LMFDB - Embedded newform 2400.4.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.32340.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 42x - 14 $ x^3 - 42*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.30716$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +27.0060 q^{7} +9.00000 q^{9} -1.66805 q^{11} -18.2287 q^{13} -39.2346 q^{17} +157.149 q^{19} -81.0179 q^{21} +115.910 q^{23} -27.0000 q^{27} -136.149 q^{29} -27.8848 q^{31} +5.00414 q^{33} +358.950 q^{37} +54.6860 q^{39} -30.3522 q^{41} +212.684 q^{43} -630.358 q^{47} +386.322 q^{49} +117.704 q^{51} +568.865 q^{53} -471.448 q^{57} +209.157 q^{59} +429.895 q^{61} +243.054 q^{63} -775.233 q^{67} -347.731 q^{69} +79.3623 q^{71} -271.435 q^{73} -45.0473 q^{77} +206.990 q^{79} +81.0000 q^{81} +354.931 q^{83} +408.448 q^{87} -529.361 q^{89} -492.282 q^{91} +83.6543 q^{93} +1255.77 q^{97} -15.0124 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 7 q^{7} + 27 q^{9} + 21 q^{13} + 32 q^{17} + 19 q^{19} - 21 q^{21} + 60 q^{23} - 81 q^{27} + 44 q^{29} - 151 q^{31} + 330 q^{37} - 63 q^{39} + 82 q^{41} + 367 q^{43} - 246 q^{47} - 42 q^{49}+ \cdots - 351 q^{97}+O(q^{100}) $ 3 * q - 9 * q^3 + 7 * q^7 + 27 * q^9 + 21 * q^13 + 32 * q^17 + 19 * q^19 - 21 * q^21 + 60 * q^23 - 81 * q^27 + 44 * q^29 - 151 * q^31 + 330 * q^37 - 63 * q^39 + 82 * q^41 + 367 * q^43 - 246 * q^47 - 42 * q^49 - 96 * q^51 + 554 * q^53 - 57 * q^57 + 42 * q^59 - 27 * q^61 + 63 * q^63 - 837 * q^67 - 180 * q^69 - 1422 * q^71 + 498 * q^73 + 560 * q^77 - 688 * q^79 + 243 * q^81 + 702 * q^83 - 132 * q^87 - 1488 * q^89 - 735 * q^91 + 453 * q^93 - 351 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	27.0060	1.45819	0.729093	−	0.684415i	$-0.239942\pi$
$7$	27.0060	1.45819	0.729093	+	0.684415i	$0.239942\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−1.66805	−0.0457214	−0.0228607	−	0.999739i	$-0.507277\pi$
$11$	−1.66805	−0.0457214	−0.0228607	+	0.999739i	$0.507277\pi$
$12$	0	0
$13$	−18.2287	−0.388901	−0.194451	−	0.980912i	$-0.562292\pi$
$13$	−18.2287	−0.388901	−0.194451	+	0.980912i	$0.562292\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−39.2346	−0.559753	−0.279876	−	0.960036i	$-0.590293\pi$
$17$	−39.2346	−0.559753	−0.279876	+	0.960036i	$0.590293\pi$
$18$	0	0
$19$	157.149	1.89750	0.948750	−	0.316027i	$-0.102349\pi$
$19$	157.149	1.89750	0.948750	+	0.316027i	$0.102349\pi$
$20$	0	0
$21$	−81.0179	−0.841884
$22$	0	0
$23$	115.910	1.05083	0.525413	−	0.850847i	$-0.323911\pi$
$23$	115.910	1.05083	0.525413	+	0.850847i	$0.323911\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−136.149	−0.871803	−0.435901	−	0.899994i	$-0.643570\pi$
$29$	−136.149	−0.871803	−0.435901	+	0.899994i	$0.643570\pi$
$30$	0	0
$31$	−27.8848	−0.161556	−0.0807782	−	0.996732i	$-0.525741\pi$
$31$	−27.8848	−0.161556	−0.0807782	+	0.996732i	$0.525741\pi$
$32$	0	0
$33$	5.00414	0.0263973
$34$	0	0
$35$	0	0
$36$	0	0
$37$	358.950	1.59489	0.797447	−	0.603389i	$-0.206183\pi$
$37$	358.950	1.59489	0.797447	+	0.603389i	$0.206183\pi$
$38$	0	0
$39$	54.6860	0.224532
$40$	0	0
$41$	−30.3522	−0.115615	−0.0578075	−	0.998328i	$-0.518411\pi$
$41$	−30.3522	−0.115615	−0.0578075	+	0.998328i	$0.518411\pi$
$42$	0	0
$43$	212.684	0.754278	0.377139	−	0.926157i	$-0.376908\pi$
$43$	212.684	0.754278	0.377139	+	0.926157i	$0.376908\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−630.358	−1.95632	−0.978162	−	0.207846i	$-0.933355\pi$
$47$	−630.358	−1.95632	−0.978162	+	0.207846i	$0.933355\pi$
$48$	0	0
$49$	386.322	1.12630
$50$	0	0
$51$	117.704	0.323173
$52$	0	0
$53$	568.865	1.47433	0.737166	−	0.675712i	$-0.236164\pi$
$53$	568.865	1.47433	0.737166	+	0.675712i	$0.236164\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−471.448	−1.09552
$58$	0	0
$59$	209.157	0.461524	0.230762	−	0.973010i	$-0.425878\pi$
$59$	209.157	0.461524	0.230762	+	0.973010i	$0.425878\pi$
$60$	0	0
$61$	429.895	0.902335	0.451168	−	0.892439i	$-0.351008\pi$
$61$	429.895	0.902335	0.451168	+	0.892439i	$0.351008\pi$
$62$	0	0
$63$	243.054	0.486062
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−775.233	−1.41358	−0.706789	−	0.707424i	$-0.749857\pi$
$67$	−775.233	−1.41358	−0.706789	+	0.707424i	$0.749857\pi$
$68$	0	0
$69$	−347.731	−0.606695
$70$	0	0
$71$	79.3623	0.132656	0.0663279	−	0.997798i	$-0.478872\pi$
$71$	79.3623	0.132656	0.0663279	+	0.997798i	$0.478872\pi$
$72$	0	0
$73$	−271.435	−0.435193	−0.217597	−	0.976039i	$-0.569822\pi$
$73$	−271.435	−0.435193	−0.217597	+	0.976039i	$0.569822\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−45.0473	−0.0666703
$78$	0	0
$79$	206.990	0.294787	0.147394	−	0.989078i	$-0.452912\pi$
$79$	206.990	0.294787	0.147394	+	0.989078i	$0.452912\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	354.931	0.469383	0.234691	−	0.972070i	$-0.424592\pi$
$83$	354.931	0.469383	0.234691	+	0.972070i	$0.424592\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	408.448	0.503335
$88$	0	0
$89$	−529.361	−0.630474	−0.315237	−	0.949013i	$-0.602084\pi$
$89$	−529.361	−0.630474	−0.315237	+	0.949013i	$0.602084\pi$
$90$	0	0
$91$	−492.282	−0.567090
$92$	0	0
$93$	83.6543	0.0932747
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1255.77	1.31448	0.657238	−	0.753683i	$-0.271724\pi$
$97$	1255.77	1.31448	0.657238	+	0.753683i	$0.271724\pi$
$98$	0	0
$99$	−15.0124	−0.0152405
$100$	0	0
$101$	633.505	0.624119	0.312060	−	0.950062i	$-0.398981\pi$
$101$	633.505	0.624119	0.312060	+	0.950062i	$0.398981\pi$
$102$	0	0
$103$	13.2297	0.0126559	0.00632795	−	0.999980i	$-0.497986\pi$
$103$	13.2297	0.0126559	0.00632795	+	0.999980i	$0.497986\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1339.38	1.21012	0.605059	−	0.796181i	$-0.293149\pi$
$107$	1339.38	1.21012	0.605059	+	0.796181i	$0.293149\pi$
$108$	0	0
$109$	352.397	0.309665	0.154833	−	0.987941i	$-0.450516\pi$
$109$	352.397	0.309665	0.154833	+	0.987941i	$0.450516\pi$
$110$	0	0
$111$	−1076.85	−0.920813
$112$	0	0
$113$	−1040.37	−0.866108	−0.433054	−	0.901368i	$-0.642564\pi$
$113$	−1040.37	−0.866108	−0.433054	+	0.901368i	$0.642564\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−164.058	−0.129634
$118$	0	0
$119$	−1059.57	−0.816223
$120$	0	0
$121$	−1328.22	−0.997910
$122$	0	0
$123$	91.0565	0.0667503
$124$	0	0
$125$	0	0
$126$	0	0
$127$	600.967	0.419899	0.209950	−	0.977712i	$-0.432670\pi$
$127$	600.967	0.419899	0.209950	+	0.977712i	$0.432670\pi$
$128$	0	0
$129$	−638.051	−0.435483
$130$	0	0
$131$	−1388.30	−0.925924	−0.462962	−	0.886378i	$-0.653213\pi$
$131$	−1388.30	−0.925924	−0.462962	+	0.886378i	$0.653213\pi$
$132$	0	0
$133$	4243.97	2.76691
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2037.68	−1.27074	−0.635369	−	0.772208i	$-0.719152\pi$
$137$	−2037.68	−1.27074	−0.635369	+	0.772208i	$0.719152\pi$
$138$	0	0
$139$	−1060.28	−0.646992	−0.323496	−	0.946229i	$-0.604858\pi$
$139$	−1060.28	−0.646992	−0.323496	+	0.946229i	$0.604858\pi$
$140$	0	0
$141$	1891.07	1.12948
$142$	0	0
$143$	30.4063	0.0177811
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1158.97	−0.650272
$148$	0	0
$149$	−1792.57	−0.985589	−0.492795	−	0.870146i	$-0.664025\pi$
$149$	−1792.57	−0.985589	−0.492795	+	0.870146i	$0.664025\pi$
$150$	0	0
$151$	−2151.86	−1.15971	−0.579855	−	0.814720i	$-0.696891\pi$
$151$	−2151.86	−1.15971	−0.579855	+	0.814720i	$0.696891\pi$
$152$	0	0
$153$	−353.112	−0.186584
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2935.85	1.49239	0.746197	−	0.665725i	$-0.231877\pi$
$157$	2935.85	1.49239	0.746197	+	0.665725i	$0.231877\pi$
$158$	0	0
$159$	−1706.59	−0.851205
$160$	0	0
$161$	3130.27	1.53230
$162$	0	0
$163$	−308.456	−0.148222	−0.0741109	−	0.997250i	$-0.523612\pi$
$163$	−308.456	−0.148222	−0.0741109	+	0.997250i	$0.523612\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1895.45	−0.878290	−0.439145	−	0.898416i	$-0.644718\pi$
$167$	−1895.45	−0.878290	−0.439145	+	0.898416i	$0.644718\pi$
$168$	0	0
$169$	−1864.72	−0.848756
$170$	0	0
$171$	1414.34	0.632500
$172$	0	0
$173$	1101.94	0.484269	0.242135	−	0.970243i	$-0.422152\pi$
$173$	1101.94	0.484269	0.242135	+	0.970243i	$0.422152\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−627.471	−0.266461
$178$	0	0
$179$	−3262.39	−1.36225	−0.681125	−	0.732167i	$-0.738509\pi$
$179$	−3262.39	−1.36225	−0.681125	+	0.732167i	$0.738509\pi$
$180$	0	0
$181$	−2421.01	−0.994212	−0.497106	−	0.867690i	$-0.665604\pi$
$181$	−2421.01	−0.994212	−0.497106	+	0.867690i	$0.665604\pi$
$182$	0	0
$183$	−1289.69	−0.520964
$184$	0	0
$185$	0	0
$186$	0	0
$187$	65.4452	0.0255927
$188$	0	0
$189$	−729.161	−0.280628
$190$	0	0
$191$	−1303.37	−0.493761	−0.246881	−	0.969046i	$-0.579406\pi$
$191$	−1303.37	−0.493761	−0.246881	+	0.969046i	$0.579406\pi$
$192$	0	0
$193$	4312.21	1.60829	0.804144	−	0.594434i	$-0.202624\pi$
$193$	4312.21	1.60829	0.804144	+	0.594434i	$0.202624\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2549.49	0.922050	0.461025	−	0.887387i	$-0.347482\pi$
$197$	2549.49	0.922050	0.461025	+	0.887387i	$0.347482\pi$
$198$	0	0
$199$	−2220.15	−0.790865	−0.395433	−	0.918495i	$-0.629405\pi$
$199$	−2220.15	−0.790865	−0.395433	+	0.918495i	$0.629405\pi$
$200$	0	0
$201$	2325.70	0.816130
$202$	0	0
$203$	−3676.84	−1.27125
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1043.19	0.350275
$208$	0	0
$209$	−262.132	−0.0867564
$210$	0	0
$211$	3440.21	1.12243	0.561217	−	0.827668i	$-0.310333\pi$
$211$	3440.21	1.12243	0.561217	+	0.827668i	$0.310333\pi$
$212$	0	0
$213$	−238.087	−0.0765889
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−753.055	−0.235579
$218$	0	0
$219$	814.306	0.251259
$220$	0	0
$221$	715.194	0.217689
$222$	0	0
$223$	2113.47	0.634655	0.317328	−	0.948316i	$-0.397215\pi$
$223$	2113.47	0.634655	0.317328	+	0.948316i	$0.397215\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3561.84	1.04144	0.520721	−	0.853727i	$-0.325663\pi$
$227$	3561.84	1.04144	0.520721	+	0.853727i	$0.325663\pi$
$228$	0	0
$229$	1267.88	0.365870	0.182935	−	0.983125i	$-0.441440\pi$
$229$	1267.88	0.365870	0.182935	+	0.983125i	$0.441440\pi$
$230$	0	0
$231$	135.142	0.0384921
$232$	0	0
$233$	6588.59	1.85250	0.926251	−	0.376906i	$-0.123012\pi$
$233$	6588.59	1.85250	0.926251	+	0.376906i	$0.123012\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−620.970	−0.170195
$238$	0	0
$239$	6253.76	1.69256	0.846280	−	0.532738i	$-0.178837\pi$
$239$	6253.76	1.69256	0.846280	+	0.532738i	$0.178837\pi$
$240$	0	0
$241$	2916.51	0.779539	0.389770	−	0.920912i	$-0.372555\pi$
$241$	2916.51	0.779539	0.389770	+	0.920912i	$0.372555\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2864.62	−0.737940
$248$	0	0
$249$	−1064.79	−0.270998
$250$	0	0
$251$	−111.138	−0.0279480	−0.0139740	−	0.999902i	$-0.504448\pi$
$251$	−111.138	−0.0279480	−0.0139740	+	0.999902i	$0.504448\pi$
$252$	0	0
$253$	−193.344	−0.0480452
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4396.61	1.06713	0.533566	−	0.845759i	$-0.320852\pi$
$257$	4396.61	1.06713	0.533566	+	0.845759i	$0.320852\pi$
$258$	0	0
$259$	9693.80	2.32565
$260$	0	0
$261$	−1225.34	−0.290601
$262$	0	0
$263$	7892.81	1.85054	0.925269	−	0.379311i	$-0.123839\pi$
$263$	7892.81	1.85054	0.925269	+	0.379311i	$0.123839\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1588.08	0.364004
$268$	0	0
$269$	−2774.18	−0.628791	−0.314395	−	0.949292i	$-0.601802\pi$
$269$	−2774.18	−0.628791	−0.314395	+	0.949292i	$0.601802\pi$
$270$	0	0
$271$	1260.84	0.282623	0.141311	−	0.989965i	$-0.454868\pi$
$271$	1260.84	0.282623	0.141311	+	0.989965i	$0.454868\pi$
$272$	0	0
$273$	1476.85	0.327410
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7001.56	1.51871	0.759356	−	0.650676i	$-0.225514\pi$
$277$	7001.56	1.51871	0.759356	+	0.650676i	$0.225514\pi$
$278$	0	0
$279$	−250.963	−0.0538522
$280$	0	0
$281$	6544.58	1.38938	0.694692	−	0.719307i	$-0.255541\pi$
$281$	6544.58	1.38938	0.694692	+	0.719307i	$0.255541\pi$
$282$	0	0
$283$	2671.27	0.561097	0.280548	−	0.959840i	$-0.409484\pi$
$283$	2671.27	0.561097	0.280548	+	0.959840i	$0.409484\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−819.690	−0.168588
$288$	0	0
$289$	−3373.64	−0.686677
$290$	0	0
$291$	−3767.31	−0.758914
$292$	0	0
$293$	4921.09	0.981204	0.490602	−	0.871384i	$-0.336777\pi$
$293$	4921.09	0.981204	0.490602	+	0.871384i	$0.336777\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	45.0373	0.00879909
$298$	0	0
$299$	−2112.89	−0.408668
$300$	0	0
$301$	5743.73	1.09988
$302$	0	0
$303$	−1900.51	−0.360336
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−849.353	−0.157899	−0.0789497	−	0.996879i	$-0.525157\pi$
$307$	−849.353	−0.157899	−0.0789497	+	0.996879i	$0.525157\pi$
$308$	0	0
$309$	−39.6890	−0.00730688
$310$	0	0
$311$	1484.27	0.270628	0.135314	−	0.990803i	$-0.456796\pi$
$311$	1484.27	0.270628	0.135314	+	0.990803i	$0.456796\pi$
$312$	0	0
$313$	4592.01	0.829251	0.414625	−	0.909992i	$-0.363913\pi$
$313$	4592.01	0.829251	0.414625	+	0.909992i	$0.363913\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4853.05	−0.859857	−0.429928	−	0.902863i	$-0.641461\pi$
$317$	−4853.05	−0.859857	−0.429928	+	0.902863i	$0.641461\pi$
$318$	0	0
$319$	227.103	0.0398600
$320$	0	0
$321$	−4018.14	−0.698662
$322$	0	0
$323$	−6165.69	−1.06213
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1057.19	−0.178785
$328$	0	0
$329$	−17023.4	−2.85268
$330$	0	0
$331$	9865.09	1.63817	0.819085	−	0.573672i	$-0.194482\pi$
$331$	9865.09	1.63817	0.819085	+	0.573672i	$0.194482\pi$
$332$	0	0
$333$	3230.55	0.531631
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4453.35	0.719850	0.359925	−	0.932981i	$-0.382802\pi$
$337$	4453.35	0.719850	0.359925	+	0.932981i	$0.382802\pi$
$338$	0	0
$339$	3121.12	0.500048
$340$	0	0
$341$	46.5131	0.00738659
$342$	0	0
$343$	1169.96	0.184175
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2101.65	−0.325137	−0.162569	−	0.986697i	$-0.551978\pi$
$347$	−2101.65	−0.325137	−0.162569	+	0.986697i	$0.551978\pi$
$348$	0	0
$349$	−5443.35	−0.834888	−0.417444	−	0.908703i	$-0.637074\pi$
$349$	−5443.35	−0.834888	−0.417444	+	0.908703i	$0.637074\pi$
$350$	0	0
$351$	492.174	0.0748441
$352$	0	0
$353$	10730.2	1.61788	0.808938	−	0.587894i	$-0.200043\pi$
$353$	10730.2	1.61788	0.808938	+	0.587894i	$0.200043\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3178.71	0.471246
$358$	0	0
$359$	4030.33	0.592514	0.296257	−	0.955108i	$-0.404262\pi$
$359$	4030.33	0.592514	0.296257	+	0.955108i	$0.404262\pi$
$360$	0	0
$361$	17836.9	2.60051
$362$	0	0
$363$	3984.65	0.576143
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6297.32	−0.895688	−0.447844	−	0.894112i	$-0.647808\pi$
$367$	−6297.32	−0.895688	−0.447844	+	0.894112i	$0.647808\pi$
$368$	0	0
$369$	−273.170	−0.0385383
$370$	0	0
$371$	15362.7	2.14985
$372$	0	0
$373$	4574.31	0.634984	0.317492	−	0.948261i	$-0.397159\pi$
$373$	4574.31	0.634984	0.317492	+	0.948261i	$0.397159\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2481.82	0.339045
$378$	0	0
$379$	9053.69	1.22706	0.613531	−	0.789670i	$-0.289748\pi$
$379$	9053.69	1.22706	0.613531	+	0.789670i	$0.289748\pi$
$380$	0	0
$381$	−1802.90	−0.242429
$382$	0	0
$383$	−6892.26	−0.919525	−0.459763	−	0.888042i	$-0.652065\pi$
$383$	−6892.26	−0.919525	−0.459763	+	0.888042i	$0.652065\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1914.15	0.251426
$388$	0	0
$389$	14484.3	1.88787	0.943936	−	0.330128i	$-0.107092\pi$
$389$	14484.3	1.88787	0.943936	+	0.330128i	$0.107092\pi$
$390$	0	0
$391$	−4547.70	−0.588203
$392$	0	0
$393$	4164.89	0.534582
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12584.4	−1.59091	−0.795456	−	0.606011i	$-0.792769\pi$
$397$	−12584.4	−1.59091	−0.795456	+	0.606011i	$0.792769\pi$
$398$	0	0
$399$	−12731.9	−1.59747
$400$	0	0
$401$	−5566.51	−0.693212	−0.346606	−	0.938011i	$-0.612666\pi$
$401$	−5566.51	−0.693212	−0.346606	+	0.938011i	$0.612666\pi$
$402$	0	0
$403$	508.302	0.0628295
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−598.747	−0.0729208
$408$	0	0
$409$	−7892.04	−0.954123	−0.477061	−	0.878870i	$-0.658298\pi$
$409$	−7892.04	−0.954123	−0.477061	+	0.878870i	$0.658298\pi$
$410$	0	0
$411$	6113.05	0.733661
$412$	0	0
$413$	5648.49	0.672988
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3180.85	0.373541
$418$	0	0
$419$	−11875.8	−1.38465	−0.692327	−	0.721584i	$-0.743414\pi$
$419$	−11875.8	−1.38465	−0.692327	+	0.721584i	$0.743414\pi$
$420$	0	0
$421$	7770.34	0.899533	0.449767	−	0.893146i	$-0.351507\pi$
$421$	7770.34	0.899533	0.449767	+	0.893146i	$0.351507\pi$
$422$	0	0
$423$	−5673.22	−0.652108
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11609.7	1.31577
$428$	0	0
$429$	−91.2188	−0.0102659
$430$	0	0
$431$	−8405.48	−0.939391	−0.469695	−	0.882828i	$-0.655636\pi$
$431$	−8405.48	−0.939391	−0.469695	+	0.882828i	$0.655636\pi$
$432$	0	0
$433$	3626.22	0.402460	0.201230	−	0.979544i	$-0.435506\pi$
$433$	3626.22	0.402460	0.201230	+	0.979544i	$0.435506\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18215.2	1.99394
$438$	0	0
$439$	5034.00	0.547288	0.273644	−	0.961831i	$-0.411771\pi$
$439$	5034.00	0.547288	0.273644	+	0.961831i	$0.411771\pi$
$440$	0	0
$441$	3476.90	0.375435
$442$	0	0
$443$	−14078.8	−1.50994	−0.754970	−	0.655760i	$-0.772348\pi$
$443$	−14078.8	−1.50994	−0.754970	+	0.655760i	$0.772348\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5377.70	0.569030
$448$	0	0
$449$	4853.93	0.510181	0.255091	−	0.966917i	$-0.417895\pi$
$449$	4853.93	0.510181	0.255091	+	0.966917i	$0.417895\pi$
$450$	0	0
$451$	50.6289	0.00528608
$452$	0	0
$453$	6455.59	0.669559
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12007.5	−1.22908	−0.614538	−	0.788888i	$-0.710657\pi$
$457$	−12007.5	−1.22908	−0.614538	+	0.788888i	$0.710657\pi$
$458$	0	0
$459$	1059.33	0.107724
$460$	0	0
$461$	651.462	0.0658169	0.0329085	−	0.999458i	$-0.489523\pi$
$461$	651.462	0.0658169	0.0329085	+	0.999458i	$0.489523\pi$
$462$	0	0
$463$	7378.45	0.740617	0.370308	−	0.928909i	$-0.379252\pi$
$463$	7378.45	0.740617	0.370308	+	0.928909i	$0.379252\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18671.3	1.85011	0.925057	−	0.379829i	$-0.124017\pi$
$467$	18671.3	1.85011	0.925057	+	0.379829i	$0.124017\pi$
$468$	0	0
$469$	−20935.9	−2.06126
$470$	0	0
$471$	−8807.54	−0.861635
$472$	0	0
$473$	−354.767	−0.0344866
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5119.78	0.491444
$478$	0	0
$479$	−12448.6	−1.18746	−0.593729	−	0.804665i	$-0.702345\pi$
$479$	−12448.6	−1.18746	−0.593729	+	0.804665i	$0.702345\pi$
$480$	0	0
$481$	−6543.18	−0.620257
$482$	0	0
$483$	−9390.82	−0.884673
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−786.923	−0.0732216	−0.0366108	−	0.999330i	$-0.511656\pi$
$487$	−786.923	−0.0732216	−0.0366108	+	0.999330i	$0.511656\pi$
$488$	0	0
$489$	925.368	0.0855759
$490$	0	0
$491$	−7315.99	−0.672436	−0.336218	−	0.941784i	$-0.609148\pi$
$491$	−7315.99	−0.672436	−0.336218	+	0.941784i	$0.609148\pi$
$492$	0	0
$493$	5341.76	0.487994
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2143.25	0.193437
$498$	0	0
$499$	6696.78	0.600779	0.300390	−	0.953817i	$-0.402883\pi$
$499$	6696.78	0.600779	0.300390	+	0.953817i	$0.402883\pi$
$500$	0	0
$501$	5686.35	0.507081
$502$	0	0
$503$	18197.2	1.61307	0.806533	−	0.591188i	$-0.201341\pi$
$503$	18197.2	1.61307	0.806533	+	0.591188i	$0.201341\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5594.15	0.490029
$508$	0	0
$509$	−18891.5	−1.64509	−0.822544	−	0.568702i	$-0.807446\pi$
$509$	−18891.5	−1.64509	−0.822544	+	0.568702i	$0.807446\pi$
$510$	0	0
$511$	−7330.37	−0.634592
$512$	0	0
$513$	−4243.03	−0.365174
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1051.47	0.0894458
$518$	0	0
$519$	−3305.81	−0.279593
$520$	0	0
$521$	14531.9	1.22199	0.610993	−	0.791636i	$-0.290770\pi$
$521$	14531.9	1.22199	0.610993	+	0.791636i	$0.290770\pi$
$522$	0	0
$523$	−12281.0	−1.02679	−0.513395	−	0.858153i	$-0.671612\pi$
$523$	−12281.0	−1.02679	−0.513395	+	0.858153i	$0.671612\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1094.05	0.0904317
$528$	0	0
$529$	1268.23	0.104236
$530$	0	0
$531$	1882.41	0.153841
$532$	0	0
$533$	553.279	0.0449628
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9787.18	0.786495
$538$	0	0
$539$	−644.404	−0.0514962
$540$	0	0
$541$	6062.86	0.481817	0.240908	−	0.970548i	$-0.422555\pi$
$541$	6062.86	0.481817	0.240908	+	0.970548i	$0.422555\pi$
$542$	0	0
$543$	7263.03	0.574008
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−19428.1	−1.51862	−0.759309	−	0.650730i	$-0.774463\pi$
$547$	−19428.1	−1.51862	−0.759309	+	0.650730i	$0.774463\pi$
$548$	0	0
$549$	3869.06	0.300778
$550$	0	0
$551$	−21395.7	−1.65425
$552$	0	0
$553$	5589.96	0.429854
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3044.47	0.231595	0.115797	−	0.993273i	$-0.463058\pi$
$557$	3044.47	0.231595	0.115797	+	0.993273i	$0.463058\pi$
$558$	0	0
$559$	−3876.94	−0.293340
$560$	0	0
$561$	−196.336	−0.0147759
$562$	0	0
$563$	534.536	0.0400142	0.0200071	−	0.999800i	$-0.493631\pi$
$563$	534.536	0.0400142	0.0200071	+	0.999800i	$0.493631\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2187.48	0.162021
$568$	0	0
$569$	17651.6	1.30051	0.650257	−	0.759715i	$-0.274661\pi$
$569$	17651.6	1.30051	0.650257	+	0.759715i	$0.274661\pi$
$570$	0	0
$571$	25083.6	1.83838	0.919189	−	0.393816i	$-0.128845\pi$
$571$	25083.6	1.83838	0.919189	+	0.393816i	$0.128845\pi$
$572$	0	0
$573$	3910.10	0.285073
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1471.14	−0.106143	−0.0530713	−	0.998591i	$-0.516901\pi$
$577$	−1471.14	−0.106143	−0.0530713	+	0.998591i	$0.516901\pi$
$578$	0	0
$579$	−12936.6	−0.928546
$580$	0	0
$581$	9585.26	0.684447
$582$	0	0
$583$	−948.893	−0.0674085
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9374.53	0.659162	0.329581	−	0.944127i	$-0.393092\pi$
$587$	9374.53	0.659162	0.329581	+	0.944127i	$0.393092\pi$
$588$	0	0
$589$	−4382.07	−0.306553
$590$	0	0
$591$	−7648.48	−0.532346
$592$	0	0
$593$	17508.2	1.21244	0.606218	−	0.795298i	$-0.292686\pi$
$593$	17508.2	1.21244	0.606218	+	0.795298i	$0.292686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6660.44	0.456606
$598$	0	0
$599$	−23357.9	−1.59329	−0.796644	−	0.604449i	$-0.793393\pi$
$599$	−23357.9	−1.59329	−0.796644	+	0.604449i	$0.793393\pi$
$600$	0	0
$601$	−14646.9	−0.994109	−0.497055	−	0.867719i	$-0.665585\pi$
$601$	−14646.9	−0.994109	−0.497055	+	0.867719i	$0.665585\pi$
$602$	0	0
$603$	−6977.09	−0.471193
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18575.7	−1.24212	−0.621059	−	0.783764i	$-0.713297\pi$
$607$	−18575.7	−1.24212	−0.621059	+	0.783764i	$0.713297\pi$
$608$	0	0
$609$	11030.5	0.733956
$610$	0	0
$611$	11490.6	0.760817
$612$	0	0
$613$	21371.4	1.40813	0.704065	−	0.710135i	$-0.251366\pi$
$613$	21371.4	1.40813	0.704065	+	0.710135i	$0.251366\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14646.6	0.955672	0.477836	−	0.878449i	$-0.341421\pi$
$617$	14646.6	0.955672	0.477836	+	0.878449i	$0.341421\pi$
$618$	0	0
$619$	9217.63	0.598527	0.299263	−	0.954171i	$-0.403259\pi$
$619$	9217.63	0.598527	0.299263	+	0.954171i	$0.403259\pi$
$620$	0	0
$621$	−3129.58	−0.202232
$622$	0	0
$623$	−14295.9	−0.919347
$624$	0	0
$625$	0	0
$626$	0	0
$627$	786.397	0.0500888
$628$	0	0
$629$	−14083.3	−0.892746
$630$	0	0
$631$	2272.47	0.143368	0.0716842	−	0.997427i	$-0.477163\pi$
$631$	2272.47	0.143368	0.0716842	+	0.997427i	$0.477163\pi$
$632$	0	0
$633$	−10320.6	−0.648038
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7042.13	−0.438021
$638$	0	0
$639$	714.260	0.0442186
$640$	0	0
$641$	−14679.5	−0.904535	−0.452267	−	0.891882i	$-0.649385\pi$
$641$	−14679.5	−0.904535	−0.452267	+	0.891882i	$0.649385\pi$
$642$	0	0
$643$	23423.6	1.43660	0.718301	−	0.695733i	$-0.244920\pi$
$643$	23423.6	1.43660	0.718301	+	0.695733i	$0.244920\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24072.1	−1.46271	−0.731355	−	0.681997i	$-0.761112\pi$
$647$	−24072.1	−1.46271	−0.731355	+	0.681997i	$0.761112\pi$
$648$	0	0
$649$	−348.884	−0.0211015
$650$	0	0
$651$	2259.17	0.136012
$652$	0	0
$653$	−13053.1	−0.782248	−0.391124	−	0.920338i	$-0.627914\pi$
$653$	−13053.1	−0.782248	−0.391124	+	0.920338i	$0.627914\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2442.92	−0.145064
$658$	0	0
$659$	29076.5	1.71876	0.859378	−	0.511340i	$-0.170851\pi$
$659$	29076.5	1.71876	0.859378	+	0.511340i	$0.170851\pi$
$660$	0	0
$661$	7462.35	0.439110	0.219555	−	0.975600i	$-0.429539\pi$
$661$	7462.35	0.439110	0.219555	+	0.975600i	$0.429539\pi$
$662$	0	0
$663$	−2145.58	−0.125683
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15781.1	−0.916113
$668$	0	0
$669$	−6340.40	−0.366418
$670$	0	0
$671$	−717.086	−0.0412560
$672$	0	0
$673$	−33073.2	−1.89432	−0.947161	−	0.320758i	$-0.896062\pi$
$673$	−33073.2	−1.89432	−0.947161	+	0.320758i	$0.896062\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23156.5	−1.31459	−0.657295	−	0.753633i	$-0.728300\pi$
$677$	−23156.5	−1.31459	−0.657295	+	0.753633i	$0.728300\pi$
$678$	0	0
$679$	33913.3	1.91675
$680$	0	0
$681$	−10685.5	−0.601277
$682$	0	0
$683$	−6453.28	−0.361534	−0.180767	−	0.983526i	$-0.557858\pi$
$683$	−6453.28	−0.361534	−0.180767	+	0.983526i	$0.557858\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3803.65	−0.211235
$688$	0	0
$689$	−10369.6	−0.573369
$690$	0	0
$691$	3552.92	0.195600	0.0977998	−	0.995206i	$-0.468820\pi$
$691$	3552.92	0.195600	0.0977998	+	0.995206i	$0.468820\pi$
$692$	0	0
$693$	−405.425	−0.0222234
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1190.86	0.0647158
$698$	0	0
$699$	−19765.8	−1.06954
$700$	0	0
$701$	6727.75	0.362487	0.181244	−	0.983438i	$-0.441988\pi$
$701$	6727.75	0.362487	0.181244	+	0.983438i	$0.441988\pi$
$702$	0	0
$703$	56408.8	3.02631
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17108.4	0.910082
$708$	0	0
$709$	14027.2	0.743021	0.371510	−	0.928429i	$-0.378840\pi$
$709$	14027.2	0.743021	0.371510	+	0.928429i	$0.378840\pi$
$710$	0	0
$711$	1862.91	0.0982624
$712$	0	0
$713$	−3232.14	−0.169768
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−18761.3	−0.977200
$718$	0	0
$719$	−7002.59	−0.363217	−0.181608	−	0.983371i	$-0.558130\pi$
$719$	−7002.59	−0.363217	−0.181608	+	0.983371i	$0.558130\pi$
$720$	0	0
$721$	357.280	0.0184546
$722$	0	0
$723$	−8749.53	−0.450067
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−20405.7	−1.04100	−0.520498	−	0.853863i	$-0.674254\pi$
$727$	−20405.7	−1.04100	−0.520498	+	0.853863i	$0.674254\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−8344.56	−0.422209
$732$	0	0
$733$	−6494.30	−0.327248	−0.163624	−	0.986523i	$-0.552318\pi$
$733$	−6494.30	−0.327248	−0.163624	+	0.986523i	$0.552318\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1293.13	0.0646308
$738$	0	0
$739$	36785.9	1.83111	0.915556	−	0.402191i	$-0.131751\pi$
$739$	36785.9	1.83111	0.915556	+	0.402191i	$0.131751\pi$
$740$	0	0
$741$	8593.85	0.426050
$742$	0	0
$743$	16321.4	0.805889	0.402945	−	0.915224i	$-0.367987\pi$
$743$	16321.4	0.805889	0.402945	+	0.915224i	$0.367987\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3194.38	0.156461
$748$	0	0
$749$	36171.2	1.76458
$750$	0	0
$751$	−32467.5	−1.57757	−0.788786	−	0.614668i	$-0.789290\pi$
$751$	−32467.5	−1.57757	−0.788786	+	0.614668i	$0.789290\pi$
$752$	0	0
$753$	333.413	0.0161358
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9188.72	0.441175	0.220588	−	0.975367i	$-0.429203\pi$
$757$	9188.72	0.441175	0.220588	+	0.975367i	$0.429203\pi$
$758$	0	0
$759$	580.033	0.0277389
$760$	0	0
$761$	−10574.6	−0.503715	−0.251858	−	0.967764i	$-0.581041\pi$
$761$	−10574.6	−0.503715	−0.251858	+	0.967764i	$0.581041\pi$
$762$	0	0
$763$	9516.82	0.451549
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3812.65	−0.179487
$768$	0	0
$769$	26600.8	1.24740	0.623699	−	0.781665i	$-0.285629\pi$
$769$	26600.8	1.24740	0.623699	+	0.781665i	$0.285629\pi$
$770$	0	0
$771$	−13189.8	−0.616109
$772$	0	0
$773$	17113.4	0.796281	0.398140	−	0.917325i	$-0.369656\pi$
$773$	17113.4	0.796281	0.398140	+	0.917325i	$0.369656\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−29081.4	−1.34272
$778$	0	0
$779$	−4769.82	−0.219379
$780$	0	0
$781$	−132.380	−0.00606521
$782$	0	0
$783$	3676.03	0.167778
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12415.2	0.562332	0.281166	−	0.959659i	$-0.409279\pi$
$787$	12415.2	0.562332	0.281166	+	0.959659i	$0.409279\pi$
$788$	0	0
$789$	−23678.4	−1.06841
$790$	0	0
$791$	−28096.3	−1.26295
$792$	0	0
$793$	−7836.41	−0.350919
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17080.1	−0.759106	−0.379553	−	0.925170i	$-0.623922\pi$
$797$	−17080.1	−0.759106	−0.379553	+	0.925170i	$0.623922\pi$
$798$	0	0
$799$	24731.9	1.09506
$800$	0	0
$801$	−4764.25	−0.210158
$802$	0	0
$803$	452.767	0.0198976
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8322.54	0.363033
$808$	0	0
$809$	−32230.3	−1.40069	−0.700344	−	0.713806i	$-0.746970\pi$
$809$	−32230.3	−1.40069	−0.700344	+	0.713806i	$0.746970\pi$
$810$	0	0
$811$	17394.1	0.753130	0.376565	−	0.926390i	$-0.377105\pi$
$811$	17394.1	0.753130	0.376565	+	0.926390i	$0.377105\pi$
$812$	0	0
$813$	−3782.53	−0.163172
$814$	0	0
$815$	0	0
$816$	0	0
$817$	33423.1	1.43124
$818$	0	0
$819$	−4430.54	−0.189030
$820$	0	0
$821$	−31848.8	−1.35387	−0.676937	−	0.736041i	$-0.736693\pi$
$821$	−31848.8	−1.35387	−0.676937	+	0.736041i	$0.736693\pi$
$822$	0	0
$823$	31497.6	1.33407	0.667033	−	0.745028i	$-0.267564\pi$
$823$	31497.6	1.33407	0.667033	+	0.745028i	$0.267564\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5371.24	0.225848	0.112924	−	0.993604i	$-0.463978\pi$
$827$	5371.24	0.225848	0.112924	+	0.993604i	$0.463978\pi$
$828$	0	0
$829$	20701.9	0.867320	0.433660	−	0.901077i	$-0.357222\pi$
$829$	20701.9	0.867320	0.433660	+	0.901077i	$0.357222\pi$
$830$	0	0
$831$	−21004.7	−0.876829
$832$	0	0
$833$	−15157.2	−0.630452
$834$	0	0
$835$	0	0
$836$	0	0
$837$	752.889	0.0310916
$838$	0	0
$839$	−38044.1	−1.56547	−0.782734	−	0.622357i	$-0.786175\pi$
$839$	−38044.1	−1.56547	−0.782734	+	0.622357i	$0.786175\pi$
$840$	0	0
$841$	−5852.39	−0.239960
$842$	0	0
$843$	−19633.7	−0.802161
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−35869.8	−1.45514
$848$	0	0
$849$	−8013.80	−0.323949
$850$	0	0
$851$	41606.1	1.67596
$852$	0	0
$853$	−22333.4	−0.896461	−0.448231	−	0.893918i	$-0.647946\pi$
$853$	−22333.4	−0.896461	−0.448231	+	0.893918i	$0.647946\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16346.4	−0.651554	−0.325777	−	0.945447i	$-0.605626\pi$
$857$	−16346.4	−0.651554	−0.325777	+	0.945447i	$0.605626\pi$
$858$	0	0
$859$	714.274	0.0283710	0.0141855	−	0.999899i	$-0.495484\pi$
$859$	714.274	0.0283710	0.0141855	+	0.999899i	$0.495484\pi$
$860$	0	0
$861$	2459.07	0.0973343
$862$	0	0
$863$	−21577.0	−0.851089	−0.425544	−	0.904938i	$-0.639917\pi$
$863$	−21577.0	−0.851089	−0.425544	+	0.904938i	$0.639917\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10120.9	0.396453
$868$	0	0
$869$	−345.269	−0.0134781
$870$	0	0
$871$	14131.4	0.549743
$872$	0	0
$873$	11301.9	0.438159
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28659.7	1.10350	0.551750	−	0.834010i	$-0.313960\pi$
$877$	28659.7	1.10350	0.551750	+	0.834010i	$0.313960\pi$
$878$	0	0
$879$	−14763.3	−0.566499
$880$	0	0
$881$	29709.1	1.13612	0.568062	−	0.822986i	$-0.307693\pi$
$881$	29709.1	1.13612	0.568062	+	0.822986i	$0.307693\pi$
$882$	0	0
$883$	−30726.3	−1.17103	−0.585517	−	0.810660i	$-0.699108\pi$
$883$	−30726.3	−1.17103	−0.585517	+	0.810660i	$0.699108\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1675.35	−0.0634192	−0.0317096	−	0.999497i	$-0.510095\pi$
$887$	−1675.35	−0.0634192	−0.0317096	+	0.999497i	$0.510095\pi$
$888$	0	0
$889$	16229.7	0.612291
$890$	0	0
$891$	−135.112	−0.00508016
$892$	0	0
$893$	−99060.3	−3.71212
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6338.67	0.235944
$898$	0	0
$899$	3796.49	0.140845
$900$	0	0
$901$	−22319.2	−0.825261
$902$	0	0
$903$	−17231.2	−0.635014
$904$	0	0
$905$	0	0
$906$	0	0
$907$	27201.4	0.995817	0.497909	−	0.867230i	$-0.334102\pi$
$907$	27201.4	0.995817	0.497909	+	0.867230i	$0.334102\pi$
$908$	0	0
$909$	5701.54	0.208040
$910$	0	0
$911$	−22130.5	−0.804846	−0.402423	−	0.915454i	$-0.631832\pi$
$911$	−22130.5	−0.804846	−0.402423	+	0.915454i	$0.631832\pi$
$912$	0	0
$913$	−592.042	−0.0214608
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−37492.3	−1.35017
$918$	0	0
$919$	19655.9	0.705536	0.352768	−	0.935711i	$-0.385240\pi$
$919$	19655.9	0.705536	0.352768	+	0.935711i	$0.385240\pi$
$920$	0	0
$921$	2548.06	0.0911633
$922$	0	0
$923$	−1446.67	−0.0515901
$924$	0	0
$925$	0	0
$926$	0	0
$927$	119.067	0.00421863
$928$	0	0
$929$	23519.5	0.830624	0.415312	−	0.909679i	$-0.363672\pi$
$929$	23519.5	0.830624	0.415312	+	0.909679i	$0.363672\pi$
$930$	0	0
$931$	60710.3	2.13716
$932$	0	0
$933$	−4452.82	−0.156247
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27724.7	0.966623	0.483312	−	0.875448i	$-0.339434\pi$
$937$	27724.7	0.966623	0.483312	+	0.875448i	$0.339434\pi$
$938$	0	0
$939$	−13776.0	−0.478768
$940$	0	0
$941$	−32499.2	−1.12587	−0.562936	−	0.826501i	$-0.690328\pi$
$941$	−32499.2	−1.12587	−0.562936	+	0.826501i	$0.690328\pi$
$942$	0	0
$943$	−3518.13	−0.121491
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27339.4	−0.938133	−0.469067	−	0.883163i	$-0.655410\pi$
$947$	−27339.4	−0.938133	−0.469067	+	0.883163i	$0.655410\pi$
$948$	0	0
$949$	4947.90	0.169247
$950$	0	0
$951$	14559.2	0.496438
$952$	0	0
$953$	28992.7	0.985484	0.492742	−	0.870175i	$-0.335995\pi$
$953$	28992.7	0.985484	0.492742	+	0.870175i	$0.335995\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−681.310	−0.0230132
$958$	0	0
$959$	−55029.7	−1.85297
$960$	0	0
$961$	−29013.4	−0.973900
$962$	0	0
$963$	12054.4	0.403373
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−59944.6	−1.99347	−0.996737	−	0.0807168i	$-0.974279\pi$
$967$	−59944.6	−1.99347	−0.996737	+	0.0807168i	$0.974279\pi$
$968$	0	0
$969$	18497.1	0.613221
$970$	0	0
$971$	−18266.4	−0.603703	−0.301851	−	0.953355i	$-0.597605\pi$
$971$	−18266.4	−0.603703	−0.301851	+	0.953355i	$0.597605\pi$
$972$	0	0
$973$	−28633.9	−0.943435
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26506.0	−0.867964	−0.433982	−	0.900922i	$-0.642892\pi$
$977$	−26506.0	−0.867964	−0.433982	+	0.900922i	$0.642892\pi$
$978$	0	0
$979$	882.999	0.0288261
$980$	0	0
$981$	3171.57	0.103222
$982$	0	0
$983$	−41980.3	−1.36212	−0.681060	−	0.732228i	$-0.738481\pi$
$983$	−41980.3	−1.36212	−0.681060	+	0.732228i	$0.738481\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	51070.3	1.64700
$988$	0	0
$989$	24652.3	0.792615
$990$	0	0
$991$	7778.08	0.249323	0.124661	−	0.992199i	$-0.460216\pi$
$991$	7778.08	0.249323	0.124661	+	0.992199i	$0.460216\pi$
$992$	0	0
$993$	−29595.3	−0.945798
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−25875.3	−0.821944	−0.410972	−	0.911648i	$-0.634811\pi$
$997$	−25875.3	−0.821944	−0.410972	+	0.911648i	$0.634811\pi$
$998$	0	0
$999$	−9691.66	−0.306938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bl.1.3	yes	3
4.3	odd	2		2400.4.a.br.1.1	yes	3
5.4	even	2		2400.4.a.bq.1.1	yes	3
20.19	odd	2		2400.4.a.bk.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bk.1.3	✓	3	20.19	odd	2
2400.4.a.bl.1.3	yes	3	1.1	even	1	trivial
2400.4.a.bq.1.1	yes	3	5.4	even	2
2400.4.a.br.1.1	yes	3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +27.0060 q^{7} +9.00000 q^{9} -1.66805 q^{11} -18.2287 q^{13} -39.2346 q^{17} +157.149 q^{19} -81.0179 q^{21} +115.910 q^{23} -27.0000 q^{27} -136.149 q^{29} -27.8848 q^{31} +5.00414 q^{33} +358.950 q^{37} +54.6860 q^{39} -30.3522 q^{41} +212.684 q^{43} -630.358 q^{47} +386.322 q^{49} +117.704 q^{51} +568.865 q^{53} -471.448 q^{57} +209.157 q^{59} +429.895 q^{61} +243.054 q^{63} -775.233 q^{67} -347.731 q^{69} +79.3623 q^{71} -271.435 q^{73} -45.0473 q^{77} +206.990 q^{79} +81.0000 q^{81} +354.931 q^{83} +408.448 q^{87} -529.361 q^{89} -492.282 q^{91} +83.6543 q^{93} +1255.77 q^{97} -15.0124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 7 q^{7} + 27 q^{9} + 21 q^{13} + 32 q^{17} + 19 q^{19} - 21 q^{21} + 60 q^{23} - 81 q^{27} + 44 q^{29} - 151 q^{31} + 330 q^{37} - 63 q^{39} + 82 q^{41} + 367 q^{43} - 246 q^{47} - 42 q^{49}+ \cdots - 351 q^{97}+O(q^{100}) \) 3 * q - 9 * q^3 + 7 * q^7 + 27 * q^9 + 21 * q^13 + 32 * q^17 + 19 * q^19 - 21 * q^21 + 60 * q^23 - 81 * q^27 + 44 * q^29 - 151 * q^31 + 330 * q^37 - 63 * q^39 + 82 * q^41 + 367 * q^43 - 246 * q^47 - 42 * q^49 - 96 * q^51 + 554 * q^53 - 57 * q^57 + 42 * q^59 - 27 * q^61 + 63 * q^63 - 837 * q^67 - 180 * q^69 - 1422 * q^71 + 498 * q^73 + 560 * q^77 - 688 * q^79 + 243 * q^81 + 702 * q^83 - 132 * q^87 - 1488 * q^89 - 735 * q^91 + 453 * q^93 - 351 * q^97

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