LMFDB - Embedded newform 1200.3.j.a.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(799,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1200.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.j (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1200.799
Dual form			1200.3.j.a.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +6.92820 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +6.92820 q^{7} +3.00000 q^{9} +20.7846i q^{11} +14.0000i q^{13} -6.00000i q^{17} -6.92820i q^{19} -12.0000 q^{21} -5.19615 q^{27} -30.0000 q^{29} -20.7846i q^{31} -36.0000i q^{33} +26.0000i q^{37} -24.2487i q^{39} -54.0000 q^{41} +20.7846 q^{43} +41.5692 q^{47} -1.00000 q^{49} +10.3923i q^{51} +18.0000i q^{53} +12.0000i q^{57} +20.7846i q^{59} -70.0000 q^{61} +20.7846 q^{63} +117.779 q^{67} +83.1384i q^{71} -82.0000i q^{73} +144.000i q^{77} +76.2102i q^{79} +9.00000 q^{81} -20.7846 q^{83} +51.9615 q^{87} -114.000 q^{89} +96.9948i q^{91} +36.0000i q^{93} +34.0000i q^{97} +62.3538i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} - 48 q^{21} - 120 q^{29} - 216 q^{41} - 4 q^{49} - 280 q^{61} + 36 q^{81} - 456 q^{89}+O(q^{100}) $ 4 * q + 12 * q^9 - 48 * q^21 - 120 * q^29 - 216 * q^41 - 4 * q^49 - 280 * q^61 + 36 * q^81 - 456 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$7$	6.92820	0.989743	0.494872	−	0.868966i	$-0.335215\pi$
$7$	6.92820	0.989743	0.494872	+	0.868966i	$0.335215\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	20.7846i	1.88951i	0.327777	+	0.944755i	$0.393700\pi$
$11$	20.7846i	1.88951i	−0.327777	+	0.944755i	$0.606300\pi$
$12$	0	0
$13$	14.0000i	1.07692i	0.842650	+	0.538462i	$0.180994\pi$
$13$	14.0000i	1.07692i	−0.842650	+	0.538462i	$0.819006\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.00000i	− 0.352941i	−0.984306	−	0.176471i	$-0.943532\pi$
$17$	− 6.00000i	− 0.352941i	0.984306	−	0.176471i	$-0.0564680\pi$
$18$	0	0
$19$	− 6.92820i	− 0.364642i	−0.983239	−	0.182321i	$-0.941639\pi$
$19$	− 6.92820i	− 0.364642i	0.983239	−	0.182321i	$-0.0583610\pi$
$20$	0	0
$21$	−12.0000	−0.571429
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	−30.0000	−1.03448	−0.517241	−	0.855840i	$-0.673041\pi$
$29$	−30.0000	−1.03448	−0.517241	+	0.855840i	$0.673041\pi$
$30$	0	0
$31$	− 20.7846i	− 0.670471i	−0.942134	−	0.335236i	$-0.891184\pi$
$31$	− 20.7846i	− 0.670471i	0.942134	−	0.335236i	$-0.108816\pi$
$32$	0	0
$33$	− 36.0000i	− 1.09091i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	26.0000i	0.702703i	0.936244	+	0.351351i	$0.114278\pi$
$37$	26.0000i	0.702703i	−0.936244	+	0.351351i	$0.885722\pi$
$38$	0	0
$39$	− 24.2487i	− 0.621762i
$40$	0	0
$41$	−54.0000	−1.31707	−0.658537	−	0.752549i	$-0.728824\pi$
$41$	−54.0000	−1.31707	−0.658537	+	0.752549i	$0.728824\pi$
$42$	0	0
$43$	20.7846	0.483363	0.241682	−	0.970356i	$-0.422301\pi$
$43$	20.7846	0.483363	0.241682	+	0.970356i	$0.422301\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	41.5692	0.884451	0.442226	−	0.896904i	$-0.354189\pi$
$47$	41.5692	0.884451	0.442226	+	0.896904i	$0.354189\pi$
$48$	0	0
$49$	−1.00000	−0.0204082
$50$	0	0
$51$	10.3923i	0.203771i
$52$	0	0
$53$	18.0000i	0.339623i	0.985477	+	0.169811i	$0.0543158\pi$
$53$	18.0000i	0.339623i	−0.985477	+	0.169811i	$0.945684\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.0000i	0.210526i
$58$	0	0
$59$	20.7846i	0.352282i	0.984365	+	0.176141i	$0.0563614\pi$
$59$	20.7846i	0.352282i	−0.984365	+	0.176141i	$0.943639\pi$
$60$	0	0
$61$	−70.0000	−1.14754	−0.573770	−	0.819016i	$-0.694520\pi$
$61$	−70.0000	−1.14754	−0.573770	+	0.819016i	$0.694520\pi$
$62$	0	0
$63$	20.7846	0.329914
$64$	0	0
$65$	0	0
$66$	0	0
$67$	117.779	1.75790	0.878951	−	0.476912i	$-0.158244\pi$
$67$	117.779	1.75790	0.878951	+	0.476912i	$0.158244\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	83.1384i	1.17096i	0.810685	+	0.585482i	$0.199095\pi$
$71$	83.1384i	1.17096i	−0.810685	+	0.585482i	$0.800905\pi$
$72$	0	0
$73$	− 82.0000i	− 1.12329i	−0.827379	−	0.561644i	$-0.810169\pi$
$73$	− 82.0000i	− 1.12329i	0.827379	−	0.561644i	$-0.189831\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	144.000i	1.87013i
$78$	0	0
$79$	76.2102i	0.964687i	0.875982	+	0.482343i	$0.160214\pi$
$79$	76.2102i	0.964687i	−0.875982	+	0.482343i	$0.839786\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−20.7846	−0.250417	−0.125208	−	0.992130i	$-0.539960\pi$
$83$	−20.7846	−0.250417	−0.125208	+	0.992130i	$0.539960\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	51.9615	0.597259
$88$	0	0
$89$	−114.000	−1.28090	−0.640449	−	0.768000i	$-0.721252\pi$
$89$	−114.000	−1.28090	−0.640449	+	0.768000i	$0.721252\pi$
$90$	0	0
$91$	96.9948i	1.06588i
$92$	0	0
$93$	36.0000i	0.387097i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	34.0000i	0.350515i	0.984523	+	0.175258i	$0.0560759\pi$
$97$	34.0000i	0.350515i	−0.984523	+	0.175258i	$0.943924\pi$
$98$	0	0
$99$	62.3538i	0.629837i
$100$	0	0
$101$	−18.0000	−0.178218	−0.0891089	−	0.996022i	$-0.528402\pi$
$101$	−18.0000	−0.178218	−0.0891089	+	0.996022i	$0.528402\pi$
$102$	0	0
$103$	−131.636	−1.27802	−0.639009	−	0.769199i	$-0.720655\pi$
$103$	−131.636	−1.27802	−0.639009	+	0.769199i	$0.720655\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−145.492	−1.35974	−0.679870	−	0.733332i	$-0.737964\pi$
$107$	−145.492	−1.35974	−0.679870	+	0.733332i	$0.737964\pi$
$108$	0	0
$109$	−34.0000	−0.311927	−0.155963	−	0.987763i	$-0.549848\pi$
$109$	−34.0000	−0.311927	−0.155963	+	0.987763i	$0.549848\pi$
$110$	0	0
$111$	− 45.0333i	− 0.405706i
$112$	0	0
$113$	78.0000i	0.690265i	0.938554	+	0.345133i	$0.112166\pi$
$113$	78.0000i	0.690265i	−0.938554	+	0.345133i	$0.887834\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	42.0000i	0.358974i
$118$	0	0
$119$	− 41.5692i	− 0.349321i
$120$	0	0
$121$	−311.000	−2.57025
$122$	0	0
$123$	93.5307	0.760413
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−103.923	−0.818292	−0.409146	−	0.912469i	$-0.634173\pi$
$127$	−103.923	−0.818292	−0.409146	+	0.912469i	$0.634173\pi$
$128$	0	0
$129$	−36.0000	−0.279070
$130$	0	0
$131$	103.923i	0.793306i	0.917969	+	0.396653i	$0.129828\pi$
$131$	103.923i	0.793306i	−0.917969	+	0.396653i	$0.870172\pi$
$132$	0	0
$133$	− 48.0000i	− 0.360902i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	186.000i	1.35766i	0.734294	+	0.678832i	$0.237514\pi$
$137$	186.000i	1.35766i	−0.734294	+	0.678832i	$0.762486\pi$
$138$	0	0
$139$	48.4974i	0.348902i	0.984666	+	0.174451i	$0.0558151\pi$
$139$	48.4974i	0.348902i	−0.984666	+	0.174451i	$0.944185\pi$
$140$	0	0
$141$	−72.0000	−0.510638
$142$	0	0
$143$	−290.985	−2.03486
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.73205	0.0117827
$148$	0	0
$149$	186.000	1.24832	0.624161	−	0.781296i	$-0.285441\pi$
$149$	186.000	1.24832	0.624161	+	0.781296i	$0.285441\pi$
$150$	0	0
$151$	− 34.6410i	− 0.229411i	−0.993400	−	0.114705i	$-0.963408\pi$
$151$	− 34.6410i	− 0.229411i	0.993400	−	0.114705i	$-0.0365924\pi$
$152$	0	0
$153$	− 18.0000i	− 0.117647i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	170.000i	1.08280i	0.840764	+	0.541401i	$0.182106\pi$
$157$	170.000i	1.08280i	−0.840764	+	0.541401i	$0.817894\pi$
$158$	0	0
$159$	− 31.1769i	− 0.196081i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	284.056	1.74268	0.871338	−	0.490683i	$-0.163253\pi$
$163$	284.056	1.74268	0.871338	+	0.490683i	$0.163253\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	207.846	1.24459	0.622294	−	0.782784i	$-0.286201\pi$
$167$	207.846	1.24459	0.622294	+	0.782784i	$0.286201\pi$
$168$	0	0
$169$	−27.0000	−0.159763
$170$	0	0
$171$	− 20.7846i	− 0.121547i
$172$	0	0
$173$	42.0000i	0.242775i	0.992605	+	0.121387i	$0.0387343\pi$
$173$	42.0000i	0.242775i	−0.992605	+	0.121387i	$0.961266\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 36.0000i	− 0.203390i
$178$	0	0
$179$	145.492i	0.812806i	0.913694	+	0.406403i	$0.133217\pi$
$179$	145.492i	0.812806i	−0.913694	+	0.406403i	$0.866783\pi$
$180$	0	0
$181$	82.0000	0.453039	0.226519	−	0.974007i	$-0.427265\pi$
$181$	82.0000	0.453039	0.226519	+	0.974007i	$0.427265\pi$
$182$	0	0
$183$	121.244	0.662533
$184$	0	0
$185$	0	0
$186$	0	0
$187$	124.708	0.666886
$188$	0	0
$189$	−36.0000	−0.190476
$190$	0	0
$191$	332.554i	1.74112i	0.492063	+	0.870560i	$0.336243\pi$
$191$	332.554i	1.74112i	−0.492063	+	0.870560i	$0.663757\pi$
$192$	0	0
$193$	94.0000i	0.487047i	0.969895	+	0.243523i	$0.0783032\pi$
$193$	94.0000i	0.487047i	−0.969895	+	0.243523i	$0.921697\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 258.000i	− 1.30964i	−0.755783	−	0.654822i	$-0.772743\pi$
$197$	− 258.000i	− 1.30964i	0.755783	−	0.654822i	$-0.227257\pi$
$198$	0	0
$199$	− 117.779i	− 0.591857i	−0.955210	−	0.295928i	$-0.904371\pi$
$199$	− 117.779i	− 0.591857i	0.955210	−	0.295928i	$-0.0956289\pi$
$200$	0	0
$201$	−204.000	−1.01493
$202$	0	0
$203$	−207.846	−1.02387
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	144.000	0.688995
$210$	0	0
$211$	− 90.0666i	− 0.426856i	−0.976959	−	0.213428i	$-0.931537\pi$
$211$	− 90.0666i	− 0.426856i	0.976959	−	0.213428i	$-0.0684629\pi$
$212$	0	0
$213$	− 144.000i	− 0.676056i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 144.000i	− 0.663594i
$218$	0	0
$219$	142.028i	0.648530i
$220$	0	0
$221$	84.0000	0.380090
$222$	0	0
$223$	353.338	1.58448	0.792238	−	0.610212i	$-0.208916\pi$
$223$	353.338	1.58448	0.792238	+	0.610212i	$0.208916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−145.492	−0.640935	−0.320468	−	0.947259i	$-0.603840\pi$
$227$	−145.492	−0.640935	−0.320468	+	0.947259i	$0.603840\pi$
$228$	0	0
$229$	−226.000	−0.986900	−0.493450	−	0.869774i	$-0.664264\pi$
$229$	−226.000	−0.986900	−0.493450	+	0.869774i	$0.664264\pi$
$230$	0	0
$231$	− 249.415i	− 1.07972i
$232$	0	0
$233$	− 114.000i	− 0.489270i	−0.969615	−	0.244635i	$-0.921332\pi$
$233$	− 114.000i	− 0.489270i	0.969615	−	0.244635i	$-0.0786682\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 132.000i	− 0.556962i
$238$	0	0
$239$	− 332.554i	− 1.39144i	−0.718314	−	0.695719i	$-0.755086\pi$
$239$	− 332.554i	− 1.39144i	0.718314	−	0.695719i	$-0.244914\pi$
$240$	0	0
$241$	178.000	0.738589	0.369295	−	0.929312i	$-0.379599\pi$
$241$	178.000	0.738589	0.369295	+	0.929312i	$0.379599\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	96.9948	0.392692
$248$	0	0
$249$	36.0000	0.144578
$250$	0	0
$251$	103.923i	0.414036i	0.978337	+	0.207018i	$0.0663759\pi$
$251$	103.923i	0.414036i	−0.978337	+	0.207018i	$0.933624\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	258.000i	1.00389i	0.864899	+	0.501946i	$0.167382\pi$
$257$	258.000i	1.00389i	−0.864899	+	0.501946i	$0.832618\pi$
$258$	0	0
$259$	180.133i	0.695495i
$260$	0	0
$261$	−90.0000	−0.344828
$262$	0	0
$263$	−374.123	−1.42252	−0.711260	−	0.702929i	$-0.751875\pi$
$263$	−374.123	−1.42252	−0.711260	+	0.702929i	$0.751875\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	197.454	0.739527
$268$	0	0
$269$	−510.000	−1.89591	−0.947955	−	0.318403i	$-0.896853\pi$
$269$	−510.000	−1.89591	−0.947955	+	0.318403i	$0.896853\pi$
$270$	0	0
$271$	− 450.333i	− 1.66175i	−0.556462	−	0.830873i	$-0.687842\pi$
$271$	− 450.333i	− 1.66175i	0.556462	−	0.830873i	$-0.312158\pi$
$272$	0	0
$273$	− 168.000i	− 0.615385i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 14.0000i	− 0.0505415i	−0.999681	−	0.0252708i	$-0.991955\pi$
$277$	− 14.0000i	− 0.0505415i	0.999681	−	0.0252708i	$-0.00804479\pi$
$278$	0	0
$279$	− 62.3538i	− 0.223490i
$280$	0	0
$281$	354.000	1.25979	0.629893	−	0.776682i	$-0.283099\pi$
$281$	354.000	1.25979	0.629893	+	0.776682i	$0.283099\pi$
$282$	0	0
$283$	145.492	0.514107	0.257053	−	0.966397i	$-0.417248\pi$
$283$	145.492	0.514107	0.257053	+	0.966397i	$0.417248\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−374.123	−1.30356
$288$	0	0
$289$	253.000	0.875433
$290$	0	0
$291$	− 58.8897i	− 0.202370i
$292$	0	0
$293$	498.000i	1.69966i	0.527058	+	0.849829i	$0.323295\pi$
$293$	498.000i	1.69966i	−0.527058	+	0.849829i	$0.676705\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 108.000i	− 0.363636i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	144.000	0.478405
$302$	0	0
$303$	31.1769	0.102894
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−187.061	−0.609321	−0.304660	−	0.952461i	$-0.598543\pi$
$307$	−187.061	−0.609321	−0.304660	+	0.952461i	$0.598543\pi$
$308$	0	0
$309$	228.000	0.737864
$310$	0	0
$311$	− 41.5692i	− 0.133663i	−0.997764	−	0.0668315i	$-0.978711\pi$
$311$	− 41.5692i	− 0.133663i	0.997764	−	0.0668315i	$-0.0212890\pi$
$312$	0	0
$313$	− 290.000i	− 0.926518i	−0.886223	−	0.463259i	$-0.846680\pi$
$313$	− 290.000i	− 0.926518i	0.886223	−	0.463259i	$-0.153320\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 210.000i	− 0.662461i	−0.943550	−	0.331230i	$-0.892536\pi$
$317$	− 210.000i	− 0.662461i	0.943550	−	0.331230i	$-0.107464\pi$
$318$	0	0
$319$	− 623.538i	− 1.95467i
$320$	0	0
$321$	252.000	0.785047
$322$	0	0
$323$	−41.5692	−0.128697
$324$	0	0
$325$	0	0
$326$	0	0
$327$	58.8897	0.180091
$328$	0	0
$329$	288.000	0.875380
$330$	0	0
$331$	200.918i	0.607003i	0.952831	+	0.303501i	$0.0981557\pi$
$331$	200.918i	0.607003i	−0.952831	+	0.303501i	$0.901844\pi$
$332$	0	0
$333$	78.0000i	0.234234i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 302.000i	− 0.896142i	−0.893998	−	0.448071i	$-0.852111\pi$
$337$	− 302.000i	− 0.896142i	0.893998	−	0.448071i	$-0.147889\pi$
$338$	0	0
$339$	− 135.100i	− 0.398525i
$340$	0	0
$341$	432.000	1.26686
$342$	0	0
$343$	−346.410	−1.00994
$344$	0	0
$345$	0	0
$346$	0	0
$347$	62.3538	0.179694	0.0898470	−	0.995956i	$-0.471362\pi$
$347$	62.3538	0.179694	0.0898470	+	0.995956i	$0.471362\pi$
$348$	0	0
$349$	358.000	1.02579	0.512894	−	0.858452i	$-0.328573\pi$
$349$	358.000	1.02579	0.512894	+	0.858452i	$0.328573\pi$
$350$	0	0
$351$	− 72.7461i	− 0.207254i
$352$	0	0
$353$	558.000i	1.58074i	0.612632	+	0.790368i	$0.290111\pi$
$353$	558.000i	1.58074i	−0.612632	+	0.790368i	$0.709889\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	72.0000i	0.201681i
$358$	0	0
$359$	83.1384i	0.231583i	0.993274	+	0.115792i	$0.0369405\pi$
$359$	83.1384i	0.231583i	−0.993274	+	0.115792i	$0.963059\pi$
$360$	0	0
$361$	313.000	0.867036
$362$	0	0
$363$	538.668	1.48393
$364$	0	0
$365$	0	0
$366$	0	0
$367$	214.774	0.585216	0.292608	−	0.956232i	$-0.405477\pi$
$367$	214.774	0.585216	0.292608	+	0.956232i	$0.405477\pi$
$368$	0	0
$369$	−162.000	−0.439024
$370$	0	0
$371$	124.708i	0.336139i
$372$	0	0
$373$	− 554.000i	− 1.48525i	−0.669705	−	0.742627i	$-0.733579\pi$
$373$	− 554.000i	− 1.48525i	0.669705	−	0.742627i	$-0.266421\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 420.000i	− 1.11406i
$378$	0	0
$379$	533.472i	1.40758i	0.710410	+	0.703788i	$0.248510\pi$
$379$	533.472i	1.40758i	−0.710410	+	0.703788i	$0.751490\pi$
$380$	0	0
$381$	180.000	0.472441
$382$	0	0
$383$	−498.831	−1.30243	−0.651215	−	0.758893i	$-0.725740\pi$
$383$	−498.831	−1.30243	−0.651215	+	0.758893i	$0.725740\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	62.3538	0.161121
$388$	0	0
$389$	−198.000	−0.508997	−0.254499	−	0.967073i	$-0.581910\pi$
$389$	−198.000	−0.508997	−0.254499	+	0.967073i	$0.581910\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 180.000i	− 0.458015i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 646.000i	− 1.62720i	−0.581422	−	0.813602i	$-0.697504\pi$
$397$	− 646.000i	− 1.62720i	0.581422	−	0.813602i	$-0.302496\pi$
$398$	0	0
$399$	83.1384i	0.208367i
$400$	0	0
$401$	330.000	0.822943	0.411471	−	0.911423i	$-0.365015\pi$
$401$	330.000	0.822943	0.411471	+	0.911423i	$0.365015\pi$
$402$	0	0
$403$	290.985	0.722046
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−540.400	−1.32776
$408$	0	0
$409$	−130.000	−0.317848	−0.158924	−	0.987291i	$-0.550803\pi$
$409$	−130.000	−0.317848	−0.158924	+	0.987291i	$0.550803\pi$
$410$	0	0
$411$	− 322.161i	− 0.783848i
$412$	0	0
$413$	144.000i	0.348668i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 84.0000i	− 0.201439i
$418$	0	0
$419$	353.338i	0.843290i	0.906761	+	0.421645i	$0.138547\pi$
$419$	353.338i	0.843290i	−0.906761	+	0.421645i	$0.861453\pi$
$420$	0	0
$421$	−398.000	−0.945368	−0.472684	−	0.881232i	$-0.656715\pi$
$421$	−398.000	−0.945368	−0.472684	+	0.881232i	$0.656715\pi$
$422$	0	0
$423$	124.708	0.294817
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−484.974	−1.13577
$428$	0	0
$429$	504.000	1.17483
$430$	0	0
$431$	− 124.708i	− 0.289345i	−0.989480	−	0.144672i	$-0.953787\pi$
$431$	− 124.708i	− 0.289345i	0.989480	−	0.144672i	$-0.0462128\pi$
$432$	0	0
$433$	142.000i	0.327945i	0.986465	+	0.163972i	$0.0524308\pi$
$433$	142.000i	0.327945i	−0.986465	+	0.163972i	$0.947569\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 561.184i	− 1.27832i	−0.769072	−	0.639162i	$-0.779281\pi$
$439$	− 561.184i	− 1.27832i	0.769072	−	0.639162i	$-0.220719\pi$
$440$	0	0
$441$	−3.00000	−0.00680272
$442$	0	0
$443$	436.477	0.985275	0.492637	−	0.870235i	$-0.336033\pi$
$443$	436.477	0.985275	0.492637	+	0.870235i	$0.336033\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−322.161	−0.720719
$448$	0	0
$449$	198.000	0.440980	0.220490	−	0.975389i	$-0.429234\pi$
$449$	198.000	0.440980	0.220490	+	0.975389i	$0.429234\pi$
$450$	0	0
$451$	− 1122.37i	− 2.48862i
$452$	0	0
$453$	60.0000i	0.132450i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 446.000i	− 0.975930i	−0.872863	−	0.487965i	$-0.837739\pi$
$457$	− 446.000i	− 0.975930i	0.872863	−	0.487965i	$-0.162261\pi$
$458$	0	0
$459$	31.1769i	0.0679236i
$460$	0	0
$461$	342.000	0.741866	0.370933	−	0.928660i	$-0.379038\pi$
$461$	342.000	0.741866	0.370933	+	0.928660i	$0.379038\pi$
$462$	0	0
$463$	159.349	0.344166	0.172083	−	0.985082i	$-0.444950\pi$
$463$	159.349	0.344166	0.172083	+	0.985082i	$0.444950\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−394.908	−0.845627	−0.422813	−	0.906217i	$-0.638957\pi$
$467$	−394.908	−0.845627	−0.422813	+	0.906217i	$0.638957\pi$
$468$	0	0
$469$	816.000	1.73987
$470$	0	0
$471$	− 294.449i	− 0.625156i
$472$	0	0
$473$	432.000i	0.913319i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	54.0000i	0.113208i
$478$	0	0
$479$	− 789.815i	− 1.64888i	−0.565947	−	0.824442i	$-0.691489\pi$
$479$	− 789.815i	− 1.64888i	0.565947	−	0.824442i	$-0.308511\pi$
$480$	0	0
$481$	−364.000	−0.756757
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.92820	0.0142263	0.00711315	−	0.999975i	$-0.497736\pi$
$487$	6.92820	0.0142263	0.00711315	+	0.999975i	$0.497736\pi$
$488$	0	0
$489$	−492.000	−1.00613
$490$	0	0
$491$	644.323i	1.31227i	0.754645	+	0.656133i	$0.227809\pi$
$491$	644.323i	1.31227i	−0.754645	+	0.656133i	$0.772191\pi$
$492$	0	0
$493$	180.000i	0.365112i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	576.000i	1.15895i
$498$	0	0
$499$	810.600i	1.62445i	0.583345	+	0.812224i	$0.301743\pi$
$499$	810.600i	1.62445i	−0.583345	+	0.812224i	$0.698257\pi$
$500$	0	0
$501$	−360.000	−0.718563
$502$	0	0
$503$	−332.554	−0.661141	−0.330570	−	0.943781i	$-0.607241\pi$
$503$	−332.554	−0.661141	−0.330570	+	0.943781i	$0.607241\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	46.7654	0.0922394
$508$	0	0
$509$	306.000	0.601179	0.300589	−	0.953754i	$-0.402817\pi$
$509$	306.000	0.601179	0.300589	+	0.953754i	$0.402817\pi$
$510$	0	0
$511$	− 568.113i	− 1.11177i
$512$	0	0
$513$	36.0000i	0.0701754i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	864.000i	1.67118i
$518$	0	0
$519$	− 72.7461i	− 0.140166i
$520$	0	0
$521$	522.000	1.00192	0.500960	−	0.865471i	$-0.332980\pi$
$521$	522.000	1.00192	0.500960	+	0.865471i	$0.332980\pi$
$522$	0	0
$523$	48.4974	0.0927293	0.0463646	−	0.998925i	$-0.485236\pi$
$523$	48.4974	0.0927293	0.0463646	+	0.998925i	$0.485236\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−124.708	−0.236637
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	62.3538i	0.117427i
$532$	0	0
$533$	− 756.000i	− 1.41839i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 252.000i	− 0.469274i
$538$	0	0
$539$	− 20.7846i	− 0.0385614i
$540$	0	0
$541$	802.000	1.48244	0.741220	−	0.671262i	$-0.234248\pi$
$541$	802.000	1.48244	0.741220	+	0.671262i	$0.234248\pi$
$542$	0	0
$543$	−142.028	−0.261562
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.6410	−0.0633291	−0.0316645	−	0.999499i	$-0.510081\pi$
$547$	−34.6410	−0.0633291	−0.0316645	+	0.999499i	$0.510081\pi$
$548$	0	0
$549$	−210.000	−0.382514
$550$	0	0
$551$	207.846i	0.377216i
$552$	0	0
$553$	528.000i	0.954792i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 474.000i	− 0.850987i	−0.904961	−	0.425494i	$-0.860100\pi$
$557$	− 474.000i	− 0.850987i	0.904961	−	0.425494i	$-0.139900\pi$
$558$	0	0
$559$	290.985i	0.520545i
$560$	0	0
$561$	−216.000	−0.385027
$562$	0	0
$563$	−685.892	−1.21828	−0.609140	−	0.793062i	$-0.708485\pi$
$563$	−685.892	−1.21828	−0.609140	+	0.793062i	$0.708485\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	62.3538	0.109971
$568$	0	0
$569$	150.000	0.263620	0.131810	−	0.991275i	$-0.457921\pi$
$569$	150.000	0.263620	0.131810	+	0.991275i	$0.457921\pi$
$570$	0	0
$571$	672.036i	1.17695i	0.808517	+	0.588473i	$0.200271\pi$
$571$	672.036i	1.17695i	−0.808517	+	0.588473i	$0.799729\pi$
$572$	0	0
$573$	− 576.000i	− 1.00524i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 46.0000i	− 0.0797227i	−0.999205	−	0.0398614i	$-0.987308\pi$
$577$	− 46.0000i	− 0.0797227i	0.999205	−	0.0398614i	$-0.0126916\pi$
$578$	0	0
$579$	− 162.813i	− 0.281197i
$580$	0	0
$581$	−144.000	−0.247849
$582$	0	0
$583$	−374.123	−0.641720
$584$	0	0
$585$	0	0
$586$	0	0
$587$	353.338	0.601939	0.300970	−	0.953634i	$-0.402690\pi$
$587$	353.338	0.601939	0.300970	+	0.953634i	$0.402690\pi$
$588$	0	0
$589$	−144.000	−0.244482
$590$	0	0
$591$	446.869i	0.756124i
$592$	0	0
$593$	− 114.000i	− 0.192243i	−0.995370	−	0.0961214i	$-0.969356\pi$
$593$	− 114.000i	− 0.192243i	0.995370	−	0.0961214i	$-0.0306437\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	204.000i	0.341709i
$598$	0	0
$599$	249.415i	0.416386i	0.978088	+	0.208193i	$0.0667582\pi$
$599$	249.415i	0.416386i	−0.978088	+	0.208193i	$0.933242\pi$
$600$	0	0
$601$	626.000	1.04160	0.520799	−	0.853680i	$-0.325634\pi$
$601$	626.000	1.04160	0.520799	+	0.853680i	$0.325634\pi$
$602$	0	0
$603$	353.338	0.585967
$604$	0	0
$605$	0	0
$606$	0	0
$607$	672.036	1.10714	0.553571	−	0.832802i	$-0.313265\pi$
$607$	672.036	1.10714	0.553571	+	0.832802i	$0.313265\pi$
$608$	0	0
$609$	360.000	0.591133
$610$	0	0
$611$	581.969i	0.952486i
$612$	0	0
$613$	694.000i	1.13214i	0.824358	+	0.566069i	$0.191536\pi$
$613$	694.000i	1.13214i	−0.824358	+	0.566069i	$0.808464\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 0.0486224i	−0.999704	−	0.0243112i	$-0.992261\pi$
$617$	− 30.0000i	− 0.0486224i	0.999704	−	0.0243112i	$-0.00773925\pi$
$618$	0	0
$619$	− 339.482i	− 0.548436i	−0.961668	−	0.274218i	$-0.911581\pi$
$619$	− 339.482i	− 0.548436i	0.961668	−	0.274218i	$-0.0884190\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−789.815	−1.26776
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−249.415	−0.397792
$628$	0	0
$629$	156.000	0.248013
$630$	0	0
$631$	464.190i	0.735641i	0.929897	+	0.367821i	$0.119896\pi$
$631$	464.190i	0.735641i	−0.929897	+	0.367821i	$0.880104\pi$
$632$	0	0
$633$	156.000i	0.246445i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 14.0000i	− 0.0219780i
$638$	0	0
$639$	249.415i	0.390321i
$640$	0	0
$641$	−390.000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$641$	−390.000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$642$	0	0
$643$	810.600	1.26065	0.630326	−	0.776330i	$-0.282921\pi$
$643$	810.600	1.26065	0.630326	+	0.776330i	$0.282921\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	581.969	0.899489	0.449744	−	0.893157i	$-0.351515\pi$
$647$	581.969	0.899489	0.449744	+	0.893157i	$0.351515\pi$
$648$	0	0
$649$	−432.000	−0.665639
$650$	0	0
$651$	249.415i	0.383126i
$652$	0	0
$653$	− 774.000i	− 1.18530i	−0.805461	−	0.592649i	$-0.798082\pi$
$653$	− 774.000i	− 1.18530i	0.805461	−	0.592649i	$-0.201918\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 246.000i	− 0.374429i
$658$	0	0
$659$	228.631i	0.346936i	0.984840	+	0.173468i	$0.0554973\pi$
$659$	228.631i	0.346936i	−0.984840	+	0.173468i	$0.944503\pi$
$660$	0	0
$661$	−454.000	−0.686838	−0.343419	−	0.939182i	$-0.611585\pi$
$661$	−454.000	−0.686838	−0.343419	+	0.939182i	$0.611585\pi$
$662$	0	0
$663$	−145.492	−0.219445
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−612.000	−0.914798
$670$	0	0
$671$	− 1454.92i	− 2.16829i
$672$	0	0
$673$	− 434.000i	− 0.644874i	−0.946591	−	0.322437i	$-0.895498\pi$
$673$	− 434.000i	− 0.644874i	0.946591	−	0.322437i	$-0.104502\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 234.000i	− 0.345643i	−0.984953	−	0.172821i	$-0.944712\pi$
$677$	− 234.000i	− 0.345643i	0.984953	−	0.172821i	$-0.0552883\pi$
$678$	0	0
$679$	235.559i	0.346920i
$680$	0	0
$681$	252.000	0.370044
$682$	0	0
$683$	270.200	0.395608	0.197804	−	0.980242i	$-0.436619\pi$
$683$	270.200	0.395608	0.197804	+	0.980242i	$0.436619\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	391.443	0.569787
$688$	0	0
$689$	−252.000	−0.365747
$690$	0	0
$691$	− 20.7846i	− 0.0300790i	−0.999887	−	0.0150395i	$-0.995213\pi$
$691$	− 20.7846i	− 0.0300790i	0.999887	−	0.0150395i	$-0.00478741\pi$
$692$	0	0
$693$	432.000i	0.623377i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	324.000i	0.464849i
$698$	0	0
$699$	197.454i	0.282480i
$700$	0	0
$701$	−1074.00	−1.53210	−0.766049	−	0.642783i	$-0.777780\pi$
$701$	−1074.00	−1.53210	−0.766049	+	0.642783i	$0.777780\pi$
$702$	0	0
$703$	180.133	0.256235
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−124.708	−0.176390
$708$	0	0
$709$	−898.000	−1.26657	−0.633286	−	0.773918i	$-0.718294\pi$
$709$	−898.000	−1.26657	−0.633286	+	0.773918i	$0.718294\pi$
$710$	0	0
$711$	228.631i	0.321562i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	576.000i	0.803347i
$718$	0	0
$719$	− 956.092i	− 1.32975i	−0.746954	−	0.664876i	$-0.768484\pi$
$719$	− 956.092i	− 1.32975i	0.746954	−	0.664876i	$-0.231516\pi$
$720$	0	0
$721$	−912.000	−1.26491
$722$	0	0
$723$	−308.305	−0.426425
$724$	0	0
$725$	0	0
$726$	0	0
$727$	810.600	1.11499	0.557496	−	0.830179i	$-0.311762\pi$
$727$	810.600	1.11499	0.557496	+	0.830179i	$0.311762\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	− 124.708i	− 0.170599i
$732$	0	0
$733$	− 370.000i	− 0.504775i	−0.967626	−	0.252387i	$-0.918784\pi$
$733$	− 370.000i	− 0.504775i	0.967626	−	0.252387i	$-0.0812157\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2448.00i	3.32157i
$738$	0	0
$739$	− 852.169i	− 1.15314i	−0.817048	−	0.576569i	$-0.804391\pi$
$739$	− 852.169i	− 1.15314i	0.817048	−	0.576569i	$-0.195609\pi$
$740$	0	0
$741$	−168.000	−0.226721
$742$	0	0
$743$	1371.78	1.84628	0.923139	−	0.384467i	$-0.125615\pi$
$743$	1371.78	1.84628	0.923139	+	0.384467i	$0.125615\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−62.3538	−0.0834723
$748$	0	0
$749$	−1008.00	−1.34579
$750$	0	0
$751$	− 76.2102i	− 0.101478i	−0.998712	−	0.0507392i	$-0.983842\pi$
$751$	− 76.2102i	− 0.101478i	0.998712	−	0.0507392i	$-0.0161577\pi$
$752$	0	0
$753$	− 180.000i	− 0.239044i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	514.000i	0.678996i	0.940607	+	0.339498i	$0.110257\pi$
$757$	514.000i	0.678996i	−0.940607	+	0.339498i	$0.889743\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−966.000	−1.26938	−0.634691	−	0.772766i	$-0.718873\pi$
$761$	−966.000	−1.26938	−0.634691	+	0.772766i	$0.718873\pi$
$762$	0	0
$763$	−235.559	−0.308727
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−290.985	−0.379380
$768$	0	0
$769$	958.000	1.24577	0.622887	−	0.782312i	$-0.285960\pi$
$769$	958.000	1.24577	0.622887	+	0.782312i	$0.285960\pi$
$770$	0	0
$771$	− 446.869i	− 0.579597i
$772$	0	0
$773$	546.000i	0.706339i	0.935559	+	0.353169i	$0.114896\pi$
$773$	546.000i	0.706339i	−0.935559	+	0.353169i	$0.885104\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 312.000i	− 0.401544i
$778$	0	0
$779$	374.123i	0.480261i
$780$	0	0
$781$	−1728.00	−2.21255
$782$	0	0
$783$	155.885	0.199086
$784$	0	0
$785$	0	0
$786$	0	0
$787$	242.487	0.308116	0.154058	−	0.988062i	$-0.450766\pi$
$787$	242.487	0.308116	0.154058	+	0.988062i	$0.450766\pi$
$788$	0	0
$789$	648.000	0.821293
$790$	0	0
$791$	540.400i	0.683186i
$792$	0	0
$793$	− 980.000i	− 1.23581i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1338.00i	− 1.67880i	−0.543518	−	0.839398i	$-0.682908\pi$
$797$	− 1338.00i	− 1.67880i	0.543518	−	0.839398i	$-0.317092\pi$
$798$	0	0
$799$	− 249.415i	− 0.312159i
$800$	0	0
$801$	−342.000	−0.426966
$802$	0	0
$803$	1704.34	2.12246
$804$	0	0
$805$	0	0
$806$	0	0
$807$	883.346	1.09460
$808$	0	0
$809$	966.000	1.19407	0.597033	−	0.802216i	$-0.296346\pi$
$809$	966.000	1.19407	0.597033	+	0.802216i	$0.296346\pi$
$810$	0	0
$811$	1517.28i	1.87087i	0.353497	+	0.935436i	$0.384992\pi$
$811$	1517.28i	1.87087i	−0.353497	+	0.935436i	$0.615008\pi$
$812$	0	0
$813$	780.000i	0.959410i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 144.000i	− 0.176255i
$818$	0	0
$819$	290.985i	0.355292i
$820$	0	0
$821$	222.000	0.270402	0.135201	−	0.990818i	$-0.456832\pi$
$821$	222.000	0.270402	0.135201	+	0.990818i	$0.456832\pi$
$822$	0	0
$823$	−1281.72	−1.55737	−0.778686	−	0.627413i	$-0.784114\pi$
$823$	−1281.72	−1.55737	−0.778686	+	0.627413i	$0.784114\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1434.14	1.73415	0.867073	−	0.498182i	$-0.165999\pi$
$827$	1434.14	1.73415	0.867073	+	0.498182i	$0.165999\pi$
$828$	0	0
$829$	−226.000	−0.272618	−0.136309	−	0.990666i	$-0.543524\pi$
$829$	−226.000	−0.272618	−0.136309	+	0.990666i	$0.543524\pi$
$830$	0	0
$831$	24.2487i	0.0291802i
$832$	0	0
$833$	6.00000i	0.00720288i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	108.000i	0.129032i
$838$	0	0
$839$	498.831i	0.594554i	0.954791	+	0.297277i	$0.0960784\pi$
$839$	498.831i	0.594554i	−0.954791	+	0.297277i	$0.903922\pi$
$840$	0	0
$841$	59.0000	0.0701546
$842$	0	0
$843$	−613.146	−0.727338
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2154.67	−2.54389
$848$	0	0
$849$	−252.000	−0.296820
$850$	0	0
$851$	0	0
$852$	0	0
$853$	70.0000i	0.0820633i	0.999158	+	0.0410317i	$0.0130644\pi$
$853$	70.0000i	0.0820633i	−0.999158	+	0.0410317i	$0.986936\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.0000i	0.0490082i	0.999700	+	0.0245041i	$0.00780067\pi$
$857$	42.0000i	0.0490082i	−0.999700	+	0.0245041i	$0.992199\pi$
$858$	0	0
$859$	921.451i	1.07270i	0.843995	+	0.536351i	$0.180198\pi$
$859$	921.451i	1.07270i	−0.843995	+	0.536351i	$0.819802\pi$
$860$	0	0
$861$	648.000	0.752613
$862$	0	0
$863$	166.277	0.192673	0.0963365	−	0.995349i	$-0.469287\pi$
$863$	166.277	0.192673	0.0963365	+	0.995349i	$0.469287\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−438.209	−0.505431
$868$	0	0
$869$	−1584.00	−1.82278
$870$	0	0
$871$	1648.91i	1.89313i
$872$	0	0
$873$	102.000i	0.116838i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 166.000i	− 0.189282i	−0.995511	−	0.0946408i	$-0.969830\pi$
$877$	− 166.000i	− 0.189282i	0.995511	−	0.0946408i	$-0.0301703\pi$
$878$	0	0
$879$	− 862.561i	− 0.981298i
$880$	0	0
$881$	−702.000	−0.796822	−0.398411	−	0.917207i	$-0.630438\pi$
$881$	−702.000	−0.796822	−0.398411	+	0.917207i	$0.630438\pi$
$882$	0	0
$883$	−630.466	−0.714005	−0.357003	−	0.934103i	$-0.616201\pi$
$883$	−630.466	−0.714005	−0.357003	+	0.934103i	$0.616201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1288.65	1.45281	0.726407	−	0.687265i	$-0.241189\pi$
$887$	1288.65	1.45281	0.726407	+	0.687265i	$0.241189\pi$
$888$	0	0
$889$	−720.000	−0.809899
$890$	0	0
$891$	187.061i	0.209946i
$892$	0	0
$893$	− 288.000i	− 0.322508i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	623.538i	0.693591i
$900$	0	0
$901$	108.000	0.119867
$902$	0	0
$903$	−249.415	−0.276207
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1447.99	1.59647	0.798233	−	0.602349i	$-0.205768\pi$
$907$	1447.99	1.59647	0.798233	+	0.602349i	$0.205768\pi$
$908$	0	0
$909$	−54.0000	−0.0594059
$910$	0	0
$911$	332.554i	0.365043i	0.983202	+	0.182521i	$0.0584258\pi$
$911$	332.554i	0.365043i	−0.983202	+	0.182521i	$0.941574\pi$
$912$	0	0
$913$	− 432.000i	− 0.473165i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	720.000i	0.785169i
$918$	0	0
$919$	561.184i	0.610647i	0.952249	+	0.305323i	$0.0987646\pi$
$919$	561.184i	0.610647i	−0.952249	+	0.305323i	$0.901235\pi$
$920$	0	0
$921$	324.000	0.351792
$922$	0	0
$923$	−1163.94	−1.26104
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−394.908	−0.426006
$928$	0	0
$929$	438.000	0.471475	0.235737	−	0.971817i	$-0.424249\pi$
$929$	438.000	0.471475	0.235737	+	0.971817i	$0.424249\pi$
$930$	0	0
$931$	6.92820i	0.00744168i
$932$	0	0
$933$	72.0000i	0.0771704i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1826.00i	1.94877i	0.224881	+	0.974386i	$0.427801\pi$
$937$	1826.00i	1.94877i	−0.224881	+	0.974386i	$0.572199\pi$
$938$	0	0
$939$	502.295i	0.534925i
$940$	0	0
$941$	−330.000	−0.350691	−0.175345	−	0.984507i	$-0.556104\pi$
$941$	−330.000	−0.350691	−0.175345	+	0.984507i	$0.556104\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−187.061	−0.197531	−0.0987653	−	0.995111i	$-0.531489\pi$
$947$	−187.061	−0.197531	−0.0987653	+	0.995111i	$0.531489\pi$
$948$	0	0
$949$	1148.00	1.20969
$950$	0	0
$951$	363.731i	0.382472i
$952$	0	0
$953$	1110.00i	1.16474i	0.812923	+	0.582371i	$0.197875\pi$
$953$	1110.00i	1.16474i	−0.812923	+	0.582371i	$0.802125\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1080.00i	1.12853i
$958$	0	0
$959$	1288.65i	1.34374i
$960$	0	0
$961$	529.000	0.550468
$962$	0	0
$963$	−436.477	−0.453247
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−755.174	−0.780945	−0.390473	−	0.920615i	$-0.627688\pi$
$967$	−755.174	−0.780945	−0.390473	+	0.920615i	$0.627688\pi$
$968$	0	0
$969$	72.0000	0.0743034
$970$	0	0
$971$	− 394.908i	− 0.406702i	−0.979106	−	0.203351i	$-0.934817\pi$
$971$	− 394.908i	− 0.406702i	0.979106	−	0.203351i	$-0.0651832\pi$
$972$	0	0
$973$	336.000i	0.345324i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 918.000i	− 0.939611i	−0.882770	−	0.469806i	$-0.844324\pi$
$977$	− 918.000i	− 0.939611i	0.882770	−	0.469806i	$-0.155676\pi$
$978$	0	0
$979$	− 2369.45i	− 2.42027i
$980$	0	0
$981$	−102.000	−0.103976
$982$	0	0
$983$	41.5692	0.0422881	0.0211441	−	0.999776i	$-0.493269\pi$
$983$	41.5692	0.0422881	0.0211441	+	0.999776i	$0.493269\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−498.831	−0.505401
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 48.4974i	− 0.0489379i	−0.999701	−	0.0244689i	$-0.992211\pi$
$991$	− 48.4974i	− 0.0489379i	0.999701	−	0.0244689i	$-0.00778948\pi$
$992$	0	0
$993$	− 348.000i	− 0.350453i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	554.000i	0.555667i	0.960629	+	0.277834i	$0.0896164\pi$
$997$	554.000i	0.555667i	−0.960629	+	0.277834i	$0.910384\pi$
$998$	0	0
$999$	− 135.100i	− 0.135235i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.j.a.799.2		4
3.2	odd	2		3600.3.j.i.1999.3		4
4.3	odd	2	inner	1200.3.j.a.799.3		4
5.2	odd	4		1200.3.e.h.751.2		2
5.3	odd	4		48.3.g.a.31.1	✓	2
5.4	even	2	inner	1200.3.j.a.799.4		4
12.11	even	2		3600.3.j.i.1999.2		4
15.2	even	4		3600.3.e.t.3151.2		2
15.8	even	4		144.3.g.b.127.1		2
15.14	odd	2		3600.3.j.i.1999.1		4
20.3	even	4		48.3.g.a.31.2	yes	2
20.7	even	4		1200.3.e.h.751.1		2
20.19	odd	2	inner	1200.3.j.a.799.1		4
35.13	even	4		2352.3.m.a.1471.2		2
40.3	even	4		192.3.g.a.127.1		2
40.13	odd	4		192.3.g.a.127.2		2
45.13	odd	12		1296.3.o.c.703.1		2
45.23	even	12		1296.3.o.p.703.1		2
45.38	even	12		1296.3.o.n.271.1		2
45.43	odd	12		1296.3.o.a.271.1		2
60.23	odd	4		144.3.g.b.127.2		2
60.47	odd	4		3600.3.e.t.3151.1		2
60.59	even	2		3600.3.j.i.1999.4		4
80.3	even	4		768.3.b.b.127.1		4
80.13	odd	4		768.3.b.b.127.3		4
80.43	even	4		768.3.b.b.127.4		4
80.53	odd	4		768.3.b.b.127.2		4
120.53	even	4		576.3.g.i.127.1		2
120.83	odd	4		576.3.g.i.127.2		2
140.83	odd	4		2352.3.m.a.1471.1		2
180.23	odd	12		1296.3.o.n.703.1		2
180.43	even	12		1296.3.o.c.271.1		2
180.83	odd	12		1296.3.o.p.271.1		2
180.103	even	12		1296.3.o.a.703.1		2
240.53	even	4		2304.3.b.n.127.2		4
240.83	odd	4		2304.3.b.n.127.3		4
240.173	even	4		2304.3.b.n.127.4		4
240.203	odd	4		2304.3.b.n.127.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.g.a.31.1	✓	2	5.3	odd	4
48.3.g.a.31.2	yes	2	20.3	even	4
144.3.g.b.127.1		2	15.8	even	4
144.3.g.b.127.2		2	60.23	odd	4
192.3.g.a.127.1		2	40.3	even	4
192.3.g.a.127.2		2	40.13	odd	4
576.3.g.i.127.1		2	120.53	even	4
576.3.g.i.127.2		2	120.83	odd	4
768.3.b.b.127.1		4	80.3	even	4
768.3.b.b.127.2		4	80.53	odd	4
768.3.b.b.127.3		4	80.13	odd	4
768.3.b.b.127.4		4	80.43	even	4
1200.3.e.h.751.1		2	20.7	even	4
1200.3.e.h.751.2		2	5.2	odd	4
1200.3.j.a.799.1		4	20.19	odd	2	inner
1200.3.j.a.799.2		4	1.1	even	1	trivial
1200.3.j.a.799.3		4	4.3	odd	2	inner
1200.3.j.a.799.4		4	5.4	even	2	inner
1296.3.o.a.271.1		2	45.43	odd	12
1296.3.o.a.703.1		2	180.103	even	12
1296.3.o.c.271.1		2	180.43	even	12
1296.3.o.c.703.1		2	45.13	odd	12
1296.3.o.n.271.1		2	45.38	even	12
1296.3.o.n.703.1		2	180.23	odd	12
1296.3.o.p.271.1		2	180.83	odd	12
1296.3.o.p.703.1		2	45.23	even	12
2304.3.b.n.127.1		4	240.203	odd	4
2304.3.b.n.127.2		4	240.53	even	4
2304.3.b.n.127.3		4	240.83	odd	4
2304.3.b.n.127.4		4	240.173	even	4
2352.3.m.a.1471.1		2	140.83	odd	4
2352.3.m.a.1471.2		2	35.13	even	4
3600.3.e.t.3151.1		2	60.47	odd	4
3600.3.e.t.3151.2		2	15.2	even	4
3600.3.j.i.1999.1		4	15.14	odd	2
3600.3.j.i.1999.2		4	12.11	even	2
3600.3.j.i.1999.3		4	3.2	odd	2
3600.3.j.i.1999.4		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +6.92820 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +6.92820 q^{7} +3.00000 q^{9} +20.7846i q^{11} +14.0000i q^{13} -6.00000i q^{17} -6.92820i q^{19} -12.0000 q^{21} -5.19615 q^{27} -30.0000 q^{29} -20.7846i q^{31} -36.0000i q^{33} +26.0000i q^{37} -24.2487i q^{39} -54.0000 q^{41} +20.7846 q^{43} +41.5692 q^{47} -1.00000 q^{49} +10.3923i q^{51} +18.0000i q^{53} +12.0000i q^{57} +20.7846i q^{59} -70.0000 q^{61} +20.7846 q^{63} +117.779 q^{67} +83.1384i q^{71} -82.0000i q^{73} +144.000i q^{77} +76.2102i q^{79} +9.00000 q^{81} -20.7846 q^{83} +51.9615 q^{87} -114.000 q^{89} +96.9948i q^{91} +36.0000i q^{93} +34.0000i q^{97} +62.3538i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} - 48 q^{21} - 120 q^{29} - 216 q^{41} - 4 q^{49} - 280 q^{61} + 36 q^{81} - 456 q^{89}+O(q^{100}) \) 4 * q + 12 * q^9 - 48 * q^21 - 120 * q^29 - 216 * q^41 - 4 * q^49 - 280 * q^61 + 36 * q^81 - 456 * q^89

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