LMFDB - Embedded newform 1008.2.s.n.865.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(289,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1008 = 2^{4} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1008.s (of order $3$ , 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:	$8.04892052375$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1008.865
Dual form			1008.2.s.n.289.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 2.59808i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(-1.50000 + 2.59808i) q^{11} -4.00000 q^{13} +(-2.00000 - 3.46410i) q^{19} +(-2.00000 + 3.46410i) q^{25} -9.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-6.00000 - 5.19615i) q^{35} +(-4.00000 - 6.92820i) q^{37} +10.0000 q^{43} +(3.00000 + 5.19615i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-1.50000 + 2.59808i) q^{53} -9.00000 q^{55} +(-1.50000 + 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(-5.00000 + 8.66025i) q^{67} -6.00000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(1.50000 - 7.79423i) q^{77} +(-0.500000 - 0.866025i) q^{79} -9.00000 q^{83} +(3.00000 + 5.19615i) q^{89} +(10.0000 - 3.46410i) q^{91} +(6.00000 - 10.3923i) q^{95} -1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 3 q^{5} - 5 q^{7} - 3 q^{11} - 8 q^{13} - 4 q^{19} - 4 q^{25} - 18 q^{29} - q^{31} - 12 q^{35} - 8 q^{37} + 20 q^{43} + 6 q^{47} + 11 q^{49} - 3 q^{53} - 18 q^{55} - 3 q^{59} + 10 q^{61} - 12 q^{65}+ \cdots - 2 q^{97}+O(q^{100})$ 2 * q + 3 * q^5 - 5 * q^7 - 3 * q^11 - 8 * q^13 - 4 * q^19 - 4 * q^25 - 18 * q^29 - q^31 - 12 * q^35 - 8 * q^37 + 20 * q^43 + 6 * q^47 + 11 * q^49 - 3 * q^53 - 18 * q^55 - 3 * q^59 + 10 * q^61 - 12 * q^65 - 10 * q^67 - 12 * q^71 - 2 * q^73 + 3 * q^77 - q^79 - 18 * q^83 + 6 * q^89 + 20 * q^91 + 12 * q^95 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$ .

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000	−	5.19615i	−1.01419	−	0.878310i
$36$		0			0
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	0.323640	−	0.946180i	$-0.395093\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$		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
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.50000	+	2.59808i	−0.206041	+	0.356873i	−0.950464	−	0.310835i	$-0.899391\pi$
$53$	−1.50000	+	2.59808i	−0.206041	+	0.356873i	0.744423	+	0.667708i	$0.232725\pi$
$54$		0			0
$55$	−9.00000			−1.21356
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	−0.946993	−	0.321253i	$-0.895896\pi$
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	0.751710	+	0.659494i	$0.229229\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	−	10.3923i	−0.744208	−	1.28901i
$66$		0			0
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	0.380251	+	0.924883i	$0.375838\pi$
$67$	−5.00000	+	8.66025i	−0.610847	+	1.05802i	−0.991098	+	0.133135i	$0.957496\pi$
$68$		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
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	0.801553	+	0.597924i	$0.204008\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.50000	−	7.79423i	0.170941	−	0.888235i
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.00000			−0.987878			−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000			−0.987878			−0.493939	+	0.869496i	$0.664443\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	0.980071	−	0.198650i	$-0.0636557\pi$
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	−0.662071	+	0.749441i	$0.730322\pi$
$90$		0			0
$91$	10.0000	−	3.46410i	1.04828	−	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	−	10.3923i	0.615587	−	1.06623i
$96$		0			0
$97$	−1.00000			−0.101535			−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000			−0.101535			−0.0507673	+	0.998711i	$0.516167\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.0623905	+	0.998052i	$0.519872\pi$
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.833143	+	0.553058i	$0.813461\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.50000	+	2.59808i	0.145010	+	0.251166i	0.929377	−	0.369132i	$-0.120345\pi$
$107$	1.50000	+	2.59808i	0.145010	+	0.251166i	−0.784366	+	0.620298i	$0.787012\pi$
$108$		0			0
$109$	−7.00000	+	12.1244i	−0.670478	+	1.16130i	0.307290	+	0.951616i	$0.400578\pi$
$109$	−7.00000	+	12.1244i	−0.670478	+	1.16130i	−0.977769	+	0.209687i	$0.932756\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	−5.00000			−0.443678			−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000			−0.443678			−0.221839	+	0.975083i	$0.571206\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.50000	+	7.79423i	0.393167	+	0.680985i	0.992865	−	0.119241i	$-0.0380462\pi$
$131$	4.50000	+	7.79423i	0.393167	+	0.680985i	−0.599699	+	0.800226i	$0.704713\pi$
$132$		0			0
$133$	8.00000	+	6.92820i	0.693688	+	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	−2.00000			−0.169638			−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000			−0.169638			−0.0848189	+	0.996396i	$0.527031\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000	−	10.3923i	0.501745	−	0.869048i
$144$		0			0
$145$	−13.5000	−	23.3827i	−1.12111	−	1.94183i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	0.953703	+	0.300750i	$0.0972370\pi$
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	−0.216394	+	0.976306i	$0.569430\pi$
$150$		0			0
$151$	−0.500000	+	0.866025i	−0.0406894	+	0.0704761i	−0.885653	−	0.464348i	$-0.846289\pi$
$151$	−0.500000	+	0.866025i	−0.0406894	+	0.0704761i	0.844963	+	0.534824i	$0.179622\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.00000			−0.240966
$156$		0			0
$157$	2.00000	−	3.46410i	0.159617	−	0.276465i	−0.775113	−	0.631822i	$-0.782307\pi$
$157$	2.00000	−	3.46410i	0.159617	−	0.276465i	0.934731	+	0.355357i	$0.115641\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.00000	−	13.8564i	−0.626608	−	1.08532i	−0.988227	−	0.152992i	$-0.951109\pi$
$163$	−8.00000	−	13.8564i	−0.626608	−	1.08532i	0.361619	−	0.932326i	$-0.382224\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000			0.464294			0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000			0.464294			0.232147	+	0.972681i	$0.425425\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	0.973670	+	0.227964i	$0.0732068\pi$
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	−0.289412	+	0.957205i	$0.593460\pi$
$174$		0			0
$175$	2.00000	−	10.3923i	0.151186	−	0.785584i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	−0.998286	−	0.0585225i	$-0.981361\pi$
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	0.549825	+	0.835280i	$0.314694\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.0000	−	20.7846i	0.882258	−	1.52811i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	9.50000	−	16.4545i	0.683825	−	1.18442i	−0.289980	−	0.957033i	$-0.593649\pi$
$193$	9.50000	−	16.4545i	0.683825	−	1.18442i	0.973805	−	0.227387i	$-0.0730182\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	10.0000	−	17.3205i	0.708881	−	1.22782i	−0.256391	−	0.966573i	$-0.582534\pi$
$199$	10.0000	−	17.3205i	0.708881	−	1.22782i	0.965272	−	0.261245i	$-0.0841331\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	22.5000	−	7.79423i	1.57919	−	0.547048i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−14.0000			−0.963800			−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000			−0.963800			−0.481900	+	0.876226i	$0.660053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.0000	+	25.9808i	1.02299	+	1.77187i
$216$		0			0
$217$	0.500000	−	2.59808i	0.0339422	−	0.176369i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	19.0000			1.27233			0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000			1.27233			0.636167	+	0.771551i	$0.280519\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.5000	−	23.3827i	0.896026	−	1.55196i	0.0634974	−	0.997982i	$-0.479775\pi$
$227$	13.5000	−	23.3827i	0.896026	−	1.55196i	0.832529	−	0.553981i	$-0.186892\pi$
$228$		0			0
$229$	2.00000	+	3.46410i	0.132164	+	0.228914i	0.924510	−	0.381157i	$-0.124474\pi$
$229$	2.00000	+	3.46410i	0.132164	+	0.228914i	−0.792347	+	0.610071i	$0.791141\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.0000	−	20.7846i	−0.786146	−	1.36165i	−0.928312	−	0.371802i	$-0.878740\pi$
$233$	−12.0000	−	20.7846i	−0.786146	−	1.36165i	0.142166	−	0.989843i	$-0.454593\pi$
$234$		0			0
$235$	−9.00000	+	15.5885i	−0.587095	+	1.01688i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	−0.849472	−	0.527633i	$-0.823079\pi$
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	0.881680	+	0.471848i	$0.156413\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	19.5000	+	7.79423i	1.24581	+	0.497955i
$246$		0			0
$247$	8.00000	+	13.8564i	0.509028	+	0.881662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.0000			1.70422			0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000			1.70422			0.852112	+	0.523359i	$0.175321\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	0.944294	−	0.329104i	$-0.106747\pi$
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	−0.757159	+	0.653231i	$0.773413\pi$
$258$		0			0
$259$	16.0000	+	13.8564i	0.994192	+	0.860995i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.00000	−	5.19615i	0.184988	−	0.320408i	−0.758585	−	0.651575i	$-0.774109\pi$
$263$	3.00000	−	5.19615i	0.184988	−	0.320408i	0.943572	+	0.331166i	$0.107442\pi$
$264$		0			0
$265$	−9.00000			−0.552866
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.5000	−	18.1865i	0.640196	−	1.10885i	−0.345192	−	0.938532i	$-0.612186\pi$
$269$	10.5000	−	18.1865i	0.640196	−	1.10885i	0.985389	−	0.170321i	$-0.0544803\pi$
$270$		0			0
$271$	5.50000	+	9.52628i	0.334101	+	0.578680i	0.983312	−	0.181928i	$-0.0582339\pi$
$271$	5.50000	+	9.52628i	0.334101	+	0.578680i	−0.649211	+	0.760609i	$0.724901\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.00000	−	10.3923i	−0.361814	−	0.626680i
$276$		0			0
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	−0.960810	−	0.277207i	$-0.910591\pi$
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	0.720473	+	0.693482i	$0.243925\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	−0.579437	−	0.815017i	$-0.696728\pi$
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	0.995544	+	0.0942988i	$0.0300609\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−33.0000			−1.92788			−0.963940	−	0.266119i	$-0.914259\pi$
$293$	−33.0000			−1.92788			−0.963940	+	0.266119i	$0.914259\pi$
$294$		0			0
$295$	−9.00000			−0.524000
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−25.0000	+	8.66025i	−1.44098	+	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−15.0000	+	25.9808i	−0.858898	+	1.48765i
$306$		0			0
$307$	−8.00000			−0.456584			−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000			−0.456584			−0.228292	+	0.973593i	$0.573314\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	0.294384	+	0.955687i	$0.404886\pi$
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	−0.974841	+	0.222900i	$0.928448\pi$
$312$		0			0
$313$	15.5000	+	26.8468i	0.876112	+	1.51747i	0.855574	+	0.517681i	$0.173205\pi$
$313$	15.5000	+	26.8468i	0.876112	+	1.51747i	0.0205381	+	0.999789i	$0.493462\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.50000	+	7.79423i	0.252745	+	0.437767i	0.964281	−	0.264883i	$-0.0853332\pi$
$317$	4.50000	+	7.79423i	0.252745	+	0.437767i	−0.711535	+	0.702650i	$0.752000\pi$
$318$		0			0
$319$	13.5000	−	23.3827i	0.755855	−	1.30918i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	8.00000	−	13.8564i	0.443760	−	0.768615i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.0000	−	10.3923i	−0.661581	−	0.572946i
$330$		0			0
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	0.998298	+	0.0583130i	$0.0185721\pi$
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	−0.448649	+	0.893708i	$0.648095\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−30.0000			−1.63908
$336$		0			0
$337$	−7.00000			−0.381314			−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000			−0.381314			−0.190657	+	0.981657i	$0.561062\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.50000	−	2.59808i	−0.0812296	−	0.140694i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.0000	−	20.7846i	0.638696	−	1.10625i	−0.347024	−	0.937856i	$-0.612808\pi$
$353$	12.0000	−	20.7846i	0.638696	−	1.10625i	0.985719	−	0.168397i	$-0.0538590\pi$
$354$		0			0
$355$	−9.00000	−	15.5885i	−0.477670	−	0.827349i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.0000	−	25.9808i	−0.791670	−	1.37121i	−0.924932	−	0.380131i	$-0.875879\pi$
$359$	−15.0000	−	25.9808i	−0.791670	−	1.37121i	0.133263	−	0.991081i	$-0.457455\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.00000			−0.314054
$366$		0			0
$367$	−9.50000	+	16.4545i	−0.495896	+	0.858917i	−0.999989	−	0.00473247i	$-0.998494\pi$
$367$	−9.50000	+	16.4545i	−0.495896	+	0.858917i	0.504093	+	0.863649i	$0.331827\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.50000	−	7.79423i	0.0778761	−	0.404656i
$372$		0			0
$373$	−4.00000	−	6.92820i	−0.207112	−	0.358729i	0.743691	−	0.668523i	$-0.233073\pi$
$373$	−4.00000	−	6.92820i	−0.207112	−	0.358729i	−0.950804	+	0.309794i	$0.899740\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	36.0000			1.85409
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	0.539076	−	0.842257i	$-0.318774\pi$
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	−0.998954	+	0.0457244i	$0.985440\pi$
$384$		0			0
$385$	22.5000	−	7.79423i	1.14671	−	0.397231i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	−0.932002	−	0.362454i	$-0.881939\pi$
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	0.779895	+	0.625910i	$0.215272\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.50000	−	2.59808i	0.0754732	−	0.130723i
$396$		0			0
$397$	2.00000	+	3.46410i	0.100377	+	0.173858i	0.911840	−	0.410546i	$-0.134662\pi$
$397$	2.00000	+	3.46410i	0.100377	+	0.173858i	−0.811463	+	0.584404i	$0.801328\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	0.992932	+	0.118686i	$0.0378683\pi$
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	−0.393680	+	0.919247i	$0.628798\pi$
$402$		0			0
$403$	2.00000	−	3.46410i	0.0996271	−	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	12.5000	−	21.6506i	0.618085	−	1.07056i	−0.371750	−	0.928333i	$-0.621242\pi$
$409$	12.5000	−	21.6506i	0.618085	−	1.07056i	0.989835	−	0.142222i	$-0.0454247\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.50000	−	7.79423i	0.0738102	−	0.383529i
$414$		0			0
$415$	−13.5000	−	23.3827i	−0.662689	−	1.14781i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−20.0000	−	17.3205i	−0.967868	−	0.838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.00000	+	10.3923i	−0.289010	+	0.500580i	−0.973574	−	0.228373i	$-0.926659\pi$
$431$	−6.00000	+	10.3923i	−0.289010	+	0.500580i	0.684564	+	0.728953i	$0.259993\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	17.5000	+	30.3109i	0.835229	+	1.44666i	0.893843	+	0.448379i	$0.147999\pi$
$439$	17.5000	+	30.3109i	0.835229	+	1.44666i	−0.0586141	+	0.998281i	$0.518668\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.5000	+	28.5788i	0.783939	+	1.35782i	0.929631	+	0.368492i	$0.120126\pi$
$443$	16.5000	+	28.5788i	0.783939	+	1.35782i	−0.145692	+	0.989330i	$0.546541\pi$
$444$		0			0
$445$	−9.00000	+	15.5885i	−0.426641	+	0.738964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	24.0000	+	20.7846i	1.12514	+	0.974398i
$456$		0			0
$457$	0.500000	+	0.866025i	0.0233890	+	0.0405110i	0.877483	−	0.479608i	$-0.159221\pi$
$457$	0.500000	+	0.866025i	0.0233890	+	0.0405110i	−0.854094	+	0.520119i	$0.825888\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−30.0000			−1.39724			−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000			−1.39724			−0.698620	+	0.715493i	$0.746202\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.0000	−	31.1769i	−0.832941	−	1.44270i	−0.895696	−	0.444667i	$-0.853322\pi$
$467$	−18.0000	−	31.1769i	−0.832941	−	1.44270i	0.0627555	−	0.998029i	$-0.480011\pi$
$468$		0			0
$469$	5.00000	−	25.9808i	0.230879	−	1.19968i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−15.0000	+	25.9808i	−0.689701	+	1.19460i
$474$		0			0
$475$	16.0000			0.734130
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	−0.583803	−	0.811895i	$-0.698436\pi$
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	0.995023	+	0.0996406i	$0.0317693\pi$
$480$		0			0
$481$	16.0000	+	27.7128i	0.729537	+	1.26360i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.50000	−	2.59808i	−0.0681115	−	0.117973i
$486$		0			0
$487$	20.5000	−	35.5070i	0.928944	−	1.60898i	0.143851	−	0.989599i	$-0.454051\pi$
$487$	20.5000	−	35.5070i	0.928944	−	1.60898i	0.785093	−	0.619378i	$-0.212615\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−33.0000			−1.48927			−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000			−1.48927			−0.744635	+	0.667472i	$0.767376\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.0000	−	5.19615i	0.672842	−	0.233079i
$498$		0			0
$499$	1.00000	+	1.73205i	0.0447661	+	0.0775372i	0.887540	−	0.460730i	$-0.152412\pi$
$499$	1.00000	+	1.73205i	0.0447661	+	0.0775372i	−0.842774	+	0.538267i	$0.819079\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	−54.0000			−2.40297
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.50000	−	2.59808i	−0.0664863	−	0.115158i	0.830866	−	0.556473i	$-0.187846\pi$
$509$	−1.50000	−	2.59808i	−0.0664863	−	0.115158i	−0.897352	+	0.441315i	$0.854512\pi$
$510$		0			0
$511$	1.00000	−	5.19615i	0.0442374	−	0.229864i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.0000	+	20.7846i	−0.528783	+	0.915879i
$516$		0			0
$517$	−18.0000			−0.791639
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	−0.993011	−	0.118020i	$-0.962345\pi$
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	0.598714	+	0.800963i	$0.295679\pi$
$522$		0			0
$523$	−2.00000	−	3.46410i	−0.0874539	−	0.151475i	0.818980	−	0.573822i	$-0.194540\pi$
$523$	−2.00000	−	3.46410i	−0.0874539	−	0.151475i	−0.906434	+	0.422347i	$0.861206\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−4.50000	+	7.79423i	−0.194552	+	0.336974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.00000	+	20.7846i	0.129219	+	0.895257i
$540$		0			0
$541$	−13.0000	−	22.5167i	−0.558914	−	0.968067i	−0.997587	−	0.0694205i	$-0.977885\pi$
$541$	−13.0000	−	22.5167i	−0.558914	−	0.968067i	0.438674	−	0.898646i	$-0.355448\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−42.0000			−1.79908
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	18.0000	+	31.1769i	0.766826	+	1.32818i
$552$		0			0
$553$	2.00000	+	1.73205i	0.0850487	+	0.0736543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.50000	−	2.59808i	0.0635570	−	0.110084i	−0.832496	−	0.554031i	$-0.813089\pi$
$557$	1.50000	−	2.59808i	0.0635570	−	0.110084i	0.896053	+	0.443947i	$0.146422\pi$
$558$		0			0
$559$	−40.0000			−1.69182
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.5000	+	33.7750i	−0.821827	+	1.42345i	0.0824933	+	0.996592i	$0.473712\pi$
$563$	−19.5000	+	33.7750i	−0.821827	+	1.42345i	−0.904320	+	0.426855i	$0.859622\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.0000	−	31.1769i	−0.754599	−	1.30700i	−0.945573	−	0.325409i	$-0.894498\pi$
$569$	−18.0000	−	31.1769i	−0.754599	−	1.30700i	0.190974	−	0.981595i	$-0.438835\pi$
$570$		0			0
$571$	−17.0000	+	29.4449i	−0.711428	+	1.23223i	0.252893	+	0.967494i	$0.418618\pi$
$571$	−17.0000	+	29.4449i	−0.711428	+	1.23223i	−0.964321	+	0.264735i	$0.914716\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−11.5000	+	19.9186i	−0.478751	+	0.829222i	−0.999703	−	0.0243645i	$-0.992244\pi$
$577$	−11.5000	+	19.9186i	−0.478751	+	0.829222i	0.520952	+	0.853586i	$0.325577\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	22.5000	−	7.79423i	0.933457	−	0.323359i
$582$		0			0
$583$	−4.50000	−	7.79423i	−0.186371	−	0.322804i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.0000			0.866763			0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000			0.866763			0.433381	+	0.901211i	$0.357320\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12.0000	+	20.7846i	0.492781	+	0.853522i	0.999965	−	0.00831589i	$-0.00264706\pi$
$593$	12.0000	+	20.7846i	0.492781	+	0.853522i	−0.507184	+	0.861838i	$0.669314\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.00000	−	15.5885i	0.367730	−	0.636927i	−0.621480	−	0.783430i	$-0.713468\pi$
$599$	9.00000	−	15.5885i	0.367730	−	0.636927i	0.989210	+	0.146503i	$0.0468017\pi$
$600$		0			0
$601$	11.0000			0.448699			0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000			0.448699			0.224350	+	0.974509i	$0.427974\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.00000	+	5.19615i	−0.121967	+	0.211254i
$606$		0			0
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	0.786212	−	0.617957i	$-0.212039\pi$
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	−0.928272	+	0.371901i	$0.878706\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	−	20.7846i	−0.485468	−	0.840855i
$612$		0			0
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	−0.658012	−	0.753007i	$-0.728603\pi$
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	0.981129	+	0.193352i	$0.0619359\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$	−17.0000	+	29.4449i	−0.683288	+	1.18349i	0.290684	+	0.956819i	$0.406117\pi$
$619$	−17.0000	+	29.4449i	−0.683288	+	1.18349i	−0.973972	+	0.226670i	$0.927216\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000	−	10.3923i	−0.480770	−	0.416359i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	7.00000			0.278666			0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000			0.278666			0.139333	+	0.990246i	$0.455504\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.50000	−	12.9904i	−0.297628	−	0.515508i
$636$		0			0
$637$	−22.0000	+	17.3205i	−0.871672	+	0.686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.0000	+	25.9808i	−0.592464	+	1.02618i	0.401435	+	0.915888i	$0.368512\pi$
$641$	−15.0000	+	25.9808i	−0.592464	+	1.02618i	−0.993899	+	0.110291i	$0.964822\pi$
$642$		0			0
$643$	34.0000			1.34083			0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000			1.34083			0.670415	+	0.741987i	$0.266116\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	−0.633090	−	0.774078i	$-0.718214\pi$
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	0.986916	+	0.161233i	$0.0515470\pi$
$648$		0			0
$649$	−4.50000	−	7.79423i	−0.176640	−	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.50000	+	2.59808i	0.0586995	+	0.101671i	0.893882	−	0.448303i	$-0.147971\pi$
$653$	1.50000	+	2.59808i	0.0586995	+	0.101671i	−0.835182	+	0.549973i	$0.814638\pi$
$654$		0			0
$655$	−13.5000	+	23.3827i	−0.527489	+	0.913637i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−24.0000			−0.934907			−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000			−0.934907			−0.467454	+	0.884018i	$0.654829\pi$
$660$		0			0
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	−0.969442	−	0.245319i	$-0.921107\pi$
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	0.697174	+	0.716902i	$0.254441\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000	+	31.1769i	−0.232670	+	1.20899i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−30.0000			−1.15814
$672$		0			0
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.5000	−	28.5788i	−0.634147	−	1.09837i	−0.986695	−	0.162581i	$-0.948018\pi$
$677$	−16.5000	−	28.5788i	−0.634147	−	1.09837i	0.352549	−	0.935793i	$-0.385315\pi$
$678$		0			0
$679$	2.50000	−	0.866025i	0.0959412	−	0.0332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.5000	+	28.5788i	−0.631355	+	1.09354i	0.355920	+	0.934516i	$0.384168\pi$
$683$	−16.5000	+	28.5788i	−0.631355	+	1.09354i	−0.987275	+	0.159022i	$0.949166\pi$
$684$		0			0
$685$	54.0000			2.06323
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000	−	10.3923i	0.228582	−	0.395915i
$690$		0			0
$691$	4.00000	+	6.92820i	0.152167	+	0.263561i	0.932024	−	0.362397i	$-0.118041\pi$
$691$	4.00000	+	6.92820i	0.152167	+	0.263561i	−0.779857	+	0.625958i	$0.784708\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.00000	−	5.19615i	−0.113796	−	0.197101i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.0000			0.566542			0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000			0.566542			0.283271	+	0.959040i	$0.408580\pi$
$702$		0			0
$703$	−16.0000	+	27.7128i	−0.603451	+	1.04521i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.00000	−	46.7654i	0.338480	−	1.75879i
$708$		0			0
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	0.944509	−	0.328484i	$-0.106538\pi$
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	−0.756730	+	0.653727i	$0.773204\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	36.0000			1.34632
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.00000	+	15.5885i	0.335643	+	0.581351i	0.983608	−	0.180319i	$-0.0577130\pi$
$719$	9.00000	+	15.5885i	0.335643	+	0.581351i	−0.647965	+	0.761670i	$0.724380\pi$
$720$		0			0
$721$	−16.0000	−	13.8564i	−0.595871	−	0.516040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.0000	−	31.1769i	0.668503	−	1.15788i
$726$		0			0
$727$	13.0000			0.482143			0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000			0.482143			0.241072	+	0.970507i	$0.422501\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	5.00000	+	8.66025i	0.184679	+	0.319874i	0.943468	−	0.331463i	$-0.107542\pi$
$733$	5.00000	+	8.66025i	0.184679	+	0.319874i	−0.758789	+	0.651336i	$0.774209\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15.0000	−	25.9808i	−0.552532	−	0.957014i
$738$		0			0
$739$	25.0000	−	43.3013i	0.919640	−	1.59286i	0.119677	−	0.992813i	$-0.461814\pi$
$739$	25.0000	−	43.3013i	0.919640	−	1.59286i	0.799962	−	0.600050i	$-0.204853\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	42.0000			1.54083			0.770415	−	0.637542i	$-0.220049\pi$
$743$	42.0000			1.54083			0.770415	+	0.637542i	$0.220049\pi$
$744$		0			0
$745$	−27.0000	+	46.7654i	−0.989203	+	1.71335i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.00000	−	5.19615i	−0.219235	−	0.189863i
$750$		0			0
$751$	−3.50000	−	6.06218i	−0.127717	−	0.221212i	0.795075	−	0.606511i	$-0.207432\pi$
$751$	−3.50000	−	6.06218i	−0.127717	−	0.221212i	−0.922792	+	0.385299i	$0.874098\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.00000			−0.109181
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000	+	10.3923i	0.217500	+	0.376721i	0.954043	−	0.299670i	$-0.0968765\pi$
$761$	6.00000	+	10.3923i	0.217500	+	0.376721i	−0.736543	+	0.676391i	$0.763543\pi$
$762$		0			0
$763$	7.00000	−	36.3731i	0.253417	−	1.31679i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.00000	−	10.3923i	0.216647	−	0.375244i
$768$		0			0
$769$	−19.0000			−0.685158			−0.342579	−	0.939489i	$-0.611300\pi$
$769$	−19.0000			−0.685158			−0.342579	+	0.939489i	$0.611300\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.00000	−	5.19615i	0.107903	−	0.186893i	−0.807018	−	0.590527i	$-0.798920\pi$
$773$	3.00000	−	5.19615i	0.107903	−	0.186893i	0.914920	+	0.403634i	$0.132253\pi$
$774$		0			0
$775$	−2.00000	−	3.46410i	−0.0718421	−	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	9.00000	−	15.5885i	0.322045	−	0.557799i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.0000			0.428298
$786$		0			0
$787$	25.0000	−	43.3013i	0.891154	−	1.54352i	0.0526599	−	0.998613i	$-0.483230\pi$
$787$	25.0000	−	43.3013i	0.891154	−	1.54352i	0.838494	−	0.544911i	$-0.183437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−20.0000	−	34.6410i	−0.710221	−	1.23014i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	33.0000			1.16892			0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000			1.16892			0.584460	+	0.811423i	$0.301306\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.00000	−	5.19615i	−0.105868	−	0.183368i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.0000	−	41.5692i	0.840683	−	1.45611i
$816$		0			0
$817$	−20.0000	−	34.6410i	−0.699711	−	1.21194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.50000	−	2.59808i	−0.0523504	−	0.0906735i	0.838663	−	0.544651i	$-0.183338\pi$
$821$	−1.50000	−	2.59808i	−0.0523504	−	0.0906735i	−0.891013	+	0.453978i	$0.850005\pi$
$822$		0			0
$823$	−20.0000	+	34.6410i	−0.697156	+	1.20751i	0.272292	+	0.962215i	$0.412218\pi$
$823$	−20.0000	+	34.6410i	−0.697156	+	1.20751i	−0.969448	+	0.245295i	$0.921115\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.0000			0.521601			0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000			0.521601			0.260801	+	0.965393i	$0.416014\pi$
$828$		0			0
$829$	2.00000	−	3.46410i	0.0694629	−	0.120313i	−0.829202	−	0.558949i	$-0.811205\pi$
$829$	2.00000	−	3.46410i	0.0694629	−	0.120313i	0.898665	+	0.438636i	$0.144538\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	9.00000	+	15.5885i	0.311458	+	0.539461i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.50000	+	7.79423i	0.154805	+	0.268130i
$846$		0			0
$847$	−4.00000	−	3.46410i	−0.137442	−	0.119028i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	0.244701	+	0.969599i	$0.421310\pi$
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	−0.962048	+	0.272882i	$0.912023\pi$
$858$		0			0
$859$	25.0000	+	43.3013i	0.852989	+	1.47742i	0.878498	+	0.477746i	$0.158546\pi$
$859$	25.0000	+	43.3013i	0.852989	+	1.47742i	−0.0255092	+	0.999675i	$0.508121\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.00000	−	5.19615i	−0.102121	−	0.176879i	0.810437	−	0.585826i	$-0.199230\pi$
$863$	−3.00000	−	5.19615i	−0.102121	−	0.176879i	−0.912558	+	0.408946i	$0.865896\pi$
$864$		0			0
$865$	−27.0000	+	46.7654i	−0.918028	+	1.59007i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.00000			0.101768
$870$		0			0
$871$	20.0000	−	34.6410i	0.677674	−	1.17377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.50000	+	2.59808i	−0.253546	+	0.0878310i
$876$		0			0
$877$	−16.0000	−	27.7128i	−0.540282	−	0.935795i	−0.998888	−	0.0471555i	$-0.984984\pi$
$877$	−16.0000	−	27.7128i	−0.540282	−	0.935795i	0.458606	−	0.888640i	$-0.348349\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6.00000			0.202145			0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000			0.202145			0.101073	+	0.994879i	$0.467773\pi$
$882$		0			0
$883$	−32.0000			−1.07689			−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000			−1.07689			−0.538443	+	0.842662i	$0.680987\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000	+	20.7846i	0.402921	+	0.697879i	0.994077	−	0.108678i	$-0.0346618\pi$
$887$	12.0000	+	20.7846i	0.402921	+	0.697879i	−0.591156	+	0.806557i	$0.701328\pi$
$888$		0			0
$889$	12.5000	−	4.33013i	0.419237	−	0.145228i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12.0000	−	20.7846i	0.401565	−	0.695530i
$894$		0			0
$895$	−36.0000			−1.20335
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.50000	−	7.79423i	0.150083	−	0.259952i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.0000	+	20.7846i	0.398893	+	0.690904i
$906$		0			0
$907$	4.00000	−	6.92820i	0.132818	−	0.230047i	−0.791944	−	0.610594i	$-0.790931\pi$
$907$	4.00000	−	6.92820i	0.132818	−	0.230047i	0.924762	+	0.380547i	$0.124264\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$	13.5000	−	23.3827i	0.446785	−	0.773854i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.0000	−	15.5885i	−0.594412	−	0.514776i
$918$		0			0
$919$	4.00000	+	6.92820i	0.131948	+	0.228540i	0.924427	−	0.381358i	$-0.124544\pi$
$919$	4.00000	+	6.92820i	0.131948	+	0.228540i	−0.792480	+	0.609898i	$0.791210\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	24.0000			0.789970
$924$		0			0
$925$	32.0000			1.05215
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	0.812607	−	0.582812i	$-0.198048\pi$
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	−0.911034	+	0.412332i	$0.864714\pi$
$930$		0			0
$931$	−26.0000	−	10.3923i	−0.852116	−	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	35.0000			1.14340			0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000			1.14340			0.571700	+	0.820463i	$0.306284\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.50000	+	7.79423i	−0.146696	+	0.254085i	−0.930004	−	0.367549i	$-0.880197\pi$
$941$	−4.50000	+	7.79423i	−0.146696	+	0.254085i	0.783309	+	0.621633i	$0.213531\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$	4.00000	−	6.92820i	0.129845	−	0.224899i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.00000	+	46.7654i	−0.290625	+	1.51013i
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	57.0000			1.83489
$966$		0			0
$967$	1.00000			0.0321578			0.0160789	−	0.999871i	$-0.494882\pi$
$967$	1.00000			0.0321578			0.0160789	+	0.999871i	$0.494882\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.5000	+	33.7750i	0.625785	+	1.08389i	0.988389	+	0.151948i	$0.0485545\pi$
$971$	19.5000	+	33.7750i	0.625785	+	1.08389i	−0.362604	+	0.931943i	$0.618112\pi$
$972$		0			0
$973$	5.00000	−	1.73205i	0.160293	−	0.0555270i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.0000	−	36.3731i	0.671850	−	1.16368i	−0.305530	−	0.952183i	$-0.598833\pi$
$977$	21.0000	−	36.3731i	0.671850	−	1.16368i	0.977379	−	0.211495i	$-0.0678332\pi$
$978$		0			0
$979$	−18.0000			−0.575282
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	0.422027	+	0.906583i	$0.361319\pi$
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	−0.996138	+	0.0878058i	$0.972015\pi$
$984$		0			0
$985$	−9.00000	−	15.5885i	−0.286764	−	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−6.50000	+	11.2583i	−0.206479	+	0.357633i	−0.950603	−	0.310409i	$-0.899534\pi$
$991$	−6.50000	+	11.2583i	−0.206479	+	0.357633i	0.744124	+	0.668042i	$0.232867\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	60.0000			1.90213
$996$		0			0
$997$	−7.00000	+	12.1244i	−0.221692	+	0.383982i	−0.955322	−	0.295567i	$-0.904491\pi$
$997$	−7.00000	+	12.1244i	−0.221692	+	0.383982i	0.733630	+	0.679549i	$0.237825\pi$
$998$		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	1008.2.s.n.865.1		2
3.2	odd	2		336.2.q.d.193.1		2
4.3	odd	2		126.2.g.b.109.1		2
7.2	even	3	inner	1008.2.s.n.289.1		2
7.3	odd	6		7056.2.a.bz.1.1		1
7.4	even	3		7056.2.a.g.1.1		1
12.11	even	2		42.2.e.b.25.1	✓	2
21.2	odd	6		336.2.q.d.289.1		2
21.5	even	6		2352.2.q.m.961.1		2
21.11	odd	6		2352.2.a.m.1.1		1
21.17	even	6		2352.2.a.n.1.1		1
21.20	even	2		2352.2.q.m.1537.1		2
24.5	odd	2		1344.2.q.j.193.1		2
24.11	even	2		1344.2.q.v.193.1		2
28.3	even	6		882.2.a.k.1.1		1
28.11	odd	6		882.2.a.g.1.1		1
28.19	even	6		882.2.g.b.667.1		2
28.23	odd	6		126.2.g.b.37.1		2
28.27	even	2		882.2.g.b.361.1		2
36.7	odd	6		1134.2.h.a.109.1		2
36.11	even	6		1134.2.h.p.109.1		2
36.23	even	6		1134.2.e.a.865.1		2
36.31	odd	6		1134.2.e.p.865.1		2
60.23	odd	4		1050.2.o.b.949.2		4
60.47	odd	4		1050.2.o.b.949.1		4
60.59	even	2		1050.2.i.e.151.1		2
84.11	even	6		294.2.a.d.1.1		1
84.23	even	6		42.2.e.b.37.1	yes	2
84.47	odd	6		294.2.e.f.79.1		2
84.59	odd	6		294.2.a.a.1.1		1
84.83	odd	2		294.2.e.f.67.1		2
168.11	even	6		9408.2.a.d.1.1		1
168.53	odd	6		9408.2.a.bu.1.1		1
168.59	odd	6		9408.2.a.db.1.1		1
168.101	even	6		9408.2.a.bm.1.1		1
168.107	even	6		1344.2.q.v.961.1		2
168.149	odd	6		1344.2.q.j.961.1		2
252.23	even	6		1134.2.h.p.541.1		2
252.79	odd	6		1134.2.e.p.919.1		2
252.191	even	6		1134.2.e.a.919.1		2
252.247	odd	6		1134.2.h.a.541.1		2
420.23	odd	12		1050.2.o.b.499.1		4
420.59	odd	6		7350.2.a.cw.1.1		1
420.107	odd	12		1050.2.o.b.499.2		4
420.179	even	6		7350.2.a.ce.1.1		1
420.359	even	6		1050.2.i.e.751.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.e.b.25.1	✓	2	12.11	even	2
42.2.e.b.37.1	yes	2	84.23	even	6
126.2.g.b.37.1		2	28.23	odd	6
126.2.g.b.109.1		2	4.3	odd	2
294.2.a.a.1.1		1	84.59	odd	6
294.2.a.d.1.1		1	84.11	even	6
294.2.e.f.67.1		2	84.83	odd	2
294.2.e.f.79.1		2	84.47	odd	6
336.2.q.d.193.1		2	3.2	odd	2
336.2.q.d.289.1		2	21.2	odd	6
882.2.a.g.1.1		1	28.11	odd	6
882.2.a.k.1.1		1	28.3	even	6
882.2.g.b.361.1		2	28.27	even	2
882.2.g.b.667.1		2	28.19	even	6
1008.2.s.n.289.1		2	7.2	even	3	inner
1008.2.s.n.865.1		2	1.1	even	1	trivial
1050.2.i.e.151.1		2	60.59	even	2
1050.2.i.e.751.1		2	420.359	even	6
1050.2.o.b.499.1		4	420.23	odd	12
1050.2.o.b.499.2		4	420.107	odd	12
1050.2.o.b.949.1		4	60.47	odd	4
1050.2.o.b.949.2		4	60.23	odd	4
1134.2.e.a.865.1		2	36.23	even	6
1134.2.e.a.919.1		2	252.191	even	6
1134.2.e.p.865.1		2	36.31	odd	6
1134.2.e.p.919.1		2	252.79	odd	6
1134.2.h.a.109.1		2	36.7	odd	6
1134.2.h.a.541.1		2	252.247	odd	6
1134.2.h.p.109.1		2	36.11	even	6
1134.2.h.p.541.1		2	252.23	even	6
1344.2.q.j.193.1		2	24.5	odd	2
1344.2.q.j.961.1		2	168.149	odd	6
1344.2.q.v.193.1		2	24.11	even	2
1344.2.q.v.961.1		2	168.107	even	6
2352.2.a.m.1.1		1	21.11	odd	6
2352.2.a.n.1.1		1	21.17	even	6
2352.2.q.m.961.1		2	21.5	even	6
2352.2.q.m.1537.1		2	21.20	even	2
7056.2.a.g.1.1		1	7.4	even	3
7056.2.a.bz.1.1		1	7.3	odd	6
7350.2.a.ce.1.1		1	420.179	even	6
7350.2.a.cw.1.1		1	420.59	odd	6
9408.2.a.d.1.1		1	168.11	even	6
9408.2.a.bm.1.1		1	168.101	even	6
9408.2.a.bu.1.1		1	168.53	odd	6
9408.2.a.db.1.1		1	168.59	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

Related objects

Downloads

Learn more

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	1008.2.s.n.865.1

Level	$1008$
Weight	$2$
Character	1008.865
Analytic conductor	$8.049$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$