LMFDB - Embedded newform 1512.2.s.h.1297.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(865,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("1512.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.s (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.0733807856$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1297.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1512.1297
Dual form			1512.2.s.h.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+(1.00000 - 1.73205i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-2.00000 - 3.46410i) q^{11} -1.00000 q^{13} +(2.00000 - 3.46410i) q^{19} +(3.00000 - 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-2.50000 - 4.33013i) q^{31} +(5.00000 - 1.73205i) q^{35} +(-0.500000 + 0.866025i) q^{37} +4.00000 q^{41} -1.00000 q^{43} +(2.00000 - 3.46410i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-3.00000 - 5.19615i) q^{53} -8.00000 q^{55} +(1.50000 - 2.59808i) q^{61} +(-1.00000 + 1.73205i) q^{65} +(-5.50000 - 9.52628i) q^{67} +14.0000 q^{71} +(7.00000 + 12.1244i) q^{73} +(2.00000 - 10.3923i) q^{77} +(-6.50000 + 11.2583i) q^{79} -14.0000 q^{83} +(3.00000 - 5.19615i) q^{89} +(-2.00000 - 1.73205i) q^{91} +(-4.00000 - 6.92820i) q^{95} +9.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 4 * q^7	$ 2 q + 2 q^{5} + 4 q^{7} - 4 q^{11} - 2 q^{13} + 4 q^{19} + 6 q^{23} + q^{25} - 5 q^{31} + 10 q^{35} - q^{37} + 8 q^{41} - 2 q^{43} + 4 q^{47} + 2 q^{49} - 6 q^{53} - 16 q^{55} + 3 q^{61} - 2 q^{65} - 11 q^{67} + 28 q^{71} + 14 q^{73} + 4 q^{77} - 13 q^{79} - 28 q^{83} + 6 q^{89} - 4 q^{91} - 8 q^{95} + 18 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 4 * q^7 - 4 * q^11 - 2 * q^13 + 4 * q^19 + 6 * q^23 + q^25 - 5 * q^31 + 10 * q^35 - q^37 + 8 * q^41 - 2 * q^43 + 4 * q^47 + 2 * q^49 - 6 * q^53 - 16 * q^55 + 3 * q^61 - 2 * q^65 - 11 * q^67 + 28 * q^71 + 14 * q^73 + 4 * q^77 - 13 * q^79 - 28 * q^83 + 6 * q^89 - 4 * q^91 - 8 * q^95 + 18 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	−0.992361	−	0.123371i	$-0.960630\pi$
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	0.389338	−	0.921095i	$-0.372704\pi$
$12$		0			0
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	−0.540068	−	0.841621i	$-0.681602\pi$
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	0.998899	+	0.0469020i	$0.0149348\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	0.549309	−	0.835619i	$-0.314891\pi$
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	−0.998322	+	0.0579057i	$0.981558\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.00000	−	1.73205i	0.845154	−	0.292770i
$36$		0			0
$37$	−0.500000	+	0.866025i	−0.0821995	+	0.142374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−0.500000	+	0.866025i	−0.0821995	+	0.142374i	0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.00000			0.624695			0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000			0.624695			0.312348	+	0.949968i	$0.398885\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	−0.682489	−	0.730896i	$-0.739102\pi$
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	0.974219	+	0.225605i	$0.0724358\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	1.50000	−	2.59808i	0.192055	−	0.332650i	−0.753876	−	0.657017i	$-0.771818\pi$
$61$	1.50000	−	2.59808i	0.192055	−	0.332650i	0.945931	+	0.324367i	$0.105151\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.00000	+	1.73205i	−0.124035	+	0.214834i
$66$		0			0
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	0.305424	−	0.952217i	$-0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$	7.00000	+	12.1244i	0.819288	+	1.41905i	0.906208	+	0.422833i	$0.138964\pi$
$73$	7.00000	+	12.1244i	0.819288	+	1.41905i	−0.0869195	+	0.996215i	$0.527702\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.00000	−	10.3923i	0.227921	−	1.18431i
$78$		0			0
$79$	−6.50000	+	11.2583i	−0.731307	+	1.26666i	0.225018	+	0.974355i	$0.427756\pi$
$79$	−6.50000	+	11.2583i	−0.731307	+	1.26666i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−14.0000			−1.53670			−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000			−1.53670			−0.768350	+	0.640030i	$0.778922\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	−0.662071	−	0.749441i	$-0.730322\pi$
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	0.980071	+	0.198650i	$0.0636557\pi$
$90$		0			0
$91$	−2.00000	−	1.73205i	−0.209657	−	0.181568i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.00000	−	6.92820i	−0.410391	−	0.710819i
$96$		0			0
$97$	9.00000			0.913812			0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000			0.913812			0.456906	+	0.889515i	$0.348958\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	0.502477	−	0.864590i	$-0.332422\pi$
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	−0.999996	+	0.00286291i	$0.999089\pi$
$102$		0			0
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.00813215	−	0.999967i	$-0.497411\pi$
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.861931	−	0.507026i	$-0.169255\pi$
$108$		0			0
$109$	0.500000	+	0.866025i	0.0478913	+	0.0829502i	0.888977	−	0.457951i	$-0.151417\pi$
$109$	0.500000	+	0.866025i	0.0478913	+	0.0829502i	−0.841086	+	0.540901i	$0.818083\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.0000			0.940721			0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000			0.940721			0.470360	+	0.882474i	$0.344124\pi$
$114$		0			0
$115$	−6.00000	−	10.3923i	−0.559503	−	0.969087i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	−11.0000			−0.976092			−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000			−0.976092			−0.488046	+	0.872818i	$0.662290\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	0.269863	+	0.962899i	$0.413022\pi$
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	−0.968826	+	0.247741i	$0.920312\pi$
$132$		0			0
$133$	10.0000	−	3.46410i	0.867110	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	−0.938148	−	0.346235i	$-0.887460\pi$
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	0.169226	−	0.985577i	$-0.445873\pi$
$138$		0			0
$139$	1.00000			0.0848189			0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000			0.0848189			0.0424094	+	0.999100i	$0.486497\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000	+	3.46410i	0.167248	+	0.289683i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	−1.50000	−	2.59808i	−0.122068	−	0.211428i	0.798515	−	0.601975i	$-0.205619\pi$
$151$	−1.50000	−	2.59808i	−0.122068	−	0.211428i	−0.920583	+	0.390547i	$0.872286\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.0000			−0.803219
$156$		0			0
$157$	9.00000	+	15.5885i	0.718278	+	1.24409i	0.961681	+	0.274169i	$0.0884028\pi$
$157$	9.00000	+	15.5885i	0.718278	+	1.24409i	−0.243403	+	0.969925i	$0.578264\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	15.0000	−	5.19615i	1.18217	−	0.409514i
$162$		0			0
$163$	10.5000	−	18.1865i	0.822423	−	1.42448i	−0.0814491	−	0.996678i	$-0.525955\pi$
$163$	10.5000	−	18.1865i	0.822423	−	1.42448i	0.903873	−	0.427802i	$-0.140712\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.00000			0.154765			0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000			0.154765			0.0773823	+	0.997001i	$0.475344\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.0000	+	20.7846i	−0.912343	+	1.58022i	−0.101598	+	0.994826i	$0.532395\pi$
$173$	−12.0000	+	20.7846i	−0.912343	+	1.58022i	−0.810745	+	0.585399i	$0.800938\pi$
$174$		0			0
$175$	−0.500000	+	2.59808i	−0.0377964	+	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.00000	+	1.73205i	0.0747435	+	0.129460i	0.900975	−	0.433872i	$-0.142853\pi$
$179$	1.00000	+	1.73205i	0.0747435	+	0.129460i	−0.826231	+	0.563331i	$0.809520\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.00000	+	1.73205i	0.0735215	+	0.127343i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.0000	+	19.0526i	−0.795932	+	1.37859i	0.126314	+	0.991990i	$0.459685\pi$
$191$	−11.0000	+	19.0526i	−0.795932	+	1.37859i	−0.922246	+	0.386604i	$0.873648\pi$
$192$		0			0
$193$	8.50000	+	14.7224i	0.611843	+	1.05974i	0.990930	+	0.134382i	$0.0429051\pi$
$193$	8.50000	+	14.7224i	0.611843	+	1.05974i	−0.379086	+	0.925361i	$0.623762\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	7.50000	+	12.9904i	0.531661	+	0.920864i	0.999317	+	0.0369532i	$0.0117652\pi$
$199$	7.50000	+	12.9904i	0.531661	+	0.920864i	−0.467656	+	0.883911i	$0.654901\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.00000	−	6.92820i	0.279372	−	0.483887i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.0000			−1.10674
$210$		0			0
$211$	15.0000			1.03264			0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000			1.03264			0.516321	+	0.856395i	$0.327301\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.00000	+	1.73205i	−0.0681994	+	0.118125i
$216$		0			0
$217$	2.50000	−	12.9904i	0.169711	−	0.881845i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.0000			−0.803579			−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000			−0.803579			−0.401790	+	0.915732i	$0.631612\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.0000	+	19.0526i	0.730096	+	1.26456i	0.956842	+	0.290609i	$0.0938578\pi$
$227$	11.0000	+	19.0526i	0.730096	+	1.26456i	−0.226746	+	0.973954i	$0.572809\pi$
$228$		0			0
$229$	−9.50000	+	16.4545i	−0.627778	+	1.08734i	0.360219	+	0.932868i	$0.382702\pi$
$229$	−9.50000	+	16.4545i	−0.627778	+	1.08734i	−0.987997	+	0.154475i	$0.950631\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.00000	+	13.8564i	−0.524097	+	0.907763i	0.475509	+	0.879711i	$0.342264\pi$
$233$	−8.00000	+	13.8564i	−0.524097	+	0.907763i	−0.999606	+	0.0280525i	$0.991069\pi$
$234$		0			0
$235$	−4.00000	−	6.92820i	−0.260931	−	0.451946i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22.0000			1.42306			0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000			1.42306			0.711531	+	0.702655i	$0.248002\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	0.849472	−	0.527633i	$-0.176921\pi$
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	−0.881680	+	0.471848i	$0.843587\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.0000	+	5.19615i	0.830540	+	0.331970i
$246$		0			0
$247$	−2.00000	+	3.46410i	−0.127257	+	0.220416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.00000			−0.378717			−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000			−0.378717			−0.189358	+	0.981908i	$0.560641\pi$
$252$		0			0
$253$	−24.0000			−1.50887
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.0000	+	19.0526i	−0.686161	+	1.18847i	0.286909	+	0.957958i	$0.407372\pi$
$257$	−11.0000	+	19.0526i	−0.686161	+	1.18847i	−0.973070	+	0.230508i	$0.925961\pi$
$258$		0			0
$259$	−2.50000	+	0.866025i	−0.155342	+	0.0538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.0000	−	20.7846i	−0.739952	−	1.28163i	−0.952517	−	0.304487i	$-0.901515\pi$
$263$	−12.0000	−	20.7846i	−0.739952	−	1.28163i	0.212565	−	0.977147i	$-0.431818\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.00000	+	13.8564i	0.487769	+	0.844840i	0.999901	−	0.0140665i	$-0.00447764\pi$
$269$	8.00000	+	13.8564i	0.487769	+	0.844840i	−0.512132	+	0.858906i	$0.671144\pi$
$270$		0			0
$271$	−7.50000	+	12.9904i	−0.455593	+	0.789109i	−0.998722	−	0.0505395i	$-0.983906\pi$
$271$	−7.50000	+	12.9904i	−0.455593	+	0.789109i	0.543130	+	0.839649i	$0.317239\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	−	3.46410i	0.120605	−	0.208893i
$276$		0			0
$277$	2.50000	+	4.33013i	0.150210	+	0.260172i	0.931305	−	0.364241i	$-0.118672\pi$
$277$	2.50000	+	4.33013i	0.150210	+	0.260172i	−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.797283	−	0.603606i	$-0.793730\pi$
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.124096	−	0.992270i	$-0.539603\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.00000	+	6.92820i	0.472225	+	0.408959i
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.0000			−1.40209			−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000			−1.40209			−0.701047	+	0.713115i	$0.747284\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	−2.00000	−	1.73205i	−0.115278	−	0.0998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.00000	−	5.19615i	−0.171780	−	0.297531i
$306$		0			0
$307$	1.00000			0.0570730			0.0285365	−	0.999593i	$-0.490915\pi$
$307$	1.00000			0.0570730			0.0285365	+	0.999593i	$0.490915\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.00000	+	1.73205i	0.0567048	+	0.0982156i	0.892984	−	0.450088i	$-0.148607\pi$
$311$	1.00000	+	1.73205i	0.0567048	+	0.0982156i	−0.836280	+	0.548303i	$0.815274\pi$
$312$		0			0
$313$	1.00000	−	1.73205i	0.0565233	−	0.0979013i	−0.836379	−	0.548151i	$-0.815332\pi$
$313$	1.00000	−	1.73205i	0.0565233	−	0.0979013i	0.892903	+	0.450250i	$0.148665\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.00000	+	10.3923i	−0.336994	+	0.583690i	−0.983866	−	0.178908i	$-0.942743\pi$
$317$	−6.00000	+	10.3923i	−0.336994	+	0.583690i	0.646872	+	0.762598i	$0.276077\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.500000	−	0.866025i	−0.0277350	−	0.0480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	10.0000	−	3.46410i	0.551318	−	0.190982i
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−22.0000			−1.20199
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.0000	+	17.3205i	−0.541530	+	0.937958i
$342$		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	0.658824	−	0.752297i	$-0.271054\pi$
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	−0.980921	+	0.194409i	$0.937721\pi$
$348$		0			0
$349$	9.00000			0.481759			0.240879	−	0.970555i	$-0.422564\pi$
$349$	9.00000			0.481759			0.240879	+	0.970555i	$0.422564\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.00000	+	13.8564i	0.425797	+	0.737502i	0.996495	−	0.0836583i	$-0.0266604\pi$
$353$	8.00000	+	13.8564i	0.425797	+	0.737502i	−0.570697	+	0.821160i	$0.693327\pi$
$354$		0			0
$355$	14.0000	−	24.2487i	0.743043	−	1.28699i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.00000	+	15.5885i	−0.475002	+	0.822727i	−0.999590	−	0.0286287i	$-0.990886\pi$
$359$	−9.00000	+	15.5885i	−0.475002	+	0.822727i	0.524588	+	0.851356i	$0.324219\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	28.0000			1.46559
$366$		0			0
$367$	4.00000	+	6.92820i	0.208798	+	0.361649i	0.951336	−	0.308155i	$-0.0997115\pi$
$367$	4.00000	+	6.92820i	0.208798	+	0.361649i	−0.742538	+	0.669804i	$0.766378\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.00000	−	15.5885i	0.155752	−	0.809312i
$372$		0			0
$373$	−5.00000	+	8.66025i	−0.258890	+	0.448411i	−0.965945	−	0.258748i	$-0.916690\pi$
$373$	−5.00000	+	8.66025i	−0.258890	+	0.448411i	0.707055	+	0.707159i	$0.250023\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	23.0000			1.18143			0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000			1.18143			0.590715	+	0.806880i	$0.298846\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.0000	+	25.9808i	−0.766464	+	1.32755i	0.173005	+	0.984921i	$0.444652\pi$
$383$	−15.0000	+	25.9808i	−0.766464	+	1.32755i	−0.939469	+	0.342634i	$0.888681\pi$
$384$		0			0
$385$	−16.0000	−	13.8564i	−0.815436	−	0.706188i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.00000	+	3.46410i	0.101404	+	0.175637i	0.912263	−	0.409604i	$-0.134333\pi$
$389$	2.00000	+	3.46410i	0.101404	+	0.175637i	−0.810859	+	0.585241i	$0.801000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.0000	+	22.5167i	0.654101	+	1.13294i
$396$		0			0
$397$	−12.5000	+	21.6506i	−0.627357	+	1.08661i	0.360723	+	0.932673i	$0.382530\pi$
$397$	−12.5000	+	21.6506i	−0.627357	+	1.08661i	−0.988080	+	0.153941i	$0.950803\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	−0.976050	−	0.217545i	$-0.930195\pi$
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	0.676425	+	0.736512i	$0.263528\pi$
$402$		0			0
$403$	2.50000	+	4.33013i	0.124534	+	0.215699i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.00000			0.198273
$408$		0			0
$409$	−0.500000	−	0.866025i	−0.0247234	−	0.0428222i	0.853399	−	0.521258i	$-0.174537\pi$
$409$	−0.500000	−	0.866025i	−0.0247234	−	0.0428222i	−0.878122	+	0.478436i	$0.841204\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−14.0000	+	24.2487i	−0.687233	+	1.19032i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.0000			0.488532			0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000			0.488532			0.244266	+	0.969708i	$0.421453\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	7.50000	−	2.59808i	0.362950	−	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.0000	−	17.3205i	−0.481683	−	0.834300i	0.518096	−	0.855323i	$-0.326641\pi$
$431$	−10.0000	−	17.3205i	−0.481683	−	0.834300i	−0.999779	+	0.0210230i	$0.993308\pi$
$432$		0			0
$433$	31.0000			1.48976			0.744882	−	0.667196i	$-0.232506\pi$
$433$	31.0000			1.48976			0.744882	+	0.667196i	$0.232506\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0000	−	20.7846i	−0.574038	−	0.994263i
$438$		0			0
$439$	−12.0000	+	20.7846i	−0.572729	+	0.991995i	0.423556	+	0.905870i	$0.360782\pi$
$439$	−12.0000	+	20.7846i	−0.572729	+	0.991995i	−0.996284	+	0.0861252i	$0.972552\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	0.426414	+	0.904528i	$0.359777\pi$
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	−0.996551	+	0.0829786i	$0.973557\pi$
$444$		0			0
$445$	−6.00000	−	10.3923i	−0.284427	−	0.492642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.0000			0.660701			0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000			0.660701			0.330350	+	0.943858i	$0.392833\pi$
$450$		0			0
$451$	−8.00000	−	13.8564i	−0.376705	−	0.652473i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.00000	+	1.73205i	−0.234404	+	0.0811998i
$456$		0			0
$457$	11.5000	−	19.9186i	0.537947	−	0.931752i	−0.461067	−	0.887365i	$-0.652533\pi$
$457$	11.5000	−	19.9186i	0.537947	−	0.931752i	0.999014	−	0.0443868i	$-0.0141334\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.0000	+	22.5167i	−0.601568	+	1.04195i	0.391015	+	0.920384i	$0.372124\pi$
$467$	−13.0000	+	22.5167i	−0.601568	+	1.04195i	−0.992584	+	0.121563i	$0.961209\pi$
$468$		0			0
$469$	5.50000	−	28.5788i	0.253966	−	1.31965i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.00000	+	3.46410i	0.0919601	+	0.159280i
$474$		0			0
$475$	4.00000			0.183533
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	0.995023	−	0.0996406i	$-0.0317693\pi$
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	−0.583803	+	0.811895i	$0.698436\pi$
$480$		0			0
$481$	0.500000	−	0.866025i	0.0227980	−	0.0394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.00000	−	15.5885i	0.408669	−	0.707835i
$486$		0			0
$487$	8.00000	+	13.8564i	0.362515	+	0.627894i	0.988374	−	0.152042i	$-0.0485850\pi$
$487$	8.00000	+	13.8564i	0.362515	+	0.627894i	−0.625859	+	0.779936i	$0.715252\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	30.0000			1.35388			0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000			1.35388			0.676941	+	0.736038i	$0.263305\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	28.0000	+	24.2487i	1.25597	+	1.08770i
$498$		0			0
$499$	0.500000	−	0.866025i	0.0223831	−	0.0387686i	−0.854617	−	0.519259i	$-0.826208\pi$
$499$	0.500000	−	0.866025i	0.0223831	−	0.0387686i	0.877000	+	0.480490i	$0.159541\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−22.0000			−0.980932			−0.490466	−	0.871460i	$-0.663173\pi$
$503$	−22.0000			−0.980932			−0.490466	+	0.871460i	$0.663173\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.0000	+	32.9090i	−0.842160	+	1.45866i	0.0459045	+	0.998946i	$0.485383\pi$
$509$	−19.0000	+	32.9090i	−0.842160	+	1.45866i	−0.888065	+	0.459718i	$0.847950\pi$
$510$		0			0
$511$	−7.00000	+	36.3731i	−0.309662	+	1.60905i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.0000	−	22.5167i	−0.572848	−	0.992203i
$516$		0			0
$517$	−16.0000			−0.703679
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000	+	5.19615i	0.131432	+	0.227648i	0.924229	−	0.381839i	$-0.124709\pi$
$521$	3.00000	+	5.19615i	0.131432	+	0.227648i	−0.792797	+	0.609486i	$0.791376\pi$
$522$		0			0
$523$	8.50000	−	14.7224i	0.371679	−	0.643767i	−0.618145	−	0.786064i	$-0.712116\pi$
$523$	8.50000	−	14.7224i	0.371679	−	0.643767i	0.989824	+	0.142297i	$0.0454489\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.00000			−0.173259
$534$		0			0
$535$	−18.0000	−	31.1769i	−0.778208	−	1.34790i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22.0000	−	17.3205i	0.947607	−	0.746047i
$540$		0			0
$541$	−1.00000	+	1.73205i	−0.0429934	+	0.0744667i	−0.886721	−	0.462304i	$-0.847023\pi$
$541$	−1.00000	+	1.73205i	−0.0429934	+	0.0744667i	0.843728	+	0.536771i	$0.180356\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.00000			0.0856706
$546$		0			0
$547$	39.0000			1.66752			0.833760	−	0.552127i	$-0.186184\pi$
$547$	39.0000			1.66752			0.833760	+	0.552127i	$0.186184\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−32.5000	+	11.2583i	−1.38204	+	0.478753i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.0000	−	29.4449i	−0.720313	−	1.24762i	−0.960874	−	0.276985i	$-0.910665\pi$
$557$	−17.0000	−	29.4449i	−0.720313	−	1.24762i	0.240561	−	0.970634i	$-0.422669\pi$
$558$		0			0
$559$	1.00000			0.0422955
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.00000	−	8.66025i	−0.210725	−	0.364986i	0.741217	−	0.671266i	$-0.234249\pi$
$563$	−5.00000	−	8.66025i	−0.210725	−	0.364986i	−0.951942	+	0.306280i	$0.900916\pi$
$564$		0			0
$565$	10.0000	−	17.3205i	0.420703	−	0.728679i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.0000	−	32.9090i	0.796521	−	1.37962i	−0.125347	−	0.992113i	$-0.540004\pi$
$569$	19.0000	−	32.9090i	0.796521	−	1.37962i	0.921869	−	0.387503i	$-0.126662\pi$
$570$		0			0
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	0.821138	−	0.570730i	$-0.193340\pi$
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	−0.904835	+	0.425762i	$0.860006\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$	−9.50000	−	16.4545i	−0.395490	−	0.685009i	0.597673	−	0.801740i	$-0.296092\pi$
$577$	−9.50000	−	16.4545i	−0.395490	−	0.685009i	−0.993164	+	0.116731i	$0.962759\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−28.0000	−	24.2487i	−1.16164	−	1.00601i
$582$		0			0
$583$	−12.0000	+	20.7846i	−0.496989	+	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	30.0000			1.23823			0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000			1.23823			0.619116	+	0.785299i	$0.287491\pi$
$588$		0			0
$589$	−20.0000			−0.824086
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.00000	−	8.66025i	0.205325	−	0.355634i	−0.744911	−	0.667164i	$-0.767508\pi$
$593$	5.00000	−	8.66025i	0.205325	−	0.355634i	0.950236	+	0.311530i	$0.100841\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−37.0000			−1.50926			−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000			−1.50926			−0.754631	+	0.656150i	$0.772184\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.00000	+	8.66025i	0.203279	+	0.352089i
$606$		0			0
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	0.333842	+	0.942629i	$0.391655\pi$
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	−0.983262	+	0.182199i	$0.941678\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.00000	+	3.46410i	−0.0809113	+	0.140143i
$612$		0			0
$613$	1.50000	+	2.59808i	0.0605844	+	0.104935i	0.894727	−	0.446614i	$-0.147370\pi$
$613$	1.50000	+	2.59808i	0.0605844	+	0.104935i	−0.834142	+	0.551549i	$0.814037\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.00000			0.322068			0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000			0.322068			0.161034	+	0.986949i	$0.448517\pi$
$618$		0			0
$619$	2.50000	+	4.33013i	0.100483	+	0.174042i	0.911884	−	0.410448i	$-0.134628\pi$
$619$	2.50000	+	4.33013i	0.100483	+	0.174042i	−0.811400	+	0.584491i	$0.801294\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	15.0000	−	5.19615i	0.600962	−	0.208179i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	27.0000			1.07485			0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000			1.07485			0.537427	+	0.843311i	$0.319397\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.0000	+	19.0526i	−0.436522	+	0.756078i
$636$		0			0
$637$	−1.00000	−	6.92820i	−0.0396214	−	0.274505i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000	−	31.1769i	−0.710957	−	1.23141i	−0.964498	−	0.264089i	$-0.914929\pi$
$641$	−18.0000	−	31.1769i	−0.710957	−	1.23141i	0.253541	−	0.967325i	$-0.418405\pi$
$642$		0			0
$643$	19.0000			0.749287			0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000			0.749287			0.374643	+	0.927169i	$0.377765\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.0000	−	27.7128i	−0.629025	−	1.08950i	−0.987748	−	0.156059i	$-0.950121\pi$
$647$	−16.0000	−	27.7128i	−0.629025	−	1.08950i	0.358723	−	0.933444i	$-0.383212\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.0000	−	29.4449i	0.665261	−	1.15227i	−0.313953	−	0.949439i	$-0.601653\pi$
$653$	17.0000	−	29.4449i	0.665261	−	1.15227i	0.979214	−	0.202828i	$-0.0650132\pi$
$654$		0			0
$655$	16.0000	+	27.7128i	0.625172	+	1.08283i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.0000			0.701180			0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000			0.701180			0.350590	+	0.936529i	$0.385981\pi$
$660$		0			0
$661$	11.0000	+	19.0526i	0.427850	+	0.741059i	0.996682	−	0.0813955i	$-0.0259377\pi$
$661$	11.0000	+	19.0526i	0.427850	+	0.741059i	−0.568831	+	0.822454i	$0.692604\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.00000	−	20.7846i	0.155113	−	0.805993i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$	−50.0000			−1.92736			−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000			−1.92736			−0.963679	+	0.267063i	$0.913947\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	−0.279453	−	0.960159i	$-0.590153\pi$
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	0.971249	−	0.238067i	$-0.0765137\pi$
$678$		0			0
$679$	18.0000	+	15.5885i	0.690777	+	0.598230i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.00000	−	15.5885i	−0.344375	−	0.596476i	0.640865	−	0.767654i	$-0.278576\pi$
$683$	−9.00000	−	15.5885i	−0.344375	−	0.596476i	−0.985240	+	0.171178i	$0.945243\pi$
$684$		0			0
$685$	−36.0000			−1.37549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.00000	+	5.19615i	0.114291	+	0.197958i
$690$		0			0
$691$	−0.500000	+	0.866025i	−0.0190209	+	0.0329452i	−0.875379	−	0.483437i	$-0.839388\pi$
$691$	−0.500000	+	0.866025i	−0.0190209	+	0.0329452i	0.856358	+	0.516382i	$0.172722\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.00000	−	1.73205i	0.0379322	−	0.0657004i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$	2.00000	+	3.46410i	0.0754314	+	0.130651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.00000	−	25.9808i	0.188044	−	0.977107i
$708$		0			0
$709$	22.5000	−	38.9711i	0.845005	−	1.46359i	−0.0406114	−	0.999175i	$-0.512931\pi$
$709$	22.5000	−	38.9711i	0.845005	−	1.46359i	0.885617	−	0.464417i	$-0.153736\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−30.0000			−1.12351
$714$		0			0
$715$	8.00000			0.299183
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.0000	−	22.5167i	0.484818	−	0.839730i	−0.515030	−	0.857172i	$-0.672219\pi$
$719$	13.0000	−	22.5167i	0.484818	−	0.839730i	0.999848	+	0.0174426i	$0.00555244\pi$
$720$		0			0
$721$	32.5000	−	11.2583i	1.21036	−	0.419282i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−5.00000			−0.185440			−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000			−0.185440			−0.0927199	+	0.995692i	$0.529556\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−17.5000	+	30.3109i	−0.646377	+	1.11956i	0.337604	+	0.941288i	$0.390383\pi$
$733$	−17.5000	+	30.3109i	−0.646377	+	1.11956i	−0.983982	+	0.178270i	$0.942950\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22.0000	+	38.1051i	−0.810380	+	1.40362i
$738$		0			0
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	0.908337	−	0.418238i	$-0.137352\pi$
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	−0.816373	+	0.577524i	$0.804019\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.00000			−0.293492			−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000			−0.293492			−0.146746	+	0.989174i	$0.546880\pi$
$744$		0			0
$745$	6.00000	+	10.3923i	0.219823	+	0.380745i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	45.0000	−	15.5885i	1.64426	−	0.569590i
$750$		0			0
$751$	−14.0000	+	24.2487i	−0.510867	+	0.884848i	0.489053	+	0.872254i	$0.337342\pi$
$751$	−14.0000	+	24.2487i	−0.510867	+	0.884848i	−0.999921	+	0.0125942i	$0.995991\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.00000			−0.218362
$756$		0			0
$757$	−29.0000			−1.05402			−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−29.0000			−1.05402			−0.527011	+	0.849858i	$0.676688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−0.500000	+	2.59808i	−0.0181012	+	0.0940567i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	38.0000			1.37032			0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000			1.37032			0.685158	+	0.728395i	$0.259733\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.00000	−	3.46410i	−0.0719350	−	0.124595i	0.827814	−	0.561002i	$-0.189584\pi$
$773$	−2.00000	−	3.46410i	−0.0719350	−	0.124595i	−0.899749	+	0.436407i	$0.856251\pi$
$774$		0			0
$775$	2.50000	−	4.33013i	0.0898027	−	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.00000	−	13.8564i	0.286630	−	0.496457i
$780$		0			0
$781$	−28.0000	−	48.4974i	−1.00192	−	1.73537i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	36.0000			1.28490
$786$		0			0
$787$	−2.50000	−	4.33013i	−0.0891154	−	0.154352i	0.818022	−	0.575187i	$-0.195071\pi$
$787$	−2.50000	−	4.33013i	−0.0891154	−	0.154352i	−0.907137	+	0.420834i	$0.861737\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.0000	+	17.3205i	0.711118	+	0.615846i
$792$		0			0
$793$	−1.50000	+	2.59808i	−0.0532666	+	0.0922604i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.0000			0.425062			0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000			0.425062			0.212531	+	0.977154i	$0.431829\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	28.0000	−	48.4974i	0.988099	−	1.71144i
$804$		0			0
$805$	6.00000	−	31.1769i	0.211472	−	1.09884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.0000	−	19.0526i	−0.386739	−	0.669852i	0.605269	−	0.796021i	$-0.293065\pi$
$809$	−11.0000	−	19.0526i	−0.386739	−	0.669852i	−0.992009	+	0.126168i	$0.959732\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21.0000	−	36.3731i	−0.735598	−	1.27409i
$816$		0			0
$817$	−2.00000	+	3.46410i	−0.0699711	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.00000	+	8.66025i	−0.174501	+	0.302245i	−0.939989	−	0.341206i	$-0.889165\pi$
$821$	−5.00000	+	8.66025i	−0.174501	+	0.302245i	0.765487	+	0.643451i	$0.222498\pi$
$822$		0			0
$823$	10.5000	+	18.1865i	0.366007	+	0.633943i	0.988937	−	0.148335i	$-0.0473913\pi$
$823$	10.5000	+	18.1865i	0.366007	+	0.633943i	−0.622930	+	0.782277i	$0.714058\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−44.0000			−1.53003			−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000			−1.53003			−0.765015	+	0.644013i	$0.777268\pi$
$828$		0			0
$829$	−13.0000	−	22.5167i	−0.451509	−	0.782036i	0.546971	−	0.837151i	$-0.315781\pi$
$829$	−13.0000	−	22.5167i	−0.451509	−	0.782036i	−0.998480	+	0.0551154i	$0.982447\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	2.00000	−	3.46410i	0.0692129	−	0.119880i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	42.0000			1.45000			0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000			1.45000			0.725001	+	0.688748i	$0.241839\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0000	+	20.7846i	−0.412813	+	0.715012i
$846$		0			0
$847$	−12.5000	+	4.33013i	−0.429505	+	0.148785i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.00000	+	5.19615i	0.102839	+	0.178122i
$852$		0			0
$853$	38.0000			1.30110			0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000			1.30110			0.650548	+	0.759465i	$0.274539\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.0000	+	36.3731i	0.717346	+	1.24248i	0.962048	+	0.272882i	$0.0879768\pi$
$857$	21.0000	+	36.3731i	0.717346	+	1.24248i	−0.244701	+	0.969599i	$0.578690\pi$
$858$		0			0
$859$	24.5000	−	42.4352i	0.835929	−	1.44787i	−0.0573424	−	0.998355i	$-0.518263\pi$
$859$	24.5000	−	42.4352i	0.835929	−	1.44787i	0.893272	−	0.449517i	$-0.148404\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.0000	−	24.2487i	0.476566	−	0.825436i	−0.523074	−	0.852287i	$-0.675215\pi$
$863$	14.0000	−	24.2487i	0.476566	−	0.825436i	0.999639	+	0.0268516i	$0.00854816\pi$
$864$		0			0
$865$	24.0000	+	41.5692i	0.816024	+	1.41340i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	52.0000			1.76398
$870$		0			0
$871$	5.50000	+	9.52628i	0.186360	+	0.322786i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.0000	+	20.7846i	0.811348	+	0.702648i
$876$		0			0
$877$	13.5000	−	23.3827i	0.455863	−	0.789577i	−0.542875	−	0.839814i	$-0.682664\pi$
$877$	13.5000	−	23.3827i	0.455863	−	0.789577i	0.998737	+	0.0502365i	$0.0159975\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−23.0000	+	39.8372i	−0.772264	+	1.33760i	0.164055	+	0.986451i	$0.447543\pi$
$887$	−23.0000	+	39.8372i	−0.772264	+	1.33760i	−0.936319	+	0.351150i	$0.885791\pi$
$888$		0			0
$889$	−22.0000	−	19.0526i	−0.737856	−	0.639002i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.00000	−	13.8564i	−0.267710	−	0.463687i
$894$		0			0
$895$	4.00000			0.133705
$896$		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
$907$	−11.5000	−	19.9186i	−0.381851	−	0.661386i	0.609476	−	0.792805i	$-0.291380\pi$
$907$	−11.5000	−	19.9186i	−0.381851	−	0.661386i	−0.991327	+	0.131419i	$0.958047\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	28.0000	+	48.4974i	0.926665	+	1.60503i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−40.0000	+	13.8564i	−1.32092	+	0.457579i
$918$		0			0
$919$	−5.50000	+	9.52628i	−0.181428	+	0.314243i	−0.942367	−	0.334581i	$-0.891405\pi$
$919$	−5.50000	+	9.52628i	−0.181428	+	0.314243i	0.760939	+	0.648824i	$0.224739\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−14.0000			−0.460816
$924$		0			0
$925$	−1.00000			−0.0328798
$926$		0			0
$927$		0			0
$928$		0			0
$929$	29.0000	−	50.2295i	0.951459	−	1.64798i	0.209189	−	0.977875i	$-0.432918\pi$
$929$	29.0000	−	50.2295i	0.951459	−	1.64798i	0.742271	−	0.670100i	$-0.233749\pi$
$930$		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
$937$	43.0000			1.40475			0.702374	−	0.711808i	$-0.252123\pi$
$937$	43.0000			1.40475			0.702374	+	0.711808i	$0.252123\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.00000	+	10.3923i	0.195594	+	0.338779i	0.947095	−	0.320953i	$-0.104003\pi$
$941$	6.00000	+	10.3923i	0.195594	+	0.338779i	−0.751501	+	0.659732i	$0.770670\pi$
$942$		0			0
$943$	12.0000	−	20.7846i	0.390774	−	0.676840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.00000	−	3.46410i	0.0649913	−	0.112568i	−0.831699	−	0.555227i	$-0.812631\pi$
$947$	2.00000	−	3.46410i	0.0649913	−	0.112568i	0.896690	+	0.442659i	$0.145965\pi$
$948$		0			0
$949$	−7.00000	−	12.1244i	−0.227230	−	0.393573i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$	22.0000	+	38.1051i	0.711903	+	1.23305i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.00000	−	46.7654i	0.290625	−	1.51013i
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	34.0000			1.09450
$966$		0			0
$967$	−19.0000			−0.610999			−0.305499	−	0.952192i	$-0.598823\pi$
$967$	−19.0000			−0.610999			−0.305499	+	0.952192i	$0.598823\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$	2.00000	+	1.73205i	0.0641171	+	0.0555270i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−29.0000	−	50.2295i	−0.927792	−	1.60698i	−0.787008	−	0.616943i	$-0.788371\pi$
$977$	−29.0000	−	50.2295i	−0.927792	−	1.60698i	−0.140784	−	0.990040i	$-0.544962\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.00000	+	5.19615i	0.0956851	+	0.165732i	0.909894	−	0.414840i	$-0.136162\pi$
$983$	3.00000	+	5.19615i	0.0956851	+	0.165732i	−0.814209	+	0.580572i	$0.802829\pi$
$984$		0			0
$985$	12.0000	−	20.7846i	0.382352	−	0.662253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.00000	+	5.19615i	−0.0953945	+	0.165228i
$990$		0			0
$991$	−23.5000	−	40.7032i	−0.746502	−	1.29298i	−0.949490	−	0.313798i	$-0.898398\pi$
$991$	−23.5000	−	40.7032i	−0.746502	−	1.29298i	0.202988	−	0.979181i	$-0.434935\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	30.0000			0.951064
$996$		0			0
$997$	23.5000	+	40.7032i	0.744252	+	1.28908i	0.950543	+	0.310592i	$0.100527\pi$
$997$	23.5000	+	40.7032i	0.744252	+	1.28908i	−0.206291	+	0.978491i	$0.566139\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.s.h.1297.1	yes	2
3.2	odd	2		1512.2.s.d.1297.1	yes	2
7.4	even	3	inner	1512.2.s.h.865.1	yes	2
21.11	odd	6		1512.2.s.d.865.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.s.d.865.1	✓	2	21.11	odd	6
1512.2.s.d.1297.1	yes	2	3.2	odd	2
1512.2.s.h.865.1	yes	2	7.4	even	3	inner
1512.2.s.h.1297.1	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-2.00000 - 3.46410i) q^{11} -1.00000 q^{13} +(2.00000 - 3.46410i) q^{19} +(3.00000 - 5.19615i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-2.50000 - 4.33013i) q^{31} +(5.00000 - 1.73205i) q^{35} +(-0.500000 + 0.866025i) q^{37} +4.00000 q^{41} -1.00000 q^{43} +(2.00000 - 3.46410i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-3.00000 - 5.19615i) q^{53} -8.00000 q^{55} +(1.50000 - 2.59808i) q^{61} +(-1.00000 + 1.73205i) q^{65} +(-5.50000 - 9.52628i) q^{67} +14.0000 q^{71} +(7.00000 + 12.1244i) q^{73} +(2.00000 - 10.3923i) q^{77} +(-6.50000 + 11.2583i) q^{79} -14.0000 q^{83} +(3.00000 - 5.19615i) q^{89} +(-2.00000 - 1.73205i) q^{91} +(-4.00000 - 6.92820i) q^{95} +9.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 4 * q^7	\( 2 q + 2 q^{5} + 4 q^{7} - 4 q^{11} - 2 q^{13} + 4 q^{19} + 6 q^{23} + q^{25} - 5 q^{31} + 10 q^{35} - q^{37} + 8 q^{41} - 2 q^{43} + 4 q^{47} + 2 q^{49} - 6 q^{53} - 16 q^{55} + 3 q^{61} - 2 q^{65} - 11 q^{67} + 28 q^{71} + 14 q^{73} + 4 q^{77} - 13 q^{79} - 28 q^{83} + 6 q^{89} - 4 q^{91} - 8 q^{95} + 18 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 4 * q^7 - 4 * q^11 - 2 * q^13 + 4 * q^19 + 6 * q^23 + q^25 - 5 * q^31 + 10 * q^35 - q^37 + 8 * q^41 - 2 * q^43 + 4 * q^47 + 2 * q^49 - 6 * q^53 - 16 * q^55 + 3 * q^61 - 2 * q^65 - 11 * q^67 + 28 * q^71 + 14 * q^73 + 4 * q^77 - 13 * q^79 - 28 * q^83 + 6 * q^89 - 4 * q^91 - 8 * q^95 + 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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