LMFDB - Embedded newform 3675.1.cy.a.857.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,1,Mod(143,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(84))

chi = DirichletCharacter(H, H._module([42, 63, 62]))

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

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

chi := DirichletCharacter("3675.143");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3675.cy (of order $84$, degree $24$, 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:	$1.83406392143$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$2$ over $\Q(\zeta_{84})$
Coefficient field:	$\Q(\zeta_{168})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{44} - x^{36} - x^{32} + x^{24} - x^{16} - x^{12} + x^{4} + 1 $ x^48 + x^44 - x^36 - x^32 + x^24 - 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:	$D_{42}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{42} + \cdots)$

Embedding invariants

Embedding label			857.1
Root			$-0.467269 + 0.884115i$ of defining polynomial
Character	$\chi$	$=$	3675.857
Dual form			3675.1.cy.a.2693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.399892 + 0.916562i) q^{3} +(0.563320 + 0.826239i) q^{4} +(-0.999301 + 0.0373912i) q^{7} +(-0.680173 - 0.733052i) q^{9} +O(q^{10})$	$q+(-0.399892 + 0.916562i) q^{3} +(0.563320 + 0.826239i) q^{4} +(-0.999301 + 0.0373912i) q^{7} +(-0.680173 - 0.733052i) q^{9} +(-0.982566 + 0.185912i) q^{12} +(-1.67453 + 1.05218i) q^{13} +(-0.365341 + 0.930874i) q^{16} +(-0.680173 - 1.17809i) q^{19} +(0.365341 - 0.930874i) q^{21} +(0.943883 - 0.330279i) q^{27} +(-0.593820 - 0.804598i) q^{28} +(-0.751509 - 0.433884i) q^{31} +(0.222521 - 0.974928i) q^{36} +(-0.370784 - 1.95964i) q^{37} +(-0.294755 - 1.95557i) q^{39} +(1.93760 + 0.218315i) q^{43} +(-0.707107 - 0.707107i) q^{48} +(0.997204 - 0.0747301i) q^{49} +(-1.81265 - 0.790851i) q^{52} +(1.35179 - 0.152310i) q^{57} +(-0.634659 + 0.930874i) q^{61} +(0.707107 + 0.707107i) q^{63} +(-0.974928 + 0.222521i) q^{64} +(-1.79831 + 0.481856i) q^{67} +(0.0698381 + 0.132140i) q^{73} +(0.590232 - 1.22563i) q^{76} +(-1.43109 + 0.826239i) q^{79} +(-0.0747301 + 0.997204i) q^{81} +(0.974928 - 0.222521i) q^{84} +(1.63402 - 1.11406i) q^{91} +(0.698204 - 0.515298i) q^{93} +(-1.35138 + 1.35138i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 4 q^{16} + 4 q^{21} + 8 q^{36} - 44 q^{61} - 4 q^{81} + 4 q^{91}+O(q^{100}) $ 48 * q - 4 * q^16 + 4 * q^21 + 8 * q^36 - 44 * q^61 - 4 * q^81 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times$.

$n$	$1177$	$1226$	$2551$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{37}{42}\right)$

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 $

$2$		0			0		−0.884115	−	0.467269i	$-0.845238\pi$
$2$		0			0		0.884115	+	0.467269i	$0.154762\pi$
$3$	−0.399892	+	0.916562i	−0.399892	+	0.916562i
$3$	−0.399892	+	0.916562i	−0.399892	+	0.916562i
$4$	0.563320	+	0.826239i	0.563320	+	0.826239i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.999301	+	0.0373912i	−0.999301	+	0.0373912i
$7$	−0.999301	+	0.0373912i	−0.999301	+	0.0373912i
$8$		0			0
$9$	−0.680173	−	0.733052i	−0.680173	−	0.733052i
$10$		0			0
$11$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$11$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$12$	−0.982566	+	0.185912i	−0.982566	+	0.185912i
$13$	−1.67453	+	1.05218i	−1.67453	+	1.05218i	−0.757972	+	0.652287i	$0.773810\pi$
$13$	−1.67453	+	1.05218i	−1.67453	+	1.05218i	−0.916562	+	0.399892i	$0.869048\pi$
$14$		0			0
$15$		0			0
$16$	−0.365341	+	0.930874i	−0.365341	+	0.930874i
$17$		0			0		−0.652287	−	0.757972i	$-0.726190\pi$
$17$		0			0		0.652287	+	0.757972i	$0.273810\pi$
$18$		0			0
$19$	−0.680173	−	1.17809i	−0.680173	−	1.17809i	−0.974928	−	0.222521i	$-0.928571\pi$
$19$	−0.680173	−	1.17809i	−0.680173	−	1.17809i	0.294755	−	0.955573i	$-0.404762\pi$
$20$		0			0
$21$	0.365341	−	0.930874i	0.365341	−	0.930874i
$22$		0			0
$23$		0			0		−0.757972	−	0.652287i	$-0.773810\pi$
$23$		0			0		0.757972	+	0.652287i	$0.226190\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.943883	−	0.330279i	0.943883	−	0.330279i
$28$	−0.593820	−	0.804598i	−0.593820	−	0.804598i
$29$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$29$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$30$		0			0
$31$	−0.751509	−	0.433884i	−0.751509	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$31$	−0.751509	−	0.433884i	−0.751509	−	0.433884i	−0.826239	+	0.563320i	$0.809524\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.222521	−	0.974928i	0.222521	−	0.974928i
$37$	−0.370784	−	1.95964i	−0.370784	−	1.95964i	−0.258819	−	0.965926i	$-0.583333\pi$
$37$	−0.370784	−	1.95964i	−0.370784	−	1.95964i	−0.111964	−	0.993712i	$-0.535714\pi$
$38$		0			0
$39$	−0.294755	−	1.95557i	−0.294755	−	1.95557i
$40$		0			0
$41$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$41$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$42$		0			0
$43$	1.93760	+	0.218315i	1.93760	+	0.218315i	0.993712	−	0.111964i	$-0.0357143\pi$
$43$	1.93760	+	0.218315i	1.93760	+	0.218315i	0.943883	+	0.330279i	$0.107143\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.467269	−	0.884115i	$-0.345238\pi$
$47$		0			0		−0.467269	+	0.884115i	$0.654762\pi$
$48$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$49$	0.997204	−	0.0747301i	0.997204	−	0.0747301i
$50$		0			0
$51$		0			0
$52$	−1.81265	−	0.790851i	−1.81265	−	0.790851i
$53$		0			0		0.185912	−	0.982566i	$-0.440476\pi$
$53$		0			0		−0.185912	+	0.982566i	$0.559524\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.35179	−	0.152310i	1.35179	−	0.152310i
$58$		0			0
$59$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$59$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$60$		0			0
$61$	−0.634659	+	0.930874i	−0.634659	+	0.930874i	0.365341	+	0.930874i	$0.380952\pi$
$61$	−0.634659	+	0.930874i	−0.634659	+	0.930874i	−1.00000			$\pi$
$62$		0			0
$63$	0.707107	+	0.707107i	0.707107	+	0.707107i
$64$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$65$		0			0
$66$		0			0
$67$	−1.79831	+	0.481856i	−1.79831	+	0.481856i	−0.993712	−	0.111964i	$-0.964286\pi$
$67$	−1.79831	+	0.481856i	−1.79831	+	0.481856i	−0.804598	+	0.593820i	$0.797619\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$71$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$72$		0			0
$73$	0.0698381	+	0.132140i	0.0698381	+	0.132140i	0.916562	−	0.399892i	$-0.130952\pi$
$73$	0.0698381	+	0.132140i	0.0698381	+	0.132140i	−0.846724	+	0.532032i	$0.821429\pi$
$74$		0			0
$75$		0			0
$76$	0.590232	−	1.22563i	0.590232	−	1.22563i
$77$		0			0
$78$		0			0
$79$	−1.43109	+	0.826239i	−1.43109	+	0.826239i	−0.997204	−	0.0747301i	$-0.976190\pi$
$79$	−1.43109	+	0.826239i	−1.43109	+	0.826239i	−0.433884	+	0.900969i	$0.642857\pi$
$80$		0			0
$81$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$82$		0			0
$83$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$83$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$84$	0.974928	−	0.222521i	0.974928	−	0.222521i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$89$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$90$		0			0
$91$	1.63402	−	1.11406i	1.63402	−	1.11406i
$92$		0			0
$93$	0.698204	−	0.515298i	0.698204	−	0.515298i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.35138	+	1.35138i	−1.35138	+	1.35138i	−0.467269	+	0.884115i	$0.654762\pi$
$97$	−1.35138	+	1.35138i	−1.35138	+	1.35138i	−0.884115	+	0.467269i	$0.845238\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$101$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$102$		0			0
$103$	−0.870602	−	1.17962i	−0.870602	−	1.17962i	−0.982566	−	0.185912i	$-0.940476\pi$
$103$	−0.870602	−	1.17962i	−0.870602	−	1.17962i	0.111964	−	0.993712i	$-0.464286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.999301	−	0.0373912i	$-0.988095\pi$
$107$		0			0		0.999301	+	0.0373912i	$0.0119048\pi$
$108$	0.804598	+	0.593820i	0.804598	+	0.593820i
$109$	0.496990	−	0.535628i	0.496990	−	0.535628i	−0.433884	−	0.900969i	$-0.642857\pi$
$109$	0.496990	−	0.535628i	0.496990	−	0.535628i	0.930874	+	0.365341i	$0.119048\pi$
$110$		0			0
$111$	1.94440	+	0.443797i	1.94440	+	0.443797i
$112$	0.330279	−	0.943883i	0.330279	−	0.943883i
$113$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$113$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.91027	+	0.511857i	1.91027	+	0.511857i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$122$		0			0
$123$		0			0
$124$	−0.0648483	−	0.865341i	−0.0648483	−	0.865341i
$125$		0			0
$126$		0			0
$127$	−0.556429	−	0.194703i	−0.556429	−	0.194703i	0.0373912	−	0.999301i	$-0.488095\pi$
$127$	−0.556429	−	0.194703i	−0.556429	−	0.194703i	−0.593820	+	0.804598i	$0.702381\pi$
$128$		0			0
$129$	−0.974928	+	1.68862i	−0.974928	+	1.68862i
$130$		0			0
$131$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$131$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$132$		0			0
$133$	0.723747	+	1.15184i	0.723747	+	1.15184i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.804598	−	0.593820i	$-0.797619\pi$
$137$		0			0		0.804598	+	0.593820i	$0.202381\pi$
$138$		0			0
$139$	−0.185853	+	0.233052i	−0.185853	+	0.233052i	−0.866025	−	0.500000i	$-0.833333\pi$
$139$	−0.185853	+	0.233052i	−0.185853	+	0.233052i	0.680173	+	0.733052i	$0.261905\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.930874	−	0.365341i	0.930874	−	0.365341i
$145$		0			0
$146$		0			0
$147$	−0.330279	+	0.943883i	−0.330279	+	0.943883i
$148$	1.41026	−	1.41026i	1.41026	−	1.41026i
$149$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$149$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$150$		0			0
$151$	−1.21135	+	0.825886i	−1.21135	+	0.825886i	−0.988831	−	0.149042i	$-0.952381\pi$
$151$	−1.21135	+	0.825886i	−1.21135	+	0.825886i	−0.222521	+	0.974928i	$0.571429\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.44973	−	1.34515i	1.44973	−	1.34515i
$157$	0.433894	−	0.587905i	0.433894	−	0.587905i	−0.532032	−	0.846724i	$-0.678571\pi$
$157$	0.433894	−	0.587905i	0.433894	−	0.587905i	0.965926	+	0.258819i	$0.0833333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.119202	−	0.273213i	−0.119202	−	0.273213i	0.846724	−	0.532032i	$-0.178571\pi$
$163$	−0.119202	−	0.273213i	−0.119202	−	0.273213i	−0.965926	+	0.258819i	$0.916667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$167$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$168$		0			0
$169$	1.26310	−	2.62285i	1.26310	−	2.62285i
$170$		0			0
$171$	−0.400969	+	1.29991i	−0.400969	+	1.29991i
$172$	0.911106	+	1.72390i	0.911106	+	1.72390i
$173$		0			0		0.652287	−	0.757972i	$-0.273810\pi$
$173$		0			0		−0.652287	+	0.757972i	$0.726190\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$179$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$180$		0			0
$181$	−1.52446	+	0.347948i	−1.52446	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$181$	−1.52446	+	0.347948i	−1.52446	+	0.347948i	−0.623490	+	0.781831i	$0.714286\pi$
$182$		0			0
$183$	−0.599409	−	0.953953i	−0.599409	−	0.953953i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.930874	+	0.365341i	−0.930874	+	0.365341i
$190$		0			0
$191$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$191$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$192$	0.185912	−	0.982566i	0.185912	−	0.982566i
$193$	1.43319	+	0.625296i	1.43319	+	0.625296i	0.965926	−	0.258819i	$-0.0833333\pi$
$193$	1.43319	+	0.625296i	1.43319	+	0.625296i	0.467269	+	0.884115i	$0.345238\pi$
$194$		0			0
$195$		0			0
$196$	0.623490	+	0.781831i	0.623490	+	0.781831i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$	−0.215372	−	0.548760i	−0.215372	−	0.548760i	0.781831	−	0.623490i	$-0.214286\pi$
$199$	−0.215372	−	0.548760i	−0.215372	−	0.548760i	−0.997204	+	0.0747301i	$0.976190\pi$
$200$		0			0
$201$	0.277479	−	1.84095i	0.277479	−	1.84095i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.367670	−	1.94318i	−0.367670	−	1.94318i
$209$		0			0
$210$		0			0
$211$	0.0332580	+	0.145713i	0.0332580	+	0.145713i	0.988831	−	0.149042i	$-0.0476190\pi$
$211$	0.0332580	+	0.145713i	0.0332580	+	0.145713i	−0.955573	+	0.294755i	$0.904762\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.767207	+	0.405481i	0.767207	+	0.405481i
$218$		0			0
$219$	−0.149042	+	0.0111692i	−0.149042	+	0.0111692i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.545779	−	1.55975i	−0.545779	−	1.55975i	−0.804598	−	0.593820i	$-0.797619\pi$
$223$	−0.545779	−	1.55975i	−0.545779	−	1.55975i	0.258819	−	0.965926i	$-0.416667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$227$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$228$	0.887336	+	1.03110i	0.887336	+	1.03110i
$229$	0.215372	−	0.548760i	0.215372	−	0.548760i	−0.781831	−	0.623490i	$-0.785714\pi$
$229$	0.215372	−	0.548760i	0.215372	−	0.548760i	0.997204	+	0.0747301i	$0.0238095\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.982566	−	0.185912i	$-0.0595238\pi$
$233$		0			0		−0.982566	+	0.185912i	$0.940476\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.185019	−	1.64209i	−0.185019	−	1.64209i
$238$		0			0
$239$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$239$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$240$		0			0
$241$	1.04876	+	1.53825i	1.04876	+	1.53825i	0.826239	+	0.563320i	$0.190476\pi$
$241$	1.04876	+	1.53825i	1.04876	+	1.53825i	0.222521	+	0.974928i	$0.428571\pi$
$242$		0			0
$243$	−0.884115	−	0.467269i	−0.884115	−	0.467269i
$244$	−1.12664			−1.12664
$245$		0			0
$246$		0			0
$247$	2.37854	+	1.25709i	2.37854	+	1.25709i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$251$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$252$	−0.185912	+	0.982566i	−0.185912	+	0.982566i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$257$		0			0		0.982566	−	0.185912i	$-0.0595238\pi$
$257$		0			0		−0.982566	+	0.185912i	$0.940476\pi$
$258$		0			0
$259$	0.443797	+	1.94440i	0.443797	+	1.94440i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$263$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.41115	−	1.21439i	−1.41115	−	1.21439i
$269$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$269$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$270$		0			0
$271$	−1.55929	+	0.116853i	−1.55929	+	0.116853i	−0.826239	−	0.563320i	$-0.809524\pi$
$271$	−1.55929	+	0.116853i	−1.55929	+	0.116853i	−0.733052	+	0.680173i	$0.761905\pi$
$272$		0			0
$273$	0.367670	+	1.94318i	0.367670	+	1.94318i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.225940	+	0.194437i	−0.225940	+	0.194437i	−0.757972	−	0.652287i	$-0.773810\pi$
$277$	−0.225940	+	0.194437i	−0.225940	+	0.194437i	0.532032	+	0.846724i	$0.321429\pi$
$278$		0			0
$279$	0.193096	+	0.846011i	0.193096	+	0.846011i
$280$		0			0
$281$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$281$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$282$		0			0
$283$	0.0617881	+	1.65132i	0.0617881	+	1.65132i	0.593820	+	0.804598i	$0.297619\pi$
$283$	0.0617881	+	1.65132i	0.0617881	+	1.65132i	−0.532032	+	0.846724i	$0.678571\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.149042	+	0.988831i	−0.149042	+	0.988831i
$290$		0			0
$291$	−0.698220	−	1.77904i	−0.698220	−	1.77904i
$292$	−0.0698381	+	0.132140i	−0.0698381	+	0.132140i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.94440	−	0.145713i	−1.94440	−	0.145713i
$302$		0			0
$303$		0			0
$304$	1.34515	−	0.202749i	1.34515	−	0.202749i
$305$		0			0
$306$		0			0
$307$	−0.236777	−	0.376828i	−0.236777	−	0.376828i	0.707107	−	0.707107i	$-0.250000\pi$
$307$	−0.236777	−	0.376828i	−0.236777	−	0.376828i	−0.943883	+	0.330279i	$0.892857\pi$
$308$		0			0
$309$	1.42935	−	0.326239i	1.42935	−	0.326239i
$310$		0			0
$311$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$311$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$312$		0			0
$313$	−0.466376	+	1.74054i	−0.466376	+	1.74054i	0.185912	+	0.982566i	$0.440476\pi$
$313$	−0.466376	+	1.74054i	−0.466376	+	1.74054i	−0.652287	+	0.757972i	$0.726190\pi$
$314$		0			0
$315$		0			0
$316$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$317$		0			0		0.652287	−	0.757972i	$-0.273810\pi$
$317$		0			0		−0.652287	+	0.757972i	$0.726190\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$325$		0			0
$326$		0			0
$327$	0.292194	+	0.669716i	0.292194	+	0.669716i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.57906	−	1.07659i	−1.57906	−	1.07659i	−0.955573	−	0.294755i	$-0.904762\pi$
$331$	−1.57906	−	1.07659i	−1.57906	−	1.07659i	−0.623490	−	0.781831i	$-0.714286\pi$
$332$		0			0
$333$	−1.18432	+	1.60470i	−1.18432	+	1.60470i
$334$		0			0
$335$		0			0
$336$	0.733052	+	0.680173i	0.733052	+	0.680173i
$337$	−0.175075	+	1.55383i	−0.175075	+	1.55383i	0.532032	+	0.846724i	$0.321429\pi$
$337$	−0.175075	+	1.55383i	−0.175075	+	1.55383i	−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.993712	+	0.111964i	−0.993712	+	0.111964i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.982566	−	0.185912i	$-0.940476\pi$
$347$		0			0		0.982566	+	0.185912i	$0.0595238\pi$
$348$		0			0
$349$	1.21572	+	1.52446i	1.21572	+	1.52446i	0.781831	+	0.623490i	$0.214286\pi$
$349$	1.21572	+	1.52446i	1.21572	+	1.52446i	0.433884	+	0.900969i	$0.357143\pi$
$350$		0			0
$351$	−1.23305	+	1.54620i	−1.23305	+	1.54620i
$352$		0			0
$353$		0			0		−0.804598	−	0.593820i	$-0.797619\pi$
$353$		0			0		0.804598	+	0.593820i	$0.202381\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$359$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$360$		0			0
$361$	−0.425270	+	0.736589i	−0.425270	+	0.736589i
$362$		0			0
$363$	−0.943883	−	0.330279i	−0.943883	−	0.330279i
$364$	1.84095	+	0.722521i	1.84095	+	0.722521i
$365$		0			0
$366$		0			0
$367$	−1.10247	+	0.582674i	−1.10247	+	0.582674i	−0.916562	−	0.399892i	$-0.869048\pi$
$367$	−1.10247	+	0.582674i	−1.10247	+	0.582674i	−0.185912	+	0.982566i	$0.559524\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.819071	+	0.286605i	0.819071	+	0.286605i
$373$	0.569423	+	0.152577i	0.569423	+	0.152577i	0.532032	−	0.846724i	$-0.321429\pi$
$373$	0.569423	+	0.152577i	0.569423	+	0.152577i	0.0373912	+	0.999301i	$0.488095\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.42935	−	0.326239i	−1.42935	−	0.326239i	−0.563320	−	0.826239i	$-0.690476\pi$
$379$	−1.42935	−	0.326239i	−1.42935	−	0.326239i	−0.866025	+	0.500000i	$0.833333\pi$
$380$		0			0
$381$	0.400969	−	0.432142i	0.400969	−	0.432142i
$382$		0			0
$383$		0			0		−0.999301	−	0.0373912i	$-0.988095\pi$
$383$		0			0		0.999301	+	0.0373912i	$0.0119048\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.15786	−	1.56885i	−1.15786	−	1.56885i
$388$	−1.87783	−	0.355304i	−1.87783	−	0.355304i
$389$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$389$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.44984	−	1.07003i	1.44984	−	1.07003i	0.467269	−	0.884115i	$-0.345238\pi$
$397$	1.44984	−	1.07003i	1.44984	−	1.07003i	0.982566	−	0.185912i	$-0.0595238\pi$
$398$		0			0
$399$	−1.34515	+	0.202749i	−1.34515	+	0.202749i
$400$		0			0
$401$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$401$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$402$		0			0
$403$	1.71495	−	0.0641689i	1.71495	−	0.0641689i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.149042	+	1.98883i	−0.149042	+	1.98883i			1.00000i	$0.5\pi$
$409$	−0.149042	+	1.98883i	−0.149042	+	1.98883i	−0.149042	+	0.988831i	$0.547619\pi$
$410$		0			0
$411$		0			0
$412$	0.484223	−	1.38383i	0.484223	−	1.38383i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.139286	−	0.263541i	−0.139286	−	0.263541i
$418$		0			0
$419$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$419$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$420$		0			0
$421$	0.658322	−	0.317031i	0.658322	−	0.317031i	−0.0747301	−	0.997204i	$-0.523810\pi$
$421$	0.658322	−	0.317031i	0.658322	−	0.317031i	0.733052	+	0.680173i	$0.238095\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.599409	−	0.953953i	0.599409	−	0.953953i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$431$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$432$	−0.0373912	+	0.999301i	−0.0373912	+	0.999301i
$433$	−0.442244	+	0.0498289i	−0.442244	+	0.0498289i	−0.330279	−	0.943883i	$-0.607143\pi$
$433$	−0.442244	+	0.0498289i	−0.442244	+	0.0498289i	−0.111964	+	0.993712i	$0.535714\pi$
$434$		0			0
$435$		0			0
$436$	0.722521	+	0.108903i	0.722521	+	0.108903i
$437$		0			0
$438$		0			0
$439$	0.284841	−	0.0878620i	0.284841	−	0.0878620i	−0.149042	−	0.988831i	$-0.547619\pi$
$439$	0.284841	−	0.0878620i	0.284841	−	0.0878620i	0.433884	+	0.900969i	$0.357143\pi$
$440$		0			0
$441$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$442$		0			0
$443$		0			0		0.467269	−	0.884115i	$-0.345238\pi$
$443$		0			0		−0.467269	+	0.884115i	$0.654762\pi$
$444$	0.728639	+	1.85654i	0.728639	+	1.85654i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.965926	−	0.258819i	0.965926	−	0.258819i
$449$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$449$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.272566	−	1.44054i	−0.272566	−	1.44054i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.03264	−	0.450534i	1.03264	−	0.450534i	0.185912	−	0.982566i	$-0.440476\pi$
$457$	1.03264	−	0.450534i	1.03264	−	0.450534i	0.846724	+	0.532032i	$0.178571\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$461$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$462$		0			0
$463$	−0.281357	+	0.0984511i	−0.281357	+	0.0984511i	−0.467269	−	0.884115i	$-0.654762\pi$
$463$	−0.281357	+	0.0984511i	−0.281357	+	0.0984511i	0.185912	+	0.982566i	$0.440476\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.757972	−	0.652287i	$-0.773810\pi$
$467$		0			0		0.757972	+	0.652287i	$0.226190\pi$
$468$	0.653180	+	1.86668i	0.653180	+	1.86668i
$469$	1.77904	−	0.548760i	1.77904	−	0.548760i
$470$		0			0
$471$	0.365341	+	0.632789i	0.365341	+	0.632789i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$479$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$480$		0			0
$481$	2.68278	+	2.89135i	2.68278	+	2.89135i
$482$		0			0
$483$		0			0
$484$	−0.781831	+	0.623490i	−0.781831	+	0.623490i
$485$		0			0
$486$		0			0
$487$	0.347013	−	0.795363i	0.347013	−	0.795363i	−0.652287	−	0.757972i	$-0.726190\pi$
$487$	0.347013	−	0.795363i	0.347013	−	0.795363i	0.999301	−	0.0373912i	$-0.0119048\pi$
$488$		0			0
$489$	0.298085			0.298085
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.678448	−	0.541044i	0.678448	−	0.541044i
$497$		0			0
$498$		0			0
$499$	−0.101659	−	0.109562i	−0.101659	−	0.109562i	0.680173	−	0.733052i	$-0.261905\pi$
$499$	−0.101659	−	0.109562i	−0.101659	−	0.109562i	−0.781831	+	0.623490i	$0.785714\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$503$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.89890	+	2.20657i	1.89890	+	2.20657i
$508$	−0.152577	−	0.569423i	−0.152577	−	0.569423i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i
$512$		0			0
$513$	−1.03110	−	0.887336i	−1.03110	−	0.887336i
$514$		0			0
$515$		0			0
$516$	−1.94440	+	0.145713i	−1.94440	+	0.145713i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	1.65159	−	0.720581i	1.65159	−	0.720581i	0.652287	−	0.757972i	$-0.273810\pi$
$523$	1.65159	−	0.720581i	1.65159	−	0.720581i	0.999301	+	0.0373912i	$0.0119048\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.149042	+	0.988831i	0.149042	+	0.988831i
$530$		0			0
$531$		0			0
$532$	−0.543991	+	1.24684i	−0.543991	+	1.24684i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.40097	+	0.432142i	−1.40097	+	0.432142i	−0.900969	−	0.433884i	$-0.857143\pi$
$541$	−1.40097	+	0.432142i	−1.40097	+	0.432142i	−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	0.290703	−	1.53640i	0.290703	−	1.53640i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.862311	+	0.0971591i	−0.862311	+	0.0971591i	−0.532032	−	0.846724i	$-0.678571\pi$
$547$	−0.862311	+	0.0971591i	−0.862311	+	0.0971591i	−0.330279	+	0.943883i	$0.607143\pi$
$548$		0			0
$549$	1.11406	−	0.167917i	1.11406	−	0.167917i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.39919	−	0.879171i	1.39919	−	0.879171i
$554$		0			0
$555$		0			0
$556$	−0.297251	−	0.0222759i	−0.297251	−	0.0222759i
$557$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$557$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$558$		0			0
$559$	−3.47428	+	1.67312i	−3.47428	+	1.67312i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.467269	−	0.884115i	$-0.654762\pi$
$563$		0			0		0.467269	+	0.884115i	$0.345238\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.0373912	−	0.999301i	0.0373912	−	0.999301i
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	0.123490	−	1.64786i	0.123490	−	1.64786i	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	0.123490	−	1.64786i	0.123490	−	1.64786i	0.623490	−	0.781831i	$-0.285714\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.826239	+	0.563320i	0.826239	+	0.563320i
$577$	−1.24611	+	0.0466261i	−1.24611	+	0.0466261i	−0.652287	−	0.757972i	$-0.726190\pi$
$577$	−1.24611	+	0.0466261i	−1.24611	+	0.0466261i	−0.593820	+	0.804598i	$0.702381\pi$
$578$		0			0
$579$	−1.14625	+	1.06356i	−1.14625	+	1.06356i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$589$			1.18046i			1.18046i
$590$		0			0
$591$		0			0
$592$	1.95964	+	0.370784i	1.95964	+	0.370784i
$593$		0			0		−0.593820	−	0.804598i	$-0.702381\pi$
$593$		0			0		0.593820	+	0.804598i	$0.297619\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.589098	+	0.0220425i	0.589098	+	0.0220425i
$598$		0			0
$599$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$599$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$600$		0			0
$601$	−1.68862	−	0.385418i	−1.68862	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$601$	−1.68862	−	0.385418i	−1.68862	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$602$		0			0
$603$	1.57639	+	0.990509i	1.57639	+	0.990509i
$604$	−1.36476	−	0.535628i	−1.36476	−	0.535628i
$605$		0			0
$606$		0			0
$607$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$607$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.767207	−	0.405481i	0.767207	−	0.405481i	−0.0373912	−	0.999301i	$-0.511905\pi$
$613$	0.767207	−	0.405481i	0.767207	−	0.405481i	0.804598	+	0.593820i	$0.202381\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$617$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.92808	+	0.440071i	1.92808	+	0.440071i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	0.730171	+	0.0273211i	0.730171	+	0.0273211i
$629$		0			0
$630$		0			0
$631$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.955573	+	0.294755i	$0.0952381\pi$
$631$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.0747301	+	0.997204i	$0.476190\pi$
$632$		0			0
$633$	−0.146855	−	0.0277864i	−0.146855	−	0.0277864i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.59122	+	1.17438i	−1.59122	+	1.17438i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$641$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$642$		0			0
$643$	−0.0818104	+	0.726088i	−0.0818104	+	0.726088i	0.884115	+	0.467269i	$0.154762\pi$
$643$	−0.0818104	+	0.726088i	−0.0818104	+	0.726088i	−0.965926	+	0.258819i	$0.916667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.593820	−	0.804598i	$-0.297619\pi$
$647$		0			0		−0.593820	+	0.804598i	$0.702381\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.678448	+	0.541044i	−0.678448	+	0.541044i
$652$	0.158591	−	0.252395i	0.158591	−	0.252395i
$653$		0			0		−0.399892	−	0.916562i	$-0.630952\pi$
$653$		0			0		0.399892	+	0.916562i	$0.369048\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.0493636	−	0.141073i	0.0493636	−	0.141073i
$658$		0			0
$659$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$659$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$660$		0			0
$661$	0.510531	−	1.65510i	0.510531	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$661$	0.510531	−	1.65510i	0.510531	−	1.65510i	0.733052	−	0.680173i	$-0.238095\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.64786	+	0.123490i	1.64786	+	0.123490i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.990509	−	1.57639i	−0.990509	−	1.57639i	−0.804598	−	0.593820i	$-0.797619\pi$
$673$	−0.990509	−	1.57639i	−0.990509	−	1.57639i	−0.185912	−	0.982566i	$-0.559524\pi$
$674$		0			0
$675$		0			0
$676$	2.87863	−	0.433884i	2.87863	−	0.433884i
$677$		0			0		0.0373912	−	0.999301i	$-0.488095\pi$
$677$		0			0		−0.0373912	+	0.999301i	$0.511905\pi$
$678$		0			0
$679$	1.29991	−	1.40097i	1.29991	−	1.40097i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.916562	−	0.399892i	$-0.869048\pi$
$683$		0			0		0.916562	+	0.399892i	$0.130952\pi$
$684$	−1.29991	+	0.400969i	−1.29991	+	0.400969i
$685$		0			0
$686$		0			0
$687$	0.416847	+	0.416847i	0.416847	+	0.416847i
$688$	−0.911106	+	1.72390i	−0.911106	+	1.72390i
$689$		0			0
$690$		0			0
$691$	0.258149	−	1.71271i	0.258149	−	1.71271i	−0.365341	−	0.930874i	$-0.619048\pi$
$691$	0.258149	−	1.71271i	0.258149	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$701$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$702$		0			0
$703$	−2.05644	+	1.76971i	−2.05644	+	1.76971i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.90580	−	0.142820i	1.90580	−	0.142820i	0.930874	−	0.365341i	$-0.119048\pi$
$709$	1.90580	−	0.142820i	1.90580	−	0.142820i	0.974928	+	0.222521i	$0.0714286\pi$
$710$		0			0
$711$	1.57906	+	0.487076i	1.57906	+	0.487076i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$719$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$720$		0			0
$721$	0.914101	+	1.14625i	0.914101	+	1.14625i
$722$		0			0
$723$	−1.82929	+	0.346120i	−1.82929	+	0.346120i
$724$	−1.14625	−	1.06356i	−1.14625	−	1.06356i
$725$		0			0
$726$		0			0
$727$	0.111964	+	0.993712i	0.111964	+	0.993712i	0.916562	+	0.399892i	$0.130952\pi$
$727$	0.111964	+	0.993712i	0.111964	+	0.993712i	−0.804598	+	0.593820i	$0.797619\pi$
$728$		0			0
$729$	0.781831	−	0.623490i	0.781831	−	0.623490i
$730$		0			0
$731$		0			0
$732$	0.450534	−	1.03264i	0.450534	−	1.03264i
$733$	1.68967	+	0.893018i	1.68967	+	0.893018i	0.982566	+	0.185912i	$0.0595238\pi$
$733$	1.68967	+	0.893018i	1.68967	+	0.893018i	0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.411608	+	0.603718i	0.411608	+	0.603718i	0.974928	−	0.222521i	$-0.0714286\pi$
$739$	0.411608	+	0.603718i	0.411608	+	0.603718i	−0.563320	+	0.826239i	$0.690476\pi$
$740$		0			0
$741$	−2.10336	+	1.67738i	−2.10336	+	1.67738i
$742$		0			0
$743$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$743$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	0.365341	+	0.930874i	$0.380952\pi$
$751$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	−0.900969	+	0.433884i	$0.857143\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$757$	0.614896	+	1.75727i	0.614896	+	1.75727i	0.652287	+	0.757972i	$0.273810\pi$
$757$	0.614896	+	1.75727i	0.614896	+	1.75727i	−0.0373912	+	0.999301i	$0.511905\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$761$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$762$		0			0
$763$	−0.476615	+	0.553836i	−0.476615	+	0.553836i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.916562	−	0.399892i	0.916562	−	0.399892i
$769$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$769$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$770$		0			0
$771$		0			0
$772$	0.290703	+	1.53640i	0.290703	+	1.53640i
$773$		0			0		−0.0373912	−	0.999301i	$-0.511905\pi$
$773$		0			0		0.0373912	+	0.999301i	$0.488095\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.95964	−	0.370784i	−1.95964	−	0.370784i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.294755	+	0.955573i	−0.294755	+	0.955573i
$785$		0			0
$786$		0			0
$787$	0.136990	+	0.0597679i	0.136990	+	0.0597679i	0.467269	−	0.884115i	$-0.345238\pi$
$787$	0.136990	+	0.0597679i	0.136990	+	0.0597679i	−0.330279	+	0.943883i	$0.607143\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.0833118	−	2.22655i	0.0833118	−	2.22655i
$794$		0			0
$795$		0			0
$796$	0.332083	−	0.487076i	0.332083	−	0.487076i
$797$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$797$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.67738	−	0.807782i	1.67738	−	0.807782i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$809$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$810$		0			0
$811$	−0.807782	+	1.67738i	−0.807782	+	1.67738i	−0.0747301	+	0.997204i	$0.523810\pi$
$811$	−0.807782	+	1.67738i	−0.807782	+	1.67738i	−0.733052	+	0.680173i	$0.761905\pi$
$812$		0			0
$813$	0.516445	−	1.47592i	0.516445	−	1.47592i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.06070	−	2.43116i	−1.06070	−	2.43116i
$818$		0			0
$819$	−1.92808	−	0.440071i	−1.92808	−	0.440071i
$820$		0			0
$821$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$821$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$822$		0			0
$823$	0.807801	−	1.09453i	0.807801	−	1.09453i	−0.185912	−	0.982566i	$-0.559524\pi$
$823$	0.807801	−	1.09453i	0.807801	−	1.09453i	0.993712	−	0.111964i	$-0.0357143\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$827$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$828$		0			0
$829$	0.930874	−	0.634659i	0.930874	−	0.634659i		−	1.00000i	$-0.5\pi$
$829$	0.930874	−	0.634659i	0.930874	−	0.634659i	0.930874	+	0.365341i	$0.119048\pi$
$830$		0			0
$831$	−0.0878620	−	0.284841i	−0.0878620	−	0.284841i
$832$	1.39842	−	1.39842i	1.39842	−	1.39842i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.852639	−	0.161328i	−0.852639	−	0.161328i
$838$		0			0
$839$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$839$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$840$		0			0
$841$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$842$		0			0
$843$		0			0
$844$	−0.101659	+	0.109562i	−0.101659	+	0.109562i
$845$		0			0
$846$		0			0
$847$	−0.111964	−	0.993712i	−0.111964	−	0.993712i
$848$		0			0
$849$	−1.53825	−	0.603718i	−1.53825	−	0.603718i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.420068	−	0.146988i	−0.420068	−	0.146988i	0.111964	−	0.993712i	$-0.464286\pi$
$853$	−0.420068	−	0.146988i	−0.420068	−	0.146988i	−0.532032	+	0.846724i	$0.678571\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.884115	−	0.467269i	$-0.154762\pi$
$857$		0			0		−0.884115	+	0.467269i	$0.845238\pi$
$858$		0			0
$859$	−0.145713	−	1.94440i	−0.145713	−	1.94440i	−0.294755	−	0.955573i	$-0.595238\pi$
$859$	−0.145713	−	1.94440i	−0.145713	−	1.94440i	0.149042	−	0.988831i	$-0.452381\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$863$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.846724	−	0.532032i	−0.846724	−	0.532032i
$868$	0.0971591	+	0.862311i	0.0971591	+	0.862311i
$869$		0			0
$870$		0			0
$871$	2.50433	−	2.69903i	2.50433	−	2.69903i
$872$		0			0
$873$	1.90981	+	0.0714600i	1.90981	+	0.0714600i
$874$		0			0
$875$		0			0
$876$	−0.0931869	−	0.116853i	−0.0931869	−	0.116853i
$877$	0.807801	+	1.09453i	0.807801	+	1.09453i	0.993712	+	0.111964i	$0.0357143\pi$
$877$	0.807801	+	1.09453i	0.807801	+	1.09453i	−0.185912	+	0.982566i	$0.559524\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	0.210778	−	0.210778i	0.210778	−	0.210778i	−0.593820	−	0.804598i	$-0.702381\pi$
$883$	0.210778	−	0.210778i	0.210778	−	0.210778i	0.804598	+	0.593820i	$0.202381\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.804598	−	0.593820i	$-0.202381\pi$
$887$		0			0		−0.804598	+	0.593820i	$0.797619\pi$
$888$		0			0
$889$	0.563320	+	0.173761i	0.563320	+	0.173761i
$890$		0			0
$891$		0			0
$892$	0.981275	−	1.32958i	0.981275	−	1.32958i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	0.911106	−	1.72390i	0.911106	−	1.72390i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.869936	+	1.64600i	0.869936	+	1.64600i	0.757972	+	0.652287i	$0.226190\pi$
$907$	0.869936	+	1.64600i	0.869936	+	1.64600i	0.111964	+	0.993712i	$0.464286\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$911$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$912$	−0.352083	+	1.31399i	−0.352083	+	1.31399i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.574730	−	0.131178i	0.574730	−	0.131178i
$917$		0			0
$918$		0			0
$919$	0.702449	−	1.03030i	0.702449	−	1.03030i	−0.294755	−	0.955573i	$-0.595238\pi$
$919$	0.702449	−	1.03030i	0.702449	−	1.03030i	0.997204	−	0.0747301i	$-0.0238095\pi$
$920$		0			0
$921$	0.440071	−	0.0663300i	0.440071	−	0.0663300i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.272566	+	1.44054i	−0.272566	+	1.44054i
$928$		0			0
$929$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$929$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$930$		0			0
$931$	−0.766310	−	1.12397i	−0.766310	−	1.12397i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.442244	+	0.0498289i	0.442244	+	0.0498289i	0.330279	−	0.943883i	$-0.392857\pi$
$937$	0.442244	+	0.0498289i	0.442244	+	0.0498289i	0.111964	+	0.993712i	$0.464286\pi$
$938$		0			0
$939$	−1.40881	−	1.12349i	−1.40881	−	1.12349i
$940$		0			0
$941$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$941$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.916562	−	0.399892i	$-0.130952\pi$
$947$		0			0		−0.916562	+	0.399892i	$0.869048\pi$
$948$	1.25253	−	1.07789i	1.25253	−	1.07789i
$949$	−0.255981	−	0.147791i	−0.255981	−	0.147791i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$953$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.123490	−	0.213891i	−0.123490	−	0.213891i
$962$		0			0
$963$		0			0
$964$	−0.680173	+	1.73305i	−0.680173	+	1.73305i
$965$		0			0
$966$		0			0
$967$	−1.15184	+	0.723747i	−1.15184	+	0.723747i	−0.965926	−	0.258819i	$-0.916667\pi$
$967$	−1.15184	+	0.723747i	−1.15184	+	0.723747i	−0.185912	+	0.982566i	$0.559524\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$971$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$972$	−0.111964	−	0.993712i	−0.111964	−	0.993712i
$973$	0.177009	−	0.239838i	0.177009	−	0.239838i
$974$		0			0
$975$		0			0
$976$	−0.634659	−	0.930874i	−0.634659	−	0.930874i
$977$		0			0		0.399892	−	0.916562i	$-0.369048\pi$
$977$		0			0		−0.399892	+	0.916562i	$0.630952\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.730682			−0.730682
$982$		0			0
$983$		0			0		0.399892	−	0.916562i	$-0.369048\pi$
$983$		0			0		−0.399892	+	0.916562i	$0.630952\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.301218	+	2.67339i	0.301218	+	2.67339i
$989$		0			0
$990$		0			0
$991$	−0.326239	−	0.302705i	−0.326239	−	0.302705i	0.500000	−	0.866025i	$-0.333333\pi$
$991$	−0.326239	−	0.302705i	−0.326239	−	0.302705i	−0.826239	+	0.563320i	$0.809524\pi$
$992$		0			0
$993$	1.61821	−	1.01679i	1.61821	−	1.01679i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.652287	−	0.757972i	−0.652287	−	0.757972i	0.330279	−	0.943883i	$-0.392857\pi$
$997$	−0.652287	−	0.757972i	−0.652287	−	0.757972i	−0.982566	+	0.185912i	$0.940476\pi$
$998$		0			0
$999$	−0.997204	−	1.72721i	−0.997204	−	1.72721i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.1.cy.a.857.1	yes	48
3.2	odd	2	CM	3675.1.cy.a.857.1	yes	48
5.2	odd	4	inner	3675.1.cy.a.2768.2	yes	48
5.3	odd	4	inner	3675.1.cy.a.2768.1	yes	48
5.4	even	2	inner	3675.1.cy.a.857.2	yes	48
15.2	even	4	inner	3675.1.cy.a.2768.2	yes	48
15.8	even	4	inner	3675.1.cy.a.2768.1	yes	48
15.14	odd	2	inner	3675.1.cy.a.857.2	yes	48
49.47	odd	42	inner	3675.1.cy.a.782.1	✓	48
147.47	even	42	inner	3675.1.cy.a.782.1	✓	48
245.47	even	84	inner	3675.1.cy.a.2693.2	yes	48
245.194	odd	42	inner	3675.1.cy.a.782.2	yes	48
245.243	even	84	inner	3675.1.cy.a.2693.1	yes	48
735.47	odd	84	inner	3675.1.cy.a.2693.2	yes	48
735.194	even	42	inner	3675.1.cy.a.782.2	yes	48
735.488	odd	84	inner	3675.1.cy.a.2693.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3675.1.cy.a.782.1	✓	48	49.47	odd	42	inner
3675.1.cy.a.782.1	✓	48	147.47	even	42	inner
3675.1.cy.a.782.2	yes	48	245.194	odd	42	inner
3675.1.cy.a.782.2	yes	48	735.194	even	42	inner
3675.1.cy.a.857.1	yes	48	1.1	even	1	trivial
3675.1.cy.a.857.1	yes	48	3.2	odd	2	CM
3675.1.cy.a.857.2	yes	48	5.4	even	2	inner
3675.1.cy.a.857.2	yes	48	15.14	odd	2	inner
3675.1.cy.a.2693.1	yes	48	245.243	even	84	inner
3675.1.cy.a.2693.1	yes	48	735.488	odd	84	inner
3675.1.cy.a.2693.2	yes	48	245.47	even	84	inner
3675.1.cy.a.2693.2	yes	48	735.47	odd	84	inner
3675.1.cy.a.2768.1	yes	48	5.3	odd	4	inner
3675.1.cy.a.2768.1	yes	48	15.8	even	4	inner
3675.1.cy.a.2768.2	yes	48	5.2	odd	4	inner
3675.1.cy.a.2768.2	yes	48	15.2	even	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:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3675.cy (of order \(84\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.399892 + 0.916562i) q^{3} +(0.563320 + 0.826239i) q^{4} +(-0.999301 + 0.0373912i) q^{7} +(-0.680173 - 0.733052i) q^{9} +O(q^{10})\)	\(q+(-0.399892 + 0.916562i) q^{3} +(0.563320 + 0.826239i) q^{4} +(-0.999301 + 0.0373912i) q^{7} +(-0.680173 - 0.733052i) q^{9} +(-0.982566 + 0.185912i) q^{12} +(-1.67453 + 1.05218i) q^{13} +(-0.365341 + 0.930874i) q^{16} +(-0.680173 - 1.17809i) q^{19} +(0.365341 - 0.930874i) q^{21} +(0.943883 - 0.330279i) q^{27} +(-0.593820 - 0.804598i) q^{28} +(-0.751509 - 0.433884i) q^{31} +(0.222521 - 0.974928i) q^{36} +(-0.370784 - 1.95964i) q^{37} +(-0.294755 - 1.95557i) q^{39} +(1.93760 + 0.218315i) q^{43} +(-0.707107 - 0.707107i) q^{48} +(0.997204 - 0.0747301i) q^{49} +(-1.81265 - 0.790851i) q^{52} +(1.35179 - 0.152310i) q^{57} +(-0.634659 + 0.930874i) q^{61} +(0.707107 + 0.707107i) q^{63} +(-0.974928 + 0.222521i) q^{64} +(-1.79831 + 0.481856i) q^{67} +(0.0698381 + 0.132140i) q^{73} +(0.590232 - 1.22563i) q^{76} +(-1.43109 + 0.826239i) q^{79} +(-0.0747301 + 0.997204i) q^{81} +(0.974928 - 0.222521i) q^{84} +(1.63402 - 1.11406i) q^{91} +(0.698204 - 0.515298i) q^{93} +(-1.35138 + 1.35138i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 4 q^{16} + 4 q^{21} + 8 q^{36} - 44 q^{61} - 4 q^{81} + 4 q^{91}+O(q^{100}) \) 48 * q - 4 * q^16 + 4 * q^21 + 8 * q^36 - 44 * q^61 - 4 * q^81 + 4 * q^91

Introduction

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