LMFDB - Embedded newform 2720.2.a.q.1.3

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 = [4,0,3,0,4,0,6,0,5] 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:	$4$
Coefficient field:	4.4.15188.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 2 $ x^4 - x^3 - 7*x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.490689$ 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)

$f(q)$	$=$	$q+1.49069 q^{3} +1.00000 q^{5} +4.58521 q^{7} -0.777846 q^{9} -2.58521 q^{11} +2.75922 q^{13} +1.49069 q^{15} +1.00000 q^{17} -2.07590 q^{19} +6.83513 q^{21} +4.58521 q^{23} +1.00000 q^{25} -5.63159 q^{27} +4.07590 q^{29} +0.509311 q^{31} -3.85375 q^{33} +4.58521 q^{35} -0.981378 q^{37} +4.11315 q^{39} +9.13318 q^{41} +8.53707 q^{43} -0.777846 q^{45} -6.61297 q^{47} +14.0242 q^{49} +1.49069 q^{51} -13.2463 q^{53} -2.58521 q^{55} -3.09452 q^{57} +11.5943 q^{59} -6.61297 q^{61} -3.56659 q^{63} +2.75922 q^{65} +2.00000 q^{67} +6.83513 q^{69} +12.2168 q^{71} -2.11315 q^{73} +1.49069 q^{75} -11.8537 q^{77} -2.01090 q^{79} -6.06142 q^{81} +7.01862 q^{83} +1.00000 q^{85} +6.07590 q^{87} -6.33671 q^{89} +12.6516 q^{91} +0.759224 q^{93} -2.07590 q^{95} +8.65022 q^{97} +2.01090 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} + 4 q^{5} + 6 q^{7} + 5 q^{9} + 2 q^{11} - 3 q^{13} + 3 q^{15} + 4 q^{17} + 7 q^{19} - 2 q^{21} + 6 q^{23} + 4 q^{25} + 9 q^{27} + q^{29} + 5 q^{31} + 8 q^{33} + 6 q^{35} + 2 q^{37} + 13 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 + 4 * q^5 + 6 * q^7 + 5 * q^9 + 2 * q^11 - 3 * q^13 + 3 * q^15 + 4 * q^17 + 7 * q^19 - 2 * q^21 + 6 * q^23 + 4 * q^25 + 9 * q^27 + q^29 + 5 * q^31 + 8 * q^33 + 6 * q^35 + 2 * q^37 + 13 * q^39 + 12 * q^43 + 5 * q^45 + 11 * q^47 + 8 * q^49 + 3 * q^51 - 13 * q^53 + 2 * q^55 - 3 * q^57 + 3 * q^59 + 11 * q^61 + 4 * q^63 - 3 * q^65 + 8 * q^67 - 2 * q^69 + 5 * q^71 - 5 * q^73 + 3 * q^75 - 24 * q^77 - 6 * q^79 + 12 * q^81 + 34 * q^83 + 4 * q^85 + 9 * q^87 + 9 * q^89 - 14 * q^91 - 11 * q^93 + 7 * q^95 + 9 * q^97 + 6 * 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
$3$	1.49069	0.860650	0.430325	−	0.902674i	$-0.358399\pi$
$3$	1.49069	0.860650	0.430325	+	0.902674i	$0.358399\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.58521	1.73305	0.866524	−	0.499136i	$-0.166349\pi$
$7$	4.58521	1.73305	0.866524	+	0.499136i	$0.166349\pi$
$8$	0	0
$9$	−0.777846	−0.259282
$10$	0	0
$11$	−2.58521	−0.779471	−0.389735	−	0.920927i	$-0.627434\pi$
$11$	−2.58521	−0.779471	−0.389735	+	0.920927i	$0.627434\pi$
$12$	0	0
$13$	2.75922	0.765271	0.382636	−	0.923899i	$-0.375016\pi$
$13$	2.75922	0.765271	0.382636	+	0.923899i	$0.375016\pi$
$14$	0	0
$15$	1.49069	0.384894
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.07590	−0.476244	−0.238122	−	0.971235i	$-0.576532\pi$
$19$	−2.07590	−0.476244	−0.238122	+	0.971235i	$0.576532\pi$
$20$	0	0
$21$	6.83513	1.49155
$22$	0	0
$23$	4.58521	0.956083	0.478041	−	0.878337i	$-0.341347\pi$
$23$	4.58521	0.956083	0.478041	+	0.878337i	$0.341347\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.63159	−1.08380
$28$	0	0
$29$	4.07590	0.756876	0.378438	−	0.925627i	$-0.376461\pi$
$29$	4.07590	0.756876	0.378438	+	0.925627i	$0.376461\pi$
$30$	0	0
$31$	0.509311	0.0914750	0.0457375	−	0.998953i	$-0.485436\pi$
$31$	0.509311	0.0914750	0.0457375	+	0.998953i	$0.485436\pi$
$32$	0	0
$33$	−3.85375	−0.670851
$34$	0	0
$35$	4.58521	0.775042
$36$	0	0
$37$	−0.981378	−0.161338	−0.0806688	−	0.996741i	$-0.525706\pi$
$37$	−0.981378	−0.161338	−0.0806688	+	0.996741i	$0.525706\pi$
$38$	0	0
$39$	4.11315	0.658630
$40$	0	0
$41$	9.13318	1.42636	0.713182	−	0.700979i	$-0.247253\pi$
$41$	9.13318	1.42636	0.713182	+	0.700979i	$0.247253\pi$
$42$	0	0
$43$	8.53707	1.30189	0.650946	−	0.759124i	$-0.274373\pi$
$43$	8.53707	1.30189	0.650946	+	0.759124i	$0.274373\pi$
$44$	0	0
$45$	−0.777846	−0.115954
$46$	0	0
$47$	−6.61297	−0.964601	−0.482301	−	0.876006i	$-0.660199\pi$
$47$	−6.61297	−0.964601	−0.482301	+	0.876006i	$0.660199\pi$
$48$	0	0
$49$	14.0242	2.00345
$50$	0	0
$51$	1.49069	0.208738
$52$	0	0
$53$	−13.2463	−1.81952	−0.909761	−	0.415132i	$-0.863736\pi$
$53$	−13.2463	−1.81952	−0.909761	+	0.415132i	$0.863736\pi$
$54$	0	0
$55$	−2.58521	−0.348590
$56$	0	0
$57$	−3.09452	−0.409880
$58$	0	0
$59$	11.5943	1.50946	0.754728	−	0.656038i	$-0.227769\pi$
$59$	11.5943	1.50946	0.754728	+	0.656038i	$0.227769\pi$
$60$	0	0
$61$	−6.61297	−0.846704	−0.423352	−	0.905965i	$-0.639147\pi$
$61$	−6.61297	−0.846704	−0.423352	+	0.905965i	$0.639147\pi$
$62$	0	0
$63$	−3.56659	−0.449348
$64$	0	0
$65$	2.75922	0.342240
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	6.83513	0.822852
$70$	0	0
$71$	12.2168	1.44987	0.724934	−	0.688818i	$-0.241870\pi$
$71$	12.2168	1.44987	0.724934	+	0.688818i	$0.241870\pi$
$72$	0	0
$73$	−2.11315	−0.247325	−0.123662	−	0.992324i	$-0.539464\pi$
$73$	−2.11315	−0.247325	−0.123662	+	0.992324i	$0.539464\pi$
$74$	0	0
$75$	1.49069	0.172130
$76$	0	0
$77$	−11.8537	−1.35086
$78$	0	0
$79$	−2.01090	−0.226244	−0.113122	−	0.993581i	$-0.536085\pi$
$79$	−2.01090	−0.226244	−0.113122	+	0.993581i	$0.536085\pi$
$80$	0	0
$81$	−6.06142	−0.673491
$82$	0	0
$83$	7.01862	0.770394	0.385197	−	0.922834i	$-0.374134\pi$
$83$	7.01862	0.770394	0.385197	+	0.922834i	$0.374134\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	6.07590	0.651405
$88$	0	0
$89$	−6.33671	−0.671690	−0.335845	−	0.941917i	$-0.609022\pi$
$89$	−6.33671	−0.671690	−0.335845	+	0.941917i	$0.609022\pi$
$90$	0	0
$91$	12.6516	1.32625
$92$	0	0
$93$	0.759224	0.0787279
$94$	0	0
$95$	−2.07590	−0.212983
$96$	0	0
$97$	8.65022	0.878296	0.439148	−	0.898415i	$-0.355280\pi$
$97$	8.65022	0.878296	0.439148	+	0.898415i	$0.355280\pi$
$98$	0	0
$99$	2.01090	0.202103
$100$	0	0
$101$	−3.31668	−0.330022	−0.165011	−	0.986292i	$-0.552766\pi$
$101$	−3.31668	−0.330022	−0.165011	+	0.986292i	$0.552766\pi$
$102$	0	0
$103$	−2.15180	−0.212023	−0.106012	−	0.994365i	$-0.533808\pi$
$103$	−2.15180	−0.212023	−0.106012	+	0.994365i	$0.533808\pi$
$104$	0	0
$105$	6.83513	0.667040
$106$	0	0
$107$	−11.2741	−1.08991	−0.544953	−	0.838466i	$-0.683453\pi$
$107$	−11.2741	−1.08991	−0.544953	+	0.838466i	$0.683453\pi$
$108$	0	0
$109$	3.72788	0.357066	0.178533	−	0.983934i	$-0.442865\pi$
$109$	3.72788	0.357066	0.178533	+	0.983934i	$0.442865\pi$
$110$	0	0
$111$	−1.46293	−0.138855
$112$	0	0
$113$	−19.9352	−1.87535	−0.937673	−	0.347519i	$-0.887024\pi$
$113$	−19.9352	−1.87535	−0.937673	+	0.347519i	$0.887024\pi$
$114$	0	0
$115$	4.58521	0.427573
$116$	0	0
$117$	−2.14625	−0.198421
$118$	0	0
$119$	4.58521	0.420326
$120$	0	0
$121$	−4.31668	−0.392425
$122$	0	0
$123$	13.6147	1.22760
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.03866	0.535844	0.267922	−	0.963441i	$-0.413663\pi$
$127$	6.03866	0.535844	0.267922	+	0.963441i	$0.413663\pi$
$128$	0	0
$129$	12.7261	1.12047
$130$	0	0
$131$	−19.2368	−1.68073	−0.840365	−	0.542020i	$-0.817660\pi$
$131$	−19.2368	−1.68073	−0.840365	+	0.542020i	$0.817660\pi$
$132$	0	0
$133$	−9.51845	−0.825354
$134$	0	0
$135$	−5.63159	−0.484690
$136$	0	0
$137$	−7.81650	−0.667809	−0.333904	−	0.942607i	$-0.608366\pi$
$137$	−7.81650	−0.667809	−0.333904	+	0.942607i	$0.608366\pi$
$138$	0	0
$139$	−3.75564	−0.318549	−0.159274	−	0.987234i	$-0.550915\pi$
$139$	−3.75564	−0.318549	−0.159274	+	0.987234i	$0.550915\pi$
$140$	0	0
$141$	−9.85788	−0.830184
$142$	0	0
$143$	−7.13318	−0.596507
$144$	0	0
$145$	4.07590	0.338485
$146$	0	0
$147$	20.9057	1.72427
$148$	0	0
$149$	−12.1890	−0.998566	−0.499283	−	0.866439i	$-0.666403\pi$
$149$	−12.1890	−0.998566	−0.499283	+	0.866439i	$0.666403\pi$
$150$	0	0
$151$	−4.72612	−0.384606	−0.192303	−	0.981336i	$-0.561596\pi$
$151$	−4.72612	−0.384606	−0.192303	+	0.981336i	$0.561596\pi$
$152$	0	0
$153$	−0.777846	−0.0628851
$154$	0	0
$155$	0.509311	0.0409088
$156$	0	0
$157$	−9.32940	−0.744567	−0.372284	−	0.928119i	$-0.621425\pi$
$157$	−9.32940	−0.744567	−0.372284	+	0.928119i	$0.621425\pi$
$158$	0	0
$159$	−19.7462	−1.56597
$160$	0	0
$161$	21.0242	1.65694
$162$	0	0
$163$	18.1999	1.42553	0.712765	−	0.701403i	$-0.247443\pi$
$163$	18.1999	1.42553	0.712765	+	0.701403i	$0.247443\pi$
$164$	0	0
$165$	−3.85375	−0.300014
$166$	0	0
$167$	15.7184	1.21633	0.608163	−	0.793812i	$-0.291907\pi$
$167$	15.7184	1.21633	0.608163	+	0.793812i	$0.291907\pi$
$168$	0	0
$169$	−5.38668	−0.414360
$170$	0	0
$171$	1.61473	0.123482
$172$	0	0
$173$	−5.42569	−0.412507	−0.206254	−	0.978499i	$-0.566127\pi$
$173$	−5.42569	−0.412507	−0.206254	+	0.978499i	$0.566127\pi$
$174$	0	0
$175$	4.58521	0.346609
$176$	0	0
$177$	17.2836	1.29911
$178$	0	0
$179$	16.2073	1.21139	0.605696	−	0.795696i	$-0.292895\pi$
$179$	16.2073	1.21139	0.605696	+	0.795696i	$0.292895\pi$
$180$	0	0
$181$	20.2446	1.50477	0.752383	−	0.658725i	$-0.228904\pi$
$181$	20.2446	1.50477	0.752383	+	0.658725i	$0.228904\pi$
$182$	0	0
$183$	−9.85788	−0.728716
$184$	0	0
$185$	−0.981378	−0.0721523
$186$	0	0
$187$	−2.58521	−0.189049
$188$	0	0
$189$	−25.8221	−1.87828
$190$	0	0
$191$	−9.07414	−0.656582	−0.328291	−	0.944577i	$-0.606473\pi$
$191$	−9.07414	−0.656582	−0.328291	+	0.944577i	$0.606473\pi$
$192$	0	0
$193$	16.9152	1.21758	0.608790	−	0.793331i	$-0.291655\pi$
$193$	16.9152	1.21758	0.608790	+	0.793331i	$0.291655\pi$
$194$	0	0
$195$	4.11315	0.294548
$196$	0	0
$197$	−12.3108	−0.877107	−0.438553	−	0.898705i	$-0.644509\pi$
$197$	−12.3108	−0.877107	−0.438553	+	0.898705i	$0.644509\pi$
$198$	0	0
$199$	24.9647	1.76970	0.884851	−	0.465874i	$-0.154260\pi$
$199$	24.9647	1.76970	0.884851	+	0.465874i	$0.154260\pi$
$200$	0	0
$201$	2.98138	0.210290
$202$	0	0
$203$	18.6889	1.31170
$204$	0	0
$205$	9.13318	0.637889
$206$	0	0
$207$	−3.56659	−0.247895
$208$	0	0
$209$	5.36665	0.371219
$210$	0	0
$211$	−0.992276	−0.0683111	−0.0341555	−	0.999417i	$-0.510874\pi$
$211$	−0.992276	−0.0683111	−0.0341555	+	0.999417i	$0.510874\pi$
$212$	0	0
$213$	18.2115	1.24783
$214$	0	0
$215$	8.53707	0.582223
$216$	0	0
$217$	2.33530	0.158530
$218$	0	0
$219$	−3.15004	−0.212860
$220$	0	0
$221$	2.75922	0.185606
$222$	0	0
$223$	19.1873	1.28488	0.642438	−	0.766338i	$-0.277923\pi$
$223$	19.1873	1.28488	0.642438	+	0.766338i	$0.277923\pi$
$224$	0	0
$225$	−0.777846	−0.0518564
$226$	0	0
$227$	−27.7571	−1.84230	−0.921150	−	0.389208i	$-0.872749\pi$
$227$	−27.7571	−1.84230	−0.921150	+	0.389208i	$0.872749\pi$
$228$	0	0
$229$	11.6948	0.772812	0.386406	−	0.922329i	$-0.373716\pi$
$229$	11.6948	0.772812	0.386406	+	0.922329i	$0.373716\pi$
$230$	0	0
$231$	−17.6703	−1.16262
$232$	0	0
$233$	17.8424	1.16890	0.584448	−	0.811431i	$-0.301311\pi$
$233$	17.8424	1.16890	0.584448	+	0.811431i	$0.301311\pi$
$234$	0	0
$235$	−6.61297	−0.431383
$236$	0	0
$237$	−2.99762	−0.194717
$238$	0	0
$239$	−14.7852	−0.956372	−0.478186	−	0.878259i	$-0.658706\pi$
$239$	−14.7852	−0.956372	−0.478186	+	0.878259i	$0.658706\pi$
$240$	0	0
$241$	−12.2446	−0.788742	−0.394371	−	0.918951i	$-0.629037\pi$
$241$	−12.2446	−0.788742	−0.394371	+	0.918951i	$0.629037\pi$
$242$	0	0
$243$	7.85910	0.504161
$244$	0	0
$245$	14.0242	0.895972
$246$	0	0
$247$	−5.72788	−0.364456
$248$	0	0
$249$	10.4626	0.663039
$250$	0	0
$251$	−5.44113	−0.343441	−0.171721	−	0.985146i	$-0.554933\pi$
$251$	−5.44113	−0.343441	−0.171721	+	0.985146i	$0.554933\pi$
$252$	0	0
$253$	−11.8537	−0.745239
$254$	0	0
$255$	1.49069	0.0933506
$256$	0	0
$257$	−26.7535	−1.66884	−0.834418	−	0.551133i	$-0.814196\pi$
$257$	−26.7535	−1.66884	−0.834418	+	0.551133i	$0.814196\pi$
$258$	0	0
$259$	−4.49983	−0.279606
$260$	0	0
$261$	−3.17042	−0.196244
$262$	0	0
$263$	7.92410	0.488621	0.244310	−	0.969697i	$-0.421438\pi$
$263$	7.92410	0.488621	0.244310	+	0.969697i	$0.421438\pi$
$264$	0	0
$265$	−13.2463	−0.813715
$266$	0	0
$267$	−9.44607	−0.578090
$268$	0	0
$269$	−24.6502	−1.50295	−0.751475	−	0.659761i	$-0.770657\pi$
$269$	−24.6502	−1.50295	−0.751475	+	0.659761i	$0.770657\pi$
$270$	0	0
$271$	0.670598	0.0407359	0.0203680	−	0.999793i	$-0.493516\pi$
$271$	0.670598	0.0407359	0.0203680	+	0.999793i	$0.493516\pi$
$272$	0	0
$273$	18.8596	1.14144
$274$	0	0
$275$	−2.58521	−0.155894
$276$	0	0
$277$	−22.0856	−1.32699	−0.663497	−	0.748179i	$-0.730929\pi$
$277$	−22.0856	−1.32699	−0.663497	+	0.748179i	$0.730929\pi$
$278$	0	0
$279$	−0.396166	−0.0237178
$280$	0	0
$281$	−2.46117	−0.146821	−0.0734105	−	0.997302i	$-0.523388\pi$
$281$	−2.46117	−0.146821	−0.0734105	+	0.997302i	$0.523388\pi$
$282$	0	0
$283$	12.4721	0.741387	0.370694	−	0.928755i	$-0.379120\pi$
$283$	12.4721	0.741387	0.370694	+	0.928755i	$0.379120\pi$
$284$	0	0
$285$	−3.09452	−0.183304
$286$	0	0
$287$	41.8776	2.47195
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.8948	0.755905
$292$	0	0
$293$	−26.3205	−1.53766	−0.768829	−	0.639454i	$-0.779160\pi$
$293$	−26.3205	−1.53766	−0.768829	+	0.639454i	$0.779160\pi$
$294$	0	0
$295$	11.5943	0.675049
$296$	0	0
$297$	14.5589	0.844791
$298$	0	0
$299$	12.6516	0.731663
$300$	0	0
$301$	39.1443	2.25624
$302$	0	0
$303$	−4.94413	−0.284033
$304$	0	0
$305$	−6.61297	−0.378658
$306$	0	0
$307$	−3.73647	−0.213251	−0.106626	−	0.994299i	$-0.534005\pi$
$307$	−3.73647	−0.213251	−0.106626	+	0.994299i	$0.534005\pi$
$308$	0	0
$309$	−3.20767	−0.182478
$310$	0	0
$311$	9.71839	0.551080	0.275540	−	0.961290i	$-0.411143\pi$
$311$	9.71839	0.551080	0.275540	+	0.961290i	$0.411143\pi$
$312$	0	0
$313$	27.9521	1.57994	0.789972	−	0.613143i	$-0.210095\pi$
$313$	27.9521	1.57994	0.789972	+	0.613143i	$0.210095\pi$
$314$	0	0
$315$	−3.56659	−0.200955
$316$	0	0
$317$	−16.7852	−0.942748	−0.471374	−	0.881933i	$-0.656242\pi$
$317$	−16.7852	−0.942748	−0.471374	+	0.881933i	$0.656242\pi$
$318$	0	0
$319$	−10.5371	−0.589963
$320$	0	0
$321$	−16.8062	−0.938028
$322$	0	0
$323$	−2.07590	−0.115506
$324$	0	0
$325$	2.75922	0.153054
$326$	0	0
$327$	5.55711	0.307309
$328$	0	0
$329$	−30.3219	−1.67170
$330$	0	0
$331$	−25.9352	−1.42553	−0.712764	−	0.701404i	$-0.752557\pi$
$331$	−25.9352	−1.42553	−0.712764	+	0.701404i	$0.752557\pi$
$332$	0	0
$333$	0.763361	0.0418319
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	24.1314	1.31452	0.657261	−	0.753663i	$-0.271715\pi$
$337$	24.1314	1.31452	0.657261	+	0.753663i	$0.271715\pi$
$338$	0	0
$339$	−29.7172	−1.61402
$340$	0	0
$341$	−1.31668	−0.0713021
$342$	0	0
$343$	32.2073	1.73903
$344$	0	0
$345$	6.83513	0.367991
$346$	0	0
$347$	−11.6797	−0.627001	−0.313501	−	0.949588i	$-0.601502\pi$
$347$	−11.6797	−0.627001	−0.313501	+	0.949588i	$0.601502\pi$
$348$	0	0
$349$	−25.3813	−1.35863	−0.679314	−	0.733848i	$-0.737723\pi$
$349$	−25.3813	−1.35863	−0.679314	+	0.733848i	$0.737723\pi$
$350$	0	0
$351$	−15.5388	−0.829401
$352$	0	0
$353$	15.9628	0.849612	0.424806	−	0.905284i	$-0.360342\pi$
$353$	15.9628	0.849612	0.424806	+	0.905284i	$0.360342\pi$
$354$	0	0
$355$	12.2168	0.648401
$356$	0	0
$357$	6.83513	0.361753
$358$	0	0
$359$	7.57749	0.399925	0.199962	−	0.979804i	$-0.435918\pi$
$359$	7.57749	0.399925	0.199962	+	0.979804i	$0.435918\pi$
$360$	0	0
$361$	−14.6906	−0.773191
$362$	0	0
$363$	−6.43482	−0.337741
$364$	0	0
$365$	−2.11315	−0.110607
$366$	0	0
$367$	−8.48893	−0.443118	−0.221559	−	0.975147i	$-0.571115\pi$
$367$	−8.48893	−0.443118	−0.221559	+	0.975147i	$0.571115\pi$
$368$	0	0
$369$	−7.10421	−0.369830
$370$	0	0
$371$	−60.7372	−3.15332
$372$	0	0
$373$	−4.58704	−0.237508	−0.118754	−	0.992924i	$-0.537890\pi$
$373$	−4.58704	−0.237508	−0.118754	+	0.992924i	$0.537890\pi$
$374$	0	0
$375$	1.49069	0.0769788
$376$	0	0
$377$	11.2463	0.579215
$378$	0	0
$379$	37.3296	1.91749	0.958747	−	0.284262i	$-0.0917487\pi$
$379$	37.3296	1.91749	0.958747	+	0.284262i	$0.0917487\pi$
$380$	0	0
$381$	9.00176	0.461174
$382$	0	0
$383$	0.746499	0.0381443	0.0190722	−	0.999818i	$-0.493929\pi$
$383$	0.746499	0.0381443	0.0190722	+	0.999818i	$0.493929\pi$
$384$	0	0
$385$	−11.8537	−0.604123
$386$	0	0
$387$	−6.64053	−0.337557
$388$	0	0
$389$	−14.7979	−0.750282	−0.375141	−	0.926968i	$-0.622406\pi$
$389$	−14.7979	−0.750282	−0.375141	+	0.926968i	$0.622406\pi$
$390$	0	0
$391$	4.58521	0.231884
$392$	0	0
$393$	−28.6761	−1.44652
$394$	0	0
$395$	−2.01090	−0.101179
$396$	0	0
$397$	−30.4554	−1.52851	−0.764257	−	0.644912i	$-0.776894\pi$
$397$	−30.4554	−1.52851	−0.764257	+	0.644912i	$0.776894\pi$
$398$	0	0
$399$	−14.1890	−0.710341
$400$	0	0
$401$	37.5668	1.87600	0.937998	−	0.346640i	$-0.112678\pi$
$401$	37.5668	1.87600	0.937998	+	0.346640i	$0.112678\pi$
$402$	0	0
$403$	1.40530	0.0700031
$404$	0	0
$405$	−6.06142	−0.301194
$406$	0	0
$407$	2.53707	0.125758
$408$	0	0
$409$	6.76477	0.334497	0.167248	−	0.985915i	$-0.446512\pi$
$409$	6.76477	0.334497	0.167248	+	0.985915i	$0.446512\pi$
$410$	0	0
$411$	−11.6520	−0.574750
$412$	0	0
$413$	53.1626	2.61596
$414$	0	0
$415$	7.01862	0.344531
$416$	0	0
$417$	−5.59849	−0.274159
$418$	0	0
$419$	−7.72557	−0.377419	−0.188709	−	0.982033i	$-0.560430\pi$
$419$	−7.72557	−0.377419	−0.188709	+	0.982033i	$0.560430\pi$
$420$	0	0
$421$	−7.84657	−0.382419	−0.191209	−	0.981549i	$-0.561241\pi$
$421$	−7.84657	−0.382419	−0.191209	+	0.981549i	$0.561241\pi$
$422$	0	0
$423$	5.14388	0.250104
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	−30.3219	−1.46738
$428$	0	0
$429$	−10.6334	−0.513383
$430$	0	0
$431$	16.5591	0.797622	0.398811	−	0.917033i	$-0.369423\pi$
$431$	16.5591	0.797622	0.398811	+	0.917033i	$0.369423\pi$
$432$	0	0
$433$	12.7051	0.610569	0.305285	−	0.952261i	$-0.401248\pi$
$433$	12.7051	0.610569	0.305285	+	0.952261i	$0.401248\pi$
$434$	0	0
$435$	6.07590	0.291317
$436$	0	0
$437$	−9.51845	−0.455329
$438$	0	0
$439$	−20.7335	−0.989556	−0.494778	−	0.869019i	$-0.664751\pi$
$439$	−20.7335	−0.989556	−0.494778	+	0.869019i	$0.664751\pi$
$440$	0	0
$441$	−10.9087	−0.519460
$442$	0	0
$443$	10.7479	0.510649	0.255324	−	0.966855i	$-0.417818\pi$
$443$	10.7479	0.510649	0.255324	+	0.966855i	$0.417818\pi$
$444$	0	0
$445$	−6.33671	−0.300389
$446$	0	0
$447$	−18.1701	−0.859415
$448$	0	0
$449$	2.34802	0.110810	0.0554050	−	0.998464i	$-0.482355\pi$
$449$	2.34802	0.110810	0.0554050	+	0.998464i	$0.482355\pi$
$450$	0	0
$451$	−23.6112	−1.11181
$452$	0	0
$453$	−7.04517	−0.331011
$454$	0	0
$455$	12.6516	0.593118
$456$	0	0
$457$	−36.9017	−1.72619	−0.863095	−	0.505041i	$-0.831477\pi$
$457$	−36.9017	−1.72619	−0.863095	+	0.505041i	$0.831477\pi$
$458$	0	0
$459$	−5.63159	−0.262860
$460$	0	0
$461$	11.2667	0.524743	0.262371	−	0.964967i	$-0.415495\pi$
$461$	11.2667	0.524743	0.262371	+	0.964967i	$0.415495\pi$
$462$	0	0
$463$	−9.72436	−0.451929	−0.225965	−	0.974136i	$-0.572553\pi$
$463$	−9.72436	−0.451929	−0.225965	+	0.974136i	$0.572553\pi$
$464$	0	0
$465$	0.759224	0.0352082
$466$	0	0
$467$	−13.6774	−0.632916	−0.316458	−	0.948607i	$-0.602494\pi$
$467$	−13.6774	−0.632916	−0.316458	+	0.948607i	$0.602494\pi$
$468$	0	0
$469$	9.17042	0.423451
$470$	0	0
$471$	−13.9072	−0.640812
$472$	0	0
$473$	−22.0701	−1.01479
$474$	0	0
$475$	−2.07590	−0.0952489
$476$	0	0
$477$	10.3036	0.471770
$478$	0	0
$479$	−6.45027	−0.294720	−0.147360	−	0.989083i	$-0.547078\pi$
$479$	−6.45027	−0.294720	−0.147360	+	0.989083i	$0.547078\pi$
$480$	0	0
$481$	−2.70784	−0.123467
$482$	0	0
$483$	31.3405	1.42604
$484$	0	0
$485$	8.65022	0.392786
$486$	0	0
$487$	−36.4746	−1.65282	−0.826410	−	0.563068i	$-0.809621\pi$
$487$	−36.4746	−1.65282	−0.826410	+	0.563068i	$0.809621\pi$
$488$	0	0
$489$	27.1305	1.22688
$490$	0	0
$491$	3.89513	0.175785	0.0878923	−	0.996130i	$-0.471987\pi$
$491$	3.89513	0.175785	0.0878923	+	0.996130i	$0.471987\pi$
$492$	0	0
$493$	4.07590	0.183569
$494$	0	0
$495$	2.01090	0.0903831
$496$	0	0
$497$	56.0167	2.51269
$498$	0	0
$499$	−27.2150	−1.21831	−0.609156	−	0.793050i	$-0.708492\pi$
$499$	−27.2150	−1.21831	−0.609156	+	0.793050i	$0.708492\pi$
$500$	0	0
$501$	23.4312	1.04683
$502$	0	0
$503$	−17.9819	−0.801775	−0.400887	−	0.916127i	$-0.631298\pi$
$503$	−17.9819	−0.801775	−0.400887	+	0.916127i	$0.631298\pi$
$504$	0	0
$505$	−3.31668	−0.147590
$506$	0	0
$507$	−8.02987	−0.356619
$508$	0	0
$509$	−8.77406	−0.388903	−0.194452	−	0.980912i	$-0.562293\pi$
$509$	−8.77406	−0.388903	−0.194452	+	0.980912i	$0.562293\pi$
$510$	0	0
$511$	−9.68922	−0.428626
$512$	0	0
$513$	11.6906	0.516154
$514$	0	0
$515$	−2.15180	−0.0948198
$516$	0	0
$517$	17.0959	0.751878
$518$	0	0
$519$	−8.08801	−0.355024
$520$	0	0
$521$	24.7963	1.08634	0.543172	−	0.839622i	$-0.317223\pi$
$521$	24.7963	1.08634	0.543172	+	0.839622i	$0.317223\pi$
$522$	0	0
$523$	−10.9997	−0.480981	−0.240491	−	0.970651i	$-0.577308\pi$
$523$	−10.9997	−0.480981	−0.240491	+	0.970651i	$0.577308\pi$
$524$	0	0
$525$	6.83513	0.298309
$526$	0	0
$527$	0.509311	0.0221859
$528$	0	0
$529$	−1.97583	−0.0859055
$530$	0	0
$531$	−9.01862	−0.391375
$532$	0	0
$533$	25.2005	1.09155
$534$	0	0
$535$	−11.2741	−0.487421
$536$	0	0
$537$	24.1601	1.04258
$538$	0	0
$539$	−36.2555	−1.56163
$540$	0	0
$541$	2.79233	0.120052	0.0600259	−	0.998197i	$-0.480882\pi$
$541$	2.79233	0.120052	0.0600259	+	0.998197i	$0.480882\pi$
$542$	0	0
$543$	30.1784	1.29508
$544$	0	0
$545$	3.72788	0.159685
$546$	0	0
$547$	−15.1309	−0.646949	−0.323475	−	0.946237i	$-0.604851\pi$
$547$	−15.1309	−0.646949	−0.323475	+	0.946237i	$0.604851\pi$
$548$	0	0
$549$	5.14388	0.219535
$550$	0	0
$551$	−8.46117	−0.360458
$552$	0	0
$553$	−9.22039	−0.392091
$554$	0	0
$555$	−1.46293	−0.0620979
$556$	0	0
$557$	1.02559	0.0434555	0.0217277	−	0.999764i	$-0.493083\pi$
$557$	1.02559	0.0434555	0.0217277	+	0.999764i	$0.493083\pi$
$558$	0	0
$559$	23.5557	0.996300
$560$	0	0
$561$	−3.85375	−0.162705
$562$	0	0
$563$	−31.7776	−1.33927	−0.669634	−	0.742691i	$-0.733549\pi$
$563$	−31.7776	−1.33927	−0.669634	+	0.742691i	$0.733549\pi$
$564$	0	0
$565$	−19.9352	−0.838680
$566$	0	0
$567$	−27.7929	−1.16719
$568$	0	0
$569$	18.6130	0.780296	0.390148	−	0.920752i	$-0.372424\pi$
$569$	18.6130	0.780296	0.390148	+	0.920752i	$0.372424\pi$
$570$	0	0
$571$	33.1996	1.38936	0.694680	−	0.719319i	$-0.255546\pi$
$571$	33.1996	1.38936	0.694680	+	0.719319i	$0.255546\pi$
$572$	0	0
$573$	−13.5267	−0.565087
$574$	0	0
$575$	4.58521	0.191217
$576$	0	0
$577$	−7.59021	−0.315985	−0.157992	−	0.987440i	$-0.550502\pi$
$577$	−7.59021	−0.315985	−0.157992	+	0.987440i	$0.550502\pi$
$578$	0	0
$579$	25.2152	1.04791
$580$	0	0
$581$	32.1819	1.33513
$582$	0	0
$583$	34.2446	1.41827
$584$	0	0
$585$	−2.14625	−0.0887366
$586$	0	0
$587$	1.80378	0.0744499	0.0372250	−	0.999307i	$-0.488148\pi$
$587$	1.80378	0.0744499	0.0372250	+	0.999307i	$0.488148\pi$
$588$	0	0
$589$	−1.05728	−0.0435644
$590$	0	0
$591$	−18.3515	−0.754882
$592$	0	0
$593$	40.5264	1.66422	0.832109	−	0.554612i	$-0.187133\pi$
$593$	40.5264	1.66422	0.832109	+	0.554612i	$0.187133\pi$
$594$	0	0
$595$	4.58521	0.187975
$596$	0	0
$597$	37.2146	1.52309
$598$	0	0
$599$	−26.0146	−1.06293	−0.531464	−	0.847081i	$-0.678358\pi$
$599$	−26.0146	−1.06293	−0.531464	+	0.847081i	$0.678358\pi$
$600$	0	0
$601$	−38.0555	−1.55232	−0.776159	−	0.630538i	$-0.782834\pi$
$601$	−38.0555	−1.55232	−0.776159	+	0.630538i	$0.782834\pi$
$602$	0	0
$603$	−1.55569	−0.0633527
$604$	0	0
$605$	−4.31668	−0.175498
$606$	0	0
$607$	23.9475	0.971999	0.486000	−	0.873959i	$-0.338456\pi$
$607$	23.9475	0.971999	0.486000	+	0.873959i	$0.338456\pi$
$608$	0	0
$609$	27.8593	1.12892
$610$	0	0
$611$	−18.2467	−0.738181
$612$	0	0
$613$	21.3208	0.861140	0.430570	−	0.902557i	$-0.358313\pi$
$613$	21.3208	0.861140	0.430570	+	0.902557i	$0.358313\pi$
$614$	0	0
$615$	13.6147	0.548999
$616$	0	0
$617$	7.40530	0.298126	0.149063	−	0.988828i	$-0.452374\pi$
$617$	7.40530	0.298126	0.149063	+	0.988828i	$0.452374\pi$
$618$	0	0
$619$	4.41796	0.177573	0.0887864	−	0.996051i	$-0.471701\pi$
$619$	4.41796	0.177573	0.0887864	+	0.996051i	$0.471701\pi$
$620$	0	0
$621$	−25.8221	−1.03620
$622$	0	0
$623$	−29.0552	−1.16407
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	−0.981378	−0.0391301
$630$	0	0
$631$	−39.8894	−1.58797	−0.793985	−	0.607937i	$-0.791997\pi$
$631$	−39.8894	−1.58797	−0.793985	+	0.607937i	$0.791997\pi$
$632$	0	0
$633$	−1.47917	−0.0587919
$634$	0	0
$635$	6.03866	0.239637
$636$	0	0
$637$	38.6958	1.53318
$638$	0	0
$639$	−9.50280	−0.375925
$640$	0	0
$641$	27.1815	1.07361	0.536803	−	0.843708i	$-0.319632\pi$
$641$	27.1815	1.07361	0.536803	+	0.843708i	$0.319632\pi$
$642$	0	0
$643$	9.98558	0.393793	0.196896	−	0.980424i	$-0.436914\pi$
$643$	9.98558	0.393793	0.196896	+	0.980424i	$0.436914\pi$
$644$	0	0
$645$	12.7261	0.501090
$646$	0	0
$647$	−37.9506	−1.49199	−0.745997	−	0.665949i	$-0.768027\pi$
$647$	−37.9506	−1.49199	−0.745997	+	0.665949i	$0.768027\pi$
$648$	0	0
$649$	−29.9739	−1.17658
$650$	0	0
$651$	3.48120	0.136439
$652$	0	0
$653$	−25.2188	−0.986887	−0.493443	−	0.869778i	$-0.664262\pi$
$653$	−25.2188	−0.986887	−0.493443	+	0.869778i	$0.664262\pi$
$654$	0	0
$655$	−19.2368	−0.751646
$656$	0	0
$657$	1.64370	0.0641269
$658$	0	0
$659$	−14.0204	−0.546157	−0.273078	−	0.961992i	$-0.588042\pi$
$659$	−14.0204	−0.546157	−0.273078	+	0.961992i	$0.588042\pi$
$660$	0	0
$661$	−17.3068	−0.673156	−0.336578	−	0.941656i	$-0.609270\pi$
$661$	−17.3068	−0.673156	−0.336578	+	0.941656i	$0.609270\pi$
$662$	0	0
$663$	4.11315	0.159741
$664$	0	0
$665$	−9.51845	−0.369110
$666$	0	0
$667$	18.6889	0.723636
$668$	0	0
$669$	28.6023	1.10583
$670$	0	0
$671$	17.0959	0.659981
$672$	0	0
$673$	−39.3910	−1.51841	−0.759205	−	0.650852i	$-0.774412\pi$
$673$	−39.3910	−1.51841	−0.759205	+	0.650852i	$0.774412\pi$
$674$	0	0
$675$	−5.63159	−0.216760
$676$	0	0
$677$	10.5216	0.404379	0.202189	−	0.979346i	$-0.435194\pi$
$677$	10.5216	0.404379	0.202189	+	0.979346i	$0.435194\pi$
$678$	0	0
$679$	39.6631	1.52213
$680$	0	0
$681$	−41.3771	−1.58558
$682$	0	0
$683$	−3.32344	−0.127168	−0.0635839	−	0.997976i	$-0.520253\pi$
$683$	−3.32344	−0.127168	−0.0635839	+	0.997976i	$0.520253\pi$
$684$	0	0
$685$	−7.81650	−0.298653
$686$	0	0
$687$	17.4333	0.665121
$688$	0	0
$689$	−36.5496	−1.39243
$690$	0	0
$691$	34.4001	1.30864	0.654321	−	0.756217i	$-0.272955\pi$
$691$	34.4001	1.30864	0.654321	+	0.756217i	$0.272955\pi$
$692$	0	0
$693$	9.22039	0.350254
$694$	0	0
$695$	−3.75564	−0.142459
$696$	0	0
$697$	9.13318	0.345944
$698$	0	0
$699$	26.5975	1.00601
$700$	0	0
$701$	−10.1273	−0.382502	−0.191251	−	0.981541i	$-0.561254\pi$
$701$	−10.1273	−0.382502	−0.191251	+	0.981541i	$0.561254\pi$
$702$	0	0
$703$	2.03724	0.0768361
$704$	0	0
$705$	−9.85788	−0.371269
$706$	0	0
$707$	−15.2077	−0.571943
$708$	0	0
$709$	20.7719	0.780107	0.390053	−	0.920792i	$-0.372457\pi$
$709$	20.7719	0.780107	0.390053	+	0.920792i	$0.372457\pi$
$710$	0	0
$711$	1.56417	0.0586609
$712$	0	0
$713$	2.33530	0.0874576
$714$	0	0
$715$	−7.13318	−0.266766
$716$	0	0
$717$	−22.0401	−0.823102
$718$	0	0
$719$	−31.6944	−1.18200	−0.591000	−	0.806672i	$-0.701267\pi$
$719$	−31.6944	−1.18200	−0.591000	+	0.806672i	$0.701267\pi$
$720$	0	0
$721$	−9.86647	−0.367447
$722$	0	0
$723$	−18.2528	−0.678830
$724$	0	0
$725$	4.07590	0.151375
$726$	0	0
$727$	−40.6613	−1.50804	−0.754022	−	0.656849i	$-0.771889\pi$
$727$	−40.6613	−1.50804	−0.754022	+	0.656849i	$0.771889\pi$
$728$	0	0
$729$	29.8997	1.10740
$730$	0	0
$731$	8.53707	0.315755
$732$	0	0
$733$	−39.0108	−1.44090	−0.720448	−	0.693509i	$-0.756064\pi$
$733$	−39.0108	−1.44090	−0.720448	+	0.693509i	$0.756064\pi$
$734$	0	0
$735$	20.9057	0.771118
$736$	0	0
$737$	−5.17042	−0.190455
$738$	0	0
$739$	13.2618	0.487842	0.243921	−	0.969795i	$-0.421566\pi$
$739$	13.2618	0.487842	0.243921	+	0.969795i	$0.421566\pi$
$740$	0	0
$741$	−8.53848	−0.313669
$742$	0	0
$743$	9.75564	0.357900	0.178950	−	0.983858i	$-0.442730\pi$
$743$	9.75564	0.357900	0.178950	+	0.983858i	$0.442730\pi$
$744$	0	0
$745$	−12.1890	−0.446572
$746$	0	0
$747$	−5.45941	−0.199749
$748$	0	0
$749$	−51.6941	−1.88886
$750$	0	0
$751$	15.2200	0.555385	0.277692	−	0.960670i	$-0.410430\pi$
$751$	15.2200	0.555385	0.277692	+	0.960670i	$0.410430\pi$
$752$	0	0
$753$	−8.11104	−0.295583
$754$	0	0
$755$	−4.72612	−0.172001
$756$	0	0
$757$	−4.25222	−0.154550	−0.0772748	−	0.997010i	$-0.524622\pi$
$757$	−4.25222	−0.154550	−0.0772748	+	0.997010i	$0.524622\pi$
$758$	0	0
$759$	−17.6703	−0.641389
$760$	0	0
$761$	−41.6686	−1.51049	−0.755243	−	0.655445i	$-0.772481\pi$
$761$	−41.6686	−1.51049	−0.755243	+	0.655445i	$0.772481\pi$
$762$	0	0
$763$	17.0931	0.618812
$764$	0	0
$765$	−0.777846	−0.0281231
$766$	0	0
$767$	31.9914	1.15514
$768$	0	0
$769$	−42.5996	−1.53618	−0.768090	−	0.640342i	$-0.778792\pi$
$769$	−42.5996	−1.53618	−0.768090	+	0.640342i	$0.778792\pi$
$770$	0	0
$771$	−39.8811	−1.43628
$772$	0	0
$773$	−53.0943	−1.90967	−0.954835	−	0.297137i	$-0.903968\pi$
$773$	−53.0943	−1.90967	−0.954835	+	0.297137i	$0.903968\pi$
$774$	0	0
$775$	0.509311	0.0182950
$776$	0	0
$777$	−6.70784	−0.240642
$778$	0	0
$779$	−18.9596	−0.679298
$780$	0	0
$781$	−31.5830	−1.13013
$782$	0	0
$783$	−22.9538	−0.820303
$784$	0	0
$785$	−9.32940	−0.332981
$786$	0	0
$787$	54.1724	1.93104	0.965519	−	0.260334i	$-0.0838328\pi$
$787$	54.1724	1.93104	0.965519	+	0.260334i	$0.0838328\pi$
$788$	0	0
$789$	11.8124	0.420531
$790$	0	0
$791$	−91.4071	−3.25006
$792$	0	0
$793$	−18.2467	−0.647958
$794$	0	0
$795$	−19.7462	−0.700324
$796$	0	0
$797$	−3.25561	−0.115320	−0.0576598	−	0.998336i	$-0.518364\pi$
$797$	−3.25561	−0.115320	−0.0576598	+	0.998336i	$0.518364\pi$
$798$	0	0
$799$	−6.61297	−0.233950
$800$	0	0
$801$	4.92899	0.174157
$802$	0	0
$803$	5.46293	0.192783
$804$	0	0
$805$	21.0242	0.741005
$806$	0	0
$807$	−36.7458	−1.29351
$808$	0	0
$809$	−13.5149	−0.475160	−0.237580	−	0.971368i	$-0.576354\pi$
$809$	−13.5149	−0.475160	−0.237580	+	0.971368i	$0.576354\pi$
$810$	0	0
$811$	−10.2626	−0.360370	−0.180185	−	0.983633i	$-0.557670\pi$
$811$	−10.2626	−0.360370	−0.180185	+	0.983633i	$0.557670\pi$
$812$	0	0
$813$	0.999653	0.0350594
$814$	0	0
$815$	18.1999	0.637516
$816$	0	0
$817$	−17.7221	−0.620018
$818$	0	0
$819$	−9.84102	−0.343873
$820$	0	0
$821$	45.3236	1.58181	0.790903	−	0.611942i	$-0.209611\pi$
$821$	45.3236	1.58181	0.790903	+	0.611942i	$0.209611\pi$
$822$	0	0
$823$	37.7338	1.31532	0.657660	−	0.753315i	$-0.271547\pi$
$823$	37.7338	1.31532	0.657660	+	0.753315i	$0.271547\pi$
$824$	0	0
$825$	−3.85375	−0.134170
$826$	0	0
$827$	16.1627	0.562032	0.281016	−	0.959703i	$-0.409329\pi$
$827$	16.1627	0.562032	0.281016	+	0.959703i	$0.409329\pi$
$828$	0	0
$829$	30.7590	1.06830	0.534152	−	0.845388i	$-0.320631\pi$
$829$	30.7590	1.06830	0.534152	+	0.845388i	$0.320631\pi$
$830$	0	0
$831$	−32.9227	−1.14208
$832$	0	0
$833$	14.0242	0.485909
$834$	0	0
$835$	15.7184	0.543957
$836$	0	0
$837$	−2.86823	−0.0991406
$838$	0	0
$839$	−13.8760	−0.479051	−0.239526	−	0.970890i	$-0.576992\pi$
$839$	−13.8760	−0.479051	−0.239526	+	0.970890i	$0.576992\pi$
$840$	0	0
$841$	−12.3870	−0.427139
$842$	0	0
$843$	−3.66884	−0.126361
$844$	0	0
$845$	−5.38668	−0.185307
$846$	0	0
$847$	−19.7929	−0.680091
$848$	0	0
$849$	18.5920	0.638075
$850$	0	0
$851$	−4.49983	−0.154252
$852$	0	0
$853$	14.9034	0.510282	0.255141	−	0.966904i	$-0.417878\pi$
$853$	14.9034	0.510282	0.255141	+	0.966904i	$0.417878\pi$
$854$	0	0
$855$	1.61473	0.0552227
$856$	0	0
$857$	−53.9835	−1.84404	−0.922021	−	0.387139i	$-0.873463\pi$
$857$	−53.9835	−1.84404	−0.922021	+	0.387139i	$0.873463\pi$
$858$	0	0
$859$	7.50876	0.256196	0.128098	−	0.991762i	$-0.459113\pi$
$859$	7.50876	0.256196	0.128098	+	0.991762i	$0.459113\pi$
$860$	0	0
$861$	62.4264	2.12749
$862$	0	0
$863$	4.97068	0.169204	0.0846020	−	0.996415i	$-0.473038\pi$
$863$	4.97068	0.169204	0.0846020	+	0.996415i	$0.473038\pi$
$864$	0	0
$865$	−5.42569	−0.184479
$866$	0	0
$867$	1.49069	0.0506265
$868$	0	0
$869$	5.19860	0.176350
$870$	0	0
$871$	5.51845	0.186985
$872$	0	0
$873$	−6.72854	−0.227727
$874$	0	0
$875$	4.58521	0.155008
$876$	0	0
$877$	−12.9596	−0.437614	−0.218807	−	0.975768i	$-0.570217\pi$
$877$	−12.9596	−0.437614	−0.218807	+	0.975768i	$0.570217\pi$
$878$	0	0
$879$	−39.2356	−1.32338
$880$	0	0
$881$	−20.5482	−0.692285	−0.346143	−	0.938182i	$-0.612509\pi$
$881$	−20.5482	−0.692285	−0.346143	+	0.938182i	$0.612509\pi$
$882$	0	0
$883$	2.61046	0.0878489	0.0439245	−	0.999035i	$-0.486014\pi$
$883$	2.61046	0.0878489	0.0439245	+	0.999035i	$0.486014\pi$
$884$	0	0
$885$	17.2836	0.580981
$886$	0	0
$887$	−37.1488	−1.24734	−0.623668	−	0.781689i	$-0.714358\pi$
$887$	−37.1488	−1.24734	−0.623668	+	0.781689i	$0.714358\pi$
$888$	0	0
$889$	27.6885	0.928643
$890$	0	0
$891$	15.6700	0.524966
$892$	0	0
$893$	13.7279	0.459386
$894$	0	0
$895$	16.2073	0.541751
$896$	0	0
$897$	18.8596	0.629705
$898$	0	0
$899$	2.07590	0.0692352
$900$	0	0
$901$	−13.2463	−0.441299
$902$	0	0
$903$	58.3520	1.94183
$904$	0	0
$905$	20.2446	0.672952
$906$	0	0
$907$	19.8126	0.657866	0.328933	−	0.944353i	$-0.393311\pi$
$907$	19.8126	0.657866	0.328933	+	0.944353i	$0.393311\pi$
$908$	0	0
$909$	2.57986	0.0855687
$910$	0	0
$911$	−10.3034	−0.341367	−0.170683	−	0.985326i	$-0.554598\pi$
$911$	−10.3034	−0.341367	−0.170683	+	0.985326i	$0.554598\pi$
$912$	0	0
$913$	−18.1446	−0.600500
$914$	0	0
$915$	−9.85788	−0.325892
$916$	0	0
$917$	−88.2050	−2.91279
$918$	0	0
$919$	2.11456	0.0697529	0.0348764	−	0.999392i	$-0.488896\pi$
$919$	2.11456	0.0697529	0.0348764	+	0.999392i	$0.488896\pi$
$920$	0	0
$921$	−5.56991	−0.183535
$922$	0	0
$923$	33.7089	1.10954
$924$	0	0
$925$	−0.981378	−0.0322675
$926$	0	0
$927$	1.67377	0.0549739
$928$	0	0
$929$	9.43679	0.309611	0.154805	−	0.987945i	$-0.450525\pi$
$929$	9.43679	0.309611	0.154805	+	0.987945i	$0.450525\pi$
$930$	0	0
$931$	−29.1128	−0.954133
$932$	0	0
$933$	14.4871	0.474286
$934$	0	0
$935$	−2.58521	−0.0845455
$936$	0	0
$937$	38.7590	1.26620	0.633101	−	0.774069i	$-0.281782\pi$
$937$	38.7590	1.26620	0.633101	+	0.774069i	$0.281782\pi$
$938$	0	0
$939$	41.6678	1.35978
$940$	0	0
$941$	−28.0787	−0.915340	−0.457670	−	0.889122i	$-0.651316\pi$
$941$	−28.0787	−0.915340	−0.457670	+	0.889122i	$0.651316\pi$
$942$	0	0
$943$	41.8776	1.36372
$944$	0	0
$945$	−25.8221	−0.839991
$946$	0	0
$947$	−23.7353	−0.771292	−0.385646	−	0.922647i	$-0.626021\pi$
$947$	−23.7353	−0.771292	−0.385646	+	0.922647i	$0.626021\pi$
$948$	0	0
$949$	−5.83064	−0.189271
$950$	0	0
$951$	−25.0214	−0.811376
$952$	0	0
$953$	20.4899	0.663734	0.331867	−	0.943326i	$-0.392321\pi$
$953$	20.4899	0.663734	0.331867	+	0.943326i	$0.392321\pi$
$954$	0	0
$955$	−9.07414	−0.293632
$956$	0	0
$957$	−15.7075	−0.507751
$958$	0	0
$959$	−35.8403	−1.15734
$960$	0	0
$961$	−30.7406	−0.991632
$962$	0	0
$963$	8.76951	0.282593
$964$	0	0
$965$	16.9152	0.544518
$966$	0	0
$967$	28.7888	0.925786	0.462893	−	0.886414i	$-0.346811\pi$
$967$	28.7888	0.925786	0.462893	+	0.886414i	$0.346811\pi$
$968$	0	0
$969$	−3.09452	−0.0994104
$970$	0	0
$971$	51.1385	1.64111	0.820557	−	0.571565i	$-0.193664\pi$
$971$	51.1385	1.64111	0.820557	+	0.571565i	$0.193664\pi$
$972$	0	0
$973$	−17.2204	−0.552060
$974$	0	0
$975$	4.11315	0.131726
$976$	0	0
$977$	53.8103	1.72154	0.860771	−	0.508992i	$-0.169982\pi$
$977$	53.8103	1.72154	0.860771	+	0.508992i	$0.169982\pi$
$978$	0	0
$979$	16.3817	0.523563
$980$	0	0
$981$	−2.89972	−0.0925808
$982$	0	0
$983$	50.8520	1.62193	0.810963	−	0.585097i	$-0.198944\pi$
$983$	50.8520	1.62193	0.810963	+	0.585097i	$0.198944\pi$
$984$	0	0
$985$	−12.3108	−0.392254
$986$	0	0
$987$	−45.2005	−1.43875
$988$	0	0
$989$	39.1443	1.24472
$990$	0	0
$991$	−21.8426	−0.693854	−0.346927	−	0.937892i	$-0.612775\pi$
$991$	−21.8426	−0.693854	−0.346927	+	0.937892i	$0.612775\pi$
$992$	0	0
$993$	−38.6613	−1.22688
$994$	0	0
$995$	24.9647	0.791435
$996$	0	0
$997$	49.9893	1.58318	0.791589	−	0.611054i	$-0.209254\pi$
$997$	49.9893	1.58318	0.791589	+	0.611054i	$0.209254\pi$
$998$	0	0
$999$	5.52672	0.174858

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2720.2.a.q.1.3	yes	4
4.3	odd	2		2720.2.a.k.1.2	✓	4
8.3	odd	2		5440.2.a.cc.1.3		4
8.5	even	2		5440.2.a.bw.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2720.2.a.k.1.2	✓	4	4.3	odd	2
2720.2.a.q.1.3	yes	4	1.1	even	1	trivial
5440.2.a.bw.1.2		4	8.5	even	2
5440.2.a.cc.1.3		4	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)

\(f(q)\)	\(=\)	\(q+1.49069 q^{3} +1.00000 q^{5} +4.58521 q^{7} -0.777846 q^{9} -2.58521 q^{11} +2.75922 q^{13} +1.49069 q^{15} +1.00000 q^{17} -2.07590 q^{19} +6.83513 q^{21} +4.58521 q^{23} +1.00000 q^{25} -5.63159 q^{27} +4.07590 q^{29} +0.509311 q^{31} -3.85375 q^{33} +4.58521 q^{35} -0.981378 q^{37} +4.11315 q^{39} +9.13318 q^{41} +8.53707 q^{43} -0.777846 q^{45} -6.61297 q^{47} +14.0242 q^{49} +1.49069 q^{51} -13.2463 q^{53} -2.58521 q^{55} -3.09452 q^{57} +11.5943 q^{59} -6.61297 q^{61} -3.56659 q^{63} +2.75922 q^{65} +2.00000 q^{67} +6.83513 q^{69} +12.2168 q^{71} -2.11315 q^{73} +1.49069 q^{75} -11.8537 q^{77} -2.01090 q^{79} -6.06142 q^{81} +7.01862 q^{83} +1.00000 q^{85} +6.07590 q^{87} -6.33671 q^{89} +12.6516 q^{91} +0.759224 q^{93} -2.07590 q^{95} +8.65022 q^{97} +2.01090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} + 4 q^{5} + 6 q^{7} + 5 q^{9} + 2 q^{11} - 3 q^{13} + 3 q^{15} + 4 q^{17} + 7 q^{19} - 2 q^{21} + 6 q^{23} + 4 q^{25} + 9 q^{27} + q^{29} + 5 q^{31} + 8 q^{33} + 6 q^{35} + 2 q^{37} + 13 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 + 4 * q^5 + 6 * q^7 + 5 * q^9 + 2 * q^11 - 3 * q^13 + 3 * q^15 + 4 * q^17 + 7 * q^19 - 2 * q^21 + 6 * q^23 + 4 * q^25 + 9 * q^27 + q^29 + 5 * q^31 + 8 * q^33 + 6 * q^35 + 2 * q^37 + 13 * q^39 + 12 * q^43 + 5 * q^45 + 11 * q^47 + 8 * q^49 + 3 * q^51 - 13 * q^53 + 2 * q^55 - 3 * q^57 + 3 * q^59 + 11 * q^61 + 4 * q^63 - 3 * q^65 + 8 * q^67 - 2 * q^69 + 5 * q^71 - 5 * q^73 + 3 * q^75 - 24 * q^77 - 6 * q^79 + 12 * q^81 + 34 * q^83 + 4 * q^85 + 9 * q^87 + 9 * q^89 - 14 * q^91 - 11 * q^93 + 7 * q^95 + 9 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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