LMFDB - Embedded newform 864.2.r.a.145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(145,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("864.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$864 = 2^{5} \cdot 3^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	864.r (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	864.145
Dual form			864.2.r.a.721.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 - 1.00000i) q^{5} +(2.00000 + 3.46410i) q^{7} +(-2.59808 + 1.50000i) q^{11} +(-1.73205 - 1.00000i) q^{13} -5.00000 q^{17} -1.00000i q^{19} +(-1.00000 + 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{31} -8.00000i q^{35} +2.00000i q^{37} +(-2.50000 + 4.33013i) q^{41} +(-9.52628 + 5.50000i) q^{43} +(3.00000 + 5.19615i) q^{47} +(-4.50000 + 7.79423i) q^{49} +6.00000 q^{55} +(0.866025 + 0.500000i) q^{59} +(-10.3923 + 6.00000i) q^{61} +(2.00000 + 3.46410i) q^{65} +(2.59808 + 1.50000i) q^{67} -6.00000 q^{71} +9.00000 q^{73} +(-10.3923 - 6.00000i) q^{77} +(-7.00000 - 12.1244i) q^{79} +(-3.46410 + 2.00000i) q^{83} +(8.66025 + 5.00000i) q^{85} +14.0000 q^{89} -8.00000i q^{91} +(-1.00000 + 1.73205i) q^{95} +(-0.500000 - 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 8 q^{7} - 20 q^{17} - 4 q^{23} - 2 q^{25} - 8 q^{31} - 10 q^{41} + 12 q^{47} - 18 q^{49} + 24 q^{55} + 8 q^{65} - 24 q^{71} + 36 q^{73} - 28 q^{79} + 56 q^{89} - 4 q^{95} - 2 q^{97}+O(q^{100})$ 4 * q + 8 * q^7 - 20 * q^17 - 4 * q^23 - 2 * q^25 - 8 * q^31 - 10 * q^41 + 12 * q^47 - 18 * q^49 + 24 * q^55 + 8 * q^65 - 24 * q^71 + 36 * q^73 - 28 * q^79 + 56 * q^89 - 4 * q^95 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$ .

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−1.73205	−	1.00000i	−0.774597	−	0.447214i	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	−1.73205	−	1.00000i	−0.774597	−	0.447214i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	0.944911	+	0.327327i	$0.106148\pi$
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.59808	+	1.50000i	−0.783349	+	0.452267i	−0.837616	−	0.546259i	$-0.816051\pi$
$11$	−2.59808	+	1.50000i	−0.783349	+	0.452267i	0.0542666	+	0.998526i	$0.482718\pi$
$12$		0			0
$13$	−1.73205	−	1.00000i	−0.480384	−	0.277350i	0.240192	−	0.970725i	$-0.422790\pi$
$13$	−1.73205	−	1.00000i	−0.480384	−	0.277350i	−0.720577	+	0.693375i	$0.756123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.00000			−1.21268			−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000			−1.21268			−0.606339	+	0.795206i	$0.707363\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000	+	1.73205i	−0.208514	+	0.361158i	−0.951247	−	0.308431i	$-0.900196\pi$
$23$	−1.00000	+	1.73205i	−0.208514	+	0.361158i	0.742732	+	0.669588i	$0.233529\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	8.00000i		−	1.35225i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.50000	+	4.33013i	−0.390434	+	0.676252i	−0.992507	−	0.122189i	$-0.961009\pi$
$41$	−2.50000	+	4.33013i	−0.390434	+	0.676252i	0.602072	+	0.798441i	$0.294342\pi$
$42$		0			0
$43$	−9.52628	+	5.50000i	−1.45274	+	0.838742i	−0.998636	−	0.0522047i	$-0.983375\pi$
$43$	−9.52628	+	5.50000i	−1.45274	+	0.838742i	−0.454108	+	0.890947i	$0.650042\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.866025	+	0.500000i	0.112747	+	0.0650945i	0.555313	−	0.831641i	$-0.312598\pi$
$59$	0.866025	+	0.500000i	0.112747	+	0.0650945i	−0.442566	+	0.896736i	$0.645932\pi$
$60$		0			0
$61$	−10.3923	+	6.00000i	−1.33060	+	0.768221i	−0.985391	−	0.170305i	$-0.945525\pi$
$61$	−10.3923	+	6.00000i	−1.33060	+	0.768221i	−0.345207	+	0.938527i	$0.612191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	+	3.46410i	0.248069	+	0.429669i
$66$		0			0
$67$	2.59808	+	1.50000i	0.317406	+	0.183254i	0.650236	−	0.759733i	$-0.274670\pi$
$67$	2.59808	+	1.50000i	0.317406	+	0.183254i	−0.332830	+	0.942987i	$0.608004\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	9.00000			1.05337			0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000			1.05337			0.526685	+	0.850060i	$0.323435\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.3923	−	6.00000i	−1.18431	−	0.683763i
$78$		0			0
$79$	−7.00000	−	12.1244i	−0.787562	−	1.36410i	−0.927457	−	0.373930i	$-0.878010\pi$
$79$	−7.00000	−	12.1244i	−0.787562	−	1.36410i	0.139895	−	0.990166i	$-0.455323\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.46410	+	2.00000i	−0.380235	+	0.219529i	−0.677920	−	0.735135i	$-0.737119\pi$
$83$	−3.46410	+	2.00000i	−0.380235	+	0.219529i	0.297686	+	0.954664i	$0.403785\pi$
$84$		0			0
$85$	8.66025	+	5.00000i	0.939336	+	0.542326i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$		−	8.00000i		−	0.838628i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	−0.500000	−	0.866025i	−0.0507673	−	0.0879316i	0.839525	−	0.543321i	$-0.182833\pi$
$97$	−0.500000	−	0.866025i	−0.0507673	−	0.0879316i	−0.890292	+	0.455389i	$0.849500\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.1244	−	7.00000i	1.20642	−	0.696526i	0.244443	−	0.969664i	$-0.421395\pi$
$101$	12.1244	−	7.00000i	1.20642	−	0.696526i	0.961975	+	0.273138i	$0.0880614\pi$
$102$		0			0
$103$	−3.00000	+	5.19615i	−0.295599	+	0.511992i	−0.975124	−	0.221660i	$-0.928852\pi$
$103$	−3.00000	+	5.19615i	−0.295599	+	0.511992i	0.679525	+	0.733652i	$0.262186\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		0			0
$109$		−	20.0000i		−	1.91565i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$		−	20.0000i		−	1.91565i	0.287348	−	0.957826i	$-0.407226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	3.46410	−	2.00000i	0.323029	−	0.186501i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.0000	−	17.3205i	−0.916698	−	1.58777i
$120$		0			0
$121$	−1.00000	+	1.73205i	−0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.0000i			1.07331i
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.46410	+	2.00000i	0.302660	+	0.174741i	0.643637	−	0.765331i	$-0.277425\pi$
$131$	3.46410	+	2.00000i	0.302660	+	0.174741i	−0.340977	+	0.940072i	$0.610758\pi$
$132$		0			0
$133$	3.46410	−	2.00000i	0.300376	−	0.173422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	0.607233	−	0.794524i	$-0.292279\pi$
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	−0.991694	+	0.128618i	$0.958946\pi$
$138$		0			0
$139$	11.2583	+	6.50000i	0.954919	+	0.551323i	0.894606	−	0.446857i	$-0.147457\pi$
$139$	11.2583	+	6.50000i	0.954919	+	0.551323i	0.0603135	+	0.998179i	$0.480790\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000			0.501745
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.5885	+	9.00000i	1.27706	+	0.737309i	0.976306	−	0.216394i	$-0.0694297\pi$
$149$	15.5885	+	9.00000i	1.27706	+	0.737309i	0.300750	+	0.953703i	$0.402763\pi$
$150$		0			0
$151$	−3.00000	−	5.19615i	−0.244137	−	0.422857i	0.717752	−	0.696299i	$-0.245171\pi$
$151$	−3.00000	−	5.19615i	−0.244137	−	0.422857i	−0.961888	+	0.273442i	$0.911838\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.92820	−	4.00000i	0.556487	−	0.321288i
$156$		0			0
$157$	3.46410	+	2.00000i	0.276465	+	0.159617i	0.631822	−	0.775113i	$-0.282307\pi$
$157$	3.46410	+	2.00000i	0.276465	+	0.159617i	−0.355357	+	0.934731i	$0.615641\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.00000			−0.630488
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000	−	1.73205i	0.0773823	−	0.134030i	−0.824737	−	0.565516i	$-0.808677\pi$
$167$	1.00000	−	1.73205i	0.0773823	−	0.134030i	0.902120	+	0.431486i	$0.142010\pi$
$168$		0			0
$169$	−4.50000	−	7.79423i	−0.346154	−	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−20.7846	+	12.0000i	−1.58022	+	0.912343i	−0.585399	+	0.810745i	$0.699062\pi$
$173$	−20.7846	+	12.0000i	−1.58022	+	0.912343i	−0.994826	+	0.101598i	$0.967605\pi$
$174$		0			0
$175$	2.00000	−	3.46410i	0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			20.0000i			1.49487i	0.664335	+	0.747435i	$0.268715\pi$
$179$			20.0000i			1.49487i	−0.664335	+	0.747435i	$0.731285\pi$
$180$		0			0
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.00000	−	3.46410i	0.147043	−	0.254686i
$186$		0			0
$187$	12.9904	−	7.50000i	0.949951	−	0.548454i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000	+	13.8564i	0.578860	+	1.00261i	0.995610	+	0.0935936i	$0.0298354\pi$
$191$	8.00000	+	13.8564i	0.578860	+	1.00261i	−0.416751	+	0.909021i	$0.636831\pi$
$192$		0			0
$193$	7.50000	−	12.9904i	0.539862	−	0.935068i	−0.459049	−	0.888411i	$-0.651810\pi$
$193$	7.50000	−	12.9904i	0.539862	−	0.935068i	0.998911	−	0.0466572i	$-0.0148568\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	8.66025	−	5.00000i	0.604858	−	0.349215i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.50000	+	2.59808i	0.103757	+	0.179713i
$210$		0			0
$211$	−13.8564	−	8.00000i	−0.953914	−	0.550743i	−0.0596196	−	0.998221i	$-0.518989\pi$
$211$	−13.8564	−	8.00000i	−0.953914	−	0.550743i	−0.894295	+	0.447478i	$0.852322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	22.0000			1.50039
$216$		0			0
$217$	−16.0000			−1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.66025	+	5.00000i	0.582552	+	0.336336i
$222$		0			0
$223$	1.00000	+	1.73205i	0.0669650	+	0.115987i	0.897564	−	0.440884i	$-0.145335\pi$
$223$	1.00000	+	1.73205i	0.0669650	+	0.115987i	−0.830599	+	0.556871i	$0.812002\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.06218	−	3.50000i	0.402361	−	0.232303i	−0.285141	−	0.958485i	$-0.592041\pi$
$227$	6.06218	−	3.50000i	0.402361	−	0.232303i	0.687502	+	0.726182i	$0.258707\pi$
$228$		0			0
$229$	17.3205	+	10.0000i	1.14457	+	0.660819i	0.947559	−	0.319582i	$-0.103543\pi$
$229$	17.3205	+	10.0000i	1.14457	+	0.660819i	0.197013	+	0.980401i	$0.436876\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.0000			−0.851658			−0.425829	−	0.904804i	$-0.640018\pi$
$233$	−13.0000			−0.851658			−0.425829	+	0.904804i	$0.640018\pi$
$234$		0			0
$235$		−	12.0000i		−	0.782794i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	15.0000	−	25.9808i	0.970269	−	1.68056i	0.275533	−	0.961292i	$-0.411146\pi$
$239$	15.0000	−	25.9808i	0.970269	−	1.68056i	0.694737	−	0.719264i	$-0.255521\pi$
$240$		0			0
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	0.450910	−	0.892570i	$-0.351100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.5885	−	9.00000i	0.995910	−	0.574989i
$246$		0			0
$247$	−1.00000	+	1.73205i	−0.0636285	+	0.110208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.0000i		−	0.946792i	−0.880850	−	0.473396i	$-0.843028\pi$
$251$		−	15.0000i		−	0.946792i	0.880850	−	0.473396i	$-0.156972\pi$
$252$		0			0
$253$		−	6.00000i		−	0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	−0.531487	−	0.847066i	$-0.678367\pi$
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	0.999325	+	0.0367485i	$0.0117000\pi$
$258$		0			0
$259$	−6.92820	+	4.00000i	−0.430498	+	0.248548i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	0.962594	−	0.270947i	$-0.0873367\pi$
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	−0.715944	+	0.698158i	$0.754003\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	12.0000i		−	0.731653i	−0.930683	−	0.365826i	$-0.880786\pi$
$269$		−	12.0000i		−	0.731653i	0.930683	−	0.365826i	$-0.119214\pi$
$270$		0			0
$271$	−12.0000			−0.728948			−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000			−0.728948			−0.364474	+	0.931214i	$0.618751\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.59808	+	1.50000i	0.156670	+	0.0904534i
$276$		0			0
$277$	−1.73205	+	1.00000i	−0.104069	+	0.0600842i	−0.551131	−	0.834419i	$-0.685804\pi$
$277$	−1.73205	+	1.00000i	−0.104069	+	0.0600842i	0.447062	+	0.894503i	$0.352470\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	0.762561	−	0.646916i	$-0.223942\pi$
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	−0.941526	+	0.336939i	$0.890608\pi$
$282$		0			0
$283$	17.3205	+	10.0000i	1.02960	+	0.594438i	0.916869	−	0.399188i	$-0.130708\pi$
$283$	17.3205	+	10.0000i	1.02960	+	0.594438i	0.112728	+	0.993626i	$0.464041\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−20.0000			−1.18056
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.3923	+	6.00000i	0.607125	+	0.350524i	0.771839	−	0.635818i	$-0.219337\pi$
$293$	10.3923	+	6.00000i	0.607125	+	0.350524i	−0.164714	+	0.986341i	$0.552670\pi$
$294$		0			0
$295$	−1.00000	−	1.73205i	−0.0582223	−	0.100844i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.46410	−	2.00000i	0.200334	−	0.115663i
$300$		0			0
$301$	−38.1051	−	22.0000i	−2.19634	−	1.26806i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.0000			1.37424
$306$		0			0
$307$			9.00000i			0.513657i	0.966457	+	0.256829i	$0.0826776\pi$
$307$			9.00000i			0.513657i	−0.966457	+	0.256829i	$0.917322\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.0000	+	17.3205i	−0.567048	+	0.982156i	0.429808	+	0.902920i	$0.358581\pi$
$311$	−10.0000	+	17.3205i	−0.567048	+	0.982156i	−0.996856	+	0.0792356i	$0.974752\pi$
$312$		0			0
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	0.879810	−	0.475325i	$-0.157669\pi$
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	−0.851549	+	0.524276i	$0.824336\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−19.0526	+	11.0000i	−1.07010	+	0.617822i	−0.928208	−	0.372061i	$-0.878651\pi$
$317$	−19.0526	+	11.0000i	−1.07010	+	0.617822i	−0.141890	+	0.989882i	$0.545318\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$			5.00000i			0.278207i
$324$		0			0
$325$			2.00000i			0.110940i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.0000	+	20.7846i	−0.661581	+	1.14589i
$330$		0			0
$331$	−17.3205	+	10.0000i	−0.952021	+	0.549650i	−0.893708	−	0.448649i	$-0.851905\pi$
$331$	−17.3205	+	10.0000i	−0.952021	+	0.549650i	−0.0583130	+	0.998298i	$0.518572\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.00000	−	5.19615i	−0.163908	−	0.283896i
$336$		0			0
$337$	−3.50000	+	6.06218i	−0.190657	+	0.330228i	−0.945468	−	0.325714i	$-0.894395\pi$
$337$	−3.50000	+	6.06218i	−0.190657	+	0.330228i	0.754811	+	0.655942i	$0.227729\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	12.0000i		−	0.649836i
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.2583	−	6.50000i	−0.604379	−	0.348938i	0.166383	−	0.986061i	$-0.446791\pi$
$347$	−11.2583	−	6.50000i	−0.604379	−	0.348938i	−0.770762	+	0.637123i	$0.780124\pi$
$348$		0			0
$349$	−13.8564	+	8.00000i	−0.741716	+	0.428230i	−0.822693	−	0.568486i	$-0.807529\pi$
$349$	−13.8564	+	8.00000i	−0.741716	+	0.428230i	0.0809766	+	0.996716i	$0.474196\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.50000	+	12.9904i	0.399185	+	0.691408i	0.993626	−	0.112731i	$-0.0359599\pi$
$353$	7.50000	+	12.9904i	0.399185	+	0.691408i	−0.594441	+	0.804139i	$0.702627\pi$
$354$		0			0
$355$	10.3923	+	6.00000i	0.551566	+	0.318447i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.00000			−0.211112			−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000			−0.211112			−0.105556	+	0.994413i	$0.533662\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.5885	−	9.00000i	−0.815937	−	0.471082i
$366$		0			0
$367$	9.00000	+	15.5885i	0.469796	+	0.813711i	0.999404	−	0.0345320i	$-0.0109941\pi$
$367$	9.00000	+	15.5885i	0.469796	+	0.813711i	−0.529607	+	0.848243i	$0.677661\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	22.5167	+	13.0000i	1.16587	+	0.673114i	0.952703	−	0.303902i	$-0.0982894\pi$
$373$	22.5167	+	13.0000i	1.16587	+	0.673114i	0.213165	+	0.977016i	$0.431623\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			5.00000i			0.256833i	0.991720	+	0.128416i	$0.0409894\pi$
$379$			5.00000i			0.256833i	−0.991720	+	0.128416i	$0.959011\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	−0.998954	−	0.0457244i	$-0.985440\pi$
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	0.539076	+	0.842257i	$0.318774\pi$
$384$		0			0
$385$	12.0000	+	20.7846i	0.611577	+	1.05928i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.92820	+	4.00000i	−0.351274	+	0.202808i	−0.665246	−	0.746624i	$-0.731673\pi$
$389$	−6.92820	+	4.00000i	−0.351274	+	0.202808i	0.313972	+	0.949432i	$0.398340\pi$
$390$		0			0
$391$	5.00000	−	8.66025i	0.252861	−	0.437968i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			28.0000i			1.40883i
$396$		0			0
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	−0.990257	−	0.139253i	$-0.955530\pi$
$401$	−7.50000	+	12.9904i	−0.374532	+	0.648709i	0.615725	+	0.787961i	$0.288863\pi$
$402$		0			0
$403$	6.92820	−	4.00000i	0.345118	−	0.199254i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.00000	−	5.19615i	−0.148704	−	0.257564i
$408$		0			0
$409$	−15.5000	+	26.8468i	−0.766426	+	1.32749i	0.173064	+	0.984911i	$0.444633\pi$
$409$	−15.5000	+	26.8468i	−0.766426	+	1.32749i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			4.00000i			0.196827i
$414$		0			0
$415$	8.00000			0.392705
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−31.1769	−	18.0000i	−1.52309	−	0.879358i	−0.999627	−	0.0273103i	$-0.991306\pi$
$419$	−31.1769	−	18.0000i	−1.52309	−	0.879358i	−0.523465	−	0.852047i	$-0.675361\pi$
$420$		0			0
$421$	19.0526	−	11.0000i	0.928565	−	0.536107i	0.0422075	−	0.999109i	$-0.486561\pi$
$421$	19.0526	−	11.0000i	0.928565	−	0.536107i	0.886357	+	0.463002i	$0.153228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.50000	+	4.33013i	0.121268	+	0.210042i
$426$		0			0
$427$	−41.5692	−	24.0000i	−2.01168	−	1.16144i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−36.0000			−1.73406			−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000			−1.73406			−0.867029	+	0.498257i	$0.833974\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.73205	+	1.00000i	0.0828552	+	0.0478365i
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.79423	+	4.50000i	−0.370315	+	0.213801i	−0.673596	−	0.739100i	$-0.735251\pi$
$443$	−7.79423	+	4.50000i	−0.370315	+	0.213801i	0.303281	+	0.952901i	$0.401918\pi$
$444$		0			0
$445$	−24.2487	−	14.0000i	−1.14950	−	0.663664i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.0000			−0.991051			−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000			−0.991051			−0.495526	+	0.868593i	$0.665025\pi$
$450$		0			0
$451$		−	15.0000i		−	0.706322i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.00000	+	13.8564i	−0.375046	+	0.649598i
$456$		0			0
$457$	18.5000	+	32.0429i	0.865393	+	1.49891i	0.866656	+	0.498906i	$0.166265\pi$
$457$	18.5000	+	32.0429i	0.865393	+	1.49891i	−0.00126243	+	0.999999i	$0.500402\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.19615	−	3.00000i	0.242009	−	0.139724i	−0.374091	−	0.927392i	$-0.622045\pi$
$461$	5.19615	−	3.00000i	0.242009	−	0.139724i	0.616100	+	0.787668i	$0.288712\pi$
$462$		0			0
$463$	5.00000	−	8.66025i	0.232370	−	0.402476i	−0.726135	−	0.687552i	$-0.758685\pi$
$463$	5.00000	−	8.66025i	0.232370	−	0.402476i	0.958505	+	0.285076i	$0.0920187\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	29.0000i		−	1.34196i	−0.741475	−	0.670980i	$-0.765874\pi$
$467$		−	29.0000i		−	1.34196i	0.741475	−	0.670980i	$-0.234126\pi$
$468$		0			0
$469$			12.0000i			0.554109i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16.5000	−	28.5788i	0.758671	−	1.31406i
$474$		0			0
$475$	−0.866025	+	0.500000i	−0.0397360	+	0.0229416i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000	−	13.8564i	−0.365529	−	0.633115i	0.623332	−	0.781958i	$-0.285779\pi$
$479$	−8.00000	−	13.8564i	−0.365529	−	0.633115i	−0.988861	+	0.148842i	$0.952445\pi$
$480$		0			0
$481$	2.00000	−	3.46410i	0.0911922	−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			2.00000i			0.0908153i
$486$		0			0
$487$	−20.0000			−0.906287			−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000			−0.906287			−0.453143	+	0.891438i	$0.649697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	16.4545	+	9.50000i	0.742580	+	0.428729i	0.823007	−	0.568032i	$-0.192295\pi$
$491$	16.4545	+	9.50000i	0.742580	+	0.428729i	−0.0804264	+	0.996761i	$0.525628\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	−	20.7846i	−0.538274	−	0.932317i
$498$		0			0
$499$	−12.9904	−	7.50000i	−0.581529	−	0.335746i	0.180212	−	0.983628i	$-0.442322\pi$
$499$	−12.9904	−	7.50000i	−0.581529	−	0.335746i	−0.761741	+	0.647882i	$0.775655\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$	−28.0000			−1.24598
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−31.1769	−	18.0000i	−1.38189	−	0.797836i	−0.389509	−	0.921023i	$-0.627355\pi$
$509$	−31.1769	−	18.0000i	−1.38189	−	0.797836i	−0.992384	+	0.123187i	$0.960689\pi$
$510$		0			0
$511$	18.0000	+	31.1769i	0.796273	+	1.37919i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.3923	−	6.00000i	0.457940	−	0.264392i
$516$		0			0
$517$	−15.5885	−	9.00000i	−0.685580	−	0.395820i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$			36.0000i			1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$			36.0000i			1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.0000	−	17.3205i	0.435607	−	0.754493i
$528$		0			0
$529$	9.50000	+	16.4545i	0.413043	+	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	8.66025	−	5.00000i	0.375117	−	0.216574i
$534$		0			0
$535$	3.00000	−	5.19615i	0.129701	−	0.224649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	27.0000i		−	1.16297i
$540$		0			0
$541$		−	22.0000i		−	0.945854i	−0.881102	−	0.472927i	$-0.843197\pi$
$541$		−	22.0000i		−	0.945854i	0.881102	−	0.472927i	$-0.156803\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.0000	+	34.6410i	−0.856706	+	1.48386i
$546$		0			0
$547$	30.3109	−	17.5000i	1.29600	−	0.748246i	0.316289	−	0.948663i	$-0.397563\pi$
$547$	30.3109	−	17.5000i	1.29600	−	0.748246i	0.979711	+	0.200417i	$0.0642296\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	28.0000	−	48.4974i	1.19068	−	2.06232i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			24.0000i			1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$			24.0000i			1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$		0			0
$559$	22.0000			0.930501
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.79423	+	4.50000i	0.328488	+	0.189652i	0.655169	−	0.755482i	$-0.272597\pi$
$563$	7.79423	+	4.50000i	0.328488	+	0.189652i	−0.326682	+	0.945134i	$0.605931\pi$
$564$		0			0
$565$	−10.3923	+	6.00000i	−0.437208	+	0.252422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.5000	+	33.7750i	0.817483	+	1.41592i	0.907532	+	0.419984i	$0.137964\pi$
$569$	19.5000	+	33.7750i	0.817483	+	1.41592i	−0.0900490	+	0.995937i	$0.528702\pi$
$570$		0			0
$571$	−4.33013	−	2.50000i	−0.181210	−	0.104622i	0.406651	−	0.913584i	$-0.366697\pi$
$571$	−4.33013	−	2.50000i	−0.181210	−	0.104622i	−0.587861	+	0.808962i	$0.700030\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.00000			0.0834058
$576$		0			0
$577$	−33.0000			−1.37381			−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000			−1.37381			−0.686904	+	0.726748i	$0.741031\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13.8564	−	8.00000i	−0.574861	−	0.331896i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−32.0429	+	18.5000i	−1.32255	+	0.763577i	−0.984135	−	0.177419i	$-0.943225\pi$
$587$	−32.0429	+	18.5000i	−1.32255	+	0.763577i	−0.338418	+	0.940996i	$0.609892\pi$
$588$		0			0
$589$	3.46410	+	2.00000i	0.142736	+	0.0824086i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$		0			0
$595$			40.0000i			1.63984i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.0000	+	32.9090i	−0.776319	+	1.34462i	0.157731	+	0.987482i	$0.449582\pi$
$599$	−19.0000	+	32.9090i	−0.776319	+	1.34462i	−0.934050	+	0.357142i	$0.883751\pi$
$600$		0			0
$601$	−2.50000	−	4.33013i	−0.101977	−	0.176630i	0.810522	−	0.585708i	$-0.199184\pi$
$601$	−2.50000	−	4.33013i	−0.101977	−	0.176630i	−0.912499	+	0.409079i	$0.865850\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.46410	−	2.00000i	0.140836	−	0.0813116i
$606$		0			0
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	0.588537	+	0.808470i	$0.299704\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	12.0000i		−	0.485468i
$612$		0			0
$613$			34.0000i			1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$			34.0000i			1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.50000	−	4.33013i	0.100646	−	0.174324i	−0.811305	−	0.584623i	$-0.801242\pi$
$617$	2.50000	−	4.33013i	0.100646	−	0.174324i	0.911951	+	0.410299i	$0.134576\pi$
$618$		0			0
$619$	−2.59808	+	1.50000i	−0.104425	+	0.0602901i	−0.551303	−	0.834305i	$-0.685869\pi$
$619$	−2.59808	+	1.50000i	−0.104425	+	0.0602901i	0.446878	+	0.894595i	$0.352536\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	28.0000	+	48.4974i	1.12180	+	1.94301i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	10.0000i		−	0.398726i
$630$		0			0
$631$	−34.0000			−1.35352			−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000			−1.35352			−0.676759	+	0.736204i	$0.736616\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.46410	−	2.00000i	−0.137469	−	0.0793676i
$636$		0			0
$637$	15.5885	−	9.00000i	0.617637	−	0.356593i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.5000	−	26.8468i	−0.612213	−	1.06038i	−0.990867	−	0.134846i	$-0.956946\pi$
$641$	−15.5000	−	26.8468i	−0.612213	−	1.06038i	0.378653	−	0.925539i	$-0.376387\pi$
$642$		0			0
$643$	4.33013	+	2.50000i	0.170764	+	0.0985904i	0.582946	−	0.812511i	$-0.301900\pi$
$643$	4.33013	+	2.50000i	0.170764	+	0.0985904i	−0.412182	+	0.911101i	$0.635233\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	−3.00000			−0.117760
$650$		0			0
$651$		0			0
$652$		0			0
$653$	31.1769	+	18.0000i	1.22005	+	0.704394i	0.964928	−	0.262515i	$-0.0845520\pi$
$653$	31.1769	+	18.0000i	1.22005	+	0.704394i	0.255119	+	0.966910i	$0.417885\pi$
$654$		0			0
$655$	−4.00000	−	6.92820i	−0.156293	−	0.270707i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.3923	+	6.00000i	−0.404827	+	0.233727i	−0.688565	−	0.725175i	$-0.741759\pi$
$659$	−10.3923	+	6.00000i	−0.404827	+	0.233727i	0.283738	+	0.958902i	$0.408425\pi$
$660$		0			0
$661$	−12.1244	−	7.00000i	−0.471583	−	0.272268i	0.245319	−	0.969442i	$-0.421107\pi$
$661$	−12.1244	−	7.00000i	−0.471583	−	0.272268i	−0.716902	+	0.697174i	$0.754441\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.00000			−0.310227
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	18.0000	−	31.1769i	0.694882	−	1.20357i
$672$		0			0
$673$	−13.0000	−	22.5167i	−0.501113	−	0.867953i	−0.999999	−	0.00128586i	$-0.999591\pi$
$673$	−13.0000	−	22.5167i	−0.501113	−	0.867953i	0.498886	−	0.866668i	$-0.333743\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	41.5692	−	24.0000i	1.59763	−	0.922395i	0.605693	−	0.795698i	$-0.292896\pi$
$677$	41.5692	−	24.0000i	1.59763	−	0.922395i	0.991941	−	0.126697i	$-0.0404375\pi$
$678$		0			0
$679$	2.00000	−	3.46410i	0.0767530	−	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			35.0000i			1.33924i	0.742705	+	0.669619i	$0.233543\pi$
$683$			35.0000i			1.33924i	−0.742705	+	0.669619i	$0.766457\pi$
$684$		0			0
$685$			18.0000i			0.687745i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−24.2487	+	14.0000i	−0.922464	+	0.532585i	−0.884420	−	0.466691i	$-0.845446\pi$
$691$	−24.2487	+	14.0000i	−0.922464	+	0.532585i	−0.0380440	+	0.999276i	$0.512113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.0000	−	22.5167i	−0.493118	−	0.854106i
$696$		0			0
$697$	12.5000	−	21.6506i	0.473471	−	0.820076i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	10.0000i		−	0.377695i	−0.982006	−	0.188847i	$-0.939525\pi$
$701$		−	10.0000i		−	0.377695i	0.982006	−	0.188847i	$-0.0604752\pi$
$702$		0			0
$703$	2.00000			0.0754314
$704$		0			0
$705$		0			0
$706$		0			0
$707$	48.4974	+	28.0000i	1.82393	+	1.05305i
$708$		0			0
$709$	−6.92820	+	4.00000i	−0.260194	+	0.150223i	−0.624423	−	0.781086i	$-0.714666\pi$
$709$	−6.92820	+	4.00000i	−0.260194	+	0.150223i	0.364229	+	0.931309i	$0.381333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.00000	−	6.92820i	−0.149801	−	0.259463i
$714$		0			0
$715$	−10.3923	−	6.00000i	−0.388650	−	0.224387i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−34.0000			−1.26799			−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000			−1.26799			−0.633993	+	0.773339i	$0.718585\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	3.00000	+	5.19615i	0.111264	+	0.192715i	0.916280	−	0.400538i	$-0.131177\pi$
$727$	3.00000	+	5.19615i	0.111264	+	0.192715i	−0.805016	+	0.593253i	$0.797843\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	47.6314	−	27.5000i	1.76171	−	1.01712i
$732$		0			0
$733$	24.2487	+	14.0000i	0.895647	+	0.517102i	0.875785	−	0.482701i	$-0.160344\pi$
$733$	24.2487	+	14.0000i	0.895647	+	0.517102i	0.0198613	+	0.999803i	$0.493678\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.00000			−0.331519
$738$		0			0
$739$		−	41.0000i		−	1.50821i	−0.656754	−	0.754105i	$-0.728071\pi$
$739$		−	41.0000i		−	1.50821i	0.656754	−	0.754105i	$-0.271929\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.0000	−	38.1051i	0.807102	−	1.39794i	−0.107761	−	0.994177i	$-0.534368\pi$
$743$	22.0000	−	38.1051i	0.807102	−	1.39794i	0.914863	−	0.403764i	$-0.132298\pi$
$744$		0			0
$745$	−18.0000	−	31.1769i	−0.659469	−	1.14223i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10.3923	+	6.00000i	−0.379727	+	0.219235i
$750$		0			0
$751$	1.00000	−	1.73205i	0.0364905	−	0.0632034i	−0.847203	−	0.531269i	$-0.821715\pi$
$751$	1.00000	−	1.73205i	0.0364905	−	0.0632034i	0.883694	+	0.468065i	$0.155049\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			12.0000i			0.436725i
$756$		0			0
$757$		−	36.0000i		−	1.30844i	−0.756303	−	0.654221i	$-0.772997\pi$
$757$		−	36.0000i		−	1.30844i	0.756303	−	0.654221i	$-0.227003\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.00000	−	12.1244i	0.253750	−	0.439508i	−0.710805	−	0.703389i	$-0.751669\pi$
$761$	7.00000	−	12.1244i	0.253750	−	0.439508i	0.964555	+	0.263881i	$0.0850027\pi$
$762$		0			0
$763$	69.2820	−	40.0000i	2.50818	−	1.44810i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.00000	−	1.73205i	−0.0361079	−	0.0625407i
$768$		0			0
$769$	−17.0000	+	29.4449i	−0.613036	+	1.06181i	0.377690	+	0.925932i	$0.376718\pi$
$769$	−17.0000	+	29.4449i	−0.613036	+	1.06181i	−0.990726	+	0.135877i	$0.956615\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			38.0000i			1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$			38.0000i			1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$		0			0
$775$	4.00000			0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.33013	+	2.50000i	0.155143	+	0.0895718i
$780$		0			0
$781$	15.5885	−	9.00000i	0.557799	−	0.322045i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.00000	−	6.92820i	−0.142766	−	0.247278i
$786$		0			0
$787$	10.3923	+	6.00000i	0.370446	+	0.213877i	0.673653	−	0.739048i	$-0.264724\pi$
$787$	10.3923	+	6.00000i	0.370446	+	0.213877i	−0.303207	+	0.952925i	$0.598058\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$	24.0000			0.852265
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.0526	−	11.0000i	−0.674876	−	0.389640i	0.123045	−	0.992401i	$-0.460734\pi$
$797$	−19.0526	−	11.0000i	−0.674876	−	0.389640i	−0.797922	+	0.602761i	$0.794067\pi$
$798$		0			0
$799$	−15.0000	−	25.9808i	−0.530662	−	0.919133i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−23.3827	+	13.5000i	−0.825157	+	0.476405i
$804$		0			0
$805$	13.8564	+	8.00000i	0.488374	+	0.281963i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.00000			0.316423			0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000			0.316423			0.158212	+	0.987405i	$0.449427\pi$
$810$		0			0
$811$			7.00000i			0.245803i	0.992419	+	0.122902i	$0.0392200\pi$
$811$			7.00000i			0.245803i	−0.992419	+	0.122902i	$0.960780\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.00000	−	6.92820i	0.140114	−	0.242684i
$816$		0			0
$817$	5.50000	+	9.52628i	0.192421	+	0.333282i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.2487	+	14.0000i	−0.846286	+	0.488603i	−0.859396	−	0.511311i	$-0.829160\pi$
$821$	−24.2487	+	14.0000i	−0.846286	+	0.488603i	0.0131101	+	0.999914i	$0.495827\pi$
$822$		0			0
$823$	13.0000	−	22.5167i	0.453152	−	0.784881i	−0.545428	−	0.838157i	$-0.683633\pi$
$823$	13.0000	−	22.5167i	0.453152	−	0.784881i	0.998580	+	0.0532760i	$0.0169663\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	48.0000i		−	1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$		−	48.0000i		−	1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$		0			0
$829$		−	32.0000i		−	1.11141i	−0.831381	−	0.555703i	$-0.812449\pi$
$829$		−	32.0000i		−	1.11141i	0.831381	−	0.555703i	$-0.187551\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	22.5000	−	38.9711i	0.779579	−	1.35027i
$834$		0			0
$835$	−3.46410	+	2.00000i	−0.119880	+	0.0692129i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.00000	−	3.46410i	−0.0690477	−	0.119594i	0.829435	−	0.558604i	$-0.188663\pi$
$839$	−2.00000	−	3.46410i	−0.0690477	−	0.119594i	−0.898482	+	0.439010i	$0.855329\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			18.0000i			0.619219i
$846$		0			0
$847$	−8.00000			−0.274883
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.46410	−	2.00000i	−0.118748	−	0.0685591i
$852$		0			0
$853$	36.3731	−	21.0000i	1.24539	−	0.719026i	0.275204	−	0.961386i	$-0.411255\pi$
$853$	36.3731	−	21.0000i	1.24539	−	0.719026i	0.970186	+	0.242360i	$0.0779214\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.00000	+	5.19615i	0.102478	+	0.177497i	0.912705	−	0.408619i	$-0.133990\pi$
$857$	3.00000	+	5.19615i	0.102478	+	0.177497i	−0.810227	+	0.586116i	$0.800656\pi$
$858$		0			0
$859$	30.3109	+	17.5000i	1.03419	+	0.597092i	0.918183	−	0.396156i	$-0.129656\pi$
$859$	30.3109	+	17.5000i	1.03419	+	0.597092i	0.116011	+	0.993248i	$0.462989\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.00000			0.204242			0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000			0.204242			0.102121	+	0.994772i	$0.467437\pi$
$864$		0			0
$865$	48.0000			1.63205
$866$		0			0
$867$		0			0
$868$		0			0
$869$	36.3731	+	21.0000i	1.23387	+	0.712376i
$870$		0			0
$871$	−3.00000	−	5.19615i	−0.101651	−	0.176065i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−41.5692	+	24.0000i	−1.40530	+	0.811348i
$876$		0			0
$877$	6.92820	+	4.00000i	0.233949	+	0.135070i	0.612392	−	0.790554i	$-0.290207\pi$
$877$	6.92820	+	4.00000i	0.233949	+	0.135070i	−0.378444	+	0.925624i	$0.623541\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$		−	23.0000i		−	0.774012i	−0.922077	−	0.387006i	$-0.873509\pi$
$883$		−	23.0000i		−	0.774012i	0.922077	−	0.387006i	$-0.126491\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.0000	+	25.9808i	−0.503651	+	0.872349i	0.496340	+	0.868128i	$0.334677\pi$
$887$	−15.0000	+	25.9808i	−0.503651	+	0.872349i	−0.999991	+	0.00422062i	$0.998657\pi$
$888$		0			0
$889$	4.00000	+	6.92820i	0.134156	+	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.19615	−	3.00000i	0.173883	−	0.100391i
$894$		0			0
$895$	20.0000	−	34.6410i	0.668526	−	1.15792i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.0000	−	17.3205i	0.332411	−	0.575753i
$906$		0			0
$907$	−21.6506	+	12.5000i	−0.718898	+	0.415056i	−0.814347	−	0.580379i	$-0.802905\pi$
$907$	−21.6506	+	12.5000i	−0.718898	+	0.415056i	0.0954492	+	0.995434i	$0.469571\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.00000	−	10.3923i	−0.198789	−	0.344312i	0.749347	−	0.662177i	$-0.230367\pi$
$911$	−6.00000	−	10.3923i	−0.198789	−	0.344312i	−0.948136	+	0.317865i	$0.897034\pi$
$912$		0			0
$913$	6.00000	−	10.3923i	0.198571	−	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			16.0000i			0.528367i
$918$		0			0
$919$	42.0000			1.38545			0.692726	−	0.721201i	$-0.256409\pi$
$919$	42.0000			1.38545			0.692726	+	0.721201i	$0.256409\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.3923	+	6.00000i	0.342067	+	0.197492i
$924$		0			0
$925$	1.73205	−	1.00000i	0.0569495	−	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.00000	+	12.1244i	0.229663	+	0.397787i	0.957708	−	0.287742i	$-0.0929044\pi$
$929$	7.00000	+	12.1244i	0.229663	+	0.397787i	−0.728046	+	0.685529i	$0.759571\pi$
$930$		0			0
$931$	7.79423	+	4.50000i	0.255446	+	0.147482i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−30.0000			−0.981105
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.7846	+	12.0000i	0.677559	+	0.391189i	0.798935	−	0.601418i	$-0.205397\pi$
$941$	20.7846	+	12.0000i	0.677559	+	0.391189i	−0.121376	+	0.992607i	$0.538731\pi$
$942$		0			0
$943$	−5.00000	−	8.66025i	−0.162822	−	0.282017i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.59808	+	1.50000i	−0.0844261	+	0.0487435i	−0.541619	−	0.840624i	$-0.682188\pi$
$947$	−2.59808	+	1.50000i	−0.0844261	+	0.0487435i	0.457193	+	0.889368i	$0.348855\pi$
$948$		0			0
$949$	−15.5885	−	9.00000i	−0.506023	−	0.292152i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.0000			0.680257			0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000			0.680257			0.340128	+	0.940379i	$0.389529\pi$
$954$		0			0
$955$		−	32.0000i		−	1.03550i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.0000	−	31.1769i	0.581250	−	1.00676i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−25.9808	+	15.0000i	−0.836350	+	0.482867i
$966$		0			0
$967$	−2.00000	+	3.46410i	−0.0643157	+	0.111398i	−0.896390	−	0.443266i	$-0.853820\pi$
$967$	−2.00000	+	3.46410i	−0.0643157	+	0.111398i	0.832075	+	0.554664i	$0.187153\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			12.0000i			0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$			12.0000i			0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$		0			0
$973$			52.0000i			1.66704i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−15.5000	+	26.8468i	−0.495889	+	0.858905i	−0.999989	−	0.00474056i	$-0.998491\pi$
$977$	−15.5000	+	26.8468i	−0.495889	+	0.858905i	0.504100	+	0.863645i	$0.331824\pi$
$978$		0			0
$979$	−36.3731	+	21.0000i	−1.16249	+	0.671163i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−23.0000	−	39.8372i	−0.733586	−	1.27061i	−0.955341	−	0.295506i	$-0.904512\pi$
$983$	−23.0000	−	39.8372i	−0.733586	−	1.27061i	0.221755	−	0.975102i	$-0.428822\pi$
$984$		0			0
$985$	−8.00000	+	13.8564i	−0.254901	+	0.441502i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	22.0000i		−	0.699559i
$990$		0			0
$991$	−56.0000			−1.77890			−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000			−1.77890			−0.889449	+	0.457034i	$0.848912\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.8564	−	8.00000i	−0.439278	−	0.253617i
$996$		0			0
$997$	−10.3923	+	6.00000i	−0.329128	+	0.190022i	−0.655454	−	0.755235i	$-0.727523\pi$
$997$	−10.3923	+	6.00000i	−0.329128	+	0.190022i	0.326326	+	0.945257i	$0.394189\pi$
$998$		0			0
$999$		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	864.2.r.a.145.1		4
3.2	odd	2		288.2.r.a.49.1		4
4.3	odd	2		216.2.n.a.37.1		4
8.3	odd	2		216.2.n.a.37.2		4
8.5	even	2	inner	864.2.r.a.145.2		4
9.2	odd	6		288.2.r.a.241.2		4
9.4	even	3		2592.2.d.a.1297.2		2
9.5	odd	6		2592.2.d.b.1297.1		2
9.7	even	3	inner	864.2.r.a.721.2		4
12.11	even	2		72.2.n.a.13.2	yes	4
24.5	odd	2		288.2.r.a.49.2		4
24.11	even	2		72.2.n.a.13.1	✓	4
36.7	odd	6		216.2.n.a.181.2		4
36.11	even	6		72.2.n.a.61.1	yes	4
36.23	even	6		648.2.d.d.325.2		2
36.31	odd	6		648.2.d.a.325.1		2
72.5	odd	6		2592.2.d.b.1297.2		2
72.11	even	6		72.2.n.a.61.2	yes	4
72.13	even	6		2592.2.d.a.1297.1		2
72.29	odd	6		288.2.r.a.241.1		4
72.43	odd	6		216.2.n.a.181.1		4
72.59	even	6		648.2.d.d.325.1		2
72.61	even	6	inner	864.2.r.a.721.1		4
72.67	odd	6		648.2.d.a.325.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.n.a.13.1	✓	4	24.11	even	2
72.2.n.a.13.2	yes	4	12.11	even	2
72.2.n.a.61.1	yes	4	36.11	even	6
72.2.n.a.61.2	yes	4	72.11	even	6
216.2.n.a.37.1		4	4.3	odd	2
216.2.n.a.37.2		4	8.3	odd	2
216.2.n.a.181.1		4	72.43	odd	6
216.2.n.a.181.2		4	36.7	odd	6
288.2.r.a.49.1		4	3.2	odd	2
288.2.r.a.49.2		4	24.5	odd	2
288.2.r.a.241.1		4	72.29	odd	6
288.2.r.a.241.2		4	9.2	odd	6
648.2.d.a.325.1		2	36.31	odd	6
648.2.d.a.325.2		2	72.67	odd	6
648.2.d.d.325.1		2	72.59	even	6
648.2.d.d.325.2		2	36.23	even	6
864.2.r.a.145.1		4	1.1	even	1	trivial
864.2.r.a.145.2		4	8.5	even	2	inner
864.2.r.a.721.1		4	72.61	even	6	inner
864.2.r.a.721.2		4	9.7	even	3	inner
2592.2.d.a.1297.1		2	72.13	even	6
2592.2.d.a.1297.2		2	9.4	even	3
2592.2.d.b.1297.1		2	9.5	odd	6
2592.2.d.b.1297.2		2	72.5	odd	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 $\F_{q}$

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	864.2.r.a.145.1

Level	$864$
Weight	$2$
Character	864.145
Analytic conductor	$6.899$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$