LMFDB - Embedded newform 4840.2.a.bd.1.3

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:	6.6.22733568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.45825$ 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)

$f(q)$	$=$	$q-0.873518 q^{3} +1.00000 q^{5} -3.64048 q^{7} -2.23697 q^{9} -1.06751 q^{13} -0.873518 q^{15} +0.252963 q^{17} +3.06751 q^{19} +3.18003 q^{21} -7.04399 q^{23} +1.00000 q^{25} +4.57459 q^{27} +1.61202 q^{29} -0.788991 q^{31} -3.64048 q^{35} -4.91208 q^{37} +0.932490 q^{39} +6.39545 q^{41} -2.63065 q^{43} -2.23697 q^{45} -7.78559 q^{47} +6.25309 q^{49} -0.220968 q^{51} -6.32291 q^{53} -2.67953 q^{57} -5.48214 q^{59} -6.80160 q^{61} +8.14363 q^{63} -1.06751 q^{65} -9.49860 q^{67} +6.15306 q^{69} +7.53405 q^{71} +16.4419 q^{73} -0.873518 q^{75} -3.02596 q^{79} +2.71491 q^{81} +13.1746 q^{83} +0.252963 q^{85} -1.40813 q^{87} +12.4460 q^{89} +3.88625 q^{91} +0.689198 q^{93} +3.06751 q^{95} +8.77719 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43}+ \cdots + 12 q^{95}+O(q^{100}) $ 6 * q - 2 * q^3 + 6 * q^5 + 4 * q^7 + 4 * q^9 - 2 * q^15 + 8 * q^17 + 12 * q^19 - 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 + 16 * q^29 - 4 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 + 32 * q^41 - 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 + 40 * q^51 + 8 * q^53 - 16 * q^57 + 4 * q^59 + 16 * q^61 + 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 + 16 * q^73 - 2 * q^75 - 10 * q^81 + 12 * q^83 + 8 * q^85 - 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 + 12 * q^95

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
$3$	−0.873518	−0.504326	−0.252163	−	0.967685i	$-0.581142\pi$
$3$	−0.873518	−0.504326	−0.252163	+	0.967685i	$0.581142\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.64048	−1.37597	−0.687986	−	0.725724i	$-0.741505\pi$
$7$	−3.64048	−1.37597	−0.687986	+	0.725724i	$0.741505\pi$
$8$	0	0
$9$	−2.23697	−0.745655
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.06751	−0.296074	−0.148037	−	0.988982i	$-0.547295\pi$
$13$	−1.06751	−0.296074	−0.148037	+	0.988982i	$0.547295\pi$
$14$	0	0
$15$	−0.873518	−0.225541
$16$	0	0
$17$	0.252963	0.0613526	0.0306763	−	0.999529i	$-0.490234\pi$
$17$	0.252963	0.0613526	0.0306763	+	0.999529i	$0.490234\pi$
$18$	0	0
$19$	3.06751	0.703735	0.351868	−	0.936050i	$-0.385547\pi$
$19$	3.06751	0.703735	0.351868	+	0.936050i	$0.385547\pi$
$20$	0	0
$21$	3.18003	0.693939
$22$	0	0
$23$	−7.04399	−1.46877	−0.734387	−	0.678731i	$-0.762530\pi$
$23$	−7.04399	−1.46877	−0.734387	+	0.678731i	$0.762530\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.57459	0.880379
$28$	0	0
$29$	1.61202	0.299344	0.149672	−	0.988736i	$-0.452178\pi$
$29$	1.61202	0.299344	0.149672	+	0.988736i	$0.452178\pi$
$30$	0	0
$31$	−0.788991	−0.141707	−0.0708535	−	0.997487i	$-0.522572\pi$
$31$	−0.788991	−0.141707	−0.0708535	+	0.997487i	$0.522572\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.64048	−0.615353
$36$	0	0
$37$	−4.91208	−0.807540	−0.403770	−	0.914860i	$-0.632300\pi$
$37$	−4.91208	−0.807540	−0.403770	+	0.914860i	$0.632300\pi$
$38$	0	0
$39$	0.932490	0.149318
$40$	0	0
$41$	6.39545	0.998802	0.499401	−	0.866371i	$-0.333554\pi$
$41$	6.39545	0.998802	0.499401	+	0.866371i	$0.333554\pi$
$42$	0	0
$43$	−2.63065	−0.401170	−0.200585	−	0.979676i	$-0.564284\pi$
$43$	−2.63065	−0.401170	−0.200585	+	0.979676i	$0.564284\pi$
$44$	0	0
$45$	−2.23697	−0.333467
$46$	0	0
$47$	−7.78559	−1.13565	−0.567823	−	0.823151i	$-0.692214\pi$
$47$	−7.78559	−1.13565	−0.567823	+	0.823151i	$0.692214\pi$
$48$	0	0
$49$	6.25309	0.893299
$50$	0	0
$51$	−0.220968	−0.0309417
$52$	0	0
$53$	−6.32291	−0.868519	−0.434259	−	0.900788i	$-0.642990\pi$
$53$	−6.32291	−0.868519	−0.434259	+	0.900788i	$0.642990\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.67953	−0.354912
$58$	0	0
$59$	−5.48214	−0.713714	−0.356857	−	0.934159i	$-0.616152\pi$
$59$	−5.48214	−0.713714	−0.356857	+	0.934159i	$0.616152\pi$
$60$	0	0
$61$	−6.80160	−0.870856	−0.435428	−	0.900224i	$-0.643403\pi$
$61$	−6.80160	−0.870856	−0.435428	+	0.900224i	$0.643403\pi$
$62$	0	0
$63$	8.14363	1.02600
$64$	0	0
$65$	−1.06751	−0.132408
$66$	0	0
$67$	−9.49860	−1.16044	−0.580219	−	0.814460i	$-0.697033\pi$
$67$	−9.49860	−1.16044	−0.580219	+	0.814460i	$0.697033\pi$
$68$	0	0
$69$	6.15306	0.740741
$70$	0	0
$71$	7.53405	0.894127	0.447064	−	0.894502i	$-0.352470\pi$
$71$	7.53405	0.894127	0.447064	+	0.894502i	$0.352470\pi$
$72$	0	0
$73$	16.4419	1.92438	0.962192	−	0.272374i	$-0.0878087\pi$
$73$	16.4419	1.92438	0.962192	+	0.272374i	$0.0878087\pi$
$74$	0	0
$75$	−0.873518	−0.100865
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.02596	−0.340447	−0.170223	−	0.985406i	$-0.554449\pi$
$79$	−3.02596	−0.340447	−0.170223	+	0.985406i	$0.554449\pi$
$80$	0	0
$81$	2.71491	0.301657
$82$	0	0
$83$	13.1746	1.44610	0.723050	−	0.690796i	$-0.242740\pi$
$83$	13.1746	1.44610	0.723050	+	0.690796i	$0.242740\pi$
$84$	0	0
$85$	0.252963	0.0274377
$86$	0	0
$87$	−1.40813	−0.150967
$88$	0	0
$89$	12.4460	1.31927	0.659634	−	0.751587i	$-0.270711\pi$
$89$	12.4460	1.31927	0.659634	+	0.751587i	$0.270711\pi$
$90$	0	0
$91$	3.88625	0.407390
$92$	0	0
$93$	0.689198	0.0714665
$94$	0	0
$95$	3.06751	0.314720
$96$	0	0
$97$	8.77719	0.891188	0.445594	−	0.895235i	$-0.352992\pi$
$97$	8.77719	0.891188	0.445594	+	0.895235i	$0.352992\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.1579	1.60777	0.803884	−	0.594787i	$-0.202763\pi$
$101$	16.1579	1.60777	0.803884	+	0.594787i	$0.202763\pi$
$102$	0	0
$103$	4.58972	0.452239	0.226119	−	0.974100i	$-0.427396\pi$
$103$	4.58972	0.452239	0.226119	+	0.974100i	$0.427396\pi$
$104$	0	0
$105$	3.18003	0.310339
$106$	0	0
$107$	0.823893	0.0796487	0.0398244	−	0.999207i	$-0.487320\pi$
$107$	0.823893	0.0796487	0.0398244	+	0.999207i	$0.487320\pi$
$108$	0	0
$109$	−6.43124	−0.616001	−0.308000	−	0.951386i	$-0.599660\pi$
$109$	−6.43124	−0.616001	−0.308000	+	0.951386i	$0.599660\pi$
$110$	0	0
$111$	4.29079	0.407264
$112$	0	0
$113$	−5.96800	−0.561423	−0.280711	−	0.959792i	$-0.590570\pi$
$113$	−5.96800	−0.561423	−0.280711	+	0.959792i	$0.590570\pi$
$114$	0	0
$115$	−7.04399	−0.656856
$116$	0	0
$117$	2.38798	0.220769
$118$	0	0
$119$	−0.920908	−0.0844195
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−5.58654	−0.503722
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.41320	−0.835287	−0.417643	−	0.908611i	$-0.637144\pi$
$127$	−9.41320	−0.835287	−0.417643	+	0.908611i	$0.637144\pi$
$128$	0	0
$129$	2.29792	0.202321
$130$	0	0
$131$	−14.2507	−1.24509	−0.622545	−	0.782584i	$-0.713901\pi$
$131$	−14.2507	−1.24509	−0.622545	+	0.782584i	$0.713901\pi$
$132$	0	0
$133$	−11.1672	−0.968320
$134$	0	0
$135$	4.57459	0.393718
$136$	0	0
$137$	21.6893	1.85304	0.926519	−	0.376247i	$-0.122786\pi$
$137$	21.6893	1.85304	0.926519	+	0.376247i	$0.122786\pi$
$138$	0	0
$139$	8.26252	0.700818	0.350409	−	0.936597i	$-0.386043\pi$
$139$	8.26252	0.700818	0.350409	+	0.936597i	$0.386043\pi$
$140$	0	0
$141$	6.80086	0.572736
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.61202	0.133871
$146$	0	0
$147$	−5.46219	−0.450514
$148$	0	0
$149$	6.82953	0.559497	0.279749	−	0.960073i	$-0.409749\pi$
$149$	6.82953	0.559497	0.279749	+	0.960073i	$0.409749\pi$
$150$	0	0
$151$	17.0217	1.38520	0.692602	−	0.721320i	$-0.256464\pi$
$151$	17.0217	1.38520	0.692602	+	0.721320i	$0.256464\pi$
$152$	0	0
$153$	−0.565870	−0.0457479
$154$	0	0
$155$	−0.788991	−0.0633733
$156$	0	0
$157$	20.4419	1.63144	0.815722	−	0.578444i	$-0.196340\pi$
$157$	20.4419	1.63144	0.815722	+	0.578444i	$0.196340\pi$
$158$	0	0
$159$	5.52318	0.438017
$160$	0	0
$161$	25.6435	2.02099
$162$	0	0
$163$	11.3516	0.889129	0.444565	−	0.895747i	$-0.353358\pi$
$163$	11.3516	0.889129	0.444565	+	0.895747i	$0.353358\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.6105	0.898449	0.449225	−	0.893419i	$-0.351700\pi$
$167$	11.6105	0.898449	0.449225	+	0.893419i	$0.351700\pi$
$168$	0	0
$169$	−11.8604	−0.912340
$170$	0	0
$171$	−6.86192	−0.524744
$172$	0	0
$173$	−14.5576	−1.10680	−0.553398	−	0.832917i	$-0.686669\pi$
$173$	−14.5576	−1.10680	−0.553398	+	0.832917i	$0.686669\pi$
$174$	0	0
$175$	−3.64048	−0.275194
$176$	0	0
$177$	4.78875	0.359944
$178$	0	0
$179$	−15.3520	−1.14746	−0.573730	−	0.819044i	$-0.694504\pi$
$179$	−15.3520	−1.14746	−0.573730	+	0.819044i	$0.694504\pi$
$180$	0	0
$181$	19.4760	1.44764	0.723819	−	0.689990i	$-0.242385\pi$
$181$	19.4760	1.44764	0.723819	+	0.689990i	$0.242385\pi$
$182$	0	0
$183$	5.94132	0.439195
$184$	0	0
$185$	−4.91208	−0.361143
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−16.6537	−1.21138
$190$	0	0
$191$	−23.2241	−1.68044	−0.840220	−	0.542246i	$-0.817574\pi$
$191$	−23.2241	−1.68044	−0.840220	+	0.542246i	$0.817574\pi$
$192$	0	0
$193$	18.3389	1.32006	0.660032	−	0.751237i	$-0.270543\pi$
$193$	18.3389	1.32006	0.660032	+	0.751237i	$0.270543\pi$
$194$	0	0
$195$	0.932490	0.0667770
$196$	0	0
$197$	−10.5899	−0.754496	−0.377248	−	0.926112i	$-0.623129\pi$
$197$	−10.5899	−0.754496	−0.377248	+	0.926112i	$0.623129\pi$
$198$	0	0
$199$	−14.7745	−1.04733	−0.523667	−	0.851923i	$-0.675437\pi$
$199$	−14.7745	−1.04733	−0.523667	+	0.851923i	$0.675437\pi$
$200$	0	0
$201$	8.29720	0.585239
$202$	0	0
$203$	−5.86851	−0.411889
$204$	0	0
$205$	6.39545	0.446678
$206$	0	0
$207$	15.7572	1.09520
$208$	0	0
$209$	0	0
$210$	0	0
$211$	23.1907	1.59652	0.798258	−	0.602316i	$-0.205755\pi$
$211$	23.1907	1.59652	0.798258	+	0.602316i	$0.205755\pi$
$212$	0	0
$213$	−6.58113	−0.450932
$214$	0	0
$215$	−2.63065	−0.179409
$216$	0	0
$217$	2.87230	0.194985
$218$	0	0
$219$	−14.3623	−0.970517
$220$	0	0
$221$	−0.270041	−0.0181649
$222$	0	0
$223$	−1.51136	−0.101208	−0.0506041	−	0.998719i	$-0.516115\pi$
$223$	−1.51136	−0.101208	−0.0506041	+	0.998719i	$0.516115\pi$
$224$	0	0
$225$	−2.23697	−0.149131
$226$	0	0
$227$	12.9716	0.860955	0.430478	−	0.902601i	$-0.358345\pi$
$227$	12.9716	0.860955	0.430478	+	0.902601i	$0.358345\pi$
$228$	0	0
$229$	24.8265	1.64058	0.820289	−	0.571949i	$-0.193813\pi$
$229$	24.8265	1.64058	0.820289	+	0.571949i	$0.193813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.4206	−1.07575	−0.537874	−	0.843025i	$-0.680772\pi$
$233$	−16.4206	−1.07575	−0.537874	+	0.843025i	$0.680772\pi$
$234$	0	0
$235$	−7.78559	−0.507876
$236$	0	0
$237$	2.64323	0.171696
$238$	0	0
$239$	22.2830	1.44137	0.720683	−	0.693265i	$-0.243828\pi$
$239$	22.2830	1.44137	0.720683	+	0.693265i	$0.243828\pi$
$240$	0	0
$241$	22.5887	1.45507	0.727534	−	0.686071i	$-0.240666\pi$
$241$	22.5887	1.45507	0.727534	+	0.686071i	$0.240666\pi$
$242$	0	0
$243$	−16.0953	−1.03251
$244$	0	0
$245$	6.25309	0.399495
$246$	0	0
$247$	−3.27460	−0.208358
$248$	0	0
$249$	−11.5083	−0.729306
$250$	0	0
$251$	−13.6602	−0.862227	−0.431113	−	0.902298i	$-0.641879\pi$
$251$	−13.6602	−0.862227	−0.431113	+	0.902298i	$0.641879\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.220968	−0.0138376
$256$	0	0
$257$	−14.2393	−0.888221	−0.444110	−	0.895972i	$-0.646480\pi$
$257$	−14.2393	−0.888221	−0.444110	+	0.895972i	$0.646480\pi$
$258$	0	0
$259$	17.8823	1.11115
$260$	0	0
$261$	−3.60603	−0.223207
$262$	0	0
$263$	6.28952	0.387828	0.193914	−	0.981018i	$-0.437882\pi$
$263$	6.28952	0.387828	0.193914	+	0.981018i	$0.437882\pi$
$264$	0	0
$265$	−6.32291	−0.388413
$266$	0	0
$267$	−10.8718	−0.665341
$268$	0	0
$269$	21.6774	1.32169	0.660847	−	0.750520i	$-0.270197\pi$
$269$	21.6774	1.32169	0.660847	+	0.750520i	$0.270197\pi$
$270$	0	0
$271$	−3.40417	−0.206789	−0.103394	−	0.994640i	$-0.532970\pi$
$271$	−3.40417	−0.206789	−0.103394	+	0.994640i	$0.532970\pi$
$272$	0	0
$273$	−3.39471	−0.205457
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.7849	−1.12868	−0.564338	−	0.825544i	$-0.690868\pi$
$277$	−18.7849	−1.12868	−0.564338	+	0.825544i	$0.690868\pi$
$278$	0	0
$279$	1.76495	0.105664
$280$	0	0
$281$	15.1832	0.905754	0.452877	−	0.891573i	$-0.350398\pi$
$281$	15.1832	0.905754	0.452877	+	0.891573i	$0.350398\pi$
$282$	0	0
$283$	10.2021	0.606450	0.303225	−	0.952919i	$-0.401937\pi$
$283$	10.2021	0.606450	0.303225	+	0.952919i	$0.401937\pi$
$284$	0	0
$285$	−2.67953	−0.158721
$286$	0	0
$287$	−23.2825	−1.37432
$288$	0	0
$289$	−16.9360	−0.996236
$290$	0	0
$291$	−7.66703	−0.449449
$292$	0	0
$293$	−2.84914	−0.166449	−0.0832243	−	0.996531i	$-0.526522\pi$
$293$	−2.84914	−0.166449	−0.0832243	+	0.996531i	$0.526522\pi$
$294$	0	0
$295$	−5.48214	−0.319182
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.51954	0.434866
$300$	0	0
$301$	9.57683	0.551999
$302$	0	0
$303$	−14.1142	−0.810839
$304$	0	0
$305$	−6.80160	−0.389459
$306$	0	0
$307$	9.11918	0.520459	0.260230	−	0.965547i	$-0.416202\pi$
$307$	9.11918	0.520459	0.260230	+	0.965547i	$0.416202\pi$
$308$	0	0
$309$	−4.00921	−0.228076
$310$	0	0
$311$	14.9333	0.846792	0.423396	−	0.905945i	$-0.360838\pi$
$311$	14.9333	0.846792	0.423396	+	0.905945i	$0.360838\pi$
$312$	0	0
$313$	13.9590	0.789012	0.394506	−	0.918893i	$-0.370916\pi$
$313$	13.9590	0.789012	0.394506	+	0.918893i	$0.370916\pi$
$314$	0	0
$315$	8.14363	0.458841
$316$	0	0
$317$	15.2502	0.856534	0.428267	−	0.903652i	$-0.359124\pi$
$317$	15.2502	0.856534	0.428267	+	0.903652i	$0.359124\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−0.719686	−0.0401689
$322$	0	0
$323$	0.775967	0.0431760
$324$	0	0
$325$	−1.06751	−0.0592148
$326$	0	0
$327$	5.61780	0.310665
$328$	0	0
$329$	28.3433	1.56262
$330$	0	0
$331$	17.1961	0.945183	0.472591	−	0.881282i	$-0.343319\pi$
$331$	17.1961	0.945183	0.472591	+	0.881282i	$0.343319\pi$
$332$	0	0
$333$	10.9881	0.602147
$334$	0	0
$335$	−9.49860	−0.518964
$336$	0	0
$337$	−27.4167	−1.49348	−0.746742	−	0.665114i	$-0.768383\pi$
$337$	−27.4167	−1.49348	−0.746742	+	0.665114i	$0.768383\pi$
$338$	0	0
$339$	5.21316	0.283140
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.71910	0.146818
$344$	0	0
$345$	6.15306	0.331269
$346$	0	0
$347$	−1.52482	−0.0818565	−0.0409283	−	0.999162i	$-0.513032\pi$
$347$	−1.52482	−0.0818565	−0.0409283	+	0.999162i	$0.513032\pi$
$348$	0	0
$349$	8.48480	0.454181	0.227091	−	0.973874i	$-0.427079\pi$
$349$	8.48480	0.454181	0.227091	+	0.973874i	$0.427079\pi$
$350$	0	0
$351$	−4.88342	−0.260658
$352$	0	0
$353$	3.14700	0.167498	0.0837490	−	0.996487i	$-0.473311\pi$
$353$	3.14700	0.167498	0.0837490	+	0.996487i	$0.473311\pi$
$354$	0	0
$355$	7.53405	0.399866
$356$	0	0
$357$	0.804430	0.0425749
$358$	0	0
$359$	−22.0212	−1.16223	−0.581117	−	0.813820i	$-0.697384\pi$
$359$	−22.0212	−1.16223	−0.581117	+	0.813820i	$0.697384\pi$
$360$	0	0
$361$	−9.59038	−0.504757
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16.4419	0.860610
$366$	0	0
$367$	−15.8706	−0.828438	−0.414219	−	0.910177i	$-0.635945\pi$
$367$	−15.8706	−0.828438	−0.414219	+	0.910177i	$0.635945\pi$
$368$	0	0
$369$	−14.3064	−0.744762
$370$	0	0
$371$	23.0184	1.19506
$372$	0	0
$373$	9.75306	0.504994	0.252497	−	0.967598i	$-0.418748\pi$
$373$	9.75306	0.504994	0.252497	+	0.967598i	$0.418748\pi$
$374$	0	0
$375$	−0.873518	−0.0451083
$376$	0	0
$377$	−1.72084	−0.0886280
$378$	0	0
$379$	−7.56281	−0.388475	−0.194238	−	0.980955i	$-0.562223\pi$
$379$	−7.56281	−0.388475	−0.194238	+	0.980955i	$0.562223\pi$
$380$	0	0
$381$	8.22260	0.421257
$382$	0	0
$383$	−8.24307	−0.421201	−0.210601	−	0.977572i	$-0.567542\pi$
$383$	−8.24307	−0.421201	−0.210601	+	0.977572i	$0.567542\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.88467	0.299135
$388$	0	0
$389$	−24.6089	−1.24772	−0.623860	−	0.781536i	$-0.714436\pi$
$389$	−24.6089	−1.24772	−0.623860	+	0.781536i	$0.714436\pi$
$390$	0	0
$391$	−1.78187	−0.0901131
$392$	0	0
$393$	12.4483	0.627932
$394$	0	0
$395$	−3.02596	−0.152252
$396$	0	0
$397$	−24.6461	−1.23695	−0.618476	−	0.785804i	$-0.712250\pi$
$397$	−24.6461	−1.23695	−0.618476	+	0.785804i	$0.712250\pi$
$398$	0	0
$399$	9.75476	0.488349
$400$	0	0
$401$	−3.71811	−0.185673	−0.0928367	−	0.995681i	$-0.529593\pi$
$401$	−3.71811	−0.185673	−0.0928367	+	0.995681i	$0.529593\pi$
$402$	0	0
$403$	0.842256	0.0419557
$404$	0	0
$405$	2.71491	0.134905
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−5.95988	−0.294697	−0.147348	−	0.989085i	$-0.547074\pi$
$409$	−5.95988	−0.294697	−0.147348	+	0.989085i	$0.547074\pi$
$410$	0	0
$411$	−18.9460	−0.934536
$412$	0	0
$413$	19.9576	0.982050
$414$	0	0
$415$	13.1746	0.646716
$416$	0	0
$417$	−7.21746	−0.353441
$418$	0	0
$419$	−7.88194	−0.385058	−0.192529	−	0.981291i	$-0.561669\pi$
$419$	−7.88194	−0.385058	−0.192529	+	0.981291i	$0.561669\pi$
$420$	0	0
$421$	31.5497	1.53764	0.768819	−	0.639467i	$-0.220845\pi$
$421$	31.5497	1.53764	0.768819	+	0.639467i	$0.220845\pi$
$422$	0	0
$423$	17.4161	0.846800
$424$	0	0
$425$	0.252963	0.0122705
$426$	0	0
$427$	24.7611	1.19827
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.21311	0.395612	0.197806	−	0.980241i	$-0.436618\pi$
$431$	8.21311	0.395612	0.197806	+	0.980241i	$0.436618\pi$
$432$	0	0
$433$	−26.4771	−1.27241	−0.636203	−	0.771522i	$-0.719496\pi$
$433$	−26.4771	−1.27241	−0.636203	+	0.771522i	$0.719496\pi$
$434$	0	0
$435$	−1.40813	−0.0675145
$436$	0	0
$437$	−21.6075	−1.03363
$438$	0	0
$439$	6.90682	0.329645	0.164822	−	0.986323i	$-0.447295\pi$
$439$	6.90682	0.329645	0.164822	+	0.986323i	$0.447295\pi$
$440$	0	0
$441$	−13.9880	−0.666093
$442$	0	0
$443$	−15.3657	−0.730048	−0.365024	−	0.930998i	$-0.618939\pi$
$443$	−15.3657	−0.730048	−0.365024	+	0.930998i	$0.618939\pi$
$444$	0	0
$445$	12.4460	0.589995
$446$	0	0
$447$	−5.96572	−0.282169
$448$	0	0
$449$	−28.9971	−1.36846	−0.684230	−	0.729267i	$-0.739861\pi$
$449$	−28.9971	−1.36846	−0.684230	+	0.729267i	$0.739861\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−14.8687	−0.698594
$454$	0	0
$455$	3.88625	0.182190
$456$	0	0
$457$	11.6957	0.547102	0.273551	−	0.961858i	$-0.411802\pi$
$457$	11.6957	0.547102	0.273551	+	0.961858i	$0.411802\pi$
$458$	0	0
$459$	1.15720	0.0540136
$460$	0	0
$461$	14.1531	0.659177	0.329589	−	0.944125i	$-0.393090\pi$
$461$	14.1531	0.659177	0.329589	+	0.944125i	$0.393090\pi$
$462$	0	0
$463$	18.1587	0.843906	0.421953	−	0.906618i	$-0.361345\pi$
$463$	18.1587	0.843906	0.421953	+	0.906618i	$0.361345\pi$
$464$	0	0
$465$	0.689198	0.0319608
$466$	0	0
$467$	23.3999	1.08282	0.541408	−	0.840760i	$-0.317891\pi$
$467$	23.3999	1.08282	0.541408	+	0.840760i	$0.317891\pi$
$468$	0	0
$469$	34.5794	1.59673
$470$	0	0
$471$	−17.8564	−0.822780
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3.06751	0.140747
$476$	0	0
$477$	14.1441	0.647616
$478$	0	0
$479$	28.3210	1.29402	0.647010	−	0.762481i	$-0.276019\pi$
$479$	28.3210	1.29402	0.647010	+	0.762481i	$0.276019\pi$
$480$	0	0
$481$	5.24369	0.239092
$482$	0	0
$483$	−22.4001	−1.01924
$484$	0	0
$485$	8.77719	0.398551
$486$	0	0
$487$	−34.4523	−1.56118	−0.780592	−	0.625041i	$-0.785082\pi$
$487$	−34.4523	−1.56118	−0.780592	+	0.625041i	$0.785082\pi$
$488$	0	0
$489$	−9.91587	−0.448411
$490$	0	0
$491$	28.0569	1.26619	0.633096	−	0.774074i	$-0.281784\pi$
$491$	28.0569	1.26619	0.633096	+	0.774074i	$0.281784\pi$
$492$	0	0
$493$	0.407781	0.0183655
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.4276	−1.23029
$498$	0	0
$499$	40.6597	1.82018	0.910089	−	0.414413i	$-0.136013\pi$
$499$	40.6597	1.82018	0.910089	+	0.414413i	$0.136013\pi$
$500$	0	0
$501$	−10.1420	−0.453111
$502$	0	0
$503$	35.5947	1.58709	0.793544	−	0.608513i	$-0.208234\pi$
$503$	35.5947	1.58709	0.793544	+	0.608513i	$0.208234\pi$
$504$	0	0
$505$	16.1579	0.719015
$506$	0	0
$507$	10.3603	0.460117
$508$	0	0
$509$	−20.1064	−0.891202	−0.445601	−	0.895232i	$-0.647010\pi$
$509$	−20.1064	−0.891202	−0.445601	+	0.895232i	$0.647010\pi$
$510$	0	0
$511$	−59.8565	−2.64790
$512$	0	0
$513$	14.0326	0.619554
$514$	0	0
$515$	4.58972	0.202247
$516$	0	0
$517$	0	0
$518$	0	0
$519$	12.7164	0.558186
$520$	0	0
$521$	23.0122	1.00818	0.504091	−	0.863651i	$-0.331828\pi$
$521$	23.0122	1.00818	0.504091	+	0.863651i	$0.331828\pi$
$522$	0	0
$523$	1.59213	0.0696191	0.0348096	−	0.999394i	$-0.488918\pi$
$523$	1.59213	0.0696191	0.0348096	+	0.999394i	$0.488918\pi$
$524$	0	0
$525$	3.18003	0.138788
$526$	0	0
$527$	−0.199586	−0.00869409
$528$	0	0
$529$	26.6178	1.15730
$530$	0	0
$531$	12.2634	0.532184
$532$	0	0
$533$	−6.82721	−0.295719
$534$	0	0
$535$	0.823893	0.0356200
$536$	0	0
$537$	13.4102	0.578694
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.4539	0.707410	0.353705	−	0.935357i	$-0.384922\pi$
$541$	16.4539	0.707410	0.353705	+	0.935357i	$0.384922\pi$
$542$	0	0
$543$	−17.0126	−0.730081
$544$	0	0
$545$	−6.43124	−0.275484
$546$	0	0
$547$	−36.0646	−1.54201	−0.771005	−	0.636829i	$-0.780246\pi$
$547$	−36.0646	−1.54201	−0.771005	+	0.636829i	$0.780246\pi$
$548$	0	0
$549$	15.2149	0.649358
$550$	0	0
$551$	4.94488	0.210659
$552$	0	0
$553$	11.0159	0.468445
$554$	0	0
$555$	4.29079	0.182134
$556$	0	0
$557$	−24.9630	−1.05772	−0.528859	−	0.848710i	$-0.677380\pi$
$557$	−24.9630	−1.05772	−0.528859	+	0.848710i	$0.677380\pi$
$558$	0	0
$559$	2.80825	0.118776
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.6458	1.37586	0.687929	−	0.725778i	$-0.258520\pi$
$563$	32.6458	1.37586	0.687929	+	0.725778i	$0.258520\pi$
$564$	0	0
$565$	−5.96800	−0.251076
$566$	0	0
$567$	−9.88358	−0.415072
$568$	0	0
$569$	−17.2126	−0.721590	−0.360795	−	0.932645i	$-0.617495\pi$
$569$	−17.2126	−0.721590	−0.360795	+	0.932645i	$0.617495\pi$
$570$	0	0
$571$	−38.8975	−1.62781	−0.813905	−	0.580997i	$-0.802663\pi$
$571$	−38.8975	−1.62781	−0.813905	+	0.580997i	$0.802663\pi$
$572$	0	0
$573$	20.2867	0.847489
$574$	0	0
$575$	−7.04399	−0.293755
$576$	0	0
$577$	13.8651	0.577212	0.288606	−	0.957448i	$-0.406808\pi$
$577$	13.8651	0.577212	0.288606	+	0.957448i	$0.406808\pi$
$578$	0	0
$579$	−16.0194	−0.665743
$580$	0	0
$581$	−47.9619	−1.98979
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.38798	0.0987310
$586$	0	0
$587$	−6.62340	−0.273377	−0.136688	−	0.990614i	$-0.543646\pi$
$587$	−6.62340	−0.273377	−0.136688	+	0.990614i	$0.543646\pi$
$588$	0	0
$589$	−2.42024	−0.0997241
$590$	0	0
$591$	9.25043	0.380512
$592$	0	0
$593$	11.2348	0.461357	0.230679	−	0.973030i	$-0.425905\pi$
$593$	11.2348	0.461357	0.230679	+	0.973030i	$0.425905\pi$
$594$	0	0
$595$	−0.920908	−0.0377535
$596$	0	0
$597$	12.9058	0.528198
$598$	0	0
$599$	11.6028	0.474077	0.237038	−	0.971500i	$-0.423823\pi$
$599$	11.6028	0.474077	0.237038	+	0.971500i	$0.423823\pi$
$600$	0	0
$601$	47.2759	1.92843	0.964213	−	0.265129i	$-0.0854147\pi$
$601$	47.2759	1.92843	0.964213	+	0.265129i	$0.0854147\pi$
$602$	0	0
$603$	21.2480	0.865287
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−38.1056	−1.54666	−0.773330	−	0.634004i	$-0.781410\pi$
$607$	−38.1056	−1.54666	−0.773330	+	0.634004i	$0.781410\pi$
$608$	0	0
$609$	5.12625	0.207726
$610$	0	0
$611$	8.31120	0.336235
$612$	0	0
$613$	−17.0761	−0.689698	−0.344849	−	0.938658i	$-0.612070\pi$
$613$	−17.0761	−0.689698	−0.344849	+	0.938658i	$0.612070\pi$
$614$	0	0
$615$	−5.58654	−0.225271
$616$	0	0
$617$	12.5333	0.504572	0.252286	−	0.967653i	$-0.418818\pi$
$617$	12.5333	0.504572	0.252286	+	0.967653i	$0.418818\pi$
$618$	0	0
$619$	−15.8379	−0.636577	−0.318289	−	0.947994i	$-0.603108\pi$
$619$	−15.8379	−0.636577	−0.318289	+	0.947994i	$0.603108\pi$
$620$	0	0
$621$	−32.2234	−1.29308
$622$	0	0
$623$	−45.3092	−1.81528
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.24258	−0.0495447
$630$	0	0
$631$	34.7403	1.38299	0.691494	−	0.722382i	$-0.256953\pi$
$631$	34.7403	1.38299	0.691494	+	0.722382i	$0.256953\pi$
$632$	0	0
$633$	−20.2575	−0.805164
$634$	0	0
$635$	−9.41320	−0.373552
$636$	0	0
$637$	−6.67524	−0.264483
$638$	0	0
$639$	−16.8534	−0.666711
$640$	0	0
$641$	3.72059	0.146954	0.0734772	−	0.997297i	$-0.476590\pi$
$641$	3.72059	0.146954	0.0734772	+	0.997297i	$0.476590\pi$
$642$	0	0
$643$	−9.61585	−0.379212	−0.189606	−	0.981860i	$-0.560721\pi$
$643$	−9.61585	−0.379212	−0.189606	+	0.981860i	$0.560721\pi$
$644$	0	0
$645$	2.29792	0.0904806
$646$	0	0
$647$	18.1587	0.713892	0.356946	−	0.934125i	$-0.383818\pi$
$647$	18.1587	0.713892	0.356946	+	0.934125i	$0.383818\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.50901	−0.0983359
$652$	0	0
$653$	24.9382	0.975907	0.487954	−	0.872870i	$-0.337744\pi$
$653$	24.9382	0.975907	0.487954	+	0.872870i	$0.337744\pi$
$654$	0	0
$655$	−14.2507	−0.556822
$656$	0	0
$657$	−36.7800	−1.43493
$658$	0	0
$659$	−17.2527	−0.672071	−0.336035	−	0.941849i	$-0.609086\pi$
$659$	−17.2527	−0.672071	−0.336035	+	0.941849i	$0.609086\pi$
$660$	0	0
$661$	17.4811	0.679936	0.339968	−	0.940437i	$-0.389584\pi$
$661$	17.4811	0.679936	0.339968	+	0.940437i	$0.389584\pi$
$662$	0	0
$663$	0.235886	0.00916104
$664$	0	0
$665$	−11.1672	−0.433046
$666$	0	0
$667$	−11.3550	−0.439669
$668$	0	0
$669$	1.32020	0.0510420
$670$	0	0
$671$	0	0
$672$	0	0
$673$	24.7907	0.955611	0.477806	−	0.878466i	$-0.341432\pi$
$673$	24.7907	0.955611	0.477806	+	0.878466i	$0.341432\pi$
$674$	0	0
$675$	4.57459	0.176076
$676$	0	0
$677$	42.4611	1.63191	0.815957	−	0.578112i	$-0.196210\pi$
$677$	42.4611	1.63191	0.815957	+	0.578112i	$0.196210\pi$
$678$	0	0
$679$	−31.9532	−1.22625
$680$	0	0
$681$	−11.3309	−0.434202
$682$	0	0
$683$	48.3820	1.85128	0.925642	−	0.378401i	$-0.123526\pi$
$683$	48.3820	1.85128	0.925642	+	0.378401i	$0.123526\pi$
$684$	0	0
$685$	21.6893	0.828704
$686$	0	0
$687$	−21.6864	−0.827386
$688$	0	0
$689$	6.74977	0.257146
$690$	0	0
$691$	11.3397	0.431382	0.215691	−	0.976462i	$-0.430800\pi$
$691$	11.3397	0.431382	0.215691	+	0.976462i	$0.430800\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.26252	0.313415
$696$	0	0
$697$	1.61781	0.0612791
$698$	0	0
$699$	14.3437	0.542527
$700$	0	0
$701$	4.06603	0.153572	0.0767859	−	0.997048i	$-0.475534\pi$
$701$	4.06603	0.153572	0.0767859	+	0.997048i	$0.475534\pi$
$702$	0	0
$703$	−15.0678	−0.568295
$704$	0	0
$705$	6.80086	0.256135
$706$	0	0
$707$	−58.8224	−2.21224
$708$	0	0
$709$	39.9682	1.50104	0.750518	−	0.660850i	$-0.229804\pi$
$709$	39.9682	1.50104	0.750518	+	0.660850i	$0.229804\pi$
$710$	0	0
$711$	6.76896	0.253856
$712$	0	0
$713$	5.55765	0.208135
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.4646	−0.726918
$718$	0	0
$719$	27.8865	1.03999	0.519996	−	0.854169i	$-0.325934\pi$
$719$	27.8865	1.03999	0.519996	+	0.854169i	$0.325934\pi$
$720$	0	0
$721$	−16.7088	−0.622268
$722$	0	0
$723$	−19.7317	−0.733829
$724$	0	0
$725$	1.61202	0.0598688
$726$	0	0
$727$	−51.5024	−1.91012	−0.955059	−	0.296417i	$-0.904208\pi$
$727$	−51.5024	−1.91012	−0.955059	+	0.296417i	$0.904208\pi$
$728$	0	0
$729$	5.91479	0.219066
$730$	0	0
$731$	−0.665458	−0.0246128
$732$	0	0
$733$	−36.3991	−1.34443	−0.672215	−	0.740356i	$-0.734657\pi$
$733$	−36.3991	−1.34443	−0.672215	+	0.740356i	$0.734657\pi$
$734$	0	0
$735$	−5.46219	−0.201476
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−10.1048	−0.371710	−0.185855	−	0.982577i	$-0.559505\pi$
$739$	−10.1048	−0.371710	−0.185855	+	0.982577i	$0.559505\pi$
$740$	0	0
$741$	2.86042	0.105080
$742$	0	0
$743$	−33.0572	−1.21275	−0.606375	−	0.795179i	$-0.707377\pi$
$743$	−33.0572	−1.21275	−0.606375	+	0.795179i	$0.707377\pi$
$744$	0	0
$745$	6.82953	0.250215
$746$	0	0
$747$	−29.4711	−1.07829
$748$	0	0
$749$	−2.99937	−0.109594
$750$	0	0
$751$	13.1368	0.479369	0.239684	−	0.970851i	$-0.422956\pi$
$751$	13.1368	0.479369	0.239684	+	0.970851i	$0.422956\pi$
$752$	0	0
$753$	11.9325	0.434843
$754$	0	0
$755$	17.0217	0.619482
$756$	0	0
$757$	−14.2945	−0.519541	−0.259770	−	0.965670i	$-0.583647\pi$
$757$	−14.2945	−0.519541	−0.259770	+	0.965670i	$0.583647\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.8121	−0.573187	−0.286594	−	0.958052i	$-0.592523\pi$
$761$	−15.8121	−0.573187	−0.286594	+	0.958052i	$0.592523\pi$
$762$	0	0
$763$	23.4128	0.847600
$764$	0	0
$765$	−0.565870	−0.0204591
$766$	0	0
$767$	5.85224	0.211312
$768$	0	0
$769$	12.3972	0.447053	0.223527	−	0.974698i	$-0.428243\pi$
$769$	12.3972	0.447053	0.223527	+	0.974698i	$0.428243\pi$
$770$	0	0
$771$	12.4383	0.447953
$772$	0	0
$773$	−17.2672	−0.621060	−0.310530	−	0.950564i	$-0.600506\pi$
$773$	−17.2672	−0.621060	−0.310530	+	0.950564i	$0.600506\pi$
$774$	0	0
$775$	−0.788991	−0.0283414
$776$	0	0
$777$	−15.6205	−0.560383
$778$	0	0
$779$	19.6181	0.702892
$780$	0	0
$781$	0	0
$782$	0	0
$783$	7.37431	0.263536
$784$	0	0
$785$	20.4419	0.729604
$786$	0	0
$787$	−22.0893	−0.787398	−0.393699	−	0.919240i	$-0.628805\pi$
$787$	−22.0893	−0.787398	−0.393699	+	0.919240i	$0.628805\pi$
$788$	0	0
$789$	−5.49401	−0.195592
$790$	0	0
$791$	21.7264	0.772502
$792$	0	0
$793$	7.26078	0.257838
$794$	0	0
$795$	5.52318	0.195887
$796$	0	0
$797$	12.7250	0.450742	0.225371	−	0.974273i	$-0.427641\pi$
$797$	12.7250	0.450742	0.225371	+	0.974273i	$0.427641\pi$
$798$	0	0
$799$	−1.96947	−0.0696748
$800$	0	0
$801$	−27.8412	−0.983719
$802$	0	0
$803$	0	0
$804$	0	0
$805$	25.6435	0.903815
$806$	0	0
$807$	−18.9356	−0.666565
$808$	0	0
$809$	−39.0381	−1.37251	−0.686254	−	0.727362i	$-0.740746\pi$
$809$	−39.0381	−1.37251	−0.686254	+	0.727362i	$0.740746\pi$
$810$	0	0
$811$	−22.3975	−0.786484	−0.393242	−	0.919435i	$-0.628647\pi$
$811$	−22.3975	−0.786484	−0.393242	+	0.919435i	$0.628647\pi$
$812$	0	0
$813$	2.97361	0.104289
$814$	0	0
$815$	11.3516	0.397631
$816$	0	0
$817$	−8.06955	−0.282318
$818$	0	0
$819$	−8.69341	−0.303772
$820$	0	0
$821$	6.19209	0.216105	0.108053	−	0.994145i	$-0.465538\pi$
$821$	6.19209	0.216105	0.108053	+	0.994145i	$0.465538\pi$
$822$	0	0
$823$	13.0630	0.455349	0.227674	−	0.973737i	$-0.426888\pi$
$823$	13.0630	0.455349	0.227674	+	0.973737i	$0.426888\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.4130	−0.988017	−0.494009	−	0.869457i	$-0.664469\pi$
$827$	−28.4130	−0.988017	−0.494009	+	0.869457i	$0.664469\pi$
$828$	0	0
$829$	−15.2772	−0.530599	−0.265300	−	0.964166i	$-0.585471\pi$
$829$	−15.2772	−0.530599	−0.265300	+	0.964166i	$0.585471\pi$
$830$	0	0
$831$	16.4090	0.569221
$832$	0	0
$833$	1.58180	0.0548062
$834$	0	0
$835$	11.6105	0.401799
$836$	0	0
$837$	−3.60931	−0.124756
$838$	0	0
$839$	−26.0887	−0.900684	−0.450342	−	0.892856i	$-0.648698\pi$
$839$	−26.0887	−0.900684	−0.450342	+	0.892856i	$0.648698\pi$
$840$	0	0
$841$	−26.4014	−0.910393
$842$	0	0
$843$	−13.2628	−0.456795
$844$	0	0
$845$	−11.8604	−0.408011
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.91169	−0.305848
$850$	0	0
$851$	34.6006	1.18609
$852$	0	0
$853$	−54.5956	−1.86932	−0.934659	−	0.355545i	$-0.884295\pi$
$853$	−54.5956	−1.86932	−0.934659	+	0.355545i	$0.884295\pi$
$854$	0	0
$855$	−6.86192	−0.234673
$856$	0	0
$857$	−6.40938	−0.218940	−0.109470	−	0.993990i	$-0.534915\pi$
$857$	−6.40938	−0.218940	−0.109470	+	0.993990i	$0.534915\pi$
$858$	0	0
$859$	−4.08632	−0.139423	−0.0697117	−	0.997567i	$-0.522208\pi$
$859$	−4.08632	−0.139423	−0.0697117	+	0.997567i	$0.522208\pi$
$860$	0	0
$861$	20.3377	0.693107
$862$	0	0
$863$	27.9434	0.951205	0.475602	−	0.879660i	$-0.342230\pi$
$863$	27.9434	0.951205	0.475602	+	0.879660i	$0.342230\pi$
$864$	0	0
$865$	−14.5576	−0.494974
$866$	0	0
$867$	14.7939	0.502428
$868$	0	0
$869$	0	0
$870$	0	0
$871$	10.1398	0.343576
$872$	0	0
$873$	−19.6343	−0.664519
$874$	0	0
$875$	−3.64048	−0.123071
$876$	0	0
$877$	15.0039	0.506645	0.253323	−	0.967382i	$-0.418477\pi$
$877$	15.0039	0.506645	0.253323	+	0.967382i	$0.418477\pi$
$878$	0	0
$879$	2.48878	0.0839443
$880$	0	0
$881$	−6.47635	−0.218194	−0.109097	−	0.994031i	$-0.534796\pi$
$881$	−6.47635	−0.218194	−0.109097	+	0.994031i	$0.534796\pi$
$882$	0	0
$883$	−7.24979	−0.243975	−0.121987	−	0.992532i	$-0.538927\pi$
$883$	−7.24979	−0.243975	−0.121987	+	0.992532i	$0.538927\pi$
$884$	0	0
$885$	4.78875	0.160972
$886$	0	0
$887$	−19.4233	−0.652170	−0.326085	−	0.945340i	$-0.605730\pi$
$887$	−19.4233	−0.652170	−0.326085	+	0.945340i	$0.605730\pi$
$888$	0	0
$889$	34.2686	1.14933
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−23.8824	−0.799194
$894$	0	0
$895$	−15.3520	−0.513160
$896$	0	0
$897$	−6.56845	−0.219314
$898$	0	0
$899$	−1.27187	−0.0424191
$900$	0	0
$901$	−1.59946	−0.0532859
$902$	0	0
$903$	−8.36553	−0.278388
$904$	0	0
$905$	19.4760	0.647403
$906$	0	0
$907$	−25.1730	−0.835854	−0.417927	−	0.908481i	$-0.637243\pi$
$907$	−25.1730	−0.835854	−0.417927	+	0.908481i	$0.637243\pi$
$908$	0	0
$909$	−36.1446	−1.19884
$910$	0	0
$911$	−52.5919	−1.74245	−0.871224	−	0.490885i	$-0.836674\pi$
$911$	−52.5919	−1.74245	−0.871224	+	0.490885i	$0.836674\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	5.94132	0.196414
$916$	0	0
$917$	51.8794	1.71321
$918$	0	0
$919$	13.9381	0.459774	0.229887	−	0.973217i	$-0.426164\pi$
$919$	13.9381	0.459774	0.229887	+	0.973217i	$0.426164\pi$
$920$	0	0
$921$	−7.96577	−0.262481
$922$	0	0
$923$	−8.04268	−0.264728
$924$	0	0
$925$	−4.91208	−0.161508
$926$	0	0
$927$	−10.2671	−0.337214
$928$	0	0
$929$	−41.5841	−1.36433	−0.682164	−	0.731199i	$-0.738961\pi$
$929$	−41.5841	−1.36433	−0.682164	+	0.731199i	$0.738961\pi$
$930$	0	0
$931$	19.1814	0.628646
$932$	0	0
$933$	−13.0446	−0.427059
$934$	0	0
$935$	0	0
$936$	0	0
$937$	23.0636	0.753455	0.376727	−	0.926324i	$-0.377049\pi$
$937$	23.0636	0.753455	0.376727	+	0.926324i	$0.377049\pi$
$938$	0	0
$939$	−12.1935	−0.397919
$940$	0	0
$941$	27.2409	0.888027	0.444014	−	0.896020i	$-0.353554\pi$
$941$	27.2409	0.888027	0.444014	+	0.896020i	$0.353554\pi$
$942$	0	0
$943$	−45.0495	−1.46701
$944$	0	0
$945$	−16.6537	−0.541744
$946$	0	0
$947$	44.2870	1.43913	0.719566	−	0.694424i	$-0.244341\pi$
$947$	44.2870	1.43913	0.719566	+	0.694424i	$0.244341\pi$
$948$	0	0
$949$	−17.5519	−0.569760
$950$	0	0
$951$	−13.3213	−0.431972
$952$	0	0
$953$	−1.93291	−0.0626132	−0.0313066	−	0.999510i	$-0.509967\pi$
$953$	−1.93291	−0.0626132	−0.0313066	+	0.999510i	$0.509967\pi$
$954$	0	0
$955$	−23.2241	−0.751515
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−78.9593	−2.54973
$960$	0	0
$961$	−30.3775	−0.979919
$962$	0	0
$963$	−1.84302	−0.0593905
$964$	0	0
$965$	18.3389	0.590351
$966$	0	0
$967$	43.0790	1.38533	0.692663	−	0.721261i	$-0.256437\pi$
$967$	43.0790	1.38533	0.692663	+	0.721261i	$0.256437\pi$
$968$	0	0
$969$	−0.677822	−0.0217748
$970$	0	0
$971$	27.2811	0.875493	0.437746	−	0.899099i	$-0.355777\pi$
$971$	27.2811	0.875493	0.437746	+	0.899099i	$0.355777\pi$
$972$	0	0
$973$	−30.0795	−0.964306
$974$	0	0
$975$	0.932490	0.0298636
$976$	0	0
$977$	5.21603	0.166876	0.0834378	−	0.996513i	$-0.473410\pi$
$977$	5.21603	0.166876	0.0834378	+	0.996513i	$0.473410\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	14.3865	0.459324
$982$	0	0
$983$	17.7199	0.565176	0.282588	−	0.959241i	$-0.408807\pi$
$983$	17.7199	0.565176	0.282588	+	0.959241i	$0.408807\pi$
$984$	0	0
$985$	−10.5899	−0.337421
$986$	0	0
$987$	−24.7584	−0.788068
$988$	0	0
$989$	18.5303	0.589229
$990$	0	0
$991$	−39.3005	−1.24842	−0.624211	−	0.781256i	$-0.714579\pi$
$991$	−39.3005	−1.24842	−0.624211	+	0.781256i	$0.714579\pi$
$992$	0	0
$993$	−15.0211	−0.476680
$994$	0	0
$995$	−14.7745	−0.468382
$996$	0	0
$997$	−59.0721	−1.87083	−0.935416	−	0.353550i	$-0.884974\pi$
$997$	−59.0721	−1.87083	−0.935416	+	0.353550i	$0.884974\pi$
$998$	0	0
$999$	−22.4707	−0.710942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bd.1.3	yes	6
4.3	odd	2		9680.2.a.da.1.4		6
11.10	odd	2		4840.2.a.bc.1.3	✓	6
44.43	even	2		9680.2.a.db.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.3	✓	6	11.10	odd	2
4840.2.a.bd.1.3	yes	6	1.1	even	1	trivial
9680.2.a.da.1.4		6	4.3	odd	2
9680.2.a.db.1.4		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.873518 q^{3} +1.00000 q^{5} -3.64048 q^{7} -2.23697 q^{9} -1.06751 q^{13} -0.873518 q^{15} +0.252963 q^{17} +3.06751 q^{19} +3.18003 q^{21} -7.04399 q^{23} +1.00000 q^{25} +4.57459 q^{27} +1.61202 q^{29} -0.788991 q^{31} -3.64048 q^{35} -4.91208 q^{37} +0.932490 q^{39} +6.39545 q^{41} -2.63065 q^{43} -2.23697 q^{45} -7.78559 q^{47} +6.25309 q^{49} -0.220968 q^{51} -6.32291 q^{53} -2.67953 q^{57} -5.48214 q^{59} -6.80160 q^{61} +8.14363 q^{63} -1.06751 q^{65} -9.49860 q^{67} +6.15306 q^{69} +7.53405 q^{71} +16.4419 q^{73} -0.873518 q^{75} -3.02596 q^{79} +2.71491 q^{81} +13.1746 q^{83} +0.252963 q^{85} -1.40813 q^{87} +12.4460 q^{89} +3.88625 q^{91} +0.689198 q^{93} +3.06751 q^{95} +8.77719 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43}+ \cdots + 12 q^{95}+O(q^{100}) \) 6 * q - 2 * q^3 + 6 * q^5 + 4 * q^7 + 4 * q^9 - 2 * q^15 + 8 * q^17 + 12 * q^19 - 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 + 16 * q^29 - 4 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 + 32 * q^41 - 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 + 40 * q^51 + 8 * q^53 - 16 * q^57 + 4 * q^59 + 16 * q^61 + 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 + 16 * q^73 - 2 * q^75 - 10 * q^81 + 12 * q^83 + 8 * q^85 - 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 + 12 * q^95

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