LMFDB - Embedded newform 4840.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.45753625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 $ x^6 - x^5 - 13x^4 + 11x^3 + 41x^2 - 30x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87511$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.03399 q^{3} -1.00000 q^{5} -0.865927 q^{7} +6.20511 q^{9} +4.43509 q^{13} +3.03399 q^{15} +0.550031 q^{17} +0.258962 q^{19} +2.62722 q^{21} +7.92701 q^{23} +1.00000 q^{25} -9.72427 q^{27} +9.36210 q^{29} -5.31588 q^{31} +0.865927 q^{35} +6.38739 q^{37} -13.4560 q^{39} -8.74218 q^{41} -6.38622 q^{43} -6.20511 q^{45} +0.823208 q^{47} -6.25017 q^{49} -1.66879 q^{51} +2.40110 q^{53} -0.785690 q^{57} -13.5148 q^{59} +6.60761 q^{61} -5.37317 q^{63} -4.43509 q^{65} +7.17725 q^{67} -24.0505 q^{69} +2.79910 q^{71} +2.89074 q^{73} -3.03399 q^{75} +5.91452 q^{79} +10.8880 q^{81} +7.09821 q^{83} -0.550031 q^{85} -28.4046 q^{87} -16.6470 q^{89} -3.84047 q^{91} +16.1283 q^{93} -0.258962 q^{95} -7.83395 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 6 q^{5} - 6 q^{7} + 10 q^{9} + 6 q^{13} - 2 q^{15} - 11 q^{17} + 11 q^{19} - 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} + 6 q^{29} + q^{31} + 6 q^{35} + 4 q^{37} - 27 q^{39} + 4 q^{41} - 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q + 2 * q^3 - 6 * q^5 - 6 * q^7 + 10 * q^9 + 6 * q^13 - 2 * q^15 - 11 * q^17 + 11 * q^19 - 2 * q^21 + 18 * q^23 + 6 * q^25 - q^27 + 6 * q^29 + q^31 + 6 * q^35 + 4 * q^37 - 27 * q^39 + 4 * q^41 - 3 * q^43 - 10 * q^45 + 14 * q^47 + 8 * q^49 - 31 * q^51 + 14 * q^53 + 5 * q^57 + 2 * q^59 + 4 * q^61 + 16 * q^63 - 6 * q^65 + 11 * q^67 + 8 * q^69 + 7 * q^71 + 9 * q^73 + 2 * q^75 - 36 * q^79 + 30 * q^81 + 45 * q^83 + 11 * q^85 - 25 * q^87 + q^89 - 8 * q^91 + 55 * q^93 - 11 * q^95 + 20 * q^97

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
$3$	−3.03399	−1.75168	−0.875838	−	0.482605i	$-0.839691\pi$
$3$	−3.03399	−1.75168	−0.875838	+	0.482605i	$0.839691\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.865927	−0.327290	−0.163645	−	0.986519i	$-0.552325\pi$
$7$	−0.865927	−0.327290	−0.163645	+	0.986519i	$0.552325\pi$
$8$	0	0
$9$	6.20511	2.06837
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.43509	1.23007	0.615037	−	0.788499i	$-0.289141\pi$
$13$	4.43509	1.23007	0.615037	+	0.788499i	$0.289141\pi$
$14$	0	0
$15$	3.03399	0.783373
$16$	0	0
$17$	0.550031	0.133402	0.0667010	−	0.997773i	$-0.478753\pi$
$17$	0.550031	0.133402	0.0667010	+	0.997773i	$0.478753\pi$
$18$	0	0
$19$	0.258962	0.0594100	0.0297050	−	0.999559i	$-0.490543\pi$
$19$	0.258962	0.0594100	0.0297050	+	0.999559i	$0.490543\pi$
$20$	0	0
$21$	2.62722	0.573306
$22$	0	0
$23$	7.92701	1.65290	0.826448	−	0.563013i	$-0.190358\pi$
$23$	7.92701	1.65290	0.826448	+	0.563013i	$0.190358\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−9.72427	−1.87144
$28$	0	0
$29$	9.36210	1.73850	0.869250	−	0.494374i	$-0.164602\pi$
$29$	9.36210	1.73850	0.869250	+	0.494374i	$0.164602\pi$
$30$	0	0
$31$	−5.31588	−0.954760	−0.477380	−	0.878697i	$-0.658413\pi$
$31$	−5.31588	−0.954760	−0.477380	+	0.878697i	$0.658413\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.865927	0.146368
$36$	0	0
$37$	6.38739	1.05008	0.525040	−	0.851077i	$-0.324050\pi$
$37$	6.38739	1.05008	0.525040	+	0.851077i	$0.324050\pi$
$38$	0	0
$39$	−13.4560	−2.15469
$40$	0	0
$41$	−8.74218	−1.36530	−0.682650	−	0.730746i	$-0.739173\pi$
$41$	−8.74218	−1.36530	−0.682650	+	0.730746i	$0.739173\pi$
$42$	0	0
$43$	−6.38622	−0.973890	−0.486945	−	0.873433i	$-0.661889\pi$
$43$	−6.38622	−0.973890	−0.486945	+	0.873433i	$0.661889\pi$
$44$	0	0
$45$	−6.20511	−0.925003
$46$	0	0
$47$	0.823208	0.120077	0.0600386	−	0.998196i	$-0.480878\pi$
$47$	0.823208	0.120077	0.0600386	+	0.998196i	$0.480878\pi$
$48$	0	0
$49$	−6.25017	−0.892881
$50$	0	0
$51$	−1.66879	−0.233677
$52$	0	0
$53$	2.40110	0.329816	0.164908	−	0.986309i	$-0.447267\pi$
$53$	2.40110	0.329816	0.164908	+	0.986309i	$0.447267\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.785690	−0.104067
$58$	0	0
$59$	−13.5148	−1.75948	−0.879741	−	0.475453i	$-0.842284\pi$
$59$	−13.5148	−1.75948	−0.879741	+	0.475453i	$0.842284\pi$
$60$	0	0
$61$	6.60761	0.846018	0.423009	−	0.906126i	$-0.360974\pi$
$61$	6.60761	0.846018	0.423009	+	0.906126i	$0.360974\pi$
$62$	0	0
$63$	−5.37317	−0.676956
$64$	0	0
$65$	−4.43509	−0.550105
$66$	0	0
$67$	7.17725	0.876840	0.438420	−	0.898770i	$-0.355538\pi$
$67$	7.17725	0.876840	0.438420	+	0.898770i	$0.355538\pi$
$68$	0	0
$69$	−24.0505	−2.89534
$70$	0	0
$71$	2.79910	0.332192	0.166096	−	0.986110i	$-0.446884\pi$
$71$	2.79910	0.332192	0.166096	+	0.986110i	$0.446884\pi$
$72$	0	0
$73$	2.89074	0.338335	0.169168	−	0.985587i	$-0.445892\pi$
$73$	2.89074	0.338335	0.169168	+	0.985587i	$0.445892\pi$
$74$	0	0
$75$	−3.03399	−0.350335
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.91452	0.665436	0.332718	−	0.943026i	$-0.392034\pi$
$79$	5.91452	0.665436	0.332718	+	0.943026i	$0.392034\pi$
$80$	0	0
$81$	10.8880	1.20978
$82$	0	0
$83$	7.09821	0.779129	0.389565	−	0.920999i	$-0.372625\pi$
$83$	7.09821	0.779129	0.389565	+	0.920999i	$0.372625\pi$
$84$	0	0
$85$	−0.550031	−0.0596592
$86$	0	0
$87$	−28.4046	−3.04529
$88$	0	0
$89$	−16.6470	−1.76458	−0.882291	−	0.470704i	$-0.844000\pi$
$89$	−16.6470	−1.76458	−0.882291	+	0.470704i	$0.844000\pi$
$90$	0	0
$91$	−3.84047	−0.402590
$92$	0	0
$93$	16.1283	1.67243
$94$	0	0
$95$	−0.258962	−0.0265690
$96$	0	0
$97$	−7.83395	−0.795417	−0.397708	−	0.917512i	$-0.630194\pi$
$97$	−7.83395	−0.795417	−0.397708	+	0.917512i	$0.630194\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.1989	1.81086	0.905431	−	0.424493i	$-0.139548\pi$
$101$	18.1989	1.81086	0.905431	+	0.424493i	$0.139548\pi$
$102$	0	0
$103$	11.8025	1.16294	0.581469	−	0.813569i	$-0.302478\pi$
$103$	11.8025	1.16294	0.581469	+	0.813569i	$0.302478\pi$
$104$	0	0
$105$	−2.62722	−0.256390
$106$	0	0
$107$	0.520965	0.0503636	0.0251818	−	0.999683i	$-0.491984\pi$
$107$	0.520965	0.0503636	0.0251818	+	0.999683i	$0.491984\pi$
$108$	0	0
$109$	6.22545	0.596290	0.298145	−	0.954521i	$-0.403632\pi$
$109$	6.22545	0.596290	0.298145	+	0.954521i	$0.403632\pi$
$110$	0	0
$111$	−19.3793	−1.83940
$112$	0	0
$113$	−9.87578	−0.929035	−0.464518	−	0.885564i	$-0.653772\pi$
$113$	−9.87578	−0.929035	−0.464518	+	0.885564i	$0.653772\pi$
$114$	0	0
$115$	−7.92701	−0.739198
$116$	0	0
$117$	27.5202	2.54425
$118$	0	0
$119$	−0.476287	−0.0436611
$120$	0	0
$121$	0	0
$122$	0	0
$123$	26.5237	2.39156
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.71934	−0.330038	−0.165019	−	0.986290i	$-0.552769\pi$
$127$	−3.71934	−0.330038	−0.165019	+	0.986290i	$0.552769\pi$
$128$	0	0
$129$	19.3758	1.70594
$130$	0	0
$131$	−20.4259	−1.78462	−0.892308	−	0.451426i	$-0.850915\pi$
$131$	−20.4259	−1.78462	−0.892308	+	0.451426i	$0.850915\pi$
$132$	0	0
$133$	−0.224243	−0.0194443
$134$	0	0
$135$	9.72427	0.836932
$136$	0	0
$137$	13.4216	1.14669	0.573344	−	0.819315i	$-0.305646\pi$
$137$	13.4216	1.14669	0.573344	+	0.819315i	$0.305646\pi$
$138$	0	0
$139$	−14.0856	−1.19473	−0.597364	−	0.801970i	$-0.703785\pi$
$139$	−14.0856	−1.19473	−0.597364	+	0.801970i	$0.703785\pi$
$140$	0	0
$141$	−2.49761	−0.210336
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.36210	−0.777480
$146$	0	0
$147$	18.9630	1.56404
$148$	0	0
$149$	−1.79173	−0.146784	−0.0733920	−	0.997303i	$-0.523382\pi$
$149$	−1.79173	−0.146784	−0.0733920	+	0.997303i	$0.523382\pi$
$150$	0	0
$151$	−19.0871	−1.55328	−0.776641	−	0.629943i	$-0.783078\pi$
$151$	−19.0871	−1.55328	−0.776641	+	0.629943i	$0.783078\pi$
$152$	0	0
$153$	3.41300	0.275925
$154$	0	0
$155$	5.31588	0.426982
$156$	0	0
$157$	9.78753	0.781130	0.390565	−	0.920575i	$-0.372280\pi$
$157$	9.78753	0.781130	0.390565	+	0.920575i	$0.372280\pi$
$158$	0	0
$159$	−7.28492	−0.577732
$160$	0	0
$161$	−6.86422	−0.540976
$162$	0	0
$163$	0.396843	0.0310831	0.0155416	−	0.999879i	$-0.495053\pi$
$163$	0.396843	0.0310831	0.0155416	+	0.999879i	$0.495053\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.9140	−1.07670	−0.538349	−	0.842722i	$-0.680952\pi$
$167$	−13.9140	−1.07670	−0.538349	+	0.842722i	$0.680952\pi$
$168$	0	0
$169$	6.67004	0.513080
$170$	0	0
$171$	1.60689	0.122882
$172$	0	0
$173$	−5.33390	−0.405529	−0.202765	−	0.979228i	$-0.564993\pi$
$173$	−5.33390	−0.405529	−0.202765	+	0.979228i	$0.564993\pi$
$174$	0	0
$175$	−0.865927	−0.0654580
$176$	0	0
$177$	41.0039	3.08204
$178$	0	0
$179$	7.63478	0.570650	0.285325	−	0.958431i	$-0.407898\pi$
$179$	7.63478	0.570650	0.285325	+	0.958431i	$0.407898\pi$
$180$	0	0
$181$	1.30453	0.0969647	0.0484824	−	0.998824i	$-0.484562\pi$
$181$	1.30453	0.0969647	0.0484824	+	0.998824i	$0.484562\pi$
$182$	0	0
$183$	−20.0474	−1.48195
$184$	0	0
$185$	−6.38739	−0.469610
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.42052	0.612502
$190$	0	0
$191$	25.0833	1.81497	0.907483	−	0.420088i	$-0.138001\pi$
$191$	25.0833	1.81497	0.907483	+	0.420088i	$0.138001\pi$
$192$	0	0
$193$	19.9561	1.43647	0.718235	−	0.695801i	$-0.244950\pi$
$193$	19.9561	1.43647	0.718235	+	0.695801i	$0.244950\pi$
$194$	0	0
$195$	13.4560	0.963607
$196$	0	0
$197$	1.31088	0.0933960	0.0466980	−	0.998909i	$-0.485130\pi$
$197$	1.31088	0.0933960	0.0466980	+	0.998909i	$0.485130\pi$
$198$	0	0
$199$	−4.99115	−0.353813	−0.176907	−	0.984228i	$-0.556609\pi$
$199$	−4.99115	−0.353813	−0.176907	+	0.984228i	$0.556609\pi$
$200$	0	0
$201$	−21.7757	−1.53594
$202$	0	0
$203$	−8.10690	−0.568993
$204$	0	0
$205$	8.74218	0.610581
$206$	0	0
$207$	49.1880	3.41880
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.41108	0.441357	0.220678	−	0.975347i	$-0.429173\pi$
$211$	6.41108	0.441357	0.220678	+	0.975347i	$0.429173\pi$
$212$	0	0
$213$	−8.49245	−0.581893
$214$	0	0
$215$	6.38622	0.435537
$216$	0	0
$217$	4.60316	0.312483
$218$	0	0
$219$	−8.77047	−0.592654
$220$	0	0
$221$	2.43944	0.164094
$222$	0	0
$223$	−14.8484	−0.994321	−0.497160	−	0.867659i	$-0.665624\pi$
$223$	−14.8484	−0.994321	−0.497160	+	0.867659i	$0.665624\pi$
$224$	0	0
$225$	6.20511	0.413674
$226$	0	0
$227$	23.4761	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$227$	23.4761	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$228$	0	0
$229$	19.6330	1.29739	0.648693	−	0.761050i	$-0.275316\pi$
$229$	19.6330	1.29739	0.648693	+	0.761050i	$0.275316\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.9946	−1.37541	−0.687703	−	0.725993i	$-0.741381\pi$
$233$	−20.9946	−1.37541	−0.687703	+	0.725993i	$0.741381\pi$
$234$	0	0
$235$	−0.823208	−0.0537002
$236$	0	0
$237$	−17.9446	−1.16563
$238$	0	0
$239$	18.8533	1.21952	0.609761	−	0.792586i	$-0.291266\pi$
$239$	18.8533	1.21952	0.609761	+	0.792586i	$0.291266\pi$
$240$	0	0
$241$	13.8318	0.890987	0.445494	−	0.895285i	$-0.353028\pi$
$241$	13.8318	0.890987	0.445494	+	0.895285i	$0.353028\pi$
$242$	0	0
$243$	−3.86143	−0.247711
$244$	0	0
$245$	6.25017	0.399309
$246$	0	0
$247$	1.14852	0.0730787
$248$	0	0
$249$	−21.5359	−1.36478
$250$	0	0
$251$	15.2719	0.963956	0.481978	−	0.876183i	$-0.339919\pi$
$251$	15.2719	0.963956	0.481978	+	0.876183i	$0.339919\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.66879	0.104504
$256$	0	0
$257$	28.7880	1.79575	0.897873	−	0.440254i	$-0.145112\pi$
$257$	28.7880	1.79575	0.897873	+	0.440254i	$0.145112\pi$
$258$	0	0
$259$	−5.53101	−0.343681
$260$	0	0
$261$	58.0929	3.59586
$262$	0	0
$263$	9.23218	0.569281	0.284640	−	0.958634i	$-0.408126\pi$
$263$	9.23218	0.569281	0.284640	+	0.958634i	$0.408126\pi$
$264$	0	0
$265$	−2.40110	−0.147498
$266$	0	0
$267$	50.5070	3.09098
$268$	0	0
$269$	26.8324	1.63600	0.817999	−	0.575220i	$-0.195084\pi$
$269$	26.8324	1.63600	0.817999	+	0.575220i	$0.195084\pi$
$270$	0	0
$271$	−23.5612	−1.43124	−0.715620	−	0.698490i	$-0.753856\pi$
$271$	−23.5612	−1.43124	−0.715620	+	0.698490i	$0.753856\pi$
$272$	0	0
$273$	11.6519	0.705208
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9.99241	−0.600386	−0.300193	−	0.953879i	$-0.597051\pi$
$277$	−9.99241	−0.600386	−0.300193	+	0.953879i	$0.597051\pi$
$278$	0	0
$279$	−32.9856	−1.97480
$280$	0	0
$281$	−4.09042	−0.244014	−0.122007	−	0.992529i	$-0.538933\pi$
$281$	−4.09042	−0.244014	−0.122007	+	0.992529i	$0.538933\pi$
$282$	0	0
$283$	−4.77268	−0.283706	−0.141853	−	0.989888i	$-0.545306\pi$
$283$	−4.77268	−0.283706	−0.141853	+	0.989888i	$0.545306\pi$
$284$	0	0
$285$	0.785690	0.0465402
$286$	0	0
$287$	7.57010	0.446849
$288$	0	0
$289$	−16.6975	−0.982204
$290$	0	0
$291$	23.7681	1.39331
$292$	0	0
$293$	−0.144012	−0.00841327	−0.00420664	−	0.999991i	$-0.501339\pi$
$293$	−0.144012	−0.00841327	−0.00420664	+	0.999991i	$0.501339\pi$
$294$	0	0
$295$	13.5148	0.786864
$296$	0	0
$297$	0	0
$298$	0	0
$299$	35.1570	2.03318
$300$	0	0
$301$	5.53001	0.318744
$302$	0	0
$303$	−55.2154	−3.17204
$304$	0	0
$305$	−6.60761	−0.378351
$306$	0	0
$307$	25.7576	1.47006	0.735031	−	0.678034i	$-0.237168\pi$
$307$	25.7576	1.47006	0.735031	+	0.678034i	$0.237168\pi$
$308$	0	0
$309$	−35.8088	−2.03709
$310$	0	0
$311$	16.3915	0.929475	0.464738	−	0.885448i	$-0.346149\pi$
$311$	16.3915	0.929475	0.464738	+	0.885448i	$0.346149\pi$
$312$	0	0
$313$	26.5238	1.49921	0.749606	−	0.661885i	$-0.230243\pi$
$313$	26.5238	1.49921	0.749606	+	0.661885i	$0.230243\pi$
$314$	0	0
$315$	5.37317	0.302744
$316$	0	0
$317$	16.7158	0.938852	0.469426	−	0.882972i	$-0.344461\pi$
$317$	16.7158	0.938852	0.469426	+	0.882972i	$0.344461\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.58060	−0.0882206
$322$	0	0
$323$	0.142437	0.00792542
$324$	0	0
$325$	4.43509	0.246015
$326$	0	0
$327$	−18.8880	−1.04451
$328$	0	0
$329$	−0.712839	−0.0393001
$330$	0	0
$331$	4.39584	0.241617	0.120809	−	0.992676i	$-0.461451\pi$
$331$	4.39584	0.241617	0.120809	+	0.992676i	$0.461451\pi$
$332$	0	0
$333$	39.6344	2.17195
$334$	0	0
$335$	−7.17725	−0.392135
$336$	0	0
$337$	−12.6035	−0.686558	−0.343279	−	0.939233i	$-0.611538\pi$
$337$	−12.6035	−0.686558	−0.343279	+	0.939233i	$0.611538\pi$
$338$	0	0
$339$	29.9630	1.62737
$340$	0	0
$341$	0	0
$342$	0	0
$343$	11.4737	0.619521
$344$	0	0
$345$	24.0505	1.29484
$346$	0	0
$347$	23.9827	1.28746	0.643731	−	0.765252i	$-0.277386\pi$
$347$	23.9827	1.28746	0.643731	+	0.765252i	$0.277386\pi$
$348$	0	0
$349$	−23.5901	−1.26275	−0.631376	−	0.775477i	$-0.717509\pi$
$349$	−23.5901	−1.26275	−0.631376	+	0.775477i	$0.717509\pi$
$350$	0	0
$351$	−43.1281	−2.30201
$352$	0	0
$353$	6.46751	0.344231	0.172115	−	0.985077i	$-0.444940\pi$
$353$	6.46751	0.344231	0.172115	+	0.985077i	$0.444940\pi$
$354$	0	0
$355$	−2.79910	−0.148561
$356$	0	0
$357$	1.44505	0.0764802
$358$	0	0
$359$	−11.8241	−0.624053	−0.312026	−	0.950073i	$-0.601008\pi$
$359$	−11.8241	−0.624053	−0.312026	+	0.950073i	$0.601008\pi$
$360$	0	0
$361$	−18.9329	−0.996470
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.89074	−0.151308
$366$	0	0
$367$	8.38713	0.437805	0.218902	−	0.975747i	$-0.429752\pi$
$367$	8.38713	0.437805	0.218902	+	0.975747i	$0.429752\pi$
$368$	0	0
$369$	−54.2462	−2.82394
$370$	0	0
$371$	−2.07918	−0.107946
$372$	0	0
$373$	−34.6263	−1.79288	−0.896441	−	0.443163i	$-0.853856\pi$
$373$	−34.6263	−1.79288	−0.896441	+	0.443163i	$0.853856\pi$
$374$	0	0
$375$	3.03399	0.156675
$376$	0	0
$377$	41.5218	2.13848
$378$	0	0
$379$	16.0726	0.825592	0.412796	−	0.910823i	$-0.364552\pi$
$379$	16.0726	0.825592	0.412796	+	0.910823i	$0.364552\pi$
$380$	0	0
$381$	11.2844	0.578120
$382$	0	0
$383$	8.41569	0.430022	0.215011	−	0.976612i	$-0.431021\pi$
$383$	8.41569	0.430022	0.215011	+	0.976612i	$0.431021\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−39.6272	−2.01436
$388$	0	0
$389$	18.5917	0.942636	0.471318	−	0.881963i	$-0.343778\pi$
$389$	18.5917	0.942636	0.471318	+	0.881963i	$0.343778\pi$
$390$	0	0
$391$	4.36010	0.220500
$392$	0	0
$393$	61.9719	3.12607
$394$	0	0
$395$	−5.91452	−0.297592
$396$	0	0
$397$	−11.9897	−0.601743	−0.300872	−	0.953665i	$-0.597278\pi$
$397$	−11.9897	−0.601743	−0.300872	+	0.953665i	$0.597278\pi$
$398$	0	0
$399$	0.680350	0.0340601
$400$	0	0
$401$	18.7991	0.938781	0.469391	−	0.882991i	$-0.344474\pi$
$401$	18.7991	0.938781	0.469391	+	0.882991i	$0.344474\pi$
$402$	0	0
$403$	−23.5764	−1.17442
$404$	0	0
$405$	−10.8880	−0.541031
$406$	0	0
$407$	0	0
$408$	0	0
$409$	9.50429	0.469957	0.234978	−	0.972001i	$-0.424498\pi$
$409$	9.50429	0.469957	0.234978	+	0.972001i	$0.424498\pi$
$410$	0	0
$411$	−40.7211	−2.00863
$412$	0	0
$413$	11.7029	0.575860
$414$	0	0
$415$	−7.09821	−0.348437
$416$	0	0
$417$	42.7357	2.09278
$418$	0	0
$419$	32.9152	1.60801	0.804007	−	0.594620i	$-0.202698\pi$
$419$	32.9152	1.60801	0.804007	+	0.594620i	$0.202698\pi$
$420$	0	0
$421$	−1.03559	−0.0504717	−0.0252359	−	0.999682i	$-0.508034\pi$
$421$	−1.03559	−0.0504717	−0.0252359	+	0.999682i	$0.508034\pi$
$422$	0	0
$423$	5.10810	0.248364
$424$	0	0
$425$	0.550031	0.0266804
$426$	0	0
$427$	−5.72171	−0.276893
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.7613	−1.24088	−0.620439	−	0.784255i	$-0.713046\pi$
$431$	−25.7613	−1.24088	−0.620439	+	0.784255i	$0.713046\pi$
$432$	0	0
$433$	−18.4406	−0.886201	−0.443100	−	0.896472i	$-0.646121\pi$
$433$	−18.4406	−0.886201	−0.443100	+	0.896472i	$0.646121\pi$
$434$	0	0
$435$	28.4046	1.36189
$436$	0	0
$437$	2.05280	0.0981986
$438$	0	0
$439$	4.44724	0.212255	0.106128	−	0.994353i	$-0.466155\pi$
$439$	4.44724	0.212255	0.106128	+	0.994353i	$0.466155\pi$
$440$	0	0
$441$	−38.7830	−1.84681
$442$	0	0
$443$	−25.0738	−1.19129	−0.595646	−	0.803247i	$-0.703104\pi$
$443$	−25.0738	−1.19129	−0.595646	+	0.803247i	$0.703104\pi$
$444$	0	0
$445$	16.6470	0.789145
$446$	0	0
$447$	5.43609	0.257118
$448$	0	0
$449$	−18.6745	−0.881306	−0.440653	−	0.897678i	$-0.645253\pi$
$449$	−18.6745	−0.881306	−0.440653	+	0.897678i	$0.645253\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	57.9100	2.72085
$454$	0	0
$455$	3.84047	0.180044
$456$	0	0
$457$	14.5569	0.680941	0.340471	−	0.940255i	$-0.389414\pi$
$457$	14.5569	0.680941	0.340471	+	0.940255i	$0.389414\pi$
$458$	0	0
$459$	−5.34865	−0.249654
$460$	0	0
$461$	−20.6525	−0.961884	−0.480942	−	0.876752i	$-0.659705\pi$
$461$	−20.6525	−0.961884	−0.480942	+	0.876752i	$0.659705\pi$
$462$	0	0
$463$	−21.8605	−1.01594	−0.507971	−	0.861374i	$-0.669604\pi$
$463$	−21.8605	−1.01594	−0.507971	+	0.861374i	$0.669604\pi$
$464$	0	0
$465$	−16.1283	−0.747933
$466$	0	0
$467$	20.2984	0.939296	0.469648	−	0.882854i	$-0.344381\pi$
$467$	20.2984	0.939296	0.469648	+	0.882854i	$0.344381\pi$
$468$	0	0
$469$	−6.21498	−0.286981
$470$	0	0
$471$	−29.6953	−1.36829
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.258962	0.0118820
$476$	0	0
$477$	14.8991	0.682182
$478$	0	0
$479$	9.89152	0.451955	0.225978	−	0.974132i	$-0.427442\pi$
$479$	9.89152	0.451955	0.225978	+	0.974132i	$0.427442\pi$
$480$	0	0
$481$	28.3287	1.29168
$482$	0	0
$483$	20.8260	0.947615
$484$	0	0
$485$	7.83395	0.355721
$486$	0	0
$487$	5.93879	0.269112	0.134556	−	0.990906i	$-0.457039\pi$
$487$	5.93879	0.269112	0.134556	+	0.990906i	$0.457039\pi$
$488$	0	0
$489$	−1.20402	−0.0544476
$490$	0	0
$491$	−27.3348	−1.23360	−0.616801	−	0.787119i	$-0.711572\pi$
$491$	−27.3348	−1.23360	−0.616801	+	0.787119i	$0.711572\pi$
$492$	0	0
$493$	5.14945	0.231919
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.42382	−0.108723
$498$	0	0
$499$	−12.1678	−0.544705	−0.272353	−	0.962198i	$-0.587802\pi$
$499$	−12.1678	−0.544705	−0.272353	+	0.962198i	$0.587802\pi$
$500$	0	0
$501$	42.2150	1.88603
$502$	0	0
$503$	9.49180	0.423218	0.211609	−	0.977354i	$-0.432130\pi$
$503$	9.49180	0.423218	0.211609	+	0.977354i	$0.432130\pi$
$504$	0	0
$505$	−18.1989	−0.809842
$506$	0	0
$507$	−20.2369	−0.898750
$508$	0	0
$509$	−0.957696	−0.0424491	−0.0212246	−	0.999775i	$-0.506756\pi$
$509$	−0.957696	−0.0424491	−0.0212246	+	0.999775i	$0.506756\pi$
$510$	0	0
$511$	−2.50317	−0.110734
$512$	0	0
$513$	−2.51822	−0.111182
$514$	0	0
$515$	−11.8025	−0.520082
$516$	0	0
$517$	0	0
$518$	0	0
$519$	16.1830	0.710356
$520$	0	0
$521$	−18.2415	−0.799176	−0.399588	−	0.916695i	$-0.630847\pi$
$521$	−18.2415	−0.799176	−0.399588	+	0.916695i	$0.630847\pi$
$522$	0	0
$523$	−28.3274	−1.23867	−0.619335	−	0.785127i	$-0.712598\pi$
$523$	−28.3274	−1.23867	−0.619335	+	0.785127i	$0.712598\pi$
$524$	0	0
$525$	2.62722	0.114661
$526$	0	0
$527$	−2.92390	−0.127367
$528$	0	0
$529$	39.8375	1.73207
$530$	0	0
$531$	−83.8610	−3.63926
$532$	0	0
$533$	−38.7724	−1.67942
$534$	0	0
$535$	−0.520965	−0.0225233
$536$	0	0
$537$	−23.1639	−0.999594
$538$	0	0
$539$	0	0
$540$	0	0
$541$	31.7569	1.36534	0.682668	−	0.730729i	$-0.260819\pi$
$541$	31.7569	1.36534	0.682668	+	0.730729i	$0.260819\pi$
$542$	0	0
$543$	−3.95792	−0.169851
$544$	0	0
$545$	−6.22545	−0.266669
$546$	0	0
$547$	−3.58429	−0.153253	−0.0766265	−	0.997060i	$-0.524415\pi$
$547$	−3.58429	−0.153253	−0.0766265	+	0.997060i	$0.524415\pi$
$548$	0	0
$549$	41.0009	1.74988
$550$	0	0
$551$	2.42443	0.103284
$552$	0	0
$553$	−5.12155	−0.217790
$554$	0	0
$555$	19.3793	0.822605
$556$	0	0
$557$	16.6556	0.705719	0.352860	−	0.935676i	$-0.385209\pi$
$557$	16.6556	0.705719	0.352860	+	0.935676i	$0.385209\pi$
$558$	0	0
$559$	−28.3235	−1.19796
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.8978	−1.68149	−0.840745	−	0.541431i	$-0.817883\pi$
$563$	−39.8978	−1.68149	−0.840745	+	0.541431i	$0.817883\pi$
$564$	0	0
$565$	9.87578	0.415477
$566$	0	0
$567$	−9.42826	−0.395950
$568$	0	0
$569$	−45.4644	−1.90597	−0.952983	−	0.303024i	$-0.902004\pi$
$569$	−45.4644	−1.90597	−0.952983	+	0.303024i	$0.902004\pi$
$570$	0	0
$571$	15.8877	0.664880	0.332440	−	0.943124i	$-0.392128\pi$
$571$	15.8877	0.664880	0.332440	+	0.943124i	$0.392128\pi$
$572$	0	0
$573$	−76.1027	−3.17923
$574$	0	0
$575$	7.92701	0.330579
$576$	0	0
$577$	−13.1413	−0.547078	−0.273539	−	0.961861i	$-0.588194\pi$
$577$	−13.1413	−0.547078	−0.273539	+	0.961861i	$0.588194\pi$
$578$	0	0
$579$	−60.5466	−2.51623
$580$	0	0
$581$	−6.14653	−0.255001
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−27.5202	−1.13782
$586$	0	0
$587$	14.9640	0.617629	0.308814	−	0.951122i	$-0.400068\pi$
$587$	14.9640	0.617629	0.308814	+	0.951122i	$0.400068\pi$
$588$	0	0
$589$	−1.37661	−0.0567223
$590$	0	0
$591$	−3.97719	−0.163600
$592$	0	0
$593$	13.7967	0.566564	0.283282	−	0.959037i	$-0.408577\pi$
$593$	13.7967	0.566564	0.283282	+	0.959037i	$0.408577\pi$
$594$	0	0
$595$	0.476287	0.0195259
$596$	0	0
$597$	15.1431	0.619766
$598$	0	0
$599$	31.3893	1.28253	0.641267	−	0.767318i	$-0.278409\pi$
$599$	31.3893	1.28253	0.641267	+	0.767318i	$0.278409\pi$
$600$	0	0
$601$	2.55635	0.104276	0.0521379	−	0.998640i	$-0.483396\pi$
$601$	2.55635	0.104276	0.0521379	+	0.998640i	$0.483396\pi$
$602$	0	0
$603$	44.5356	1.81363
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.8289	1.53543	0.767713	−	0.640794i	$-0.221395\pi$
$607$	37.8289	1.53543	0.767713	+	0.640794i	$0.221395\pi$
$608$	0	0
$609$	24.5963	0.996692
$610$	0	0
$611$	3.65100	0.147704
$612$	0	0
$613$	−11.6599	−0.470940	−0.235470	−	0.971882i	$-0.575663\pi$
$613$	−11.6599	−0.470940	−0.235470	+	0.971882i	$0.575663\pi$
$614$	0	0
$615$	−26.5237	−1.06954
$616$	0	0
$617$	38.5910	1.55361	0.776807	−	0.629739i	$-0.216838\pi$
$617$	38.5910	1.55361	0.776807	+	0.629739i	$0.216838\pi$
$618$	0	0
$619$	−8.16365	−0.328125	−0.164062	−	0.986450i	$-0.552460\pi$
$619$	−8.16365	−0.328125	−0.164062	+	0.986450i	$0.552460\pi$
$620$	0	0
$621$	−77.0844	−3.09329
$622$	0	0
$623$	14.4151	0.577530
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.51326	0.140083
$630$	0	0
$631$	−13.0813	−0.520759	−0.260379	−	0.965506i	$-0.583848\pi$
$631$	−13.0813	−0.520759	−0.260379	+	0.965506i	$0.583848\pi$
$632$	0	0
$633$	−19.4512	−0.773114
$634$	0	0
$635$	3.71934	0.147597
$636$	0	0
$637$	−27.7201	−1.09831
$638$	0	0
$639$	17.3687	0.687096
$640$	0	0
$641$	35.6499	1.40809	0.704044	−	0.710156i	$-0.251376\pi$
$641$	35.6499	1.40809	0.704044	+	0.710156i	$0.251376\pi$
$642$	0	0
$643$	−7.65614	−0.301929	−0.150964	−	0.988539i	$-0.548238\pi$
$643$	−7.65614	−0.301929	−0.150964	+	0.988539i	$0.548238\pi$
$644$	0	0
$645$	−19.3758	−0.762919
$646$	0	0
$647$	9.18246	0.361000	0.180500	−	0.983575i	$-0.442228\pi$
$647$	9.18246	0.361000	0.180500	+	0.983575i	$0.442228\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−13.9660	−0.547369
$652$	0	0
$653$	−5.85282	−0.229039	−0.114519	−	0.993421i	$-0.536533\pi$
$653$	−5.85282	−0.229039	−0.114519	+	0.993421i	$0.536533\pi$
$654$	0	0
$655$	20.4259	0.798105
$656$	0	0
$657$	17.9373	0.699802
$658$	0	0
$659$	22.6530	0.882434	0.441217	−	0.897400i	$-0.354547\pi$
$659$	22.6530	0.882434	0.441217	+	0.897400i	$0.354547\pi$
$660$	0	0
$661$	1.68482	0.0655320	0.0327660	−	0.999463i	$-0.489568\pi$
$661$	1.68482	0.0655320	0.0327660	+	0.999463i	$0.489568\pi$
$662$	0	0
$663$	−7.40123	−0.287440
$664$	0	0
$665$	0.224243	0.00869575
$666$	0	0
$667$	74.2135	2.87356
$668$	0	0
$669$	45.0499	1.74173
$670$	0	0
$671$	0	0
$672$	0	0
$673$	9.48493	0.365617	0.182809	−	0.983149i	$-0.441481\pi$
$673$	9.48493	0.365617	0.182809	+	0.983149i	$0.441481\pi$
$674$	0	0
$675$	−9.72427	−0.374288
$676$	0	0
$677$	4.11952	0.158326	0.0791631	−	0.996862i	$-0.474775\pi$
$677$	4.11952	0.158326	0.0791631	+	0.996862i	$0.474775\pi$
$678$	0	0
$679$	6.78363	0.260332
$680$	0	0
$681$	−71.2262	−2.72940
$682$	0	0
$683$	29.0594	1.11193	0.555963	−	0.831207i	$-0.312349\pi$
$683$	29.0594	1.11193	0.555963	+	0.831207i	$0.312349\pi$
$684$	0	0
$685$	−13.4216	−0.512814
$686$	0	0
$687$	−59.5664	−2.27260
$688$	0	0
$689$	10.6491	0.405698
$690$	0	0
$691$	19.4633	0.740418	0.370209	−	0.928949i	$-0.379286\pi$
$691$	19.4633	0.740418	0.370209	+	0.928949i	$0.379286\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.0856	0.534298
$696$	0	0
$697$	−4.80847	−0.182134
$698$	0	0
$699$	63.6976	2.40926
$700$	0	0
$701$	15.1359	0.571675	0.285838	−	0.958278i	$-0.407728\pi$
$701$	15.1359	0.571675	0.285838	+	0.958278i	$0.407728\pi$
$702$	0	0
$703$	1.65409	0.0623853
$704$	0	0
$705$	2.49761	0.0940653
$706$	0	0
$707$	−15.7590	−0.592677
$708$	0	0
$709$	−27.0377	−1.01542	−0.507712	−	0.861527i	$-0.669508\pi$
$709$	−27.0377	−1.01542	−0.507712	+	0.861527i	$0.669508\pi$
$710$	0	0
$711$	36.7003	1.37637
$712$	0	0
$713$	−42.1390	−1.57812
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−57.2009	−2.13621
$718$	0	0
$719$	3.74967	0.139839	0.0699195	−	0.997553i	$-0.477726\pi$
$719$	3.74967	0.139839	0.0699195	+	0.997553i	$0.477726\pi$
$720$	0	0
$721$	−10.2201	−0.380618
$722$	0	0
$723$	−41.9657	−1.56072
$724$	0	0
$725$	9.36210	0.347700
$726$	0	0
$727$	45.5685	1.69004	0.845022	−	0.534732i	$-0.179587\pi$
$727$	45.5685	1.69004	0.845022	+	0.534732i	$0.179587\pi$
$728$	0	0
$729$	−20.9486	−0.775874
$730$	0	0
$731$	−3.51262	−0.129919
$732$	0	0
$733$	37.1793	1.37325	0.686625	−	0.727012i	$-0.259091\pi$
$733$	37.1793	1.37325	0.686625	+	0.727012i	$0.259091\pi$
$734$	0	0
$735$	−18.9630	−0.699460
$736$	0	0
$737$	0	0
$738$	0	0
$739$	3.86380	0.142132	0.0710661	−	0.997472i	$-0.477360\pi$
$739$	3.86380	0.142132	0.0710661	+	0.997472i	$0.477360\pi$
$740$	0	0
$741$	−3.48461	−0.128010
$742$	0	0
$743$	7.73510	0.283773	0.141887	−	0.989883i	$-0.454683\pi$
$743$	7.73510	0.283773	0.141887	+	0.989883i	$0.454683\pi$
$744$	0	0
$745$	1.79173	0.0656438
$746$	0	0
$747$	44.0451	1.61153
$748$	0	0
$749$	−0.451118	−0.0164835
$750$	0	0
$751$	−19.7386	−0.720271	−0.360135	−	0.932900i	$-0.617269\pi$
$751$	−19.7386	−0.720271	−0.360135	+	0.932900i	$0.617269\pi$
$752$	0	0
$753$	−46.3349	−1.68854
$754$	0	0
$755$	19.0871	0.694649
$756$	0	0
$757$	10.5828	0.384637	0.192319	−	0.981333i	$-0.438399\pi$
$757$	10.5828	0.384637	0.192319	+	0.981333i	$0.438399\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−50.6698	−1.83678	−0.918390	−	0.395677i	$-0.870510\pi$
$761$	−50.6698	−1.83678	−0.918390	+	0.395677i	$0.870510\pi$
$762$	0	0
$763$	−5.39079	−0.195160
$764$	0	0
$765$	−3.41300	−0.123397
$766$	0	0
$767$	−59.9395	−2.16429
$768$	0	0
$769$	16.2588	0.586306	0.293153	−	0.956066i	$-0.405295\pi$
$769$	16.2588	0.586306	0.293153	+	0.956066i	$0.405295\pi$
$770$	0	0
$771$	−87.3426	−3.14557
$772$	0	0
$773$	16.4897	0.593095	0.296547	−	0.955018i	$-0.404165\pi$
$773$	16.4897	0.593095	0.296547	+	0.955018i	$0.404165\pi$
$774$	0	0
$775$	−5.31588	−0.190952
$776$	0	0
$777$	16.7811	0.602017
$778$	0	0
$779$	−2.26390	−0.0811125
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−91.0397	−3.25349
$784$	0	0
$785$	−9.78753	−0.349332
$786$	0	0
$787$	−25.7184	−0.916762	−0.458381	−	0.888756i	$-0.651570\pi$
$787$	−25.7184	−0.916762	−0.458381	+	0.888756i	$0.651570\pi$
$788$	0	0
$789$	−28.0104	−0.997196
$790$	0	0
$791$	8.55171	0.304064
$792$	0	0
$793$	29.3054	1.04066
$794$	0	0
$795$	7.28492	0.258369
$796$	0	0
$797$	5.59752	0.198274	0.0991372	−	0.995074i	$-0.468392\pi$
$797$	5.59752	0.198274	0.0991372	+	0.995074i	$0.468392\pi$
$798$	0	0
$799$	0.452790	0.0160186
$800$	0	0
$801$	−103.297	−3.64981
$802$	0	0
$803$	0	0
$804$	0	0
$805$	6.86422	0.241932
$806$	0	0
$807$	−81.4092	−2.86574
$808$	0	0
$809$	−19.6788	−0.691871	−0.345936	−	0.938258i	$-0.612438\pi$
$809$	−19.6788	−0.691871	−0.345936	+	0.938258i	$0.612438\pi$
$810$	0	0
$811$	7.41499	0.260375	0.130188	−	0.991489i	$-0.458442\pi$
$811$	7.41499	0.260375	0.130188	+	0.991489i	$0.458442\pi$
$812$	0	0
$813$	71.4844	2.50707
$814$	0	0
$815$	−0.396843	−0.0139008
$816$	0	0
$817$	−1.65379	−0.0578588
$818$	0	0
$819$	−23.8305	−0.832706
$820$	0	0
$821$	5.30773	0.185241	0.0926205	−	0.995701i	$-0.470476\pi$
$821$	5.30773	0.185241	0.0926205	+	0.995701i	$0.470476\pi$
$822$	0	0
$823$	−16.1952	−0.564528	−0.282264	−	0.959337i	$-0.591085\pi$
$823$	−16.1952	−0.564528	−0.282264	+	0.959337i	$0.591085\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.97372	0.346820	0.173410	−	0.984850i	$-0.444521\pi$
$827$	9.97372	0.346820	0.173410	+	0.984850i	$0.444521\pi$
$828$	0	0
$829$	−26.4859	−0.919893	−0.459947	−	0.887947i	$-0.652131\pi$
$829$	−26.4859	−0.919893	−0.459947	+	0.887947i	$0.652131\pi$
$830$	0	0
$831$	30.3169	1.05168
$832$	0	0
$833$	−3.43779	−0.119112
$834$	0	0
$835$	13.9140	0.481514
$836$	0	0
$837$	51.6931	1.78677
$838$	0	0
$839$	24.0591	0.830613	0.415307	−	0.909681i	$-0.363674\pi$
$839$	24.0591	0.830613	0.415307	+	0.909681i	$0.363674\pi$
$840$	0	0
$841$	58.6490	2.02238
$842$	0	0
$843$	12.4103	0.427433
$844$	0	0
$845$	−6.67004	−0.229456
$846$	0	0
$847$	0	0
$848$	0	0
$849$	14.4803	0.496962
$850$	0	0
$851$	50.6329	1.73567
$852$	0	0
$853$	−22.5471	−0.771997	−0.385999	−	0.922499i	$-0.626143\pi$
$853$	−22.5471	−0.771997	−0.385999	+	0.922499i	$0.626143\pi$
$854$	0	0
$855$	−1.60689	−0.0549545
$856$	0	0
$857$	−10.0453	−0.343143	−0.171571	−	0.985172i	$-0.554884\pi$
$857$	−10.0453	−0.343143	−0.171571	+	0.985172i	$0.554884\pi$
$858$	0	0
$859$	15.7921	0.538819	0.269409	−	0.963026i	$-0.413172\pi$
$859$	15.7921	0.538819	0.269409	+	0.963026i	$0.413172\pi$
$860$	0	0
$861$	−22.9676	−0.782734
$862$	0	0
$863$	−22.6412	−0.770717	−0.385358	−	0.922767i	$-0.625922\pi$
$863$	−22.6412	−0.770717	−0.385358	+	0.922767i	$0.625922\pi$
$864$	0	0
$865$	5.33390	0.181358
$866$	0	0
$867$	50.6600	1.72050
$868$	0	0
$869$	0	0
$870$	0	0
$871$	31.8318	1.07858
$872$	0	0
$873$	−48.6105	−1.64522
$874$	0	0
$875$	0.865927	0.0292737
$876$	0	0
$877$	−23.7591	−0.802288	−0.401144	−	0.916015i	$-0.631387\pi$
$877$	−23.7591	−0.802288	−0.401144	+	0.916015i	$0.631387\pi$
$878$	0	0
$879$	0.436931	0.0147373
$880$	0	0
$881$	−21.6772	−0.730323	−0.365161	−	0.930944i	$-0.618986\pi$
$881$	−21.6772	−0.730323	−0.365161	+	0.930944i	$0.618986\pi$
$882$	0	0
$883$	−40.6198	−1.36697	−0.683483	−	0.729967i	$-0.739535\pi$
$883$	−40.6198	−1.36697	−0.683483	+	0.729967i	$0.739535\pi$
$884$	0	0
$885$	−41.0039	−1.37833
$886$	0	0
$887$	−34.1509	−1.14668	−0.573338	−	0.819319i	$-0.694352\pi$
$887$	−34.1509	−1.14668	−0.573338	+	0.819319i	$0.694352\pi$
$888$	0	0
$889$	3.22068	0.108018
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.213180	0.00713379
$894$	0	0
$895$	−7.63478	−0.255202
$896$	0	0
$897$	−106.666	−3.56148
$898$	0	0
$899$	−49.7678	−1.65985
$900$	0	0
$901$	1.32068	0.0439982
$902$	0	0
$903$	−16.7780	−0.558337
$904$	0	0
$905$	−1.30453	−0.0433639
$906$	0	0
$907$	11.5541	0.383647	0.191824	−	0.981429i	$-0.438560\pi$
$907$	11.5541	0.383647	0.191824	+	0.981429i	$0.438560\pi$
$908$	0	0
$909$	112.926	3.74553
$910$	0	0
$911$	−14.4057	−0.477284	−0.238642	−	0.971108i	$-0.576702\pi$
$911$	−14.4057	−0.477284	−0.238642	+	0.971108i	$0.576702\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	20.0474	0.662748
$916$	0	0
$917$	17.6873	0.584087
$918$	0	0
$919$	49.1155	1.62017	0.810085	−	0.586312i	$-0.199421\pi$
$919$	49.1155	1.62017	0.810085	+	0.586312i	$0.199421\pi$
$920$	0	0
$921$	−78.1482	−2.57507
$922$	0	0
$923$	12.4143	0.408621
$924$	0	0
$925$	6.38739	0.210016
$926$	0	0
$927$	73.2360	2.40539
$928$	0	0
$929$	23.9888	0.787048	0.393524	−	0.919314i	$-0.371256\pi$
$929$	23.9888	0.787048	0.393524	+	0.919314i	$0.371256\pi$
$930$	0	0
$931$	−1.61856	−0.0530461
$932$	0	0
$933$	−49.7316	−1.62814
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.1022	−0.428029	−0.214014	−	0.976831i	$-0.568654\pi$
$937$	−13.1022	−0.428029	−0.214014	+	0.976831i	$0.568654\pi$
$938$	0	0
$939$	−80.4729	−2.62613
$940$	0	0
$941$	36.6826	1.19582	0.597909	−	0.801564i	$-0.295998\pi$
$941$	36.6826	1.19582	0.597909	+	0.801564i	$0.295998\pi$
$942$	0	0
$943$	−69.2994	−2.25670
$944$	0	0
$945$	−8.42052	−0.273919
$946$	0	0
$947$	20.6112	0.669774	0.334887	−	0.942258i	$-0.391302\pi$
$947$	20.6112	0.669774	0.334887	+	0.942258i	$0.391302\pi$
$948$	0	0
$949$	12.8207	0.416177
$950$	0	0
$951$	−50.7156	−1.64457
$952$	0	0
$953$	−17.7949	−0.576433	−0.288216	−	0.957565i	$-0.593062\pi$
$953$	−17.7949	−0.576433	−0.288216	+	0.957565i	$0.593062\pi$
$954$	0	0
$955$	−25.0833	−0.811678
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.6222	−0.375299
$960$	0	0
$961$	−2.74144	−0.0884337
$962$	0	0
$963$	3.23264	0.104170
$964$	0	0
$965$	−19.9561	−0.642409
$966$	0	0
$967$	0.828077	0.0266292	0.0133146	−	0.999911i	$-0.495762\pi$
$967$	0.828077	0.0266292	0.0133146	+	0.999911i	$0.495762\pi$
$968$	0	0
$969$	−0.432154	−0.0138828
$970$	0	0
$971$	−0.124535	−0.00399653	−0.00199827	−	0.999998i	$-0.500636\pi$
$971$	−0.124535	−0.00399653	−0.00199827	+	0.999998i	$0.500636\pi$
$972$	0	0
$973$	12.1971	0.391022
$974$	0	0
$975$	−13.4560	−0.430938
$976$	0	0
$977$	−21.4664	−0.686770	−0.343385	−	0.939195i	$-0.611573\pi$
$977$	−21.4664	−0.686770	−0.343385	+	0.939195i	$0.611573\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	38.6296	1.23335
$982$	0	0
$983$	31.7566	1.01288	0.506440	−	0.862275i	$-0.330961\pi$
$983$	31.7566	1.01288	0.506440	+	0.862275i	$0.330961\pi$
$984$	0	0
$985$	−1.31088	−0.0417680
$986$	0	0
$987$	2.16275	0.0688410
$988$	0	0
$989$	−50.6237	−1.60974
$990$	0	0
$991$	46.7101	1.48379	0.741897	−	0.670514i	$-0.233926\pi$
$991$	46.7101	1.48379	0.741897	+	0.670514i	$0.233926\pi$
$992$	0	0
$993$	−13.3370	−0.423236
$994$	0	0
$995$	4.99115	0.158230
$996$	0	0
$997$	−51.6548	−1.63592	−0.817962	−	0.575273i	$-0.804896\pi$
$997$	−51.6548	−1.63592	−0.817962	+	0.575273i	$0.804896\pi$
$998$	0	0
$999$	−62.1127	−1.96516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.be.1.1		6
4.3	odd	2		9680.2.a.cy.1.6		6
11.7	odd	10		440.2.y.b.401.3	yes	12
11.8	odd	10		440.2.y.b.361.3	✓	12
11.10	odd	2		4840.2.a.bf.1.1		6
44.7	even	10		880.2.bo.j.401.1		12
44.19	even	10		880.2.bo.j.801.1		12
44.43	even	2		9680.2.a.cx.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.b.361.3	✓	12	11.8	odd	10
440.2.y.b.401.3	yes	12	11.7	odd	10
880.2.bo.j.401.1		12	44.7	even	10
880.2.bo.j.801.1		12	44.19	even	10
4840.2.a.be.1.1		6	1.1	even	1	trivial
4840.2.a.bf.1.1		6	11.10	odd	2
9680.2.a.cx.1.6		6	44.43	even	2
9680.2.a.cy.1.6		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q-3.03399 q^{3} -1.00000 q^{5} -0.865927 q^{7} +6.20511 q^{9} +4.43509 q^{13} +3.03399 q^{15} +0.550031 q^{17} +0.258962 q^{19} +2.62722 q^{21} +7.92701 q^{23} +1.00000 q^{25} -9.72427 q^{27} +9.36210 q^{29} -5.31588 q^{31} +0.865927 q^{35} +6.38739 q^{37} -13.4560 q^{39} -8.74218 q^{41} -6.38622 q^{43} -6.20511 q^{45} +0.823208 q^{47} -6.25017 q^{49} -1.66879 q^{51} +2.40110 q^{53} -0.785690 q^{57} -13.5148 q^{59} +6.60761 q^{61} -5.37317 q^{63} -4.43509 q^{65} +7.17725 q^{67} -24.0505 q^{69} +2.79910 q^{71} +2.89074 q^{73} -3.03399 q^{75} +5.91452 q^{79} +10.8880 q^{81} +7.09821 q^{83} -0.550031 q^{85} -28.4046 q^{87} -16.6470 q^{89} -3.84047 q^{91} +16.1283 q^{93} -0.258962 q^{95} -7.83395 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 6 q^{5} - 6 q^{7} + 10 q^{9} + 6 q^{13} - 2 q^{15} - 11 q^{17} + 11 q^{19} - 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} + 6 q^{29} + q^{31} + 6 q^{35} + 4 q^{37} - 27 q^{39} + 4 q^{41} - 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q + 2 * q^3 - 6 * q^5 - 6 * q^7 + 10 * q^9 + 6 * q^13 - 2 * q^15 - 11 * q^17 + 11 * q^19 - 2 * q^21 + 18 * q^23 + 6 * q^25 - q^27 + 6 * q^29 + q^31 + 6 * q^35 + 4 * q^37 - 27 * q^39 + 4 * q^41 - 3 * q^43 - 10 * q^45 + 14 * q^47 + 8 * q^49 - 31 * q^51 + 14 * q^53 + 5 * q^57 + 2 * q^59 + 4 * q^61 + 16 * q^63 - 6 * q^65 + 11 * q^67 + 8 * q^69 + 7 * q^71 + 9 * q^73 + 2 * q^75 - 36 * q^79 + 30 * q^81 + 45 * q^83 + 11 * q^85 - 25 * q^87 + q^89 - 8 * q^91 + 55 * q^93 - 11 * q^95 + 20 * q^97

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