LMFDB - Embedded newform 1305.2.c.a.784.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(784,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1305.784");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.c (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:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			784.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1305.784
Dual form			1305.2.c.a.784.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -2.00000 q^{4} +(2.00000 + 1.00000i) q^{5} +2.00000i q^{7} +(2.00000 - 4.00000i) q^{10} +3.00000 q^{11} +4.00000i q^{13} +4.00000 q^{14} -4.00000 q^{16} +8.00000i q^{17} +(-4.00000 - 2.00000i) q^{20} -6.00000i q^{22} +1.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +8.00000 q^{26} -4.00000i q^{28} +1.00000 q^{29} -8.00000 q^{31} +8.00000i q^{32} +16.0000 q^{34} +(-2.00000 + 4.00000i) q^{35} +7.00000i q^{37} -7.00000 q^{41} +9.00000i q^{43} -6.00000 q^{44} +2.00000 q^{46} -12.0000i q^{47} +3.00000 q^{49} +(8.00000 - 6.00000i) q^{50} -8.00000i q^{52} -9.00000i q^{53} +(6.00000 + 3.00000i) q^{55} -2.00000i q^{58} +10.0000 q^{59} +2.00000 q^{61} +16.0000i q^{62} +8.00000 q^{64} +(-4.00000 + 8.00000i) q^{65} -8.00000i q^{67} -16.0000i q^{68} +(8.00000 + 4.00000i) q^{70} +8.00000 q^{71} -1.00000i q^{73} +14.0000 q^{74} +6.00000i q^{77} +10.0000 q^{79} +(-8.00000 - 4.00000i) q^{80} +14.0000i q^{82} -9.00000i q^{83} +(-8.00000 + 16.0000i) q^{85} +18.0000 q^{86} +10.0000 q^{89} -8.00000 q^{91} -2.00000i q^{92} -24.0000 q^{94} -13.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 4 q^{5} + 4 q^{10} + 6 q^{11} + 8 q^{14} - 8 q^{16} - 8 q^{20} + 6 q^{25} + 16 q^{26} + 2 q^{29} - 16 q^{31} + 32 q^{34} - 4 q^{35} - 14 q^{41} - 12 q^{44} + 4 q^{46} + 6 q^{49} + 16 q^{50}+ \cdots - 48 q^{94}+O(q^{100}) $ 2 * q - 4 * q^4 + 4 * q^5 + 4 * q^10 + 6 * q^11 + 8 * q^14 - 8 * q^16 - 8 * q^20 + 6 * q^25 + 16 * q^26 + 2 * q^29 - 16 * q^31 + 32 * q^34 - 4 * q^35 - 14 * q^41 - 12 * q^44 + 4 * q^46 + 6 * q^49 + 16 * q^50 + 12 * q^55 + 20 * q^59 + 4 * q^61 + 16 * q^64 - 8 * q^65 + 16 * q^70 + 16 * q^71 + 28 * q^74 + 20 * q^79 - 16 * q^80 - 16 * q^85 + 36 * q^86 + 20 * q^89 - 16 * q^91 - 48 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		−	2.00000i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$		−	2.00000i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0
$3$		0			0
$4$	−2.00000			−1.00000
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$		0			0
$10$	2.00000	−	4.00000i	0.632456	−	1.26491i
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	4.00000			1.06904
$15$		0			0
$16$	−4.00000			−1.00000
$17$			8.00000i			1.94029i	0.242536	+	0.970143i	$0.422021\pi$
$17$			8.00000i			1.94029i	−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−4.00000	−	2.00000i	−0.894427	−	0.447214i
$21$		0			0
$22$		−	6.00000i		−	1.27920i
$23$			1.00000i			0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$			1.00000i			0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$	8.00000			1.56893
$27$		0			0
$28$		−	4.00000i		−	0.755929i
$29$	1.00000			0.185695
$29$	1.00000			0.185695
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$			8.00000i			1.41421i
$33$		0			0
$34$	16.0000			2.74398
$35$	−2.00000	+	4.00000i	−0.338062	+	0.676123i
$36$		0			0
$37$			7.00000i			1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$			7.00000i			1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.00000			−1.09322			−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000			−1.09322			−0.546608	+	0.837389i	$0.684081\pi$
$42$		0			0
$43$			9.00000i			1.37249i	0.727372	+	0.686244i	$0.240742\pi$
$43$			9.00000i			1.37249i	−0.727372	+	0.686244i	$0.759258\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$	2.00000			0.294884
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	8.00000	−	6.00000i	1.13137	−	0.848528i
$51$		0			0
$52$		−	8.00000i		−	1.10940i
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$		0			0
$55$	6.00000	+	3.00000i	0.809040	+	0.404520i
$56$		0			0
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$			16.0000i			2.03200i
$63$		0			0
$64$	8.00000			1.00000
$65$	−4.00000	+	8.00000i	−0.496139	+	0.992278i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		−	16.0000i		−	1.94029i
$69$		0			0
$70$	8.00000	+	4.00000i	0.956183	+	0.478091i
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$		−	1.00000i		−	0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$		−	1.00000i		−	0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	14.0000			1.62747
$75$		0			0
$76$		0			0
$77$			6.00000i			0.683763i
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$	−8.00000	−	4.00000i	−0.894427	−	0.447214i
$81$		0			0
$82$			14.0000i			1.54604i
$83$		−	9.00000i		−	0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$		−	9.00000i		−	0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$		0			0
$85$	−8.00000	+	16.0000i	−0.867722	+	1.73544i
$86$	18.0000			1.94099
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		−	2.00000i		−	0.208514i
$93$		0			0
$94$	−24.0000			−2.47541
$95$		0			0
$96$		0			0
$97$		−	13.0000i		−	1.31995i	−0.751288	−	0.659975i	$-0.770567\pi$
$97$		−	13.0000i		−	1.31995i	0.751288	−	0.659975i	$-0.229433\pi$
$98$		−	6.00000i		−	0.606092i
$99$		0			0
$100$	−6.00000	−	8.00000i	−0.600000	−	0.800000i
$101$	−17.0000			−1.69156			−0.845782	−	0.533529i	$-0.820865\pi$
$101$	−17.0000			−1.69156			−0.845782	+	0.533529i	$0.820865\pi$
$102$		0			0
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$		0			0
$106$	−18.0000			−1.74831
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$	6.00000	−	12.0000i	0.572078	−	1.14416i
$111$		0			0
$112$		−	8.00000i		−	0.755929i
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$		0			0
$115$	−1.00000	+	2.00000i	−0.0932505	+	0.186501i
$116$	−2.00000			−0.185695
$117$		0			0
$118$		−	20.0000i		−	1.84115i
$119$	−16.0000			−1.46672
$120$		0			0
$121$	−2.00000			−0.181818
$122$		−	4.00000i		−	0.362143i
$123$		0			0
$124$	16.0000			1.43684
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$		−	3.00000i		−	0.266207i	−0.991102	−	0.133103i	$-0.957506\pi$
$127$		−	3.00000i		−	0.266207i	0.991102	−	0.133103i	$-0.0424943\pi$
$128$		0			0
$129$		0			0
$130$	16.0000	+	8.00000i	1.40329	+	0.701646i
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$	−16.0000			−1.38219
$135$		0			0
$136$		0			0
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		0			0
$139$	−5.00000			−0.424094			−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000			−0.424094			−0.212047	+	0.977259i	$0.568013\pi$
$140$	4.00000	−	8.00000i	0.338062	−	0.676123i
$141$		0			0
$142$		−	16.0000i		−	1.34269i
$143$			12.0000i			1.00349i
$144$		0			0
$145$	2.00000	+	1.00000i	0.166091	+	0.0830455i
$146$	−2.00000			−0.165521
$147$		0			0
$148$		−	14.0000i		−	1.15079i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−3.00000			−0.244137			−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000			−0.244137			−0.122068	+	0.992522i	$0.538953\pi$
$152$		0			0
$153$		0			0
$154$	12.0000			0.966988
$155$	−16.0000	−	8.00000i	−1.28515	−	0.642575i
$156$		0			0
$157$			2.00000i			0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$			2.00000i			0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$		−	20.0000i		−	1.59111i
$159$		0			0
$160$	−8.00000	+	16.0000i	−0.632456	+	1.26491i
$161$	−2.00000			−0.157622
$162$		0			0
$163$		−	1.00000i		−	0.0783260i	−0.999233	−	0.0391630i	$-0.987531\pi$
$163$		−	1.00000i		−	0.0783260i	0.999233	−	0.0391630i	$-0.0124692\pi$
$164$	14.0000			1.09322
$165$		0			0
$166$	−18.0000			−1.39707
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	32.0000	+	16.0000i	2.45429	+	1.22714i
$171$		0			0
$172$		−	18.0000i		−	1.37249i
$173$		−	9.00000i		−	0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$		−	9.00000i		−	0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$		0			0
$175$	−8.00000	+	6.00000i	−0.604743	+	0.453557i
$176$	−12.0000			−0.904534
$177$		0			0
$178$		−	20.0000i		−	1.49906i
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	17.0000			1.26360			0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000			1.26360			0.631800	+	0.775131i	$0.282316\pi$
$182$			16.0000i			1.18600i
$183$		0			0
$184$		0			0
$185$	−7.00000	+	14.0000i	−0.514650	+	1.02930i
$186$		0			0
$187$			24.0000i			1.75505i
$188$			24.0000i			1.75038i
$189$		0			0
$190$		0			0
$191$	3.00000			0.217072			0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000			0.217072			0.108536	+	0.994092i	$0.465384\pi$
$192$		0			0
$193$		−	6.00000i		−	0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$		−	6.00000i		−	0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	−26.0000			−1.86669
$195$		0			0
$196$	−6.00000			−0.428571
$197$			13.0000i			0.926212i	0.886303	+	0.463106i	$0.153265\pi$
$197$			13.0000i			0.926212i	−0.886303	+	0.463106i	$0.846735\pi$
$198$		0			0
$199$	25.0000			1.77220			0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000			1.77220			0.886102	+	0.463491i	$0.153403\pi$
$200$		0			0
$201$		0			0
$202$			34.0000i			2.39223i
$203$			2.00000i			0.140372i
$204$		0			0
$205$	−14.0000	−	7.00000i	−0.977802	−	0.488901i
$206$	8.00000			0.557386
$207$		0			0
$208$		−	16.0000i		−	1.10940i
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$			18.0000i			1.23625i
$213$		0			0
$214$	−24.0000			−1.64061
$215$	−9.00000	+	18.0000i	−0.613795	+	1.22759i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		−	10.0000i		−	0.677285i
$219$		0			0
$220$	−12.0000	−	6.00000i	−0.809040	−	0.404520i
$221$	−32.0000			−2.15255
$222$		0			0
$223$		−	6.00000i		−	0.401790i	−0.979613	−	0.200895i	$-0.935615\pi$
$223$		−	6.00000i		−	0.401790i	0.979613	−	0.200895i	$-0.0643850\pi$
$224$	−16.0000			−1.06904
$225$		0			0
$226$	−28.0000			−1.86253
$227$			3.00000i			0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$			3.00000i			0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$		0			0
$229$	−20.0000			−1.32164			−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000			−1.32164			−0.660819	+	0.750546i	$0.729791\pi$
$230$	4.00000	+	2.00000i	0.263752	+	0.131876i
$231$		0			0
$232$		0			0
$233$			1.00000i			0.0655122i	0.999463	+	0.0327561i	$0.0104285\pi$
$233$			1.00000i			0.0655122i	−0.999463	+	0.0327561i	$0.989572\pi$
$234$		0			0
$235$	12.0000	−	24.0000i	0.782794	−	1.56559i
$236$	−20.0000			−1.30189
$237$		0			0
$238$			32.0000i			2.07425i
$239$	30.0000			1.94054			0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000			1.94054			0.970269	+	0.242028i	$0.0778125\pi$
$240$		0			0
$241$	−3.00000			−0.193247			−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000			−0.193247			−0.0966235	+	0.995321i	$0.530804\pi$
$242$			4.00000i			0.257130i
$243$		0			0
$244$	−4.00000			−0.256074
$245$	6.00000	+	3.00000i	0.383326	+	0.191663i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	22.0000	−	4.00000i	1.39140	−	0.252982i
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			3.00000i			0.188608i
$254$	−6.00000			−0.376473
$255$		0			0
$256$	16.0000			1.00000
$257$			23.0000i			1.43470i	0.696713	+	0.717350i	$0.254645\pi$
$257$			23.0000i			1.43470i	−0.696713	+	0.717350i	$0.745355\pi$
$258$		0			0
$259$	−14.0000			−0.869918
$260$	8.00000	−	16.0000i	0.496139	−	0.992278i
$261$		0			0
$262$			24.0000i			1.48272i
$263$		−	4.00000i		−	0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$		−	4.00000i		−	0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$		0			0
$265$	9.00000	−	18.0000i	0.552866	−	1.10573i
$266$		0			0
$267$		0			0
$268$			16.0000i			0.977356i
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$	12.0000			0.728948			0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000			0.728948			0.364474	+	0.931214i	$0.381249\pi$
$272$		−	32.0000i		−	1.94029i
$273$		0			0
$274$	−4.00000			−0.241649
$275$	9.00000	+	12.0000i	0.542720	+	0.723627i
$276$		0			0
$277$		−	28.0000i		−	1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$		−	28.0000i		−	1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$			10.0000i			0.599760i
$279$		0			0
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$			14.0000i			0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$			14.0000i			0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	−16.0000			−0.949425
$285$		0			0
$286$	24.0000			1.41915
$287$		−	14.0000i		−	0.826394i
$288$		0			0
$289$	−47.0000			−2.76471
$290$	2.00000	−	4.00000i	0.117444	−	0.234888i
$291$		0			0
$292$			2.00000i			0.117041i
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$	20.0000	+	10.0000i	1.16445	+	0.582223i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.00000			−0.231326
$300$		0			0
$301$	−18.0000			−1.03750
$302$			6.00000i			0.345261i
$303$		0			0
$304$		0			0
$305$	4.00000	+	2.00000i	0.229039	+	0.114520i
$306$		0			0
$307$			17.0000i			0.970241i	0.874447	+	0.485121i	$0.161224\pi$
$307$			17.0000i			0.970241i	−0.874447	+	0.485121i	$0.838776\pi$
$308$		−	12.0000i		−	0.683763i
$309$		0			0
$310$	−16.0000	+	32.0000i	−0.908739	+	1.81748i
$311$	−17.0000			−0.963982			−0.481991	−	0.876176i	$-0.660086\pi$
$311$	−17.0000			−0.963982			−0.481991	+	0.876176i	$0.660086\pi$
$312$		0			0
$313$			4.00000i			0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$			4.00000i			0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$	4.00000			0.225733
$315$		0			0
$316$	−20.0000			−1.12509
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$	3.00000			0.167968
$320$	16.0000	+	8.00000i	0.894427	+	0.447214i
$321$		0			0
$322$			4.00000i			0.222911i
$323$		0			0
$324$		0			0
$325$	−16.0000	+	12.0000i	−0.887520	+	0.665640i
$326$	−2.00000			−0.110770
$327$		0			0
$328$		0			0
$329$	24.0000			1.32316
$330$		0			0
$331$	32.0000			1.75888			0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000			1.75888			0.879440	+	0.476011i	$0.157918\pi$
$332$			18.0000i			0.987878i
$333$		0			0
$334$	16.0000			0.875481
$335$	8.00000	−	16.0000i	0.437087	−	0.874173i
$336$		0			0
$337$			22.0000i			1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$			22.0000i			1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$			6.00000i			0.326357i
$339$		0			0
$340$	16.0000	−	32.0000i	0.867722	−	1.73544i
$341$	−24.0000			−1.29967
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$	−18.0000			−0.967686
$347$			3.00000i			0.161048i	0.996753	+	0.0805242i	$0.0256594\pi$
$347$			3.00000i			0.161048i	−0.996753	+	0.0805242i	$0.974341\pi$
$348$		0			0
$349$	5.00000			0.267644			0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000			0.267644			0.133822	+	0.991005i	$0.457275\pi$
$350$	12.0000	+	16.0000i	0.641427	+	0.855236i
$351$		0			0
$352$			24.0000i			1.27920i
$353$			26.0000i			1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$		0			0
$355$	16.0000	+	8.00000i	0.849192	+	0.424596i
$356$	−20.0000			−1.06000
$357$		0			0
$358$		−	20.0000i		−	1.05703i
$359$	−15.0000			−0.791670			−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000			−0.791670			−0.395835	+	0.918322i	$0.629545\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		−	34.0000i		−	1.78700i
$363$		0			0
$364$	16.0000			0.838628
$365$	1.00000	−	2.00000i	0.0523424	−	0.104685i
$366$		0			0
$367$		−	3.00000i		−	0.156599i	−0.996930	−	0.0782994i	$-0.975051\pi$
$367$		−	3.00000i		−	0.156599i	0.996930	−	0.0782994i	$-0.0249490\pi$
$368$		−	4.00000i		−	0.208514i
$369$		0			0
$370$	28.0000	+	14.0000i	1.45565	+	0.727825i
$371$	18.0000			0.934513
$372$		0			0
$373$			4.00000i			0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$			4.00000i			0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	48.0000			2.48202
$375$		0			0
$376$		0			0
$377$			4.00000i			0.206010i
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$		0			0
$381$		0			0
$382$		−	6.00000i		−	0.306987i
$383$		−	39.0000i		−	1.99281i	−0.0847358	−	0.996403i	$-0.527005\pi$
$383$		−	39.0000i		−	1.99281i	0.0847358	−	0.996403i	$-0.472995\pi$
$384$		0			0
$385$	−6.00000	+	12.0000i	−0.305788	+	0.611577i
$386$	−12.0000			−0.610784
$387$		0			0
$388$			26.0000i			1.31995i
$389$	−25.0000			−1.26755			−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000			−1.26755			−0.633775	+	0.773517i	$0.718496\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$		0			0
$393$		0			0
$394$	26.0000			1.30986
$395$	20.0000	+	10.0000i	1.00631	+	0.503155i
$396$		0			0
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		−	50.0000i		−	2.50627i
$399$		0			0
$400$	−12.0000	−	16.0000i	−0.600000	−	0.800000i
$401$	28.0000			1.39825			0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000			1.39825			0.699127	+	0.714998i	$0.253572\pi$
$402$		0			0
$403$		−	32.0000i		−	1.59403i
$404$	34.0000			1.69156
$405$		0			0
$406$	4.00000			0.198517
$407$			21.0000i			1.04093i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	−14.0000	+	28.0000i	−0.691411	+	1.38282i
$411$		0			0
$412$		−	8.00000i		−	0.394132i
$413$			20.0000i			0.984136i
$414$		0			0
$415$	9.00000	−	18.0000i	0.441793	−	0.883585i
$416$	−32.0000			−1.56893
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		−	24.0000i		−	1.16830i
$423$		0			0
$424$		0			0
$425$	−32.0000	+	24.0000i	−1.55223	+	1.16417i
$426$		0			0
$427$			4.00000i			0.193574i
$428$			24.0000i			1.16008i
$429$		0			0
$430$	36.0000	+	18.0000i	1.73607	+	0.868037i
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		0			0
$433$			19.0000i			0.913082i	0.889702	+	0.456541i	$0.150912\pi$
$433$			19.0000i			0.913082i	−0.889702	+	0.456541i	$0.849088\pi$
$434$	−32.0000			−1.53605
$435$		0			0
$436$	−10.0000			−0.478913
$437$		0			0
$438$		0			0
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$		0			0
$441$		0			0
$442$			64.0000i			3.04417i
$443$			6.00000i			0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$			6.00000i			0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$		0			0
$445$	20.0000	+	10.0000i	0.948091	+	0.474045i
$446$	−12.0000			−0.568216
$447$		0			0
$448$			16.0000i			0.755929i
$449$	5.00000			0.235965			0.117982	−	0.993016i	$-0.462357\pi$
$449$	5.00000			0.235965			0.117982	+	0.993016i	$0.462357\pi$
$450$		0			0
$451$	−21.0000			−0.988851
$452$			28.0000i			1.31701i
$453$		0			0
$454$	6.00000			0.281594
$455$	−16.0000	−	8.00000i	−0.750092	−	0.375046i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$			40.0000i			1.86908i
$459$		0			0
$460$	2.00000	−	4.00000i	0.0932505	−	0.186501i
$461$	13.0000			0.605470			0.302735	−	0.953075i	$-0.402100\pi$
$461$	13.0000			0.605470			0.302735	+	0.953075i	$0.402100\pi$
$462$		0			0
$463$		−	6.00000i		−	0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$		−	6.00000i		−	0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$	−4.00000			−0.185695
$465$		0			0
$466$	2.00000			0.0926482
$467$		−	42.0000i		−	1.94353i	−0.235954	−	0.971764i	$-0.575822\pi$
$467$		−	42.0000i		−	1.94353i	0.235954	−	0.971764i	$-0.424178\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$	−48.0000	−	24.0000i	−2.21407	−	1.10704i
$471$		0			0
$472$		0			0
$473$			27.0000i			1.24146i
$474$		0			0
$475$		0			0
$476$	32.0000			1.46672
$477$		0			0
$478$		−	60.0000i		−	2.74434i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−28.0000			−1.27669
$482$			6.00000i			0.273293i
$483$		0			0
$484$	4.00000			0.181818
$485$	13.0000	−	26.0000i	0.590300	−	1.18060i
$486$		0			0
$487$			22.0000i			0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$			22.0000i			0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$		0			0
$489$		0			0
$490$	6.00000	−	12.0000i	0.271052	−	0.542105i
$491$	28.0000			1.26362			0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000			1.26362			0.631811	+	0.775122i	$0.282312\pi$
$492$		0			0
$493$			8.00000i			0.360302i
$494$		0			0
$495$		0			0
$496$	32.0000			1.43684
$497$			16.0000i			0.717698i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$	−4.00000	−	22.0000i	−0.178885	−	0.983870i
$501$		0			0
$502$			24.0000i			1.07117i
$503$		−	4.00000i		−	0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$		−	4.00000i		−	0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$		0			0
$505$	−34.0000	−	17.0000i	−1.51298	−	0.756490i
$506$	6.00000			0.266733
$507$		0			0
$508$			6.00000i			0.266207i
$509$	−40.0000			−1.77297			−0.886484	−	0.462758i	$-0.846860\pi$
$509$	−40.0000			−1.77297			−0.886484	+	0.462758i	$0.846860\pi$
$510$		0			0
$511$	2.00000			0.0884748
$512$		−	32.0000i		−	1.41421i
$513$		0			0
$514$	46.0000			2.02897
$515$	−4.00000	+	8.00000i	−0.176261	+	0.352522i
$516$		0			0
$517$		−	36.0000i		−	1.58328i
$518$			28.0000i			1.23025i
$519$		0			0
$520$		0			0
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	24.0000			1.04844
$525$		0			0
$526$	−8.00000			−0.348817
$527$		−	64.0000i		−	2.78788i
$528$		0			0
$529$	22.0000			0.956522
$530$	−36.0000	−	18.0000i	−1.56374	−	0.781870i
$531$		0			0
$532$		0			0
$533$		−	28.0000i		−	1.21281i
$534$		0			0
$535$	12.0000	−	24.0000i	0.518805	−	1.03761i
$536$		0			0
$537$		0			0
$538$			20.0000i			0.862261i
$539$	9.00000			0.387657
$540$		0			0
$541$	−8.00000			−0.343947			−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000			−0.343947			−0.171973	+	0.985102i	$0.555014\pi$
$542$		−	24.0000i		−	1.03089i
$543$		0			0
$544$	−64.0000			−2.74398
$545$	10.0000	+	5.00000i	0.428353	+	0.214176i
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$			4.00000i			0.170872i
$549$		0			0
$550$	24.0000	−	18.0000i	1.02336	−	0.767523i
$551$		0			0
$552$		0			0
$553$			20.0000i			0.850487i
$554$	−56.0000			−2.37921
$555$		0			0
$556$	10.0000			0.424094
$557$		−	47.0000i		−	1.99145i	−0.0923462	−	0.995727i	$-0.529437\pi$
$557$		−	47.0000i		−	1.99145i	0.0923462	−	0.995727i	$-0.470563\pi$
$558$		0			0
$559$	−36.0000			−1.52264
$560$	8.00000	−	16.0000i	0.338062	−	0.676123i
$561$		0			0
$562$		−	36.0000i		−	1.51857i
$563$			46.0000i			1.93867i	0.245745	+	0.969334i	$0.420967\pi$
$563$			46.0000i			1.93867i	−0.245745	+	0.969334i	$0.579033\pi$
$564$		0			0
$565$	14.0000	−	28.0000i	0.588984	−	1.17797i
$566$	28.0000			1.17693
$567$		0			0
$568$		0			0
$569$	−10.0000			−0.419222			−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000			−0.419222			−0.209611	+	0.977785i	$0.567220\pi$
$570$		0			0
$571$	−23.0000			−0.962520			−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000			−0.962520			−0.481260	+	0.876578i	$0.659821\pi$
$572$		−	24.0000i		−	1.00349i
$573$		0			0
$574$	−28.0000			−1.16870
$575$	−4.00000	+	3.00000i	−0.166812	+	0.125109i
$576$		0			0
$577$		−	18.0000i		−	0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$		−	18.0000i		−	0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$			94.0000i			3.90988i
$579$		0			0
$580$	−4.00000	−	2.00000i	−0.166091	−	0.0830455i
$581$	18.0000			0.746766
$582$		0			0
$583$		−	27.0000i		−	1.11823i
$584$		0			0
$585$		0			0
$586$	12.0000			0.495715
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$		0			0
$590$	20.0000	−	40.0000i	0.823387	−	1.64677i
$591$		0			0
$592$		−	28.0000i		−	1.15079i
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$		0			0
$595$	−32.0000	−	16.0000i	−1.31187	−	0.655936i
$596$		0			0
$597$		0			0
$598$			8.00000i			0.327144i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−8.00000			−0.326327			−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$			36.0000i			1.46725i
$603$		0			0
$604$	6.00000			0.244137
$605$	−4.00000	−	2.00000i	−0.162623	−	0.0813116i
$606$		0			0
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$		0			0
$609$		0			0
$610$	4.00000	−	8.00000i	0.161955	−	0.323911i
$611$	48.0000			1.94187
$612$		0			0
$613$			34.0000i			1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$			34.0000i			1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	34.0000			1.37213
$615$		0			0
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$	30.0000			1.20580			0.602901	−	0.797816i	$-0.294011\pi$
$619$	30.0000			1.20580			0.602901	+	0.797816i	$0.294011\pi$
$620$	32.0000	+	16.0000i	1.28515	+	0.642575i
$621$		0			0
$622$			34.0000i			1.36328i
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	8.00000			0.319744
$627$		0			0
$628$		−	4.00000i		−	0.159617i
$629$	−56.0000			−2.23287
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$	−24.0000			−0.953162
$635$	3.00000	−	6.00000i	0.119051	−	0.238103i
$636$		0			0
$637$			12.0000i			0.475457i
$638$		−	6.00000i		−	0.237542i
$639$		0			0
$640$		0			0
$641$	43.0000			1.69840			0.849199	−	0.528073i	$-0.177085\pi$
$641$	43.0000			1.69840			0.849199	+	0.528073i	$0.177085\pi$
$642$		0			0
$643$		−	16.0000i		−	0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$		−	16.0000i		−	0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	4.00000			0.157622
$645$		0			0
$646$		0			0
$647$			33.0000i			1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$			33.0000i			1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$		0			0
$649$	30.0000			1.17760
$650$	24.0000	+	32.0000i	0.941357	+	1.25514i
$651$		0			0
$652$			2.00000i			0.0783260i
$653$			36.0000i			1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$			36.0000i			1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$		0			0
$655$	−24.0000	−	12.0000i	−0.937758	−	0.468879i
$656$	28.0000			1.09322
$657$		0			0
$658$		−	48.0000i		−	1.87123i
$659$	25.0000			0.973862			0.486931	−	0.873441i	$-0.338116\pi$
$659$	25.0000			0.973862			0.486931	+	0.873441i	$0.338116\pi$
$660$		0			0
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$		−	64.0000i		−	2.48743i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			1.00000i			0.0387202i
$668$		−	16.0000i		−	0.619059i
$669$		0			0
$670$	−32.0000	−	16.0000i	−1.23627	−	0.618134i
$671$	6.00000			0.231627
$672$		0			0
$673$		−	6.00000i		−	0.231283i	−0.993291	−	0.115642i	$-0.963108\pi$
$673$		−	6.00000i		−	0.231283i	0.993291	−	0.115642i	$-0.0368924\pi$
$674$	44.0000			1.69482
$675$		0			0
$676$	6.00000			0.230769
$677$		−	22.0000i		−	0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$		−	22.0000i		−	0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$		0			0
$679$	26.0000			0.997788
$680$		0			0
$681$		0			0
$682$			48.0000i			1.83801i
$683$		−	39.0000i		−	1.49229i	−0.665782	−	0.746147i	$-0.731902\pi$
$683$		−	39.0000i		−	1.49229i	0.665782	−	0.746147i	$-0.268098\pi$
$684$		0			0
$685$	2.00000	−	4.00000i	0.0764161	−	0.152832i
$686$	40.0000			1.52721
$687$		0			0
$688$		−	36.0000i		−	1.37249i
$689$	36.0000			1.37149
$690$		0			0
$691$	−8.00000			−0.304334			−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000			−0.304334			−0.152167	+	0.988355i	$0.548625\pi$
$692$			18.0000i			0.684257i
$693$		0			0
$694$	6.00000			0.227757
$695$	−10.0000	−	5.00000i	−0.379322	−	0.189661i
$696$		0			0
$697$		−	56.0000i		−	2.12115i
$698$		−	10.0000i		−	0.378506i
$699$		0			0
$700$	16.0000	−	12.0000i	0.604743	−	0.453557i
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$		0			0
$704$	24.0000			0.904534
$705$		0			0
$706$	52.0000			1.95705
$707$		−	34.0000i		−	1.27870i
$708$		0			0
$709$	45.0000			1.69001			0.845005	−	0.534758i	$-0.179597\pi$
$709$	45.0000			1.69001			0.845005	+	0.534758i	$0.179597\pi$
$710$	16.0000	−	32.0000i	0.600469	−	1.20094i
$711$		0			0
$712$		0			0
$713$		−	8.00000i		−	0.299602i
$714$		0			0
$715$	−12.0000	+	24.0000i	−0.448775	+	0.897549i
$716$	−20.0000			−0.747435
$717$		0			0
$718$			30.0000i			1.11959i
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$			38.0000i			1.41421i
$723$		0			0
$724$	−34.0000			−1.26360
$725$	3.00000	+	4.00000i	0.111417	+	0.148556i
$726$		0			0
$727$		−	8.00000i		−	0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$		−	8.00000i		−	0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$		0			0
$729$		0			0
$730$	−4.00000	−	2.00000i	−0.148047	−	0.0740233i
$731$	−72.0000			−2.66302
$732$		0			0
$733$			34.0000i			1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$			34.0000i			1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	−6.00000			−0.221464
$735$		0			0
$736$	−8.00000			−0.294884
$737$		−	24.0000i		−	0.884051i
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$	14.0000	−	28.0000i	0.514650	−	1.02930i
$741$		0			0
$742$		−	36.0000i		−	1.32160i
$743$		−	44.0000i		−	1.61420i	−0.590412	−	0.807102i	$-0.701035\pi$
$743$		−	44.0000i		−	1.61420i	0.590412	−	0.807102i	$-0.298965\pi$
$744$		0			0
$745$		0			0
$746$	8.00000			0.292901
$747$		0			0
$748$		−	48.0000i		−	1.75505i
$749$	24.0000			0.876941
$750$		0			0
$751$	22.0000			0.802791			0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000			0.802791			0.401396	+	0.915905i	$0.368525\pi$
$752$			48.0000i			1.75038i
$753$		0			0
$754$	8.00000			0.291343
$755$	−6.00000	−	3.00000i	−0.218362	−	0.109181i
$756$		0			0
$757$		−	3.00000i		−	0.109037i	−0.998513	−	0.0545184i	$-0.982638\pi$
$757$		−	3.00000i		−	0.109037i	0.998513	−	0.0545184i	$-0.0173624\pi$
$758$		−	20.0000i		−	0.726433i
$759$		0			0
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$			10.0000i			0.362024i
$764$	−6.00000			−0.217072
$765$		0			0
$766$	−78.0000			−2.81825
$767$			40.0000i			1.44432i
$768$		0			0
$769$	40.0000			1.44244			0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000			1.44244			0.721218	+	0.692708i	$0.243582\pi$
$770$	24.0000	+	12.0000i	0.864900	+	0.432450i
$771$		0			0
$772$			12.0000i			0.431889i
$773$		−	24.0000i		−	0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$		−	24.0000i		−	0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$		0			0
$775$	−24.0000	−	32.0000i	−0.862105	−	1.14947i
$776$		0			0
$777$		0			0
$778$			50.0000i			1.79259i
$779$		0			0
$780$		0			0
$781$	24.0000			0.858788
$782$			16.0000i			0.572159i
$783$		0			0
$784$	−12.0000			−0.428571
$785$	−2.00000	+	4.00000i	−0.0713831	+	0.142766i
$786$		0			0
$787$		−	38.0000i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$		−	26.0000i		−	0.926212i
$789$		0			0
$790$	20.0000	−	40.0000i	0.711568	−	1.42314i
$791$	28.0000			0.995565
$792$		0			0
$793$			8.00000i			0.284088i
$794$	−16.0000			−0.567819
$795$		0			0
$796$	−50.0000			−1.77220
$797$			8.00000i			0.283375i	0.989911	+	0.141687i	$0.0452527\pi$
$797$			8.00000i			0.283375i	−0.989911	+	0.141687i	$0.954747\pi$
$798$		0			0
$799$	96.0000			3.39624
$800$	−32.0000	+	24.0000i	−1.13137	+	0.848528i
$801$		0			0
$802$		−	56.0000i		−	1.97743i
$803$		−	3.00000i		−	0.105868i
$804$		0			0
$805$	−4.00000	−	2.00000i	−0.140981	−	0.0704907i
$806$	−64.0000			−2.25430
$807$		0			0
$808$		0			0
$809$	−45.0000			−1.58212			−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000			−1.58212			−0.791058	+	0.611741i	$0.790469\pi$
$810$		0			0
$811$	−13.0000			−0.456492			−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000			−0.456492			−0.228246	+	0.973604i	$0.573299\pi$
$812$		−	4.00000i		−	0.140372i
$813$		0			0
$814$	42.0000			1.47210
$815$	1.00000	−	2.00000i	0.0350285	−	0.0700569i
$816$		0			0
$817$		0			0
$818$		−	20.0000i		−	0.699284i
$819$		0			0
$820$	28.0000	+	14.0000i	0.977802	+	0.488901i
$821$	48.0000			1.67521			0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000			1.67521			0.837606	+	0.546275i	$0.183955\pi$
$822$		0			0
$823$		−	16.0000i		−	0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$		−	16.0000i		−	0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$		0			0
$825$		0			0
$826$	40.0000			1.39178
$827$		−	22.0000i		−	0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$		−	22.0000i		−	0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$		0			0
$829$	−50.0000			−1.73657			−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000			−1.73657			−0.868286	+	0.496064i	$0.834778\pi$
$830$	−36.0000	−	18.0000i	−1.24958	−	0.624789i
$831$		0			0
$832$			32.0000i			1.10940i
$833$			24.0000i			0.831551i
$834$		0			0
$835$	−8.00000	+	16.0000i	−0.276851	+	0.553703i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−40.0000			−1.38095			−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000			−1.38095			−0.690477	+	0.723355i	$0.742599\pi$
$840$		0			0
$841$	1.00000			0.0344828
$842$			56.0000i			1.92989i
$843$		0			0
$844$	−24.0000			−0.826114
$845$	−6.00000	−	3.00000i	−0.206406	−	0.103203i
$846$		0			0
$847$		−	4.00000i		−	0.137442i
$848$			36.0000i			1.23625i
$849$		0			0
$850$	48.0000	+	64.0000i	1.64639	+	2.19518i
$851$	−7.00000			−0.239957
$852$		0			0
$853$			19.0000i			0.650548i	0.945620	+	0.325274i	$0.105456\pi$
$853$			19.0000i			0.650548i	−0.945620	+	0.325274i	$0.894544\pi$
$854$	8.00000			0.273754
$855$		0			0
$856$		0			0
$857$		−	7.00000i		−	0.239115i	−0.992827	−	0.119558i	$-0.961852\pi$
$857$		−	7.00000i		−	0.239115i	0.992827	−	0.119558i	$-0.0381477\pi$
$858$		0			0
$859$	30.0000			1.02359			0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000			1.02359			0.511793	+	0.859109i	$0.328981\pi$
$860$	18.0000	−	36.0000i	0.613795	−	1.22759i
$861$		0			0
$862$			24.0000i			0.817443i
$863$		−	44.0000i		−	1.49778i	−0.662696	−	0.748889i	$-0.730588\pi$
$863$		−	44.0000i		−	1.49778i	0.662696	−	0.748889i	$-0.269412\pi$
$864$		0			0
$865$	9.00000	−	18.0000i	0.306009	−	0.612018i
$866$	38.0000			1.29129
$867$		0			0
$868$			32.0000i			1.08615i
$869$	30.0000			1.01768
$870$		0			0
$871$	32.0000			1.08428
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.0000	+	4.00000i	−0.743736	+	0.135225i
$876$		0			0
$877$			52.0000i			1.75592i	0.478738	+	0.877958i	$0.341094\pi$
$877$			52.0000i			1.75592i	−0.478738	+	0.877958i	$0.658906\pi$
$878$			40.0000i			1.34993i
$879$		0			0
$880$	−24.0000	−	12.0000i	−0.809040	−	0.404520i
$881$	−57.0000			−1.92038			−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000			−1.92038			−0.960189	+	0.279350i	$0.909881\pi$
$882$		0			0
$883$			4.00000i			0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$			4.00000i			0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	64.0000			2.15255
$885$		0			0
$886$	12.0000			0.403148
$887$		−	22.0000i		−	0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$		−	22.0000i		−	0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$		0			0
$889$	6.00000			0.201234
$890$	20.0000	−	40.0000i	0.670402	−	1.34080i
$891$		0			0
$892$			12.0000i			0.401790i
$893$		0			0
$894$		0			0
$895$	20.0000	+	10.0000i	0.668526	+	0.334263i
$896$		0			0
$897$		0			0
$898$		−	10.0000i		−	0.333704i
$899$	−8.00000			−0.266815
$900$		0			0
$901$	72.0000			2.39867
$902$			42.0000i			1.39845i
$903$		0			0
$904$		0			0
$905$	34.0000	+	17.0000i	1.13020	+	0.565099i
$906$		0			0
$907$			37.0000i			1.22856i	0.789086	+	0.614282i	$0.210554\pi$
$907$			37.0000i			1.22856i	−0.789086	+	0.614282i	$0.789446\pi$
$908$		−	6.00000i		−	0.199117i
$909$		0			0
$910$	−16.0000	+	32.0000i	−0.530395	+	1.06079i
$911$	−27.0000			−0.894550			−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000			−0.894550			−0.447275	+	0.894397i	$0.647605\pi$
$912$		0			0
$913$		−	27.0000i		−	0.893570i
$914$	−56.0000			−1.85232
$915$		0			0
$916$	40.0000			1.32164
$917$		−	24.0000i		−	0.792550i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		−	26.0000i		−	0.856264i
$923$			32.0000i			1.05329i
$924$		0			0
$925$	−28.0000	+	21.0000i	−0.920634	+	0.690476i
$926$	−12.0000			−0.394344
$927$		0			0
$928$			8.00000i			0.262613i
$929$	20.0000			0.656179			0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000			0.656179			0.328089	+	0.944647i	$0.393595\pi$
$930$		0			0
$931$		0			0
$932$		−	2.00000i		−	0.0655122i
$933$		0			0
$934$	−84.0000			−2.74856
$935$	−24.0000	+	48.0000i	−0.784884	+	1.56977i
$936$		0			0
$937$			22.0000i			0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$			22.0000i			0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$		−	32.0000i		−	1.04484i
$939$		0			0
$940$	−24.0000	+	48.0000i	−0.782794	+	1.56559i
$941$	−42.0000			−1.36916			−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000			−1.36916			−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$		−	7.00000i		−	0.227951i
$944$	−40.0000			−1.30189
$945$		0			0
$946$	54.0000			1.75569
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	4.00000			0.129845
$950$		0			0
$951$		0			0
$952$		0			0
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$		0			0
$955$	6.00000	+	3.00000i	0.194155	+	0.0970777i
$956$	−60.0000			−1.94054
$957$		0			0
$958$		0			0
$959$	4.00000			0.129167
$960$		0			0
$961$	33.0000			1.06452
$962$			56.0000i			1.80551i
$963$		0			0
$964$	6.00000			0.193247
$965$	6.00000	−	12.0000i	0.193147	−	0.386294i
$966$		0			0
$967$		−	3.00000i		−	0.0964735i	−0.998836	−	0.0482367i	$-0.984640\pi$
$967$		−	3.00000i		−	0.0964735i	0.998836	−	0.0482367i	$-0.0153602\pi$
$968$		0			0
$969$		0			0
$970$	−52.0000	−	26.0000i	−1.66962	−	0.834810i
$971$	3.00000			0.0962746			0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000			0.0962746			0.0481373	+	0.998841i	$0.484672\pi$
$972$		0			0
$973$		−	10.0000i		−	0.320585i
$974$	44.0000			1.40985
$975$		0			0
$976$	−8.00000			−0.256074
$977$			13.0000i			0.415907i	0.978139	+	0.207953i	$0.0666802\pi$
$977$			13.0000i			0.415907i	−0.978139	+	0.207953i	$0.933320\pi$
$978$		0			0
$979$	30.0000			0.958804
$980$	−12.0000	−	6.00000i	−0.383326	−	0.191663i
$981$		0			0
$982$		−	56.0000i		−	1.78703i
$983$			36.0000i			1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$983$			36.0000i			1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$984$		0			0
$985$	−13.0000	+	26.0000i	−0.414214	+	0.828429i
$986$	16.0000			0.509544
$987$		0			0
$988$		0			0
$989$	−9.00000			−0.286183
$990$		0			0
$991$	−33.0000			−1.04828			−0.524140	−	0.851632i	$-0.675613\pi$
$991$	−33.0000			−1.04828			−0.524140	+	0.851632i	$0.675613\pi$
$992$		−	64.0000i		−	2.03200i
$993$		0			0
$994$	32.0000			1.01498
$995$	50.0000	+	25.0000i	1.58511	+	0.792553i
$996$		0			0
$997$			7.00000i			0.221692i	0.993838	+	0.110846i	$0.0353561\pi$
$997$			7.00000i			0.221692i	−0.993838	+	0.110846i	$0.964644\pi$
$998$			40.0000i			1.26618i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.c.a.784.1		2
3.2	odd	2		435.2.c.a.349.2	yes	2
5.2	odd	4		6525.2.a.l.1.1		1
5.3	odd	4		6525.2.a.b.1.1		1
5.4	even	2	inner	1305.2.c.a.784.2		2
15.2	even	4		2175.2.a.a.1.1		1
15.8	even	4		2175.2.a.j.1.1		1
15.14	odd	2		435.2.c.a.349.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.a.349.1	✓	2	15.14	odd	2
435.2.c.a.349.2	yes	2	3.2	odd	2
1305.2.c.a.784.1		2	1.1	even	1	trivial
1305.2.c.a.784.2		2	5.4	even	2	inner
2175.2.a.a.1.1		1	15.2	even	4
2175.2.a.j.1.1		1	15.8	even	4
6525.2.a.b.1.1		1	5.3	odd	4
6525.2.a.l.1.1		1	5.2	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000 q^{4} +(2.00000 + 1.00000i) q^{5} +2.00000i q^{7} +(2.00000 - 4.00000i) q^{10} +3.00000 q^{11} +4.00000i q^{13} +4.00000 q^{14} -4.00000 q^{16} +8.00000i q^{17} +(-4.00000 - 2.00000i) q^{20} -6.00000i q^{22} +1.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +8.00000 q^{26} -4.00000i q^{28} +1.00000 q^{29} -8.00000 q^{31} +8.00000i q^{32} +16.0000 q^{34} +(-2.00000 + 4.00000i) q^{35} +7.00000i q^{37} -7.00000 q^{41} +9.00000i q^{43} -6.00000 q^{44} +2.00000 q^{46} -12.0000i q^{47} +3.00000 q^{49} +(8.00000 - 6.00000i) q^{50} -8.00000i q^{52} -9.00000i q^{53} +(6.00000 + 3.00000i) q^{55} -2.00000i q^{58} +10.0000 q^{59} +2.00000 q^{61} +16.0000i q^{62} +8.00000 q^{64} +(-4.00000 + 8.00000i) q^{65} -8.00000i q^{67} -16.0000i q^{68} +(8.00000 + 4.00000i) q^{70} +8.00000 q^{71} -1.00000i q^{73} +14.0000 q^{74} +6.00000i q^{77} +10.0000 q^{79} +(-8.00000 - 4.00000i) q^{80} +14.0000i q^{82} -9.00000i q^{83} +(-8.00000 + 16.0000i) q^{85} +18.0000 q^{86} +10.0000 q^{89} -8.00000 q^{91} -2.00000i q^{92} -24.0000 q^{94} -13.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 4 q^{5} + 4 q^{10} + 6 q^{11} + 8 q^{14} - 8 q^{16} - 8 q^{20} + 6 q^{25} + 16 q^{26} + 2 q^{29} - 16 q^{31} + 32 q^{34} - 4 q^{35} - 14 q^{41} - 12 q^{44} + 4 q^{46} + 6 q^{49} + 16 q^{50}+ \cdots - 48 q^{94}+O(q^{100}) \) 2 * q - 4 * q^4 + 4 * q^5 + 4 * q^10 + 6 * q^11 + 8 * q^14 - 8 * q^16 - 8 * q^20 + 6 * q^25 + 16 * q^26 + 2 * q^29 - 16 * q^31 + 32 * q^34 - 4 * q^35 - 14 * q^41 - 12 * q^44 + 4 * q^46 + 6 * q^49 + 16 * q^50 + 12 * q^55 + 20 * q^59 + 4 * q^61 + 16 * q^64 - 8 * q^65 + 16 * q^70 + 16 * q^71 + 28 * q^74 + 20 * q^79 - 16 * q^80 - 16 * q^85 + 36 * q^86 + 20 * q^89 - 16 * q^91 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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