LMFDB - Embedded newform 2016.2.s.r.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(289,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("2016.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.s (of order $3$, degree $2$, 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:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2016.289
Dual form			2016.2.s.r.865.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(1.73205 - 2.00000i) q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(1.73205 - 2.00000i) q^{7} +(2.59808 + 4.50000i) q^{11} +(2.50000 + 4.33013i) q^{17} +(-0.866025 + 1.50000i) q^{19} +(-0.866025 + 1.50000i) q^{23} +(2.00000 + 3.46410i) q^{25} -8.00000 q^{29} +(4.33013 + 7.50000i) q^{31} +(-0.866025 - 2.50000i) q^{35} +(2.50000 - 4.33013i) q^{37} -4.00000 q^{41} +6.92820 q^{43} +(4.33013 - 7.50000i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(-0.500000 - 0.866025i) q^{53} +5.19615 q^{55} +(0.866025 + 1.50000i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(6.06218 + 10.5000i) q^{67} +13.8564 q^{71} +(-7.50000 - 12.9904i) q^{73} +(13.5000 + 2.59808i) q^{77} +(0.866025 - 1.50000i) q^{79} -6.92820 q^{83} +5.00000 q^{85} +(3.50000 - 6.06218i) q^{89} +(0.866025 + 1.50000i) q^{95} +12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5}+O(q^{10}) $ 4 * q + 2 * q^5	$ 4 q + 2 q^{5} + 10 q^{17} + 8 q^{25} - 32 q^{29} + 10 q^{37} - 16 q^{41} - 4 q^{49} - 2 q^{53} - 22 q^{61} - 30 q^{73} + 54 q^{77} + 20 q^{85} + 14 q^{89} + 48 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 10 * q^17 + 8 * q^25 - 32 * q^29 + 10 * q^37 - 16 * q^41 - 4 * q^49 - 2 * q^53 - 22 * q^61 - 30 * q^73 + 54 * q^77 + 20 * q^85 + 14 * q^89 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	0.955901	+	0.293691i	$0.0948835\pi$
$6$		0			0
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.59808	+	4.50000i	0.783349	+	1.35680i	0.929980	+	0.367610i	$0.119824\pi$
$11$	2.59808	+	4.50000i	0.783349	+	1.35680i	−0.146631	+	0.989191i	$0.546843\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.50000	+	4.33013i	0.606339	+	1.05021i	0.991838	+	0.127502i	$0.0406959\pi$
$17$	2.50000	+	4.33013i	0.606339	+	1.05021i	−0.385499	+	0.922708i	$0.625971\pi$
$18$		0			0
$19$	−0.866025	+	1.50000i	−0.198680	+	0.344124i	−0.948101	−	0.317970i	$-0.896999\pi$
$19$	−0.866025	+	1.50000i	−0.198680	+	0.344124i	0.749421	+	0.662094i	$0.230332\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.866025	+	1.50000i	−0.180579	+	0.312772i	−0.942078	−	0.335394i	$-0.891130\pi$
$23$	−0.866025	+	1.50000i	−0.180579	+	0.312772i	0.761499	+	0.648166i	$0.224464\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$	4.33013	+	7.50000i	0.777714	+	1.34704i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.33013	+	7.50000i	0.777714	+	1.34704i	−0.155543	+	0.987829i	$0.549713\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.866025	−	2.50000i	−0.146385	−	0.422577i
$36$		0			0
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	−0.584002	−	0.811752i	$-0.698514\pi$
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	0.994999	+	0.0998840i	$0.0318472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		0			0
$43$	6.92820			1.05654			0.528271	−	0.849076i	$-0.322841\pi$
$43$	6.92820			1.05654			0.528271	+	0.849076i	$0.322841\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.33013	−	7.50000i	0.631614	−	1.09399i	−0.355608	−	0.934635i	$-0.615726\pi$
$47$	4.33013	−	7.50000i	0.631614	−	1.09399i	0.987222	−	0.159352i	$-0.0509405\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.500000	−	0.866025i	−0.0686803	−	0.118958i	0.829640	−	0.558298i	$-0.188546\pi$
$53$	−0.500000	−	0.866025i	−0.0686803	−	0.118958i	−0.898321	+	0.439340i	$0.855212\pi$
$54$		0			0
$55$	5.19615			0.700649
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.866025	+	1.50000i	0.112747	+	0.195283i	0.916877	−	0.399170i	$-0.130702\pi$
$59$	0.866025	+	1.50000i	0.112747	+	0.195283i	−0.804130	+	0.594454i	$0.797368\pi$
$60$		0			0
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	0.262776	+	0.964857i	$0.415362\pi$
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	−0.966978	+	0.254858i	$0.917971\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.06218	+	10.5000i	0.740613	+	1.28278i	0.952217	+	0.305424i	$0.0987981\pi$
$67$	6.06218	+	10.5000i	0.740613	+	1.28278i	−0.211604	+	0.977356i	$0.567869\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	13.8564			1.64445			0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564			1.64445			0.822226	+	0.569160i	$0.192732\pi$
$72$		0			0
$73$	−7.50000	−	12.9904i	−0.877809	−	1.52041i	−0.853740	−	0.520699i	$-0.825671\pi$
$73$	−7.50000	−	12.9904i	−0.877809	−	1.52041i	−0.0240681	−	0.999710i	$-0.507662\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.5000	+	2.59808i	1.53847	+	0.296078i
$78$		0			0
$79$	0.866025	−	1.50000i	0.0974355	−	0.168763i	−0.813187	−	0.582003i	$-0.802269\pi$
$79$	0.866025	−	1.50000i	0.0974355	−	0.168763i	0.910622	+	0.413239i	$0.135603\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.92820			−0.760469			−0.380235	−	0.924890i	$-0.624157\pi$
$83$	−6.92820			−0.760469			−0.380235	+	0.924890i	$0.624157\pi$
$84$		0			0
$85$	5.00000			0.542326
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.50000	−	6.06218i	0.370999	−	0.642590i	−0.618720	−	0.785611i	$-0.712349\pi$
$89$	3.50000	−	6.06218i	0.370999	−	0.642590i	0.989720	+	0.143022i	$0.0456819\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.866025	+	1.50000i	0.0888523	+	0.153897i
$96$		0			0
$97$	12.0000			1.21842			0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000			1.21842			0.609208	+	0.793011i	$0.291488\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.50000	−	9.52628i	−0.547270	−	0.947900i	−0.998460	−	0.0554722i	$-0.982334\pi$
$101$	−5.50000	−	9.52628i	−0.547270	−	0.947900i	0.451190	−	0.892428i	$-0.351000\pi$
$102$		0			0
$103$	7.79423	−	13.5000i	0.767988	−	1.33019i	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	7.79423	−	13.5000i	0.767988	−	1.33019i	0.938652	−	0.344865i	$-0.112075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.06218	+	10.5000i	−0.586053	+	1.01507i	0.408690	+	0.912673i	$0.365986\pi$
$107$	−6.06218	+	10.5000i	−0.586053	+	1.01507i	−0.994743	+	0.102400i	$0.967348\pi$
$108$		0			0
$109$	1.50000	+	2.59808i	0.143674	+	0.248851i	0.928877	−	0.370387i	$-0.120775\pi$
$109$	1.50000	+	2.59808i	0.143674	+	0.248851i	−0.785203	+	0.619238i	$0.787442\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.00000			−0.376288			−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000			−0.376288			−0.188144	+	0.982141i	$0.560247\pi$
$114$		0			0
$115$	0.866025	+	1.50000i	0.0807573	+	0.139876i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.9904	+	2.50000i	1.19083	+	0.229175i
$120$		0			0
$121$	−8.00000	+	13.8564i	−0.727273	+	1.25967i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.33013	+	7.50000i	−0.378325	+	0.655278i	−0.990819	−	0.135197i	$-0.956833\pi$
$131$	−4.33013	+	7.50000i	−0.378325	+	0.655278i	0.612494	+	0.790475i	$0.290166\pi$
$132$		0			0
$133$	1.50000	+	4.33013i	0.130066	+	0.375470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.50000	+	4.33013i	0.213589	+	0.369948i	0.952835	−	0.303488i	$-0.0981512\pi$
$137$	2.50000	+	4.33013i	0.213589	+	0.369948i	−0.739246	+	0.673436i	$0.764818\pi$
$138$		0			0
$139$	6.92820			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−4.00000	+	6.92820i	−0.332182	+	0.575356i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.50000	+	6.06218i	−0.286731	+	0.496633i	−0.973028	−	0.230689i	$-0.925902\pi$
$149$	−3.50000	+	6.06218i	−0.286731	+	0.496633i	0.686296	+	0.727322i	$0.259235\pi$
$150$		0			0
$151$	4.33013	+	7.50000i	0.352381	+	0.610341i	0.986666	−	0.162758i	$-0.0520389\pi$
$151$	4.33013	+	7.50000i	0.352381	+	0.610341i	−0.634285	+	0.773099i	$0.718706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.66025			0.695608
$156$		0			0
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	0.691888	−	0.722005i	$-0.256779\pi$
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	−0.971219	+	0.238190i	$0.923446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.50000	+	4.33013i	0.118217	+	0.341262i
$162$		0			0
$163$	4.33013	−	7.50000i	0.339162	−	0.587445i	−0.645114	−	0.764087i	$-0.723190\pi$
$163$	4.33013	−	7.50000i	0.339162	−	0.587445i	0.984275	+	0.176641i	$0.0565233\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.46410			0.268060			0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410			0.268060			0.134030	+	0.990977i	$0.457208\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.50000	+	4.33013i	−0.190071	+	0.329213i	−0.945274	−	0.326278i	$-0.894205\pi$
$173$	−2.50000	+	4.33013i	−0.190071	+	0.329213i	0.755202	+	0.655492i	$0.227539\pi$
$174$		0			0
$175$	10.3923	+	2.00000i	0.785584	+	0.151186i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.33013	−	7.50000i	−0.323649	−	0.560576i	0.657589	−	0.753377i	$-0.271576\pi$
$179$	−4.33013	−	7.50000i	−0.323649	−	0.560576i	−0.981238	+	0.192800i	$0.938243\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.50000	−	4.33013i	−0.183804	−	0.318357i
$186$		0			0
$187$	−12.9904	+	22.5000i	−0.949951	+	1.64536i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.2583	−	19.5000i	0.814624	−	1.41097i	−0.0949733	−	0.995480i	$-0.530277\pi$
$191$	11.2583	−	19.5000i	0.814624	−	1.41097i	0.909597	−	0.415491i	$-0.136390\pi$
$192$		0			0
$193$	5.50000	+	9.52628i	0.395899	+	0.685717i	0.993215	−	0.116289i	$-0.0370998\pi$
$193$	5.50000	+	9.52628i	0.395899	+	0.685717i	−0.597317	+	0.802005i	$0.703766\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.00000			0.569976			0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000			0.569976			0.284988	+	0.958531i	$0.408010\pi$
$198$		0			0
$199$	4.33013	+	7.50000i	0.306955	+	0.531661i	0.977695	−	0.210032i	$-0.0673567\pi$
$199$	4.33013	+	7.50000i	0.306955	+	0.531661i	−0.670740	+	0.741693i	$0.734023\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.8564	+	16.0000i	−0.972529	+	1.12298i
$204$		0			0
$205$	−2.00000	+	3.46410i	−0.139686	+	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.00000			−0.622543
$210$		0			0
$211$	−20.7846			−1.43087			−0.715436	−	0.698679i	$-0.753772\pi$
$211$	−20.7846			−1.43087			−0.715436	+	0.698679i	$0.753772\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.46410	−	6.00000i	0.236250	−	0.409197i
$216$		0			0
$217$	22.5000	+	4.33013i	1.52740	+	0.293948i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	13.8564			0.927894			0.463947	−	0.885863i	$-0.346433\pi$
$223$	13.8564			0.927894			0.463947	+	0.885863i	$0.346433\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.33013	−	7.50000i	−0.287401	−	0.497792i	0.685788	−	0.727801i	$-0.259458\pi$
$227$	−4.33013	−	7.50000i	−0.287401	−	0.497792i	−0.973189	+	0.230009i	$0.926124\pi$
$228$		0			0
$229$	7.50000	−	12.9904i	0.495614	−	0.858429i	−0.504373	−	0.863486i	$-0.668276\pi$
$229$	7.50000	−	12.9904i	0.495614	−	0.858429i	0.999987	+	0.00505719i	$0.00160976\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.50000	−	4.33013i	0.163780	−	0.283676i	−0.772441	−	0.635086i	$-0.780964\pi$
$233$	2.50000	−	4.33013i	0.163780	−	0.283676i	0.936222	+	0.351410i	$0.114298\pi$
$234$		0			0
$235$	−4.33013	−	7.50000i	−0.282466	−	0.489246i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−17.3205			−1.12037			−0.560185	−	0.828367i	$-0.689270\pi$
$239$	−17.3205			−1.12037			−0.560185	+	0.828367i	$0.689270\pi$
$240$		0			0
$241$	−7.50000	−	12.9904i	−0.483117	−	0.836784i	0.516695	−	0.856170i	$-0.327162\pi$
$241$	−7.50000	−	12.9904i	−0.483117	−	0.836784i	−0.999812	+	0.0193858i	$0.993829\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.50000	−	2.59808i	−0.415270	−	0.165985i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.3205			1.09326			0.546630	−	0.837374i	$-0.315910\pi$
$251$	17.3205			1.09326			0.546630	+	0.837374i	$0.315910\pi$
$252$		0			0
$253$	−9.00000			−0.565825
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.50000	−	4.33013i	0.155946	−	0.270106i	−0.777457	−	0.628936i	$-0.783491\pi$
$257$	2.50000	−	4.33013i	0.155946	−	0.270106i	0.933403	+	0.358830i	$0.116824\pi$
$258$		0			0
$259$	−4.33013	−	12.5000i	−0.269061	−	0.776712i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.866025	+	1.50000i	0.0534014	+	0.0924940i	0.891490	−	0.453040i	$-0.149660\pi$
$263$	0.866025	+	1.50000i	0.0534014	+	0.0924940i	−0.838089	+	0.545534i	$0.816327\pi$
$264$		0			0
$265$	−1.00000			−0.0614295
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.50000	−	14.7224i	−0.518254	−	0.897643i	−0.999775	−	0.0212079i	$-0.993249\pi$
$269$	−8.50000	−	14.7224i	−0.518254	−	0.897643i	0.481521	−	0.876435i	$-0.340085\pi$
$270$		0			0
$271$	−6.06218	+	10.5000i	−0.368251	+	0.637830i	−0.989292	−	0.145948i	$-0.953377\pi$
$271$	−6.06218	+	10.5000i	−0.368251	+	0.637830i	0.621041	+	0.783778i	$0.286710\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.3923	+	18.0000i	−0.626680	+	1.08544i
$276$		0			0
$277$	−7.50000	−	12.9904i	−0.450631	−	0.780516i	0.547794	−	0.836613i	$-0.315468\pi$
$277$	−7.50000	−	12.9904i	−0.450631	−	0.780516i	−0.998425	+	0.0560969i	$0.982134\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	20.0000			1.19310			0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000			1.19310			0.596550	+	0.802576i	$0.296538\pi$
$282$		0			0
$283$	4.33013	+	7.50000i	0.257399	+	0.445829i	0.965544	−	0.260238i	$-0.0838011\pi$
$283$	4.33013	+	7.50000i	0.257399	+	0.445829i	−0.708145	+	0.706067i	$0.750468\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.92820	+	8.00000i	−0.408959	+	0.472225i
$288$		0			0
$289$	−4.00000	+	6.92820i	−0.235294	+	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.0000			0.584206			0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000			0.584206			0.292103	+	0.956387i	$0.405645\pi$
$294$		0			0
$295$	1.73205			0.100844
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	12.0000	−	13.8564i	0.691669	−	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.50000	+	9.52628i	0.314929	+	0.545473i
$306$		0			0
$307$	−20.7846			−1.18624			−0.593120	−	0.805114i	$-0.702104\pi$
$307$	−20.7846			−1.18624			−0.593120	+	0.805114i	$0.702104\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.06218	−	10.5000i	−0.343755	−	0.595400i	0.641372	−	0.767230i	$-0.278365\pi$
$311$	−6.06218	−	10.5000i	−0.343755	−	0.595400i	−0.985127	+	0.171830i	$0.945032\pi$
$312$		0			0
$313$	12.5000	−	21.6506i	0.706542	−	1.22377i	−0.259590	−	0.965719i	$-0.583588\pi$
$313$	12.5000	−	21.6506i	0.706542	−	1.22377i	0.966132	−	0.258047i	$-0.0830791\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.50000	+	4.33013i	−0.140414	+	0.243204i	−0.927653	−	0.373444i	$-0.878177\pi$
$317$	−2.50000	+	4.33013i	−0.140414	+	0.243204i	0.787239	+	0.616649i	$0.211510\pi$
$318$		0			0
$319$	−20.7846	−	36.0000i	−1.16371	−	2.01561i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.66025			−0.481869
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.50000	−	21.6506i	−0.413488	−	1.19364i
$330$		0			0
$331$	4.33013	−	7.50000i	0.238005	−	0.412237i	−0.722137	−	0.691751i	$-0.756840\pi$
$331$	4.33013	−	7.50000i	0.238005	−	0.412237i	0.960142	+	0.279513i	$0.0901731\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.1244			0.662424
$336$		0			0
$337$	−12.0000			−0.653682			−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000			−0.653682			−0.326841	+	0.945079i	$0.605984\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−22.5000	+	38.9711i	−1.21844	+	2.11041i
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.06218	−	10.5000i	−0.325435	−	0.563670i	0.656165	−	0.754617i	$-0.272177\pi$
$347$	−6.06218	−	10.5000i	−0.325435	−	0.563670i	−0.981600	+	0.190947i	$0.938844\pi$
$348$		0			0
$349$	32.0000			1.71292			0.856460	−	0.516213i	$-0.172659\pi$
$349$	32.0000			1.71292			0.856460	+	0.516213i	$0.172659\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.50000	−	9.52628i	−0.292735	−	0.507033i	0.681720	−	0.731613i	$-0.261232\pi$
$353$	−5.50000	−	9.52628i	−0.292735	−	0.507033i	−0.974456	+	0.224580i	$0.927899\pi$
$354$		0			0
$355$	6.92820	−	12.0000i	0.367711	−	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.33013	−	7.50000i	0.228535	−	0.395835i	−0.728839	−	0.684685i	$-0.759940\pi$
$359$	4.33013	−	7.50000i	0.228535	−	0.395835i	0.957374	+	0.288850i	$0.0932730\pi$
$360$		0			0
$361$	8.00000	+	13.8564i	0.421053	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.0000			−0.785136
$366$		0			0
$367$	−14.7224	−	25.5000i	−0.768505	−	1.33109i	−0.938374	−	0.345623i	$-0.887668\pi$
$367$	−14.7224	−	25.5000i	−0.768505	−	1.33109i	0.169869	−	0.985467i	$-0.445666\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.59808	−	0.500000i	−0.134885	−	0.0259587i
$372$		0			0
$373$	−17.5000	+	30.3109i	−0.906116	+	1.56944i	−0.0867031	+	0.996234i	$0.527633\pi$
$373$	−17.5000	+	30.3109i	−0.906116	+	1.56944i	−0.819413	+	0.573204i	$0.805700\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−10.3923			−0.533817			−0.266908	−	0.963722i	$-0.586002\pi$
$379$	−10.3923			−0.533817			−0.266908	+	0.963722i	$0.586002\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.52628	+	16.5000i	−0.486770	+	0.843111i	−0.999884	−	0.0152097i	$-0.995158\pi$
$383$	−9.52628	+	16.5000i	−0.486770	+	0.843111i	0.513114	+	0.858320i	$0.328492\pi$
$384$		0			0
$385$	9.00000	−	10.3923i	0.458682	−	0.529641i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.50000	+	4.33013i	0.126755	+	0.219546i	0.922418	−	0.386194i	$-0.126210\pi$
$389$	2.50000	+	4.33013i	0.126755	+	0.219546i	−0.795663	+	0.605740i	$0.792877\pi$
$390$		0			0
$391$	−8.66025			−0.437968
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.866025	−	1.50000i	−0.0435745	−	0.0754732i
$396$		0			0
$397$	−8.50000	+	14.7224i	−0.426603	+	0.738898i	−0.996569	−	0.0827707i	$-0.973623\pi$
$397$	−8.50000	+	14.7224i	−0.426603	+	0.738898i	0.569966	+	0.821668i	$0.306956\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.50000	+	16.4545i	−0.474407	+	0.821698i	−0.999571	−	0.0293039i	$-0.990671\pi$
$401$	−9.50000	+	16.4545i	−0.474407	+	0.821698i	0.525163	+	0.851002i	$0.324004\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	25.9808			1.28782
$408$		0			0
$409$	1.50000	+	2.59808i	0.0741702	+	0.128467i	0.900725	−	0.434389i	$-0.143036\pi$
$409$	1.50000	+	2.59808i	0.0741702	+	0.128467i	−0.826555	+	0.562856i	$0.809703\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.50000	+	0.866025i	0.221431	+	0.0426143i
$414$		0			0
$415$	−3.46410	+	6.00000i	−0.170046	+	0.294528i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−34.6410			−1.69232			−0.846162	−	0.532925i	$-0.821093\pi$
$419$	−34.6410			−1.69232			−0.846162	+	0.532925i	$0.821093\pi$
$420$		0			0
$421$	−24.0000			−1.16969			−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000			−1.16969			−0.584844	+	0.811146i	$0.698844\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.0000	+	17.3205i	−0.485071	+	0.840168i
$426$		0			0
$427$	9.52628	+	27.5000i	0.461009	+	1.33082i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.2583	−	19.5000i	−0.542295	−	0.939282i	−0.998772	−	0.0495468i	$-0.984222\pi$
$431$	−11.2583	−	19.5000i	−0.542295	−	0.939282i	0.456477	−	0.889735i	$-0.349111\pi$
$432$		0			0
$433$	4.00000			0.192228			0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000			0.192228			0.0961139	+	0.995370i	$0.469359\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.50000	−	2.59808i	−0.0717547	−	0.124283i
$438$		0			0
$439$	−4.33013	+	7.50000i	−0.206666	+	0.357955i	−0.950662	−	0.310228i	$-0.899595\pi$
$439$	−4.33013	+	7.50000i	−0.206666	+	0.357955i	0.743996	+	0.668184i	$0.232928\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.79423	−	13.5000i	0.370315	−	0.641404i	−0.619299	−	0.785155i	$-0.712583\pi$
$443$	7.79423	−	13.5000i	0.370315	−	0.641404i	0.989614	+	0.143751i	$0.0459164\pi$
$444$		0			0
$445$	−3.50000	−	6.06218i	−0.165916	−	0.287375i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−28.0000			−1.32140			−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000			−1.32140			−0.660701	+	0.750649i	$0.729741\pi$
$450$		0			0
$451$	−10.3923	−	18.0000i	−0.489355	−	0.847587i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.50000	+	12.9904i	−0.350835	+	0.607664i	−0.986396	−	0.164386i	$-0.947436\pi$
$457$	−7.50000	+	12.9904i	−0.350835	+	0.607664i	0.635561	+	0.772051i	$0.280769\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	40.0000			1.86299			0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000			1.86299			0.931493	+	0.363760i	$0.118507\pi$
$462$		0			0
$463$	−27.7128			−1.28792			−0.643962	−	0.765058i	$-0.722710\pi$
$463$	−27.7128			−1.28792			−0.643962	+	0.765058i	$0.722710\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.59808	−	4.50000i	0.120225	−	0.208235i	−0.799632	−	0.600491i	$-0.794972\pi$
$467$	2.59808	−	4.50000i	0.120225	−	0.208235i	0.919856	+	0.392256i	$0.128305\pi$
$468$		0			0
$469$	31.5000	+	6.06218i	1.45453	+	0.279925i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.0000	+	31.1769i	0.827641	+	1.43352i
$474$		0			0
$475$	−6.92820			−0.317888
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.9904	−	22.5000i	−0.593546	−	1.02805i	−0.993750	−	0.111625i	$-0.964394\pi$
$479$	−12.9904	−	22.5000i	−0.593546	−	1.02805i	0.400205	−	0.916426i	$-0.368939\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.00000	−	10.3923i	0.272446	−	0.471890i
$486$		0			0
$487$	12.9904	+	22.5000i	0.588650	+	1.01957i	0.994410	+	0.105592i	$0.0336738\pi$
$487$	12.9904	+	22.5000i	0.588650	+	1.01957i	−0.405759	+	0.913980i	$0.632993\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.3205			−0.781664			−0.390832	−	0.920462i	$-0.627813\pi$
$491$	−17.3205			−0.781664			−0.390832	+	0.920462i	$0.627813\pi$
$492$		0			0
$493$	−20.0000	−	34.6410i	−0.900755	−	1.56015i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.0000	−	27.7128i	1.07655	−	1.24309i
$498$		0			0
$499$	−9.52628	+	16.5000i	−0.426455	+	0.738641i	−0.996555	−	0.0829337i	$-0.973571\pi$
$499$	−9.52628	+	16.5000i	−0.426455	+	0.738641i	0.570100	+	0.821575i	$0.306904\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−27.7128			−1.23565			−0.617827	−	0.786314i	$-0.711987\pi$
$503$	−27.7128			−1.23565			−0.617827	+	0.786314i	$0.711987\pi$
$504$		0			0
$505$	−11.0000			−0.489494
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.50000	−	14.7224i	0.376756	−	0.652560i	−0.613832	−	0.789436i	$-0.710373\pi$
$509$	8.50000	−	14.7224i	0.376756	−	0.652560i	0.990588	+	0.136876i	$0.0437062\pi$
$510$		0			0
$511$	−38.9711	−	7.50000i	−1.72398	−	0.331780i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.79423	−	13.5000i	−0.343455	−	0.594881i
$516$		0			0
$517$	45.0000			1.97910
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.5000	−	21.6506i	−0.547635	−	0.948532i	−0.998436	−	0.0559071i	$-0.982195\pi$
$521$	−12.5000	−	21.6506i	−0.547635	−	0.948532i	0.450801	−	0.892624i	$-0.351138\pi$
$522$		0			0
$523$	12.9904	−	22.5000i	0.568030	−	0.983856i	−0.428731	−	0.903432i	$-0.641039\pi$
$523$	12.9904	−	22.5000i	0.568030	−	0.983856i	0.996761	−	0.0804241i	$-0.0256275\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.6506	+	37.5000i	−0.943116	+	1.63353i
$528$		0			0
$529$	10.0000	+	17.3205i	0.434783	+	0.753066i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	6.06218	+	10.5000i	0.262091	+	0.453955i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	28.5788	−	22.5000i	1.23098	−	0.969144i
$540$		0			0
$541$	−4.50000	+	7.79423i	−0.193470	+	0.335100i	−0.946398	−	0.323003i	$-0.895308\pi$
$541$	−4.50000	+	7.79423i	−0.193470	+	0.335100i	0.752928	+	0.658103i	$0.228641\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.00000			0.128506
$546$		0			0
$547$	−3.46410			−0.148114			−0.0740571	−	0.997254i	$-0.523595\pi$
$547$	−3.46410			−0.148114			−0.0740571	+	0.997254i	$0.523595\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.92820	−	12.0000i	0.295151	−	0.511217i
$552$		0			0
$553$	−1.50000	−	4.33013i	−0.0637865	−	0.184136i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.5000	−	37.2391i	−0.910984	−	1.57787i	−0.812677	−	0.582715i	$-0.801990\pi$
$557$	−21.5000	−	37.2391i	−0.910984	−	1.57787i	−0.0983076	−	0.995156i	$-0.531343\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.79423	+	13.5000i	0.328488	+	0.568957i	0.982212	−	0.187776i	$-0.0601278\pi$
$563$	7.79423	+	13.5000i	0.328488	+	0.568957i	−0.653724	+	0.756733i	$0.726794\pi$
$564$		0			0
$565$	−2.00000	+	3.46410i	−0.0841406	+	0.145736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.5000	+	21.6506i	−0.524027	+	0.907642i	0.475581	+	0.879672i	$0.342238\pi$
$569$	−12.5000	+	21.6506i	−0.524027	+	0.907642i	−0.999609	+	0.0279702i	$0.991096\pi$
$570$		0			0
$571$	−2.59808	−	4.50000i	−0.108726	−	0.188319i	0.806528	−	0.591195i	$-0.201344\pi$
$571$	−2.59808	−	4.50000i	−0.108726	−	0.188319i	−0.915254	+	0.402876i	$0.868010\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.92820			−0.288926
$576$		0			0
$577$	−11.5000	−	19.9186i	−0.478751	−	0.829222i	0.520952	−	0.853586i	$-0.325577\pi$
$577$	−11.5000	−	19.9186i	−0.478751	−	0.829222i	−0.999703	+	0.0243645i	$0.992244\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000	+	13.8564i	−0.497844	+	0.574861i
$582$		0			0
$583$	2.59808	−	4.50000i	0.107601	−	0.186371i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	34.6410			1.42979			0.714894	−	0.699233i	$-0.246475\pi$
$587$	34.6410			1.42979			0.714894	+	0.699233i	$0.246475\pi$
$588$		0			0
$589$	−15.0000			−0.618064
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.5000	−	25.1147i	0.595444	−	1.03134i	−0.398040	−	0.917368i	$-0.630310\pi$
$593$	14.5000	−	25.1147i	0.595444	−	1.03134i	0.993484	−	0.113971i	$-0.0363570\pi$
$594$		0			0
$595$	8.66025	−	10.0000i	0.355036	−	0.409960i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.6506	+	37.5000i	0.884621	+	1.53221i	0.846147	+	0.532949i	$0.178916\pi$
$599$	21.6506	+	37.5000i	0.884621	+	1.53221i	0.0384735	+	0.999260i	$0.487750\pi$
$600$		0			0
$601$	−20.0000			−0.815817			−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000			−0.815817			−0.407909	+	0.913023i	$0.633742\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.00000	+	13.8564i	0.325246	+	0.563343i
$606$		0			0
$607$	2.59808	−	4.50000i	0.105453	−	0.182649i	−0.808470	−	0.588537i	$-0.799704\pi$
$607$	2.59808	−	4.50000i	0.105453	−	0.182649i	0.913923	+	0.405887i	$0.133038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−2.50000	−	4.33013i	−0.100974	−	0.174892i	0.811112	−	0.584891i	$-0.198863\pi$
$613$	−2.50000	−	4.33013i	−0.100974	−	0.174892i	−0.912086	+	0.409998i	$0.865529\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	20.0000			0.805170			0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000			0.805170			0.402585	+	0.915383i	$0.368112\pi$
$618$		0			0
$619$	−7.79423	−	13.5000i	−0.313276	−	0.542611i	0.665793	−	0.746136i	$-0.268093\pi$
$619$	−7.79423	−	13.5000i	−0.313276	−	0.542611i	−0.979070	+	0.203526i	$0.934760\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.06218	−	17.5000i	−0.242876	−	0.701123i
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	25.0000			0.996815
$630$		0			0
$631$	13.8564			0.551615			0.275807	−	0.961213i	$-0.411055\pi$
$631$	13.8564			0.551615			0.275807	+	0.961213i	$0.411055\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.500000	−	0.866025i	−0.0197488	−	0.0342059i	0.855982	−	0.517005i	$-0.172953\pi$
$641$	−0.500000	−	0.866025i	−0.0197488	−	0.0342059i	−0.875731	+	0.482800i	$0.839620\pi$
$642$		0			0
$643$	−34.6410			−1.36611			−0.683054	−	0.730368i	$-0.739349\pi$
$643$	−34.6410			−1.36611			−0.683054	+	0.730368i	$0.739349\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.7224	+	25.5000i	0.578799	+	1.00251i	0.995618	+	0.0935189i	$0.0298116\pi$
$647$	14.7224	+	25.5000i	0.578799	+	1.00251i	−0.416819	+	0.908990i	$0.636855\pi$
$648$		0			0
$649$	−4.50000	+	7.79423i	−0.176640	+	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.50000	+	4.33013i	−0.0978326	+	0.169451i	−0.910787	−	0.412876i	$-0.864524\pi$
$653$	−2.50000	+	4.33013i	−0.0978326	+	0.169451i	0.812955	+	0.582327i	$0.197858\pi$
$654$		0			0
$655$	4.33013	+	7.50000i	0.169192	+	0.293049i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.92820			0.269884			0.134942	−	0.990853i	$-0.456915\pi$
$659$	6.92820			0.269884			0.134942	+	0.990853i	$0.456915\pi$
$660$		0			0
$661$	−15.5000	−	26.8468i	−0.602880	−	1.04422i	−0.992383	−	0.123194i	$-0.960686\pi$
$661$	−15.5000	−	26.8468i	−0.602880	−	1.04422i	0.389503	−	0.921025i	$-0.372647\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.50000	+	0.866025i	0.174503	+	0.0335830i
$666$		0			0
$667$	6.92820	−	12.0000i	0.268261	−	0.464642i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−57.1577			−2.20655
$672$		0			0
$673$	20.0000			0.770943			0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000			0.770943			0.385472	+	0.922720i	$0.374039\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.5000	−	21.6506i	0.480414	−	0.832102i	−0.519333	−	0.854572i	$-0.673820\pi$
$677$	12.5000	−	21.6506i	0.480414	−	0.832102i	0.999748	+	0.0224702i	$0.00715308\pi$
$678$		0			0
$679$	20.7846	−	24.0000i	0.797640	−	0.921035i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.4545	+	28.5000i	0.629613	+	1.09052i	0.987629	+	0.156807i	$0.0501200\pi$
$683$	16.4545	+	28.5000i	0.629613	+	1.09052i	−0.358016	+	0.933715i	$0.616547\pi$
$684$		0			0
$685$	5.00000			0.191040
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−2.59808	+	4.50000i	−0.0988355	+	0.171188i	−0.911203	−	0.411958i	$-0.864845\pi$
$691$	−2.59808	+	4.50000i	−0.0988355	+	0.171188i	0.812367	+	0.583146i	$0.198178\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.46410	−	6.00000i	0.131401	−	0.227593i
$696$		0			0
$697$	−10.0000	−	17.3205i	−0.378777	−	0.656061i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−14.0000			−0.528773			−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000			−0.528773			−0.264386	+	0.964417i	$0.585169\pi$
$702$		0			0
$703$	4.33013	+	7.50000i	0.163314	+	0.282868i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.5788	−	5.50000i	−1.07482	−	0.206849i
$708$		0			0
$709$	−13.5000	+	23.3827i	−0.507003	+	0.878155i	0.492964	+	0.870050i	$0.335913\pi$
$709$	−13.5000	+	23.3827i	−0.507003	+	0.878155i	−0.999967	+	0.00810550i	$0.997420\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15.0000			−0.561754
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.79423	+	13.5000i	−0.290676	+	0.503465i	−0.973970	−	0.226678i	$-0.927213\pi$
$719$	−7.79423	+	13.5000i	−0.290676	+	0.503465i	0.683294	+	0.730143i	$0.260547\pi$
$720$		0			0
$721$	−13.5000	−	38.9711i	−0.502766	−	1.45136i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16.0000	−	27.7128i	−0.594225	−	1.02923i
$726$		0			0
$727$	−13.8564			−0.513906			−0.256953	−	0.966424i	$-0.582719\pi$
$727$	−13.8564			−0.513906			−0.256953	+	0.966424i	$0.582719\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	17.3205	+	30.0000i	0.640622	+	1.10959i
$732$		0			0
$733$	−25.5000	+	44.1673i	−0.941864	+	1.63136i	−0.179952	+	0.983675i	$0.557594\pi$
$733$	−25.5000	+	44.1673i	−0.941864	+	1.63136i	−0.761912	+	0.647681i	$0.775739\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−31.5000	+	54.5596i	−1.16032	+	2.00973i
$738$		0			0
$739$	−9.52628	−	16.5000i	−0.350430	−	0.606962i	0.635895	−	0.771776i	$-0.280631\pi$
$739$	−9.52628	−	16.5000i	−0.350430	−	0.606962i	−0.986325	+	0.164813i	$0.947298\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−27.7128			−1.01668			−0.508342	−	0.861155i	$-0.669742\pi$
$743$	−27.7128			−1.01668			−0.508342	+	0.861155i	$0.669742\pi$
$744$		0			0
$745$	3.50000	+	6.06218i	0.128230	+	0.222101i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.5000	+	30.3109i	0.383662	+	1.10754i
$750$		0			0
$751$	−4.33013	+	7.50000i	−0.158009	+	0.273679i	−0.934150	−	0.356879i	$-0.883841\pi$
$751$	−4.33013	+	7.50000i	−0.158009	+	0.273679i	0.776142	+	0.630558i	$0.217174\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.66025			0.315179
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.50000	−	11.2583i	0.235625	−	0.408114i	−0.723829	−	0.689979i	$-0.757620\pi$
$761$	6.50000	−	11.2583i	0.235625	−	0.408114i	0.959454	+	0.281865i	$0.0909530\pi$
$762$		0			0
$763$	7.79423	+	1.50000i	0.282170	+	0.0543036i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	20.0000			0.721218			0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000			0.721218			0.360609	+	0.932717i	$0.382569\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.50000	+	4.33013i	0.0899188	+	0.155744i	0.907477	−	0.420103i	$-0.138006\pi$
$773$	2.50000	+	4.33013i	0.0899188	+	0.155744i	−0.817558	+	0.575846i	$0.804673\pi$
$774$		0			0
$775$	−17.3205	+	30.0000i	−0.622171	+	1.07763i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.46410	−	6.00000i	0.124114	−	0.214972i
$780$		0			0
$781$	36.0000	+	62.3538i	1.28818	+	2.23120i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.00000			−0.249841
$786$		0			0
$787$	19.9186	+	34.5000i	0.710021	+	1.22979i	0.964849	+	0.262806i	$0.0846478\pi$
$787$	19.9186	+	34.5000i	0.710021	+	1.22979i	−0.254828	+	0.966986i	$0.582019\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.92820	+	8.00000i	−0.246339	+	0.284447i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−40.0000			−1.41687			−0.708436	−	0.705775i	$-0.750599\pi$
$797$	−40.0000			−1.41687			−0.708436	+	0.705775i	$0.750599\pi$
$798$		0			0
$799$	43.3013			1.53189
$800$		0			0
$801$		0			0
$802$		0			0
$803$	38.9711	−	67.5000i	1.37526	−	2.38202i
$804$		0			0
$805$	4.50000	+	0.866025i	0.158604	+	0.0305234i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.50000	+	6.06218i	0.123053	+	0.213135i	0.920970	−	0.389633i	$-0.127398\pi$
$809$	3.50000	+	6.06218i	0.123053	+	0.213135i	−0.797917	+	0.602767i	$0.794065\pi$
$810$		0			0
$811$	34.6410			1.21641			0.608205	−	0.793780i	$-0.291890\pi$
$811$	34.6410			1.21641			0.608205	+	0.793780i	$0.291890\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.33013	−	7.50000i	−0.151678	−	0.262714i
$816$		0			0
$817$	−6.00000	+	10.3923i	−0.209913	+	0.363581i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.5000	−	21.6506i	0.436253	−	0.755612i	−0.561144	−	0.827718i	$-0.689639\pi$
$821$	12.5000	−	21.6506i	0.436253	−	0.755612i	0.997397	+	0.0721058i	$0.0229719\pi$
$822$		0			0
$823$	−7.79423	−	13.5000i	−0.271690	−	0.470580i	0.697605	−	0.716483i	$-0.254249\pi$
$823$	−7.79423	−	13.5000i	−0.271690	−	0.470580i	−0.969295	+	0.245902i	$0.920916\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.7846			0.722752			0.361376	−	0.932420i	$-0.382307\pi$
$827$	20.7846			0.722752			0.361376	+	0.932420i	$0.382307\pi$
$828$		0			0
$829$	−22.5000	−	38.9711i	−0.781457	−	1.35352i	−0.931093	−	0.364783i	$-0.881143\pi$
$829$	−22.5000	−	38.9711i	−0.781457	−	1.35352i	0.149635	−	0.988741i	$-0.452190\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	27.5000	−	21.6506i	0.952819	−	0.750150i
$834$		0			0
$835$	1.73205	−	3.00000i	0.0599401	−	0.103819i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	41.5692			1.43513			0.717564	−	0.696492i	$-0.245257\pi$
$839$	41.5692			1.43513			0.717564	+	0.696492i	$0.245257\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.50000	+	11.2583i	−0.223607	+	0.387298i
$846$		0			0
$847$	13.8564	+	40.0000i	0.476112	+	1.37442i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.33013	+	7.50000i	0.148435	+	0.257097i
$852$		0			0
$853$	16.0000			0.547830			0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000			0.547830			0.273915	+	0.961754i	$0.411681\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.50000	+	6.06218i	0.119558	+	0.207080i	0.919592	−	0.392874i	$-0.128519\pi$
$857$	3.50000	+	6.06218i	0.119558	+	0.207080i	−0.800035	+	0.599954i	$0.795186\pi$
$858$		0			0
$859$	12.9904	−	22.5000i	0.443226	−	0.767690i	−0.554701	−	0.832050i	$-0.687167\pi$
$859$	12.9904	−	22.5000i	0.443226	−	0.767690i	0.997927	+	0.0643599i	$0.0205006\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.8468	−	46.5000i	0.913875	−	1.58288i	0.105336	−	0.994437i	$-0.466408\pi$
$863$	26.8468	−	46.5000i	0.913875	−	1.58288i	0.808540	−	0.588442i	$-0.200258\pi$
$864$		0			0
$865$	2.50000	+	4.33013i	0.0850026	+	0.147229i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.00000			0.305304
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	15.5885	−	18.0000i	0.526986	−	0.608511i
$876$		0			0
$877$	3.50000	−	6.06218i	0.118187	−	0.204705i	−0.800862	−	0.598848i	$-0.795625\pi$
$877$	3.50000	−	6.06218i	0.118187	−	0.204705i	0.919049	+	0.394143i	$0.128959\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34.0000			1.14549			0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000			1.14549			0.572745	+	0.819734i	$0.305879\pi$
$882$		0			0
$883$	34.6410			1.16576			0.582882	−	0.812557i	$-0.301925\pi$
$883$	34.6410			1.16576			0.582882	+	0.812557i	$0.301925\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−23.3827	+	40.5000i	−0.785114	+	1.35986i	0.143817	+	0.989604i	$0.454062\pi$
$887$	−23.3827	+	40.5000i	−0.785114	+	1.35986i	−0.928931	+	0.370253i	$0.879271\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.50000	+	12.9904i	0.250978	+	0.434707i
$894$		0			0
$895$	−8.66025			−0.289480
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−34.6410	−	60.0000i	−1.15534	−	2.00111i
$900$		0			0
$901$	2.50000	−	4.33013i	0.0832871	−	0.144257i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.00000	+	5.19615i	−0.0997234	+	0.172726i
$906$		0			0
$907$	19.9186	+	34.5000i	0.661386	+	1.14555i	0.980252	+	0.197754i	$0.0633647\pi$
$907$	19.9186	+	34.5000i	0.661386	+	1.14555i	−0.318866	+	0.947800i	$0.603302\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−13.8564			−0.459083			−0.229542	−	0.973299i	$-0.573723\pi$
$911$	−13.8564			−0.459083			−0.229542	+	0.973299i	$0.573723\pi$
$912$		0			0
$913$	−18.0000	−	31.1769i	−0.595713	−	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.50000	+	21.6506i	0.247672	+	0.714967i
$918$		0			0
$919$	−18.1865	+	31.5000i	−0.599918	+	1.03909i	0.392914	+	0.919575i	$0.371467\pi$
$919$	−18.1865	+	31.5000i	−0.599918	+	1.03909i	−0.992833	+	0.119514i	$0.961866\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	20.0000			0.657596
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.5000	−	40.7032i	0.771010	−	1.33543i	−0.166000	−	0.986126i	$-0.553085\pi$
$929$	23.5000	−	40.7032i	0.771010	−	1.33543i	0.937010	−	0.349303i	$-0.113582\pi$
$930$		0			0
$931$	11.2583	+	4.50000i	0.368977	+	0.147482i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.9904	+	22.5000i	0.424831	+	0.735829i
$936$		0			0
$937$	10.0000			0.326686			0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000			0.326686			0.163343	+	0.986569i	$0.447772\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.5000	−	21.6506i	−0.407488	−	0.705791i	0.587119	−	0.809500i	$-0.300262\pi$
$941$	−12.5000	−	21.6506i	−0.407488	−	0.705791i	−0.994608	+	0.103710i	$0.966929\pi$
$942$		0			0
$943$	3.46410	−	6.00000i	0.112807	−	0.195387i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.59808	−	4.50000i	0.0844261	−	0.146230i	−0.820720	−	0.571330i	$-0.806428\pi$
$947$	2.59808	−	4.50000i	0.0844261	−	0.146230i	0.905147	+	0.425100i	$0.139761\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.0000			−0.647864			−0.323932	−	0.946080i	$-0.605005\pi$
$953$	−20.0000			−0.647864			−0.323932	+	0.946080i	$0.605005\pi$
$954$		0			0
$955$	−11.2583	−	19.5000i	−0.364311	−	0.631005i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.9904	+	2.50000i	0.419481	+	0.0807292i
$960$		0			0
$961$	−22.0000	+	38.1051i	−0.709677	+	1.22920i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.0000			0.354103
$966$		0			0
$967$	13.8564			0.445592			0.222796	−	0.974865i	$-0.428482\pi$
$967$	13.8564			0.445592			0.222796	+	0.974865i	$0.428482\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.33013	+	7.50000i	−0.138960	+	0.240686i	−0.927103	−	0.374806i	$-0.877709\pi$
$971$	−4.33013	+	7.50000i	−0.138960	+	0.240686i	0.788143	+	0.615492i	$0.211043\pi$
$972$		0			0
$973$	12.0000	−	13.8564i	0.384702	−	0.444216i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.5000	+	32.0429i	0.591867	+	1.02514i	0.993981	+	0.109555i	$0.0349424\pi$
$977$	18.5000	+	32.0429i	0.591867	+	1.02514i	−0.402113	+	0.915590i	$0.631724\pi$
$978$		0			0
$979$	36.3731			1.16249
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.4545	+	28.5000i	0.524816	+	0.909009i	0.999582	+	0.0288967i	$0.00919939\pi$
$983$	16.4545	+	28.5000i	0.524816	+	0.909009i	−0.474766	+	0.880112i	$0.657467\pi$
$984$		0			0
$985$	4.00000	−	6.92820i	0.127451	−	0.220751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6.00000	+	10.3923i	−0.190789	+	0.330456i
$990$		0			0
$991$	12.9904	+	22.5000i	0.412653	+	0.714736i	0.995179	−	0.0980761i	$-0.0312689\pi$
$991$	12.9904	+	22.5000i	0.412653	+	0.714736i	−0.582526	+	0.812812i	$0.697936\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.66025			0.274549
$996$		0			0
$997$	−11.5000	−	19.9186i	−0.364209	−	0.630828i	0.624440	−	0.781073i	$-0.285327\pi$
$997$	−11.5000	−	19.9186i	−0.364209	−	0.630828i	−0.988649	+	0.150245i	$0.951994\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.s.r.289.2		4
3.2	odd	2		224.2.i.b.65.1	✓	4
4.3	odd	2	inner	2016.2.s.r.289.1		4
7.4	even	3	inner	2016.2.s.r.865.2		4
12.11	even	2		224.2.i.b.65.2	yes	4
21.2	odd	6		1568.2.a.s.1.2		2
21.5	even	6		1568.2.a.n.1.1		2
21.11	odd	6		224.2.i.b.193.1	yes	4
21.17	even	6		1568.2.i.u.1537.2		4
21.20	even	2		1568.2.i.u.961.2		4
24.5	odd	2		448.2.i.i.65.2		4
24.11	even	2		448.2.i.i.65.1		4
28.11	odd	6	inner	2016.2.s.r.865.1		4
84.11	even	6		224.2.i.b.193.2	yes	4
84.23	even	6		1568.2.a.s.1.1		2
84.47	odd	6		1568.2.a.n.1.2		2
84.59	odd	6		1568.2.i.u.1537.1		4
84.83	odd	2		1568.2.i.u.961.1		4
168.5	even	6		3136.2.a.bu.1.2		2
168.11	even	6		448.2.i.i.193.1		4
168.53	odd	6		448.2.i.i.193.2		4
168.107	even	6		3136.2.a.bh.1.2		2
168.131	odd	6		3136.2.a.bu.1.1		2
168.149	odd	6		3136.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.i.b.65.1	✓	4	3.2	odd	2
224.2.i.b.65.2	yes	4	12.11	even	2
224.2.i.b.193.1	yes	4	21.11	odd	6
224.2.i.b.193.2	yes	4	84.11	even	6
448.2.i.i.65.1		4	24.11	even	2
448.2.i.i.65.2		4	24.5	odd	2
448.2.i.i.193.1		4	168.11	even	6
448.2.i.i.193.2		4	168.53	odd	6
1568.2.a.n.1.1		2	21.5	even	6
1568.2.a.n.1.2		2	84.47	odd	6
1568.2.a.s.1.1		2	84.23	even	6
1568.2.a.s.1.2		2	21.2	odd	6
1568.2.i.u.961.1		4	84.83	odd	2
1568.2.i.u.961.2		4	21.20	even	2
1568.2.i.u.1537.1		4	84.59	odd	6
1568.2.i.u.1537.2		4	21.17	even	6
2016.2.s.r.289.1		4	4.3	odd	2	inner
2016.2.s.r.289.2		4	1.1	even	1	trivial
2016.2.s.r.865.1		4	28.11	odd	6	inner
2016.2.s.r.865.2		4	7.4	even	3	inner
3136.2.a.bh.1.1		2	168.149	odd	6
3136.2.a.bh.1.2		2	168.107	even	6
3136.2.a.bu.1.1		2	168.131	odd	6
3136.2.a.bu.1.2		2	168.5	even	6

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.73205 - 2.00000i) q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(1.73205 - 2.00000i) q^{7} +(2.59808 + 4.50000i) q^{11} +(2.50000 + 4.33013i) q^{17} +(-0.866025 + 1.50000i) q^{19} +(-0.866025 + 1.50000i) q^{23} +(2.00000 + 3.46410i) q^{25} -8.00000 q^{29} +(4.33013 + 7.50000i) q^{31} +(-0.866025 - 2.50000i) q^{35} +(2.50000 - 4.33013i) q^{37} -4.00000 q^{41} +6.92820 q^{43} +(4.33013 - 7.50000i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(-0.500000 - 0.866025i) q^{53} +5.19615 q^{55} +(0.866025 + 1.50000i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(6.06218 + 10.5000i) q^{67} +13.8564 q^{71} +(-7.50000 - 12.9904i) q^{73} +(13.5000 + 2.59808i) q^{77} +(0.866025 - 1.50000i) q^{79} -6.92820 q^{83} +5.00000 q^{85} +(3.50000 - 6.06218i) q^{89} +(0.866025 + 1.50000i) q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5}+O(q^{10}) \) 4 * q + 2 * q^5	\( 4 q + 2 q^{5} + 10 q^{17} + 8 q^{25} - 32 q^{29} + 10 q^{37} - 16 q^{41} - 4 q^{49} - 2 q^{53} - 22 q^{61} - 30 q^{73} + 54 q^{77} + 20 q^{85} + 14 q^{89} + 48 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 10 * q^17 + 8 * q^25 - 32 * q^29 + 10 * q^37 - 16 * q^41 - 4 * q^49 - 2 * q^53 - 22 * q^61 - 30 * q^73 + 54 * q^77 + 20 * q^85 + 14 * q^89 + 48 * q^97

Introduction

L-functions

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