LMFDB - Embedded newform 3525.1.l.d.1268.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,1,Mod(1268,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3525.1268");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3525.l (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:	$1.75920416953$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{120})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 $ x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{30}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{30} - \cdots)$

Embedding invariants

Embedding label			1268.16
Root			$-0.0523360 + 0.998630i$ of defining polynomial
Character	$\chi$	$=$	3525.1268
Dual form			3525.1.l.d.1832.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.38331 + 1.38331i) q^{2} +(0.777146 + 0.629320i) q^{3} +2.82709i q^{4} +(0.204489 + 1.94558i) q^{6} +(1.05097 - 1.05097i) q^{7} +(-2.52743 + 2.52743i) q^{8} +(0.207912 + 0.978148i) q^{9} +O(q^{10})$	\(q+(1.38331 + 1.38331i) q^{2} +(0.777146 + 0.629320i) q^{3} +2.82709i q^{4} +(0.204489 + 1.94558i) q^{6} +(1.05097 - 1.05097i) q^{7} +(-2.52743 + 2.52743i) q^{8} +(0.207912 + 0.978148i) q^{9} +(-1.77915 + 2.19706i) q^{12} +2.90762 q^{14} -4.16535 q^{16} +(-1.29195 - 1.29195i) q^{17} +(-1.06547 + 1.64069i) q^{18} +(1.47815 - 0.155360i) q^{21} +(-3.55475 + 0.373619i) q^{24} +(-0.453990 + 0.891007i) q^{27} +(2.97117 + 2.97117i) q^{28} +(-3.23454 - 3.23454i) q^{32} -3.57433i q^{34} +(-2.76531 + 0.587785i) q^{36} +(0.831254 - 0.831254i) q^{37} +(2.25965 + 1.82983i) q^{42} +(0.707107 + 0.707107i) q^{47} +(-3.23709 - 2.62134i) q^{48} -1.20906i q^{49} +(-0.190983 - 1.81708i) q^{51} +(-0.437016 + 0.437016i) q^{53} +(-1.86055 + 0.604528i) q^{54} +5.31249i q^{56} -1.48629 q^{59} -0.618034 q^{61} +(1.24651 + 0.809491i) q^{63} -4.78339i q^{64} +(3.65246 - 3.65246i) q^{68} -1.98904i q^{71} +(-2.99768 - 1.94672i) q^{72} +2.29976 q^{74} +1.61803i q^{79} +(-0.913545 + 0.406737i) q^{81} +(0.707107 - 0.707107i) q^{83} +(0.439216 + 4.17886i) q^{84} +1.90211 q^{89} +1.95630i q^{94} +(-0.478148 - 4.54927i) q^{96} +(1.34500 - 1.34500i) q^{97} +(1.67250 - 1.67250i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 4 q^{6}+O(q^{10}) $ 32 * q - 4 * q^6	$ 32 q - 4 q^{6} - 48 q^{16} + 12 q^{21} - 24 q^{51} + 16 q^{61} - 4 q^{81} + 20 q^{96}+O(q^{100}) $ 32 * q - 4 * q^6 - 48 * q^16 + 12 * q^21 - 24 * q^51 + 16 * q^61 - 4 * q^81 + 20 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times$.

$n$	$1552$	$2026$	$2351$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

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 $

$2$	1.38331	+	1.38331i	1.38331	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$2$	1.38331	+	1.38331i	1.38331	+	1.38331i	0.544639	+	0.838671i	$0.316667\pi$
$3$	0.777146	+	0.629320i	0.777146	+	0.629320i
$3$	0.777146	+	0.629320i	0.777146	+	0.629320i
$4$			2.82709i			2.82709i
$5$		0			0
$5$		0			0
$6$	0.204489	+	1.94558i	0.204489	+	1.94558i
$7$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.0523360	−	0.998630i	$-0.483333\pi$
$7$	1.05097	−	1.05097i	1.05097	−	1.05097i	0.998630	−	0.0523360i	$-0.0166667\pi$
$8$	−2.52743	+	2.52743i	−2.52743	+	2.52743i
$9$	0.207912	+	0.978148i	0.207912	+	0.978148i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−1.77915	+	2.19706i	−1.77915	+	2.19706i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	2.90762			2.90762
$15$		0			0
$16$	−4.16535			−4.16535
$17$	−1.29195	−	1.29195i	−1.29195	−	1.29195i	−0.933580	−	0.358368i	$-0.883333\pi$
$17$	−1.29195	−	1.29195i	−1.29195	−	1.29195i	−0.358368	−	0.933580i	$-0.616667\pi$
$18$	−1.06547	+	1.64069i	−1.06547	+	1.64069i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	1.47815	−	0.155360i	1.47815	−	0.155360i
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	−3.55475	+	0.373619i	−3.55475	+	0.373619i
$25$		0			0
$26$		0			0
$27$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$28$	2.97117	+	2.97117i	2.97117	+	2.97117i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−3.23454	−	3.23454i	−3.23454	−	3.23454i
$33$		0			0
$34$		−	3.57433i		−	3.57433i
$35$		0			0
$36$	−2.76531	+	0.587785i	−2.76531	+	0.587785i
$37$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$37$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	2.25965	+	1.82983i	2.25965	+	1.82983i
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.707107	+	0.707107i	0.707107	+	0.707107i
$47$	0.707107	+	0.707107i	0.707107	+	0.707107i
$48$	−3.23709	−	2.62134i	−3.23709	−	2.62134i
$49$		−	1.20906i		−	1.20906i
$50$		0			0
$51$	−0.190983	−	1.81708i	−0.190983	−	1.81708i
$52$		0			0
$53$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$53$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$54$	−1.86055	+	0.604528i	−1.86055	+	0.604528i
$55$		0			0
$56$			5.31249i			5.31249i
$57$		0			0
$58$		0			0
$59$	−1.48629			−1.48629			−0.743145	−	0.669131i	$-0.766667\pi$
$59$	−1.48629			−1.48629			−0.743145	+	0.669131i	$0.766667\pi$
$60$		0			0
$61$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$61$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$62$		0			0
$63$	1.24651	+	0.809491i	1.24651	+	0.809491i
$64$		−	4.78339i		−	4.78339i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	3.65246	−	3.65246i	3.65246	−	3.65246i
$69$		0			0
$70$		0			0
$71$		−	1.98904i		−	1.98904i	−0.104528	−	0.994522i	$-0.533333\pi$
$71$		−	1.98904i		−	1.98904i	0.104528	−	0.994522i	$-0.466667\pi$
$72$	−2.99768	−	1.94672i	−2.99768	−	1.94672i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	2.29976			2.29976
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$79$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$80$		0			0
$81$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$82$		0			0
$83$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$83$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$84$	0.439216	+	4.17886i	0.439216	+	4.17886i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$89$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$			1.95630i			1.95630i
$95$		0			0
$96$	−0.478148	−	4.54927i	−0.478148	−	4.54927i
$97$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$97$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$98$	1.67250	−	1.67250i	1.67250	−	1.67250i
$99$		0			0
$100$		0			0
$101$			0.813473i			0.813473i	0.913545	+	0.406737i	$0.133333\pi$
$101$			0.813473i			0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$102$	2.24940	−	2.77778i	2.24940	−	2.77778i
$103$	−0.575212	−	0.575212i	−0.575212	−	0.575212i	0.358368	−	0.933580i	$-0.383333\pi$
$103$	−0.575212	−	0.575212i	−0.575212	−	0.575212i	−0.933580	+	0.358368i	$0.883333\pi$
$104$		0			0
$105$		0			0
$106$	−1.20906			−1.20906
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	−2.51896	−	1.28347i	−2.51896	−	1.28347i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	1.16913	−	0.122881i	1.16913	−	0.122881i
$112$	−4.37764	+	4.37764i	−4.37764	+	4.37764i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−2.05600	−	2.05600i	−2.05600	−	2.05600i
$119$	−2.71559			−2.71559
$120$		0			0
$121$	−1.00000			−1.00000
$122$	−0.854932	−	0.854932i	−0.854932	−	0.854932i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.604528	+	2.84408i	0.604528	+	2.84408i
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	3.38236	−	3.38236i	3.38236	−	3.38236i
$129$		0			0
$130$		0			0
$131$		−	0.415823i		−	0.415823i	−0.978148	−	0.207912i	$-0.933333\pi$
$131$		−	0.415823i		−	0.415823i	0.978148	−	0.207912i	$-0.0666667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	6.53062			6.53062
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	0.104528	+	0.994522i	0.104528	+	0.994522i
$142$	2.75146	−	2.75146i	2.75146	−	2.75146i
$143$		0			0
$144$	−0.866025	−	4.07433i	−0.866025	−	4.07433i
$145$		0			0
$146$		0			0
$147$	0.760884	−	0.939614i	0.760884	−	0.939614i
$148$	2.35003	+	2.35003i	2.35003	+	2.35003i
$149$	−1.98904			−1.98904			−0.994522	−	0.104528i	$-0.966667\pi$
$149$	−1.98904			−1.98904			−0.994522	+	0.104528i	$0.966667\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	0.995105	−	1.53233i	0.995105	−	1.53233i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.294032	−	0.294032i	0.294032	−	0.294032i	−0.544639	−	0.838671i	$-0.683333\pi$
$157$	0.294032	−	0.294032i	0.294032	−	0.294032i	0.838671	+	0.544639i	$0.183333\pi$
$158$	−2.23824	+	2.23824i	−2.23824	+	2.23824i
$159$	−0.614648	+	0.0646021i	−0.614648	+	0.0646021i
$160$		0			0
$161$		0			0
$162$	−1.82636	−	0.701074i	−1.82636	−	0.701074i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	1.95630			1.95630
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$	−3.34326	+	4.12858i	−3.34326	+	4.12858i
$169$			1.00000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.946294	+	0.946294i	−0.946294	+	0.946294i	−0.998630	−	0.0523360i	$-0.983333\pi$
$173$	−0.946294	+	0.946294i	−0.946294	+	0.946294i	0.0523360	+	0.998630i	$0.483333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.15506	−	0.935352i	−1.15506	−	0.935352i
$178$	2.63121	+	2.63121i	2.63121	+	2.63121i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−0.480303	−	0.388941i	−0.480303	−	0.388941i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−1.99906	+	1.99906i	−1.99906	+	1.99906i
$189$	0.459289	+	1.41355i	0.459289	+	1.41355i
$190$		0			0
$191$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$191$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$192$	3.01028	−	3.71739i	3.01028	−	3.71739i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$	3.72109			3.72109
$195$		0			0
$196$	3.41811			3.41811
$197$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$197$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$	−1.12529	+	1.12529i	−1.12529	+	1.12529i
$203$		0			0
$204$	5.13706	−	0.539926i	5.13706	−	0.539926i
$205$		0			0
$206$		−	1.59139i		−	1.59139i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−1.23548	−	1.23548i	−1.23548	−	1.23548i
$213$	1.25175	−	1.54578i	1.25175	−	1.54578i
$214$		0			0
$215$		0			0
$216$	−1.10453	−	3.39939i	−1.10453	−	3.39939i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	1.78725	+	1.44729i	1.78725	+	1.44729i
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$	−6.79878			−6.79878
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		−	4.20188i		−	4.20188i
$237$	−1.01826	+	1.25745i	−1.01826	+	1.25745i
$238$	−3.75650	−	3.75650i	−3.75650	−	3.75650i
$239$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$239$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$240$		0			0
$241$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$241$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$242$	−1.38331	−	1.38331i	−1.38331	−	1.38331i
$243$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$244$		−	1.74724i		−	1.74724i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.994522	−	0.104528i	0.994522	−	0.104528i
$250$		0			0
$251$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$251$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$252$	−2.28851	+	3.52399i	−2.28851	+	3.52399i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.57433			4.57433
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		−	1.74724i		−	1.74724i
$260$		0			0
$261$		0			0
$262$	0.575212	−	0.575212i	0.575212	−	0.575212i
$263$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$263$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.47822	+	1.19704i	1.47822	+	1.19704i
$268$		0			0
$269$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$269$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	0.209057			0.209057			0.104528	−	0.994522i	$-0.466667\pi$
$271$	0.209057			0.209057			0.104528	+	0.994522i	$0.466667\pi$
$272$	5.38142	+	5.38142i	5.38142	+	5.38142i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	−0.933580	−	0.358368i	$-0.883333\pi$
$277$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	0.358368	+	0.933580i	$0.383333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−1.23114	+	1.52033i	−1.23114	+	1.52033i
$283$	−0.294032	−	0.294032i	−0.294032	−	0.294032i	0.544639	−	0.838671i	$-0.316667\pi$
$283$	−0.294032	−	0.294032i	−0.294032	−	0.294032i	−0.838671	+	0.544639i	$0.816667\pi$
$284$	5.62321			5.62321
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.49136	−	3.83636i	2.49136	−	3.83636i
$289$			2.33826i			2.33826i
$290$		0			0
$291$	1.89169	−	0.198825i	1.89169	−	0.198825i
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	2.35232	−	0.247238i	2.35232	−	0.247238i
$295$		0			0
$296$			4.20188i			4.20188i
$297$		0			0
$298$	−2.75146	−	2.75146i	−2.75146	−	2.75146i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−0.511935	+	0.632187i	−0.511935	+	0.632187i
$304$		0			0
$305$		0			0
$306$	3.49622	−	0.743145i	3.49622	−	0.743145i
$307$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$307$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$308$		0			0
$309$	−0.0850311	−	0.809017i	−0.0850311	−	0.809017i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$	0.813473			0.813473
$315$		0			0
$316$	−4.57433			−4.57433
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$	−0.939614	−	0.760884i	−0.939614	−	0.760884i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.14988	−	2.58268i	−1.14988	−	2.58268i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.48629			1.48629
$330$		0			0
$331$	−1.33826			−1.33826			−0.669131	−	0.743145i	$-0.733333\pi$
$331$	−1.33826			−1.33826			−0.669131	+	0.743145i	$0.733333\pi$
$332$	1.99906	+	1.99906i	1.99906	+	1.99906i
$333$	0.985916	+	0.640262i	0.985916	+	0.640262i
$334$		0			0
$335$		0			0
$336$	−6.15701	+	0.647127i	−6.15701	+	0.647127i
$337$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	−0.838671	−	0.544639i	$-0.816667\pi$
$337$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	0.544639	+	0.838671i	$0.316667\pi$
$338$	−1.38331	+	1.38331i	−1.38331	+	1.38331i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.219712	−	0.219712i	−0.219712	−	0.219712i
$344$		0			0
$345$		0			0
$346$	−2.61803			−2.61803
$347$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$347$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.946294	−	0.946294i	0.946294	−	0.946294i	−0.0523360	−	0.998630i	$-0.516667\pi$
$353$	0.946294	−	0.946294i	0.946294	−	0.946294i	0.998630	+	0.0523360i	$0.0166667\pi$
$354$	−0.303929	−	2.89169i	−0.303929	−	2.89169i
$355$		0			0
$356$			5.37745i			5.37745i
$357$	−2.11041	−	1.70897i	−2.11041	−	1.70897i
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$	−0.777146	−	0.629320i	−0.777146	−	0.629320i
$364$		0			0
$365$		0			0
$366$	−0.126381	−	1.20243i	−0.126381	−	1.20243i
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			0.918578i			0.918578i
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	−3.57433			−3.57433
$377$		0			0
$378$	−1.32003	+	2.59071i	−1.32003	+	2.59071i
$379$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$379$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$380$		0			0
$381$		0			0
$382$	−2.39596	+	2.39596i	−2.39596	+	2.39596i
$383$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$383$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$384$	4.75718	−	0.500000i	4.75718	−	0.500000i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	3.80243	+	3.80243i	3.80243	+	3.80243i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	3.05581	+	3.05581i	3.05581	+	3.05581i
$393$	0.261686	−	0.323155i	0.261686	−	0.323155i
$394$			1.95630i			1.95630i
$395$		0			0
$396$		0			0
$397$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.629320	−	0.777146i	$-0.283333\pi$
$397$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.777146	−	0.629320i	$-0.216667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$401$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$402$		0			0
$403$		0			0
$404$	−2.29976			−2.29976
$405$		0			0
$406$		0			0
$407$		0			0
$408$	5.07525	+	4.10986i	5.07525	+	4.10986i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	1.62618	−	1.62618i	1.62618	−	1.62618i
$413$	−1.56204	+	1.56204i	−1.56204	+	1.56204i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	−0.544639	+	0.838671i	−0.544639	+	0.838671i
$424$		−	2.20906i		−	2.20906i
$425$		0			0
$426$	3.86984	−	0.406737i	3.86984	−	0.406737i
$427$	−0.649532	+	0.649532i	−0.649532	+	0.649532i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			0.415823i			0.415823i	0.978148	+	0.207912i	$0.0666667\pi$
$431$			0.415823i			0.415823i	−0.978148	+	0.207912i	$0.933333\pi$
$432$	1.89103	−	3.71136i	1.89103	−	3.71136i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$439$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	1.18264	−	0.251377i	1.18264	−	0.251377i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$	0.347395	+	3.30524i	0.347395	+	3.30524i
$445$		0			0
$446$		0			0
$447$	−1.54578	−	1.25175i	−1.54578	−	1.25175i
$448$	−5.02717	−	5.02717i	−5.02717	−	5.02717i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.629320	+	0.777146i	$0.716667\pi$
$457$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.777146	+	0.629320i	$0.783333\pi$
$458$		0			0
$459$	1.73767	−	0.564602i	1.73767	−	0.564602i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.413545	−	0.0434654i	0.413545	−	0.0434654i
$472$	3.75650	−	3.75650i	3.75650	−	3.75650i
$473$		0			0
$474$	−3.14801	+	0.330869i	−3.14801	+	0.330869i
$475$		0			0
$476$		−	7.67721i		−	7.67721i
$477$	−0.518327	−	0.336605i	−0.518327	−	0.336605i
$478$	−1.62618	−	1.62618i	−1.62618	−	1.62618i
$479$	0.415823			0.415823			0.207912	−	0.978148i	$-0.433333\pi$
$479$	0.415823			0.415823			0.207912	+	0.978148i	$0.433333\pi$
$480$		0			0
$481$		0			0
$482$	2.23824	+	2.23824i	2.23824	+	2.23824i
$483$		0			0
$484$		−	2.82709i		−	2.82709i
$485$		0			0
$486$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$487$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$487$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$488$	1.56204	−	1.56204i	1.56204	−	1.56204i
$489$		0			0
$490$		0			0
$491$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$491$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.09042	−	2.09042i	−2.09042	−	2.09042i
$498$	1.52033	+	1.23114i	1.52033	+	1.23114i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$	1.62618	−	1.62618i	1.62618	−	1.62618i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$	−5.19640	+	1.10453i	−5.19640	+	1.10453i
$505$		0			0
$506$		0			0
$507$	−0.629320	+	0.777146i	−0.629320	+	0.777146i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	2.94535	+	2.94535i	2.94535	+	2.94535i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	2.41697	−	2.41697i	2.41697	−	2.41697i
$519$	−1.33093	+	0.139886i	−1.33093	+	0.139886i
$520$		0			0
$521$			1.48629i			1.48629i	0.669131	+	0.743145i	$0.266667\pi$
$521$			1.48629i			1.48629i	−0.669131	+	0.743145i	$0.733333\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$	1.17557			1.17557
$525$		0			0
$526$	1.20906			1.20906
$527$		0			0
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$		0			0
$531$	−0.309017	−	1.45381i	−0.309017	−	1.45381i
$532$		0			0
$533$		0			0
$534$	0.388960	+	3.70071i	0.388960	+	3.70071i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−2.39596	−	2.39596i	−2.39596	−	2.39596i
$539$		0			0
$540$		0			0
$541$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$541$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$542$	0.289190	+	0.289190i	0.289190	+	0.289190i
$543$		0			0
$544$			8.35772i			8.35772i
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$	−0.128496	−	0.604528i	−0.128496	−	0.604528i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.70050	+	1.70050i	1.70050	+	1.70050i
$554$	−1.59139			−1.59139
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$	−2.81160	+	0.295511i	−2.81160	+	0.295511i
$565$		0			0
$566$		−	0.813473i		−	0.813473i
$567$	−0.532639	+	1.38757i	−0.532639	+	1.38757i
$568$	5.02717	+	5.02717i	5.02717	+	5.02717i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$571$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$572$		0			0
$573$	−1.09001	+	1.34606i	−1.09001	+	1.34606i
$574$		0			0
$575$		0			0
$576$	4.67886	−	0.994522i	4.67886	−	0.994522i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−3.23454	+	3.23454i	−3.23454	+	3.23454i
$579$		0			0
$580$		0			0
$581$		−	1.48629i		−	1.48629i
$582$	2.89183	+	2.34176i	2.89183	+	2.34176i
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	2.65637	+	2.15109i	2.65637	+	2.15109i
$589$		0			0
$590$		0			0
$591$	0.104528	+	0.994522i	0.104528	+	0.994522i
$592$	−3.46247	+	3.46247i	−3.46247	+	3.46247i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		−	5.62321i		−	5.62321i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−1.33826			−1.33826			−0.669131	−	0.743145i	$-0.733333\pi$
$601$	−1.33826			−1.33826			−0.669131	+	0.743145i	$0.733333\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−1.58268	+	0.166346i	−1.58268	+	0.166346i
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	4.33203	+	2.81325i	4.33203	+	2.81325i
$613$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$613$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$614$	−2.29976			−2.29976
$615$		0			0
$616$		0			0
$617$	0.147826	+	0.147826i	0.147826	+	0.147826i	0.777146	−	0.629320i	$-0.216667\pi$
$617$	0.147826	+	0.147826i	0.147826	+	0.147826i	−0.629320	+	0.777146i	$0.716667\pi$
$618$	1.00150	−	1.23675i	1.00150	−	1.23675i
$619$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$619$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.99906	−	1.99906i	1.99906	−	1.99906i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	0.831254	+	0.831254i	0.831254	+	0.831254i
$629$	−2.14787			−2.14787
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−4.08947	−	4.08947i	−4.08947	−	4.08947i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−0.182636	−	1.73767i	−0.182636	−	1.73767i
$637$		0			0
$638$		0			0
$639$	1.94558	−	0.413545i	1.94558	−	0.413545i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	0.294032	+	0.294032i	0.294032	+	0.294032i	0.838671	−	0.544639i	$-0.183333\pi$
$643$	0.294032	+	0.294032i	0.294032	+	0.294032i	−0.544639	+	0.838671i	$0.683333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$647$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$648$	1.28093	−	3.33692i	1.28093	−	3.33692i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.147826	−	0.147826i	0.147826	−	0.147826i	−0.629320	−	0.777146i	$-0.716667\pi$
$653$	0.147826	−	0.147826i	0.147826	−	0.147826i	0.777146	+	0.629320i	$0.216667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	2.05600	+	2.05600i	2.05600	+	2.05600i
$659$	1.98904			1.98904			0.994522	−	0.104528i	$-0.0333333\pi$
$659$	1.98904			1.98904			0.994522	+	0.104528i	$0.0333333\pi$
$660$		0			0
$661$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$661$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$662$	−1.85123	−	1.85123i	−1.85123	−	1.85123i
$663$		0			0
$664$			3.57433i			3.57433i
$665$		0			0
$666$	0.478148	+	2.24951i	0.478148	+	2.24951i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−5.28364	−	4.27861i	−5.28364	−	4.27861i
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$	−0.813473			−0.813473
$675$		0			0
$676$	−2.82709			−2.82709
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		−	2.82709i		−	2.82709i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.946294	−	0.946294i	0.946294	−	0.946294i	−0.0523360	−	0.998630i	$-0.516667\pi$
$683$	0.946294	−	0.946294i	0.946294	−	0.946294i	0.998630	+	0.0523360i	$0.0166667\pi$
$684$		0			0
$685$		0			0
$686$		−	0.607858i		−	0.607858i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−2.67526	−	2.67526i	−2.67526	−	2.67526i
$693$		0			0
$694$			3.16535i			3.16535i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	2.61803			2.61803
$707$	0.854932	+	0.854932i	0.854932	+	0.854932i
$708$	2.64433	−	3.26547i	2.64433	−	3.26547i
$709$		−	1.95630i		−	1.95630i	−0.207912	−	0.978148i	$-0.566667\pi$
$709$		−	1.95630i		−	1.95630i	0.207912	−	0.978148i	$-0.433333\pi$
$710$		0			0
$711$	−1.58268	+	0.336408i	−1.58268	+	0.336408i
$712$	−4.80746	+	4.80746i	−4.80746	+	4.80746i
$713$		0			0
$714$	−0.555306	−	5.28339i	−0.555306	−	5.28339i
$715$		0			0
$716$		0			0
$717$	−0.913590	−	0.739810i	−0.913590	−	0.739810i
$718$		0			0
$719$	1.48629			1.48629			0.743145	−	0.669131i	$-0.233333\pi$
$719$	1.48629			1.48629			0.743145	+	0.669131i	$0.233333\pi$
$720$		0			0
$721$	−1.20906			−1.20906
$722$	−1.38331	−	1.38331i	−1.38331	−	1.38331i
$723$	1.25745	+	1.01826i	1.25745	+	1.01826i
$724$		0			0
$725$		0			0
$726$	−0.204489	−	1.94558i	−0.204489	−	1.94558i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$730$		0			0
$731$		0			0
$732$	1.09957	−	1.35786i	1.09957	−	1.35786i
$733$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.777146	−	0.629320i	$-0.783333\pi$
$733$	−1.40647	−	1.40647i	−1.40647	−	1.40647i	−0.629320	−	0.777146i	$-0.716667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$			1.95630i			1.95630i	0.207912	+	0.978148i	$0.433333\pi$
$739$			1.95630i			1.95630i	−0.207912	+	0.978148i	$0.566667\pi$
$740$		0			0
$741$		0			0
$742$	−1.27068	+	1.27068i	−1.27068	+	1.27068i
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.838671	+	0.544639i	0.838671	+	0.544639i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−2.94535	−	2.94535i	−2.94535	−	2.94535i
$753$	0.739810	−	0.913590i	0.739810	−	0.913590i
$754$		0			0
$755$		0			0
$756$	−3.99622	+	1.29845i	−3.99622	+	1.29845i
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	−0.854932	+	0.854932i	−0.854932	+	0.854932i
$759$		0			0
$760$		0			0
$761$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$761$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$762$		0			0
$763$		0			0
$764$	−4.89667			−4.89667
$765$		0			0
$766$	3.16535			3.16535
$767$		0			0
$768$	3.55492	+	2.87872i	3.55492	+	2.87872i
$769$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$769$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$773$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$774$		0			0
$775$		0			0
$776$			6.79878i			6.79878i
$777$	1.09957	−	1.35786i	1.09957	−	1.35786i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			5.03615i			5.03615i
$785$		0			0
$786$	0.809017	−	0.0850311i	0.809017	−	0.0850311i
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−1.99906	+	1.99906i	−1.99906	+	1.99906i
$789$	0.614648	−	0.0646021i	0.614648	−	0.0646021i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	3.89116			3.89116
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		−	1.82709i		−	1.82709i
$800$		0			0
$801$	0.395472	+	1.86055i	0.395472	+	1.86055i
$802$	−1.62618	+	1.62618i	−1.62618	+	1.62618i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.34606	−	1.09001i	−1.34606	−	1.09001i
$808$	−2.05600	−	2.05600i	−2.05600	−	2.05600i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$811$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$812$		0			0
$813$	0.162468	+	0.131564i	0.162468	+	0.131564i
$814$		0			0
$815$		0			0
$816$	0.795511	+	7.56879i	0.795511	+	7.56879i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$	2.90762			2.90762
$825$		0			0
$826$	−4.32157			−4.32157
$827$	0.946294	+	0.946294i	0.946294	+	0.946294i	0.998630	−	0.0523360i	$-0.0166667\pi$
$827$	0.946294	+	0.946294i	0.946294	+	0.946294i	−0.0523360	+	0.998630i	$0.516667\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−0.809017	+	0.0850311i	−0.809017	+	0.0850311i
$832$		0			0
$833$	−1.56204	+	1.56204i	−1.56204	+	1.56204i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	−1.91355	+	0.406737i	−1.91355	+	0.406737i
$847$	−1.05097	+	1.05097i	−1.05097	+	1.05097i
$848$	1.82033	−	1.82033i	1.82033	−	1.82033i
$849$	−0.0434654	−	0.413545i	−0.0434654	−	0.413545i
$850$		0			0
$851$		0			0
$852$	4.37005	+	3.53880i	4.37005	+	3.53880i
$853$	0.575212	+	0.575212i	0.575212	+	0.575212i	0.933580	−	0.358368i	$-0.116667\pi$
$853$	0.575212	+	0.575212i	0.575212	+	0.575212i	−0.358368	+	0.933580i	$0.616667\pi$
$854$	−1.79701			−1.79701
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−0.575212	+	0.575212i	−0.575212	+	0.575212i
$863$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.358368	−	0.933580i	$-0.383333\pi$
$863$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$864$	4.35045	−	1.41355i	4.35045	−	1.41355i
$865$		0			0
$866$		0			0
$867$	−1.47152	+	1.81717i	−1.47152	+	1.81717i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.59525	+	1.03597i	1.59525	+	1.03597i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	−1.38331	+	1.38331i	−1.38331	+	1.38331i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	1.98368	+	1.28822i	1.98368	+	1.28822i
$883$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$883$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$	−2.64433	+	3.26547i	−2.64433	+	3.26547i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−0.406737	−	3.86984i	−0.406737	−	3.86984i
$895$		0			0
$896$		−	7.10950i		−	7.10950i
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	1.12920			1.12920
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$907$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$908$		0			0
$909$	−0.795697	+	0.169131i	−0.795697	+	0.169131i
$910$		0			0
$911$		−	1.48629i		−	1.48629i	−0.669131	−	0.743145i	$-0.733333\pi$
$911$		−	1.48629i		−	1.48629i	0.669131	−	0.743145i	$-0.266667\pi$
$912$		0			0
$913$		0			0
$914$	−3.89116			−3.89116
$915$		0			0
$916$		0			0
$917$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$918$	3.18475	+	1.62271i	3.18475	+	1.62271i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−1.16913	+	0.122881i	−1.16913	+	0.122881i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	0.443049	−	0.682236i	0.443049	−	0.682236i
$928$		0			0
$929$	0.813473			0.813473			0.406737	−	0.913545i	$-0.366667\pi$
$929$	0.813473			0.813473			0.406737	+	0.913545i	$0.366667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.48629i		−	1.48629i	−0.669131	−	0.743145i	$-0.733333\pi$
$941$		−	1.48629i		−	1.48629i	0.669131	−	0.743145i	$-0.266667\pi$
$942$	0.632187	+	0.511935i	0.632187	+	0.511935i
$943$		0			0
$944$	6.19092			6.19092
$945$		0			0
$946$		0			0
$947$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$947$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$948$	−3.55492	−	2.87872i	−3.55492	−	2.87872i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	6.86346	−	6.86346i	6.86346	−	6.86346i
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$	−0.251377	−	1.18264i	−0.251377	−	1.18264i
$955$		0			0
$956$		−	3.32344i		−	3.32344i
$957$		0			0
$958$	0.575212	+	0.575212i	0.575212	+	0.575212i
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$			4.57433i			4.57433i
$965$		0			0
$966$		0			0
$967$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	−0.838671	−	0.544639i	$-0.816667\pi$
$967$	−0.294032	+	0.294032i	−0.294032	+	0.294032i	0.544639	+	0.838671i	$0.316667\pi$
$968$	2.52743	−	2.52743i	2.52743	−	2.52743i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	0.731705	−	2.73076i	0.731705	−	2.73076i
$973$		0			0
$974$	3.38840			3.38840
$975$		0			0
$976$	2.57433			2.57433
$977$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$977$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−2.63121	+	2.63121i	−2.63121	+	2.63121i
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1.15506	+	0.935352i	1.15506	+	0.935352i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$991$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$992$		0			0
$993$	−1.04002	−	0.842195i	−1.04002	−	0.842195i
$994$		−	5.78339i		−	5.78339i
$995$		0			0
$996$	0.295511	+	2.81160i	0.295511	+	2.81160i
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$	0.363271	+	1.11803i	0.363271	+	1.11803i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.1.l.d.1268.16	yes	32
3.2	odd	2	inner	3525.1.l.d.1268.2	yes	32
5.2	odd	4	inner	3525.1.l.d.1832.2	yes	32
5.3	odd	4	inner	3525.1.l.d.1832.15	yes	32
5.4	even	2	inner	3525.1.l.d.1268.1	✓	32
15.2	even	4	inner	3525.1.l.d.1832.16	yes	32
15.8	even	4	inner	3525.1.l.d.1832.1	yes	32
15.14	odd	2	inner	3525.1.l.d.1268.15	yes	32
47.46	odd	2	CM	3525.1.l.d.1268.16	yes	32
141.140	even	2	inner	3525.1.l.d.1268.2	yes	32
235.93	even	4	inner	3525.1.l.d.1832.15	yes	32
235.187	even	4	inner	3525.1.l.d.1832.2	yes	32
235.234	odd	2	inner	3525.1.l.d.1268.1	✓	32
705.422	odd	4	inner	3525.1.l.d.1832.16	yes	32
705.563	odd	4	inner	3525.1.l.d.1832.1	yes	32
705.704	even	2	inner	3525.1.l.d.1268.15	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3525.1.l.d.1268.1	✓	32	5.4	even	2	inner
3525.1.l.d.1268.1	✓	32	235.234	odd	2	inner
3525.1.l.d.1268.2	yes	32	3.2	odd	2	inner
3525.1.l.d.1268.2	yes	32	141.140	even	2	inner
3525.1.l.d.1268.15	yes	32	15.14	odd	2	inner
3525.1.l.d.1268.15	yes	32	705.704	even	2	inner
3525.1.l.d.1268.16	yes	32	1.1	even	1	trivial
3525.1.l.d.1268.16	yes	32	47.46	odd	2	CM
3525.1.l.d.1832.1	yes	32	15.8	even	4	inner
3525.1.l.d.1832.1	yes	32	705.563	odd	4	inner
3525.1.l.d.1832.2	yes	32	5.2	odd	4	inner
3525.1.l.d.1832.2	yes	32	235.187	even	4	inner
3525.1.l.d.1832.15	yes	32	5.3	odd	4	inner
3525.1.l.d.1832.15	yes	32	235.93	even	4	inner
3525.1.l.d.1832.16	yes	32	15.2	even	4	inner
3525.1.l.d.1832.16	yes	32	705.422	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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3525.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.38331 + 1.38331i) q^{2} +(0.777146 + 0.629320i) q^{3} +2.82709i q^{4} +(0.204489 + 1.94558i) q^{6} +(1.05097 - 1.05097i) q^{7} +(-2.52743 + 2.52743i) q^{8} +(0.207912 + 0.978148i) q^{9} +O(q^{10})\)	\(q+(1.38331 + 1.38331i) q^{2} +(0.777146 + 0.629320i) q^{3} +2.82709i q^{4} +(0.204489 + 1.94558i) q^{6} +(1.05097 - 1.05097i) q^{7} +(-2.52743 + 2.52743i) q^{8} +(0.207912 + 0.978148i) q^{9} +(-1.77915 + 2.19706i) q^{12} +2.90762 q^{14} -4.16535 q^{16} +(-1.29195 - 1.29195i) q^{17} +(-1.06547 + 1.64069i) q^{18} +(1.47815 - 0.155360i) q^{21} +(-3.55475 + 0.373619i) q^{24} +(-0.453990 + 0.891007i) q^{27} +(2.97117 + 2.97117i) q^{28} +(-3.23454 - 3.23454i) q^{32} -3.57433i q^{34} +(-2.76531 + 0.587785i) q^{36} +(0.831254 - 0.831254i) q^{37} +(2.25965 + 1.82983i) q^{42} +(0.707107 + 0.707107i) q^{47} +(-3.23709 - 2.62134i) q^{48} -1.20906i q^{49} +(-0.190983 - 1.81708i) q^{51} +(-0.437016 + 0.437016i) q^{53} +(-1.86055 + 0.604528i) q^{54} +5.31249i q^{56} -1.48629 q^{59} -0.618034 q^{61} +(1.24651 + 0.809491i) q^{63} -4.78339i q^{64} +(3.65246 - 3.65246i) q^{68} -1.98904i q^{71} +(-2.99768 - 1.94672i) q^{72} +2.29976 q^{74} +1.61803i q^{79} +(-0.913545 + 0.406737i) q^{81} +(0.707107 - 0.707107i) q^{83} +(0.439216 + 4.17886i) q^{84} +1.90211 q^{89} +1.95630i q^{94} +(-0.478148 - 4.54927i) q^{96} +(1.34500 - 1.34500i) q^{97} +(1.67250 - 1.67250i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{6}+O(q^{10}) \) 32 * q - 4 * q^6	\( 32 q - 4 q^{6} - 48 q^{16} + 12 q^{21} - 24 q^{51} + 16 q^{61} - 4 q^{81} + 20 q^{96}+O(q^{100}) \) 32 * q - 4 * q^6 - 48 * q^16 + 12 * q^21 - 24 * q^51 + 16 * q^61 - 4 * q^81 + 20 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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