LMFDB - Embedded newform 1216.3.d.f.191.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,3,Mod(191,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, 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("1216.191"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 124 x^{18} + 6294 x^{16} + 169580 x^{14} + 2633777 x^{12} + 23965840 x^{10} + 123396288 x^{8} + \cdots + 6553600 $ x^20 + 124x^18 + 6294x^16 + 169580x^14 + 2633777x^12 + 23965840x^10 + 123396288x^8 + 323597312x^6 + 340403200x^4 + 121339904*x^2 + 6553600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 608)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.8
Root			$1.05616i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.f.191.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05616i q^{3} +9.08870 q^{5} +10.7326i q^{7} +7.88453 q^{9} -10.6397i q^{11} +1.41113 q^{13} -9.59909i q^{15} -5.11408 q^{17} +4.35890i q^{19} +11.3353 q^{21} -28.0907i q^{23} +57.6044 q^{25} -17.8327i q^{27} +47.6305 q^{29} -16.9716i q^{31} -11.2372 q^{33} +97.5452i q^{35} +37.9673 q^{37} -1.49037i q^{39} -65.9127 q^{41} +24.4023i q^{43} +71.6601 q^{45} +15.3535i q^{47} -66.1883 q^{49} +5.40127i q^{51} -41.7427 q^{53} -96.7007i q^{55} +4.60368 q^{57} +59.7473i q^{59} -16.6603 q^{61} +84.6214i q^{63} +12.8253 q^{65} -21.1960i q^{67} -29.6681 q^{69} +45.3051i q^{71} +17.0782 q^{73} -60.8393i q^{75} +114.191 q^{77} +59.9085i q^{79} +52.1266 q^{81} +157.492i q^{83} -46.4803 q^{85} -50.3053i q^{87} +119.385 q^{89} +15.1450i q^{91} -17.9247 q^{93} +39.6167i q^{95} +10.8878 q^{97} -83.8888i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 68 q^{9} + 16 q^{13} + 8 q^{17} - 64 q^{21} + 196 q^{25} + 88 q^{29} - 184 q^{33} - 16 q^{37} - 16 q^{41} - 16 q^{45} + 52 q^{49} - 88 q^{53} + 208 q^{61} - 192 q^{65} - 248 q^{69} - 152 q^{73} - 312 q^{77}+ \cdots - 72 q^{97}+O(q^{100}) $ 20 * q - 68 * q^9 + 16 * q^13 + 8 * q^17 - 64 * q^21 + 196 * q^25 + 88 * q^29 - 184 * q^33 - 16 * q^37 - 16 * q^41 - 16 * q^45 + 52 * q^49 - 88 * q^53 + 208 * q^61 - 192 * q^65 - 248 * q^69 - 152 * q^73 - 312 * q^77 + 500 * q^81 + 488 * q^85 + 304 * q^89 + 240 * q^93 - 72 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 1.05616i	− 0.352052i	−0.984385	−	0.176026i	$-0.943676\pi$
$3$	− 1.05616i	− 0.352052i	0.984385	−	0.176026i	$-0.0563243\pi$
$4$	0	0
$5$	9.08870	1.81774	0.908870	−	0.417081i	$-0.136947\pi$
$5$	9.08870	1.81774	0.908870	+	0.417081i	$0.136947\pi$
$6$	0	0
$7$	10.7326i	1.53323i	0.642110	+	0.766613i	$0.278059\pi$
$7$	10.7326i	1.53323i	−0.642110	+	0.766613i	$0.721941\pi$
$8$	0	0
$9$	7.88453	0.876059
$10$	0	0
$11$	− 10.6397i	− 0.967243i	−0.875277	−	0.483622i	$-0.839321\pi$
$11$	− 10.6397i	− 0.967243i	0.875277	−	0.483622i	$-0.160679\pi$
$12$	0	0
$13$	1.41113	0.108548	0.0542741	−	0.998526i	$-0.482716\pi$
$13$	1.41113	0.108548	0.0542741	+	0.998526i	$0.482716\pi$
$14$	0	0
$15$	− 9.59909i	− 0.639939i
$16$	0	0
$17$	−5.11408	−0.300828	−0.150414	−	0.988623i	$-0.548061\pi$
$17$	−5.11408	−0.300828	−0.150414	+	0.988623i	$0.548061\pi$
$18$	0	0
$19$	4.35890i	0.229416i
$19$	4.35890i	0.229416i
$20$	0	0
$21$	11.3353	0.539776
$22$	0	0
$23$	− 28.0907i	− 1.22133i	−0.791888	−	0.610666i	$-0.790902\pi$
$23$	− 28.0907i	− 1.22133i	0.791888	−	0.610666i	$-0.209098\pi$
$24$	0	0
$25$	57.6044	2.30418
$26$	0	0
$27$	− 17.8327i	− 0.660471i
$28$	0	0
$29$	47.6305	1.64243	0.821215	−	0.570618i	$-0.193296\pi$
$29$	47.6305	1.64243	0.821215	+	0.570618i	$0.193296\pi$
$30$	0	0
$31$	− 16.9716i	− 0.547472i	−0.961805	−	0.273736i	$-0.911741\pi$
$31$	− 16.9716i	− 0.547472i	0.961805	−	0.273736i	$-0.0882594\pi$
$32$	0	0
$33$	−11.2372	−0.340520
$34$	0	0
$35$	97.5452i	2.78700i
$36$	0	0
$37$	37.9673	1.02614	0.513072	−	0.858345i	$-0.328507\pi$
$37$	37.9673	1.02614	0.513072	+	0.858345i	$0.328507\pi$
$38$	0	0
$39$	− 1.49037i	− 0.0382147i
$40$	0	0
$41$	−65.9127	−1.60763	−0.803813	−	0.594882i	$-0.797199\pi$
$41$	−65.9127	−1.60763	−0.803813	+	0.594882i	$0.797199\pi$
$42$	0	0
$43$	24.4023i	0.567496i	0.958899	+	0.283748i	$0.0915779\pi$
$43$	24.4023i	0.567496i	−0.958899	+	0.283748i	$0.908422\pi$
$44$	0	0
$45$	71.6601	1.59245
$46$	0	0
$47$	15.3535i	0.326670i	0.986571	+	0.163335i	$0.0522251\pi$
$47$	15.3535i	0.326670i	−0.986571	+	0.163335i	$0.947775\pi$
$48$	0	0
$49$	−66.1883	−1.35078
$50$	0	0
$51$	5.40127i	0.105907i
$52$	0	0
$53$	−41.7427	−0.787599	−0.393799	−	0.919196i	$-0.628840\pi$
$53$	−41.7427	−0.787599	−0.393799	+	0.919196i	$0.628840\pi$
$54$	0	0
$55$	− 96.7007i	− 1.75820i
$56$	0	0
$57$	4.60368	0.0807664
$58$	0	0
$59$	59.7473i	1.01267i	0.862338	+	0.506333i	$0.168999\pi$
$59$	59.7473i	1.01267i	−0.862338	+	0.506333i	$0.831001\pi$
$60$	0	0
$61$	−16.6603	−0.273120	−0.136560	−	0.990632i	$-0.543605\pi$
$61$	−16.6603	−0.273120	−0.136560	+	0.990632i	$0.543605\pi$
$62$	0	0
$63$	84.6214i	1.34320i
$64$	0	0
$65$	12.8253	0.197312
$66$	0	0
$67$	− 21.1960i	− 0.316359i	−0.987410	−	0.158179i	$-0.949438\pi$
$67$	− 21.1960i	− 0.316359i	0.987410	−	0.158179i	$-0.0505624\pi$
$68$	0	0
$69$	−29.6681	−0.429973
$70$	0	0
$71$	45.3051i	0.638101i	0.947738	+	0.319050i	$0.103364\pi$
$71$	45.3051i	0.638101i	−0.947738	+	0.319050i	$0.896636\pi$
$72$	0	0
$73$	17.0782	0.233948	0.116974	−	0.993135i	$-0.462681\pi$
$73$	17.0782	0.233948	0.116974	+	0.993135i	$0.462681\pi$
$74$	0	0
$75$	− 60.8393i	− 0.811190i
$76$	0	0
$77$	114.191	1.48300
$78$	0	0
$79$	59.9085i	0.758336i	0.925328	+	0.379168i	$0.123790\pi$
$79$	59.9085i	0.758336i	−0.925328	+	0.379168i	$0.876210\pi$
$80$	0	0
$81$	52.1266	0.643539
$82$	0	0
$83$	157.492i	1.89750i	0.316032	+	0.948748i	$0.397649\pi$
$83$	157.492i	1.89750i	−0.316032	+	0.948748i	$0.602351\pi$
$84$	0	0
$85$	−46.4803	−0.546827
$86$	0	0
$87$	− 50.3053i	− 0.578222i
$88$	0	0
$89$	119.385	1.34140	0.670700	−	0.741729i	$-0.265994\pi$
$89$	119.385	1.34140	0.670700	+	0.741729i	$0.265994\pi$
$90$	0	0
$91$	15.1450i	0.166429i
$92$	0	0
$93$	−17.9247	−0.192739
$94$	0	0
$95$	39.6167i	0.417018i
$96$	0	0
$97$	10.8878	0.112246	0.0561228	−	0.998424i	$-0.482126\pi$
$97$	10.8878	0.112246	0.0561228	+	0.998424i	$0.482126\pi$
$98$	0	0
$99$	− 83.8888i	− 0.847362i
$100$	0	0
$101$	119.173	1.17993	0.589967	−	0.807428i	$-0.299141\pi$
$101$	119.173	1.17993	0.589967	+	0.807428i	$0.299141\pi$
$102$	0	0
$103$	− 160.655i	− 1.55976i	−0.625929	−	0.779880i	$-0.715280\pi$
$103$	− 160.655i	− 1.55976i	0.625929	−	0.779880i	$-0.284720\pi$
$104$	0	0
$105$	103.023	0.981172
$106$	0	0
$107$	− 121.599i	− 1.13644i	−0.822877	−	0.568220i	$-0.807632\pi$
$107$	− 121.599i	− 1.13644i	0.822877	−	0.568220i	$-0.192368\pi$
$108$	0	0
$109$	−146.538	−1.34438	−0.672192	−	0.740377i	$-0.734647\pi$
$109$	−146.538	−1.34438	−0.672192	+	0.740377i	$0.734647\pi$
$110$	0	0
$111$	− 40.0995i	− 0.361257i
$112$	0	0
$113$	−121.020	−1.07098	−0.535489	−	0.844543i	$-0.679872\pi$
$113$	−121.020	−1.07098	−0.535489	+	0.844543i	$0.679872\pi$
$114$	0	0
$115$	− 255.307i	− 2.22006i
$116$	0	0
$117$	11.1261	0.0950946
$118$	0	0
$119$	− 54.8873i	− 0.461237i
$120$	0	0
$121$	7.79736	0.0644410
$122$	0	0
$123$	69.6142i	0.565969i
$124$	0	0
$125$	296.331	2.37065
$126$	0	0
$127$	− 101.781i	− 0.801427i	−0.916203	−	0.400713i	$-0.868762\pi$
$127$	− 101.781i	− 0.801427i	0.916203	−	0.400713i	$-0.131238\pi$
$128$	0	0
$129$	25.7727	0.199788
$130$	0	0
$131$	211.601i	1.61528i	0.589679	+	0.807638i	$0.299254\pi$
$131$	211.601i	1.61528i	−0.589679	+	0.807638i	$0.700746\pi$
$132$	0	0
$133$	−46.7822	−0.351746
$134$	0	0
$135$	− 162.076i	− 1.20056i
$136$	0	0
$137$	−246.389	−1.79846	−0.899231	−	0.437474i	$-0.855873\pi$
$137$	−246.389	−1.79846	−0.899231	+	0.437474i	$0.855873\pi$
$138$	0	0
$139$	− 106.610i	− 0.766982i	−0.923545	−	0.383491i	$-0.874722\pi$
$139$	− 106.610i	− 0.766982i	0.923545	−	0.383491i	$-0.125278\pi$
$140$	0	0
$141$	16.2157	0.115005
$142$	0	0
$143$	− 15.0139i	− 0.104992i
$144$	0	0
$145$	432.899	2.98551
$146$	0	0
$147$	69.9052i	0.475546i
$148$	0	0
$149$	247.376	1.66024	0.830120	−	0.557585i	$-0.188272\pi$
$149$	247.376	1.66024	0.830120	+	0.557585i	$0.188272\pi$
$150$	0	0
$151$	212.829i	1.40947i	0.709472	+	0.704733i	$0.248933\pi$
$151$	212.829i	1.40947i	−0.709472	+	0.704733i	$0.751067\pi$
$152$	0	0
$153$	−40.3221	−0.263543
$154$	0	0
$155$	− 154.250i	− 0.995161i
$156$	0	0
$157$	64.5993	0.411461	0.205730	−	0.978609i	$-0.434043\pi$
$157$	64.5993	0.411461	0.205730	+	0.978609i	$0.434043\pi$
$158$	0	0
$159$	44.0869i	0.277276i
$160$	0	0
$161$	301.485	1.87258
$162$	0	0
$163$	− 63.9954i	− 0.392610i	−0.980543	−	0.196305i	$-0.937106\pi$
$163$	− 63.9954i	− 0.392610i	0.980543	−	0.196305i	$-0.0628942\pi$
$164$	0	0
$165$	−102.131	−0.618977
$166$	0	0
$167$	21.2326i	0.127142i	0.997977	+	0.0635708i	$0.0202489\pi$
$167$	21.2326i	0.127142i	−0.997977	+	0.0635708i	$0.979751\pi$
$168$	0	0
$169$	−167.009	−0.988217
$170$	0	0
$171$	34.3679i	0.200982i
$172$	0	0
$173$	20.9025	0.120824	0.0604119	−	0.998174i	$-0.480759\pi$
$173$	20.9025	0.120824	0.0604119	+	0.998174i	$0.480759\pi$
$174$	0	0
$175$	618.244i	3.53282i
$176$	0	0
$177$	63.1026	0.356512
$178$	0	0
$179$	− 284.510i	− 1.58944i	−0.606976	−	0.794720i	$-0.707618\pi$
$179$	− 284.510i	− 1.58944i	0.606976	−	0.794720i	$-0.292382\pi$
$180$	0	0
$181$	201.589	1.11375	0.556876	−	0.830596i	$-0.312000\pi$
$181$	201.589	1.11375	0.556876	+	0.830596i	$0.312000\pi$
$182$	0	0
$183$	17.5959i	0.0961524i
$184$	0	0
$185$	345.074	1.86526
$186$	0	0
$187$	54.4121i	0.290974i
$188$	0	0
$189$	191.391	1.01265
$190$	0	0
$191$	71.0568i	0.372025i	0.982547	+	0.186013i	$0.0595565\pi$
$191$	71.0568i	0.372025i	−0.982547	+	0.186013i	$0.940443\pi$
$192$	0	0
$193$	−50.5535	−0.261935	−0.130968	−	0.991387i	$-0.541808\pi$
$193$	−50.5535	−0.261935	−0.130968	+	0.991387i	$0.541808\pi$
$194$	0	0
$195$	− 13.5455i	− 0.0694643i
$196$	0	0
$197$	−291.498	−1.47969	−0.739843	−	0.672780i	$-0.765100\pi$
$197$	−291.498	−1.47969	−0.739843	+	0.672780i	$0.765100\pi$
$198$	0	0
$199$	− 169.601i	− 0.852266i	−0.904660	−	0.426133i	$-0.859876\pi$
$199$	− 169.601i	− 0.852266i	0.904660	−	0.426133i	$-0.140124\pi$
$200$	0	0
$201$	−22.3864	−0.111375
$202$	0	0
$203$	511.198i	2.51822i
$204$	0	0
$205$	−599.060	−2.92225
$206$	0	0
$207$	− 221.482i	− 1.06996i
$208$	0	0
$209$	46.3773	0.221901
$210$	0	0
$211$	− 292.244i	− 1.38504i	−0.721399	−	0.692520i	$-0.756500\pi$
$211$	− 292.244i	− 1.38504i	0.721399	−	0.692520i	$-0.243500\pi$
$212$	0	0
$213$	47.8494	0.224645
$214$	0	0
$215$	221.785i	1.03156i
$216$	0	0
$217$	182.149	0.839398
$218$	0	0
$219$	− 18.0373i	− 0.0823620i
$220$	0	0
$221$	−7.21661	−0.0326544
$222$	0	0
$223$	168.114i	0.753873i	0.926239	+	0.376937i	$0.123023\pi$
$223$	168.114i	0.753873i	−0.926239	+	0.376937i	$0.876977\pi$
$224$	0	0
$225$	454.184	2.01859
$226$	0	0
$227$	− 132.903i	− 0.585476i	−0.956193	−	0.292738i	$-0.905434\pi$
$227$	− 132.903i	− 0.585476i	0.956193	−	0.292738i	$-0.0945664\pi$
$228$	0	0
$229$	−65.7878	−0.287283	−0.143642	−	0.989630i	$-0.545881\pi$
$229$	−65.7878	−0.287283	−0.143642	+	0.989630i	$0.545881\pi$
$230$	0	0
$231$	− 120.604i	− 0.522094i
$232$	0	0
$233$	192.686	0.826978	0.413489	−	0.910509i	$-0.364310\pi$
$233$	192.686	0.826978	0.413489	+	0.910509i	$0.364310\pi$
$234$	0	0
$235$	139.543i	0.593801i
$236$	0	0
$237$	63.2728	0.266974
$238$	0	0
$239$	457.827i	1.91559i	0.287445	+	0.957797i	$0.407194\pi$
$239$	457.827i	1.91559i	−0.287445	+	0.957797i	$0.592806\pi$
$240$	0	0
$241$	−231.163	−0.959181	−0.479590	−	0.877493i	$-0.659215\pi$
$241$	−231.163	−0.959181	−0.479590	+	0.877493i	$0.659215\pi$
$242$	0	0
$243$	− 215.548i	− 0.887030i
$244$	0	0
$245$	−601.565	−2.45537
$246$	0	0
$247$	6.15096i	0.0249027i
$248$	0	0
$249$	166.337	0.668018
$250$	0	0
$251$	336.349i	1.34003i	0.742346	+	0.670017i	$0.233713\pi$
$251$	336.349i	1.34003i	−0.742346	+	0.670017i	$0.766287\pi$
$252$	0	0
$253$	−298.875	−1.18133
$254$	0	0
$255$	49.0905i	0.192512i
$256$	0	0
$257$	−431.835	−1.68029	−0.840146	−	0.542361i	$-0.817531\pi$
$257$	−431.835	−1.68029	−0.840146	+	0.542361i	$0.817531\pi$
$258$	0	0
$259$	407.488i	1.57331i
$260$	0	0
$261$	375.544	1.43887
$262$	0	0
$263$	− 403.178i	− 1.53299i	−0.642248	−	0.766497i	$-0.721998\pi$
$263$	− 403.178i	− 1.53299i	0.642248	−	0.766497i	$-0.278002\pi$
$264$	0	0
$265$	−379.387	−1.43165
$266$	0	0
$267$	− 126.089i	− 0.472243i
$268$	0	0
$269$	−60.9622	−0.226625	−0.113313	−	0.993559i	$-0.536146\pi$
$269$	−60.9622	−0.226625	−0.113313	+	0.993559i	$0.536146\pi$
$270$	0	0
$271$	102.993i	0.380048i	0.981779	+	0.190024i	$0.0608566\pi$
$271$	102.993i	0.380048i	−0.981779	+	0.190024i	$0.939143\pi$
$272$	0	0
$273$	15.9955	0.0585917
$274$	0	0
$275$	− 612.892i	− 2.22870i
$276$	0	0
$277$	258.876	0.934571	0.467286	−	0.884106i	$-0.345232\pi$
$277$	258.876	0.934571	0.467286	+	0.884106i	$0.345232\pi$
$278$	0	0
$279$	− 133.813i	− 0.479618i
$280$	0	0
$281$	−210.378	−0.748677	−0.374339	−	0.927292i	$-0.622130\pi$
$281$	−210.378	−0.748677	−0.374339	+	0.927292i	$0.622130\pi$
$282$	0	0
$283$	− 112.401i	− 0.397178i	−0.980083	−	0.198589i	$-0.936364\pi$
$283$	− 112.401i	− 0.397178i	0.980083	−	0.198589i	$-0.0636358\pi$
$284$	0	0
$285$	41.8415	0.146812
$286$	0	0
$287$	− 707.413i	− 2.46485i
$288$	0	0
$289$	−262.846	−0.909502
$290$	0	0
$291$	− 11.4993i	− 0.0395164i
$292$	0	0
$293$	103.310	0.352596	0.176298	−	0.984337i	$-0.443588\pi$
$293$	103.310	0.352596	0.176298	+	0.984337i	$0.443588\pi$
$294$	0	0
$295$	543.025i	1.84076i
$296$	0	0
$297$	−189.734	−0.638836
$298$	0	0
$299$	− 39.6395i	− 0.132573i
$300$	0	0
$301$	−261.900	−0.870099
$302$	0	0
$303$	− 125.866i	− 0.415398i
$304$	0	0
$305$	−151.420	−0.496460
$306$	0	0
$307$	− 570.333i	− 1.85776i	−0.370378	−	0.928881i	$-0.620772\pi$
$307$	− 570.333i	− 1.85776i	0.370378	−	0.928881i	$-0.379228\pi$
$308$	0	0
$309$	−169.677	−0.549117
$310$	0	0
$311$	550.030i	1.76859i	0.466933	+	0.884293i	$0.345359\pi$
$311$	550.030i	1.76859i	−0.466933	+	0.884293i	$0.654641\pi$
$312$	0	0
$313$	−376.747	−1.20367	−0.601833	−	0.798622i	$-0.705563\pi$
$313$	−376.747	−1.20367	−0.601833	+	0.798622i	$0.705563\pi$
$314$	0	0
$315$	769.098i	2.44158i
$316$	0	0
$317$	−505.206	−1.59371	−0.796855	−	0.604170i	$-0.793505\pi$
$317$	−505.206	−1.59371	−0.796855	+	0.604170i	$0.793505\pi$
$318$	0	0
$319$	− 506.773i	− 1.58863i
$320$	0	0
$321$	−128.428	−0.400086
$322$	0	0
$323$	− 22.2918i	− 0.0690147i
$324$	0	0
$325$	81.2871	0.250114
$326$	0	0
$327$	154.767i	0.473294i
$328$	0	0
$329$	−164.783	−0.500859
$330$	0	0
$331$	− 188.498i	− 0.569480i	−0.958605	−	0.284740i	$-0.908093\pi$
$331$	− 188.498i	− 0.569480i	0.958605	−	0.284740i	$-0.0919073\pi$
$332$	0	0
$333$	299.355	0.898963
$334$	0	0
$335$	− 192.644i	− 0.575058i
$336$	0	0
$337$	−400.785	−1.18927	−0.594636	−	0.803995i	$-0.702704\pi$
$337$	−400.785	−1.18927	−0.594636	+	0.803995i	$0.702704\pi$
$338$	0	0
$339$	127.817i	0.377040i
$340$	0	0
$341$	−180.573	−0.529538
$342$	0	0
$343$	− 184.475i	− 0.537827i
$344$	0	0
$345$	−269.645	−0.781579
$346$	0	0
$347$	526.018i	1.51590i	0.652312	+	0.757951i	$0.273799\pi$
$347$	526.018i	1.51590i	−0.652312	+	0.757951i	$0.726201\pi$
$348$	0	0
$349$	70.6106	0.202323	0.101161	−	0.994870i	$-0.467744\pi$
$349$	70.6106	0.202323	0.101161	+	0.994870i	$0.467744\pi$
$350$	0	0
$351$	− 25.1642i	− 0.0716930i
$352$	0	0
$353$	−147.236	−0.417100	−0.208550	−	0.978012i	$-0.566874\pi$
$353$	−147.236	−0.417100	−0.208550	+	0.978012i	$0.566874\pi$
$354$	0	0
$355$	411.765i	1.15990i
$356$	0	0
$357$	−57.9696	−0.162380
$358$	0	0
$359$	− 301.319i	− 0.839329i	−0.907679	−	0.419665i	$-0.862148\pi$
$359$	− 301.319i	− 0.839329i	0.907679	−	0.419665i	$-0.137852\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	− 8.23524i	− 0.0226866i
$364$	0	0
$365$	155.219	0.425256
$366$	0	0
$367$	− 389.154i	− 1.06037i	−0.847883	−	0.530183i	$-0.822123\pi$
$367$	− 389.154i	− 1.06037i	0.847883	−	0.530183i	$-0.177877\pi$
$368$	0	0
$369$	−519.691	−1.40838
$370$	0	0
$371$	− 448.007i	− 1.20757i
$372$	0	0
$373$	−172.585	−0.462694	−0.231347	−	0.972871i	$-0.574313\pi$
$373$	−172.585	−0.462694	−0.231347	+	0.972871i	$0.574313\pi$
$374$	0	0
$375$	− 312.972i	− 0.834593i
$376$	0	0
$377$	67.2127	0.178283
$378$	0	0
$379$	− 301.787i	− 0.796271i	−0.917327	−	0.398136i	$-0.869657\pi$
$379$	− 301.787i	− 0.796271i	0.917327	−	0.398136i	$-0.130343\pi$
$380$	0	0
$381$	−107.497	−0.282144
$382$	0	0
$383$	11.8009i	0.0308118i	0.999881	+	0.0154059i	$0.00490405\pi$
$383$	11.8009i	0.0308118i	−0.999881	+	0.0154059i	$0.995096\pi$
$384$	0	0
$385$	1037.85	2.69571
$386$	0	0
$387$	192.401i	0.497160i
$388$	0	0
$389$	67.1649	0.172660	0.0863302	−	0.996267i	$-0.472486\pi$
$389$	67.1649	0.172660	0.0863302	+	0.996267i	$0.472486\pi$
$390$	0	0
$391$	143.658i	0.367411i
$392$	0	0
$393$	223.484	0.568662
$394$	0	0
$395$	544.490i	1.37846i
$396$	0	0
$397$	147.524	0.371596	0.185798	−	0.982588i	$-0.440513\pi$
$397$	147.524	0.371596	0.185798	+	0.982588i	$0.440513\pi$
$398$	0	0
$399$	49.4094i	0.123833i
$400$	0	0
$401$	162.234	0.404573	0.202286	−	0.979326i	$-0.435163\pi$
$401$	162.234	0.404573	0.202286	+	0.979326i	$0.435163\pi$
$402$	0	0
$403$	− 23.9491i	− 0.0594271i
$404$	0	0
$405$	473.763	1.16979
$406$	0	0
$407$	− 403.960i	− 0.992531i
$408$	0	0
$409$	199.679	0.488213	0.244107	−	0.969748i	$-0.421505\pi$
$409$	199.679	0.488213	0.244107	+	0.969748i	$0.421505\pi$
$410$	0	0
$411$	260.226i	0.633153i
$412$	0	0
$413$	−641.243	−1.55265
$414$	0	0
$415$	1431.40i	3.44915i
$416$	0	0
$417$	−112.597	−0.270018
$418$	0	0
$419$	64.3969i	0.153692i	0.997043	+	0.0768459i	$0.0244849\pi$
$419$	64.3969i	0.153692i	−0.997043	+	0.0768459i	$0.975515\pi$
$420$	0	0
$421$	−701.790	−1.66696	−0.833479	−	0.552550i	$-0.813655\pi$
$421$	−701.790	−1.66696	−0.833479	+	0.552550i	$0.813655\pi$
$422$	0	0
$423$	121.055i	0.286182i
$424$	0	0
$425$	−294.593	−0.693161
$426$	0	0
$427$	− 178.808i	− 0.418754i
$428$	0	0
$429$	−15.8571	−0.0369629
$430$	0	0
$431$	580.968i	1.34795i	0.738753	+	0.673977i	$0.235415\pi$
$431$	580.968i	1.34795i	−0.738753	+	0.673977i	$0.764585\pi$
$432$	0	0
$433$	−115.234	−0.266129	−0.133064	−	0.991107i	$-0.542482\pi$
$433$	−115.234	−0.266129	−0.133064	+	0.991107i	$0.542482\pi$
$434$	0	0
$435$	− 457.209i	− 1.05106i
$436$	0	0
$437$	122.444	0.280193
$438$	0	0
$439$	− 299.467i	− 0.682157i	−0.940035	−	0.341078i	$-0.889208\pi$
$439$	− 299.467i	− 0.682157i	0.940035	−	0.341078i	$-0.110792\pi$
$440$	0	0
$441$	−521.864	−1.18336
$442$	0	0
$443$	222.362i	0.501946i	0.967994	+	0.250973i	$0.0807506\pi$
$443$	222.362i	0.501946i	−0.967994	+	0.250973i	$0.919249\pi$
$444$	0	0
$445$	1085.05	2.43831
$446$	0	0
$447$	− 261.268i	− 0.584491i
$448$	0	0
$449$	−696.538	−1.55131	−0.775655	−	0.631157i	$-0.782580\pi$
$449$	−696.538	−1.55131	−0.775655	+	0.631157i	$0.782580\pi$
$450$	0	0
$451$	701.289i	1.55497i
$452$	0	0
$453$	224.781	0.496206
$454$	0	0
$455$	137.649i	0.302524i
$456$	0	0
$457$	597.764	1.30802	0.654008	−	0.756487i	$-0.273086\pi$
$457$	597.764	1.30802	0.654008	+	0.756487i	$0.273086\pi$
$458$	0	0
$459$	91.1979i	0.198688i
$460$	0	0
$461$	493.391	1.07026	0.535131	−	0.844769i	$-0.320262\pi$
$461$	493.391	1.07026	0.535131	+	0.844769i	$0.320262\pi$
$462$	0	0
$463$	334.964i	0.723464i	0.932282	+	0.361732i	$0.117814\pi$
$463$	334.964i	0.723464i	−0.932282	+	0.361732i	$0.882186\pi$
$464$	0	0
$465$	−162.912	−0.350349
$466$	0	0
$467$	− 92.0212i	− 0.197047i	−0.995135	−	0.0985237i	$-0.968588\pi$
$467$	− 92.0212i	− 0.197047i	0.995135	−	0.0985237i	$-0.0314120\pi$
$468$	0	0
$469$	227.488	0.485049
$470$	0	0
$471$	− 68.2270i	− 0.144856i
$472$	0	0
$473$	259.633	0.548906
$474$	0	0
$475$	251.092i	0.528614i
$476$	0	0
$477$	−329.122	−0.689983
$478$	0	0
$479$	187.369i	0.391167i	0.980687	+	0.195584i	$0.0626601\pi$
$479$	187.369i	0.391167i	−0.980687	+	0.195584i	$0.937340\pi$
$480$	0	0
$481$	53.5767	0.111386
$482$	0	0
$483$	− 318.416i	− 0.659246i
$484$	0	0
$485$	98.9562	0.204033
$486$	0	0
$487$	− 906.793i	− 1.86200i	−0.365022	−	0.930999i	$-0.618939\pi$
$487$	− 906.793i	− 1.86200i	0.365022	−	0.930999i	$-0.381061\pi$
$488$	0	0
$489$	−67.5892	−0.138219
$490$	0	0
$491$	− 459.761i	− 0.936376i	−0.883629	−	0.468188i	$-0.844907\pi$
$491$	− 459.761i	− 0.936376i	0.883629	−	0.468188i	$-0.155093\pi$
$492$	0	0
$493$	−243.586	−0.494089
$494$	0	0
$495$	− 762.440i	− 1.54028i
$496$	0	0
$497$	−486.241	−0.978352
$498$	0	0
$499$	557.911i	1.11806i	0.829148	+	0.559029i	$0.188826\pi$
$499$	557.911i	1.11806i	−0.829148	+	0.559029i	$0.811174\pi$
$500$	0	0
$501$	22.4250	0.0447605
$502$	0	0
$503$	− 535.885i	− 1.06538i	−0.846311	−	0.532689i	$-0.821182\pi$
$503$	− 535.885i	− 1.06538i	0.846311	−	0.532689i	$-0.178818\pi$
$504$	0	0
$505$	1083.13	2.14481
$506$	0	0
$507$	176.387i	0.347904i
$508$	0	0
$509$	92.2346	0.181207	0.0906037	−	0.995887i	$-0.471120\pi$
$509$	92.2346	0.181207	0.0906037	+	0.995887i	$0.471120\pi$
$510$	0	0
$511$	183.293i	0.358695i
$512$	0	0
$513$	77.7310	0.151522
$514$	0	0
$515$	− 1460.15i	− 2.83524i
$516$	0	0
$517$	163.356	0.315969
$518$	0	0
$519$	− 22.0763i	− 0.0425363i
$520$	0	0
$521$	−222.114	−0.426322	−0.213161	−	0.977017i	$-0.568376\pi$
$521$	−222.114	−0.426322	−0.213161	+	0.977017i	$0.568376\pi$
$522$	0	0
$523$	− 207.757i	− 0.397241i	−0.980076	−	0.198621i	$-0.936354\pi$
$523$	− 207.757i	− 0.397241i	0.980076	−	0.198621i	$-0.0636462\pi$
$524$	0	0
$525$	652.963	1.24374
$526$	0	0
$527$	86.7942i	0.164695i
$528$	0	0
$529$	−260.085	−0.491654
$530$	0	0
$531$	471.080i	0.887156i
$532$	0	0
$533$	−93.0112	−0.174505
$534$	0	0
$535$	− 1105.18i	− 2.06575i
$536$	0	0
$537$	−300.487	−0.559566
$538$	0	0
$539$	704.222i	1.30653i
$540$	0	0
$541$	−177.653	−0.328379	−0.164189	−	0.986429i	$-0.552501\pi$
$541$	−177.653	−0.328379	−0.164189	+	0.986429i	$0.552501\pi$
$542$	0	0
$543$	− 212.910i	− 0.392099i
$544$	0	0
$545$	−1331.84	−2.44374
$546$	0	0
$547$	217.862i	0.398286i	0.979970	+	0.199143i	$0.0638158\pi$
$547$	217.862i	0.398286i	−0.979970	+	0.199143i	$0.936184\pi$
$548$	0	0
$549$	−131.359	−0.239269
$550$	0	0
$551$	207.616i	0.376799i
$552$	0	0
$553$	−642.973	−1.16270
$554$	0	0
$555$	− 364.452i	− 0.656670i
$556$	0	0
$557$	69.8765	0.125452	0.0627258	−	0.998031i	$-0.480021\pi$
$557$	69.8765	0.125452	0.0627258	+	0.998031i	$0.480021\pi$
$558$	0	0
$559$	34.4348i	0.0616006i
$560$	0	0
$561$	57.4678	0.102438
$562$	0	0
$563$	− 202.033i	− 0.358851i	−0.983772	−	0.179426i	$-0.942576\pi$
$563$	− 202.033i	− 0.358851i	0.983772	−	0.179426i	$-0.0574239\pi$
$564$	0	0
$565$	−1099.92	−1.94676
$566$	0	0
$567$	559.453i	0.986690i
$568$	0	0
$569$	−31.6232	−0.0555769	−0.0277884	−	0.999614i	$-0.508846\pi$
$569$	−31.6232	−0.0555769	−0.0277884	+	0.999614i	$0.508846\pi$
$570$	0	0
$571$	983.070i	1.72166i	0.508889	+	0.860832i	$0.330056\pi$
$571$	983.070i	1.72166i	−0.508889	+	0.860832i	$0.669944\pi$
$572$	0	0
$573$	75.0472	0.130972
$574$	0	0
$575$	− 1618.14i	− 2.81417i
$576$	0	0
$577$	361.922	0.627248	0.313624	−	0.949547i	$-0.398457\pi$
$577$	361.922	0.627248	0.313624	+	0.949547i	$0.398457\pi$
$578$	0	0
$579$	53.3924i	0.0922149i
$580$	0	0
$581$	−1690.30	−2.90929
$582$	0	0
$583$	444.129i	0.761799i
$584$	0	0
$585$	101.121	0.172857
$586$	0	0
$587$	580.530i	0.988978i	0.869184	+	0.494489i	$0.164645\pi$
$587$	580.530i	0.988978i	−0.869184	+	0.494489i	$0.835355\pi$
$588$	0	0
$589$	73.9776	0.125599
$590$	0	0
$591$	307.868i	0.520927i
$592$	0	0
$593$	−797.055	−1.34411	−0.672053	−	0.740503i	$-0.734587\pi$
$593$	−797.055	−1.34411	−0.672053	+	0.740503i	$0.734587\pi$
$594$	0	0
$595$	− 498.854i	− 0.838409i
$596$	0	0
$597$	−179.125	−0.300042
$598$	0	0
$599$	− 1105.87i	− 1.84619i	−0.384568	−	0.923097i	$-0.625650\pi$
$599$	− 1105.87i	− 1.84619i	0.384568	−	0.923097i	$-0.374350\pi$
$600$	0	0
$601$	353.190	0.587670	0.293835	−	0.955856i	$-0.405068\pi$
$601$	353.190	0.587670	0.293835	+	0.955856i	$0.405068\pi$
$602$	0	0
$603$	− 167.121i	− 0.277149i
$604$	0	0
$605$	70.8678	0.117137
$606$	0	0
$607$	890.397i	1.46688i	0.679754	+	0.733440i	$0.262087\pi$
$607$	890.397i	1.46688i	−0.679754	+	0.733440i	$0.737913\pi$
$608$	0	0
$609$	539.906	0.886544
$610$	0	0
$611$	21.6657i	0.0354594i
$612$	0	0
$613$	−105.121	−0.171486	−0.0857432	−	0.996317i	$-0.527326\pi$
$613$	−105.121	−0.171486	−0.0857432	+	0.996317i	$0.527326\pi$
$614$	0	0
$615$	632.702i	1.02878i
$616$	0	0
$617$	−199.151	−0.322773	−0.161386	−	0.986891i	$-0.551597\pi$
$617$	−199.151	−0.322773	−0.161386	+	0.986891i	$0.551597\pi$
$618$	0	0
$619$	− 488.557i	− 0.789268i	−0.918838	−	0.394634i	$-0.870871\pi$
$619$	− 488.557i	− 0.789268i	0.918838	−	0.394634i	$-0.129129\pi$
$620$	0	0
$621$	−500.933	−0.806655
$622$	0	0
$623$	1281.30i	2.05667i
$624$	0	0
$625$	1253.16	2.00505
$626$	0	0
$627$	− 48.9817i	− 0.0781207i
$628$	0	0
$629$	−194.168	−0.308693
$630$	0	0
$631$	180.553i	0.286138i	0.989713	+	0.143069i	$0.0456971\pi$
$631$	180.553i	0.286138i	−0.989713	+	0.143069i	$0.954303\pi$
$632$	0	0
$633$	−308.655	−0.487607
$634$	0	0
$635$	− 925.058i	− 1.45678i
$636$	0	0
$637$	−93.4001	−0.146625
$638$	0	0
$639$	357.210i	0.559014i
$640$	0	0
$641$	704.076	1.09840	0.549201	−	0.835690i	$-0.314932\pi$
$641$	704.076	1.09840	0.549201	+	0.835690i	$0.314932\pi$
$642$	0	0
$643$	472.324i	0.734562i	0.930110	+	0.367281i	$0.119711\pi$
$643$	472.324i	0.734562i	−0.930110	+	0.367281i	$0.880289\pi$
$644$	0	0
$645$	234.240	0.363163
$646$	0	0
$647$	− 694.047i	− 1.07271i	−0.843991	−	0.536357i	$-0.819800\pi$
$647$	− 694.047i	− 1.07271i	0.843991	−	0.536357i	$-0.180200\pi$
$648$	0	0
$649$	635.692	0.979494
$650$	0	0
$651$	− 192.378i	− 0.295512i
$652$	0	0
$653$	−61.1281	−0.0936112	−0.0468056	−	0.998904i	$-0.514904\pi$
$653$	−61.1281	−0.0936112	−0.0468056	+	0.998904i	$0.514904\pi$
$654$	0	0
$655$	1923.18i	2.93615i
$656$	0	0
$657$	134.654	0.204952
$658$	0	0
$659$	− 572.595i	− 0.868884i	−0.900700	−	0.434442i	$-0.856946\pi$
$659$	− 572.595i	− 0.868884i	0.900700	−	0.434442i	$-0.143054\pi$
$660$	0	0
$661$	1204.00	1.82149	0.910745	−	0.412969i	$-0.135508\pi$
$661$	1204.00	1.82149	0.910745	+	0.412969i	$0.135508\pi$
$662$	0	0
$663$	7.62188i	0.0114960i
$664$	0	0
$665$	−425.189	−0.639383
$666$	0	0
$667$	− 1337.97i	− 2.00595i
$668$	0	0
$669$	177.555	0.265403
$670$	0	0
$671$	177.260i	0.264173i
$672$	0	0
$673$	−521.086	−0.774274	−0.387137	−	0.922022i	$-0.626536\pi$
$673$	−521.086	−0.774274	−0.387137	+	0.922022i	$0.626536\pi$
$674$	0	0
$675$	− 1027.24i	− 1.52184i
$676$	0	0
$677$	−868.153	−1.28235	−0.641176	−	0.767393i	$-0.721553\pi$
$677$	−868.153	−1.28235	−0.641176	+	0.767393i	$0.721553\pi$
$678$	0	0
$679$	116.855i	0.172098i
$680$	0	0
$681$	−140.367	−0.206118
$682$	0	0
$683$	220.439i	0.322752i	0.986893	+	0.161376i	$0.0515931\pi$
$683$	220.439i	0.322752i	−0.986893	+	0.161376i	$0.948407\pi$
$684$	0	0
$685$	−2239.36	−3.26913
$686$	0	0
$687$	69.4823i	0.101139i
$688$	0	0
$689$	−58.9043	−0.0854924
$690$	0	0
$691$	− 868.756i	− 1.25724i	−0.777711	−	0.628622i	$-0.783619\pi$
$691$	− 868.756i	− 1.25724i	0.777711	−	0.628622i	$-0.216381\pi$
$692$	0	0
$693$	900.344	1.29920
$694$	0	0
$695$	− 968.950i	− 1.39417i
$696$	0	0
$697$	337.083	0.483619
$698$	0	0
$699$	− 203.506i	− 0.291139i
$700$	0	0
$701$	−788.469	−1.12478	−0.562389	−	0.826873i	$-0.690118\pi$
$701$	−788.469	−1.12478	−0.562389	+	0.826873i	$0.690118\pi$
$702$	0	0
$703$	165.496i	0.235414i
$704$	0	0
$705$	147.380	0.209049
$706$	0	0
$707$	1279.04i	1.80910i
$708$	0	0
$709$	−467.071	−0.658774	−0.329387	−	0.944195i	$-0.606842\pi$
$709$	−467.071	−0.658774	−0.329387	+	0.944195i	$0.606842\pi$
$710$	0	0
$711$	472.351i	0.664347i
$712$	0	0
$713$	−476.744	−0.668645
$714$	0	0
$715$	− 136.457i	− 0.190849i
$716$	0	0
$717$	483.537	0.674390
$718$	0	0
$719$	883.729i	1.22911i	0.788875	+	0.614554i	$0.210664\pi$
$719$	883.729i	1.22911i	−0.788875	+	0.614554i	$0.789336\pi$
$720$	0	0
$721$	1724.25	2.39146
$722$	0	0
$723$	244.144i	0.337682i
$724$	0	0
$725$	2743.73	3.78445
$726$	0	0
$727$	− 259.838i	− 0.357412i	−0.983903	−	0.178706i	$-0.942809\pi$
$727$	− 259.838i	− 0.357412i	0.983903	−	0.178706i	$-0.0571911\pi$
$728$	0	0
$729$	241.487	0.331257
$730$	0	0
$731$	− 124.795i	− 0.170719i
$732$	0	0
$733$	−34.1773	−0.0466265	−0.0233133	−	0.999728i	$-0.507422\pi$
$733$	−34.1773	−0.0466265	−0.0233133	+	0.999728i	$0.507422\pi$
$734$	0	0
$735$	635.347i	0.864418i
$736$	0	0
$737$	−225.519	−0.305996
$738$	0	0
$739$	− 352.235i	− 0.476637i	−0.971187	−	0.238318i	$-0.923404\pi$
$739$	− 352.235i	− 0.476637i	0.971187	−	0.238318i	$-0.0765962\pi$
$740$	0	0
$741$	6.49638	0.00876704
$742$	0	0
$743$	1174.36i	1.58057i	0.612739	+	0.790285i	$0.290068\pi$
$743$	1174.36i	1.58057i	−0.612739	+	0.790285i	$0.709932\pi$
$744$	0	0
$745$	2248.32	3.01788
$746$	0	0
$747$	1241.75i	1.66232i
$748$	0	0
$749$	1305.07	1.74242
$750$	0	0
$751$	778.648i	1.03681i	0.855134	+	0.518407i	$0.173475\pi$
$751$	778.648i	1.03681i	−0.855134	+	0.518407i	$0.826525\pi$
$752$	0	0
$753$	355.237	0.471762
$754$	0	0
$755$	1934.34i	2.56204i
$756$	0	0
$757$	788.148	1.04115	0.520573	−	0.853817i	$-0.325718\pi$
$757$	788.148	1.04115	0.520573	+	0.853817i	$0.325718\pi$
$758$	0	0
$759$	315.659i	0.415889i
$760$	0	0
$761$	98.7509	0.129765	0.0648823	−	0.997893i	$-0.479333\pi$
$761$	98.7509	0.129765	0.0648823	+	0.997893i	$0.479333\pi$
$762$	0	0
$763$	− 1572.73i	− 2.06124i
$764$	0	0
$765$	−366.475	−0.479053
$766$	0	0
$767$	84.3110i	0.109923i
$768$	0	0
$769$	−103.589	−0.134706	−0.0673530	−	0.997729i	$-0.521455\pi$
$769$	−103.589	−0.134706	−0.0673530	+	0.997729i	$0.521455\pi$
$770$	0	0
$771$	456.085i	0.591551i
$772$	0	0
$773$	33.3757	0.0431769	0.0215884	−	0.999767i	$-0.493128\pi$
$773$	33.3757	0.0431769	0.0215884	+	0.999767i	$0.493128\pi$
$774$	0	0
$775$	− 977.640i	− 1.26147i
$776$	0	0
$777$	430.371	0.553888
$778$	0	0
$779$	− 287.307i	− 0.368815i
$780$	0	0
$781$	482.032	0.617198
$782$	0	0
$783$	− 849.381i	− 1.08478i
$784$	0	0
$785$	587.124	0.747928
$786$	0	0
$787$	352.700i	0.448157i	0.974571	+	0.224079i	$0.0719373\pi$
$787$	352.700i	0.448157i	−0.974571	+	0.224079i	$0.928063\pi$
$788$	0	0
$789$	−425.819	−0.539694
$790$	0	0
$791$	− 1298.86i	− 1.64205i
$792$	0	0
$793$	−23.5098	−0.0296467
$794$	0	0
$795$	400.692i	0.504015i
$796$	0	0
$797$	92.2273	0.115718	0.0578591	−	0.998325i	$-0.481573\pi$
$797$	92.2273	0.115718	0.0578591	+	0.998325i	$0.481573\pi$
$798$	0	0
$799$	− 78.5189i	− 0.0982715i
$800$	0	0
$801$	941.291	1.17515
$802$	0	0
$803$	− 181.707i	− 0.226285i
$804$	0	0
$805$	2740.11	3.40386
$806$	0	0
$807$	64.3857i	0.0797840i
$808$	0	0
$809$	−492.297	−0.608525	−0.304263	−	0.952588i	$-0.598410\pi$
$809$	−492.297	−0.608525	−0.304263	+	0.952588i	$0.598410\pi$
$810$	0	0
$811$	− 523.571i	− 0.645587i	−0.946469	−	0.322793i	$-0.895378\pi$
$811$	− 523.571i	− 0.645587i	0.946469	−	0.322793i	$-0.104622\pi$
$812$	0	0
$813$	108.777	0.133797
$814$	0	0
$815$	− 581.634i	− 0.713662i
$816$	0	0
$817$	−106.367	−0.130192
$818$	0	0
$819$	119.411i	0.145802i
$820$	0	0
$821$	−1176.68	−1.43323	−0.716617	−	0.697467i	$-0.754310\pi$
$821$	−1176.68	−1.43323	−0.716617	+	0.697467i	$0.754310\pi$
$822$	0	0
$823$	− 342.603i	− 0.416286i	−0.978098	−	0.208143i	$-0.933258\pi$
$823$	− 342.603i	− 0.416286i	0.978098	−	0.208143i	$-0.0667419\pi$
$824$	0	0
$825$	−647.310	−0.784618
$826$	0	0
$827$	695.311i	0.840763i	0.907347	+	0.420382i	$0.138104\pi$
$827$	695.311i	0.840763i	−0.907347	+	0.420382i	$0.861896\pi$
$828$	0	0
$829$	885.118	1.06769	0.533847	−	0.845581i	$-0.320746\pi$
$829$	885.118	1.06769	0.533847	+	0.845581i	$0.320746\pi$
$830$	0	0
$831$	− 273.414i	− 0.329018i
$832$	0	0
$833$	338.492	0.406353
$834$	0	0
$835$	192.977i	0.231110i
$836$	0	0
$837$	−302.650	−0.361589
$838$	0	0
$839$	556.979i	0.663861i	0.943304	+	0.331930i	$0.107700\pi$
$839$	556.979i	0.663861i	−0.943304	+	0.331930i	$0.892300\pi$
$840$	0	0
$841$	1427.66	1.69758
$842$	0	0
$843$	222.193i	0.263574i
$844$	0	0
$845$	−1517.89	−1.79632
$846$	0	0
$847$	83.6858i	0.0988026i
$848$	0	0
$849$	−118.713	−0.139827
$850$	0	0
$851$	− 1066.53i	− 1.25326i
$852$	0	0
$853$	811.661	0.951536	0.475768	−	0.879571i	$-0.342170\pi$
$853$	811.661	0.951536	0.475768	+	0.879571i	$0.342170\pi$
$854$	0	0
$855$	312.359i	0.365332i
$856$	0	0
$857$	−1253.44	−1.46259	−0.731294	−	0.682062i	$-0.761083\pi$
$857$	−1253.44	−1.46259	−0.731294	+	0.682062i	$0.761083\pi$
$858$	0	0
$859$	− 576.904i	− 0.671599i	−0.941933	−	0.335800i	$-0.890994\pi$
$859$	− 576.904i	− 0.671599i	0.941933	−	0.335800i	$-0.109006\pi$
$860$	0	0
$861$	−747.140	−0.867758
$862$	0	0
$863$	1208.60i	1.40047i	0.713913	+	0.700234i	$0.246921\pi$
$863$	1208.60i	1.40047i	−0.713913	+	0.700234i	$0.753079\pi$
$864$	0	0
$865$	189.977	0.219626
$866$	0	0
$867$	277.607i	0.320193i
$868$	0	0
$869$	637.407	0.733495
$870$	0	0
$871$	− 29.9103i	− 0.0343402i
$872$	0	0
$873$	85.8454	0.0983338
$874$	0	0
$875$	3180.40i	3.63474i
$876$	0	0
$877$	−181.176	−0.206586	−0.103293	−	0.994651i	$-0.532938\pi$
$877$	−181.176	−0.206586	−0.103293	+	0.994651i	$0.532938\pi$
$878$	0	0
$879$	− 109.112i	− 0.124132i
$880$	0	0
$881$	−640.333	−0.726825	−0.363412	−	0.931628i	$-0.618388\pi$
$881$	−640.333	−0.726825	−0.363412	+	0.931628i	$0.618388\pi$
$882$	0	0
$883$	− 132.874i	− 0.150480i	−0.997165	−	0.0752398i	$-0.976028\pi$
$883$	− 132.874i	− 0.150480i	0.997165	−	0.0752398i	$-0.0239722\pi$
$884$	0	0
$885$	573.520	0.648045
$886$	0	0
$887$	1239.47i	1.39738i	0.715425	+	0.698689i	$0.246233\pi$
$887$	1239.47i	1.39738i	−0.715425	+	0.698689i	$0.753767\pi$
$888$	0	0
$889$	1092.37	1.22877
$890$	0	0
$891$	− 554.610i	− 0.622458i
$892$	0	0
$893$	−66.9243	−0.0749432
$894$	0	0
$895$	− 2585.82i	− 2.88919i
$896$	0	0
$897$	−41.8655	−0.0466728
$898$	0	0
$899$	− 808.367i	− 0.899184i
$900$	0	0
$901$	213.476	0.236932
$902$	0	0
$903$	276.607i	0.306320i
$904$	0	0
$905$	1832.18	2.02451
$906$	0	0
$907$	− 611.509i	− 0.674211i	−0.941467	−	0.337105i	$-0.890552\pi$
$907$	− 611.509i	− 0.674211i	0.941467	−	0.337105i	$-0.109448\pi$
$908$	0	0
$909$	939.625	1.03369
$910$	0	0
$911$	896.546i	0.984134i	0.870557	+	0.492067i	$0.163758\pi$
$911$	896.546i	0.984134i	−0.870557	+	0.492067i	$0.836242\pi$
$912$	0	0
$913$	1675.67	1.83534
$914$	0	0
$915$	159.924i	0.174780i
$916$	0	0
$917$	−2271.03	−2.47658
$918$	0	0
$919$	− 886.935i	− 0.965108i	−0.875866	−	0.482554i	$-0.839709\pi$
$919$	− 886.935i	− 0.965108i	0.875866	−	0.482554i	$-0.160291\pi$
$920$	0	0
$921$	−602.361	−0.654030
$922$	0	0
$923$	63.9313i	0.0692647i
$924$	0	0
$925$	2187.09	2.36442
$926$	0	0
$927$	− 1266.69i	− 1.36644i
$928$	0	0
$929$	368.049	0.396178	0.198089	−	0.980184i	$-0.436527\pi$
$929$	368.049	0.396178	0.198089	+	0.980184i	$0.436527\pi$
$930$	0	0
$931$	− 288.508i	− 0.309891i
$932$	0	0
$933$	580.918	0.622635
$934$	0	0
$935$	494.535i	0.528915i
$936$	0	0
$937$	−1349.93	−1.44070	−0.720348	−	0.693613i	$-0.756018\pi$
$937$	−1349.93	−1.44070	−0.720348	+	0.693613i	$0.756018\pi$
$938$	0	0
$939$	397.904i	0.423753i
$940$	0	0
$941$	443.896	0.471728	0.235864	−	0.971786i	$-0.424208\pi$
$941$	443.896	0.471728	0.235864	+	0.971786i	$0.424208\pi$
$942$	0	0
$943$	1851.53i	1.96345i
$944$	0	0
$945$	1739.50	1.84074
$946$	0	0
$947$	350.509i	0.370126i	0.982727	+	0.185063i	$0.0592489\pi$
$947$	350.509i	0.370126i	−0.982727	+	0.185063i	$0.940751\pi$
$948$	0	0
$949$	24.0995	0.0253946
$950$	0	0
$951$	533.577i	0.561070i
$952$	0	0
$953$	−263.837	−0.276849	−0.138424	−	0.990373i	$-0.544204\pi$
$953$	−263.837	−0.276849	−0.138424	+	0.990373i	$0.544204\pi$
$954$	0	0
$955$	645.814i	0.676245i
$956$	0	0
$957$	−535.232	−0.559281
$958$	0	0
$959$	− 2644.39i	− 2.75745i
$960$	0	0
$961$	672.964	0.700275
$962$	0	0
$963$	− 958.752i	− 0.995588i
$964$	0	0
$965$	−459.465	−0.476130
$966$	0	0
$967$	− 1063.16i	− 1.09944i	−0.835349	−	0.549720i	$-0.814734\pi$
$967$	− 1063.16i	− 1.09944i	0.835349	−	0.549720i	$-0.185266\pi$
$968$	0	0
$969$	−23.5436	−0.0242968
$970$	0	0
$971$	− 1134.08i	− 1.16795i	−0.811772	−	0.583975i	$-0.801497\pi$
$971$	− 1134.08i	− 1.16795i	0.811772	−	0.583975i	$-0.198503\pi$
$972$	0	0
$973$	1144.21	1.17596
$974$	0	0
$975$	− 85.8519i	− 0.0880533i
$976$	0	0
$977$	−652.300	−0.667657	−0.333828	−	0.942634i	$-0.608341\pi$
$977$	−652.300	−0.667657	−0.333828	+	0.942634i	$0.608341\pi$
$978$	0	0
$979$	− 1270.21i	− 1.29746i
$980$	0	0
$981$	−1155.38	−1.17776
$982$	0	0
$983$	− 1237.86i	− 1.25926i	−0.776894	−	0.629631i	$-0.783206\pi$
$983$	− 1237.86i	− 1.25926i	0.776894	−	0.629631i	$-0.216794\pi$
$984$	0	0
$985$	−2649.34	−2.68968
$986$	0	0
$987$	174.036i	0.176329i
$988$	0	0
$989$	685.477	0.693101
$990$	0	0
$991$	230.602i	0.232696i	0.993209	+	0.116348i	$0.0371188\pi$
$991$	230.602i	0.232696i	−0.993209	+	0.116348i	$0.962881\pi$
$992$	0	0
$993$	−199.083	−0.200487
$994$	0	0
$995$	− 1541.45i	− 1.54920i
$996$	0	0
$997$	1023.39	1.02647	0.513234	−	0.858248i	$-0.328447\pi$
$997$	1023.39	1.02647	0.513234	+	0.858248i	$0.328447\pi$
$998$	0	0
$999$	− 677.061i	− 0.677739i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.f.191.8		20
4.3	odd	2	inner	1216.3.d.f.191.13		20
8.3	odd	2		608.3.d.b.191.8	✓	20
8.5	even	2		608.3.d.b.191.13	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.d.b.191.8	✓	20	8.3	odd	2
608.3.d.b.191.13	yes	20	8.5	even	2
1216.3.d.f.191.8		20	1.1	even	1	trivial
1216.3.d.f.191.13		20	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.05616i q^{3} +9.08870 q^{5} +10.7326i q^{7} +7.88453 q^{9} -10.6397i q^{11} +1.41113 q^{13} -9.59909i q^{15} -5.11408 q^{17} +4.35890i q^{19} +11.3353 q^{21} -28.0907i q^{23} +57.6044 q^{25} -17.8327i q^{27} +47.6305 q^{29} -16.9716i q^{31} -11.2372 q^{33} +97.5452i q^{35} +37.9673 q^{37} -1.49037i q^{39} -65.9127 q^{41} +24.4023i q^{43} +71.6601 q^{45} +15.3535i q^{47} -66.1883 q^{49} +5.40127i q^{51} -41.7427 q^{53} -96.7007i q^{55} +4.60368 q^{57} +59.7473i q^{59} -16.6603 q^{61} +84.6214i q^{63} +12.8253 q^{65} -21.1960i q^{67} -29.6681 q^{69} +45.3051i q^{71} +17.0782 q^{73} -60.8393i q^{75} +114.191 q^{77} +59.9085i q^{79} +52.1266 q^{81} +157.492i q^{83} -46.4803 q^{85} -50.3053i q^{87} +119.385 q^{89} +15.1450i q^{91} -17.9247 q^{93} +39.6167i q^{95} +10.8878 q^{97} -83.8888i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 68 q^{9} + 16 q^{13} + 8 q^{17} - 64 q^{21} + 196 q^{25} + 88 q^{29} - 184 q^{33} - 16 q^{37} - 16 q^{41} - 16 q^{45} + 52 q^{49} - 88 q^{53} + 208 q^{61} - 192 q^{65} - 248 q^{69} - 152 q^{73} - 312 q^{77}+ \cdots - 72 q^{97}+O(q^{100}) \) 20 * q - 68 * q^9 + 16 * q^13 + 8 * q^17 - 64 * q^21 + 196 * q^25 + 88 * q^29 - 184 * q^33 - 16 * q^37 - 16 * q^41 - 16 * q^45 + 52 * q^49 - 88 * q^53 + 208 * q^61 - 192 * q^65 - 248 * q^69 - 152 * q^73 - 312 * q^77 + 500 * q^81 + 488 * q^85 + 304 * q^89 + 240 * q^93 - 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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