LMFDB - Embedded newform 306.2.b.a.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [306,2,Mod(271,306)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(306, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("306.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$306 = 2 \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	306.b (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:	$2.44342230185$
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:	$2$
Twist minimal:	no (minimal twist has level 102)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	306.271
Dual form			306.2.b.a.271.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.00000i q^{5} -2.00000i q^{7} -1.00000 q^{8} +2.00000i q^{10} -6.00000 q^{13} +2.00000i q^{14} +1.00000 q^{16} +(1.00000 - 4.00000i) q^{17} -2.00000i q^{20} -6.00000i q^{23} +1.00000 q^{25} +6.00000 q^{26} -2.00000i q^{28} +6.00000i q^{29} -10.0000i q^{31} -1.00000 q^{32} +(-1.00000 + 4.00000i) q^{34} -4.00000 q^{35} -2.00000i q^{37} +2.00000i q^{40} +4.00000 q^{43} +6.00000i q^{46} -8.00000 q^{47} +3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} +6.00000 q^{53} +2.00000i q^{56} -6.00000i q^{58} +10.0000i q^{61} +10.0000i q^{62} +1.00000 q^{64} +12.0000i q^{65} +8.00000 q^{67} +(1.00000 - 4.00000i) q^{68} +4.00000 q^{70} +10.0000i q^{71} +16.0000i q^{73} +2.00000i q^{74} -6.00000i q^{79} -2.00000i q^{80} +16.0000 q^{83} +(-8.00000 - 2.00000i) q^{85} -4.00000 q^{86} -10.0000 q^{89} +12.0000i q^{91} -6.00000i q^{92} +8.00000 q^{94} -12.0000i q^{97} -3.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 12 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{25} + 12 q^{26} - 2 q^{32} - 2 q^{34} - 8 q^{35} + 8 q^{43} - 16 q^{47} + 6 q^{49} - 2 q^{50} - 12 q^{52} + 12 q^{53} + 2 q^{64}+ \cdots - 6 q^{98}+O(q^{100})$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 12 * q^13 + 2 * q^16 + 2 * q^17 + 2 * q^25 + 12 * q^26 - 2 * q^32 - 2 * q^34 - 8 * q^35 + 8 * q^43 - 16 * q^47 + 6 * q^49 - 2 * q^50 - 12 * q^52 + 12 * q^53 + 2 * q^64 + 16 * q^67 + 2 * q^68 + 8 * q^70 + 32 * q^83 - 16 * q^85 - 8 * q^86 - 20 * q^89 + 16 * q^94 - 6 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/306\mathbb{Z}\right)^\times$ .

$n$	$37$	$137$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$			2.00000i			0.632456i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−6.00000			−1.66410			−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000			−1.66410			−0.832050	+	0.554700i	$0.812833\pi$
$14$			2.00000i			0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.00000	−	4.00000i	0.242536	−	0.970143i
$17$	1.00000	−	4.00000i	0.242536	−	0.970143i
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		−	2.00000i		−	0.447214i
$21$		0			0
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$	6.00000			1.17670
$27$		0			0
$28$		−	2.00000i		−	0.377964i
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	−1.00000	+	4.00000i	−0.171499	+	0.685994i
$35$	−4.00000			−0.676123
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$			2.00000i			0.316228i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$			6.00000i			0.884652i
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	−1.00000			−0.141421
$51$		0			0
$52$	−6.00000			−0.832050
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$			2.00000i			0.267261i
$57$		0			0
$58$		−	6.00000i		−	0.787839i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			10.0000i			1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$			10.0000i			1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$			10.0000i			1.27000i
$63$		0			0
$64$	1.00000			0.125000
$65$			12.0000i			1.48842i
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$	1.00000	−	4.00000i	0.121268	−	0.485071i
$69$		0			0
$70$	4.00000			0.478091
$71$			10.0000i			1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$			10.0000i			1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$		0			0
$73$			16.0000i			1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$			16.0000i			1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$			2.00000i			0.232495i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	6.00000i		−	0.675053i	−0.941316	−	0.337526i	$-0.890410\pi$
$79$		−	6.00000i		−	0.675053i	0.941316	−	0.337526i	$-0.109590\pi$
$80$		−	2.00000i		−	0.223607i
$81$		0			0
$82$		0			0
$83$	16.0000			1.75623			0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000			1.75623			0.878114	+	0.478451i	$0.158802\pi$
$84$		0			0
$85$	−8.00000	−	2.00000i	−0.867722	−	0.216930i
$86$	−4.00000			−0.431331
$87$		0			0
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$			12.0000i			1.25794i
$92$		−	6.00000i		−	0.625543i
$93$		0			0
$94$	8.00000			0.825137
$95$		0			0
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$	−3.00000			−0.303046
$99$		0			0
$100$	1.00000			0.100000
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$	6.00000			0.588348
$105$		0			0
$106$	−6.00000			−0.582772
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$			14.0000i			1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$			14.0000i			1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$		0			0
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$			4.00000i			0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$			4.00000i			0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$		0			0
$115$	−12.0000			−1.11901
$116$			6.00000i			0.557086i
$117$		0			0
$118$		0			0
$119$	−8.00000	−	2.00000i	−0.733359	−	0.183340i
$120$		0			0
$121$	11.0000			1.00000
$122$		−	10.0000i		−	0.905357i
$123$		0			0
$124$		−	10.0000i		−	0.898027i
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$		−	12.0000i		−	1.05247i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	−8.00000			−0.691095
$135$		0			0
$136$	−1.00000	+	4.00000i	−0.0857493	+	0.342997i
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$		0			0
$139$		−	16.0000i		−	1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$		−	16.0000i		−	1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$	−4.00000			−0.338062
$141$		0			0
$142$		−	10.0000i		−	0.839181i
$143$		0			0
$144$		0			0
$145$	12.0000			0.996546
$146$		−	16.0000i		−	1.32417i
$147$		0			0
$148$		−	2.00000i		−	0.164399i
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−20.0000			−1.60644
$156$		0			0
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$			6.00000i			0.477334i
$159$		0			0
$160$			2.00000i			0.158114i
$161$	−12.0000			−0.945732
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$		0			0
$166$	−16.0000			−1.24184
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$	8.00000	+	2.00000i	0.613572	+	0.153393i
$171$		0			0
$172$	4.00000			0.304997
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$		−	2.00000i		−	0.151186i
$176$		0			0
$177$		0			0
$178$	10.0000			0.749532
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$		−	12.0000i		−	0.889499i
$183$		0			0
$184$			6.00000i			0.442326i
$185$	−4.00000			−0.294086
$186$		0			0
$187$		0			0
$188$	−8.00000			−0.583460
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$			12.0000i			0.861550i
$195$		0			0
$196$	3.00000			0.214286
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$		−	6.00000i		−	0.425329i	−0.977125	−	0.212664i	$-0.931786\pi$
$199$		−	6.00000i		−	0.425329i	0.977125	−	0.212664i	$-0.0682141\pi$
$200$	−1.00000			−0.0707107
$201$		0			0
$202$	−18.0000			−1.26648
$203$	12.0000			0.842235
$204$		0			0
$205$		0			0
$206$	16.0000			1.11477
$207$		0			0
$208$	−6.00000			−0.416025
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	6.00000			0.412082
$213$		0			0
$214$			8.00000i			0.546869i
$215$		−	8.00000i		−	0.545595i
$216$		0			0
$217$	−20.0000			−1.35769
$218$		−	14.0000i		−	0.948200i
$219$		0			0
$220$		0			0
$221$	−6.00000	+	24.0000i	−0.403604	+	1.61441i
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$			2.00000i			0.133631i
$225$		0			0
$226$		−	4.00000i		−	0.266076i
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$	12.0000			0.791257
$231$		0			0
$232$		−	6.00000i		−	0.393919i
$233$			4.00000i			0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$			4.00000i			0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$		0			0
$235$			16.0000i			1.04372i
$236$		0			0
$237$		0			0
$238$	8.00000	+	2.00000i	0.518563	+	0.129641i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$			20.0000i			1.28831i	0.764894	+	0.644157i	$0.222792\pi$
$241$			20.0000i			1.28831i	−0.764894	+	0.644157i	$0.777208\pi$
$242$	−11.0000			−0.707107
$243$		0			0
$244$			10.0000i			0.640184i
$245$		−	6.00000i		−	0.383326i
$246$		0			0
$247$		0			0
$248$			10.0000i			0.635001i
$249$		0			0
$250$			12.0000i			0.758947i
$251$	−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
$253$		0			0
$254$	−8.00000			−0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$			12.0000i			0.744208i
$261$		0			0
$262$		0			0
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$		−	12.0000i		−	0.737154i
$266$		0			0
$267$		0			0
$268$	8.00000			0.488678
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$		0			0
$271$	32.0000			1.94386			0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000			1.94386			0.971931	+	0.235267i	$0.0755965\pi$
$272$	1.00000	−	4.00000i	0.0606339	−	0.242536i
$273$		0			0
$274$	−2.00000			−0.120824
$275$		0			0
$276$		0			0
$277$		−	22.0000i		−	1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$		−	22.0000i		−	1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$			16.0000i			0.959616i
$279$		0			0
$280$	4.00000			0.239046
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$			10.0000i			0.593391i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.0000	−	8.00000i	−0.882353	−	0.470588i
$290$	−12.0000			−0.704664
$291$		0			0
$292$			16.0000i			0.936329i
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		0			0
$296$			2.00000i			0.116248i
$297$		0			0
$298$	10.0000			0.579284
$299$			36.0000i			2.08193i
$300$		0			0
$301$		−	8.00000i		−	0.461112i
$302$	8.00000			0.460348
$303$		0			0
$304$		0			0
$305$	20.0000			1.14520
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$		0			0
$310$	20.0000			1.13592
$311$		−	10.0000i		−	0.567048i	−0.958965	−	0.283524i	$-0.908496\pi$
$311$		−	10.0000i		−	0.567048i	0.958965	−	0.283524i	$-0.0915036\pi$
$312$		0			0
$313$		−	4.00000i		−	0.226093i	−0.993590	−	0.113047i	$-0.963939\pi$
$313$		−	4.00000i		−	0.226093i	0.993590	−	0.113047i	$-0.0360610\pi$
$314$	2.00000			0.112867
$315$		0			0
$316$		−	6.00000i		−	0.337526i
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		0			0
$320$		−	2.00000i		−	0.111803i
$321$		0			0
$322$	12.0000			0.668734
$323$		0			0
$324$		0			0
$325$	−6.00000			−0.332820
$326$			4.00000i			0.221540i
$327$		0			0
$328$		0			0
$329$			16.0000i			0.882109i
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$	16.0000			0.878114
$333$		0			0
$334$			18.0000i			0.984916i
$335$		−	16.0000i		−	0.874173i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	−23.0000			−1.25104
$339$		0			0
$340$	−8.00000	−	2.00000i	−0.433861	−	0.108465i
$341$		0			0
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$	−4.00000			−0.215666
$345$		0			0
$346$		−	14.0000i		−	0.752645i
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$			2.00000i			0.106904i
$351$		0			0
$352$		0			0
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$	20.0000			1.06149
$356$	−10.0000			−0.529999
$357$		0			0
$358$	−20.0000			−1.05703
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		−	10.0000i		−	0.525588i
$363$		0			0
$364$			12.0000i			0.628971i
$365$	32.0000			1.67496
$366$		0			0
$367$		−	22.0000i		−	1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$		−	22.0000i		−	1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$		−	6.00000i		−	0.312772i
$369$		0			0
$370$	4.00000			0.207950
$371$		−	12.0000i		−	0.623009i
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$		0			0
$376$	8.00000			0.412568
$377$		−	36.0000i		−	1.85409i
$378$		0			0
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$		0			0
$381$		0			0
$382$	−8.00000			−0.409316
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$		0			0
$386$			4.00000i			0.203595i
$387$		0			0
$388$		−	12.0000i		−	0.609208i
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$	−24.0000	−	6.00000i	−1.21373	−	0.303433i
$392$	−3.00000			−0.151523
$393$		0			0
$394$			18.0000i			0.906827i
$395$	−12.0000			−0.603786
$396$		0			0
$397$		−	22.0000i		−	1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$		−	22.0000i		−	1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$			6.00000i			0.300753i
$399$		0			0
$400$	1.00000			0.0500000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$			60.0000i			2.98881i
$404$	18.0000			0.895533
$405$		0			0
$406$	−12.0000			−0.595550
$407$		0			0
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$	−16.0000			−0.788263
$413$		0			0
$414$		0			0
$415$		−	32.0000i		−	1.57082i
$416$	6.00000			0.294174
$417$		0			0
$418$		0			0
$419$		−	4.00000i		−	0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$		−	4.00000i		−	0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$		0			0
$423$		0			0
$424$	−6.00000			−0.291386
$425$	1.00000	−	4.00000i	0.0485071	−	0.194029i
$426$		0			0
$427$	20.0000			0.967868
$428$		−	8.00000i		−	0.386695i
$429$		0			0
$430$			8.00000i			0.385794i
$431$			10.0000i			0.481683i	0.970564	+	0.240842i	$0.0774234\pi$
$431$			10.0000i			0.481683i	−0.970564	+	0.240842i	$0.922577\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$	20.0000			0.960031
$435$		0			0
$436$			14.0000i			0.670478i
$437$		0			0
$438$		0			0
$439$		−	6.00000i		−	0.286364i	−0.989696	−	0.143182i	$-0.954267\pi$
$439$		−	6.00000i		−	0.286364i	0.989696	−	0.143182i	$-0.0457335\pi$
$440$		0			0
$441$		0			0
$442$	6.00000	−	24.0000i	0.285391	−	1.14156i
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$			20.0000i			0.948091i
$446$	16.0000			0.757622
$447$		0			0
$448$		−	2.00000i		−	0.0944911i
$449$		−	24.0000i		−	1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$		−	24.0000i		−	1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$		0			0
$451$		0			0
$452$			4.00000i			0.188144i
$453$		0			0
$454$		−	12.0000i		−	0.563188i
$455$	24.0000			1.12514
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$	−10.0000			−0.467269
$459$		0			0
$460$	−12.0000			−0.559503
$461$	−2.00000			−0.0931493			−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000			−0.0931493			−0.0465746	+	0.998915i	$0.514831\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$			6.00000i			0.278543i
$465$		0			0
$466$		−	4.00000i		−	0.185296i
$467$	−28.0000			−1.29569			−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000			−1.29569			−0.647843	+	0.761774i	$0.724329\pi$
$468$		0			0
$469$		−	16.0000i		−	0.738811i
$470$		−	16.0000i		−	0.738025i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−8.00000	−	2.00000i	−0.366679	−	0.0916698i
$477$		0			0
$478$		0			0
$479$		−	14.0000i		−	0.639676i	−0.947472	−	0.319838i	$-0.896371\pi$
$479$		−	14.0000i		−	0.639676i	0.947472	−	0.319838i	$-0.103629\pi$
$480$		0			0
$481$			12.0000i			0.547153i
$482$		−	20.0000i		−	0.910975i
$483$		0			0
$484$	11.0000			0.500000
$485$	−24.0000			−1.08978
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$		−	10.0000i		−	0.452679i
$489$		0			0
$490$			6.00000i			0.271052i
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	24.0000	+	6.00000i	1.08091	+	0.270226i
$494$		0			0
$495$		0			0
$496$		−	10.0000i		−	0.449013i
$497$	20.0000			0.897123
$498$		0			0
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$		−	12.0000i		−	0.536656i
$501$		0			0
$502$	12.0000			0.535586
$503$			34.0000i			1.51599i	0.652263	+	0.757993i	$0.273820\pi$
$503$			34.0000i			1.51599i	−0.652263	+	0.757993i	$0.726180\pi$
$504$		0			0
$505$		−	36.0000i		−	1.60198i
$506$		0			0
$507$		0			0
$508$	8.00000			0.354943
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$	32.0000			1.41560
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−2.00000			−0.0882162
$515$			32.0000i			1.41009i
$516$		0			0
$517$		0			0
$518$	4.00000			0.175750
$519$		0			0
$520$		−	12.0000i		−	0.526235i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$		0			0
$526$	−16.0000			−0.697633
$527$	−40.0000	−	10.0000i	−1.74243	−	0.435607i
$528$		0			0
$529$	−13.0000			−0.565217
$530$			12.0000i			0.521247i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−16.0000			−0.691740
$536$	−8.00000			−0.345547
$537$		0			0
$538$		−	6.00000i		−	0.258678i
$539$		0			0
$540$		0			0
$541$		−	10.0000i		−	0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$		−	10.0000i		−	0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	−32.0000			−1.37452
$543$		0			0
$544$	−1.00000	+	4.00000i	−0.0428746	+	0.171499i
$545$	28.0000			1.19939
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	2.00000			0.0854358
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−12.0000			−0.510292
$554$			22.0000i			0.934690i
$555$		0			0
$556$		−	16.0000i		−	0.678551i
$557$	−38.0000			−1.61011			−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000			−1.61011			−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$	−4.00000			−0.169031
$561$		0			0
$562$	22.0000			0.928014
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$	8.00000			0.336563
$566$		−	16.0000i		−	0.672530i
$567$		0			0
$568$		−	10.0000i		−	0.419591i
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	6.00000i		−	0.250217i
$576$		0			0
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$	15.0000	+	8.00000i	0.623918	+	0.332756i
$579$		0			0
$580$	12.0000			0.498273
$581$		−	32.0000i		−	1.32758i
$582$		0			0
$583$		0			0
$584$		−	16.0000i		−	0.662085i
$585$		0			0
$586$	−6.00000			−0.247858
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		−	2.00000i		−	0.0821995i
$593$	46.0000			1.88899			0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000			1.88899			0.944497	+	0.328521i	$0.106550\pi$
$594$		0			0
$595$	−4.00000	+	16.0000i	−0.163984	+	0.655936i
$596$	−10.0000			−0.409616
$597$		0			0
$598$		−	36.0000i		−	1.47215i
$599$	40.0000			1.63436			0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000			1.63436			0.817178	+	0.576386i	$0.195537\pi$
$600$		0			0
$601$		−	40.0000i		−	1.63163i	−0.578310	−	0.815817i	$-0.696288\pi$
$601$		−	40.0000i		−	1.63163i	0.578310	−	0.815817i	$-0.303712\pi$
$602$			8.00000i			0.326056i
$603$		0			0
$604$	−8.00000			−0.325515
$605$		−	22.0000i		−	0.894427i
$606$		0			0
$607$			38.0000i			1.54237i	0.636610	+	0.771186i	$0.280336\pi$
$607$			38.0000i			1.54237i	−0.636610	+	0.771186i	$0.719664\pi$
$608$		0			0
$609$		0			0
$610$	−20.0000			−0.809776
$611$	48.0000			1.94187
$612$		0			0
$613$	14.0000			0.565455			0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000			0.565455			0.282727	+	0.959200i	$0.408761\pi$
$614$	12.0000			0.484281
$615$		0			0
$616$		0			0
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$		0			0
$619$			4.00000i			0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$			4.00000i			0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	−20.0000			−0.803219
$621$		0			0
$622$			10.0000i			0.400963i
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−19.0000			−0.760000
$626$			4.00000i			0.159872i
$627$		0			0
$628$	−2.00000			−0.0798087
$629$	−8.00000	−	2.00000i	−0.318981	−	0.0797452i
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$			6.00000i			0.238667i
$633$		0			0
$634$			18.0000i			0.714871i
$635$		−	16.0000i		−	0.634941i
$636$		0			0
$637$	−18.0000			−0.713186
$638$		0			0
$639$		0			0
$640$			2.00000i			0.0790569i
$641$			40.0000i			1.57991i	0.613168	+	0.789953i	$0.289895\pi$
$641$			40.0000i			1.57991i	−0.613168	+	0.789953i	$0.710105\pi$
$642$		0			0
$643$			36.0000i			1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$			36.0000i			1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$	−12.0000			−0.472866
$645$		0			0
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$		0			0
$650$	6.00000			0.235339
$651$		0			0
$652$		−	4.00000i		−	0.156652i
$653$			34.0000i			1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$			34.0000i			1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		−	16.0000i		−	0.623745i
$659$	40.0000			1.55818			0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000			1.55818			0.779089	+	0.626913i	$0.215682\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$	−12.0000			−0.466393
$663$		0			0
$664$	−16.0000			−0.620920
$665$		0			0
$666$		0			0
$667$	36.0000			1.39393
$668$		−	18.0000i		−	0.696441i
$669$		0			0
$670$			16.0000i			0.618134i
$671$		0			0
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$		−	8.00000i		−	0.308148i
$675$		0			0
$676$	23.0000			0.884615
$677$			42.0000i			1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$			42.0000i			1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$		0			0
$679$	−24.0000			−0.921035
$680$	8.00000	+	2.00000i	0.306786	+	0.0766965i
$681$		0			0
$682$		0			0
$683$			44.0000i			1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$			44.0000i			1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$		0			0
$685$		−	4.00000i		−	0.152832i
$686$			20.0000i			0.763604i
$687$		0			0
$688$	4.00000			0.152499
$689$	−36.0000			−1.37149
$690$		0			0
$691$		−	40.0000i		−	1.52167i	−0.648944	−	0.760836i	$-0.724789\pi$
$691$		−	40.0000i		−	1.52167i	0.648944	−	0.760836i	$-0.275211\pi$
$692$			14.0000i			0.532200i
$693$		0			0
$694$		−	12.0000i		−	0.455514i
$695$	−32.0000			−1.21383
$696$		0			0
$697$		0			0
$698$	10.0000			0.378506
$699$		0			0
$700$		−	2.00000i		−	0.0755929i
$701$	−22.0000			−0.830929			−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	14.0000			0.526897
$707$		−	36.0000i		−	1.35392i
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$	−20.0000			−0.750587
$711$		0			0
$712$	10.0000			0.374766
$713$	−60.0000			−2.24702
$714$		0			0
$715$		0			0
$716$	20.0000			0.747435
$717$		0			0
$718$		0			0
$719$			26.0000i			0.969636i	0.874615	+	0.484818i	$0.161114\pi$
$719$			26.0000i			0.969636i	−0.874615	+	0.484818i	$0.838886\pi$
$720$		0			0
$721$			32.0000i			1.19174i
$722$	19.0000			0.707107
$723$		0			0
$724$			10.0000i			0.371647i
$725$			6.00000i			0.222834i
$726$		0			0
$727$	48.0000			1.78022			0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000			1.78022			0.890111	+	0.455744i	$0.150627\pi$
$728$		−	12.0000i		−	0.444750i
$729$		0			0
$730$	−32.0000			−1.18437
$731$	4.00000	−	16.0000i	0.147945	−	0.591781i
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$			22.0000i			0.812035i
$735$		0			0
$736$			6.00000i			0.221163i
$737$		0			0
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$	−4.00000			−0.147043
$741$		0			0
$742$			12.0000i			0.440534i
$743$		−	26.0000i		−	0.953847i	−0.878945	−	0.476924i	$-0.841752\pi$
$743$		−	26.0000i		−	0.953847i	0.878945	−	0.476924i	$-0.158248\pi$
$744$		0			0
$745$			20.0000i			0.732743i
$746$	6.00000			0.219676
$747$		0			0
$748$		0			0
$749$	−16.0000			−0.584627
$750$		0			0
$751$			10.0000i			0.364905i	0.983215	+	0.182453i	$0.0584036\pi$
$751$			10.0000i			0.364905i	−0.983215	+	0.182453i	$0.941596\pi$
$752$	−8.00000			−0.291730
$753$		0			0
$754$			36.0000i			1.31104i
$755$			16.0000i			0.582300i
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		−	4.00000i		−	0.145287i
$759$		0			0
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$	28.0000			1.01367
$764$	8.00000			0.289430
$765$		0			0
$766$	24.0000			0.867155
$767$		0			0
$768$		0			0
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$		0			0
$771$		0			0
$772$		−	4.00000i		−	0.143963i
$773$	−14.0000			−0.503545			−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000			−0.503545			−0.251773	+	0.967786i	$0.581013\pi$
$774$		0			0
$775$		−	10.0000i		−	0.359211i
$776$			12.0000i			0.430775i
$777$		0			0
$778$	−10.0000			−0.358517
$779$		0			0
$780$		0			0
$781$		0			0
$782$	24.0000	+	6.00000i	0.858238	+	0.214560i
$783$		0			0
$784$	3.00000			0.107143
$785$			4.00000i			0.142766i
$786$		0			0
$787$		−	32.0000i		−	1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$		−	32.0000i		−	1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$		−	18.0000i		−	0.641223i
$789$		0			0
$790$	12.0000			0.426941
$791$	8.00000			0.284447
$792$		0			0
$793$		−	60.0000i		−	2.13066i
$794$			22.0000i			0.780751i
$795$		0			0
$796$		−	6.00000i		−	0.212664i
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	−8.00000	+	32.0000i	−0.283020	+	1.13208i
$800$	−1.00000			−0.0353553
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$			24.0000i			0.845889i
$806$		−	60.0000i		−	2.11341i
$807$		0			0
$808$	−18.0000			−0.633238
$809$		−	24.0000i		−	0.843795i	−0.906644	−	0.421898i	$-0.861364\pi$
$809$		−	24.0000i		−	0.843795i	0.906644	−	0.421898i	$-0.138636\pi$
$810$		0			0
$811$		−	20.0000i		−	0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$		−	20.0000i		−	0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	12.0000			0.421117
$813$		0			0
$814$		0			0
$815$	−8.00000			−0.280228
$816$		0			0
$817$		0			0
$818$	10.0000			0.349642
$819$		0			0
$820$		0			0
$821$			10.0000i			0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$			10.0000i			0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$		0			0
$823$		−	14.0000i		−	0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$		−	14.0000i		−	0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	16.0000			0.557386
$825$		0			0
$826$		0			0
$827$			32.0000i			1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$			32.0000i			1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$			32.0000i			1.11074i
$831$		0			0
$832$	−6.00000			−0.208013
$833$	3.00000	−	12.0000i	0.103944	−	0.415775i
$834$		0			0
$835$	−36.0000			−1.24583
$836$		0			0
$837$		0			0
$838$			4.00000i			0.138178i
$839$		−	34.0000i		−	1.17381i	−0.809656	−	0.586905i	$-0.800346\pi$
$839$		−	34.0000i		−	1.17381i	0.809656	−	0.586905i	$-0.199654\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$	−2.00000			−0.0689246
$843$		0			0
$844$		0			0
$845$		−	46.0000i		−	1.58245i
$846$		0			0
$847$		−	22.0000i		−	0.755929i
$848$	6.00000			0.206041
$849$		0			0
$850$	−1.00000	+	4.00000i	−0.0342997	+	0.137199i
$851$	−12.0000			−0.411355
$852$		0			0
$853$		−	14.0000i		−	0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$		−	14.0000i		−	0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	−20.0000			−0.684386
$855$		0			0
$856$			8.00000i			0.273434i
$857$		−	48.0000i		−	1.63965i	−0.572615	−	0.819824i	$-0.694071\pi$
$857$		−	48.0000i		−	1.63965i	0.572615	−	0.819824i	$-0.305929\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$		−	8.00000i		−	0.272798i
$861$		0			0
$862$		−	10.0000i		−	0.340601i
$863$	16.0000			0.544646			0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000			0.544646			0.272323	+	0.962206i	$0.412208\pi$
$864$		0			0
$865$	28.0000			0.952029
$866$	−34.0000			−1.15537
$867$		0			0
$868$	−20.0000			−0.678844
$869$		0			0
$870$		0			0
$871$	−48.0000			−1.62642
$872$		−	14.0000i		−	0.474100i
$873$		0			0
$874$		0			0
$875$	−24.0000			−0.811348
$876$		0			0
$877$			38.0000i			1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$			38.0000i			1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$			6.00000i			0.202490i
$879$		0			0
$880$		0			0
$881$		−	20.0000i		−	0.673817i	−0.941537	−	0.336909i	$-0.890619\pi$
$881$		−	20.0000i		−	0.673817i	0.941537	−	0.336909i	$-0.109381\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$	−6.00000	+	24.0000i	−0.201802	+	0.807207i
$885$		0			0
$886$	4.00000			0.134383
$887$			42.0000i			1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$			42.0000i			1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$		0			0
$889$		−	16.0000i		−	0.536623i
$890$		−	20.0000i		−	0.670402i
$891$		0			0
$892$	−16.0000			−0.535720
$893$		0			0
$894$		0			0
$895$		−	40.0000i		−	1.33705i
$896$			2.00000i			0.0668153i
$897$		0			0
$898$			24.0000i			0.800890i
$899$	60.0000			2.00111
$900$		0			0
$901$	6.00000	−	24.0000i	0.199889	−	0.799556i
$902$		0			0
$903$		0			0
$904$		−	4.00000i		−	0.133038i
$905$	20.0000			0.664822
$906$		0			0
$907$			8.00000i			0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$			8.00000i			0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$			12.0000i			0.398234i
$909$		0			0
$910$	−24.0000			−0.795592
$911$		−	50.0000i		−	1.65657i	−0.560304	−	0.828287i	$-0.689316\pi$
$911$		−	50.0000i		−	1.65657i	0.560304	−	0.828287i	$-0.310684\pi$
$912$		0			0
$913$		0			0
$914$	22.0000			0.727695
$915$		0			0
$916$	10.0000			0.330409
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	12.0000			0.395628
$921$		0			0
$922$	2.00000			0.0658665
$923$		−	60.0000i		−	1.97492i
$924$		0			0
$925$		−	2.00000i		−	0.0657596i
$926$	−24.0000			−0.788689
$927$		0			0
$928$		−	6.00000i		−	0.196960i
$929$			16.0000i			0.524943i	0.964940	+	0.262471i	$0.0845376\pi$
$929$			16.0000i			0.524943i	−0.964940	+	0.262471i	$0.915462\pi$
$930$		0			0
$931$		0			0
$932$			4.00000i			0.131024i
$933$		0			0
$934$	28.0000			0.916188
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$			16.0000i			0.522419i
$939$		0			0
$940$			16.0000i			0.521862i
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	8.00000i		−	0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$		−	8.00000i		−	0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$		0			0
$949$		−	96.0000i		−	3.11629i
$950$		0			0
$951$		0			0
$952$	8.00000	+	2.00000i	0.259281	+	0.0648204i
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$		−	16.0000i		−	0.517748i
$956$		0			0
$957$		0			0
$958$			14.0000i			0.452319i
$959$		−	4.00000i		−	0.129167i
$960$		0			0
$961$	−69.0000			−2.22581
$962$		−	12.0000i		−	0.386896i
$963$		0			0
$964$			20.0000i			0.644157i
$965$	−8.00000			−0.257529
$966$		0			0
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$	−11.0000			−0.353553
$969$		0			0
$970$	24.0000			0.770594
$971$	−32.0000			−1.02693			−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000			−1.02693			−0.513464	+	0.858111i	$0.671638\pi$
$972$		0			0
$973$	−32.0000			−1.02587
$974$		−	18.0000i		−	0.576757i
$975$		0			0
$976$			10.0000i			0.320092i
$977$	22.0000			0.703842			0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000			0.703842			0.351921	+	0.936030i	$0.385529\pi$
$978$		0			0
$979$		0			0
$980$		−	6.00000i		−	0.191663i
$981$		0			0
$982$	12.0000			0.382935
$983$		−	46.0000i		−	1.46717i	−0.679597	−	0.733586i	$-0.737845\pi$
$983$		−	46.0000i		−	1.46717i	0.679597	−	0.733586i	$-0.262155\pi$
$984$		0			0
$985$	−36.0000			−1.14706
$986$	−24.0000	−	6.00000i	−0.764316	−	0.191079i
$987$		0			0
$988$		0			0
$989$		−	24.0000i		−	0.763156i
$990$		0			0
$991$			10.0000i			0.317660i	0.987306	+	0.158830i	$0.0507723\pi$
$991$			10.0000i			0.317660i	−0.987306	+	0.158830i	$0.949228\pi$
$992$			10.0000i			0.317500i
$993$		0			0
$994$	−20.0000			−0.634361
$995$	−12.0000			−0.380426
$996$		0			0
$997$			18.0000i			0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$			18.0000i			0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$		−	24.0000i		−	0.759707i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	306.2.b.a.271.1		2
3.2	odd	2		102.2.b.a.67.1	✓	2
4.3	odd	2		2448.2.c.a.577.1		2
12.11	even	2		816.2.c.a.577.2		2
15.2	even	4		2550.2.f.e.1699.2		2
15.8	even	4		2550.2.f.j.1699.1		2
15.14	odd	2		2550.2.c.f.1801.2		2
17.4	even	4		5202.2.a.h.1.1		1
17.13	even	4		5202.2.a.n.1.1		1
17.16	even	2	inner	306.2.b.a.271.2		2
24.5	odd	2		3264.2.c.j.577.2		2
24.11	even	2		3264.2.c.i.577.1		2
51.2	odd	8		1734.2.f.h.829.1		4
51.8	odd	8		1734.2.f.h.1483.1		4
51.26	odd	8		1734.2.f.h.1483.2		4
51.32	odd	8		1734.2.f.h.829.2		4
51.38	odd	4		1734.2.a.d.1.1		1
51.47	odd	4		1734.2.a.e.1.1		1
51.50	odd	2		102.2.b.a.67.2	yes	2
68.67	odd	2		2448.2.c.a.577.2		2
204.203	even	2		816.2.c.a.577.1		2
255.152	even	4		2550.2.f.j.1699.2		2
255.203	even	4		2550.2.f.e.1699.1		2
255.254	odd	2		2550.2.c.f.1801.1		2
408.101	odd	2		3264.2.c.j.577.1		2
408.203	even	2		3264.2.c.i.577.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.b.a.67.1	✓	2	3.2	odd	2
102.2.b.a.67.2	yes	2	51.50	odd	2
306.2.b.a.271.1		2	1.1	even	1	trivial
306.2.b.a.271.2		2	17.16	even	2	inner
816.2.c.a.577.1		2	204.203	even	2
816.2.c.a.577.2		2	12.11	even	2
1734.2.a.d.1.1		1	51.38	odd	4
1734.2.a.e.1.1		1	51.47	odd	4
1734.2.f.h.829.1		4	51.2	odd	8
1734.2.f.h.829.2		4	51.32	odd	8
1734.2.f.h.1483.1		4	51.8	odd	8
1734.2.f.h.1483.2		4	51.26	odd	8
2448.2.c.a.577.1		2	4.3	odd	2
2448.2.c.a.577.2		2	68.67	odd	2
2550.2.c.f.1801.1		2	255.254	odd	2
2550.2.c.f.1801.2		2	15.14	odd	2
2550.2.f.e.1699.1		2	255.203	even	4
2550.2.f.e.1699.2		2	15.2	even	4
2550.2.f.j.1699.1		2	15.8	even	4
2550.2.f.j.1699.2		2	255.152	even	4
3264.2.c.i.577.1		2	24.11	even	2
3264.2.c.i.577.2		2	408.203	even	2
3264.2.c.j.577.1		2	408.101	odd	2
3264.2.c.j.577.2		2	24.5	odd	2
5202.2.a.h.1.1		1	17.4	even	4
5202.2.a.n.1.1		1	17.13	even	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	306.2.b.a.271.1

Level	$306$
Weight	$2$
Character	306.271
Analytic conductor	$2.443$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$