LMFDB - Embedded newform 425.2.c.c.424.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(424,425)]

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([1, 1]))

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

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

chi := DirichletCharacter("425.424");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.c (of order $2$, degree $1$, not 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 16 x^{10} - 4 x^{9} + 111 x^{8} + 4 x^{7} + 394 x^{6} + 236 x^{5} + 581 x^{4} + 1000 x^{3} + \cdots + 845 $ x^12 + 16x^10 - 4x^9 + 111x^8 + 4x^7 + 394x^6 + 236x^5 + 581x^4 + 1000x^3 + 962x^2 + 1504x + 845
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			424.7
Root			$1.10716 + 1.97403i$ of defining polynomial
Character	$\chi$	$=$	425.424
Dual form			425.2.c.c.424.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.311108i q^{2} -1.12261 q^{3} +1.90321 q^{4} -0.349253i q^{6} -3.60843 q^{7} +1.21432i q^{8} -1.73975 q^{9} +O(q^{10})$	\(q+0.311108i q^{2} -1.12261 q^{3} +1.90321 q^{4} -0.349253i q^{6} -3.60843 q^{7} +1.21432i q^{8} -1.73975 q^{9} +5.74499i q^{11} -2.13656 q^{12} +2.59210i q^{13} -1.12261i q^{14} +3.42864 q^{16} +(3.25917 + 2.52543i) q^{17} -0.541249i q^{18} -4.28100 q^{19} +4.05086 q^{21} -1.78731 q^{22} +1.47186 q^{23} -1.36321i q^{24} -0.806424 q^{26} +5.32089 q^{27} -6.86760 q^{28} -5.50439i q^{29} +8.33946i q^{31} +3.49532i q^{32} -6.44938i q^{33} +(-0.785680 + 1.01395i) q^{34} -3.31111 q^{36} -6.20290 q^{37} -1.33185i q^{38} -2.90992i q^{39} -3.95768i q^{41} +1.26025i q^{42} +9.76049i q^{43} +10.9339i q^{44} +0.457908i q^{46} -6.62222i q^{47} -3.84902 q^{48} +6.02074 q^{49} +(-3.65878 - 2.83507i) q^{51} +4.93332i q^{52} +0.658781i q^{53} +1.65537i q^{54} -4.38178i q^{56} +4.80589 q^{57} +1.71246 q^{58} -5.95407 q^{59} -8.76357i q^{61} -2.59447 q^{62} +6.27775 q^{63} +5.76986 q^{64} +2.00645 q^{66} +0.428639i q^{67} +(6.20290 + 4.80642i) q^{68} -1.65233 q^{69} +1.36321i q^{71} -2.11261i q^{72} +3.25917 q^{73} -1.92977i q^{74} -8.14764 q^{76} -20.7304i q^{77} +0.905299 q^{78} +1.89597i q^{79} -0.754037 q^{81} +1.23127 q^{82} -16.6637i q^{83} +7.70964 q^{84} -3.03657 q^{86} +6.17929i q^{87} -6.97626 q^{88} +3.52543 q^{89} -9.35342i q^{91} +2.80127 q^{92} -9.36196i q^{93} +2.06022 q^{94} -3.92388i q^{96} +11.7073 q^{97} +1.87310i q^{98} -9.99483i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{4} + 32 q^{9}+O(q^{10}) $ 12 * q - 4 * q^4 + 32 * q^9	$ 12 q - 4 q^{4} + 32 q^{9} - 12 q^{16} - 24 q^{19} - 4 q^{21} + 44 q^{26} - 36 q^{34} - 40 q^{36} - 8 q^{49} - 16 q^{51} + 8 q^{59} + 44 q^{64} - 32 q^{66} - 48 q^{69} - 72 q^{76} + 68 q^{81} + 12 q^{84} - 8 q^{86} + 16 q^{89} + 80 q^{94}+O(q^{100}) $ 12 * q - 4 * q^4 + 32 * q^9 - 12 * q^16 - 24 * q^19 - 4 * q^21 + 44 * q^26 - 36 * q^34 - 40 * q^36 - 8 * q^49 - 16 * q^51 + 8 * q^59 + 44 * q^64 - 32 * q^66 - 48 * q^69 - 72 * q^76 + 68 * q^81 + 12 * q^84 - 8 * q^86 + 16 * q^89 + 80 * q^94

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 $

$2$			0.311108i			0.219986i	0.993932	+	0.109993i	$0.0350829\pi$
$2$			0.311108i			0.219986i	−0.993932	+	0.109993i	$0.964917\pi$
$3$	−1.12261			−0.648139			−0.324070	−	0.946033i	$-0.605051\pi$
$3$	−1.12261			−0.648139			−0.324070	+	0.946033i	$0.605051\pi$
$4$	1.90321			0.951606
$5$		0			0
$5$		0			0
$6$		−	0.349253i		−	0.142582i
$7$	−3.60843			−1.36386			−0.681929	−	0.731419i	$-0.738859\pi$
$7$	−3.60843			−1.36386			−0.681929	+	0.731419i	$0.738859\pi$
$8$			1.21432i			0.429327i
$9$	−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	$0.666626\pi$
$12$	−2.13656			−0.616773
$13$			2.59210i			0.718920i	0.933160	+	0.359460i	$0.117039\pi$
$13$			2.59210i			0.718920i	−0.933160	+	0.359460i	$0.882961\pi$
$14$		−	1.12261i		−	0.300030i
$15$		0			0
$16$	3.42864			0.857160
$17$	3.25917	+	2.52543i	0.790466	+	0.612506i
$17$	3.25917	+	2.52543i	0.790466	+	0.612506i
$18$		−	0.541249i		−	0.127574i
$19$	−4.28100			−0.982128			−0.491064	−	0.871124i	$-0.663392\pi$
$19$	−4.28100			−0.982128			−0.491064	+	0.871124i	$0.663392\pi$
$20$		0			0
$21$	4.05086			0.883969
$22$	−1.78731			−0.381056
$23$	1.47186			0.306905			0.153452	−	0.988156i	$-0.450961\pi$
$23$	1.47186			0.306905			0.153452	+	0.988156i	$0.450961\pi$
$24$		−	1.36321i		−	0.278264i
$25$		0			0
$26$	−0.806424			−0.158153
$27$	5.32089			1.02401
$28$	−6.86760			−1.29785
$29$		−	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	$0.269421\pi$
$31$			8.33946i			1.49781i	−0.662676	+	0.748906i	$0.730579\pi$
$32$			3.49532i			0.617890i
$33$		−	6.44938i		−	1.12269i
$34$	−0.785680	+	1.01395i	−0.134743	+	0.173892i
$35$		0			0
$36$	−3.31111			−0.551851
$37$	−6.20290			−1.01975			−0.509875	−	0.860248i	$-0.670308\pi$
$37$	−6.20290			−1.01975			−0.509875	+	0.860248i	$0.670308\pi$
$38$		−	1.33185i		−	0.216055i
$39$		−	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	$-0.100009\pi$
$42$			1.26025i			0.194461i
$43$			9.76049i			1.48846i	0.667923	+	0.744230i	$0.267184\pi$
$43$			9.76049i			1.48846i	−0.667923	+	0.744230i	$0.732816\pi$
$44$			10.9339i			1.64835i
$45$		0			0
$46$			0.457908i			0.0675148i
$47$		−	6.62222i		−	0.965949i	−0.875634	−	0.482975i	$-0.839556\pi$
$47$		−	6.62222i		−	0.965949i	0.875634	−	0.482975i	$-0.160444\pi$
$48$	−3.84902			−0.555559
$49$	6.02074			0.860106
$50$		0			0
$51$	−3.65878	−	2.83507i	−0.512332	−	0.396989i
$52$			4.93332i			0.684129i
$53$			0.658781i			0.0904905i	0.998976	+	0.0452452i	$0.0144069\pi$
$53$			0.658781i			0.0904905i	−0.998976	+	0.0452452i	$0.985593\pi$
$54$			1.65537i			0.225267i
$55$		0			0
$56$		−	4.38178i		−	0.585540i
$57$	4.80589			0.636555
$58$	1.71246			0.224857
$59$	−5.95407			−0.775154			−0.387577	−	0.921837i	$-0.626688\pi$
$59$	−5.95407			−0.775154			−0.387577	+	0.921837i	$0.626688\pi$
$60$		0			0
$61$		−	8.76357i		−	1.12206i	−0.827796	−	0.561030i	$-0.810405\pi$
$61$		−	8.76357i		−	1.12206i	0.827796	−	0.561030i	$-0.189595\pi$
$62$	−2.59447			−0.329498
$63$	6.27775			0.790922
$64$	5.76986			0.721232
$65$		0			0
$66$	2.00645			0.246977
$67$			0.428639i			0.0523666i	0.999657	+	0.0261833i	$0.00833536\pi$
$67$			0.428639i			0.0523666i	−0.999657	+	0.0261833i	$0.991665\pi$
$68$	6.20290	+	4.80642i	0.752212	+	0.582865i
$69$	−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	$0.974223\pi$
$72$		−	2.11261i		−	0.248973i
$73$	3.25917			0.381457			0.190729	−	0.981643i	$-0.438915\pi$
$73$	3.25917			0.381457			0.190729	+	0.981643i	$0.438915\pi$
$74$		−	1.92977i		−	0.224331i
$75$		0			0
$76$	−8.14764			−0.934599
$77$		−	20.7304i		−	2.36245i
$78$	0.905299			0.102505
$79$			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
$82$	1.23127			0.135970
$83$		−	16.6637i		−	1.82908i	−0.404497	−	0.914540i	$-0.632553\pi$
$83$		−	16.6637i		−	1.82908i	0.404497	−	0.914540i	$-0.367447\pi$
$84$	7.70964			0.841190
$85$		0			0
$86$	−3.03657			−0.327441
$87$			6.17929i			0.662489i
$88$	−6.97626			−0.743671
$89$	3.52543			0.373695			0.186847	−	0.982389i	$-0.440173\pi$
$89$	3.52543			0.373695			0.186847	+	0.982389i	$0.440173\pi$
$90$		0			0
$91$		−	9.35342i		−	0.980505i
$92$	2.80127			0.292052
$93$		−	9.36196i		−	0.970790i
$94$	2.06022			0.212496
$95$		0			0
$96$		−	3.92388i		−	0.400479i
$97$	11.7073			1.18870			0.594348	−	0.804208i	$-0.297410\pi$
$97$	11.7073			1.18870			0.594348	+	0.804208i	$0.297410\pi$
$98$			1.87310i			0.189212i
$99$		−	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	$0.436715\pi$
$102$	0.882012	−	1.13828i	0.0873322	−	0.112706i
$103$		−	4.23506i		−	0.417293i	−0.977991	−	0.208647i	$-0.933094\pi$
$103$		−	4.23506i		−	0.417293i	0.977991	−	0.208647i	$-0.0669058\pi$
$104$	−3.14764			−0.308652
$105$		0			0
$106$	−0.204952			−0.0199067
$107$	11.8392			1.14454			0.572271	−	0.820065i	$-0.306063\pi$
$107$	11.8392			1.14454			0.572271	+	0.820065i	$0.306063\pi$
$108$	10.1268			0.974449
$109$			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
$112$	−12.3720			−1.16904
$113$	10.7915			1.01518			0.507588	−	0.861600i	$-0.330537\pi$
$113$	10.7915			1.01518			0.507588	+	0.861600i	$0.330537\pi$
$114$			1.49515i			0.140034i
$115$		0			0
$116$		−	10.4760i		−	0.972675i
$117$		−	4.50961i		−	0.416913i
$118$		−	1.85236i		−	0.170523i
$119$	−11.7605	−	9.11282i	−1.07808	−	0.835371i
$120$		0			0
$121$	−22.0049			−2.00045
$122$	2.72641			0.246838
$123$			4.44293i			0.400605i
$124$			15.8718i			1.42533i
$125$		0			0
$126$			1.95306i			0.173992i
$127$			17.1383i			1.52078i	0.649469	+	0.760388i	$0.274991\pi$
$127$			17.1383i			1.52078i	−0.649469	+	0.760388i	$0.725009\pi$
$128$			8.78568i			0.776552i
$129$		−	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	$0.872958\pi$
$132$		−	12.2745i		−	1.06836i
$133$	15.4477			1.33948
$134$	−0.133353			−0.0115200
$135$		0			0
$136$	−3.06668	+	3.95768i	−0.262965	+	0.339368i
$137$			14.0716i			1.20222i	0.799167	+	0.601109i	$0.205274\pi$
$137$			14.0716i			1.20222i	−0.799167	+	0.601109i	$0.794726\pi$
$138$		−	0.514052i		−	0.0437590i
$139$			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
$142$	−0.424104			−0.0355901
$143$	−14.8916			−1.24530
$144$	−5.96497			−0.497081
$145$		0			0
$146$			1.01395i			0.0839155i
$147$	−6.75895			−0.557468
$148$	−11.8054			−0.970400
$149$	4.76494			0.390359			0.195179	−	0.980768i	$-0.437471\pi$
$149$	4.76494			0.390359			0.195179	+	0.980768i	$0.437471\pi$
$150$		0			0
$151$	2.99063			0.243374			0.121687	−	0.992569i	$-0.461170\pi$
$151$	2.99063			0.243374			0.121687	+	0.992569i	$0.461170\pi$
$152$		−	5.19850i		−	0.421654i
$153$	−5.67014	−	4.39361i	−0.458404	−	0.355202i
$154$	6.44938			0.519706
$155$		0			0
$156$		−	5.53820i		−	0.443411i
$157$		−	13.9447i		−	1.11291i	−0.830878	−	0.556454i	$-0.812162\pi$
$157$		−	13.9447i		−	1.11291i	0.830878	−	0.556454i	$-0.187838\pi$
$158$	−0.589850			−0.0469260
$159$		−	0.739554i		−	0.0586504i
$160$		0			0
$161$	−5.31111			−0.418574
$162$		−	0.234587i		−	0.0184309i
$163$	−8.65491			−0.677905			−0.338953	−	0.940803i	$-0.610073\pi$
$163$	−8.65491			−0.677905			−0.338953	+	0.940803i	$0.610073\pi$
$164$		−	7.53230i		−	0.588174i
$165$		0			0
$166$	5.18421			0.402373
$167$	0.773357			0.0598442			0.0299221	−	0.999552i	$-0.490474\pi$
$167$	0.773357			0.0598442			0.0299221	+	0.999552i	$0.490474\pi$
$168$			4.91903i			0.379512i
$169$	6.28100			0.483154
$170$		0			0
$171$	7.44785			0.569551
$172$			18.5763i			1.41643i
$173$	2.41097			0.183302			0.0916511	−	0.995791i	$-0.470786\pi$
$173$	2.41097			0.183302			0.0916511	+	0.995791i	$0.470786\pi$
$174$	−1.92242			−0.145739
$175$		0			0
$176$			19.6975i			1.48476i
$177$	6.68409			0.502407
$178$			1.09679i			0.0822077i
$179$	18.4558			1.37945			0.689727	−	0.724070i	$-0.257731\pi$
$179$	18.4558			1.37945			0.689727	+	0.724070i	$0.257731\pi$
$180$		0			0
$181$		−	20.6366i		−	1.53391i	−0.641703	−	0.766953i	$-0.721772\pi$
$181$		−	20.6366i		−	1.53391i	0.641703	−	0.766953i	$-0.278228\pi$
$182$	2.90992			0.215698
$183$			9.83807i			0.727251i
$184$			1.78731i			0.131762i
$185$		0			0
$186$	2.91258			0.213561
$187$	−14.5086	+	18.7239i	−1.06097	+	1.36923i
$188$		−	12.6035i		−	0.919203i
$189$	−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	$0.191014\pi$
$192$	−6.47730			−0.467459
$193$	−19.7208			−1.41953			−0.709767	−	0.704437i	$-0.751200\pi$
$193$	−19.7208			−1.41953			−0.709767	+	0.704437i	$0.751200\pi$
$194$			3.64223i			0.261497i
$195$		0			0
$196$	11.4588			0.818482
$197$	−12.8870			−0.918160			−0.459080	−	0.888395i	$-0.651821\pi$
$197$	−12.8870			−0.918160			−0.459080	+	0.888395i	$0.651821\pi$
$198$	3.10947			0.220980
$199$		−	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
$202$			1.23506i			0.0868988i
$203$			19.8622i			1.39405i
$204$	−6.96343	−	5.39574i	−0.487538	−	0.377777i
$205$		0			0
$206$	1.31756			0.0917988
$207$	−2.56067			−0.177979
$208$			8.88739i			0.616230i
$209$		−	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	$0.872143\pi$
$212$			1.25380i			0.0861113i
$213$		−	1.53035i		−	0.104858i
$214$			3.68328i			0.251784i
$215$		0			0
$216$			6.46126i			0.439633i
$217$		−	30.0923i		−	2.04280i
$218$	−3.74037			−0.253330
$219$	−3.65878			−0.247237
$220$		0			0
$221$	−6.54617	+	8.44812i	−0.440343	+	0.568282i
$222$			2.16638i			0.145398i
$223$		−	0.888922i		−	0.0595266i	−0.999557	−	0.0297633i	$-0.990525\pi$
$223$		−	0.888922i		−	0.0595266i	0.999557	−	0.0297633i	$-0.00947535\pi$
$224$		−	12.6126i		−	0.842714i
$225$		0			0
$226$			3.35731i			0.223325i
$227$	−12.9280			−0.858064			−0.429032	−	0.903289i	$-0.641145\pi$
$227$	−12.9280			−0.858064			−0.429032	+	0.903289i	$0.641145\pi$
$228$	9.14662			0.605750
$229$	−2.83161			−0.187118			−0.0935591	−	0.995614i	$-0.529824\pi$
$229$	−2.83161			−0.187118			−0.0935591	+	0.995614i	$0.529824\pi$
$230$		0			0
$231$			23.2721i			1.53119i
$232$	6.68409			0.438832
$233$	19.9381			1.30619			0.653094	−	0.757277i	$-0.273471\pi$
$233$	19.9381			1.30619			0.653094	+	0.757277i	$0.273471\pi$
$234$	1.40297			0.0917153
$235$		0			0
$236$	−11.3319			−0.737641
$237$		−	2.12843i		−	0.138256i
$238$	2.83507	−	3.65878i	0.183770	−	0.237164i
$239$	−27.2859			−1.76498			−0.882490	−	0.470332i	$-0.844134\pi$
$239$	−27.2859			−1.76498			−0.882490	+	0.470332i	$0.844134\pi$
$240$		0			0
$241$		−	3.57462i		−	0.230262i	−0.993350	−	0.115131i	$-0.963271\pi$
$241$		−	3.57462i		−	0.230262i	0.993350	−	0.115131i	$-0.0367287\pi$
$242$		−	6.84590i		−	0.440071i
$243$	−15.1162			−0.969703
$244$		−	16.6789i		−	1.06776i
$245$		0			0
$246$	−1.38223			−0.0881278
$247$		−	11.0968i		−	0.706072i
$248$	−10.1268			−0.643051
$249$			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	$0.312390\pi$
$252$	11.9479			0.752646
$253$			8.45584i			0.531614i
$254$	−5.33185			−0.334550
$255$		0			0
$256$	8.80642			0.550401
$257$			20.5462i			1.28163i	0.767693	+	0.640817i	$0.221404\pi$
$257$			20.5462i			1.28163i	−0.767693	+	0.640817i	$0.778596\pi$
$258$	3.40888			0.212227
$259$	22.3827			1.39079
$260$		0			0
$261$			9.57625i			0.592755i
$262$	−2.76746			−0.170974
$263$		−	8.38715i		−	0.517174i	−0.965988	−	0.258587i	$-0.916743\pi$
$263$		−	8.38715i		−	0.517174i	0.965988	−	0.258587i	$-0.0832569\pi$
$264$	7.83161			0.482002
$265$		0			0
$266$			4.80589i			0.294668i
$267$	−3.95768			−0.242206
$268$			0.815792i			0.0498324i
$269$			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	$0.569565\pi$
$272$	11.1745	+	8.65878i	0.677556	+	0.525016i
$273$			10.5002i			0.635503i
$274$	−4.37778			−0.264472
$275$		0			0
$276$	−3.14473			−0.189290
$277$	−20.4193			−1.22688			−0.613438	−	0.789743i	$-0.710214\pi$
$277$	−20.4193			−1.22688			−0.613438	+	0.789743i	$0.710214\pi$
$278$	−5.22275			−0.313240
$279$		−	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	$0.164590\pi$
$282$	−2.31283			−0.137727
$283$	−9.16991			−0.545095			−0.272547	−	0.962142i	$-0.587866\pi$
$283$	−9.16991			−0.545095			−0.272547	+	0.962142i	$0.587866\pi$
$284$			2.59447i			0.153954i
$285$		0			0
$286$		−	4.63290i		−	0.273949i
$287$			14.2810i			0.842981i
$288$		−	6.08097i		−	0.358324i
$289$	4.24443	+	16.4616i	0.249672	+	0.968330i
$290$		0			0
$291$	−13.1427			−0.770440
$292$	6.20290			0.362997
$293$			16.2351i			0.948463i	0.880400	+	0.474231i	$0.157274\pi$
$293$			16.2351i			0.948463i	−0.880400	+	0.474231i	$0.842726\pi$
$294$		−	2.10276i		−	0.122636i
$295$		0			0
$296$		−	7.53230i		−	0.437806i
$297$			30.5684i			1.77376i
$298$			1.48241i			0.0858737i
$299$			3.81522i			0.220640i
$300$		0			0
$301$		−	35.2200i		−	2.03005i
$302$			0.930409i			0.0535390i
$303$	−4.45664			−0.256027
$304$	−14.6780			−0.841841
$305$		0			0
$306$	1.36689	−	1.76402i	0.0781396	−	0.100843i
$307$			3.37778i			0.192780i	0.995344	+	0.0963902i	$0.0307297\pi$
$307$			3.37778i			0.192780i	−0.995344	+	0.0963902i	$0.969270\pi$
$308$		−	39.4543i		−	2.24812i
$309$			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	$0.927571\pi$
$312$	3.53358			0.200049
$313$	6.83380			0.386269			0.193135	−	0.981172i	$-0.438135\pi$
$313$	6.83380			0.386269			0.193135	+	0.981172i	$0.438135\pi$
$314$	4.33830			0.244825
$315$		0			0
$316$			3.60843i			0.202990i
$317$	27.1550			1.52517			0.762587	−	0.646886i	$-0.223929\pi$
$317$	27.1550			1.52517			0.762587	+	0.646886i	$0.223929\pi$
$318$	0.230081			0.0129023
$319$	31.6227			1.77053
$320$		0			0
$321$	−13.2908			−0.741822
$322$		−	1.65233i		−	0.0920806i
$323$	−13.9525	−	10.8113i	−0.776339	−	0.601559i
$324$	−1.43509			−0.0797274
$325$		0			0
$326$		−	2.69261i		−	0.149130i
$327$		−	13.4968i		−	0.746377i
$328$	4.80589			0.265361
$329$			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	$0.689216\pi$
$332$		−	31.7146i		−	1.74056i
$333$	10.7915			0.591369
$334$			0.240597i			0.0131649i
$335$		0			0
$336$	13.8889			0.757703
$337$	−4.70775			−0.256447			−0.128224	−	0.991745i	$-0.540928\pi$
$337$	−4.70775			−0.256447			−0.128224	+	0.991745i	$0.540928\pi$
$338$			1.95407i			0.106287i
$339$	−12.1146			−0.657976
$340$		0			0
$341$	−47.9101			−2.59448
$342$			2.31708i			0.125294i
$343$	3.53358			0.190795
$344$	−11.8524			−0.639036
$345$		0			0
$346$			0.750070i			0.0403240i
$347$	−35.3520			−1.89779			−0.948896	−	0.315588i	$-0.897798\pi$
$347$	−35.3520			−1.89779			−0.948896	+	0.315588i	$0.897798\pi$
$348$			11.7605i			0.630428i
$349$	25.0558			1.34121			0.670603	−	0.741817i	$-0.266036\pi$
$349$	25.0558			1.34121			0.670603	+	0.741817i	$0.266036\pi$
$350$		0			0
$351$			13.7923i			0.736178i
$352$	−20.0806			−1.07030
$353$		−	13.6953i		−	0.728930i	−0.931217	−	0.364465i	$-0.881252\pi$
$353$		−	13.6953i		−	0.728930i	0.931217	−	0.364465i	$-0.118748\pi$
$354$			2.07947i			0.110523i
$355$		0			0
$356$	6.70964			0.355610
$357$	13.2024	+	10.2301i	0.698747	+	0.541436i
$358$			5.74176i			0.303461i
$359$	−2.53480			−0.133781			−0.0668907	−	0.997760i	$-0.521308\pi$
$359$	−2.53480			−0.133781			−0.0668907	+	0.997760i	$0.521308\pi$
$360$		0			0
$361$	−0.673071			−0.0354248
$362$	6.42021			0.337439
$363$	24.7029			1.29657
$364$		−	17.8015i		−	0.933054i
$365$		0			0
$366$	−3.06070			−0.159985
$367$	36.3426			1.89707			0.948535	−	0.316673i	$-0.102566\pi$
$367$	36.3426			1.89707			0.948535	+	0.316673i	$0.102566\pi$
$368$	5.04649			0.263066
$369$			6.88536i			0.358438i
$370$		0			0
$371$		−	2.37716i		−	0.123416i
$372$		−	17.8178i		−	0.923810i
$373$			12.7714i			0.661278i	0.943757	+	0.330639i	$0.107264\pi$
$373$			12.7714i			0.661278i	−0.943757	+	0.330639i	$0.892736\pi$
$374$	−5.82516	−	4.51373i	−0.301212	−	0.233399i
$375$		0			0
$376$	8.04149			0.414708
$377$	14.2680			0.734837
$378$		−	5.97328i		−	0.307232i
$379$			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
$382$			7.09679i			0.363103i
$383$			1.95407i			0.0998482i	0.998753	+	0.0499241i	$0.0158979\pi$
$383$			1.95407i			0.0998482i	−0.998753	+	0.0499241i	$0.984102\pi$
$384$		−	9.86289i		−	0.503314i
$385$		0			0
$386$		−	6.13529i		−	0.312278i
$387$		−	16.9808i		−	0.863182i
$388$	22.2815			1.13117
$389$	7.91258			0.401184			0.200592	−	0.979675i	$-0.435713\pi$
$389$	7.91258			0.401184			0.200592	+	0.979675i	$0.435713\pi$
$390$		0			0
$391$	4.79706	+	3.71708i	0.242598	+	0.187981i
$392$			7.31111i			0.369267i
$393$		−	9.98619i		−	0.503736i
$394$		−	4.00924i		−	0.201983i
$395$		0			0
$396$		−	19.0223i		−	0.955906i
$397$	9.24476			0.463981			0.231991	−	0.972718i	$-0.425476\pi$
$397$	9.24476			0.463981			0.231991	+	0.972718i	$0.425476\pi$
$398$	2.00462			0.100483
$399$	−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	$-0.180235\pi$
$402$	0.149703			0.00746653
$403$	−21.6168			−1.07681
$404$	7.55554			0.375902
$405$		0			0
$406$	−6.17929			−0.306673
$407$		−	35.6356i		−	1.76639i
$408$	3.44268	−	4.44293i	0.170438	−	0.219958i
$409$	−18.1669			−0.898293			−0.449147	−	0.893458i	$-0.648272\pi$
$409$	−18.1669			−0.898293			−0.449147	+	0.893458i	$0.648272\pi$
$410$		0			0
$411$		−	15.7969i		−	0.779204i
$412$		−	8.06022i		−	0.397099i
$413$	21.4848			1.05720
$414$		−	0.796644i		−	0.0391529i
$415$		0			0
$416$	−9.06022			−0.444214
$417$		−	18.8459i		−	0.922888i
$418$	7.65147			0.374246
$419$		−	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	$0.469158\pi$
$422$	−3.53358			−0.172012
$423$			11.5210i			0.560169i
$424$	−0.799970			−0.0388500
$425$		0			0
$426$	0.476104			0.0230673
$427$			31.6227i			1.53033i
$428$	22.5326			1.08915
$429$	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	$-0.291577\pi$
$432$	18.2434			0.877736
$433$		−	38.2306i		−	1.83725i	−0.395135	−	0.918623i	$-0.629302\pi$
$433$		−	38.2306i		−	1.83725i	0.395135	−	0.918623i	$-0.370698\pi$
$434$	9.36196			0.449389
$435$		0			0
$436$			22.8818i			1.09584i
$437$	−6.30104			−0.301420
$438$		−	1.13828i		−	0.0543889i
$439$			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
$442$	−2.62828	−	2.03657i	−0.125014	−	0.0968695i
$443$			27.5669i			1.30974i	0.755740	+	0.654872i	$0.227277\pi$
$443$			27.5669i			1.30974i	−0.755740	+	0.654872i	$0.772723\pi$
$444$	13.2529			0.628954
$445$		0			0
$446$	0.276551			0.0130950
$447$	−5.34916			−0.253007
$448$	−20.8201			−0.983658
$449$			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
$452$	20.5385			0.966048
$453$	−3.35731			−0.157740
$454$		−	4.02201i		−	0.188762i
$455$		0			0
$456$			5.83589i			0.273290i
$457$		−	13.0350i		−	0.609753i	−0.952392	−	0.304877i	$-0.901385\pi$
$457$		−	13.0350i		−	0.609753i	0.952392	−	0.304877i	$-0.0986152\pi$
$458$		−	0.880937i		−	0.0411635i
$459$	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	$0.582068\pi$
$462$	−7.24014			−0.336842
$463$		−	17.2400i		−	0.801210i	−0.916251	−	0.400605i	$-0.868800\pi$
$463$		−	17.2400i		−	0.801210i	0.916251	−	0.400605i	$-0.131200\pi$
$464$		−	18.8726i		−	0.876138i
$465$		0			0
$466$			6.20290i			0.287344i
$467$		−	0.0414872i		−	0.00191980i	−1.00000		0.000959899i	$-0.999694\pi$
$467$		−	0.0414872i		−	0.00191980i	1.00000		0.000959899i	$-0.000305545\pi$
$468$		−	8.58274i		−	0.396737i
$469$		−	1.54671i		−	0.0714206i
$470$		0			0
$471$			15.6545i			0.721319i
$472$		−	7.23014i		−	0.332794i
$473$	−56.0739			−2.57828
$474$	0.662171			0.0304145
$475$		0			0
$476$	−22.3827	−	17.3436i	−1.02591	−	0.794944i
$477$		−	1.14611i		−	0.0524769i
$478$		−	8.48886i		−	0.388272i
$479$		−	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
$482$	1.11209			0.0506545
$483$	5.96230			0.271294
$484$	−41.8800			−1.90364
$485$		0			0
$486$		−	4.70276i		−	0.213321i
$487$	−24.6691			−1.11787			−0.558933	−	0.829213i	$-0.688789\pi$
$487$	−24.6691			−1.11787			−0.558933	+	0.829213i	$0.688789\pi$
$488$	10.6418			0.481730
$489$	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	$0.437977\pi$
$492$			8.45584i			0.381219i
$493$	13.9009	−	17.9398i	0.626067	−	0.807967i
$494$	3.45230			0.155326
$495$		0			0
$496$			28.5930i			1.28386i
$497$		−	4.91903i		−	0.220649i
$498$	−5.81984			−0.260793
$499$			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
$502$			5.47949i			0.244562i
$503$	24.1879			1.07849			0.539244	−	0.842150i	$-0.318710\pi$
$503$	24.1879			1.07849			0.539244	+	0.842150i	$0.318710\pi$
$504$			7.62320i			0.339564i
$505$		0			0
$506$	−2.63068			−0.116948
$507$	−7.05111			−0.313151
$508$			32.6178i			1.44718i
$509$	25.8671			1.14654			0.573270	−	0.819366i	$-0.305675\pi$
$509$	25.8671			1.14654			0.573270	+	0.819366i	$0.305675\pi$
$510$		0			0
$511$	−11.7605			−0.520253
$512$			20.3111i			0.897633i
$513$	−22.7787			−1.00570
$514$	−6.39207			−0.281942
$515$		0			0
$516$		−	20.8539i		−	0.918042i
$517$	38.0446			1.67320
$518$			6.96343i			0.305956i
$519$	−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	$-0.0809872\pi$
$522$	−2.97925			−0.130398
$523$			33.0509i			1.44521i	0.691260	+	0.722606i	$0.257056\pi$
$523$			33.0509i			1.44521i	−0.691260	+	0.722606i	$0.742944\pi$
$524$			16.9300i			0.739592i
$525$		0			0
$526$	2.60931			0.113771
$527$	−21.0607	+	27.1798i	−0.917419	+	1.18397i
$528$		−	22.1126i		−	0.962328i
$529$	−20.8336			−0.905810
$530$		0			0
$531$	10.3586			0.449524
$532$	29.4002			1.27466
$533$	10.2587			0.444354
$534$		−	1.23127i		−	0.0532820i
$535$		0			0
$536$	−0.520505			−0.0224824
$537$	−20.7187			−0.894078
$538$	−3.85954			−0.166397
$539$			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	$0.860139\pi$
$542$		−	2.22077i		−	0.0953904i
$543$			23.1669i			0.994185i
$544$	−8.82717	+	11.3918i	−0.378462	+	0.488421i
$545$		0			0
$546$	−3.26671			−0.139802
$547$	−15.4815			−0.661940			−0.330970	−	0.943641i	$-0.607376\pi$
$547$	−15.4815			−0.661940			−0.330970	+	0.943641i	$0.607376\pi$
$548$			26.7812i			1.14404i
$549$			15.2464i			0.650700i
$550$		0			0
$551$			23.5643i			1.00387i
$552$		−	2.00645i		−	0.0854003i
$553$		−	6.84146i		−	0.290928i
$554$		−	6.35260i		−	0.269896i
$555$		0			0
$556$			31.9503i			1.35500i
$557$			15.3477i			0.650302i	0.945662	+	0.325151i	$0.105415\pi$
$557$			15.3477i			0.650302i	−0.945662	+	0.325151i	$0.894585\pi$
$558$	4.51373			0.191081
$559$	−25.3002			−1.07008
$560$		0			0
$561$	16.2874	−	21.0197i	0.687657	−	0.887451i
$562$			9.06668i			0.382455i
$563$			14.3827i			0.606159i	0.952965	+	0.303079i	$0.0980147\pi$
$563$			14.3827i			0.606159i	−0.952965	+	0.303079i	$0.901985\pi$
$564$			14.1488i			0.595772i
$565$		0			0
$566$		−	2.85283i		−	0.119913i
$567$	2.72089			0.114267
$568$	−1.65537			−0.0694578
$569$	12.1432			0.509069			0.254535	−	0.967064i	$-0.418078\pi$
$569$	12.1432			0.509069			0.254535	+	0.967064i	$0.418078\pi$
$570$		0			0
$571$		−	2.88663i		−	0.120802i	−0.998174	−	0.0604009i	$-0.980762\pi$
$571$		−	2.88663i		−	0.120802i	0.998174	−	0.0604009i	$-0.0192379\pi$
$572$	−28.3419			−1.18503
$573$	−25.6082			−1.06980
$574$	−4.44293			−0.185444
$575$		0			0
$576$	−10.0381			−0.418254
$577$			3.10970i			0.129458i	0.997903	+	0.0647291i	$0.0206183\pi$
$577$			3.10970i			0.129458i	−0.997903	+	0.0647291i	$0.979382\pi$
$578$	−5.12134	+	1.32048i	−0.213020	+	0.0549246i
$579$	22.1388			0.920055
$580$		0			0
$581$			60.1298i			2.49460i
$582$		−	4.08880i		−	0.169486i
$583$	−3.78469			−0.156746
$584$			3.95768i			0.163770i
$585$		0			0
$586$	−5.05086			−0.208649
$587$		−	10.2034i		−	0.421140i	−0.977579	−	0.210570i	$-0.932468\pi$
$587$		−	10.2034i		−	0.421140i	0.977579	−	0.210570i	$-0.0675320\pi$
$588$	−12.8637			−0.530490
$589$		−	35.7012i		−	1.47104i
$590$		0			0
$591$	14.4671			0.595095
$592$	−21.2675			−0.874089
$593$		−	16.2301i		−	0.666492i	−0.942840	−	0.333246i	$-0.891856\pi$
$593$		−	16.2301i		−	0.666492i	0.942840	−	0.333246i	$-0.108144\pi$
$594$	−9.51008			−0.390203
$595$		0			0
$596$	9.06868			0.371468
$597$			7.23353i			0.296049i
$598$	−1.18694			−0.0485378
$599$	8.43309			0.344567			0.172283	−	0.985047i	$-0.444886\pi$
$599$	8.43309			0.344567			0.172283	+	0.985047i	$0.444886\pi$
$600$		0			0
$601$			20.2535i			0.826160i	0.910695	+	0.413080i	$0.135547\pi$
$601$			20.2535i			0.826160i	−0.910695	+	0.413080i	$0.864453\pi$
$602$	10.9572			0.446583
$603$		−	0.745724i		−	0.0303682i
$604$	5.69181			0.231596
$605$		0			0
$606$		−	1.38649i		−	0.0563225i
$607$	22.7732			0.924334			0.462167	−	0.886793i	$-0.347072\pi$
$607$	22.7732			0.924334			0.462167	+	0.886793i	$0.347072\pi$
$608$		−	14.9634i		−	0.606847i
$609$		−	22.2975i		−	0.903540i
$610$		0			0
$611$	17.1655			0.694441
$612$	−10.7915	−	8.36196i	−0.436220	−	0.338012i
$613$		−	8.40636i		−	0.339530i	−0.985485	−	0.169765i	$-0.945699\pi$
$613$		−	8.40636i		−	0.339530i	0.985485	−	0.169765i	$-0.0543008\pi$
$614$	−1.05086			−0.0424091
$615$		0			0
$616$	25.1733			1.01426
$617$	14.8168			0.596500			0.298250	−	0.954488i	$-0.403597\pi$
$617$	14.8168			0.596500			0.298250	+	0.954488i	$0.403597\pi$
$618$	−1.47911			−0.0594984
$619$		−	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
$622$	−2.47530			−0.0992505
$623$	−12.7212			−0.509666
$624$		−	9.97707i		−	0.399403i
$625$		0			0
$626$			2.12605i			0.0849740i
$627$			27.6098i			1.10263i
$628$		−	26.5397i		−	1.05905i
$629$	−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	$0.612396\pi$
$632$	−2.30231			−0.0915810
$633$		−	12.7506i		−	0.506793i
$634$			8.44812i			0.335518i
$635$		0			0
$636$		−	1.40753i		−	0.0558121i
$637$			15.6064i			0.618348i
$638$			9.83807i			0.389493i
$639$		−	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	$-0.0458494\pi$
$642$		−	4.13488i		−	0.163191i
$643$	−4.40507			−0.173719			−0.0868595	−	0.996221i	$-0.527683\pi$
$643$	−4.40507			−0.173719			−0.0868595	+	0.996221i	$0.527683\pi$
$644$	−10.1082			−0.398317
$645$		0			0
$646$	3.36349	−	4.34074i	0.132335	−	0.170784i
$647$		−	0.198022i		−	0.00778504i	−0.999992	−	0.00389252i	$-0.998761\pi$
$647$		−	0.198022i		−	0.00778504i	0.999992	−	0.00389252i	$-0.00123903\pi$
$648$		−	0.915643i		−	0.0359698i
$649$		−	34.2061i		−	1.34271i
$650$		0			0
$651$			33.7820i			1.32402i
$652$	−16.4721			−0.645098
$653$	−9.92723			−0.388482			−0.194241	−	0.980954i	$-0.562224\pi$
$653$	−9.92723			−0.388482			−0.194241	+	0.980954i	$0.562224\pi$
$654$	4.19897			0.164193
$655$		0			0
$656$		−	13.5695i		−	0.529798i
$657$	−5.67014			−0.221213
$658$	−7.43416			−0.289814
$659$	−3.00492			−0.117055			−0.0585276	−	0.998286i	$-0.518641\pi$
$659$	−3.00492			−0.117055			−0.0585276	+	0.998286i	$0.518641\pi$
$660$		0			0
$661$	23.6923			0.921523			0.460762	−	0.887524i	$-0.347576\pi$
$661$	23.6923			0.921523			0.460762	+	0.887524i	$0.347576\pi$
$662$		−	6.33984i		−	0.246405i
$663$	7.34880	−	9.48394i	0.285404	−	0.368326i
$664$	20.2351			0.785273
$665$		0			0
$666$			3.35731i			0.130093i
$667$		−	8.10171i		−	0.313699i
$668$	1.47186			0.0569481
$669$			0.997912i			0.0385815i
$670$		0			0
$671$	50.3466			1.94361
$672$			14.1590i			0.546196i
$673$	30.1968			1.16400			0.582001	−	0.813188i	$-0.302270\pi$
$673$	30.1968			1.16400			0.582001	+	0.813188i	$0.302270\pi$
$674$		−	1.46462i		−	0.0564150i
$675$		0			0
$676$	11.9541			0.459772
$677$	−47.9413			−1.84253			−0.921266	−	0.388933i	$-0.872844\pi$
$677$	−47.9413			−1.84253			−0.921266	+	0.388933i	$0.872844\pi$
$678$		−	3.76895i		−	0.144746i
$679$	−42.2449			−1.62121
$680$		0			0
$681$	14.5131			0.556145
$682$		−	14.9052i		−	0.570750i
$683$	−1.41477			−0.0541347			−0.0270674	−	0.999634i	$-0.508617\pi$
$683$	−1.41477			−0.0541347			−0.0270674	+	0.999634i	$0.508617\pi$
$684$	14.1748			0.541989
$685$		0			0
$686$			1.09932i			0.0419723i
$687$	3.17880			0.121279
$688$			33.4652i			1.27585i
$689$	−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	$-0.00825452\pi$
$692$	4.58858			0.174432
$693$			36.0656i			1.37002i
$694$		−	10.9983i		−	0.417489i
$695$		0			0
$696$	−7.50363			−0.284424
$697$	9.99483	−	12.8988i	0.378581	−	0.488576i
$698$			7.79505i			0.295047i
$699$	−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	$0.774384\pi$
$702$	−4.29089			−0.161949
$703$	26.5546			1.00153
$704$			33.1478i			1.24930i
$705$		0			0
$706$	4.26073			0.160355
$707$	−14.3251			−0.538749
$708$	12.7212			0.478094
$709$			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
$712$			4.28100i			0.160437i
$713$			12.2745i			0.459685i
$714$	−3.18268	+	4.10738i	−0.119109	+	0.153715i
$715$		0			0
$716$	35.1254			1.31270
$717$	30.6314			1.14395
$718$		−	0.788595i		−	0.0294301i
$719$			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
$722$		−	0.209398i		−	0.00779297i
$723$			4.01291i			0.149242i
$724$		−	39.2758i		−	1.45967i
$725$		0			0
$726$			7.68528i			0.285227i
$727$		−	36.7007i		−	1.36116i	−0.732676	−	0.680578i	$-0.761729\pi$
$727$		−	36.7007i		−	1.36116i	0.732676	−	0.680578i	$-0.238271\pi$
$728$	11.3580			0.420957
$729$	19.2317			0.712284
$730$		0			0
$731$	−24.6494	+	31.8111i	−0.911691	+	1.17658i
$732$			18.7239i			0.692056i
$733$			24.8666i			0.918471i	0.888315	+	0.459235i	$0.151877\pi$
$733$			24.8666i			0.918471i	−0.888315	+	0.459235i	$0.848123\pi$
$734$			11.3065i			0.417330i
$735$		0			0
$736$			5.14462i			0.189633i
$737$	−2.46253			−0.0907085
$738$	−2.14209			−0.0788514
$739$	32.7368			1.20424			0.602122	−	0.798404i	$-0.294322\pi$
$739$	32.7368			1.20424			0.602122	+	0.798404i	$0.294322\pi$
$740$		0			0
$741$			12.4574i			0.457633i
$742$	0.739554			0.0271499
$743$	−21.3762			−0.784215			−0.392108	−	0.919919i	$-0.628254\pi$
$743$	−21.3762			−0.784215			−0.392108	+	0.919919i	$0.628254\pi$
$744$	11.3684			0.416786
$745$		0			0
$746$	−3.97328			−0.145472
$747$			28.9906i			1.06071i
$748$	−27.6129	+	35.6356i	−1.00963	+	1.30297i
$749$	−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	$-0.302129\pi$
$752$		−	22.7052i		−	0.827973i
$753$	−19.7724			−0.720545
$754$			4.43887i			0.161654i
$755$		0			0
$756$	−36.5417			−1.32901
$757$		−	12.1793i		−	0.442664i	−0.975199	−	0.221332i	$-0.928960\pi$
$757$		−	12.1793i		−	0.442664i	0.975199	−	0.221332i	$-0.0710404\pi$
$758$	−6.89089			−0.250288
$759$		−	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	$0.458417\pi$
$762$	5.98559			0.216835
$763$		−	43.3832i		−	1.57058i
$764$	43.4148			1.57069
$765$		0			0
$766$	−0.607926			−0.0219652
$767$		−	15.4336i		−	0.557274i
$768$	−9.88618			−0.356737
$769$	−14.9748			−0.540005			−0.270003	−	0.962860i	$-0.587025\pi$
$769$	−14.9748			−0.540005			−0.270003	+	0.962860i	$0.587025\pi$
$770$		0			0
$771$		−	23.0653i		−	0.830678i
$772$	−37.5328			−1.35084
$773$		−	26.2938i		−	0.945721i	−0.881137	−	0.472860i	$-0.843222\pi$
$773$		−	26.2938i		−	0.945721i	0.881137	−	0.472860i	$-0.156778\pi$
$774$	5.28286			0.189888
$775$		0			0
$776$			14.2164i			0.510339i
$777$	−25.1270			−0.901428
$778$			2.46167i			0.0882550i
$779$			16.9428i			0.607039i
$780$		0			0
$781$	−7.83161			−0.280237
$782$	−1.15641	+	1.49240i	−0.0413533	+	0.0533682i
$783$		−	29.2883i		−	1.04668i
$784$	20.6430			0.737249
$785$		0			0
$786$	3.10678			0.110815
$787$	−9.35342			−0.333413			−0.166707	−	0.986007i	$-0.553313\pi$
$787$	−9.35342			−0.333413			−0.166707	+	0.986007i	$0.553313\pi$
$788$	−24.5267			−0.873727
$789$			9.41550i			0.335201i
$790$		0			0
$791$	−38.9403			−1.38456
$792$	12.1369			0.431267
$793$	22.7161			0.806672
$794$			2.87612i			0.102070i
$795$		0			0
$796$		−	12.2633i		−	0.434663i
$797$			29.4800i			1.04423i	0.852874	+	0.522117i	$0.174858\pi$
$797$			29.4800i			1.04423i	−0.852874	+	0.522117i	$0.825142\pi$
$798$		−	5.39514i		−	0.190986i
$799$	16.7239	−	21.5830i	0.591650	−	0.763550i
$800$		0			0
$801$	−6.13335			−0.216711
$802$	6.68409			0.236024
$803$			18.7239i			0.660753i
$804$		−	0.915816i		−	0.0322983i
$805$		0			0
$806$		−	6.72514i		−	0.236883i
$807$		−	13.9269i		−	0.490249i
$808$			4.82071i			0.169592i
$809$			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	$-0.900189\pi$
$811$		−	17.5682i		−	0.616902i	0.951240	−	0.308451i	$-0.0998106\pi$
$812$			37.8020i			1.32659i
$813$	8.01350			0.281046
$814$	11.0865			0.388582
$815$		0			0
$816$	−12.5446	−	9.72043i	−0.439150	−	0.340283i
$817$		−	41.7846i		−	1.46186i
$818$		−	5.65185i		−	0.197612i
$819$			16.2726i			0.568610i
$820$		0			0
$821$		−	13.1043i		−	0.457343i	−0.973504	−	0.228672i	$-0.926562\pi$
$821$		−	13.1043i		−	0.457343i	0.973504	−	0.228672i	$-0.0734382\pi$
$822$	4.91454			0.171414
$823$	23.7638			0.828355			0.414178	−	0.910196i	$-0.364069\pi$
$823$	23.7638			0.828355			0.414178	+	0.910196i	$0.364069\pi$
$824$	5.14272			0.179155
$825$		0			0
$826$			6.68409i			0.232569i
$827$	28.9423			1.00642			0.503211	−	0.864164i	$-0.332152\pi$
$827$	28.9423			1.00642			0.503211	+	0.864164i	$0.332152\pi$
$828$	−4.87350			−0.169366
$829$	−48.3131			−1.67798			−0.838992	−	0.544144i	$-0.816855\pi$
$829$	−48.3131			−1.67798			−0.838992	+	0.544144i	$0.816855\pi$
$830$		0			0
$831$	22.9229			0.795187
$832$			14.9561i			0.518509i
$833$	19.6227	+	15.2050i	0.679885	+	0.526820i
$834$	5.86311			0.203023
$835$		0			0
$836$		−	46.8081i		−	1.61889i
$837$			44.3733i			1.53377i
$838$	4.46715			0.154315
$839$		−	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
$842$			1.23506i			0.0425631i
$843$	−32.7164			−1.12681
$844$			21.6168i			0.744079i
$845$		0			0
$846$	−3.58427			−0.123230
$847$	79.4031			2.72832
$848$			2.25872i			0.0775648i
$849$	10.2942			0.353297
$850$		0			0
$851$	−9.12981			−0.312966
$852$		−	2.91258i		−	0.0997833i
$853$	29.8138			1.02080			0.510402	−	0.859936i	$-0.329497\pi$
$853$	29.8138			1.02080			0.510402	+	0.859936i	$0.329497\pi$
$854$	−9.83807			−0.336652
$855$		0			0
$856$			14.3766i			0.491383i
$857$	1.71246			0.0584965			0.0292483	−	0.999572i	$-0.490689\pi$
$857$	1.71246			0.0584965			0.0292483	+	0.999572i	$0.490689\pi$
$858$			5.20094i			0.177557i
$859$	−22.5749			−0.770246			−0.385123	−	0.922865i	$-0.625841\pi$
$859$	−22.5749			−0.770246			−0.385123	+	0.922865i	$0.625841\pi$
$860$		0			0
$861$		−	16.0320i		−	0.546369i
$862$	10.2459			0.348978
$863$		−	54.2578i		−	1.84696i	−0.383650	−	0.923479i	$-0.625333\pi$
$863$		−	54.2578i		−	1.84696i	0.383650	−	0.923479i	$-0.374667\pi$
$864$			18.5982i			0.632723i
$865$		0			0
$866$	11.8938			0.404169
$867$	−4.76484	−	18.4800i	−0.161822	−	0.627613i
$868$		−	57.2721i		−	1.94394i
$869$	−10.8923			−0.369496
$870$		0			0
$871$	−1.11108			−0.0376474
$872$	−14.5995			−0.494400
$873$	−20.3677			−0.689343
$874$		−	1.96030i		−	0.0663082i
$875$		0			0
$876$	−6.96343			−0.235273
$877$	14.8989			0.503099			0.251549	−	0.967844i	$-0.419060\pi$
$877$	14.8989			0.503099			0.251549	+	0.967844i	$0.419060\pi$
$878$	−0.206794			−0.00697896
$879$		−	18.2256i		−	0.614736i
$880$		0			0
$881$		−	22.3330i		−	0.752419i	−0.926535	−	0.376209i	$-0.877227\pi$
$881$		−	22.3330i		−	0.752419i	0.926535	−	0.376209i	$-0.122773\pi$
$882$		−	3.25872i		−	0.109727i
$883$		−	25.3319i		−	0.852485i	−0.904609	−	0.426242i	$-0.859837\pi$
$883$		−	25.3319i		−	0.852485i	0.904609	−	0.426242i	$-0.140163\pi$
$884$	−12.4588	+	16.0786i	−0.419033	+	0.540780i
$885$		0			0
$886$	−8.57628			−0.288126
$887$	6.35813			0.213485			0.106743	−	0.994287i	$-0.465958\pi$
$887$	6.35813			0.213485			0.106743	+	0.994287i	$0.465958\pi$
$888$			8.45584i			0.283759i
$889$		−	61.8422i		−	2.07412i
$890$		0			0
$891$		−	4.33194i		−	0.145125i
$892$		−	1.69181i		−	0.0566459i
$893$			28.3497i			0.948686i
$894$		−	1.66417i		−	0.0556581i
$895$		0			0
$896$		−	31.7025i		−	1.05911i
$897$		−	4.28300i		−	0.143005i
$898$	−5.43679			−0.181428
$899$	45.9037			1.53097
$900$		0			0
$901$	−1.66370	+	2.14708i	−0.0554260	+	0.0715296i
$902$			7.07361i			0.235525i
$903$			39.5383i			1.31575i
$904$			13.1043i			0.435843i
$905$		0			0
$906$		−	1.04449i		−	0.0347007i
$907$	13.0295			0.432636			0.216318	−	0.976323i	$-0.430595\pi$
$907$	13.0295			0.432636			0.216318	+	0.976323i	$0.430595\pi$
$908$	−24.6048			−0.816539
$909$	−6.90660			−0.229078
$910$		0			0
$911$			20.7685i			0.688093i	0.938953	+	0.344046i	$0.111798\pi$
$911$			20.7685i			0.688093i	−0.938953	+	0.344046i	$0.888202\pi$
$912$	16.4777			0.545630
$913$	95.7328			3.16829
$914$	4.05530			0.134137
$915$		0			0
$916$	−5.38916			−0.178063
$917$		−	32.0988i		−	1.06000i
$918$	−4.18052	+	5.39514i	−0.137978	+	0.178066i
$919$	5.10970			0.168553			0.0842766	−	0.996442i	$-0.473142\pi$
$919$	5.10970			0.168553			0.0842766	+	0.996442i	$0.473142\pi$
$920$		0			0
$921$		−	3.79193i		−	0.124948i
$922$		−	3.40636i		−	0.112183i
$923$	−3.53358			−0.116309
$924$			44.2918i			1.45709i
$925$		0			0
$926$	5.36349			0.176255
$927$			7.36794i			0.241995i
$928$	19.2396			0.631571
$929$			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
$932$	37.9464			1.24298
$933$		−	8.93194i		−	0.292419i
$934$	0.0129070			0.000422330
$935$		0			0
$936$	5.47610			0.178992
$937$		−	39.5575i		−	1.29229i	−0.763215	−	0.646144i	$-0.776380\pi$
$937$		−	39.5575i		−	1.29229i	0.763215	−	0.646144i	$-0.223620\pi$
$938$	0.481195			0.0157116
$939$	−7.67169			−0.250356
$940$		0			0
$941$			53.2283i			1.73519i	0.497268	+	0.867597i	$0.334337\pi$
$941$			53.2283i			1.73519i	−0.497268	+	0.867597i	$0.665663\pi$
$942$	−4.87022			−0.158680
$943$		−	5.82516i		−	0.189693i
$944$	−20.4143			−0.664430
$945$		0			0
$946$		−	17.4450i		−	0.567187i
$947$	0.773357			0.0251307			0.0125654	−	0.999921i	$-0.496000\pi$
$947$	0.773357			0.0251307			0.0125654	+	0.999921i	$0.496000\pi$
$948$		−	4.05086i		−	0.131566i
$949$			8.44812i			0.274238i
$950$		0			0
$951$	−30.4844			−0.988525
$952$	11.0659	−	14.2810i	0.358647	−	0.462850i
$953$			36.8731i			1.19444i	0.802079	+	0.597218i	$0.203727\pi$
$953$			36.8731i			1.19444i	−0.802079	+	0.597218i	$0.796273\pi$
$954$	0.356564			0.0115442
$955$		0			0
$956$	−51.9309			−1.67956
$957$	−35.4999			−1.14755
$958$	11.0321			0.356430
$959$		−	50.7763i		−	1.63965i
$960$		0			0
$961$	−38.5466			−1.24344
$962$	5.00217			0.161276
$963$	−20.5973			−0.663738
$964$		−	6.80327i		−	0.219118i
$965$		0			0
$966$			1.85492i			0.0596810i
$967$		−	17.8306i		−	0.573392i	−0.958022	−	0.286696i	$-0.907443\pi$
$967$		−	17.8306i		−	0.573392i	0.958022	−	0.286696i	$-0.0925570\pi$
$968$		−	26.7210i		−	0.858846i
$969$	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	$0.606033\pi$
$972$	−28.7693			−0.922775
$973$		−	60.5768i		−	1.94200i
$974$		−	7.67476i		−	0.245915i
$975$		0			0
$976$		−	30.0471i		−	0.961785i
$977$			8.64587i			0.276606i	0.990390	+	0.138303i	$0.0441648\pi$
$977$			8.64587i			0.276606i	−0.990390	+	0.138303i	$0.955835\pi$
$978$			3.02275i			0.0966569i
$979$			20.2535i			0.647306i
$980$		0			0
$981$		−	20.9165i		−	0.667813i
$982$			2.66953i			0.0851882i
$983$	61.1615			1.95075			0.975374	−	0.220558i	$-0.0707877\pi$
$983$	61.1615			1.95075			0.975374	+	0.220558i	$0.0707877\pi$
$984$	−5.39514			−0.171991
$985$		0			0
$986$	5.58120	+	4.32469i	0.177742	+	0.137726i
$987$		−	26.8256i		−	0.853869i
$988$		−	21.1195i		−	0.671902i
$989$			14.3661i			0.456815i
$990$		0			0
$991$		−	27.0979i		−	0.860792i	−0.902640	−	0.430396i	$-0.858374\pi$
$991$		−	27.0979i		−	0.860792i	0.902640	−	0.430396i	$-0.141626\pi$
$992$	−29.1491			−0.925484
$993$	22.8768			0.725974
$994$	1.53035			0.0485397
$995$		0			0
$996$			35.6031i			1.12813i
$997$	48.3759			1.53208			0.766040	−	0.642793i	$-0.222225\pi$
$997$	48.3759			1.53208			0.766040	+	0.642793i	$0.222225\pi$
$998$	−2.64105			−0.0836009
$999$	−33.0049			−1.04423

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.c.c.424.5		12	85.84	even	2	inner
425.2.c.c.424.6		12	5.4	even	2	inner
425.2.c.c.424.7		12	1.1	even	1	trivial
425.2.c.c.424.8		12	17.16	even	2	inner
425.2.d.a.101.3	✓	6	85.67	odd	4
425.2.d.a.101.4	yes	6	5.2	odd	4
425.2.d.b.101.3	yes	6	5.3	odd	4
425.2.d.b.101.4	yes	6	85.33	odd	4
7225.2.a.ba.1.3		6	85.38	odd	4
7225.2.a.ba.1.4		6	85.13	odd	4
7225.2.a.bg.1.3		6	85.47	odd	4
7225.2.a.bg.1.4		6	85.72	odd	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.311108i q^{2} -1.12261 q^{3} +1.90321 q^{4} -0.349253i q^{6} -3.60843 q^{7} +1.21432i q^{8} -1.73975 q^{9} +O(q^{10})\)	\(q+0.311108i q^{2} -1.12261 q^{3} +1.90321 q^{4} -0.349253i q^{6} -3.60843 q^{7} +1.21432i q^{8} -1.73975 q^{9} +5.74499i q^{11} -2.13656 q^{12} +2.59210i q^{13} -1.12261i q^{14} +3.42864 q^{16} +(3.25917 + 2.52543i) q^{17} -0.541249i q^{18} -4.28100 q^{19} +4.05086 q^{21} -1.78731 q^{22} +1.47186 q^{23} -1.36321i q^{24} -0.806424 q^{26} +5.32089 q^{27} -6.86760 q^{28} -5.50439i q^{29} +8.33946i q^{31} +3.49532i q^{32} -6.44938i q^{33} +(-0.785680 + 1.01395i) q^{34} -3.31111 q^{36} -6.20290 q^{37} -1.33185i q^{38} -2.90992i q^{39} -3.95768i q^{41} +1.26025i q^{42} +9.76049i q^{43} +10.9339i q^{44} +0.457908i q^{46} -6.62222i q^{47} -3.84902 q^{48} +6.02074 q^{49} +(-3.65878 - 2.83507i) q^{51} +4.93332i q^{52} +0.658781i q^{53} +1.65537i q^{54} -4.38178i q^{56} +4.80589 q^{57} +1.71246 q^{58} -5.95407 q^{59} -8.76357i q^{61} -2.59447 q^{62} +6.27775 q^{63} +5.76986 q^{64} +2.00645 q^{66} +0.428639i q^{67} +(6.20290 + 4.80642i) q^{68} -1.65233 q^{69} +1.36321i q^{71} -2.11261i q^{72} +3.25917 q^{73} -1.92977i q^{74} -8.14764 q^{76} -20.7304i q^{77} +0.905299 q^{78} +1.89597i q^{79} -0.754037 q^{81} +1.23127 q^{82} -16.6637i q^{83} +7.70964 q^{84} -3.03657 q^{86} +6.17929i q^{87} -6.97626 q^{88} +3.52543 q^{89} -9.35342i q^{91} +2.80127 q^{92} -9.36196i q^{93} +2.06022 q^{94} -3.92388i q^{96} +11.7073 q^{97} +1.87310i q^{98} -9.99483i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{4} + 32 q^{9}+O(q^{10}) \) 12 * q - 4 * q^4 + 32 * q^9	\( 12 q - 4 q^{4} + 32 q^{9} - 12 q^{16} - 24 q^{19} - 4 q^{21} + 44 q^{26} - 36 q^{34} - 40 q^{36} - 8 q^{49} - 16 q^{51} + 8 q^{59} + 44 q^{64} - 32 q^{66} - 48 q^{69} - 72 q^{76} + 68 q^{81} + 12 q^{84} - 8 q^{86} + 16 q^{89} + 80 q^{94}+O(q^{100}) \) 12 * q - 4 * q^4 + 32 * q^9 - 12 * q^16 - 24 * q^19 - 4 * q^21 + 44 * q^26 - 36 * q^34 - 40 * q^36 - 8 * q^49 - 16 * q^51 + 8 * q^59 + 44 * q^64 - 32 * q^66 - 48 * q^69 - 72 * q^76 + 68 * q^81 + 12 * q^84 - 8 * q^86 + 16 * q^89 + 80 * q^94

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