LMFDB - Embedded newform 2960.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.935504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2 $ x^5 - x^4 - 8x^3 + 4x^2 + 8*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.3
Root			$-0.355205$ 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)

$f(q)$	$=$	$q-0.355205 q^{3} -1.00000 q^{5} -3.27534 q^{7} -2.87383 q^{9} +O(q^{10})$	$q-0.355205 q^{3} -1.00000 q^{5} -3.27534 q^{7} -2.87383 q^{9} -0.833725 q^{11} -6.22100 q^{13} +0.355205 q^{15} -1.04010 q^{17} +5.33257 q^{19} +1.16342 q^{21} +2.71041 q^{23} +1.00000 q^{25} +2.08641 q^{27} -8.55069 q^{29} -8.47656 q^{31} +0.296144 q^{33} +3.27534 q^{35} +1.00000 q^{37} +2.20973 q^{39} +1.00655 q^{41} +3.19697 q^{43} +2.87383 q^{45} +6.78879 q^{47} +3.72788 q^{49} +0.369450 q^{51} +8.23847 q^{53} +0.833725 q^{55} -1.89416 q^{57} -5.18697 q^{59} -9.54839 q^{61} +9.41278 q^{63} +6.22100 q^{65} +6.62216 q^{67} -0.962752 q^{69} +6.97737 q^{71} +9.68778 q^{73} -0.355205 q^{75} +2.73074 q^{77} -1.15203 q^{79} +7.88038 q^{81} +14.2465 q^{83} +1.04010 q^{85} +3.03725 q^{87} -14.2078 q^{89} +20.3759 q^{91} +3.01092 q^{93} -5.33257 q^{95} +3.94898 q^{97} +2.39598 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9	$ 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100}) $ 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 4 * q^13 - q^15 + 10 * q^19 - 5 * q^21 + 8 * q^23 + 5 * q^25 + 7 * q^27 - 8 * q^29 + 2 * q^31 - 11 * q^33 - q^35 + 5 * q^37 + 2 * q^39 - 13 * q^41 + 18 * q^43 - 2 * q^45 + 9 * q^47 - 6 * q^49 + 22 * q^51 - 13 * q^53 - 7 * q^55 + 4 * q^57 + 24 * q^59 - 2 * q^61 + 20 * q^63 - 4 * q^65 + 22 * q^67 - 32 * q^69 + 21 * q^71 + 29 * q^73 + q^75 + 11 * q^77 + 6 * q^79 + 5 * q^81 + 33 * q^83 - 12 * q^87 - 26 * q^89 + 38 * q^91 + 14 * q^93 - 10 * q^95 + 26 * q^97 + 42 * 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
$3$	−0.355205	−0.205078	−0.102539	−	0.994729i	$-0.532697\pi$
$3$	−0.355205	−0.205078	−0.102539	+	0.994729i	$0.532697\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.27534	−1.23796	−0.618982	−	0.785405i	$-0.712455\pi$
$7$	−3.27534	−1.23796	−0.618982	+	0.785405i	$0.712455\pi$
$8$	0	0
$9$	−2.87383	−0.957943
$10$	0	0
$11$	−0.833725	−0.251378	−0.125689	−	0.992070i	$-0.540114\pi$
$11$	−0.833725	−0.251378	−0.125689	+	0.992070i	$0.540114\pi$
$12$	0	0
$13$	−6.22100	−1.72539	−0.862697	−	0.505721i	$-0.831226\pi$
$13$	−6.22100	−1.72539	−0.862697	+	0.505721i	$0.831226\pi$
$14$	0	0
$15$	0.355205	0.0917136
$16$	0	0
$17$	−1.04010	−0.252262	−0.126131	−	0.992014i	$-0.540256\pi$
$17$	−1.04010	−0.252262	−0.126131	+	0.992014i	$0.540256\pi$
$18$	0	0
$19$	5.33257	1.22338	0.611688	−	0.791099i	$-0.290491\pi$
$19$	5.33257	1.22338	0.611688	+	0.791099i	$0.290491\pi$
$20$	0	0
$21$	1.16342	0.253879
$22$	0	0
$23$	2.71041	0.565160	0.282580	−	0.959244i	$-0.408810\pi$
$23$	2.71041	0.565160	0.282580	+	0.959244i	$0.408810\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.08641	0.401531
$28$	0	0
$29$	−8.55069	−1.58782	−0.793912	−	0.608033i	$-0.791959\pi$
$29$	−8.55069	−1.58782	−0.793912	+	0.608033i	$0.791959\pi$
$30$	0	0
$31$	−8.47656	−1.52244	−0.761218	−	0.648496i	$-0.775398\pi$
$31$	−8.47656	−1.52244	−0.761218	+	0.648496i	$0.775398\pi$
$32$	0	0
$33$	0.296144	0.0515520
$34$	0	0
$35$	3.27534	0.553634
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	2.20973	0.353840
$40$	0	0
$41$	1.00655	0.157197	0.0785986	−	0.996906i	$-0.474955\pi$
$41$	1.00655	0.157197	0.0785986	+	0.996906i	$0.474955\pi$
$42$	0	0
$43$	3.19697	0.487533	0.243767	−	0.969834i	$-0.421617\pi$
$43$	3.19697	0.487533	0.243767	+	0.969834i	$0.421617\pi$
$44$	0	0
$45$	2.87383	0.428405
$46$	0	0
$47$	6.78879	0.990246	0.495123	−	0.868823i	$-0.335123\pi$
$47$	6.78879	0.990246	0.495123	+	0.868823i	$0.335123\pi$
$48$	0	0
$49$	3.72788	0.532555
$50$	0	0
$51$	0.369450	0.0517334
$52$	0	0
$53$	8.23847	1.13164	0.565820	−	0.824529i	$-0.308560\pi$
$53$	8.23847	1.13164	0.565820	+	0.824529i	$0.308560\pi$
$54$	0	0
$55$	0.833725	0.112420
$56$	0	0
$57$	−1.89416	−0.250887
$58$	0	0
$59$	−5.18697	−0.675287	−0.337643	−	0.941274i	$-0.609630\pi$
$59$	−5.18697	−0.675287	−0.337643	+	0.941274i	$0.609630\pi$
$60$	0	0
$61$	−9.54839	−1.22255	−0.611273	−	0.791420i	$-0.709342\pi$
$61$	−9.54839	−1.22255	−0.611273	+	0.791420i	$0.709342\pi$
$62$	0	0
$63$	9.41278	1.18590
$64$	0	0
$65$	6.22100	0.771620
$66$	0	0
$67$	6.62216	0.809026	0.404513	−	0.914532i	$-0.367441\pi$
$67$	6.62216	0.809026	0.404513	+	0.914532i	$0.367441\pi$
$68$	0	0
$69$	−0.962752	−0.115902
$70$	0	0
$71$	6.97737	0.828061	0.414031	−	0.910263i	$-0.364121\pi$
$71$	6.97737	0.828061	0.414031	+	0.910263i	$0.364121\pi$
$72$	0	0
$73$	9.68778	1.13387	0.566934	−	0.823763i	$-0.308129\pi$
$73$	9.68778	1.13387	0.566934	+	0.823763i	$0.308129\pi$
$74$	0	0
$75$	−0.355205	−0.0410156
$76$	0	0
$77$	2.73074	0.311197
$78$	0	0
$79$	−1.15203	−0.129613	−0.0648067	−	0.997898i	$-0.520643\pi$
$79$	−1.15203	−0.129613	−0.0648067	+	0.997898i	$0.520643\pi$
$80$	0	0
$81$	7.88038	0.875598
$82$	0	0
$83$	14.2465	1.56376	0.781879	−	0.623431i	$-0.214262\pi$
$83$	14.2465	1.56376	0.781879	+	0.623431i	$0.214262\pi$
$84$	0	0
$85$	1.04010	0.112815
$86$	0	0
$87$	3.03725	0.325627
$88$	0	0
$89$	−14.2078	−1.50602	−0.753011	−	0.658008i	$-0.771399\pi$
$89$	−14.2078	−1.50602	−0.753011	+	0.658008i	$0.771399\pi$
$90$	0	0
$91$	20.3759	2.13598
$92$	0	0
$93$	3.01092	0.312218
$94$	0	0
$95$	−5.33257	−0.547111
$96$	0	0
$97$	3.94898	0.400958	0.200479	−	0.979698i	$-0.435750\pi$
$97$	3.94898	0.400958	0.200479	+	0.979698i	$0.435750\pi$
$98$	0	0
$99$	2.39598	0.240806
$100$	0	0
$101$	13.2494	1.31836	0.659182	−	0.751984i	$-0.270903\pi$
$101$	13.2494	1.31836	0.659182	+	0.751984i	$0.270903\pi$
$102$	0	0
$103$	6.57987	0.648334	0.324167	−	0.946000i	$-0.394916\pi$
$103$	6.57987	0.648334	0.324167	+	0.946000i	$0.394916\pi$
$104$	0	0
$105$	−1.16342	−0.113538
$106$	0	0
$107$	6.62481	0.640445	0.320222	−	0.947342i	$-0.396242\pi$
$107$	6.62481	0.640445	0.320222	+	0.947342i	$0.396242\pi$
$108$	0	0
$109$	0.367495	0.0351996	0.0175998	−	0.999845i	$-0.494398\pi$
$109$	0.367495	0.0351996	0.0175998	+	0.999845i	$0.494398\pi$
$110$	0	0
$111$	−0.355205	−0.0337146
$112$	0	0
$113$	−16.1757	−1.52169	−0.760843	−	0.648937i	$-0.775214\pi$
$113$	−16.1757	−1.52169	−0.760843	+	0.648937i	$0.775214\pi$
$114$	0	0
$115$	−2.71041	−0.252747
$116$	0	0
$117$	17.8781	1.65283
$118$	0	0
$119$	3.40670	0.312292
$120$	0	0
$121$	−10.3049	−0.936809
$122$	0	0
$123$	−0.357533	−0.0322377
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.77031	−0.866975	−0.433488	−	0.901160i	$-0.642717\pi$
$127$	−9.77031	−0.866975	−0.433488	+	0.901160i	$0.642717\pi$
$128$	0	0
$129$	−1.13558	−0.0999822
$130$	0	0
$131$	−2.26684	−0.198054	−0.0990272	−	0.995085i	$-0.531573\pi$
$131$	−2.26684	−0.198054	−0.0990272	+	0.995085i	$0.531573\pi$
$132$	0	0
$133$	−17.4660	−1.51450
$134$	0	0
$135$	−2.08641	−0.179570
$136$	0	0
$137$	0.515744	0.0440630	0.0220315	−	0.999757i	$-0.492987\pi$
$137$	0.515744	0.0440630	0.0220315	+	0.999757i	$0.492987\pi$
$138$	0	0
$139$	−9.70240	−0.822947	−0.411473	−	0.911422i	$-0.634986\pi$
$139$	−9.70240	−0.822947	−0.411473	+	0.911422i	$0.634986\pi$
$140$	0	0
$141$	−2.41141	−0.203077
$142$	0	0
$143$	5.18660	0.433726
$144$	0	0
$145$	8.55069	0.710096
$146$	0	0
$147$	−1.32416	−0.109215
$148$	0	0
$149$	−14.6826	−1.20285	−0.601424	−	0.798930i	$-0.705400\pi$
$149$	−14.6826	−1.20285	−0.601424	+	0.798930i	$0.705400\pi$
$150$	0	0
$151$	7.32454	0.596062	0.298031	−	0.954556i	$-0.403670\pi$
$151$	7.32454	0.596062	0.298031	+	0.954556i	$0.403670\pi$
$152$	0	0
$153$	2.98908	0.241653
$154$	0	0
$155$	8.47656	0.680854
$156$	0	0
$157$	0.644770	0.0514582	0.0257291	−	0.999669i	$-0.491809\pi$
$157$	0.644770	0.0514582	0.0257291	+	0.999669i	$0.491809\pi$
$158$	0	0
$159$	−2.92635	−0.232074
$160$	0	0
$161$	−8.87753	−0.699647
$162$	0	0
$163$	−16.7424	−1.31136	−0.655682	−	0.755037i	$-0.727619\pi$
$163$	−16.7424	−1.31136	−0.655682	+	0.755037i	$0.727619\pi$
$164$	0	0
$165$	−0.296144	−0.0230547
$166$	0	0
$167$	−16.9474	−1.31143	−0.655713	−	0.755010i	$-0.727632\pi$
$167$	−16.9474	−1.31143	−0.655713	+	0.755010i	$0.727632\pi$
$168$	0	0
$169$	25.7008	1.97698
$170$	0	0
$171$	−15.3249	−1.17192
$172$	0	0
$173$	10.6220	0.807579	0.403789	−	0.914852i	$-0.367693\pi$
$173$	10.6220	0.807579	0.403789	+	0.914852i	$0.367693\pi$
$174$	0	0
$175$	−3.27534	−0.247593
$176$	0	0
$177$	1.84244	0.138486
$178$	0	0
$179$	6.95471	0.519820	0.259910	−	0.965633i	$-0.416307\pi$
$179$	6.95471	0.519820	0.259910	+	0.965633i	$0.416307\pi$
$180$	0	0
$181$	8.97451	0.667070	0.333535	−	0.942738i	$-0.391758\pi$
$181$	8.97451	0.667070	0.333535	+	0.942738i	$0.391758\pi$
$182$	0	0
$183$	3.39164	0.250717
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	0.867161	0.0634131
$188$	0	0
$189$	−6.83373	−0.497080
$190$	0	0
$191$	17.6290	1.27559	0.637794	−	0.770207i	$-0.279847\pi$
$191$	17.6290	1.27559	0.637794	+	0.770207i	$0.279847\pi$
$192$	0	0
$193$	26.6374	1.91740	0.958699	−	0.284422i	$-0.0918016\pi$
$193$	26.6374	1.91740	0.958699	+	0.284422i	$0.0918016\pi$
$194$	0	0
$195$	−2.20973	−0.158242
$196$	0	0
$197$	7.48070	0.532978	0.266489	−	0.963838i	$-0.414136\pi$
$197$	7.48070	0.532978	0.266489	+	0.963838i	$0.414136\pi$
$198$	0	0
$199$	2.70237	0.191566	0.0957830	−	0.995402i	$-0.469465\pi$
$199$	2.70237	0.191566	0.0957830	+	0.995402i	$0.469465\pi$
$200$	0	0
$201$	−2.35223	−0.165913
$202$	0	0
$203$	28.0065	1.96567
$204$	0	0
$205$	−1.00655	−0.0703008
$206$	0	0
$207$	−7.78926	−0.541391
$208$	0	0
$209$	−4.44590	−0.307530
$210$	0	0
$211$	21.6988	1.49381	0.746905	−	0.664931i	$-0.231539\pi$
$211$	21.6988	1.49381	0.746905	+	0.664931i	$0.231539\pi$
$212$	0	0
$213$	−2.47840	−0.169817
$214$	0	0
$215$	−3.19697	−0.218031
$216$	0	0
$217$	27.7637	1.88472
$218$	0	0
$219$	−3.44115	−0.232531
$220$	0	0
$221$	6.47048	0.435252
$222$	0	0
$223$	−17.4056	−1.16556	−0.582781	−	0.812629i	$-0.698036\pi$
$223$	−17.4056	−1.16556	−0.582781	+	0.812629i	$0.698036\pi$
$224$	0	0
$225$	−2.87383	−0.191589
$226$	0	0
$227$	11.6353	0.772262	0.386131	−	0.922444i	$-0.373811\pi$
$227$	11.6353	0.772262	0.386131	+	0.922444i	$0.373811\pi$
$228$	0	0
$229$	7.40049	0.489038	0.244519	−	0.969644i	$-0.421370\pi$
$229$	7.40049	0.489038	0.244519	+	0.969644i	$0.421370\pi$
$230$	0	0
$231$	−0.969972	−0.0638195
$232$	0	0
$233$	7.43119	0.486833	0.243417	−	0.969922i	$-0.421732\pi$
$233$	7.43119	0.486833	0.243417	+	0.969922i	$0.421732\pi$
$234$	0	0
$235$	−6.78879	−0.442852
$236$	0	0
$237$	0.409207	0.0265808
$238$	0	0
$239$	−6.54196	−0.423164	−0.211582	−	0.977360i	$-0.567862\pi$
$239$	−6.54196	−0.423164	−0.211582	+	0.977360i	$0.567862\pi$
$240$	0	0
$241$	−1.97722	−0.127364	−0.0636820	−	0.997970i	$-0.520284\pi$
$241$	−1.97722	−0.127364	−0.0636820	+	0.997970i	$0.520284\pi$
$242$	0	0
$243$	−9.05839	−0.581096
$244$	0	0
$245$	−3.72788	−0.238166
$246$	0	0
$247$	−33.1739	−2.11081
$248$	0	0
$249$	−5.06043	−0.320692
$250$	0	0
$251$	27.7330	1.75049	0.875245	−	0.483680i	$-0.160700\pi$
$251$	27.7330	1.75049	0.875245	+	0.483680i	$0.160700\pi$
$252$	0	0
$253$	−2.25974	−0.142069
$254$	0	0
$255$	−0.369450	−0.0231359
$256$	0	0
$257$	−11.3952	−0.710812	−0.355406	−	0.934712i	$-0.615657\pi$
$257$	−11.3952	−0.710812	−0.355406	+	0.934712i	$0.615657\pi$
$258$	0	0
$259$	−3.27534	−0.203520
$260$	0	0
$261$	24.5732	1.52104
$262$	0	0
$263$	−11.0079	−0.678779	−0.339389	−	0.940646i	$-0.610220\pi$
$263$	−11.0079	−0.678779	−0.339389	+	0.940646i	$0.610220\pi$
$264$	0	0
$265$	−8.23847	−0.506085
$266$	0	0
$267$	5.04667	0.308851
$268$	0	0
$269$	−28.1123	−1.71404	−0.857019	−	0.515286i	$-0.827686\pi$
$269$	−28.1123	−1.71404	−0.857019	+	0.515286i	$0.827686\pi$
$270$	0	0
$271$	−21.3267	−1.29551	−0.647754	−	0.761850i	$-0.724291\pi$
$271$	−21.3267	−1.29551	−0.647754	+	0.761850i	$0.724291\pi$
$272$	0	0
$273$	−7.23763	−0.438041
$274$	0	0
$275$	−0.833725	−0.0502755
$276$	0	0
$277$	23.3100	1.40056	0.700280	−	0.713868i	$-0.253059\pi$
$277$	23.3100	1.40056	0.700280	+	0.713868i	$0.253059\pi$
$278$	0	0
$279$	24.3602	1.45841
$280$	0	0
$281$	−1.59091	−0.0949056	−0.0474528	−	0.998873i	$-0.515110\pi$
$281$	−1.59091	−0.0949056	−0.0474528	+	0.998873i	$0.515110\pi$
$282$	0	0
$283$	12.2756	0.729707	0.364853	−	0.931065i	$-0.381119\pi$
$283$	12.2756	0.729707	0.364853	+	0.931065i	$0.381119\pi$
$284$	0	0
$285$	1.89416	0.112200
$286$	0	0
$287$	−3.29681	−0.194605
$288$	0	0
$289$	−15.9182	−0.936364
$290$	0	0
$291$	−1.40270	−0.0822275
$292$	0	0
$293$	7.62288	0.445334	0.222667	−	0.974895i	$-0.428524\pi$
$293$	7.62288	0.445334	0.222667	+	0.974895i	$0.428524\pi$
$294$	0	0
$295$	5.18697	0.301997
$296$	0	0
$297$	−1.73950	−0.100936
$298$	0	0
$299$	−16.8615	−0.975123
$300$	0	0
$301$	−10.4712	−0.603549
$302$	0	0
$303$	−4.70625	−0.270367
$304$	0	0
$305$	9.54839	0.546739
$306$	0	0
$307$	−16.4402	−0.938289	−0.469145	−	0.883121i	$-0.655438\pi$
$307$	−16.4402	−0.938289	−0.469145	+	0.883121i	$0.655438\pi$
$308$	0	0
$309$	−2.33720	−0.132959
$310$	0	0
$311$	−8.13826	−0.461478	−0.230739	−	0.973016i	$-0.574114\pi$
$311$	−8.13826	−0.461478	−0.230739	+	0.973016i	$0.574114\pi$
$312$	0	0
$313$	−1.33764	−0.0756080	−0.0378040	−	0.999285i	$-0.512036\pi$
$313$	−1.33764	−0.0756080	−0.0378040	+	0.999285i	$0.512036\pi$
$314$	0	0
$315$	−9.41278	−0.530350
$316$	0	0
$317$	25.9534	1.45769	0.728844	−	0.684680i	$-0.240058\pi$
$317$	25.9534	1.45769	0.728844	+	0.684680i	$0.240058\pi$
$318$	0	0
$319$	7.12893	0.399143
$320$	0	0
$321$	−2.35317	−0.131341
$322$	0	0
$323$	−5.54643	−0.308612
$324$	0	0
$325$	−6.22100	−0.345079
$326$	0	0
$327$	−0.130536	−0.00721865
$328$	0	0
$329$	−22.2356	−1.22589
$330$	0	0
$331$	23.6148	1.29799	0.648995	−	0.760793i	$-0.275190\pi$
$331$	23.6148	1.29799	0.648995	+	0.760793i	$0.275190\pi$
$332$	0	0
$333$	−2.87383	−0.157485
$334$	0	0
$335$	−6.62216	−0.361807
$336$	0	0
$337$	−30.6425	−1.66921	−0.834603	−	0.550852i	$-0.814303\pi$
$337$	−30.6425	−1.66921	−0.834603	+	0.550852i	$0.814303\pi$
$338$	0	0
$339$	5.74570	0.312064
$340$	0	0
$341$	7.06713	0.382706
$342$	0	0
$343$	10.7173	0.578680
$344$	0	0
$345$	0.962752	0.0518328
$346$	0	0
$347$	−8.39027	−0.450414	−0.225207	−	0.974311i	$-0.572306\pi$
$347$	−8.39027	−0.450414	−0.225207	+	0.974311i	$0.572306\pi$
$348$	0	0
$349$	24.6127	1.31749	0.658743	−	0.752368i	$-0.271089\pi$
$349$	24.6127	1.31749	0.658743	+	0.752368i	$0.271089\pi$
$350$	0	0
$351$	−12.9796	−0.692798
$352$	0	0
$353$	5.62219	0.299239	0.149619	−	0.988744i	$-0.452195\pi$
$353$	5.62219	0.299239	0.149619	+	0.988744i	$0.452195\pi$
$354$	0	0
$355$	−6.97737	−0.370320
$356$	0	0
$357$	−1.21008	−0.0640441
$358$	0	0
$359$	27.1969	1.43540	0.717699	−	0.696354i	$-0.245196\pi$
$359$	27.1969	1.43540	0.717699	+	0.696354i	$0.245196\pi$
$360$	0	0
$361$	9.43635	0.496650
$362$	0	0
$363$	3.66035	0.192119
$364$	0	0
$365$	−9.68778	−0.507082
$366$	0	0
$367$	−15.5440	−0.811391	−0.405695	−	0.914008i	$-0.632971\pi$
$367$	−15.5440	−0.811391	−0.405695	+	0.914008i	$0.632971\pi$
$368$	0	0
$369$	−2.89266	−0.150586
$370$	0	0
$371$	−26.9838	−1.40093
$372$	0	0
$373$	−9.54419	−0.494179	−0.247090	−	0.968993i	$-0.579474\pi$
$373$	−9.54419	−0.494179	−0.247090	+	0.968993i	$0.579474\pi$
$374$	0	0
$375$	0.355205	0.0183427
$376$	0	0
$377$	53.1938	2.73962
$378$	0	0
$379$	−6.01462	−0.308950	−0.154475	−	0.987997i	$-0.549369\pi$
$379$	−6.01462	−0.308950	−0.154475	+	0.987997i	$0.549369\pi$
$380$	0	0
$381$	3.47047	0.177797
$382$	0	0
$383$	−19.7447	−1.00891	−0.504453	−	0.863439i	$-0.668306\pi$
$383$	−19.7447	−1.00891	−0.504453	+	0.863439i	$0.668306\pi$
$384$	0	0
$385$	−2.73074	−0.139171
$386$	0	0
$387$	−9.18754	−0.467029
$388$	0	0
$389$	−0.704650	−0.0357272	−0.0178636	−	0.999840i	$-0.505686\pi$
$389$	−0.704650	−0.0357272	−0.0178636	+	0.999840i	$0.505686\pi$
$390$	0	0
$391$	−2.81911	−0.142568
$392$	0	0
$393$	0.805191	0.0406165
$394$	0	0
$395$	1.15203	0.0579649
$396$	0	0
$397$	−5.73309	−0.287736	−0.143868	−	0.989597i	$-0.545954\pi$
$397$	−5.73309	−0.287736	−0.143868	+	0.989597i	$0.545954\pi$
$398$	0	0
$399$	6.20402	0.310589
$400$	0	0
$401$	−29.5900	−1.47766	−0.738828	−	0.673894i	$-0.764620\pi$
$401$	−29.5900	−1.47766	−0.738828	+	0.673894i	$0.764620\pi$
$402$	0	0
$403$	52.7327	2.62680
$404$	0	0
$405$	−7.88038	−0.391579
$406$	0	0
$407$	−0.833725	−0.0413262
$408$	0	0
$409$	12.9984	0.642729	0.321365	−	0.946956i	$-0.395859\pi$
$409$	12.9984	0.642729	0.321365	+	0.946956i	$0.395859\pi$
$410$	0	0
$411$	−0.183195	−0.00903634
$412$	0	0
$413$	16.9891	0.835980
$414$	0	0
$415$	−14.2465	−0.699334
$416$	0	0
$417$	3.44634	0.168768
$418$	0	0
$419$	−23.8358	−1.16445	−0.582227	−	0.813026i	$-0.697819\pi$
$419$	−23.8358	−1.16445	−0.582227	+	0.813026i	$0.697819\pi$
$420$	0	0
$421$	25.4981	1.24270	0.621351	−	0.783532i	$-0.286584\pi$
$421$	25.4981	1.24270	0.621351	+	0.783532i	$0.286584\pi$
$422$	0	0
$423$	−19.5098	−0.948599
$424$	0	0
$425$	−1.04010	−0.0504524
$426$	0	0
$427$	31.2743	1.51347
$428$	0	0
$429$	−1.84231	−0.0889475
$430$	0	0
$431$	−37.6977	−1.81583	−0.907917	−	0.419150i	$-0.862328\pi$
$431$	−37.6977	−1.81583	−0.907917	+	0.419150i	$0.862328\pi$
$432$	0	0
$433$	27.0376	1.29934	0.649672	−	0.760215i	$-0.274906\pi$
$433$	27.0376	1.29934	0.649672	+	0.760215i	$0.274906\pi$
$434$	0	0
$435$	−3.03725	−0.145625
$436$	0	0
$437$	14.4535	0.691403
$438$	0	0
$439$	5.88623	0.280935	0.140467	−	0.990085i	$-0.455140\pi$
$439$	5.88623	0.280935	0.140467	+	0.990085i	$0.455140\pi$
$440$	0	0
$441$	−10.7133	−0.510157
$442$	0	0
$443$	28.2228	1.34090	0.670452	−	0.741953i	$-0.266100\pi$
$443$	28.2228	1.34090	0.670452	+	0.741953i	$0.266100\pi$
$444$	0	0
$445$	14.2078	0.673513
$446$	0	0
$447$	5.21534	0.246677
$448$	0	0
$449$	−25.9839	−1.22626	−0.613129	−	0.789983i	$-0.710089\pi$
$449$	−25.9839	−1.22626	−0.613129	+	0.789983i	$0.710089\pi$
$450$	0	0
$451$	−0.839189	−0.0395159
$452$	0	0
$453$	−2.60171	−0.122239
$454$	0	0
$455$	−20.3759	−0.955237
$456$	0	0
$457$	27.5463	1.28856	0.644282	−	0.764788i	$-0.277156\pi$
$457$	27.5463	1.28856	0.644282	+	0.764788i	$0.277156\pi$
$458$	0	0
$459$	−2.17009	−0.101291
$460$	0	0
$461$	2.29765	0.107012	0.0535062	−	0.998568i	$-0.482960\pi$
$461$	2.29765	0.107012	0.0535062	+	0.998568i	$0.482960\pi$
$462$	0	0
$463$	27.8146	1.29265	0.646327	−	0.763061i	$-0.276304\pi$
$463$	27.8146	1.29265	0.646327	+	0.763061i	$0.276304\pi$
$464$	0	0
$465$	−3.01092	−0.139628
$466$	0	0
$467$	−29.5705	−1.36836	−0.684180	−	0.729313i	$-0.739840\pi$
$467$	−29.5705	−1.36836	−0.684180	+	0.729313i	$0.739840\pi$
$468$	0	0
$469$	−21.6899	−1.00155
$470$	0	0
$471$	−0.229026	−0.0105529
$472$	0	0
$473$	−2.66539	−0.122555
$474$	0	0
$475$	5.33257	0.244675
$476$	0	0
$477$	−23.6760	−1.08405
$478$	0	0
$479$	−31.5250	−1.44041	−0.720206	−	0.693760i	$-0.755953\pi$
$479$	−31.5250	−1.44041	−0.720206	+	0.693760i	$0.755953\pi$
$480$	0	0
$481$	−6.22100	−0.283653
$482$	0	0
$483$	3.15334	0.143482
$484$	0	0
$485$	−3.94898	−0.179314
$486$	0	0
$487$	−4.50840	−0.204295	−0.102147	−	0.994769i	$-0.532571\pi$
$487$	−4.50840	−0.204295	−0.102147	+	0.994769i	$0.532571\pi$
$488$	0	0
$489$	5.94698	0.268932
$490$	0	0
$491$	8.23993	0.371863	0.185931	−	0.982563i	$-0.440470\pi$
$491$	8.23993	0.371863	0.185931	+	0.982563i	$0.440470\pi$
$492$	0	0
$493$	8.89361	0.400548
$494$	0	0
$495$	−2.39598	−0.107692
$496$	0	0
$497$	−22.8533	−1.02511
$498$	0	0
$499$	−5.61516	−0.251369	−0.125685	−	0.992070i	$-0.540113\pi$
$499$	−5.61516	−0.251369	−0.125685	+	0.992070i	$0.540113\pi$
$500$	0	0
$501$	6.01979	0.268944
$502$	0	0
$503$	14.6840	0.654726	0.327363	−	0.944899i	$-0.393840\pi$
$503$	14.6840	0.654726	0.327363	+	0.944899i	$0.393840\pi$
$504$	0	0
$505$	−13.2494	−0.589590
$506$	0	0
$507$	−9.12905	−0.405435
$508$	0	0
$509$	−16.4046	−0.727121	−0.363560	−	0.931571i	$-0.618439\pi$
$509$	−16.4046	−0.727121	−0.363560	+	0.931571i	$0.618439\pi$
$510$	0	0
$511$	−31.7308	−1.40369
$512$	0	0
$513$	11.1260	0.491223
$514$	0	0
$515$	−6.57987	−0.289944
$516$	0	0
$517$	−5.65998	−0.248926
$518$	0	0
$519$	−3.77300	−0.165616
$520$	0	0
$521$	−4.50770	−0.197486	−0.0987429	−	0.995113i	$-0.531482\pi$
$521$	−4.50770	−0.197486	−0.0987429	+	0.995113i	$0.531482\pi$
$522$	0	0
$523$	−17.0541	−0.745722	−0.372861	−	0.927887i	$-0.621623\pi$
$523$	−17.0541	−0.745722	−0.372861	+	0.927887i	$0.621623\pi$
$524$	0	0
$525$	1.16342	0.0507758
$526$	0	0
$527$	8.81651	0.384053
$528$	0	0
$529$	−15.6537	−0.680595
$530$	0	0
$531$	14.9065	0.646886
$532$	0	0
$533$	−6.26177	−0.271227
$534$	0	0
$535$	−6.62481	−0.286416
$536$	0	0
$537$	−2.47035	−0.106603
$538$	0	0
$539$	−3.10803	−0.133872
$540$	0	0
$541$	28.5592	1.22786	0.613928	−	0.789362i	$-0.289589\pi$
$541$	28.5592	1.22786	0.613928	+	0.789362i	$0.289589\pi$
$542$	0	0
$543$	−3.18779	−0.136801
$544$	0	0
$545$	−0.367495	−0.0157417
$546$	0	0
$547$	−22.7233	−0.971579	−0.485790	−	0.874076i	$-0.661468\pi$
$547$	−22.7233	−0.971579	−0.485790	+	0.874076i	$0.661468\pi$
$548$	0	0
$549$	27.4404	1.17113
$550$	0	0
$551$	−45.5972	−1.94251
$552$	0	0
$553$	3.77329	0.160457
$554$	0	0
$555$	0.355205	0.0150776
$556$	0	0
$557$	−2.67832	−0.113484	−0.0567420	−	0.998389i	$-0.518071\pi$
$557$	−2.67832	−0.113484	−0.0567420	+	0.998389i	$0.518071\pi$
$558$	0	0
$559$	−19.8883	−0.841187
$560$	0	0
$561$	−0.308020	−0.0130046
$562$	0	0
$563$	36.9927	1.55906	0.779528	−	0.626368i	$-0.215459\pi$
$563$	36.9927	1.55906	0.779528	+	0.626368i	$0.215459\pi$
$564$	0	0
$565$	16.1757	0.680518
$566$	0	0
$567$	−25.8110	−1.08396
$568$	0	0
$569$	3.19306	0.133860	0.0669300	−	0.997758i	$-0.478680\pi$
$569$	3.19306	0.133860	0.0669300	+	0.997758i	$0.478680\pi$
$570$	0	0
$571$	45.4872	1.90358	0.951789	−	0.306753i	$-0.0992426\pi$
$571$	45.4872	1.90358	0.951789	+	0.306753i	$0.0992426\pi$
$572$	0	0
$573$	−6.26190	−0.261595
$574$	0	0
$575$	2.71041	0.113032
$576$	0	0
$577$	−13.9337	−0.580068	−0.290034	−	0.957016i	$-0.593667\pi$
$577$	−13.9337	−0.580068	−0.290034	+	0.957016i	$0.593667\pi$
$578$	0	0
$579$	−9.46172	−0.393216
$580$	0	0
$581$	−46.6622	−1.93588
$582$	0	0
$583$	−6.86862	−0.284469
$584$	0	0
$585$	−17.8781	−0.739168
$586$	0	0
$587$	21.1920	0.874686	0.437343	−	0.899295i	$-0.355920\pi$
$587$	21.1920	0.874686	0.437343	+	0.899295i	$0.355920\pi$
$588$	0	0
$589$	−45.2019	−1.86251
$590$	0	0
$591$	−2.65718	−0.109302
$592$	0	0
$593$	18.4557	0.757883	0.378942	−	0.925421i	$-0.376288\pi$
$593$	18.4557	0.757883	0.378942	+	0.925421i	$0.376288\pi$
$594$	0	0
$595$	−3.40670	−0.139661
$596$	0	0
$597$	−0.959896	−0.0392859
$598$	0	0
$599$	2.36598	0.0966715	0.0483358	−	0.998831i	$-0.484608\pi$
$599$	2.36598	0.0966715	0.0483358	+	0.998831i	$0.484608\pi$
$600$	0	0
$601$	15.3982	0.628107	0.314053	−	0.949405i	$-0.398313\pi$
$601$	15.3982	0.628107	0.314053	+	0.949405i	$0.398313\pi$
$602$	0	0
$603$	−19.0310	−0.775001
$604$	0	0
$605$	10.3049	0.418954
$606$	0	0
$607$	44.3290	1.79926	0.899629	−	0.436656i	$-0.143837\pi$
$607$	44.3290	1.79926	0.899629	+	0.436656i	$0.143837\pi$
$608$	0	0
$609$	−9.94804	−0.403115
$610$	0	0
$611$	−42.2330	−1.70856
$612$	0	0
$613$	−14.3603	−0.580008	−0.290004	−	0.957025i	$-0.593657\pi$
$613$	−14.3603	−0.580008	−0.290004	+	0.957025i	$0.593657\pi$
$614$	0	0
$615$	0.357533	0.0144171
$616$	0	0
$617$	−24.5325	−0.987640	−0.493820	−	0.869564i	$-0.664400\pi$
$617$	−24.5325	−0.987640	−0.493820	+	0.869564i	$0.664400\pi$
$618$	0	0
$619$	44.2574	1.77885	0.889427	−	0.457077i	$-0.151104\pi$
$619$	44.2574	1.77885	0.889427	+	0.457077i	$0.151104\pi$
$620$	0	0
$621$	5.65504	0.226929
$622$	0	0
$623$	46.5354	1.86440
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.57921	0.0630675
$628$	0	0
$629$	−1.04010	−0.0414717
$630$	0	0
$631$	−0.955654	−0.0380440	−0.0190220	−	0.999819i	$-0.506055\pi$
$631$	−0.955654	−0.0380440	−0.0190220	+	0.999819i	$0.506055\pi$
$632$	0	0
$633$	−7.70754	−0.306347
$634$	0	0
$635$	9.77031	0.387723
$636$	0	0
$637$	−23.1911	−0.918867
$638$	0	0
$639$	−20.0518	−0.793236
$640$	0	0
$641$	−19.9219	−0.786867	−0.393434	−	0.919353i	$-0.628713\pi$
$641$	−19.9219	−0.786867	−0.393434	+	0.919353i	$0.628713\pi$
$642$	0	0
$643$	31.3988	1.23825	0.619124	−	0.785294i	$-0.287488\pi$
$643$	31.3988	1.23825	0.619124	+	0.785294i	$0.287488\pi$
$644$	0	0
$645$	1.13558	0.0447134
$646$	0	0
$647$	15.3464	0.603328	0.301664	−	0.953414i	$-0.402458\pi$
$647$	15.3464	0.603328	0.301664	+	0.953414i	$0.402458\pi$
$648$	0	0
$649$	4.32451	0.169752
$650$	0	0
$651$	−9.86180	−0.386514
$652$	0	0
$653$	−36.7847	−1.43949	−0.719747	−	0.694236i	$-0.755742\pi$
$653$	−36.7847	−1.43949	−0.719747	+	0.694236i	$0.755742\pi$
$654$	0	0
$655$	2.26684	0.0885726
$656$	0	0
$657$	−27.8410	−1.08618
$658$	0	0
$659$	40.2392	1.56750	0.783748	−	0.621079i	$-0.213306\pi$
$659$	40.2392	1.56750	0.783748	+	0.621079i	$0.213306\pi$
$660$	0	0
$661$	−31.4907	−1.22485	−0.612423	−	0.790530i	$-0.709805\pi$
$661$	−31.4907	−1.22485	−0.612423	+	0.790530i	$0.709805\pi$
$662$	0	0
$663$	−2.29835	−0.0892604
$664$	0	0
$665$	17.4660	0.677303
$666$	0	0
$667$	−23.1759	−0.897373
$668$	0	0
$669$	6.18254	0.239031
$670$	0	0
$671$	7.96073	0.307321
$672$	0	0
$673$	8.42092	0.324603	0.162301	−	0.986741i	$-0.448108\pi$
$673$	8.42092	0.324603	0.162301	+	0.986741i	$0.448108\pi$
$674$	0	0
$675$	2.08641	0.0803061
$676$	0	0
$677$	−43.1215	−1.65729	−0.828647	−	0.559771i	$-0.810889\pi$
$677$	−43.1215	−1.65729	−0.828647	+	0.559771i	$0.810889\pi$
$678$	0	0
$679$	−12.9343	−0.496371
$680$	0	0
$681$	−4.13292	−0.158374
$682$	0	0
$683$	−21.2535	−0.813243	−0.406622	−	0.913597i	$-0.633293\pi$
$683$	−21.2535	−0.813243	−0.406622	+	0.913597i	$0.633293\pi$
$684$	0	0
$685$	−0.515744	−0.0197056
$686$	0	0
$687$	−2.62869	−0.100291
$688$	0	0
$689$	−51.2515	−1.95253
$690$	0	0
$691$	45.1869	1.71899	0.859496	−	0.511142i	$-0.170778\pi$
$691$	45.1869	1.71899	0.859496	+	0.511142i	$0.170778\pi$
$692$	0	0
$693$	−7.84768	−0.298109
$694$	0	0
$695$	9.70240	0.368033
$696$	0	0
$697$	−1.04692	−0.0396549
$698$	0	0
$699$	−2.63960	−0.0998387
$700$	0	0
$701$	−12.7577	−0.481852	−0.240926	−	0.970543i	$-0.577451\pi$
$701$	−12.7577	−0.481852	−0.240926	+	0.970543i	$0.577451\pi$
$702$	0	0
$703$	5.33257	0.201122
$704$	0	0
$705$	2.41141	0.0908190
$706$	0	0
$707$	−43.3963	−1.63209
$708$	0	0
$709$	2.13422	0.0801522	0.0400761	−	0.999197i	$-0.487240\pi$
$709$	2.13422	0.0801522	0.0400761	+	0.999197i	$0.487240\pi$
$710$	0	0
$711$	3.31074	0.124162
$712$	0	0
$713$	−22.9750	−0.860419
$714$	0	0
$715$	−5.18660	−0.193968
$716$	0	0
$717$	2.32374	0.0867815
$718$	0	0
$719$	11.2587	0.419879	0.209939	−	0.977714i	$-0.432673\pi$
$719$	11.2587	0.419879	0.209939	+	0.977714i	$0.432673\pi$
$720$	0	0
$721$	−21.5514	−0.802614
$722$	0	0
$723$	0.702319	0.0261195
$724$	0	0
$725$	−8.55069	−0.317565
$726$	0	0
$727$	−43.0230	−1.59564	−0.797818	−	0.602899i	$-0.794012\pi$
$727$	−43.0230	−1.59564	−0.797818	+	0.602899i	$0.794012\pi$
$728$	0	0
$729$	−20.4236	−0.756428
$730$	0	0
$731$	−3.32518	−0.122986
$732$	0	0
$733$	−7.92474	−0.292707	−0.146353	−	0.989232i	$-0.546754\pi$
$733$	−7.92474	−0.292707	−0.146353	+	0.989232i	$0.546754\pi$
$734$	0	0
$735$	1.32416	0.0488425
$736$	0	0
$737$	−5.52107	−0.203371
$738$	0	0
$739$	−30.4670	−1.12075	−0.560373	−	0.828240i	$-0.689342\pi$
$739$	−30.4670	−1.12075	−0.560373	+	0.828240i	$0.689342\pi$
$740$	0	0
$741$	11.7835	0.432879
$742$	0	0
$743$	−12.4976	−0.458494	−0.229247	−	0.973368i	$-0.573626\pi$
$743$	−12.4976	−0.458494	−0.229247	+	0.973368i	$0.573626\pi$
$744$	0	0
$745$	14.6826	0.537930
$746$	0	0
$747$	−40.9420	−1.49799
$748$	0	0
$749$	−21.6985	−0.792848
$750$	0	0
$751$	48.5497	1.77160	0.885802	−	0.464063i	$-0.153609\pi$
$751$	48.5497	1.77160	0.885802	+	0.464063i	$0.153609\pi$
$752$	0	0
$753$	−9.85089	−0.358986
$754$	0	0
$755$	−7.32454	−0.266567
$756$	0	0
$757$	13.8481	0.503317	0.251659	−	0.967816i	$-0.419024\pi$
$757$	13.8481	0.503317	0.251659	+	0.967816i	$0.419024\pi$
$758$	0	0
$759$	0.802670	0.0291351
$760$	0	0
$761$	42.7372	1.54922	0.774612	−	0.632437i	$-0.217945\pi$
$761$	42.7372	1.54922	0.774612	+	0.632437i	$0.217945\pi$
$762$	0	0
$763$	−1.20367	−0.0435758
$764$	0	0
$765$	−2.98908	−0.108070
$766$	0	0
$767$	32.2681	1.16514
$768$	0	0
$769$	−2.13928	−0.0771446	−0.0385723	−	0.999256i	$-0.512281\pi$
$769$	−2.13928	−0.0771446	−0.0385723	+	0.999256i	$0.512281\pi$
$770$	0	0
$771$	4.04763	0.145772
$772$	0	0
$773$	3.28813	0.118266	0.0591329	−	0.998250i	$-0.481166\pi$
$773$	3.28813	0.118266	0.0591329	+	0.998250i	$0.481166\pi$
$774$	0	0
$775$	−8.47656	−0.304487
$776$	0	0
$777$	1.16342	0.0417374
$778$	0	0
$779$	5.36752	0.192311
$780$	0	0
$781$	−5.81721	−0.208156
$782$	0	0
$783$	−17.8403	−0.637560
$784$	0	0
$785$	−0.644770	−0.0230128
$786$	0	0
$787$	16.6077	0.592000	0.296000	−	0.955188i	$-0.404347\pi$
$787$	16.6077	0.592000	0.296000	+	0.955188i	$0.404347\pi$
$788$	0	0
$789$	3.91008	0.139202
$790$	0	0
$791$	52.9811	1.88379
$792$	0	0
$793$	59.4005	2.10937
$794$	0	0
$795$	2.92635	0.103787
$796$	0	0
$797$	−2.94266	−0.104235	−0.0521173	−	0.998641i	$-0.516597\pi$
$797$	−2.94266	−0.104235	−0.0521173	+	0.998641i	$0.516597\pi$
$798$	0	0
$799$	−7.06104	−0.249802
$800$	0	0
$801$	40.8307	1.44268
$802$	0	0
$803$	−8.07695	−0.285029
$804$	0	0
$805$	8.87753	0.312892
$806$	0	0
$807$	9.98563	0.351511
$808$	0	0
$809$	−51.4232	−1.80794	−0.903972	−	0.427592i	$-0.859362\pi$
$809$	−51.4232	−1.80794	−0.903972	+	0.427592i	$0.859362\pi$
$810$	0	0
$811$	42.1996	1.48183	0.740914	−	0.671599i	$-0.234392\pi$
$811$	42.1996	1.48183	0.740914	+	0.671599i	$0.234392\pi$
$812$	0	0
$813$	7.57537	0.265680
$814$	0	0
$815$	16.7424	0.586460
$816$	0	0
$817$	17.0481	0.596437
$818$	0	0
$819$	−58.5569	−2.04614
$820$	0	0
$821$	10.5538	0.368330	0.184165	−	0.982895i	$-0.441042\pi$
$821$	10.5538	0.368330	0.184165	+	0.982895i	$0.441042\pi$
$822$	0	0
$823$	−44.9468	−1.56675	−0.783374	−	0.621550i	$-0.786503\pi$
$823$	−44.9468	−1.56675	−0.783374	+	0.621550i	$0.786503\pi$
$824$	0	0
$825$	0.296144	0.0103104
$826$	0	0
$827$	−3.85269	−0.133971	−0.0669856	−	0.997754i	$-0.521338\pi$
$827$	−3.85269	−0.133971	−0.0669856	+	0.997754i	$0.521338\pi$
$828$	0	0
$829$	32.6783	1.13496	0.567482	−	0.823386i	$-0.307918\pi$
$829$	32.6783	1.13496	0.567482	+	0.823386i	$0.307918\pi$
$830$	0	0
$831$	−8.27982	−0.287224
$832$	0	0
$833$	−3.87739	−0.134343
$834$	0	0
$835$	16.9474	0.586488
$836$	0	0
$837$	−17.6856	−0.611305
$838$	0	0
$839$	−33.3524	−1.15145	−0.575727	−	0.817642i	$-0.695281\pi$
$839$	−33.3524	−1.15145	−0.575727	+	0.817642i	$0.695281\pi$
$840$	0	0
$841$	44.1143	1.52118
$842$	0	0
$843$	0.565099	0.0194630
$844$	0	0
$845$	−25.7008	−0.884134
$846$	0	0
$847$	33.7521	1.15974
$848$	0	0
$849$	−4.36034	−0.149647
$850$	0	0
$851$	2.71041	0.0929117
$852$	0	0
$853$	33.2386	1.13807	0.569034	−	0.822314i	$-0.307317\pi$
$853$	33.2386	1.13807	0.569034	+	0.822314i	$0.307317\pi$
$854$	0	0
$855$	15.3249	0.524101
$856$	0	0
$857$	−23.8828	−0.815820	−0.407910	−	0.913022i	$-0.633742\pi$
$857$	−23.8828	−0.815820	−0.407910	+	0.913022i	$0.633742\pi$
$858$	0	0
$859$	2.05437	0.0700940	0.0350470	−	0.999386i	$-0.488842\pi$
$859$	2.05437	0.0700940	0.0350470	+	0.999386i	$0.488842\pi$
$860$	0	0
$861$	1.17104	0.0399091
$862$	0	0
$863$	−15.1823	−0.516811	−0.258405	−	0.966037i	$-0.583197\pi$
$863$	−15.1823	−0.516811	−0.258405	+	0.966037i	$0.583197\pi$
$864$	0	0
$865$	−10.6220	−0.361160
$866$	0	0
$867$	5.65422	0.192027
$868$	0	0
$869$	0.960476	0.0325819
$870$	0	0
$871$	−41.1965	−1.39589
$872$	0	0
$873$	−11.3487	−0.384095
$874$	0	0
$875$	3.27534	0.110727
$876$	0	0
$877$	44.6038	1.50616	0.753082	−	0.657926i	$-0.228566\pi$
$877$	44.6038	1.50616	0.753082	+	0.657926i	$0.228566\pi$
$878$	0	0
$879$	−2.70769	−0.0913280
$880$	0	0
$881$	−20.0969	−0.677081	−0.338540	−	0.940952i	$-0.609933\pi$
$881$	−20.0969	−0.677081	−0.338540	+	0.940952i	$0.609933\pi$
$882$	0	0
$883$	−0.115153	−0.00387521	−0.00193761	−	0.999998i	$-0.500617\pi$
$883$	−0.115153	−0.00387521	−0.00193761	+	0.999998i	$0.500617\pi$
$884$	0	0
$885$	−1.84244	−0.0619329
$886$	0	0
$887$	17.6123	0.591363	0.295682	−	0.955287i	$-0.404453\pi$
$887$	17.6123	0.591363	0.295682	+	0.955287i	$0.404453\pi$
$888$	0	0
$889$	32.0011	1.07328
$890$	0	0
$891$	−6.57008	−0.220106
$892$	0	0
$893$	36.2017	1.21144
$894$	0	0
$895$	−6.95471	−0.232470
$896$	0	0
$897$	5.98927	0.199976
$898$	0	0
$899$	72.4805	2.41736
$900$	0	0
$901$	−8.56886	−0.285470
$902$	0	0
$903$	3.71942	0.123774
$904$	0	0
$905$	−8.97451	−0.298323
$906$	0	0
$907$	−28.3335	−0.940800	−0.470400	−	0.882453i	$-0.655890\pi$
$907$	−28.3335	−0.940800	−0.470400	+	0.882453i	$0.655890\pi$
$908$	0	0
$909$	−38.0765	−1.26292
$910$	0	0
$911$	13.5531	0.449035	0.224517	−	0.974470i	$-0.427919\pi$
$911$	13.5531	0.449035	0.224517	+	0.974470i	$0.427919\pi$
$912$	0	0
$913$	−11.8777	−0.393094
$914$	0	0
$915$	−3.39164	−0.112124
$916$	0	0
$917$	7.42467	0.245184
$918$	0	0
$919$	14.0420	0.463202	0.231601	−	0.972811i	$-0.425604\pi$
$919$	14.0420	0.463202	0.231601	+	0.972811i	$0.425604\pi$
$920$	0	0
$921$	5.83963	0.192422
$922$	0	0
$923$	−43.4062	−1.42873
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−18.9094	−0.621067
$928$	0	0
$929$	29.4110	0.964942	0.482471	−	0.875912i	$-0.339739\pi$
$929$	29.4110	0.964942	0.482471	+	0.875912i	$0.339739\pi$
$930$	0	0
$931$	19.8792	0.651515
$932$	0	0
$933$	2.89075	0.0946389
$934$	0	0
$935$	−0.867161	−0.0283592
$936$	0	0
$937$	−10.3150	−0.336977	−0.168488	−	0.985704i	$-0.553889\pi$
$937$	−10.3150	−0.336977	−0.168488	+	0.985704i	$0.553889\pi$
$938$	0	0
$939$	0.475138	0.0155055
$940$	0	0
$941$	−59.2466	−1.93138	−0.965692	−	0.259689i	$-0.916380\pi$
$941$	−59.2466	−1.93138	−0.965692	+	0.259689i	$0.916380\pi$
$942$	0	0
$943$	2.72817	0.0888416
$944$	0	0
$945$	6.83373	0.222301
$946$	0	0
$947$	16.1400	0.524479	0.262239	−	0.965003i	$-0.415539\pi$
$947$	16.1400	0.524479	0.262239	+	0.965003i	$0.415539\pi$
$948$	0	0
$949$	−60.2676	−1.95637
$950$	0	0
$951$	−9.21878	−0.298939
$952$	0	0
$953$	43.7604	1.41754	0.708769	−	0.705440i	$-0.249251\pi$
$953$	43.7604	1.41754	0.708769	+	0.705440i	$0.249251\pi$
$954$	0	0
$955$	−17.6290	−0.570460
$956$	0	0
$957$	−2.53223	−0.0818554
$958$	0	0
$959$	−1.68924	−0.0545484
$960$	0	0
$961$	40.8521	1.31781
$962$	0	0
$963$	−19.0386	−0.613510
$964$	0	0
$965$	−26.6374	−0.857487
$966$	0	0
$967$	46.6498	1.50016	0.750078	−	0.661350i	$-0.230016\pi$
$967$	46.6498	1.50016	0.750078	+	0.661350i	$0.230016\pi$
$968$	0	0
$969$	1.97012	0.0632894
$970$	0	0
$971$	14.2880	0.458523	0.229262	−	0.973365i	$-0.426369\pi$
$971$	14.2880	0.458523	0.229262	+	0.973365i	$0.426369\pi$
$972$	0	0
$973$	31.7787	1.01878
$974$	0	0
$975$	2.20973	0.0707680
$976$	0	0
$977$	12.4259	0.397540	0.198770	−	0.980046i	$-0.436305\pi$
$977$	12.4259	0.397540	0.198770	+	0.980046i	$0.436305\pi$
$978$	0	0
$979$	11.8454	0.378580
$980$	0	0
$981$	−1.05612	−0.0337192
$982$	0	0
$983$	22.5455	0.719091	0.359546	−	0.933127i	$-0.382932\pi$
$983$	22.5455	0.719091	0.359546	+	0.933127i	$0.382932\pi$
$984$	0	0
$985$	−7.48070	−0.238355
$986$	0	0
$987$	7.89820	0.251403
$988$	0	0
$989$	8.66510	0.275534
$990$	0	0
$991$	47.3400	1.50380	0.751902	−	0.659275i	$-0.229137\pi$
$991$	47.3400	1.50380	0.751902	+	0.659275i	$0.229137\pi$
$992$	0	0
$993$	−8.38811	−0.266189
$994$	0	0
$995$	−2.70237	−0.0856709
$996$	0	0
$997$	−50.4407	−1.59747	−0.798736	−	0.601681i	$-0.794498\pi$
$997$	−50.4407	−1.59747	−0.798736	+	0.601681i	$0.794498\pi$
$998$	0	0
$999$	2.08641	0.0660112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.z.1.3		5
4.3	odd	2		1480.2.a.h.1.3	✓	5
20.19	odd	2		7400.2.a.q.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.h.1.3	✓	5	4.3	odd	2
2960.2.a.z.1.3		5	1.1	even	1	trivial
7400.2.a.q.1.3		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)

\(f(q)\)	\(=\)	\(q-0.355205 q^{3} -1.00000 q^{5} -3.27534 q^{7} -2.87383 q^{9} +O(q^{10})\)	\(q-0.355205 q^{3} -1.00000 q^{5} -3.27534 q^{7} -2.87383 q^{9} -0.833725 q^{11} -6.22100 q^{13} +0.355205 q^{15} -1.04010 q^{17} +5.33257 q^{19} +1.16342 q^{21} +2.71041 q^{23} +1.00000 q^{25} +2.08641 q^{27} -8.55069 q^{29} -8.47656 q^{31} +0.296144 q^{33} +3.27534 q^{35} +1.00000 q^{37} +2.20973 q^{39} +1.00655 q^{41} +3.19697 q^{43} +2.87383 q^{45} +6.78879 q^{47} +3.72788 q^{49} +0.369450 q^{51} +8.23847 q^{53} +0.833725 q^{55} -1.89416 q^{57} -5.18697 q^{59} -9.54839 q^{61} +9.41278 q^{63} +6.22100 q^{65} +6.62216 q^{67} -0.962752 q^{69} +6.97737 q^{71} +9.68778 q^{73} -0.355205 q^{75} +2.73074 q^{77} -1.15203 q^{79} +7.88038 q^{81} +14.2465 q^{83} +1.04010 q^{85} +3.03725 q^{87} -14.2078 q^{89} +20.3759 q^{91} +3.01092 q^{93} -5.33257 q^{95} +3.94898 q^{97} +2.39598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9	\( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100}) \) 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 4 * q^13 - q^15 + 10 * q^19 - 5 * q^21 + 8 * q^23 + 5 * q^25 + 7 * q^27 - 8 * q^29 + 2 * q^31 - 11 * q^33 - q^35 + 5 * q^37 + 2 * q^39 - 13 * q^41 + 18 * q^43 - 2 * q^45 + 9 * q^47 - 6 * q^49 + 22 * q^51 - 13 * q^53 - 7 * q^55 + 4 * q^57 + 24 * q^59 - 2 * q^61 + 20 * q^63 - 4 * q^65 + 22 * q^67 - 32 * q^69 + 21 * q^71 + 29 * q^73 + q^75 + 11 * q^77 + 6 * q^79 + 5 * q^81 + 33 * q^83 - 12 * q^87 - 26 * q^89 + 38 * q^91 + 14 * q^93 - 10 * q^95 + 26 * q^97 + 42 * q^99

Introduction

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