LMFDB - Embedded newform 1210.3.d.f.241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1210,3,Mod(241,1210)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$32.9701119876$
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} - 28 x^{14} + 336 x^{13} + 362 x^{12} - 6904 x^{11} - 3132 x^{10} + 87908 x^{9} + \cdots + 24267881 $ x^16 - 8x^15 - 28x^14 + 336x^13 + 362x^12 - 6904x^11 - 3132x^10 + 87908x^9 + 30150x^8 - 739436x^7 - 337806x^6 + 4062836x^5 + 2750033x^4 - 13356124x^3 - 12538020x^2 + 20049832*x + 24267881
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			241.1
Root			$3.46412 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1210.241
Dual form			1210.3.d.f.241.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.80911 q^{3} -2.00000 q^{4} +2.23607 q^{5} +3.97268i q^{6} -1.46309i q^{7} +2.82843i q^{8} -1.10889 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -2.80911 q^{3} -2.00000 q^{4} +2.23607 q^{5} +3.97268i q^{6} -1.46309i q^{7} +2.82843i q^{8} -1.10889 q^{9} -3.16228i q^{10} +5.61822 q^{12} +7.89390i q^{13} -2.06912 q^{14} -6.28137 q^{15} +4.00000 q^{16} -7.54998i q^{17} +1.56820i q^{18} -0.541631i q^{19} -4.47214 q^{20} +4.10999i q^{21} -21.5560 q^{23} -7.94537i q^{24} +5.00000 q^{25} +11.1637 q^{26} +28.3970 q^{27} +2.92618i q^{28} +24.9375i q^{29} +8.88319i q^{30} +5.90165 q^{31} -5.65685i q^{32} -10.6773 q^{34} -3.27157i q^{35} +2.21777 q^{36} +3.36409 q^{37} -0.765982 q^{38} -22.1748i q^{39} +6.32456i q^{40} -35.8406i q^{41} +5.81240 q^{42} +65.2759i q^{43} -2.47955 q^{45} +30.4848i q^{46} +30.7870 q^{47} -11.2364 q^{48} +46.8594 q^{49} -7.07107i q^{50} +21.2087i q^{51} -15.7878i q^{52} +56.1720 q^{53} -40.1594i q^{54} +4.13825 q^{56} +1.52150i q^{57} +35.2669 q^{58} -111.028 q^{59} +12.5627 q^{60} -7.82896i q^{61} -8.34619i q^{62} +1.62240i q^{63} -8.00000 q^{64} +17.6513i q^{65} +36.7187 q^{67} +15.1000i q^{68} +60.5532 q^{69} -4.62670 q^{70} +107.123 q^{71} -3.13641i q^{72} -138.355i q^{73} -4.75754i q^{74} -14.0456 q^{75} +1.08326i q^{76} -31.3600 q^{78} -93.9368i q^{79} +8.94427 q^{80} -69.7904 q^{81} -50.6863 q^{82} +2.67818i q^{83} -8.21998i q^{84} -16.8823i q^{85} +92.3141 q^{86} -70.0522i q^{87} -3.30755 q^{89} +3.50661i q^{90} +11.5495 q^{91} +43.1120 q^{92} -16.5784 q^{93} -43.5394i q^{94} -1.21112i q^{95} +15.8907i q^{96} +154.052 q^{97} -66.2691i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{3} - 32 q^{4} + 40 q^{9}+O(q^{10}) $ 16 * q + 24 * q^3 - 32 * q^4 + 40 * q^9	$ 16 q + 24 q^{3} - 32 q^{4} + 40 q^{9} - 48 q^{12} + 16 q^{14} + 64 q^{16} - 12 q^{23} + 80 q^{25} + 228 q^{27} + 212 q^{31} - 176 q^{34} - 80 q^{36} + 164 q^{37} + 120 q^{38} - 16 q^{42} + 164 q^{47} + 96 q^{48} + 396 q^{49} + 244 q^{53} - 32 q^{56} - 288 q^{58} - 180 q^{59} - 128 q^{64} + 148 q^{67} + 784 q^{69} + 228 q^{71} + 120 q^{75} - 392 q^{78} + 56 q^{81} + 176 q^{82} - 496 q^{86} - 16 q^{89} - 160 q^{91} + 24 q^{92} - 324 q^{93} + 1056 q^{97}+O(q^{100}) $ 16 * q + 24 * q^3 - 32 * q^4 + 40 * q^9 - 48 * q^12 + 16 * q^14 + 64 * q^16 - 12 * q^23 + 80 * q^25 + 228 * q^27 + 212 * q^31 - 176 * q^34 - 80 * q^36 + 164 * q^37 + 120 * q^38 - 16 * q^42 + 164 * q^47 + 96 * q^48 + 396 * q^49 + 244 * q^53 - 32 * q^56 - 288 * q^58 - 180 * q^59 - 128 * q^64 + 148 * q^67 + 784 * q^69 + 228 * q^71 + 120 * q^75 - 392 * q^78 + 56 * q^81 + 176 * q^82 - 496 * q^86 - 16 * q^89 - 160 * q^91 + 24 * q^92 - 324 * q^93 + 1056 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−2.80911	−0.936371	−0.468185	−	0.883630i	$-0.655092\pi$
$3$	−2.80911	−0.936371	−0.468185	+	0.883630i	$0.655092\pi$
$4$	−2.00000	−0.500000
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	3.97268i	0.662114i
$7$	− 1.46309i	− 0.209013i	−0.994524	−	0.104507i	$-0.966674\pi$
$7$	− 1.46309i	− 0.209013i	0.994524	−	0.104507i	$-0.0333263\pi$
$8$	2.82843i	0.353553i
$9$	−1.10889	−0.123210
$10$	− 3.16228i	− 0.316228i
$11$	0	0
$11$	0	0
$12$	5.61822	0.468185
$13$	7.89390i	0.607223i	0.952796	+	0.303611i	$0.0981924\pi$
$13$	7.89390i	0.607223i	−0.952796	+	0.303611i	$0.901808\pi$
$14$	−2.06912	−0.147795
$15$	−6.28137	−0.418758
$16$	4.00000	0.250000
$17$	− 7.54998i	− 0.444116i	−0.975033	−	0.222058i	$-0.928723\pi$
$17$	− 7.54998i	− 0.444116i	0.975033	−	0.222058i	$-0.0712775\pi$
$18$	1.56820i	0.0871224i
$19$	− 0.541631i	− 0.0285069i	−0.999898	−	0.0142534i	$-0.995463\pi$
$19$	− 0.541631i	− 0.0285069i	0.999898	−	0.0142534i	$-0.00453717\pi$
$20$	−4.47214	−0.223607
$21$	4.10999i	0.195714i
$22$	0	0
$23$	−21.5560	−0.937217	−0.468609	−	0.883406i	$-0.655245\pi$
$23$	−21.5560	−0.937217	−0.468609	+	0.883406i	$0.655245\pi$
$24$	− 7.94537i	− 0.331057i
$25$	5.00000	0.200000
$26$	11.1637	0.429371
$27$	28.3970	1.05174
$28$	2.92618i	0.104507i
$29$	24.9375i	0.859913i	0.902850	+	0.429957i	$0.141471\pi$
$29$	24.9375i	0.859913i	−0.902850	+	0.429957i	$0.858529\pi$
$30$	8.88319i	0.296106i
$31$	5.90165	0.190376	0.0951879	−	0.995459i	$-0.469655\pi$
$31$	5.90165	0.190376	0.0951879	+	0.995459i	$0.469655\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−10.6773	−0.314038
$35$	− 3.27157i	− 0.0934735i
$36$	2.21777	0.0616049
$37$	3.36409	0.0909213	0.0454606	−	0.998966i	$-0.485524\pi$
$37$	3.36409	0.0909213	0.0454606	+	0.998966i	$0.485524\pi$
$38$	−0.765982	−0.0201574
$39$	− 22.1748i	− 0.568586i
$40$	6.32456i	0.158114i
$41$	− 35.8406i	− 0.874161i	−0.899422	−	0.437081i	$-0.856012\pi$
$41$	− 35.8406i	− 0.874161i	0.899422	−	0.437081i	$-0.143988\pi$
$42$	5.81240	0.138391
$43$	65.2759i	1.51804i	0.651065	+	0.759022i	$0.274323\pi$
$43$	65.2759i	1.51804i	−0.651065	+	0.759022i	$0.725677\pi$
$44$	0	0
$45$	−2.47955	−0.0551011
$46$	30.4848i	0.662713i
$47$	30.7870	0.655042	0.327521	−	0.944844i	$-0.393787\pi$
$47$	30.7870	0.655042	0.327521	+	0.944844i	$0.393787\pi$
$48$	−11.2364	−0.234093
$49$	46.8594	0.956314
$50$	− 7.07107i	− 0.141421i
$51$	21.2087i	0.415858i
$52$	− 15.7878i	− 0.303611i
$53$	56.1720	1.05985	0.529925	−	0.848045i	$-0.322220\pi$
$53$	56.1720	1.05985	0.529925	+	0.848045i	$0.322220\pi$
$54$	− 40.1594i	− 0.743693i
$55$	0	0
$56$	4.13825	0.0738973
$57$	1.52150i	0.0266930i
$58$	35.2669	0.608050
$59$	−111.028	−1.88182	−0.940911	−	0.338654i	$-0.890029\pi$
$59$	−111.028	−1.88182	−0.940911	+	0.338654i	$0.890029\pi$
$60$	12.5627	0.209379
$61$	− 7.82896i	− 0.128344i	−0.997939	−	0.0641718i	$-0.979559\pi$
$61$	− 7.82896i	− 0.128344i	0.997939	−	0.0641718i	$-0.0204406\pi$
$62$	− 8.34619i	− 0.134616i
$63$	1.62240i	0.0257525i
$64$	−8.00000	−0.125000
$65$	17.6513i	0.271558i
$66$	0	0
$67$	36.7187	0.548041	0.274021	−	0.961724i	$-0.411646\pi$
$67$	36.7187	0.548041	0.274021	+	0.961724i	$0.411646\pi$
$68$	15.1000i	0.222058i
$69$	60.5532	0.877583
$70$	−4.62670	−0.0660958
$71$	107.123	1.50878	0.754389	−	0.656427i	$-0.227933\pi$
$71$	107.123	1.50878	0.754389	+	0.656427i	$0.227933\pi$
$72$	− 3.13641i	− 0.0435612i
$73$	− 138.355i	− 1.89528i	−0.319344	−	0.947639i	$-0.603463\pi$
$73$	− 138.355i	− 1.89528i	0.319344	−	0.947639i	$-0.396537\pi$
$74$	− 4.75754i	− 0.0642911i
$75$	−14.0456	−0.187274
$76$	1.08326i	0.0142534i
$77$	0	0
$78$	−31.3600	−0.402051
$79$	− 93.9368i	− 1.18907i	−0.804069	−	0.594536i	$-0.797336\pi$
$79$	− 93.9368i	− 1.18907i	0.804069	−	0.594536i	$-0.202664\pi$
$80$	8.94427	0.111803
$81$	−69.7904	−0.861610
$82$	−50.6863	−0.618125
$83$	2.67818i	0.0322672i	0.999870	+	0.0161336i	$0.00513571\pi$
$83$	2.67818i	0.0322672i	−0.999870	+	0.0161336i	$0.994864\pi$
$84$	− 8.21998i	− 0.0978569i
$85$	− 16.8823i	− 0.198615i
$86$	92.3141	1.07342
$87$	− 70.0522i	− 0.805197i
$88$	0	0
$89$	−3.30755	−0.0371635	−0.0185817	−	0.999827i	$-0.505915\pi$
$89$	−3.30755	−0.0371635	−0.0185817	+	0.999827i	$0.505915\pi$
$90$	3.50661i	0.0389623i
$91$	11.5495	0.126918
$92$	43.1120	0.468609
$93$	−16.5784	−0.178262
$94$	− 43.5394i	− 0.463185i
$95$	− 1.21112i	− 0.0127487i
$96$	15.8907i	0.165529i
$97$	154.052	1.58816	0.794081	−	0.607812i	$-0.207953\pi$
$97$	154.052	1.58816	0.794081	+	0.607812i	$0.207953\pi$
$98$	− 66.2691i	− 0.676216i
$99$	0	0
$100$	−10.0000	−0.100000
$101$	− 25.3551i	− 0.251041i	−0.992091	−	0.125520i	$-0.959940\pi$
$101$	− 25.3551i	− 0.251041i	0.992091	−	0.125520i	$-0.0400601\pi$
$102$	29.9937	0.294056
$103$	−105.433	−1.02362	−0.511810	−	0.859099i	$-0.671025\pi$
$103$	−105.433	−1.02362	−0.511810	+	0.859099i	$0.671025\pi$
$104$	−22.3273	−0.214686
$105$	9.19022i	0.0875259i
$106$	− 79.4393i	− 0.749427i
$107$	− 58.5281i	− 0.546992i	−0.961873	−	0.273496i	$-0.911820\pi$
$107$	− 58.5281i	− 0.546992i	0.961873	−	0.273496i	$-0.0881800\pi$
$108$	−56.7940	−0.525870
$109$	46.0539i	0.422513i	0.977431	+	0.211257i	$0.0677555\pi$
$109$	46.0539i	0.422513i	−0.977431	+	0.211257i	$0.932244\pi$
$110$	0	0
$111$	−9.45010	−0.0851360
$112$	− 5.85237i	− 0.0522533i
$113$	185.075	1.63783	0.818916	−	0.573914i	$-0.194576\pi$
$113$	185.075	1.63783	0.818916	+	0.573914i	$0.194576\pi$
$114$	2.15173	0.0188748
$115$	−48.2007	−0.419136
$116$	− 49.8750i	− 0.429957i
$117$	− 8.75344i	− 0.0748157i
$118$	157.017i	1.33065i
$119$	−11.0463	−0.0928261
$120$	− 17.7664i	− 0.148053i
$121$	0	0
$122$	−11.0718	−0.0907526
$123$	100.680i	0.818539i
$124$	−11.8033	−0.0951879
$125$	11.1803	0.0894427
$126$	2.29443	0.0182097
$127$	− 153.147i	− 1.20588i	−0.797786	−	0.602940i	$-0.793996\pi$
$127$	− 153.147i	− 1.20588i	0.797786	−	0.602940i	$-0.206004\pi$
$128$	11.3137i	0.0883883i
$129$	− 183.367i	− 1.42145i
$130$	24.9627	0.192021
$131$	− 158.760i	− 1.21191i	−0.795501	−	0.605953i	$-0.792792\pi$
$131$	− 158.760i	− 1.21191i	0.795501	−	0.605953i	$-0.207208\pi$
$132$	0	0
$133$	−0.792456	−0.00595832
$134$	− 51.9282i	− 0.387524i
$135$	63.4976	0.470353
$136$	21.3546	0.157019
$137$	−19.6172	−0.143191	−0.0715957	−	0.997434i	$-0.522809\pi$
$137$	−19.6172	−0.143191	−0.0715957	+	0.997434i	$0.522809\pi$
$138$	− 85.6352i	− 0.620545i
$139$	− 161.183i	− 1.15959i	−0.814764	−	0.579793i	$-0.803133\pi$
$139$	− 161.183i	− 1.15959i	0.814764	−	0.579793i	$-0.196867\pi$
$140$	6.54315i	0.0467368i
$141$	−86.4841	−0.613362
$142$	− 151.495i	− 1.06687i
$143$	0	0
$144$	−4.43555	−0.0308024
$145$	55.7619i	0.384565i
$146$	−195.664	−1.34016
$147$	−131.633	−0.895464
$148$	−6.72818	−0.0454606
$149$	− 235.764i	− 1.58231i	−0.611618	−	0.791153i	$-0.709481\pi$
$149$	− 235.764i	− 1.58231i	0.611618	−	0.791153i	$-0.290519\pi$
$150$	19.8634i	0.132423i
$151$	− 200.919i	− 1.33059i	−0.746582	−	0.665294i	$-0.768306\pi$
$151$	− 200.919i	− 1.33059i	0.746582	−	0.665294i	$-0.231694\pi$
$152$	1.53196	0.0100787
$153$	8.37207i	0.0547194i
$154$	0	0
$155$	13.1965	0.0851386
$156$	44.3497i	0.284293i
$157$	27.3761	0.174370	0.0871851	−	0.996192i	$-0.472213\pi$
$157$	27.3761	0.174370	0.0871851	+	0.996192i	$0.472213\pi$
$158$	−132.847	−0.840801
$159$	−157.794	−0.992413
$160$	− 12.6491i	− 0.0790569i
$161$	31.5384i	0.195891i
$162$	98.6985i	0.609250i
$163$	144.596	0.887094	0.443547	−	0.896251i	$-0.353720\pi$
$163$	144.596	0.887094	0.443547	+	0.896251i	$0.353720\pi$
$164$	71.6812i	0.437081i
$165$	0	0
$166$	3.78752	0.0228164
$167$	83.2517i	0.498513i	0.968437	+	0.249257i	$0.0801863\pi$
$167$	83.2517i	0.498513i	−0.968437	+	0.249257i	$0.919814\pi$
$168$	−11.6248	−0.0691953
$169$	106.686	0.631281
$170$	−23.8751	−0.140442
$171$	0.600608i	0.00351233i
$172$	− 130.552i	− 0.759022i
$173$	130.375i	0.753613i	0.926292	+	0.376807i	$0.122978\pi$
$173$	130.375i	0.753613i	−0.926292	+	0.376807i	$0.877022\pi$
$174$	−99.0687	−0.569361
$175$	− 7.31546i	− 0.0418026i
$176$	0	0
$177$	311.889	1.76208
$178$	4.67758i	0.0262786i
$179$	−93.7684	−0.523846	−0.261923	−	0.965089i	$-0.584357\pi$
$179$	−93.7684	−0.523846	−0.261923	+	0.965089i	$0.584357\pi$
$180$	4.95910	0.0275505
$181$	165.058	0.911921	0.455961	−	0.890000i	$-0.349296\pi$
$181$	165.058	0.911921	0.455961	+	0.890000i	$0.349296\pi$
$182$	− 16.3335i	− 0.0897442i
$183$	21.9924i	0.120177i
$184$	− 60.9696i	− 0.331356i
$185$	7.52233	0.0406612
$186$	23.4454i	0.126050i
$187$	0	0
$188$	−61.5739	−0.327521
$189$	− 41.5474i	− 0.219828i
$190$	−1.71279	−0.00901467
$191$	−178.305	−0.933534	−0.466767	−	0.884380i	$-0.654581\pi$
$191$	−178.305	−0.933534	−0.466767	+	0.884380i	$0.654581\pi$
$192$	22.4729	0.117046
$193$	254.038i	1.31626i	0.752905	+	0.658130i	$0.228652\pi$
$193$	254.038i	1.31626i	−0.752905	+	0.658130i	$0.771348\pi$
$194$	− 217.862i	− 1.12300i
$195$	− 49.5844i	− 0.254279i
$196$	−93.7187	−0.478157
$197$	− 269.113i	− 1.36606i	−0.730391	−	0.683029i	$-0.760662\pi$
$197$	− 269.113i	− 1.36606i	0.730391	−	0.683029i	$-0.239338\pi$
$198$	0	0
$199$	141.583	0.711473	0.355737	−	0.934586i	$-0.384230\pi$
$199$	141.583	0.711473	0.355737	+	0.934586i	$0.384230\pi$
$200$	14.1421i	0.0707107i
$201$	−103.147	−0.513170
$202$	−35.8576	−0.177513
$203$	36.4858	0.179733
$204$	− 42.4175i	− 0.207929i
$205$	− 80.1420i	− 0.390937i
$206$	149.105i	0.723809i
$207$	23.9032	0.115474
$208$	31.5756i	0.151806i
$209$	0	0
$210$	12.9969	0.0618901
$211$	16.9923i	0.0805323i	0.999189	+	0.0402662i	$0.0128206\pi$
$211$	16.9923i	0.0805323i	−0.999189	+	0.0402662i	$0.987179\pi$
$212$	−112.344	−0.529925
$213$	−300.921	−1.41278
$214$	−82.7712	−0.386781
$215$	145.961i	0.678890i
$216$	80.3188i	0.371847i
$217$	− 8.63465i	− 0.0397910i
$218$	65.1301	0.298762
$219$	388.655i	1.77468i
$220$	0	0
$221$	59.5987	0.269677
$222$	13.3645i	0.0602003i
$223$	78.7461	0.353122	0.176561	−	0.984290i	$-0.443503\pi$
$223$	78.7461	0.353122	0.176561	+	0.984290i	$0.443503\pi$
$224$	−8.27650	−0.0369487
$225$	−5.54444	−0.0246419
$226$	− 261.736i	− 1.15812i
$227$	314.132i	1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$227$	314.132i	1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$228$	− 3.04301i	− 0.0133465i
$229$	−249.814	−1.09089	−0.545445	−	0.838147i	$-0.683639\pi$
$229$	−249.814	−1.09089	−0.545445	+	0.838147i	$0.683639\pi$
$230$	68.1661i	0.296374i
$231$	0	0
$232$	−70.5338	−0.304025
$233$	282.076i	1.21063i	0.795987	+	0.605314i	$0.206952\pi$
$233$	282.076i	1.21063i	−0.795987	+	0.605314i	$0.793048\pi$
$234$	−12.3792	−0.0529027
$235$	68.8418	0.292944
$236$	222.055	0.940911
$237$	263.879i	1.11341i
$238$	15.6218i	0.0656380i
$239$	− 413.931i	− 1.73193i	−0.500106	−	0.865964i	$-0.666706\pi$
$239$	− 413.931i	− 1.73193i	0.500106	−	0.865964i	$-0.333294\pi$
$240$	−25.1255	−0.104689
$241$	− 141.848i	− 0.588581i	−0.955716	−	0.294291i	$-0.904917\pi$
$241$	− 141.848i	− 0.588581i	0.955716	−	0.294291i	$-0.0950833\pi$
$242$	0	0
$243$	−59.5240	−0.244955
$244$	15.6579i	0.0641718i
$245$	104.781	0.427676
$246$	142.383	0.578795
$247$	4.27558	0.0173100
$248$	16.6924i	0.0673080i
$249$	− 7.52330i	− 0.0302141i
$250$	− 15.8114i	− 0.0632456i
$251$	197.920	0.788525	0.394262	−	0.918998i	$-0.371000\pi$
$251$	197.920	0.788525	0.394262	+	0.918998i	$0.371000\pi$
$252$	− 3.24481i	− 0.0128762i
$253$	0	0
$254$	−216.582	−0.852686
$255$	47.4242i	0.185977i
$256$	16.0000	0.0625000
$257$	468.627	1.82345	0.911726	−	0.410799i	$-0.134750\pi$
$257$	468.627	1.82345	0.911726	+	0.410799i	$0.134750\pi$
$258$	−259.321	−1.00512
$259$	− 4.92197i	− 0.0190037i
$260$	− 35.3026i	− 0.135779i
$261$	− 27.6529i	− 0.105950i
$262$	−224.520	−0.856947
$263$	231.641i	0.880766i	0.897810	+	0.440383i	$0.145157\pi$
$263$	231.641i	0.880766i	−0.897810	+	0.440383i	$0.854843\pi$
$264$	0	0
$265$	125.605	0.473979
$266$	1.12070i	0.00421317i
$267$	9.29128	0.0347988
$268$	−73.4375	−0.274021
$269$	461.660	1.71621	0.858104	−	0.513476i	$-0.171643\pi$
$269$	461.660	1.71621	0.858104	+	0.513476i	$0.171643\pi$
$270$	− 89.7992i	− 0.332590i
$271$	253.602i	0.935799i	0.883782	+	0.467900i	$0.154989\pi$
$271$	253.602i	0.935799i	−0.883782	+	0.467900i	$0.845011\pi$
$272$	− 30.1999i	− 0.111029i
$273$	−32.4438	−0.118842
$274$	27.7429i	0.101252i
$275$	0	0
$276$	−121.106	−0.438791
$277$	232.424i	0.839077i	0.907738	+	0.419538i	$0.137808\pi$
$277$	232.424i	0.839077i	−0.907738	+	0.419538i	$0.862192\pi$
$278$	−227.947	−0.819952
$279$	−6.54426	−0.0234561
$280$	9.25341	0.0330479
$281$	458.923i	1.63318i	0.577220	+	0.816588i	$0.304137\pi$
$281$	458.923i	1.63318i	−0.577220	+	0.816588i	$0.695863\pi$
$282$	122.307i	0.433713i
$283$	222.307i	0.785537i	0.919637	+	0.392768i	$0.128483\pi$
$283$	222.307i	0.785537i	−0.919637	+	0.392768i	$0.871517\pi$
$284$	−214.247	−0.754389
$285$	3.40218i	0.0119375i
$286$	0	0
$287$	−52.4381	−0.182711
$288$	6.27281i	0.0217806i
$289$	231.998	0.802761
$290$	78.8592	0.271928
$291$	−432.749	−1.48711
$292$	276.711i	0.947639i
$293$	292.763i	0.999190i	0.866259	+	0.499595i	$0.166518\pi$
$293$	292.763i	0.999190i	−0.866259	+	0.499595i	$0.833482\pi$
$294$	186.157i	0.633189i
$295$	−248.265	−0.841577
$296$	9.51508i	0.0321455i
$297$	0	0
$298$	−333.420	−1.11886
$299$	− 170.161i	− 0.569100i
$300$	28.0911	0.0936371
$301$	95.5047	0.317291
$302$	−284.142	−0.940867
$303$	71.2254i	0.235067i
$304$	− 2.16652i	− 0.00712672i
$305$	− 17.5061i	− 0.0573970i
$306$	11.8399	0.0386925
$307$	− 24.3489i	− 0.0793122i	−0.999213	−	0.0396561i	$-0.987374\pi$
$307$	− 24.3489i	− 0.0793122i	0.999213	−	0.0396561i	$-0.0126262\pi$
$308$	0	0
$309$	296.173	0.958488
$310$	− 18.6626i	− 0.0602021i
$311$	−531.951	−1.71045	−0.855226	−	0.518255i	$-0.826582\pi$
$311$	−531.951	−1.71045	−0.855226	+	0.518255i	$0.826582\pi$
$312$	62.7199	0.201025
$313$	110.791	0.353965	0.176983	−	0.984214i	$-0.443366\pi$
$313$	110.791	0.353965	0.176983	+	0.984214i	$0.443366\pi$
$314$	− 38.7157i	− 0.123298i
$315$	3.62781i	0.0115168i
$316$	187.874i	0.594536i
$317$	354.519	1.11836	0.559178	−	0.829047i	$-0.311117\pi$
$317$	354.519	1.11836	0.559178	+	0.829047i	$0.311117\pi$
$318$	223.154i	0.701742i
$319$	0	0
$320$	−17.8885	−0.0559017
$321$	164.412i	0.512187i
$322$	44.6020	0.138516
$323$	−4.08930	−0.0126604
$324$	139.581	0.430805
$325$	39.4695i	0.121445i
$326$	− 204.490i	− 0.627270i
$327$	− 129.371i	− 0.395629i
$328$	101.373	0.309063
$329$	− 45.0442i	− 0.136912i
$330$	0	0
$331$	−398.211	−1.20306	−0.601528	−	0.798852i	$-0.705441\pi$
$331$	−398.211	−1.20306	−0.601528	+	0.798852i	$0.705441\pi$
$332$	− 5.35636i	− 0.0161336i
$333$	−3.73039	−0.0112024
$334$	117.736	0.352502
$335$	82.1056	0.245091
$336$	16.4400i	0.0489285i
$337$	− 107.257i	− 0.318269i	−0.987257	−	0.159134i	$-0.949130\pi$
$337$	− 107.257i	− 0.318269i	0.987257	−	0.159134i	$-0.0508703\pi$
$338$	− 150.877i	− 0.446383i
$339$	−519.896	−1.53362
$340$	33.7645i	0.0993074i
$341$	0	0
$342$	0.849388	0.00248359
$343$	− 140.251i	− 0.408895i
$344$	−184.628	−0.536710
$345$	135.401	0.392467
$346$	184.378	0.532885
$347$	− 114.816i	− 0.330881i	−0.986220	−	0.165440i	$-0.947095\pi$
$347$	− 114.816i	− 0.330881i	0.986220	−	0.165440i	$-0.0529045\pi$
$348$	140.104i	0.402599i
$349$	355.581i	1.01886i	0.860513	+	0.509428i	$0.170143\pi$
$349$	355.581i	1.01886i	−0.860513	+	0.509428i	$0.829857\pi$
$350$	−10.3456	−0.0295589
$351$	224.163i	0.638641i
$352$	0	0
$353$	−275.130	−0.779406	−0.389703	−	0.920941i	$-0.627422\pi$
$353$	−275.130	−0.779406	−0.389703	+	0.920941i	$0.627422\pi$
$354$	− 441.077i	− 1.24598i
$355$	239.535	0.674746
$356$	6.61510	0.0185817
$357$	31.0303	0.0869197
$358$	132.609i	0.370415i
$359$	− 153.054i	− 0.426335i	−0.977016	−	0.213167i	$-0.931622\pi$
$359$	− 153.054i	− 0.426335i	0.977016	−	0.213167i	$-0.0683780\pi$
$360$	− 7.01322i	− 0.0194812i
$361$	360.707	0.999187
$362$	− 233.427i	− 0.644826i
$363$	0	0
$364$	−23.0990	−0.0634588
$365$	− 309.372i	− 0.847594i
$366$	31.1020	0.0849781
$367$	360.602	0.982567	0.491283	−	0.871000i	$-0.336528\pi$
$367$	360.602	0.982567	0.491283	+	0.871000i	$0.336528\pi$
$368$	−86.2240	−0.234304
$369$	39.7432i	0.107705i
$370$	− 10.6382i	− 0.0287518i
$371$	− 82.1849i	− 0.221523i
$372$	33.1568	0.0891311
$373$	− 48.5421i	− 0.130140i	−0.997881	−	0.0650698i	$-0.979273\pi$
$373$	− 48.5421i	− 0.130140i	0.997881	−	0.0650698i	$-0.0207270\pi$
$374$	0	0
$375$	−31.4068	−0.0837516
$376$	87.0787i	0.231592i
$377$	−196.854	−0.522159
$378$	−58.7569	−0.155442
$379$	289.368	0.763504	0.381752	−	0.924265i	$-0.375321\pi$
$379$	289.368	0.763504	0.381752	+	0.924265i	$0.375321\pi$
$380$	2.42225i	0.00637434i
$381$	430.207i	1.12915i
$382$	252.161i	0.660108i
$383$	546.519	1.42694	0.713471	−	0.700684i	$-0.247122\pi$
$383$	546.519	1.42694	0.713471	+	0.700684i	$0.247122\pi$
$384$	− 31.7815i	− 0.0827643i
$385$	0	0
$386$	359.264	0.930736
$387$	− 72.3837i	− 0.187038i
$388$	−308.103	−0.794081
$389$	747.706	1.92212	0.961062	−	0.276334i	$-0.0891196\pi$
$389$	747.706	1.92212	0.961062	+	0.276334i	$0.0891196\pi$
$390$	−70.1230	−0.179803
$391$	162.747i	0.416233i
$392$	132.538i	0.338108i
$393$	445.974i	1.13479i
$394$	−380.584	−0.965949
$395$	− 210.049i	− 0.531770i
$396$	0	0
$397$	277.155	0.698123	0.349061	−	0.937100i	$-0.386500\pi$
$397$	277.155	0.698123	0.349061	+	0.937100i	$0.386500\pi$
$398$	− 200.229i	− 0.503088i
$399$	2.22610	0.00557919
$400$	20.0000	0.0500000
$401$	−642.558	−1.60239	−0.801194	−	0.598404i	$-0.795802\pi$
$401$	−642.558	−1.60239	−0.801194	+	0.598404i	$0.795802\pi$
$402$	145.872i	0.362866i
$403$	46.5870i	0.115600i
$404$	50.7103i	0.125520i
$405$	−156.056	−0.385324
$406$	− 51.5988i	− 0.127091i
$407$	0	0
$408$	−59.9874	−0.147028
$409$	− 692.557i	− 1.69329i	−0.532155	−	0.846647i	$-0.678617\pi$
$409$	− 692.557i	− 1.69329i	0.532155	−	0.846647i	$-0.321383\pi$
$410$	−113.338	−0.276434
$411$	55.1070	0.134080
$412$	210.866	0.511810
$413$	162.443i	0.393326i
$414$	− 33.8042i	− 0.0816527i
$415$	5.98859i	0.0144303i
$416$	44.6546	0.107343
$417$	452.780i	1.08580i
$418$	0	0
$419$	−36.7232	−0.0876448	−0.0438224	−	0.999039i	$-0.513954\pi$
$419$	−36.7232	−0.0876448	−0.0438224	+	0.999039i	$0.513954\pi$
$420$	− 18.3804i	− 0.0437629i
$421$	492.255	1.16925	0.584625	−	0.811303i	$-0.301242\pi$
$421$	492.255	1.16925	0.584625	+	0.811303i	$0.301242\pi$
$422$	24.0308	0.0569449
$423$	−34.1393	−0.0807075
$424$	158.879i	0.374714i
$425$	− 37.7499i	− 0.0888233i
$426$	425.567i	0.998984i
$427$	−11.4545	−0.0268255
$428$	117.056i	0.273496i
$429$	0	0
$430$	206.421	0.480048
$431$	− 793.034i	− 1.83999i	−0.391935	−	0.919993i	$-0.628194\pi$
$431$	− 793.034i	− 1.83999i	0.391935	−	0.919993i	$-0.371806\pi$
$432$	113.588	0.262935
$433$	479.867	1.10824	0.554119	−	0.832438i	$-0.313055\pi$
$433$	479.867	1.10824	0.554119	+	0.832438i	$0.313055\pi$
$434$	−12.2112	−0.0281365
$435$	− 156.641i	− 0.360095i
$436$	− 92.1078i	− 0.211257i
$437$	11.6754i	0.0267172i
$438$	549.642	1.25489
$439$	759.484i	1.73003i	0.501745	+	0.865016i	$0.332692\pi$
$439$	759.484i	1.73003i	−0.501745	+	0.865016i	$0.667308\pi$
$440$	0	0
$441$	−51.9618	−0.117827
$442$	− 84.2853i	− 0.190691i
$443$	−708.821	−1.60005	−0.800024	−	0.599968i	$-0.795180\pi$
$443$	−708.821	−1.60005	−0.800024	+	0.599968i	$0.795180\pi$
$444$	18.9002	0.0425680
$445$	−7.39591	−0.0166200
$446$	− 111.364i	− 0.249695i
$447$	662.287i	1.48163i
$448$	11.7047i	0.0261266i
$449$	−547.462	−1.21929	−0.609646	−	0.792674i	$-0.708688\pi$
$449$	−547.462	−1.21929	−0.609646	+	0.792674i	$0.708688\pi$
$450$	7.84102i	0.0174245i
$451$	0	0
$452$	−370.150	−0.818916
$453$	564.403i	1.24592i
$454$	444.250	0.978523
$455$	25.8255	0.0567592
$456$	−4.30346	−0.00943741
$457$	248.405i	0.543556i	0.962360	+	0.271778i	$0.0876116\pi$
$457$	248.405i	0.543556i	−0.962360	+	0.271778i	$0.912388\pi$
$458$	353.290i	0.771376i
$459$	− 214.397i	− 0.467095i
$460$	96.4014	0.209568
$461$	− 244.791i	− 0.531000i	−0.964111	−	0.265500i	$-0.914463\pi$
$461$	− 244.791i	− 0.531000i	0.964111	−	0.265500i	$-0.0855370\pi$
$462$	0	0
$463$	−136.860	−0.295594	−0.147797	−	0.989018i	$-0.547218\pi$
$463$	−136.860	−0.295594	−0.147797	+	0.989018i	$0.547218\pi$
$464$	99.7499i	0.214978i
$465$	−37.0704	−0.0797213
$466$	398.916	0.856043
$467$	−72.3385	−0.154900	−0.0774502	−	0.996996i	$-0.524678\pi$
$467$	−72.3385	−0.154900	−0.0774502	+	0.996996i	$0.524678\pi$
$468$	17.5069i	0.0374079i
$469$	− 53.7229i	− 0.114548i
$470$	− 97.3570i	− 0.207142i
$471$	−76.9026	−0.163275
$472$	− 314.033i	− 0.665325i
$473$	0	0
$474$	373.181	0.787302
$475$	− 2.70816i	− 0.00570138i
$476$	22.0926	0.0464131
$477$	−62.2885	−0.130584
$478$	−585.387	−1.22466
$479$	− 466.291i	− 0.973467i	−0.873550	−	0.486734i	$-0.838188\pi$
$479$	− 466.291i	− 0.973467i	0.873550	−	0.486734i	$-0.161812\pi$
$480$	35.5328i	0.0740266i
$481$	26.5558i	0.0552095i
$482$	−200.603	−0.416190
$483$	− 88.5949i	− 0.183426i
$484$	0	0
$485$	344.470	0.710248
$486$	84.1796i	0.173209i
$487$	103.235	0.211982	0.105991	−	0.994367i	$-0.466199\pi$
$487$	103.235	0.211982	0.105991	+	0.994367i	$0.466199\pi$
$488$	22.1436	0.0453763
$489$	−406.187	−0.830649
$490$	− 148.182i	− 0.302413i
$491$	127.057i	0.258772i	0.991594	+	0.129386i	$0.0413006\pi$
$491$	127.057i	0.258772i	−0.991594	+	0.129386i	$0.958699\pi$
$492$	− 201.361i	− 0.409270i
$493$	188.277	0.381901
$494$	− 6.04658i	− 0.0122400i
$495$	0	0
$496$	23.6066	0.0475939
$497$	− 156.731i	− 0.315355i
$498$	−10.6396	−0.0213646
$499$	147.609	0.295810	0.147905	−	0.989002i	$-0.452747\pi$
$499$	147.609	0.295810	0.147905	+	0.989002i	$0.452747\pi$
$500$	−22.3607	−0.0447214
$501$	− 233.863i	− 0.466793i
$502$	− 279.901i	− 0.557571i
$503$	223.441i	0.444217i	0.975022	+	0.222108i	$0.0712938\pi$
$503$	223.441i	0.444217i	−0.975022	+	0.222108i	$0.928706\pi$
$504$	−4.58885	−0.00910487
$505$	− 56.6958i	− 0.112269i
$506$	0	0
$507$	−299.694	−0.591113
$508$	306.294i	0.602940i
$509$	212.621	0.417723	0.208861	−	0.977945i	$-0.433024\pi$
$509$	212.621	0.417723	0.208861	+	0.977945i	$0.433024\pi$
$510$	67.0679	0.131506
$511$	−202.426	−0.396138
$512$	− 22.6274i	− 0.0441942i
$513$	− 15.3807i	− 0.0299819i
$514$	− 662.739i	− 1.28937i
$515$	−235.755	−0.457777
$516$	366.735i	0.710726i
$517$	0	0
$518$	−6.96072	−0.0134377
$519$	− 366.238i	− 0.705661i
$520$	−49.9254	−0.0960103
$521$	−530.677	−1.01857	−0.509287	−	0.860597i	$-0.670091\pi$
$521$	−530.677	−1.01857	−0.509287	+	0.860597i	$0.670091\pi$
$522$	−39.1070	−0.0749177
$523$	267.265i	0.511024i	0.966806	+	0.255512i	$0.0822440\pi$
$523$	267.265i	0.511024i	−0.966806	+	0.255512i	$0.917756\pi$
$524$	317.519i	0.605953i
$525$	20.5500i	0.0391428i
$526$	327.590	0.622795
$527$	− 44.5573i	− 0.0845489i
$528$	0	0
$529$	−64.3389	−0.121624
$530$	− 177.632i	− 0.335154i
$531$	123.117	0.231859
$532$	1.58491	0.00297916
$533$	282.922	0.530811
$534$	− 13.1399i	− 0.0246065i
$535$	− 130.873i	− 0.244622i
$536$	103.856i	0.193762i
$537$	263.406	0.490514
$538$	− 652.886i	− 1.21354i
$539$	0	0
$540$	−126.995	−0.235176
$541$	317.702i	0.587249i	0.955921	+	0.293624i	$0.0948615\pi$
$541$	317.702i	0.587249i	−0.955921	+	0.293624i	$0.905139\pi$
$542$	358.647	0.661710
$543$	−463.666	−0.853896
$544$	−42.7091	−0.0785094
$545$	102.980i	0.188954i
$546$	45.8825i	0.0840339i
$547$	836.627i	1.52948i	0.644338	+	0.764741i	$0.277133\pi$
$547$	836.627i	1.52948i	−0.644338	+	0.764741i	$0.722867\pi$
$548$	39.2344	0.0715957
$549$	8.68143i	0.0158132i
$550$	0	0
$551$	13.5069	0.0245135
$552$	171.270i	0.310272i
$553$	−137.438	−0.248532
$554$	328.698	0.593317
$555$	−21.1311	−0.0380740
$556$	322.365i	0.579793i
$557$	107.569i	0.193123i	0.995327	+	0.0965614i	$0.0307844\pi$
$557$	107.569i	0.193123i	−0.995327	+	0.0965614i	$0.969216\pi$
$558$	9.25498i	0.0165860i
$559$	−515.281	−0.921791
$560$	− 13.0863i	− 0.0233684i
$561$	0	0
$562$	649.015	1.15483
$563$	− 493.566i	− 0.876671i	−0.898812	−	0.438335i	$-0.855568\pi$
$563$	− 493.566i	− 0.876671i	0.898812	−	0.438335i	$-0.144432\pi$
$564$	172.968	0.306681
$565$	413.840	0.732461
$566$	314.389	0.555458
$567$	102.110i	0.180088i
$568$	302.990i	0.533434i
$569$	− 995.783i	− 1.75006i	−0.484070	−	0.875029i	$-0.660842\pi$
$569$	− 995.783i	− 1.75006i	0.484070	−	0.875029i	$-0.339158\pi$
$570$	4.81141	0.00844108
$571$	706.234i	1.23684i	0.785849	+	0.618419i	$0.212226\pi$
$571$	706.234i	1.23684i	−0.785849	+	0.618419i	$0.787774\pi$
$572$	0	0
$573$	500.879	0.874134
$574$	74.1587i	0.129196i
$575$	−107.780	−0.187443
$576$	8.87110	0.0154012
$577$	−273.794	−0.474512	−0.237256	−	0.971447i	$-0.576248\pi$
$577$	−273.794	−0.474512	−0.237256	+	0.971447i	$0.576248\pi$
$578$	− 328.095i	− 0.567638i
$579$	− 713.621i	− 1.23251i
$580$	− 111.524i	− 0.192282i
$581$	3.91842	0.00674427
$582$	611.999i	1.05154i
$583$	0	0
$584$	391.328	0.670082
$585$	− 19.5733i	− 0.0334586i
$586$	414.029	0.706534
$587$	−695.118	−1.18419	−0.592094	−	0.805869i	$-0.701698\pi$
$587$	−695.118	−1.18419	−0.592094	+	0.805869i	$0.701698\pi$
$588$	263.266	0.447732
$589$	− 3.19652i	− 0.00542702i
$590$	351.100i	0.595084i
$591$	755.970i	1.27914i
$592$	13.4564	0.0227303
$593$	− 1075.43i	− 1.81354i	−0.421626	−	0.906770i	$-0.638541\pi$
$593$	− 1075.43i	− 1.81354i	0.421626	−	0.906770i	$-0.361459\pi$
$594$	0	0
$595$	−24.7003	−0.0415131
$596$	471.527i	0.791153i
$597$	−397.723	−0.666203
$598$	−240.644	−0.402414
$599$	−358.122	−0.597867	−0.298934	−	0.954274i	$-0.596631\pi$
$599$	−358.122	−0.597867	−0.298934	+	0.954274i	$0.596631\pi$
$600$	− 39.7268i	− 0.0662114i
$601$	− 431.905i	− 0.718644i	−0.933214	−	0.359322i	$-0.883008\pi$
$601$	− 431.905i	− 0.718644i	0.933214	−	0.359322i	$-0.116992\pi$
$602$	− 135.064i	− 0.224359i
$603$	−40.7170	−0.0675240
$604$	401.837i	0.665294i
$605$	0	0
$606$	100.728	0.166218
$607$	237.515i	0.391293i	0.980675	+	0.195646i	$0.0626805\pi$
$607$	237.515i	0.391293i	−0.980675	+	0.195646i	$0.937320\pi$
$608$	−3.06393	−0.00503936
$609$	−102.493	−0.168297
$610$	−24.7573	−0.0405858
$611$	243.029i	0.397756i
$612$	− 16.7441i	− 0.0273597i
$613$	668.573i	1.09066i	0.838222	+	0.545329i	$0.183595\pi$
$613$	668.573i	1.09066i	−0.838222	+	0.545329i	$0.816405\pi$
$614$	−34.4345	−0.0560822
$615$	225.128i	0.366062i
$616$	0	0
$617$	−348.051	−0.564103	−0.282051	−	0.959399i	$-0.591015\pi$
$617$	−348.051	−0.564103	−0.282051	+	0.959399i	$0.591015\pi$
$618$	− 418.852i	− 0.677753i
$619$	−420.360	−0.679096	−0.339548	−	0.940589i	$-0.610274\pi$
$619$	−420.360	−0.679096	−0.339548	+	0.940589i	$0.610274\pi$
$620$	−26.3930	−0.0425693
$621$	−612.126	−0.985710
$622$	752.292i	1.20947i
$623$	4.83925i	0.00776766i
$624$	− 88.6994i	− 0.142146i
$625$	25.0000	0.0400000
$626$	− 156.682i	− 0.250291i
$627$	0	0
$628$	−54.7523	−0.0871851
$629$	− 25.3988i	− 0.0403796i
$630$	5.13049	0.00814364
$631$	−1099.75	−1.74287	−0.871434	−	0.490513i	$-0.836810\pi$
$631$	−1099.75	−1.74287	−0.871434	+	0.490513i	$0.836810\pi$
$632$	265.693	0.420401
$633$	− 47.7333i	− 0.0754081i
$634$	− 501.366i	− 0.790797i
$635$	− 342.447i	− 0.539286i
$636$	315.587	0.496206
$637$	369.903i	0.580695i
$638$	0	0
$639$	−118.788	−0.185896
$640$	25.2982i	0.0395285i
$641$	−299.687	−0.467530	−0.233765	−	0.972293i	$-0.575105\pi$
$641$	−299.687	−0.467530	−0.233765	+	0.972293i	$0.575105\pi$
$642$	232.514	0.362171
$643$	1249.01	1.94247	0.971233	−	0.238132i	$-0.0765351\pi$
$643$	1249.01	1.94247	0.971233	+	0.238132i	$0.0765351\pi$
$644$	− 63.0768i	− 0.0979454i
$645$	− 410.022i	− 0.635693i
$646$	5.78315i	0.00895224i
$647$	−333.967	−0.516178	−0.258089	−	0.966121i	$-0.583093\pi$
$647$	−333.967	−0.516178	−0.258089	+	0.966121i	$0.583093\pi$
$648$	− 197.397i	− 0.304625i
$649$	0	0
$650$	55.8183	0.0858743
$651$	24.2557i	0.0372592i
$652$	−289.193	−0.443547
$653$	526.840	0.806799	0.403400	−	0.915024i	$-0.367828\pi$
$653$	526.840	0.806799	0.403400	+	0.915024i	$0.367828\pi$
$654$	−182.958	−0.279752
$655$	− 354.997i	− 0.541981i
$656$	− 143.362i	− 0.218540i
$657$	153.420i	0.233517i
$658$	−63.7021	−0.0968117
$659$	− 13.3676i	− 0.0202847i	−0.999949	−	0.0101424i	$-0.996772\pi$
$659$	− 13.3676i	− 0.0202847i	0.999949	−	0.0101424i	$-0.00322847\pi$
$660$	0	0
$661$	−421.870	−0.638231	−0.319115	−	0.947716i	$-0.603386\pi$
$661$	−421.870	−0.638231	−0.319115	+	0.947716i	$0.603386\pi$
$662$	563.156i	0.850689i
$663$	−167.420	−0.252518
$664$	−7.57503	−0.0114082
$665$	−1.77199	−0.00266464
$666$	5.27558i	0.00792128i
$667$	− 537.552i	− 0.805925i
$668$	− 166.503i	− 0.249257i
$669$	−221.207	−0.330653
$670$	− 116.115i	− 0.173306i
$671$	0	0
$672$	23.2496	0.0345976
$673$	1235.10i	1.83522i	0.397479	+	0.917611i	$0.369885\pi$
$673$	1235.10i	1.83522i	−0.397479	+	0.917611i	$0.630115\pi$
$674$	−151.684	−0.225050
$675$	141.985	0.210348
$676$	−213.373	−0.315640
$677$	− 873.072i	− 1.28962i	−0.764343	−	0.644810i	$-0.776937\pi$
$677$	− 873.072i	− 1.28962i	0.764343	−	0.644810i	$-0.223063\pi$
$678$	735.245i	1.08443i
$679$	− 225.392i	− 0.331947i
$680$	47.7502	0.0702209
$681$	− 882.432i	− 1.29579i
$682$	0	0
$683$	−90.8680	−0.133043	−0.0665213	−	0.997785i	$-0.521190\pi$
$683$	−90.8680	−0.133043	−0.0665213	+	0.997785i	$0.521190\pi$
$684$	− 1.20122i	− 0.00175616i
$685$	−43.8654	−0.0640371
$686$	−198.345	−0.289133
$687$	701.755	1.02148
$688$	261.104i	0.379511i
$689$	443.416i	0.643565i
$690$	− 191.486i	− 0.277516i
$691$	−145.003	−0.209845	−0.104922	−	0.994480i	$-0.533459\pi$
$691$	−145.003	−0.209845	−0.104922	+	0.994480i	$0.533459\pi$
$692$	− 260.750i	− 0.376807i
$693$	0	0
$694$	−162.374	−0.233968
$695$	− 360.415i	− 0.518583i
$696$	198.137	0.284680
$697$	−270.596	−0.388229
$698$	502.867	0.720440
$699$	− 792.383i	− 1.13360i
$700$	14.6309i	0.0209013i
$701$	74.4465i	0.106200i	0.998589	+	0.0531002i	$0.0169103\pi$
$701$	74.4465i	0.106200i	−0.998589	+	0.0531002i	$0.983090\pi$
$702$	317.014	0.451587
$703$	− 1.82209i	− 0.00259188i
$704$	0	0
$705$	−193.384	−0.274304
$706$	389.093i	0.551123i
$707$	−37.0969	−0.0524708
$708$	−623.778	−0.881042
$709$	−183.712	−0.259114	−0.129557	−	0.991572i	$-0.541355\pi$
$709$	−183.712	−0.259114	−0.129557	+	0.991572i	$0.541355\pi$
$710$	− 338.754i	− 0.477118i
$711$	104.165i	0.146505i
$712$	− 9.35516i	− 0.0131393i
$713$	−127.216	−0.178423
$714$	− 43.8835i	− 0.0614615i
$715$	0	0
$716$	187.537	0.261923
$717$	1162.78i	1.62173i
$718$	−216.451	−0.301464
$719$	182.690	0.254089	0.127045	−	0.991897i	$-0.459451\pi$
$719$	182.690	0.254089	0.127045	+	0.991897i	$0.459451\pi$
$720$	−9.91819	−0.0137753
$721$	154.258i	0.213950i
$722$	− 510.116i	− 0.706532i
$723$	398.467i	0.551130i
$724$	−330.115	−0.455961
$725$	124.687i	0.171983i
$726$	0	0
$727$	−1204.32	−1.65655	−0.828277	−	0.560318i	$-0.810679\pi$
$727$	−1204.32	−1.65655	−0.828277	+	0.560318i	$0.810679\pi$
$728$	32.6669i	0.0448721i
$729$	795.323	1.09098
$730$	−437.518	−0.599339
$731$	492.832	0.674188
$732$	− 43.9848i	− 0.0600886i
$733$	− 1031.77i	− 1.40760i	−0.710396	−	0.703802i	$-0.751484\pi$
$733$	− 1031.77i	− 1.40760i	0.710396	−	0.703802i	$-0.248516\pi$
$734$	− 509.968i	− 0.694779i
$735$	−294.341	−0.400464
$736$	121.939i	0.165678i
$737$	0	0
$738$	56.2054	0.0761591
$739$	589.384i	0.797543i	0.917050	+	0.398771i	$0.130563\pi$
$739$	589.384i	0.797543i	−0.917050	+	0.398771i	$0.869437\pi$
$740$	−15.0447	−0.0203306
$741$	−12.0106	−0.0162086
$742$	−116.227	−0.156640
$743$	− 85.0983i	− 0.114533i	−0.998359	−	0.0572667i	$-0.981761\pi$
$743$	− 85.0983i	− 0.114533i	0.998359	−	0.0572667i	$-0.0182385\pi$
$744$	− 46.8908i	− 0.0630252i
$745$	− 527.184i	− 0.707629i
$746$	−68.6489	−0.0920226
$747$	− 2.96980i	− 0.00397563i
$748$	0	0
$749$	−85.6320	−0.114328
$750$	44.4160i	0.0592213i
$751$	547.171	0.728590	0.364295	−	0.931284i	$-0.381310\pi$
$751$	547.171	0.728590	0.364295	+	0.931284i	$0.381310\pi$
$752$	123.148	0.163760
$753$	−555.979	−0.738352
$754$	278.393i	0.369222i
$755$	− 449.268i	− 0.595057i
$756$	83.0949i	0.109914i
$757$	370.910	0.489973	0.244987	−	0.969526i	$-0.421216\pi$
$757$	370.910	0.489973	0.244987	+	0.969526i	$0.421216\pi$
$758$	− 409.228i	− 0.539879i
$759$	0	0
$760$	3.42558	0.00450734
$761$	751.173i	0.987087i	0.869721	+	0.493544i	$0.164299\pi$
$761$	751.173i	0.987087i	−0.869721	+	0.493544i	$0.835701\pi$
$762$	608.404	0.798430
$763$	67.3811	0.0883108
$764$	356.610	0.466767
$765$	18.7205i	0.0244713i
$766$	− 772.895i	− 1.00900i
$767$	− 876.440i	− 1.14269i
$768$	−44.9458	−0.0585232
$769$	− 369.638i	− 0.480674i	−0.970690	−	0.240337i	$-0.922742\pi$
$769$	− 369.638i	− 0.480674i	0.970690	−	0.240337i	$-0.0772580\pi$
$770$	0	0
$771$	−1316.43	−1.70743
$772$	− 508.076i	− 0.658130i
$773$	−1316.58	−1.70321	−0.851606	−	0.524182i	$-0.824371\pi$
$773$	−1316.58	−1.70321	−0.851606	+	0.524182i	$0.824371\pi$
$774$	−102.366	−0.132256
$775$	29.5082	0.0380751
$776$	435.724i	0.561500i
$777$	13.8264i	0.0177946i
$778$	− 1057.42i	− 1.35915i
$779$	−19.4124	−0.0249196
$780$	99.1689i	0.127140i
$781$	0	0
$782$	230.159	0.294322
$783$	708.150i	0.904406i
$784$	187.437	0.239078
$785$	61.2149	0.0779807
$786$	630.702	0.802420
$787$	− 737.823i	− 0.937513i	−0.883327	−	0.468757i	$-0.844702\pi$
$787$	− 737.823i	− 0.937513i	0.883327	−	0.468757i	$-0.155298\pi$
$788$	538.227i	0.683029i
$789$	− 650.707i	− 0.824723i
$790$	−297.054	−0.376018
$791$	− 270.782i	− 0.342328i
$792$	0	0
$793$	61.8010	0.0779331
$794$	− 391.956i	− 0.493647i
$795$	−352.837	−0.443820
$796$	−283.166	−0.355737
$797$	−90.9571	−0.114124	−0.0570622	−	0.998371i	$-0.518173\pi$
$797$	−90.9571	−0.114124	−0.0570622	+	0.998371i	$0.518173\pi$
$798$	− 3.14818i	− 0.00394509i
$799$	− 232.441i	− 0.290915i
$800$	− 28.2843i	− 0.0353553i
$801$	3.66770	0.00457890
$802$	908.714i	1.13306i
$803$	0	0
$804$	206.294	0.256585
$805$	70.5220i	0.0876050i
$806$	65.8839	0.0817419
$807$	−1296.85	−1.60701
$808$	71.7151	0.0887564
$809$	− 78.3659i	− 0.0968676i	−0.998826	−	0.0484338i	$-0.984577\pi$
$809$	− 78.3659i	− 0.0968676i	0.998826	−	0.0484338i	$-0.0154230\pi$
$810$	220.697i	0.272465i
$811$	422.180i	0.520567i	0.965532	+	0.260284i	$0.0838161\pi$
$811$	422.180i	0.520567i	−0.965532	+	0.260284i	$0.916184\pi$
$812$	−72.9717	−0.0898666
$813$	− 712.395i	− 0.876255i
$814$	0	0
$815$	323.327	0.396720
$816$	84.8349i	0.103964i
$817$	35.3555	0.0432747
$818$	−979.424	−1.19734
$819$	−12.8071	−0.0156375
$820$	160.284i	0.195468i
$821$	884.667i	1.07755i	0.842450	+	0.538774i	$0.181112\pi$
$821$	884.667i	1.07755i	−0.842450	+	0.538774i	$0.818888\pi$
$822$	− 77.9330i	− 0.0948090i
$823$	926.015	1.12517	0.562585	−	0.826739i	$-0.309807\pi$
$823$	926.015	1.12517	0.562585	+	0.826739i	$0.309807\pi$
$824$	− 298.209i	− 0.361904i
$825$	0	0
$826$	229.730	0.278123
$827$	− 1069.26i	− 1.29294i	−0.762940	−	0.646469i	$-0.776245\pi$
$827$	− 1069.26i	− 1.29294i	0.762940	−	0.646469i	$-0.223755\pi$
$828$	−47.8064	−0.0577371
$829$	184.934	0.223081	0.111540	−	0.993760i	$-0.464422\pi$
$829$	184.934	0.223081	0.111540	+	0.993760i	$0.464422\pi$
$830$	8.46914	0.0102038
$831$	− 652.906i	− 0.785687i
$832$	− 63.1512i	− 0.0759028i
$833$	− 353.787i	− 0.424714i
$834$	640.327	0.767779
$835$	186.157i	0.222942i
$836$	0	0
$837$	167.589	0.200226
$838$	51.9344i	0.0619742i
$839$	−469.478	−0.559568	−0.279784	−	0.960063i	$-0.590263\pi$
$839$	−469.478	−0.559568	−0.279784	+	0.960063i	$0.590263\pi$
$840$	−25.9939	−0.0309451
$841$	219.122	0.260550
$842$	− 696.153i	− 0.826785i
$843$	− 1289.17i	− 1.52926i
$844$	− 33.9846i	− 0.0402662i
$845$	238.558	0.282317
$846$	48.2802i	0.0570688i
$847$	0	0
$848$	224.688	0.264962
$849$	− 624.485i	− 0.735554i
$850$	−53.3864	−0.0628075
$851$	−72.5163	−0.0852130
$852$	601.843	0.706388
$853$	346.679i	0.406423i	0.979135	+	0.203212i	$0.0651380\pi$
$853$	346.679i	0.406423i	−0.979135	+	0.203212i	$0.934862\pi$
$854$	16.1991i	0.0189685i
$855$	1.34300i	0.00157076i
$856$	165.542	0.193391
$857$	− 1210.35i	− 1.41231i	−0.708057	−	0.706155i	$-0.750428\pi$
$857$	− 1210.35i	− 1.41231i	0.708057	−	0.706155i	$-0.249572\pi$
$858$	0	0
$859$	1346.17	1.56714	0.783570	−	0.621304i	$-0.213397\pi$
$859$	1346.17	1.56714	0.783570	+	0.621304i	$0.213397\pi$
$860$	− 291.923i	− 0.339445i
$861$	147.305	0.171085
$862$	−1121.52	−1.30107
$863$	785.108	0.909743	0.454872	−	0.890557i	$-0.349685\pi$
$863$	785.108	0.909743	0.454872	+	0.890557i	$0.349685\pi$
$864$	− 160.638i	− 0.185923i
$865$	291.528i	0.337026i
$866$	− 678.634i	− 0.783642i
$867$	−651.708	−0.751682
$868$	17.2693i	0.0198955i
$869$	0	0
$870$	−221.524	−0.254626
$871$	289.854i	0.332783i
$872$	−130.260	−0.149381
$873$	−170.826	−0.195677
$874$	16.5115	0.0188919
$875$	− 16.3579i	− 0.0186947i
$876$	− 777.311i	− 0.887341i
$877$	498.952i	0.568930i	0.958686	+	0.284465i	$0.0918160\pi$
$877$	498.952i	0.568930i	−0.958686	+	0.284465i	$0.908184\pi$
$878$	1074.07	1.22332
$879$	− 822.403i	− 0.935613i
$880$	0	0
$881$	772.928	0.877330	0.438665	−	0.898651i	$-0.355451\pi$
$881$	772.928	0.877330	0.438665	+	0.898651i	$0.355451\pi$
$882$	73.4850i	0.0833164i
$883$	807.206	0.914163	0.457082	−	0.889425i	$-0.348895\pi$
$883$	807.206	0.914163	0.457082	+	0.889425i	$0.348895\pi$
$884$	−119.197	−0.134839
$885$	697.405	0.788028
$886$	1002.42i	1.13140i
$887$	− 1009.38i	− 1.13797i	−0.822347	−	0.568986i	$-0.807336\pi$
$887$	− 1009.38i	− 1.13797i	0.822347	−	0.568986i	$-0.192664\pi$
$888$	− 26.7289i	− 0.0301001i
$889$	−224.068	−0.252045
$890$	10.4594i	0.0117521i
$891$	0	0
$892$	−157.492	−0.176561
$893$	− 16.6752i	− 0.0186732i
$894$	936.615	1.04767
$895$	−209.673	−0.234271
$896$	16.5530	0.0184743
$897$	478.001i	0.532888i
$898$	774.228i	0.862169i
$899$	147.172i	0.163707i
$900$	11.0889	0.0123210
$901$	− 424.098i	− 0.470697i
$902$	0	0
$903$	−268.283	−0.297102
$904$	523.471i	0.579061i
$905$	369.080	0.407824
$906$	798.187	0.881001
$907$	−1070.61	−1.18038	−0.590191	−	0.807263i	$-0.700948\pi$
$907$	−1070.61	−1.18038	−0.590191	+	0.807263i	$0.700948\pi$
$908$	− 628.264i	− 0.691920i
$909$	28.1160i	0.0309307i
$910$	− 36.5227i	− 0.0401348i
$911$	439.312	0.482230	0.241115	−	0.970497i	$-0.422487\pi$
$911$	439.312	0.482230	0.241115	+	0.970497i	$0.422487\pi$
$912$	6.08601i	0.00667326i
$913$	0	0
$914$	351.298	0.384352
$915$	49.1765i	0.0537449i
$916$	499.628	0.545445
$917$	−232.280	−0.253304
$918$	−303.203	−0.330286
$919$	− 341.898i	− 0.372032i	−0.982547	−	0.186016i	$-0.940442\pi$
$919$	− 341.898i	− 0.372032i	0.982547	−	0.186016i	$-0.0595577\pi$
$920$	− 136.332i	− 0.148187i
$921$	68.3987i	0.0742657i
$922$	−346.187	−0.375474
$923$	845.620i	0.916164i
$924$	0	0
$925$	16.8204	0.0181843
$926$	193.549i	0.209017i
$927$	116.913	0.126120
$928$	141.068	0.152013
$929$	159.146	0.171309	0.0856547	−	0.996325i	$-0.472702\pi$
$929$	159.146	0.171309	0.0856547	+	0.996325i	$0.472702\pi$
$930$	52.4255i	0.0563715i
$931$	− 25.3805i	− 0.0272615i
$932$	− 564.152i	− 0.605314i
$933$	1494.31	1.60162
$934$	102.302i	0.109531i
$935$	0	0
$936$	24.7585	0.0264514
$937$	− 1143.75i	− 1.22065i	−0.792152	−	0.610324i	$-0.791039\pi$
$937$	− 1143.75i	− 1.22065i	0.792152	−	0.610324i	$-0.208961\pi$
$938$	−75.9757	−0.0809975
$939$	−311.225	−0.331443
$940$	−137.684	−0.146472
$941$	− 1153.63i	− 1.22596i	−0.790098	−	0.612981i	$-0.789970\pi$
$941$	− 1153.63i	− 1.22596i	0.790098	−	0.612981i	$-0.210030\pi$
$942$	108.757i	0.115453i
$943$	772.580i	0.819279i
$944$	−444.110	−0.470456
$945$	− 92.9029i	− 0.0983099i
$946$	0	0
$947$	681.693	0.719845	0.359922	−	0.932982i	$-0.382803\pi$
$947$	681.693	0.719845	0.359922	+	0.932982i	$0.382803\pi$
$948$	− 527.758i	− 0.556707i
$949$	1092.16	1.15086
$950$	−3.82991	−0.00403148
$951$	−995.884	−1.04720
$952$	− 31.2437i	− 0.0328190i
$953$	202.806i	0.212808i	0.994323	+	0.106404i	$0.0339336\pi$
$953$	202.806i	0.212808i	−0.994323	+	0.106404i	$0.966066\pi$
$954$	88.0892i	0.0923367i
$955$	−398.702	−0.417489
$956$	827.862i	0.865964i
$957$	0	0
$958$	−659.435	−0.688345
$959$	28.7018i	0.0299289i
$960$	50.2509	0.0523447
$961$	−926.171	−0.963757
$962$	37.5555	0.0390390
$963$	64.9011i	0.0673947i
$964$	283.696i	0.294291i
$965$	568.046i	0.588649i
$966$	−125.292	−0.129702
$967$	1229.94i	1.27192i	0.771723	+	0.635958i	$0.219395\pi$
$967$	1229.94i	1.27192i	−0.771723	+	0.635958i	$0.780605\pi$
$968$	0	0
$969$	11.4873	0.0118548
$970$	− 487.154i	− 0.502221i
$971$	−734.904	−0.756853	−0.378427	−	0.925631i	$-0.623535\pi$
$971$	−734.904	−0.756853	−0.378427	+	0.925631i	$0.623535\pi$
$972$	119.048	0.122477
$973$	−235.825	−0.242369
$974$	− 145.997i	− 0.149894i
$975$	− 110.874i	− 0.113717i
$976$	− 31.3158i	− 0.0320859i
$977$	−473.118	−0.484256	−0.242128	−	0.970244i	$-0.577845\pi$
$977$	−473.118	−0.484256	−0.242128	+	0.970244i	$0.577845\pi$
$978$	574.435i	0.587357i
$979$	0	0
$980$	−209.561	−0.213838
$981$	− 51.0686i	− 0.0520577i
$982$	179.686	0.182980
$983$	1100.05	1.11907	0.559535	−	0.828806i	$-0.310980\pi$
$983$	1100.05	1.11907	0.559535	+	0.828806i	$0.310980\pi$
$984$	−284.767	−0.289397
$985$	− 601.756i	− 0.610920i
$986$	− 266.264i	− 0.270045i
$987$	126.534i	0.128201i
$988$	−8.55116	−0.00865502
$989$	− 1407.09i	− 1.42274i
$990$	0	0
$991$	−129.253	−0.130427	−0.0652133	−	0.997871i	$-0.520773\pi$
$991$	−129.253	−0.130427	−0.0652133	+	0.997871i	$0.520773\pi$
$992$	− 33.3848i	− 0.0336540i
$993$	1118.62	1.12651
$994$	−221.651	−0.222989
$995$	316.590	0.318181
$996$	15.0466i	0.0151070i
$997$	935.069i	0.937883i	0.883229	+	0.468941i	$0.155364\pi$
$997$	935.069i	0.937883i	−0.883229	+	0.468941i	$0.844636\pi$
$998$	− 208.751i	− 0.209169i
$999$	95.5300	0.0956256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1210.3.d.f.241.1		16
11.4	even	5		110.3.h.a.61.4	✓	16
11.8	odd	10		110.3.h.a.101.4	yes	16
11.10	odd	2	inner	1210.3.d.f.241.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.3.h.a.61.4	✓	16	11.4	even	5
110.3.h.a.101.4	yes	16	11.8	odd	10
1210.3.d.f.241.1		16	1.1	even	1	trivial
1210.3.d.f.241.9		16	11.10	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.80911 q^{3} -2.00000 q^{4} +2.23607 q^{5} +3.97268i q^{6} -1.46309i q^{7} +2.82843i q^{8} -1.10889 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -2.80911 q^{3} -2.00000 q^{4} +2.23607 q^{5} +3.97268i q^{6} -1.46309i q^{7} +2.82843i q^{8} -1.10889 q^{9} -3.16228i q^{10} +5.61822 q^{12} +7.89390i q^{13} -2.06912 q^{14} -6.28137 q^{15} +4.00000 q^{16} -7.54998i q^{17} +1.56820i q^{18} -0.541631i q^{19} -4.47214 q^{20} +4.10999i q^{21} -21.5560 q^{23} -7.94537i q^{24} +5.00000 q^{25} +11.1637 q^{26} +28.3970 q^{27} +2.92618i q^{28} +24.9375i q^{29} +8.88319i q^{30} +5.90165 q^{31} -5.65685i q^{32} -10.6773 q^{34} -3.27157i q^{35} +2.21777 q^{36} +3.36409 q^{37} -0.765982 q^{38} -22.1748i q^{39} +6.32456i q^{40} -35.8406i q^{41} +5.81240 q^{42} +65.2759i q^{43} -2.47955 q^{45} +30.4848i q^{46} +30.7870 q^{47} -11.2364 q^{48} +46.8594 q^{49} -7.07107i q^{50} +21.2087i q^{51} -15.7878i q^{52} +56.1720 q^{53} -40.1594i q^{54} +4.13825 q^{56} +1.52150i q^{57} +35.2669 q^{58} -111.028 q^{59} +12.5627 q^{60} -7.82896i q^{61} -8.34619i q^{62} +1.62240i q^{63} -8.00000 q^{64} +17.6513i q^{65} +36.7187 q^{67} +15.1000i q^{68} +60.5532 q^{69} -4.62670 q^{70} +107.123 q^{71} -3.13641i q^{72} -138.355i q^{73} -4.75754i q^{74} -14.0456 q^{75} +1.08326i q^{76} -31.3600 q^{78} -93.9368i q^{79} +8.94427 q^{80} -69.7904 q^{81} -50.6863 q^{82} +2.67818i q^{83} -8.21998i q^{84} -16.8823i q^{85} +92.3141 q^{86} -70.0522i q^{87} -3.30755 q^{89} +3.50661i q^{90} +11.5495 q^{91} +43.1120 q^{92} -16.5784 q^{93} -43.5394i q^{94} -1.21112i q^{95} +15.8907i q^{96} +154.052 q^{97} -66.2691i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{3} - 32 q^{4} + 40 q^{9}+O(q^{10}) \) 16 * q + 24 * q^3 - 32 * q^4 + 40 * q^9	\( 16 q + 24 q^{3} - 32 q^{4} + 40 q^{9} - 48 q^{12} + 16 q^{14} + 64 q^{16} - 12 q^{23} + 80 q^{25} + 228 q^{27} + 212 q^{31} - 176 q^{34} - 80 q^{36} + 164 q^{37} + 120 q^{38} - 16 q^{42} + 164 q^{47} + 96 q^{48} + 396 q^{49} + 244 q^{53} - 32 q^{56} - 288 q^{58} - 180 q^{59} - 128 q^{64} + 148 q^{67} + 784 q^{69} + 228 q^{71} + 120 q^{75} - 392 q^{78} + 56 q^{81} + 176 q^{82} - 496 q^{86} - 16 q^{89} - 160 q^{91} + 24 q^{92} - 324 q^{93} + 1056 q^{97}+O(q^{100}) \) 16 * q + 24 * q^3 - 32 * q^4 + 40 * q^9 - 48 * q^12 + 16 * q^14 + 64 * q^16 - 12 * q^23 + 80 * q^25 + 228 * q^27 + 212 * q^31 - 176 * q^34 - 80 * q^36 + 164 * q^37 + 120 * q^38 - 16 * q^42 + 164 * q^47 + 96 * q^48 + 396 * q^49 + 244 * q^53 - 32 * q^56 - 288 * q^58 - 180 * q^59 - 128 * q^64 + 148 * q^67 + 784 * q^69 + 228 * q^71 + 120 * q^75 - 392 * q^78 + 56 * q^81 + 176 * q^82 - 496 * q^86 - 16 * q^89 - 160 * q^91 + 24 * q^92 - 324 * q^93 + 1056 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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