LMFDB - Embedded newform 3528.2.s.be.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, 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("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.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:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 392)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3528.361
Dual form			3528.2.s.be.3313.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 2.44949i) q^{5} +O(q^{10})$	$q+(-1.41421 - 2.44949i) q^{5} +(-2.00000 + 3.46410i) q^{11} +2.82843 q^{13} +(2.82843 - 4.89898i) q^{17} +(-1.41421 - 2.44949i) q^{19} +(-1.50000 + 2.59808i) q^{25} -2.00000 q^{29} +(-2.82843 + 4.89898i) q^{31} +(-5.00000 - 8.66025i) q^{37} +5.65685 q^{41} -4.00000 q^{43} +(-2.82843 - 4.89898i) q^{47} +(3.00000 - 5.19615i) q^{53} +11.3137 q^{55} +(1.41421 - 2.44949i) q^{59} +(7.07107 + 12.2474i) q^{61} +(-4.00000 - 6.92820i) q^{65} +(-6.00000 + 10.3923i) q^{67} +(-4.00000 - 6.92820i) q^{79} -14.1421 q^{83} -16.0000 q^{85} +(-4.00000 + 6.92820i) q^{95} -5.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{11} - 6 q^{25} - 8 q^{29} - 20 q^{37} - 16 q^{43} + 12 q^{53} - 16 q^{65} - 24 q^{67} - 16 q^{79} - 64 q^{85} - 16 q^{95}+O(q^{100}) $ 4 * q - 8 * q^11 - 6 * q^25 - 8 * q^29 - 20 * q^37 - 16 * q^43 + 12 * q^53 - 16 * q^65 - 24 * q^67 - 16 * q^79 - 64 * q^85 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$	−1.41421	−	2.44949i	−0.632456	−	1.09545i	−0.987048	−	0.160424i	$-0.948714\pi$
$5$	−1.41421	−	2.44949i	−0.632456	−	1.09545i	0.354593	−	0.935021i	$-0.384620\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	0.389338	+	0.921095i	$0.372704\pi$
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	−0.992361	+	0.123371i	$0.960630\pi$
$12$		0			0
$13$	2.82843			0.784465			0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843			0.784465			0.392232	+	0.919866i	$0.371703\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.82843	−	4.89898i	0.685994	−	1.18818i	−0.287129	−	0.957892i	$-0.592701\pi$
$17$	2.82843	−	4.89898i	0.685994	−	1.18818i	0.973123	−	0.230285i	$-0.0739659\pi$
$18$		0			0
$19$	−1.41421	−	2.44949i	−0.324443	−	0.561951i	0.656957	−	0.753928i	$-0.271843\pi$
$19$	−1.41421	−	2.44949i	−0.324443	−	0.561951i	−0.981399	+	0.191977i	$0.938510\pi$
$20$		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
$25$	−1.50000	+	2.59808i	−0.300000	+	0.519615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	−2.82843	+	4.89898i	−0.508001	+	0.879883i	0.491957	+	0.870620i	$0.336282\pi$
$31$	−2.82843	+	4.89898i	−0.508001	+	0.879883i	−0.999957	+	0.00926296i	$0.997051\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	0.0821995	−	0.996616i	$-0.473806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.65685			0.883452			0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685			0.883452			0.441726	+	0.897150i	$0.354366\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.82843	−	4.89898i	−0.412568	−	0.714590i	0.582601	−	0.812758i	$-0.302035\pi$
$47$	−2.82843	−	4.89898i	−0.412568	−	0.714590i	−0.995170	+	0.0981685i	$0.968702\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$	11.3137			1.52554
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.41421	−	2.44949i	0.184115	−	0.318896i	−0.759163	−	0.650901i	$-0.774391\pi$
$59$	1.41421	−	2.44949i	0.184115	−	0.318896i	0.943278	+	0.332004i	$0.107725\pi$
$60$		0			0
$61$	7.07107	+	12.2474i	0.905357	+	1.56813i	0.820437	+	0.571737i	$0.193730\pi$
$61$	7.07107	+	12.2474i	0.905357	+	1.56813i	0.0849208	+	0.996388i	$0.472936\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	−	6.92820i	−0.496139	−	0.859338i
$66$		0			0
$67$	−6.00000	+	10.3923i	−0.733017	+	1.26962i	0.222571	+	0.974916i	$0.428555\pi$
$67$	−6.00000	+	10.3923i	−0.733017	+	1.26962i	−0.955588	+	0.294706i	$0.904778\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−14.1421			−1.55230			−0.776151	−	0.630548i	$-0.782830\pi$
$83$	−14.1421			−1.55230			−0.776151	+	0.630548i	$0.782830\pi$
$84$		0			0
$85$	−16.0000			−1.73544
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		0			0
$97$	−5.65685			−0.574367			−0.287183	−	0.957876i	$-0.592719\pi$
$97$	−5.65685			−0.574367			−0.287183	+	0.957876i	$0.592719\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.24264	−	7.34847i	0.422159	−	0.731200i	−0.573992	−	0.818861i	$-0.694606\pi$
$101$	4.24264	−	7.34847i	0.422159	−	0.731200i	0.996150	+	0.0876610i	$0.0279392\pi$
$102$		0			0
$103$	−8.48528	−	14.6969i	−0.836080	−	1.44813i	−0.893148	−	0.449762i	$-0.851509\pi$
$103$	−8.48528	−	14.6969i	−0.836080	−	1.44813i	0.0570688	−	0.998370i	$-0.481825\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	−0.995474	−	0.0950377i	$-0.969703\pi$
$107$	−6.00000	−	10.3923i	−0.580042	−	1.00466i	0.415432	−	0.909624i	$-0.363630\pi$
$108$		0			0
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	0.814152	+	0.580651i	$0.197202\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$		0			0
$115$		0			0
$116$		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
$125$	−5.65685			−0.505964
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.07107	+	12.2474i	0.617802	+	1.07006i	0.989886	+	0.141865i	$0.0453100\pi$
$131$	7.07107	+	12.2474i	0.617802	+	1.07006i	−0.372084	+	0.928199i	$0.621357\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.00000	+	8.66025i	−0.427179	+	0.739895i	−0.996621	−	0.0821359i	$-0.973826\pi$
$137$	−5.00000	+	8.66025i	−0.427179	+	0.739895i	0.569442	+	0.822031i	$0.307159\pi$
$138$		0			0
$139$	−14.1421			−1.19952			−0.599760	−	0.800180i	$-0.704737\pi$
$139$	−14.1421			−1.19952			−0.599760	+	0.800180i	$0.704737\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.65685	+	9.79796i	−0.473050	+	0.819346i
$144$		0			0
$145$	2.82843	+	4.89898i	0.234888	+	0.406838i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	0.585231	−	0.810867i	$-0.301004\pi$
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	−0.994847	+	0.101391i	$0.967671\pi$
$150$		0			0
$151$	−8.00000	+	13.8564i	−0.651031	+	1.12762i	0.331842	+	0.943335i	$0.392330\pi$
$151$	−8.00000	+	13.8564i	−0.651031	+	1.12762i	−0.982873	+	0.184284i	$0.941004\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.0000			1.28515
$156$		0			0
$157$	−7.07107	+	12.2474i	−0.564333	+	0.977453i	0.432779	+	0.901500i	$0.357533\pi$
$157$	−7.07107	+	12.2474i	−0.564333	+	0.977453i	−0.997111	+	0.0759527i	$0.975800\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.0000	−	17.3205i	−0.783260	−	1.35665i	−0.930033	−	0.367477i	$-0.880222\pi$
$163$	−10.0000	−	17.3205i	−0.783260	−	1.35665i	0.146772	−	0.989170i	$-0.453112\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.65685			−0.437741			−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685			−0.437741			−0.218870	+	0.975754i	$0.570237\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.41421	+	2.44949i	0.107521	+	0.186231i	0.914765	−	0.403986i	$-0.132375\pi$
$173$	1.41421	+	2.44949i	0.107521	+	0.186231i	−0.807245	+	0.590217i	$0.799042\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.0000	+	17.3205i	−0.747435	+	1.29460i	0.201613	+	0.979465i	$0.435382\pi$
$179$	−10.0000	+	17.3205i	−0.747435	+	1.29460i	−0.949048	+	0.315130i	$0.897952\pi$
$180$		0			0
$181$	−8.48528			−0.630706			−0.315353	−	0.948974i	$-0.602123\pi$
$181$	−8.48528			−0.630706			−0.315353	+	0.948974i	$0.602123\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.1421	+	24.4949i	−1.03975	+	1.80090i
$186$		0			0
$187$	11.3137	+	19.5959i	0.827340	+	1.43300i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.0000	+	20.7846i	0.868290	+	1.50392i	0.863743	+	0.503932i	$0.168114\pi$
$191$	12.0000	+	20.7846i	0.868290	+	1.50392i	0.00454614	+	0.999990i	$0.498553\pi$
$192$		0			0
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	−0.987945	−	0.154805i	$-0.950525\pi$
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	0.628037	+	0.778183i	$0.283859\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.0000			−1.56744			−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000			−1.56744			−0.783718	+	0.621117i	$0.786679\pi$
$198$		0			0
$199$	8.48528	−	14.6969i	0.601506	−	1.04184i	−0.391088	−	0.920353i	$-0.627901\pi$
$199$	8.48528	−	14.6969i	0.601506	−	1.04184i	0.992593	−	0.121485i	$-0.0387656\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−8.00000	−	13.8564i	−0.558744	−	0.967773i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.3137			0.782586
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.65685	+	9.79796i	0.385794	+	0.668215i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.00000	−	13.8564i	0.538138	−	0.932083i
$222$		0			0
$223$	11.3137			0.757622			0.378811	−	0.925474i	$-0.376333\pi$
$223$	11.3137			0.757622			0.378811	+	0.925474i	$0.376333\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.24264	+	7.34847i	−0.281594	+	0.487735i	−0.971778	−	0.235899i	$-0.924196\pi$
$227$	−4.24264	+	7.34847i	−0.281594	+	0.487735i	0.690184	+	0.723634i	$0.257530\pi$
$228$		0			0
$229$	−12.7279	−	22.0454i	−0.841085	−	1.45680i	−0.888978	−	0.457949i	$-0.848584\pi$
$229$	−12.7279	−	22.0454i	−0.841085	−	1.45680i	0.0478936	−	0.998852i	$-0.484749\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.00000	−	8.66025i	−0.327561	−	0.567352i	0.654466	−	0.756091i	$-0.272893\pi$
$233$	−5.00000	−	8.66025i	−0.327561	−	0.567352i	−0.982027	+	0.188739i	$0.939560\pi$
$234$		0			0
$235$	−8.00000	+	13.8564i	−0.521862	+	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	14.1421	−	24.4949i	0.910975	−	1.57786i	0.0982854	−	0.995158i	$-0.468664\pi$
$241$	14.1421	−	24.4949i	0.910975	−	1.57786i	0.812690	−	0.582697i	$-0.198002\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.00000	−	6.92820i	−0.254514	−	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.1421			0.892644			0.446322	−	0.894873i	$-0.352734\pi$
$251$	14.1421			0.892644			0.446322	+	0.894873i	$0.352734\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	−0.999970	−	0.00771799i	$-0.997543\pi$
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	0.506669	+	0.862141i	$0.330877\pi$
$264$		0			0
$265$	−16.9706			−1.04249
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.07107	−	12.2474i	0.431131	−	0.746740i	−0.565840	−	0.824515i	$-0.691448\pi$
$269$	7.07107	−	12.2474i	0.431131	−	0.746740i	0.996971	+	0.0777747i	$0.0247815\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.00000	−	10.3923i	−0.361814	−	0.626680i
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$		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
$283$	−1.41421	+	2.44949i	−0.0840663	+	0.145607i	−0.904993	−	0.425427i	$-0.860124\pi$
$283$	−1.41421	+	2.44949i	−0.0840663	+	0.145607i	0.820927	+	0.571034i	$0.193457\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.50000	−	12.9904i	−0.441176	−	0.764140i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.82843			0.165238			0.0826192	−	0.996581i	$-0.473671\pi$
$293$	2.82843			0.165238			0.0826192	+	0.996581i	$0.473671\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.0000	−	34.6410i	1.14520	−	1.98354i
$306$		0			0
$307$	−8.48528			−0.484281			−0.242140	−	0.970241i	$-0.577849\pi$
$307$	−8.48528			−0.484281			−0.242140	+	0.970241i	$0.577849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.3137	−	19.5959i	0.641542	−	1.11118i	−0.343547	−	0.939135i	$-0.611629\pi$
$311$	11.3137	−	19.5959i	0.641542	−	1.11118i	0.985089	−	0.172047i	$-0.0550381\pi$
$312$		0			0
$313$	−8.48528	−	14.6969i	−0.479616	−	0.830720i	0.520110	−	0.854099i	$-0.325891\pi$
$313$	−8.48528	−	14.6969i	−0.479616	−	0.830720i	−0.999727	+	0.0233791i	$0.992558\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	−0.999980	−	0.00635137i	$-0.997978\pi$
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	0.494489	−	0.869184i	$-0.335355\pi$
$318$		0			0
$319$	4.00000	−	6.92820i	0.223957	−	0.387905i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.0000			−0.890264
$324$		0			0
$325$	−4.24264	+	7.34847i	−0.235339	+	0.407620i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		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
$335$	33.9411			1.85440
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−11.3137	−	19.5959i	−0.612672	−	1.06118i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.0000	+	17.3205i	−0.536828	+	0.929814i	0.462244	+	0.886753i	$0.347044\pi$
$347$	−10.0000	+	17.3205i	−0.536828	+	0.929814i	−0.999072	+	0.0430610i	$0.986289\pi$
$348$		0			0
$349$	25.4558			1.36262			0.681310	−	0.731995i	$-0.261411\pi$
$349$	25.4558			1.36262			0.681310	+	0.731995i	$0.261411\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.65685	+	9.79796i	−0.301084	+	0.521493i	−0.976382	−	0.216052i	$-0.930682\pi$
$353$	−5.65685	+	9.79796i	−0.301084	+	0.521493i	0.675298	+	0.737545i	$0.264015\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	5.50000	−	9.52628i	0.289474	−	0.501383i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	16.9706	−	29.3939i	0.885856	−	1.53435i	0.0411270	−	0.999154i	$-0.486905\pi$
$367$	16.9706	−	29.3939i	0.885856	−	1.53435i	0.844729	−	0.535194i	$-0.179761\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.65685			−0.291343
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.1421	−	24.4949i	−0.722629	−	1.25163i	−0.959942	−	0.280198i	$-0.909600\pi$
$383$	−14.1421	−	24.4949i	−0.722629	−	1.25163i	0.237313	−	0.971433i	$-0.423733\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.0000	+	19.0526i	−0.557722	+	0.966003i	0.439964	+	0.898015i	$0.354991\pi$
$389$	−11.0000	+	19.0526i	−0.557722	+	0.966003i	−0.997686	+	0.0679877i	$0.978342\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.3137	+	19.5959i	−0.569254	+	0.985978i
$396$		0			0
$397$	4.24264	+	7.34847i	0.212932	+	0.368809i	0.952631	−	0.304129i	$-0.0983654\pi$
$397$	4.24264	+	7.34847i	0.212932	+	0.368809i	−0.739699	+	0.672938i	$0.765032\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	0.931158	−	0.364615i	$-0.118800\pi$
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$	−8.00000	+	13.8564i	−0.398508	+	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	40.0000			1.98273
$408$		0			0
$409$	−8.48528	+	14.6969i	−0.419570	+	0.726717i	−0.995896	−	0.0905030i	$-0.971153\pi$
$409$	−8.48528	+	14.6969i	−0.419570	+	0.726717i	0.576326	+	0.817220i	$0.304486\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	20.0000	+	34.6410i	0.981761	+	1.70046i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.82843			0.138178			0.0690889	−	0.997611i	$-0.477991\pi$
$419$	2.82843			0.138178			0.0690889	+	0.997611i	$0.477991\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.48528	+	14.6969i	0.411597	+	0.712906i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000	−	20.7846i	0.578020	−	1.00116i	−0.417687	−	0.908591i	$-0.637159\pi$
$431$	12.0000	−	20.7846i	0.578020	−	1.00116i	0.995706	−	0.0925683i	$-0.0295076\pi$
$432$		0			0
$433$	−5.65685			−0.271851			−0.135926	−	0.990719i	$-0.543401\pi$
$433$	−5.65685			−0.271851			−0.135926	+	0.990719i	$0.543401\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.00000	+	3.46410i	0.0950229	+	0.164584i	0.909618	−	0.415445i	$-0.136374\pi$
$443$	2.00000	+	3.46410i	0.0950229	+	0.164584i	−0.814595	+	0.580030i	$0.803041\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	−11.3137	+	19.5959i	−0.532742	+	0.922736i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−9.00000	−	15.5885i	−0.421002	−	0.729197i	0.575036	−	0.818128i	$-0.304988\pi$
$457$	−9.00000	−	15.5885i	−0.421002	−	0.729197i	−0.996038	+	0.0889312i	$0.971655\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.1421			−0.658665			−0.329332	−	0.944214i	$-0.606824\pi$
$461$	−14.1421			−0.658665			−0.329332	+	0.944214i	$0.606824\pi$
$462$		0			0
$463$	40.0000			1.85896			0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000			1.85896			0.929479	+	0.368875i	$0.120257\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.24264	−	7.34847i	−0.196326	−	0.340047i	0.751008	−	0.660293i	$-0.229568\pi$
$467$	−4.24264	−	7.34847i	−0.196326	−	0.340047i	−0.947334	+	0.320246i	$0.896234\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.00000	−	13.8564i	0.367840	−	0.637118i
$474$		0			0
$475$	8.48528			0.389331
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.1421	−	24.4949i	0.646171	−	1.11920i	−0.337859	−	0.941197i	$-0.609703\pi$
$479$	14.1421	−	24.4949i	0.646171	−	1.11920i	0.984030	−	0.178004i	$-0.0569639\pi$
$480$		0			0
$481$	−14.1421	−	24.4949i	−0.644826	−	1.11687i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000	+	13.8564i	0.363261	+	0.629187i
$486$		0			0
$487$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$487$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.00000			−0.180517			−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000			−0.180517			−0.0902587	+	0.995918i	$0.528769\pi$
$492$		0			0
$493$	−5.65685	+	9.79796i	−0.254772	+	0.441278i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	−0.988204	−	0.153141i	$-0.951061\pi$
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	0.361478	−	0.932381i	$-0.382272\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11.3137			0.504453			0.252227	−	0.967668i	$-0.418837\pi$
$503$	11.3137			0.504453			0.252227	+	0.967668i	$0.418837\pi$
$504$		0			0
$505$	−24.0000			−1.06799
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.07107	+	12.2474i	0.313420	+	0.542859i	0.979100	−	0.203378i	$-0.0651920\pi$
$509$	7.07107	+	12.2474i	0.313420	+	0.542859i	−0.665681	+	0.746237i	$0.731859\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.0000	+	41.5692i	−1.05757	+	1.83176i
$516$		0			0
$517$	22.6274			0.995153
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.82843	−	4.89898i	0.123916	−	0.214628i	−0.797393	−	0.603460i	$-0.793788\pi$
$521$	2.82843	−	4.89898i	0.123916	−	0.214628i	0.921309	+	0.388832i	$0.127121\pi$
$522$		0			0
$523$	9.89949	+	17.1464i	0.432875	+	0.749761i	0.997119	−	0.0758466i	$-0.0241659\pi$
$523$	9.89949	+	17.1464i	0.432875	+	0.749761i	−0.564245	+	0.825608i	$0.690833\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.0000	+	27.7128i	0.696971	+	1.20719i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	16.0000			0.693037
$534$		0			0
$535$	−16.9706	+	29.3939i	−0.733701	+	1.27081i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	0.956504	+	0.291718i	$0.0942267\pi$
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	−0.225617	+	0.974216i	$0.572440\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.65685			0.242313
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.82843	+	4.89898i	0.120495	+	0.208704i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.0000	−	25.9808i	0.635570	−	1.10084i	−0.350824	−	0.936442i	$-0.614098\pi$
$557$	15.0000	−	25.9808i	0.635570	−	1.10084i	0.986394	−	0.164399i	$-0.0525683\pi$
$558$		0			0
$559$	−11.3137			−0.478519
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.2132	−	36.7423i	0.894030	−	1.54851i	0.0590293	−	0.998256i	$-0.481199\pi$
$563$	21.2132	−	36.7423i	0.894030	−	1.54851i	0.835001	−	0.550249i	$-0.185467\pi$
$564$		0			0
$565$	14.1421	+	24.4949i	0.594964	+	1.03051i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.0000	+	25.9808i	0.628833	+	1.08917i	0.987786	+	0.155815i	$0.0498003\pi$
$569$	15.0000	+	25.9808i	0.628833	+	1.08917i	−0.358954	+	0.933355i	$0.616866\pi$
$570$		0			0
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	−0.577301	−	0.816532i	$-0.695894\pi$
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	0.995788	+	0.0916910i	$0.0292272\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	16.9706	−	29.3939i	0.706494	−	1.22368i	−0.259656	−	0.965701i	$-0.583609\pi$
$577$	16.9706	−	29.3939i	0.706494	−	1.22368i	0.966150	−	0.257982i	$-0.0830575\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.0000	+	20.7846i	0.496989	+	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−42.4264			−1.75113			−0.875563	−	0.483105i	$-0.839509\pi$
$587$	−42.4264			−1.75113			−0.875563	+	0.483105i	$0.839509\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.65685	+	9.79796i	0.232299	+	0.402354i	0.958484	−	0.285145i	$-0.0920418\pi$
$593$	5.65685	+	9.79796i	0.232299	+	0.402354i	−0.726185	+	0.687499i	$0.758708\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.0000	−	27.7128i	0.653742	−	1.13231i	−0.328465	−	0.944516i	$-0.606531\pi$
$599$	16.0000	−	27.7128i	0.653742	−	1.13231i	0.982208	−	0.187799i	$-0.0601353\pi$
$600$		0			0
$601$	22.6274			0.922992			0.461496	−	0.887142i	$-0.347313\pi$
$601$	22.6274			0.922992			0.461496	+	0.887142i	$0.347313\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.07107	+	12.2474i	−0.287480	+	0.497930i
$606$		0			0
$607$	11.3137	+	19.5959i	0.459209	+	0.795374i	0.998919	−	0.0464772i	$-0.0147995\pi$
$607$	11.3137	+	19.5959i	0.459209	+	0.795374i	−0.539710	+	0.841851i	$0.681466\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.00000	−	13.8564i	−0.323645	−	0.560570i
$612$		0			0
$613$	3.00000	−	5.19615i	0.121169	−	0.209871i	−0.799060	−	0.601251i	$-0.794669\pi$
$613$	3.00000	−	5.19615i	0.121169	−	0.209871i	0.920229	+	0.391381i	$0.128002\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	−7.07107	+	12.2474i	−0.284210	+	0.492267i	−0.972417	−	0.233248i	$-0.925065\pi$
$619$	−7.07107	+	12.2474i	−0.284210	+	0.492267i	0.688207	+	0.725514i	$0.258398\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−56.5685			−2.25554
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.3137	−	19.5959i	−0.448971	−	0.777640i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.00000	+	8.66025i	−0.197488	+	0.342059i	−0.947713	−	0.319123i	$-0.896612\pi$
$641$	−5.00000	+	8.66025i	−0.197488	+	0.342059i	0.750225	+	0.661182i	$0.229945\pi$
$642$		0			0
$643$	−25.4558			−1.00388			−0.501940	−	0.864902i	$-0.667380\pi$
$643$	−25.4558			−1.00388			−0.501940	+	0.864902i	$0.667380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.48528	+	14.6969i	−0.333591	+	0.577796i	−0.983213	−	0.182461i	$-0.941594\pi$
$647$	−8.48528	+	14.6969i	−0.333591	+	0.577796i	0.649622	+	0.760257i	$0.274927\pi$
$648$		0			0
$649$	5.65685	+	9.79796i	0.222051	+	0.384604i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	0.695934	−	0.718106i	$-0.254991\pi$
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	−0.969865	+	0.243643i	$0.921657\pi$
$654$		0			0
$655$	20.0000	−	34.6410i	0.781465	−	1.35354i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	7.07107	−	12.2474i	0.275033	−	0.476371i	−0.695111	−	0.718903i	$-0.744645\pi$
$661$	7.07107	−	12.2474i	0.275033	−	0.476371i	0.970143	+	0.242532i	$0.0779780\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−56.5685			−2.18380
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.24264	−	7.34847i	−0.163058	−	0.282425i	0.772906	−	0.634521i	$-0.218802\pi$
$677$	−4.24264	−	7.34847i	−0.163058	−	0.282425i	−0.935964	+	0.352096i	$0.885469\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.0000	−	17.3205i	0.382639	−	0.662751i	−0.608799	−	0.793324i	$-0.708349\pi$
$683$	10.0000	−	17.3205i	0.382639	−	0.662751i	0.991439	+	0.130573i	$0.0416818\pi$
$684$		0			0
$685$	28.2843			1.08069
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.48528	−	14.6969i	0.323263	−	0.559909i
$690$		0			0
$691$	−7.07107	−	12.2474i	−0.268996	−	0.465915i	0.699607	−	0.714528i	$-0.253359\pi$
$691$	−7.07107	−	12.2474i	−0.268996	−	0.465915i	−0.968603	+	0.248613i	$0.920025\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	20.0000	+	34.6410i	0.758643	+	1.31401i
$696$		0			0
$697$	16.0000	−	27.7128i	0.606043	−	1.04970i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−34.0000			−1.28416			−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000			−1.28416			−0.642081	+	0.766637i	$0.721929\pi$
$702$		0			0
$703$	−14.1421	+	24.4949i	−0.533381	+	0.923843i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	0.995228	−	0.0975728i	$-0.0311079\pi$
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	−0.582115	+	0.813107i	$0.697775\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	32.0000			1.19673
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14.1421	+	24.4949i	0.527413	+	0.913506i	0.999490	+	0.0319481i	$0.0101711\pi$
$719$	14.1421	+	24.4949i	0.527413	+	0.913506i	−0.472077	+	0.881557i	$0.656496\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.00000	−	5.19615i	0.111417	−	0.192980i
$726$		0			0
$727$	−28.2843			−1.04901			−0.524503	−	0.851409i	$-0.675749\pi$
$727$	−28.2843			−1.04901			−0.524503	+	0.851409i	$0.675749\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−11.3137	+	19.5959i	−0.418453	+	0.724781i
$732$		0			0
$733$	12.7279	+	22.0454i	0.470117	+	0.814266i	0.999416	−	0.0341693i	$-0.0108786\pi$
$733$	12.7279	+	22.0454i	0.470117	+	0.814266i	−0.529300	+	0.848435i	$0.677545\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.0000	−	41.5692i	−0.884051	−	1.53122i
$738$		0			0
$739$	6.00000	−	10.3923i	0.220714	−	0.382287i	−0.734311	−	0.678813i	$-0.762495\pi$
$739$	6.00000	−	10.3923i	0.220714	−	0.382287i	0.955025	+	0.296526i	$0.0958281\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.0000			0.586983			0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000			0.586983			0.293492	+	0.955962i	$0.405183\pi$
$744$		0			0
$745$	−14.1421	+	24.4949i	−0.518128	+	0.897424i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	12.0000	+	20.7846i	0.437886	+	0.758441i	0.997526	−	0.0702946i	$-0.0223939\pi$
$751$	12.0000	+	20.7846i	0.437886	+	0.758441i	−0.559640	+	0.828736i	$0.689061\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	45.2548			1.64699
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14.1421	+	24.4949i	0.512652	+	0.887939i	0.999892	+	0.0146714i	$0.00467023\pi$
$761$	14.1421	+	24.4949i	0.512652	+	0.887939i	−0.487240	+	0.873268i	$0.661996\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.00000	−	6.92820i	0.144432	−	0.250163i
$768$		0			0
$769$	−16.9706			−0.611974			−0.305987	−	0.952036i	$-0.598986\pi$
$769$	−16.9706			−0.611974			−0.305987	+	0.952036i	$0.598986\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	26.8701	−	46.5403i	0.966449	−	1.67394i	0.260778	−	0.965399i	$-0.416021\pi$
$773$	26.8701	−	46.5403i	0.966449	−	1.67394i	0.705671	−	0.708540i	$-0.250646\pi$
$774$		0			0
$775$	−8.48528	−	14.6969i	−0.304800	−	0.527930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.00000	−	13.8564i	−0.286630	−	0.496457i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	40.0000			1.42766
$786$		0			0
$787$	4.24264	−	7.34847i	0.151234	−	0.261945i	−0.780447	−	0.625221i	$-0.785009\pi$
$787$	4.24264	−	7.34847i	0.151234	−	0.261945i	0.931681	+	0.363277i	$0.118342\pi$
$788$		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
$797$	8.48528			0.300564			0.150282	−	0.988643i	$-0.451982\pi$
$797$	8.48528			0.300564			0.150282	+	0.988643i	$0.451982\pi$
$798$		0			0
$799$	−32.0000			−1.13208
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.00000	−	15.5885i	0.316423	−	0.548061i	−0.663316	−	0.748340i	$-0.730851\pi$
$809$	9.00000	−	15.5885i	0.316423	−	0.548061i	0.979739	+	0.200279i	$0.0641847\pi$
$810$		0			0
$811$	14.1421			0.496598			0.248299	−	0.968683i	$-0.420129\pi$
$811$	14.1421			0.496598			0.248299	+	0.968683i	$0.420129\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−28.2843	+	48.9898i	−0.990755	+	1.71604i
$816$		0			0
$817$	5.65685	+	9.79796i	0.197908	+	0.342787i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	27.0000	+	46.7654i	0.942306	+	1.63212i	0.761056	+	0.648686i	$0.224681\pi$
$821$	27.0000	+	46.7654i	0.942306	+	1.63212i	0.181250	+	0.983437i	$0.441986\pi$
$822$		0			0
$823$	8.00000	−	13.8564i	0.278862	−	0.483004i	−0.692240	−	0.721668i	$-0.743376\pi$
$823$	8.00000	−	13.8564i	0.278862	−	0.483004i	0.971102	+	0.238664i	$0.0767093\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.0000			−0.695468			−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000			−0.695468			−0.347734	+	0.937593i	$0.613049\pi$
$828$		0			0
$829$	−15.5563	+	26.9444i	−0.540294	+	0.935817i	0.458593	+	0.888647i	$0.348354\pi$
$829$	−15.5563	+	26.9444i	−0.540294	+	0.935817i	−0.998887	+	0.0471706i	$0.984980\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	8.00000	+	13.8564i	0.276851	+	0.479521i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28.2843			0.976481			0.488241	−	0.872709i	$-0.337639\pi$
$839$	28.2843			0.976481			0.488241	+	0.872709i	$0.337639\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.07107	+	12.2474i	0.243252	+	0.421325i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−42.4264			−1.45265			−0.726326	−	0.687350i	$-0.758774\pi$
$853$	−42.4264			−1.45265			−0.726326	+	0.687350i	$0.758774\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.1421	+	24.4949i	−0.483086	+	0.836730i	−0.999811	−	0.0194215i	$-0.993818\pi$
$857$	−14.1421	+	24.4949i	−0.483086	+	0.836730i	0.516725	+	0.856151i	$0.327151\pi$
$858$		0			0
$859$	−21.2132	−	36.7423i	−0.723785	−	1.25363i	−0.959472	−	0.281804i	$-0.909067\pi$
$859$	−21.2132	−	36.7423i	−0.723785	−	1.25363i	0.235687	−	0.971829i	$-0.424266\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	12.0000	+	20.7846i	0.408485	+	0.707516i	0.994720	−	0.102624i	$-0.0327240\pi$
$863$	12.0000	+	20.7846i	0.408485	+	0.707516i	−0.586235	+	0.810141i	$0.699391\pi$
$864$		0			0
$865$	4.00000	−	6.92820i	0.136004	−	0.235566i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	32.0000			1.08553
$870$		0			0
$871$	−16.9706	+	29.3939i	−0.575026	+	0.995974i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	0.848648	−	0.528958i	$-0.177417\pi$
$877$	−1.00000	−	1.73205i	−0.0337676	−	0.0584872i	−0.882415	+	0.470471i	$0.844084\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.1421	−	24.4949i	−0.474846	−	0.822458i	0.524739	−	0.851263i	$-0.324163\pi$
$887$	−14.1421	−	24.4949i	−0.474846	−	0.822458i	−0.999585	+	0.0288053i	$0.990830\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.00000	+	13.8564i	−0.267710	+	0.463687i
$894$		0			0
$895$	56.5685			1.89088
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.65685	−	9.79796i	0.188667	−	0.326780i
$900$		0			0
$901$	−16.9706	−	29.3939i	−0.565371	−	0.979252i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.0000	+	20.7846i	0.398893	+	0.690904i
$906$		0			0
$907$	−10.0000	+	17.3205i	−0.332045	+	0.575118i	−0.982913	−	0.184073i	$-0.941072\pi$
$907$	−10.0000	+	17.3205i	−0.332045	+	0.575118i	0.650868	+	0.759191i	$0.274405\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	40.0000			1.32526			0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000			1.32526			0.662630	+	0.748947i	$0.269440\pi$
$912$		0			0
$913$	28.2843	−	48.9898i	0.936073	−	1.62133i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	0.999475	+	0.0323936i	$0.0103130\pi$
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	−0.471684	+	0.881768i	$0.656354\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	30.0000			0.986394
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.7990	−	34.2929i	−0.649584	−	1.12511i	−0.983222	−	0.182411i	$-0.941610\pi$
$929$	−19.7990	−	34.2929i	−0.649584	−	1.12511i	0.333639	−	0.942701i	$-0.391723\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	32.0000	−	55.4256i	1.04651	−	1.81261i
$936$		0			0
$937$	22.6274			0.739205			0.369603	−	0.929190i	$-0.379494\pi$
$937$	22.6274			0.739205			0.369603	+	0.929190i	$0.379494\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.07107	+	12.2474i	−0.230510	+	0.399255i	−0.957958	−	0.286907i	$-0.907373\pi$
$941$	−7.07107	+	12.2474i	−0.230510	+	0.399255i	0.727448	+	0.686163i	$0.240706\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.00000	−	10.3923i	−0.194974	−	0.337705i	0.751918	−	0.659256i	$-0.229129\pi$
$947$	−6.00000	−	10.3923i	−0.194974	−	0.337705i	−0.946892	+	0.321552i	$0.895796\pi$
$948$		0			0
$949$		0			0
$950$		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
$955$	33.9411	−	58.7878i	1.09831	−	1.90233i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.0161290	−	0.0279363i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	28.2843			0.910503
$966$		0			0
$967$	48.0000			1.54358			0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000			1.54358			0.771788	+	0.635880i	$0.219363\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.89949	−	17.1464i	−0.317690	−	0.550255i	0.662316	−	0.749225i	$-0.269574\pi$
$971$	−9.89949	−	17.1464i	−0.317690	−	0.550255i	−0.980006	+	0.198970i	$0.936240\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.00000	+	15.5885i	−0.287936	+	0.498719i	−0.973317	−	0.229465i	$-0.926302\pi$
$977$	−9.00000	+	15.5885i	−0.287936	+	0.498719i	0.685381	+	0.728184i	$0.259636\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19.7990	−	34.2929i	0.631490	−	1.09377i	−0.355758	−	0.934578i	$-0.615777\pi$
$983$	19.7990	−	34.2929i	0.631490	−	1.09377i	0.987247	−	0.159194i	$-0.0508895\pi$
$984$		0			0
$985$	31.1127	+	53.8888i	0.991333	+	1.71704i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−4.00000	+	6.92820i	−0.127064	+	0.220082i	−0.922538	−	0.385906i	$-0.873889\pi$
$991$	−4.00000	+	6.92820i	−0.127064	+	0.220082i	0.795474	+	0.605988i	$0.207222\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−48.0000			−1.52170
$996$		0			0
$997$	18.3848	−	31.8434i	0.582252	−	1.00849i	−0.412960	−	0.910749i	$-0.635505\pi$
$997$	18.3848	−	31.8434i	0.582252	−	1.00849i	0.995212	−	0.0977405i	$-0.0311615\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.be.361.1		4
3.2	odd	2		392.2.i.g.361.2		4
7.2	even	3	inner	3528.2.s.be.3313.1		4
7.3	odd	6		3528.2.a.bj.1.1		2
7.4	even	3		3528.2.a.bj.1.2		2
7.5	odd	6	inner	3528.2.s.be.3313.2		4
7.6	odd	2	inner	3528.2.s.be.361.2		4
12.11	even	2		784.2.i.k.753.1		4
21.2	odd	6		392.2.i.g.177.2		4
21.5	even	6		392.2.i.g.177.1		4
21.11	odd	6		392.2.a.h.1.1	✓	2
21.17	even	6		392.2.a.h.1.2	yes	2
21.20	even	2		392.2.i.g.361.1		4
28.3	even	6		7056.2.a.cj.1.1		2
28.11	odd	6		7056.2.a.cj.1.2		2
84.11	even	6		784.2.a.n.1.2		2
84.23	even	6		784.2.i.k.177.1		4
84.47	odd	6		784.2.i.k.177.2		4
84.59	odd	6		784.2.a.n.1.1		2
84.83	odd	2		784.2.i.k.753.2		4
105.59	even	6		9800.2.a.bw.1.1		2
105.74	odd	6		9800.2.a.bw.1.2		2
168.11	even	6		3136.2.a.bq.1.1		2
168.53	odd	6		3136.2.a.bt.1.2		2
168.59	odd	6		3136.2.a.bq.1.2		2
168.101	even	6		3136.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.a.h.1.1	✓	2	21.11	odd	6
392.2.a.h.1.2	yes	2	21.17	even	6
392.2.i.g.177.1		4	21.5	even	6
392.2.i.g.177.2		4	21.2	odd	6
392.2.i.g.361.1		4	21.20	even	2
392.2.i.g.361.2		4	3.2	odd	2
784.2.a.n.1.1		2	84.59	odd	6
784.2.a.n.1.2		2	84.11	even	6
784.2.i.k.177.1		4	84.23	even	6
784.2.i.k.177.2		4	84.47	odd	6
784.2.i.k.753.1		4	12.11	even	2
784.2.i.k.753.2		4	84.83	odd	2
3136.2.a.bq.1.1		2	168.11	even	6
3136.2.a.bq.1.2		2	168.59	odd	6
3136.2.a.bt.1.1		2	168.101	even	6
3136.2.a.bt.1.2		2	168.53	odd	6
3528.2.a.bj.1.1		2	7.3	odd	6
3528.2.a.bj.1.2		2	7.4	even	3
3528.2.s.be.361.1		4	1.1	even	1	trivial
3528.2.s.be.361.2		4	7.6	odd	2	inner
3528.2.s.be.3313.1		4	7.2	even	3	inner
3528.2.s.be.3313.2		4	7.5	odd	6	inner
7056.2.a.cj.1.1		2	28.3	even	6
7056.2.a.cj.1.2		2	28.11	odd	6
9800.2.a.bw.1.1		2	105.59	even	6
9800.2.a.bw.1.2		2	105.74	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 2.44949i) q^{5} +O(q^{10})\)	\(q+(-1.41421 - 2.44949i) q^{5} +(-2.00000 + 3.46410i) q^{11} +2.82843 q^{13} +(2.82843 - 4.89898i) q^{17} +(-1.41421 - 2.44949i) q^{19} +(-1.50000 + 2.59808i) q^{25} -2.00000 q^{29} +(-2.82843 + 4.89898i) q^{31} +(-5.00000 - 8.66025i) q^{37} +5.65685 q^{41} -4.00000 q^{43} +(-2.82843 - 4.89898i) q^{47} +(3.00000 - 5.19615i) q^{53} +11.3137 q^{55} +(1.41421 - 2.44949i) q^{59} +(7.07107 + 12.2474i) q^{61} +(-4.00000 - 6.92820i) q^{65} +(-6.00000 + 10.3923i) q^{67} +(-4.00000 - 6.92820i) q^{79} -14.1421 q^{83} -16.0000 q^{85} +(-4.00000 + 6.92820i) q^{95} -5.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{11} - 6 q^{25} - 8 q^{29} - 20 q^{37} - 16 q^{43} + 12 q^{53} - 16 q^{65} - 24 q^{67} - 16 q^{79} - 64 q^{85} - 16 q^{95}+O(q^{100}) \) 4 * q - 8 * q^11 - 6 * q^25 - 8 * q^29 - 20 * q^37 - 16 * q^43 + 12 * q^53 - 16 * q^65 - 24 * q^67 - 16 * q^79 - 64 * q^85 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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