	−42.9509	−1.55697	−0.778485	+	0.627663i	$0.784011\pi$
$762$	0	0
$763$	−12.5528	−0.454441
$764$	0	0
$765$	−7.11636	−0.257292
$766$	0	0
$767$	−1.16149	−0.0419389
$768$	0	0
$769$	−13.0633	−0.471076	−0.235538	−	0.971865i	$-0.575685\pi$
$769$	−13.0633	−0.471076	−0.235538	+	0.971865i	$0.575685\pi$
$770$	0	0
$771$	−11.0348	−0.397409
$772$	0	0
$773$	−15.3662	−0.552685	−0.276343	−	0.961059i	$-0.589122\pi$
$773$	−15.3662	−0.552685	−0.276343	+	0.961059i	$0.589122\pi$
$774$	0	0
$775$	8.75186	0.314376
$776$	0	0
$777$	−1.18654	−0.0425670
$778$	0	0
$779$	−2.58074	−0.0924648
$780$	0	0
$781$	95.3668	3.41249
$782$	0	0
$783$	−12.6343	−0.451515
$784$	0	0
$785$	−5.32401	−0.190022
$786$	0	0
$787$	−10.9316	−0.389668	−0.194834	−	0.980836i	$-0.562417\pi$
$787$	−10.9316	−0.389668	−0.194834	+	0.980836i	$0.562417\pi$
$788$	0	0
$789$	−8.22134	−0.292688
$790$	0	0
$791$	26.7182	0.949990
$792$	0	0
$793$	1.07703	0.0382464
$794$	0	0
$795$	−0.953831	−0.0338289
$796$	0	0
$797$	−24.2921	−0.860471	−0.430235	−	0.902717i	$-0.641569\pi$
$797$	−24.2921	−0.860471	−0.430235	+	0.902717i	$0.641569\pi$
$798$	0	0
$799$	−13.6269	−0.482086
$800$	0	0
$801$	−17.0074	−0.600928
$802$	0	0
$803$	71.7536	2.53213
$804$	0	0
$805$	−14.6480	−0.516275
$806$	0	0
$807$	0.0844605	0.00297315
$808$	0	0
$809$	−40.0422	−1.40781	−0.703905	−	0.710294i	$-0.748562\pi$
$809$	−40.0422	−1.40781	−0.703905	+	0.710294i	$0.748562\pi$
$810$	0	0
$811$	40.6343	1.42686	0.713432	−	0.700724i	$-0.247140\pi$
$811$	40.6343	1.42686	0.713432	+	0.700724i	$0.247140\pi$
$812$	0	0
$813$	−12.4654	−0.437182
$814$	0	0
$815$	15.2008	0.532461
$816$	0	0
$817$	−2.97888	−0.104218
$818$	0	0
$819$	−6.72668	−0.235049
$820$	0	0
$821$	−41.4787	−1.44762	−0.723808	−	0.690002i	$-0.757610\pi$
$821$	−41.4787	−1.44762	−0.723808	+	0.690002i	$0.757610\pi$
$822$	0	0
$823$	5.27610	0.183913	0.0919566	−	0.995763i	$-0.470688\pi$
$823$	5.27610	0.183913	0.0919566	+	0.995763i	$0.470688\pi$
$824$	0	0
$825$	2.64803	0.0921925
$826$	0	0
$827$	−1.76438	−0.0613537	−0.0306768	−	0.999529i	$-0.509766\pi$
$827$	−1.76438	−0.0613537	−0.0306768	+	0.999529i	$0.509766\pi$
$828$	0	0
$829$	16.3029	0.566223	0.283112	−	0.959087i	$-0.408633\pi$
$829$	16.3029	0.566223	0.283112	+	0.959087i	$0.408633\pi$
$830$	0	0
$831$	6.83851	0.237225
$832$	0	0
$833$	−3.79234	−0.131397
$834$	0	0
$835$	−12.4826	−0.431978
$836$	0	0
$837$	20.7689	0.717879
$838$	0	0
$839$	−14.3731	−0.496214	−0.248107	−	0.968733i	$-0.579808\pi$
$839$	−14.3731	−0.496214	−0.248107	+	0.968733i	$0.579808\pi$
$840$	0	0
$841$	−0.654870	−0.0225817
$842$	0	0
$843$	5.80602	0.199970
$844$	0	0
$845$	12.3383	0.424450
$846$	0	0
$847$	−91.5659	−3.14624
$848$	0	0
$849$	−5.38052	−0.184659
$850$	0	0
$851$	5.02112	0.172122
$852$	0	0
$853$	43.2961	1.48243	0.741214	−	0.671268i	$-0.234250\pi$
$853$	43.2961	1.48243	0.741214	+	0.671268i	$0.234250\pi$
$854$	0	0
$855$	−1.15290	−0.0394284
$856$	0	0
$857$	−49.0776	−1.67646	−0.838230	−	0.545317i	$-0.816409\pi$
$857$	−49.0776	−1.67646	−0.838230	+	0.545317i	$0.816409\pi$
$858$	0	0
$859$	−15.5374	−0.530128	−0.265064	−	0.964231i	$-0.585393\pi$
$859$	−15.5374	−0.530128	−0.265064	+	0.964231i	$0.585393\pi$
$860$	0	0
$861$	7.52877	0.256580
$862$	0	0
$863$	36.0901	1.22852	0.614261	−	0.789103i	$-0.289454\pi$
$863$	36.0901	1.22852	0.614261	+	0.789103i	$0.289454\pi$
$864$	0	0
$865$	−23.0354	−0.783227
$866$	0	0
$867$	4.35081	0.147761
$868$	0	0
$869$	−11.4501	−0.388419
$870$	0	0
$871$	−4.41532	−0.149607
$872$	0	0
$873$	−6.64744	−0.224982
$874$	0	0
$875$	2.91729	0.0986223
$876$	0	0
$877$	29.3240	0.990202	0.495101	−	0.868836i	$-0.335131\pi$
$877$	29.3240	0.990202	0.495101	+	0.868836i	$0.335131\pi$
$878$	0	0
$879$	−0.953831	−0.0321719
$880$	0	0
$881$	23.8066	0.802065	0.401033	−	0.916064i	$-0.368651\pi$
$881$	23.8066	0.802065	0.401033	+	0.916064i	$0.368651\pi$
$882$	0	0
$883$	3.94699	0.132827	0.0664134	−	0.997792i	$-0.478844\pi$
$883$	3.94699	0.132827	0.0664134	+	0.997792i	$0.478844\pi$
$884$	0	0
$885$	0.580745	0.0195215
$886$	0	0
$887$	−1.96346	−0.0659264	−0.0329632	−	0.999457i	$-0.510494\pi$
$887$	−1.96346	−0.0659264	−0.0329632	+	0.999457i	$0.510494\pi$
$888$	0	0
$889$	−25.1306	−0.842854
$890$	0	0
$891$	−49.0799	−1.64424
$892$	0	0
$893$	−2.20766	−0.0738765
$894$	0	0
$895$	24.2835	0.811709
$896$	0	0
$897$	−1.66127	−0.0554681
$898$	0	0
$899$	−46.5950	−1.55403
$900$	0	0
$901$	5.88758	0.196144
$902$	0	0
$903$	8.69026	0.289194
$904$	0	0
$905$	−2.16543	−0.0719813
$906$	0	0
$907$	−5.89049	−0.195590	−0.0977952	−	0.995207i	$-0.531179\pi$
$907$	−5.89049	−0.195590	−0.0977952	+	0.995207i	$0.531179\pi$
$908$	0	0
$909$	23.2652	0.771658
$910$	0	0
$911$	27.9527	0.926113	0.463057	−	0.886329i	$-0.346753\pi$
$911$	27.9527	0.926113	0.463057	+	0.886329i	$0.346753\pi$
$912$	0	0
$913$	−45.9304	−1.52007
$914$	0	0
$915$	−0.538514	−0.0178027
$916$	0	0
$917$	39.1729	1.29360
$918$	0	0
$919$	−17.7587	−0.585805	−0.292903	−	0.956142i	$-0.594621\pi$
$919$	−17.7587	−0.585805	−0.292903	+	0.956142i	$0.594621\pi$
$920$	0	0
$921$	−9.46149	−0.311767
$922$	0	0
$923$	11.9155	0.392205
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−24.9824	−0.820529
$928$	0	0
$929$	34.2892	1.12499	0.562496	−	0.826800i	$-0.309841\pi$
$929$	34.2892	1.12499	0.562496	+	0.826800i	$0.309841\pi$
$930$	0	0
$931$	−0.614387	−0.0201357
$932$	0	0
$933$	−12.7712	−0.418111
$934$	0	0
$935$	−16.3451	−0.534543
$936$	0	0
$937$	−14.0845	−0.460119	−0.230060	−	0.973177i	$-0.573892\pi$
$937$	−14.0845	−0.460119	−0.230060	+	0.973177i	$0.573892\pi$
$938$	0	0
$939$	3.27100	0.106745
$940$	0	0
$941$	14.2499	0.464533	0.232267	−	0.972652i	$-0.425386\pi$
$941$	14.2499	0.464533	0.232267	+	0.972652i	$0.425386\pi$
$942$	0	0
$943$	−31.8596	−1.03749
$944$	0	0
$945$	6.92297	0.225204
$946$	0	0
$947$	29.8203	0.969029	0.484515	−	0.874783i	$-0.338996\pi$
$947$	29.8203	0.969029	0.484515	+	0.874783i	$0.338996\pi$
$948$	0	0
$949$	8.96520	0.291023
$950$	0	0
$951$	5.00394	0.162264
$952$	0	0
$953$	8.64803	0.280137	0.140069	−	0.990142i	$-0.455268\pi$
$953$	8.64803	0.280137	0.140069	+	0.990142i	$0.455268\pi$
$954$	0	0
$955$	−19.3326	−0.625588
$956$	0	0
$957$	−14.0981	−0.455728
$958$	0	0
$959$	42.8557	1.38388
$960$	0	0
$961$	45.5950	1.47081
$962$	0	0
$963$	43.2624	1.39411
$964$	0	0
$965$	20.0422	0.645182
$966$	0	0
$967$	45.0325	1.44815	0.724074	−	0.689723i	$-0.242268\pi$
$967$	45.0325	1.44815	0.724074	+	0.689723i	$0.242268\pi$
$968$	0	0
$969$	−0.415317	−0.0133419
$970$	0	0
$971$	−26.3029	−0.844100	−0.422050	−	0.906573i	$-0.638689\pi$
$971$	−26.3029	−0.844100	−0.422050	+	0.906573i	$0.638689\pi$
$972$	0	0
$973$	50.6623	1.62416
$974$	0	0
$975$	0.330856	0.0105959
$976$	0	0
$977$	25.6549	0.820772	0.410386	−	0.911912i	$-0.365394\pi$
$977$	25.6549	0.820772	0.410386	+	0.911912i	$0.365394\pi$
$978$	0	0
$979$	−39.0633	−1.24847
$980$	0	0
$981$	−12.1969	−0.389416
$982$	0	0
$983$	55.3611	1.76575	0.882873	−	0.469611i	$-0.155606\pi$
$983$	55.3611	1.76575	0.882873	+	0.469611i	$0.155606\pi$
$984$	0	0
$985$	−15.6269	−0.497915
$986$	0	0
$987$	6.44037	0.204999
$988$	0	0
$989$	−36.7747	−1.16937
$990$	0	0
$991$	3.24814	0.103181	0.0515903	−	0.998668i	$-0.483571\pi$
$991$	3.24814	0.103181	0.0515903	+	0.998668i	$0.483571\pi$
$992$	0	0
$993$	−2.75754	−0.0875080
$994$	0	0
$995$	20.2835	0.643031
$996$	0	0
$997$	−12.6480	−0.400567	−0.200284	−	0.979738i	$-0.564186\pi$
$997$	−12.6480	−0.400567	−0.200284	+	0.979738i	$0.564186\pi$
$998$	0	0
$999$	−2.37309	−0.0750811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.u.1.2		3
4.3	odd	2		370.2.a.g.1.2	✓	3
12.11	even	2		3330.2.a.bg.1.3		3
20.3	even	4		1850.2.b.o.149.2		6
20.7	even	4		1850.2.b.o.149.5		6
20.19	odd	2		1850.2.a.z.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.g.1.2	✓	3	4.3	odd	2
1850.2.a.z.1.2		3	20.19	odd	2
1850.2.b.o.149.2		6	20.3	even	4
1850.2.b.o.149.5		6	20.7	even	4
2960.2.a.u.1.2		3	1.1	even	1	trivial
3330.2.a.bg.1.3		3	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)

\(f(q)\)	\(=\)	\(q-0.406728 q^{3} -1.00000 q^{5} -2.91729 q^{7} -2.83457 q^{9} -6.51056 q^{11} -0.813457 q^{13} +0.406728 q^{15} -2.51056 q^{17} -0.406728 q^{19} +1.18654 q^{21} -5.02112 q^{23} +1.00000 q^{25} +2.37309 q^{27} -5.32401 q^{29} +8.75186 q^{31} +2.64803 q^{33} +2.91729 q^{35} -1.00000 q^{37} +0.330856 q^{39} +6.34513 q^{41} +7.32401 q^{43} +2.83457 q^{45} +5.42784 q^{47} +1.51056 q^{49} +1.02112 q^{51} -2.34513 q^{53} +6.51056 q^{55} +0.165428 q^{57} +1.42784 q^{59} -1.32401 q^{61} +8.26926 q^{63} +0.813457 q^{65} +5.42784 q^{67} +2.04223 q^{69} -14.6480 q^{71} -11.0211 q^{73} -0.406728 q^{75} +18.9932 q^{77} +1.75870 q^{79} +7.53851 q^{81} +7.05476 q^{83} +2.51056 q^{85} +2.16543 q^{87} +6.00000 q^{89} +2.37309 q^{91} -3.55963 q^{93} +0.406728 q^{95} +2.34513 q^{97} +18.4546 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + q^{7} + 11 q^{9} - 11 q^{11} + q^{17} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 5 q^{29} - 3 q^{31} - 14 q^{33} - q^{35} - 3 q^{37} + 40 q^{39} - 9 q^{41} + 11 q^{43} - 11 q^{45}+ \cdots - 19 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + q^7 + 11 * q^9 - 11 * q^11 + q^17 + 6 * q^21 + 2 * q^23 + 3 * q^25 + 12 * q^27 - 5 * q^29 - 3 * q^31 - 14 * q^33 - q^35 - 3 * q^37 + 40 * q^39 - 9 * q^41 + 11 * q^43 - 11 * q^45 - 2 * q^47 - 4 * q^49 - 14 * q^51 + 21 * q^53 + 11 * q^55 + 20 * q^57 - 14 * q^59 + 7 * q^61 + 37 * q^63 - 2 * q^67 - 28 * q^69 - 22 * q^71 - 16 * q^73 + 7 * q^77 + 26 * q^79 + 47 * q^81 - 2 * q^83 - q^85 + 26 * q^87 + 18 * q^89 + 12 * q^91 - 18 * q^93 - 21 * q^97 - 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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