LMFDB - Embedded newform 225.2.f.b.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(107,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("225.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	225.f (of order $4$, 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:	$1.79663404548$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.1
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	225.107
Dual form			225.2.f.b.143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 + 1.73205i) q^{2} -4.00000i q^{4} +(-1.22474 - 1.22474i) q^{7} +(3.46410 + 3.46410i) q^{8} +O(q^{10})$	\(q+(-1.73205 + 1.73205i) q^{2} -4.00000i q^{4} +(-1.22474 - 1.22474i) q^{7} +(3.46410 + 3.46410i) q^{8} -4.24264i q^{11} +(3.67423 - 3.67423i) q^{13} +4.24264 q^{14} -4.00000 q^{16} +(1.73205 - 1.73205i) q^{17} +5.00000i q^{19} +(7.34847 + 7.34847i) q^{22} +(-1.73205 - 1.73205i) q^{23} +12.7279i q^{26} +(-4.89898 + 4.89898i) q^{28} +4.24264 q^{29} +1.00000 q^{31} +6.00000i q^{34} +(-2.44949 - 2.44949i) q^{37} +(-8.66025 - 8.66025i) q^{38} -8.48528i q^{41} +(1.22474 - 1.22474i) q^{43} -16.9706 q^{44} +6.00000 q^{46} +(5.19615 - 5.19615i) q^{47} -4.00000i q^{49} +(-14.6969 - 14.6969i) q^{52} +(6.92820 + 6.92820i) q^{53} -8.48528i q^{56} +(-7.34847 + 7.34847i) q^{58} -12.7279 q^{59} -7.00000 q^{61} +(-1.73205 + 1.73205i) q^{62} -8.00000i q^{64} +(-3.67423 - 3.67423i) q^{67} +(-6.92820 - 6.92820i) q^{68} +8.48528i q^{71} +(-2.44949 + 2.44949i) q^{73} +8.48528 q^{74} +20.0000 q^{76} +(-5.19615 + 5.19615i) q^{77} -2.00000i q^{79} +(14.6969 + 14.6969i) q^{82} +(-1.73205 - 1.73205i) q^{83} +4.24264i q^{86} +(14.6969 - 14.6969i) q^{88} +8.48528 q^{89} -9.00000 q^{91} +(-6.92820 + 6.92820i) q^{92} +18.0000i q^{94} +(8.57321 + 8.57321i) q^{97} +(6.92820 + 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 32 q^{16} + 8 q^{31} + 48 q^{46} - 56 q^{61} + 160 q^{76} - 72 q^{91}+O(q^{100}) $ 8 * q - 32 * q^16 + 8 * q^31 + 48 * q^46 - 56 * q^61 + 160 * q^76 - 72 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.73205	+	1.73205i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$2$	−1.73205	+	1.73205i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$3$		0			0
$3$		0			0
$4$		−	4.00000i		−	2.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.22474	−	1.22474i	−0.462910	−	0.462910i	0.436698	−	0.899608i	$-0.356148\pi$
$7$	−1.22474	−	1.22474i	−0.462910	−	0.462910i	−0.899608	+	0.436698i	$0.856148\pi$
$8$	3.46410	+	3.46410i	1.22474	+	1.22474i
$9$		0			0
$10$		0			0
$11$		−	4.24264i		−	1.27920i	−0.768706	−	0.639602i	$-0.779099\pi$
$11$		−	4.24264i		−	1.27920i	0.768706	−	0.639602i	$-0.220901\pi$
$12$		0			0
$13$	3.67423	−	3.67423i	1.01905	−	1.01905i	0.0192343	−	0.999815i	$-0.493877\pi$
$13$	3.67423	−	3.67423i	1.01905	−	1.01905i	0.999815	−	0.0192343i	$-0.00612285\pi$
$14$	4.24264			1.13389
$15$		0			0
$16$	−4.00000			−1.00000
$17$	1.73205	−	1.73205i	0.420084	−	0.420084i	−0.465149	−	0.885233i	$-0.653999\pi$
$17$	1.73205	−	1.73205i	0.420084	−	0.420084i	0.885233	+	0.465149i	$0.153999\pi$
$18$		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
$22$	7.34847	+	7.34847i	1.56670	+	1.56670i
$23$	−1.73205	−	1.73205i	−0.361158	−	0.361158i	0.503081	−	0.864239i	$-0.332200\pi$
$23$	−1.73205	−	1.73205i	−0.361158	−	0.361158i	−0.864239	+	0.503081i	$0.832200\pi$
$24$		0			0
$25$		0			0
$26$			12.7279i			2.49615i
$27$		0			0
$28$	−4.89898	+	4.89898i	−0.925820	+	0.925820i
$29$	4.24264			0.787839			0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264			0.787839			0.393919	+	0.919145i	$0.371119\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$		0			0
$34$			6.00000i			1.02899i
$35$		0			0
$36$		0			0
$37$	−2.44949	−	2.44949i	−0.402694	−	0.402694i	0.476488	−	0.879181i	$-0.341910\pi$
$37$	−2.44949	−	2.44949i	−0.402694	−	0.402694i	−0.879181	+	0.476488i	$0.841910\pi$
$38$	−8.66025	−	8.66025i	−1.40488	−	1.40488i
$39$		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	−0.607527	−	0.794299i	$-0.707838\pi$
$43$	1.22474	−	1.22474i	0.186772	−	0.186772i	0.794299	+	0.607527i	$0.207838\pi$
$44$	−16.9706			−2.55841
$45$		0			0
$46$	6.00000			0.884652
$47$	5.19615	−	5.19615i	0.757937	−	0.757937i	−0.218010	−	0.975947i	$-0.569956\pi$
$47$	5.19615	−	5.19615i	0.757937	−	0.757937i	0.975947	+	0.218010i	$0.0699565\pi$
$48$		0			0
$49$		−	4.00000i		−	0.571429i
$50$		0			0
$51$		0			0
$52$	−14.6969	−	14.6969i	−2.03810	−	2.03810i
$53$	6.92820	+	6.92820i	0.951662	+	0.951662i	0.998884	−	0.0472225i	$-0.0150370\pi$
$53$	6.92820	+	6.92820i	0.951662	+	0.951662i	−0.0472225	+	0.998884i	$0.515037\pi$
$54$		0			0
$55$		0			0
$56$		−	8.48528i		−	1.13389i
$57$		0			0
$58$	−7.34847	+	7.34847i	−0.964901	+	0.964901i
$59$	−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	$0.647910\pi$
$62$	−1.73205	+	1.73205i	−0.219971	+	0.219971i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−3.67423	−	3.67423i	−0.448879	−	0.448879i	0.446103	−	0.894982i	$-0.352812\pi$
$67$	−3.67423	−	3.67423i	−0.448879	−	0.448879i	−0.894982	+	0.446103i	$0.852812\pi$
$68$	−6.92820	−	6.92820i	−0.840168	−	0.840168i
$69$		0			0
$70$		0			0
$71$			8.48528i			1.00702i	0.863990	+	0.503509i	$0.167958\pi$
$71$			8.48528i			1.00702i	−0.863990	+	0.503509i	$0.832042\pi$
$72$		0			0
$73$	−2.44949	+	2.44949i	−0.286691	+	0.286691i	−0.835770	−	0.549079i	$-0.814979\pi$
$73$	−2.44949	+	2.44949i	−0.286691	+	0.286691i	0.549079	+	0.835770i	$0.314979\pi$
$74$	8.48528			0.986394
$75$		0			0
$76$	20.0000			2.29416
$77$	−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
$82$	14.6969	+	14.6969i	1.62301	+	1.62301i
$83$	−1.73205	−	1.73205i	−0.190117	−	0.190117i	0.605629	−	0.795747i	$-0.292921\pi$
$83$	−1.73205	−	1.73205i	−0.190117	−	0.190117i	−0.795747	+	0.605629i	$0.792921\pi$
$84$		0			0
$85$		0			0
$86$			4.24264i			0.457496i
$87$		0			0
$88$	14.6969	−	14.6969i	1.56670	−	1.56670i
$89$	8.48528			0.899438			0.449719	−	0.893170i	$-0.351524\pi$
$89$	8.48528			0.899438			0.449719	+	0.893170i	$0.351524\pi$
$90$		0			0
$91$	−9.00000			−0.943456
$92$	−6.92820	+	6.92820i	−0.722315	+	0.722315i
$93$		0			0
$94$			18.0000i			1.85656i
$95$		0			0
$96$		0			0
$97$	8.57321	+	8.57321i	0.870478	+	0.870478i	0.992524	−	0.122046i	$-0.0389457\pi$
$97$	8.57321	+	8.57321i	0.870478	+	0.870478i	−0.122046	+	0.992524i	$0.538946\pi$
$98$	6.92820	+	6.92820i	0.699854	+	0.699854i
$99$		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	−0.234712	+	0.972065i	$0.575415\pi$
$103$	−12.2474	+	12.2474i	−1.20678	+	1.20678i	−0.972065	+	0.234712i	$0.924585\pi$
$104$	25.4558			2.49615
$105$		0			0
$106$	−24.0000			−2.33109
$107$	10.3923	−	10.3923i	1.00466	−	1.00466i	0.00467295	−	0.999989i	$-0.498513\pi$
$107$	10.3923	−	10.3923i	1.00466	−	1.00466i	0.999989	−	0.00467295i	$-0.00148745\pi$
$108$		0			0
$109$		−	5.00000i		−	0.478913i	−0.970907	−	0.239457i	$-0.923031\pi$
$109$		−	5.00000i		−	0.478913i	0.970907	−	0.239457i	$-0.0769693\pi$
$110$		0			0
$111$		0			0
$112$	4.89898	+	4.89898i	0.462910	+	0.462910i
$113$	10.3923	+	10.3923i	0.977626	+	0.977626i	0.999755	−	0.0221293i	$-0.00704455\pi$
$113$	10.3923	+	10.3923i	0.977626	+	0.977626i	−0.0221293	+	0.999755i	$0.507045\pi$
$114$		0			0
$115$		0			0
$116$		−	16.9706i		−	1.57568i
$117$		0			0
$118$	22.0454	−	22.0454i	2.02944	−	2.02944i
$119$	−4.24264			−0.388922
$120$		0			0
$121$	−7.00000			−0.636364
$122$	12.1244	−	12.1244i	1.09769	−	1.09769i
$123$		0			0
$124$		−	4.00000i		−	0.359211i
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	13.8564	+	13.8564i	1.22474	+	1.22474i
$129$		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
$134$	12.7279			1.09952
$135$		0			0
$136$	12.0000			1.02899
$137$	−8.66025	+	8.66025i	−0.739895	+	0.739895i	−0.972558	−	0.232662i	$-0.925256\pi$
$137$	−8.66025	+	8.66025i	−0.739895	+	0.739895i	0.232662	+	0.972558i	$0.425256\pi$
$138$		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
$142$	−14.6969	−	14.6969i	−1.23334	−	1.23334i
$143$	−15.5885	−	15.5885i	−1.30357	−	1.30357i
$144$		0			0
$145$		0			0
$146$		−	8.48528i		−	0.702247i
$147$		0			0
$148$	−9.79796	+	9.79796i	−0.805387	+	0.805387i
$149$	−4.24264			−0.347571			−0.173785	−	0.984784i	$-0.555600\pi$
$149$	−4.24264			−0.347571			−0.173785	+	0.984784i	$0.555600\pi$
$150$		0			0
$151$	13.0000			1.05792			0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000			1.05792			0.528962	+	0.848645i	$0.322581\pi$
$152$	−17.3205	+	17.3205i	−1.40488	+	1.40488i
$153$		0			0
$154$		−	18.0000i		−	1.45048i
$155$		0			0
$156$		0			0
$157$	6.12372	+	6.12372i	0.488726	+	0.488726i	0.907904	−	0.419178i	$-0.137682\pi$
$157$	6.12372	+	6.12372i	0.488726	+	0.488726i	−0.419178	+	0.907904i	$0.637682\pi$
$158$	3.46410	+	3.46410i	0.275589	+	0.275589i
$159$		0			0
$160$		0			0
$161$			4.24264i			0.334367i
$162$		0			0
$163$	−11.0227	+	11.0227i	−0.863365	+	0.863365i	−0.991727	−	0.128363i	$-0.959028\pi$
$163$	−11.0227	+	11.0227i	−0.863365	+	0.863365i	0.128363	+	0.991727i	$0.459028\pi$
$164$	−33.9411			−2.65036
$165$		0			0
$166$	6.00000			0.465690
$167$	−17.3205	+	17.3205i	−1.34030	+	1.34030i	−0.444544	+	0.895757i	$0.646634\pi$
$167$	−17.3205	+	17.3205i	−1.34030	+	1.34030i	−0.895757	+	0.444544i	$0.853366\pi$
$168$		0			0
$169$		−	14.0000i		−	1.07692i
$170$		0			0
$171$		0			0
$172$	−4.89898	−	4.89898i	−0.373544	−	0.373544i
$173$	1.73205	+	1.73205i	0.131685	+	0.131685i	0.769877	−	0.638192i	$-0.220317\pi$
$173$	1.73205	+	1.73205i	0.131685	+	0.131685i	−0.638192	+	0.769877i	$0.720317\pi$
$174$		0			0
$175$		0			0
$176$			16.9706i			1.27920i
$177$		0			0
$178$	−14.6969	+	14.6969i	−1.10158	+	1.10158i
$179$	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	$0.250440\pi$
$182$	15.5885	−	15.5885i	1.15549	−	1.15549i
$183$		0			0
$184$		−	12.0000i		−	0.884652i
$185$		0			0
$186$		0			0
$187$	−7.34847	−	7.34847i	−0.537373	−	0.537373i
$188$	−20.7846	−	20.7846i	−1.51587	−	1.51587i
$189$		0			0
$190$		0			0
$191$			12.7279i			0.920960i	0.887670	+	0.460480i	$0.152323\pi$
$191$			12.7279i			0.920960i	−0.887670	+	0.460480i	$0.847677\pi$
$192$		0			0
$193$	15.9217	−	15.9217i	1.14607	−	1.14607i	0.158749	−	0.987319i	$-0.449254\pi$
$193$	15.9217	−	15.9217i	1.14607	−	1.14607i	0.987319	−	0.158749i	$-0.0507460\pi$
$194$	−29.6985			−2.13223
$195$		0			0
$196$	−16.0000			−1.14286
$197$	−1.73205	+	1.73205i	−0.123404	+	0.123404i	−0.766111	−	0.642708i	$-0.777811\pi$
$197$	−1.73205	+	1.73205i	−0.123404	+	0.123404i	0.642708	+	0.766111i	$0.277811\pi$
$198$		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
$202$	−14.6969	−	14.6969i	−1.03407	−	1.03407i
$203$	−5.19615	−	5.19615i	−0.364698	−	0.364698i
$204$		0			0
$205$		0			0
$206$		−	42.4264i		−	2.95599i
$207$		0			0
$208$	−14.6969	+	14.6969i	−1.01905	+	1.01905i
$209$	21.2132			1.46735
$210$		0			0
$211$	11.0000			0.757271			0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000			0.757271			0.378636	+	0.925546i	$0.376393\pi$
$212$	27.7128	−	27.7128i	1.90332	−	1.90332i
$213$		0			0
$214$			36.0000i			2.46091i
$215$		0			0
$216$		0			0
$217$	−1.22474	−	1.22474i	−0.0831411	−	0.0831411i
$218$	8.66025	+	8.66025i	0.586546	+	0.586546i
$219$		0			0
$220$		0			0
$221$		−	12.7279i		−	0.856173i
$222$		0			0
$223$	6.12372	−	6.12372i	0.410075	−	0.410075i	−0.471690	−	0.881765i	$-0.656356\pi$
$223$	6.12372	−	6.12372i	0.410075	−	0.410075i	0.881765	+	0.471690i	$0.156356\pi$
$224$		0			0
$225$		0			0
$226$	−36.0000			−2.39468
$227$	13.8564	−	13.8564i	0.919682	−	0.919682i	−0.0773240	−	0.997006i	$-0.524638\pi$
$227$	13.8564	−	13.8564i	0.919682	−	0.919682i	0.997006	+	0.0773240i	$0.0246376\pi$
$228$		0			0
$229$		−	1.00000i		−	0.0660819i	−0.999454	−	0.0330409i	$-0.989481\pi$
$229$		−	1.00000i		−	0.0660819i	0.999454	−	0.0330409i	$-0.0105192\pi$
$230$		0			0
$231$		0			0
$232$	14.6969	+	14.6969i	0.964901	+	0.964901i
$233$	−10.3923	−	10.3923i	−0.680823	−	0.680823i	0.279363	−	0.960186i	$-0.409877\pi$
$233$	−10.3923	−	10.3923i	−0.680823	−	0.680823i	−0.960186	+	0.279363i	$0.909877\pi$
$234$		0			0
$235$		0			0
$236$			50.9117i			3.31407i
$237$		0			0
$238$	7.34847	−	7.34847i	0.476331	−	0.476331i
$239$	−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	$0.448515\pi$
$242$	12.1244	−	12.1244i	0.779383	−	0.779383i
$243$		0			0
$244$			28.0000i			1.79252i
$245$		0			0
$246$		0			0
$247$	18.3712	+	18.3712i	1.16893	+	1.16893i
$248$	3.46410	+	3.46410i	0.219971	+	0.219971i
$249$		0			0
$250$		0			0
$251$		−	8.48528i		−	0.535586i	−0.963476	−	0.267793i	$-0.913706\pi$
$251$		−	8.48528i		−	0.535586i	0.963476	−	0.267793i	$-0.0862944\pi$
$252$		0			0
$253$	−7.34847	+	7.34847i	−0.461994	+	0.461994i
$254$		0			0
$255$		0			0
$256$	−32.0000			−2.00000
$257$	−13.8564	+	13.8564i	−0.864339	+	0.864339i	−0.991839	−	0.127500i	$-0.959305\pi$
$257$	−13.8564	+	13.8564i	−0.864339	+	0.864339i	0.127500	+	0.991839i	$0.459305\pi$
$258$		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	$-0.399691\pi$
$263$	−10.3923	−	10.3923i	−0.640817	−	0.640817i	−0.950756	+	0.309939i	$0.899691\pi$
$264$		0			0
$265$		0			0
$266$			21.2132i			1.30066i
$267$		0			0
$268$	−14.6969	+	14.6969i	−0.897758	+	0.897758i
$269$	29.6985			1.81075			0.905374	−	0.424614i	$-0.139590\pi$
$269$	29.6985			1.81075			0.905374	+	0.424614i	$0.139590\pi$
$270$		0			0
$271$	14.0000			0.850439			0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000			0.850439			0.425220	+	0.905090i	$0.360197\pi$
$272$	−6.92820	+	6.92820i	−0.420084	+	0.420084i
$273$		0			0
$274$		−	30.0000i		−	1.81237i
$275$		0			0
$276$		0			0
$277$	8.57321	+	8.57321i	0.515115	+	0.515115i	0.916089	−	0.400975i	$-0.131328\pi$
$277$	8.57321	+	8.57321i	0.515115	+	0.515115i	−0.400975	+	0.916089i	$0.631328\pi$
$278$	−3.46410	−	3.46410i	−0.207763	−	0.207763i
$279$		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	−0.182388	−	0.983227i	$-0.558383\pi$
$283$	13.4722	−	13.4722i	0.800839	−	0.800839i	0.983227	+	0.182388i	$0.0583827\pi$
$284$	33.9411			2.01404
$285$		0			0
$286$	54.0000			3.19309
$287$	−10.3923	+	10.3923i	−0.613438	+	0.613438i
$288$		0			0
$289$			11.0000i			0.647059i
$290$		0			0
$291$		0			0
$292$	9.79796	+	9.79796i	0.573382	+	0.573382i
$293$	1.73205	+	1.73205i	0.101187	+	0.101187i	0.755888	−	0.654701i	$-0.227205\pi$
$293$	1.73205	+	1.73205i	0.101187	+	0.101187i	−0.654701	+	0.755888i	$0.727205\pi$
$294$		0			0
$295$		0			0
$296$		−	16.9706i		−	0.986394i
$297$		0			0
$298$	7.34847	−	7.34847i	0.425685	−	0.425685i
$299$	−12.7279			−0.736075
$300$		0			0
$301$	−3.00000			−0.172917
$302$	−22.5167	+	22.5167i	−1.29569	+	1.29569i
$303$		0			0
$304$		−	20.0000i		−	1.14708i
$305$		0			0
$306$		0			0
$307$	−23.2702	−	23.2702i	−1.32810	−	1.32810i	−0.907028	−	0.421069i	$-0.861655\pi$
$307$	−23.2702	−	23.2702i	−1.32810	−	1.32810i	−0.421069	−	0.907028i	$-0.638345\pi$
$308$	20.7846	+	20.7846i	1.18431	+	1.18431i
$309$		0			0
$310$		0			0
$311$			4.24264i			0.240578i	0.992739	+	0.120289i	$0.0383821\pi$
$311$			4.24264i			0.240578i	−0.992739	+	0.120289i	$0.961618\pi$
$312$		0			0
$313$	−3.67423	+	3.67423i	−0.207680	+	0.207680i	−0.803281	−	0.595601i	$-0.796914\pi$
$313$	−3.67423	+	3.67423i	−0.207680	+	0.207680i	0.595601	+	0.803281i	$0.296914\pi$
$314$	−21.2132			−1.19713
$315$		0			0
$316$	−8.00000			−0.450035
$317$	3.46410	−	3.46410i	0.194563	−	0.194563i	−0.603101	−	0.797665i	$-0.706069\pi$
$317$	3.46410	−	3.46410i	0.194563	−	0.194563i	0.797665	+	0.603101i	$0.206069\pi$
$318$		0			0
$319$		−	18.0000i		−	1.00781i
$320$		0			0
$321$		0			0
$322$	−7.34847	−	7.34847i	−0.409514	−	0.409514i
$323$	8.66025	+	8.66025i	0.481869	+	0.481869i
$324$		0			0
$325$		0			0
$326$		−	38.1838i		−	2.11480i
$327$		0			0
$328$	29.3939	−	29.3939i	1.62301	−	1.62301i
$329$	−12.7279			−0.701713
$330$		0			0
$331$	−22.0000			−1.20923			−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000			−1.20923			−0.604615	+	0.796518i	$0.706673\pi$
$332$	−6.92820	+	6.92820i	−0.380235	+	0.380235i
$333$		0			0
$334$		−	60.0000i		−	3.28305i
$335$		0			0
$336$		0			0
$337$	−18.3712	−	18.3712i	−1.00074	−	1.00074i	−1.00000		0.000741840i	$-0.999764\pi$
$337$	−18.3712	−	18.3712i	−1.00074	−	1.00074i	−0.000741840	−	1.00000i	$-0.500236\pi$
$338$	24.2487	+	24.2487i	1.31896	+	1.31896i
$339$		0			0
$340$		0			0
$341$		−	4.24264i		−	0.229752i
$342$		0			0
$343$	−13.4722	+	13.4722i	−0.727430	+	0.727430i
$344$	8.48528			0.457496
$345$		0			0
$346$	−6.00000			−0.322562
$347$	15.5885	−	15.5885i	0.836832	−	0.836832i	−0.151608	−	0.988441i	$-0.548445\pi$
$347$	15.5885	−	15.5885i	0.836832	−	0.836832i	0.988441	+	0.151608i	$0.0484453\pi$
$348$		0			0
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.5885	+	15.5885i	0.829690	+	0.829690i	0.987474	−	0.157784i	$-0.0504349\pi$
$353$	15.5885	+	15.5885i	0.829690	+	0.829690i	−0.157784	+	0.987474i	$0.550435\pi$
$354$		0			0
$355$		0			0
$356$		−	33.9411i		−	1.79888i
$357$		0			0
$358$	−7.34847	+	7.34847i	−0.388379	+	0.388379i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−6.00000			−0.315789
$362$	−32.9090	+	32.9090i	−1.72966	+	1.72966i
$363$		0			0
$364$			36.0000i			1.88691i
$365$		0			0
$366$		0			0
$367$	11.0227	+	11.0227i	0.575380	+	0.575380i	0.933627	−	0.358247i	$-0.116625\pi$
$367$	11.0227	+	11.0227i	0.575380	+	0.575380i	−0.358247	+	0.933627i	$0.616625\pi$
$368$	6.92820	+	6.92820i	0.361158	+	0.361158i
$369$		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	$-0.764278\pi$
$373$	−1.22474	+	1.22474i	−0.0634149	+	0.0634149i	0.674688	+	0.738103i	$0.264278\pi$
$374$	25.4558			1.31629
$375$		0			0
$376$	36.0000			1.85656
$377$	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
$382$	−22.0454	−	22.0454i	−1.12794	−	1.12794i
$383$	−3.46410	−	3.46410i	−0.177007	−	0.177007i	0.613043	−	0.790050i	$-0.289945\pi$
$383$	−3.46410	−	3.46410i	−0.177007	−	0.177007i	−0.790050	+	0.613043i	$0.789945\pi$
$384$		0			0
$385$		0			0
$386$			55.1543i			2.80728i
$387$		0			0
$388$	34.2929	−	34.2929i	1.74096	−	1.74096i
$389$	−16.9706			−0.860442			−0.430221	−	0.902724i	$-0.641564\pi$
$389$	−16.9706			−0.860442			−0.430221	+	0.902724i	$0.641564\pi$
$390$		0			0
$391$	−6.00000			−0.303433
$392$	13.8564	−	13.8564i	0.699854	−	0.699854i
$393$		0			0
$394$		−	6.00000i		−	0.302276i
$395$		0			0
$396$		0			0
$397$	8.57321	+	8.57321i	0.430277	+	0.430277i	0.888723	−	0.458445i	$-0.151594\pi$
$397$	8.57321	+	8.57321i	0.430277	+	0.430277i	−0.458445	+	0.888723i	$0.651594\pi$
$398$	−43.3013	−	43.3013i	−2.17050	−	2.17050i
$399$		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
$404$	33.9411			1.68863
$405$		0			0
$406$	18.0000			0.893325
$407$	−10.3923	+	10.3923i	−0.515127	+	0.515127i
$408$		0			0
$409$			11.0000i			0.543915i	0.962309	+	0.271957i	$0.0876710\pi$
$409$			11.0000i			0.543915i	−0.962309	+	0.271957i	$0.912329\pi$
$410$		0			0
$411$		0			0
$412$	48.9898	+	48.9898i	2.41355	+	2.41355i
$413$	15.5885	+	15.5885i	0.767058	+	0.767058i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−36.7423	+	36.7423i	−1.79713	+	1.79713i
$419$	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	$0.662044\pi$
$422$	−19.0526	+	19.0526i	−0.927464	+	0.927464i
$423$		0			0
$424$			48.0000i			2.33109i
$425$		0			0
$426$		0			0
$427$	8.57321	+	8.57321i	0.414887	+	0.414887i
$428$	−41.5692	−	41.5692i	−2.00932	−	2.00932i
$429$		0			0
$430$		0			0
$431$			29.6985i			1.43053i	0.698856	+	0.715263i	$0.253693\pi$
$431$			29.6985i			1.43053i	−0.698856	+	0.715263i	$0.746307\pi$
$432$		0			0
$433$	−1.22474	+	1.22474i	−0.0588575	+	0.0588575i	−0.735923	−	0.677065i	$-0.763251\pi$
$433$	−1.22474	+	1.22474i	−0.0588575	+	0.0588575i	0.677065	+	0.735923i	$0.263251\pi$
$434$	4.24264			0.203653
$435$		0			0
$436$	−20.0000			−0.957826
$437$	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
$442$	22.0454	+	22.0454i	1.04859	+	1.04859i
$443$	17.3205	+	17.3205i	0.822922	+	0.822922i	0.986526	−	0.163604i	$-0.0523119\pi$
$443$	17.3205	+	17.3205i	0.822922	+	0.822922i	−0.163604	+	0.986526i	$0.552312\pi$
$444$		0			0
$445$		0			0
$446$			21.2132i			1.00447i
$447$		0			0
$448$	−9.79796	+	9.79796i	−0.462910	+	0.462910i
$449$	−33.9411			−1.60178			−0.800890	−	0.598811i	$-0.795640\pi$
$449$	−33.9411			−1.60178			−0.800890	+	0.598811i	$0.795640\pi$
$450$		0			0
$451$	−36.0000			−1.69517
$452$	41.5692	−	41.5692i	1.95525	−	1.95525i
$453$		0			0
$454$			48.0000i			2.25275i
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$	1.73205	+	1.73205i	0.0809334	+	0.0809334i
$459$		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	−0.149633	+	0.988742i	$0.547809\pi$
$463$	−24.4949	+	24.4949i	−1.13837	+	1.13837i	−0.988742	+	0.149633i	$0.952191\pi$
$464$	−16.9706			−0.787839
$465$		0			0
$466$	36.0000			1.66767
$467$	19.0526	−	19.0526i	0.881647	−	0.881647i	−0.112055	−	0.993702i	$-0.535743\pi$
$467$	19.0526	−	19.0526i	0.881647	−	0.881647i	0.993702	+	0.112055i	$0.0357432\pi$
$468$		0			0
$469$			9.00000i			0.415581i
$470$		0			0
$471$		0			0
$472$	−44.0908	−	44.0908i	−2.02944	−	2.02944i
$473$	−5.19615	−	5.19615i	−0.238919	−	0.238919i
$474$		0			0
$475$		0			0
$476$			16.9706i			0.777844i
$477$		0			0
$478$	29.3939	−	29.3939i	1.34444	−	1.34444i
$479$	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
$482$	−8.66025	+	8.66025i	−0.394464	+	0.394464i
$483$		0			0
$484$			28.0000i			1.27273i
$485$		0			0
$486$		0			0
$487$	−18.3712	−	18.3712i	−0.832477	−	0.832477i	0.155378	−	0.987855i	$-0.450341\pi$
$487$	−18.3712	−	18.3712i	−0.832477	−	0.832477i	−0.987855	+	0.155378i	$0.950341\pi$
$488$	−24.2487	−	24.2487i	−1.09769	−	1.09769i
$489$		0			0
$490$		0			0
$491$		−	42.4264i		−	1.91468i	−0.288969	−	0.957338i	$-0.593312\pi$
$491$		−	42.4264i		−	1.91468i	0.288969	−	0.957338i	$-0.406688\pi$
$492$		0			0
$493$	7.34847	−	7.34847i	0.330958	−	0.330958i
$494$	−63.6396			−2.86328
$495$		0			0
$496$	−4.00000			−0.179605
$497$	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
$502$	14.6969	+	14.6969i	0.655956	+	0.655956i
$503$	13.8564	+	13.8564i	0.617827	+	0.617827i	0.944974	−	0.327147i	$-0.106087\pi$
$503$	13.8564	+	13.8564i	0.617827	+	0.617827i	−0.327147	+	0.944974i	$0.606087\pi$
$504$		0			0
$505$		0			0
$506$		−	25.4558i		−	1.13165i
$507$		0			0
$508$		0			0
$509$	−4.24264			−0.188052			−0.0940259	−	0.995570i	$-0.529974\pi$
$509$	−4.24264			−0.188052			−0.0940259	+	0.995570i	$0.529974\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$	27.7128	−	27.7128i	1.22474	−	1.22474i
$513$		0			0
$514$		−	48.0000i		−	2.11719i
$515$		0			0
$516$		0			0
$517$	−22.0454	−	22.0454i	−0.969556	−	0.969556i
$518$	−10.3923	−	10.3923i	−0.456612	−	0.456612i
$519$		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	−0.348289	−	0.937387i	$-0.613237\pi$
$523$	13.4722	−	13.4722i	0.589098	−	0.589098i	0.937387	+	0.348289i	$0.113237\pi$
$524$		0			0
$525$		0			0
$526$	36.0000			1.56967
$527$	1.73205	−	1.73205i	0.0754493	−	0.0754493i
$528$		0			0
$529$		−	17.0000i		−	0.739130i
$530$		0			0
$531$		0			0
$532$	−24.4949	−	24.4949i	−1.06199	−	1.06199i
$533$	−31.1769	−	31.1769i	−1.35042	−	1.35042i
$534$		0			0
$535$		0			0
$536$		−	25.4558i		−	1.09952i
$537$		0			0
$538$	−51.4393	+	51.4393i	−2.21771	+	2.21771i
$539$	−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	$0.506843\pi$
$542$	−24.2487	+	24.2487i	−1.04157	+	1.04157i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12.2474	−	12.2474i	−0.523663	−	0.523663i	0.395013	−	0.918676i	$-0.370740\pi$
$547$	−12.2474	−	12.2474i	−0.523663	−	0.523663i	−0.918676	+	0.395013i	$0.870740\pi$
$548$	34.6410	+	34.6410i	1.47979	+	1.47979i
$549$		0			0
$550$		0			0
$551$			21.2132i			0.903713i
$552$		0			0
$553$	−2.44949	+	2.44949i	−0.104163	+	0.104163i
$554$	−29.6985			−1.26177
$555$		0			0
$556$	8.00000			0.339276
$557$	−15.5885	+	15.5885i	−0.660504	+	0.660504i	−0.955499	−	0.294995i	$-0.904682\pi$
$557$	−15.5885	+	15.5885i	−0.660504	+	0.660504i	0.294995	+	0.955499i	$0.404682\pi$
$558$		0			0
$559$		−	9.00000i		−	0.380659i
$560$		0			0
$561$		0			0
$562$	−7.34847	−	7.34847i	−0.309976	−	0.309976i
$563$	25.9808	+	25.9808i	1.09496	+	1.09496i	0.994991	+	0.0999679i	$0.0318740\pi$
$563$	25.9808	+	25.9808i	1.09496	+	1.09496i	0.0999679	+	0.994991i	$0.468126\pi$
$564$		0			0
$565$		0			0
$566$			46.6690i			1.96165i
$567$		0			0
$568$	−29.3939	+	29.3939i	−1.23334	+	1.23334i
$569$	−12.7279			−0.533582			−0.266791	−	0.963754i	$-0.585963\pi$
$569$	−12.7279			−0.533582			−0.266791	+	0.963754i	$0.585963\pi$
$570$		0			0
$571$	17.0000			0.711428			0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000			0.711428			0.355714	+	0.934595i	$0.384238\pi$
$572$	−62.3538	+	62.3538i	−2.60714	+	2.60714i
$573$		0			0
$574$		−	36.0000i		−	1.50261i
$575$		0			0
$576$		0			0
$577$	23.2702	+	23.2702i	0.968749	+	0.968749i	0.999526	−	0.0307771i	$-0.00979822\pi$
$577$	23.2702	+	23.2702i	0.968749	+	0.968749i	−0.0307771	+	0.999526i	$0.509798\pi$
$578$	−19.0526	−	19.0526i	−0.792482	−	0.792482i
$579$		0			0
$580$		0			0
$581$			4.24264i			0.176014i
$582$		0			0
$583$	29.3939	−	29.3939i	1.21737	−	1.21737i
$584$	−16.9706			−0.702247
$585$		0			0
$586$	−6.00000			−0.247858
$587$	−13.8564	+	13.8564i	−0.571915	+	0.571915i	−0.932663	−	0.360748i	$-0.882521\pi$
$587$	−13.8564	+	13.8564i	−0.571915	+	0.571915i	0.360748	+	0.932663i	$0.382521\pi$
$588$		0			0
$589$			5.00000i			0.206021i
$590$		0			0
$591$		0			0
$592$	9.79796	+	9.79796i	0.402694	+	0.402694i
$593$	−6.92820	−	6.92820i	−0.284507	−	0.284507i	0.550396	−	0.834904i	$-0.314477\pi$
$593$	−6.92820	−	6.92820i	−0.284507	−	0.284507i	−0.834904	+	0.550396i	$0.814477\pi$
$594$		0			0
$595$		0			0
$596$			16.9706i			0.695141i
$597$		0			0
$598$	22.0454	−	22.0454i	0.901504	−	0.901504i
$599$	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	$0.159539\pi$
$602$	5.19615	−	5.19615i	0.211779	−	0.211779i
$603$		0			0
$604$		−	52.0000i		−	2.11585i
$605$		0			0
$606$		0			0
$607$	19.5959	+	19.5959i	0.795374	+	0.795374i	0.982362	−	0.186988i	$-0.0598727\pi$
$607$	19.5959	+	19.5959i	0.795374	+	0.795374i	−0.186988	+	0.982362i	$0.559873\pi$
$608$		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	−0.0925671	+	0.995706i	$0.529507\pi$
$613$	−26.9444	+	26.9444i	−1.08827	+	1.08827i	−0.995706	+	0.0925671i	$0.970493\pi$
$614$	80.6102			3.25316
$615$		0			0
$616$	−36.0000			−1.45048
$617$	17.3205	−	17.3205i	0.697297	−	0.697297i	−0.266529	−	0.963827i	$-0.585877\pi$
$617$	17.3205	−	17.3205i	0.697297	−	0.697297i	0.963827	+	0.266529i	$0.0858769\pi$
$618$		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
$622$	−7.34847	−	7.34847i	−0.294647	−	0.294647i
$623$	−10.3923	−	10.3923i	−0.416359	−	0.416359i
$624$		0			0
$625$		0			0
$626$		−	12.7279i		−	0.508710i
$627$		0			0
$628$	24.4949	−	24.4949i	0.977453	−	0.977453i
$629$	−8.48528			−0.338330
$630$		0			0
$631$	35.0000			1.39333			0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000			1.39333			0.696664	+	0.717398i	$0.254667\pi$
$632$	6.92820	−	6.92820i	0.275589	−	0.275589i
$633$		0			0
$634$			12.0000i			0.476581i
$635$		0			0
$636$		0			0
$637$	−14.6969	−	14.6969i	−0.582314	−	0.582314i
$638$	31.1769	+	31.1769i	1.23431	+	1.23431i
$639$		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	0.174529	−	0.984652i	$-0.444160\pi$
$643$	29.3939	−	29.3939i	1.15918	−	1.15918i	0.984652	−	0.174529i	$-0.0558404\pi$
$644$	16.9706			0.668734
$645$		0			0
$646$	−30.0000			−1.18033
$647$	−17.3205	+	17.3205i	−0.680939	+	0.680939i	−0.960212	−	0.279272i	$-0.909907\pi$
$647$	−17.3205	+	17.3205i	−0.680939	+	0.680939i	0.279272	+	0.960212i	$0.409907\pi$
$648$		0			0
$649$			54.0000i			2.11969i
$650$		0			0
$651$		0			0
$652$	44.0908	+	44.0908i	1.72673	+	1.72673i
$653$	−19.0526	−	19.0526i	−0.745584	−	0.745584i	0.228062	−	0.973647i	$-0.426761\pi$
$653$	−19.0526	−	19.0526i	−0.745584	−	0.745584i	−0.973647	+	0.228062i	$0.926761\pi$
$654$		0			0
$655$		0			0
$656$			33.9411i			1.32518i
$657$		0			0
$658$	22.0454	−	22.0454i	0.859419	−	0.859419i
$659$	−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	$0.627164\pi$
$662$	38.1051	−	38.1051i	1.48100	−	1.48100i
$663$		0			0
$664$		−	12.0000i		−	0.465690i
$665$		0			0
$666$		0			0
$667$	−7.34847	−	7.34847i	−0.284534	−	0.284534i
$668$	69.2820	+	69.2820i	2.68060	+	2.68060i
$669$		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	$-0.771268\pi$
$673$	−2.44949	+	2.44949i	−0.0944209	+	0.0944209i	0.658319	+	0.752739i	$0.271268\pi$
$674$	63.6396			2.45131
$675$		0			0
$676$	−56.0000			−2.15385
$677$	15.5885	−	15.5885i	0.599113	−	0.599113i	−0.340963	−	0.940077i	$-0.610753\pi$
$677$	15.5885	−	15.5885i	0.599113	−	0.599113i	0.940077	+	0.340963i	$0.110753\pi$
$678$		0			0
$679$		−	21.0000i		−	0.805906i
$680$		0			0
$681$		0			0
$682$	7.34847	+	7.34847i	0.281387	+	0.281387i
$683$	6.92820	+	6.92820i	0.265100	+	0.265100i	0.827122	−	0.562022i	$-0.189976\pi$
$683$	6.92820	+	6.92820i	0.265100	+	0.265100i	−0.562022	+	0.827122i	$0.689976\pi$
$684$		0			0
$685$		0			0
$686$		−	46.6690i		−	1.78183i
$687$		0			0
$688$	−4.89898	+	4.89898i	−0.186772	+	0.186772i
$689$	50.9117			1.93958
$690$		0			0
$691$	−32.0000			−1.21734			−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000			−1.21734			−0.608669	+	0.793424i	$0.708296\pi$
$692$	6.92820	−	6.92820i	0.263371	−	0.263371i
$693$		0			0
$694$			54.0000i			2.04981i
$695$		0			0
$696$		0			0
$697$	−14.6969	−	14.6969i	−0.556686	−	0.556686i
$698$	27.7128	+	27.7128i	1.04895	+	1.04895i
$699$		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
$704$	−33.9411			−1.27920
$705$		0			0
$706$	−54.0000			−2.03232
$707$	10.3923	−	10.3923i	0.390843	−	0.390843i
$708$		0			0
$709$			41.0000i			1.53979i	0.638172	+	0.769894i	$0.279691\pi$
$709$			41.0000i			1.53979i	−0.638172	+	0.769894i	$0.720309\pi$
$710$		0			0
$711$		0			0
$712$	29.3939	+	29.3939i	1.10158	+	1.10158i
$713$	−1.73205	−	1.73205i	−0.0648658	−	0.0648658i
$714$		0			0
$715$		0			0
$716$		−	16.9706i		−	0.634220i
$717$		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
$722$	10.3923	−	10.3923i	0.386762	−	0.386762i
$723$		0			0
$724$		−	76.0000i		−	2.82452i
$725$		0			0
$726$		0			0
$727$	35.5176	+	35.5176i	1.31727	+	1.31727i	0.915925	+	0.401350i	$0.131459\pi$
$727$	35.5176	+	35.5176i	1.31727	+	1.31727i	0.401350	+	0.915925i	$0.368541\pi$
$728$	−31.1769	−	31.1769i	−1.15549	−	1.15549i
$729$		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	$-0.554701\pi$
$733$	22.0454	−	22.0454i	0.814266	−	0.814266i	0.985270	+	0.171005i	$0.0547013\pi$
$734$	−38.1838			−1.40939
$735$		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
$742$	29.3939	+	29.3939i	1.07908	+	1.07908i
$743$	−20.7846	−	20.7846i	−0.762513	−	0.762513i	0.214263	−	0.976776i	$-0.431265\pi$
$743$	−20.7846	−	20.7846i	−0.762513	−	0.762513i	−0.976776	+	0.214263i	$0.931265\pi$
$744$		0			0
$745$		0			0
$746$		−	4.24264i		−	0.155334i
$747$		0			0
$748$	−29.3939	+	29.3939i	−1.07475	+	1.07475i
$749$	−25.4558			−0.930136
$750$		0			0
$751$	46.0000			1.67856			0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000			1.67856			0.839282	+	0.543696i	$0.182976\pi$
$752$	−20.7846	+	20.7846i	−0.757937	+	0.757937i
$753$		0			0
$754$			54.0000i			1.96656i
$755$		0			0
$756$		0			0
$757$	−15.9217	−	15.9217i	−0.578683	−	0.578683i	0.355857	−	0.934540i	$-0.384189\pi$
$757$	−15.9217	−	15.9217i	−0.578683	−	0.578683i	−0.934540	+	0.355857i	$0.884189\pi$
$758$	1.73205	+	1.73205i	0.0629109	+	0.0629109i
$759$		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
$764$	50.9117			1.84192
$765$		0			0
$766$	12.0000			0.433578
$767$	−46.7654	+	46.7654i	−1.68860	+	1.68860i
$768$		0			0
$769$		−	29.0000i		−	1.04577i	−0.852404	−	0.522883i	$-0.824856\pi$
$769$		−	29.0000i		−	1.04577i	0.852404	−	0.522883i	$-0.175144\pi$
$770$		0			0
$771$		0			0
$772$	−63.6867	−	63.6867i	−2.29214	−	2.29214i
$773$	−27.7128	−	27.7128i	−0.996761	−	0.996761i	0.00323417	−	0.999995i	$-0.498971\pi$
$773$	−27.7128	−	27.7128i	−0.996761	−	0.996761i	−0.999995	+	0.00323417i	$0.998971\pi$
$774$		0			0
$775$		0			0
$776$			59.3970i			2.13223i
$777$		0			0
$778$	29.3939	−	29.3939i	1.05382	−	1.05382i
$779$	42.4264			1.52008
$780$		0			0
$781$	36.0000			1.28818
$782$	10.3923	−	10.3923i	0.371628	−	0.371628i
$783$		0			0
$784$			16.0000i			0.571429i
$785$		0			0
$786$		0			0
$787$	20.8207	+	20.8207i	0.742176	+	0.742176i	0.972996	−	0.230820i	$-0.0741409\pi$
$787$	20.8207	+	20.8207i	0.742176	+	0.742176i	−0.230820	+	0.972996i	$0.574141\pi$
$788$	6.92820	+	6.92820i	0.246807	+	0.246807i
$789$		0			0
$790$		0			0
$791$		−	25.4558i		−	0.905106i
$792$		0			0

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.73205i) q^{2} -4.00000i q^{4} +(-1.22474 - 1.22474i) q^{7} +(3.46410 + 3.46410i) q^{8} +O(q^{10})\)	\(q+(-1.73205 + 1.73205i) q^{2} -4.00000i q^{4} +(-1.22474 - 1.22474i) q^{7} +(3.46410 + 3.46410i) q^{8} -4.24264i q^{11} +(3.67423 - 3.67423i) q^{13} +4.24264 q^{14} -4.00000 q^{16} +(1.73205 - 1.73205i) q^{17} +5.00000i q^{19} +(7.34847 + 7.34847i) q^{22} +(-1.73205 - 1.73205i) q^{23} +12.7279i q^{26} +(-4.89898 + 4.89898i) q^{28} +4.24264 q^{29} +1.00000 q^{31} +6.00000i q^{34} +(-2.44949 - 2.44949i) q^{37} +(-8.66025 - 8.66025i) q^{38} -8.48528i q^{41} +(1.22474 - 1.22474i) q^{43} -16.9706 q^{44} +6.00000 q^{46} +(5.19615 - 5.19615i) q^{47} -4.00000i q^{49} +(-14.6969 - 14.6969i) q^{52} +(6.92820 + 6.92820i) q^{53} -8.48528i q^{56} +(-7.34847 + 7.34847i) q^{58} -12.7279 q^{59} -7.00000 q^{61} +(-1.73205 + 1.73205i) q^{62} -8.00000i q^{64} +(-3.67423 - 3.67423i) q^{67} +(-6.92820 - 6.92820i) q^{68} +8.48528i q^{71} +(-2.44949 + 2.44949i) q^{73} +8.48528 q^{74} +20.0000 q^{76} +(-5.19615 + 5.19615i) q^{77} -2.00000i q^{79} +(14.6969 + 14.6969i) q^{82} +(-1.73205 - 1.73205i) q^{83} +4.24264i q^{86} +(14.6969 - 14.6969i) q^{88} +8.48528 q^{89} -9.00000 q^{91} +(-6.92820 + 6.92820i) q^{92} +18.0000i q^{94} +(8.57321 + 8.57321i) q^{97} +(6.92820 + 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 32 q^{16} + 8 q^{31} + 48 q^{46} - 56 q^{61} + 160 q^{76} - 72 q^{91}+O(q^{100}) \) 8 * q - 32 * q^16 + 8 * q^31 + 48 * q^46 - 56 * q^61 + 160 * q^76 - 72 * q^91

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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