LMFDB - Embedded newform 2400.4.a.bd.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{89}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 22 $ x^2 - x - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.21699$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +22.0000 q^{7} +9.00000 q^{9} -18.8680 q^{11} -18.8680 q^{13} -132.076 q^{17} -113.208 q^{19} +66.0000 q^{21} +160.000 q^{23} +27.0000 q^{27} +128.000 q^{29} +75.4718 q^{31} -56.6039 q^{33} +18.8680 q^{37} -56.6039 q^{39} -358.000 q^{41} +172.000 q^{43} -4.00000 q^{47} +141.000 q^{49} -396.227 q^{51} +660.379 q^{53} -339.623 q^{57} +735.851 q^{59} -14.0000 q^{61} +198.000 q^{63} +848.000 q^{67} +480.000 q^{69} +641.511 q^{71} +1169.81 q^{73} -415.095 q^{77} -75.4718 q^{79} +81.0000 q^{81} +596.000 q^{83} +384.000 q^{87} +750.000 q^{89} -415.095 q^{91} +226.416 q^{93} -528.303 q^{97} -169.812 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 44 q^{7} + 18 q^{9} + 132 q^{21} + 320 q^{23} + 54 q^{27} + 256 q^{29} - 716 q^{41} + 344 q^{43} - 8 q^{47} + 282 q^{49} - 28 q^{61} + 396 q^{63} + 1696 q^{67} + 960 q^{69} + 162 q^{81}+ \cdots + 1500 q^{89}+O(q^{100}) $ 2 * q + 6 * q^3 + 44 * q^7 + 18 * q^9 + 132 * q^21 + 320 * q^23 + 54 * q^27 + 256 * q^29 - 716 * q^41 + 344 * q^43 - 8 * q^47 + 282 * q^49 - 28 * q^61 + 396 * q^63 + 1696 * q^67 + 960 * q^69 + 162 * q^81 + 1192 * q^83 + 768 * q^87 + 1500 * q^89

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
$7$	22.0000	1.18789	0.593944	−	0.804506i	$-0.297570\pi$
$7$	22.0000	1.18789	0.593944	+	0.804506i	$0.297570\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−18.8680	−0.517173	−0.258587	−	0.965988i	$-0.583257\pi$
$11$	−18.8680	−0.517173	−0.258587	+	0.965988i	$0.583257\pi$
$12$	0	0
$13$	−18.8680	−0.402541	−0.201270	−	0.979536i	$-0.564507\pi$
$13$	−18.8680	−0.402541	−0.201270	+	0.979536i	$0.564507\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−132.076	−1.88430	−0.942149	−	0.335194i	$-0.891198\pi$
$17$	−132.076	−1.88430	−0.942149	+	0.335194i	$0.891198\pi$
$18$	0	0
$19$	−113.208	−1.36693	−0.683464	−	0.729984i	$-0.739528\pi$
$19$	−113.208	−1.36693	−0.683464	+	0.729984i	$0.739528\pi$
$20$	0	0
$21$	66.0000	0.685828
$22$	0	0
$23$	160.000	1.45054	0.725268	−	0.688467i	$-0.241716\pi$
$23$	160.000	1.45054	0.725268	+	0.688467i	$0.241716\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	128.000	0.819621	0.409810	−	0.912171i	$-0.365595\pi$
$29$	128.000	0.819621	0.409810	+	0.912171i	$0.365595\pi$
$30$	0	0
$31$	75.4718	0.437263	0.218631	−	0.975808i	$-0.429841\pi$
$31$	75.4718	0.437263	0.218631	+	0.975808i	$0.429841\pi$
$32$	0	0
$33$	−56.6039	−0.298590
$34$	0	0
$35$	0	0
$36$	0	0
$37$	18.8680	0.0838344	0.0419172	−	0.999121i	$-0.486653\pi$
$37$	18.8680	0.0838344	0.0419172	+	0.999121i	$0.486653\pi$
$38$	0	0
$39$	−56.6039	−0.232407
$40$	0	0
$41$	−358.000	−1.36366	−0.681832	−	0.731509i	$-0.738816\pi$
$41$	−358.000	−1.36366	−0.681832	+	0.731509i	$0.738816\pi$
$42$	0	0
$43$	172.000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	172.000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.0124140	−0.00620702	−	0.999981i	$-0.501976\pi$
$47$	−4.00000	−0.0124140	−0.00620702	+	0.999981i	$0.501976\pi$
$48$	0	0
$49$	141.000	0.411079
$50$	0	0
$51$	−396.227	−1.08790
$52$	0	0
$53$	660.379	1.71151	0.855755	−	0.517382i	$-0.173093\pi$
$53$	660.379	1.71151	0.855755	+	0.517382i	$0.173093\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−339.623	−0.789197
$58$	0	0
$59$	735.851	1.62372	0.811861	−	0.583851i	$-0.198455\pi$
$59$	735.851	1.62372	0.811861	+	0.583851i	$0.198455\pi$
$60$	0	0
$61$	−14.0000	−0.0293855	−0.0146928	−	0.999892i	$-0.504677\pi$
$61$	−14.0000	−0.0293855	−0.0146928	+	0.999892i	$0.504677\pi$
$62$	0	0
$63$	198.000	0.395963
$64$	0	0
$65$	0	0
$66$	0	0
$67$	848.000	1.54626	0.773132	−	0.634245i	$-0.218689\pi$
$67$	848.000	1.54626	0.773132	+	0.634245i	$0.218689\pi$
$68$	0	0
$69$	480.000	0.837467
$70$	0	0
$71$	641.511	1.07230	0.536150	−	0.844123i	$-0.319878\pi$
$71$	641.511	1.07230	0.536150	+	0.844123i	$0.319878\pi$
$72$	0	0
$73$	1169.81	1.87557	0.937783	−	0.347222i	$-0.112875\pi$
$73$	1169.81	1.87557	0.937783	+	0.347222i	$0.112875\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−415.095	−0.614344
$78$	0	0
$79$	−75.4718	−0.107484	−0.0537421	−	0.998555i	$-0.517115\pi$
$79$	−75.4718	−0.107484	−0.0537421	+	0.998555i	$0.517115\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	596.000	0.788187	0.394093	−	0.919070i	$-0.371059\pi$
$83$	596.000	0.788187	0.394093	+	0.919070i	$0.371059\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	384.000	0.473208
$88$	0	0
$89$	750.000	0.893257	0.446628	−	0.894720i	$-0.352625\pi$
$89$	750.000	0.893257	0.446628	+	0.894720i	$0.352625\pi$
$90$	0	0
$91$	−415.095	−0.478174
$92$	0	0
$93$	226.416	0.252454
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−528.303	−0.553000	−0.276500	−	0.961014i	$-0.589175\pi$
$97$	−528.303	−0.553000	−0.276500	+	0.961014i	$0.589175\pi$
$98$	0	0
$99$	−169.812	−0.172391
$100$	0	0
$101$	1224.00	1.20587	0.602933	−	0.797792i	$-0.293998\pi$
$101$	1224.00	1.20587	0.602933	+	0.797792i	$0.293998\pi$
$102$	0	0
$103$	14.0000	0.0133928	0.00669641	−	0.999978i	$-0.497868\pi$
$103$	14.0000	0.0133928	0.00669641	+	0.999978i	$0.497868\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	852.000	0.769775	0.384888	−	0.922963i	$-0.374240\pi$
$107$	852.000	0.769775	0.384888	+	0.922963i	$0.374240\pi$
$108$	0	0
$109$	−1094.00	−0.961341	−0.480671	−	0.876901i	$-0.659607\pi$
$109$	−1094.00	−0.961341	−0.480671	+	0.876901i	$0.659607\pi$
$110$	0	0
$111$	56.6039	0.0484018
$112$	0	0
$113$	−547.171	−0.455518	−0.227759	−	0.973718i	$-0.573140\pi$
$113$	−547.171	−0.455518	−0.227759	+	0.973718i	$0.573140\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−169.812	−0.134180
$118$	0	0
$119$	−2905.67	−2.23834
$120$	0	0
$121$	−975.000	−0.732532
$122$	0	0
$123$	−1074.00	−0.787312
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1010.00	−0.705693	−0.352846	−	0.935681i	$-0.614786\pi$
$127$	−1010.00	−0.705693	−0.352846	+	0.935681i	$0.614786\pi$
$128$	0	0
$129$	516.000	0.352180
$130$	0	0
$131$	924.530	0.616615	0.308308	−	0.951287i	$-0.400237\pi$
$131$	924.530	0.616615	0.308308	+	0.951287i	$0.400237\pi$
$132$	0	0
$133$	−2490.57	−1.62376
$134$	0	0
$135$	0	0
$136$	0	0
$137$	584.907	0.364759	0.182379	−	0.983228i	$-0.441620\pi$
$137$	584.907	0.364759	0.182379	+	0.983228i	$0.441620\pi$
$138$	0	0
$139$	2641.51	1.61187	0.805937	−	0.592002i	$-0.201662\pi$
$139$	2641.51	1.61187	0.805937	+	0.592002i	$0.201662\pi$
$140$	0	0
$141$	−12.0000	−0.00716725
$142$	0	0
$143$	356.000	0.208183
$144$	0	0
$145$	0	0
$146$	0	0
$147$	423.000	0.237336
$148$	0	0
$149$	1452.00	0.798339	0.399169	−	0.916877i	$-0.369299\pi$
$149$	1452.00	0.798339	0.399169	+	0.916877i	$0.369299\pi$
$150$	0	0
$151$	−528.303	−0.284720	−0.142360	−	0.989815i	$-0.545469\pi$
$151$	−528.303	−0.284720	−0.142360	+	0.989815i	$0.545469\pi$
$152$	0	0
$153$	−1188.68	−0.628099
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−849.058	−0.431607	−0.215803	−	0.976437i	$-0.569237\pi$
$157$	−849.058	−0.431607	−0.215803	+	0.976437i	$0.569237\pi$
$158$	0	0
$159$	1981.14	0.988140
$160$	0	0
$161$	3520.00	1.72307
$162$	0	0
$163$	−3432.00	−1.64917	−0.824586	−	0.565737i	$-0.808592\pi$
$163$	−3432.00	−1.64917	−0.824586	+	0.565737i	$0.808592\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2664.00	−1.23441	−0.617205	−	0.786802i	$-0.711735\pi$
$167$	−2664.00	−1.23441	−0.617205	+	0.786802i	$0.711735\pi$
$168$	0	0
$169$	−1841.00	−0.837961
$170$	0	0
$171$	−1018.87	−0.455643
$172$	0	0
$173$	1830.19	0.804318	0.402159	−	0.915570i	$-0.368260\pi$
$173$	1830.19	0.804318	0.402159	+	0.915570i	$0.368260\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2207.55	0.937456
$178$	0	0
$179$	−3264.16	−1.36299	−0.681493	−	0.731824i	$-0.738669\pi$
$179$	−3264.16	−1.36299	−0.681493	+	0.731824i	$0.738669\pi$
$180$	0	0
$181$	3570.00	1.46606	0.733028	−	0.680199i	$-0.238107\pi$
$181$	3570.00	1.46606	0.733028	+	0.680199i	$0.238107\pi$
$182$	0	0
$183$	−42.0000	−0.0169657
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2492.00	0.974508
$188$	0	0
$189$	594.000	0.228609
$190$	0	0
$191$	3207.55	1.21513	0.607567	−	0.794269i	$-0.292146\pi$
$191$	3207.55	1.21513	0.607567	+	0.794269i	$0.292146\pi$
$192$	0	0
$193$	264.151	0.0985183	0.0492592	−	0.998786i	$-0.484314\pi$
$193$	264.151	0.0985183	0.0492592	+	0.998786i	$0.484314\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2471.70	0.893917	0.446958	−	0.894555i	$-0.352507\pi$
$197$	2471.70	0.893917	0.446958	+	0.894555i	$0.352507\pi$
$198$	0	0
$199$	679.247	0.241962	0.120981	−	0.992655i	$-0.461396\pi$
$199$	679.247	0.241962	0.120981	+	0.992655i	$0.461396\pi$
$200$	0	0
$201$	2544.00	0.892736
$202$	0	0
$203$	2816.00	0.973618
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1440.00	0.483512
$208$	0	0
$209$	2136.00	0.706939
$210$	0	0
$211$	−3396.23	−1.10809	−0.554043	−	0.832488i	$-0.686916\pi$
$211$	−3396.23	−1.10809	−0.554043	+	0.832488i	$0.686916\pi$
$212$	0	0
$213$	1924.53	0.619093
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1660.38	0.519419
$218$	0	0
$219$	3509.44	1.08286
$220$	0	0
$221$	2492.00	0.758507
$222$	0	0
$223$	318.000	0.0954926	0.0477463	−	0.998859i	$-0.484796\pi$
$223$	318.000	0.0954926	0.0477463	+	0.998859i	$0.484796\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	204.000	0.0596474	0.0298237	−	0.999555i	$-0.490505\pi$
$227$	204.000	0.0596474	0.0298237	+	0.999555i	$0.490505\pi$
$228$	0	0
$229$	−3450.00	−0.995556	−0.497778	−	0.867304i	$-0.665851\pi$
$229$	−3450.00	−0.995556	−0.497778	+	0.867304i	$0.665851\pi$
$230$	0	0
$231$	−1245.29	−0.354692
$232$	0	0
$233$	−3452.84	−0.970828	−0.485414	−	0.874284i	$-0.661331\pi$
$233$	−3452.84	−0.970828	−0.485414	+	0.874284i	$0.661331\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−226.416	−0.0620560
$238$	0	0
$239$	−3207.55	−0.868115	−0.434057	−	0.900885i	$-0.642918\pi$
$239$	−3207.55	−0.868115	−0.434057	+	0.900885i	$0.642918\pi$
$240$	0	0
$241$	6806.00	1.81914	0.909571	−	0.415550i	$-0.136411\pi$
$241$	6806.00	1.81914	0.909571	+	0.415550i	$0.136411\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2136.00	0.550245
$248$	0	0
$249$	1788.00	0.455060
$250$	0	0
$251$	−2018.87	−0.507690	−0.253845	−	0.967245i	$-0.581695\pi$
$251$	−2018.87	−0.507690	−0.253845	+	0.967245i	$0.581695\pi$
$252$	0	0
$253$	−3018.87	−0.750178
$254$	0	0
$255$	0	0
$256$	0	0
$257$	735.851	0.178603	0.0893017	−	0.996005i	$-0.471536\pi$
$257$	735.851	0.178603	0.0893017	+	0.996005i	$0.471536\pi$
$258$	0	0
$259$	415.095	0.0995859
$260$	0	0
$261$	1152.00	0.273207
$262$	0	0
$263$	−1072.00	−0.251340	−0.125670	−	0.992072i	$-0.540108\pi$
$263$	−1072.00	−0.251340	−0.125670	+	0.992072i	$0.540108\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2250.00	0.515722
$268$	0	0
$269$	−2368.00	−0.536727	−0.268363	−	0.963318i	$-0.586483\pi$
$269$	−2368.00	−0.536727	−0.268363	+	0.963318i	$0.586483\pi$
$270$	0	0
$271$	1358.49	0.304511	0.152256	−	0.988341i	$-0.451346\pi$
$271$	1358.49	0.304511	0.152256	+	0.988341i	$0.451346\pi$
$272$	0	0
$273$	−1245.29	−0.276074
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6962.28	1.51019	0.755095	−	0.655615i	$-0.227591\pi$
$277$	6962.28	1.51019	0.755095	+	0.655615i	$0.227591\pi$
$278$	0	0
$279$	679.247	0.145754
$280$	0	0
$281$	6894.00	1.46356	0.731782	−	0.681539i	$-0.238689\pi$
$281$	6894.00	1.46356	0.731782	+	0.681539i	$0.238689\pi$
$282$	0	0
$283$	−2456.00	−0.515880	−0.257940	−	0.966161i	$-0.583044\pi$
$283$	−2456.00	−0.515880	−0.257940	+	0.966161i	$0.583044\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7876.00	−1.61988
$288$	0	0
$289$	12531.0	2.55058
$290$	0	0
$291$	−1584.91	−0.319275
$292$	0	0
$293$	−9301.91	−1.85469	−0.927343	−	0.374212i	$-0.877913\pi$
$293$	−9301.91	−1.85469	−0.927343	+	0.374212i	$0.877913\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−509.435	−0.0995300
$298$	0	0
$299$	−3018.87	−0.583900
$300$	0	0
$301$	3784.00	0.724605
$302$	0	0
$303$	3672.00	0.696208
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4964.00	−0.922835	−0.461418	−	0.887183i	$-0.652659\pi$
$307$	−4964.00	−0.922835	−0.461418	+	0.887183i	$0.652659\pi$
$308$	0	0
$309$	42.0000	0.00773235
$310$	0	0
$311$	5018.88	0.915095	0.457547	−	0.889185i	$-0.348728\pi$
$311$	5018.88	0.915095	0.457547	+	0.889185i	$0.348728\pi$
$312$	0	0
$313$	2830.19	0.511093	0.255546	−	0.966797i	$-0.417745\pi$
$313$	2830.19	0.511093	0.255546	+	0.966797i	$0.417745\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3000.01	−0.531537	−0.265768	−	0.964037i	$-0.585626\pi$
$317$	−3000.01	−0.531537	−0.265768	+	0.964037i	$0.585626\pi$
$318$	0	0
$319$	−2415.10	−0.423886
$320$	0	0
$321$	2556.00	0.444430
$322$	0	0
$323$	14952.0	2.57570
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3282.00	−0.555031
$328$	0	0
$329$	−88.0000	−0.0147465
$330$	0	0
$331$	6000.01	0.996346	0.498173	−	0.867078i	$-0.334004\pi$
$331$	6000.01	0.996346	0.498173	+	0.867078i	$0.334004\pi$
$332$	0	0
$333$	169.812	0.0279448
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8415.11	−1.36024	−0.680119	−	0.733102i	$-0.738072\pi$
$337$	−8415.11	−1.36024	−0.680119	+	0.733102i	$0.738072\pi$
$338$	0	0
$339$	−1641.51	−0.262993
$340$	0	0
$341$	−1424.00	−0.226141
$342$	0	0
$343$	−4444.00	−0.699573
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7500.00	1.16029	0.580146	−	0.814513i	$-0.302996\pi$
$347$	7500.00	1.16029	0.580146	+	0.814513i	$0.302996\pi$
$348$	0	0
$349$	−1814.00	−0.278227	−0.139113	−	0.990276i	$-0.544425\pi$
$349$	−1814.00	−0.278227	−0.139113	+	0.990276i	$0.544425\pi$
$350$	0	0
$351$	−509.435	−0.0774690
$352$	0	0
$353$	5905.67	0.890445	0.445223	−	0.895420i	$-0.353125\pi$
$353$	5905.67	0.890445	0.445223	+	0.895420i	$0.353125\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−8717.00	−1.29230
$358$	0	0
$359$	13283.0	1.95279	0.976396	−	0.215988i	$-0.0692972\pi$
$359$	13283.0	1.95279	0.976396	+	0.215988i	$0.0692972\pi$
$360$	0	0
$361$	5957.00	0.868494
$362$	0	0
$363$	−2925.00	−0.422928
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6834.00	−0.972022	−0.486011	−	0.873953i	$-0.661548\pi$
$367$	−6834.00	−0.972022	−0.486011	+	0.873953i	$0.661548\pi$
$368$	0	0
$369$	−3222.00	−0.454555
$370$	0	0
$371$	14528.3	2.03308
$372$	0	0
$373$	11377.4	1.57935	0.789676	−	0.613524i	$-0.210249\pi$
$373$	11377.4	1.57935	0.789676	+	0.613524i	$0.210249\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2415.10	−0.329931
$378$	0	0
$379$	−1358.49	−0.184119	−0.0920595	−	0.995754i	$-0.529345\pi$
$379$	−1358.49	−0.184119	−0.0920595	+	0.995754i	$0.529345\pi$
$380$	0	0
$381$	−3030.00	−0.407432
$382$	0	0
$383$	−4644.00	−0.619575	−0.309788	−	0.950806i	$-0.600258\pi$
$383$	−4644.00	−0.619575	−0.309788	+	0.950806i	$0.600258\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1548.00	0.203331
$388$	0	0
$389$	8820.00	1.14959	0.574797	−	0.818296i	$-0.305081\pi$
$389$	8820.00	1.14959	0.574797	+	0.818296i	$0.305081\pi$
$390$	0	0
$391$	−21132.1	−2.73324
$392$	0	0
$393$	2773.59	0.356003
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6584.92	0.832462	0.416231	−	0.909259i	$-0.363351\pi$
$397$	6584.92	0.832462	0.416231	+	0.909259i	$0.363351\pi$
$398$	0	0
$399$	−7471.71	−0.937477
$400$	0	0
$401$	8030.00	0.999998	0.499999	−	0.866026i	$-0.333334\pi$
$401$	8030.00	0.999998	0.499999	+	0.866026i	$0.333334\pi$
$402$	0	0
$403$	−1424.00	−0.176016
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−356.000	−0.0433569
$408$	0	0
$409$	7670.00	0.927279	0.463639	−	0.886024i	$-0.346543\pi$
$409$	7670.00	0.927279	0.463639	+	0.886024i	$0.346543\pi$
$410$	0	0
$411$	1754.72	0.210594
$412$	0	0
$413$	16188.7	1.92880
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7924.54	0.930615
$418$	0	0
$419$	1226.42	0.142994	0.0714969	−	0.997441i	$-0.477222\pi$
$419$	1226.42	0.142994	0.0714969	+	0.997441i	$0.477222\pi$
$420$	0	0
$421$	6038.00	0.698988	0.349494	−	0.936939i	$-0.386353\pi$
$421$	6038.00	0.698988	0.349494	+	0.936939i	$0.386353\pi$
$422$	0	0
$423$	−36.0000	−0.00413801
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−308.000	−0.0349067
$428$	0	0
$429$	1068.00	0.120195
$430$	0	0
$431$	−12377.4	−1.38329	−0.691645	−	0.722238i	$-0.743113\pi$
$431$	−12377.4	−1.38329	−0.691645	+	0.722238i	$0.743113\pi$
$432$	0	0
$433$	3735.86	0.414628	0.207314	−	0.978274i	$-0.433528\pi$
$433$	3735.86	0.414628	0.207314	+	0.978274i	$0.433528\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−18113.2	−1.98278
$438$	0	0
$439$	10339.6	1.12411	0.562055	−	0.827100i	$-0.310011\pi$
$439$	10339.6	1.12411	0.562055	+	0.827100i	$0.310011\pi$
$440$	0	0
$441$	1269.00	0.137026
$442$	0	0
$443$	−3156.00	−0.338479	−0.169239	−	0.985575i	$-0.554131\pi$
$443$	−3156.00	−0.338479	−0.169239	+	0.985575i	$0.554131\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4356.00	0.460921
$448$	0	0
$449$	2830.00	0.297452	0.148726	−	0.988878i	$-0.452483\pi$
$449$	2830.00	0.297452	0.148726	+	0.988878i	$0.452483\pi$
$450$	0	0
$451$	6754.73	0.705250
$452$	0	0
$453$	−1584.91	−0.164383
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12226.4	−1.25149	−0.625743	−	0.780030i	$-0.715204\pi$
$457$	−12226.4	−1.25149	−0.625743	+	0.780030i	$0.715204\pi$
$458$	0	0
$459$	−3566.04	−0.362633
$460$	0	0
$461$	13724.0	1.38653	0.693265	−	0.720683i	$-0.256172\pi$
$461$	13724.0	1.38653	0.693265	+	0.720683i	$0.256172\pi$
$462$	0	0
$463$	−19110.0	−1.91818	−0.959090	−	0.283103i	$-0.908636\pi$
$463$	−19110.0	−1.91818	−0.959090	+	0.283103i	$0.908636\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1524.00	−0.151011	−0.0755057	−	0.997145i	$-0.524057\pi$
$467$	−1524.00	−0.151011	−0.0755057	+	0.997145i	$0.524057\pi$
$468$	0	0
$469$	18656.0	1.83679
$470$	0	0
$471$	−2547.17	−0.249188
$472$	0	0
$473$	−3245.29	−0.315473
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5943.41	0.570503
$478$	0	0
$479$	−4264.16	−0.406752	−0.203376	−	0.979101i	$-0.565191\pi$
$479$	−4264.16	−0.406752	−0.203376	+	0.979101i	$0.565191\pi$
$480$	0	0
$481$	−356.000	−0.0337468
$482$	0	0
$483$	10560.0	0.994817
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15838.0	−1.47369	−0.736846	−	0.676060i	$-0.763686\pi$
$487$	−15838.0	−1.47369	−0.736846	+	0.676060i	$0.763686\pi$
$488$	0	0
$489$	−10296.0	−0.952150
$490$	0	0
$491$	−15603.8	−1.43420	−0.717098	−	0.696973i	$-0.754530\pi$
$491$	−15603.8	−1.43420	−0.717098	+	0.696973i	$0.754530\pi$
$492$	0	0
$493$	−16905.7	−1.54441
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14113.2	1.27377
$498$	0	0
$499$	−14301.9	−1.28305	−0.641525	−	0.767102i	$-0.721698\pi$
$499$	−14301.9	−1.28305	−0.641525	+	0.767102i	$0.721698\pi$
$500$	0	0
$501$	−7992.00	−0.712687
$502$	0	0
$503$	−19140.0	−1.69664	−0.848320	−	0.529483i	$-0.822386\pi$
$503$	−19140.0	−1.69664	−0.848320	+	0.529483i	$0.822386\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5523.00	−0.483797
$508$	0	0
$509$	3072.00	0.267513	0.133756	−	0.991014i	$-0.457296\pi$
$509$	3072.00	0.267513	0.133756	+	0.991014i	$0.457296\pi$
$510$	0	0
$511$	25735.9	2.22796
$512$	0	0
$513$	−3056.61	−0.263066
$514$	0	0
$515$	0	0
$516$	0	0
$517$	75.4718	0.00642021
$518$	0	0
$519$	5490.58	0.464373
$520$	0	0
$521$	−11850.0	−0.996464	−0.498232	−	0.867044i	$-0.666017\pi$
$521$	−11850.0	−0.996464	−0.498232	+	0.867044i	$0.666017\pi$
$522$	0	0
$523$	−5256.00	−0.439443	−0.219722	−	0.975563i	$-0.570515\pi$
$523$	−5256.00	−0.439443	−0.219722	+	0.975563i	$0.570515\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9968.00	−0.823933
$528$	0	0
$529$	13433.0	1.10405
$530$	0	0
$531$	6622.65	0.541241
$532$	0	0
$533$	6754.73	0.548930
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9792.47	−0.786921
$538$	0	0
$539$	−2660.38	−0.212599
$540$	0	0
$541$	19898.0	1.58130	0.790649	−	0.612270i	$-0.209743\pi$
$541$	19898.0	1.58130	0.790649	+	0.612270i	$0.209743\pi$
$542$	0	0
$543$	10710.0	0.846427
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11844.0	0.925800	0.462900	−	0.886410i	$-0.346809\pi$
$547$	11844.0	0.925800	0.462900	+	0.886410i	$0.346809\pi$
$548$	0	0
$549$	−126.000	−0.00979517
$550$	0	0
$551$	−14490.6	−1.12036
$552$	0	0
$553$	−1660.38	−0.127679
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9641.53	0.733437	0.366719	−	0.930332i	$-0.380481\pi$
$557$	9641.53	0.733437	0.366719	+	0.930332i	$0.380481\pi$
$558$	0	0
$559$	−3245.29	−0.245548
$560$	0	0
$561$	7476.00	0.562633
$562$	0	0
$563$	8596.00	0.643478	0.321739	−	0.946828i	$-0.395733\pi$
$563$	8596.00	0.643478	0.321739	+	0.946828i	$0.395733\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1782.00	0.131988
$568$	0	0
$569$	−15234.0	−1.12239	−0.561197	−	0.827682i	$-0.689659\pi$
$569$	−15234.0	−1.12239	−0.561197	+	0.827682i	$0.689659\pi$
$570$	0	0
$571$	−14830.2	−1.08691	−0.543455	−	0.839438i	$-0.682884\pi$
$571$	−14830.2	−1.08691	−0.543455	+	0.839438i	$0.682884\pi$
$572$	0	0
$573$	9622.66	0.701557
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15547.2	−1.12173	−0.560865	−	0.827907i	$-0.689531\pi$
$577$	−15547.2	−1.12173	−0.560865	+	0.827907i	$0.689531\pi$
$578$	0	0
$579$	792.454	0.0568796
$580$	0	0
$581$	13112.0	0.936278
$582$	0	0
$583$	−12460.0	−0.885147
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25908.0	1.82170	0.910850	−	0.412738i	$-0.135428\pi$
$587$	25908.0	1.82170	0.910850	+	0.412738i	$0.135428\pi$
$588$	0	0
$589$	−8544.00	−0.597707
$590$	0	0
$591$	7415.11	0.516103
$592$	0	0
$593$	−14434.0	−0.999550	−0.499775	−	0.866155i	$-0.666584\pi$
$593$	−14434.0	−0.999550	−0.499775	+	0.866155i	$0.666584\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2037.74	0.139697
$598$	0	0
$599$	12377.4	0.844284	0.422142	−	0.906530i	$-0.361278\pi$
$599$	12377.4	0.844284	0.422142	+	0.906530i	$0.361278\pi$
$600$	0	0
$601$	−3630.00	−0.246374	−0.123187	−	0.992383i	$-0.539312\pi$
$601$	−3630.00	−0.246374	−0.123187	+	0.992383i	$0.539312\pi$
$602$	0	0
$603$	7632.00	0.515421
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19710.0	−1.31796	−0.658982	−	0.752159i	$-0.729013\pi$
$607$	−19710.0	−1.31796	−0.658982	+	0.752159i	$0.729013\pi$
$608$	0	0
$609$	8448.00	0.562119
$610$	0	0
$611$	75.4718	0.00499716
$612$	0	0
$613$	−8283.04	−0.545756	−0.272878	−	0.962049i	$-0.587976\pi$
$613$	−8283.04	−0.545756	−0.272878	+	0.962049i	$0.587976\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14509.5	0.946724	0.473362	−	0.880868i	$-0.343040\pi$
$617$	14509.5	0.946724	0.473362	+	0.880868i	$0.343040\pi$
$618$	0	0
$619$	12226.4	0.793897	0.396948	−	0.917841i	$-0.370069\pi$
$619$	12226.4	0.793897	0.396948	+	0.917841i	$0.370069\pi$
$620$	0	0
$621$	4320.00	0.279156
$622$	0	0
$623$	16500.0	1.06109
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6408.00	0.408151
$628$	0	0
$629$	−2492.00	−0.157969
$630$	0	0
$631$	29811.4	1.88078	0.940390	−	0.340098i	$-0.110460\pi$
$631$	29811.4	1.88078	0.940390	+	0.340098i	$0.110460\pi$
$632$	0	0
$633$	−10188.7	−0.639754
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2660.38	−0.165476
$638$	0	0
$639$	5773.60	0.357433
$640$	0	0
$641$	2690.00	0.165754	0.0828772	−	0.996560i	$-0.473589\pi$
$641$	2690.00	0.165754	0.0828772	+	0.996560i	$0.473589\pi$
$642$	0	0
$643$	1216.00	0.0745791	0.0372895	−	0.999305i	$-0.488128\pi$
$643$	1216.00	0.0745791	0.0372895	+	0.999305i	$0.488128\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22396.0	1.36086	0.680431	−	0.732812i	$-0.261793\pi$
$647$	22396.0	1.36086	0.680431	+	0.732812i	$0.261793\pi$
$648$	0	0
$649$	−13884.0	−0.839745
$650$	0	0
$651$	4981.14	0.299887
$652$	0	0
$653$	−11226.4	−0.672779	−0.336389	−	0.941723i	$-0.609206\pi$
$653$	−11226.4	−0.672779	−0.336389	+	0.941723i	$0.609206\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10528.3	0.625189
$658$	0	0
$659$	13377.4	0.790757	0.395379	−	0.918518i	$-0.370613\pi$
$659$	13377.4	0.790757	0.395379	+	0.918518i	$0.370613\pi$
$660$	0	0
$661$	5794.00	0.340939	0.170469	−	0.985363i	$-0.445472\pi$
$661$	5794.00	0.340939	0.170469	+	0.985363i	$0.445472\pi$
$662$	0	0
$663$	7476.00	0.437924
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20480.0	1.18889
$668$	0	0
$669$	954.000	0.0551327
$670$	0	0
$671$	264.151	0.0151974
$672$	0	0
$673$	21622.7	1.23847	0.619237	−	0.785204i	$-0.287442\pi$
$673$	21622.7	1.23847	0.619237	+	0.785204i	$0.287442\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11641.5	0.660887	0.330443	−	0.943826i	$-0.392802\pi$
$677$	11641.5	0.660887	0.330443	+	0.943826i	$0.392802\pi$
$678$	0	0
$679$	−11622.7	−0.656903
$680$	0	0
$681$	612.000	0.0344374
$682$	0	0
$683$	14924.0	0.836092	0.418046	−	0.908426i	$-0.362715\pi$
$683$	14924.0	0.836092	0.418046	+	0.908426i	$0.362715\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10350.0	−0.574785
$688$	0	0
$689$	−12460.0	−0.688952
$690$	0	0
$691$	17698.1	0.974341	0.487170	−	0.873307i	$-0.338029\pi$
$691$	17698.1	0.974341	0.487170	+	0.873307i	$0.338029\pi$
$692$	0	0
$693$	−3735.86	−0.204781
$694$	0	0
$695$	0	0
$696$	0	0
$697$	47283.1	2.56955
$698$	0	0
$699$	−10358.5	−0.560508
$700$	0	0
$701$	−1792.00	−0.0965519	−0.0482760	−	0.998834i	$-0.515373\pi$
$701$	−1792.00	−0.0965519	−0.0482760	+	0.998834i	$0.515373\pi$
$702$	0	0
$703$	−2136.00	−0.114596
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26928.0	1.43244
$708$	0	0
$709$	−562.000	−0.0297692	−0.0148846	−	0.999889i	$-0.504738\pi$
$709$	−562.000	−0.0297692	−0.0148846	+	0.999889i	$0.504738\pi$
$710$	0	0
$711$	−679.247	−0.0358280
$712$	0	0
$713$	12075.5	0.634265
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9622.66	−0.501206
$718$	0	0
$719$	−20905.7	−1.08435	−0.542177	−	0.840264i	$-0.682400\pi$
$719$	−20905.7	−1.08435	−0.542177	+	0.840264i	$0.682400\pi$
$720$	0	0
$721$	308.000	0.0159092
$722$	0	0
$723$	20418.0	1.05028
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−21006.0	−1.07162	−0.535811	−	0.844338i	$-0.679994\pi$
$727$	−21006.0	−1.07162	−0.535811	+	0.844338i	$0.679994\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−22717.0	−1.14941
$732$	0	0
$733$	−30622.7	−1.54308	−0.771538	−	0.636183i	$-0.780512\pi$
$733$	−30622.7	−1.54308	−0.771538	+	0.636183i	$0.780512\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16000.0	−0.799686
$738$	0	0
$739$	−21320.8	−1.06130	−0.530648	−	0.847592i	$-0.678051\pi$
$739$	−21320.8	−1.06130	−0.530648	+	0.847592i	$0.678051\pi$
$740$	0	0
$741$	6408.00	0.317684
$742$	0	0
$743$	32972.0	1.62803	0.814014	−	0.580845i	$-0.197278\pi$
$743$	32972.0	1.62803	0.814014	+	0.580845i	$0.197278\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5364.00	0.262729
$748$	0	0
$749$	18744.0	0.914407
$750$	0	0
$751$	−33509.5	−1.62820	−0.814101	−	0.580724i	$-0.802770\pi$
$751$	−33509.5	−1.62820	−0.814101	+	0.580724i	$0.802770\pi$
$752$	0	0
$753$	−6056.62	−0.293115
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23717.0	1.13872	0.569359	−	0.822089i	$-0.307191\pi$
$757$	23717.0	1.13872	0.569359	+	0.822089i	$0.307191\pi$
$758$	0	0
$759$	−9056.62	−0.433115
$760$	0	0
$761$	−38530.0	−1.83536	−0.917682	−	0.397317i	$-0.869941\pi$
$761$	−38530.0	−1.83536	−0.917682	+	0.397317i	$0.869941\pi$
$762$	0	0
$763$	−24068.0	−1.14197
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13884.0	−0.653614
$768$	0	0
$769$	−24926.0	−1.16886	−0.584431	−	0.811444i	$-0.698682\pi$
$769$	−24926.0	−1.16886	−0.584431	+	0.811444i	$0.698682\pi$
$770$	0	0
$771$	2207.55	0.103117
$772$	0	0
$773$	−3377.37	−0.157148	−0.0785740	−	0.996908i	$-0.525037\pi$
$773$	−3377.37	−0.157148	−0.0785740	+	0.996908i	$0.525037\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1245.29	0.0574960
$778$	0	0
$779$	40528.4	1.86403
$780$	0	0
$781$	−12104.0	−0.554565
$782$	0	0
$783$	3456.00	0.157736
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7144.00	0.323578	0.161789	−	0.986825i	$-0.448274\pi$
$787$	7144.00	0.323578	0.161789	+	0.986825i	$0.448274\pi$
$788$	0	0
$789$	−3216.00	−0.145111
$790$	0	0
$791$	−12037.8	−0.541104
$792$	0	0
$793$	264.151	0.0118289
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12207.6	−0.542553	−0.271276	−	0.962502i	$-0.587446\pi$
$797$	−12207.6	−0.542553	−0.271276	+	0.962502i	$0.587446\pi$
$798$	0	0
$799$	528.303	0.0233918
$800$	0	0
$801$	6750.00	0.297752
$802$	0	0
$803$	−22072.0	−0.969992
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7104.00	−0.309879
$808$	0	0
$809$	−15246.0	−0.662572	−0.331286	−	0.943530i	$-0.607483\pi$
$809$	−15246.0	−0.662572	−0.331286	+	0.943530i	$0.607483\pi$
$810$	0	0
$811$	−15622.7	−0.676432	−0.338216	−	0.941069i	$-0.609823\pi$
$811$	−15622.7	−0.676432	−0.338216	+	0.941069i	$0.609823\pi$
$812$	0	0
$813$	4075.48	0.175810
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−19471.7	−0.833819
$818$	0	0
$819$	−3735.86	−0.159391
$820$	0	0
$821$	13736.0	0.583910	0.291955	−	0.956432i	$-0.405694\pi$
$821$	13736.0	0.583910	0.291955	+	0.956432i	$0.405694\pi$
$822$	0	0
$823$	−22222.0	−0.941203	−0.470602	−	0.882346i	$-0.655963\pi$
$823$	−22222.0	−0.941203	−0.470602	+	0.882346i	$0.655963\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41700.0	1.75339	0.876694	−	0.481049i	$-0.159744\pi$
$827$	41700.0	1.75339	0.876694	+	0.481049i	$0.159744\pi$
$828$	0	0
$829$	−20206.0	−0.846542	−0.423271	−	0.906003i	$-0.639118\pi$
$829$	−20206.0	−0.846542	−0.423271	+	0.906003i	$0.639118\pi$
$830$	0	0
$831$	20886.8	0.871909
$832$	0	0
$833$	−18622.7	−0.774595
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2037.74	0.0841512
$838$	0	0
$839$	5811.33	0.239129	0.119565	−	0.992826i	$-0.461850\pi$
$839$	5811.33	0.239129	0.119565	+	0.992826i	$0.461850\pi$
$840$	0	0
$841$	−8005.00	−0.328222
$842$	0	0
$843$	20682.0	0.844989
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21450.0	−0.870166
$848$	0	0
$849$	−7368.00	−0.297843
$850$	0	0
$851$	3018.87	0.121605
$852$	0	0
$853$	13981.2	0.561203	0.280601	−	0.959824i	$-0.409466\pi$
$853$	13981.2	0.561203	0.280601	+	0.959824i	$0.409466\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34773.7	−1.38605	−0.693025	−	0.720913i	$-0.743723\pi$
$857$	−34773.7	−1.38605	−0.693025	+	0.720913i	$0.743723\pi$
$858$	0	0
$859$	32000.1	1.27105	0.635523	−	0.772082i	$-0.280785\pi$
$859$	32000.1	1.27105	0.635523	+	0.772082i	$0.280785\pi$
$860$	0	0
$861$	−23628.0	−0.935238
$862$	0	0
$863$	−7424.00	−0.292834	−0.146417	−	0.989223i	$-0.546774\pi$
$863$	−7424.00	−0.292834	−0.146417	+	0.989223i	$0.546774\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	37593.0	1.47258
$868$	0	0
$869$	1424.00	0.0555879
$870$	0	0
$871$	−16000.0	−0.622434
$872$	0	0
$873$	−4754.73	−0.184333
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−11717.0	−0.451146	−0.225573	−	0.974226i	$-0.572425\pi$
$877$	−11717.0	−0.451146	−0.225573	+	0.974226i	$0.572425\pi$
$878$	0	0
$879$	−27905.7	−1.07080
$880$	0	0
$881$	−7722.00	−0.295302	−0.147651	−	0.989040i	$-0.547171\pi$
$881$	−7722.00	−0.295302	−0.147651	+	0.989040i	$0.547171\pi$
$882$	0	0
$883$	24400.0	0.929927	0.464963	−	0.885330i	$-0.346067\pi$
$883$	24400.0	0.929927	0.464963	+	0.885330i	$0.346067\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1032.00	−0.0390656	−0.0195328	−	0.999809i	$-0.506218\pi$
$887$	−1032.00	−0.0390656	−0.0195328	+	0.999809i	$0.506218\pi$
$888$	0	0
$889$	−22220.0	−0.838284
$890$	0	0
$891$	−1528.30	−0.0574637
$892$	0	0
$893$	452.831	0.0169691
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9056.62	−0.337115
$898$	0	0
$899$	9660.40	0.358390
$900$	0	0
$901$	−87220.0	−3.22499
$902$	0	0
$903$	11352.0	0.418351
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−25380.0	−0.929139	−0.464569	−	0.885537i	$-0.653791\pi$
$907$	−25380.0	−0.929139	−0.464569	+	0.885537i	$0.653791\pi$
$908$	0	0
$909$	11016.0	0.401956
$910$	0	0
$911$	39245.4	1.42729	0.713643	−	0.700510i	$-0.247044\pi$
$911$	39245.4	1.42729	0.713643	+	0.700510i	$0.247044\pi$
$912$	0	0
$913$	−11245.3	−0.407629
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20339.7	0.732470
$918$	0	0
$919$	−30641.6	−1.09986	−0.549930	−	0.835210i	$-0.685346\pi$
$919$	−30641.6	−1.09986	−0.549930	+	0.835210i	$0.685346\pi$
$920$	0	0
$921$	−14892.0	−0.532799
$922$	0	0
$923$	−12104.0	−0.431645
$924$	0	0
$925$	0	0
$926$	0	0
$927$	126.000	0.00446428
$928$	0	0
$929$	25246.0	0.891598	0.445799	−	0.895133i	$-0.352920\pi$
$929$	25246.0	0.891598	0.445799	+	0.895133i	$0.352920\pi$
$930$	0	0
$931$	−15962.3	−0.561915
$932$	0	0
$933$	15056.6	0.528330
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16528.3	0.576262	0.288131	−	0.957591i	$-0.406966\pi$
$937$	16528.3	0.576262	0.288131	+	0.957591i	$0.406966\pi$
$938$	0	0
$939$	8490.58	0.295080
$940$	0	0
$941$	−13372.0	−0.463246	−0.231623	−	0.972806i	$-0.574404\pi$
$941$	−13372.0	−0.463246	−0.231623	+	0.972806i	$0.574404\pi$
$942$	0	0
$943$	−57280.0	−1.97804
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19564.0	−0.671325	−0.335662	−	0.941982i	$-0.608960\pi$
$947$	−19564.0	−0.671325	−0.335662	+	0.941982i	$0.608960\pi$
$948$	0	0
$949$	−22072.0	−0.754992
$950$	0	0
$951$	−9000.02	−0.306883
$952$	0	0
$953$	13566.1	0.461121	0.230560	−	0.973058i	$-0.425944\pi$
$953$	13566.1	0.461121	0.230560	+	0.973058i	$0.425944\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7245.30	−0.244731
$958$	0	0
$959$	12868.0	0.433293
$960$	0	0
$961$	−24095.0	−0.808801
$962$	0	0
$963$	7668.00	0.256592
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1874.00	−0.0623203	−0.0311602	−	0.999514i	$-0.509920\pi$
$967$	−1874.00	−0.0623203	−0.0311602	+	0.999514i	$0.509920\pi$
$968$	0	0
$969$	44856.0	1.48708
$970$	0	0
$971$	24849.1	0.821262	0.410631	−	0.911802i	$-0.365308\pi$
$971$	24849.1	0.821262	0.410631	+	0.911802i	$0.365308\pi$
$972$	0	0
$973$	58113.3	1.91473
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22283.1	0.729681	0.364841	−	0.931070i	$-0.381124\pi$
$977$	22283.1	0.729681	0.364841	+	0.931070i	$0.381124\pi$
$978$	0	0
$979$	−14151.0	−0.461968
$980$	0	0
$981$	−9846.00	−0.320447
$982$	0	0
$983$	53892.0	1.74861	0.874307	−	0.485373i	$-0.161316\pi$
$983$	53892.0	1.74861	0.874307	+	0.485373i	$0.161316\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−264.000	−0.00851389
$988$	0	0
$989$	27520.0	0.884818
$990$	0	0
$991$	60981.3	1.95473	0.977363	−	0.211570i	$-0.0678575\pi$
$991$	60981.3	1.95473	0.977363	+	0.211570i	$0.0678575\pi$
$992$	0	0
$993$	18000.0	0.575241
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−60019.0	−1.90654	−0.953270	−	0.302120i	$-0.902306\pi$
$997$	−60019.0	−1.90654	−0.953270	+	0.302120i	$0.902306\pi$
$998$	0	0
$999$	509.435	0.0161339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bd.1.1		2
4.3	odd	2		2400.4.a.w.1.2		2
5.2	odd	4		480.4.f.c.289.1	✓	4
5.3	odd	4		480.4.f.c.289.4	yes	4
5.4	even	2		2400.4.a.w.1.1		2
15.2	even	4		1440.4.f.f.289.4		4
15.8	even	4		1440.4.f.f.289.1		4
20.3	even	4		480.4.f.c.289.2	yes	4
20.7	even	4		480.4.f.c.289.3	yes	4
20.19	odd	2	inner	2400.4.a.bd.1.2		2
40.3	even	4		960.4.f.m.769.3		4
40.13	odd	4		960.4.f.m.769.1		4
40.27	even	4		960.4.f.m.769.2		4
40.37	odd	4		960.4.f.m.769.4		4
60.23	odd	4		1440.4.f.f.289.2		4
60.47	odd	4		1440.4.f.f.289.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.c.289.1	✓	4	5.2	odd	4
480.4.f.c.289.2	yes	4	20.3	even	4
480.4.f.c.289.3	yes	4	20.7	even	4
480.4.f.c.289.4	yes	4	5.3	odd	4
960.4.f.m.769.1		4	40.13	odd	4
960.4.f.m.769.2		4	40.27	even	4
960.4.f.m.769.3		4	40.3	even	4
960.4.f.m.769.4		4	40.37	odd	4
1440.4.f.f.289.1		4	15.8	even	4
1440.4.f.f.289.2		4	60.23	odd	4
1440.4.f.f.289.3		4	60.47	odd	4
1440.4.f.f.289.4		4	15.2	even	4
2400.4.a.w.1.1		2	5.4	even	2
2400.4.a.w.1.2		2	4.3	odd	2
2400.4.a.bd.1.1		2	1.1	even	1	trivial
2400.4.a.bd.1.2		2	20.19	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +22.0000 q^{7} +9.00000 q^{9} -18.8680 q^{11} -18.8680 q^{13} -132.076 q^{17} -113.208 q^{19} +66.0000 q^{21} +160.000 q^{23} +27.0000 q^{27} +128.000 q^{29} +75.4718 q^{31} -56.6039 q^{33} +18.8680 q^{37} -56.6039 q^{39} -358.000 q^{41} +172.000 q^{43} -4.00000 q^{47} +141.000 q^{49} -396.227 q^{51} +660.379 q^{53} -339.623 q^{57} +735.851 q^{59} -14.0000 q^{61} +198.000 q^{63} +848.000 q^{67} +480.000 q^{69} +641.511 q^{71} +1169.81 q^{73} -415.095 q^{77} -75.4718 q^{79} +81.0000 q^{81} +596.000 q^{83} +384.000 q^{87} +750.000 q^{89} -415.095 q^{91} +226.416 q^{93} -528.303 q^{97} -169.812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 44 q^{7} + 18 q^{9} + 132 q^{21} + 320 q^{23} + 54 q^{27} + 256 q^{29} - 716 q^{41} + 344 q^{43} - 8 q^{47} + 282 q^{49} - 28 q^{61} + 396 q^{63} + 1696 q^{67} + 960 q^{69} + 162 q^{81}+ \cdots + 1500 q^{89}+O(q^{100}) \) 2 * q + 6 * q^3 + 44 * q^7 + 18 * q^9 + 132 * q^21 + 320 * q^23 + 54 * q^27 + 256 * q^29 - 716 * q^41 + 344 * q^43 - 8 * q^47 + 282 * q^49 - 28 * q^61 + 396 * q^63 + 1696 * q^67 + 960 * q^69 + 162 * q^81 + 1192 * q^83 + 768 * q^87 + 1500 * q^89

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