LMFDB - Embedded newform 2400.3.j.d.799.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,3,Mod(799,2400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2400.799"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2400.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-16,0,24,0,0,0,0,0,0,0,0,0,0,0,96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$65.3952634465$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.8
Root			$0.535233 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	2400.799
Dual form			2400.3.j.d.799.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +9.40034 q^{7} +3.00000 q^{9} +10.2658i q^{11} -22.1543i q^{13} +30.7181i q^{17} +34.0627i q^{19} +16.2819 q^{21} -10.3735 q^{23} +5.19615 q^{27} +25.4164 q^{29} +33.3338i q^{31} +17.7809i q^{33} -10.5670i q^{37} -38.3724i q^{39} -62.2047 q^{41} -54.6732 q^{43} -37.8419 q^{47} +39.3664 q^{49} +53.2053i q^{51} +82.6051i q^{53} +58.9984i q^{57} +33.7021i q^{59} +78.0215 q^{61} +28.2010 q^{63} +131.316 q^{67} -17.9674 q^{69} +22.6109i q^{71} -60.9732i q^{73} +96.5020i q^{77} -27.8901i q^{79} +9.00000 q^{81} +70.7268 q^{83} +44.0225 q^{87} -28.7655 q^{89} -208.258i q^{91} +57.7358i q^{93} -10.5744i q^{97} +30.7974i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{7} + 24 q^{9} + 96 q^{21} - 16 q^{23} + 96 q^{29} + 112 q^{41} - 128 q^{43} - 272 q^{47} + 184 q^{49} + 80 q^{61} - 48 q^{63} + 320 q^{67} + 144 q^{69} + 72 q^{81} - 320 q^{83} + 48 q^{89}+O(q^{100}) $ 8 * q - 16 * q^7 + 24 * q^9 + 96 * q^21 - 16 * q^23 + 96 * q^29 + 112 * q^41 - 128 * q^43 - 272 * q^47 + 184 * q^49 + 80 * q^61 - 48 * q^63 + 320 * q^67 + 144 * q^69 + 72 * q^81 - 320 * q^83 + 48 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.40034	1.34291	0.671453	−	0.741047i	$-0.265671\pi$
$7$	9.40034	1.34291	0.671453	+	0.741047i	$0.265671\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	10.2658i	0.933254i	0.884454	+	0.466627i	$0.154531\pi$
$11$	10.2658i	0.933254i	−0.884454	+	0.466627i	$0.845469\pi$
$12$	0	0
$13$	− 22.1543i	− 1.70418i	−0.523395	−	0.852090i	$-0.675335\pi$
$13$	− 22.1543i	− 1.70418i	0.523395	−	0.852090i	$-0.324665\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.7181i	1.80695i	0.428646	+	0.903473i	$0.358991\pi$
$17$	30.7181i	1.80695i	−0.428646	+	0.903473i	$0.641009\pi$
$18$	0	0
$19$	34.0627i	1.79278i	0.443270	+	0.896388i	$0.353818\pi$
$19$	34.0627i	1.79278i	−0.443270	+	0.896388i	$0.646182\pi$
$20$	0	0
$21$	16.2819	0.775327
$22$	0	0
$23$	−10.3735	−0.451020	−0.225510	−	0.974241i	$-0.572405\pi$
$23$	−10.3735	−0.451020	−0.225510	+	0.974241i	$0.572405\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	25.4164	0.876428	0.438214	−	0.898871i	$-0.355611\pi$
$29$	25.4164	0.876428	0.438214	+	0.898871i	$0.355611\pi$
$30$	0	0
$31$	33.3338i	1.07528i	0.843173	+	0.537642i	$0.180685\pi$
$31$	33.3338i	1.07528i	−0.843173	+	0.537642i	$0.819315\pi$
$32$	0	0
$33$	17.7809i	0.538815i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.5670i	− 0.285595i	−0.989752	−	0.142798i	$-0.954390\pi$
$37$	− 10.5670i	− 0.285595i	0.989752	−	0.142798i	$-0.0456098\pi$
$38$	0	0
$39$	− 38.3724i	− 0.983909i
$40$	0	0
$41$	−62.2047	−1.51719	−0.758594	−	0.651564i	$-0.774113\pi$
$41$	−62.2047	−1.51719	−0.758594	+	0.651564i	$0.774113\pi$
$42$	0	0
$43$	−54.6732	−1.27147	−0.635734	−	0.771908i	$-0.719303\pi$
$43$	−54.6732	−1.27147	−0.635734	+	0.771908i	$0.719303\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−37.8419	−0.805146	−0.402573	−	0.915388i	$-0.631884\pi$
$47$	−37.8419	−0.805146	−0.402573	+	0.915388i	$0.631884\pi$
$48$	0	0
$49$	39.3664	0.803395
$50$	0	0
$51$	53.2053i	1.04324i
$52$	0	0
$53$	82.6051i	1.55859i	0.626659	+	0.779293i	$0.284422\pi$
$53$	82.6051i	1.55859i	−0.626659	+	0.779293i	$0.715578\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	58.9984i	1.03506i
$58$	0	0
$59$	33.7021i	0.571221i	0.958346	+	0.285611i	$0.0921964\pi$
$59$	33.7021i	0.571221i	−0.958346	+	0.285611i	$0.907804\pi$
$60$	0	0
$61$	78.0215	1.27904	0.639520	−	0.768774i	$-0.279133\pi$
$61$	78.0215	1.27904	0.639520	+	0.768774i	$0.279133\pi$
$62$	0	0
$63$	28.2010	0.447635
$64$	0	0
$65$	0	0
$66$	0	0
$67$	131.316	1.95994	0.979971	−	0.199142i	$-0.0638154\pi$
$67$	131.316	1.95994	0.979971	+	0.199142i	$0.0638154\pi$
$68$	0	0
$69$	−17.9674	−0.260397
$70$	0	0
$71$	22.6109i	0.318464i	0.987241	+	0.159232i	$0.0509017\pi$
$71$	22.6109i	0.318464i	−0.987241	+	0.159232i	$0.949098\pi$
$72$	0	0
$73$	− 60.9732i	− 0.835250i	−0.908620	−	0.417625i	$-0.862863\pi$
$73$	− 60.9732i	− 0.835250i	0.908620	−	0.417625i	$-0.137137\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	96.5020i	1.25327i
$78$	0	0
$79$	− 27.8901i	− 0.353039i	−0.984297	−	0.176519i	$-0.943516\pi$
$79$	− 27.8901i	− 0.353039i	0.984297	−	0.176519i	$-0.0564838\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	70.7268	0.852130	0.426065	−	0.904693i	$-0.359899\pi$
$83$	70.7268	0.852130	0.426065	+	0.904693i	$0.359899\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	44.0225	0.506006
$88$	0	0
$89$	−28.7655	−0.323208	−0.161604	−	0.986856i	$-0.551667\pi$
$89$	−28.7655	−0.323208	−0.161604	+	0.986856i	$0.551667\pi$
$90$	0	0
$91$	− 208.258i	− 2.28855i
$92$	0	0
$93$	57.7358i	0.620815i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.5744i	− 0.109014i	−0.998513	−	0.0545071i	$-0.982641\pi$
$97$	− 10.5744i	− 0.109014i	0.998513	−	0.0545071i	$-0.0173587\pi$
$98$	0	0
$99$	30.7974i	0.311085i
$100$	0	0
$101$	−10.7519	−0.106454	−0.0532271	−	0.998582i	$-0.516951\pi$
$101$	−10.7519	−0.106454	−0.0532271	+	0.998582i	$0.516951\pi$
$102$	0	0
$103$	−70.8109	−0.687485	−0.343742	−	0.939064i	$-0.611695\pi$
$103$	−70.8109	−0.687485	−0.343742	+	0.939064i	$0.611695\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	38.3936	0.358818	0.179409	−	0.983775i	$-0.442581\pi$
$107$	38.3936	0.358818	0.179409	+	0.983775i	$0.442581\pi$
$108$	0	0
$109$	−156.704	−1.43765	−0.718826	−	0.695190i	$-0.755320\pi$
$109$	−156.704	−1.43765	−0.718826	+	0.695190i	$0.755320\pi$
$110$	0	0
$111$	− 18.3026i	− 0.164888i
$112$	0	0
$113$	− 148.344i	− 1.31278i	−0.754422	−	0.656389i	$-0.772083\pi$
$113$	− 148.344i	− 1.31278i	0.754422	−	0.656389i	$-0.227917\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 66.4630i	− 0.568060i
$118$	0	0
$119$	288.760i	2.42656i
$120$	0	0
$121$	15.6134	0.129037
$122$	0	0
$123$	−107.742	−0.875949
$124$	0	0
$125$	0	0
$126$	0	0
$127$	109.508	0.862267	0.431133	−	0.902288i	$-0.358114\pi$
$127$	109.508	0.862267	0.431133	+	0.902288i	$0.358114\pi$
$128$	0	0
$129$	−94.6967	−0.734083
$130$	0	0
$131$	− 48.7899i	− 0.372442i	−0.982508	−	0.186221i	$-0.940376\pi$
$131$	− 48.7899i	− 0.372442i	0.982508	−	0.186221i	$-0.0596241\pi$
$132$	0	0
$133$	320.201i	2.40753i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	44.6433i	0.325863i	0.986637	+	0.162932i	$0.0520950\pi$
$137$	44.6433i	0.325863i	−0.986637	+	0.162932i	$0.947905\pi$
$138$	0	0
$139$	111.352i	0.801094i	0.916276	+	0.400547i	$0.131180\pi$
$139$	111.352i	0.801094i	−0.916276	+	0.400547i	$0.868820\pi$
$140$	0	0
$141$	−65.5440	−0.464851
$142$	0	0
$143$	227.432	1.59043
$144$	0	0
$145$	0	0
$146$	0	0
$147$	68.1846	0.463841
$148$	0	0
$149$	41.7071	0.279914	0.139957	−	0.990158i	$-0.455304\pi$
$149$	41.7071	0.279914	0.139957	+	0.990158i	$0.455304\pi$
$150$	0	0
$151$	− 53.3242i	− 0.353140i	−0.984288	−	0.176570i	$-0.943500\pi$
$151$	− 53.3242i	− 0.353140i	0.984288	−	0.176570i	$-0.0565002\pi$
$152$	0	0
$153$	92.1542i	0.602315i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 86.5304i	− 0.551149i	−0.961280	−	0.275574i	$-0.911132\pi$
$157$	− 86.5304i	− 0.551149i	0.961280	−	0.275574i	$-0.0888680\pi$
$158$	0	0
$159$	143.076i	0.899850i
$160$	0	0
$161$	−97.5140	−0.605677
$162$	0	0
$163$	36.4719	0.223754	0.111877	−	0.993722i	$-0.464314\pi$
$163$	36.4719	0.223754	0.111877	+	0.993722i	$0.464314\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	219.793	1.31612	0.658062	−	0.752963i	$-0.271376\pi$
$167$	219.793	1.31612	0.658062	+	0.752963i	$0.271376\pi$
$168$	0	0
$169$	−321.815	−1.90423
$170$	0	0
$171$	102.188i	0.597592i
$172$	0	0
$173$	− 51.3508i	− 0.296825i	−0.988926	−	0.148413i	$-0.952584\pi$
$173$	− 51.3508i	− 0.296825i	0.988926	−	0.148413i	$-0.0474164\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	58.3737i	0.329795i
$178$	0	0
$179$	80.2049i	0.448072i	0.974581	+	0.224036i	$0.0719233\pi$
$179$	80.2049i	0.448072i	−0.974581	+	0.224036i	$0.928077\pi$
$180$	0	0
$181$	−114.578	−0.633027	−0.316514	−	0.948588i	$-0.602512\pi$
$181$	−114.578	−0.633027	−0.316514	+	0.948588i	$0.602512\pi$
$182$	0	0
$183$	135.137	0.738455
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−315.345	−1.68634
$188$	0	0
$189$	48.8456	0.258442
$190$	0	0
$191$	− 71.0845i	− 0.372170i	−0.982534	−	0.186085i	$-0.940420\pi$
$191$	− 71.0845i	− 0.372170i	0.982534	−	0.186085i	$-0.0595801\pi$
$192$	0	0
$193$	25.1660i	0.130394i	0.997872	+	0.0651970i	$0.0207676\pi$
$193$	25.1660i	0.130394i	−0.997872	+	0.0651970i	$0.979232\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	247.751i	1.25762i	0.777559	+	0.628810i	$0.216458\pi$
$197$	247.751i	1.25762i	−0.777559	+	0.628810i	$0.783542\pi$
$198$	0	0
$199$	− 72.5171i	− 0.364407i	−0.983261	−	0.182204i	$-0.941677\pi$
$199$	− 72.5171i	− 0.364407i	0.983261	−	0.182204i	$-0.0583230\pi$
$200$	0	0
$201$	227.446	1.13157
$202$	0	0
$203$	238.923	1.17696
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−31.1204	−0.150340
$208$	0	0
$209$	−349.681	−1.67312
$210$	0	0
$211$	67.7838i	0.321250i	0.987015	+	0.160625i	$0.0513510\pi$
$211$	67.7838i	0.321250i	−0.987015	+	0.160625i	$0.948649\pi$
$212$	0	0
$213$	39.1632i	0.183865i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	313.349i	1.44400i
$218$	0	0
$219$	− 105.609i	− 0.482232i
$220$	0	0
$221$	680.539	3.07936
$222$	0	0
$223$	133.216	0.597381	0.298690	−	0.954350i	$-0.403450\pi$
$223$	133.216	0.597381	0.298690	+	0.954350i	$0.403450\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	100.300	0.441851	0.220926	−	0.975291i	$-0.429092\pi$
$227$	100.300	0.441851	0.220926	+	0.975291i	$0.429092\pi$
$228$	0	0
$229$	−14.8076	−0.0646622	−0.0323311	−	0.999477i	$-0.510293\pi$
$229$	−14.8076	−0.0646622	−0.0323311	+	0.999477i	$0.510293\pi$
$230$	0	0
$231$	167.146i	0.723577i
$232$	0	0
$233$	267.030i	1.14605i	0.819538	+	0.573025i	$0.194230\pi$
$233$	267.030i	1.14605i	−0.819538	+	0.573025i	$0.805770\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 48.3070i	− 0.203827i
$238$	0	0
$239$	− 21.4827i	− 0.0898858i	−0.998990	−	0.0449429i	$-0.985689\pi$
$239$	− 21.4827i	− 0.0898858i	0.998990	−	0.0449429i	$-0.0143106\pi$
$240$	0	0
$241$	12.7498	0.0529039	0.0264520	−	0.999650i	$-0.491579\pi$
$241$	12.7498	0.0529039	0.0264520	+	0.999650i	$0.491579\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	754.638	3.05521
$248$	0	0
$249$	122.502	0.491977
$250$	0	0
$251$	− 198.528i	− 0.790950i	−0.918477	−	0.395475i	$-0.870580\pi$
$251$	− 198.528i	− 0.790950i	0.918477	−	0.395475i	$-0.129420\pi$
$252$	0	0
$253$	− 106.492i	− 0.420916i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	417.753i	1.62550i	0.582615	+	0.812748i	$0.302029\pi$
$257$	417.753i	1.62550i	−0.582615	+	0.812748i	$0.697971\pi$
$258$	0	0
$259$	− 99.3336i	− 0.383527i
$260$	0	0
$261$	76.2492	0.292143
$262$	0	0
$263$	61.0555	0.232150	0.116075	−	0.993240i	$-0.462969\pi$
$263$	61.0555	0.232150	0.116075	+	0.993240i	$0.462969\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−49.8233	−0.186604
$268$	0	0
$269$	459.039	1.70647	0.853233	−	0.521530i	$-0.174639\pi$
$269$	459.039	1.70647	0.853233	+	0.521530i	$0.174639\pi$
$270$	0	0
$271$	92.9034i	0.342817i	0.985200	+	0.171408i	$0.0548318\pi$
$271$	92.9034i	0.342817i	−0.985200	+	0.171408i	$0.945168\pi$
$272$	0	0
$273$	− 360.714i	− 1.32130i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	415.661i	1.50058i	0.661108	+	0.750291i	$0.270087\pi$
$277$	415.661i	1.50058i	−0.661108	+	0.750291i	$0.729913\pi$
$278$	0	0
$279$	100.001i	0.358428i
$280$	0	0
$281$	−271.730	−0.967011	−0.483506	−	0.875341i	$-0.660637\pi$
$281$	−271.730	−0.967011	−0.483506	+	0.875341i	$0.660637\pi$
$282$	0	0
$283$	208.707	0.737482	0.368741	−	0.929532i	$-0.379789\pi$
$283$	208.707	0.737482	0.368741	+	0.929532i	$0.379789\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−584.745	−2.03744
$288$	0	0
$289$	−654.600	−2.26505
$290$	0	0
$291$	− 18.3154i	− 0.0629394i
$292$	0	0
$293$	− 27.7933i	− 0.0948577i	−0.998875	−	0.0474288i	$-0.984897\pi$
$293$	− 27.7933i	− 0.0948577i	0.998875	−	0.0474288i	$-0.0151027\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	53.3426i	0.179605i
$298$	0	0
$299$	229.817i	0.768619i
$300$	0	0
$301$	−513.946	−1.70746
$302$	0	0
$303$	−18.6228	−0.0614613
$304$	0	0
$305$	0	0
$306$	0	0
$307$	236.018	0.768789	0.384394	−	0.923169i	$-0.374410\pi$
$307$	236.018	0.768789	0.384394	+	0.923169i	$0.374410\pi$
$308$	0	0
$309$	−122.648	−0.396919
$310$	0	0
$311$	87.7923i	0.282290i	0.989989	+	0.141145i	$0.0450784\pi$
$311$	87.7923i	0.282290i	−0.989989	+	0.141145i	$0.954922\pi$
$312$	0	0
$313$	− 177.589i	− 0.567376i	−0.958917	−	0.283688i	$-0.908442\pi$
$313$	− 177.589i	− 0.567376i	0.958917	−	0.283688i	$-0.0915581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	437.410i	1.37984i	0.723885	+	0.689921i	$0.242355\pi$
$317$	437.410i	1.37984i	−0.723885	+	0.689921i	$0.757645\pi$
$318$	0	0
$319$	260.920i	0.817930i
$320$	0	0
$321$	66.4996	0.207164
$322$	0	0
$323$	−1046.34	−3.23945
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−271.419	−0.830028
$328$	0	0
$329$	−355.726	−1.08124
$330$	0	0
$331$	− 198.207i	− 0.598814i	−0.954125	−	0.299407i	$-0.903211\pi$
$331$	− 198.207i	− 0.598814i	0.954125	−	0.299407i	$-0.0967889\pi$
$332$	0	0
$333$	− 31.7011i	− 0.0951984i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 541.738i	− 1.60753i	−0.594946	−	0.803766i	$-0.702827\pi$
$337$	− 541.738i	− 1.60753i	0.594946	−	0.803766i	$-0.297173\pi$
$338$	0	0
$339$	− 256.939i	− 0.757933i
$340$	0	0
$341$	−342.198	−1.00351
$342$	0	0
$343$	−90.5593	−0.264021
$344$	0	0
$345$	0	0
$346$	0	0
$347$	654.377	1.88581	0.942906	−	0.333059i	$-0.108081\pi$
$347$	654.377	1.88581	0.942906	+	0.333059i	$0.108081\pi$
$348$	0	0
$349$	363.418	1.04131	0.520655	−	0.853767i	$-0.325688\pi$
$349$	363.418	1.04131	0.520655	+	0.853767i	$0.325688\pi$
$350$	0	0
$351$	− 115.117i	− 0.327970i
$352$	0	0
$353$	− 335.772i	− 0.951195i	−0.879663	−	0.475597i	$-0.842232\pi$
$353$	− 335.772i	− 0.951195i	0.879663	−	0.475597i	$-0.157768\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	500.147i	1.40097i
$358$	0	0
$359$	− 479.525i	− 1.33573i	−0.744285	−	0.667863i	$-0.767209\pi$
$359$	− 479.525i	− 1.33573i	0.744285	−	0.667863i	$-0.232791\pi$
$360$	0	0
$361$	−799.271	−2.21405
$362$	0	0
$363$	27.0432	0.0744993
$364$	0	0
$365$	0	0
$366$	0	0
$367$	690.724	1.88208	0.941041	−	0.338292i	$-0.109849\pi$
$367$	690.724	1.88208	0.941041	+	0.338292i	$0.109849\pi$
$368$	0	0
$369$	−186.614	−0.505729
$370$	0	0
$371$	776.516i	2.09303i
$372$	0	0
$373$	295.228i	0.791496i	0.918359	+	0.395748i	$0.129515\pi$
$373$	295.228i	0.791496i	−0.918359	+	0.395748i	$0.870485\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 563.084i	− 1.49359i
$378$	0	0
$379$	314.112i	0.828791i	0.910097	+	0.414396i	$0.136007\pi$
$379$	314.112i	0.828791i	−0.910097	+	0.414396i	$0.863993\pi$
$380$	0	0
$381$	189.673	0.497830
$382$	0	0
$383$	−68.8331	−0.179721	−0.0898605	−	0.995954i	$-0.528642\pi$
$383$	−68.8331	−0.179721	−0.0898605	+	0.995954i	$0.528642\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−164.019	−0.423823
$388$	0	0
$389$	−357.730	−0.919613	−0.459807	−	0.888019i	$-0.652081\pi$
$389$	−357.730	−0.919613	−0.459807	+	0.888019i	$0.652081\pi$
$390$	0	0
$391$	− 318.653i	− 0.814968i
$392$	0	0
$393$	− 84.5066i	− 0.215030i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 715.056i	− 1.80115i	−0.434703	−	0.900574i	$-0.643146\pi$
$397$	− 715.056i	− 1.80115i	0.434703	−	0.900574i	$-0.356854\pi$
$398$	0	0
$399$	554.605i	1.38999i
$400$	0	0
$401$	624.362	1.55701	0.778506	−	0.627637i	$-0.215978\pi$
$401$	624.362	1.55701	0.778506	+	0.627637i	$0.215978\pi$
$402$	0	0
$403$	738.488	1.83248
$404$	0	0
$405$	0	0
$406$	0	0
$407$	108.479	0.266533
$408$	0	0
$409$	183.591	0.448877	0.224438	−	0.974488i	$-0.427945\pi$
$409$	183.591	0.448877	0.224438	+	0.974488i	$0.427945\pi$
$410$	0	0
$411$	77.3244i	0.188137i
$412$	0	0
$413$	316.811i	0.767096i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	192.868i	0.462512i
$418$	0	0
$419$	130.389i	0.311192i	0.987821	+	0.155596i	$0.0497298\pi$
$419$	130.389i	0.311192i	−0.987821	+	0.155596i	$0.950270\pi$
$420$	0	0
$421$	−194.504	−0.462005	−0.231002	−	0.972953i	$-0.574201\pi$
$421$	−194.504	−0.462005	−0.231002	+	0.972953i	$0.574201\pi$
$422$	0	0
$423$	−113.526	−0.268382
$424$	0	0
$425$	0	0
$426$	0	0
$427$	733.429	1.71763
$428$	0	0
$429$	393.924	0.918237
$430$	0	0
$431$	− 172.160i	− 0.399444i	−0.979853	−	0.199722i	$-0.935996\pi$
$431$	− 172.160i	− 0.399444i	0.979853	−	0.199722i	$-0.0640038\pi$
$432$	0	0
$433$	161.682i	0.373400i	0.982417	+	0.186700i	$0.0597793\pi$
$433$	161.682i	0.373400i	−0.982417	+	0.186700i	$0.940221\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 353.349i	− 0.808578i
$438$	0	0
$439$	− 820.549i	− 1.86913i	−0.355789	−	0.934566i	$-0.615788\pi$
$439$	− 820.549i	− 1.86913i	0.355789	−	0.934566i	$-0.384212\pi$
$440$	0	0
$441$	118.099	0.267798
$442$	0	0
$443$	772.107	1.74291	0.871453	−	0.490480i	$-0.163178\pi$
$443$	772.107	1.74291	0.871453	+	0.490480i	$0.163178\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	72.2388	0.161608
$448$	0	0
$449$	−367.146	−0.817696	−0.408848	−	0.912602i	$-0.634069\pi$
$449$	−367.146	−0.817696	−0.408848	+	0.912602i	$0.634069\pi$
$450$	0	0
$451$	− 638.581i	− 1.41592i
$452$	0	0
$453$	− 92.3602i	− 0.203886i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	122.079i	0.267131i	0.991040	+	0.133565i	$0.0426426\pi$
$457$	122.079i	0.267131i	−0.991040	+	0.133565i	$0.957357\pi$
$458$	0	0
$459$	159.616i	0.347747i
$460$	0	0
$461$	233.565	0.506650	0.253325	−	0.967381i	$-0.418476\pi$
$461$	233.565	0.506650	0.253325	+	0.967381i	$0.418476\pi$
$462$	0	0
$463$	16.6739	0.0360127	0.0180064	−	0.999838i	$-0.494268\pi$
$463$	16.6739	0.0360127	0.0180064	+	0.999838i	$0.494268\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−119.734	−0.256391	−0.128195	−	0.991749i	$-0.540918\pi$
$467$	−119.734	−0.256391	−0.128195	+	0.991749i	$0.540918\pi$
$468$	0	0
$469$	1234.42	2.63202
$470$	0	0
$471$	− 149.875i	− 0.318206i
$472$	0	0
$473$	− 561.263i	− 1.18660i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	247.815i	0.519529i
$478$	0	0
$479$	− 359.494i	− 0.750510i	−0.926922	−	0.375255i	$-0.877555\pi$
$479$	− 359.494i	− 0.750510i	0.926922	−	0.375255i	$-0.122445\pi$
$480$	0	0
$481$	−234.105	−0.486705
$482$	0	0
$483$	−168.899	−0.349688
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−573.644	−1.17791	−0.588957	−	0.808164i	$-0.700461\pi$
$487$	−573.644	−1.17791	−0.588957	+	0.808164i	$0.700461\pi$
$488$	0	0
$489$	63.1711	0.129184
$490$	0	0
$491$	338.696i	0.689809i	0.938638	+	0.344904i	$0.112089\pi$
$491$	338.696i	0.689809i	−0.938638	+	0.344904i	$0.887911\pi$
$492$	0	0
$493$	780.743i	1.58366i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	212.550i	0.427666i
$498$	0	0
$499$	− 468.632i	− 0.939142i	−0.882895	−	0.469571i	$-0.844409\pi$
$499$	− 468.632i	− 0.939142i	0.882895	−	0.469571i	$-0.155591\pi$
$500$	0	0
$501$	380.692	0.759865
$502$	0	0
$503$	−627.373	−1.24726	−0.623631	−	0.781719i	$-0.714343\pi$
$503$	−627.373	−1.24726	−0.623631	+	0.781719i	$0.714343\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−557.400	−1.09941
$508$	0	0
$509$	501.143	0.984564	0.492282	−	0.870436i	$-0.336163\pi$
$509$	501.143	0.984564	0.492282	+	0.870436i	$0.336163\pi$
$510$	0	0
$511$	− 573.169i	− 1.12166i
$512$	0	0
$513$	176.995i	0.345020i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 388.477i	− 0.751406i
$518$	0	0
$519$	− 88.9422i	− 0.171372i
$520$	0	0
$521$	535.165	1.02719	0.513594	−	0.858033i	$-0.328314\pi$
$521$	535.165	1.02719	0.513594	+	0.858033i	$0.328314\pi$
$522$	0	0
$523$	−291.291	−0.556963	−0.278481	−	0.960442i	$-0.589831\pi$
$523$	−291.291	−0.556963	−0.278481	+	0.960442i	$0.589831\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1023.95	−1.94298
$528$	0	0
$529$	−421.391	−0.796581
$530$	0	0
$531$	101.106i	0.190407i
$532$	0	0
$533$	1378.10i	2.58556i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	138.919i	0.258695i
$538$	0	0
$539$	404.127i	0.749772i
$540$	0	0
$541$	160.869	0.297355	0.148678	−	0.988886i	$-0.452498\pi$
$541$	160.869	0.297355	0.148678	+	0.988886i	$0.452498\pi$
$542$	0	0
$543$	−198.455	−0.365478
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−543.958	−0.994439	−0.497220	−	0.867625i	$-0.665646\pi$
$547$	−543.958	−0.994439	−0.497220	+	0.867625i	$0.665646\pi$
$548$	0	0
$549$	234.064	0.426347
$550$	0	0
$551$	865.753i	1.57124i
$552$	0	0
$553$	− 262.176i	− 0.474098i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 682.329i	− 1.22501i	−0.790468	−	0.612503i	$-0.790163\pi$
$557$	− 682.329i	− 1.22501i	0.790468	−	0.612503i	$-0.209837\pi$
$558$	0	0
$559$	1211.25i	2.16681i
$560$	0	0
$561$	−546.194	−0.973608
$562$	0	0
$563$	635.777	1.12927	0.564633	−	0.825342i	$-0.309018\pi$
$563$	635.777	1.12927	0.564633	+	0.825342i	$0.309018\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	84.6031	0.149212
$568$	0	0
$569$	−202.561	−0.355995	−0.177998	−	0.984031i	$-0.556962\pi$
$569$	−202.561	−0.355995	−0.177998	+	0.984031i	$0.556962\pi$
$570$	0	0
$571$	− 911.868i	− 1.59697i	−0.602017	−	0.798483i	$-0.705636\pi$
$571$	− 911.868i	− 1.59697i	0.602017	−	0.798483i	$-0.294364\pi$
$572$	0	0
$573$	− 123.122i	− 0.214873i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	641.689i	1.11211i	0.831145	+	0.556056i	$0.187686\pi$
$577$	641.689i	1.11211i	−0.831145	+	0.556056i	$0.812314\pi$
$578$	0	0
$579$	43.5889i	0.0752830i
$580$	0	0
$581$	664.856	1.14433
$582$	0	0
$583$	−848.007	−1.45456
$584$	0	0
$585$	0	0
$586$	0	0
$587$	155.622	0.265115	0.132557	−	0.991175i	$-0.457681\pi$
$587$	155.622	0.265115	0.132557	+	0.991175i	$0.457681\pi$
$588$	0	0
$589$	−1135.44	−1.92774
$590$	0	0
$591$	429.117i	0.726087i
$592$	0	0
$593$	− 202.248i	− 0.341060i	−0.985353	−	0.170530i	$-0.945452\pi$
$593$	− 202.248i	− 0.341060i	0.985353	−	0.170530i	$-0.0545479\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 125.603i	− 0.210391i
$598$	0	0
$599$	256.049i	0.427460i	0.976893	+	0.213730i	$0.0685613\pi$
$599$	256.049i	0.427460i	−0.976893	+	0.213730i	$0.931439\pi$
$600$	0	0
$601$	−723.075	−1.20312	−0.601560	−	0.798828i	$-0.705454\pi$
$601$	−723.075	−1.20312	−0.601560	+	0.798828i	$0.705454\pi$
$602$	0	0
$603$	393.948	0.653314
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−871.890	−1.43639	−0.718196	−	0.695841i	$-0.755032\pi$
$607$	−871.890	−1.43639	−0.718196	+	0.695841i	$0.755032\pi$
$608$	0	0
$609$	413.827	0.679518
$610$	0	0
$611$	838.362i	1.37211i
$612$	0	0
$613$	− 1027.53i	− 1.67623i	−0.545491	−	0.838117i	$-0.683657\pi$
$613$	− 1027.53i	− 1.67623i	0.545491	−	0.838117i	$-0.316343\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 250.905i	− 0.406652i	−0.979111	−	0.203326i	$-0.934825\pi$
$617$	− 250.905i	− 0.406652i	0.979111	−	0.203326i	$-0.0651752\pi$
$618$	0	0
$619$	− 704.772i	− 1.13857i	−0.822142	−	0.569283i	$-0.807221\pi$
$619$	− 704.772i	− 1.13857i	0.822142	−	0.569283i	$-0.192779\pi$
$620$	0	0
$621$	−53.9021	−0.0867988
$622$	0	0
$623$	−270.405	−0.434037
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−605.666	−0.965974
$628$	0	0
$629$	324.598	0.516055
$630$	0	0
$631$	− 481.066i	− 0.762387i	−0.924495	−	0.381193i	$-0.875513\pi$
$631$	− 481.066i	− 0.762387i	0.924495	−	0.381193i	$-0.124487\pi$
$632$	0	0
$633$	117.405i	0.185474i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 872.136i	− 1.36913i
$638$	0	0
$639$	67.8327i	0.106155i
$640$	0	0
$641$	509.057	0.794161	0.397080	−	0.917784i	$-0.370023\pi$
$641$	509.057	0.794161	0.397080	+	0.917784i	$0.370023\pi$
$642$	0	0
$643$	−943.939	−1.46802	−0.734012	−	0.679137i	$-0.762354\pi$
$643$	−943.939	−1.46802	−0.734012	+	0.679137i	$0.762354\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−787.201	−1.21669	−0.608347	−	0.793671i	$-0.708167\pi$
$647$	−787.201	−1.21669	−0.608347	+	0.793671i	$0.708167\pi$
$648$	0	0
$649$	−345.979	−0.533095
$650$	0	0
$651$	542.736i	0.833697i
$652$	0	0
$653$	− 184.869i	− 0.283107i	−0.989931	−	0.141553i	$-0.954790\pi$
$653$	− 184.869i	− 0.283107i	0.989931	−	0.141553i	$-0.0452097\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 182.920i	− 0.278417i
$658$	0	0
$659$	928.276i	1.40861i	0.709896	+	0.704307i	$0.248742\pi$
$659$	928.276i	1.40861i	−0.709896	+	0.704307i	$0.751258\pi$
$660$	0	0
$661$	948.317	1.43467	0.717335	−	0.696728i	$-0.245362\pi$
$661$	948.317	1.43467	0.717335	+	0.696728i	$0.245362\pi$
$662$	0	0
$663$	1178.73	1.77787
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−263.656	−0.395286
$668$	0	0
$669$	230.737	0.344898
$670$	0	0
$671$	800.953i	1.19367i
$672$	0	0
$673$	− 355.386i	− 0.528063i	−0.964514	−	0.264031i	$-0.914948\pi$
$673$	− 355.386i	− 0.528063i	0.964514	−	0.264031i	$-0.0850522\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 107.844i	− 0.159296i	−0.996823	−	0.0796481i	$-0.974620\pi$
$677$	− 107.844i	− 0.159296i	0.996823	−	0.0796481i	$-0.0253797\pi$
$678$	0	0
$679$	− 99.4027i	− 0.146396i
$680$	0	0
$681$	173.725	0.255103
$682$	0	0
$683$	−229.914	−0.336624	−0.168312	−	0.985734i	$-0.553832\pi$
$683$	−229.914	−0.336624	−0.168312	+	0.985734i	$0.553832\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−25.6476	−0.0373327
$688$	0	0
$689$	1830.06	2.65611
$690$	0	0
$691$	998.502i	1.44501i	0.691366	+	0.722505i	$0.257009\pi$
$691$	998.502i	1.44501i	−0.691366	+	0.722505i	$0.742991\pi$
$692$	0	0
$693$	289.506i	0.417757i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1910.81i	− 2.74147i
$698$	0	0
$699$	462.509i	0.661672i
$700$	0	0
$701$	−563.962	−0.804510	−0.402255	−	0.915528i	$-0.631774\pi$
$701$	−563.962	−0.804510	−0.402255	+	0.915528i	$0.631774\pi$
$702$	0	0
$703$	359.942	0.512008
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−101.071	−0.142958
$708$	0	0
$709$	54.7374	0.0772037	0.0386018	−	0.999255i	$-0.487710\pi$
$709$	54.7374	0.0772037	0.0386018	+	0.999255i	$0.487710\pi$
$710$	0	0
$711$	− 83.6702i	− 0.117680i
$712$	0	0
$713$	− 345.787i	− 0.484974i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 37.2091i	− 0.0518956i
$718$	0	0
$719$	− 1026.78i	− 1.42807i	−0.700108	−	0.714037i	$-0.746865\pi$
$719$	− 1026.78i	− 1.42807i	0.700108	−	0.714037i	$-0.253135\pi$
$720$	0	0
$721$	−665.647	−0.923227
$722$	0	0
$723$	22.0834	0.0305441
$724$	0	0
$725$	0	0
$726$	0	0
$727$	761.761	1.04781	0.523907	−	0.851775i	$-0.324474\pi$
$727$	761.761	1.04781	0.523907	+	0.851775i	$0.324474\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	− 1679.45i	− 2.29747i
$732$	0	0
$733$	264.657i	0.361060i	0.983570	+	0.180530i	$0.0577812\pi$
$733$	264.657i	0.361060i	−0.983570	+	0.180530i	$0.942219\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1348.06i	1.82912i
$738$	0	0
$739$	− 669.696i	− 0.906219i	−0.891455	−	0.453109i	$-0.850315\pi$
$739$	− 669.696i	− 0.906219i	0.891455	−	0.453109i	$-0.149685\pi$
$740$	0	0
$741$	1307.07	1.76393
$742$	0	0
$743$	−771.458	−1.03830	−0.519151	−	0.854683i	$-0.673752\pi$
$743$	−771.458	−1.03830	−0.519151	+	0.854683i	$0.673752\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	212.180	0.284043
$748$	0	0
$749$	360.912	0.481859
$750$	0	0
$751$	335.971i	0.447365i	0.974662	+	0.223683i	$0.0718079\pi$
$751$	335.971i	0.447365i	−0.974662	+	0.223683i	$0.928192\pi$
$752$	0	0
$753$	− 343.861i	− 0.456655i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	174.316i	0.230272i	0.993350	+	0.115136i	$0.0367304\pi$
$757$	174.316i	0.230272i	−0.993350	+	0.115136i	$0.963270\pi$
$758$	0	0
$759$	− 184.449i	− 0.243016i
$760$	0	0
$761$	−471.375	−0.619415	−0.309708	−	0.950832i	$-0.600231\pi$
$761$	−471.375	−0.619415	−0.309708	+	0.950832i	$0.600231\pi$
$762$	0	0
$763$	−1473.07	−1.93063
$764$	0	0
$765$	0	0
$766$	0	0
$767$	746.647	0.973464
$768$	0	0
$769$	−110.326	−0.143467	−0.0717334	−	0.997424i	$-0.522853\pi$
$769$	−110.326	−0.143467	−0.0717334	+	0.997424i	$0.522853\pi$
$770$	0	0
$771$	723.569i	0.938481i
$772$	0	0
$773$	− 166.817i	− 0.215804i	−0.994162	−	0.107902i	$-0.965587\pi$
$773$	− 166.817i	− 0.215804i	0.994162	−	0.107902i	$-0.0344133\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 172.051i	− 0.221430i
$778$	0	0
$779$	− 2118.86i	− 2.71998i
$780$	0	0
$781$	−232.119	−0.297207
$782$	0	0
$783$	132.068	0.168669
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−368.824	−0.468646	−0.234323	−	0.972159i	$-0.575287\pi$
$787$	−368.824	−0.468646	−0.234323	+	0.972159i	$0.575287\pi$
$788$	0	0
$789$	105.751	0.134032
$790$	0	0
$791$	− 1394.48i	− 1.76294i
$792$	0	0
$793$	− 1728.51i	− 2.17972i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 883.998i	− 1.10916i	−0.832132	−	0.554578i	$-0.812880\pi$
$797$	− 883.998i	− 1.10916i	0.832132	−	0.554578i	$-0.187120\pi$
$798$	0	0
$799$	− 1162.43i	− 1.45485i
$800$	0	0
$801$	−86.2965	−0.107736
$802$	0	0
$803$	625.939	0.779500
$804$	0	0
$805$	0	0
$806$	0	0
$807$	795.079	0.985228
$808$	0	0
$809$	1035.45	1.27991	0.639955	−	0.768412i	$-0.278953\pi$
$809$	1035.45	1.27991	0.639955	+	0.768412i	$0.278953\pi$
$810$	0	0
$811$	− 765.157i	− 0.943473i	−0.881740	−	0.471737i	$-0.843627\pi$
$811$	− 765.157i	− 0.943473i	0.881740	−	0.471737i	$-0.156373\pi$
$812$	0	0
$813$	160.913i	0.197925i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1862.32i	− 2.27946i
$818$	0	0
$819$	− 624.775i	− 0.762851i
$820$	0	0
$821$	−64.7641	−0.0788844	−0.0394422	−	0.999222i	$-0.512558\pi$
$821$	−64.7641	−0.0788844	−0.0394422	+	0.999222i	$0.512558\pi$
$822$	0	0
$823$	597.516	0.726022	0.363011	−	0.931785i	$-0.381749\pi$
$823$	597.516	0.726022	0.363011	+	0.931785i	$0.381749\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−937.277	−1.13335	−0.566673	−	0.823943i	$-0.691770\pi$
$827$	−937.277	−1.13335	−0.566673	+	0.823943i	$0.691770\pi$
$828$	0	0
$829$	1053.44	1.27074	0.635371	−	0.772207i	$-0.280847\pi$
$829$	1053.44	1.27074	0.635371	+	0.772207i	$0.280847\pi$
$830$	0	0
$831$	719.946i	0.866362i
$832$	0	0
$833$	1209.26i	1.45169i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	173.208i	0.206938i
$838$	0	0
$839$	− 312.763i	− 0.372781i	−0.982476	−	0.186390i	$-0.940321\pi$
$839$	− 312.763i	− 0.372781i	0.982476	−	0.186390i	$-0.0596789\pi$
$840$	0	0
$841$	−195.006	−0.231874
$842$	0	0
$843$	−470.650	−0.558304
$844$	0	0
$845$	0	0
$846$	0	0
$847$	146.771	0.173284
$848$	0	0
$849$	361.492	0.425785
$850$	0	0
$851$	109.617i	0.128809i
$852$	0	0
$853$	57.0533i	0.0668855i	0.999441	+	0.0334428i	$0.0106471\pi$
$853$	57.0533i	0.0668855i	−0.999441	+	0.0334428i	$0.989353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	969.837i	1.13166i	0.824520	+	0.565832i	$0.191445\pi$
$857$	969.837i	1.13166i	−0.824520	+	0.565832i	$0.808555\pi$
$858$	0	0
$859$	− 1321.85i	− 1.53882i	−0.638755	−	0.769410i	$-0.720550\pi$
$859$	− 1321.85i	− 1.53882i	0.638755	−	0.769410i	$-0.279450\pi$
$860$	0	0
$861$	−1012.81	−1.17632
$862$	0	0
$863$	384.652	0.445715	0.222858	−	0.974851i	$-0.428462\pi$
$863$	384.652	0.445715	0.222858	+	0.974851i	$0.428462\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1133.80	−1.30773
$868$	0	0
$869$	286.314	0.329475
$870$	0	0
$871$	− 2909.22i	− 3.34009i
$872$	0	0
$873$	− 31.7231i	− 0.0363381i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 138.409i	− 0.157821i	−0.996882	−	0.0789106i	$-0.974856\pi$
$877$	− 138.409i	− 0.157821i	0.996882	−	0.0789106i	$-0.0251442\pi$
$878$	0	0
$879$	− 48.1394i	− 0.0547661i
$880$	0	0
$881$	−1462.25	−1.65976	−0.829882	−	0.557939i	$-0.811592\pi$
$881$	−1462.25	−1.65976	−0.829882	+	0.557939i	$0.811592\pi$
$882$	0	0
$883$	−372.564	−0.421929	−0.210965	−	0.977494i	$-0.567660\pi$
$883$	−372.564	−0.421929	−0.210965	+	0.977494i	$0.567660\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1152.51	−1.29934	−0.649668	−	0.760218i	$-0.725092\pi$
$887$	−1152.51	−1.29934	−0.649668	+	0.760218i	$0.725092\pi$
$888$	0	0
$889$	1029.41	1.15794
$890$	0	0
$891$	92.3922i	0.103695i
$892$	0	0
$893$	− 1289.00i	− 1.44345i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	398.055i	0.443763i
$898$	0	0
$899$	847.225i	0.942409i
$900$	0	0
$901$	−2537.47	−2.81628
$902$	0	0
$903$	−890.181	−0.985804
$904$	0	0
$905$	0	0
$906$	0	0
$907$	741.083	0.817071	0.408535	−	0.912742i	$-0.366040\pi$
$907$	741.083	0.817071	0.408535	+	0.912742i	$0.366040\pi$
$908$	0	0
$909$	−32.2556	−0.0354847
$910$	0	0
$911$	1000.10i	1.09781i	0.835886	+	0.548903i	$0.184954\pi$
$911$	1000.10i	1.09781i	−0.835886	+	0.548903i	$0.815046\pi$
$912$	0	0
$913$	726.067i	0.795254i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 458.642i	− 0.500155i
$918$	0	0
$919$	130.002i	0.141460i	0.997495	+	0.0707300i	$0.0225329\pi$
$919$	130.002i	0.141460i	−0.997495	+	0.0707300i	$0.977467\pi$
$920$	0	0
$921$	408.795	0.443860
$922$	0	0
$923$	500.930	0.542719
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−212.433	−0.229162
$928$	0	0
$929$	−1556.51	−1.67547	−0.837734	−	0.546078i	$-0.816120\pi$
$929$	−1556.51	−1.67547	−0.837734	+	0.546078i	$0.816120\pi$
$930$	0	0
$931$	1340.93i	1.44031i
$932$	0	0
$933$	152.061i	0.162980i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 19.4457i	− 0.0207531i	−0.999946	−	0.0103765i	$-0.996697\pi$
$937$	− 19.4457i	− 0.0207531i	0.999946	−	0.0103765i	$-0.00330302\pi$
$938$	0	0
$939$	− 307.593i	− 0.327575i
$940$	0	0
$941$	−1063.66	−1.13035	−0.565176	−	0.824971i	$-0.691192\pi$
$941$	−1063.66	−1.13035	−0.565176	+	0.824971i	$0.691192\pi$
$942$	0	0
$943$	645.278	0.684282
$944$	0	0
$945$	0	0
$946$	0	0
$947$	343.394	0.362612	0.181306	−	0.983427i	$-0.441968\pi$
$947$	343.394	0.362612	0.181306	+	0.983427i	$0.441968\pi$
$948$	0	0
$949$	−1350.82	−1.42342
$950$	0	0
$951$	757.616i	0.796652i
$952$	0	0
$953$	− 1575.10i	− 1.65279i	−0.563094	−	0.826393i	$-0.690389\pi$
$953$	− 1575.10i	− 1.65279i	0.563094	−	0.826393i	$-0.309611\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	451.926i	0.472232i
$958$	0	0
$959$	419.662i	0.437604i
$960$	0	0
$961$	−150.142	−0.156235
$962$	0	0
$963$	115.181	0.119606
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1823.70	1.88593	0.942967	−	0.332886i	$-0.108022\pi$
$967$	1823.70	1.88593	0.942967	+	0.332886i	$0.108022\pi$
$968$	0	0
$969$	−1812.32	−1.87030
$970$	0	0
$971$	731.769i	0.753624i	0.926290	+	0.376812i	$0.122980\pi$
$971$	731.769i	0.753624i	−0.926290	+	0.376812i	$0.877020\pi$
$972$	0	0
$973$	1046.75i	1.07579i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 382.708i	− 0.391717i	−0.980632	−	0.195859i	$-0.937251\pi$
$977$	− 382.708i	− 0.391717i	0.980632	−	0.195859i	$-0.0627493\pi$
$978$	0	0
$979$	− 295.301i	− 0.301635i
$980$	0	0
$981$	−470.112	−0.479217
$982$	0	0
$983$	−649.146	−0.660372	−0.330186	−	0.943916i	$-0.607111\pi$
$983$	−649.146	−0.660372	−0.330186	+	0.943916i	$0.607111\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−616.136	−0.624251
$988$	0	0
$989$	567.150	0.573458
$990$	0	0
$991$	702.986i	0.709370i	0.934986	+	0.354685i	$0.115412\pi$
$991$	702.986i	0.709370i	−0.934986	+	0.354685i	$0.884588\pi$
$992$	0	0
$993$	− 343.305i	− 0.345725i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1221.41i	− 1.22508i	−0.790439	−	0.612541i	$-0.790147\pi$
$997$	− 1221.41i	− 1.22508i	0.790439	−	0.612541i	$-0.209853\pi$
$998$	0	0
$999$	− 54.9078i	− 0.0549628i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.3.j.d.799.8		8
4.3	odd	2		2400.3.j.j.799.1		8
5.2	odd	4		480.3.e.a.31.4	✓	8
5.3	odd	4		2400.3.e.g.1951.6		8
5.4	even	2		2400.3.j.j.799.2		8
15.2	even	4		1440.3.e.e.991.4		8
20.3	even	4		2400.3.e.g.1951.3		8
20.7	even	4		480.3.e.a.31.7	yes	8
20.19	odd	2	inner	2400.3.j.d.799.7		8
40.27	even	4		960.3.e.d.511.1		8
40.37	odd	4		960.3.e.d.511.6		8
60.47	odd	4		1440.3.e.e.991.1		8
120.77	even	4		2880.3.e.n.2431.8		8
120.107	odd	4		2880.3.e.n.2431.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.e.a.31.4	✓	8	5.2	odd	4
480.3.e.a.31.7	yes	8	20.7	even	4
960.3.e.d.511.1		8	40.27	even	4
960.3.e.d.511.6		8	40.37	odd	4
1440.3.e.e.991.1		8	60.47	odd	4
1440.3.e.e.991.4		8	15.2	even	4
2400.3.e.g.1951.3		8	20.3	even	4
2400.3.e.g.1951.6		8	5.3	odd	4
2400.3.j.d.799.7		8	20.19	odd	2	inner
2400.3.j.d.799.8		8	1.1	even	1	trivial
2400.3.j.j.799.1		8	4.3	odd	2
2400.3.j.j.799.2		8	5.4	even	2
2880.3.e.n.2431.5		8	120.107	odd	4
2880.3.e.n.2431.8		8	120.77	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2400.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +9.40034 q^{7} +3.00000 q^{9} +10.2658i q^{11} -22.1543i q^{13} +30.7181i q^{17} +34.0627i q^{19} +16.2819 q^{21} -10.3735 q^{23} +5.19615 q^{27} +25.4164 q^{29} +33.3338i q^{31} +17.7809i q^{33} -10.5670i q^{37} -38.3724i q^{39} -62.2047 q^{41} -54.6732 q^{43} -37.8419 q^{47} +39.3664 q^{49} +53.2053i q^{51} +82.6051i q^{53} +58.9984i q^{57} +33.7021i q^{59} +78.0215 q^{61} +28.2010 q^{63} +131.316 q^{67} -17.9674 q^{69} +22.6109i q^{71} -60.9732i q^{73} +96.5020i q^{77} -27.8901i q^{79} +9.00000 q^{81} +70.7268 q^{83} +44.0225 q^{87} -28.7655 q^{89} -208.258i q^{91} +57.7358i q^{93} -10.5744i q^{97} +30.7974i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7} + 24 q^{9} + 96 q^{21} - 16 q^{23} + 96 q^{29} + 112 q^{41} - 128 q^{43} - 272 q^{47} + 184 q^{49} + 80 q^{61} - 48 q^{63} + 320 q^{67} + 144 q^{69} + 72 q^{81} - 320 q^{83} + 48 q^{89}+O(q^{100}) \) 8 * q - 16 * q^7 + 24 * q^9 + 96 * q^21 - 16 * q^23 + 96 * q^29 + 112 * q^41 - 128 * q^43 - 272 * q^47 + 184 * q^49 + 80 * q^61 - 48 * q^63 + 320 * q^67 + 144 * q^69 + 72 * q^81 - 320 * q^83 + 48 * q^89

Introduction

L-functions

Modular forms

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Database

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