LMFDB - Embedded newform 425.2.d.a.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.a.101.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	$-0.535083\pi$
$2$	−0.311108			−0.219986			−0.109993	+	0.993932i	$0.535083\pi$
$3$			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	$0.333374\pi$
$11$			5.74499i			1.73218i	−0.499888	+	0.866090i	$0.666626\pi$
$12$		−	2.13656i		−	0.616773i
$13$	2.59210			0.718920			0.359460	−	0.933160i	$-0.382961\pi$
$13$	2.59210			0.718920			0.359460	+	0.933160i	$0.382961\pi$
$14$			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	$0.170751\pi$
$29$			5.50439i			1.02214i	−0.859539	+	0.511070i	$0.829249\pi$
$30$		0			0
$31$			8.33946i			1.49781i	0.662676	+	0.748906i	$0.269421\pi$
$31$			8.33946i			1.49781i	−0.662676	+	0.748906i	$0.730579\pi$
$32$	−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	$-0.899991\pi$
$41$		−	3.95768i		−	0.618086i	0.951048	−	0.309043i	$-0.100009\pi$
$42$	−1.26025			−0.194461
$43$	9.76049			1.48846			0.744230	−	0.667923i	$-0.232816\pi$
$43$	9.76049			1.48846			0.744230	+	0.667923i	$0.232816\pi$
$44$		−	10.9339i		−	1.64835i
$45$		0			0
$46$			0.457908i			0.0675148i
$47$	6.62222			0.965949			0.482975	−	0.875634i	$-0.339556\pi$
$47$	6.62222			0.965949			0.482975	+	0.875634i	$0.339556\pi$
$48$			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	$-0.485593\pi$
$53$	0.658781			0.0904905			0.0452452	+	0.998976i	$0.485593\pi$
$54$		−	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	$-0.810405\pi$
$61$		−	8.76357i		−	1.12206i	0.827796	−	0.561030i	$-0.189595\pi$
$62$		−	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	$-0.508335\pi$
$67$	−0.428639			−0.0523666			−0.0261833	+	0.999657i	$0.508335\pi$
$68$	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	$0.0257767\pi$
$71$			1.36321i			0.161783i	−0.996723	+	0.0808915i	$0.974223\pi$
$72$	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	$-0.965985\pi$
$79$		−	1.89597i		−	0.213313i	0.994296	−	0.106656i	$-0.0340145\pi$
$80$		0			0
$81$	−0.754037			−0.0837819
$82$			1.23127i			0.135970i
$83$	−16.6637			−1.82908			−0.914540	−	0.404497i	$-0.867447\pi$
$83$	−16.6637			−1.82908			−0.914540	+	0.404497i	$0.867447\pi$
$84$	−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	$-0.566906\pi$
$103$	−4.23506			−0.417293			−0.208647	+	0.977991i	$0.566906\pi$
$104$	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	$-0.804697\pi$
$109$		−	12.0227i		−	1.15157i	0.817601	−	0.575785i	$-0.195303\pi$
$110$		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	$-0.774991\pi$
$127$	−17.1383			−1.52078			−0.760388	+	0.649469i	$0.774991\pi$
$128$	8.78568			0.776552
$129$			10.9572i			0.964730i
$130$		0			0
$131$			8.89551i			0.777204i	0.921406	+	0.388602i	$0.127042\pi$
$131$			8.89551i			0.777204i	−0.921406	+	0.388602i	$0.872958\pi$
$132$	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	$-0.705274\pi$
$137$	−14.0716			−1.20222			−0.601109	+	0.799167i	$0.705274\pi$
$138$	−0.514052			−0.0437590
$139$		−	16.7876i		−	1.42390i	−0.702228	−	0.711952i	$-0.747811\pi$
$139$		−	16.7876i		−	1.42390i	0.702228	−	0.711952i	$-0.252189\pi$
$140$		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	$-0.312162\pi$
$157$	13.9447			1.11291			0.556454	+	0.830878i	$0.312162\pi$
$158$			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	$-0.721772\pi$
$181$		−	20.6366i		−	1.53391i	0.641703	−	0.766953i	$-0.278228\pi$
$182$			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	$0.0733441\pi$
$199$			6.44350i			0.456767i	−0.973571	+	0.228384i	$0.926656\pi$
$200$		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	$0.127857\pi$
$211$			11.3580i			0.781920i	−0.920408	+	0.390960i	$0.872143\pi$
$212$	−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	$-0.509475\pi$
$223$	−0.888922			−0.0595266			−0.0297633	+	0.999557i	$0.509475\pi$
$224$			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	$-0.963271\pi$
$241$		−	3.57462i		−	0.230262i	0.993350	−	0.115131i	$-0.0367287\pi$
$242$	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	$-0.721404\pi$
$257$	−20.5462			−1.28163			−0.640817	+	0.767693i	$0.721404\pi$
$258$		−	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	$-0.583257\pi$
$263$	−8.38715			−0.517174			−0.258587	+	0.965988i	$0.583257\pi$
$264$	−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	$-0.876544\pi$
$269$		−	12.4058i		−	0.756395i	0.925725	−	0.378197i	$-0.123456\pi$
$270$		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	$-0.342726\pi$
$293$	16.2351			0.948463			0.474231	+	0.880400i	$0.342726\pi$
$294$			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	$-0.530730\pi$
$307$	−3.37778			−0.192780			−0.0963902	+	0.995344i	$0.530730\pi$
$308$	−39.4543			−2.24812
$309$		−	4.75432i		−	0.270464i
$310$		0			0
$311$			7.95641i			0.451166i	0.974224	+	0.225583i	$0.0724288\pi$
$311$			7.95641i			0.451166i	−0.974224	+	0.225583i	$0.927571\pi$
$312$			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	$-0.618748\pi$
$353$	−13.6953			−0.728930			−0.364465	+	0.931217i	$0.618748\pi$
$354$		−	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	$-0.392736\pi$
$373$	12.7714			0.661278			0.330639	+	0.943757i	$0.392736\pi$
$374$	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	$-0.807380\pi$
$379$		−	22.1495i		−	1.13774i	0.822426	−	0.568872i	$-0.192620\pi$
$380$		0			0
$381$		−	19.2396i		−	0.985674i
$382$	−7.09679			−0.363103
$383$	1.95407			0.0998482			0.0499241	−	0.998753i	$-0.484102\pi$
$383$	1.95407			0.0998482			0.0499241	+	0.998753i	$0.484102\pi$
$384$			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	$-0.819765\pi$
$401$		−	21.4848i		−	1.07290i	0.843932	−	0.536450i	$-0.180235\pi$
$402$			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	$0.114069\pi$
$419$			14.3589i			0.701476i	−0.936474	+	0.350738i	$0.885931\pi$
$420$		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	$-0.708423\pi$
$431$		−	32.9337i		−	1.58636i	0.608985	−	0.793181i	$-0.291577\pi$
$432$			18.2434i			0.877736i
$433$	−38.2306			−1.83725			−0.918623	−	0.395135i	$-0.870698\pi$
$433$	−38.2306			−1.83725			−0.918623	+	0.395135i	$0.870698\pi$
$434$	−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	$-0.994951\pi$
$439$		−	0.664702i		−	0.0317245i	0.999874	−	0.0158622i	$-0.00504932\pi$
$440$		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	$-0.272723\pi$
$443$	27.5669			1.30974			0.654872	+	0.755740i	$0.272723\pi$
$444$	−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	$-0.864704\pi$
$449$		−	17.4756i		−	0.824723i	0.911020	−	0.412362i	$-0.135296\pi$
$450$		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	$-0.401385\pi$
$457$	13.0350			0.609753			0.304877	+	0.952392i	$0.401385\pi$
$458$	−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	$-0.631200\pi$
$463$	−17.2400			−0.801210			−0.400605	+	0.916251i	$0.631200\pi$
$464$			18.8726i			0.876138i
$465$		0			0
$466$			6.20290i			0.287344i
$467$	0.0414872			0.00191980			0.000959899	−	1.00000i	$-0.499694\pi$
$467$	0.0414872			0.00191980			0.000959899		1.00000i	$0.499694\pi$
$468$	−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	$0.300597\pi$
$479$			35.4606i			1.62024i	−0.586266	+	0.810118i	$0.699403\pi$
$480$		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	$-0.939147\pi$
$499$		−	8.48917i		−	0.380027i	0.981781	−	0.190014i	$-0.0608532\pi$
$500$		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	$-0.919013\pi$
$521$		−	11.4900i		−	0.503385i	0.967807	−	0.251693i	$-0.0809872\pi$
$522$		−	2.97925i		−	0.130398i
$523$	33.0509			1.44521			0.722606	−	0.691260i	$-0.242944\pi$
$523$	33.0509			1.44521			0.722606	+	0.691260i	$0.242944\pi$
$524$		−	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	$0.139861\pi$
$541$			19.7884i			0.850770i	−0.905013	+	0.425385i	$0.860139\pi$
$542$	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	$-0.605415\pi$
$557$	−15.3477			−0.650302			−0.325151	+	0.945662i	$0.605415\pi$
$558$		−	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	$-0.401985\pi$
$563$	14.3827			0.606159			0.303079	+	0.952965i	$0.401985\pi$
$564$		−	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	$-0.980762\pi$
$571$		−	2.88663i		−	0.120802i	0.998174	−	0.0604009i	$-0.0192379\pi$
$572$		−	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	$-0.520618\pi$
$577$	−3.10970			−0.129458			−0.0647291	+	0.997903i	$0.520618\pi$
$578$	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	$-0.432468\pi$
$587$	10.2034			0.421140			0.210570	+	0.977579i	$0.432468\pi$
$588$			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	$-0.608144\pi$
$593$	−16.2301			−0.666492			−0.333246	+	0.942840i	$0.608144\pi$
$594$	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	$0.135547\pi$
$601$			20.2535i			0.826160i	−0.910695	+	0.413080i	$0.864453\pi$
$602$			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	$-0.554301\pi$
$613$	−8.40636			−0.339530			−0.169765	+	0.985485i	$0.554301\pi$
$614$	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	$0.219206\pi$
$619$			31.6221i			1.27100i	−0.772101	+	0.635500i	$0.780794\pi$
$620$		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	$-0.954151\pi$
$641$		−	7.26842i		−	0.287085i	0.989644	−	0.143543i	$-0.0458494\pi$
$642$	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	$-0.498761\pi$
$647$	0.198022			0.00778504			0.00389252	+	0.999992i	$0.498761\pi$
$648$	−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	$-0.991745\pi$
$691$		−	1.36321i		−	0.0518588i	0.999664	−	0.0259294i	$-0.00825452\pi$
$692$			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	$-0.880843\pi$
$709$		−	19.4729i		−	0.731322i	0.930748	−	0.365661i	$-0.119157\pi$
$710$		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	$-0.865601\pi$
$719$		−	21.9765i		−	0.819586i	0.912179	−	0.409793i	$-0.134399\pi$
$720$		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	$-0.261729\pi$
$727$	36.7007			1.36116			0.680578	+	0.732676i	$0.261729\pi$
$728$		−	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	$-0.348123\pi$
$733$	24.8666			0.918471			0.459235	+	0.888315i	$0.348123\pi$
$734$		−	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	$-0.697871\pi$
$751$		−	44.5557i		−	1.62586i	0.582362	−	0.812930i	$-0.302129\pi$
$752$	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	$-0.428960\pi$
$757$	12.1793			0.442664			0.221332	+	0.975199i	$0.428960\pi$
$758$			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	$-0.656778\pi$
$773$	−26.2938			−0.945721			−0.472860	+	0.881137i	$0.656778\pi$
$774$	−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

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)

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

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Representations

Groups

Database

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