LMFDB - Embedded newform 1210.3.d.c.241.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1210,3,Mod(241,1210)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1210.241"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1210 = 2 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1210.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$32.9701119876$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3078 x^{12} - 12280 x^{11} + 49836 x^{10} - 147672 x^{9} + \cdots + 339856 $ x^16 - 8x^15 + 92x^14 - 504x^13 + 3078x^12 - 12280x^11 + 49836x^10 - 147672x^9 + 423553x^8 - 924384x^7 + 1876224x^6 - 2898272x^5 + 3947928x^4 - 3906048x^3 + 3026408x^2 - 1437952*x + 339856
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.13
Root			$0.500000 - 3.96822i$ of defining polynomial
Character	$\chi$	$=$	1210.241
Dual form			1210.3.d.c.241.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} +0.468308 q^{3} -2.00000 q^{4} +2.23607 q^{5} +0.662288i q^{6} -13.4610i q^{7} -2.82843i q^{8} -8.78069 q^{9} +3.16228i q^{10} -0.936617 q^{12} +7.43101i q^{13} +19.0367 q^{14} +1.04717 q^{15} +4.00000 q^{16} -12.1368i q^{17} -12.4178i q^{18} +4.76968i q^{19} -4.47214 q^{20} -6.30390i q^{21} -9.96131 q^{23} -1.32458i q^{24} +5.00000 q^{25} -10.5090 q^{26} -8.32685 q^{27} +26.9220i q^{28} +28.8922i q^{29} +1.48092i q^{30} -50.3194 q^{31} +5.65685i q^{32} +17.1640 q^{34} -30.0997i q^{35} +17.5614 q^{36} +48.5268 q^{37} -6.74534 q^{38} +3.48001i q^{39} -6.32456i q^{40} +53.4434i q^{41} +8.91506 q^{42} -15.5167i q^{43} -19.6342 q^{45} -14.0874i q^{46} -29.7533 q^{47} +1.87323 q^{48} -132.198 q^{49} +7.07107i q^{50} -5.68377i q^{51} -14.8620i q^{52} -15.0761 q^{53} -11.7759i q^{54} -38.0735 q^{56} +2.23368i q^{57} -40.8598 q^{58} -106.522 q^{59} -2.09434 q^{60} +53.8738i q^{61} -71.1624i q^{62} +118.197i q^{63} -8.00000 q^{64} +16.6162i q^{65} -82.5702 q^{67} +24.2736i q^{68} -4.66497 q^{69} +42.5674 q^{70} +58.4935 q^{71} +24.8355i q^{72} -40.4706i q^{73} +68.6272i q^{74} +2.34154 q^{75} -9.53935i q^{76} -4.92147 q^{78} +100.411i q^{79} +8.94427 q^{80} +75.1267 q^{81} -75.5804 q^{82} +27.1970i q^{83} +12.6078i q^{84} -27.1387i q^{85} +21.9439 q^{86} +13.5305i q^{87} -22.2491 q^{89} -27.7670i q^{90} +100.029 q^{91} +19.9226 q^{92} -23.5650 q^{93} -42.0776i q^{94} +10.6653i q^{95} +2.64915i q^{96} -103.175 q^{97} -186.957i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} - 32 q^{4} + 40 q^{9} + 32 q^{12} + 16 q^{14} + 40 q^{15} + 64 q^{16} + 108 q^{23} + 80 q^{25} - 292 q^{27} - 268 q^{31} - 16 q^{34} - 80 q^{36} + 44 q^{37} - 280 q^{38} - 16 q^{42} - 476 q^{47}+ \cdots + 296 q^{97}+O(q^{100}) $ 16 * q - 16 * q^3 - 32 * q^4 + 40 * q^9 + 32 * q^12 + 16 * q^14 + 40 * q^15 + 64 * q^16 + 108 * q^23 + 80 * q^25 - 292 * q^27 - 268 * q^31 - 16 * q^34 - 80 * q^36 + 44 * q^37 - 280 * q^38 - 16 * q^42 - 476 * q^47 - 64 * q^48 - 644 * q^49 - 396 * q^53 - 32 * q^56 + 192 * q^58 - 180 * q^59 - 80 * q^60 - 128 * q^64 + 188 * q^67 - 416 * q^69 + 308 * q^71 - 80 * q^75 + 8 * q^78 + 536 * q^81 + 16 * q^82 + 304 * q^86 - 176 * q^89 + 80 * q^91 - 216 * q^92 + 436 * q^93 + 296 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times$.

$n$	$727$	$1091$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0.468308	0.156103	0.0780514	−	0.996949i	$-0.475130\pi$
$3$	0.468308	0.156103	0.0780514	+	0.996949i	$0.475130\pi$
$4$	−2.00000	−0.500000
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0.662288i	0.110381i
$7$	− 13.4610i	− 1.92300i	−0.274805	−	0.961500i	$-0.588613\pi$
$7$	− 13.4610i	− 1.92300i	0.274805	−	0.961500i	$-0.411387\pi$
$8$	− 2.82843i	− 0.353553i
$9$	−8.78069	−0.975632
$10$	3.16228i	0.316228i
$11$	0	0
$11$	0	0
$12$	−0.936617	−0.0780514
$13$	7.43101i	0.571616i	0.958287	+	0.285808i	$0.0922620\pi$
$13$	7.43101i	0.571616i	−0.958287	+	0.285808i	$0.907738\pi$
$14$	19.0367	1.35977
$15$	1.04717	0.0698113
$16$	4.00000	0.250000
$17$	− 12.1368i	− 0.713930i	−0.934118	−	0.356965i	$-0.883812\pi$
$17$	− 12.1368i	− 0.713930i	0.934118	−	0.356965i	$-0.116188\pi$
$18$	− 12.4178i	− 0.689876i
$19$	4.76968i	0.251036i	0.992091	+	0.125518i	$0.0400592\pi$
$19$	4.76968i	0.251036i	−0.992091	+	0.125518i	$0.959941\pi$
$20$	−4.47214	−0.223607
$21$	− 6.30390i	− 0.300186i
$22$	0	0
$23$	−9.96131	−0.433101	−0.216550	−	0.976271i	$-0.569481\pi$
$23$	−9.96131	−0.433101	−0.216550	+	0.976271i	$0.569481\pi$
$24$	− 1.32458i	− 0.0551907i
$25$	5.00000	0.200000
$26$	−10.5090	−0.404194
$27$	−8.32685	−0.308402
$28$	26.9220i	0.961500i
$29$	28.8922i	0.996284i	0.867095	+	0.498142i	$0.165984\pi$
$29$	28.8922i	0.996284i	−0.867095	+	0.498142i	$0.834016\pi$
$30$	1.48092i	0.0493640i
$31$	−50.3194	−1.62321	−0.811603	−	0.584209i	$-0.801405\pi$
$31$	−50.3194	−1.62321	−0.811603	+	0.584209i	$0.801405\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	17.1640	0.504825
$35$	− 30.0997i	− 0.859992i
$36$	17.5614	0.487816
$37$	48.5268	1.31153	0.655767	−	0.754963i	$-0.272345\pi$
$37$	48.5268	1.31153	0.655767	+	0.754963i	$0.272345\pi$
$38$	−6.74534	−0.177509
$39$	3.48001i	0.0892309i
$40$	− 6.32456i	− 0.158114i
$41$	53.4434i	1.30350i	0.758435	+	0.651749i	$0.225964\pi$
$41$	53.4434i	1.30350i	−0.758435	+	0.651749i	$0.774036\pi$
$42$	8.91506	0.212263
$43$	− 15.5167i	− 0.360853i	−0.983588	−	0.180427i	$-0.942252\pi$
$43$	− 15.5167i	− 0.360853i	0.983588	−	0.180427i	$-0.0577478\pi$
$44$	0	0
$45$	−19.6342	−0.436316
$46$	− 14.0874i	− 0.306248i
$47$	−29.7533	−0.633049	−0.316525	−	0.948584i	$-0.602516\pi$
$47$	−29.7533	−0.633049	−0.316525	+	0.948584i	$0.602516\pi$
$48$	1.87323	0.0390257
$49$	−132.198	−2.69793
$50$	7.07107i	0.141421i
$51$	− 5.68377i	− 0.111446i
$52$	− 14.8620i	− 0.285808i
$53$	−15.0761	−0.284455	−0.142228	−	0.989834i	$-0.545426\pi$
$53$	−15.0761	−0.284455	−0.142228	+	0.989834i	$0.545426\pi$
$54$	− 11.7759i	− 0.218073i
$55$	0	0
$56$	−38.0735	−0.679883
$57$	2.23368i	0.0391874i
$58$	−40.8598	−0.704479
$59$	−106.522	−1.80546	−0.902729	−	0.430210i	$-0.858440\pi$
$59$	−106.522	−1.80546	−0.902729	+	0.430210i	$0.858440\pi$
$60$	−2.09434	−0.0349057
$61$	53.8738i	0.883177i	0.897218	+	0.441588i	$0.145585\pi$
$61$	53.8738i	0.883177i	−0.897218	+	0.441588i	$0.854415\pi$
$62$	− 71.1624i	− 1.14778i
$63$	118.197i	1.87614i
$64$	−8.00000	−0.125000
$65$	16.6162i	0.255635i
$66$	0	0
$67$	−82.5702	−1.23239	−0.616196	−	0.787593i	$-0.711327\pi$
$67$	−82.5702	−1.23239	−0.616196	+	0.787593i	$0.711327\pi$
$68$	24.2736i	0.356965i
$69$	−4.66497	−0.0676082
$70$	42.5674	0.608106
$71$	58.4935	0.823852	0.411926	−	0.911217i	$-0.364856\pi$
$71$	58.4935	0.823852	0.411926	+	0.911217i	$0.364856\pi$
$72$	24.8355i	0.344938i
$73$	− 40.4706i	− 0.554391i	−0.960813	−	0.277196i	$-0.910595\pi$
$73$	− 40.4706i	− 0.554391i	0.960813	−	0.277196i	$-0.0894050\pi$
$74$	68.6272i	0.927395i
$75$	2.34154	0.0312206
$76$	− 9.53935i	− 0.125518i
$77$	0	0
$78$	−4.92147	−0.0630958
$79$	100.411i	1.27102i	0.772092	+	0.635510i	$0.219210\pi$
$79$	100.411i	1.27102i	−0.772092	+	0.635510i	$0.780790\pi$
$80$	8.94427	0.111803
$81$	75.1267	0.927490
$82$	−75.5804	−0.921712
$83$	27.1970i	0.327675i	0.986487	+	0.163838i	$0.0523873\pi$
$83$	27.1970i	0.327675i	−0.986487	+	0.163838i	$0.947613\pi$
$84$	12.6078i	0.150093i
$85$	− 27.1387i	− 0.319279i
$86$	21.9439	0.255162
$87$	13.5305i	0.155523i
$88$	0	0
$89$	−22.2491	−0.249990	−0.124995	−	0.992157i	$-0.539891\pi$
$89$	−22.2491	−0.249990	−0.124995	+	0.992157i	$0.539891\pi$
$90$	− 27.7670i	− 0.308522i
$91$	100.029	1.09922
$92$	19.9226	0.216550
$93$	−23.5650	−0.253387
$94$	− 42.0776i	− 0.447634i
$95$	10.6653i	0.112267i
$96$	2.64915i	0.0275953i
$97$	−103.175	−1.06366	−0.531828	−	0.846853i	$-0.678495\pi$
$97$	−103.175	−1.06366	−0.531828	+	0.846853i	$0.678495\pi$
$98$	− 186.957i	− 1.90772i
$99$	0	0
$100$	−10.0000	−0.100000
$101$	81.1684i	0.803648i	0.915717	+	0.401824i	$0.131624\pi$
$101$	81.1684i	0.803648i	−0.915717	+	0.401824i	$0.868376\pi$
$102$	8.03806	0.0788046
$103$	−122.413	−1.18848	−0.594240	−	0.804288i	$-0.702547\pi$
$103$	−122.413	−1.18848	−0.594240	+	0.804288i	$0.702547\pi$
$104$	21.0181	0.202097
$105$	− 14.0959i	− 0.134247i
$106$	− 21.3208i	− 0.201140i
$107$	− 99.2627i	− 0.927688i	−0.885917	−	0.463844i	$-0.846470\pi$
$107$	− 99.2627i	− 0.927688i	0.885917	−	0.463844i	$-0.153530\pi$
$108$	16.6537	0.154201
$109$	− 122.026i	− 1.11951i	−0.828660	−	0.559753i	$-0.810896\pi$
$109$	− 122.026i	− 1.11951i	0.828660	−	0.559753i	$-0.189104\pi$
$110$	0	0
$111$	22.7255	0.204734
$112$	− 53.8440i	− 0.480750i
$113$	200.014	1.77004	0.885019	−	0.465555i	$-0.154145\pi$
$113$	200.014	1.77004	0.885019	+	0.465555i	$0.154145\pi$
$114$	−3.15890	−0.0277096
$115$	−22.2742	−0.193688
$116$	− 57.7845i	− 0.498142i
$117$	− 65.2494i	− 0.557687i
$118$	− 150.645i	− 1.27665i
$119$	−163.374	−1.37289
$120$	− 2.96184i	− 0.0246820i
$121$	0	0
$122$	−76.1890	−0.624500
$123$	25.0280i	0.203480i
$124$	100.639	0.811603
$125$	11.1803	0.0894427
$126$	−167.156	−1.32663
$127$	7.08770i	0.0558086i	0.999611	+	0.0279043i	$0.00888337\pi$
$127$	7.08770i	0.0558086i	−0.999611	+	0.0279043i	$0.991117\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	− 7.26660i	− 0.0563302i
$130$	−23.4989	−0.180761
$131$	− 195.917i	− 1.49555i	−0.663954	−	0.747774i	$-0.731123\pi$
$131$	− 195.917i	− 1.49555i	0.663954	−	0.747774i	$-0.268877\pi$
$132$	0	0
$133$	64.2046	0.482741
$134$	− 116.772i	− 0.871432i
$135$	−18.6194	−0.137921
$136$	−34.3281	−0.252412
$137$	140.711	1.02708	0.513542	−	0.858064i	$-0.328333\pi$
$137$	140.711	1.02708	0.513542	+	0.858064i	$0.328333\pi$
$138$	− 6.59726i	− 0.0478062i
$139$	− 17.6148i	− 0.126725i	−0.997991	−	0.0633626i	$-0.979818\pi$
$139$	− 17.6148i	− 0.126725i	0.997991	−	0.0633626i	$-0.0201825\pi$
$140$	60.1994i	0.429996i
$141$	−13.9337	−0.0988208
$142$	82.7223i	0.582551i
$143$	0	0
$144$	−35.1227	−0.243908
$145$	64.6050i	0.445552i
$146$	57.2340	0.392014
$147$	−61.9097	−0.421154
$148$	−97.0536	−0.655767
$149$	− 166.939i	− 1.12040i	−0.828359	−	0.560198i	$-0.810725\pi$
$149$	− 166.939i	− 1.12040i	0.828359	−	0.560198i	$-0.189275\pi$
$150$	3.31144i	0.0220763i
$151$	103.199i	0.683438i	0.939802	+	0.341719i	$0.111009\pi$
$151$	103.199i	0.683438i	−0.939802	+	0.341719i	$0.888991\pi$
$152$	13.4907	0.0887545
$153$	106.570i	0.696533i
$154$	0	0
$155$	−112.518	−0.725920
$156$	− 6.96001i	− 0.0446155i
$157$	−184.395	−1.17449	−0.587247	−	0.809408i	$-0.699788\pi$
$157$	−184.395	−1.17449	−0.587247	+	0.809408i	$0.699788\pi$
$158$	−142.002	−0.898747
$159$	−7.06027	−0.0444042
$160$	12.6491i	0.0790569i
$161$	134.089i	0.832852i
$162$	106.245i	0.655834i
$163$	−273.017	−1.67495	−0.837477	−	0.546473i	$-0.815970\pi$
$163$	−273.017	−1.67495	−0.837477	+	0.546473i	$0.815970\pi$
$164$	− 106.887i	− 0.651749i
$165$	0	0
$166$	−38.4624	−0.231701
$167$	− 211.623i	− 1.26720i	−0.773659	−	0.633602i	$-0.781576\pi$
$167$	− 211.623i	− 1.26720i	0.773659	−	0.633602i	$-0.218424\pi$
$168$	−17.8301	−0.106132
$169$	113.780	0.673255
$170$	38.3800	0.225764
$171$	− 41.8810i	− 0.244918i
$172$	31.0334i	0.180427i
$173$	17.2637i	0.0997903i	0.998754	+	0.0498951i	$0.0158887\pi$
$173$	17.2637i	0.0997903i	−0.998754	+	0.0498951i	$0.984111\pi$
$174$	−19.1350	−0.109971
$175$	− 67.3050i	− 0.384600i
$176$	0	0
$177$	−49.8852	−0.281837
$178$	− 31.4650i	− 0.176770i
$179$	−214.183	−1.19655	−0.598276	−	0.801290i	$-0.704147\pi$
$179$	−214.183	−1.19655	−0.598276	+	0.801290i	$0.704147\pi$
$180$	39.2684	0.218158
$181$	22.5829	0.124767	0.0623836	−	0.998052i	$-0.480130\pi$
$181$	22.5829	0.124767	0.0623836	+	0.998052i	$0.480130\pi$
$182$	141.462i	0.777265i
$183$	25.2295i	0.137866i
$184$	28.1749i	0.153124i
$185$	108.509	0.586536
$186$	− 33.3259i	− 0.179172i
$187$	0	0
$188$	59.5066	0.316525
$189$	112.088i	0.593056i
$190$	−15.0830	−0.0793844
$191$	−33.7016	−0.176448	−0.0882240	−	0.996101i	$-0.528119\pi$
$191$	−33.7016	−0.176448	−0.0882240	+	0.996101i	$0.528119\pi$
$192$	−3.74647	−0.0195129
$193$	24.4296i	0.126578i	0.997995	+	0.0632891i	$0.0201590\pi$
$193$	24.4296i	0.126578i	−0.997995	+	0.0632891i	$0.979841\pi$
$194$	− 145.911i	− 0.752118i
$195$	7.78153i	0.0399053i
$196$	264.397	1.34896
$197$	− 120.534i	− 0.611848i	−0.952056	−	0.305924i	$-0.901035\pi$
$197$	− 120.534i	− 0.611848i	0.952056	−	0.305924i	$-0.0989654\pi$
$198$	0	0
$199$	−300.037	−1.50772	−0.753861	−	0.657034i	$-0.771811\pi$
$199$	−300.037	−1.50772	−0.753861	+	0.657034i	$0.771811\pi$
$200$	− 14.1421i	− 0.0707107i
$201$	−38.6683	−0.192380
$202$	−114.789	−0.568265
$203$	388.918	1.91585
$204$	11.3675i	0.0557232i
$205$	119.503i	0.582942i
$206$	− 173.119i	− 0.840382i
$207$	87.4672	0.422547
$208$	29.7240i	0.142904i
$209$	0	0
$210$	19.9347	0.0949270
$211$	− 26.6669i	− 0.126383i	−0.998001	−	0.0631917i	$-0.979872\pi$
$211$	− 26.6669i	− 0.126383i	0.998001	−	0.0631917i	$-0.0201280\pi$
$212$	30.1522	0.142228
$213$	27.3930	0.128606
$214$	140.379	0.655975
$215$	− 34.6964i	− 0.161378i
$216$	23.5519i	0.109036i
$217$	677.349i	3.12143i
$218$	172.571	0.791610
$219$	− 18.9527i	− 0.0865421i
$220$	0	0
$221$	90.1888	0.408094
$222$	32.1387i	0.144769i
$223$	309.146	1.38631	0.693153	−	0.720790i	$-0.256221\pi$
$223$	309.146	1.38631	0.693153	+	0.720790i	$0.256221\pi$
$224$	76.1469	0.339942
$225$	−43.9034	−0.195126
$226$	282.863i	1.25161i
$227$	356.181i	1.56908i	0.620079	+	0.784540i	$0.287101\pi$
$227$	356.181i	1.56908i	−0.620079	+	0.784540i	$0.712899\pi$
$228$	− 4.46736i	− 0.0195937i
$229$	−95.0338	−0.414995	−0.207497	−	0.978236i	$-0.566532\pi$
$229$	−95.0338	−0.414995	−0.207497	+	0.978236i	$0.566532\pi$
$230$	− 31.5004i	− 0.136958i
$231$	0	0
$232$	81.7196	0.352240
$233$	− 170.668i	− 0.732482i	−0.930520	−	0.366241i	$-0.880645\pi$
$233$	− 170.668i	− 0.732482i	0.930520	−	0.366241i	$-0.119355\pi$
$234$	92.2766	0.394344
$235$	−66.5305	−0.283108
$236$	213.044	0.902729
$237$	47.0231i	0.198410i
$238$	− 231.045i	− 0.970778i
$239$	− 387.419i	− 1.62100i	−0.585738	−	0.810500i	$-0.699195\pi$
$239$	− 387.419i	− 1.62100i	0.585738	−	0.810500i	$-0.300805\pi$
$240$	4.18868	0.0174528
$241$	231.343i	0.959930i	0.877288	+	0.479965i	$0.159351\pi$
$241$	231.343i	0.959930i	−0.877288	+	0.479965i	$0.840649\pi$
$242$	0	0
$243$	110.124	0.453185
$244$	− 107.748i	− 0.441588i
$245$	−295.605	−1.20655
$246$	−35.3949	−0.143882
$247$	−35.4435	−0.143496
$248$	142.325i	0.573890i
$249$	12.7366i	0.0511510i
$250$	15.8114i	0.0632456i
$251$	−88.9283	−0.354296	−0.177148	−	0.984184i	$-0.556687\pi$
$251$	−88.9283	−0.354296	−0.177148	+	0.984184i	$0.556687\pi$
$252$	− 236.394i	− 0.938070i
$253$	0	0
$254$	−10.0235	−0.0394627
$255$	− 12.7093i	− 0.0498404i
$256$	16.0000	0.0625000
$257$	−231.107	−0.899247	−0.449624	−	0.893218i	$-0.648442\pi$
$257$	−231.107	−0.899247	−0.449624	+	0.893218i	$0.648442\pi$
$258$	10.2765	0.0398315
$259$	− 653.219i	− 2.52208i
$260$	− 33.2325i	− 0.127817i
$261$	− 253.694i	− 0.972007i
$262$	277.068	1.05751
$263$	190.148i	0.722997i	0.932373	+	0.361498i	$0.117735\pi$
$263$	190.148i	0.722997i	−0.932373	+	0.361498i	$0.882265\pi$
$264$	0	0
$265$	−33.7112	−0.127212
$266$	90.7990i	0.341350i
$267$	−10.4194	−0.0390241
$268$	165.140	0.616196
$269$	−234.323	−0.871090	−0.435545	−	0.900167i	$-0.643444\pi$
$269$	−234.323	−0.871090	−0.435545	+	0.900167i	$0.643444\pi$
$270$	− 26.3318i	− 0.0975252i
$271$	− 133.619i	− 0.493059i	−0.969135	−	0.246529i	$-0.920710\pi$
$271$	− 133.619i	− 0.493059i	0.969135	−	0.246529i	$-0.0792902\pi$
$272$	− 48.5472i	− 0.178482i
$273$	46.8444	0.171591
$274$	198.995i	0.726258i
$275$	0	0
$276$	9.32993	0.0338041
$277$	346.017i	1.24916i	0.780961	+	0.624580i	$0.214730\pi$
$277$	346.017i	1.24916i	−0.780961	+	0.624580i	$0.785270\pi$
$278$	24.9111	0.0896082
$279$	441.839	1.58365
$280$	−85.1348	−0.304053
$281$	− 89.0883i	− 0.317040i	−0.987356	−	0.158520i	$-0.949328\pi$
$281$	− 89.0883i	− 0.317040i	0.987356	−	0.158520i	$-0.0506723\pi$
$282$	− 19.7053i	− 0.0698769i
$283$	− 443.364i	− 1.56666i	−0.621608	−	0.783329i	$-0.713520\pi$
$283$	− 443.364i	− 1.56666i	0.621608	−	0.783329i	$-0.286480\pi$
$284$	−116.987	−0.411926
$285$	4.99466i	0.0175251i
$286$	0	0
$287$	719.401	2.50663
$288$	− 49.6711i	− 0.172469i
$289$	141.698	0.490304
$290$	−91.3653	−0.315053
$291$	−48.3175	−0.166040
$292$	80.9411i	0.277196i
$293$	− 84.3268i	− 0.287805i	−0.989592	−	0.143902i	$-0.954035\pi$
$293$	− 84.3268i	− 0.287805i	0.989592	−	0.143902i	$-0.0459651\pi$
$294$	− 87.5535i	− 0.297801i
$295$	−238.190	−0.807425
$296$	− 137.254i	− 0.463697i
$297$	0	0
$298$	236.088	0.792240
$299$	− 74.0226i	− 0.247567i
$300$	−4.68308	−0.0156103
$301$	−208.870	−0.693921
$302$	−145.946	−0.483263
$303$	38.0119i	0.125452i
$304$	19.0787i	0.0627589i
$305$	120.465i	0.394969i
$306$	−150.712	−0.492523
$307$	370.937i	1.20826i	0.796884	+	0.604132i	$0.206480\pi$
$307$	370.937i	1.20826i	−0.796884	+	0.604132i	$0.793520\pi$
$308$	0	0
$309$	−57.3273	−0.185525
$310$	− 159.124i	− 0.513303i
$311$	154.761	0.497623	0.248812	−	0.968552i	$-0.419960\pi$
$311$	154.761	0.497623	0.248812	+	0.968552i	$0.419960\pi$
$312$	9.84294	0.0315479
$313$	410.624	1.31190	0.655949	−	0.754805i	$-0.272268\pi$
$313$	410.624	1.31190	0.655949	+	0.754805i	$0.272268\pi$
$314$	− 260.774i	− 0.830492i
$315$	264.296i	0.839035i
$316$	− 200.821i	− 0.635510i
$317$	34.9608	0.110287	0.0551433	−	0.998478i	$-0.482438\pi$
$317$	34.9608	0.110287	0.0551433	+	0.998478i	$0.482438\pi$
$318$	− 9.98473i	− 0.0313985i
$319$	0	0
$320$	−17.8885	−0.0559017
$321$	− 46.4855i	− 0.144815i
$322$	−189.631	−0.588916
$323$	57.8886	0.179222
$324$	−150.253	−0.463745
$325$	37.1551i	0.114323i
$326$	− 386.105i	− 1.18437i
$327$	− 57.1458i	− 0.174758i
$328$	151.161	0.460856
$329$	400.509i	1.21735i
$330$	0	0
$331$	83.8861	0.253432	0.126716	−	0.991939i	$-0.459556\pi$
$331$	83.8861	0.253432	0.126716	+	0.991939i	$0.459556\pi$
$332$	− 54.3941i	− 0.163838i
$333$	−426.098	−1.27957
$334$	299.280	0.896049
$335$	−184.633	−0.551142
$336$	− 25.2156i	− 0.0750464i
$337$	201.420i	0.597685i	0.954303	+	0.298842i	$0.0966005\pi$
$337$	201.420i	0.597685i	−0.954303	+	0.298842i	$0.903400\pi$
$338$	160.909i	0.476063i
$339$	93.6684	0.276308
$340$	54.2775i	0.159640i
$341$	0	0
$342$	59.2287	0.173183
$343$	1119.93i	3.26512i
$344$	−43.8878	−0.127581
$345$	−10.4312	−0.0302353
$346$	−24.4146	−0.0705624
$347$	303.238i	0.873884i	0.899490	+	0.436942i	$0.143938\pi$
$347$	303.238i	0.873884i	−0.899490	+	0.436942i	$0.856062\pi$
$348$	− 27.0610i	− 0.0777614i
$349$	− 144.865i	− 0.415086i	−0.978226	−	0.207543i	$-0.933453\pi$
$349$	− 144.865i	− 0.415086i	0.978226	−	0.207543i	$-0.0665466\pi$
$350$	95.1836	0.271953
$351$	− 61.8769i	− 0.176287i
$352$	0	0
$353$	−50.4196	−0.142832	−0.0714159	−	0.997447i	$-0.522752\pi$
$353$	−50.4196	−0.142832	−0.0714159	+	0.997447i	$0.522752\pi$
$354$	− 70.5483i	− 0.199289i
$355$	130.795	0.368438
$356$	44.4982	0.124995
$357$	−76.5092	−0.214312
$358$	− 302.900i	− 0.846090i
$359$	37.1468i	0.103473i	0.998661	+	0.0517365i	$0.0164756\pi$
$359$	37.1468i	0.103473i	−0.998661	+	0.0517365i	$0.983524\pi$
$360$	55.5339i	0.154261i
$361$	338.250	0.936981
$362$	31.9370i	0.0882238i
$363$	0	0
$364$	−200.058	−0.549609
$365$	− 90.4949i	− 0.247931i
$366$	−35.6800	−0.0974863
$367$	181.999	0.495910	0.247955	−	0.968772i	$-0.420241\pi$
$367$	181.999	0.495910	0.247955	+	0.968772i	$0.420241\pi$
$368$	−39.8453	−0.108275
$369$	− 469.270i	− 1.27173i
$370$	153.455i	0.414744i
$371$	202.940i	0.547007i
$372$	47.1300	0.126694
$373$	13.6655i	0.0366368i	0.999832	+	0.0183184i	$0.00583126\pi$
$373$	13.6655i	0.0366368i	−0.999832	+	0.0183184i	$0.994169\pi$
$374$	0	0
$375$	5.23585	0.0139623
$376$	84.1551i	0.223817i
$377$	−214.699	−0.569492
$378$	−158.516	−0.419354
$379$	−58.9190	−0.155459	−0.0777295	−	0.996974i	$-0.524767\pi$
$379$	−58.9190	−0.155459	−0.0777295	+	0.996974i	$0.524767\pi$
$380$	− 21.3306i	− 0.0561333i
$381$	3.31923i	0.00871189i
$382$	− 47.6612i	− 0.124768i
$383$	202.084	0.527634	0.263817	−	0.964573i	$-0.415019\pi$
$383$	202.084	0.527634	0.263817	+	0.964573i	$0.415019\pi$
$384$	− 5.29831i	− 0.0137977i
$385$	0	0
$386$	−34.5486	−0.0895043
$387$	136.247i	0.352060i
$388$	206.349	0.531828
$389$	121.038	0.311153	0.155576	−	0.987824i	$-0.450277\pi$
$389$	121.038	0.311153	0.155576	+	0.987824i	$0.450277\pi$
$390$	−11.0047	−0.0282173
$391$	120.899i	0.309203i
$392$	373.914i	0.953862i
$393$	− 91.7494i	− 0.233459i
$394$	170.461	0.432642
$395$	224.525i	0.568418i
$396$	0	0
$397$	−545.583	−1.37426	−0.687132	−	0.726532i	$-0.741131\pi$
$397$	−545.583	−1.37426	−0.687132	+	0.726532i	$0.741131\pi$
$398$	− 424.316i	− 1.06612i
$399$	30.0676	0.0753573
$400$	20.0000	0.0500000
$401$	731.593	1.82442	0.912210	−	0.409722i	$-0.134374\pi$
$401$	731.593	1.82442	0.912210	+	0.409722i	$0.134374\pi$
$402$	− 54.6853i	− 0.136033i
$403$	− 373.924i	− 0.927851i
$404$	− 162.337i	− 0.401824i
$405$	167.988	0.414786
$406$	550.014i	1.35471i
$407$	0	0
$408$	−16.0761	−0.0394023
$409$	− 684.162i	− 1.67277i	−0.548144	−	0.836384i	$-0.684665\pi$
$409$	− 684.162i	− 1.67277i	0.548144	−	0.836384i	$-0.315335\pi$
$410$	−169.003	−0.412202
$411$	65.8959	0.160331
$412$	244.827	0.594240
$413$	1433.89i	3.47190i
$414$	123.697i	0.298786i
$415$	60.8144i	0.146541i
$416$	−42.0362	−0.101048
$417$	− 8.24916i	− 0.0197822i
$418$	0	0
$419$	211.603	0.505020	0.252510	−	0.967594i	$-0.418744\pi$
$419$	211.603	0.505020	0.252510	+	0.967594i	$0.418744\pi$
$420$	28.1919i	0.0671236i
$421$	−735.584	−1.74723	−0.873615	−	0.486617i	$-0.838231\pi$
$421$	−735.584	−1.74723	−0.873615	+	0.486617i	$0.838231\pi$
$422$	37.7127	0.0893666
$423$	261.255	0.617623
$424$	42.6417i	0.100570i
$425$	− 60.6840i	− 0.142786i
$426$	38.7396i	0.0909379i
$427$	725.195	1.69835
$428$	198.525i	0.463844i
$429$	0	0
$430$	49.0681	0.114112
$431$	42.1914i	0.0978920i	0.998801	+	0.0489460i	$0.0155862\pi$
$431$	42.1914i	0.0978920i	−0.998801	+	0.0489460i	$0.984414\pi$
$432$	−33.3074	−0.0771004
$433$	−423.211	−0.977393	−0.488696	−	0.872454i	$-0.662527\pi$
$433$	−423.211	−0.977393	−0.488696	+	0.872454i	$0.662527\pi$
$434$	−957.917	−2.20718
$435$	30.2551i	0.0695519i
$436$	244.052i	0.559753i
$437$	− 47.5122i	− 0.108724i
$438$	26.8032	0.0611945
$439$	192.175i	0.437755i	0.975752	+	0.218878i	$0.0702396\pi$
$439$	192.175i	0.437755i	−0.975752	+	0.218878i	$0.929760\pi$
$440$	0	0
$441$	1160.79	2.63219
$442$	127.546i	0.288566i
$443$	46.8395	0.105732	0.0528662	−	0.998602i	$-0.483164\pi$
$443$	46.8395	0.105732	0.0528662	+	0.998602i	$0.483164\pi$
$444$	−45.4510	−0.102367
$445$	−49.7505	−0.111799
$446$	437.199i	0.980267i
$447$	− 78.1790i	− 0.174897i
$448$	107.688i	0.240375i
$449$	−577.728	−1.28670	−0.643350	−	0.765573i	$-0.722456\pi$
$449$	−577.728	−1.28670	−0.643350	+	0.765573i	$0.722456\pi$
$450$	− 62.0888i	− 0.137975i
$451$	0	0
$452$	−400.029	−0.885019
$453$	48.3290i	0.106687i
$454$	−503.716	−1.10951
$455$	223.671	0.491585
$456$	6.31780	0.0138548
$457$	696.802i	1.52473i	0.647146	+	0.762366i	$0.275962\pi$
$457$	696.802i	1.52473i	−0.647146	+	0.762366i	$0.724038\pi$
$458$	− 134.398i	− 0.293446i
$459$	101.061i	0.220177i
$460$	44.5483	0.0968442
$461$	623.859i	1.35327i	0.736317	+	0.676637i	$0.236563\pi$
$461$	623.859i	1.35327i	−0.736317	+	0.676637i	$0.763437\pi$
$462$	0	0
$463$	627.644	1.35560	0.677801	−	0.735245i	$-0.262933\pi$
$463$	627.644	1.35560	0.677801	+	0.735245i	$0.262933\pi$
$464$	115.569i	0.249071i
$465$	−52.6929	−0.113318
$466$	241.361	0.517943
$467$	50.8874	0.108967	0.0544833	−	0.998515i	$-0.482649\pi$
$467$	50.8874	0.108967	0.0544833	+	0.998515i	$0.482649\pi$
$468$	130.499i	0.278844i
$469$	1111.48i	2.36989i
$470$	− 94.0883i	− 0.200188i
$471$	−86.3539	−0.183342
$472$	301.290i	0.638326i
$473$	0	0
$474$	−66.5008	−0.140297
$475$	23.8484i	0.0502071i
$476$	326.747	0.686444
$477$	132.379	0.277523
$478$	547.893	1.14622
$479$	− 341.082i	− 0.712070i	−0.934473	−	0.356035i	$-0.884128\pi$
$479$	− 341.082i	− 0.712070i	0.934473	−	0.356035i	$-0.115872\pi$
$480$	5.92369i	0.0123410i
$481$	360.603i	0.749695i
$482$	−327.168	−0.678773
$483$	62.7951i	0.130011i
$484$	0	0
$485$	−230.705	−0.475681
$486$	155.739i	0.320450i
$487$	−245.581	−0.504273	−0.252137	−	0.967692i	$-0.581133\pi$
$487$	−245.581	−0.504273	−0.252137	+	0.967692i	$0.581133\pi$
$488$	152.378	0.312250
$489$	−127.856	−0.261465
$490$	− 418.048i	− 0.853160i
$491$	− 828.775i	− 1.68793i	−0.536397	−	0.843966i	$-0.680215\pi$
$491$	− 828.775i	− 1.68793i	0.536397	−	0.843966i	$-0.319785\pi$
$492$	− 50.0560i	− 0.101740i
$493$	350.660	0.711277
$494$	− 50.1247i	− 0.101467i
$495$	0	0
$496$	−201.278	−0.405802
$497$	− 787.381i	− 1.58427i
$498$	−18.0123	−0.0361692
$499$	101.091	0.202587	0.101293	−	0.994857i	$-0.467702\pi$
$499$	101.091	0.202587	0.101293	+	0.994857i	$0.467702\pi$
$500$	−22.3607	−0.0447214
$501$	− 99.1049i	− 0.197814i
$502$	− 125.764i	− 0.250525i
$503$	− 138.778i	− 0.275900i	−0.990439	−	0.137950i	$-0.955949\pi$
$503$	− 138.778i	− 0.275900i	0.990439	−	0.137950i	$-0.0440513\pi$
$504$	334.311	0.663316
$505$	181.498i	0.359402i
$506$	0	0
$507$	53.2842	0.105097
$508$	− 14.1754i	− 0.0279043i
$509$	−417.255	−0.819755	−0.409878	−	0.912141i	$-0.634429\pi$
$509$	−417.255	−0.819755	−0.409878	+	0.912141i	$0.634429\pi$
$510$	17.9737	0.0352425
$511$	−544.774	−1.06609
$512$	22.6274i	0.0441942i
$513$	− 39.7164i	− 0.0774198i
$514$	− 326.834i	− 0.635864i
$515$	−273.725	−0.531504
$516$	14.5332i	0.0281651i
$517$	0	0
$518$	923.791	1.78338
$519$	8.08474i	0.0155775i
$520$	46.9978	0.0903805
$521$	128.293	0.246244	0.123122	−	0.992392i	$-0.460709\pi$
$521$	128.293	0.246244	0.123122	+	0.992392i	$0.460709\pi$
$522$	358.777	0.687313
$523$	− 244.529i	− 0.467551i	−0.972291	−	0.233775i	$-0.924892\pi$
$523$	− 244.529i	− 0.467551i	0.972291	−	0.233775i	$-0.0751080\pi$
$524$	391.833i	0.747774i
$525$	− 31.5195i	− 0.0600371i
$526$	−268.910	−0.511236
$527$	610.717i	1.15886i
$528$	0	0
$529$	−429.772	−0.812424
$530$	− 47.6749i	− 0.0899526i
$531$	935.337	1.76146
$532$	−128.409	−0.241371
$533$	−397.139	−0.745100
$534$	− 14.7353i	− 0.0275942i
$535$	− 221.958i	− 0.414875i
$536$	233.544i	0.435716i
$537$	−100.304	−0.186785
$538$	− 331.383i	− 0.615954i
$539$	0	0
$540$	37.2388	0.0689607
$541$	− 119.924i	− 0.221671i	−0.993839	−	0.110836i	$-0.964647\pi$
$541$	− 119.924i	− 0.221671i	0.993839	−	0.110836i	$-0.0353527\pi$
$542$	188.966	0.348645
$543$	10.5758	0.0194765
$544$	68.6562	0.126206
$545$	− 272.859i	− 0.500658i
$546$	66.2479i	0.121333i
$547$	− 898.038i	− 1.64175i	−0.571107	−	0.820875i	$-0.693486\pi$
$547$	− 898.038i	− 1.64175i	0.571107	−	0.820875i	$-0.306514\pi$
$548$	−281.421	−0.513542
$549$	− 473.049i	− 0.861656i
$550$	0	0
$551$	−137.807	−0.250103
$552$	13.1945i	0.0239031i
$553$	1351.63	2.44417
$554$	−489.342	−0.883290
$555$	50.8158	0.0915599
$556$	35.2296i	0.0633626i
$557$	− 588.320i	− 1.05623i	−0.849173	−	0.528115i	$-0.822899\pi$
$557$	− 588.320i	− 1.05623i	0.849173	−	0.528115i	$-0.177101\pi$
$558$	624.854i	1.11981i
$559$	115.305	0.206270
$560$	− 120.399i	− 0.214998i
$561$	0	0
$562$	125.990	0.224181
$563$	270.290i	0.480089i	0.970762	+	0.240045i	$0.0771620\pi$
$563$	270.290i	0.480089i	−0.970762	+	0.240045i	$0.922838\pi$
$564$	27.8675	0.0494104
$565$	447.245	0.791585
$566$	627.011	1.10779
$567$	− 1011.28i	− 1.78356i
$568$	− 165.445i	− 0.291276i
$569$	737.253i	1.29570i	0.761768	+	0.647850i	$0.224332\pi$
$569$	737.253i	1.29570i	−0.761768	+	0.647850i	$0.775668\pi$
$570$	−7.06352	−0.0123921
$571$	− 860.510i	− 1.50702i	−0.657435	−	0.753512i	$-0.728358\pi$
$571$	− 860.510i	− 1.50702i	0.657435	−	0.753512i	$-0.271642\pi$
$572$	0	0
$573$	−15.7827	−0.0275440
$574$	1017.39i	1.77245i
$575$	−49.8066	−0.0866201
$576$	70.2455	0.121954
$577$	90.2786	0.156462	0.0782311	−	0.996935i	$-0.475073\pi$
$577$	90.2786	0.156462	0.0782311	+	0.996935i	$0.475073\pi$
$578$	200.391i	0.346697i
$579$	11.4406i	0.0197592i
$580$	− 129.210i	− 0.222776i
$581$	366.099	0.630119
$582$	− 68.3313i	− 0.117408i
$583$	0	0
$584$	−114.468	−0.196007
$585$	− 145.902i	− 0.249405i
$586$	119.256	0.203509
$587$	722.379	1.23063	0.615315	−	0.788282i	$-0.289029\pi$
$587$	722.379	1.23063	0.615315	+	0.788282i	$0.289029\pi$
$588$	123.819	0.210577
$589$	− 240.007i	− 0.407483i
$590$	− 336.852i	− 0.570936i
$591$	− 56.4471i	− 0.0955112i
$592$	194.107	0.327884
$593$	15.2461i	0.0257102i	0.999917	+	0.0128551i	$0.00409201\pi$
$593$	15.2461i	0.0257102i	−0.999917	+	0.0128551i	$0.995908\pi$
$594$	0	0
$595$	−365.314	−0.613974
$596$	333.878i	0.560198i
$597$	−140.510	−0.235360
$598$	104.684	0.175057
$599$	5.05884	0.00844548	0.00422274	−	0.999991i	$-0.498656\pi$
$599$	5.05884	0.00844548	0.00422274	+	0.999991i	$0.498656\pi$
$600$	− 6.62288i	− 0.0110381i
$601$	791.103i	1.31631i	0.752882	+	0.658156i	$0.228663\pi$
$601$	791.103i	1.31631i	−0.752882	+	0.658156i	$0.771337\pi$
$602$	− 295.387i	− 0.490676i
$603$	725.023	1.20236
$604$	− 206.398i	− 0.341719i
$605$	0	0
$606$	−53.7569	−0.0887077
$607$	− 221.379i	− 0.364711i	−0.983233	−	0.182355i	$-0.941628\pi$
$607$	− 221.379i	− 0.364711i	0.983233	−	0.182355i	$-0.0583721\pi$
$608$	−26.9814	−0.0443772
$609$	182.134	0.299070
$610$	−170.364	−0.279285
$611$	− 221.097i	− 0.361861i
$612$	− 213.139i	− 0.348266i
$613$	− 1155.83i	− 1.88554i	−0.333450	−	0.942768i	$-0.608213\pi$
$613$	− 1155.83i	− 1.88554i	0.333450	−	0.942768i	$-0.391787\pi$
$614$	−524.585	−0.854372
$615$	55.9643i	0.0909989i
$616$	0	0
$617$	−509.365	−0.825551	−0.412775	−	0.910833i	$-0.635441\pi$
$617$	−509.365	−0.825551	−0.412775	+	0.910833i	$0.635441\pi$
$618$	− 81.0730i	− 0.131186i
$619$	56.2366	0.0908507	0.0454254	−	0.998968i	$-0.485536\pi$
$619$	56.2366	0.0908507	0.0454254	+	0.998968i	$0.485536\pi$
$620$	225.035	0.362960
$621$	82.9463	0.133569
$622$	218.865i	0.351873i
$623$	299.495i	0.480731i
$624$	13.9200i	0.0223077i
$625$	25.0000	0.0400000
$626$	580.711i	0.927653i
$627$	0	0
$628$	368.791	0.587247
$629$	− 588.960i	− 0.936344i
$630$	−373.771	−0.593288
$631$	−792.217	−1.25549	−0.627747	−	0.778417i	$-0.716023\pi$
$631$	−792.217	−1.25549	−0.627747	+	0.778417i	$0.716023\pi$
$632$	284.004	0.449374
$633$	− 12.4883i	− 0.0197288i
$634$	49.4421i	0.0779844i
$635$	15.8486i	0.0249584i
$636$	14.1205	0.0222021
$637$	− 982.369i	− 1.54218i
$638$	0	0
$639$	−513.613	−0.803776
$640$	− 25.2982i	− 0.0395285i
$641$	−1022.18	−1.59466	−0.797332	−	0.603541i	$-0.793756\pi$
$641$	−1022.18	−1.59466	−0.797332	+	0.603541i	$0.793756\pi$
$642$	65.7405	0.102400
$643$	−123.563	−0.192166	−0.0960831	−	0.995373i	$-0.530631\pi$
$643$	−123.563	−0.192166	−0.0960831	+	0.995373i	$0.530631\pi$
$644$	− 268.178i	− 0.416426i
$645$	− 16.2486i	− 0.0251916i
$646$	81.8669i	0.126729i
$647$	−861.871	−1.33210	−0.666052	−	0.745906i	$-0.732017\pi$
$647$	−861.871	−1.33210	−0.666052	+	0.745906i	$0.732017\pi$
$648$	− 212.490i	− 0.327917i
$649$	0	0
$650$	−52.5452	−0.0808388
$651$	317.208i	0.487263i
$652$	546.035	0.837477
$653$	−1162.24	−1.77985	−0.889925	−	0.456106i	$-0.849244\pi$
$653$	−1162.24	−1.77985	−0.889925	+	0.456106i	$0.849244\pi$
$654$	80.8164	0.123572
$655$	− 438.083i	− 0.668829i
$656$	213.774i	0.325874i
$657$	355.359i	0.540882i
$658$	−566.406	−0.860799
$659$	− 708.972i	− 1.07583i	−0.842999	−	0.537915i	$-0.819212\pi$
$659$	− 708.972i	− 1.07583i	0.842999	−	0.537915i	$-0.180788\pi$
$660$	0	0
$661$	600.497	0.908468	0.454234	−	0.890882i	$-0.349913\pi$
$661$	600.497	0.908468	0.454234	+	0.890882i	$0.349913\pi$
$662$	118.633i	0.179204i
$663$	42.2362	0.0637046
$664$	76.9249	0.115851
$665$	143.566	0.215889
$666$	− 602.594i	− 0.904796i
$667$	− 287.805i	− 0.431491i
$668$	423.246i	0.633602i
$669$	144.776	0.216406
$670$	− 261.110i	− 0.389716i
$671$	0	0
$672$	35.6602	0.0530658
$673$	435.294i	0.646797i	0.946263	+	0.323398i	$0.104825\pi$
$673$	435.294i	0.646797i	−0.946263	+	0.323398i	$0.895175\pi$
$674$	−284.850	−0.422627
$675$	−41.6342	−0.0616803
$676$	−227.560	−0.336627
$677$	− 160.653i	− 0.237302i	−0.992936	−	0.118651i	$-0.962143\pi$
$677$	− 160.653i	− 0.237302i	0.992936	−	0.118651i	$-0.0378569\pi$
$678$	132.467i	0.195379i
$679$	1388.83i	2.04541i
$680$	−76.7599	−0.112882
$681$	166.803i	0.244938i
$682$	0	0
$683$	−581.794	−0.851822	−0.425911	−	0.904765i	$-0.640046\pi$
$683$	−581.794	−0.851822	−0.425911	+	0.904765i	$0.640046\pi$
$684$	83.7621i	0.122459i
$685$	314.638	0.459326
$686$	−1583.83	−2.30879
$687$	−44.5051	−0.0647819
$688$	− 62.0668i	− 0.0902133i
$689$	− 112.031i	− 0.162599i
$690$	− 14.7519i	− 0.0213796i
$691$	−280.643	−0.406141	−0.203070	−	0.979164i	$-0.565092\pi$
$691$	−280.643	−0.406141	−0.203070	+	0.979164i	$0.565092\pi$
$692$	− 34.5274i	− 0.0498951i
$693$	0	0
$694$	−428.843	−0.617929
$695$	− 39.3879i	− 0.0566732i
$696$	38.2700	0.0549856
$697$	648.632	0.930606
$698$	204.870	0.293510
$699$	− 79.9254i	− 0.114343i
$700$	134.610i	0.192300i
$701$	− 661.193i	− 0.943214i	−0.881809	−	0.471607i	$-0.843674\pi$
$701$	− 661.193i	− 0.943214i	0.881809	−	0.471607i	$-0.156326\pi$
$702$	87.5071	0.124654
$703$	231.457i	0.329242i
$704$	0	0
$705$	−31.1568	−0.0441940
$706$	− 71.3041i	− 0.100997i
$707$	1092.61	1.54541
$708$	99.7703	0.140919
$709$	−439.220	−0.619492	−0.309746	−	0.950819i	$-0.600244\pi$
$709$	−439.220	−0.619492	−0.309746	+	0.950819i	$0.600244\pi$
$710$	184.973i	0.260525i
$711$	− 881.674i	− 1.24005i
$712$	62.9300i	0.0883848i
$713$	501.247	0.703012
$714$	− 108.200i	− 0.151541i
$715$	0	0
$716$	428.366	0.598276
$717$	− 181.432i	− 0.253043i
$718$	−52.5335	−0.0731665
$719$	−36.3206	−0.0505154	−0.0252577	−	0.999681i	$-0.508041\pi$
$719$	−36.3206	−0.0505154	−0.0252577	+	0.999681i	$0.508041\pi$
$720$	−78.5369	−0.109079
$721$	1647.81i	2.28545i
$722$	478.358i	0.662546i
$723$	108.340i	0.149848i
$724$	−45.1658	−0.0623836
$725$	144.461i	0.199257i
$726$	0	0
$727$	720.834	0.991519	0.495760	−	0.868460i	$-0.334890\pi$
$727$	720.834	0.991519	0.495760	+	0.868460i	$0.334890\pi$
$728$	− 282.924i	− 0.388632i
$729$	−624.568	−0.856746
$730$	127.979	0.175314
$731$	−188.323	−0.257624
$732$	− 50.4591i	− 0.0689332i
$733$	− 178.472i	− 0.243481i	−0.992562	−	0.121740i	$-0.961152\pi$
$733$	− 178.472i	− 0.243481i	0.992562	−	0.121740i	$-0.0388476\pi$
$734$	257.386i	0.350662i
$735$	−138.434	−0.188346
$736$	− 56.3497i	− 0.0765621i
$737$	0	0
$738$	663.648	0.899252
$739$	489.257i	0.662052i	0.943622	+	0.331026i	$0.107395\pi$
$739$	489.257i	0.662052i	−0.943622	+	0.331026i	$0.892605\pi$
$740$	−217.018	−0.293268
$741$	−16.5985	−0.0224001
$742$	−287.000	−0.386792
$743$	1173.01i	1.57875i	0.613913	+	0.789373i	$0.289594\pi$
$743$	1173.01i	1.57875i	−0.613913	+	0.789373i	$0.710406\pi$
$744$	66.6519i	0.0895859i
$745$	− 373.287i	− 0.501057i
$746$	−19.3260	−0.0259061
$747$	− 238.809i	− 0.319690i
$748$	0	0
$749$	−1336.17	−1.78394
$750$	7.40461i	0.00987281i
$751$	−200.689	−0.267229	−0.133614	−	0.991033i	$-0.542658\pi$
$751$	−200.689	−0.267229	−0.133614	+	0.991033i	$0.542658\pi$
$752$	−119.013	−0.158262
$753$	−41.6459	−0.0553066
$754$	− 303.630i	− 0.402692i
$755$	230.760i	0.305643i
$756$	− 224.175i	− 0.296528i
$757$	402.491	0.531692	0.265846	−	0.964015i	$-0.414349\pi$
$757$	402.491	0.531692	0.265846	+	0.964015i	$0.414349\pi$
$758$	− 83.3240i	− 0.109926i
$759$	0	0
$760$	30.1661	0.0396922
$761$	− 1253.11i	− 1.64666i	−0.567562	−	0.823331i	$-0.692113\pi$
$761$	− 1253.11i	− 1.64666i	0.567562	−	0.823331i	$-0.307887\pi$
$762$	−4.69410	−0.00616023
$763$	−1642.59	−2.15281
$764$	67.4031	0.0882240
$765$	238.297i	0.311499i
$766$	285.789i	0.373093i
$767$	− 791.566i	− 1.03203i
$768$	7.49294	0.00975643
$769$	− 89.5678i	− 0.116473i	−0.998303	−	0.0582365i	$-0.981452\pi$
$769$	− 89.5678i	− 0.116473i	0.998303	−	0.0582365i	$-0.0185478\pi$
$770$	0	0
$771$	−108.229	−0.140375
$772$	− 48.8592i	− 0.0632891i
$773$	−533.020	−0.689547	−0.344774	−	0.938686i	$-0.612044\pi$
$773$	−533.020	−0.689547	−0.344774	+	0.938686i	$0.612044\pi$
$774$	−192.683	−0.248944
$775$	−251.597	−0.324641
$776$	291.822i	0.376059i
$777$	− 305.908i	− 0.393704i
$778$	171.174i	0.220018i
$779$	−254.908	−0.327224
$780$	− 15.5631i	− 0.0199526i
$781$	0	0
$782$	−170.976	−0.218640
$783$	− 240.581i	− 0.307256i
$784$	−528.794	−0.674482
$785$	−412.321	−0.525249
$786$	129.753	0.165081
$787$	− 82.9231i	− 0.105366i	−0.998611	−	0.0526831i	$-0.983223\pi$
$787$	− 82.9231i	− 0.105366i	0.998611	−	0.0526831i	$-0.0167773\pi$
$788$	241.068i	0.305924i
$789$	89.0480i	0.112862i
$790$	−317.526	−0.401932
$791$	− 2692.39i	− 3.40378i
$792$	0	0
$793$	−400.337	−0.504838
$794$	− 771.571i	− 0.971752i
$795$	−15.7872	−0.0198582
$796$	600.073	0.753861
$797$	143.634	0.180218	0.0901088	−	0.995932i	$-0.471279\pi$
$797$	143.634	0.180218	0.0901088	+	0.995932i	$0.471279\pi$
$798$	42.5220i	0.0532857i
$799$	361.110i	0.451953i
$800$	28.2843i	0.0353553i
$801$	195.362	0.243898
$802$	1034.63i	1.29006i
$803$	0	0
$804$	77.3366	0.0961899
$805$	299.833i	0.372463i
$806$	528.808	0.656090
$807$	−109.736	−0.135980
$808$	229.579	0.284132
$809$	− 58.7770i	− 0.0726539i	−0.999340	−	0.0363269i	$-0.988434\pi$
$809$	− 58.7770i	− 0.0726539i	0.999340	−	0.0363269i	$-0.0115658\pi$
$810$	237.571i	0.293298i
$811$	79.9991i	0.0986426i	0.998783	+	0.0493213i	$0.0157058\pi$
$811$	79.9991i	0.0986426i	−0.998783	+	0.0493213i	$0.984294\pi$
$812$	−777.837	−0.957927
$813$	− 62.5749i	− 0.0769679i
$814$	0	0
$815$	−610.486	−0.749062
$816$	− 22.7351i	− 0.0278616i
$817$	74.0096	0.0905870
$818$	967.552	1.18283
$819$	−878.322	−1.07243
$820$	− 239.006i	− 0.291471i
$821$	− 631.160i	− 0.768770i	−0.923173	−	0.384385i	$-0.874413\pi$
$821$	− 631.160i	− 0.768770i	0.923173	−	0.384385i	$-0.125587\pi$
$822$	93.1909i	0.113371i
$823$	−200.492	−0.243611	−0.121806	−	0.992554i	$-0.538868\pi$
$823$	−200.492	−0.243611	−0.121806	+	0.992554i	$0.538868\pi$
$824$	346.238i	0.420191i
$825$	0	0
$826$	−2027.83	−2.45500
$827$	− 380.936i	− 0.460624i	−0.973117	−	0.230312i	$-0.926025\pi$
$827$	− 380.936i	− 0.460624i	0.973117	−	0.230312i	$-0.0739747\pi$
$828$	−174.934	−0.211273
$829$	1059.57	1.27813	0.639067	−	0.769151i	$-0.279321\pi$
$829$	1059.57	1.27813	0.639067	+	0.769151i	$0.279321\pi$
$830$	−86.0046	−0.103620
$831$	162.043i	0.194997i
$832$	− 59.4481i	− 0.0714520i
$833$	1604.47i	1.92613i
$834$	11.6661	0.0139881
$835$	− 473.204i	− 0.566711i
$836$	0	0
$837$	419.002	0.500600
$838$	299.252i	0.357103i
$839$	3.94706	0.00470448	0.00235224	−	0.999997i	$-0.499251\pi$
$839$	3.94706	0.00470448	0.00235224	+	0.999997i	$0.499251\pi$
$840$	−39.8694	−0.0474635
$841$	6.23825	0.00741766
$842$	− 1040.27i	− 1.23548i
$843$	− 41.7208i	− 0.0494909i
$844$	53.3338i	0.0631917i
$845$	254.420	0.301089
$846$	369.470i	0.436726i
$847$	0	0
$848$	−60.3045	−0.0711138
$849$	− 207.631i	− 0.244560i
$850$	85.8202	0.100965
$851$	−483.390	−0.568026
$852$	−54.7860	−0.0643028
$853$	418.055i	0.490099i	0.969511	+	0.245050i	$0.0788043\pi$
$853$	418.055i	0.490099i	−0.969511	+	0.245050i	$0.921196\pi$
$854$	1025.58i	1.20091i
$855$	− 93.6488i	− 0.109531i
$856$	−280.757	−0.327987
$857$	− 1487.56i	− 1.73578i	−0.496757	−	0.867890i	$-0.665476\pi$
$857$	− 1487.56i	− 1.73578i	0.496757	−	0.867890i	$-0.334524\pi$
$858$	0	0
$859$	590.367	0.687272	0.343636	−	0.939103i	$-0.388341\pi$
$859$	590.367	0.687272	0.343636	+	0.939103i	$0.388341\pi$
$860$	69.3927i	0.0806892i
$861$	336.902	0.391291
$862$	−59.6677	−0.0692201
$863$	772.532	0.895170	0.447585	−	0.894241i	$-0.352284\pi$
$863$	772.532	0.895170	0.447585	+	0.894241i	$0.352284\pi$
$864$	− 47.1038i	− 0.0545182i
$865$	38.6028i	0.0446276i
$866$	− 598.511i	− 0.691121i
$867$	66.3583	0.0765379
$868$	− 1354.70i	− 1.56071i
$869$	0	0
$870$	−42.7871	−0.0491806
$871$	− 613.580i	− 0.704455i
$872$	−345.142	−0.395805
$873$	905.943	1.03774
$874$	67.1925	0.0768792
$875$	− 150.499i	− 0.171998i
$876$	37.9054i	0.0432710i
$877$	1369.31i	1.56136i	0.624932	+	0.780679i	$0.285127\pi$
$877$	1369.31i	1.56136i	−0.624932	+	0.780679i	$0.714873\pi$
$878$	−271.776	−0.309540
$879$	− 39.4910i	− 0.0449271i
$880$	0	0
$881$	−684.598	−0.777069	−0.388535	−	0.921434i	$-0.627019\pi$
$881$	−684.598	−0.777069	−0.388535	+	0.921434i	$0.627019\pi$
$882$	1641.61i	1.86124i
$883$	1691.93	1.91612	0.958058	−	0.286574i	$-0.0925165\pi$
$883$	1691.93	1.91612	0.958058	+	0.286574i	$0.0925165\pi$
$884$	−180.378	−0.204047
$885$	−111.547	−0.126041
$886$	66.2410i	0.0747642i
$887$	− 1275.52i	− 1.43802i	−0.695002	−	0.719008i	$-0.744597\pi$
$887$	− 1275.52i	− 1.43802i	0.695002	−	0.719008i	$-0.255403\pi$
$888$	− 64.2774i	− 0.0723845i
$889$	95.4075	0.107320
$890$	− 70.3578i	− 0.0790537i
$891$	0	0
$892$	−618.293	−0.693153
$893$	− 141.914i	− 0.158918i
$894$	110.562	0.123671
$895$	−478.927	−0.535114
$896$	−152.294	−0.169971
$897$	− 34.6654i	− 0.0386460i
$898$	− 817.031i	− 0.909834i
$899$	− 1453.84i	− 1.61717i
$900$	87.8069	0.0975632
$901$	182.976i	0.203081i
$902$	0	0
$903$	−97.8157	−0.108323
$904$	− 565.726i	− 0.625803i
$905$	50.4969	0.0557976
$906$	−68.3475	−0.0754388
$907$	428.826	0.472796	0.236398	−	0.971656i	$-0.424033\pi$
$907$	428.826	0.472796	0.236398	+	0.971656i	$0.424033\pi$
$908$	− 712.362i	− 0.784540i
$909$	− 712.715i	− 0.784064i
$910$	316.319i	0.347603i
$911$	−772.047	−0.847472	−0.423736	−	0.905786i	$-0.639281\pi$
$911$	−772.047	−0.847472	−0.423736	+	0.905786i	$0.639281\pi$
$912$	8.93472i	0.00979684i
$913$	0	0
$914$	−985.427	−1.07815
$915$	56.4150i	0.0616557i
$916$	190.068	0.207497
$917$	−2637.23	−2.87594
$918$	−142.922	−0.155689
$919$	− 377.767i	− 0.411063i	−0.978650	−	0.205532i	$-0.934108\pi$
$919$	− 377.767i	− 0.411063i	0.978650	−	0.205532i	$-0.0658923\pi$
$920$	63.0009i	0.0684792i
$921$	173.713i	0.188614i
$922$	−882.270	−0.956909
$923$	434.666i	0.470927i
$924$	0	0
$925$	242.634	0.262307
$926$	887.623i	0.958556i
$927$	1074.87	1.15952
$928$	−163.439	−0.176120
$929$	−750.854	−0.808239	−0.404120	−	0.914706i	$-0.632422\pi$
$929$	−750.854	−0.808239	−0.404120	+	0.914706i	$0.632422\pi$
$930$	− 74.5191i	− 0.0801280i
$931$	− 630.544i	− 0.677276i
$932$	341.337i	0.366241i
$933$	72.4758	0.0776804
$934$	71.9656i	0.0770510i
$935$	0	0
$936$	−184.553	−0.197172
$937$	1637.28i	1.74736i	0.486499	+	0.873681i	$0.338274\pi$
$937$	1637.28i	1.74736i	−0.486499	+	0.873681i	$0.661726\pi$
$938$	−1571.87	−1.67576
$939$	192.299	0.204791
$940$	133.061	0.141554
$941$	− 1521.21i	− 1.61659i	−0.588779	−	0.808294i	$-0.700391\pi$
$941$	− 1521.21i	− 1.61659i	0.588779	−	0.808294i	$-0.299609\pi$
$942$	− 122.123i	− 0.129642i
$943$	− 532.366i	− 0.564546i
$944$	−426.088	−0.451365
$945$	250.636i	0.265223i
$946$	0	0
$947$	149.269	0.157623	0.0788114	−	0.996890i	$-0.474888\pi$
$947$	149.269	0.157623	0.0788114	+	0.996890i	$0.474888\pi$
$948$	− 94.0463i	− 0.0992050i
$949$	300.737	0.316899
$950$	−33.7267	−0.0355018
$951$	16.3725	0.0172160
$952$	462.090i	0.485389i
$953$	− 956.094i	− 1.00325i	−0.865086	−	0.501623i	$-0.832736\pi$
$953$	− 956.094i	− 1.00325i	0.865086	−	0.501623i	$-0.167264\pi$
$954$	187.212i	0.196239i
$955$	−75.3590	−0.0789099
$956$	774.838i	0.810500i
$957$	0	0
$958$	482.362	0.503510
$959$	− 1894.10i	− 1.97508i
$960$	−8.37736	−0.00872641
$961$	1571.04	1.63480
$962$	−509.970	−0.530114
$963$	871.594i	0.905082i
$964$	− 462.686i	− 0.479965i
$965$	54.6262i	0.0566075i
$966$	−88.8057	−0.0919314
$967$	− 664.517i	− 0.687194i	−0.939117	−	0.343597i	$-0.888355\pi$
$967$	− 664.517i	− 0.687194i	0.939117	−	0.343597i	$-0.111645\pi$
$968$	0	0
$969$	27.1097	0.0279770
$970$	− 326.267i	− 0.336357i
$971$	337.668	0.347753	0.173876	−	0.984768i	$-0.444371\pi$
$971$	337.668	0.347753	0.173876	+	0.984768i	$0.444371\pi$
$972$	−220.248	−0.226593
$973$	−237.113	−0.243693
$974$	− 347.304i	− 0.356575i
$975$	17.4000i	0.0178462i
$976$	215.495i	0.220794i
$977$	−953.101	−0.975539	−0.487769	−	0.872973i	$-0.662189\pi$
$977$	−953.101	−0.975539	−0.487769	+	0.872973i	$0.662189\pi$
$978$	− 180.816i	− 0.184884i
$979$	0	0
$980$	591.210	0.603275
$981$	1071.47i	1.09222i
$982$	1172.06	1.19355
$983$	412.745	0.419883	0.209941	−	0.977714i	$-0.432673\pi$
$983$	412.745	0.419883	0.209941	+	0.977714i	$0.432673\pi$
$984$	70.7899	0.0719409
$985$	− 269.522i	− 0.273627i
$986$	495.908i	0.502949i
$987$	187.562i	0.190032i
$988$	70.8870	0.0717480
$989$	154.567i	0.156286i
$990$	0	0
$991$	1106.98	1.11703	0.558515	−	0.829495i	$-0.311371\pi$
$991$	1106.98	1.11703	0.558515	+	0.829495i	$0.311371\pi$
$992$	− 284.649i	− 0.286945i
$993$	39.2846	0.0395615
$994$	1113.52	1.12025
$995$	−670.902	−0.674273
$996$	− 25.4732i	− 0.0255755i
$997$	1273.05i	1.27688i	0.769673	+	0.638439i	$0.220420\pi$
$997$	1273.05i	1.27688i	−0.769673	+	0.638439i	$0.779580\pi$
$998$	142.964i	0.143250i
$999$	−404.075	−0.404479

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1210.3.d.c.241.13		16
11.2	odd	10		110.3.h.b.51.1	yes	16
11.5	even	5		110.3.h.b.41.1	✓	16
11.10	odd	2	inner	1210.3.d.c.241.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.3.h.b.41.1	✓	16	11.5	even	5
110.3.h.b.51.1	yes	16	11.2	odd	10
1210.3.d.c.241.5		16	11.10	odd	2	inner
1210.3.d.c.241.13		16	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:	\( N \)	\(=\)	\( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1210.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +0.468308 q^{3} -2.00000 q^{4} +2.23607 q^{5} +0.662288i q^{6} -13.4610i q^{7} -2.82843i q^{8} -8.78069 q^{9} +3.16228i q^{10} -0.936617 q^{12} +7.43101i q^{13} +19.0367 q^{14} +1.04717 q^{15} +4.00000 q^{16} -12.1368i q^{17} -12.4178i q^{18} +4.76968i q^{19} -4.47214 q^{20} -6.30390i q^{21} -9.96131 q^{23} -1.32458i q^{24} +5.00000 q^{25} -10.5090 q^{26} -8.32685 q^{27} +26.9220i q^{28} +28.8922i q^{29} +1.48092i q^{30} -50.3194 q^{31} +5.65685i q^{32} +17.1640 q^{34} -30.0997i q^{35} +17.5614 q^{36} +48.5268 q^{37} -6.74534 q^{38} +3.48001i q^{39} -6.32456i q^{40} +53.4434i q^{41} +8.91506 q^{42} -15.5167i q^{43} -19.6342 q^{45} -14.0874i q^{46} -29.7533 q^{47} +1.87323 q^{48} -132.198 q^{49} +7.07107i q^{50} -5.68377i q^{51} -14.8620i q^{52} -15.0761 q^{53} -11.7759i q^{54} -38.0735 q^{56} +2.23368i q^{57} -40.8598 q^{58} -106.522 q^{59} -2.09434 q^{60} +53.8738i q^{61} -71.1624i q^{62} +118.197i q^{63} -8.00000 q^{64} +16.6162i q^{65} -82.5702 q^{67} +24.2736i q^{68} -4.66497 q^{69} +42.5674 q^{70} +58.4935 q^{71} +24.8355i q^{72} -40.4706i q^{73} +68.6272i q^{74} +2.34154 q^{75} -9.53935i q^{76} -4.92147 q^{78} +100.411i q^{79} +8.94427 q^{80} +75.1267 q^{81} -75.5804 q^{82} +27.1970i q^{83} +12.6078i q^{84} -27.1387i q^{85} +21.9439 q^{86} +13.5305i q^{87} -22.2491 q^{89} -27.7670i q^{90} +100.029 q^{91} +19.9226 q^{92} -23.5650 q^{93} -42.0776i q^{94} +10.6653i q^{95} +2.64915i q^{96} -103.175 q^{97} -186.957i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} - 32 q^{4} + 40 q^{9} + 32 q^{12} + 16 q^{14} + 40 q^{15} + 64 q^{16} + 108 q^{23} + 80 q^{25} - 292 q^{27} - 268 q^{31} - 16 q^{34} - 80 q^{36} + 44 q^{37} - 280 q^{38} - 16 q^{42} - 476 q^{47}+ \cdots + 296 q^{97}+O(q^{100}) \) 16 * q - 16 * q^3 - 32 * q^4 + 40 * q^9 + 32 * q^12 + 16 * q^14 + 40 * q^15 + 64 * q^16 + 108 * q^23 + 80 * q^25 - 292 * q^27 - 268 * q^31 - 16 * q^34 - 80 * q^36 + 44 * q^37 - 280 * q^38 - 16 * q^42 - 476 * q^47 - 64 * q^48 - 644 * q^49 - 396 * q^53 - 32 * q^56 + 192 * q^58 - 180 * q^59 - 80 * q^60 - 128 * q^64 + 188 * q^67 - 416 * q^69 + 308 * q^71 - 80 * q^75 + 8 * q^78 + 536 * q^81 + 16 * q^82 + 304 * q^86 - 176 * q^89 + 80 * q^91 - 216 * q^92 + 436 * q^93 + 296 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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