LMFDB - Embedded newform 5824.2.a.cc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5824,2,Mod(1,5824)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5824, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5824.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5824 = 2^{6} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5824.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-1,0,0,0,4,0,9,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.5048741372$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.183064.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 6x + 8 $ x^4 - x^3 - 10x^2 + 6x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.669601$ of defining polynomial
Character	$\chi$	$=$	5824.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.669601 q^{3} +4.30411 q^{5} +1.00000 q^{7} -2.55163 q^{9} +0.669601 q^{11} -1.00000 q^{13} +2.88203 q^{15} +6.22124 q^{17} -6.30411 q^{19} +0.669601 q^{21} -5.63451 q^{23} +13.5253 q^{25} -3.71738 q^{27} +1.42207 q^{29} +4.09167 q^{31} +0.448365 q^{33} +4.30411 q^{35} -2.66960 q^{37} -0.669601 q^{39} +7.05658 q^{41} +10.5253 q^{43} -10.9825 q^{45} +4.09167 q^{47} +1.00000 q^{49} +4.16574 q^{51} -5.42207 q^{53} +2.88203 q^{55} -4.22124 q^{57} +7.26901 q^{59} -2.28262 q^{61} -2.55163 q^{63} -4.30411 q^{65} -2.21243 q^{67} -3.77287 q^{69} +5.56044 q^{71} +11.8557 q^{73} +9.05658 q^{75} +0.669601 q^{77} +0.312910 q^{79} +5.16574 q^{81} +3.03509 q^{83} +26.7769 q^{85} +0.952222 q^{87} +4.12076 q^{89} -1.00000 q^{91} +2.73979 q^{93} -27.1336 q^{95} +7.43088 q^{97} -1.70858 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} - 4 q^{15} + 2 q^{17} - 8 q^{19} - q^{21} - 9 q^{23} + 14 q^{25} - 7 q^{27} + 4 q^{29} + 11 q^{31} + 21 q^{33} - 7 q^{37} + q^{39} + 13 q^{41} + 2 q^{43}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^7 + 9 * q^9 - q^11 - 4 * q^13 - 4 * q^15 + 2 * q^17 - 8 * q^19 - q^21 - 9 * q^23 + 14 * q^25 - 7 * q^27 + 4 * q^29 + 11 * q^31 + 21 * q^33 - 7 * q^37 + q^39 + 13 * q^41 + 2 * q^43 - 12 * q^45 + 11 * q^47 + 4 * q^49 + 28 * q^51 - 20 * q^53 - 4 * q^55 + 6 * q^57 + 2 * q^59 - 17 * q^61 + 9 * q^63 + 3 * q^67 + 27 * q^69 - 8 * q^71 + 11 * q^73 + 21 * q^75 - q^77 - 27 * q^79 + 32 * q^81 + 22 * q^83 + 8 * q^85 + 8 * q^87 + 10 * q^89 - 4 * q^91 + 27 * q^93 - 34 * q^95 + 17 * q^97 - 10 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.669601	0.386594	0.193297	−	0.981140i	$-0.438082\pi$
$3$	0.669601	0.386594	0.193297	+	0.981140i	$0.438082\pi$
$4$	0	0
$5$	4.30411	1.92486	0.962428	−	0.271538i	$-0.0875322\pi$
$5$	4.30411	1.92486	0.962428	+	0.271538i	$0.0875322\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.55163	−0.850545
$10$	0	0
$11$	0.669601	0.201892	0.100946	−	0.994892i	$-0.467813\pi$
$11$	0.669601	0.201892	0.100946	+	0.994892i	$0.467813\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.88203	0.744138
$16$	0	0
$17$	6.22124	1.50887	0.754436	−	0.656374i	$-0.227911\pi$
$17$	6.22124	1.50887	0.754436	+	0.656374i	$0.227911\pi$
$18$	0	0
$19$	−6.30411	−1.44626	−0.723131	−	0.690711i	$-0.757298\pi$
$19$	−6.30411	−1.44626	−0.723131	+	0.690711i	$0.757298\pi$
$20$	0	0
$21$	0.669601	0.146119
$22$	0	0
$23$	−5.63451	−1.17488	−0.587438	−	0.809269i	$-0.699863\pi$
$23$	−5.63451	−1.17488	−0.587438	+	0.809269i	$0.699863\pi$
$24$	0	0
$25$	13.5253	2.70507
$26$	0	0
$27$	−3.71738	−0.715410
$28$	0	0
$29$	1.42207	0.264072	0.132036	−	0.991245i	$-0.457848\pi$
$29$	1.42207	0.264072	0.132036	+	0.991245i	$0.457848\pi$
$30$	0	0
$31$	4.09167	0.734886	0.367443	−	0.930046i	$-0.380233\pi$
$31$	4.09167	0.734886	0.367443	+	0.930046i	$0.380233\pi$
$32$	0	0
$33$	0.448365	0.0780504
$34$	0	0
$35$	4.30411	0.727527
$36$	0	0
$37$	−2.66960	−0.438880	−0.219440	−	0.975626i	$-0.570423\pi$
$37$	−2.66960	−0.438880	−0.219440	+	0.975626i	$0.570423\pi$
$38$	0	0
$39$	−0.669601	−0.107222
$40$	0	0
$41$	7.05658	1.10205	0.551026	−	0.834488i	$-0.314236\pi$
$41$	7.05658	1.10205	0.551026	+	0.834488i	$0.314236\pi$
$42$	0	0
$43$	10.5253	1.60510	0.802550	−	0.596585i	$-0.203476\pi$
$43$	10.5253	1.60510	0.802550	+	0.596585i	$0.203476\pi$
$44$	0	0
$45$	−10.9825	−1.63718
$46$	0	0
$47$	4.09167	0.596832	0.298416	−	0.954436i	$-0.403542\pi$
$47$	4.09167	0.596832	0.298416	+	0.954436i	$0.403542\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.16574	0.583321
$52$	0	0
$53$	−5.42207	−0.744779	−0.372390	−	0.928077i	$-0.621461\pi$
$53$	−5.42207	−0.744779	−0.372390	+	0.928077i	$0.621461\pi$
$54$	0	0
$55$	2.88203	0.388613
$56$	0	0
$57$	−4.22124	−0.559116
$58$	0	0
$59$	7.26901	0.946345	0.473172	−	0.880970i	$-0.343109\pi$
$59$	7.26901	0.946345	0.473172	+	0.880970i	$0.343109\pi$
$60$	0	0
$61$	−2.28262	−0.292260	−0.146130	−	0.989265i	$-0.546682\pi$
$61$	−2.28262	−0.292260	−0.146130	+	0.989265i	$0.546682\pi$
$62$	0	0
$63$	−2.55163	−0.321476
$64$	0	0
$65$	−4.30411	−0.533859
$66$	0	0
$67$	−2.21243	−0.270292	−0.135146	−	0.990826i	$-0.543150\pi$
$67$	−2.21243	−0.270292	−0.135146	+	0.990826i	$0.543150\pi$
$68$	0	0
$69$	−3.77287	−0.454200
$70$	0	0
$71$	5.56044	0.659902	0.329951	−	0.943998i	$-0.392968\pi$
$71$	5.56044	0.659902	0.329951	+	0.943998i	$0.392968\pi$
$72$	0	0
$73$	11.8557	1.38761	0.693805	−	0.720163i	$-0.255933\pi$
$73$	11.8557	1.38761	0.693805	+	0.720163i	$0.255933\pi$
$74$	0	0
$75$	9.05658	1.04576
$76$	0	0
$77$	0.669601	0.0763081
$78$	0	0
$79$	0.312910	0.0352051	0.0176026	−	0.999845i	$-0.494397\pi$
$79$	0.312910	0.0352051	0.0176026	+	0.999845i	$0.494397\pi$
$80$	0	0
$81$	5.16574	0.573972
$82$	0	0
$83$	3.03509	0.333145	0.166572	−	0.986029i	$-0.446730\pi$
$83$	3.03509	0.333145	0.166572	+	0.986029i	$0.446730\pi$
$84$	0	0
$85$	26.7769	2.90436
$86$	0	0
$87$	0.952222	0.102089
$88$	0	0
$89$	4.12076	0.436800	0.218400	−	0.975859i	$-0.429916\pi$
$89$	4.12076	0.436800	0.218400	+	0.975859i	$0.429916\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.73979	0.284103
$94$	0	0
$95$	−27.1336	−2.78384
$96$	0	0
$97$	7.43088	0.754491	0.377246	−	0.926113i	$-0.376871\pi$
$97$	7.43088	0.754491	0.377246	+	0.926113i	$0.376871\pi$
$98$	0	0
$99$	−1.70858	−0.171718
$100$	0	0
$101$	11.0945	1.10394	0.551970	−	0.833864i	$-0.313876\pi$
$101$	11.0945	1.10394	0.551970	+	0.833864i	$0.313876\pi$
$102$	0	0
$103$	−4.42487	−0.435995	−0.217998	−	0.975949i	$-0.569952\pi$
$103$	−4.42487	−0.435995	−0.217998	+	0.975949i	$0.569952\pi$
$104$	0	0
$105$	2.88203	0.281258
$106$	0	0
$107$	16.6082	1.60558	0.802788	−	0.596264i	$-0.203349\pi$
$107$	16.6082	1.60558	0.802788	+	0.596264i	$0.203349\pi$
$108$	0	0
$109$	1.79637	0.172061	0.0860305	−	0.996293i	$-0.472582\pi$
$109$	1.79637	0.172061	0.0860305	+	0.996293i	$0.472582\pi$
$110$	0	0
$111$	−1.78757	−0.169668
$112$	0	0
$113$	7.63451	0.718194	0.359097	−	0.933300i	$-0.383085\pi$
$113$	7.63451	0.718194	0.359097	+	0.933300i	$0.383085\pi$
$114$	0	0
$115$	−24.2515	−2.26147
$116$	0	0
$117$	2.55163	0.235899
$118$	0	0
$119$	6.22124	0.570300
$120$	0	0
$121$	−10.5516	−0.959240
$122$	0	0
$123$	4.72509	0.426047
$124$	0	0
$125$	36.6940	3.28201
$126$	0	0
$127$	20.3432	1.80517	0.902584	−	0.430515i	$-0.141668\pi$
$127$	20.3432	1.80517	0.902584	+	0.430515i	$0.141668\pi$
$128$	0	0
$129$	7.04778	0.620522
$130$	0	0
$131$	−21.6589	−1.89235	−0.946174	−	0.323660i	$-0.895087\pi$
$131$	−21.6589	−1.89235	−0.946174	+	0.323660i	$0.895087\pi$
$132$	0	0
$133$	−6.30411	−0.546635
$134$	0	0
$135$	−16.0000	−1.37706
$136$	0	0
$137$	8.71629	0.744683	0.372341	−	0.928096i	$-0.378555\pi$
$137$	8.71629	0.744683	0.372341	+	0.928096i	$0.378555\pi$
$138$	0	0
$139$	−12.6461	−1.07263	−0.536314	−	0.844018i	$-0.680184\pi$
$139$	−12.6461	−1.07263	−0.536314	+	0.844018i	$0.680184\pi$
$140$	0	0
$141$	2.73979	0.230732
$142$	0	0
$143$	−0.669601	−0.0559948
$144$	0	0
$145$	6.12076	0.508301
$146$	0	0
$147$	0.669601	0.0552277
$148$	0	0
$149$	−15.7203	−1.28786	−0.643928	−	0.765086i	$-0.722696\pi$
$149$	−15.7203	−1.28786	−0.643928	+	0.765086i	$0.722696\pi$
$150$	0	0
$151$	2.88203	0.234537	0.117268	−	0.993100i	$-0.462586\pi$
$151$	2.88203	0.234537	0.117268	+	0.993100i	$0.462586\pi$
$152$	0	0
$153$	−15.8743	−1.28336
$154$	0	0
$155$	17.6110	1.41455
$156$	0	0
$157$	17.0419	1.36009	0.680045	−	0.733170i	$-0.261960\pi$
$157$	17.0419	1.36009	0.680045	+	0.733170i	$0.261960\pi$
$158$	0	0
$159$	−3.63062	−0.287927
$160$	0	0
$161$	−5.63451	−0.444061
$162$	0	0
$163$	−15.4524	−1.21032	−0.605161	−	0.796103i	$-0.706891\pi$
$163$	−15.4524	−1.21032	−0.605161	+	0.796103i	$0.706891\pi$
$164$	0	0
$165$	1.92981	0.150236
$166$	0	0
$167$	−22.0156	−1.70362	−0.851809	−	0.523853i	$-0.824494\pi$
$167$	−22.0156	−1.70362	−0.851809	+	0.523853i	$0.824494\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	16.0858	1.23011
$172$	0	0
$173$	−12.3344	−0.937767	−0.468883	−	0.883260i	$-0.655344\pi$
$173$	−12.3344	−0.937767	−0.468883	+	0.883260i	$0.655344\pi$
$174$	0	0
$175$	13.5253	1.02242
$176$	0	0
$177$	4.86734	0.365851
$178$	0	0
$179$	0.743671	0.0555845	0.0277923	−	0.999614i	$-0.491152\pi$
$179$	0.743671	0.0555845	0.0277923	+	0.999614i	$0.491152\pi$
$180$	0	0
$181$	13.7729	1.02373	0.511865	−	0.859066i	$-0.328955\pi$
$181$	13.7729	1.02373	0.511865	+	0.859066i	$0.328955\pi$
$182$	0	0
$183$	−1.52844	−0.112986
$184$	0	0
$185$	−11.4902	−0.844780
$186$	0	0
$187$	4.16574	0.304629
$188$	0	0
$189$	−3.71738	−0.270400
$190$	0	0
$191$	−17.5983	−1.27337	−0.636685	−	0.771124i	$-0.719695\pi$
$191$	−17.5983	−1.27337	−0.636685	+	0.771124i	$0.719695\pi$
$192$	0	0
$193$	8.66080	0.623418	0.311709	−	0.950178i	$-0.399099\pi$
$193$	8.66080	0.623418	0.311709	+	0.950178i	$0.399099\pi$
$194$	0	0
$195$	−2.88203	−0.206387
$196$	0	0
$197$	−21.6122	−1.53981	−0.769903	−	0.638161i	$-0.779696\pi$
$197$	−21.6122	−1.53981	−0.769903	+	0.638161i	$0.779696\pi$
$198$	0	0
$199$	−4.40458	−0.312233	−0.156116	−	0.987739i	$-0.549897\pi$
$199$	−4.40458	−0.312233	−0.156116	+	0.987739i	$0.549897\pi$
$200$	0	0
$201$	−1.48145	−0.104493
$202$	0	0
$203$	1.42207	0.0998100
$204$	0	0
$205$	30.3723	2.12129
$206$	0	0
$207$	14.3772	0.999285
$208$	0	0
$209$	−4.22124	−0.291989
$210$	0	0
$211$	13.7944	0.949643	0.474821	−	0.880082i	$-0.342513\pi$
$211$	13.7944	0.949643	0.474821	+	0.880082i	$0.342513\pi$
$212$	0	0
$213$	3.72327	0.255114
$214$	0	0
$215$	45.3022	3.08959
$216$	0	0
$217$	4.09167	0.277761
$218$	0	0
$219$	7.93861	0.536442
$220$	0	0
$221$	−6.22124	−0.418486
$222$	0	0
$223$	−27.5146	−1.84252	−0.921258	−	0.388951i	$-0.872838\pi$
$223$	−27.5146	−1.84252	−0.921258	+	0.388951i	$0.872838\pi$
$224$	0	0
$225$	−34.5117	−2.30078
$226$	0	0
$227$	−20.4425	−1.35681	−0.678407	−	0.734686i	$-0.737329\pi$
$227$	−20.4425	−1.35681	−0.678407	+	0.734686i	$0.737329\pi$
$228$	0	0
$229$	13.7115	0.906080	0.453040	−	0.891490i	$-0.350339\pi$
$229$	13.7115	0.906080	0.453040	+	0.891490i	$0.350339\pi$
$230$	0	0
$231$	0.448365	0.0295003
$232$	0	0
$233$	0.531237	0.0348025	0.0174013	−	0.999849i	$-0.494461\pi$
$233$	0.531237	0.0348025	0.0174013	+	0.999849i	$0.494461\pi$
$234$	0	0
$235$	17.6110	1.14882
$236$	0	0
$237$	0.209525	0.0136101
$238$	0	0
$239$	2.55563	0.165310	0.0826551	−	0.996578i	$-0.473660\pi$
$239$	2.55563	0.165310	0.0826551	+	0.996578i	$0.473660\pi$
$240$	0	0
$241$	−8.48746	−0.546725	−0.273363	−	0.961911i	$-0.588136\pi$
$241$	−8.48746	−0.546725	−0.273363	+	0.961911i	$0.588136\pi$
$242$	0	0
$243$	14.6111	0.937304
$244$	0	0
$245$	4.30411	0.274979
$246$	0	0
$247$	6.30411	0.401121
$248$	0	0
$249$	2.03230	0.128792
$250$	0	0
$251$	6.73979	0.425412	0.212706	−	0.977116i	$-0.431772\pi$
$251$	6.73979	0.425412	0.212706	+	0.977116i	$0.431772\pi$
$252$	0	0
$253$	−3.77287	−0.237198
$254$	0	0
$255$	17.9298	1.12281
$256$	0	0
$257$	−3.37709	−0.210657	−0.105328	−	0.994437i	$-0.533589\pi$
$257$	−3.37709	−0.210657	−0.105328	+	0.994437i	$0.533589\pi$
$258$	0	0
$259$	−2.66960	−0.165881
$260$	0	0
$261$	−3.62861	−0.224605
$262$	0	0
$263$	−18.7263	−1.15471	−0.577356	−	0.816492i	$-0.695916\pi$
$263$	−18.7263	−1.15471	−0.577356	+	0.816492i	$0.695916\pi$
$264$	0	0
$265$	−23.3372	−1.43359
$266$	0	0
$267$	2.75926	0.168864
$268$	0	0
$269$	−29.4289	−1.79431	−0.897155	−	0.441716i	$-0.854370\pi$
$269$	−29.4289	−1.79431	−0.897155	+	0.441716i	$0.854370\pi$
$270$	0	0
$271$	0.707487	0.0429768	0.0214884	−	0.999769i	$-0.493160\pi$
$271$	0.707487	0.0429768	0.0214884	+	0.999769i	$0.493160\pi$
$272$	0	0
$273$	−0.669601	−0.0405261
$274$	0	0
$275$	9.05658	0.546132
$276$	0	0
$277$	−18.2136	−1.09435	−0.547176	−	0.837018i	$-0.684297\pi$
$277$	−18.2136	−1.09435	−0.547176	+	0.837018i	$0.684297\pi$
$278$	0	0
$279$	−10.4405	−0.625054
$280$	0	0
$281$	26.1161	1.55795	0.778977	−	0.627052i	$-0.215739\pi$
$281$	26.1161	1.55795	0.778977	+	0.627052i	$0.215739\pi$
$282$	0	0
$283$	−31.2631	−1.85840	−0.929200	−	0.369578i	$-0.879502\pi$
$283$	−31.2631	−1.85840	−0.929200	+	0.369578i	$0.879502\pi$
$284$	0	0
$285$	−18.1687	−1.07622
$286$	0	0
$287$	7.05658	0.416537
$288$	0	0
$289$	21.7038	1.27669
$290$	0	0
$291$	4.97572	0.291682
$292$	0	0
$293$	−24.6238	−1.43854	−0.719269	−	0.694732i	$-0.755523\pi$
$293$	−24.6238	−1.43854	−0.719269	+	0.694732i	$0.755523\pi$
$294$	0	0
$295$	31.2866	1.82158
$296$	0	0
$297$	−2.48916	−0.144436
$298$	0	0
$299$	5.63451	0.325852
$300$	0	0
$301$	10.5253	0.606671
$302$	0	0
$303$	7.42886	0.426777
$304$	0	0
$305$	−9.82465	−0.562558
$306$	0	0
$307$	−31.6433	−1.80598	−0.902990	−	0.429663i	$-0.858633\pi$
$307$	−31.6433	−1.80598	−0.902990	+	0.429663i	$0.858633\pi$
$308$	0	0
$309$	−2.96289	−0.168553
$310$	0	0
$311$	−18.3520	−1.04065	−0.520323	−	0.853969i	$-0.674189\pi$
$311$	−18.3520	−1.04065	−0.520323	+	0.853969i	$0.674189\pi$
$312$	0	0
$313$	8.71629	0.492674	0.246337	−	0.969184i	$-0.420773\pi$
$313$	8.71629	0.492674	0.246337	+	0.969184i	$0.420773\pi$
$314$	0	0
$315$	−10.9825	−0.618794
$316$	0	0
$317$	13.1268	0.737273	0.368636	−	0.929574i	$-0.379825\pi$
$317$	13.1268	0.737273	0.368636	+	0.929574i	$0.379825\pi$
$318$	0	0
$319$	0.952222	0.0533142
$320$	0	0
$321$	11.1209	0.620707
$322$	0	0
$323$	−39.2193	−2.18222
$324$	0	0
$325$	−13.5253	−0.750251
$326$	0	0
$327$	1.20285	0.0665178
$328$	0	0
$329$	4.09167	0.225581
$330$	0	0
$331$	9.23993	0.507872	0.253936	−	0.967221i	$-0.418275\pi$
$331$	9.23993	0.507872	0.253936	+	0.967221i	$0.418275\pi$
$332$	0	0
$333$	6.81185	0.373287
$334$	0	0
$335$	−9.52255	−0.520273
$336$	0	0
$337$	19.1626	1.04386	0.521928	−	0.852990i	$-0.325213\pi$
$337$	19.1626	1.04386	0.521928	+	0.852990i	$0.325213\pi$
$338$	0	0
$339$	5.11207	0.277650
$340$	0	0
$341$	2.73979	0.148368
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−16.2388	−0.874270
$346$	0	0
$347$	−10.5380	−0.565711	−0.282855	−	0.959163i	$-0.591282\pi$
$347$	−10.5380	−0.565711	−0.282855	+	0.959163i	$0.591282\pi$
$348$	0	0
$349$	2.55764	0.136908	0.0684538	−	0.997654i	$-0.478193\pi$
$349$	2.55764	0.136908	0.0684538	+	0.997654i	$0.478193\pi$
$350$	0	0
$351$	3.71738	0.198419
$352$	0	0
$353$	−5.14437	−0.273807	−0.136904	−	0.990584i	$-0.543715\pi$
$353$	−5.14437	−0.273807	−0.136904	+	0.990584i	$0.543715\pi$
$354$	0	0
$355$	23.9327	1.27022
$356$	0	0
$357$	4.16574	0.220475
$358$	0	0
$359$	−2.88203	−0.152108	−0.0760540	−	0.997104i	$-0.524232\pi$
$359$	−2.88203	−0.152108	−0.0760540	+	0.997104i	$0.524232\pi$
$360$	0	0
$361$	20.7418	1.09167
$362$	0	0
$363$	−7.06538	−0.370836
$364$	0	0
$365$	51.0284	2.67095
$366$	0	0
$367$	35.3245	1.84392	0.921962	−	0.387280i	$-0.126585\pi$
$367$	35.3245	1.84392	0.921962	+	0.387280i	$0.126585\pi$
$368$	0	0
$369$	−18.0058	−0.937345
$370$	0	0
$371$	−5.42207	−0.281500
$372$	0	0
$373$	6.04297	0.312893	0.156447	−	0.987686i	$-0.449996\pi$
$373$	6.04297	0.312893	0.156447	+	0.987686i	$0.449996\pi$
$374$	0	0
$375$	24.5703	1.26881
$376$	0	0
$377$	−1.42207	−0.0732405
$378$	0	0
$379$	−10.2738	−0.527731	−0.263865	−	0.964560i	$-0.584997\pi$
$379$	−10.2738	−0.527731	−0.263865	+	0.964560i	$0.584997\pi$
$380$	0	0
$381$	13.6218	0.697867
$382$	0	0
$383$	14.8030	0.756400	0.378200	−	0.925724i	$-0.376543\pi$
$383$	14.8030	0.756400	0.378200	+	0.925724i	$0.376543\pi$
$384$	0	0
$385$	2.88203	0.146882
$386$	0	0
$387$	−26.8568	−1.36521
$388$	0	0
$389$	−8.05258	−0.408282	−0.204141	−	0.978941i	$-0.565440\pi$
$389$	−8.05258	−0.408282	−0.204141	+	0.978941i	$0.565440\pi$
$390$	0	0
$391$	−35.0536	−1.77274
$392$	0	0
$393$	−14.5028	−0.731570
$394$	0	0
$395$	1.34680	0.0677648
$396$	0	0
$397$	−29.6863	−1.48991	−0.744956	−	0.667114i	$-0.767530\pi$
$397$	−29.6863	−1.48991	−0.744956	+	0.667114i	$0.767530\pi$
$398$	0	0
$399$	−4.22124	−0.211326
$400$	0	0
$401$	−21.1638	−1.05687	−0.528436	−	0.848973i	$-0.677221\pi$
$401$	−21.1638	−1.05687	−0.528436	+	0.848973i	$0.677221\pi$
$402$	0	0
$403$	−4.09167	−0.203821
$404$	0	0
$405$	22.2339	1.10481
$406$	0	0
$407$	−1.78757	−0.0886064
$408$	0	0
$409$	−19.3022	−0.954433	−0.477216	−	0.878786i	$-0.658354\pi$
$409$	−19.3022	−0.954433	−0.477216	+	0.878786i	$0.658354\pi$
$410$	0	0
$411$	5.83643	0.287890
$412$	0	0
$413$	7.26901	0.357685
$414$	0	0
$415$	13.0634	0.641256
$416$	0	0
$417$	−8.46784	−0.414672
$418$	0	0
$419$	−28.7478	−1.40442	−0.702211	−	0.711969i	$-0.747803\pi$
$419$	−28.7478	−1.40442	−0.702211	+	0.711969i	$0.747803\pi$
$420$	0	0
$421$	−23.9036	−1.16499	−0.582496	−	0.812834i	$-0.697924\pi$
$421$	−23.9036	−1.16499	−0.582496	+	0.812834i	$0.697924\pi$
$422$	0	0
$423$	−10.4405	−0.507632
$424$	0	0
$425$	84.1443	4.08160
$426$	0	0
$427$	−2.28262	−0.110464
$428$	0	0
$429$	−0.448365	−0.0216473
$430$	0	0
$431$	−2.05258	−0.0988694	−0.0494347	−	0.998777i	$-0.515742\pi$
$431$	−2.05258	−0.0988694	−0.0494347	+	0.998777i	$0.515742\pi$
$432$	0	0
$433$	12.3723	0.594574	0.297287	−	0.954788i	$-0.403918\pi$
$433$	12.3723	0.594574	0.297287	+	0.954788i	$0.403918\pi$
$434$	0	0
$435$	4.09846	0.196506
$436$	0	0
$437$	35.5205	1.69918
$438$	0	0
$439$	−39.6063	−1.89031	−0.945153	−	0.326627i	$-0.894088\pi$
$439$	−39.6063	−1.89031	−0.945153	+	0.326627i	$0.894088\pi$
$440$	0	0
$441$	−2.55163	−0.121506
$442$	0	0
$443$	31.9075	1.51597	0.757986	−	0.652271i	$-0.226184\pi$
$443$	31.9075	1.51597	0.757986	+	0.652271i	$0.226184\pi$
$444$	0	0
$445$	17.7362	0.840776
$446$	0	0
$447$	−10.5263	−0.497878
$448$	0	0
$449$	−42.0312	−1.98357	−0.991787	−	0.127899i	$-0.959177\pi$
$449$	−42.0312	−1.98357	−0.991787	+	0.127899i	$0.959177\pi$
$450$	0	0
$451$	4.72509	0.222496
$452$	0	0
$453$	1.92981	0.0906705
$454$	0	0
$455$	−4.30411	−0.201780
$456$	0	0
$457$	−16.5001	−0.771844	−0.385922	−	0.922531i	$-0.626117\pi$
$457$	−16.5001	−0.771844	−0.385922	+	0.922531i	$0.626117\pi$
$458$	0	0
$459$	−23.1267	−1.07946
$460$	0	0
$461$	−2.34909	−0.109408	−0.0547041	−	0.998503i	$-0.517422\pi$
$461$	−2.34909	−0.109408	−0.0547041	+	0.998503i	$0.517422\pi$
$462$	0	0
$463$	−21.6210	−1.00481	−0.502407	−	0.864631i	$-0.667552\pi$
$463$	−21.6210	−1.00481	−0.502407	+	0.864631i	$0.667552\pi$
$464$	0	0
$465$	11.7923	0.546857
$466$	0	0
$467$	26.3344	1.21861	0.609305	−	0.792936i	$-0.291448\pi$
$467$	26.3344	1.21861	0.609305	+	0.792936i	$0.291448\pi$
$468$	0	0
$469$	−2.21243	−0.102161
$470$	0	0
$471$	11.4113	0.525803
$472$	0	0
$473$	7.04778	0.324057
$474$	0	0
$475$	−85.2652	−3.91224
$476$	0	0
$477$	13.8352	0.633468
$478$	0	0
$479$	10.5632	0.482646	0.241323	−	0.970445i	$-0.422419\pi$
$479$	10.5632	0.482646	0.241323	+	0.970445i	$0.422419\pi$
$480$	0	0
$481$	2.66960	0.121723
$482$	0	0
$483$	−3.77287	−0.171672
$484$	0	0
$485$	31.9833	1.45229
$486$	0	0
$487$	22.3547	1.01299	0.506494	−	0.862244i	$-0.330941\pi$
$487$	22.3547	1.01299	0.506494	+	0.862244i	$0.330941\pi$
$488$	0	0
$489$	−10.3469	−0.467904
$490$	0	0
$491$	10.4774	0.472841	0.236420	−	0.971651i	$-0.424026\pi$
$491$	10.4774	0.472841	0.236420	+	0.971651i	$0.424026\pi$
$492$	0	0
$493$	8.84706	0.398451
$494$	0	0
$495$	−7.35390	−0.330533
$496$	0	0
$497$	5.56044	0.249420
$498$	0	0
$499$	34.9191	1.56319	0.781597	−	0.623784i	$-0.214405\pi$
$499$	34.9191	1.56319	0.781597	+	0.623784i	$0.214405\pi$
$500$	0	0
$501$	−14.7417	−0.658609
$502$	0	0
$503$	35.2572	1.57204	0.786021	−	0.618200i	$-0.212138\pi$
$503$	35.2572	1.57204	0.786021	+	0.618200i	$0.212138\pi$
$504$	0	0
$505$	47.7518	2.12493
$506$	0	0
$507$	0.669601	0.0297380
$508$	0	0
$509$	−15.4832	−0.686279	−0.343139	−	0.939285i	$-0.611490\pi$
$509$	−15.4832	−0.686279	−0.343139	+	0.939285i	$0.611490\pi$
$510$	0	0
$511$	11.8557	0.524467
$512$	0	0
$513$	23.4348	1.03467
$514$	0	0
$515$	−19.0451	−0.839227
$516$	0	0
$517$	2.73979	0.120496
$518$	0	0
$519$	−8.25912	−0.362535
$520$	0	0
$521$	−14.0606	−0.616005	−0.308003	−	0.951386i	$-0.599661\pi$
$521$	−14.0606	−0.616005	−0.308003	+	0.951386i	$0.599661\pi$
$522$	0	0
$523$	−11.0216	−0.481941	−0.240971	−	0.970532i	$-0.577466\pi$
$523$	−11.0216	−0.481941	−0.240971	+	0.970532i	$0.577466\pi$
$524$	0	0
$525$	9.05658	0.395262
$526$	0	0
$527$	25.4553	1.10885
$528$	0	0
$529$	8.74767	0.380333
$530$	0	0
$531$	−18.5479	−0.804909
$532$	0	0
$533$	−7.05658	−0.305654
$534$	0	0
$535$	71.4835	3.09050
$536$	0	0
$537$	0.497962	0.0214887
$538$	0	0
$539$	0.669601	0.0288417
$540$	0	0
$541$	34.3400	1.47639	0.738196	−	0.674587i	$-0.235678\pi$
$541$	34.3400	1.47639	0.738196	+	0.674587i	$0.235678\pi$
$542$	0	0
$543$	9.22232	0.395768
$544$	0	0
$545$	7.73177	0.331192
$546$	0	0
$547$	25.8373	1.10472	0.552362	−	0.833604i	$-0.313727\pi$
$547$	25.8373	1.10472	0.552362	+	0.833604i	$0.313727\pi$
$548$	0	0
$549$	5.82441	0.248580
$550$	0	0
$551$	−8.96491	−0.381918
$552$	0	0
$553$	0.312910	0.0133063
$554$	0	0
$555$	−7.69388	−0.326587
$556$	0	0
$557$	−21.5516	−0.913172	−0.456586	−	0.889679i	$-0.650928\pi$
$557$	−21.5516	−0.913172	−0.456586	+	0.889679i	$0.650928\pi$
$558$	0	0
$559$	−10.5253	−0.445175
$560$	0	0
$561$	2.78939	0.117768
$562$	0	0
$563$	29.0742	1.22533	0.612665	−	0.790342i	$-0.290097\pi$
$563$	29.0742	1.22533	0.612665	+	0.790342i	$0.290097\pi$
$564$	0	0
$565$	32.8597	1.38242
$566$	0	0
$567$	5.16574	0.216941
$568$	0	0
$569$	−26.3559	−1.10490	−0.552448	−	0.833547i	$-0.686306\pi$
$569$	−26.3559	−1.10490	−0.552448	+	0.833547i	$0.686306\pi$
$570$	0	0
$571$	34.3420	1.43717	0.718584	−	0.695440i	$-0.244791\pi$
$571$	34.3420	1.43717	0.718584	+	0.695440i	$0.244791\pi$
$572$	0	0
$573$	−11.7839	−0.492277
$574$	0	0
$575$	−76.2086	−3.17812
$576$	0	0
$577$	3.21643	0.133902	0.0669509	−	0.997756i	$-0.478673\pi$
$577$	3.21643	0.133902	0.0669509	+	0.997756i	$0.478673\pi$
$578$	0	0
$579$	5.79928	0.241010
$580$	0	0
$581$	3.03509	0.125917
$582$	0	0
$583$	−3.63062	−0.150365
$584$	0	0
$585$	10.9825	0.454071
$586$	0	0
$587$	−14.3041	−0.590394	−0.295197	−	0.955436i	$-0.595385\pi$
$587$	−14.3041	−0.590394	−0.295197	+	0.955436i	$0.595385\pi$
$588$	0	0
$589$	−25.7944	−1.06284
$590$	0	0
$591$	−14.4716	−0.595280
$592$	0	0
$593$	6.72339	0.276096	0.138048	−	0.990426i	$-0.455917\pi$
$593$	6.72339	0.276096	0.138048	+	0.990426i	$0.455917\pi$
$594$	0	0
$595$	26.7769	1.09774
$596$	0	0
$597$	−2.94931	−0.120707
$598$	0	0
$599$	−19.2152	−0.785113	−0.392556	−	0.919728i	$-0.628409\pi$
$599$	−19.2152	−0.785113	−0.392556	+	0.919728i	$0.628409\pi$
$600$	0	0
$601$	39.4428	1.60890	0.804452	−	0.594018i	$-0.202459\pi$
$601$	39.4428	1.60890	0.804452	+	0.594018i	$0.202459\pi$
$602$	0	0
$603$	5.64532	0.229895
$604$	0	0
$605$	−45.4154	−1.84640
$606$	0	0
$607$	−4.99520	−0.202749	−0.101374	−	0.994848i	$-0.532324\pi$
$607$	−4.99520	−0.202749	−0.101374	+	0.994848i	$0.532324\pi$
$608$	0	0
$609$	0.952222	0.0385860
$610$	0	0
$611$	−4.09167	−0.165531
$612$	0	0
$613$	31.7582	1.28270	0.641350	−	0.767248i	$-0.278375\pi$
$613$	31.7582	1.28270	0.641350	+	0.767248i	$0.278375\pi$
$614$	0	0
$615$	20.3373	0.820079
$616$	0	0
$617$	35.2164	1.41776	0.708880	−	0.705329i	$-0.249201\pi$
$617$	35.2164	1.41776	0.708880	+	0.705329i	$0.249201\pi$
$618$	0	0
$619$	27.9882	1.12494	0.562471	−	0.826817i	$-0.309851\pi$
$619$	27.9882	1.12494	0.562471	+	0.826817i	$0.309851\pi$
$620$	0	0
$621$	20.9456	0.840518
$622$	0	0
$623$	4.12076	0.165095
$624$	0	0
$625$	90.3082	3.61233
$626$	0	0
$627$	−2.82654	−0.112881
$628$	0	0
$629$	−16.6082	−0.662213
$630$	0	0
$631$	−15.1638	−0.603663	−0.301832	−	0.953361i	$-0.597598\pi$
$631$	−15.1638	−0.603663	−0.301832	+	0.953361i	$0.597598\pi$
$632$	0	0
$633$	9.23671	0.367126
$634$	0	0
$635$	87.5593	3.47469
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−14.1882	−0.561277
$640$	0	0
$641$	1.35560	0.0535430	0.0267715	−	0.999642i	$-0.491477\pi$
$641$	1.35560	0.0535430	0.0267715	+	0.999642i	$0.491477\pi$
$642$	0	0
$643$	10.1308	0.399518	0.199759	−	0.979845i	$-0.435984\pi$
$643$	10.1308	0.399518	0.199759	+	0.979845i	$0.435984\pi$
$644$	0	0
$645$	30.3344	1.19442
$646$	0	0
$647$	10.4396	0.410422	0.205211	−	0.978718i	$-0.434212\pi$
$647$	10.4396	0.410422	0.205211	+	0.978718i	$0.434212\pi$
$648$	0	0
$649$	4.86734	0.191060
$650$	0	0
$651$	2.73979	0.107381
$652$	0	0
$653$	−6.56524	−0.256918	−0.128459	−	0.991715i	$-0.541003\pi$
$653$	−6.56524	−0.256918	−0.128459	+	0.991715i	$0.541003\pi$
$654$	0	0
$655$	−93.2222	−3.64249
$656$	0	0
$657$	−30.2515	−1.18022
$658$	0	0
$659$	−5.42766	−0.211432	−0.105716	−	0.994396i	$-0.533713\pi$
$659$	−5.42766	−0.211432	−0.105716	+	0.994396i	$0.533713\pi$
$660$	0	0
$661$	−4.88695	−0.190080	−0.0950402	−	0.995473i	$-0.530298\pi$
$661$	−4.88695	−0.190080	−0.0950402	+	0.995473i	$0.530298\pi$
$662$	0	0
$663$	−4.16574	−0.161784
$664$	0	0
$665$	−27.1336	−1.05219
$666$	0	0
$667$	−8.01268	−0.310252
$668$	0	0
$669$	−18.4238	−0.712306
$670$	0	0
$671$	−1.52844	−0.0590050
$672$	0	0
$673$	−33.8157	−1.30350	−0.651749	−	0.758434i	$-0.725965\pi$
$673$	−33.8157	−1.30350	−0.651749	+	0.758434i	$0.725965\pi$
$674$	0	0
$675$	−50.2788	−1.93523
$676$	0	0
$677$	−5.27223	−0.202628	−0.101314	−	0.994854i	$-0.532305\pi$
$677$	−5.27223	−0.202628	−0.101314	+	0.994854i	$0.532305\pi$
$678$	0	0
$679$	7.43088	0.285171
$680$	0	0
$681$	−13.6883	−0.524537
$682$	0	0
$683$	49.4115	1.89068	0.945339	−	0.326089i	$-0.105731\pi$
$683$	49.4115	1.89068	0.945339	+	0.326089i	$0.105731\pi$
$684$	0	0
$685$	37.5158	1.43341
$686$	0	0
$687$	9.18122	0.350285
$688$	0	0
$689$	5.42207	0.206565
$690$	0	0
$691$	−6.04499	−0.229962	−0.114981	−	0.993368i	$-0.536681\pi$
$691$	−6.04499	−0.229962	−0.114981	+	0.993368i	$0.536681\pi$
$692$	0	0
$693$	−1.70858	−0.0649035
$694$	0	0
$695$	−54.4302	−2.06465
$696$	0	0
$697$	43.9007	1.66286
$698$	0	0
$699$	0.355717	0.0134545
$700$	0	0
$701$	30.8395	1.16479	0.582395	−	0.812906i	$-0.302116\pi$
$701$	30.8395	1.16479	0.582395	+	0.812906i	$0.302116\pi$
$702$	0	0
$703$	16.8295	0.634735
$704$	0	0
$705$	11.7923	0.444125
$706$	0	0
$707$	11.0945	0.417250
$708$	0	0
$709$	6.69497	0.251435	0.125717	−	0.992066i	$-0.459877\pi$
$709$	6.69497	0.251435	0.125717	+	0.992066i	$0.459877\pi$
$710$	0	0
$711$	−0.798432	−0.0299436
$712$	0	0
$713$	−23.0546	−0.863400
$714$	0	0
$715$	−2.88203	−0.107782
$716$	0	0
$717$	1.71125	0.0639079
$718$	0	0
$719$	−3.44728	−0.128562	−0.0642808	−	0.997932i	$-0.520475\pi$
$719$	−3.44728	−0.128562	−0.0642808	+	0.997932i	$0.520475\pi$
$720$	0	0
$721$	−4.42487	−0.164791
$722$	0	0
$723$	−5.68321	−0.211361
$724$	0	0
$725$	19.2340	0.714334
$726$	0	0
$727$	−29.9826	−1.11199	−0.555997	−	0.831184i	$-0.687663\pi$
$727$	−29.9826	−1.11199	−0.555997	+	0.831184i	$0.687663\pi$
$728$	0	0
$729$	−5.71361	−0.211615
$730$	0	0
$731$	65.4806	2.42189
$732$	0	0
$733$	−18.5400	−0.684792	−0.342396	−	0.939556i	$-0.611238\pi$
$733$	−18.5400	−0.684792	−0.342396	+	0.939556i	$0.611238\pi$
$734$	0	0
$735$	2.88203	0.106305
$736$	0	0
$737$	−1.48145	−0.0545698
$738$	0	0
$739$	−17.2543	−0.634710	−0.317355	−	0.948307i	$-0.602795\pi$
$739$	−17.2543	−0.634710	−0.317355	+	0.948307i	$0.602795\pi$
$740$	0	0
$741$	4.22124	0.155071
$742$	0	0
$743$	34.7644	1.27538	0.637690	−	0.770293i	$-0.279890\pi$
$743$	34.7644	1.27538	0.637690	+	0.770293i	$0.279890\pi$
$744$	0	0
$745$	−67.6618	−2.47894
$746$	0	0
$747$	−7.74445	−0.283355
$748$	0	0
$749$	16.6082	0.606851
$750$	0	0
$751$	−33.8355	−1.23467	−0.617337	−	0.786699i	$-0.711788\pi$
$751$	−33.8355	−1.23467	−0.617337	+	0.786699i	$0.711788\pi$
$752$	0	0
$753$	4.51297	0.164462
$754$	0	0
$755$	12.4046	0.451449
$756$	0	0
$757$	27.2291	0.989659	0.494830	−	0.868990i	$-0.335231\pi$
$757$	27.2291	0.989659	0.494830	+	0.868990i	$0.335231\pi$
$758$	0	0
$759$	−2.52632	−0.0916995
$760$	0	0
$761$	−15.6724	−0.568124	−0.284062	−	0.958806i	$-0.591682\pi$
$761$	−15.6724	−0.568124	−0.284062	+	0.958806i	$0.591682\pi$
$762$	0	0
$763$	1.79637	0.0650329
$764$	0	0
$765$	−68.3248	−2.47029
$766$	0	0
$767$	−7.26901	−0.262469
$768$	0	0
$769$	−0.481563	−0.0173656	−0.00868280	−	0.999962i	$-0.502764\pi$
$769$	−0.481563	−0.0173656	−0.00868280	+	0.999962i	$0.502764\pi$
$770$	0	0
$771$	−2.26130	−0.0814388
$772$	0	0
$773$	10.7740	0.387512	0.193756	−	0.981050i	$-0.437933\pi$
$773$	10.7740	0.387512	0.193756	+	0.981050i	$0.437933\pi$
$774$	0	0
$775$	55.3413	1.98792
$776$	0	0
$777$	−1.78757	−0.0641286
$778$	0	0
$779$	−44.4854	−1.59386
$780$	0	0
$781$	3.72327	0.133229
$782$	0	0
$783$	−5.28639	−0.188920
$784$	0	0
$785$	73.3501	2.61798
$786$	0	0
$787$	19.4272	0.692503	0.346252	−	0.938142i	$-0.387454\pi$
$787$	19.4272	0.692503	0.346252	+	0.938142i	$0.387454\pi$
$788$	0	0
$789$	−12.5391	−0.446405
$790$	0	0
$791$	7.63451	0.271452
$792$	0	0
$793$	2.28262	0.0810582
$794$	0	0
$795$	−15.6266	−0.554218
$796$	0	0
$797$	48.7507	1.72684	0.863419	−	0.504487i	$-0.168319\pi$
$797$	48.7507	1.72684	0.863419	+	0.504487i	$0.168319\pi$
$798$	0	0
$799$	25.4553	0.900543
$800$	0	0
$801$	−10.5147	−0.371518
$802$	0	0
$803$	7.93861	0.280148
$804$	0	0
$805$	−24.2515	−0.854754
$806$	0	0
$807$	−19.7056	−0.693670
$808$	0	0
$809$	15.3089	0.538233	0.269116	−	0.963108i	$-0.413268\pi$
$809$	15.3089	0.538233	0.269116	+	0.963108i	$0.413268\pi$
$810$	0	0
$811$	41.0761	1.44238	0.721188	−	0.692740i	$-0.243597\pi$
$811$	41.0761	1.44238	0.721188	+	0.692740i	$0.243597\pi$
$812$	0	0
$813$	0.473734	0.0166146
$814$	0	0
$815$	−66.5086	−2.32970
$816$	0	0
$817$	−66.3529	−2.32139
$818$	0	0
$819$	2.55163	0.0891613
$820$	0	0
$821$	−37.6787	−1.31500	−0.657498	−	0.753457i	$-0.728385\pi$
$821$	−37.6787	−1.31500	−0.657498	+	0.753457i	$0.728385\pi$
$822$	0	0
$823$	−23.9941	−0.836382	−0.418191	−	0.908359i	$-0.637336\pi$
$823$	−23.9941	−0.836382	−0.418191	+	0.908359i	$0.637336\pi$
$824$	0	0
$825$	6.06429	0.211132
$826$	0	0
$827$	−46.5086	−1.61726	−0.808632	−	0.588315i	$-0.799792\pi$
$827$	−46.5086	−1.61726	−0.808632	+	0.588315i	$0.799792\pi$
$828$	0	0
$829$	−25.3851	−0.881660	−0.440830	−	0.897591i	$-0.645316\pi$
$829$	−25.3851	−0.881660	−0.440830	+	0.897591i	$0.645316\pi$
$830$	0	0
$831$	−12.1959	−0.423070
$832$	0	0
$833$	6.22124	0.215553
$834$	0	0
$835$	−94.7575	−3.27922
$836$	0	0
$837$	−15.2103	−0.525745
$838$	0	0
$839$	16.6317	0.574191	0.287095	−	0.957902i	$-0.407310\pi$
$839$	16.6317	0.574191	0.287095	+	0.957902i	$0.407310\pi$
$840$	0	0
$841$	−26.9777	−0.930266
$842$	0	0
$843$	17.4873	0.602296
$844$	0	0
$845$	4.30411	0.148066
$846$	0	0
$847$	−10.5516	−0.362558
$848$	0	0
$849$	−20.9338	−0.718446
$850$	0	0
$851$	15.0419	0.515629
$852$	0	0
$853$	−51.5261	−1.76422	−0.882110	−	0.471043i	$-0.843878\pi$
$853$	−51.5261	−1.76422	−0.882110	+	0.471043i	$0.843878\pi$
$854$	0	0
$855$	69.2349	2.36778
$856$	0	0
$857$	−7.65890	−0.261623	−0.130812	−	0.991407i	$-0.541758\pi$
$857$	−7.65890	−0.261623	−0.130812	+	0.991407i	$0.541758\pi$
$858$	0	0
$859$	−19.3413	−0.659917	−0.329958	−	0.943995i	$-0.607035\pi$
$859$	−19.3413	−0.659917	−0.329958	+	0.943995i	$0.607035\pi$
$860$	0	0
$861$	4.72509	0.161031
$862$	0	0
$863$	−4.99011	−0.169865	−0.0849326	−	0.996387i	$-0.527068\pi$
$863$	−4.99011	−0.169865	−0.0849326	+	0.996387i	$0.527068\pi$
$864$	0	0
$865$	−53.0886	−1.80507
$866$	0	0
$867$	14.5329	0.493562
$868$	0	0
$869$	0.209525	0.00710764
$870$	0	0
$871$	2.21243	0.0749654
$872$	0	0
$873$	−18.9609	−0.641729
$874$	0	0
$875$	36.6940	1.24048
$876$	0	0
$877$	−41.1201	−1.38853	−0.694263	−	0.719721i	$-0.744270\pi$
$877$	−41.1201	−1.38853	−0.694263	+	0.719721i	$0.744270\pi$
$878$	0	0
$879$	−16.4881	−0.556131
$880$	0	0
$881$	−56.8753	−1.91618	−0.958089	−	0.286470i	$-0.907518\pi$
$881$	−56.8753	−1.91618	−0.958089	+	0.286470i	$0.907518\pi$
$882$	0	0
$883$	6.41928	0.216026	0.108013	−	0.994149i	$-0.465551\pi$
$883$	6.41928	0.216026	0.108013	+	0.994149i	$0.465551\pi$
$884$	0	0
$885$	20.9495	0.704211
$886$	0	0
$887$	13.2340	0.444355	0.222178	−	0.975006i	$-0.428683\pi$
$887$	13.2340	0.444355	0.222178	+	0.975006i	$0.428683\pi$
$888$	0	0
$889$	20.3432	0.682289
$890$	0	0
$891$	3.45899	0.115880
$892$	0	0
$893$	−25.7944	−0.863175
$894$	0	0
$895$	3.20084	0.106992
$896$	0	0
$897$	3.77287	0.125972
$898$	0	0
$899$	5.81866	0.194063
$900$	0	0
$901$	−33.7320	−1.12378
$902$	0	0
$903$	7.04778	0.234535
$904$	0	0
$905$	59.2799	1.97053
$906$	0	0
$907$	10.2662	0.340884	0.170442	−	0.985368i	$-0.445480\pi$
$907$	10.2662	0.340884	0.170442	+	0.985368i	$0.445480\pi$
$908$	0	0
$909$	−28.3090	−0.938951
$910$	0	0
$911$	−21.0028	−0.695854	−0.347927	−	0.937522i	$-0.613114\pi$
$911$	−21.0028	−0.695854	−0.347927	+	0.937522i	$0.613114\pi$
$912$	0	0
$913$	2.03230	0.0672594
$914$	0	0
$915$	−6.57859	−0.217482
$916$	0	0
$917$	−21.6589	−0.715240
$918$	0	0
$919$	38.1855	1.25962	0.629811	−	0.776749i	$-0.283132\pi$
$919$	38.1855	1.25962	0.629811	+	0.776749i	$0.283132\pi$
$920$	0	0
$921$	−21.1884	−0.698181
$922$	0	0
$923$	−5.56044	−0.183024
$924$	0	0
$925$	−36.1073	−1.18720
$926$	0	0
$927$	11.2906	0.370833
$928$	0	0
$929$	38.9414	1.27763	0.638813	−	0.769362i	$-0.279426\pi$
$929$	38.9414	1.27763	0.638813	+	0.769362i	$0.279426\pi$
$930$	0	0
$931$	−6.30411	−0.206609
$932$	0	0
$933$	−12.2885	−0.402308
$934$	0	0
$935$	17.9298	0.586368
$936$	0	0
$937$	51.7398	1.69026	0.845132	−	0.534557i	$-0.179522\pi$
$937$	51.7398	1.69026	0.845132	+	0.534557i	$0.179522\pi$
$938$	0	0
$939$	5.83643	0.190465
$940$	0	0
$941$	−13.1309	−0.428055	−0.214027	−	0.976828i	$-0.568658\pi$
$941$	−13.1309	−0.428055	−0.214027	+	0.976828i	$0.568658\pi$
$942$	0	0
$943$	−39.7604	−1.29478
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	11.0411	0.358787	0.179393	−	0.983777i	$-0.442587\pi$
$947$	11.0411	0.358787	0.179393	+	0.983777i	$0.442587\pi$
$948$	0	0
$949$	−11.8557	−0.384854
$950$	0	0
$951$	8.78969	0.285025
$952$	0	0
$953$	−9.90752	−0.320936	−0.160468	−	0.987041i	$-0.551300\pi$
$953$	−9.90752	−0.320936	−0.160468	+	0.987041i	$0.551300\pi$
$954$	0	0
$955$	−75.7451	−2.45105
$956$	0	0
$957$	0.637608	0.0206110
$958$	0	0
$959$	8.71629	0.281464
$960$	0	0
$961$	−14.2582	−0.459942
$962$	0	0
$963$	−42.3781	−1.36562
$964$	0	0
$965$	37.2770	1.19999
$966$	0	0
$967$	18.7339	0.602441	0.301221	−	0.953554i	$-0.402606\pi$
$967$	18.7339	0.602441	0.301221	+	0.953554i	$0.402606\pi$
$968$	0	0
$969$	−26.2613	−0.843635
$970$	0	0
$971$	−33.6824	−1.08092	−0.540460	−	0.841370i	$-0.681750\pi$
$971$	−33.6824	−1.08092	−0.540460	+	0.841370i	$0.681750\pi$
$972$	0	0
$973$	−12.6461	−0.405415
$974$	0	0
$975$	−9.05658	−0.290043
$976$	0	0
$977$	−20.0029	−0.639950	−0.319975	−	0.947426i	$-0.603674\pi$
$977$	−20.0029	−0.639950	−0.319975	+	0.947426i	$0.603674\pi$
$978$	0	0
$979$	2.75926	0.0881864
$980$	0	0
$981$	−4.58368	−0.146346
$982$	0	0
$983$	16.6334	0.530524	0.265262	−	0.964176i	$-0.414542\pi$
$983$	16.6334	0.530524	0.265262	+	0.964176i	$0.414542\pi$
$984$	0	0
$985$	−93.0213	−2.96390
$986$	0	0
$987$	2.73979	0.0872084
$988$	0	0
$989$	−59.3051	−1.88579
$990$	0	0
$991$	−3.16198	−0.100444	−0.0502218	−	0.998738i	$-0.515993\pi$
$991$	−3.16198	−0.100444	−0.0502218	+	0.998738i	$0.515993\pi$
$992$	0	0
$993$	6.18706	0.196341
$994$	0	0
$995$	−18.9578	−0.601003
$996$	0	0
$997$	19.5420	0.618902	0.309451	−	0.950915i	$-0.399855\pi$
$997$	19.5420	0.618902	0.309451	+	0.950915i	$0.399855\pi$
$998$	0	0
$999$	9.92392	0.313979

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5824.2.a.cc.1.3		4
4.3	odd	2		5824.2.a.cf.1.2		4
8.3	odd	2		1456.2.a.u.1.3		4
8.5	even	2		728.2.a.h.1.2	✓	4
24.5	odd	2		6552.2.a.bt.1.4		4
56.13	odd	2		5096.2.a.t.1.3		4
104.77	even	2		9464.2.a.ba.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.h.1.2	✓	4	8.5	even	2
1456.2.a.u.1.3		4	8.3	odd	2
5096.2.a.t.1.3		4	56.13	odd	2
5824.2.a.cc.1.3		4	1.1	even	1	trivial
5824.2.a.cf.1.2		4	4.3	odd	2
6552.2.a.bt.1.4		4	24.5	odd	2
9464.2.a.ba.1.2		4	104.77	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5824.a (trivial)

\(f(q)\)	\(=\)	\(q+0.669601 q^{3} +4.30411 q^{5} +1.00000 q^{7} -2.55163 q^{9} +0.669601 q^{11} -1.00000 q^{13} +2.88203 q^{15} +6.22124 q^{17} -6.30411 q^{19} +0.669601 q^{21} -5.63451 q^{23} +13.5253 q^{25} -3.71738 q^{27} +1.42207 q^{29} +4.09167 q^{31} +0.448365 q^{33} +4.30411 q^{35} -2.66960 q^{37} -0.669601 q^{39} +7.05658 q^{41} +10.5253 q^{43} -10.9825 q^{45} +4.09167 q^{47} +1.00000 q^{49} +4.16574 q^{51} -5.42207 q^{53} +2.88203 q^{55} -4.22124 q^{57} +7.26901 q^{59} -2.28262 q^{61} -2.55163 q^{63} -4.30411 q^{65} -2.21243 q^{67} -3.77287 q^{69} +5.56044 q^{71} +11.8557 q^{73} +9.05658 q^{75} +0.669601 q^{77} +0.312910 q^{79} +5.16574 q^{81} +3.03509 q^{83} +26.7769 q^{85} +0.952222 q^{87} +4.12076 q^{89} -1.00000 q^{91} +2.73979 q^{93} -27.1336 q^{95} +7.43088 q^{97} -1.70858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} - 4 q^{15} + 2 q^{17} - 8 q^{19} - q^{21} - 9 q^{23} + 14 q^{25} - 7 q^{27} + 4 q^{29} + 11 q^{31} + 21 q^{33} - 7 q^{37} + q^{39} + 13 q^{41} + 2 q^{43}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^7 + 9 * q^9 - q^11 - 4 * q^13 - 4 * q^15 + 2 * q^17 - 8 * q^19 - q^21 - 9 * q^23 + 14 * q^25 - 7 * q^27 + 4 * q^29 + 11 * q^31 + 21 * q^33 - 7 * q^37 + q^39 + 13 * q^41 + 2 * q^43 - 12 * q^45 + 11 * q^47 + 4 * q^49 + 28 * q^51 - 20 * q^53 - 4 * q^55 + 6 * q^57 + 2 * q^59 - 17 * q^61 + 9 * q^63 + 3 * q^67 + 27 * q^69 - 8 * q^71 + 11 * q^73 + 21 * q^75 - q^77 - 27 * q^79 + 32 * q^81 + 22 * q^83 + 8 * q^85 + 8 * q^87 + 10 * q^89 - 4 * q^91 + 27 * q^93 - 34 * q^95 + 17 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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