LMFDB - Embedded newform 425.2.d.b.101.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(101,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("425.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.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:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.93924352.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 17x^{4} + 73x^{2} + 67 $ x^6 + 17x^4 + 73x^2 + 67
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.4
Root			$-1.12261i$ of defining polynomial
Character	$\chi$	$=$	425.101
Dual form			425.2.d.b.101.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.311108 q^{2} +1.12261i q^{3} -1.90321 q^{4} +0.349253i q^{6} -3.60843i q^{7} -1.21432 q^{8} +1.73975 q^{9} +O(q^{10})$	\(q+0.311108 q^{2} +1.12261i q^{3} -1.90321 q^{4} +0.349253i q^{6} -3.60843i q^{7} -1.21432 q^{8} +1.73975 q^{9} -5.74499i q^{11} -2.13656i q^{12} -2.59210 q^{13} -1.12261i q^{14} +3.42864 q^{16} +(2.52543 + 3.25917i) q^{17} +0.541249 q^{18} +4.28100 q^{19} +4.05086 q^{21} -1.78731i q^{22} -1.47186i q^{23} -1.36321i q^{24} -0.806424 q^{26} +5.32089i q^{27} +6.86760i q^{28} -5.50439i q^{29} -8.33946i q^{31} +3.49532 q^{32} +6.44938 q^{33} +(0.785680 + 1.01395i) q^{34} -3.31111 q^{36} -6.20290i q^{37} +1.33185 q^{38} -2.90992i q^{39} +3.95768i q^{41} +1.26025 q^{42} -9.76049 q^{43} +10.9339i q^{44} -0.457908i q^{46} -6.62222 q^{47} +3.84902i q^{48} -6.02074 q^{49} +(-3.65878 + 2.83507i) q^{51} +4.93332 q^{52} -0.658781 q^{53} +1.65537i q^{54} +4.38178i q^{56} +4.80589i q^{57} -1.71246i q^{58} +5.95407 q^{59} +8.76357i q^{61} -2.59447i q^{62} -6.27775i q^{63} -5.76986 q^{64} +2.00645 q^{66} +0.428639 q^{67} +(-4.80642 - 6.20290i) q^{68} +1.65233 q^{69} -1.36321i q^{71} -2.11261 q^{72} -3.25917i q^{73} -1.92977i q^{74} -8.14764 q^{76} -20.7304 q^{77} -0.905299i q^{78} +1.89597i q^{79} -0.754037 q^{81} +1.23127i q^{82} +16.6637 q^{83} -7.70964 q^{84} -3.03657 q^{86} +6.17929 q^{87} +6.97626i q^{88} -3.52543 q^{89} +9.35342i q^{91} +2.80127i q^{92} +9.36196 q^{93} -2.06022 q^{94} +3.92388i q^{96} +11.7073i q^{97} -1.87310 q^{98} -9.99483i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} - 16 q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 - 16 * q^9	$ 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} - 16 q^{9} - 2 q^{13} - 6 q^{16} + 2 q^{17} + 16 q^{18} + 12 q^{19} - 2 q^{21} + 22 q^{26} - 6 q^{32} - 28 q^{33} + 18 q^{34} - 20 q^{36} - 32 q^{38} + 34 q^{42} + 8 q^{43} - 40 q^{47} + 4 q^{49} - 8 q^{51} + 30 q^{52} + 10 q^{53} - 4 q^{59} - 22 q^{64} - 16 q^{66} - 24 q^{67} - 2 q^{68} + 24 q^{69} - 52 q^{72} - 36 q^{76} - 44 q^{77} + 34 q^{81} + 20 q^{83} - 6 q^{84} - 4 q^{86} + 76 q^{87} - 8 q^{89} + 30 q^{93} - 40 q^{94} + 16 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 - 16 * q^9 - 2 * q^13 - 6 * q^16 + 2 * q^17 + 16 * q^18 + 12 * q^19 - 2 * q^21 + 22 * q^26 - 6 * q^32 - 28 * q^33 + 18 * q^34 - 20 * q^36 - 32 * q^38 + 34 * q^42 + 8 * q^43 - 40 * q^47 + 4 * q^49 - 8 * q^51 + 30 * q^52 + 10 * q^53 - 4 * q^59 - 22 * q^64 - 16 * q^66 - 24 * q^67 - 2 * q^68 + 24 * q^69 - 52 * q^72 - 36 * q^76 - 44 * q^77 + 34 * q^81 + 20 * q^83 - 6 * q^84 - 4 * q^86 + 76 * q^87 - 8 * q^89 + 30 * q^93 - 40 * q^94 + 16 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.311108			0.219986			0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108			0.219986			0.109993	+	0.993932i	$0.464917\pi$
$3$			1.12261i			0.648139i	0.946033	+	0.324070i	$0.105051\pi$
$3$			1.12261i			0.648139i	−0.946033	+	0.324070i	$0.894949\pi$
$4$	−1.90321			−0.951606
$5$		0			0
$5$		0			0
$6$			0.349253i			0.142582i
$7$		−	3.60843i		−	1.36386i	−0.731419	−	0.681929i	$-0.761141\pi$
$7$		−	3.60843i		−	1.36386i	0.731419	−	0.681929i	$-0.238859\pi$
$8$	−1.21432			−0.429327
$9$	1.73975			0.579916
$10$		0			0
$11$		−	5.74499i		−	1.73218i	−0.499888	−	0.866090i	$-0.666626\pi$
$11$		−	5.74499i		−	1.73218i	0.499888	−	0.866090i	$-0.333374\pi$
$12$		−	2.13656i		−	0.616773i
$13$	−2.59210			−0.718920			−0.359460	−	0.933160i	$-0.617039\pi$
$13$	−2.59210			−0.718920			−0.359460	+	0.933160i	$0.617039\pi$
$14$		−	1.12261i		−	0.300030i
$15$		0			0
$16$	3.42864			0.857160
$17$	2.52543	+	3.25917i	0.612506	+	0.790466i
$17$	2.52543	+	3.25917i	0.612506	+	0.790466i
$18$	0.541249			0.127574
$19$	4.28100			0.982128			0.491064	−	0.871124i	$-0.336608\pi$
$19$	4.28100			0.982128			0.491064	+	0.871124i	$0.336608\pi$
$20$		0			0
$21$	4.05086			0.883969
$22$		−	1.78731i		−	0.381056i
$23$		−	1.47186i		−	0.306905i	−0.988156	−	0.153452i	$-0.950961\pi$
$23$		−	1.47186i		−	0.306905i	0.988156	−	0.153452i	$-0.0490391\pi$
$24$		−	1.36321i		−	0.278264i
$25$		0			0
$26$	−0.806424			−0.158153
$27$			5.32089i			1.02401i
$28$			6.86760i			1.29785i
$29$		−	5.50439i		−	1.02214i	−0.859539	−	0.511070i	$-0.829249\pi$
$29$		−	5.50439i		−	1.02214i	0.859539	−	0.511070i	$-0.170751\pi$
$30$		0			0
$31$		−	8.33946i		−	1.49781i	−0.662676	−	0.748906i	$-0.730579\pi$
$31$		−	8.33946i		−	1.49781i	0.662676	−	0.748906i	$-0.269421\pi$
$32$	3.49532			0.617890
$33$	6.44938			1.12269
$34$	0.785680	+	1.01395i	0.134743	+	0.173892i
$35$		0			0
$36$	−3.31111			−0.551851
$37$		−	6.20290i		−	1.01975i	−0.860248	−	0.509875i	$-0.829692\pi$
$37$		−	6.20290i		−	1.01975i	0.860248	−	0.509875i	$-0.170308\pi$
$38$	1.33185			0.216055
$39$		−	2.90992i		−	0.465960i
$40$		0			0
$41$			3.95768i			0.618086i	0.951048	+	0.309043i	$0.100009\pi$
$41$			3.95768i			0.618086i	−0.951048	+	0.309043i	$0.899991\pi$
$42$	1.26025			0.194461
$43$	−9.76049			−1.48846			−0.744230	−	0.667923i	$-0.767184\pi$
$43$	−9.76049			−1.48846			−0.744230	+	0.667923i	$0.767184\pi$
$44$			10.9339i			1.64835i
$45$		0			0
$46$		−	0.457908i		−	0.0675148i
$47$	−6.62222			−0.965949			−0.482975	−	0.875634i	$-0.660444\pi$
$47$	−6.62222			−0.965949			−0.482975	+	0.875634i	$0.660444\pi$
$48$			3.84902i			0.555559i
$49$	−6.02074			−0.860106
$50$		0			0
$51$	−3.65878	+	2.83507i	−0.512332	+	0.396989i
$52$	4.93332			0.684129
$53$	−0.658781			−0.0904905			−0.0452452	−	0.998976i	$-0.514407\pi$
$53$	−0.658781			−0.0904905			−0.0452452	+	0.998976i	$0.514407\pi$
$54$			1.65537i			0.225267i
$55$		0			0
$56$			4.38178i			0.585540i
$57$			4.80589i			0.636555i
$58$		−	1.71246i		−	0.224857i
$59$	5.95407			0.775154			0.387577	−	0.921837i	$-0.373312\pi$
$59$	5.95407			0.775154			0.387577	+	0.921837i	$0.373312\pi$
$60$		0			0
$61$			8.76357i			1.12206i	0.827796	+	0.561030i	$0.189595\pi$
$61$			8.76357i			1.12206i	−0.827796	+	0.561030i	$0.810405\pi$
$62$		−	2.59447i		−	0.329498i
$63$		−	6.27775i		−	0.790922i
$64$	−5.76986			−0.721232
$65$		0			0
$66$	2.00645			0.246977
$67$	0.428639			0.0523666			0.0261833	−	0.999657i	$-0.491665\pi$
$67$	0.428639			0.0523666			0.0261833	+	0.999657i	$0.491665\pi$
$68$	−4.80642	−	6.20290i	−0.582865	−	0.752212i
$69$	1.65233			0.198917
$70$		0			0
$71$		−	1.36321i		−	0.161783i	−0.996723	−	0.0808915i	$-0.974223\pi$
$71$		−	1.36321i		−	0.161783i	0.996723	−	0.0808915i	$-0.0257767\pi$
$72$	−2.11261			−0.248973
$73$		−	3.25917i		−	0.381457i	−0.981643	−	0.190729i	$-0.938915\pi$
$73$		−	3.25917i		−	0.381457i	0.981643	−	0.190729i	$-0.0610851\pi$
$74$		−	1.92977i		−	0.224331i
$75$		0			0
$76$	−8.14764			−0.934599
$77$	−20.7304			−2.36245
$78$		−	0.905299i		−	0.102505i
$79$			1.89597i			0.213313i	0.994296	+	0.106656i	$0.0340145\pi$
$79$			1.89597i			0.213313i	−0.994296	+	0.106656i	$0.965985\pi$
$80$		0			0
$81$	−0.754037			−0.0837819
$82$			1.23127i			0.135970i
$83$	16.6637			1.82908			0.914540	−	0.404497i	$-0.132553\pi$
$83$	16.6637			1.82908			0.914540	+	0.404497i	$0.132553\pi$
$84$	−7.70964			−0.841190
$85$		0			0
$86$	−3.03657			−0.327441
$87$	6.17929			0.662489
$88$			6.97626i			0.743671i
$89$	−3.52543			−0.373695			−0.186847	−	0.982389i	$-0.559827\pi$
$89$	−3.52543			−0.373695			−0.186847	+	0.982389i	$0.559827\pi$
$90$		0			0
$91$			9.35342i			0.980505i
$92$			2.80127i			0.292052i
$93$	9.36196			0.970790
$94$	−2.06022			−0.212496
$95$		0			0
$96$			3.92388i			0.400479i
$97$			11.7073i			1.18870i	0.804208	+	0.594348i	$0.202590\pi$
$97$			11.7073i			1.18870i	−0.804208	+	0.594348i	$0.797410\pi$
$98$	−1.87310			−0.189212
$99$		−	9.99483i		−	1.00452i
$100$		0			0
$101$	3.96989			0.395019			0.197509	−	0.980301i	$-0.436715\pi$
$101$	3.96989			0.395019			0.197509	+	0.980301i	$0.436715\pi$
$102$	−1.13828	+	0.882012i	−0.112706	+	0.0873322i
$103$	4.23506			0.417293			0.208647	−	0.977991i	$-0.433094\pi$
$103$	4.23506			0.417293			0.208647	+	0.977991i	$0.433094\pi$
$104$	3.14764			0.308652
$105$		0			0
$106$	−0.204952			−0.0199067
$107$			11.8392i			1.14454i	0.820065	+	0.572271i	$0.193937\pi$
$107$			11.8392i			1.14454i	−0.820065	+	0.572271i	$0.806063\pi$
$108$		−	10.1268i		−	0.974449i
$109$			12.0227i			1.15157i	0.817601	+	0.575785i	$0.195303\pi$
$109$			12.0227i			1.15157i	−0.817601	+	0.575785i	$0.804697\pi$
$110$		0			0
$111$	6.96343			0.660940
$112$		−	12.3720i		−	1.16904i
$113$		−	10.7915i		−	1.01518i	−0.861600	−	0.507588i	$-0.830537\pi$
$113$		−	10.7915i		−	1.01518i	0.861600	−	0.507588i	$-0.169463\pi$
$114$			1.49515i			0.140034i
$115$		0			0
$116$			10.4760i			0.972675i
$117$	−4.50961			−0.416913
$118$	1.85236			0.170523
$119$	11.7605	−	9.11282i	1.07808	−	0.835371i
$120$		0			0
$121$	−22.0049			−2.00045
$122$			2.72641i			0.246838i
$123$	−4.44293			−0.400605
$124$			15.8718i			1.42533i
$125$		0			0
$126$		−	1.95306i		−	0.173992i
$127$	17.1383			1.52078			0.760388	−	0.649469i	$-0.225009\pi$
$127$	17.1383			1.52078			0.760388	+	0.649469i	$0.225009\pi$
$128$	−8.78568			−0.776552
$129$		−	10.9572i		−	0.964730i
$130$		0			0
$131$		−	8.89551i		−	0.777204i	−0.921406	−	0.388602i	$-0.872958\pi$
$131$		−	8.89551i		−	0.777204i	0.921406	−	0.388602i	$-0.127042\pi$
$132$	−12.2745			−1.06836
$133$		−	15.4477i		−	1.33948i
$134$	0.133353			0.0115200
$135$		0			0
$136$	−3.06668	−	3.95768i	−0.262965	−	0.339368i
$137$	14.0716			1.20222			0.601109	−	0.799167i	$-0.294726\pi$
$137$	14.0716			1.20222			0.601109	+	0.799167i	$0.294726\pi$
$138$	0.514052			0.0437590
$139$			16.7876i			1.42390i	0.702228	+	0.711952i	$0.252189\pi$
$139$			16.7876i			1.42390i	−0.702228	+	0.711952i	$0.747811\pi$
$140$		0			0
$141$		−	7.43416i		−	0.626070i
$142$		−	0.424104i		−	0.0355901i
$143$			14.8916i			1.24530i
$144$	5.96497			0.497081
$145$		0			0
$146$		−	1.01395i		−	0.0839155i
$147$		−	6.75895i		−	0.557468i
$148$			11.8054i			0.970400i
$149$	−4.76494			−0.390359			−0.195179	−	0.980768i	$-0.562529\pi$
$149$	−4.76494			−0.390359			−0.195179	+	0.980768i	$0.562529\pi$
$150$		0			0
$151$	2.99063			0.243374			0.121687	−	0.992569i	$-0.461170\pi$
$151$	2.99063			0.243374			0.121687	+	0.992569i	$0.461170\pi$
$152$	−5.19850			−0.421654
$153$	4.39361	+	5.67014i	0.355202	+	0.458404i
$154$	−6.44938			−0.519706
$155$		0			0
$156$			5.53820i			0.443411i
$157$	−13.9447			−1.11291			−0.556454	−	0.830878i	$-0.687838\pi$
$157$	−13.9447			−1.11291			−0.556454	+	0.830878i	$0.687838\pi$
$158$			0.589850i			0.0469260i
$159$		−	0.739554i		−	0.0586504i
$160$		0			0
$161$	−5.31111			−0.418574
$162$	−0.234587			−0.0184309
$163$			8.65491i			0.677905i	0.940803	+	0.338953i	$0.110073\pi$
$163$			8.65491i			0.677905i	−0.940803	+	0.338953i	$0.889927\pi$
$164$		−	7.53230i		−	0.588174i
$165$		0			0
$166$	5.18421			0.402373
$167$			0.773357i			0.0598442i	0.999552	+	0.0299221i	$0.00952592\pi$
$167$			0.773357i			0.0598442i	−0.999552	+	0.0299221i	$0.990474\pi$
$168$	−4.91903			−0.379512
$169$	−6.28100			−0.483154
$170$		0			0
$171$	7.44785			0.569551
$172$	18.5763			1.41643
$173$		−	2.41097i		−	0.183302i	−0.995791	−	0.0916511i	$-0.970786\pi$
$173$		−	2.41097i		−	0.183302i	0.995791	−	0.0916511i	$-0.0292145\pi$
$174$	1.92242			0.145739
$175$		0			0
$176$		−	19.6975i		−	1.48476i
$177$			6.68409i			0.502407i
$178$	−1.09679			−0.0822077
$179$	−18.4558			−1.37945			−0.689727	−	0.724070i	$-0.742269\pi$
$179$	−18.4558			−1.37945			−0.689727	+	0.724070i	$0.742269\pi$
$180$		0			0
$181$			20.6366i			1.53391i	0.641703	+	0.766953i	$0.278228\pi$
$181$			20.6366i			1.53391i	−0.641703	+	0.766953i	$0.721772\pi$
$182$			2.90992i			0.215698i
$183$	−9.83807			−0.727251
$184$			1.78731i			0.131762i
$185$		0			0
$186$	2.91258			0.213561
$187$	18.7239	−	14.5086i	1.36923	−	1.06097i
$188$	12.6035			0.919203
$189$	19.2000			1.39660
$190$		0			0
$191$	22.8113			1.65057			0.825286	−	0.564716i	$-0.191014\pi$
$191$	22.8113			1.65057			0.825286	+	0.564716i	$0.191014\pi$
$192$		−	6.47730i		−	0.467459i
$193$			19.7208i			1.41953i	0.704437	+	0.709767i	$0.251200\pi$
$193$			19.7208i			1.41953i	−0.704437	+	0.709767i	$0.748800\pi$
$194$			3.64223i			0.261497i
$195$		0			0
$196$	11.4588			0.818482
$197$		−	12.8870i		−	0.918160i	−0.888395	−	0.459080i	$-0.848179\pi$
$197$		−	12.8870i		−	0.918160i	0.888395	−	0.459080i	$-0.151821\pi$
$198$		−	3.10947i		−	0.220980i
$199$		−	6.44350i		−	0.456767i	−0.973571	−	0.228384i	$-0.926656\pi$
$199$		−	6.44350i		−	0.456767i	0.973571	−	0.228384i	$-0.0733441\pi$
$200$		0			0
$201$			0.481195i			0.0339409i
$202$	1.23506			0.0868988
$203$	−19.8622			−1.39405
$204$	6.96343	−	5.39574i	0.487538	−	0.377777i
$205$		0			0
$206$	1.31756			0.0917988
$207$		−	2.56067i		−	0.177979i
$208$	−8.88739			−0.616230
$209$		−	24.5943i		−	1.70122i
$210$		0			0
$211$		−	11.3580i		−	0.781920i	−0.920408	−	0.390960i	$-0.872143\pi$
$211$		−	11.3580i		−	0.781920i	0.920408	−	0.390960i	$-0.127857\pi$
$212$	1.25380			0.0861113
$213$	1.53035			0.104858
$214$			3.68328i			0.251784i
$215$		0			0
$216$		−	6.46126i		−	0.439633i
$217$	−30.0923			−2.04280
$218$			3.74037i			0.253330i
$219$	3.65878			0.247237
$220$		0			0
$221$	−6.54617	−	8.44812i	−0.440343	−	0.568282i
$222$	2.16638			0.145398
$223$	0.888922			0.0595266			0.0297633	−	0.999557i	$-0.490525\pi$
$223$	0.888922			0.0595266			0.0297633	+	0.999557i	$0.490525\pi$
$224$		−	12.6126i		−	0.842714i
$225$		0			0
$226$		−	3.35731i		−	0.223325i
$227$		−	12.9280i		−	0.858064i	−0.903289	−	0.429032i	$-0.858855\pi$
$227$		−	12.9280i		−	0.858064i	0.903289	−	0.429032i	$-0.141145\pi$
$228$		−	9.14662i		−	0.605750i
$229$	2.83161			0.187118			0.0935591	−	0.995614i	$-0.470176\pi$
$229$	2.83161			0.187118			0.0935591	+	0.995614i	$0.470176\pi$
$230$		0			0
$231$		−	23.2721i		−	1.53119i
$232$			6.68409i			0.438832i
$233$		−	19.9381i		−	1.30619i	−0.757277	−	0.653094i	$-0.773471\pi$
$233$		−	19.9381i		−	1.30619i	0.757277	−	0.653094i	$-0.226529\pi$
$234$	−1.40297			−0.0917153
$235$		0			0
$236$	−11.3319			−0.737641
$237$	−2.12843			−0.138256
$238$	3.65878	−	2.83507i	0.237164	−	0.183770i
$239$	27.2859			1.76498			0.882490	−	0.470332i	$-0.155866\pi$
$239$	27.2859			1.76498			0.882490	+	0.470332i	$0.155866\pi$
$240$		0			0
$241$			3.57462i			0.230262i	0.993350	+	0.115131i	$0.0367287\pi$
$241$			3.57462i			0.230262i	−0.993350	+	0.115131i	$0.963271\pi$
$242$	−6.84590			−0.440071
$243$			15.1162i			0.969703i
$244$		−	16.6789i		−	1.06776i
$245$		0			0
$246$	−1.38223			−0.0881278
$247$	−11.0968			−0.706072
$248$			10.1268i			0.643051i
$249$			18.7068i			1.18550i
$250$		0			0
$251$	17.6128			1.11171			0.555857	−	0.831278i	$-0.312390\pi$
$251$	17.6128			1.11171			0.555857	+	0.831278i	$0.312390\pi$
$252$			11.9479i			0.752646i
$253$	−8.45584			−0.531614
$254$	5.33185			0.334550
$255$		0			0
$256$	8.80642			0.550401
$257$	20.5462			1.28163			0.640817	−	0.767693i	$-0.278596\pi$
$257$	20.5462			1.28163			0.640817	+	0.767693i	$0.278596\pi$
$258$		−	3.40888i		−	0.212227i
$259$	−22.3827			−1.39079
$260$		0			0
$261$		−	9.57625i		−	0.592755i
$262$		−	2.76746i		−	0.170974i
$263$	8.38715			0.517174			0.258587	−	0.965988i	$-0.416743\pi$
$263$	8.38715			0.517174			0.258587	+	0.965988i	$0.416743\pi$
$264$	−7.83161			−0.482002
$265$		0			0
$266$		−	4.80589i		−	0.294668i
$267$		−	3.95768i		−	0.242206i
$268$	−0.815792			−0.0498324
$269$			12.4058i			0.756395i	0.925725	+	0.378197i	$0.123456\pi$
$269$			12.4058i			0.756395i	−0.925725	+	0.378197i	$0.876544\pi$
$270$		0			0
$271$	−7.13828			−0.433619			−0.216810	−	0.976214i	$-0.569565\pi$
$271$	−7.13828			−0.433619			−0.216810	+	0.976214i	$0.569565\pi$
$272$	8.65878	+	11.1745i	0.525016	+	0.677556i
$273$	−10.5002			−0.635503
$274$	4.37778			0.264472
$275$		0			0
$276$	−3.14473			−0.189290
$277$		−	20.4193i		−	1.22688i	−0.789743	−	0.613438i	$-0.789786\pi$
$277$		−	20.4193i		−	1.22688i	0.789743	−	0.613438i	$-0.210214\pi$
$278$			5.22275i			0.313240i
$279$		−	14.5086i		−	0.868605i
$280$		0			0
$281$	29.1432			1.73854			0.869269	−	0.494340i	$-0.164590\pi$
$281$	29.1432			1.73854			0.869269	+	0.494340i	$0.164590\pi$
$282$		−	2.31283i		−	0.137727i
$283$			9.16991i			0.545095i	0.962142	+	0.272547i	$0.0878661\pi$
$283$			9.16991i			0.545095i	−0.962142	+	0.272547i	$0.912134\pi$
$284$			2.59447i			0.153954i
$285$		0			0
$286$			4.63290i			0.273949i
$287$	14.2810			0.842981
$288$	6.08097			0.358324
$289$	−4.24443	+	16.4616i	−0.249672	+	0.968330i
$290$		0			0
$291$	−13.1427			−0.770440
$292$			6.20290i			0.362997i
$293$	−16.2351			−0.948463			−0.474231	−	0.880400i	$-0.657274\pi$
$293$	−16.2351			−0.948463			−0.474231	+	0.880400i	$0.657274\pi$
$294$		−	2.10276i		−	0.122636i
$295$		0			0
$296$			7.53230i			0.437806i
$297$	30.5684			1.77376
$298$	−1.48241			−0.0858737
$299$			3.81522i			0.220640i
$300$		0			0
$301$			35.2200i			2.03005i
$302$	0.930409			0.0535390
$303$			4.45664i			0.256027i
$304$	14.6780			0.841841
$305$		0			0
$306$	1.36689	+	1.76402i	0.0781396	+	0.100843i
$307$	3.37778			0.192780			0.0963902	−	0.995344i	$-0.469270\pi$
$307$	3.37778			0.192780			0.0963902	+	0.995344i	$0.469270\pi$
$308$	39.4543			2.24812
$309$			4.75432i			0.270464i
$310$		0			0
$311$		−	7.95641i		−	0.451166i	−0.974224	−	0.225583i	$-0.927571\pi$
$311$		−	7.95641i		−	0.451166i	0.974224	−	0.225583i	$-0.0724288\pi$
$312$			3.53358i			0.200049i
$313$		−	6.83380i		−	0.386269i	−0.981172	−	0.193135i	$-0.938135\pi$
$313$		−	6.83380i		−	0.386269i	0.981172	−	0.193135i	$-0.0618654\pi$
$314$	−4.33830			−0.244825
$315$		0			0
$316$		−	3.60843i		−	0.202990i
$317$			27.1550i			1.52517i	0.646886	+	0.762587i	$0.276071\pi$
$317$			27.1550i			1.52517i	−0.646886	+	0.762587i	$0.723929\pi$
$318$		−	0.230081i		−	0.0129023i
$319$	−31.6227			−1.77053
$320$		0			0
$321$	−13.2908			−0.741822
$322$	−1.65233			−0.0920806
$323$	10.8113	+	13.9525i	0.601559	+	0.776339i
$324$	1.43509			0.0797274
$325$		0			0
$326$			2.69261i			0.149130i
$327$	−13.4968			−0.746377
$328$		−	4.80589i		−	0.265361i
$329$			23.8958i			1.31742i
$330$		0			0
$331$	−20.3783			−1.12009			−0.560045	−	0.828462i	$-0.689216\pi$
$331$	−20.3783			−1.12009			−0.560045	+	0.828462i	$0.689216\pi$
$332$	−31.7146			−1.74056
$333$		−	10.7915i		−	0.591369i
$334$			0.240597i			0.0131649i
$335$		0			0
$336$	13.8889			0.757703
$337$		−	4.70775i		−	0.256447i	−0.991745	−	0.128224i	$-0.959072\pi$
$337$		−	4.70775i		−	0.256447i	0.991745	−	0.128224i	$-0.0409276\pi$
$338$	−1.95407			−0.106287
$339$	12.1146			0.657976
$340$		0			0
$341$	−47.9101			−2.59448
$342$	2.31708			0.125294
$343$		−	3.53358i		−	0.190795i
$344$	11.8524			0.639036
$345$		0			0
$346$		−	0.750070i		−	0.0403240i
$347$		−	35.3520i		−	1.89779i	−0.315588	−	0.948896i	$-0.602202\pi$
$347$		−	35.3520i		−	1.89779i	0.315588	−	0.948896i	$-0.397798\pi$
$348$	−11.7605			−0.630428
$349$	−25.0558			−1.34121			−0.670603	−	0.741817i	$-0.733964\pi$
$349$	−25.0558			−1.34121			−0.670603	+	0.741817i	$0.733964\pi$
$350$		0			0
$351$		−	13.7923i		−	0.736178i
$352$		−	20.0806i		−	1.07030i
$353$	13.6953			0.728930			0.364465	−	0.931217i	$-0.381252\pi$
$353$	13.6953			0.728930			0.364465	+	0.931217i	$0.381252\pi$
$354$			2.07947i			0.110523i
$355$		0			0
$356$	6.70964			0.355610
$357$	10.2301	+	13.2024i	0.541436	+	0.698747i
$358$	−5.74176			−0.303461
$359$	2.53480			0.133781			0.0668907	−	0.997760i	$-0.478692\pi$
$359$	2.53480			0.133781			0.0668907	+	0.997760i	$0.478692\pi$
$360$		0			0
$361$	−0.673071			−0.0354248
$362$			6.42021i			0.337439i
$363$		−	24.7029i		−	1.29657i
$364$		−	17.8015i		−	0.933054i
$365$		0			0
$366$	−3.06070			−0.159985
$367$			36.3426i			1.89707i	0.316673	+	0.948535i	$0.397434\pi$
$367$			36.3426i			1.89707i	−0.316673	+	0.948535i	$0.602566\pi$
$368$		−	5.04649i		−	0.263066i
$369$			6.88536i			0.358438i
$370$		0			0
$371$			2.37716i			0.123416i
$372$	−17.8178			−0.923810
$373$	−12.7714			−0.661278			−0.330639	−	0.943757i	$-0.607264\pi$
$373$	−12.7714			−0.661278			−0.330639	+	0.943757i	$0.607264\pi$
$374$	5.82516	−	4.51373i	0.301212	−	0.233399i
$375$		0			0
$376$	8.04149			0.414708
$377$			14.2680i			0.734837i
$378$	5.97328			0.307232
$379$			22.1495i			1.13774i	0.822426	+	0.568872i	$0.192620\pi$
$379$			22.1495i			1.13774i	−0.822426	+	0.568872i	$0.807380\pi$
$380$		0			0
$381$			19.2396i			0.985674i
$382$	7.09679			0.363103
$383$	−1.95407			−0.0998482			−0.0499241	−	0.998753i	$-0.515898\pi$
$383$	−1.95407			−0.0998482			−0.0499241	+	0.998753i	$0.515898\pi$
$384$		−	9.86289i		−	0.503314i
$385$		0			0
$386$			6.13529i			0.312278i
$387$	−16.9808			−0.863182
$388$		−	22.2815i		−	1.13117i
$389$	−7.91258			−0.401184			−0.200592	−	0.979675i	$-0.564287\pi$
$389$	−7.91258			−0.401184			−0.200592	+	0.979675i	$0.564287\pi$
$390$		0			0
$391$	4.79706	−	3.71708i	0.242598	−	0.187981i
$392$	7.31111			0.369267
$393$	9.98619			0.503736
$394$		−	4.00924i		−	0.201983i
$395$		0			0
$396$			19.0223i			0.955906i
$397$			9.24476i			0.463981i	0.972718	+	0.231991i	$0.0745239\pi$
$397$			9.24476i			0.463981i	−0.972718	+	0.231991i	$0.925476\pi$
$398$		−	2.00462i		−	0.100483i
$399$	17.3417			0.868171
$400$		0			0
$401$			21.4848i			1.07290i	0.843932	+	0.536450i	$0.180235\pi$
$401$			21.4848i			1.07290i	−0.843932	+	0.536450i	$0.819765\pi$
$402$			0.149703i			0.00746653i
$403$			21.6168i			1.07681i
$404$	−7.55554			−0.375902
$405$		0			0
$406$	−6.17929			−0.306673
$407$	−35.6356			−1.76639
$408$	4.44293	−	3.44268i	0.219958	−	0.170438i
$409$	18.1669			0.898293			0.449147	−	0.893458i	$-0.351728\pi$
$409$	18.1669			0.898293			0.449147	+	0.893458i	$0.351728\pi$
$410$		0			0
$411$			15.7969i			0.779204i
$412$	−8.06022			−0.397099
$413$		−	21.4848i		−	1.05720i
$414$		−	0.796644i		−	0.0391529i
$415$		0			0
$416$	−9.06022			−0.444214
$417$	−18.8459			−0.922888
$418$		−	7.65147i		−	0.374246i
$419$		−	14.3589i		−	0.701476i	−0.936474	−	0.350738i	$-0.885931\pi$
$419$		−	14.3589i		−	0.701476i	0.936474	−	0.350738i	$-0.114069\pi$
$420$		0			0
$421$	3.96989			0.193481			0.0967403	−	0.995310i	$-0.469158\pi$
$421$	3.96989			0.193481			0.0967403	+	0.995310i	$0.469158\pi$
$422$		−	3.53358i		−	0.172012i
$423$	−11.5210			−0.560169
$424$	0.799970			0.0388500
$425$		0			0
$426$	0.476104			0.0230673
$427$	31.6227			1.53033
$428$		−	22.5326i		−	1.08915i
$429$	−16.7175			−0.807127
$430$		0			0
$431$			32.9337i			1.58636i	0.608985	+	0.793181i	$0.291577\pi$
$431$			32.9337i			1.58636i	−0.608985	+	0.793181i	$0.708423\pi$
$432$			18.2434i			0.877736i
$433$	38.2306			1.83725			0.918623	−	0.395135i	$-0.129302\pi$
$433$	38.2306			1.83725			0.918623	+	0.395135i	$0.129302\pi$
$434$	−9.36196			−0.449389
$435$		0			0
$436$		−	22.8818i		−	1.09584i
$437$		−	6.30104i		−	0.301420i
$438$	1.13828			0.0543889
$439$			0.664702i			0.0317245i	0.999874	+	0.0158622i	$0.00504932\pi$
$439$			0.664702i			0.0317245i	−0.999874	+	0.0158622i	$0.994951\pi$
$440$		0			0
$441$	−10.4746			−0.498789
$442$	−2.03657	−	2.62828i	−0.0968695	−	0.125014i
$443$	−27.5669			−1.30974			−0.654872	−	0.755740i	$-0.727277\pi$
$443$	−27.5669			−1.30974			−0.654872	+	0.755740i	$0.727277\pi$
$444$	−13.2529			−0.628954
$445$		0			0
$446$	0.276551			0.0130950
$447$		−	5.34916i		−	0.253007i
$448$			20.8201i			0.983658i
$449$			17.4756i			0.824723i	0.911020	+	0.412362i	$0.135296\pi$
$449$			17.4756i			0.824723i	−0.911020	+	0.412362i	$0.864704\pi$
$450$		0			0
$451$	22.7368			1.07064
$452$			20.5385i			0.966048i
$453$			3.35731i			0.157740i
$454$		−	4.02201i		−	0.188762i
$455$		0			0
$456$		−	5.83589i		−	0.273290i
$457$	−13.0350			−0.609753			−0.304877	−	0.952392i	$-0.598615\pi$
$457$	−13.0350			−0.609753			−0.304877	+	0.952392i	$0.598615\pi$
$458$	0.880937			0.0411635
$459$	−17.3417	+	13.4375i	−0.809441	+	0.627209i
$460$		0			0
$461$	−10.9491			−0.509953			−0.254976	−	0.966947i	$-0.582068\pi$
$461$	−10.9491			−0.509953			−0.254976	+	0.966947i	$0.582068\pi$
$462$		−	7.24014i		−	0.336842i
$463$	17.2400			0.801210			0.400605	−	0.916251i	$-0.368800\pi$
$463$	17.2400			0.801210			0.400605	+	0.916251i	$0.368800\pi$
$464$		−	18.8726i		−	0.876138i
$465$		0			0
$466$		−	6.20290i		−	0.287344i
$467$	−0.0414872			−0.00191980			−0.000959899	−	1.00000i	$-0.500306\pi$
$467$	−0.0414872			−0.00191980			−0.000959899		1.00000i	$0.500306\pi$
$468$	8.58274			0.396737
$469$		−	1.54671i		−	0.0714206i
$470$		0			0
$471$		−	15.6545i		−	0.721319i
$472$	−7.23014			−0.332794
$473$			56.0739i			2.57828i
$474$	−0.662171			−0.0304145
$475$		0			0
$476$	−22.3827	+	17.3436i	−1.02591	+	0.794944i
$477$	−1.14611			−0.0524769
$478$	8.48886			0.388272
$479$		−	35.4606i		−	1.62024i	−0.586266	−	0.810118i	$-0.699403\pi$
$479$		−	35.4606i		−	1.62024i	0.586266	−	0.810118i	$-0.300597\pi$
$480$		0			0
$481$			16.0786i			0.733119i
$482$			1.11209i			0.0506545i
$483$		−	5.96230i		−	0.271294i
$484$	41.8800			1.90364
$485$		0			0
$486$			4.70276i			0.213321i
$487$		−	24.6691i		−	1.11787i	−0.829213	−	0.558933i	$-0.811211\pi$
$487$		−	24.6691i		−	1.11787i	0.829213	−	0.558933i	$-0.188789\pi$
$488$		−	10.6418i		−	0.481730i
$489$	−9.71609			−0.439377
$490$		0			0
$491$	8.58073			0.387243			0.193621	−	0.981076i	$-0.437977\pi$
$491$	8.58073			0.387243			0.193621	+	0.981076i	$0.437977\pi$
$492$	8.45584			0.381219
$493$	17.9398	−	13.9009i	0.807967	−	0.626067i
$494$	−3.45230			−0.155326
$495$		0			0
$496$		−	28.5930i		−	1.28386i
$497$	−4.91903			−0.220649
$498$			5.81984i			0.260793i
$499$			8.48917i			0.380027i	0.981781	+	0.190014i	$0.0608532\pi$
$499$			8.48917i			0.380027i	−0.981781	+	0.190014i	$0.939147\pi$
$500$		0			0
$501$	−0.868178			−0.0387873
$502$	5.47949			0.244562
$503$		−	24.1879i		−	1.07849i	−0.842150	−	0.539244i	$-0.818710\pi$
$503$		−	24.1879i		−	1.07849i	0.842150	−	0.539244i	$-0.181290\pi$
$504$			7.62320i			0.339564i
$505$		0			0
$506$	−2.63068			−0.116948
$507$		−	7.05111i		−	0.313151i
$508$	−32.6178			−1.44718
$509$	−25.8671			−1.14654			−0.573270	−	0.819366i	$-0.694325\pi$
$509$	−25.8671			−1.14654			−0.573270	+	0.819366i	$0.694325\pi$
$510$		0			0
$511$	−11.7605			−0.520253
$512$	20.3111			0.897633
$513$			22.7787i			1.00570i
$514$	6.39207			0.281942
$515$		0			0
$516$			20.8539i			0.918042i
$517$			38.0446i			1.67320i
$518$	−6.96343			−0.305956
$519$	2.70657			0.118805
$520$		0			0
$521$			11.4900i			0.503385i	0.967807	+	0.251693i	$0.0809872\pi$
$521$			11.4900i			0.503385i	−0.967807	+	0.251693i	$0.919013\pi$
$522$		−	2.97925i		−	0.130398i
$523$	−33.0509			−1.44521			−0.722606	−	0.691260i	$-0.757056\pi$
$523$	−33.0509			−1.44521			−0.722606	+	0.691260i	$0.757056\pi$
$524$			16.9300i			0.739592i
$525$		0			0
$526$	2.60931			0.113771
$527$	27.1798	−	21.0607i	1.18397	−	0.917419i
$528$	22.1126			0.962328
$529$	20.8336			0.905810
$530$		0			0
$531$	10.3586			0.449524
$532$			29.4002i			1.27466i
$533$		−	10.2587i		−	0.444354i
$534$		−	1.23127i		−	0.0532820i
$535$		0			0
$536$	−0.520505			−0.0224824
$537$		−	20.7187i		−	0.894078i
$538$			3.85954i			0.166397i
$539$			34.5891i			1.48986i
$540$		0			0
$541$		−	19.7884i		−	0.850770i	−0.905013	−	0.425385i	$-0.860139\pi$
$541$		−	19.7884i		−	0.850770i	0.905013	−	0.425385i	$-0.139861\pi$
$542$	−2.22077			−0.0953904
$543$	−23.1669			−0.994185
$544$	8.82717	+	11.3918i	0.378462	+	0.488421i
$545$		0			0
$546$	−3.26671			−0.139802
$547$		−	15.4815i		−	0.661940i	−0.943641	−	0.330970i	$-0.892624\pi$
$547$		−	15.4815i		−	0.661940i	0.943641	−	0.330970i	$-0.107376\pi$
$548$	−26.7812			−1.14404
$549$			15.2464i			0.650700i
$550$		0			0
$551$		−	23.5643i		−	1.00387i
$552$	−2.00645			−0.0854003
$553$	6.84146			0.290928
$554$		−	6.35260i		−	0.269896i
$555$		0			0
$556$		−	31.9503i		−	1.35500i
$557$	15.3477			0.650302			0.325151	−	0.945662i	$-0.394585\pi$
$557$	15.3477			0.650302			0.325151	+	0.945662i	$0.394585\pi$
$558$		−	4.51373i		−	0.191081i
$559$	25.3002			1.07008
$560$		0			0
$561$	16.2874	+	21.0197i	0.687657	+	0.887451i
$562$	9.06668			0.382455
$563$	−14.3827			−0.606159			−0.303079	−	0.952965i	$-0.598015\pi$
$563$	−14.3827			−0.606159			−0.303079	+	0.952965i	$0.598015\pi$
$564$			14.1488i			0.595772i
$565$		0			0
$566$			2.85283i			0.119913i
$567$			2.72089i			0.114267i
$568$			1.65537i			0.0694578i
$569$	−12.1432			−0.509069			−0.254535	−	0.967064i	$-0.581922\pi$
$569$	−12.1432			−0.509069			−0.254535	+	0.967064i	$0.581922\pi$
$570$		0			0
$571$			2.88663i			0.120802i	0.998174	+	0.0604009i	$0.0192379\pi$
$571$			2.88663i			0.120802i	−0.998174	+	0.0604009i	$0.980762\pi$
$572$		−	28.3419i		−	1.18503i
$573$			25.6082i			1.06980i
$574$	4.44293			0.185444
$575$		0			0
$576$	−10.0381			−0.418254
$577$	3.10970			0.129458			0.0647291	−	0.997903i	$-0.479382\pi$
$577$	3.10970			0.129458			0.0647291	+	0.997903i	$0.479382\pi$
$578$	−1.32048	+	5.12134i	−0.0549246	+	0.213020i
$579$	−22.1388			−0.920055
$580$		0			0
$581$		−	60.1298i		−	2.49460i
$582$	−4.08880			−0.169486
$583$			3.78469i			0.156746i
$584$			3.95768i			0.163770i
$585$		0			0
$586$	−5.05086			−0.208649
$587$	−10.2034			−0.421140			−0.210570	−	0.977579i	$-0.567532\pi$
$587$	−10.2034			−0.421140			−0.210570	+	0.977579i	$0.567532\pi$
$588$			12.8637i			0.530490i
$589$		−	35.7012i		−	1.47104i
$590$		0			0
$591$	14.4671			0.595095
$592$		−	21.2675i		−	0.874089i
$593$	16.2301			0.666492			0.333246	−	0.942840i	$-0.391856\pi$
$593$	16.2301			0.666492			0.333246	+	0.942840i	$0.391856\pi$
$594$	9.51008			0.390203
$595$		0			0
$596$	9.06868			0.371468
$597$	7.23353			0.296049
$598$			1.18694i			0.0485378i
$599$	−8.43309			−0.344567			−0.172283	−	0.985047i	$-0.555114\pi$
$599$	−8.43309			−0.344567			−0.172283	+	0.985047i	$0.555114\pi$
$600$		0			0
$601$		−	20.2535i		−	0.826160i	−0.910695	−	0.413080i	$-0.864453\pi$
$601$		−	20.2535i		−	0.826160i	0.910695	−	0.413080i	$-0.135547\pi$
$602$			10.9572i			0.446583i
$603$	0.745724			0.0303682
$604$	−5.69181			−0.231596
$605$		0			0
$606$			1.38649i			0.0563225i
$607$			22.7732i			0.924334i	0.886793	+	0.462167i	$0.152928\pi$
$607$			22.7732i			0.924334i	−0.886793	+	0.462167i	$0.847072\pi$
$608$	14.9634			0.606847
$609$		−	22.2975i		−	0.903540i
$610$		0			0
$611$	17.1655			0.694441
$612$	−8.36196	−	10.7915i	−0.338012	−	0.436220i
$613$	8.40636			0.339530			0.169765	−	0.985485i	$-0.445699\pi$
$613$	8.40636			0.339530			0.169765	+	0.985485i	$0.445699\pi$
$614$	1.05086			0.0424091
$615$		0			0
$616$	25.1733			1.01426
$617$			14.8168i			0.596500i	0.954488	+	0.298250i	$0.0964030\pi$
$617$			14.8168i			0.596500i	−0.954488	+	0.298250i	$0.903597\pi$
$618$			1.47911i			0.0594984i
$619$		−	31.6221i		−	1.27100i	−0.772101	−	0.635500i	$-0.780794\pi$
$619$		−	31.6221i		−	1.27100i	0.772101	−	0.635500i	$-0.219206\pi$
$620$		0			0
$621$	7.83161			0.314272
$622$		−	2.47530i		−	0.0992505i
$623$			12.7212i			0.509666i
$624$		−	9.97707i		−	0.399403i
$625$		0			0
$626$		−	2.12605i		−	0.0849740i
$627$	27.6098			1.10263
$628$	26.5397			1.05905
$629$	20.2163	−	15.6650i	0.806078	−	0.624603i
$630$		0			0
$631$	−17.3733			−0.691622			−0.345811	−	0.938304i	$-0.612396\pi$
$631$	−17.3733			−0.691622			−0.345811	+	0.938304i	$0.612396\pi$
$632$		−	2.30231i		−	0.0915810i
$633$	12.7506			0.506793
$634$			8.44812i			0.335518i
$635$		0			0
$636$			1.40753i			0.0558121i
$637$	15.6064			0.618348
$638$	−9.83807			−0.389493
$639$		−	2.37164i		−	0.0938205i
$640$		0			0
$641$			7.26842i			0.287085i	0.989644	+	0.143543i	$0.0458494\pi$
$641$			7.26842i			0.287085i	−0.989644	+	0.143543i	$0.954151\pi$
$642$	−4.13488			−0.163191
$643$			4.40507i			0.173719i	0.996221	+	0.0868595i	$0.0276831\pi$
$643$			4.40507i			0.173719i	−0.996221	+	0.0868595i	$0.972317\pi$
$644$	10.1082			0.398317
$645$		0			0
$646$	3.36349	+	4.34074i	0.132335	+	0.170784i
$647$	−0.198022			−0.00778504			−0.00389252	−	0.999992i	$-0.501239\pi$
$647$	−0.198022			−0.00778504			−0.00389252	+	0.999992i	$0.501239\pi$
$648$	0.915643			0.0359698
$649$		−	34.2061i		−	1.34271i
$650$		0			0
$651$		−	33.7820i		−	1.32402i
$652$		−	16.4721i		−	0.645098i
$653$			9.92723i			0.388482i	0.980954	+	0.194241i	$0.0622244\pi$
$653$			9.92723i			0.388482i	−0.980954	+	0.194241i	$0.937776\pi$
$654$	−4.19897			−0.164193
$655$		0			0
$656$			13.5695i			0.529798i
$657$		−	5.67014i		−	0.221213i
$658$			7.43416i			0.289814i
$659$	3.00492			0.117055			0.0585276	−	0.998286i	$-0.481359\pi$
$659$	3.00492			0.117055			0.0585276	+	0.998286i	$0.481359\pi$
$660$		0			0
$661$	23.6923			0.921523			0.460762	−	0.887524i	$-0.347576\pi$
$661$	23.6923			0.921523			0.460762	+	0.887524i	$0.347576\pi$
$662$	−6.33984			−0.246405
$663$	9.48394	−	7.34880i	0.368326	−	0.285404i
$664$	−20.2351			−0.785273
$665$		0			0
$666$		−	3.35731i		−	0.130093i
$667$	−8.10171			−0.313699
$668$		−	1.47186i		−	0.0569481i
$669$			0.997912i			0.0385815i
$670$		0			0
$671$	50.3466			1.94361
$672$	14.1590			0.546196
$673$		−	30.1968i		−	1.16400i	−0.813188	−	0.582001i	$-0.802270\pi$
$673$		−	30.1968i		−	1.16400i	0.813188	−	0.582001i	$-0.197730\pi$
$674$		−	1.46462i		−	0.0564150i
$675$		0			0
$676$	11.9541			0.459772
$677$		−	47.9413i		−	1.84253i	−0.388933	−	0.921266i	$-0.627156\pi$
$677$		−	47.9413i		−	1.84253i	0.388933	−	0.921266i	$-0.372844\pi$
$678$	3.76895			0.144746
$679$	42.2449			1.62121
$680$		0			0
$681$	14.5131			0.556145
$682$	−14.9052			−0.570750
$683$			1.41477i			0.0541347i	0.999634	+	0.0270674i	$0.00861686\pi$
$683$			1.41477i			0.0541347i	−0.999634	+	0.0270674i	$0.991383\pi$
$684$	−14.1748			−0.541989
$685$		0			0
$686$		−	1.09932i		−	0.0419723i
$687$			3.17880i			0.121279i
$688$	−33.4652			−1.27585
$689$	1.70763			0.0650554
$690$		0			0
$691$			1.36321i			0.0518588i	0.999664	+	0.0259294i	$0.00825452\pi$
$691$			1.36321i			0.0518588i	−0.999664	+	0.0259294i	$0.991745\pi$
$692$			4.58858i			0.174432i
$693$	−36.0656			−1.37002
$694$		−	10.9983i		−	0.417489i
$695$		0			0
$696$	−7.50363			−0.284424
$697$	−12.8988	+	9.99483i	−0.488576	+	0.378581i
$698$	−7.79505			−0.295047
$699$	22.3827			0.846592
$700$		0			0
$701$	−40.1990			−1.51829			−0.759147	−	0.650919i	$-0.774384\pi$
$701$	−40.1990			−1.51829			−0.759147	+	0.650919i	$0.774384\pi$
$702$		−	4.29089i		−	0.161949i
$703$		−	26.5546i		−	1.00153i
$704$			33.1478i			1.24930i
$705$		0			0
$706$	4.26073			0.160355
$707$		−	14.3251i		−	0.538749i
$708$		−	12.7212i		−	0.478094i
$709$			19.4729i			0.731322i	0.930748	+	0.365661i	$0.119157\pi$
$709$			19.4729i			0.731322i	−0.930748	+	0.365661i	$0.880843\pi$
$710$		0			0
$711$			3.29850i			0.123704i
$712$	4.28100			0.160437
$713$	−12.2745			−0.459685
$714$	3.18268	+	4.10738i	0.119109	+	0.153715i
$715$		0			0
$716$	35.1254			1.31270
$717$			30.6314i			1.14395i
$718$	0.788595			0.0294301
$719$			21.9765i			0.819586i	0.912179	+	0.409793i	$0.134399\pi$
$719$			21.9765i			0.819586i	−0.912179	+	0.409793i	$0.865601\pi$
$720$		0			0
$721$		−	15.2819i		−	0.569128i
$722$	−0.209398			−0.00779297
$723$	−4.01291			−0.149242
$724$		−	39.2758i		−	1.45967i
$725$		0			0
$726$		−	7.68528i		−	0.285227i
$727$	−36.7007			−1.36116			−0.680578	−	0.732676i	$-0.738271\pi$
$727$	−36.7007			−1.36116			−0.680578	+	0.732676i	$0.738271\pi$
$728$		−	11.3580i		−	0.420957i
$729$	−19.2317			−0.712284
$730$		0			0
$731$	−24.6494	−	31.8111i	−0.911691	−	1.17658i
$732$	18.7239			0.692056
$733$	−24.8666			−0.918471			−0.459235	−	0.888315i	$-0.651877\pi$
$733$	−24.8666			−0.918471			−0.459235	+	0.888315i	$0.651877\pi$
$734$			11.3065i			0.417330i
$735$		0			0
$736$		−	5.14462i		−	0.189633i
$737$		−	2.46253i		−	0.0907085i
$738$			2.14209i			0.0788514i
$739$	−32.7368			−1.20424			−0.602122	−	0.798404i	$-0.705678\pi$
$739$	−32.7368			−1.20424			−0.602122	+	0.798404i	$0.705678\pi$
$740$		0			0
$741$		−	12.4574i		−	0.457633i
$742$			0.739554i			0.0271499i
$743$			21.3762i			0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$743$			21.3762i			0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$744$	−11.3684			−0.416786
$745$		0			0
$746$	−3.97328			−0.145472
$747$	28.9906			1.06071
$748$	−35.6356	+	27.6129i	−1.30297	+	1.00963i
$749$	42.7210			1.56099
$750$		0			0
$751$			44.5557i			1.62586i	0.582362	+	0.812930i	$0.302129\pi$
$751$			44.5557i			1.62586i	−0.582362	+	0.812930i	$0.697871\pi$
$752$	−22.7052			−0.827973
$753$			19.7724i			0.720545i
$754$			4.43887i			0.161654i
$755$		0			0
$756$	−36.5417			−1.32901
$757$	−12.1793			−0.442664			−0.221332	−	0.975199i	$-0.571040\pi$
$757$	−12.1793			−0.442664			−0.221332	+	0.975199i	$0.571040\pi$
$758$			6.89089i			0.250288i
$759$		−	9.49260i		−	0.344560i
$760$		0			0
$761$	7.18712			0.260533			0.130266	−	0.991479i	$-0.458417\pi$
$761$	7.18712			0.260533			0.130266	+	0.991479i	$0.458417\pi$
$762$			5.98559i			0.216835i
$763$	43.3832			1.57058
$764$	−43.4148			−1.57069
$765$		0			0
$766$	−0.607926			−0.0219652
$767$	−15.4336			−0.557274
$768$			9.88618i			0.356737i
$769$	14.9748			0.540005			0.270003	−	0.962860i	$-0.412975\pi$
$769$	14.9748			0.540005			0.270003	+	0.962860i	$0.412975\pi$
$770$		0			0
$771$			23.0653i			0.830678i
$772$		−	37.5328i		−	1.35084i
$773$	26.2938			0.945721			0.472860	−	0.881137i	$-0.343222\pi$
$773$	26.2938			0.945721			0.472860	+	0.881137i	$0.343222\pi$
$774$	−5.28286			−0.189888
$775$		0			0
$776$		−	14.2164i		−	0.510339i
$777$		−	25.1270i		−	0.901428i
$778$	−2.46167			−0.0882550
$779$			16.9428i			0.607039i
$780$		0			0
$781$	−7.83161			−0.280237
$782$	1.49240	−	1.15641i	0.0533682	−	0.0413533i
$783$	29.2883			1.04668
$784$	−20.6430			−0.737249
$785$		0			0
$786$	3.10678			0.110815
$787$		−	9.35342i		−	0.333413i	−0.986007	−	0.166707i	$-0.946687\pi$
$787$		−	9.35342i		−	0.333413i	0.986007	−	0.166707i	$-0.0533133\pi$
$788$			24.5267i			0.873727i
$789$			9.41550i			0.335201i
$790$		0			0
$791$	−38.9403			−1.38456
$792$			12.1369i			0.431267i
$793$		−	22.7161i		−	0.806672i
$794$			2.87612i			0.102070i
$795$		0			0
$796$			12.2633i			0.434663i
$797$	29.4800			1.04423			0.522117	−	0.852874i	$-0.325142\pi$
$797$	29.4800			1.04423			0.522117	+	0.852874i	$0.325142\pi$
$798$	5.39514			0.190986
$799$	−16.7239	−	21.5830i	−0.591650	−	0.763550i
$800$		0			0
$801$	−6.13335			−0.216711
$802$			6.68409i			0.236024i
$803$	−18.7239			−0.660753
$804$		−	0.915816i		−	0.0322983i
$805$		0			0
$806$			6.72514i			0.236883i
$807$	−13.9269			−0.490249
$808$	−4.82071			−0.169592
$809$			35.1219i			1.23482i	0.786642	+	0.617410i	$0.211818\pi$
$809$			35.1219i			1.23482i	−0.786642	+	0.617410i	$0.788182\pi$
$810$		0			0
$811$			17.5682i			0.616902i	0.951240	+	0.308451i	$0.0998106\pi$
$811$			17.5682i			0.616902i	−0.951240	+	0.308451i	$0.900189\pi$
$812$	37.8020			1.32659
$813$		−	8.01350i		−	0.281046i
$814$	−11.0865			−0.388582
$815$		0			0
$816$	−12.5446	+	9.72043i	−0.439150	+	0.340283i
$817$	−41.7846			−1.46186
$818$	5.65185			0.197612
$819$			16.2726i			0.568610i
$820$		0			0
$821$			13.1043i			0.457343i	0.973504	+	0.228672i	$0.0734382\pi$
$821$			13.1043i			0.457343i	−0.973504	+	0.228672i	$0.926562\pi$
$822$			4.91454i			0.171414i
$823$		−	23.7638i		−	0.828355i	−0.910196	−	0.414178i	$-0.864069\pi$
$823$		−	23.7638i		−	0.828355i	0.910196	−	0.414178i	$-0.135931\pi$
$824$	−5.14272			−0.179155
$825$		0			0
$826$		−	6.68409i		−	0.232569i
$827$			28.9423i			1.00642i	0.864164	+	0.503211i	$0.167848\pi$
$827$			28.9423i			1.00642i	−0.864164	+	0.503211i	$0.832152\pi$
$828$			4.87350i			0.169366i
$829$	48.3131			1.67798			0.838992	−	0.544144i	$-0.183145\pi$
$829$	48.3131			1.67798			0.838992	+	0.544144i	$0.183145\pi$
$830$		0			0
$831$	22.9229			0.795187
$832$	14.9561			0.518509
$833$	−15.2050	−	19.6227i	−0.526820	−	0.679885i
$834$	−5.86311			−0.203023
$835$		0			0
$836$			46.8081i			1.61889i
$837$	44.3733			1.53377
$838$		−	4.46715i		−	0.154315i
$839$		−	36.8238i		−	1.27130i	−0.771978	−	0.635650i	$-0.780732\pi$
$839$		−	36.8238i		−	1.27130i	0.771978	−	0.635650i	$-0.219268\pi$
$840$		0			0
$841$	−1.29835			−0.0447707
$842$	1.23506			0.0425631
$843$			32.7164i			1.12681i
$844$			21.6168i			0.744079i
$845$		0			0
$846$	−3.58427			−0.123230
$847$			79.4031i			2.72832i
$848$	−2.25872			−0.0775648
$849$	−10.2942			−0.353297
$850$		0			0
$851$	−9.12981			−0.312966
$852$	−2.91258			−0.0997833
$853$		−	29.8138i		−	1.02080i	−0.859936	−	0.510402i	$-0.829497\pi$
$853$		−	29.8138i		−	1.02080i	0.859936	−	0.510402i	$-0.170503\pi$
$854$	9.83807			0.336652
$855$		0			0
$856$		−	14.3766i		−	0.491383i
$857$			1.71246i			0.0584965i	0.999572	+	0.0292483i	$0.00931134\pi$
$857$			1.71246i			0.0584965i	−0.999572	+	0.0292483i	$0.990689\pi$
$858$	−5.20094			−0.177557
$859$	22.5749			0.770246			0.385123	−	0.922865i	$-0.374159\pi$
$859$	22.5749			0.770246			0.385123	+	0.922865i	$0.374159\pi$
$860$		0			0
$861$			16.0320i			0.546369i
$862$			10.2459i			0.348978i
$863$	54.2578			1.84696			0.923479	−	0.383650i	$-0.125333\pi$
$863$	54.2578			1.84696			0.923479	+	0.383650i	$0.125333\pi$
$864$			18.5982i			0.632723i
$865$		0			0
$866$	11.8938			0.404169
$867$	−18.4800	−	4.76484i	−0.627613	−	0.161822i
$868$	57.2721			1.94394
$869$	10.8923			0.369496
$870$		0			0
$871$	−1.11108			−0.0376474
$872$		−	14.5995i		−	0.494400i
$873$			20.3677i			0.689343i
$874$		−	1.96030i		−	0.0663082i
$875$		0			0
$876$	−6.96343			−0.235273
$877$			14.8989i			0.503099i	0.967844	+	0.251549i	$0.0809401\pi$
$877$			14.8989i			0.503099i	−0.967844	+	0.251549i	$0.919060\pi$
$878$			0.206794i			0.00697896i
$879$		−	18.2256i		−	0.614736i
$880$		0			0
$881$			22.3330i			0.752419i	0.926535	+	0.376209i	$0.122773\pi$
$881$			22.3330i			0.752419i	−0.926535	+	0.376209i	$0.877227\pi$
$882$	−3.25872			−0.109727
$883$	25.3319			0.852485			0.426242	−	0.904609i	$-0.359837\pi$
$883$	25.3319			0.852485			0.426242	+	0.904609i	$0.359837\pi$
$884$	12.4588	+	16.0786i	0.419033	+	0.540780i
$885$		0			0
$886$	−8.57628			−0.288126
$887$			6.35813i			0.213485i	0.994287	+	0.106743i	$0.0340421\pi$
$887$			6.35813i			0.213485i	−0.994287	+	0.106743i	$0.965958\pi$
$888$	−8.45584			−0.283759
$889$		−	61.8422i		−	2.07412i
$890$		0			0
$891$			4.33194i			0.145125i
$892$	−1.69181			−0.0566459
$893$	−28.3497			−0.948686
$894$		−	1.66417i		−	0.0556581i
$895$		0			0
$896$			31.7025i			1.05911i
$897$	−4.28300			−0.143005
$898$			5.43679i			0.181428i
$899$	−45.9037			−1.53097
$900$		0			0
$901$	−1.66370	−	2.14708i	−0.0554260	−	0.0715296i
$902$	7.07361			0.235525
$903$	−39.5383			−1.31575
$904$			13.1043i			0.435843i
$905$		0			0
$906$			1.04449i			0.0347007i
$907$			13.0295i			0.432636i	0.976323	+	0.216318i	$0.0694048\pi$
$907$			13.0295i			0.432636i	−0.976323	+	0.216318i	$0.930595\pi$
$908$			24.6048i			0.816539i
$909$	6.90660			0.229078
$910$		0			0
$911$		−	20.7685i		−	0.688093i	−0.938953	−	0.344046i	$-0.888202\pi$
$911$		−	20.7685i		−	0.688093i	0.938953	−	0.344046i	$-0.111798\pi$
$912$			16.4777i			0.545630i
$913$		−	95.7328i		−	3.16829i
$914$	−4.05530			−0.134137
$915$		0			0
$916$	−5.38916			−0.178063
$917$	−32.0988			−1.06000
$918$	−5.39514	+	4.18052i	−0.178066	+	0.137978i
$919$	−5.10970			−0.168553			−0.0842766	−	0.996442i	$-0.526858\pi$
$919$	−5.10970			−0.168553			−0.0842766	+	0.996442i	$0.526858\pi$
$920$		0			0
$921$			3.79193i			0.124948i
$922$	−3.40636			−0.112183
$923$			3.53358i			0.116309i
$924$			44.2918i			1.45709i
$925$		0			0
$926$	5.36349			0.176255
$927$	7.36794			0.241995
$928$		−	19.2396i		−	0.631571i
$929$			39.6073i			1.29947i	0.760159	+	0.649737i	$0.225121\pi$
$929$			39.6073i			1.29947i	−0.760159	+	0.649737i	$0.774879\pi$
$930$		0			0
$931$	−25.7748			−0.844734
$932$			37.9464i			1.24298i
$933$	8.93194			0.292419
$934$	−0.0129070			−0.000422330
$935$		0			0
$936$	5.47610			0.178992
$937$	−39.5575			−1.29229			−0.646144	−	0.763215i	$-0.723620\pi$
$937$	−39.5575			−1.29229			−0.646144	+	0.763215i	$0.723620\pi$
$938$		−	0.481195i		−	0.0157116i
$939$	7.67169			0.250356
$940$		0			0
$941$		−	53.2283i		−	1.73519i	−0.497268	−	0.867597i	$-0.665663\pi$
$941$		−	53.2283i		−	1.73519i	0.497268	−	0.867597i	$-0.334337\pi$
$942$		−	4.87022i		−	0.158680i
$943$	5.82516			0.189693
$944$	20.4143			0.664430
$945$		0			0
$946$			17.4450i			0.567187i
$947$			0.773357i			0.0251307i	0.999921	+	0.0125654i	$0.00399978\pi$
$947$			0.773357i			0.0251307i	−0.999921	+	0.0125654i	$0.996000\pi$
$948$	4.05086			0.131566
$949$			8.44812i			0.274238i
$950$		0			0
$951$	−30.4844			−0.988525
$952$	−14.2810	+	11.0659i	−0.462850	+	0.358647i
$953$	−36.8731			−1.19444			−0.597218	−	0.802079i	$-0.703727\pi$
$953$	−36.8731			−1.19444			−0.597218	+	0.802079i	$0.703727\pi$
$954$	−0.356564			−0.0115442
$955$		0			0
$956$	−51.9309			−1.67956
$957$		−	35.4999i		−	1.14755i
$958$		−	11.0321i		−	0.356430i
$959$		−	50.7763i		−	1.63965i
$960$		0			0
$961$	−38.5466			−1.24344
$962$			5.00217i			0.161276i
$963$			20.5973i			0.663738i
$964$		−	6.80327i		−	0.219118i
$965$		0			0
$966$		−	1.85492i		−	0.0596810i
$967$	−17.8306			−0.573392			−0.286696	−	0.958022i	$-0.592557\pi$
$967$	−17.8306			−0.573392			−0.286696	+	0.958022i	$0.592557\pi$
$968$	26.7210			0.858846
$969$	−15.6632	+	12.1369i	−0.503175	+	0.389894i
$970$		0			0
$971$	−20.3783			−0.653970			−0.326985	−	0.945030i	$-0.606033\pi$
$971$	−20.3783			−0.653970			−0.326985	+	0.945030i	$0.606033\pi$
$972$		−	28.7693i		−	0.922775i
$973$	60.5768			1.94200
$974$		−	7.67476i		−	0.245915i
$975$		0			0
$976$			30.0471i			0.961785i
$977$	8.64587			0.276606			0.138303	−	0.990390i	$-0.455835\pi$
$977$	8.64587			0.276606			0.138303	+	0.990390i	$0.455835\pi$
$978$	−3.02275			−0.0966569
$979$			20.2535i			0.647306i
$980$		0			0
$981$			20.9165i			0.667813i
$982$	2.66953			0.0851882
$983$		−	61.1615i		−	1.95075i	−0.220558	−	0.975374i	$-0.570788\pi$
$983$		−	61.1615i		−	1.95075i	0.220558	−	0.975374i	$-0.429212\pi$
$984$	5.39514			0.171991
$985$		0			0
$986$	5.58120	−	4.32469i	0.177742	−	0.137726i
$987$	−26.8256			−0.853869
$988$	21.1195			0.671902
$989$			14.3661i			0.456815i
$990$		0			0
$991$			27.0979i			0.860792i	0.902640	+	0.430396i	$0.141626\pi$
$991$			27.0979i			0.860792i	−0.902640	+	0.430396i	$0.858374\pi$
$992$		−	29.1491i		−	0.925484i
$993$		−	22.8768i		−	0.725974i
$994$	−1.53035			−0.0485397
$995$		0			0
$996$		−	35.6031i		−	1.12813i
$997$			48.3759i			1.53208i	0.642793	+	0.766040i	$0.277775\pi$
$997$			48.3759i			1.53208i	−0.642793	+	0.766040i	$0.722225\pi$
$998$			2.64105i			0.0836009i
$999$	33.0049			1.04423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.d.b.101.4	yes	6
5.2	odd	4		425.2.c.c.424.8		12
5.3	odd	4		425.2.c.c.424.5		12
5.4	even	2		425.2.d.a.101.3	✓	6
17.4	even	4		7225.2.a.ba.1.4		6
17.13	even	4		7225.2.a.ba.1.3		6
17.16	even	2	inner	425.2.d.b.101.3	yes	6
85.4	even	4		7225.2.a.bg.1.3		6
85.33	odd	4		425.2.c.c.424.6		12
85.64	even	4		7225.2.a.bg.1.4		6
85.67	odd	4		425.2.c.c.424.7		12
85.84	even	2		425.2.d.a.101.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.c.c.424.5		12	5.3	odd	4
425.2.c.c.424.6		12	85.33	odd	4
425.2.c.c.424.7		12	85.67	odd	4
425.2.c.c.424.8		12	5.2	odd	4
425.2.d.a.101.3	✓	6	5.4	even	2
425.2.d.a.101.4	yes	6	85.84	even	2
425.2.d.b.101.3	yes	6	17.16	even	2	inner
425.2.d.b.101.4	yes	6	1.1	even	1	trivial
7225.2.a.ba.1.3		6	17.13	even	4
7225.2.a.ba.1.4		6	17.4	even	4
7225.2.a.bg.1.3		6	85.4	even	4
7225.2.a.bg.1.4		6	85.64	even	4

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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} +1.12261i q^{3} -1.90321 q^{4} +0.349253i q^{6} -3.60843i q^{7} -1.21432 q^{8} +1.73975 q^{9} +O(q^{10})\)	\(q+0.311108 q^{2} +1.12261i q^{3} -1.90321 q^{4} +0.349253i q^{6} -3.60843i q^{7} -1.21432 q^{8} +1.73975 q^{9} -5.74499i q^{11} -2.13656i q^{12} -2.59210 q^{13} -1.12261i q^{14} +3.42864 q^{16} +(2.52543 + 3.25917i) q^{17} +0.541249 q^{18} +4.28100 q^{19} +4.05086 q^{21} -1.78731i q^{22} -1.47186i q^{23} -1.36321i q^{24} -0.806424 q^{26} +5.32089i q^{27} +6.86760i q^{28} -5.50439i q^{29} -8.33946i q^{31} +3.49532 q^{32} +6.44938 q^{33} +(0.785680 + 1.01395i) q^{34} -3.31111 q^{36} -6.20290i q^{37} +1.33185 q^{38} -2.90992i q^{39} +3.95768i q^{41} +1.26025 q^{42} -9.76049 q^{43} +10.9339i q^{44} -0.457908i q^{46} -6.62222 q^{47} +3.84902i q^{48} -6.02074 q^{49} +(-3.65878 + 2.83507i) q^{51} +4.93332 q^{52} -0.658781 q^{53} +1.65537i q^{54} +4.38178i q^{56} +4.80589i q^{57} -1.71246i q^{58} +5.95407 q^{59} +8.76357i q^{61} -2.59447i q^{62} -6.27775i q^{63} -5.76986 q^{64} +2.00645 q^{66} +0.428639 q^{67} +(-4.80642 - 6.20290i) q^{68} +1.65233 q^{69} -1.36321i q^{71} -2.11261 q^{72} -3.25917i q^{73} -1.92977i q^{74} -8.14764 q^{76} -20.7304 q^{77} -0.905299i q^{78} +1.89597i q^{79} -0.754037 q^{81} +1.23127i q^{82} +16.6637 q^{83} -7.70964 q^{84} -3.03657 q^{86} +6.17929 q^{87} +6.97626i q^{88} -3.52543 q^{89} +9.35342i q^{91} +2.80127i q^{92} +9.36196 q^{93} -2.06022 q^{94} +3.92388i q^{96} +11.7073i q^{97} -1.87310 q^{98} -9.99483i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} - 16 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 - 16 * q^9	\( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} - 16 q^{9} - 2 q^{13} - 6 q^{16} + 2 q^{17} + 16 q^{18} + 12 q^{19} - 2 q^{21} + 22 q^{26} - 6 q^{32} - 28 q^{33} + 18 q^{34} - 20 q^{36} - 32 q^{38} + 34 q^{42} + 8 q^{43} - 40 q^{47} + 4 q^{49} - 8 q^{51} + 30 q^{52} + 10 q^{53} - 4 q^{59} - 22 q^{64} - 16 q^{66} - 24 q^{67} - 2 q^{68} + 24 q^{69} - 52 q^{72} - 36 q^{76} - 44 q^{77} + 34 q^{81} + 20 q^{83} - 6 q^{84} - 4 q^{86} + 76 q^{87} - 8 q^{89} + 30 q^{93} - 40 q^{94} + 16 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 - 16 * q^9 - 2 * q^13 - 6 * q^16 + 2 * q^17 + 16 * q^18 + 12 * q^19 - 2 * q^21 + 22 * q^26 - 6 * q^32 - 28 * q^33 + 18 * q^34 - 20 * q^36 - 32 * q^38 + 34 * q^42 + 8 * q^43 - 40 * q^47 + 4 * q^49 - 8 * q^51 + 30 * q^52 + 10 * q^53 - 4 * q^59 - 22 * q^64 - 16 * q^66 - 24 * q^67 - 2 * q^68 + 24 * q^69 - 52 * q^72 - 36 * q^76 - 44 * q^77 + 34 * q^81 + 20 * q^83 - 6 * q^84 - 4 * q^86 + 76 * q^87 - 8 * q^89 + 30 * q^93 - 40 * q^94 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more