LMFDB - Embedded newform 6080.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.428007 q^{3} +1.00000 q^{5} -4.67282 q^{7} -2.81681 q^{9} +2.67282 q^{11} +2.24482 q^{13} -0.428007 q^{15} +2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} +0.672824 q^{23} +1.00000 q^{25} +2.48963 q^{27} -1.14399 q^{29} -1.14399 q^{31} -1.14399 q^{33} -4.67282 q^{35} -4.24482 q^{37} -0.960797 q^{39} +4.85601 q^{41} -5.52884 q^{43} -2.81681 q^{45} -4.67282 q^{47} +14.8353 q^{49} -0.856013 q^{51} +0.244817 q^{53} +2.67282 q^{55} -0.428007 q^{57} +0.287973 q^{59} +8.87448 q^{61} +13.1625 q^{63} +2.24482 q^{65} -9.20561 q^{67} -0.287973 q^{69} -9.14399 q^{71} +1.14399 q^{73} -0.428007 q^{75} -12.4896 q^{77} +2.00000 q^{79} +7.38485 q^{81} -0.183190 q^{83} +2.00000 q^{85} +0.489634 q^{87} +9.14399 q^{89} -10.4896 q^{91} +0.489634 q^{93} +1.00000 q^{95} +9.10083 q^{97} -7.52884 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} - 8 q^{23} + 3 q^{25} - 14 q^{27} - 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 2 q^{37} + 10 q^{39}+ \cdots - 14 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 2 * q^11 - 4 * q^13 - 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 - 8 * q^23 + 3 * q^25 - 14 * q^27 - 2 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 - 2 * q^37 + 10 * q^39 + 16 * q^41 - 8 * q^43 + 3 * q^45 - 4 * q^47 + 3 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^55 - 2 * q^57 - 2 * q^59 - 2 * q^61 + 8 * q^63 - 4 * q^65 - 4 * q^67 + 2 * q^69 - 26 * q^71 + 2 * q^73 - 2 * q^75 - 16 * q^77 + 6 * q^79 + 15 * q^81 - 12 * q^83 + 6 * q^85 - 20 * q^87 + 26 * q^89 - 10 * q^91 - 20 * q^93 + 3 * q^95 + 18 * q^97 - 14 * 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
$3$	−0.428007	−0.247110	−0.123555	−	0.992338i	$-0.539430\pi$
$3$	−0.428007	−0.247110	−0.123555	+	0.992338i	$0.539430\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.67282	−1.76616	−0.883081	−	0.469221i	$-0.844535\pi$
$7$	−4.67282	−1.76616	−0.883081	+	0.469221i	$0.844535\pi$
$8$	0	0
$9$	−2.81681	−0.938937
$10$	0	0
$11$	2.67282	0.805887	0.402943	−	0.915225i	$-0.367987\pi$
$11$	2.67282	0.805887	0.402943	+	0.915225i	$0.367987\pi$
$12$	0	0
$13$	2.24482	0.622600	0.311300	−	0.950312i	$-0.399236\pi$
$13$	2.24482	0.622600	0.311300	+	0.950312i	$0.399236\pi$
$14$	0	0
$15$	−0.428007	−0.110511
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	0.672824	0.140293	0.0701467	−	0.997537i	$-0.477653\pi$
$23$	0.672824	0.140293	0.0701467	+	0.997537i	$0.477653\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.48963	0.479130
$28$	0	0
$29$	−1.14399	−0.212433	−0.106216	−	0.994343i	$-0.533874\pi$
$29$	−1.14399	−0.212433	−0.106216	+	0.994343i	$0.533874\pi$
$30$	0	0
$31$	−1.14399	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$31$	−1.14399	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$32$	0	0
$33$	−1.14399	−0.199142
$34$	0	0
$35$	−4.67282	−0.789851
$36$	0	0
$37$	−4.24482	−0.697844	−0.348922	−	0.937152i	$-0.613452\pi$
$37$	−4.24482	−0.697844	−0.348922	+	0.937152i	$0.613452\pi$
$38$	0	0
$39$	−0.960797	−0.153851
$40$	0	0
$41$	4.85601	0.758382	0.379191	−	0.925318i	$-0.376202\pi$
$41$	4.85601	0.758382	0.379191	+	0.925318i	$0.376202\pi$
$42$	0	0
$43$	−5.52884	−0.843140	−0.421570	−	0.906796i	$-0.638521\pi$
$43$	−5.52884	−0.843140	−0.421570	+	0.906796i	$0.638521\pi$
$44$	0	0
$45$	−2.81681	−0.419905
$46$	0	0
$47$	−4.67282	−0.681601	−0.340801	−	0.940136i	$-0.610698\pi$
$47$	−4.67282	−0.681601	−0.340801	+	0.940136i	$0.610698\pi$
$48$	0	0
$49$	14.8353	2.11933
$50$	0	0
$51$	−0.856013	−0.119866
$52$	0	0
$53$	0.244817	0.0336282	0.0168141	−	0.999859i	$-0.494648\pi$
$53$	0.244817	0.0336282	0.0168141	+	0.999859i	$0.494648\pi$
$54$	0	0
$55$	2.67282	0.360403
$56$	0	0
$57$	−0.428007	−0.0566909
$58$	0	0
$59$	0.287973	0.0374909	0.0187455	−	0.999824i	$-0.494033\pi$
$59$	0.287973	0.0374909	0.0187455	+	0.999824i	$0.494033\pi$
$60$	0	0
$61$	8.87448	1.13626	0.568131	−	0.822938i	$-0.307667\pi$
$61$	8.87448	1.13626	0.568131	+	0.822938i	$0.307667\pi$
$62$	0	0
$63$	13.1625	1.65831
$64$	0	0
$65$	2.24482	0.278435
$66$	0	0
$67$	−9.20561	−1.12464	−0.562322	−	0.826918i	$-0.690092\pi$
$67$	−9.20561	−1.12464	−0.562322	+	0.826918i	$0.690092\pi$
$68$	0	0
$69$	−0.287973	−0.0346679
$70$	0	0
$71$	−9.14399	−1.08519	−0.542596	−	0.839994i	$-0.682558\pi$
$71$	−9.14399	−1.08519	−0.542596	+	0.839994i	$0.682558\pi$
$72$	0	0
$73$	1.14399	0.133893	0.0669467	−	0.997757i	$-0.478674\pi$
$73$	1.14399	0.133893	0.0669467	+	0.997757i	$0.478674\pi$
$74$	0	0
$75$	−0.428007	−0.0494220
$76$	0	0
$77$	−12.4896	−1.42333
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	7.38485	0.820539
$82$	0	0
$83$	−0.183190	−0.0201077	−0.0100538	−	0.999949i	$-0.503200\pi$
$83$	−0.183190	−0.0201077	−0.0100538	+	0.999949i	$0.503200\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0.489634	0.0524943
$88$	0	0
$89$	9.14399	0.969261	0.484630	−	0.874719i	$-0.338954\pi$
$89$	9.14399	0.969261	0.484630	+	0.874719i	$0.338954\pi$
$90$	0	0
$91$	−10.4896	−1.09961
$92$	0	0
$93$	0.489634	0.0507727
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	9.10083	0.924049	0.462025	−	0.886867i	$-0.347123\pi$
$97$	9.10083	0.924049	0.462025	+	0.886867i	$0.347123\pi$
$98$	0	0
$99$	−7.52884	−0.756677
$100$	0	0
$101$	−2.67282	−0.265956	−0.132978	−	0.991119i	$-0.542454\pi$
$101$	−2.67282	−0.265956	−0.132978	+	0.991119i	$0.542454\pi$
$102$	0	0
$103$	6.91764	0.681615	0.340808	−	0.940133i	$-0.389299\pi$
$103$	6.91764	0.681615	0.340808	+	0.940133i	$0.389299\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	8.62967	0.834261	0.417131	−	0.908846i	$-0.363036\pi$
$107$	8.62967	0.834261	0.417131	+	0.908846i	$0.363036\pi$
$108$	0	0
$109$	−11.6336	−1.11430	−0.557149	−	0.830412i	$-0.688105\pi$
$109$	−11.6336	−1.11430	−0.557149	+	0.830412i	$0.688105\pi$
$110$	0	0
$111$	1.81681	0.172444
$112$	0	0
$113$	6.61120	0.621929	0.310965	−	0.950422i	$-0.399348\pi$
$113$	6.61120	0.621929	0.310965	+	0.950422i	$0.399348\pi$
$114$	0	0
$115$	0.672824	0.0627411
$116$	0	0
$117$	−6.32322	−0.584582
$118$	0	0
$119$	−9.34565	−0.856714
$120$	0	0
$121$	−3.85601	−0.350547
$122$	0	0
$123$	−2.07841	−0.187404
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.91764	−0.791313	−0.395656	−	0.918399i	$-0.629483\pi$
$127$	−8.91764	−0.791313	−0.395656	+	0.918399i	$0.629483\pi$
$128$	0	0
$129$	2.36638	0.208348
$130$	0	0
$131$	−6.77761	−0.592162	−0.296081	−	0.955163i	$-0.595680\pi$
$131$	−6.77761	−0.592162	−0.296081	+	0.955163i	$0.595680\pi$
$132$	0	0
$133$	−4.67282	−0.405185
$134$	0	0
$135$	2.48963	0.214274
$136$	0	0
$137$	−14.0369	−1.19926	−0.599628	−	0.800279i	$-0.704685\pi$
$137$	−14.0369	−1.19926	−0.599628	+	0.800279i	$0.704685\pi$
$138$	0	0
$139$	−4.96080	−0.420769	−0.210385	−	0.977619i	$-0.567472\pi$
$139$	−4.96080	−0.420769	−0.210385	+	0.977619i	$0.567472\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	−1.14399	−0.0950029
$146$	0	0
$147$	−6.34960	−0.523706
$148$	0	0
$149$	0.874485	0.0716406	0.0358203	−	0.999358i	$-0.488596\pi$
$149$	0.874485	0.0716406	0.0358203	+	0.999358i	$0.488596\pi$
$150$	0	0
$151$	−21.4689	−1.74711	−0.873557	−	0.486721i	$-0.838193\pi$
$151$	−21.4689	−1.74711	−0.873557	+	0.486721i	$0.838193\pi$
$152$	0	0
$153$	−5.63362	−0.455451
$154$	0	0
$155$	−1.14399	−0.0918872
$156$	0	0
$157$	−11.9216	−0.951447	−0.475723	−	0.879595i	$-0.657814\pi$
$157$	−11.9216	−0.951447	−0.475723	+	0.879595i	$0.657814\pi$
$158$	0	0
$159$	−0.104783	−0.00830986
$160$	0	0
$161$	−3.14399	−0.247781
$162$	0	0
$163$	−2.47116	−0.193556	−0.0967782	−	0.995306i	$-0.530854\pi$
$163$	−2.47116	−0.193556	−0.0967782	+	0.995306i	$0.530854\pi$
$164$	0	0
$165$	−1.14399	−0.0890592
$166$	0	0
$167$	−19.4857	−1.50785	−0.753924	−	0.656962i	$-0.771841\pi$
$167$	−19.4857	−1.50785	−0.753924	+	0.656962i	$0.771841\pi$
$168$	0	0
$169$	−7.96080	−0.612369
$170$	0	0
$171$	−2.81681	−0.215407
$172$	0	0
$173$	−21.3025	−1.61960	−0.809799	−	0.586707i	$-0.800424\pi$
$173$	−21.3025	−1.61960	−0.809799	+	0.586707i	$0.800424\pi$
$174$	0	0
$175$	−4.67282	−0.353232
$176$	0	0
$177$	−0.123254	−0.00926437
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	21.2672	1.58078	0.790391	−	0.612603i	$-0.209878\pi$
$181$	21.2672	1.58078	0.790391	+	0.612603i	$0.209878\pi$
$182$	0	0
$183$	−3.79834	−0.280781
$184$	0	0
$185$	−4.24482	−0.312085
$186$	0	0
$187$	5.34565	0.390912
$188$	0	0
$189$	−11.6336	−0.846221
$190$	0	0
$191$	−24.0369	−1.73925	−0.869626	−	0.493711i	$-0.835640\pi$
$191$	−24.0369	−1.73925	−0.869626	+	0.493711i	$0.835640\pi$
$192$	0	0
$193$	22.0801	1.58936	0.794680	−	0.607028i	$-0.207639\pi$
$193$	22.0801	1.58936	0.794680	+	0.607028i	$0.207639\pi$
$194$	0	0
$195$	−0.960797	−0.0688041
$196$	0	0
$197$	−25.5473	−1.82017	−0.910085	−	0.414421i	$-0.863984\pi$
$197$	−25.5473	−1.82017	−0.910085	+	0.414421i	$0.863984\pi$
$198$	0	0
$199$	−16.9793	−1.20363	−0.601814	−	0.798636i	$-0.705555\pi$
$199$	−16.9793	−1.20363	−0.601814	+	0.798636i	$0.705555\pi$
$200$	0	0
$201$	3.94006	0.277911
$202$	0	0
$203$	5.34565	0.375191
$204$	0	0
$205$	4.85601	0.339159
$206$	0	0
$207$	−1.89522	−0.131727
$208$	0	0
$209$	2.67282	0.184883
$210$	0	0
$211$	8.48963	0.584451	0.292225	−	0.956350i	$-0.405604\pi$
$211$	8.48963	0.584451	0.292225	+	0.956350i	$0.405604\pi$
$212$	0	0
$213$	3.91369	0.268161
$214$	0	0
$215$	−5.52884	−0.377064
$216$	0	0
$217$	5.34565	0.362886
$218$	0	0
$219$	−0.489634	−0.0330864
$220$	0	0
$221$	4.48963	0.302005
$222$	0	0
$223$	8.55126	0.572635	0.286317	−	0.958135i	$-0.407569\pi$
$223$	8.55126	0.572635	0.286317	+	0.958135i	$0.407569\pi$
$224$	0	0
$225$	−2.81681	−0.187787
$226$	0	0
$227$	−19.9832	−1.32633	−0.663166	−	0.748472i	$-0.730788\pi$
$227$	−19.9832	−1.32633	−0.663166	+	0.748472i	$0.730788\pi$
$228$	0	0
$229$	−0.751230	−0.0496427	−0.0248213	−	0.999692i	$-0.507902\pi$
$229$	−0.751230	−0.0496427	−0.0248213	+	0.999692i	$0.507902\pi$
$230$	0	0
$231$	5.34565	0.351718
$232$	0	0
$233$	−15.2593	−0.999672	−0.499836	−	0.866120i	$-0.666606\pi$
$233$	−15.2593	−0.999672	−0.499836	+	0.866120i	$0.666606\pi$
$234$	0	0
$235$	−4.67282	−0.304821
$236$	0	0
$237$	−0.856013	−0.0556040
$238$	0	0
$239$	16.5266	1.06902	0.534508	−	0.845164i	$-0.320497\pi$
$239$	16.5266	1.06902	0.534508	+	0.845164i	$0.320497\pi$
$240$	0	0
$241$	4.48963	0.289203	0.144601	−	0.989490i	$-0.453810\pi$
$241$	4.48963	0.289203	0.144601	+	0.989490i	$0.453810\pi$
$242$	0	0
$243$	−10.6297	−0.681893
$244$	0	0
$245$	14.8353	0.947791
$246$	0	0
$247$	2.24482	0.142834
$248$	0	0
$249$	0.0784065	0.00496881
$250$	0	0
$251$	20.0369	1.26472	0.632360	−	0.774674i	$-0.282086\pi$
$251$	20.0369	1.26472	0.632360	+	0.774674i	$0.282086\pi$
$252$	0	0
$253$	1.79834	0.113061
$254$	0	0
$255$	−0.856013	−0.0536056
$256$	0	0
$257$	−4.15850	−0.259400	−0.129700	−	0.991553i	$-0.541401\pi$
$257$	−4.15850	−0.259400	−0.129700	+	0.991553i	$0.541401\pi$
$258$	0	0
$259$	19.8353	1.23250
$260$	0	0
$261$	3.22239	0.199461
$262$	0	0
$263$	−17.0761	−1.05296	−0.526480	−	0.850187i	$-0.676489\pi$
$263$	−17.0761	−1.05296	−0.526480	+	0.850187i	$0.676489\pi$
$264$	0	0
$265$	0.244817	0.0150390
$266$	0	0
$267$	−3.91369	−0.239514
$268$	0	0
$269$	−29.3826	−1.79149	−0.895744	−	0.444570i	$-0.853356\pi$
$269$	−29.3826	−1.79149	−0.895744	+	0.444570i	$0.853356\pi$
$270$	0	0
$271$	16.5944	1.00804	0.504020	−	0.863692i	$-0.331854\pi$
$271$	16.5944	1.00804	0.504020	+	0.863692i	$0.331854\pi$
$272$	0	0
$273$	4.48963	0.271725
$274$	0	0
$275$	2.67282	0.161177
$276$	0	0
$277$	−6.36638	−0.382519	−0.191259	−	0.981540i	$-0.561257\pi$
$277$	−6.36638	−0.382519	−0.191259	+	0.981540i	$0.561257\pi$
$278$	0	0
$279$	3.22239	0.192920
$280$	0	0
$281$	−0.280067	−0.0167074	−0.00835371	−	0.999965i	$-0.502659\pi$
$281$	−0.280067	−0.0167074	−0.00835371	+	0.999965i	$0.502659\pi$
$282$	0	0
$283$	12.1832	0.724215	0.362108	−	0.932136i	$-0.382057\pi$
$283$	12.1832	0.724215	0.362108	+	0.932136i	$0.382057\pi$
$284$	0	0
$285$	−0.428007	−0.0253529
$286$	0	0
$287$	−22.6913	−1.33942
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−3.89522	−0.228342
$292$	0	0
$293$	−25.5042	−1.48997	−0.744984	−	0.667082i	$-0.767543\pi$
$293$	−25.5042	−1.48997	−0.744984	+	0.667082i	$0.767543\pi$
$294$	0	0
$295$	0.287973	0.0167664
$296$	0	0
$297$	6.65435	0.386125
$298$	0	0
$299$	1.51037	0.0873467
$300$	0	0
$301$	25.8353	1.48912
$302$	0	0
$303$	1.14399	0.0657203
$304$	0	0
$305$	8.87448	0.508152
$306$	0	0
$307$	−16.3496	−0.933121	−0.466560	−	0.884489i	$-0.654507\pi$
$307$	−16.3496	−0.933121	−0.466560	+	0.884489i	$0.654507\pi$
$308$	0	0
$309$	−2.96080	−0.168434
$310$	0	0
$311$	10.3064	0.584425	0.292212	−	0.956353i	$-0.405609\pi$
$311$	10.3064	0.584425	0.292212	+	0.956353i	$0.405609\pi$
$312$	0	0
$313$	−16.9714	−0.959278	−0.479639	−	0.877466i	$-0.659232\pi$
$313$	−16.9714	−0.959278	−0.479639	+	0.877466i	$0.659232\pi$
$314$	0	0
$315$	13.1625	0.741620
$316$	0	0
$317$	−31.3473	−1.76064	−0.880321	−	0.474378i	$-0.842673\pi$
$317$	−31.3473	−1.76064	−0.880321	+	0.474378i	$0.842673\pi$
$318$	0	0
$319$	−3.05767	−0.171197
$320$	0	0
$321$	−3.69356	−0.206154
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	2.24482	0.124520
$326$	0	0
$327$	4.97927	0.275354
$328$	0	0
$329$	21.8353	1.20382
$330$	0	0
$331$	−26.4112	−1.45169	−0.725846	−	0.687857i	$-0.758552\pi$
$331$	−26.4112	−1.45169	−0.725846	+	0.687857i	$0.758552\pi$
$332$	0	0
$333$	11.9568	0.655231
$334$	0	0
$335$	−9.20561	−0.502956
$336$	0	0
$337$	−23.7137	−1.29177	−0.645884	−	0.763435i	$-0.723511\pi$
$337$	−23.7137	−1.29177	−0.645884	+	0.763435i	$0.723511\pi$
$338$	0	0
$339$	−2.82964	−0.153685
$340$	0	0
$341$	−3.05767	−0.165582
$342$	0	0
$343$	−36.6129	−1.97691
$344$	0	0
$345$	−0.287973	−0.0155039
$346$	0	0
$347$	−3.69356	−0.198280	−0.0991402	−	0.995073i	$-0.531609\pi$
$347$	−3.69356	−0.198280	−0.0991402	+	0.995073i	$0.531609\pi$
$348$	0	0
$349$	24.9714	1.33669	0.668343	−	0.743853i	$-0.267004\pi$
$349$	24.9714	1.33669	0.668343	+	0.743853i	$0.267004\pi$
$350$	0	0
$351$	5.58877	0.298307
$352$	0	0
$353$	12.8639	0.684677	0.342339	−	0.939577i	$-0.388781\pi$
$353$	12.8639	0.684677	0.342339	+	0.939577i	$0.388781\pi$
$354$	0	0
$355$	−9.14399	−0.485312
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	32.0554	1.69182	0.845910	−	0.533326i	$-0.179058\pi$
$359$	32.0554	1.69182	0.845910	+	0.533326i	$0.179058\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.65040	0.0866235
$364$	0	0
$365$	1.14399	0.0598790
$366$	0	0
$367$	20.4297	1.06642	0.533211	−	0.845982i	$-0.320985\pi$
$367$	20.4297	1.06642	0.533211	+	0.845982i	$0.320985\pi$
$368$	0	0
$369$	−13.6785	−0.712073
$370$	0	0
$371$	−1.14399	−0.0593928
$372$	0	0
$373$	−5.50415	−0.284994	−0.142497	−	0.989795i	$-0.545513\pi$
$373$	−5.50415	−0.284994	−0.142497	+	0.989795i	$0.545513\pi$
$374$	0	0
$375$	−0.428007	−0.0221022
$376$	0	0
$377$	−2.56804	−0.132261
$378$	0	0
$379$	−26.6913	−1.37104	−0.685520	−	0.728054i	$-0.740425\pi$
$379$	−26.6913	−1.37104	−0.685520	+	0.728054i	$0.740425\pi$
$380$	0	0
$381$	3.81681	0.195541
$382$	0	0
$383$	7.77365	0.397215	0.198608	−	0.980079i	$-0.436358\pi$
$383$	7.77365	0.397215	0.198608	+	0.980079i	$0.436358\pi$
$384$	0	0
$385$	−12.4896	−0.636531
$386$	0	0
$387$	15.5737	0.791655
$388$	0	0
$389$	24.8145	1.25815	0.629074	−	0.777346i	$-0.283434\pi$
$389$	24.8145	1.25815	0.629074	+	0.777346i	$0.283434\pi$
$390$	0	0
$391$	1.34565	0.0680523
$392$	0	0
$393$	2.90086	0.146329
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	32.4482	1.62853	0.814263	−	0.580495i	$-0.197141\pi$
$397$	32.4482	1.62853	0.814263	+	0.580495i	$0.197141\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	0.481728	0.0240564	0.0120282	−	0.999928i	$-0.496171\pi$
$401$	0.481728	0.0240564	0.0120282	+	0.999928i	$0.496171\pi$
$402$	0	0
$403$	−2.56804	−0.127923
$404$	0	0
$405$	7.38485	0.366956
$406$	0	0
$407$	−11.3456	−0.562383
$408$	0	0
$409$	16.9793	0.839571	0.419785	−	0.907623i	$-0.362105\pi$
$409$	16.9793	0.839571	0.419785	+	0.907623i	$0.362105\pi$
$410$	0	0
$411$	6.00791	0.296348
$412$	0	0
$413$	−1.34565	−0.0662150
$414$	0	0
$415$	−0.183190	−0.00899243
$416$	0	0
$417$	2.12325	0.103976
$418$	0	0
$419$	9.43196	0.460781	0.230391	−	0.973098i	$-0.426000\pi$
$419$	9.43196	0.460781	0.230391	+	0.973098i	$0.426000\pi$
$420$	0	0
$421$	−10.6050	−0.516855	−0.258428	−	0.966031i	$-0.583204\pi$
$421$	−10.6050	−0.516855	−0.258428	+	0.966031i	$0.583204\pi$
$422$	0	0
$423$	13.1625	0.639981
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−41.4689	−2.00682
$428$	0	0
$429$	−2.56804	−0.123986
$430$	0	0
$431$	13.2593	0.638680	0.319340	−	0.947640i	$-0.396539\pi$
$431$	13.2593	0.638680	0.319340	+	0.947640i	$0.396539\pi$
$432$	0	0
$433$	20.5697	0.988518	0.494259	−	0.869315i	$-0.335439\pi$
$433$	20.5697	0.988518	0.494259	+	0.869315i	$0.335439\pi$
$434$	0	0
$435$	0.489634	0.0234762
$436$	0	0
$437$	0.672824	0.0321855
$438$	0	0
$439$	−12.0784	−0.576471	−0.288235	−	0.957560i	$-0.593069\pi$
$439$	−12.0784	−0.576471	−0.288235	+	0.957560i	$0.593069\pi$
$440$	0	0
$441$	−41.7882	−1.98991
$442$	0	0
$443$	2.01847	0.0959005	0.0479502	−	0.998850i	$-0.484731\pi$
$443$	2.01847	0.0959005	0.0479502	+	0.998850i	$0.484731\pi$
$444$	0	0
$445$	9.14399	0.433467
$446$	0	0
$447$	−0.374285	−0.0177031
$448$	0	0
$449$	−12.0369	−0.568058	−0.284029	−	0.958816i	$-0.591671\pi$
$449$	−12.0369	−0.568058	−0.284029	+	0.958816i	$0.591671\pi$
$450$	0	0
$451$	12.9793	0.611170
$452$	0	0
$453$	9.18883	0.431729
$454$	0	0
$455$	−10.4896	−0.491762
$456$	0	0
$457$	38.6498	1.80796	0.903981	−	0.427572i	$-0.140631\pi$
$457$	38.6498	1.80796	0.903981	+	0.427572i	$0.140631\pi$
$458$	0	0
$459$	4.97927	0.232412
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−12.8824	−0.598695	−0.299348	−	0.954144i	$-0.596769\pi$
$463$	−12.8824	−0.598695	−0.299348	+	0.954144i	$0.596769\pi$
$464$	0	0
$465$	0.489634	0.0227062
$466$	0	0
$467$	−13.2488	−0.613080	−0.306540	−	0.951858i	$-0.599171\pi$
$467$	−13.2488	−0.613080	−0.306540	+	0.951858i	$0.599171\pi$
$468$	0	0
$469$	43.0162	1.98630
$470$	0	0
$471$	5.10252	0.235112
$472$	0	0
$473$	−14.7776	−0.679475
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−0.689603	−0.0315747
$478$	0	0
$479$	24.9608	1.14049	0.570244	−	0.821475i	$-0.306849\pi$
$479$	24.9608	1.14049	0.570244	+	0.821475i	$0.306849\pi$
$480$	0	0
$481$	−9.52884	−0.434478
$482$	0	0
$483$	1.34565	0.0612291
$484$	0	0
$485$	9.10083	0.413247
$486$	0	0
$487$	−3.73671	−0.169327	−0.0846633	−	0.996410i	$-0.526981\pi$
$487$	−3.73671	−0.169327	−0.0846633	+	0.996410i	$0.526981\pi$
$488$	0	0
$489$	1.05767	0.0478297
$490$	0	0
$491$	−41.9955	−1.89523	−0.947615	−	0.319416i	$-0.896513\pi$
$491$	−41.9955	−1.89523	−0.947615	+	0.319416i	$0.896513\pi$
$492$	0	0
$493$	−2.28797	−0.103045
$494$	0	0
$495$	−7.52884	−0.338396
$496$	0	0
$497$	42.7282	1.91662
$498$	0	0
$499$	−12.0185	−0.538021	−0.269010	−	0.963137i	$-0.586697\pi$
$499$	−12.0185	−0.538021	−0.269010	+	0.963137i	$0.586697\pi$
$500$	0	0
$501$	8.34000	0.372604
$502$	0	0
$503$	−14.9977	−0.668716	−0.334358	−	0.942446i	$-0.608519\pi$
$503$	−14.9977	−0.668716	−0.334358	+	0.942446i	$0.608519\pi$
$504$	0	0
$505$	−2.67282	−0.118939
$506$	0	0
$507$	3.40727	0.151322
$508$	0	0
$509$	−7.79834	−0.345655	−0.172828	−	0.984952i	$-0.555290\pi$
$509$	−7.79834	−0.345655	−0.172828	+	0.984952i	$0.555290\pi$
$510$	0	0
$511$	−5.34565	−0.236478
$512$	0	0
$513$	2.48963	0.109920
$514$	0	0
$515$	6.91764	0.304828
$516$	0	0
$517$	−12.4896	−0.549293
$518$	0	0
$519$	9.11761	0.400219
$520$	0	0
$521$	−19.3456	−0.847548	−0.423774	−	0.905768i	$-0.639295\pi$
$521$	−19.3456	−0.847548	−0.423774	+	0.905768i	$0.639295\pi$
$522$	0	0
$523$	8.42801	0.368531	0.184266	−	0.982877i	$-0.441009\pi$
$523$	8.42801	0.368531	0.184266	+	0.982877i	$0.441009\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	−2.28797	−0.0996657
$528$	0	0
$529$	−22.5473	−0.980318
$530$	0	0
$531$	−0.811166	−0.0352016
$532$	0	0
$533$	10.9009	0.472169
$534$	0	0
$535$	8.62967	0.373093
$536$	0	0
$537$	2.56804	0.110819
$538$	0	0
$539$	39.6521	1.70794
$540$	0	0
$541$	40.6314	1.74688	0.873439	−	0.486933i	$-0.161884\pi$
$541$	40.6314	1.74688	0.873439	+	0.486933i	$0.161884\pi$
$542$	0	0
$543$	−9.10252	−0.390627
$544$	0	0
$545$	−11.6336	−0.498330
$546$	0	0
$547$	−35.5226	−1.51884	−0.759419	−	0.650602i	$-0.774517\pi$
$547$	−35.5226	−1.51884	−0.759419	+	0.650602i	$0.774517\pi$
$548$	0	0
$549$	−24.9977	−1.06688
$550$	0	0
$551$	−1.14399	−0.0487355
$552$	0	0
$553$	−9.34565	−0.397417
$554$	0	0
$555$	1.81681	0.0771193
$556$	0	0
$557$	−3.67508	−0.155718	−0.0778592	−	0.996964i	$-0.524808\pi$
$557$	−3.67508	−0.155718	−0.0778592	+	0.996964i	$0.524808\pi$
$558$	0	0
$559$	−12.4112	−0.524939
$560$	0	0
$561$	−2.28797	−0.0965983
$562$	0	0
$563$	10.3865	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$563$	10.3865	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$564$	0	0
$565$	6.61120	0.278135
$566$	0	0
$567$	−34.5081	−1.44920
$568$	0	0
$569$	32.4033	1.35842	0.679209	−	0.733945i	$-0.262323\pi$
$569$	32.4033	1.35842	0.679209	+	0.733945i	$0.262323\pi$
$570$	0	0
$571$	−6.70977	−0.280795	−0.140397	−	0.990095i	$-0.544838\pi$
$571$	−6.70977	−0.280795	−0.140397	+	0.990095i	$0.544838\pi$
$572$	0	0
$573$	10.2880	0.429786
$574$	0	0
$575$	0.672824	0.0280587
$576$	0	0
$577$	13.7199	0.571168	0.285584	−	0.958354i	$-0.407812\pi$
$577$	13.7199	0.571168	0.285584	+	0.958354i	$0.407812\pi$
$578$	0	0
$579$	−9.45043	−0.392746
$580$	0	0
$581$	0.856013	0.0355134
$582$	0	0
$583$	0.654353	0.0271005
$584$	0	0
$585$	−6.32322	−0.261433
$586$	0	0
$587$	−6.10478	−0.251971	−0.125986	−	0.992032i	$-0.540209\pi$
$587$	−6.10478	−0.251971	−0.125986	+	0.992032i	$0.540209\pi$
$588$	0	0
$589$	−1.14399	−0.0471371
$590$	0	0
$591$	10.9344	0.449782
$592$	0	0
$593$	−26.4033	−1.08425	−0.542127	−	0.840296i	$-0.682381\pi$
$593$	−26.4033	−1.08425	−0.542127	+	0.840296i	$0.682381\pi$
$594$	0	0
$595$	−9.34565	−0.383134
$596$	0	0
$597$	7.26724	0.297428
$598$	0	0
$599$	7.34565	0.300135	0.150068	−	0.988676i	$-0.452051\pi$
$599$	7.34565	0.300135	0.150068	+	0.988676i	$0.452051\pi$
$600$	0	0
$601$	−18.7776	−0.765955	−0.382977	−	0.923758i	$-0.625101\pi$
$601$	−18.7776	−0.765955	−0.382977	+	0.923758i	$0.625101\pi$
$602$	0	0
$603$	25.9305	1.05597
$604$	0	0
$605$	−3.85601	−0.156769
$606$	0	0
$607$	5.20561	0.211289	0.105645	−	0.994404i	$-0.466309\pi$
$607$	5.20561	0.211289	0.105645	+	0.994404i	$0.466309\pi$
$608$	0	0
$609$	−2.28797	−0.0927133
$610$	0	0
$611$	−10.4896	−0.424365
$612$	0	0
$613$	−28.8145	−1.16381	−0.581904	−	0.813257i	$-0.697692\pi$
$613$	−28.8145	−1.16381	−0.581904	+	0.813257i	$0.697692\pi$
$614$	0	0
$615$	−2.07841	−0.0838094
$616$	0	0
$617$	38.5266	1.55102	0.775511	−	0.631334i	$-0.217492\pi$
$617$	38.5266	1.55102	0.775511	+	0.631334i	$0.217492\pi$
$618$	0	0
$619$	36.4218	1.46392	0.731958	−	0.681350i	$-0.238607\pi$
$619$	36.4218	1.46392	0.731958	+	0.681350i	$0.238607\pi$
$620$	0	0
$621$	1.67508	0.0672188
$622$	0	0
$623$	−42.7282	−1.71187
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.14399	−0.0456864
$628$	0	0
$629$	−8.48963	−0.338504
$630$	0	0
$631$	10.7098	0.426349	0.213175	−	0.977014i	$-0.431620\pi$
$631$	10.7098	0.426349	0.213175	+	0.977014i	$0.431620\pi$
$632$	0	0
$633$	−3.63362	−0.144423
$634$	0	0
$635$	−8.91764	−0.353886
$636$	0	0
$637$	33.3025	1.31949
$638$	0	0
$639$	25.7569	1.01893
$640$	0	0
$641$	−30.6050	−1.20882	−0.604412	−	0.796672i	$-0.706592\pi$
$641$	−30.6050	−1.20882	−0.604412	+	0.796672i	$0.706592\pi$
$642$	0	0
$643$	−13.4425	−0.530121	−0.265061	−	0.964232i	$-0.585392\pi$
$643$	−13.4425	−0.530121	−0.265061	+	0.964232i	$0.585392\pi$
$644$	0	0
$645$	2.36638	0.0931761
$646$	0	0
$647$	−33.5288	−1.31815	−0.659077	−	0.752075i	$-0.729053\pi$
$647$	−33.5288	−1.31815	−0.659077	+	0.752075i	$0.729053\pi$
$648$	0	0
$649$	0.769701	0.0302134
$650$	0	0
$651$	−2.28797	−0.0896727
$652$	0	0
$653$	23.2593	0.910208	0.455104	−	0.890438i	$-0.349602\pi$
$653$	23.2593	0.910208	0.455104	+	0.890438i	$0.349602\pi$
$654$	0	0
$655$	−6.77761	−0.264823
$656$	0	0
$657$	−3.22239	−0.125718
$658$	0	0
$659$	−30.9793	−1.20678	−0.603390	−	0.797446i	$-0.706184\pi$
$659$	−30.9793	−1.20678	−0.603390	+	0.797446i	$0.706184\pi$
$660$	0	0
$661$	10.3743	0.403513	0.201756	−	0.979436i	$-0.435335\pi$
$661$	10.3743	0.403513	0.201756	+	0.979436i	$0.435335\pi$
$662$	0	0
$663$	−1.92159	−0.0746285
$664$	0	0
$665$	−4.67282	−0.181204
$666$	0	0
$667$	−0.769701	−0.0298030
$668$	0	0
$669$	−3.66000	−0.141504
$670$	0	0
$671$	23.7199	0.915698
$672$	0	0
$673$	−26.0880	−1.00562	−0.502809	−	0.864397i	$-0.667700\pi$
$673$	−26.0880	−1.00562	−0.502809	+	0.864397i	$0.667700\pi$
$674$	0	0
$675$	2.48963	0.0958261
$676$	0	0
$677$	−12.5249	−0.481370	−0.240685	−	0.970603i	$-0.577372\pi$
$677$	−12.5249	−0.481370	−0.240685	+	0.970603i	$0.577372\pi$
$678$	0	0
$679$	−42.5266	−1.63202
$680$	0	0
$681$	8.55295	0.327750
$682$	0	0
$683$	−25.9339	−0.992331	−0.496166	−	0.868228i	$-0.665259\pi$
$683$	−25.9339	−0.992331	−0.496166	+	0.868228i	$0.665259\pi$
$684$	0	0
$685$	−14.0369	−0.536324
$686$	0	0
$687$	0.321532	0.0122672
$688$	0	0
$689$	0.549569	0.0209369
$690$	0	0
$691$	5.52093	0.210026	0.105013	−	0.994471i	$-0.466512\pi$
$691$	5.52093	0.210026	0.105013	+	0.994471i	$0.466512\pi$
$692$	0	0
$693$	35.1809	1.33641
$694$	0	0
$695$	−4.96080	−0.188174
$696$	0	0
$697$	9.71203	0.367869
$698$	0	0
$699$	6.53110	0.247029
$700$	0	0
$701$	22.9687	0.867516	0.433758	−	0.901029i	$-0.357187\pi$
$701$	22.9687	0.867516	0.433758	+	0.901029i	$0.357187\pi$
$702$	0	0
$703$	−4.24482	−0.160096
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	12.4896	0.469721
$708$	0	0
$709$	−16.2386	−0.609854	−0.304927	−	0.952376i	$-0.598632\pi$
$709$	−16.2386	−0.609854	−0.304927	+	0.952376i	$0.598632\pi$
$710$	0	0
$711$	−5.63362	−0.211277
$712$	0	0
$713$	−0.769701	−0.0288255
$714$	0	0
$715$	6.00000	0.224387
$716$	0	0
$717$	−7.07349	−0.264164
$718$	0	0
$719$	13.1704	0.491172	0.245586	−	0.969375i	$-0.421020\pi$
$719$	13.1704	0.491172	0.245586	+	0.969375i	$0.421020\pi$
$720$	0	0
$721$	−32.3249	−1.20384
$722$	0	0
$723$	−1.92159	−0.0714648
$724$	0	0
$725$	−1.14399	−0.0424866
$726$	0	0
$727$	−40.7961	−1.51304	−0.756521	−	0.653969i	$-0.773103\pi$
$727$	−40.7961	−1.51304	−0.756521	+	0.653969i	$0.773103\pi$
$728$	0	0
$729$	−17.6050	−0.652036
$730$	0	0
$731$	−11.0577	−0.408983
$732$	0	0
$733$	27.8722	1.02948	0.514742	−	0.857345i	$-0.327888\pi$
$733$	27.8722	1.02948	0.514742	+	0.857345i	$0.327888\pi$
$734$	0	0
$735$	−6.34960	−0.234209
$736$	0	0
$737$	−24.6050	−0.906336
$738$	0	0
$739$	36.0739	1.32700	0.663500	−	0.748177i	$-0.269070\pi$
$739$	36.0739	1.32700	0.663500	+	0.748177i	$0.269070\pi$
$740$	0	0
$741$	−0.960797	−0.0352958
$742$	0	0
$743$	19.8890	0.729657	0.364828	−	0.931075i	$-0.381128\pi$
$743$	19.8890	0.729657	0.364828	+	0.931075i	$0.381128\pi$
$744$	0	0
$745$	0.874485	0.0320386
$746$	0	0
$747$	0.516011	0.0188798
$748$	0	0
$749$	−40.3249	−1.47344
$750$	0	0
$751$	17.1025	0.624080	0.312040	−	0.950069i	$-0.398988\pi$
$751$	17.1025	0.624080	0.312040	+	0.950069i	$0.398988\pi$
$752$	0	0
$753$	−8.57595	−0.312525
$754$	0	0
$755$	−21.4689	−0.781333
$756$	0	0
$757$	−10.6992	−0.388869	−0.194435	−	0.980915i	$-0.562287\pi$
$757$	−10.6992	−0.388869	−0.194435	+	0.980915i	$0.562287\pi$
$758$	0	0
$759$	−0.769701	−0.0279384
$760$	0	0
$761$	−48.4218	−1.75529	−0.877644	−	0.479312i	$-0.840886\pi$
$761$	−48.4218	−1.75529	−0.877644	+	0.479312i	$0.840886\pi$
$762$	0	0
$763$	54.3619	1.96803
$764$	0	0
$765$	−5.63362	−0.203684
$766$	0	0
$767$	0.646447	0.0233418
$768$	0	0
$769$	−49.4011	−1.78145	−0.890724	−	0.454545i	$-0.849802\pi$
$769$	−49.4011	−1.78145	−0.890724	+	0.454545i	$0.849802\pi$
$770$	0	0
$771$	1.77987	0.0641004
$772$	0	0
$773$	−32.8129	−1.18020	−0.590098	−	0.807331i	$-0.700911\pi$
$773$	−32.8129	−1.18020	−0.590098	+	0.807331i	$0.700911\pi$
$774$	0	0
$775$	−1.14399	−0.0410932
$776$	0	0
$777$	−8.48963	−0.304564
$778$	0	0
$779$	4.85601	0.173985
$780$	0	0
$781$	−24.4403	−0.874541
$782$	0	0
$783$	−2.84811	−0.101783
$784$	0	0
$785$	−11.9216	−0.425500
$786$	0	0
$787$	−30.1400	−1.07438	−0.537188	−	0.843462i	$-0.680513\pi$
$787$	−30.1400	−1.07438	−0.537188	+	0.843462i	$0.680513\pi$
$788$	0	0
$789$	7.30871	0.260197
$790$	0	0
$791$	−30.8930	−1.09843
$792$	0	0
$793$	19.9216	0.707437
$794$	0	0
$795$	−0.104783	−0.00371628
$796$	0	0
$797$	−1.31040	−0.0464166	−0.0232083	−	0.999731i	$-0.507388\pi$
$797$	−1.31040	−0.0464166	−0.0232083	+	0.999731i	$0.507388\pi$
$798$	0	0
$799$	−9.34565	−0.330625
$800$	0	0
$801$	−25.7569	−0.910074
$802$	0	0
$803$	3.05767	0.107903
$804$	0	0
$805$	−3.14399	−0.110811
$806$	0	0
$807$	12.5759	0.442694
$808$	0	0
$809$	35.5058	1.24832	0.624159	−	0.781297i	$-0.285442\pi$
$809$	35.5058	1.24832	0.624159	+	0.781297i	$0.285442\pi$
$810$	0	0
$811$	26.9299	0.945637	0.472818	−	0.881160i	$-0.343237\pi$
$811$	26.9299	0.945637	0.472818	+	0.881160i	$0.343237\pi$
$812$	0	0
$813$	−7.10252	−0.249096
$814$	0	0
$815$	−2.47116	−0.0865611
$816$	0	0
$817$	−5.52884	−0.193430
$818$	0	0
$819$	29.5473	1.03247
$820$	0	0
$821$	−13.1440	−0.458728	−0.229364	−	0.973341i	$-0.573665\pi$
$821$	−13.1440	−0.458728	−0.229364	+	0.973341i	$0.573665\pi$
$822$	0	0
$823$	40.3928	1.40800	0.704001	−	0.710198i	$-0.251395\pi$
$823$	40.3928	1.40800	0.704001	+	0.710198i	$0.251395\pi$
$824$	0	0
$825$	−1.14399	−0.0398285
$826$	0	0
$827$	27.3658	0.951602	0.475801	−	0.879553i	$-0.342158\pi$
$827$	27.3658	0.951602	0.475801	+	0.879553i	$0.342158\pi$
$828$	0	0
$829$	29.3456	1.01922	0.509608	−	0.860407i	$-0.329790\pi$
$829$	29.3456	1.01922	0.509608	+	0.860407i	$0.329790\pi$
$830$	0	0
$831$	2.72485	0.0945241
$832$	0	0
$833$	29.6706	1.02802
$834$	0	0
$835$	−19.4857	−0.674330
$836$	0	0
$837$	−2.84811	−0.0984450
$838$	0	0
$839$	49.6336	1.71354	0.856771	−	0.515696i	$-0.172467\pi$
$839$	49.6336	1.71354	0.856771	+	0.515696i	$0.172467\pi$
$840$	0	0
$841$	−27.6913	−0.954872
$842$	0	0
$843$	0.119871	0.00412857
$844$	0	0
$845$	−7.96080	−0.273860
$846$	0	0
$847$	18.0185	0.619122
$848$	0	0
$849$	−5.21449	−0.178961
$850$	0	0
$851$	−2.85601	−0.0979029
$852$	0	0
$853$	−31.1730	−1.06734	−0.533672	−	0.845692i	$-0.679188\pi$
$853$	−31.1730	−1.06734	−0.533672	+	0.845692i	$0.679188\pi$
$854$	0	0
$855$	−2.81681	−0.0963329
$856$	0	0
$857$	48.4755	1.65589	0.827946	−	0.560808i	$-0.189509\pi$
$857$	48.4755	1.65589	0.827946	+	0.560808i	$0.189509\pi$
$858$	0	0
$859$	25.8353	0.881488	0.440744	−	0.897633i	$-0.354715\pi$
$859$	25.8353	0.881488	0.440744	+	0.897633i	$0.354715\pi$
$860$	0	0
$861$	9.71203	0.330985
$862$	0	0
$863$	38.3865	1.30669	0.653347	−	0.757059i	$-0.273364\pi$
$863$	38.3865	1.30669	0.653347	+	0.757059i	$0.273364\pi$
$864$	0	0
$865$	−21.3025	−0.724306
$866$	0	0
$867$	5.56409	0.188966
$868$	0	0
$869$	5.34565	0.181339
$870$	0	0
$871$	−20.6649	−0.700204
$872$	0	0
$873$	−25.6353	−0.867624
$874$	0	0
$875$	−4.67282	−0.157970
$876$	0	0
$877$	17.2610	0.582863	0.291432	−	0.956592i	$-0.405868\pi$
$877$	17.2610	0.582863	0.291432	+	0.956592i	$0.405868\pi$
$878$	0	0
$879$	10.9159	0.368186
$880$	0	0
$881$	−13.8168	−0.465500	−0.232750	−	0.972537i	$-0.574772\pi$
$881$	−13.8168	−0.465500	−0.232750	+	0.972537i	$0.574772\pi$
$882$	0	0
$883$	−51.2073	−1.72326	−0.861632	−	0.507534i	$-0.830557\pi$
$883$	−51.2073	−1.72326	−0.861632	+	0.507534i	$0.830557\pi$
$884$	0	0
$885$	−0.123254	−0.00414315
$886$	0	0
$887$	39.6538	1.33144	0.665722	−	0.746200i	$-0.268124\pi$
$887$	39.6538	1.33144	0.665722	+	0.746200i	$0.268124\pi$
$888$	0	0
$889$	41.6706	1.39759
$890$	0	0
$891$	19.7384	0.661261
$892$	0	0
$893$	−4.67282	−0.156370
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	−0.646447	−0.0215842
$898$	0	0
$899$	1.30871	0.0436478
$900$	0	0
$901$	0.489634	0.0163121
$902$	0	0
$903$	−11.0577	−0.367976
$904$	0	0
$905$	21.2672	0.706947
$906$	0	0
$907$	4.62967	0.153726	0.0768628	−	0.997042i	$-0.475510\pi$
$907$	4.62967	0.153726	0.0768628	+	0.997042i	$0.475510\pi$
$908$	0	0
$909$	7.52884	0.249716
$910$	0	0
$911$	−15.9506	−0.528468	−0.264234	−	0.964459i	$-0.585119\pi$
$911$	−15.9506	−0.528468	−0.264234	+	0.964459i	$0.585119\pi$
$912$	0	0
$913$	−0.489634	−0.0162045
$914$	0	0
$915$	−3.79834	−0.125569
$916$	0	0
$917$	31.6706	1.04585
$918$	0	0
$919$	−32.1233	−1.05965	−0.529824	−	0.848107i	$-0.677742\pi$
$919$	−32.1233	−1.05965	−0.529824	+	0.848107i	$0.677742\pi$
$920$	0	0
$921$	6.99774	0.230583
$922$	0	0
$923$	−20.5266	−0.675640
$924$	0	0
$925$	−4.24482	−0.139569
$926$	0	0
$927$	−19.4857	−0.639994
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	14.8353	0.486207
$932$	0	0
$933$	−4.41123	−0.144417
$934$	0	0
$935$	5.34565	0.174821
$936$	0	0
$937$	−20.9872	−0.685621	−0.342811	−	0.939405i	$-0.611379\pi$
$937$	−20.9872	−0.685621	−0.342811	+	0.939405i	$0.611379\pi$
$938$	0	0
$939$	7.26386	0.237047
$940$	0	0
$941$	3.51827	0.114692	0.0573462	−	0.998354i	$-0.481736\pi$
$941$	3.51827	0.114692	0.0573462	+	0.998354i	$0.481736\pi$
$942$	0	0
$943$	3.26724	0.106396
$944$	0	0
$945$	−11.6336	−0.378442
$946$	0	0
$947$	0.112689	0.00366190	0.00183095	−	0.999998i	$-0.499417\pi$
$947$	0.112689	0.00366190	0.00183095	+	0.999998i	$0.499417\pi$
$948$	0	0
$949$	2.56804	0.0833621
$950$	0	0
$951$	13.4169	0.435072
$952$	0	0
$953$	−31.6353	−1.02477	−0.512384	−	0.858756i	$-0.671238\pi$
$953$	−31.6353	−1.02477	−0.512384	+	0.858756i	$0.671238\pi$
$954$	0	0
$955$	−24.0369	−0.777817
$956$	0	0
$957$	1.30871	0.0423044
$958$	0	0
$959$	65.5922	2.11808
$960$	0	0
$961$	−29.6913	−0.957784
$962$	0	0
$963$	−24.3081	−0.783319
$964$	0	0
$965$	22.0801	0.710784
$966$	0	0
$967$	31.6442	1.01761	0.508804	−	0.860882i	$-0.330088\pi$
$967$	31.6442	1.01761	0.508804	+	0.860882i	$0.330088\pi$
$968$	0	0
$969$	−0.856013	−0.0274991
$970$	0	0
$971$	55.4689	1.78008	0.890041	−	0.455881i	$-0.150676\pi$
$971$	55.4689	1.78008	0.890041	+	0.455881i	$0.150676\pi$
$972$	0	0
$973$	23.1809	0.743146
$974$	0	0
$975$	−0.960797	−0.0307701
$976$	0	0
$977$	−2.82076	−0.0902442	−0.0451221	−	0.998981i	$-0.514368\pi$
$977$	−2.82076	−0.0902442	−0.0451221	+	0.998981i	$0.514368\pi$
$978$	0	0
$979$	24.4403	0.781114
$980$	0	0
$981$	32.7697	1.04626
$982$	0	0
$983$	24.9467	0.795675	0.397838	−	0.917456i	$-0.369761\pi$
$983$	24.9467	0.795675	0.397838	+	0.917456i	$0.369761\pi$
$984$	0	0
$985$	−25.5473	−0.814005
$986$	0	0
$987$	−9.34565	−0.297475
$988$	0	0
$989$	−3.71993	−0.118287
$990$	0	0
$991$	31.0241	0.985514	0.492757	−	0.870167i	$-0.335989\pi$
$991$	31.0241	0.985514	0.492757	+	0.870167i	$0.335989\pi$
$992$	0	0
$993$	11.3042	0.358727
$994$	0	0
$995$	−16.9793	−0.538279
$996$	0	0
$997$	−55.7859	−1.76676	−0.883379	−	0.468660i	$-0.844737\pi$
$997$	−55.7859	−1.76676	−0.883379	+	0.468660i	$0.844737\pi$
$998$	0	0
$999$	−10.5680	−0.334358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bp.1.2		3
4.3	odd	2		6080.2.a.bz.1.2		3
8.3	odd	2		3040.2.a.k.1.2	✓	3
8.5	even	2		3040.2.a.n.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.k.1.2	✓	3	8.3	odd	2
3040.2.a.n.1.2	yes	3	8.5	even	2
6080.2.a.bp.1.2		3	1.1	even	1	trivial
6080.2.a.bz.1.2		3	4.3	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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q-0.428007 q^{3} +1.00000 q^{5} -4.67282 q^{7} -2.81681 q^{9} +2.67282 q^{11} +2.24482 q^{13} -0.428007 q^{15} +2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} +0.672824 q^{23} +1.00000 q^{25} +2.48963 q^{27} -1.14399 q^{29} -1.14399 q^{31} -1.14399 q^{33} -4.67282 q^{35} -4.24482 q^{37} -0.960797 q^{39} +4.85601 q^{41} -5.52884 q^{43} -2.81681 q^{45} -4.67282 q^{47} +14.8353 q^{49} -0.856013 q^{51} +0.244817 q^{53} +2.67282 q^{55} -0.428007 q^{57} +0.287973 q^{59} +8.87448 q^{61} +13.1625 q^{63} +2.24482 q^{65} -9.20561 q^{67} -0.287973 q^{69} -9.14399 q^{71} +1.14399 q^{73} -0.428007 q^{75} -12.4896 q^{77} +2.00000 q^{79} +7.38485 q^{81} -0.183190 q^{83} +2.00000 q^{85} +0.489634 q^{87} +9.14399 q^{89} -10.4896 q^{91} +0.489634 q^{93} +1.00000 q^{95} +9.10083 q^{97} -7.52884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} - 8 q^{23} + 3 q^{25} - 14 q^{27} - 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 2 q^{37} + 10 q^{39}+ \cdots - 14 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 2 * q^11 - 4 * q^13 - 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 - 8 * q^23 + 3 * q^25 - 14 * q^27 - 2 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 - 2 * q^37 + 10 * q^39 + 16 * q^41 - 8 * q^43 + 3 * q^45 - 4 * q^47 + 3 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^55 - 2 * q^57 - 2 * q^59 - 2 * q^61 + 8 * q^63 - 4 * q^65 - 4 * q^67 + 2 * q^69 - 26 * q^71 + 2 * q^73 - 2 * q^75 - 16 * q^77 + 6 * q^79 + 15 * q^81 - 12 * q^83 + 6 * q^85 - 20 * q^87 + 26 * q^89 - 10 * q^91 - 20 * q^93 + 3 * q^95 + 18 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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