LMFDB - Embedded newform 3600.2.w.l.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(593,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("3600.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3600.w (of order $4$, 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:	$28.7461447277$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.2
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	3600.593
Dual form			3600.2.w.l.1457.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 1.22474i) q^{7} +O(q^{10})$	$q+(-1.22474 + 1.22474i) q^{7} +4.24264i q^{11} +(-3.67423 - 3.67423i) q^{13} +(1.73205 + 1.73205i) q^{17} +5.00000i q^{19} +(1.73205 - 1.73205i) q^{23} -4.24264 q^{29} -1.00000 q^{31} +(2.44949 - 2.44949i) q^{37} -8.48528i q^{41} +(1.22474 + 1.22474i) q^{43} +(-5.19615 - 5.19615i) q^{47} +4.00000i q^{49} +(6.92820 - 6.92820i) q^{53} -12.7279 q^{59} -7.00000 q^{61} +(-3.67423 + 3.67423i) q^{67} -8.48528i q^{71} +(2.44949 + 2.44949i) q^{73} +(-5.19615 - 5.19615i) q^{77} -2.00000i q^{79} +(1.73205 - 1.73205i) q^{83} -8.48528 q^{89} +9.00000 q^{91} +(-8.57321 + 8.57321i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{31} - 56 q^{61} + 72 q^{91}+O(q^{100}) $ 8 * q - 8 * q^31 - 56 * q^61 + 72 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$1$

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
$5$		0			0
$6$		0			0
$7$	−1.22474	+	1.22474i	−0.462910	+	0.462910i	−0.899608	−	0.436698i	$-0.856148\pi$
$7$	−1.22474	+	1.22474i	−0.462910	+	0.462910i	0.436698	+	0.899608i	$0.356148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.24264i			1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$			4.24264i			1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$		0			0
$13$	−3.67423	−	3.67423i	−1.01905	−	1.01905i	−0.999815	−	0.0192343i	$-0.993877\pi$
$13$	−3.67423	−	3.67423i	−1.01905	−	1.01905i	−0.0192343	−	0.999815i	$-0.506123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.73205	+	1.73205i	0.420084	+	0.420084i	0.885233	−	0.465149i	$-0.153999\pi$
$17$	1.73205	+	1.73205i	0.420084	+	0.420084i	−0.465149	+	0.885233i	$0.653999\pi$
$18$		0			0
$19$			5.00000i			1.14708i	0.819178	+	0.573539i	$0.194430\pi$
$19$			5.00000i			1.14708i	−0.819178	+	0.573539i	$0.805570\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.73205	−	1.73205i	0.361158	−	0.361158i	−0.503081	−	0.864239i	$-0.667800\pi$
$23$	1.73205	−	1.73205i	0.361158	−	0.361158i	0.864239	+	0.503081i	$0.167800\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.24264			−0.787839			−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264			−0.787839			−0.393919	+	0.919145i	$0.628881\pi$
$30$		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
$37$	2.44949	−	2.44949i	0.402694	−	0.402694i	−0.476488	−	0.879181i	$-0.658090\pi$
$37$	2.44949	−	2.44949i	0.402694	−	0.402694i	0.879181	+	0.476488i	$0.158090\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	8.48528i		−	1.32518i	−0.748983	−	0.662589i	$-0.769458\pi$
$41$		−	8.48528i		−	1.32518i	0.748983	−	0.662589i	$-0.230542\pi$
$42$		0			0
$43$	1.22474	+	1.22474i	0.186772	+	0.186772i	0.794299	−	0.607527i	$-0.207838\pi$
$43$	1.22474	+	1.22474i	0.186772	+	0.186772i	−0.607527	+	0.794299i	$0.707838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.19615	−	5.19615i	−0.757937	−	0.757937i	0.218010	−	0.975947i	$-0.430044\pi$
$47$	−5.19615	−	5.19615i	−0.757937	−	0.757937i	−0.975947	+	0.218010i	$0.930044\pi$
$48$		0			0
$49$			4.00000i			0.571429i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.92820	−	6.92820i	0.951662	−	0.951662i	−0.0472225	−	0.998884i	$-0.515037\pi$
$53$	6.92820	−	6.92820i	0.951662	−	0.951662i	0.998884	+	0.0472225i	$0.0150370\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.7279			−1.65703			−0.828517	−	0.559964i	$-0.810815\pi$
$59$	−12.7279			−1.65703			−0.828517	+	0.559964i	$0.810815\pi$
$60$		0			0
$61$	−7.00000			−0.896258			−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000			−0.896258			−0.448129	+	0.893969i	$0.647910\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.67423	+	3.67423i	−0.448879	+	0.448879i	−0.894982	−	0.446103i	$-0.852812\pi$
$67$	−3.67423	+	3.67423i	−0.448879	+	0.448879i	0.446103	+	0.894982i	$0.352812\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.48528i		−	1.00702i	−0.863990	−	0.503509i	$-0.832042\pi$
$71$		−	8.48528i		−	1.00702i	0.863990	−	0.503509i	$-0.167958\pi$
$72$		0			0
$73$	2.44949	+	2.44949i	0.286691	+	0.286691i	0.835770	−	0.549079i	$-0.185021\pi$
$73$	2.44949	+	2.44949i	0.286691	+	0.286691i	−0.549079	+	0.835770i	$0.685021\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.19615	−	5.19615i	−0.592157	−	0.592157i
$78$		0			0
$79$		−	2.00000i		−	0.225018i	−0.993651	−	0.112509i	$-0.964111\pi$
$79$		−	2.00000i		−	0.225018i	0.993651	−	0.112509i	$-0.0358886\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.73205	−	1.73205i	0.190117	−	0.190117i	−0.605629	−	0.795747i	$-0.707079\pi$
$83$	1.73205	−	1.73205i	0.190117	−	0.190117i	0.795747	+	0.605629i	$0.207079\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.48528			−0.899438			−0.449719	−	0.893170i	$-0.648476\pi$
$89$	−8.48528			−0.899438			−0.449719	+	0.893170i	$0.648476\pi$
$90$		0			0
$91$	9.00000			0.943456
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.57321	+	8.57321i	−0.870478	+	0.870478i	−0.992524	−	0.122046i	$-0.961054\pi$
$97$	−8.57321	+	8.57321i	−0.870478	+	0.870478i	0.122046	+	0.992524i	$0.461054\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.48528i			0.844317i	0.906522	+	0.422159i	$0.138727\pi$
$101$			8.48528i			0.844317i	−0.906522	+	0.422159i	$0.861273\pi$
$102$		0			0
$103$	−12.2474	−	12.2474i	−1.20678	−	1.20678i	−0.972065	−	0.234712i	$-0.924585\pi$
$103$	−12.2474	−	12.2474i	−1.20678	−	1.20678i	−0.234712	−	0.972065i	$-0.575415\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.3923	−	10.3923i	−1.00466	−	1.00466i	−0.999989	−	0.00467295i	$-0.998513\pi$
$107$	−10.3923	−	10.3923i	−1.00466	−	1.00466i	−0.00467295	−	0.999989i	$-0.501487\pi$
$108$		0			0
$109$			5.00000i			0.478913i	0.970907	+	0.239457i	$0.0769693\pi$
$109$			5.00000i			0.478913i	−0.970907	+	0.239457i	$0.923031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.3923	−	10.3923i	0.977626	−	0.977626i	−0.0221293	−	0.999755i	$-0.507045\pi$
$113$	10.3923	−	10.3923i	0.977626	−	0.977626i	0.999755	+	0.0221293i	$0.00704455\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.24264			−0.388922
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−6.12372	−	6.12372i	−0.530994	−	0.530994i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.66025	−	8.66025i	−0.739895	−	0.739895i	0.232662	−	0.972558i	$-0.425256\pi$
$137$	−8.66025	−	8.66025i	−0.739895	−	0.739895i	−0.972558	+	0.232662i	$0.925256\pi$
$138$		0			0
$139$			2.00000i			0.169638i	0.996396	+	0.0848189i	$0.0270312\pi$
$139$			2.00000i			0.169638i	−0.996396	+	0.0848189i	$0.972969\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.5885	−	15.5885i	1.30357	−	1.30357i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.24264			0.347571			0.173785	−	0.984784i	$-0.444400\pi$
$149$	4.24264			0.347571			0.173785	+	0.984784i	$0.444400\pi$
$150$		0			0
$151$	−13.0000			−1.05792			−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000			−1.05792			−0.528962	+	0.848645i	$0.677419\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.12372	+	6.12372i	−0.488726	+	0.488726i	−0.907904	−	0.419178i	$-0.862318\pi$
$157$	−6.12372	+	6.12372i	−0.488726	+	0.488726i	0.419178	+	0.907904i	$0.362318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.24264i			0.334367i
$162$		0			0
$163$	−11.0227	−	11.0227i	−0.863365	−	0.863365i	0.128363	−	0.991727i	$-0.459028\pi$
$163$	−11.0227	−	11.0227i	−0.863365	−	0.863365i	−0.991727	+	0.128363i	$0.959028\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	17.3205	+	17.3205i	1.34030	+	1.34030i	0.895757	+	0.444544i	$0.146634\pi$
$167$	17.3205	+	17.3205i	1.34030	+	1.34030i	0.444544	+	0.895757i	$0.353366\pi$
$168$		0			0
$169$			14.0000i			1.07692i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.73205	−	1.73205i	0.131685	−	0.131685i	−0.638192	−	0.769877i	$-0.720317\pi$
$173$	1.73205	−	1.73205i	0.131685	−	0.131685i	0.769877	+	0.638192i	$0.220317\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.24264			0.317110			0.158555	−	0.987350i	$-0.449317\pi$
$179$	4.24264			0.317110			0.158555	+	0.987350i	$0.449317\pi$
$180$		0			0
$181$	19.0000			1.41226			0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000			1.41226			0.706129	+	0.708083i	$0.250440\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−7.34847	+	7.34847i	−0.537373	+	0.537373i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.7279i		−	0.920960i	−0.887670	−	0.460480i	$-0.847677\pi$
$191$		−	12.7279i		−	0.920960i	0.887670	−	0.460480i	$-0.152323\pi$
$192$		0			0
$193$	−15.9217	−	15.9217i	−1.14607	−	1.14607i	−0.987319	−	0.158749i	$-0.949254\pi$
$193$	−15.9217	−	15.9217i	−1.14607	−	1.14607i	−0.158749	−	0.987319i	$-0.550746\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.73205	−	1.73205i	−0.123404	−	0.123404i	0.642708	−	0.766111i	$-0.277811\pi$
$197$	−1.73205	−	1.73205i	−0.123404	−	0.123404i	−0.766111	+	0.642708i	$0.777811\pi$
$198$		0			0
$199$			25.0000i			1.77220i	0.463491	+	0.886102i	$0.346597\pi$
$199$			25.0000i			1.77220i	−0.463491	+	0.886102i	$0.653403\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.19615	−	5.19615i	0.364698	−	0.364698i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.2132			−1.46735
$210$		0			0
$211$	−11.0000			−0.757271			−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000			−0.757271			−0.378636	+	0.925546i	$0.623607\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.22474	−	1.22474i	0.0831411	−	0.0831411i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	12.7279i		−	0.856173i
$222$		0			0
$223$	6.12372	+	6.12372i	0.410075	+	0.410075i	0.881765	−	0.471690i	$-0.156356\pi$
$223$	6.12372	+	6.12372i	0.410075	+	0.410075i	−0.471690	+	0.881765i	$0.656356\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.8564	−	13.8564i	−0.919682	−	0.919682i	0.0773240	−	0.997006i	$-0.475362\pi$
$227$	−13.8564	−	13.8564i	−0.919682	−	0.919682i	−0.997006	+	0.0773240i	$0.975362\pi$
$228$		0			0
$229$			1.00000i			0.0660819i	0.999454	+	0.0330409i	$0.0105192\pi$
$229$			1.00000i			0.0660819i	−0.999454	+	0.0330409i	$0.989481\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.3923	+	10.3923i	−0.680823	+	0.680823i	−0.960186	−	0.279363i	$-0.909877\pi$
$233$	−10.3923	+	10.3923i	−0.680823	+	0.680823i	0.279363	+	0.960186i	$0.409877\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.9706			−1.09773			−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706			−1.09773			−0.548867	+	0.835910i	$0.684941\pi$
$240$		0			0
$241$	5.00000			0.322078			0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000			0.322078			0.161039	+	0.986948i	$0.448515\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	18.3712	−	18.3712i	1.16893	−	1.16893i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			8.48528i			0.535586i	0.963476	+	0.267793i	$0.0862944\pi$
$251$			8.48528i			0.535586i	−0.963476	+	0.267793i	$0.913706\pi$
$252$		0			0
$253$	7.34847	+	7.34847i	0.461994	+	0.461994i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.8564	−	13.8564i	−0.864339	−	0.864339i	0.127500	−	0.991839i	$-0.459305\pi$
$257$	−13.8564	−	13.8564i	−0.864339	−	0.864339i	−0.991839	+	0.127500i	$0.959305\pi$
$258$		0			0
$259$			6.00000i			0.372822i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.3923	−	10.3923i	0.640817	−	0.640817i	−0.309939	−	0.950756i	$-0.600309\pi$
$263$	10.3923	−	10.3923i	0.640817	−	0.640817i	0.950756	+	0.309939i	$0.100309\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−29.6985			−1.81075			−0.905374	−	0.424614i	$-0.860410\pi$
$269$	−29.6985			−1.81075			−0.905374	+	0.424614i	$0.860410\pi$
$270$		0			0
$271$	−14.0000			−0.850439			−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000			−0.850439			−0.425220	+	0.905090i	$0.639803\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−8.57321	+	8.57321i	−0.515115	+	0.515115i	−0.916089	−	0.400975i	$-0.868672\pi$
$277$	−8.57321	+	8.57321i	−0.515115	+	0.515115i	0.400975	+	0.916089i	$0.368672\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4.24264i			0.253095i	0.991961	+	0.126547i	$0.0403896\pi$
$281$			4.24264i			0.253095i	−0.991961	+	0.126547i	$0.959610\pi$
$282$		0			0
$283$	13.4722	+	13.4722i	0.800839	+	0.800839i	0.983227	−	0.182388i	$-0.0583827\pi$
$283$	13.4722	+	13.4722i	0.800839	+	0.800839i	−0.182388	+	0.983227i	$0.558383\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.3923	+	10.3923i	0.613438	+	0.613438i
$288$		0			0
$289$		−	11.0000i		−	0.647059i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.73205	−	1.73205i	0.101187	−	0.101187i	−0.654701	−	0.755888i	$-0.727205\pi$
$293$	1.73205	−	1.73205i	0.101187	−	0.101187i	0.755888	+	0.654701i	$0.227205\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.7279			−0.736075
$300$		0			0
$301$	−3.00000			−0.172917
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−23.2702	+	23.2702i	−1.32810	+	1.32810i	−0.421069	+	0.907028i	$0.638345\pi$
$307$	−23.2702	+	23.2702i	−1.32810	+	1.32810i	−0.907028	+	0.421069i	$0.861655\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4.24264i		−	0.240578i	−0.992739	−	0.120289i	$-0.961618\pi$
$311$		−	4.24264i		−	0.240578i	0.992739	−	0.120289i	$-0.0383821\pi$
$312$		0			0
$313$	3.67423	+	3.67423i	0.207680	+	0.207680i	0.803281	−	0.595601i	$-0.203086\pi$
$313$	3.67423	+	3.67423i	0.207680	+	0.207680i	−0.595601	+	0.803281i	$0.703086\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.46410	+	3.46410i	0.194563	+	0.194563i	0.797665	−	0.603101i	$-0.206069\pi$
$317$	3.46410	+	3.46410i	0.194563	+	0.194563i	−0.603101	+	0.797665i	$0.706069\pi$
$318$		0			0
$319$		−	18.0000i		−	1.00781i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.66025	+	8.66025i	−0.481869	+	0.481869i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.7279			0.701713
$330$		0			0
$331$	22.0000			1.20923			0.604615	−	0.796518i	$-0.293327\pi$
$331$	22.0000			1.20923			0.604615	+	0.796518i	$0.293327\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.3712	−	18.3712i	1.00074	−	1.00074i	0.000741840	−	1.00000i	$-0.499764\pi$
$337$	18.3712	−	18.3712i	1.00074	−	1.00074i	1.00000		0.000741840i	$-0.000236135\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	4.24264i		−	0.229752i
$342$		0			0
$343$	−13.4722	−	13.4722i	−0.727430	−	0.727430i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.5885	−	15.5885i	−0.836832	−	0.836832i	0.151608	−	0.988441i	$-0.451555\pi$
$347$	−15.5885	−	15.5885i	−0.836832	−	0.836832i	−0.988441	+	0.151608i	$0.951555\pi$
$348$		0			0
$349$			16.0000i			0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$			16.0000i			0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.5885	−	15.5885i	0.829690	−	0.829690i	−0.157784	−	0.987474i	$-0.550435\pi$
$353$	15.5885	−	15.5885i	0.829690	−	0.829690i	0.987474	+	0.157784i	$0.0504349\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−6.00000			−0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	11.0227	−	11.0227i	0.575380	−	0.575380i	−0.358247	−	0.933627i	$-0.616625\pi$
$367$	11.0227	−	11.0227i	0.575380	−	0.575380i	0.933627	+	0.358247i	$0.116625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			16.9706i			0.881068i
$372$		0			0
$373$	1.22474	+	1.22474i	0.0634149	+	0.0634149i	0.738103	−	0.674688i	$-0.235722\pi$
$373$	1.22474	+	1.22474i	0.0634149	+	0.0634149i	−0.674688	+	0.738103i	$0.735722\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.5885	+	15.5885i	0.802846	+	0.802846i
$378$		0			0
$379$		−	1.00000i		−	0.0513665i	−0.999670	−	0.0256833i	$-0.991824\pi$
$379$		−	1.00000i		−	0.0513665i	0.999670	−	0.0256833i	$-0.00817614\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.46410	−	3.46410i	0.177007	−	0.177007i	−0.613043	−	0.790050i	$-0.710055\pi$
$383$	3.46410	−	3.46410i	0.177007	−	0.177007i	0.790050	+	0.613043i	$0.210055\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.9706			0.860442			0.430221	−	0.902724i	$-0.358436\pi$
$389$	16.9706			0.860442			0.430221	+	0.902724i	$0.358436\pi$
$390$		0			0
$391$	6.00000			0.303433
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−8.57321	+	8.57321i	−0.430277	+	0.430277i	−0.888723	−	0.458445i	$-0.848406\pi$
$397$	−8.57321	+	8.57321i	−0.430277	+	0.430277i	0.458445	+	0.888723i	$0.348406\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.9706i			0.847469i	0.905786	+	0.423735i	$0.139281\pi$
$401$			16.9706i			0.847469i	−0.905786	+	0.423735i	$0.860719\pi$
$402$		0			0
$403$	3.67423	+	3.67423i	0.183027	+	0.183027i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.3923	+	10.3923i	0.515127	+	0.515127i
$408$		0			0
$409$		−	11.0000i		−	0.543915i	−0.962309	−	0.271957i	$-0.912329\pi$
$409$		−	11.0000i		−	0.543915i	0.962309	−	0.271957i	$-0.0876710\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	15.5885	−	15.5885i	0.767058	−	0.767058i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.48528			0.414533			0.207267	−	0.978285i	$-0.433543\pi$
$419$	8.48528			0.414533			0.207267	+	0.978285i	$0.433543\pi$
$420$		0			0
$421$	−20.0000			−0.974740			−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000			−0.974740			−0.487370	+	0.873195i	$0.662044\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.57321	−	8.57321i	0.414887	−	0.414887i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	29.6985i		−	1.43053i	−0.698856	−	0.715263i	$-0.746307\pi$
$431$		−	29.6985i		−	1.43053i	0.698856	−	0.715263i	$-0.253693\pi$
$432$		0			0
$433$	1.22474	+	1.22474i	0.0588575	+	0.0588575i	0.735923	−	0.677065i	$-0.236749\pi$
$433$	1.22474	+	1.22474i	0.0588575	+	0.0588575i	−0.677065	+	0.735923i	$0.736749\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.66025	+	8.66025i	0.414276	+	0.414276i
$438$		0			0
$439$		−	11.0000i		−	0.525001i	−0.964932	−	0.262501i	$-0.915453\pi$
$439$		−	11.0000i		−	0.525001i	0.964932	−	0.262501i	$-0.0845472\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.3205	+	17.3205i	−0.822922	+	0.822922i	−0.986526	−	0.163604i	$-0.947688\pi$
$443$	−17.3205	+	17.3205i	−0.822922	+	0.822922i	0.163604	+	0.986526i	$0.447688\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	33.9411			1.60178			0.800890	−	0.598811i	$-0.204360\pi$
$449$	33.9411			1.60178			0.800890	+	0.598811i	$0.204360\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	16.9706i		−	0.790398i	−0.918596	−	0.395199i	$-0.870676\pi$
$461$		−	16.9706i		−	0.790398i	0.918596	−	0.395199i	$-0.129324\pi$
$462$		0			0
$463$	−24.4949	−	24.4949i	−1.13837	−	1.13837i	−0.988742	−	0.149633i	$-0.952191\pi$
$463$	−24.4949	−	24.4949i	−1.13837	−	1.13837i	−0.149633	−	0.988742i	$-0.547809\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.0526	−	19.0526i	−0.881647	−	0.881647i	0.112055	−	0.993702i	$-0.464257\pi$
$467$	−19.0526	−	19.0526i	−0.881647	−	0.881647i	−0.993702	+	0.112055i	$0.964257\pi$
$468$		0			0
$469$		−	9.00000i		−	0.415581i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.19615	+	5.19615i	−0.238919	+	0.238919i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	38.1838			1.74466			0.872330	−	0.488917i	$-0.162608\pi$
$479$	38.1838			1.74466			0.872330	+	0.488917i	$0.162608\pi$
$480$		0			0
$481$	−18.0000			−0.820729
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−18.3712	+	18.3712i	−0.832477	+	0.832477i	−0.987855	−	0.155378i	$-0.950341\pi$
$487$	−18.3712	+	18.3712i	−0.832477	+	0.832477i	0.155378	+	0.987855i	$0.450341\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			42.4264i			1.91468i	0.288969	+	0.957338i	$0.406688\pi$
$491$			42.4264i			1.91468i	−0.288969	+	0.957338i	$0.593312\pi$
$492$		0			0
$493$	−7.34847	−	7.34847i	−0.330958	−	0.330958i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.3923	+	10.3923i	0.466159	+	0.466159i
$498$		0			0
$499$			1.00000i			0.0447661i	0.999749	+	0.0223831i	$0.00712535\pi$
$499$			1.00000i			0.0447661i	−0.999749	+	0.0223831i	$0.992875\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−13.8564	+	13.8564i	−0.617827	+	0.617827i	−0.944974	−	0.327147i	$-0.893913\pi$
$503$	−13.8564	+	13.8564i	−0.617827	+	0.617827i	0.327147	+	0.944974i	$0.393913\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.24264			0.188052			0.0940259	−	0.995570i	$-0.470026\pi$
$509$	4.24264			0.188052			0.0940259	+	0.995570i	$0.470026\pi$
$510$		0			0
$511$	−6.00000			−0.265424
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	22.0454	−	22.0454i	0.969556	−	0.969556i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	4.24264i		−	0.185873i	−0.995672	−	0.0929367i	$-0.970375\pi$
$521$		−	4.24264i		−	0.185873i	0.995672	−	0.0929367i	$-0.0296254\pi$
$522$		0			0
$523$	13.4722	+	13.4722i	0.589098	+	0.589098i	0.937387	−	0.348289i	$-0.113237\pi$
$523$	13.4722	+	13.4722i	0.589098	+	0.589098i	−0.348289	+	0.937387i	$0.613237\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.73205	−	1.73205i	−0.0754493	−	0.0754493i
$528$		0			0
$529$			17.0000i			0.739130i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−31.1769	+	31.1769i	−1.35042	+	1.35042i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16.9706			−0.730974
$540$		0			0
$541$	−1.00000			−0.0429934			−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000			−0.0429934			−0.0214967	+	0.999769i	$0.506843\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12.2474	+	12.2474i	−0.523663	+	0.523663i	−0.918676	−	0.395013i	$-0.870740\pi$
$547$	−12.2474	+	12.2474i	−0.523663	+	0.523663i	0.395013	+	0.918676i	$0.370740\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	21.2132i		−	0.903713i
$552$		0			0
$553$	2.44949	+	2.44949i	0.104163	+	0.104163i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.5885	−	15.5885i	−0.660504	−	0.660504i	0.294995	−	0.955499i	$-0.404682\pi$
$557$	−15.5885	−	15.5885i	−0.660504	−	0.660504i	−0.955499	+	0.294995i	$0.904682\pi$
$558$		0			0
$559$		−	9.00000i		−	0.380659i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.9808	+	25.9808i	−1.09496	+	1.09496i	−0.0999679	+	0.994991i	$0.531874\pi$
$563$	−25.9808	+	25.9808i	−1.09496	+	1.09496i	−0.994991	+	0.0999679i	$0.968126\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.7279			0.533582			0.266791	−	0.963754i	$-0.414037\pi$
$569$	12.7279			0.533582			0.266791	+	0.963754i	$0.414037\pi$
$570$		0			0
$571$	−17.0000			−0.711428			−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000			−0.711428			−0.355714	+	0.934595i	$0.615762\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−23.2702	+	23.2702i	−0.968749	+	0.968749i	−0.999526	−	0.0307771i	$-0.990202\pi$
$577$	−23.2702	+	23.2702i	−0.968749	+	0.968749i	0.0307771	+	0.999526i	$0.490202\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			4.24264i			0.176014i
$582$		0			0
$583$	29.3939	+	29.3939i	1.21737	+	1.21737i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.8564	+	13.8564i	0.571915	+	0.571915i	0.932663	−	0.360748i	$-0.117479\pi$
$587$	13.8564	+	13.8564i	0.571915	+	0.571915i	−0.360748	+	0.932663i	$0.617479\pi$
$588$		0			0
$589$		−	5.00000i		−	0.206021i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.92820	+	6.92820i	−0.284507	+	0.284507i	−0.834904	−	0.550396i	$-0.814477\pi$
$593$	−6.92820	+	6.92820i	−0.284507	+	0.284507i	0.550396	+	0.834904i	$0.314477\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	25.4558			1.04010			0.520049	−	0.854137i	$-0.325914\pi$
$599$	25.4558			1.04010			0.520049	+	0.854137i	$0.325914\pi$
$600$		0			0
$601$	43.0000			1.75401			0.877003	−	0.480484i	$-0.159539\pi$
$601$	43.0000			1.75401			0.877003	+	0.480484i	$0.159539\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	19.5959	−	19.5959i	0.795374	−	0.795374i	−0.186988	−	0.982362i	$-0.559873\pi$
$607$	19.5959	−	19.5959i	0.795374	−	0.795374i	0.982362	+	0.186988i	$0.0598727\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			38.1838i			1.54475i
$612$		0			0
$613$	26.9444	+	26.9444i	1.08827	+	1.08827i	0.995706	+	0.0925671i	$0.0295073\pi$
$613$	26.9444	+	26.9444i	1.08827	+	1.08827i	0.0925671	+	0.995706i	$0.470493\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.3205	+	17.3205i	0.697297	+	0.697297i	0.963827	−	0.266529i	$-0.0858769\pi$
$617$	17.3205	+	17.3205i	0.697297	+	0.697297i	−0.266529	+	0.963827i	$0.585877\pi$
$618$		0			0
$619$			13.0000i			0.522514i	0.965269	+	0.261257i	$0.0841370\pi$
$619$			13.0000i			0.522514i	−0.965269	+	0.261257i	$0.915863\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.3923	−	10.3923i	0.416359	−	0.416359i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.48528			0.338330
$630$		0			0
$631$	−35.0000			−1.39333			−0.696664	−	0.717398i	$-0.745333\pi$
$631$	−35.0000			−1.39333			−0.696664	+	0.717398i	$0.745333\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	14.6969	−	14.6969i	0.582314	−	0.582314i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	33.9411i		−	1.34059i	−0.742093	−	0.670297i	$-0.766167\pi$
$641$		−	33.9411i		−	1.34059i	0.742093	−	0.670297i	$-0.233833\pi$
$642$		0			0
$643$	29.3939	+	29.3939i	1.15918	+	1.15918i	0.984652	+	0.174529i	$0.0558404\pi$
$643$	29.3939	+	29.3939i	1.15918	+	1.15918i	0.174529	+	0.984652i	$0.444160\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.3205	+	17.3205i	0.680939	+	0.680939i	0.960212	−	0.279272i	$-0.0900933\pi$
$647$	17.3205	+	17.3205i	0.680939	+	0.680939i	−0.279272	+	0.960212i	$0.590093\pi$
$648$		0			0
$649$		−	54.0000i		−	2.11969i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.0526	+	19.0526i	−0.745584	+	0.745584i	−0.973647	−	0.228062i	$-0.926761\pi$
$653$	−19.0526	+	19.0526i	−0.745584	+	0.745584i	0.228062	+	0.973647i	$0.426761\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.48528			−0.330540			−0.165270	−	0.986248i	$-0.552849\pi$
$659$	−8.48528			−0.330540			−0.165270	+	0.986248i	$0.552849\pi$
$660$		0			0
$661$	−20.0000			−0.777910			−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000			−0.777910			−0.388955	+	0.921257i	$0.627164\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−7.34847	+	7.34847i	−0.284534	+	0.284534i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	29.6985i		−	1.14650i
$672$		0			0
$673$	2.44949	+	2.44949i	0.0944209	+	0.0944209i	0.752739	−	0.658319i	$-0.228732\pi$
$673$	2.44949	+	2.44949i	0.0944209	+	0.0944209i	−0.658319	+	0.752739i	$0.728732\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.5885	+	15.5885i	0.599113	+	0.599113i	0.940077	−	0.340963i	$-0.110753\pi$
$677$	15.5885	+	15.5885i	0.599113	+	0.599113i	−0.340963	+	0.940077i	$0.610753\pi$
$678$		0			0
$679$		−	21.0000i		−	0.805906i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.92820	+	6.92820i	−0.265100	+	0.265100i	−0.827122	−	0.562022i	$-0.810024\pi$
$683$	−6.92820	+	6.92820i	−0.265100	+	0.265100i	0.562022	+	0.827122i	$0.310024\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−50.9117			−1.93958
$690$		0			0
$691$	32.0000			1.21734			0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000			1.21734			0.608669	+	0.793424i	$0.291704\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	14.6969	−	14.6969i	0.556686	−	0.556686i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			21.2132i			0.801212i	0.916250	+	0.400606i	$0.131200\pi$
$701$			21.2132i			0.801212i	−0.916250	+	0.400606i	$0.868800\pi$
$702$		0			0
$703$	12.2474	+	12.2474i	0.461921	+	0.461921i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.3923	−	10.3923i	−0.390843	−	0.390843i
$708$		0			0
$709$		−	41.0000i		−	1.53979i	−0.638172	−	0.769894i	$-0.720309\pi$
$709$		−	41.0000i		−	1.53979i	0.638172	−	0.769894i	$-0.279691\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.73205	+	1.73205i	−0.0648658	+	0.0648658i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	21.2132			0.791119			0.395559	−	0.918440i	$-0.370551\pi$
$719$	21.2132			0.791119			0.395559	+	0.918440i	$0.370551\pi$
$720$		0			0
$721$	30.0000			1.11726
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	35.5176	−	35.5176i	1.31727	−	1.31727i	0.401350	−	0.915925i	$-0.368541\pi$
$727$	35.5176	−	35.5176i	1.31727	−	1.31727i	0.915925	−	0.401350i	$-0.131459\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			4.24264i			0.156920i
$732$		0			0
$733$	−22.0454	−	22.0454i	−0.814266	−	0.814266i	0.171005	−	0.985270i	$-0.445299\pi$
$733$	−22.0454	−	22.0454i	−0.814266	−	0.814266i	−0.985270	+	0.171005i	$0.945299\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15.5885	−	15.5885i	−0.574208	−	0.574208i
$738$		0			0
$739$		−	10.0000i		−	0.367856i	−0.982940	−	0.183928i	$-0.941119\pi$
$739$		−	10.0000i		−	0.367856i	0.982940	−	0.183928i	$-0.0588813\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	20.7846	−	20.7846i	0.762513	−	0.762513i	−0.214263	−	0.976776i	$-0.568735\pi$
$743$	20.7846	−	20.7846i	0.762513	−	0.762513i	0.976776	+	0.214263i	$0.0687349\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	25.4558			0.930136
$750$		0			0
$751$	−46.0000			−1.67856			−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000			−1.67856			−0.839282	+	0.543696i	$0.817024\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	15.9217	−	15.9217i	0.578683	−	0.578683i	−0.355857	−	0.934540i	$-0.615811\pi$
$757$	15.9217	−	15.9217i	0.578683	−	0.578683i	0.934540	+	0.355857i	$0.115811\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			25.4558i			0.922774i	0.887199	+	0.461387i	$0.152648\pi$
$761$			25.4558i			0.922774i	−0.887199	+	0.461387i	$0.847352\pi$
$762$		0			0
$763$	−6.12372	−	6.12372i	−0.221694	−	0.221694i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	46.7654	+	46.7654i	1.68860	+	1.68860i
$768$		0			0
$769$			29.0000i			1.04577i	0.852404	+	0.522883i	$0.175144\pi$
$769$			29.0000i			1.04577i	−0.852404	+	0.522883i	$0.824856\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.7128	+	27.7128i	−0.996761	+	0.996761i	−0.999995	−	0.00323417i	$-0.998971\pi$
$773$	−27.7128	+	27.7128i	−0.996761	+	0.996761i	0.00323417	+	0.999995i	$0.498971\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	42.4264			1.52008
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	20.8207	−	20.8207i	0.742176	−	0.742176i	−0.230820	−	0.972996i	$-0.574141\pi$
$787$	20.8207	−	20.8207i	0.742176	−	0.742176i	0.972996	+	0.230820i	$0.0741409\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			25.4558i			0.905106i
$792$		0			0
$793$	25.7196	+	25.7196i	0.913331	+	0.913331i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.46410	+	3.46410i	0.122705	+	0.122705i	0.765793	−	0.643088i	$-0.222347\pi$
$797$	3.46410	+	3.46410i	0.122705	+	0.122705i	−0.643088	+	0.765793i	$0.722347\pi$
$798$		0			0
$799$		−	18.0000i		−	0.636794i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.3923	+	10.3923i	−0.366736	+	0.366736i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	50.9117			1.78996			0.894980	−	0.446107i	$-0.147190\pi$
$809$	50.9117			1.78996			0.894980	+	0.446107i	$0.147190\pi$
$810$		0			0
$811$	35.0000			1.22902			0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000			1.22902			0.614508	+	0.788911i	$0.289355\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−6.12372	+	6.12372i	−0.214242	+	0.214242i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			21.2132i			0.740346i	0.928963	+	0.370173i	$0.120702\pi$
$821$			21.2132i			0.740346i	−0.928963	+	0.370173i	$0.879298\pi$
$822$		0			0
$823$	−18.3712	−	18.3712i	−0.640379	−	0.640379i	0.310270	−	0.950649i	$-0.399581\pi$
$823$	−18.3712	−	18.3712i	−0.640379	−	0.640379i	−0.950649	+	0.310270i	$0.899581\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	13.8564	+	13.8564i	0.481834	+	0.481834i	0.905717	−	0.423883i	$-0.139333\pi$
$827$	13.8564	+	13.8564i	0.481834	+	0.481834i	−0.423883	+	0.905717i	$0.639333\pi$
$828$		0			0
$829$			10.0000i			0.347314i	0.984806	+	0.173657i	$0.0555585\pi$
$829$			10.0000i			0.347314i	−0.984806	+	0.173657i	$0.944442\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.92820	+	6.92820i	−0.240048	+	0.240048i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.7279			−0.439417			−0.219708	−	0.975566i	$-0.570511\pi$
$839$	−12.7279			−0.439417			−0.219708	+	0.975566i	$0.570511\pi$
$840$		0			0
$841$	−11.0000			−0.379310
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	8.57321	−	8.57321i	0.294579	−	0.294579i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	8.48528i		−	0.290872i
$852$		0			0
$853$	25.7196	+	25.7196i	0.880624	+	0.880624i	0.993598	−	0.112974i	$-0.0360377\pi$
$853$	25.7196	+	25.7196i	0.880624	+	0.880624i	−0.112974	+	0.993598i	$0.536038\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.92820	+	6.92820i	0.236663	+	0.236663i	0.815467	−	0.578804i	$-0.196480\pi$
$857$	6.92820	+	6.92820i	0.236663	+	0.236663i	−0.578804	+	0.815467i	$0.696480\pi$
$858$		0			0
$859$		−	8.00000i		−	0.272956i	−0.990643	−	0.136478i	$-0.956422\pi$
$859$		−	8.00000i		−	0.272956i	0.990643	−	0.136478i	$-0.0435784\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.92820	+	6.92820i	−0.235839	+	0.235839i	−0.815125	−	0.579286i	$-0.803332\pi$
$863$	−6.92820	+	6.92820i	−0.235839	+	0.235839i	0.579286	+	0.815125i	$0.303332\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.48528			0.287843
$870$		0			0
$871$	27.0000			0.914860
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.22474	−	1.22474i	0.0413567	−	0.0413567i	−0.686126	−	0.727483i	$-0.740690\pi$
$877$	1.22474	−	1.22474i	0.0413567	−	0.0413567i	0.727483	+	0.686126i	$0.240690\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	13.4722	+	13.4722i	0.453375	+	0.453375i	0.896473	−	0.443098i	$-0.146121\pi$
$883$	13.4722	+	13.4722i	0.453375	+	0.453375i	−0.443098	+	0.896473i	$0.646121\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.66025	+	8.66025i	0.290783	+	0.290783i	0.837390	−	0.546607i	$-0.184081\pi$
$887$	8.66025	+	8.66025i	0.290783	+	0.290783i	−0.546607	+	0.837390i	$0.684081\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	25.9808	−	25.9808i	0.869413	−	0.869413i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.24264			0.141500
$900$		0			0
$901$	24.0000			0.799556
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−24.4949	+	24.4949i	−0.813340	+	0.813340i	−0.985133	−	0.171793i	$-0.945044\pi$
$907$	−24.4949	+	24.4949i	−0.813340	+	0.813340i	0.171793	+	0.985133i	$0.445044\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	12.7279i		−	0.421695i	−0.977519	−	0.210847i	$-0.932378\pi$
$911$		−	12.7279i		−	0.421695i	0.977519	−	0.210847i	$-0.0676223\pi$
$912$		0			0
$913$	7.34847	+	7.34847i	0.243199	+	0.243199i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	1.00000i		−	0.0329870i	−0.999864	−	0.0164935i	$-0.994750\pi$
$919$		−	1.00000i		−	0.0329870i	0.999864	−	0.0164935i	$-0.00525028\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−31.1769	+	31.1769i	−1.02620	+	1.02620i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.24264			0.139197			0.0695983	−	0.997575i	$-0.477828\pi$
$929$	4.24264			0.139197			0.0695983	+	0.997575i	$0.477828\pi$
$930$		0			0
$931$	−20.0000			−0.655474
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−13.4722	+	13.4722i	−0.440117	+	0.440117i	−0.892051	−	0.451934i	$-0.850734\pi$
$937$	−13.4722	+	13.4722i	−0.440117	+	0.440117i	0.451934	+	0.892051i	$0.350734\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			55.1543i			1.79798i	0.437969	+	0.898990i	$0.355698\pi$
$941$			55.1543i			1.79798i	−0.437969	+	0.898990i	$0.644302\pi$
$942$		0			0
$943$	−14.6969	−	14.6969i	−0.478598	−	0.478598i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.5167	−	22.5167i	−0.731693	−	0.731693i	0.239262	−	0.970955i	$-0.423095\pi$
$947$	−22.5167	−	22.5167i	−0.731693	−	0.731693i	−0.970955	+	0.239262i	$0.923095\pi$
$948$		0			0
$949$		−	18.0000i		−	0.584305i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−31.1769	+	31.1769i	−1.00992	+	1.00992i	−0.00996902	+	0.999950i	$0.503173\pi$
$953$	−31.1769	+	31.1769i	−1.00992	+	1.00992i	−0.999950	+	0.00996902i	$0.996827\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	21.2132			0.685010
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	19.5959	−	19.5959i	0.630162	−	0.630162i	−0.317946	−	0.948109i	$-0.602993\pi$
$967$	19.5959	−	19.5959i	0.630162	−	0.630162i	0.948109	+	0.317946i	$0.102993\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	12.7279i		−	0.408458i	−0.978923	−	0.204229i	$-0.934531\pi$
$971$		−	12.7279i		−	0.408458i	0.978923	−	0.204229i	$-0.0654688\pi$
$972$		0			0
$973$	−2.44949	−	2.44949i	−0.0785270	−	0.0785270i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.19615	+	5.19615i	0.166240	+	0.166240i	0.785324	−	0.619085i	$-0.212496\pi$
$977$	5.19615	+	5.19615i	0.166240	+	0.166240i	−0.619085	+	0.785324i	$0.712496\pi$
$978$		0			0
$979$		−	36.0000i		−	1.15056i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.8564	+	13.8564i	−0.441951	+	0.441951i	−0.892667	−	0.450716i	$-0.851169\pi$
$983$	−13.8564	+	13.8564i	−0.441951	+	0.441951i	0.450716	+	0.892667i	$0.351169\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.24264			0.134908
$990$		0			0
$991$	31.0000			0.984747			0.492374	−	0.870384i	$-0.336129\pi$
$991$	31.0000			0.984747			0.492374	+	0.870384i	$0.336129\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−36.7423	+	36.7423i	−1.16364	+	1.16364i	−0.179970	+	0.983672i	$0.557600\pi$
$997$	−36.7423	+	36.7423i	−1.16364	+	1.16364i	−0.983672	+	0.179970i	$0.942400\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.w.l.593.2		8
3.2	odd	2	inner	3600.2.w.l.593.1		8
4.3	odd	2		225.2.f.b.143.2	yes	8
5.2	odd	4	inner	3600.2.w.l.1457.2		8
5.3	odd	4	inner	3600.2.w.l.1457.4		8
5.4	even	2	inner	3600.2.w.l.593.4		8
12.11	even	2		225.2.f.b.143.4	yes	8
15.2	even	4	inner	3600.2.w.l.1457.1		8
15.8	even	4	inner	3600.2.w.l.1457.3		8
15.14	odd	2	inner	3600.2.w.l.593.3		8
20.3	even	4		225.2.f.b.107.1	✓	8
20.7	even	4		225.2.f.b.107.4	yes	8
20.19	odd	2		225.2.f.b.143.3	yes	8
60.23	odd	4		225.2.f.b.107.3	yes	8
60.47	odd	4		225.2.f.b.107.2	yes	8
60.59	even	2		225.2.f.b.143.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.f.b.107.1	✓	8	20.3	even	4
225.2.f.b.107.2	yes	8	60.47	odd	4
225.2.f.b.107.3	yes	8	60.23	odd	4
225.2.f.b.107.4	yes	8	20.7	even	4
225.2.f.b.143.1	yes	8	60.59	even	2
225.2.f.b.143.2	yes	8	4.3	odd	2
225.2.f.b.143.3	yes	8	20.19	odd	2
225.2.f.b.143.4	yes	8	12.11	even	2
3600.2.w.l.593.1		8	3.2	odd	2	inner
3600.2.w.l.593.2		8	1.1	even	1	trivial
3600.2.w.l.593.3		8	15.14	odd	2	inner
3600.2.w.l.593.4		8	5.4	even	2	inner
3600.2.w.l.1457.1		8	15.2	even	4	inner
3600.2.w.l.1457.2		8	5.2	odd	4	inner
3600.2.w.l.1457.3		8	15.8	even	4	inner
3600.2.w.l.1457.4		8	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 1.22474i) q^{7} +O(q^{10})\)	\(q+(-1.22474 + 1.22474i) q^{7} +4.24264i q^{11} +(-3.67423 - 3.67423i) q^{13} +(1.73205 + 1.73205i) q^{17} +5.00000i q^{19} +(1.73205 - 1.73205i) q^{23} -4.24264 q^{29} -1.00000 q^{31} +(2.44949 - 2.44949i) q^{37} -8.48528i q^{41} +(1.22474 + 1.22474i) q^{43} +(-5.19615 - 5.19615i) q^{47} +4.00000i q^{49} +(6.92820 - 6.92820i) q^{53} -12.7279 q^{59} -7.00000 q^{61} +(-3.67423 + 3.67423i) q^{67} -8.48528i q^{71} +(2.44949 + 2.44949i) q^{73} +(-5.19615 - 5.19615i) q^{77} -2.00000i q^{79} +(1.73205 - 1.73205i) q^{83} -8.48528 q^{89} +9.00000 q^{91} +(-8.57321 + 8.57321i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{31} - 56 q^{61} + 72 q^{91}+O(q^{100}) \) 8 * q - 8 * q^31 - 56 * q^61 + 72 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more