LMFDB - Embedded newform 2736.2.bm.b.1855.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(559,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.bm (of order $6$, 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:	$21.8470699930$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1855.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2736.1855
Dual form			2736.2.bm.b.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{7} +3.46410i q^{11} +(-4.50000 - 2.59808i) q^{13} +(3.00000 + 5.19615i) q^{17} +(4.00000 - 1.73205i) q^{19} +(-3.00000 - 1.73205i) q^{23} +(2.50000 - 4.33013i) q^{25} +(3.00000 + 1.73205i) q^{29} -1.00000 q^{31} +8.66025i q^{37} +(6.00000 - 3.46410i) q^{41} +(4.50000 - 2.59808i) q^{43} +(-9.00000 - 5.19615i) q^{47} +4.00000 q^{49} +(9.00000 + 5.19615i) q^{53} +(6.00000 + 10.3923i) q^{59} +(2.50000 - 4.33013i) q^{61} +(6.50000 - 11.2583i) q^{67} +(2.50000 + 4.33013i) q^{73} +6.00000 q^{77} +(-0.500000 - 0.866025i) q^{79} -17.3205i q^{83} +(-4.50000 + 7.79423i) q^{91} +(12.0000 - 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{13} + 6 q^{17} + 8 q^{19} - 6 q^{23} + 5 q^{25} + 6 q^{29} - 2 q^{31} + 12 q^{41} + 9 q^{43} - 18 q^{47} + 8 q^{49} + 18 q^{53} + 12 q^{59} + 5 q^{61} + 13 q^{67} + 5 q^{73} + 12 q^{77} - q^{79}+ \cdots + 24 q^{97}+O(q^{100}) $ 2 * q - 9 * q^13 + 6 * q^17 + 8 * q^19 - 6 * q^23 + 5 * q^25 + 6 * q^29 - 2 * q^31 + 12 * q^41 + 9 * q^43 - 18 * q^47 + 8 * q^49 + 18 * q^53 + 12 * q^59 + 5 * q^61 + 13 * q^67 + 5 * q^73 + 12 * q^77 - q^79 - 9 * q^91 + 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$-1$	$1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$		−	1.73205i		−	0.654654i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$		−	1.73205i		−	0.654654i	0.944911	−	0.327327i	$-0.106148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	−4.50000	−	2.59808i	−1.24808	−	0.720577i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−4.50000	−	2.59808i	−1.24808	−	0.720577i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	−0.230285	+	0.973123i	$0.573966\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	1.73205i	−0.625543	−	0.361158i	0.153481	−	0.988152i	$-0.450952\pi$
$23$	−3.00000	−	1.73205i	−0.625543	−	0.361158i	−0.779024	+	0.626994i	$0.784285\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000	+	1.73205i	0.557086	+	0.321634i	0.751975	−	0.659192i	$-0.229101\pi$
$29$	3.00000	+	1.73205i	0.557086	+	0.321634i	−0.194889	+	0.980825i	$0.562435\pi$
$30$		0			0
$31$	−1.00000			−0.179605			−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000			−0.179605			−0.0898027	+	0.995960i	$0.528624\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.66025i			1.42374i	0.702313	+	0.711868i	$0.252151\pi$
$37$			8.66025i			1.42374i	−0.702313	+	0.711868i	$0.747849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000	−	3.46410i	0.937043	−	0.541002i	0.0480106	−	0.998847i	$-0.484712\pi$
$41$	6.00000	−	3.46410i	0.937043	−	0.541002i	0.889032	+	0.457845i	$0.151379\pi$
$42$		0			0
$43$	4.50000	−	2.59808i	0.686244	−	0.396203i	−0.115960	−	0.993254i	$-0.536994\pi$
$43$	4.50000	−	2.59808i	0.686244	−	0.396203i	0.802203	+	0.597051i	$0.203661\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.00000	−	5.19615i	−1.31278	−	0.757937i	−0.330228	−	0.943901i	$-0.607126\pi$
$47$	−9.00000	−	5.19615i	−1.31278	−	0.757937i	−0.982556	+	0.185964i	$0.940459\pi$
$48$		0			0
$49$	4.00000			0.571429
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.00000	+	5.19615i	1.23625	+	0.713746i	0.968325	−	0.249695i	$-0.0803302\pi$
$53$	9.00000	+	5.19615i	1.23625	+	0.713746i	0.267920	+	0.963441i	$0.413664\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	$-0.729619\pi$
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	0.980507	+	0.196485i	$0.0629528\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.50000	−	11.2583i	0.794101	−	1.37542i	−0.129307	−	0.991605i	$-0.541275\pi$
$67$	6.50000	−	11.2583i	0.794101	−	1.37542i	0.923408	−	0.383819i	$-0.125391\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$		0			0
$73$	2.50000	+	4.33013i	0.292603	+	0.506803i	0.974424	−	0.224716i	$-0.0721453\pi$
$73$	2.50000	+	4.33013i	0.292603	+	0.506803i	−0.681822	+	0.731519i	$0.738812\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000			0.683763
$78$		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
$83$		−	17.3205i		−	1.90117i	−0.310460	−	0.950586i	$-0.600483\pi$
$83$		−	17.3205i		−	1.90117i	0.310460	−	0.950586i	$-0.399517\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−4.50000	+	7.79423i	−0.471728	+	0.817057i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.0000	−	6.92820i	1.21842	−	0.703452i	0.253837	−	0.967247i	$-0.418307\pi$
$97$	12.0000	−	6.92820i	1.21842	−	0.703452i	0.964579	+	0.263795i	$0.0849741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	1.00000			0.0985329			0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000			0.0985329			0.0492665	+	0.998786i	$0.484312\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−12.0000	+	6.92820i	−1.14939	+	0.663602i	−0.948739	−	0.316061i	$-0.897640\pi$
$109$	−12.0000	+	6.92820i	−1.14939	+	0.663602i	−0.200653	+	0.979662i	$0.564306\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.92820i		−	0.651751i	−0.945413	−	0.325875i	$-0.894341\pi$
$113$		−	6.92820i		−	0.651751i	0.945413	−	0.325875i	$-0.105659\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.00000	−	5.19615i	0.825029	−	0.476331i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.00000	−	13.8564i	0.709885	−	1.22956i	−0.255014	−	0.966937i	$-0.582080\pi$
$127$	8.00000	−	13.8564i	0.709885	−	1.22956i	0.964899	−	0.262620i	$-0.0845865\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	−3.00000	−	6.92820i	−0.260133	−	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	0.487278	+	0.873247i	$0.337990\pi$
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	−0.999893	+	0.0146279i	$0.995344\pi$
$138$		0			0
$139$	−7.50000	−	4.33013i	−0.636142	−	0.367277i	0.146985	−	0.989139i	$-0.453043\pi$
$139$	−7.50000	−	4.33013i	−0.636142	−	0.367277i	−0.783127	+	0.621862i	$0.786376\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.00000	−	15.5885i	0.752618	−	1.30357i
$144$		0			0
$145$		0			0
$146$		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
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.50000	+	14.7224i	0.678374	+	1.17498i	0.975470	+	0.220131i	$0.0706483\pi$
$157$	8.50000	+	14.7224i	0.678374	+	1.17498i	−0.297097	+	0.954847i	$0.596018\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.00000	+	5.19615i	−0.236433	+	0.409514i
$162$		0			0
$163$			22.5167i			1.76364i	0.471585	+	0.881820i	$0.343682\pi$
$163$			22.5167i			1.76364i	−0.471585	+	0.881820i	$0.656318\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	7.00000	+	12.1244i	0.538462	+	0.932643i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	21.0000	−	12.1244i	1.59660	−	0.921798i	0.604465	−	0.796632i	$-0.293387\pi$
$173$	21.0000	−	12.1244i	1.59660	−	0.921798i	0.992136	−	0.125166i	$-0.0399462\pi$
$174$		0			0
$175$	−7.50000	−	4.33013i	−0.566947	−	0.327327i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.00000			0.448461			0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000			0.448461			0.224231	+	0.974536i	$0.428013\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−18.0000	+	10.3923i	−1.31629	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			6.92820i			0.501307i	0.968077	+	0.250654i	$0.0806455\pi$
$191$			6.92820i			0.501307i	−0.968077	+	0.250654i	$0.919354\pi$
$192$		0			0
$193$	13.5000	−	7.79423i	0.971751	−	0.561041i	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	13.5000	−	7.79423i	0.971751	−	0.561041i	0.899770	+	0.436365i	$0.143734\pi$
$194$		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
$199$	10.5000	+	6.06218i	0.744325	+	0.429736i	0.823640	−	0.567113i	$-0.191940\pi$
$199$	10.5000	+	6.06218i	0.744325	+	0.429736i	−0.0793146	+	0.996850i	$0.525273\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.00000	−	5.19615i	0.210559	−	0.364698i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.00000	+	13.8564i	0.415029	+	0.958468i
$210$		0			0
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			1.73205i			0.117579i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	31.1769i		−	2.09719i
$222$		0			0
$223$	5.50000	+	9.52628i	0.368307	+	0.637927i	0.989301	−	0.145889i	$-0.0466041\pi$
$223$	5.50000	+	9.52628i	0.368307	+	0.637927i	−0.620994	+	0.783815i	$0.713271\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	23.0000			1.51988			0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000			1.51988			0.759941	+	0.649992i	$0.225228\pi$
$230$		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
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.92820i		−	0.448148i	−0.974572	−	0.224074i	$-0.928064\pi$
$239$		−	6.92820i		−	0.448148i	0.974572	−	0.224074i	$-0.0719358\pi$
$240$		0			0
$241$	7.50000	+	4.33013i	0.483117	+	0.278928i	0.721715	−	0.692191i	$-0.243354\pi$
$241$	7.50000	+	4.33013i	0.483117	+	0.278928i	−0.238597	+	0.971119i	$0.576688\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−22.5000	−	2.59808i	−1.43164	−	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.0000	−	8.66025i	−0.946792	−	0.546630i	−0.0547088	−	0.998502i	$-0.517423\pi$
$251$	−15.0000	−	8.66025i	−0.946792	−	0.546630i	−0.892083	+	0.451872i	$0.850756\pi$
$252$		0			0
$253$	6.00000	−	10.3923i	0.377217	−	0.653359i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.00000	+	1.73205i	0.187135	+	0.108042i	0.590641	−	0.806935i	$-0.298875\pi$
$257$	3.00000	+	1.73205i	0.187135	+	0.108042i	−0.403506	+	0.914977i	$0.632208\pi$
$258$		0			0
$259$	15.0000			0.932055
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.0000	+	10.3923i	−1.10993	+	0.640817i	−0.938811	−	0.344434i	$-0.888071\pi$
$263$	−18.0000	+	10.3923i	−1.10993	+	0.640817i	−0.171117	+	0.985251i	$0.554738\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000	+	3.46410i	−0.365826	+	0.211210i	−0.671634	−	0.740883i	$-0.734407\pi$
$269$	−6.00000	+	3.46410i	−0.365826	+	0.211210i	0.305807	+	0.952093i	$0.401074\pi$
$270$		0			0
$271$	−3.00000	+	1.73205i	−0.182237	+	0.105215i	−0.588343	−	0.808611i	$-0.700220\pi$
$271$	−3.00000	+	1.73205i	−0.182237	+	0.105215i	0.406106	+	0.913826i	$0.366886\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.0000	+	8.66025i	0.904534	+	0.522233i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000	+	3.46410i	0.357930	+	0.206651i	0.668172	−	0.744007i	$-0.267077\pi$
$281$	6.00000	+	3.46410i	0.357930	+	0.206651i	−0.310242	+	0.950657i	$0.600410\pi$
$282$		0			0
$283$	−3.00000	+	1.73205i	−0.178331	+	0.102960i	−0.586509	−	0.809943i	$-0.699498\pi$
$283$	−3.00000	+	1.73205i	−0.178331	+	0.102960i	0.408177	+	0.912903i	$0.366165\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000	−	10.3923i	−0.354169	−	0.613438i
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			24.2487i			1.41662i	0.705899	+	0.708312i	$0.250543\pi$
$293$			24.2487i			1.41662i	−0.705899	+	0.708312i	$0.749457\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.00000	+	15.5885i	0.520483	+	0.901504i
$300$		0			0
$301$	−4.50000	−	7.79423i	−0.259376	−	0.449252i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2.00000	−	3.46410i	−0.114146	−	0.197707i	0.803292	−	0.595585i	$-0.203080\pi$
$307$	−2.00000	−	3.46410i	−0.114146	−	0.197707i	−0.917438	+	0.397879i	$0.869747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	20.7846i		−	1.17859i	−0.807919	−	0.589294i	$-0.799406\pi$
$311$		−	20.7846i		−	1.17859i	0.807919	−	0.589294i	$-0.200594\pi$
$312$		0			0
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	0.689412	+	0.724370i	$0.257869\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−24.0000	−	13.8564i	−1.34797	−	0.778253i	−0.360012	−	0.932948i	$-0.617227\pi$
$317$	−24.0000	−	13.8564i	−1.34797	−	0.778253i	−0.987962	+	0.154694i	$0.950561\pi$
$318$		0			0
$319$	−6.00000	+	10.3923i	−0.335936	+	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	21.0000	+	15.5885i	1.16847	+	0.867365i
$324$		0			0
$325$	−22.5000	+	12.9904i	−1.24808	+	0.720577i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9.00000	+	15.5885i	−0.496186	+	0.859419i
$330$		0			0
$331$	25.0000			1.37412			0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000			1.37412			0.687062	+	0.726599i	$0.258900\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.50000	+	4.33013i	−0.408551	+	0.235877i	−0.690167	−	0.723650i	$-0.742463\pi$
$337$	−7.50000	+	4.33013i	−0.408551	+	0.235877i	0.281616	+	0.959527i	$0.409130\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	3.46410i		−	0.187592i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000	+	6.92820i	−0.644194	+	0.371925i	−0.786228	−	0.617936i	$-0.787969\pi$
$347$	−12.0000	+	6.92820i	−0.644194	+	0.371925i	0.142034	+	0.989862i	$0.454636\pi$
$348$		0			0
$349$	−19.0000			−1.01705			−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000			−1.01705			−0.508523	+	0.861048i	$0.669808\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−24.0000			−1.27739			−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000			−1.27739			−0.638696	+	0.769460i	$0.720526\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.0000	+	10.3923i	−0.950004	+	0.548485i	−0.893082	−	0.449894i	$-0.851462\pi$
$359$	−18.0000	+	10.3923i	−0.950004	+	0.548485i	−0.0569216	+	0.998379i	$0.518129\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.50000	−	0.866025i	−0.0782994	−	0.0452062i	0.460339	−	0.887743i	$-0.347728\pi$
$367$	−1.50000	−	0.866025i	−0.0782994	−	0.0452062i	−0.538639	+	0.842537i	$0.681061\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.00000	−	15.5885i	0.467257	−	0.809312i
$372$		0			0
$373$		−	13.8564i		−	0.717458i	−0.933442	−	0.358729i	$-0.883210\pi$
$373$		−	13.8564i		−	0.717458i	0.933442	−	0.358729i	$-0.116790\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9.00000	−	15.5885i	−0.463524	−	0.802846i
$378$		0			0
$379$	25.0000			1.28416			0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000			1.28416			0.642082	+	0.766636i	$0.278071\pi$
$380$		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
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$		−	20.7846i		−	1.05112i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	3.50000	+	6.06218i	0.175660	+	0.304252i	0.940389	−	0.340099i	$-0.110461\pi$
$397$	3.50000	+	6.06218i	0.175660	+	0.304252i	−0.764730	+	0.644351i	$0.777127\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−24.0000	+	13.8564i	−1.19850	+	0.691956i	−0.960221	−	0.279240i	$-0.909917\pi$
$401$	−24.0000	+	13.8564i	−1.19850	+	0.691956i	−0.238282	+	0.971196i	$0.576584\pi$
$402$		0			0
$403$	4.50000	+	2.59808i	0.224161	+	0.129419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−30.0000			−1.48704
$408$		0			0
$409$	−18.0000	−	10.3923i	−0.890043	−	0.513866i	−0.0160862	−	0.999871i	$-0.505121\pi$
$409$	−18.0000	−	10.3923i	−0.890043	−	0.513866i	−0.873956	+	0.486004i	$0.838454\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	18.0000	−	10.3923i	0.885722	−	0.511372i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			31.1769i			1.52309i	0.648111	+	0.761546i	$0.275559\pi$
$419$			31.1769i			1.52309i	−0.648111	+	0.761546i	$0.724441\pi$
$420$		0			0
$421$	18.0000	−	10.3923i	0.877266	−	0.506490i	0.00751023	−	0.999972i	$-0.497609\pi$
$421$	18.0000	−	10.3923i	0.877266	−	0.506490i	0.869756	+	0.493482i	$0.164276\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	30.0000			1.45521
$426$		0			0
$427$	−7.50000	−	4.33013i	−0.362950	−	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	0.237460	+	0.971397i	$0.423685\pi$
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	−0.959985	+	0.280052i	$0.909648\pi$
$432$		0			0
$433$	−22.5000	−	12.9904i	−1.08128	−	0.624278i	−0.150039	−	0.988680i	$-0.547940\pi$
$433$	−22.5000	−	12.9904i	−1.08128	−	0.624278i	−0.931242	+	0.364402i	$0.881273\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.0000	−	1.73205i	−0.717547	−	0.0828552i
$438$		0			0
$439$	−9.50000	−	16.4545i	−0.453410	−	0.785330i	0.545185	−	0.838316i	$-0.316459\pi$
$439$	−9.50000	−	16.4545i	−0.453410	−	0.785330i	−0.998595	+	0.0529862i	$0.983126\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.0000	+	12.1244i	0.997740	+	0.576046i	0.907579	−	0.419882i	$-0.137928\pi$
$443$	21.0000	+	12.1244i	0.997740	+	0.576046i	0.0901612	+	0.995927i	$0.471262\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	10.3923i		−	0.490443i	−0.969467	−	0.245222i	$-0.921139\pi$
$449$		−	10.3923i		−	0.490443i	0.969467	−	0.245222i	$-0.0788607\pi$
$450$		0			0
$451$	12.0000	+	20.7846i	0.565058	+	0.978709i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.00000			−0.233890			−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000			−0.233890			−0.116945	+	0.993138i	$0.537310\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.00000	+	10.3923i	0.279448	+	0.484018i	0.971248	−	0.238071i	$-0.0765153\pi$
$461$	6.00000	+	10.3923i	0.279448	+	0.484018i	−0.691800	+	0.722089i	$0.743182\pi$
$462$		0			0
$463$		−	5.19615i		−	0.241486i	−0.992684	−	0.120743i	$-0.961472\pi$
$463$		−	5.19615i		−	0.241486i	0.992684	−	0.120743i	$-0.0385276\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−19.5000	−	11.2583i	−0.900426	−	0.519861i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.00000	+	15.5885i	0.413820	+	0.716758i
$474$		0			0
$475$	2.50000	−	21.6506i	0.114708	−	0.993399i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.0000	+	6.92820i	0.548294	+	0.316558i	0.748434	−	0.663210i	$-0.230806\pi$
$479$	12.0000	+	6.92820i	0.548294	+	0.316558i	−0.200140	+	0.979767i	$0.564140\pi$
$480$		0			0
$481$	22.5000	−	38.9711i	1.02591	−	1.77693i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.0000	+	6.92820i	−0.541552	+	0.312665i	−0.745708	−	0.666273i	$-0.767889\pi$
$491$	−12.0000	+	6.92820i	−0.541552	+	0.312665i	0.204155	+	0.978938i	$0.434555\pi$
$492$		0			0
$493$			20.7846i			0.936092i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	7.50000	−	4.33013i	0.335746	−	0.193843i	−0.322643	−	0.946521i	$-0.604571\pi$
$499$	7.50000	−	4.33013i	0.335746	−	0.193843i	0.658389	+	0.752678i	$0.271238\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.0000	−	8.66025i	−0.668817	−	0.386142i	0.126811	−	0.991927i	$-0.459526\pi$
$503$	−15.0000	−	8.66025i	−0.668817	−	0.386142i	−0.795628	+	0.605785i	$0.792859\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.0000	+	12.1244i	0.930809	+	0.537403i	0.887067	−	0.461640i	$-0.152739\pi$
$509$	21.0000	+	12.1244i	0.930809	+	0.537403i	0.0437414	+	0.999043i	$0.486072\pi$
$510$		0			0
$511$	7.50000	−	4.33013i	0.331780	−	0.191554i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	18.0000	−	31.1769i	0.791639	−	1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	3.46410i		−	0.151765i	−0.997117	−	0.0758825i	$-0.975823\pi$
$521$		−	3.46410i		−	0.151765i	0.997117	−	0.0758825i	$-0.0241774\pi$
$522$		0			0
$523$	−14.5000	+	25.1147i	−0.634041	+	1.09819i	0.352677	+	0.935745i	$0.385272\pi$
$523$	−14.5000	+	25.1147i	−0.634041	+	1.09819i	−0.986718	+	0.162446i	$0.948062\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.00000	−	5.19615i	−0.130682	−	0.226348i
$528$		0			0
$529$	−5.50000	−	9.52628i	−0.239130	−	0.414186i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−36.0000			−1.55933
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			13.8564i			0.596838i
$540$		0			0
$541$	−12.5000	+	21.6506i	−0.537417	+	0.930834i	0.461625	+	0.887075i	$0.347267\pi$
$541$	−12.5000	+	21.6506i	−0.537417	+	0.930834i	−0.999042	+	0.0437584i	$0.986067\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−6.50000	+	11.2583i	−0.277920	+	0.481371i	−0.970868	−	0.239616i	$-0.922978\pi$
$547$	−6.50000	+	11.2583i	−0.277920	+	0.481371i	0.692948	+	0.720988i	$0.256312\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	15.0000	+	1.73205i	0.639021	+	0.0737878i
$552$		0			0
$553$	−1.50000	+	0.866025i	−0.0637865	+	0.0368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.0000	+	20.7846i	−0.508456	+	0.880672i	0.491496	+	0.870880i	$0.336450\pi$
$557$	−12.0000	+	20.7846i	−0.508456	+	0.880672i	−0.999952	+	0.00979220i	$0.996883\pi$
$558$		0			0
$559$	−27.0000			−1.14198
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	34.6410i		−	1.45223i	−0.687575	−	0.726113i	$-0.741325\pi$
$569$		−	34.6410i		−	1.45223i	0.687575	−	0.726113i	$-0.258675\pi$
$570$		0			0
$571$			32.9090i			1.37720i	0.725143	+	0.688599i	$0.241774\pi$
$571$			32.9090i			1.37720i	−0.725143	+	0.688599i	$0.758226\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−15.0000	+	8.66025i	−0.625543	+	0.361158i
$576$		0			0
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−30.0000			−1.24461
$582$		0			0
$583$	−18.0000	+	31.1769i	−0.745484	+	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−36.0000	+	20.7846i	−1.48588	+	0.857873i	−0.999871	−	0.0160815i	$-0.994881\pi$
$587$	−36.0000	+	20.7846i	−1.48588	+	0.857873i	−0.486008	+	0.873954i	$0.661548\pi$
$588$		0			0
$589$	−4.00000	+	1.73205i	−0.164817	+	0.0713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12.0000	+	20.7846i	−0.492781	+	0.853522i	−0.999965	−	0.00831589i	$-0.997353\pi$
$593$	−12.0000	+	20.7846i	−0.492781	+	0.853522i	0.507184	+	0.861838i	$0.330686\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−21.0000	+	36.3731i	−0.858037	+	1.48616i	0.0157622	+	0.999876i	$0.494983\pi$
$599$	−21.0000	+	36.3731i	−0.858037	+	1.48616i	−0.873799	+	0.486287i	$0.838351\pi$
$600$		0			0
$601$			46.7654i			1.90760i	0.300443	+	0.953800i	$0.402865\pi$
$601$			46.7654i			1.90760i	−0.300443	+	0.953800i	$0.597135\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	47.0000			1.90767			0.953836	−	0.300329i	$-0.0970966\pi$
$607$	47.0000			1.90767			0.953836	+	0.300329i	$0.0970966\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	27.0000	+	46.7654i	1.09230	+	1.89192i
$612$		0			0
$613$	19.0000	+	32.9090i	0.767403	+	1.32918i	0.938967	+	0.344008i	$0.111785\pi$
$613$	19.0000	+	32.9090i	0.767403	+	1.32918i	−0.171564	+	0.985173i	$0.554882\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.0000	−	36.3731i	0.845428	−	1.46432i	−0.0398207	−	0.999207i	$-0.512679\pi$
$617$	21.0000	−	36.3731i	0.845428	−	1.46432i	0.885249	−	0.465118i	$-0.153988\pi$
$618$		0			0
$619$		−	39.8372i		−	1.60119i	−0.599205	−	0.800595i	$-0.704517\pi$
$619$		−	39.8372i		−	1.60119i	0.599205	−	0.800595i	$-0.295483\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−45.0000	+	25.9808i	−1.79427	+	1.03592i
$630$		0			0
$631$	19.5000	+	11.2583i	0.776283	+	0.448187i	0.835111	−	0.550081i	$-0.185403\pi$
$631$	19.5000	+	11.2583i	0.776283	+	0.448187i	−0.0588285	+	0.998268i	$0.518737\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−18.0000	−	10.3923i	−0.713186	−	0.411758i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.00000	+	1.73205i	−0.118493	+	0.0684119i	−0.558075	−	0.829790i	$-0.688460\pi$
$641$	−3.00000	+	1.73205i	−0.118493	+	0.0684119i	0.439582	+	0.898202i	$0.355127\pi$
$642$		0			0
$643$	10.5000	−	6.06218i	0.414080	−	0.239069i	−0.278462	−	0.960447i	$-0.589824\pi$
$643$	10.5000	−	6.06218i	0.414080	−	0.239069i	0.692541	+	0.721378i	$0.256491\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	24.2487i		−	0.953315i	−0.879089	−	0.476658i	$-0.841848\pi$
$647$		−	24.2487i		−	0.953315i	0.879089	−	0.476658i	$-0.158152\pi$
$648$		0			0
$649$	−36.0000	+	20.7846i	−1.41312	+	0.815867i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−48.0000			−1.87839			−0.939193	−	0.343391i	$-0.888424\pi$
$653$	−48.0000			−1.87839			−0.939193	+	0.343391i	$0.888424\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.00000	+	15.5885i	−0.350590	+	0.607240i	−0.986353	−	0.164644i	$-0.947352\pi$
$659$	−9.00000	+	15.5885i	−0.350590	+	0.607240i	0.635763	+	0.771885i	$0.280686\pi$
$660$		0			0
$661$	−6.00000	−	3.46410i	−0.233373	−	0.134738i	0.378754	−	0.925497i	$-0.376353\pi$
$661$	−6.00000	−	3.46410i	−0.233373	−	0.134738i	−0.612127	+	0.790759i	$0.709686\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−6.00000	−	10.3923i	−0.232321	−	0.402392i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.0000	+	8.66025i	0.579069	+	0.334325i
$672$		0			0
$673$		−	8.66025i		−	0.333828i	−0.985971	−	0.166914i	$-0.946620\pi$
$673$		−	8.66025i		−	0.333828i	0.985971	−	0.166914i	$-0.0533803\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	17.3205i		−	0.665681i	−0.942983	−	0.332841i	$-0.891993\pi$
$677$		−	17.3205i		−	0.665681i	0.942983	−	0.332841i	$-0.108007\pi$
$678$		0			0
$679$	−12.0000	−	20.7846i	−0.460518	−	0.797640i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−27.0000	−	46.7654i	−1.02862	−	1.78162i
$690$		0			0
$691$			3.46410i			0.131781i	0.997827	+	0.0658903i	$0.0209887\pi$
$691$			3.46410i			0.131781i	−0.997827	+	0.0658903i	$0.979011\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	36.0000	+	20.7846i	1.36360	+	0.787273i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−12.0000	−	20.7846i	−0.453234	−	0.785024i	0.545351	−	0.838208i	$-0.316396\pi$
$701$	−12.0000	−	20.7846i	−0.453234	−	0.785024i	−0.998585	+	0.0531839i	$0.983063\pi$
$702$		0			0
$703$	15.0000	+	34.6410i	0.565736	+	1.30651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	17.5000	−	30.3109i	0.657226	−	1.13835i	−0.324104	−	0.946021i	$-0.605063\pi$
$709$	17.5000	−	30.3109i	0.657226	−	1.13835i	0.981331	−	0.192328i	$-0.0616038\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.00000	+	1.73205i	0.112351	+	0.0648658i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	39.0000	−	22.5167i	1.45445	−	0.839730i	0.455725	−	0.890121i	$-0.349380\pi$
$719$	39.0000	−	22.5167i	1.45445	−	0.839730i	0.998730	+	0.0503909i	$0.0160467\pi$
$720$		0			0
$721$		−	1.73205i		−	0.0645049i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15.0000	−	8.66025i	0.557086	−	0.321634i
$726$		0			0
$727$	−16.5000	+	9.52628i	−0.611951	+	0.353310i	−0.773729	−	0.633517i	$-0.781611\pi$
$727$	−16.5000	+	9.52628i	−0.611951	+	0.353310i	0.161778	+	0.986827i	$0.448277\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	27.0000	+	15.5885i	0.998631	+	0.576560i
$732$		0			0
$733$	22.0000			0.812589			0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000			0.812589			0.406294	+	0.913742i	$0.366821\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	39.0000	+	22.5167i	1.43658	+	0.829412i
$738$		0			0
$739$	−28.5000	+	16.4545i	−1.04839	+	0.605288i	−0.922198	−	0.386718i	$-0.873609\pi$
$739$	−28.5000	+	16.4545i	−1.04839	+	0.605288i	−0.126191	+	0.992006i	$0.540275\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.0000	−	36.3731i	−0.770415	−	1.33440i	−0.937336	−	0.348428i	$-0.886716\pi$
$743$	−21.0000	−	36.3731i	−0.770415	−	1.33440i	0.166920	−	0.985970i	$-0.446618\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6.50000	+	11.2583i	−0.237188	+	0.410822i	−0.959906	−	0.280321i	$-0.909559\pi$
$751$	−6.50000	+	11.2583i	−0.237188	+	0.410822i	0.722718	+	0.691143i	$0.242893\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−17.5000	−	30.3109i	−0.636048	−	1.10167i	−0.986292	−	0.165009i	$-0.947235\pi$
$757$	−17.5000	−	30.3109i	−0.636048	−	1.10167i	0.350244	−	0.936659i	$-0.386099\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.0000			−0.435000			−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000			−0.435000			−0.217500	+	0.976060i	$0.569790\pi$
$762$		0			0
$763$	12.0000	+	20.7846i	0.434429	+	0.752453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	62.3538i		−	2.25147i
$768$		0			0
$769$	−0.500000	+	0.866025i	−0.0180305	+	0.0312297i	−0.874900	−	0.484304i	$-0.839073\pi$
$769$	−0.500000	+	0.866025i	−0.0180305	+	0.0312297i	0.856869	+	0.515534i	$0.172406\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.00000	+	1.73205i	0.107903	+	0.0622975i	0.552980	−	0.833194i	$-0.313491\pi$
$773$	3.00000	+	1.73205i	0.107903	+	0.0622975i	−0.445078	+	0.895492i	$0.646824\pi$
$774$		0			0
$775$	−2.50000	+	4.33013i	−0.0898027	+	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18.0000	−	24.2487i	0.644917	−	0.868800i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−31.0000			−1.10503			−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−31.0000			−1.10503			−0.552515	+	0.833503i	$0.686332\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$	−22.5000	+	12.9904i	−0.798998	+	0.461302i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	48.4974i		−	1.71787i	−0.512087	−	0.858933i	$-0.671128\pi$
$797$		−	48.4974i		−	1.71787i	0.512087	−	0.858933i	$-0.328872\pi$
$798$		0			0
$799$		−	62.3538i		−	2.20592i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−15.0000	+	8.66025i	−0.529339	+	0.305614i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.0000			−0.421898			−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000			−0.421898			−0.210949	+	0.977497i	$0.567655\pi$
$810$		0			0
$811$	10.0000	−	17.3205i	0.351147	−	0.608205i	−0.635303	−	0.772263i	$-0.719125\pi$
$811$	10.0000	−	17.3205i	0.351147	−	0.608205i	0.986451	+	0.164057i	$0.0524582\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	13.5000	−	18.1865i	0.472305	−	0.636266i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.00000	+	15.5885i	−0.314102	+	0.544041i	−0.979246	−	0.202674i	$-0.935037\pi$
$821$	−9.00000	+	15.5885i	−0.314102	+	0.544041i	0.665144	+	0.746715i	$0.268370\pi$
$822$		0			0
$823$	−15.0000	−	8.66025i	−0.522867	−	0.301877i	0.215240	−	0.976561i	$-0.430947\pi$
$823$	−15.0000	−	8.66025i	−0.522867	−	0.301877i	−0.738107	+	0.674684i	$0.764280\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.00000	−	15.5885i	0.312961	−	0.542064i	−0.666041	−	0.745915i	$-0.732013\pi$
$827$	9.00000	−	15.5885i	0.312961	−	0.542064i	0.979002	+	0.203851i	$0.0653459\pi$
$828$		0			0
$829$			53.6936i			1.86485i	0.361357	+	0.932427i	$0.382314\pi$
$829$			53.6936i			1.86485i	−0.361357	+	0.932427i	$0.617686\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.0000	+	20.7846i	0.415775	+	0.720144i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.0000	+	31.1769i	0.621429	+	1.07635i	0.989220	+	0.146438i	$0.0467809\pi$
$839$	18.0000	+	31.1769i	0.621429	+	1.07635i	−0.367791	+	0.929909i	$0.619886\pi$
$840$		0			0
$841$	−8.50000	−	14.7224i	−0.293103	−	0.507670i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$			1.73205i			0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15.0000	−	25.9808i	0.514193	−	0.890609i
$852$		0			0
$853$	−14.5000	−	25.1147i	−0.496471	−	0.859912i	0.503521	−	0.863983i	$-0.332038\pi$
$853$	−14.5000	−	25.1147i	−0.496471	−	0.859912i	−0.999992	+	0.00407068i	$0.998704\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.00000	+	5.19615i	−0.307434	+	0.177497i	−0.645778	−	0.763525i	$-0.723467\pi$
$857$	−9.00000	+	5.19615i	−0.307434	+	0.177497i	0.338344	+	0.941023i	$0.390133\pi$
$858$		0			0
$859$	−19.5000	−	11.2583i	−0.665331	−	0.384129i	0.128974	−	0.991648i	$-0.458832\pi$
$859$	−19.5000	−	11.2583i	−0.665331	−	0.384129i	−0.794305	+	0.607519i	$0.792165\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.00000			0.204242			0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000			0.204242			0.102121	+	0.994772i	$0.467437\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.00000	−	1.73205i	0.101768	−	0.0587558i
$870$		0			0
$871$	−58.5000	+	33.7750i	−1.98220	+	1.14442i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	46.5000	−	26.8468i	1.57019	−	0.906552i	0.574049	−	0.818821i	$-0.305372\pi$
$877$	46.5000	−	26.8468i	1.57019	−	0.906552i	0.996144	−	0.0877308i	$-0.0279615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.0000			0.404290			0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000			0.404290			0.202145	+	0.979356i	$0.435209\pi$
$882$		0			0
$883$	22.5000	+	12.9904i	0.757185	+	0.437161i	0.828284	−	0.560308i	$-0.189317\pi$
$883$	22.5000	+	12.9904i	0.757185	+	0.437161i	−0.0710990	+	0.997469i	$0.522651\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−24.0000	−	13.8564i	−0.804934	−	0.464729i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−45.0000	−	5.19615i	−1.50587	−	0.173883i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.00000	−	1.73205i	−0.100056	−	0.0577671i
$900$		0			0
$901$			62.3538i			2.07731i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	14.0000	+	24.2487i	0.464862	+	0.805165i	0.999195	−	0.0401089i	$-0.0127705\pi$
$907$	14.0000	+	24.2487i	0.464862	+	0.805165i	−0.534333	+	0.845274i	$0.679437\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$	60.0000			1.98571
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			32.9090i			1.08557i	0.839873	+	0.542783i	$0.182630\pi$
$919$			32.9090i			1.08557i	−0.839873	+	0.542783i	$0.817370\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	37.5000	+	21.6506i	1.23299	+	0.711868i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12.0000	+	20.7846i	0.393707	+	0.681921i	0.992935	−	0.118657i	$-0.0378590\pi$
$929$	12.0000	+	20.7846i	0.393707	+	0.681921i	−0.599228	+	0.800578i	$0.704526\pi$
$930$		0			0
$931$	16.0000	−	6.92820i	0.524379	−	0.227063i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−23.5000	+	40.7032i	−0.767712	+	1.32972i	0.171089	+	0.985255i	$0.445271\pi$
$937$	−23.5000	+	40.7032i	−0.767712	+	1.32972i	−0.938801	+	0.344460i	$0.888062\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.0000	+	6.92820i	0.391189	+	0.225853i	0.682675	−	0.730722i	$-0.260816\pi$
$941$	12.0000	+	6.92820i	0.391189	+	0.225853i	−0.291486	+	0.956575i	$0.594150\pi$
$942$		0			0
$943$	−24.0000			−0.781548
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.00000	−	3.46410i	0.194974	−	0.112568i	−0.399335	−	0.916805i	$-0.630759\pi$
$947$	6.00000	−	3.46410i	0.194974	−	0.112568i	0.594309	+	0.804237i	$0.297426\pi$
$948$		0			0
$949$		−	25.9808i		−	0.843371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.00000	−	5.19615i	0.291539	−	0.168320i	−0.347097	−	0.937829i	$-0.612833\pi$
$953$	9.00000	−	5.19615i	0.291539	−	0.168320i	0.638636	+	0.769509i	$0.279499\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.0000	+	10.3923i	0.581250	+	0.335585i
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	4.50000	−	2.59808i	0.144710	−	0.0835485i	−0.425897	−	0.904772i	$-0.640041\pi$
$967$	4.50000	−	2.59808i	0.144710	−	0.0835485i	0.570607	+	0.821223i	$0.306708\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−27.0000	−	46.7654i	−0.866471	−	1.50077i	−0.865579	−	0.500773i	$-0.833049\pi$
$971$	−27.0000	−	46.7654i	−0.866471	−	1.50077i	−0.000892350	−	1.00000i	$-0.500284\pi$
$972$		0			0
$973$	−7.50000	+	12.9904i	−0.240439	+	0.416452i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	17.3205i		−	0.554132i	−0.960851	−	0.277066i	$-0.910638\pi$
$977$		−	17.3205i		−	0.554132i	0.960851	−	0.277066i	$-0.0893621\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−15.0000	−	25.9808i	−0.478426	−	0.828658i	0.521268	−	0.853393i	$-0.325459\pi$
$983$	−15.0000	−	25.9808i	−0.478426	−	0.828658i	−0.999694	+	0.0247352i	$0.992126\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$	9.50000	+	16.4545i	0.301777	+	0.522694i	0.976539	−	0.215342i	$-0.0690867\pi$
$991$	9.50000	+	16.4545i	0.301777	+	0.522694i	−0.674761	+	0.738036i	$0.735753\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	26.5000	−	45.8993i	0.839263	−	1.45365i	−0.0512480	−	0.998686i	$-0.516320\pi$
$997$	26.5000	−	45.8993i	0.839263	−	1.45365i	0.890511	−	0.454961i	$-0.150347\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.bm.b.1855.1		2
3.2	odd	2		912.2.bb.b.31.1	yes	2
4.3	odd	2		2736.2.bm.a.1855.1		2
12.11	even	2		912.2.bb.a.31.1	✓	2
19.8	odd	6		2736.2.bm.a.559.1		2
57.8	even	6		912.2.bb.a.559.1	yes	2
76.27	even	6	inner	2736.2.bm.b.559.1		2
228.179	odd	6		912.2.bb.b.559.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bb.a.31.1	✓	2	12.11	even	2
912.2.bb.a.559.1	yes	2	57.8	even	6
912.2.bb.b.31.1	yes	2	3.2	odd	2
912.2.bb.b.559.1	yes	2	228.179	odd	6
2736.2.bm.a.559.1		2	19.8	odd	6
2736.2.bm.a.1855.1		2	4.3	odd	2
2736.2.bm.b.559.1		2	76.27	even	6	inner
2736.2.bm.b.1855.1		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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.bm (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{7} +3.46410i q^{11} +(-4.50000 - 2.59808i) q^{13} +(3.00000 + 5.19615i) q^{17} +(4.00000 - 1.73205i) q^{19} +(-3.00000 - 1.73205i) q^{23} +(2.50000 - 4.33013i) q^{25} +(3.00000 + 1.73205i) q^{29} -1.00000 q^{31} +8.66025i q^{37} +(6.00000 - 3.46410i) q^{41} +(4.50000 - 2.59808i) q^{43} +(-9.00000 - 5.19615i) q^{47} +4.00000 q^{49} +(9.00000 + 5.19615i) q^{53} +(6.00000 + 10.3923i) q^{59} +(2.50000 - 4.33013i) q^{61} +(6.50000 - 11.2583i) q^{67} +(2.50000 + 4.33013i) q^{73} +6.00000 q^{77} +(-0.500000 - 0.866025i) q^{79} -17.3205i q^{83} +(-4.50000 + 7.79423i) q^{91} +(12.0000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{13} + 6 q^{17} + 8 q^{19} - 6 q^{23} + 5 q^{25} + 6 q^{29} - 2 q^{31} + 12 q^{41} + 9 q^{43} - 18 q^{47} + 8 q^{49} + 18 q^{53} + 12 q^{59} + 5 q^{61} + 13 q^{67} + 5 q^{73} + 12 q^{77} - q^{79}+ \cdots + 24 q^{97}+O(q^{100}) \) 2 * q - 9 * q^13 + 6 * q^17 + 8 * q^19 - 6 * q^23 + 5 * q^25 + 6 * q^29 - 2 * q^31 + 12 * q^41 + 9 * q^43 - 18 * q^47 + 8 * q^49 + 18 * q^53 + 12 * q^59 + 5 * q^61 + 13 * q^67 + 5 * q^73 + 12 * q^77 - q^79 - 9 * q^91 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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