LMFDB - Embedded newform 495.3.e.a.386.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,3,Mod(386,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.386"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	495.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			386.15
Character	$\chi$	$=$	495.386
Dual form			495.3.e.a.386.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.11561i q^{2} +2.75541 q^{4} -2.23607i q^{5} -12.6993 q^{7} +7.53641i q^{8} +2.49458 q^{10} +3.31662i q^{11} -9.12509 q^{13} -14.1675i q^{14} +2.61397 q^{16} -31.5229i q^{17} -9.04678 q^{19} -6.16129i q^{20} -3.70006 q^{22} +4.42287i q^{23} -5.00000 q^{25} -10.1800i q^{26} -34.9920 q^{28} -44.8219i q^{29} -27.6070 q^{31} +33.0618i q^{32} +35.1673 q^{34} +28.3966i q^{35} -60.3888 q^{37} -10.0927i q^{38} +16.8519 q^{40} +24.6869i q^{41} -77.7411 q^{43} +9.13868i q^{44} -4.93420 q^{46} -2.98217i q^{47} +112.273 q^{49} -5.57805i q^{50} -25.1434 q^{52} -20.5037i q^{53} +7.41620 q^{55} -95.7075i q^{56} +50.0038 q^{58} +41.2471i q^{59} +26.1992 q^{61} -30.7987i q^{62} -26.4282 q^{64} +20.4043i q^{65} +107.643 q^{67} -86.8586i q^{68} -31.6795 q^{70} +17.9972i q^{71} -82.8666 q^{73} -67.3703i q^{74} -24.9276 q^{76} -42.1190i q^{77} -66.0390 q^{79} -5.84501i q^{80} -27.5409 q^{82} -70.6886i q^{83} -70.4873 q^{85} -86.7288i q^{86} -24.9954 q^{88} -12.0920i q^{89} +115.883 q^{91} +12.1869i q^{92} +3.32694 q^{94} +20.2292i q^{95} +67.7241 q^{97} +125.253i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 40 q^{4} - 32 q^{7} + 40 q^{10} + 48 q^{13} - 24 q^{16} - 120 q^{25} + 128 q^{28} + 96 q^{31} + 112 q^{34} - 320 q^{37} - 120 q^{40} + 208 q^{43} - 288 q^{46} + 296 q^{49} - 192 q^{52} + 96 q^{58}+ \cdots + 576 q^{97}+O(q^{100}) $ 24 * q - 40 * q^4 - 32 * q^7 + 40 * q^10 + 48 * q^13 - 24 * q^16 - 120 * q^25 + 128 * q^28 + 96 * q^31 + 112 * q^34 - 320 * q^37 - 120 * q^40 + 208 * q^43 - 288 * q^46 + 296 * q^49 - 192 * q^52 + 96 * q^58 + 400 * q^61 + 440 * q^64 - 288 * q^67 - 480 * q^73 + 32 * q^76 - 544 * q^79 + 96 * q^82 - 320 * q^85 + 384 * q^91 + 160 * q^94 + 576 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.11561i	0.557805i	0.960319	+	0.278902i	$0.0899706\pi$
$2$	1.11561i	0.557805i	−0.960319	+	0.278902i	$0.910029\pi$
$3$	0	0
$3$	0	0
$4$	2.75541	0.688854
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	−12.6993	−1.81419	−0.907096	−	0.420923i	$-0.861706\pi$
$7$	−12.6993	−1.81419	−0.907096	+	0.420923i	$0.861706\pi$
$8$	7.53641i	0.942051i
$9$	0	0
$10$	2.49458	0.249458
$11$	3.31662i	0.301511i
$11$	3.31662i	0.301511i
$12$	0	0
$13$	−9.12509	−0.701930	−0.350965	−	0.936389i	$-0.614146\pi$
$13$	−9.12509	−0.701930	−0.350965	+	0.936389i	$0.614146\pi$
$14$	− 14.1675i	− 1.01197i
$15$	0	0
$16$	2.61397	0.163373
$17$	− 31.5229i	− 1.85429i	−0.374705	−	0.927144i	$-0.622256\pi$
$17$	− 31.5229i	− 1.85429i	0.374705	−	0.927144i	$-0.377744\pi$
$18$	0	0
$19$	−9.04678	−0.476147	−0.238073	−	0.971247i	$-0.576516\pi$
$19$	−9.04678	−0.476147	−0.238073	+	0.971247i	$0.576516\pi$
$20$	− 6.16129i	− 0.308065i
$21$	0	0
$22$	−3.70006	−0.168185
$23$	4.42287i	0.192299i	0.995367	+	0.0961494i	$0.0306527\pi$
$23$	4.42287i	0.192299i	−0.995367	+	0.0961494i	$0.969347\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	− 10.1800i	− 0.391540i
$27$	0	0
$28$	−34.9920	−1.24971
$29$	− 44.8219i	− 1.54558i	−0.634660	−	0.772792i	$-0.718860\pi$
$29$	− 44.8219i	− 1.54558i	0.634660	−	0.772792i	$-0.281140\pi$
$30$	0	0
$31$	−27.6070	−0.890550	−0.445275	−	0.895394i	$-0.646894\pi$
$31$	−27.6070	−0.890550	−0.445275	+	0.895394i	$0.646894\pi$
$32$	33.0618i	1.03318i
$33$	0	0
$34$	35.1673	1.03433
$35$	28.3966i	0.811332i
$36$	0	0
$37$	−60.3888	−1.63213	−0.816065	−	0.577960i	$-0.803849\pi$
$37$	−60.3888	−1.63213	−0.816065	+	0.577960i	$0.803849\pi$
$38$	− 10.0927i	− 0.265597i
$39$	0	0
$40$	16.8519	0.421298
$41$	24.6869i	0.602119i	0.953605	+	0.301059i	$0.0973402\pi$
$41$	24.6869i	0.602119i	−0.953605	+	0.301059i	$0.902660\pi$
$42$	0	0
$43$	−77.7411	−1.80793	−0.903966	−	0.427603i	$-0.859358\pi$
$43$	−77.7411	−1.80793	−0.903966	+	0.427603i	$0.859358\pi$
$44$	9.13868i	0.207697i
$45$	0	0
$46$	−4.93420	−0.107265
$47$	− 2.98217i	− 0.0634505i	−0.999497	−	0.0317252i	$-0.989900\pi$
$47$	− 2.98217i	− 0.0634505i	0.999497	−	0.0317252i	$-0.0101001\pi$
$48$	0	0
$49$	112.273	2.29129
$50$	− 5.57805i	− 0.111561i
$51$	0	0
$52$	−25.1434	−0.483527
$53$	− 20.5037i	− 0.386863i	−0.981114	−	0.193431i	$-0.938038\pi$
$53$	− 20.5037i	− 0.386863i	0.981114	−	0.193431i	$-0.0619616\pi$
$54$	0	0
$55$	7.41620	0.134840
$56$	− 95.7075i	− 1.70906i
$57$	0	0
$58$	50.0038	0.862134
$59$	41.2471i	0.699104i	0.936917	+	0.349552i	$0.113666\pi$
$59$	41.2471i	0.699104i	−0.936917	+	0.349552i	$0.886334\pi$
$60$	0	0
$61$	26.1992	0.429495	0.214747	−	0.976670i	$-0.431107\pi$
$61$	26.1992	0.429495	0.214747	+	0.976670i	$0.431107\pi$
$62$	− 30.7987i	− 0.496753i
$63$	0	0
$64$	−26.4282	−0.412940
$65$	20.4043i	0.313913i
$66$	0	0
$67$	107.643	1.60661	0.803307	−	0.595565i	$-0.203072\pi$
$67$	107.643	1.60661	0.803307	+	0.595565i	$0.203072\pi$
$68$	− 86.8586i	− 1.27733i
$69$	0	0
$70$	−31.6795	−0.452565
$71$	17.9972i	0.253482i	0.991936	+	0.126741i	$0.0404517\pi$
$71$	17.9972i	0.253482i	−0.991936	+	0.126741i	$0.959548\pi$
$72$	0	0
$73$	−82.8666	−1.13516	−0.567580	−	0.823319i	$-0.692120\pi$
$73$	−82.8666	−1.13516	−0.567580	+	0.823319i	$0.692120\pi$
$74$	− 67.3703i	− 0.910410i
$75$	0	0
$76$	−24.9276	−0.327995
$77$	− 42.1190i	− 0.547000i
$78$	0	0
$79$	−66.0390	−0.835937	−0.417968	−	0.908462i	$-0.637258\pi$
$79$	−66.0390	−0.835937	−0.417968	+	0.908462i	$0.637258\pi$
$80$	− 5.84501i	− 0.0730627i
$81$	0	0
$82$	−27.5409	−0.335865
$83$	− 70.6886i	− 0.851669i	−0.904801	−	0.425835i	$-0.859981\pi$
$83$	− 70.6886i	− 0.851669i	0.904801	−	0.425835i	$-0.140019\pi$
$84$	0	0
$85$	−70.4873	−0.829263
$86$	− 86.7288i	− 1.00847i
$87$	0	0
$88$	−24.9954	−0.284039
$89$	− 12.0920i	− 0.135865i	−0.997690	−	0.0679324i	$-0.978360\pi$
$89$	− 12.0920i	− 0.135865i	0.997690	−	0.0679324i	$-0.0216402\pi$
$90$	0	0
$91$	115.883	1.27344
$92$	12.1869i	0.132466i
$93$	0	0
$94$	3.32694	0.0353930
$95$	20.2292i	0.212939i
$96$	0	0
$97$	67.7241	0.698187	0.349094	−	0.937088i	$-0.386490\pi$
$97$	67.7241	0.698187	0.349094	+	0.937088i	$0.386490\pi$
$98$	125.253i	1.27810i
$99$	0	0
$100$	−13.7771	−0.137771
$101$	− 114.701i	− 1.13566i	−0.823146	−	0.567829i	$-0.807783\pi$
$101$	− 114.701i	− 1.13566i	0.823146	−	0.567829i	$-0.192217\pi$
$102$	0	0
$103$	79.4536	0.771394	0.385697	−	0.922626i	$-0.373961\pi$
$103$	79.4536	0.771394	0.385697	+	0.922626i	$0.373961\pi$
$104$	− 68.7704i	− 0.661254i
$105$	0	0
$106$	22.8741	0.215794
$107$	50.9166i	0.475856i	0.971283	+	0.237928i	$0.0764682\pi$
$107$	50.9166i	0.475856i	−0.971283	+	0.237928i	$0.923532\pi$
$108$	0	0
$109$	−66.0009	−0.605512	−0.302756	−	0.953068i	$-0.597907\pi$
$109$	−66.0009	−0.605512	−0.302756	+	0.953068i	$0.597907\pi$
$110$	8.27358i	0.0752144i
$111$	0	0
$112$	−33.1957	−0.296390
$113$	115.631i	1.02328i	0.859199	+	0.511642i	$0.170963\pi$
$113$	115.631i	1.02328i	−0.859199	+	0.511642i	$0.829037\pi$
$114$	0	0
$115$	9.88985	0.0859987
$116$	− 123.503i	− 1.06468i
$117$	0	0
$118$	−46.0157	−0.389964
$119$	400.320i	3.36404i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	29.2280i	0.239574i
$123$	0	0
$124$	−76.0689	−0.613459
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	138.530	1.09079	0.545394	−	0.838180i	$-0.316380\pi$
$127$	138.530	1.09079	0.545394	+	0.838180i	$0.316380\pi$
$128$	102.764i	0.802841i
$129$	0	0
$130$	−22.7633	−0.175102
$131$	222.548i	1.69884i	0.527715	+	0.849421i	$0.323049\pi$
$131$	222.548i	1.69884i	−0.527715	+	0.849421i	$0.676951\pi$
$132$	0	0
$133$	114.888	0.863822
$134$	120.088i	0.896178i
$135$	0	0
$136$	237.569	1.74683
$137$	25.0512i	0.182855i	0.995812	+	0.0914276i	$0.0291430\pi$
$137$	25.0512i	0.182855i	−0.995812	+	0.0914276i	$0.970857\pi$
$138$	0	0
$139$	277.204	1.99427	0.997135	−	0.0756377i	$-0.0240992\pi$
$139$	277.204	1.99427	0.997135	+	0.0756377i	$0.0240992\pi$
$140$	78.2444i	0.558889i
$141$	0	0
$142$	−20.0779	−0.141393
$143$	− 30.2645i	− 0.211640i
$144$	0	0
$145$	−100.225	−0.691206
$146$	− 92.4468i	− 0.633197i
$147$	0	0
$148$	−166.396	−1.12430
$149$	185.398i	1.24428i	0.782904	+	0.622142i	$0.213737\pi$
$149$	185.398i	1.24428i	−0.782904	+	0.622142i	$0.786263\pi$
$150$	0	0
$151$	23.6428	0.156575	0.0782873	−	0.996931i	$-0.475055\pi$
$151$	23.6428	0.156575	0.0782873	+	0.996931i	$0.475055\pi$
$152$	− 68.1802i	− 0.448554i
$153$	0	0
$154$	46.9883	0.305119
$155$	61.7312i	0.398266i
$156$	0	0
$157$	116.640	0.742927	0.371464	−	0.928448i	$-0.378856\pi$
$157$	116.640	0.742927	0.371464	+	0.928448i	$0.378856\pi$
$158$	− 73.6738i	− 0.466290i
$159$	0	0
$160$	73.9284	0.462053
$161$	− 56.1676i	− 0.348867i
$162$	0	0
$163$	−186.451	−1.14387	−0.571936	−	0.820298i	$-0.693808\pi$
$163$	−186.451	−1.14387	−0.571936	+	0.820298i	$0.693808\pi$
$164$	68.0226i	0.414772i
$165$	0	0
$166$	78.8608	0.475065
$167$	− 69.4681i	− 0.415976i	−0.978131	−	0.207988i	$-0.933308\pi$
$167$	− 69.4681i	− 0.415976i	0.978131	−	0.207988i	$-0.0666916\pi$
$168$	0	0
$169$	−85.7328	−0.507295
$170$	− 78.6364i	− 0.462567i
$171$	0	0
$172$	−214.209	−1.24540
$173$	23.4828i	0.135739i	0.997694	+	0.0678694i	$0.0216201\pi$
$173$	23.4828i	0.135739i	−0.997694	+	0.0678694i	$0.978380\pi$
$174$	0	0
$175$	63.4967	0.362839
$176$	8.66956i	0.0492588i
$177$	0	0
$178$	13.4899	0.0757860
$179$	− 124.336i	− 0.694615i	−0.937751	−	0.347308i	$-0.887096\pi$
$179$	− 124.336i	− 0.694615i	0.937751	−	0.347308i	$-0.112904\pi$
$180$	0	0
$181$	27.5614	0.152273	0.0761363	−	0.997097i	$-0.475742\pi$
$181$	27.5614	0.152273	0.0761363	+	0.997097i	$0.475742\pi$
$182$	129.280i	0.710329i
$183$	0	0
$184$	−33.3326	−0.181155
$185$	135.033i	0.729911i
$186$	0	0
$187$	104.550	0.559089
$188$	− 8.21712i	− 0.0437081i
$189$	0	0
$190$	−22.5679	−0.118779
$191$	− 197.513i	− 1.03410i	−0.855956	−	0.517049i	$-0.827030\pi$
$191$	− 197.513i	− 1.03410i	0.855956	−	0.517049i	$-0.172970\pi$
$192$	0	0
$193$	−332.235	−1.72143	−0.860713	−	0.509090i	$-0.829982\pi$
$193$	−332.235	−1.72143	−0.860713	+	0.509090i	$0.829982\pi$
$194$	75.5537i	0.389452i
$195$	0	0
$196$	309.360	1.57837
$197$	− 281.286i	− 1.42785i	−0.700224	−	0.713923i	$-0.746916\pi$
$197$	− 281.286i	− 1.42785i	0.700224	−	0.713923i	$-0.253084\pi$
$198$	0	0
$199$	13.8710	0.0697036	0.0348518	−	0.999392i	$-0.488904\pi$
$199$	13.8710	0.0697036	0.0348518	+	0.999392i	$0.488904\pi$
$200$	− 37.6820i	− 0.188410i
$201$	0	0
$202$	127.962	0.633476
$203$	569.209i	2.80399i
$204$	0	0
$205$	55.2015	0.269276
$206$	88.6392i	0.430287i
$207$	0	0
$208$	−23.8527	−0.114676
$209$	− 30.0048i	− 0.143564i
$210$	0	0
$211$	−260.034	−1.23239	−0.616193	−	0.787595i	$-0.711326\pi$
$211$	−260.034	−1.23239	−0.616193	+	0.787595i	$0.711326\pi$
$212$	− 56.4962i	− 0.266492i
$213$	0	0
$214$	−56.8030	−0.265435
$215$	173.834i	0.808532i
$216$	0	0
$217$	350.591	1.61563
$218$	− 73.6312i	− 0.337758i
$219$	0	0
$220$	20.4347	0.0928850
$221$	287.649i	1.30158i
$222$	0	0
$223$	−391.761	−1.75678	−0.878388	−	0.477947i	$-0.841381\pi$
$223$	−391.761	−1.75678	−0.878388	+	0.477947i	$0.841381\pi$
$224$	− 419.863i	− 1.87439i
$225$	0	0
$226$	−128.999	−0.570793
$227$	211.525i	0.931828i	0.884830	+	0.465914i	$0.154274\pi$
$227$	211.525i	0.931828i	−0.884830	+	0.465914i	$0.845726\pi$
$228$	0	0
$229$	−50.3417	−0.219833	−0.109916	−	0.993941i	$-0.535058\pi$
$229$	−50.3417	−0.219833	−0.109916	+	0.993941i	$0.535058\pi$
$230$	11.0332i	0.0479705i
$231$	0	0
$232$	337.796	1.45602
$233$	371.450i	1.59421i	0.603843	+	0.797104i	$0.293636\pi$
$233$	371.450i	1.59421i	−0.603843	+	0.797104i	$0.706364\pi$
$234$	0	0
$235$	−6.66834	−0.0283759
$236$	113.653i	0.481580i
$237$	0	0
$238$	−446.601	−1.87648
$239$	− 402.590i	− 1.68448i	−0.539104	−	0.842239i	$-0.681237\pi$
$239$	− 402.590i	− 1.68448i	0.539104	−	0.842239i	$-0.318763\pi$
$240$	0	0
$241$	309.549	1.28443	0.642217	−	0.766523i	$-0.278015\pi$
$241$	309.549	1.28443	0.642217	+	0.766523i	$0.278015\pi$
$242$	− 12.2717i	− 0.0507095i
$243$	0	0
$244$	72.1896	0.295859
$245$	− 251.051i	− 1.02470i
$246$	0	0
$247$	82.5527	0.334221
$248$	− 208.058i	− 0.838943i
$249$	0	0
$250$	−12.4729	−0.0498916
$251$	135.958i	0.541666i	0.962626	+	0.270833i	$0.0872991\pi$
$251$	135.958i	0.541666i	−0.962626	+	0.270833i	$0.912701\pi$
$252$	0	0
$253$	−14.6690	−0.0579803
$254$	154.545i	0.608446i
$255$	0	0
$256$	−220.357	−0.860769
$257$	− 296.823i	− 1.15495i	−0.816407	−	0.577476i	$-0.804038\pi$
$257$	− 296.823i	− 1.15495i	0.816407	−	0.577476i	$-0.195962\pi$
$258$	0	0
$259$	766.898	2.96100
$260$	56.2224i	0.216240i
$261$	0	0
$262$	−248.277	−0.947623
$263$	194.114i	0.738075i	0.929414	+	0.369038i	$0.120313\pi$
$263$	194.114i	0.738075i	−0.929414	+	0.369038i	$0.879687\pi$
$264$	0	0
$265$	−45.8477	−0.173010
$266$	128.170i	0.481844i
$267$	0	0
$268$	296.602	1.10672
$269$	− 243.609i	− 0.905608i	−0.891610	−	0.452804i	$-0.850424\pi$
$269$	− 243.609i	− 0.905608i	0.891610	−	0.452804i	$-0.149576\pi$
$270$	0	0
$271$	82.0767	0.302866	0.151433	−	0.988468i	$-0.451611\pi$
$271$	82.0767	0.302866	0.151433	+	0.988468i	$0.451611\pi$
$272$	− 82.3999i	− 0.302941i
$273$	0	0
$274$	−27.9473	−0.101998
$275$	− 16.5831i	− 0.0603023i
$276$	0	0
$277$	−375.339	−1.35501	−0.677507	−	0.735517i	$-0.736939\pi$
$277$	−375.339	−1.35501	−0.677507	+	0.735517i	$0.736939\pi$
$278$	309.251i	1.11241i
$279$	0	0
$280$	−214.008	−0.764316
$281$	− 490.945i	− 1.74713i	−0.486704	−	0.873567i	$-0.661801\pi$
$281$	− 490.945i	− 1.74713i	0.486704	−	0.873567i	$-0.338199\pi$
$282$	0	0
$283$	71.6683	0.253245	0.126623	−	0.991951i	$-0.459586\pi$
$283$	71.6683	0.253245	0.126623	+	0.991951i	$0.459586\pi$
$284$	49.5898i	0.174612i
$285$	0	0
$286$	33.7634	0.118054
$287$	− 313.507i	− 1.09236i
$288$	0	0
$289$	−704.693	−2.43838
$290$	− 111.812i	− 0.385558i
$291$	0	0
$292$	−228.332	−0.781959
$293$	− 234.351i	− 0.799832i	−0.916552	−	0.399916i	$-0.869039\pi$
$293$	− 234.351i	− 0.799832i	0.916552	−	0.399916i	$-0.130961\pi$
$294$	0	0
$295$	92.2314	0.312649
$296$	− 455.115i	− 1.53755i
$297$	0	0
$298$	−206.832	−0.694068
$299$	− 40.3591i	− 0.134980i
$300$	0	0
$301$	987.261	3.27994
$302$	26.3761i	0.0873380i
$303$	0	0
$304$	−23.6480	−0.0777895
$305$	− 58.5831i	− 0.192076i
$306$	0	0
$307$	224.422	0.731018	0.365509	−	0.930808i	$-0.380895\pi$
$307$	224.422	0.731018	0.365509	+	0.930808i	$0.380895\pi$
$308$	− 116.055i	− 0.376803i
$309$	0	0
$310$	−68.8680	−0.222155
$311$	− 195.641i	− 0.629071i	−0.949246	−	0.314536i	$-0.898151\pi$
$311$	− 195.641i	− 0.629071i	0.949246	−	0.314536i	$-0.101849\pi$
$312$	0	0
$313$	266.861	0.852592	0.426296	−	0.904584i	$-0.359818\pi$
$313$	266.861	0.852592	0.426296	+	0.904584i	$0.359818\pi$
$314$	130.124i	0.414408i
$315$	0	0
$316$	−181.965	−0.575838
$317$	365.911i	1.15429i	0.816641	+	0.577146i	$0.195834\pi$
$317$	365.911i	1.15429i	−0.816641	+	0.577146i	$0.804166\pi$
$318$	0	0
$319$	148.657	0.466011
$320$	59.0952i	0.184673i
$321$	0	0
$322$	62.6611	0.194600
$323$	285.181i	0.882913i
$324$	0	0
$325$	45.6254	0.140386
$326$	− 208.007i	− 0.638058i
$327$	0	0
$328$	−186.050	−0.567227
$329$	37.8716i	0.115111i
$330$	0	0
$331$	−156.291	−0.472177	−0.236089	−	0.971732i	$-0.575866\pi$
$331$	−156.291	−0.472177	−0.236089	+	0.971732i	$0.575866\pi$
$332$	− 194.776i	− 0.586676i
$333$	0	0
$334$	77.4993	0.232034
$335$	− 240.698i	− 0.718500i
$336$	0	0
$337$	79.3564	0.235479	0.117739	−	0.993045i	$-0.462435\pi$
$337$	79.3564	0.235479	0.117739	+	0.993045i	$0.462435\pi$
$338$	− 95.6443i	− 0.282971i
$339$	0	0
$340$	−194.222	−0.571241
$341$	− 91.5622i	− 0.268511i
$342$	0	0
$343$	−803.531	−2.34266
$344$	− 585.889i	− 1.70316i
$345$	0	0
$346$	−26.1976	−0.0757157
$347$	− 461.970i	− 1.33133i	−0.746253	−	0.665663i	$-0.768149\pi$
$347$	− 461.970i	− 1.33133i	0.746253	−	0.665663i	$-0.231851\pi$
$348$	0	0
$349$	−470.245	−1.34741	−0.673703	−	0.739002i	$-0.735297\pi$
$349$	−470.245	−1.34741	−0.673703	+	0.739002i	$0.735297\pi$
$350$	70.8376i	0.202393i
$351$	0	0
$352$	−109.654	−0.311516
$353$	97.9301i	0.277422i	0.990333	+	0.138711i	$0.0442960\pi$
$353$	97.9301i	0.277422i	−0.990333	+	0.138711i	$0.955704\pi$
$354$	0	0
$355$	40.2430	0.113360
$356$	− 33.3184i	− 0.0935909i
$357$	0	0
$358$	138.711	0.387460
$359$	318.455i	0.887063i	0.896259	+	0.443531i	$0.146274\pi$
$359$	318.455i	0.887063i	−0.896259	+	0.443531i	$0.853726\pi$
$360$	0	0
$361$	−279.156	−0.773284
$362$	30.7477i	0.0849384i
$363$	0	0
$364$	319.305	0.877211
$365$	185.295i	0.507659i
$366$	0	0
$367$	−99.3286	−0.270650	−0.135325	−	0.990801i	$-0.543208\pi$
$367$	−99.3286	−0.270650	−0.135325	+	0.990801i	$0.543208\pi$
$368$	11.5613i	0.0314165i
$369$	0	0
$370$	−150.645	−0.407148
$371$	260.384i	0.701843i
$372$	0	0
$373$	294.653	0.789953	0.394977	−	0.918691i	$-0.370753\pi$
$373$	294.653	0.789953	0.394977	+	0.918691i	$0.370753\pi$
$374$	116.637i	0.311862i
$375$	0	0
$376$	22.4749	0.0597736
$377$	409.004i	1.08489i
$378$	0	0
$379$	−187.832	−0.495599	−0.247800	−	0.968811i	$-0.579707\pi$
$379$	−187.832	−0.495599	−0.247800	+	0.968811i	$0.579707\pi$
$380$	55.7399i	0.146684i
$381$	0	0
$382$	220.347	0.576825
$383$	− 469.714i	− 1.22641i	−0.789924	−	0.613204i	$-0.789880\pi$
$383$	− 469.714i	− 1.22641i	0.789924	−	0.613204i	$-0.210120\pi$
$384$	0	0
$385$	−94.1809	−0.244626
$386$	− 370.645i	− 0.960220i
$387$	0	0
$388$	186.608	0.480949
$389$	108.264i	0.278314i	0.990270	+	0.139157i	$0.0444393\pi$
$389$	108.264i	0.278314i	−0.990270	+	0.139157i	$0.955561\pi$
$390$	0	0
$391$	139.422	0.356577
$392$	846.138i	2.15852i
$393$	0	0
$394$	313.805	0.796460
$395$	147.668i	0.373842i
$396$	0	0
$397$	−295.766	−0.745003	−0.372502	−	0.928032i	$-0.621500\pi$
$397$	−295.766	−0.745003	−0.372502	+	0.928032i	$0.621500\pi$
$398$	15.4746i	0.0388810i
$399$	0	0
$400$	−13.0698	−0.0326746
$401$	− 199.002i	− 0.496265i	−0.968726	−	0.248132i	$-0.920183\pi$
$401$	− 199.002i	− 0.496265i	0.968726	−	0.248132i	$-0.0798168\pi$
$402$	0	0
$403$	251.917	0.625104
$404$	− 316.050i	− 0.782302i
$405$	0	0
$406$	−635.015	−1.56408
$407$	− 200.287i	− 0.492106i
$408$	0	0
$409$	−180.714	−0.441843	−0.220922	−	0.975292i	$-0.570906\pi$
$409$	−180.714	−0.441843	−0.220922	+	0.975292i	$0.570906\pi$
$410$	61.5834i	0.150203i
$411$	0	0
$412$	218.928	0.531377
$413$	− 523.812i	− 1.26831i
$414$	0	0
$415$	−158.064	−0.380878
$416$	− 301.692i	− 0.725221i
$417$	0	0
$418$	33.4736	0.0800805
$419$	− 419.400i	− 1.00096i	−0.865750	−	0.500478i	$-0.833158\pi$
$419$	− 419.400i	− 1.00096i	0.865750	−	0.500478i	$-0.166842\pi$
$420$	0	0
$421$	624.508	1.48339	0.741696	−	0.670736i	$-0.234022\pi$
$421$	624.508	1.48339	0.741696	+	0.670736i	$0.234022\pi$
$422$	− 290.096i	− 0.687431i
$423$	0	0
$424$	154.524	0.364444
$425$	157.614i	0.370858i
$426$	0	0
$427$	−332.712	−0.779186
$428$	140.296i	0.327795i
$429$	0	0
$430$	−193.931	−0.451003
$431$	195.941i	0.454618i	0.973823	+	0.227309i	$0.0729928\pi$
$431$	195.941i	0.454618i	−0.973823	+	0.227309i	$0.927007\pi$
$432$	0	0
$433$	474.712	1.09633	0.548167	−	0.836369i	$-0.315326\pi$
$433$	474.712	1.09633	0.548167	+	0.836369i	$0.315326\pi$
$434$	391.123i	0.901206i
$435$	0	0
$436$	−181.860	−0.417109
$437$	− 40.0128i	− 0.0915624i
$438$	0	0
$439$	−834.138	−1.90009	−0.950044	−	0.312117i	$-0.898962\pi$
$439$	−834.138	−1.90009	−0.950044	+	0.312117i	$0.898962\pi$
$440$	55.8915i	0.127026i
$441$	0	0
$442$	−320.904	−0.726028
$443$	290.154i	0.654975i	0.944855	+	0.327488i	$0.106202\pi$
$443$	290.154i	0.654975i	−0.944855	+	0.327488i	$0.893798\pi$
$444$	0	0
$445$	−27.0384	−0.0607606
$446$	− 437.053i	− 0.979939i
$447$	0	0
$448$	335.621	0.749153
$449$	440.567i	0.981219i	0.871380	+	0.490610i	$0.163226\pi$
$449$	440.567i	0.981219i	−0.871380	+	0.490610i	$0.836774\pi$
$450$	0	0
$451$	−81.8771	−0.181546
$452$	318.612i	0.704893i
$453$	0	0
$454$	−235.979	−0.519778
$455$	− 259.121i	− 0.569498i
$456$	0	0
$457$	655.012	1.43329	0.716643	−	0.697440i	$-0.245678\pi$
$457$	655.012	1.43329	0.716643	+	0.697440i	$0.245678\pi$
$458$	− 56.1616i	− 0.122624i
$459$	0	0
$460$	27.2506	0.0592405
$461$	205.888i	0.446612i	0.974748	+	0.223306i	$0.0716849\pi$
$461$	205.888i	0.446612i	−0.974748	+	0.223306i	$0.928315\pi$
$462$	0	0
$463$	195.172	0.421538	0.210769	−	0.977536i	$-0.432403\pi$
$463$	195.172	0.421538	0.210769	+	0.977536i	$0.432403\pi$
$464$	− 117.163i	− 0.252507i
$465$	0	0
$466$	−414.394	−0.889257
$467$	− 544.431i	− 1.16580i	−0.812542	−	0.582902i	$-0.801917\pi$
$467$	− 544.431i	− 1.16580i	0.812542	−	0.582902i	$-0.198083\pi$
$468$	0	0
$469$	−1367.00	−2.91471
$470$	− 7.43926i	− 0.0158282i
$471$	0	0
$472$	−310.855	−0.658592
$473$	− 257.838i	− 0.545112i
$474$	0	0
$475$	45.2339	0.0952293
$476$	1103.05i	2.31733i
$477$	0	0
$478$	449.134	0.939610
$479$	− 492.669i	− 1.02854i	−0.857630	−	0.514268i	$-0.828064\pi$
$479$	− 492.669i	− 1.02854i	0.857630	−	0.514268i	$-0.171936\pi$
$480$	0	0
$481$	551.053	1.14564
$482$	345.336i	0.716464i
$483$	0	0
$484$	−30.3096	−0.0626231
$485$	− 151.436i	− 0.312239i
$486$	0	0
$487$	−349.610	−0.717886	−0.358943	−	0.933360i	$-0.616863\pi$
$487$	−349.610	−0.717886	−0.358943	+	0.933360i	$0.616863\pi$
$488$	197.448i	0.404606i
$489$	0	0
$490$	280.075	0.571582
$491$	135.851i	0.276683i	0.990385	+	0.138341i	$0.0441771\pi$
$491$	135.851i	0.276683i	−0.990385	+	0.138341i	$0.955823\pi$
$492$	0	0
$493$	−1412.92	−2.86596
$494$	92.0966i	0.186430i
$495$	0	0
$496$	−72.1640	−0.145492
$497$	− 228.553i	− 0.459865i
$498$	0	0
$499$	−84.6656	−0.169671	−0.0848353	−	0.996395i	$-0.527036\pi$
$499$	−84.6656	−0.169671	−0.0848353	+	0.996395i	$0.527036\pi$
$500$	30.8065i	0.0616129i
$501$	0	0
$502$	−151.676	−0.302144
$503$	405.710i	0.806581i	0.915072	+	0.403290i	$0.132134\pi$
$503$	405.710i	0.806581i	−0.915072	+	0.403290i	$0.867866\pi$
$504$	0	0
$505$	−256.480	−0.507882
$506$	− 16.3649i	− 0.0323417i
$507$	0	0
$508$	381.708	0.751393
$509$	− 947.062i	− 1.86063i	−0.366758	−	0.930317i	$-0.619532\pi$
$509$	− 947.062i	− 1.86063i	0.366758	−	0.930317i	$-0.380468\pi$
$510$	0	0
$511$	1052.35	2.05940
$512$	165.222i	0.322700i
$513$	0	0
$514$	331.138	0.644238
$515$	− 177.664i	− 0.344978i
$516$	0	0
$517$	9.89074	0.0191310
$518$	855.559i	1.65166i
$519$	0	0
$520$	−153.775	−0.295722
$521$	− 129.920i	− 0.249367i	−0.992197	−	0.124684i	$-0.960208\pi$
$521$	− 129.920i	− 0.249367i	0.992197	−	0.124684i	$-0.0397916\pi$
$522$	0	0
$523$	477.650	0.913288	0.456644	−	0.889649i	$-0.349051\pi$
$523$	477.650	0.913288	0.456644	+	0.889649i	$0.349051\pi$
$524$	613.213i	1.17025i
$525$	0	0
$526$	−216.555	−0.411702
$527$	870.254i	1.65134i
$528$	0	0
$529$	509.438	0.963021
$530$	− 51.1481i	− 0.0965059i
$531$	0	0
$532$	316.565	0.595047
$533$	− 225.270i	− 0.422645i
$534$	0	0
$535$	113.853	0.212809
$536$	811.243i	1.51351i
$537$	0	0
$538$	271.772	0.505153
$539$	372.369i	0.690851i
$540$	0	0
$541$	−963.454	−1.78088	−0.890438	−	0.455104i	$-0.849602\pi$
$541$	−963.454	−1.78088	−0.890438	+	0.455104i	$0.849602\pi$
$542$	91.5655i	0.168940i
$543$	0	0
$544$	1042.20	1.91582
$545$	147.582i	0.270793i
$546$	0	0
$547$	433.672	0.792819	0.396410	−	0.918074i	$-0.370256\pi$
$547$	433.672	0.792819	0.396410	+	0.918074i	$0.370256\pi$
$548$	69.0263i	0.125960i
$549$	0	0
$550$	18.5003	0.0336369
$551$	405.494i	0.735924i
$552$	0	0
$553$	838.652	1.51655
$554$	− 418.731i	− 0.755833i
$555$	0	0
$556$	763.811	1.37376
$557$	− 728.503i	− 1.30791i	−0.756535	−	0.653953i	$-0.773110\pi$
$557$	− 728.503i	− 1.30791i	0.756535	−	0.653953i	$-0.226890\pi$
$558$	0	0
$559$	709.395	1.26904
$560$	74.2279i	0.132550i
$561$	0	0
$562$	547.703	0.974560
$563$	468.762i	0.832615i	0.909224	+	0.416308i	$0.136676\pi$
$563$	468.762i	0.832615i	−0.909224	+	0.416308i	$0.863324\pi$
$564$	0	0
$565$	258.559	0.457627
$566$	79.9539i	0.141261i
$567$	0	0
$568$	−135.634	−0.238793
$569$	− 50.4987i	− 0.0887499i	−0.999015	−	0.0443749i	$-0.985870\pi$
$569$	− 50.4987i	− 0.0887499i	0.999015	−	0.0443749i	$-0.0141296\pi$
$570$	0	0
$571$	−135.691	−0.237638	−0.118819	−	0.992916i	$-0.537911\pi$
$571$	−135.691	−0.237638	−0.118819	+	0.992916i	$0.537911\pi$
$572$	− 83.3912i	− 0.145789i
$573$	0	0
$574$	349.752	0.609323
$575$	− 22.1144i	− 0.0384598i
$576$	0	0
$577$	−620.094	−1.07469	−0.537343	−	0.843364i	$-0.680572\pi$
$577$	−620.094	−1.07469	−0.537343	+	0.843364i	$0.680572\pi$
$578$	− 786.162i	− 1.36014i
$579$	0	0
$580$	−276.161	−0.476140
$581$	897.699i	1.54509i
$582$	0	0
$583$	68.0031	0.116643
$584$	− 624.517i	− 1.06938i
$585$	0	0
$586$	261.444	0.446150
$587$	682.341i	1.16242i	0.813753	+	0.581210i	$0.197421\pi$
$587$	682.341i	1.16242i	−0.813753	+	0.581210i	$0.802579\pi$
$588$	0	0
$589$	249.755	0.424032
$590$	102.894i	0.174397i
$591$	0	0
$592$	−157.854	−0.266646
$593$	22.8537i	0.0385392i	0.999814	+	0.0192696i	$0.00613408\pi$
$593$	22.8537i	0.0385392i	−0.999814	+	0.0192696i	$0.993866\pi$
$594$	0	0
$595$	895.143	1.50444
$596$	510.849i	0.857130i
$597$	0	0
$598$	45.0250	0.0752927
$599$	− 140.644i	− 0.234799i	−0.993085	−	0.117399i	$-0.962544\pi$
$599$	− 140.644i	− 0.234799i	0.993085	−	0.117399i	$-0.0374557\pi$
$600$	0	0
$601$	−478.918	−0.796868	−0.398434	−	0.917197i	$-0.630446\pi$
$601$	−478.918	−0.796868	−0.398434	+	0.917197i	$0.630446\pi$
$602$	1101.40i	1.82957i
$603$	0	0
$604$	65.1456	0.107857
$605$	24.5967i	0.0406558i
$606$	0	0
$607$	136.422	0.224747	0.112374	−	0.993666i	$-0.464155\pi$
$607$	136.422	0.224747	0.112374	+	0.993666i	$0.464155\pi$
$608$	− 299.103i	− 0.491946i
$609$	0	0
$610$	65.3559	0.107141
$611$	27.2126i	0.0445378i
$612$	0	0
$613$	−507.524	−0.827935	−0.413967	−	0.910292i	$-0.635857\pi$
$613$	−507.524	−0.827935	−0.413967	+	0.910292i	$0.635857\pi$
$614$	250.368i	0.407765i
$615$	0	0
$616$	317.426	0.515301
$617$	− 586.984i	− 0.951352i	−0.879620	−	0.475676i	$-0.842203\pi$
$617$	− 586.984i	− 0.951352i	0.879620	−	0.475676i	$-0.157797\pi$
$618$	0	0
$619$	845.382	1.36572	0.682861	−	0.730548i	$-0.260735\pi$
$619$	845.382	1.36572	0.682861	+	0.730548i	$0.260735\pi$
$620$	170.095i	0.274347i
$621$	0	0
$622$	218.259	0.350899
$623$	153.560i	0.246485i
$624$	0	0
$625$	25.0000	0.0400000
$626$	297.713i	0.475580i
$627$	0	0
$628$	321.390	0.511768
$629$	1903.63i	3.02644i
$630$	0	0
$631$	−137.038	−0.217175	−0.108588	−	0.994087i	$-0.534633\pi$
$631$	−137.038	−0.217175	−0.108588	+	0.994087i	$0.534633\pi$
$632$	− 497.697i	− 0.787495i
$633$	0	0
$634$	−408.213	−0.643870
$635$	− 309.762i	− 0.487815i
$636$	0	0
$637$	−1024.50	−1.60833
$638$	165.844i	0.259943i
$639$	0	0
$640$	229.787	0.359041
$641$	− 766.448i	− 1.19571i	−0.801605	−	0.597854i	$-0.796020\pi$
$641$	− 766.448i	− 1.19571i	0.801605	−	0.597854i	$-0.203980\pi$
$642$	0	0
$643$	−365.800	−0.568896	−0.284448	−	0.958691i	$-0.591810\pi$
$643$	−365.800	−0.568896	−0.284448	+	0.958691i	$0.591810\pi$
$644$	− 154.765i	− 0.240318i
$645$	0	0
$646$	−318.151	−0.492493
$647$	− 868.788i	− 1.34279i	−0.741098	−	0.671397i	$-0.765694\pi$
$647$	− 868.788i	− 1.34279i	0.741098	−	0.671397i	$-0.234306\pi$
$648$	0	0
$649$	−136.801	−0.210788
$650$	50.9002i	0.0783080i
$651$	0	0
$652$	−513.751	−0.787961
$653$	− 711.802i	− 1.09005i	−0.838420	−	0.545024i	$-0.816520\pi$
$653$	− 711.802i	− 1.09005i	0.838420	−	0.545024i	$-0.183480\pi$
$654$	0	0
$655$	497.633	0.759746
$656$	64.5307i	0.0983700i
$657$	0	0
$658$	−42.2500	−0.0642097
$659$	482.328i	0.731909i	0.930633	+	0.365955i	$0.119257\pi$
$659$	482.328i	0.731909i	−0.930633	+	0.365955i	$0.880743\pi$
$660$	0	0
$661$	−904.710	−1.36870	−0.684350	−	0.729154i	$-0.739914\pi$
$661$	−904.710	−1.36870	−0.684350	+	0.729154i	$0.739914\pi$
$662$	− 174.359i	− 0.263383i
$663$	0	0
$664$	532.738	0.802316
$665$	− 256.898i	− 0.386313i
$666$	0	0
$667$	198.242	0.297214
$668$	− 191.413i	− 0.286547i
$669$	0	0
$670$	268.525	0.400783
$671$	86.8928i	0.129497i
$672$	0	0
$673$	23.6860	0.0351947	0.0175973	−	0.999845i	$-0.494398\pi$
$673$	23.6860	0.0351947	0.0175973	+	0.999845i	$0.494398\pi$
$674$	88.5308i	0.131351i
$675$	0	0
$676$	−236.229	−0.349452
$677$	− 422.665i	− 0.624321i	−0.950029	−	0.312160i	$-0.898947\pi$
$677$	− 422.665i	− 0.624321i	0.950029	−	0.312160i	$-0.101053\pi$
$678$	0	0
$679$	−860.052	−1.26665
$680$	− 531.221i	− 0.781208i
$681$	0	0
$682$	102.148	0.149777
$683$	− 709.073i	− 1.03817i	−0.854721	−	0.519087i	$-0.826272\pi$
$683$	− 709.073i	− 1.03817i	0.854721	−	0.519087i	$-0.173728\pi$
$684$	0	0
$685$	56.0161	0.0817753
$686$	− 896.427i	− 1.30675i
$687$	0	0
$688$	−203.213	−0.295368
$689$	187.098i	0.271550i
$690$	0	0
$691$	−69.2933	−0.100280	−0.0501399	−	0.998742i	$-0.515967\pi$
$691$	−69.2933	−0.100280	−0.0501399	+	0.998742i	$0.515967\pi$
$692$	64.7049i	0.0935041i
$693$	0	0
$694$	515.378	0.742620
$695$	− 619.846i	− 0.891865i
$696$	0	0
$697$	778.202	1.11650
$698$	− 524.610i	− 0.751590i
$699$	0	0
$700$	174.960	0.249943
$701$	− 1012.90i	− 1.44494i	−0.691401	−	0.722471i	$-0.743006\pi$
$701$	− 1012.90i	− 1.44494i	0.691401	−	0.722471i	$-0.256994\pi$
$702$	0	0
$703$	546.324	0.777133
$704$	− 87.6524i	− 0.124506i
$705$	0	0
$706$	−109.252	−0.154748
$707$	1456.63i	2.06030i
$708$	0	0
$709$	−452.862	−0.638733	−0.319366	−	0.947631i	$-0.603470\pi$
$709$	−452.862	−0.638733	−0.319366	+	0.947631i	$0.603470\pi$
$710$	44.8955i	0.0632330i
$711$	0	0
$712$	91.1299	0.127991
$713$	− 122.102i	− 0.171252i
$714$	0	0
$715$	−67.6735	−0.0946482
$716$	− 342.597i	− 0.478488i
$717$	0	0
$718$	−355.272	−0.494808
$719$	240.714i	0.334790i	0.985890	+	0.167395i	$0.0535355\pi$
$719$	240.714i	0.334790i	−0.985890	+	0.167395i	$0.946465\pi$
$720$	0	0
$721$	−1009.01	−1.39946
$722$	− 311.429i	− 0.431342i
$723$	0	0
$724$	75.9430	0.104894
$725$	224.110i	0.309117i
$726$	0	0
$727$	−500.438	−0.688361	−0.344180	−	0.938904i	$-0.611843\pi$
$727$	−500.438	−0.688361	−0.344180	+	0.938904i	$0.611843\pi$
$728$	873.339i	1.19964i
$729$	0	0
$730$	−206.717	−0.283174
$731$	2450.63i	3.35243i
$732$	0	0
$733$	13.6818	0.0186655	0.00933276	−	0.999956i	$-0.497029\pi$
$733$	13.6818	0.0186655	0.00933276	+	0.999956i	$0.497029\pi$
$734$	− 110.812i	− 0.150970i
$735$	0	0
$736$	−146.228	−0.198680
$737$	357.012i	0.484413i
$738$	0	0
$739$	−91.6056	−0.123959	−0.0619795	−	0.998077i	$-0.519741\pi$
$739$	−91.6056	−0.123959	−0.0619795	+	0.998077i	$0.519741\pi$
$740$	372.073i	0.502802i
$741$	0	0
$742$	−290.487	−0.391492
$743$	1104.81i	1.48696i	0.668761	+	0.743478i	$0.266825\pi$
$743$	1104.81i	1.48696i	−0.668761	+	0.743478i	$0.733175\pi$
$744$	0	0
$745$	414.563	0.556461
$746$	328.717i	0.440640i
$747$	0	0
$748$	288.078	0.385130
$749$	− 646.607i	− 0.863294i
$750$	0	0
$751$	−1317.52	−1.75435	−0.877175	−	0.480170i	$-0.840575\pi$
$751$	−1317.52	−1.75435	−0.877175	+	0.480170i	$0.840575\pi$
$752$	− 7.79531i	− 0.0103661i
$753$	0	0
$754$	−456.289	−0.605158
$755$	− 52.8668i	− 0.0700222i
$756$	0	0
$757$	548.802	0.724970	0.362485	−	0.931990i	$-0.381928\pi$
$757$	548.802	0.724970	0.362485	+	0.931990i	$0.381928\pi$
$758$	− 209.547i	− 0.276448i
$759$	0	0
$760$	−152.456	−0.200600
$761$	− 797.390i	− 1.04782i	−0.851774	−	0.523910i	$-0.824473\pi$
$761$	− 797.390i	− 1.04782i	0.851774	−	0.523910i	$-0.175527\pi$
$762$	0	0
$763$	838.168	1.09852
$764$	− 544.230i	− 0.712342i
$765$	0	0
$766$	524.018	0.684097
$767$	− 376.384i	− 0.490722i
$768$	0	0
$769$	−631.448	−0.821129	−0.410564	−	0.911832i	$-0.634668\pi$
$769$	−631.448	−0.821129	−0.410564	+	0.911832i	$0.634668\pi$
$770$	− 105.069i	− 0.136453i
$771$	0	0
$772$	−915.446	−1.18581
$773$	182.477i	0.236063i	0.993010	+	0.118032i	$0.0376584\pi$
$773$	182.477i	0.236063i	−0.993010	+	0.118032i	$0.962342\pi$
$774$	0	0
$775$	138.035	0.178110
$776$	510.397i	0.657728i
$777$	0	0
$778$	−120.781	−0.155245
$779$	− 223.337i	− 0.286697i
$780$	0	0
$781$	−59.6900	−0.0764276
$782$	155.540i	0.198901i
$783$	0	0
$784$	293.479	0.374336
$785$	− 260.814i	− 0.332247i
$786$	0	0
$787$	1523.24	1.93550	0.967750	−	0.251911i	$-0.0810590\pi$
$787$	1523.24	1.93550	0.967750	+	0.251911i	$0.0810590\pi$
$788$	− 775.059i	− 0.983578i
$789$	0	0
$790$	−164.740	−0.208531
$791$	− 1468.44i	− 1.85644i
$792$	0	0
$793$	−239.070	−0.301475
$794$	− 329.960i	− 0.415566i
$795$	0	0
$796$	38.2204	0.0480156
$797$	− 511.144i	− 0.641335i	−0.947192	−	0.320667i	$-0.896093\pi$
$797$	− 511.144i	− 0.641335i	0.947192	−	0.320667i	$-0.103907\pi$
$798$	0	0
$799$	−94.0067	−0.117655
$800$	− 165.309i	− 0.206636i
$801$	0	0
$802$	222.009	0.276819
$803$	− 274.837i	− 0.342263i
$804$	0	0
$805$	−125.595	−0.156018
$806$	281.041i	0.348686i
$807$	0	0
$808$	864.437	1.06985
$809$	321.467i	0.397363i	0.980064	+	0.198681i	$0.0636659\pi$
$809$	321.467i	0.397363i	−0.980064	+	0.198681i	$0.936334\pi$
$810$	0	0
$811$	1.44390	0.00178039	0.000890197	−	1.00000i	$-0.499717\pi$
$811$	1.44390	0.00178039	0.000890197		1.00000i	$0.499717\pi$
$812$	1568.41i	1.93154i
$813$	0	0
$814$	223.442	0.274499
$815$	416.918i	0.511555i
$816$	0	0
$817$	703.307	0.860841
$818$	− 201.606i	− 0.246462i
$819$	0	0
$820$	152.103	0.185492
$821$	1214.75i	1.47960i	0.672828	+	0.739799i	$0.265079\pi$
$821$	1214.75i	1.47960i	−0.672828	+	0.739799i	$0.734921\pi$
$822$	0	0
$823$	759.694	0.923079	0.461540	−	0.887120i	$-0.347297\pi$
$823$	759.694	0.923079	0.461540	+	0.887120i	$0.347297\pi$
$824$	598.794i	0.726692i
$825$	0	0
$826$	584.369	0.707469
$827$	1064.46i	1.28713i	0.765390	+	0.643567i	$0.222546\pi$
$827$	1064.46i	1.28713i	−0.765390	+	0.643567i	$0.777454\pi$
$828$	0	0
$829$	−382.338	−0.461204	−0.230602	−	0.973048i	$-0.574069\pi$
$829$	−382.338	−0.461204	−0.230602	+	0.973048i	$0.574069\pi$
$830$	− 176.338i	− 0.212456i
$831$	0	0
$832$	241.160	0.289855
$833$	− 3539.18i	− 4.24872i
$834$	0	0
$835$	−155.335	−0.186030
$836$	− 82.6756i	− 0.0988943i
$837$	0	0
$838$	467.887	0.558338
$839$	945.778i	1.12727i	0.826025	+	0.563634i	$0.190597\pi$
$839$	945.778i	1.12727i	−0.826025	+	0.563634i	$0.809403\pi$
$840$	0	0
$841$	−1168.00	−1.38883
$842$	696.707i	0.827443i
$843$	0	0
$844$	−716.500	−0.848934
$845$	191.704i	0.226869i
$846$	0	0
$847$	139.693	0.164927
$848$	− 53.5961i	− 0.0632029i
$849$	0	0
$850$	−175.836	−0.206866
$851$	− 267.092i	− 0.313857i
$852$	0	0
$853$	42.2044	0.0494776	0.0247388	−	0.999694i	$-0.492125\pi$
$853$	42.2044	0.0494776	0.0247388	+	0.999694i	$0.492125\pi$
$854$	− 371.177i	− 0.434634i
$855$	0	0
$856$	−383.728	−0.448280
$857$	− 1630.88i	− 1.90301i	−0.307626	−	0.951507i	$-0.599534\pi$
$857$	− 1630.88i	− 1.90301i	0.307626	−	0.951507i	$-0.400466\pi$
$858$	0	0
$859$	−903.228	−1.05149	−0.525744	−	0.850643i	$-0.676213\pi$
$859$	−903.228	−1.05149	−0.525744	+	0.850643i	$0.676213\pi$
$860$	478.986i	0.556960i
$861$	0	0
$862$	−218.593	−0.253588
$863$	270.091i	0.312967i	0.987681	+	0.156484i	$0.0500158\pi$
$863$	270.091i	0.312967i	−0.987681	+	0.156484i	$0.949984\pi$
$864$	0	0
$865$	52.5091	0.0607042
$866$	529.594i	0.611540i
$867$	0	0
$868$	966.025	1.11293
$869$	− 219.027i	− 0.252044i
$870$	0	0
$871$	−982.254	−1.12773
$872$	− 497.409i	− 0.570423i
$873$	0	0
$874$	44.6387	0.0510740
$875$	− 141.983i	− 0.162266i
$876$	0	0
$877$	−186.858	−0.213064	−0.106532	−	0.994309i	$-0.533975\pi$
$877$	−186.858	−0.213064	−0.106532	+	0.994309i	$0.533975\pi$
$878$	− 930.573i	− 1.05988i
$879$	0	0
$880$	19.3857	0.0220292
$881$	− 1485.66i	− 1.68633i	−0.537654	−	0.843166i	$-0.680689\pi$
$881$	− 1485.66i	− 1.68633i	0.537654	−	0.843166i	$-0.319311\pi$
$882$	0	0
$883$	−497.570	−0.563499	−0.281750	−	0.959488i	$-0.590915\pi$
$883$	−497.570	−0.563499	−0.281750	+	0.959488i	$0.590915\pi$
$884$	792.593i	0.896598i
$885$	0	0
$886$	−323.699	−0.365349
$887$	164.802i	0.185797i	0.995676	+	0.0928984i	$0.0296132\pi$
$887$	164.802i	0.185797i	−0.995676	+	0.0928984i	$0.970387\pi$
$888$	0	0
$889$	−1759.24	−1.97890
$890$	− 30.1644i	− 0.0338925i
$891$	0	0
$892$	−1079.46	−1.21016
$893$	26.9791i	0.0302117i
$894$	0	0
$895$	−278.024	−0.310641
$896$	− 1305.03i	− 1.45651i
$897$	0	0
$898$	−491.501	−0.547329
$899$	1237.40i	1.37642i
$900$	0	0
$901$	−646.336	−0.717355
$902$	− 91.3429i	− 0.101267i
$903$	0	0
$904$	−871.444	−0.963986
$905$	− 61.6291i	− 0.0680984i
$906$	0	0
$907$	−1532.40	−1.68953	−0.844766	−	0.535137i	$-0.820260\pi$
$907$	−1532.40	−1.68953	−0.844766	+	0.535137i	$0.820260\pi$
$908$	582.839i	0.641893i
$909$	0	0
$910$	289.078	0.317669
$911$	1440.39i	1.58111i	0.612394	+	0.790553i	$0.290207\pi$
$911$	1440.39i	1.58111i	−0.612394	+	0.790553i	$0.709793\pi$
$912$	0	0
$913$	234.447	0.256788
$914$	730.738i	0.799494i
$915$	0	0
$916$	−138.712	−0.151432
$917$	− 2826.22i	− 3.08203i
$918$	0	0
$919$	448.482	0.488010	0.244005	−	0.969774i	$-0.421539\pi$
$919$	448.482	0.488010	0.244005	+	0.969774i	$0.421539\pi$
$920$	74.5339i	0.0810151i
$921$	0	0
$922$	−229.691	−0.249122
$923$	− 164.226i	− 0.177926i
$924$	0	0
$925$	301.944	0.326426
$926$	217.736i	0.235136i
$927$	0	0
$928$	1481.89	1.59687
$929$	320.054i	0.344515i	0.985052	+	0.172257i	$0.0551060\pi$
$929$	320.054i	0.344515i	−0.985052	+	0.172257i	$0.944894\pi$
$930$	0	0
$931$	−1015.71	−1.09099
$932$	1023.50i	1.09818i
$933$	0	0
$934$	607.372	0.650292
$935$	− 233.780i	− 0.250032i
$936$	0	0
$937$	−2.36541	−0.00252445	−0.00126223	−	0.999999i	$-0.500402\pi$
$937$	−2.36541	−0.00252445	−0.00126223	+	0.999999i	$0.500402\pi$
$938$	− 1525.04i	− 1.62584i
$939$	0	0
$940$	−18.3740	−0.0195468
$941$	− 49.6427i	− 0.0527553i	−0.999652	−	0.0263777i	$-0.991603\pi$
$941$	− 49.6427i	− 0.0527553i	0.999652	−	0.0263777i	$-0.00839724\pi$
$942$	0	0
$943$	−109.187	−0.115787
$944$	107.819i	0.114215i
$945$	0	0
$946$	287.647	0.304066
$947$	− 1088.55i	− 1.14948i	−0.818338	−	0.574738i	$-0.805104\pi$
$947$	− 1088.55i	− 1.14948i	0.818338	−	0.574738i	$-0.194896\pi$
$948$	0	0
$949$	756.165	0.796802
$950$	50.4634i	0.0531194i
$951$	0	0
$952$	−3016.98	−3.16909
$953$	− 92.5920i	− 0.0971585i	−0.998819	−	0.0485792i	$-0.984531\pi$
$953$	− 92.5920i	− 0.0971585i	0.998819	−	0.0485792i	$-0.0154693\pi$
$954$	0	0
$955$	−441.652	−0.462463
$956$	− 1109.30i	− 1.16036i
$957$	0	0
$958$	549.626	0.573722
$959$	− 318.133i	− 0.331734i
$960$	0	0
$961$	−198.851	−0.206921
$962$	614.760i	0.639044i
$963$	0	0
$964$	852.935	0.884787
$965$	742.901i	0.769845i
$966$	0	0
$967$	−168.695	−0.174452	−0.0872258	−	0.996189i	$-0.527800\pi$
$967$	−168.695	−0.174452	−0.0872258	+	0.996189i	$0.527800\pi$
$968$	− 82.9005i	− 0.0856410i
$969$	0	0
$970$	168.943	0.174168
$971$	1324.00i	1.36354i	0.731566	+	0.681770i	$0.238790\pi$
$971$	1324.00i	1.36354i	−0.731566	+	0.681770i	$0.761210\pi$
$972$	0	0
$973$	−3520.31	−3.61799
$974$	− 390.029i	− 0.400440i
$975$	0	0
$976$	68.4838	0.0701679
$977$	39.7715i	0.0407078i	0.999793	+	0.0203539i	$0.00647930\pi$
$977$	39.7715i	0.0407078i	−0.999793	+	0.0203539i	$0.993521\pi$
$978$	0	0
$979$	40.1045	0.0409648
$980$	− 691.750i	− 0.705867i
$981$	0	0
$982$	−151.557	−0.154335
$983$	− 1193.15i	− 1.21379i	−0.794783	−	0.606893i	$-0.792415\pi$
$983$	− 1193.15i	− 1.21379i	0.794783	−	0.606893i	$-0.207585\pi$
$984$	0	0
$985$	−628.974	−0.638553
$986$	− 1576.26i	− 1.59864i
$987$	0	0
$988$	227.467	0.230230
$989$	− 343.839i	− 0.347663i
$990$	0	0
$991$	995.220	1.00426	0.502129	−	0.864793i	$-0.332550\pi$
$991$	995.220	1.00426	0.502129	+	0.864793i	$0.332550\pi$
$992$	− 912.739i	− 0.920099i
$993$	0	0
$994$	254.976	0.256515
$995$	− 31.0166i	− 0.0311724i
$996$	0	0
$997$	868.526	0.871139	0.435570	−	0.900155i	$-0.356547\pi$
$997$	868.526	0.871139	0.435570	+	0.900155i	$0.356547\pi$
$998$	− 94.4538i	− 0.0946430i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.e.a.386.15	yes	24
3.2	odd	2	inner	495.3.e.a.386.10	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.3.e.a.386.10	✓	24	3.2	odd	2	inner
495.3.e.a.386.15	yes	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	495.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.11561i q^{2} +2.75541 q^{4} -2.23607i q^{5} -12.6993 q^{7} +7.53641i q^{8} +2.49458 q^{10} +3.31662i q^{11} -9.12509 q^{13} -14.1675i q^{14} +2.61397 q^{16} -31.5229i q^{17} -9.04678 q^{19} -6.16129i q^{20} -3.70006 q^{22} +4.42287i q^{23} -5.00000 q^{25} -10.1800i q^{26} -34.9920 q^{28} -44.8219i q^{29} -27.6070 q^{31} +33.0618i q^{32} +35.1673 q^{34} +28.3966i q^{35} -60.3888 q^{37} -10.0927i q^{38} +16.8519 q^{40} +24.6869i q^{41} -77.7411 q^{43} +9.13868i q^{44} -4.93420 q^{46} -2.98217i q^{47} +112.273 q^{49} -5.57805i q^{50} -25.1434 q^{52} -20.5037i q^{53} +7.41620 q^{55} -95.7075i q^{56} +50.0038 q^{58} +41.2471i q^{59} +26.1992 q^{61} -30.7987i q^{62} -26.4282 q^{64} +20.4043i q^{65} +107.643 q^{67} -86.8586i q^{68} -31.6795 q^{70} +17.9972i q^{71} -82.8666 q^{73} -67.3703i q^{74} -24.9276 q^{76} -42.1190i q^{77} -66.0390 q^{79} -5.84501i q^{80} -27.5409 q^{82} -70.6886i q^{83} -70.4873 q^{85} -86.7288i q^{86} -24.9954 q^{88} -12.0920i q^{89} +115.883 q^{91} +12.1869i q^{92} +3.32694 q^{94} +20.2292i q^{95} +67.7241 q^{97} +125.253i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 40 q^{4} - 32 q^{7} + 40 q^{10} + 48 q^{13} - 24 q^{16} - 120 q^{25} + 128 q^{28} + 96 q^{31} + 112 q^{34} - 320 q^{37} - 120 q^{40} + 208 q^{43} - 288 q^{46} + 296 q^{49} - 192 q^{52} + 96 q^{58}+ \cdots + 576 q^{97}+O(q^{100}) \) 24 * q - 40 * q^4 - 32 * q^7 + 40 * q^10 + 48 * q^13 - 24 * q^16 - 120 * q^25 + 128 * q^28 + 96 * q^31 + 112 * q^34 - 320 * q^37 - 120 * q^40 + 208 * q^43 - 288 * q^46 + 296 * q^49 - 192 * q^52 + 96 * q^58 + 400 * q^61 + 440 * q^64 - 288 * q^67 - 480 * q^73 + 32 * q^76 - 544 * q^79 + 96 * q^82 - 320 * q^85 + 384 * q^91 + 160 * q^94 + 576 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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