LMFDB - Embedded newform 2720.2.a.t.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2720,2,Mod(1,2720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2720.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2720 = 2^{5} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2720.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,3,0,5,0,8,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$21.7193093498$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.457904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 8x^{2} + 5x - 6 $ x^5 - 2x^4 - 5x^3 + 8x^2 + 5x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.788997$ of defining polynomial
Character	$\chi$	$=$	2720.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16648 q^{3} +1.00000 q^{5} +4.86012 q^{7} +7.02660 q^{9} -4.61840 q^{11} -3.02992 q^{13} +3.16648 q^{15} -1.00000 q^{17} +3.87393 q^{19} +15.3895 q^{21} -0.860121 q^{23} +1.00000 q^{25} +12.7502 q^{27} +2.84963 q^{29} -5.46803 q^{31} -14.6241 q^{33} +4.86012 q^{35} +9.90053 q^{37} -9.59417 q^{39} -12.4438 q^{41} -0.754968 q^{43} +7.02660 q^{45} -1.11233 q^{47} +16.6208 q^{49} -3.16648 q^{51} +11.3629 q^{53} -4.61840 q^{55} +12.2667 q^{57} -5.63601 q^{59} +4.81961 q^{61} +34.1501 q^{63} -3.02992 q^{65} -0.965273 q^{67} -2.72356 q^{69} +2.92476 q^{71} -6.29973 q^{73} +3.16648 q^{75} -22.4460 q^{77} +12.7711 q^{79} +19.2933 q^{81} +2.14888 q^{83} -1.00000 q^{85} +9.02329 q^{87} -14.9304 q^{89} -14.7258 q^{91} -17.3144 q^{93} +3.87393 q^{95} -12.2069 q^{97} -32.4517 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} + 5 q^{5} + 8 q^{7} + 6 q^{9} + 2 q^{11} + q^{13} + 3 q^{15} - 5 q^{17} + 11 q^{19} + 6 q^{21} + 12 q^{23} + 5 q^{25} + 15 q^{27} + 3 q^{29} + 9 q^{31} - 16 q^{33} + 8 q^{35} + 12 q^{37}+ \cdots - 14 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 + 5 * q^5 + 8 * q^7 + 6 * q^9 + 2 * q^11 + q^13 + 3 * q^15 - 5 * q^17 + 11 * q^19 + 6 * q^21 + 12 * q^23 + 5 * q^25 + 15 * q^27 + 3 * q^29 + 9 * q^31 - 16 * q^33 + 8 * q^35 + 12 * q^37 - 7 * q^39 - 10 * q^41 + 18 * q^43 + 6 * q^45 + 7 * q^47 + 13 * q^49 - 3 * q^51 + 15 * q^53 + 2 * q^55 - 5 * q^57 + 7 * q^59 - 3 * q^61 + 46 * q^63 + q^65 + 6 * q^67 + 6 * q^69 - 7 * q^71 - 9 * q^73 + 3 * q^75 + 4 * q^77 + 18 * q^79 + q^81 + 8 * q^83 - 5 * q^85 + 3 * q^87 - 21 * q^89 + 10 * q^91 - 13 * q^93 + 11 * q^95 - 27 * 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
$3$	3.16648	1.82817	0.914084	−	0.405524i	$-0.132911\pi$
$3$	3.16648	1.82817	0.914084	+	0.405524i	$0.132911\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.86012	1.83695	0.918476	−	0.395476i	$-0.129420\pi$
$7$	4.86012	1.83695	0.918476	+	0.395476i	$0.129420\pi$
$8$	0	0
$9$	7.02660	2.34220
$10$	0	0
$11$	−4.61840	−1.39250	−0.696250	−	0.717799i	$-0.745150\pi$
$11$	−4.61840	−1.39250	−0.696250	+	0.717799i	$0.745150\pi$
$12$	0	0
$13$	−3.02992	−0.840347	−0.420174	−	0.907444i	$-0.638031\pi$
$13$	−3.02992	−0.840347	−0.420174	+	0.907444i	$0.638031\pi$
$14$	0	0
$15$	3.16648	0.817582
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.87393	0.888740	0.444370	−	0.895843i	$-0.353428\pi$
$19$	3.87393	0.888740	0.444370	+	0.895843i	$0.353428\pi$
$20$	0	0
$21$	15.3895	3.35826
$22$	0	0
$23$	−0.860121	−0.179348	−0.0896738	−	0.995971i	$-0.528582\pi$
$23$	−0.860121	−0.179348	−0.0896738	+	0.995971i	$0.528582\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	12.7502	2.45377
$28$	0	0
$29$	2.84963	0.529162	0.264581	−	0.964363i	$-0.414766\pi$
$29$	2.84963	0.529162	0.264581	+	0.964363i	$0.414766\pi$
$30$	0	0
$31$	−5.46803	−0.982087	−0.491044	−	0.871135i	$-0.663384\pi$
$31$	−5.46803	−0.982087	−0.491044	+	0.871135i	$0.663384\pi$
$32$	0	0
$33$	−14.6241	−2.54573
$34$	0	0
$35$	4.86012	0.821510
$36$	0	0
$37$	9.90053	1.62764	0.813819	−	0.581119i	$-0.197385\pi$
$37$	9.90053	1.62764	0.813819	+	0.581119i	$0.197385\pi$
$38$	0	0
$39$	−9.59417	−1.53630
$40$	0	0
$41$	−12.4438	−1.94339	−0.971697	−	0.236230i	$-0.924088\pi$
$41$	−12.4438	−1.94339	−0.971697	+	0.236230i	$0.924088\pi$
$42$	0	0
$43$	−0.754968	−0.115132	−0.0575658	−	0.998342i	$-0.518334\pi$
$43$	−0.754968	−0.115132	−0.0575658	+	0.998342i	$0.518334\pi$
$44$	0	0
$45$	7.02660	1.04746
$46$	0	0
$47$	−1.11233	−0.162250	−0.0811252	−	0.996704i	$-0.525851\pi$
$47$	−1.11233	−0.162250	−0.0811252	+	0.996704i	$0.525851\pi$
$48$	0	0
$49$	16.6208	2.37440
$50$	0	0
$51$	−3.16648	−0.443396
$52$	0	0
$53$	11.3629	1.56081	0.780406	−	0.625274i	$-0.215013\pi$
$53$	11.3629	1.56081	0.780406	+	0.625274i	$0.215013\pi$
$54$	0	0
$55$	−4.61840	−0.622745
$56$	0	0
$57$	12.2667	1.62477
$58$	0	0
$59$	−5.63601	−0.733746	−0.366873	−	0.930271i	$-0.619572\pi$
$59$	−5.63601	−0.733746	−0.366873	+	0.930271i	$0.619572\pi$
$60$	0	0
$61$	4.81961	0.617088	0.308544	−	0.951210i	$-0.400158\pi$
$61$	4.81961	0.617088	0.308544	+	0.951210i	$0.400158\pi$
$62$	0	0
$63$	34.1501	4.30251
$64$	0	0
$65$	−3.02992	−0.375815
$66$	0	0
$67$	−0.965273	−0.117927	−0.0589634	−	0.998260i	$-0.518780\pi$
$67$	−0.965273	−0.117927	−0.0589634	+	0.998260i	$0.518780\pi$
$68$	0	0
$69$	−2.72356	−0.327878
$70$	0	0
$71$	2.92476	0.347106	0.173553	−	0.984825i	$-0.444475\pi$
$71$	2.92476	0.347106	0.173553	+	0.984825i	$0.444475\pi$
$72$	0	0
$73$	−6.29973	−0.737328	−0.368664	−	0.929563i	$-0.620185\pi$
$73$	−6.29973	−0.737328	−0.368664	+	0.929563i	$0.620185\pi$
$74$	0	0
$75$	3.16648	0.365634
$76$	0	0
$77$	−22.4460	−2.55796
$78$	0	0
$79$	12.7711	1.43686	0.718429	−	0.695600i	$-0.244861\pi$
$79$	12.7711	1.43686	0.718429	+	0.695600i	$0.244861\pi$
$80$	0	0
$81$	19.2933	2.14370
$82$	0	0
$83$	2.14888	0.235870	0.117935	−	0.993021i	$-0.462373\pi$
$83$	2.14888	0.235870	0.117935	+	0.993021i	$0.462373\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	9.02329	0.967398
$88$	0	0
$89$	−14.9304	−1.58262	−0.791312	−	0.611412i	$-0.790602\pi$
$89$	−14.9304	−1.58262	−0.791312	+	0.611412i	$0.790602\pi$
$90$	0	0
$91$	−14.7258	−1.54368
$92$	0	0
$93$	−17.3144	−1.79542
$94$	0	0
$95$	3.87393	0.397457
$96$	0	0
$97$	−12.2069	−1.23942	−0.619711	−	0.784830i	$-0.712750\pi$
$97$	−12.2069	−1.23942	−0.619711	+	0.784830i	$0.712750\pi$
$98$	0	0
$99$	−32.4517	−3.26152
$100$	0	0
$101$	10.4856	1.04336	0.521680	−	0.853141i	$-0.325306\pi$
$101$	10.4856	1.04336	0.521680	+	0.853141i	$0.325306\pi$
$102$	0	0
$103$	−12.5350	−1.23511	−0.617557	−	0.786526i	$-0.711878\pi$
$103$	−12.5350	−1.23511	−0.617557	+	0.786526i	$0.711878\pi$
$104$	0	0
$105$	15.3895	1.50186
$106$	0	0
$107$	−0.203025	−0.0196272	−0.00981358	−	0.999952i	$-0.503124\pi$
$107$	−0.203025	−0.0196272	−0.00981358	+	0.999952i	$0.503124\pi$
$108$	0	0
$109$	−12.5122	−1.19845	−0.599227	−	0.800579i	$-0.704525\pi$
$109$	−12.5122	−1.19845	−0.599227	+	0.800579i	$0.704525\pi$
$110$	0	0
$111$	31.3498	2.97560
$112$	0	0
$113$	−14.3662	−1.35146	−0.675729	−	0.737150i	$-0.736171\pi$
$113$	−14.3662	−1.35146	−0.675729	+	0.737150i	$0.736171\pi$
$114$	0	0
$115$	−0.860121	−0.0802067
$116$	0	0
$117$	−21.2900	−1.96826
$118$	0	0
$119$	−4.86012	−0.445527
$120$	0	0
$121$	10.3296	0.939059
$122$	0	0
$123$	−39.4030	−3.55285
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.5650	−1.73611	−0.868055	−	0.496469i	$-0.834630\pi$
$127$	−19.5650	−1.73611	−0.868055	+	0.496469i	$0.834630\pi$
$128$	0	0
$129$	−2.39059	−0.210480
$130$	0	0
$131$	10.5908	0.925321	0.462661	−	0.886535i	$-0.346895\pi$
$131$	10.5908	0.925321	0.462661	+	0.886535i	$0.346895\pi$
$132$	0	0
$133$	18.8278	1.63257
$134$	0	0
$135$	12.7502	1.09736
$136$	0	0
$137$	−19.8429	−1.69529	−0.847647	−	0.530561i	$-0.821981\pi$
$137$	−19.8429	−1.69529	−0.847647	+	0.530561i	$0.821981\pi$
$138$	0	0
$139$	17.2215	1.46071	0.730354	−	0.683069i	$-0.239355\pi$
$139$	17.2215	1.46071	0.730354	+	0.683069i	$0.239355\pi$
$140$	0	0
$141$	−3.52218	−0.296621
$142$	0	0
$143$	13.9934	1.17018
$144$	0	0
$145$	2.84963	0.236649
$146$	0	0
$147$	52.6294	4.34080
$148$	0	0
$149$	−12.4280	−1.01814	−0.509071	−	0.860725i	$-0.670011\pi$
$149$	−12.4280	−1.01814	−0.509071	+	0.860725i	$0.670011\pi$
$150$	0	0
$151$	1.48895	0.121169	0.0605845	−	0.998163i	$-0.480704\pi$
$151$	1.48895	0.121169	0.0605845	+	0.998163i	$0.480704\pi$
$152$	0	0
$153$	−7.02660	−0.568067
$154$	0	0
$155$	−5.46803	−0.439203
$156$	0	0
$157$	7.93055	0.632926	0.316463	−	0.948605i	$-0.397505\pi$
$157$	7.93055	0.632926	0.316463	+	0.948605i	$0.397505\pi$
$158$	0	0
$159$	35.9803	2.85343
$160$	0	0
$161$	−4.18029	−0.329453
$162$	0	0
$163$	−14.4809	−1.13423	−0.567115	−	0.823638i	$-0.691941\pi$
$163$	−14.4809	−1.13423	−0.567115	+	0.823638i	$0.691941\pi$
$164$	0	0
$165$	−14.6241	−1.13848
$166$	0	0
$167$	−3.23492	−0.250326	−0.125163	−	0.992136i	$-0.539945\pi$
$167$	−3.23492	−0.250326	−0.125163	+	0.992136i	$0.539945\pi$
$168$	0	0
$169$	−3.81961	−0.293816
$170$	0	0
$171$	27.2206	2.08161
$172$	0	0
$173$	−4.14604	−0.315218	−0.157609	−	0.987502i	$-0.550379\pi$
$173$	−4.14604	−0.315218	−0.157609	+	0.987502i	$0.550379\pi$
$174$	0	0
$175$	4.86012	0.367391
$176$	0	0
$177$	−17.8463	−1.34141
$178$	0	0
$179$	3.42912	0.256304	0.128152	−	0.991755i	$-0.459095\pi$
$179$	3.42912	0.256304	0.128152	+	0.991755i	$0.459095\pi$
$180$	0	0
$181$	−0.684712	−0.0508943	−0.0254471	−	0.999676i	$-0.508101\pi$
$181$	−0.684712	−0.0508943	−0.0254471	+	0.999676i	$0.508101\pi$
$182$	0	0
$183$	15.2612	1.12814
$184$	0	0
$185$	9.90053	0.727901
$186$	0	0
$187$	4.61840	0.337731
$188$	0	0
$189$	61.9673	4.50746
$190$	0	0
$191$	−0.483435	−0.0349801	−0.0174901	−	0.999847i	$-0.505568\pi$
$191$	−0.483435	−0.0349801	−0.0174901	+	0.999847i	$0.505568\pi$
$192$	0	0
$193$	18.3143	1.31829	0.659147	−	0.752015i	$-0.270918\pi$
$193$	18.3143	1.31829	0.659147	+	0.752015i	$0.270918\pi$
$194$	0	0
$195$	−9.59417	−0.687053
$196$	0	0
$197$	−21.3410	−1.52048	−0.760242	−	0.649640i	$-0.774920\pi$
$197$	−21.3410	−1.52048	−0.760242	+	0.649640i	$0.774920\pi$
$198$	0	0
$199$	18.1017	1.28320	0.641599	−	0.767040i	$-0.278271\pi$
$199$	18.1017	1.28320	0.641599	+	0.767040i	$0.278271\pi$
$200$	0	0
$201$	−3.05652	−0.215590
$202$	0	0
$203$	13.8495	0.972046
$204$	0	0
$205$	−12.4438	−0.869112
$206$	0	0
$207$	−6.04372	−0.420068
$208$	0	0
$209$	−17.8914	−1.23757
$210$	0	0
$211$	6.74106	0.464074	0.232037	−	0.972707i	$-0.425461\pi$
$211$	6.74106	0.464074	0.232037	+	0.972707i	$0.425461\pi$
$212$	0	0
$213$	9.26121	0.634568
$214$	0	0
$215$	−0.754968	−0.0514884
$216$	0	0
$217$	−26.5753	−1.80405
$218$	0	0
$219$	−19.9480	−1.34796
$220$	0	0
$221$	3.02992	0.203814
$222$	0	0
$223$	23.4565	1.57076	0.785382	−	0.619011i	$-0.212466\pi$
$223$	23.4565	1.57076	0.785382	+	0.619011i	$0.212466\pi$
$224$	0	0
$225$	7.02660	0.468440
$226$	0	0
$227$	6.93519	0.460305	0.230152	−	0.973155i	$-0.426078\pi$
$227$	6.93519	0.460305	0.230152	+	0.973155i	$0.426078\pi$
$228$	0	0
$229$	−14.4856	−0.957238	−0.478619	−	0.878023i	$-0.658862\pi$
$229$	−14.4856	−0.957238	−0.478619	+	0.878023i	$0.658862\pi$
$230$	0	0
$231$	−71.0748	−4.67638
$232$	0	0
$233$	−18.8242	−1.23321	−0.616607	−	0.787271i	$-0.711493\pi$
$233$	−18.8242	−1.23321	−0.616607	+	0.787271i	$0.711493\pi$
$234$	0	0
$235$	−1.11233	−0.0725605
$236$	0	0
$237$	40.4394	2.62682
$238$	0	0
$239$	3.45011	0.223169	0.111584	−	0.993755i	$-0.464407\pi$
$239$	3.45011	0.223169	0.111584	+	0.993755i	$0.464407\pi$
$240$	0	0
$241$	1.66924	0.107525	0.0537625	−	0.998554i	$-0.482879\pi$
$241$	1.66924	0.107525	0.0537625	+	0.998554i	$0.482879\pi$
$242$	0	0
$243$	22.8415	1.46528
$244$	0	0
$245$	16.6208	1.06186
$246$	0	0
$247$	−11.7377	−0.746851
$248$	0	0
$249$	6.80437	0.431210
$250$	0	0
$251$	−14.6593	−0.925287	−0.462643	−	0.886545i	$-0.653099\pi$
$251$	−14.6593	−0.925287	−0.462643	+	0.886545i	$0.653099\pi$
$252$	0	0
$253$	3.97238	0.249742
$254$	0	0
$255$	−3.16648	−0.198293
$256$	0	0
$257$	−3.37591	−0.210584	−0.105292	−	0.994441i	$-0.533578\pi$
$257$	−3.37591	−0.210584	−0.105292	+	0.994441i	$0.533578\pi$
$258$	0	0
$259$	48.1178	2.98989
$260$	0	0
$261$	20.0232	1.23940
$262$	0	0
$263$	3.45855	0.213263	0.106632	−	0.994299i	$-0.465993\pi$
$263$	3.45855	0.213263	0.106632	+	0.994299i	$0.465993\pi$
$264$	0	0
$265$	11.3629	0.698016
$266$	0	0
$267$	−47.2770	−2.89330
$268$	0	0
$269$	−20.0080	−1.21991	−0.609953	−	0.792437i	$-0.708812\pi$
$269$	−20.0080	−1.21991	−0.609953	+	0.792437i	$0.708812\pi$
$270$	0	0
$271$	5.08653	0.308985	0.154492	−	0.987994i	$-0.450626\pi$
$271$	5.08653	0.308985	0.154492	+	0.987994i	$0.450626\pi$
$272$	0	0
$273$	−46.6288	−2.82210
$274$	0	0
$275$	−4.61840	−0.278500
$276$	0	0
$277$	18.2217	1.09483	0.547417	−	0.836860i	$-0.315611\pi$
$277$	18.2217	1.09483	0.547417	+	0.836860i	$0.315611\pi$
$278$	0	0
$279$	−38.4217	−2.30024
$280$	0	0
$281$	16.9957	1.01388	0.506938	−	0.861982i	$-0.330777\pi$
$281$	16.9957	1.01388	0.506938	+	0.861982i	$0.330777\pi$
$282$	0	0
$283$	24.9077	1.48061	0.740305	−	0.672271i	$-0.234681\pi$
$283$	24.9077	1.48061	0.740305	+	0.672271i	$0.234681\pi$
$284$	0	0
$285$	12.2667	0.726618
$286$	0	0
$287$	−60.4783	−3.56992
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−38.6529	−2.26587
$292$	0	0
$293$	12.2697	0.716805	0.358402	−	0.933567i	$-0.383322\pi$
$293$	12.2697	0.716805	0.358402	+	0.933567i	$0.383322\pi$
$294$	0	0
$295$	−5.63601	−0.328141
$296$	0	0
$297$	−58.8854	−3.41688
$298$	0	0
$299$	2.60609	0.150714
$300$	0	0
$301$	−3.66924	−0.211491
$302$	0	0
$303$	33.2026	1.90744
$304$	0	0
$305$	4.81961	0.275970
$306$	0	0
$307$	−27.9127	−1.59306	−0.796531	−	0.604598i	$-0.793334\pi$
$307$	−27.9127	−1.59306	−0.796531	+	0.604598i	$0.793334\pi$
$308$	0	0
$309$	−39.6920	−2.25800
$310$	0	0
$311$	1.23366	0.0699545	0.0349772	−	0.999388i	$-0.488864\pi$
$311$	1.23366	0.0699545	0.0349772	+	0.999388i	$0.488864\pi$
$312$	0	0
$313$	−8.20128	−0.463564	−0.231782	−	0.972768i	$-0.574456\pi$
$313$	−8.20128	−0.463564	−0.231782	+	0.972768i	$0.574456\pi$
$314$	0	0
$315$	34.1501	1.92414
$316$	0	0
$317$	29.8675	1.67753	0.838764	−	0.544496i	$-0.183279\pi$
$317$	29.8675	1.67753	0.838764	+	0.544496i	$0.183279\pi$
$318$	0	0
$319$	−13.1607	−0.736859
$320$	0	0
$321$	−0.642874	−0.0358817
$322$	0	0
$323$	−3.87393	−0.215551
$324$	0	0
$325$	−3.02992	−0.168069
$326$	0	0
$327$	−39.6198	−2.19098
$328$	0	0
$329$	−5.40607	−0.298046
$330$	0	0
$331$	8.28771	0.455534	0.227767	−	0.973716i	$-0.426858\pi$
$331$	8.28771	0.455534	0.227767	+	0.973716i	$0.426858\pi$
$332$	0	0
$333$	69.5671	3.81225
$334$	0	0
$335$	−0.965273	−0.0527385
$336$	0	0
$337$	−25.4855	−1.38829	−0.694143	−	0.719837i	$-0.744216\pi$
$337$	−25.4855	−1.38829	−0.694143	+	0.719837i	$0.744216\pi$
$338$	0	0
$339$	−45.4903	−2.47069
$340$	0	0
$341$	25.2536	1.36756
$342$	0	0
$343$	46.7581	2.52470
$344$	0	0
$345$	−2.72356	−0.146631
$346$	0	0
$347$	35.5706	1.90953	0.954766	−	0.297358i	$-0.0961056\pi$
$347$	35.5706	1.90953	0.954766	+	0.297358i	$0.0961056\pi$
$348$	0	0
$349$	−7.19497	−0.385138	−0.192569	−	0.981283i	$-0.561682\pi$
$349$	−7.19497	−0.385138	−0.192569	+	0.981283i	$0.561682\pi$
$350$	0	0
$351$	−38.6319	−2.06202
$352$	0	0
$353$	35.0073	1.86325	0.931624	−	0.363423i	$-0.118392\pi$
$353$	35.0073	1.86325	0.931624	+	0.363423i	$0.118392\pi$
$354$	0	0
$355$	2.92476	0.155230
$356$	0	0
$357$	−15.3895	−0.814498
$358$	0	0
$359$	18.7999	0.992223	0.496112	−	0.868259i	$-0.334761\pi$
$359$	18.7999	0.992223	0.496112	+	0.868259i	$0.334761\pi$
$360$	0	0
$361$	−3.99268	−0.210141
$362$	0	0
$363$	32.7086	1.71676
$364$	0	0
$365$	−6.29973	−0.329743
$366$	0	0
$367$	−4.67067	−0.243807	−0.121903	−	0.992542i	$-0.538900\pi$
$367$	−4.67067	−0.243807	−0.121903	+	0.992542i	$0.538900\pi$
$368$	0	0
$369$	−87.4376	−4.55182
$370$	0	0
$371$	55.2250	2.86714
$372$	0	0
$373$	−23.3986	−1.21154	−0.605768	−	0.795641i	$-0.707134\pi$
$373$	−23.3986	−1.21154	−0.605768	+	0.795641i	$0.707134\pi$
$374$	0	0
$375$	3.16648	0.163516
$376$	0	0
$377$	−8.63413	−0.444680
$378$	0	0
$379$	3.80660	0.195532	0.0977661	−	0.995209i	$-0.468830\pi$
$379$	3.80660	0.195532	0.0977661	+	0.995209i	$0.468830\pi$
$380$	0	0
$381$	−61.9521	−3.17390
$382$	0	0
$383$	−20.0275	−1.02336	−0.511680	−	0.859176i	$-0.670977\pi$
$383$	−20.0275	−1.02336	−0.511680	+	0.859176i	$0.670977\pi$
$384$	0	0
$385$	−22.4460	−1.14395
$386$	0	0
$387$	−5.30486	−0.269661
$388$	0	0
$389$	−19.7015	−0.998903	−0.499452	−	0.866342i	$-0.666465\pi$
$389$	−19.7015	−0.998903	−0.499452	+	0.866342i	$0.666465\pi$
$390$	0	0
$391$	0.860121	0.0434982
$392$	0	0
$393$	33.5355	1.69164
$394$	0	0
$395$	12.7711	0.642583
$396$	0	0
$397$	−25.6133	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$397$	−25.6133	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$398$	0	0
$399$	59.6177	2.98462
$400$	0	0
$401$	7.63922	0.381485	0.190742	−	0.981640i	$-0.438911\pi$
$401$	7.63922	0.381485	0.190742	+	0.981640i	$0.438911\pi$
$402$	0	0
$403$	16.5677	0.825294
$404$	0	0
$405$	19.2933	0.958693
$406$	0	0
$407$	−45.7246	−2.26649
$408$	0	0
$409$	36.2569	1.79279	0.896394	−	0.443258i	$-0.146177\pi$
$409$	36.2569	1.79279	0.896394	+	0.443258i	$0.146177\pi$
$410$	0	0
$411$	−62.8322	−3.09928
$412$	0	0
$413$	−27.3917	−1.34786
$414$	0	0
$415$	2.14888	0.105484
$416$	0	0
$417$	54.5316	2.67042
$418$	0	0
$419$	2.18474	0.106731	0.0533657	−	0.998575i	$-0.483005\pi$
$419$	2.18474	0.106731	0.0533657	+	0.998575i	$0.483005\pi$
$420$	0	0
$421$	−13.9490	−0.679833	−0.339916	−	0.940456i	$-0.610399\pi$
$421$	−13.9490	−0.679833	−0.339916	+	0.940456i	$0.610399\pi$
$422$	0	0
$423$	−7.81591	−0.380023
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	23.4239	1.13356
$428$	0	0
$429$	44.3097	2.13929
$430$	0	0
$431$	−25.1639	−1.21210	−0.606050	−	0.795426i	$-0.707247\pi$
$431$	−25.1639	−1.21210	−0.606050	+	0.795426i	$0.707247\pi$
$432$	0	0
$433$	14.6175	0.702470	0.351235	−	0.936287i	$-0.385762\pi$
$433$	14.6175	0.702470	0.351235	+	0.936287i	$0.385762\pi$
$434$	0	0
$435$	9.02329	0.432634
$436$	0	0
$437$	−3.33205	−0.159393
$438$	0	0
$439$	−18.0707	−0.862468	−0.431234	−	0.902240i	$-0.641922\pi$
$439$	−18.0707	−0.862468	−0.431234	+	0.902240i	$0.641922\pi$
$440$	0	0
$441$	116.788	5.56131
$442$	0	0
$443$	10.1811	0.483718	0.241859	−	0.970311i	$-0.422243\pi$
$443$	10.1811	0.483718	0.241859	+	0.970311i	$0.422243\pi$
$444$	0	0
$445$	−14.9304	−0.707771
$446$	0	0
$447$	−39.3530	−1.86133
$448$	0	0
$449$	3.42372	0.161576	0.0807878	−	0.996731i	$-0.474256\pi$
$449$	3.42372	0.161576	0.0807878	+	0.996731i	$0.474256\pi$
$450$	0	0
$451$	57.4705	2.70618
$452$	0	0
$453$	4.71473	0.221517
$454$	0	0
$455$	−14.7258	−0.690354
$456$	0	0
$457$	8.04184	0.376181	0.188091	−	0.982152i	$-0.439770\pi$
$457$	8.04184	0.376181	0.188091	+	0.982152i	$0.439770\pi$
$458$	0	0
$459$	−12.7502	−0.595126
$460$	0	0
$461$	18.3053	0.852565	0.426282	−	0.904590i	$-0.359823\pi$
$461$	18.3053	0.852565	0.426282	+	0.904590i	$0.359823\pi$
$462$	0	0
$463$	12.7582	0.592923	0.296462	−	0.955045i	$-0.404193\pi$
$463$	12.7582	0.592923	0.296462	+	0.955045i	$0.404193\pi$
$464$	0	0
$465$	−17.3144	−0.802937
$466$	0	0
$467$	25.0927	1.16115	0.580575	−	0.814207i	$-0.302828\pi$
$467$	25.0927	1.16115	0.580575	+	0.814207i	$0.302828\pi$
$468$	0	0
$469$	−4.69134	−0.216626
$470$	0	0
$471$	25.1119	1.15710
$472$	0	0
$473$	3.48675	0.160321
$474$	0	0
$475$	3.87393	0.177748
$476$	0	0
$477$	79.8424	3.65573
$478$	0	0
$479$	19.3079	0.882202	0.441101	−	0.897457i	$-0.354588\pi$
$479$	19.3079	0.882202	0.441101	+	0.897457i	$0.354588\pi$
$480$	0	0
$481$	−29.9978	−1.36778
$482$	0	0
$483$	−13.2368	−0.602296
$484$	0	0
$485$	−12.2069	−0.554286
$486$	0	0
$487$	4.31386	0.195480	0.0977398	−	0.995212i	$-0.468839\pi$
$487$	4.31386	0.195480	0.0977398	+	0.995212i	$0.468839\pi$
$488$	0	0
$489$	−45.8535	−2.07357
$490$	0	0
$491$	−24.7388	−1.11645	−0.558223	−	0.829691i	$-0.688517\pi$
$491$	−24.7388	−1.11645	−0.558223	+	0.829691i	$0.688517\pi$
$492$	0	0
$493$	−2.84963	−0.128341
$494$	0	0
$495$	−32.4517	−1.45859
$496$	0	0
$497$	14.2147	0.637617
$498$	0	0
$499$	−23.2359	−1.04018	−0.520090	−	0.854111i	$-0.674102\pi$
$499$	−23.2359	−1.04018	−0.520090	+	0.854111i	$0.674102\pi$
$500$	0	0
$501$	−10.2433	−0.457638
$502$	0	0
$503$	−19.5613	−0.872193	−0.436097	−	0.899900i	$-0.643639\pi$
$503$	−19.5613	−0.872193	−0.436097	+	0.899900i	$0.643639\pi$
$504$	0	0
$505$	10.4856	0.466605
$506$	0	0
$507$	−12.0947	−0.537146
$508$	0	0
$509$	20.1018	0.890997	0.445498	−	0.895283i	$-0.353026\pi$
$509$	20.1018	0.890997	0.445498	+	0.895283i	$0.353026\pi$
$510$	0	0
$511$	−30.6175	−1.35444
$512$	0	0
$513$	49.3932	2.18076
$514$	0	0
$515$	−12.5350	−0.552360
$516$	0	0
$517$	5.13720	0.225934
$518$	0	0
$519$	−13.1284	−0.576271
$520$	0	0
$521$	20.4672	0.896684	0.448342	−	0.893862i	$-0.352015\pi$
$521$	20.4672	0.896684	0.448342	+	0.893862i	$0.352015\pi$
$522$	0	0
$523$	−5.91569	−0.258675	−0.129338	−	0.991601i	$-0.541285\pi$
$523$	−5.91569	−0.258675	−0.129338	+	0.991601i	$0.541285\pi$
$524$	0	0
$525$	15.3895	0.671652
$526$	0	0
$527$	5.46803	0.238191
$528$	0	0
$529$	−22.2602	−0.967834
$530$	0	0
$531$	−39.6020	−1.71858
$532$	0	0
$533$	37.7037	1.63313
$534$	0	0
$535$	−0.203025	−0.00877753
$536$	0	0
$537$	10.8582	0.468567
$538$	0	0
$539$	−76.7614	−3.30635
$540$	0	0
$541$	9.83088	0.422662	0.211331	−	0.977415i	$-0.432220\pi$
$541$	9.83088	0.422662	0.211331	+	0.977415i	$0.432220\pi$
$542$	0	0
$543$	−2.16813	−0.0930433
$544$	0	0
$545$	−12.5122	−0.535965
$546$	0	0
$547$	−20.5718	−0.879585	−0.439792	−	0.898100i	$-0.644948\pi$
$547$	−20.5718	−0.879585	−0.439792	+	0.898100i	$0.644948\pi$
$548$	0	0
$549$	33.8655	1.44534
$550$	0	0
$551$	11.0392	0.470288
$552$	0	0
$553$	62.0690	2.63944
$554$	0	0
$555$	31.3498	1.33073
$556$	0	0
$557$	5.27543	0.223527	0.111764	−	0.993735i	$-0.464350\pi$
$557$	5.27543	0.223527	0.111764	+	0.993735i	$0.464350\pi$
$558$	0	0
$559$	2.28749	0.0967506
$560$	0	0
$561$	14.6241	0.617429
$562$	0	0
$563$	26.0031	1.09590	0.547951	−	0.836511i	$-0.315408\pi$
$563$	26.0031	1.09590	0.547951	+	0.836511i	$0.315408\pi$
$564$	0	0
$565$	−14.3662	−0.604390
$566$	0	0
$567$	93.7679	3.93788
$568$	0	0
$569$	−0.0908612	−0.00380910	−0.00190455	−	0.999998i	$-0.500606\pi$
$569$	−0.0908612	−0.00380910	−0.00190455	+	0.999998i	$0.500606\pi$
$570$	0	0
$571$	−1.82286	−0.0762842	−0.0381421	−	0.999272i	$-0.512144\pi$
$571$	−1.82286	−0.0762842	−0.0381421	+	0.999272i	$0.512144\pi$
$572$	0	0
$573$	−1.53079	−0.0639495
$574$	0	0
$575$	−0.860121	−0.0358695
$576$	0	0
$577$	15.3139	0.637524	0.318762	−	0.947835i	$-0.396733\pi$
$577$	15.3139	0.637524	0.318762	+	0.947835i	$0.396733\pi$
$578$	0	0
$579$	57.9919	2.41006
$580$	0	0
$581$	10.4438	0.433282
$582$	0	0
$583$	−52.4783	−2.17343
$584$	0	0
$585$	−21.2900	−0.880234
$586$	0	0
$587$	−7.00412	−0.289091	−0.144545	−	0.989498i	$-0.546172\pi$
$587$	−7.00412	−0.289091	−0.144545	+	0.989498i	$0.546172\pi$
$588$	0	0
$589$	−21.1828	−0.872820
$590$	0	0
$591$	−67.5759	−2.77970
$592$	0	0
$593$	−15.8077	−0.649144	−0.324572	−	0.945861i	$-0.605220\pi$
$593$	−15.8077	−0.649144	−0.324572	+	0.945861i	$0.605220\pi$
$594$	0	0
$595$	−4.86012	−0.199246
$596$	0	0
$597$	57.3188	2.34590
$598$	0	0
$599$	2.06742	0.0844727	0.0422363	−	0.999108i	$-0.486552\pi$
$599$	2.06742	0.0844727	0.0422363	+	0.999108i	$0.486552\pi$
$600$	0	0
$601$	15.5542	0.634468	0.317234	−	0.948347i	$-0.397246\pi$
$601$	15.5542	0.634468	0.317234	+	0.948347i	$0.397246\pi$
$602$	0	0
$603$	−6.78259	−0.276208
$604$	0	0
$605$	10.3296	0.419960
$606$	0	0
$607$	26.3072	1.06778	0.533889	−	0.845555i	$-0.320730\pi$
$607$	26.3072	1.06778	0.533889	+	0.845555i	$0.320730\pi$
$608$	0	0
$609$	43.8543	1.77706
$610$	0	0
$611$	3.37027	0.136347
$612$	0	0
$613$	−49.0184	−1.97983	−0.989917	−	0.141647i	$-0.954760\pi$
$613$	−49.0184	−1.97983	−0.989917	+	0.141647i	$0.954760\pi$
$614$	0	0
$615$	−39.4030	−1.58888
$616$	0	0
$617$	−36.3463	−1.46325	−0.731624	−	0.681709i	$-0.761237\pi$
$617$	−36.3463	−1.46325	−0.731624	+	0.681709i	$0.761237\pi$
$618$	0	0
$619$	10.5308	0.423267	0.211633	−	0.977349i	$-0.432122\pi$
$619$	10.5308	0.423267	0.211633	+	0.977349i	$0.432122\pi$
$620$	0	0
$621$	−10.9667	−0.440077
$622$	0	0
$623$	−72.5638	−2.90721
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−56.6527	−2.26249
$628$	0	0
$629$	−9.90053	−0.394760
$630$	0	0
$631$	1.41490	0.0563261	0.0281631	−	0.999603i	$-0.491034\pi$
$631$	1.41490	0.0563261	0.0281631	+	0.999603i	$0.491034\pi$
$632$	0	0
$633$	21.3454	0.848405
$634$	0	0
$635$	−19.5650	−0.776412
$636$	0	0
$637$	−50.3595	−1.99532
$638$	0	0
$639$	20.5511	0.812991
$640$	0	0
$641$	−24.6339	−0.972980	−0.486490	−	0.873686i	$-0.661723\pi$
$641$	−24.6339	−0.972980	−0.486490	+	0.873686i	$0.661723\pi$
$642$	0	0
$643$	−0.792865	−0.0312676	−0.0156338	−	0.999878i	$-0.504977\pi$
$643$	−0.792865	−0.0312676	−0.0156338	+	0.999878i	$0.504977\pi$
$644$	0	0
$645$	−2.39059	−0.0941295
$646$	0	0
$647$	9.35485	0.367777	0.183889	−	0.982947i	$-0.441131\pi$
$647$	9.35485	0.367777	0.183889	+	0.982947i	$0.441131\pi$
$648$	0	0
$649$	26.0294	1.02174
$650$	0	0
$651$	−84.1501	−3.29810
$652$	0	0
$653$	14.1527	0.553837	0.276918	−	0.960893i	$-0.410687\pi$
$653$	14.1527	0.553837	0.276918	+	0.960893i	$0.410687\pi$
$654$	0	0
$655$	10.5908	0.413816
$656$	0	0
$657$	−44.2657	−1.72697
$658$	0	0
$659$	34.2505	1.33421	0.667104	−	0.744964i	$-0.267533\pi$
$659$	34.2505	1.33421	0.667104	+	0.744964i	$0.267533\pi$
$660$	0	0
$661$	13.7314	0.534090	0.267045	−	0.963684i	$-0.413953\pi$
$661$	13.7314	0.534090	0.267045	+	0.963684i	$0.413953\pi$
$662$	0	0
$663$	9.59417	0.372607
$664$	0	0
$665$	18.8278	0.730109
$666$	0	0
$667$	−2.45102	−0.0949040
$668$	0	0
$669$	74.2746	2.87162
$670$	0	0
$671$	−22.2589	−0.859296
$672$	0	0
$673$	−35.9862	−1.38716	−0.693582	−	0.720377i	$-0.743969\pi$
$673$	−35.9862	−1.38716	−0.693582	+	0.720377i	$0.743969\pi$
$674$	0	0
$675$	12.7502	0.490754
$676$	0	0
$677$	−41.8055	−1.60671	−0.803357	−	0.595497i	$-0.796955\pi$
$677$	−41.8055	−1.60671	−0.803357	+	0.595497i	$0.796955\pi$
$678$	0	0
$679$	−59.3270	−2.27676
$680$	0	0
$681$	21.9601	0.841514
$682$	0	0
$683$	−17.2585	−0.660379	−0.330190	−	0.943915i	$-0.607113\pi$
$683$	−17.2585	−0.660379	−0.330190	+	0.943915i	$0.607113\pi$
$684$	0	0
$685$	−19.8429	−0.758158
$686$	0	0
$687$	−45.8685	−1.74999
$688$	0	0
$689$	−34.4286	−1.31162
$690$	0	0
$691$	−45.4927	−1.73063	−0.865313	−	0.501232i	$-0.832880\pi$
$691$	−45.4927	−1.73063	−0.865313	+	0.501232i	$0.832880\pi$
$692$	0	0
$693$	−157.719	−5.99125
$694$	0	0
$695$	17.2215	0.653249
$696$	0	0
$697$	12.4438	0.471342
$698$	0	0
$699$	−59.6065	−2.25452
$700$	0	0
$701$	5.20380	0.196545	0.0982723	−	0.995160i	$-0.468668\pi$
$701$	5.20380	0.196545	0.0982723	+	0.995160i	$0.468668\pi$
$702$	0	0
$703$	38.3540	1.44655
$704$	0	0
$705$	−3.52218	−0.132653
$706$	0	0
$707$	50.9614	1.91660
$708$	0	0
$709$	19.8569	0.745741	0.372870	−	0.927883i	$-0.378374\pi$
$709$	19.8569	0.745741	0.372870	+	0.927883i	$0.378374\pi$
$710$	0	0
$711$	89.7373	3.36541
$712$	0	0
$713$	4.70316	0.176135
$714$	0	0
$715$	13.9934	0.523322
$716$	0	0
$717$	10.9247	0.407990
$718$	0	0
$719$	−14.7841	−0.551354	−0.275677	−	0.961250i	$-0.588902\pi$
$719$	−14.7841	−0.551354	−0.275677	+	0.961250i	$0.588902\pi$
$720$	0	0
$721$	−60.9218	−2.26885
$722$	0	0
$723$	5.28561	0.196574
$724$	0	0
$725$	2.84963	0.105832
$726$	0	0
$727$	33.4709	1.24137	0.620683	−	0.784062i	$-0.286855\pi$
$727$	33.4709	1.24137	0.620683	+	0.784062i	$0.286855\pi$
$728$	0	0
$729$	14.4471	0.535078
$730$	0	0
$731$	0.754968	0.0279235
$732$	0	0
$733$	−2.18249	−0.0806121	−0.0403060	−	0.999187i	$-0.512833\pi$
$733$	−2.18249	−0.0806121	−0.0403060	+	0.999187i	$0.512833\pi$
$734$	0	0
$735$	52.6294	1.94126
$736$	0	0
$737$	4.45802	0.164213
$738$	0	0
$739$	−17.8717	−0.657421	−0.328711	−	0.944431i	$-0.606614\pi$
$739$	−17.8717	−0.657421	−0.328711	+	0.944431i	$0.606614\pi$
$740$	0	0
$741$	−37.1671	−1.36537
$742$	0	0
$743$	−10.0969	−0.370420	−0.185210	−	0.982699i	$-0.559297\pi$
$743$	−10.0969	−0.370420	−0.185210	+	0.982699i	$0.559297\pi$
$744$	0	0
$745$	−12.4280	−0.455327
$746$	0	0
$747$	15.0993	0.552454
$748$	0	0
$749$	−0.986725	−0.0360542
$750$	0	0
$751$	11.2657	0.411090	0.205545	−	0.978648i	$-0.434103\pi$
$751$	11.2657	0.411090	0.205545	+	0.978648i	$0.434103\pi$
$752$	0	0
$753$	−46.4184	−1.69158
$754$	0	0
$755$	1.48895	0.0541884
$756$	0	0
$757$	−28.4638	−1.03453	−0.517267	−	0.855824i	$-0.673050\pi$
$757$	−28.4638	−1.03453	−0.517267	+	0.855824i	$0.673050\pi$
$758$	0	0
$759$	12.5785	0.456570
$760$	0	0
$761$	31.0603	1.12593	0.562967	−	0.826479i	$-0.309660\pi$
$761$	31.0603	1.12593	0.562967	+	0.826479i	$0.309660\pi$
$762$	0	0
$763$	−60.8110	−2.20151
$764$	0	0
$765$	−7.02660	−0.254047
$766$	0	0
$767$	17.0766	0.616601
$768$	0	0
$769$	18.1073	0.652965	0.326483	−	0.945203i	$-0.394137\pi$
$769$	18.1073	0.652965	0.326483	+	0.945203i	$0.394137\pi$
$770$	0	0
$771$	−10.6898	−0.384983
$772$	0	0
$773$	−35.5420	−1.27836	−0.639178	−	0.769059i	$-0.720725\pi$
$773$	−35.5420	−1.27836	−0.639178	+	0.769059i	$0.720725\pi$
$774$	0	0
$775$	−5.46803	−0.196417
$776$	0	0
$777$	152.364	5.46603
$778$	0	0
$779$	−48.2064	−1.72717
$780$	0	0
$781$	−13.5077	−0.483345
$782$	0	0
$783$	36.3332	1.29844
$784$	0	0
$785$	7.93055	0.283053
$786$	0	0
$787$	20.6325	0.735471	0.367735	−	0.929930i	$-0.380133\pi$
$787$	20.6325	0.735471	0.367735	+	0.929930i	$0.380133\pi$
$788$	0	0
$789$	10.9514	0.389881
$790$	0	0
$791$	−69.8214	−2.48256
$792$	0	0
$793$	−14.6030	−0.518568
$794$	0	0
$795$	35.9803	1.27609
$796$	0	0
$797$	49.5077	1.75365	0.876826	−	0.480807i	$-0.159656\pi$
$797$	49.5077	1.75365	0.876826	+	0.480807i	$0.159656\pi$
$798$	0	0
$799$	1.11233	0.0393515
$800$	0	0
$801$	−104.910	−3.70682
$802$	0	0
$803$	29.0947	1.02673
$804$	0	0
$805$	−4.18029	−0.147336
$806$	0	0
$807$	−63.3548	−2.23019
$808$	0	0
$809$	−25.6858	−0.903065	−0.451532	−	0.892255i	$-0.649122\pi$
$809$	−25.6858	−0.903065	−0.451532	+	0.892255i	$0.649122\pi$
$810$	0	0
$811$	32.4859	1.14074	0.570368	−	0.821390i	$-0.306801\pi$
$811$	32.4859	1.14074	0.570368	+	0.821390i	$0.306801\pi$
$812$	0	0
$813$	16.1064	0.564876
$814$	0	0
$815$	−14.4809	−0.507244
$816$	0	0
$817$	−2.92469	−0.102322
$818$	0	0
$819$	−103.472	−3.61560
$820$	0	0
$821$	−12.1049	−0.422463	−0.211232	−	0.977436i	$-0.567747\pi$
$821$	−12.1049	−0.422463	−0.211232	+	0.977436i	$0.567747\pi$
$822$	0	0
$823$	31.4212	1.09527	0.547637	−	0.836716i	$-0.315528\pi$
$823$	31.4212	1.09527	0.547637	+	0.836716i	$0.315528\pi$
$824$	0	0
$825$	−14.6241	−0.509145
$826$	0	0
$827$	−40.5617	−1.41047	−0.705234	−	0.708974i	$-0.749158\pi$
$827$	−40.5617	−1.41047	−0.705234	+	0.708974i	$0.749158\pi$
$828$	0	0
$829$	−11.3415	−0.393906	−0.196953	−	0.980413i	$-0.563105\pi$
$829$	−11.3415	−0.393906	−0.196953	+	0.980413i	$0.563105\pi$
$830$	0	0
$831$	57.6986	2.00154
$832$	0	0
$833$	−16.6208	−0.575876
$834$	0	0
$835$	−3.23492	−0.111949
$836$	0	0
$837$	−69.7182	−2.40981
$838$	0	0
$839$	−42.7694	−1.47656	−0.738282	−	0.674492i	$-0.764363\pi$
$839$	−42.7694	−1.47656	−0.738282	+	0.674492i	$0.764363\pi$
$840$	0	0
$841$	−20.8796	−0.719987
$842$	0	0
$843$	53.8165	1.85354
$844$	0	0
$845$	−3.81961	−0.131399
$846$	0	0
$847$	50.2033	1.72501
$848$	0	0
$849$	78.8698	2.70680
$850$	0	0
$851$	−8.51565	−0.291913
$852$	0	0
$853$	46.4207	1.58942	0.794708	−	0.606993i	$-0.207624\pi$
$853$	46.4207	1.58942	0.794708	+	0.606993i	$0.207624\pi$
$854$	0	0
$855$	27.2206	0.930923
$856$	0	0
$857$	−20.3176	−0.694036	−0.347018	−	0.937859i	$-0.612806\pi$
$857$	−20.3176	−0.694036	−0.347018	+	0.937859i	$0.612806\pi$
$858$	0	0
$859$	1.70566	0.0581964	0.0290982	−	0.999577i	$-0.490736\pi$
$859$	1.70566	0.0581964	0.0290982	+	0.999577i	$0.490736\pi$
$860$	0	0
$861$	−191.504	−6.52642
$862$	0	0
$863$	44.1789	1.50387	0.751933	−	0.659240i	$-0.229122\pi$
$863$	44.1789	1.50387	0.751933	+	0.659240i	$0.229122\pi$
$864$	0	0
$865$	−4.14604	−0.140970
$866$	0	0
$867$	3.16648	0.107539
$868$	0	0
$869$	−58.9820	−2.00083
$870$	0	0
$871$	2.92469	0.0990995
$872$	0	0
$873$	−85.7730	−2.90297
$874$	0	0
$875$	4.86012	0.164302
$876$	0	0
$877$	33.1568	1.11962	0.559812	−	0.828620i	$-0.310873\pi$
$877$	33.1568	1.11962	0.559812	+	0.828620i	$0.310873\pi$
$878$	0	0
$879$	38.8518	1.31044
$880$	0	0
$881$	−0.324134	−0.0109203	−0.00546017	−	0.999985i	$-0.501738\pi$
$881$	−0.324134	−0.0109203	−0.00546017	+	0.999985i	$0.501738\pi$
$882$	0	0
$883$	49.4398	1.66378	0.831892	−	0.554938i	$-0.187258\pi$
$883$	49.4398	1.66378	0.831892	+	0.554938i	$0.187258\pi$
$884$	0	0
$885$	−17.8463	−0.599897
$886$	0	0
$887$	12.6252	0.423913	0.211956	−	0.977279i	$-0.432017\pi$
$887$	12.6252	0.423913	0.211956	+	0.977279i	$0.432017\pi$
$888$	0	0
$889$	−95.0880	−3.18915
$890$	0	0
$891$	−89.1043	−2.98511
$892$	0	0
$893$	−4.30910	−0.144198
$894$	0	0
$895$	3.42912	0.114623
$896$	0	0
$897$	8.25214	0.275531
$898$	0	0
$899$	−15.5818	−0.519683
$900$	0	0
$901$	−11.3629	−0.378552
$902$	0	0
$903$	−11.6186	−0.386642
$904$	0	0
$905$	−0.684712	−0.0227606
$906$	0	0
$907$	41.8714	1.39032	0.695158	−	0.718857i	$-0.255334\pi$
$907$	41.8714	1.39032	0.695158	+	0.718857i	$0.255334\pi$
$908$	0	0
$909$	73.6784	2.44376
$910$	0	0
$911$	−10.7837	−0.357280	−0.178640	−	0.983915i	$-0.557170\pi$
$911$	−10.7837	−0.357280	−0.178640	+	0.983915i	$0.557170\pi$
$912$	0	0
$913$	−9.92438	−0.328449
$914$	0	0
$915$	15.2612	0.504520
$916$	0	0
$917$	51.4725	1.69977
$918$	0	0
$919$	44.6771	1.47376	0.736882	−	0.676022i	$-0.236298\pi$
$919$	44.6771	1.47376	0.736882	+	0.676022i	$0.236298\pi$
$920$	0	0
$921$	−88.3850	−2.91239
$922$	0	0
$923$	−8.86179	−0.291689
$924$	0	0
$925$	9.90053	0.325527
$926$	0	0
$927$	−88.0787	−2.89289
$928$	0	0
$929$	23.3248	0.765261	0.382630	−	0.923902i	$-0.375018\pi$
$929$	23.3248	0.765261	0.382630	+	0.923902i	$0.375018\pi$
$930$	0	0
$931$	64.3877	2.11022
$932$	0	0
$933$	3.90636	0.127889
$934$	0	0
$935$	4.61840	0.151038
$936$	0	0
$937$	32.5268	1.06260	0.531302	−	0.847182i	$-0.321703\pi$
$937$	32.5268	1.06260	0.531302	+	0.847182i	$0.321703\pi$
$938$	0	0
$939$	−25.9692	−0.847472
$940$	0	0
$941$	3.31221	0.107975	0.0539875	−	0.998542i	$-0.482807\pi$
$941$	3.31221	0.107975	0.0539875	+	0.998542i	$0.482807\pi$
$942$	0	0
$943$	10.7032	0.348543
$944$	0	0
$945$	61.9673	2.01580
$946$	0	0
$947$	15.8666	0.515597	0.257798	−	0.966199i	$-0.417003\pi$
$947$	15.8666	0.515597	0.257798	+	0.966199i	$0.417003\pi$
$948$	0	0
$949$	19.0877	0.619612
$950$	0	0
$951$	94.5749	3.06680
$952$	0	0
$953$	29.7322	0.963122	0.481561	−	0.876413i	$-0.340070\pi$
$953$	29.7322	0.963122	0.481561	+	0.876413i	$0.340070\pi$
$954$	0	0
$955$	−0.483435	−0.0156436
$956$	0	0
$957$	−41.6732	−1.34710
$958$	0	0
$959$	−96.4389	−3.11417
$960$	0	0
$961$	−1.10065	−0.0355050
$962$	0	0
$963$	−1.42657	−0.0459707
$964$	0	0
$965$	18.3143	0.589559
$966$	0	0
$967$	53.3740	1.71639	0.858196	−	0.513323i	$-0.171586\pi$
$967$	53.3740	1.71639	0.858196	+	0.513323i	$0.171586\pi$
$968$	0	0
$969$	−12.2667	−0.394064
$970$	0	0
$971$	4.96805	0.159432	0.0797162	−	0.996818i	$-0.474599\pi$
$971$	4.96805	0.159432	0.0797162	+	0.996818i	$0.474599\pi$
$972$	0	0
$973$	83.6986	2.68325
$974$	0	0
$975$	−9.59417	−0.307259
$976$	0	0
$977$	−35.6260	−1.13978	−0.569888	−	0.821723i	$-0.693013\pi$
$977$	−35.6260	−1.13978	−0.569888	+	0.821723i	$0.693013\pi$
$978$	0	0
$979$	68.9548	2.20381
$980$	0	0
$981$	−87.9185	−2.80702
$982$	0	0
$983$	55.6536	1.77508	0.887538	−	0.460735i	$-0.152414\pi$
$983$	55.6536	1.77508	0.887538	+	0.460735i	$0.152414\pi$
$984$	0	0
$985$	−21.3410	−0.679981
$986$	0	0
$987$	−17.1182	−0.544879
$988$	0	0
$989$	0.649364	0.0206486
$990$	0	0
$991$	8.84772	0.281057	0.140528	−	0.990077i	$-0.455120\pi$
$991$	8.84772	0.281057	0.140528	+	0.990077i	$0.455120\pi$
$992$	0	0
$993$	26.2429	0.832793
$994$	0	0
$995$	18.1017	0.573864
$996$	0	0
$997$	26.1990	0.829732	0.414866	−	0.909882i	$-0.363828\pi$
$997$	26.1990	0.829732	0.414866	+	0.909882i	$0.363828\pi$
$998$	0	0
$999$	126.233	3.99385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2720.2.a.t.1.5	yes	5
4.3	odd	2		2720.2.a.s.1.1	✓	5
8.3	odd	2		5440.2.a.cf.1.5		5
8.5	even	2		5440.2.a.ce.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2720.2.a.s.1.1	✓	5	4.3	odd	2
2720.2.a.t.1.5	yes	5	1.1	even	1	trivial
5440.2.a.ce.1.1		5	8.5	even	2
5440.2.a.cf.1.5		5	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2720 = 2^{5} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2720.a (trivial)

\(f(q)\)	\(=\)	\(q+3.16648 q^{3} +1.00000 q^{5} +4.86012 q^{7} +7.02660 q^{9} -4.61840 q^{11} -3.02992 q^{13} +3.16648 q^{15} -1.00000 q^{17} +3.87393 q^{19} +15.3895 q^{21} -0.860121 q^{23} +1.00000 q^{25} +12.7502 q^{27} +2.84963 q^{29} -5.46803 q^{31} -14.6241 q^{33} +4.86012 q^{35} +9.90053 q^{37} -9.59417 q^{39} -12.4438 q^{41} -0.754968 q^{43} +7.02660 q^{45} -1.11233 q^{47} +16.6208 q^{49} -3.16648 q^{51} +11.3629 q^{53} -4.61840 q^{55} +12.2667 q^{57} -5.63601 q^{59} +4.81961 q^{61} +34.1501 q^{63} -3.02992 q^{65} -0.965273 q^{67} -2.72356 q^{69} +2.92476 q^{71} -6.29973 q^{73} +3.16648 q^{75} -22.4460 q^{77} +12.7711 q^{79} +19.2933 q^{81} +2.14888 q^{83} -1.00000 q^{85} +9.02329 q^{87} -14.9304 q^{89} -14.7258 q^{91} -17.3144 q^{93} +3.87393 q^{95} -12.2069 q^{97} -32.4517 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + 5 q^{5} + 8 q^{7} + 6 q^{9} + 2 q^{11} + q^{13} + 3 q^{15} - 5 q^{17} + 11 q^{19} + 6 q^{21} + 12 q^{23} + 5 q^{25} + 15 q^{27} + 3 q^{29} + 9 q^{31} - 16 q^{33} + 8 q^{35} + 12 q^{37}+ \cdots - 14 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 + 5 * q^5 + 8 * q^7 + 6 * q^9 + 2 * q^11 + q^13 + 3 * q^15 - 5 * q^17 + 11 * q^19 + 6 * q^21 + 12 * q^23 + 5 * q^25 + 15 * q^27 + 3 * q^29 + 9 * q^31 - 16 * q^33 + 8 * q^35 + 12 * q^37 - 7 * q^39 - 10 * q^41 + 18 * q^43 + 6 * q^45 + 7 * q^47 + 13 * q^49 - 3 * q^51 + 15 * q^53 + 2 * q^55 - 5 * q^57 + 7 * q^59 - 3 * q^61 + 46 * q^63 + q^65 + 6 * q^67 + 6 * q^69 - 7 * q^71 - 9 * q^73 + 3 * q^75 + 4 * q^77 + 18 * q^79 + q^81 + 8 * q^83 - 5 * q^85 + 3 * q^87 - 21 * q^89 + 10 * q^91 - 13 * q^93 + 11 * q^95 - 27 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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