LMFDB - Embedded newform 4840.2.a.bf.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	$-0.447675\pi$
$7$	0.865927	0.327290	0.163645	+	0.986519i	$0.447675\pi$
$8$	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	$-0.710859\pi$
$13$	−4.43509	−1.23007	−0.615037	+	0.788499i	$0.710859\pi$
$14$	0	0
$15$	3.03399	0.783373
$16$	0	0
$17$	−0.550031	−0.133402	−0.0667010	−	0.997773i	$-0.521247\pi$
$17$	−0.550031	−0.133402	−0.0667010	+	0.997773i	$0.521247\pi$
$18$	0	0
$19$	−0.258962	−0.0594100	−0.0297050	−	0.999559i	$-0.509457\pi$
$19$	−0.258962	−0.0594100	−0.0297050	+	0.999559i	$0.509457\pi$
$20$	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	$-0.835398\pi$
$29$	−9.36210	−1.73850	−0.869250	+	0.494374i	$0.835398\pi$
$30$	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	$-0.260827\pi$
$41$	8.74218	1.36530	0.682650	+	0.730746i	$0.260827\pi$
$42$	0	0
$43$	6.38622	0.973890	0.486945	−	0.873433i	$-0.338111\pi$
$43$	6.38622	0.973890	0.486945	+	0.873433i	$0.338111\pi$
$44$	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	$-0.639026\pi$
$61$	−6.60761	−0.846018	−0.423009	+	0.906126i	$0.639026\pi$
$62$	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	$-0.554108\pi$
$73$	−2.89074	−0.338335	−0.169168	+	0.985587i	$0.554108\pi$
$74$	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	$-0.607966\pi$
$79$	−5.91452	−0.665436	−0.332718	+	0.943026i	$0.607966\pi$
$80$	0	0
$81$	10.8880	1.20978
$82$	0	0
$83$	−7.09821	−0.779129	−0.389565	−	0.920999i	$-0.627375\pi$
$83$	−7.09821	−0.779129	−0.389565	+	0.920999i	$0.627375\pi$
$84$	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	$-0.860452\pi$
$101$	−18.1989	−1.81086	−0.905431	+	0.424493i	$0.860452\pi$
$102$	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	$-0.508016\pi$
$107$	−0.520965	−0.0503636	−0.0251818	+	0.999683i	$0.508016\pi$
$108$	0	0
$109$	−6.22545	−0.596290	−0.298145	−	0.954521i	$-0.596368\pi$
$109$	−6.22545	−0.596290	−0.298145	+	0.954521i	$0.596368\pi$
$110$	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	$-0.447231\pi$
$127$	3.71934	0.330038	0.165019	+	0.986290i	$0.447231\pi$
$128$	0	0
$129$	−19.3758	−1.70594
$130$	0	0
$131$	20.4259	1.78462	0.892308	−	0.451426i	$-0.149085\pi$
$131$	20.4259	1.78462	0.892308	+	0.451426i	$0.149085\pi$
$132$	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	$-0.296215\pi$
$139$	14.0856	1.19473	0.597364	+	0.801970i	$0.296215\pi$
$140$	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	$-0.476618\pi$
$149$	1.79173	0.146784	0.0733920	+	0.997303i	$0.476618\pi$
$150$	0	0
$151$	19.0871	1.55328	0.776641	−	0.629943i	$-0.216922\pi$
$151$	19.0871	1.55328	0.776641	+	0.629943i	$0.216922\pi$
$152$	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	$-0.319048\pi$
$167$	13.9140	1.07670	0.538349	+	0.842722i	$0.319048\pi$
$168$	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	$-0.435007\pi$
$173$	5.33390	0.405529	0.202765	+	0.979228i	$0.435007\pi$
$174$	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	$-0.755050\pi$
$193$	−19.9561	−1.43647	−0.718235	+	0.695801i	$0.755050\pi$
$194$	0	0
$195$	−13.4560	−0.963607
$196$	0	0
$197$	−1.31088	−0.0933960	−0.0466980	−	0.998909i	$-0.514870\pi$
$197$	−1.31088	−0.0933960	−0.0466980	+	0.998909i	$0.514870\pi$
$198$	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	$-0.570827\pi$
$211$	−6.41108	−0.441357	−0.220678	+	0.975347i	$0.570827\pi$
$212$	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	$-0.784314\pi$
$227$	−23.4761	−1.55816	−0.779081	+	0.626923i	$0.784314\pi$
$228$	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	$-0.258619\pi$
$233$	20.9946	1.37541	0.687703	+	0.725993i	$0.258619\pi$
$234$	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	$-0.708734\pi$
$239$	−18.8533	−1.21952	−0.609761	+	0.792586i	$0.708734\pi$
$240$	0	0
$241$	−13.8318	−0.890987	−0.445494	−	0.895285i	$-0.646972\pi$
$241$	−13.8318	−0.890987	−0.445494	+	0.895285i	$0.646972\pi$
$242$	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	$-0.591874\pi$
$263$	−9.23218	−0.569281	−0.284640	+	0.958634i	$0.591874\pi$
$264$	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	$-0.246144\pi$
$271$	23.5612	1.43124	0.715620	+	0.698490i	$0.246144\pi$
$272$	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	$-0.402949\pi$
$277$	9.99241	0.600386	0.300193	+	0.953879i	$0.402949\pi$
$278$	0	0
$279$	−32.9856	−1.97480
$280$	0	0
$281$	4.09042	0.244014	0.122007	−	0.992529i	$-0.461067\pi$
$281$	4.09042	0.244014	0.122007	+	0.992529i	$0.461067\pi$
$282$	0	0
$283$	4.77268	0.283706	0.141853	−	0.989888i	$-0.454694\pi$
$283$	4.77268	0.283706	0.141853	+	0.989888i	$0.454694\pi$
$284$	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	$-0.498661\pi$
$293$	0.144012	0.00841327	0.00420664	+	0.999991i	$0.498661\pi$
$294$	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	$-0.762832\pi$
$307$	−25.7576	−1.47006	−0.735031	+	0.678034i	$0.762832\pi$
$308$	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	$-0.388462\pi$
$337$	12.6035	0.686558	0.343279	+	0.939233i	$0.388462\pi$
$338$	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	$-0.722614\pi$
$347$	−23.9827	−1.28746	−0.643731	+	0.765252i	$0.722614\pi$
$348$	0	0
$349$	23.5901	1.26275	0.631376	−	0.775477i	$-0.282491\pi$
$349$	23.5901	1.26275	0.631376	+	0.775477i	$0.282491\pi$
$350$	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	$-0.398992\pi$
$359$	11.8241	0.624053	0.312026	+	0.950073i	$0.398992\pi$
$360$	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	$-0.146144\pi$
$373$	34.6263	1.79288	0.896441	+	0.443163i	$0.146144\pi$
$374$	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	$-0.575502\pi$
$409$	−9.50429	−0.469957	−0.234978	+	0.972001i	$0.575502\pi$
$410$	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	$-0.286954\pi$
$431$	25.7613	1.24088	0.620439	+	0.784255i	$0.286954\pi$
$432$	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	$-0.533845\pi$
$439$	−4.44724	−0.212255	−0.106128	+	0.994353i	$0.533845\pi$
$440$	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	$-0.610586\pi$
$457$	−14.5569	−0.680941	−0.340471	+	0.940255i	$0.610586\pi$
$458$	0	0
$459$	5.34865	0.249654
$460$	0	0
$461$	20.6525	0.961884	0.480942	−	0.876752i	$-0.340295\pi$
$461$	20.6525	0.961884	0.480942	+	0.876752i	$0.340295\pi$
$462$	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	$-0.572558\pi$
$479$	−9.89152	−0.451955	−0.225978	+	0.974132i	$0.572558\pi$
$480$	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	$-0.288428\pi$
$491$	27.3348	1.23360	0.616801	+	0.787119i	$0.288428\pi$
$492$	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	$-0.567870\pi$
$503$	−9.49180	−0.423218	−0.211609	+	0.977354i	$0.567870\pi$
$504$	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	$-0.287402\pi$
$523$	28.3274	1.23867	0.619335	+	0.785127i	$0.287402\pi$
$524$	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	$-0.739181\pi$
$541$	−31.7569	−1.36534	−0.682668	+	0.730729i	$0.739181\pi$
$542$	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	$-0.475585\pi$
$547$	3.58429	0.153253	0.0766265	+	0.997060i	$0.475585\pi$
$548$	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	$-0.614791\pi$
$557$	−16.6556	−0.705719	−0.352860	+	0.935676i	$0.614791\pi$
$558$	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	$-0.182117\pi$
$563$	39.8978	1.68149	0.840745	+	0.541431i	$0.182117\pi$
$564$	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	$-0.0979964\pi$
$569$	45.4644	1.90597	0.952983	+	0.303024i	$0.0979964\pi$
$570$	0	0
$571$	−15.8877	−0.664880	−0.332440	−	0.943124i	$-0.607872\pi$
$571$	−15.8877	−0.664880	−0.332440	+	0.943124i	$0.607872\pi$
$572$	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	$-0.591423\pi$
$593$	−13.7967	−0.566564	−0.283282	+	0.959037i	$0.591423\pi$
$594$	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	$-0.516604\pi$
$601$	−2.55635	−0.104276	−0.0521379	+	0.998640i	$0.516604\pi$
$602$	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	$-0.778605\pi$
$607$	−37.8289	−1.53543	−0.767713	+	0.640794i	$0.778605\pi$
$608$	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	$-0.424337\pi$
$613$	11.6599	0.470940	0.235470	+	0.971882i	$0.424337\pi$
$614$	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	$-0.645453\pi$
$659$	−22.6530	−0.882434	−0.441217	+	0.897400i	$0.645453\pi$
$660$	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	$-0.558519\pi$
$673$	−9.48493	−0.365617	−0.182809	+	0.983149i	$0.558519\pi$
$674$	0	0
$675$	−9.72427	−0.374288
$676$	0	0
$677$	−4.11952	−0.158326	−0.0791631	−	0.996862i	$-0.525225\pi$
$677$	−4.11952	−0.158326	−0.0791631	+	0.996862i	$0.525225\pi$
$678$	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	$-0.592272\pi$
$701$	−15.1359	−0.571675	−0.285838	+	0.958278i	$0.592272\pi$
$702$	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	$-0.740909\pi$
$733$	−37.1793	−1.37325	−0.686625	+	0.727012i	$0.740909\pi$
$734$	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	$-0.522640\pi$
$739$	−3.86380	−0.142132	−0.0710661	+	0.997472i	$0.522640\pi$
$740$	0	0
$741$	−3.48461	−0.128010
$742$	0	0
$743$	−7.73510	−0.283773	−0.141887	−	0.989883i	$-0.545317\pi$
$743$	−7.73510	−0.283773	−0.141887	+	0.989883i	$0.545317\pi$
$744$	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	$-0.129490\pi$
$761$	50.6698	1.83678	0.918390	+	0.395677i	$0.129490\pi$
$762$	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	$-0.594705\pi$
$769$	−16.2588	−0.586306	−0.293153	+	0.956066i	$0.594705\pi$
$770$	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	$-0.348430\pi$
$787$	25.7184	0.916762	0.458381	+	0.888756i	$0.348430\pi$
$788$	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	$-0.387562\pi$
$809$	19.6788	0.691871	0.345936	+	0.938258i	$0.387562\pi$
$810$	0	0
$811$	−7.41499	−0.260375	−0.130188	−	0.991489i	$-0.541558\pi$
$811$	−7.41499	−0.260375	−0.130188	+	0.991489i	$0.541558\pi$
$812$	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	$-0.529524\pi$
$821$	−5.30773	−0.185241	−0.0926205	+	0.995701i	$0.529524\pi$
$822$	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	$-0.555479\pi$
$827$	−9.97372	−0.346820	−0.173410	+	0.984850i	$0.555479\pi$
$828$	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	$-0.373857\pi$
$853$	22.5471	0.771997	0.385999	+	0.922499i	$0.373857\pi$
$854$	0	0
$855$	1.60689	0.0549545
$856$	0	0
$857$	10.0453	0.343143	0.171571	−	0.985172i	$-0.445116\pi$
$857$	10.0453	0.343143	0.171571	+	0.985172i	$0.445116\pi$
$858$	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	$-0.368613\pi$
$877$	23.7591	0.802288	0.401144	+	0.916015i	$0.368613\pi$
$878$	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	$-0.305648\pi$
$887$	34.1509	1.14668	0.573338	+	0.819319i	$0.305648\pi$
$888$	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	$-0.800579\pi$
$919$	−49.1155	−1.62017	−0.810085	+	0.586312i	$0.800579\pi$
$920$	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	$-0.431346\pi$
$937$	13.1022	0.428029	0.214014	+	0.976831i	$0.431346\pi$
$938$	0	0
$939$	−80.4729	−2.62613
$940$	0	0
$941$	−36.6826	−1.19582	−0.597909	−	0.801564i	$-0.704002\pi$
$941$	−36.6826	−1.19582	−0.597909	+	0.801564i	$0.704002\pi$
$942$	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	$-0.406938\pi$
$953$	17.7949	0.576433	0.288216	+	0.957565i	$0.406938\pi$
$954$	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	$-0.504238\pi$
$967$	−0.828077	−0.0266292	−0.0133146	+	0.999911i	$0.504238\pi$
$968$	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	$-0.195104\pi$
$997$	51.6548	1.63592	0.817962	+	0.575273i	$0.195104\pi$
$998$	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.bf.1.1		6
4.3	odd	2		9680.2.a.cx.1.6		6
11.3	even	5		440.2.y.b.361.3	✓	12
11.4	even	5		440.2.y.b.401.3	yes	12
11.10	odd	2		4840.2.a.be.1.1		6
44.3	odd	10		880.2.bo.j.801.1		12
44.15	odd	10		880.2.bo.j.401.1		12
44.43	even	2		9680.2.a.cy.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.b.361.3	✓	12	11.3	even	5
440.2.y.b.401.3	yes	12	11.4	even	5
880.2.bo.j.401.1		12	44.15	odd	10
880.2.bo.j.801.1		12	44.3	odd	10
4840.2.a.be.1.1		6	11.10	odd	2
4840.2.a.bf.1.1		6	1.1	even	1	trivial
9680.2.a.cx.1.6		6	4.3	odd	2
9680.2.a.cy.1.6		6	44.43	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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
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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