LMFDB - Embedded newform 1395.1.dw.a.694.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1395,1,Mod(199,1395)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1395, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([0, 15, 11]))

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

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

chi := DirichletCharacter("1395.199");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1395 = 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1395.dw (of order $30$, degree $8$, 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:	$0.696195692623$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{30})$
Coefficient field:	$\Q(\zeta_{60})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 $ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			694.2
Root			$0.406737 - 0.913545i$ of defining polynomial
Character	$\chi$	$=$	1395.694
Dual form			1395.1.dw.a.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.786610 - 1.08268i) q^{2} +(-0.244415 - 0.752232i) q^{4} +(0.866025 + 0.500000i) q^{5} +(0.266080 + 0.0864545i) q^{8} +O(q^{10})$	$q+(0.786610 - 1.08268i) q^{2} +(-0.244415 - 0.752232i) q^{4} +(0.866025 + 0.500000i) q^{5} +(0.266080 + 0.0864545i) q^{8} +(1.22256 - 0.544320i) q^{10} +(0.942790 - 0.684977i) q^{16} +(-1.69420 + 0.360114i) q^{17} +(-0.190983 + 1.81708i) q^{19} +(0.164446 - 0.773659i) q^{20} +(0.587785 - 1.80902i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-0.809017 - 0.587785i) q^{31} -1.27977i q^{32} +(-0.942790 + 2.11754i) q^{34} +(1.81708 + 1.63611i) q^{38} +(0.187205 + 0.207912i) q^{40} +(-1.49622 - 2.05937i) q^{46} +(-1.14988 - 1.58268i) q^{47} +(-0.104528 - 0.994522i) q^{49} +(1.33093 + 0.139886i) q^{50} +(-0.786610 - 0.873619i) q^{53} +1.98904i q^{61} +(-1.27276 + 0.413545i) q^{62} +(-0.442790 - 0.321706i) q^{64} +(0.684977 + 1.18641i) q^{68} +(1.41355 - 0.300458i) q^{76} +(-0.169131 - 0.795697i) q^{79} +(1.15897 - 0.121812i) q^{80} +(-0.379874 + 0.169131i) q^{83} +(-1.64728 - 0.535233i) q^{85} -1.50446 q^{92} -2.61803 q^{94} +(-1.07394 + 1.47815i) q^{95} +(-1.15897 - 0.669131i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 2 q^{10} - 6 q^{16} - 12 q^{19} + 8 q^{25} - 4 q^{31} + 6 q^{34} - 22 q^{40} + 10 q^{46} + 2 q^{49} + 14 q^{64} + 10 q^{76} + 6 q^{79} - 24 q^{94}+O(q^{100}) $ 16 * q - 2 * q^10 - 6 * q^16 - 12 * q^19 + 8 * q^25 - 4 * q^31 + 6 * q^34 - 22 * q^40 + 10 * q^46 + 2 * q^49 + 14 * q^64 + 10 * q^76 + 6 * q^79 - 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1395\mathbb{Z}\right)^\times$.

$n$	$406$	$776$	$1117$
$\chi(n)$	$e\left(\frac{19}{30}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.786610	−	1.08268i	0.786610	−	1.08268i	−0.207912	−	0.978148i	$-0.566667\pi$
$2$	0.786610	−	1.08268i	0.786610	−	1.08268i	0.994522	−	0.104528i	$-0.0333333\pi$
$3$		0			0
$3$		0			0
$4$	−0.244415	−	0.752232i	−0.244415	−	0.752232i
$5$	0.866025	+	0.500000i	0.866025	+	0.500000i
$5$	0.866025	+	0.500000i	0.866025	+	0.500000i
$6$		0			0
$7$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$7$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$8$	0.266080	+	0.0864545i	0.266080	+	0.0864545i
$9$		0			0
$10$	1.22256	−	0.544320i	1.22256	−	0.544320i
$11$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$11$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$12$		0			0
$13$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$13$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$14$		0			0
$15$		0			0
$16$	0.942790	−	0.684977i	0.942790	−	0.684977i
$17$	−1.69420	+	0.360114i	−1.69420	+	0.360114i	−0.951057	−	0.309017i	$-0.900000\pi$
$17$	−1.69420	+	0.360114i	−1.69420	+	0.360114i	−0.743145	+	0.669131i	$0.766667\pi$
$18$		0			0
$19$	−0.190983	+	1.81708i	−0.190983	+	1.81708i	0.309017	+	0.951057i	$0.400000\pi$
$19$	−0.190983	+	1.81708i	−0.190983	+	1.81708i	−0.500000	+	0.866025i	$0.666667\pi$
$20$	0.164446	−	0.773659i	0.164446	−	0.773659i
$21$		0			0
$22$		0			0
$23$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$23$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$29$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$30$		0			0
$31$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$31$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$32$		−	1.27977i		−	1.27977i
$33$		0			0
$34$	−0.942790	+	2.11754i	−0.942790	+	2.11754i
$35$		0			0
$36$		0			0
$37$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$37$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$38$	1.81708	+	1.63611i	1.81708	+	1.63611i
$39$		0			0
$40$	0.187205	+	0.207912i	0.187205	+	0.207912i
$41$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$41$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$42$		0			0
$43$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$43$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$44$		0			0
$45$		0			0
$46$	−1.49622	−	2.05937i	−1.49622	−	2.05937i
$47$	−1.14988	−	1.58268i	−1.14988	−	1.58268i	−0.743145	−	0.669131i	$-0.766667\pi$
$47$	−1.14988	−	1.58268i	−1.14988	−	1.58268i	−0.406737	−	0.913545i	$-0.633333\pi$
$48$		0			0
$49$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$50$	1.33093	+	0.139886i	1.33093	+	0.139886i
$51$		0			0
$52$		0			0
$53$	−0.786610	−	0.873619i	−0.786610	−	0.873619i	0.207912	−	0.978148i	$-0.433333\pi$
$53$	−0.786610	−	0.873619i	−0.786610	−	0.873619i	−0.994522	+	0.104528i	$0.966667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$59$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$60$		0			0
$61$			1.98904i			1.98904i	0.104528	+	0.994522i	$0.466667\pi$
$61$			1.98904i			1.98904i	−0.104528	+	0.994522i	$0.533333\pi$
$62$	−1.27276	+	0.413545i	−1.27276	+	0.413545i
$63$		0			0
$64$	−0.442790	−	0.321706i	−0.442790	−	0.321706i
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	0.684977	+	1.18641i	0.684977	+	1.18641i
$69$		0			0
$70$		0			0
$71$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$71$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$72$		0			0
$73$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$73$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$74$		0			0
$75$		0			0
$76$	1.41355	−	0.300458i	1.41355	−	0.300458i
$77$		0			0
$78$		0			0
$79$	−0.169131	−	0.795697i	−0.169131	−	0.795697i	−0.978148	−	0.207912i	$-0.933333\pi$
$79$	−0.169131	−	0.795697i	−0.169131	−	0.795697i	0.809017	−	0.587785i	$-0.200000\pi$
$80$	1.15897	−	0.121812i	1.15897	−	0.121812i
$81$		0			0
$82$		0			0
$83$	−0.379874	+	0.169131i	−0.379874	+	0.169131i	−0.587785	−	0.809017i	$-0.700000\pi$
$83$	−0.379874	+	0.169131i	−0.379874	+	0.169131i	0.207912	+	0.978148i	$0.433333\pi$
$84$		0			0
$85$	−1.64728	−	0.535233i	−1.64728	−	0.535233i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$89$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$90$		0			0
$91$		0			0
$92$	−1.50446			−1.50446
$93$		0			0
$94$	−2.61803			−2.61803
$95$	−1.07394	+	1.47815i	−1.07394	+	1.47815i
$96$		0			0
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$	−1.15897	−	0.669131i	−1.15897	−	0.669131i
$99$		0			0
$100$	0.529244	−	0.587785i	0.529244	−	0.587785i
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$		0			0
$103$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$103$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$104$		0			0
$105$		0			0
$106$	−1.56460	+	0.164446i	−1.56460	+	0.164446i
$107$	0.128496	+	0.604528i	0.128496	+	0.604528i	0.994522	+	0.104528i	$0.0333333\pi$
$107$	0.128496	+	0.604528i	0.128496	+	0.604528i	−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$109$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.207912	+	0.978148i	−0.207912	+	0.978148i	0.743145	+	0.669131i	$0.233333\pi$
$113$	−0.207912	+	0.978148i	−0.207912	+	0.978148i	−0.951057	+	0.309017i	$0.900000\pi$
$114$		0			0
$115$	1.41355	−	1.27276i	1.41355	−	1.27276i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.913545	+	0.406737i	0.913545	+	0.406737i
$122$	2.15349	+	1.56460i	2.15349	+	1.56460i
$123$		0			0
$124$	−0.244415	+	0.752232i	−0.244415	+	0.752232i
$125$			1.00000i			1.00000i
$126$		0			0
$127$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$127$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$128$	0.520530	−	0.169131i	0.520530	−	0.169131i
$129$		0			0
$130$		0			0
$131$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$131$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.481926	−	0.0506525i	−0.481926	−	0.0506525i
$137$	0.155360	+	1.47815i	0.155360	+	1.47815i	0.743145	+	0.669131i	$0.233333\pi$
$137$	0.155360	+	1.47815i	0.155360	+	1.47815i	−0.587785	+	0.809017i	$0.700000\pi$
$138$		0			0
$139$	−0.244415	−	0.336408i	−0.244415	−	0.336408i	0.669131	−	0.743145i	$-0.266667\pi$
$139$	−0.244415	−	0.336408i	−0.244415	−	0.336408i	−0.913545	+	0.406737i	$0.866667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$151$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$152$	−0.207912	+	0.466977i	−0.207912	+	0.466977i
$153$		0			0
$154$		0			0
$155$	−0.406737	−	0.913545i	−0.406737	−	0.913545i
$156$		0			0
$157$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$157$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$158$	−0.994522	−	0.442790i	−0.994522	−	0.442790i
$159$		0			0
$160$	0.639886	−	1.10832i	0.639886	−	1.10832i
$161$		0			0
$162$		0			0
$163$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$163$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$164$		0			0
$165$		0			0
$166$	−0.115699	+	0.544320i	−0.115699	+	0.544320i
$167$	0.155360	−	1.47815i	0.155360	−	1.47815i	−0.587785	−	0.809017i	$-0.700000\pi$
$167$	0.155360	−	1.47815i	0.155360	−	1.47815i	0.743145	−	0.669131i	$-0.233333\pi$
$168$		0			0
$169$	0.978148	−	0.207912i	0.978148	−	0.207912i
$170$	−1.87525	+	1.36245i	−1.87525	+	1.36245i
$171$		0			0
$172$		0			0
$173$	0.614648	−	0.0646021i	0.614648	−	0.0646021i	0.207912	−	0.978148i	$-0.433333\pi$
$173$	0.614648	−	0.0646021i	0.614648	−	0.0646021i	0.406737	+	0.913545i	$0.366667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$179$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$180$		0			0
$181$	1.64728	+	0.951057i	1.64728	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$181$	1.64728	+	0.951057i	1.64728	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$182$		0			0
$183$		0			0
$184$	0.312795	−	0.430526i	0.312795	−	0.430526i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.909491	+	1.25181i	−0.909491	+	1.25181i
$189$		0			0
$190$	0.755585	+	2.32545i	0.755585	+	2.32545i
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$193$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$194$		0			0
$195$		0			0
$196$	−0.722562	+	0.321706i	−0.722562	+	0.321706i
$197$	0.406737	+	0.0864545i	0.406737	+	0.0864545i	0.406737	−	0.913545i	$-0.366667\pi$
$197$	0.406737	+	0.0864545i	0.406737	+	0.0864545i			1.00000i	$0.5\pi$
$198$		0			0
$199$	−1.72256	+	0.181049i	−1.72256	+	0.181049i	−0.913545	−	0.406737i	$-0.866667\pi$
$199$	−1.72256	+	0.181049i	−1.72256	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$200$	0.0581680	+	0.273659i	0.0581680	+	0.273659i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.309017	+	0.535233i	0.309017	+	0.535233i	0.978148	−	0.207912i	$-0.0666667\pi$
$211$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$212$	−0.464905	+	0.805239i	−0.464905	+	0.805239i
$213$		0			0
$214$	0.755585	+	0.336408i	0.755585	+	0.336408i
$215$		0			0
$216$		0			0
$217$		0			0
$218$			2.16535i			2.16535i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$223$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$224$		0			0
$225$		0			0
$226$	0.895472	+	0.994522i	0.895472	+	0.994522i
$227$	−0.406737	−	0.913545i	−0.406737	−	0.913545i	−0.994522	−	0.104528i	$-0.966667\pi$
$227$	−0.406737	−	0.913545i	−0.406737	−	0.913545i	0.587785	−	0.809017i	$-0.300000\pi$
$228$		0			0
$229$	−1.16913	−	0.122881i	−1.16913	−	0.122881i	−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.16913	−	0.122881i	−1.16913	−	0.122881i	−0.669131	+	0.743145i	$0.733333\pi$
$230$	−0.266080	−	2.53158i	−0.266080	−	2.53158i
$231$		0			0
$232$		0			0
$233$	0.122881	+	0.169131i	0.122881	+	0.169131i	0.866025	−	0.500000i	$-0.166667\pi$
$233$	0.122881	+	0.169131i	0.122881	+	0.169131i	−0.743145	+	0.669131i	$0.766667\pi$
$234$		0			0
$235$	−0.204489	−	1.94558i	−0.204489	−	1.94558i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$239$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$240$		0			0
$241$	−0.604528	−	0.544320i	−0.604528	−	0.544320i	0.309017	−	0.951057i	$-0.400000\pi$
$241$	−0.604528	−	0.544320i	−0.604528	−	0.544320i	−0.913545	+	0.406737i	$0.866667\pi$
$242$	1.15897	−	0.669131i	1.15897	−	0.669131i
$243$		0			0
$244$	1.49622	−	0.486152i	1.49622	−	0.486152i
$245$	0.406737	−	0.913545i	0.406737	−	0.913545i
$246$		0			0
$247$		0			0
$248$	−0.164446	−	0.226341i	−0.164446	−	0.226341i
$249$		0			0
$250$	1.08268	+	0.786610i	1.08268	+	0.786610i
$251$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$251$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.395472	−	1.21714i	0.395472	−	1.21714i
$257$	−1.35779	+	1.22256i	−1.35779	+	1.22256i	−0.406737	+	0.913545i	$0.633333\pi$
$257$	−1.35779	+	1.22256i	−1.35779	+	1.22256i	−0.951057	+	0.309017i	$0.900000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.658114	+	0.478148i	−0.658114	+	0.478148i	−0.866025	−	0.500000i	$-0.833333\pi$
$263$	−0.658114	+	0.478148i	−0.658114	+	0.478148i	0.207912	+	0.978148i	$0.433333\pi$
$264$		0			0
$265$	−0.244415	−	1.14988i	−0.244415	−	1.14988i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$269$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$270$		0			0
$271$	1.89169	+	0.614648i	1.89169	+	0.614648i	0.978148	+	0.207912i	$0.0666667\pi$
$271$	1.89169	+	0.614648i	1.89169	+	0.614648i	0.913545	+	0.406737i	$0.133333\pi$
$272$	−1.35061	+	1.50000i	−1.35061	+	1.50000i
$273$		0			0
$274$	1.72256	+	0.994522i	1.72256	+	0.994522i
$275$		0			0
$276$		0			0
$277$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$277$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$278$	−0.556480			−0.556480
$279$		0			0
$280$		0			0
$281$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$281$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$282$		0			0
$283$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$283$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.82709	−	0.813473i	1.82709	−	0.813473i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.0434654	−	0.204489i	−0.0434654	−	0.204489i	0.951057	−	0.309017i	$-0.100000\pi$
$293$	−0.0434654	−	0.204489i	−0.0434654	−	0.204489i	−0.994522	+	0.104528i	$0.966667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	0.486152	−	1.49622i	0.486152	−	1.49622i
$303$		0			0
$304$	1.06460	+	1.84395i	1.06460	+	1.84395i
$305$	−0.994522	+	1.72256i	−0.994522	+	1.72256i
$306$		0			0
$307$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$307$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$308$		0			0
$309$		0			0
$310$	−1.30902	−	0.278240i	−1.30902	−	0.278240i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$313$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$314$		0			0
$315$		0			0
$316$	−0.557210	+	0.321706i	−0.557210	+	0.321706i
$317$	1.45381	+	1.30902i	1.45381	+	1.30902i	0.866025	+	0.500000i	$0.166667\pi$
$317$	1.45381	+	1.30902i	1.45381	+	1.30902i	0.587785	+	0.809017i	$0.300000\pi$
$318$		0			0
$319$		0			0
$320$	−0.222614	−	0.500000i	−0.222614	−	0.500000i
$321$		0			0
$322$		0			0
$323$	−0.330792	−	3.14728i	−0.330792	−	3.14728i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.809017	−	1.81708i	−0.809017	−	1.81708i	−0.500000	−	0.866025i	$-0.666667\pi$
$331$	−0.809017	−	1.81708i	−0.809017	−	1.81708i	−0.309017	−	0.951057i	$-0.600000\pi$
$332$	0.220072	+	0.244415i	0.220072	+	0.244415i
$333$		0			0
$334$	−1.47815	−	1.33093i	−1.47815	−	1.33093i
$335$		0			0
$336$		0			0
$337$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$337$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$338$	0.544320	−	1.22256i	0.544320	−	1.22256i
$339$		0			0
$340$			1.36995i			1.36995i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.413545	−	0.716282i	0.413545	−	0.716282i
$347$	0.743145	+	1.28716i	0.743145	+	1.28716i	0.951057	+	0.309017i	$0.100000\pi$
$347$	0.743145	+	1.28716i	0.743145	+	1.28716i	−0.207912	+	0.978148i	$0.566667\pi$
$348$		0			0
$349$	0.604528	−	1.86055i	0.604528	−	1.86055i	0.104528	−	0.994522i	$-0.466667\pi$
$349$	0.604528	−	1.86055i	0.604528	−	1.86055i	0.500000	−	0.866025i	$-0.333333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.0434654	−	0.413545i	0.0434654	−	0.413545i	−0.951057	−	0.309017i	$-0.900000\pi$
$353$	0.0434654	−	0.413545i	0.0434654	−	0.413545i	0.994522	−	0.104528i	$-0.0333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$359$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$360$		0			0
$361$	−2.28716	−	0.486152i	−2.28716	−	0.486152i
$362$	2.32545	−	1.03536i	2.32545	−	1.03536i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$	−0.684977	−	2.10814i	−0.684977	−	2.10814i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−0.169131	−	0.520530i	−0.169131	−	0.520530i
$377$		0			0
$378$		0			0
$379$	−1.08268	+	1.20243i	−1.08268	+	1.20243i	−0.104528	+	0.994522i	$0.533333\pi$
$379$	−1.08268	+	1.20243i	−1.08268	+	1.20243i	−0.978148	+	0.207912i	$0.933333\pi$
$380$	1.37440	+	0.446568i	1.37440	+	0.446568i
$381$		0			0
$382$		0			0
$383$	1.94558	+	0.413545i	1.94558	+	0.413545i	0.994522	+	0.104528i	$0.0333333\pi$
$383$	1.94558	+	0.413545i	1.94558	+	0.413545i	0.951057	+	0.309017i	$0.100000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$389$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$390$		0			0
$391$	−0.344375	+	3.27651i	−0.344375	+	3.27651i
$392$	0.0581680	−	0.273659i	0.0581680	−	0.273659i
$393$		0			0
$394$	0.413545	−	0.372358i	0.413545	−	0.372358i
$395$	0.251377	−	0.773659i	0.251377	−	0.773659i
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	−1.15897	+	2.00739i	−1.15897	+	2.00739i
$399$		0			0
$400$	1.06460	+	0.473991i	1.06460	+	0.473991i
$401$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$401$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.360114	−	0.207912i	0.360114	−	0.207912i	−0.309017	−	0.951057i	$-0.600000\pi$
$409$	0.360114	−	0.207912i	0.360114	−	0.207912i	0.669131	+	0.743145i	$0.266667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.413545	−	0.0434654i	−0.413545	−	0.0434654i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$419$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$420$		0			0
$421$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	−0.913545	−	0.406737i	$-0.866667\pi$
$421$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	0.809017	−	0.587785i	$-0.200000\pi$
$422$	0.822560	+	0.0864545i	0.822560	+	0.0864545i
$423$		0			0
$424$	−0.133773	−	0.300458i	−0.133773	−	0.300458i
$425$	−1.15897	−	1.28716i	−1.15897	−	1.28716i
$426$		0			0
$427$		0			0
$428$	0.423339	−	0.244415i	0.423339	−	0.244415i
$429$		0			0
$430$		0			0
$431$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$431$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	1.03536	+	0.752232i	1.03536	+	0.752232i
$437$	3.17488	+	1.41355i	3.17488	+	1.41355i
$438$		0			0
$439$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$439$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.155360	+	0.139886i	−0.155360	+	0.139886i	−0.743145	−	0.669131i	$-0.766667\pi$
$443$	−0.155360	+	0.139886i	−0.155360	+	0.139886i	0.587785	+	0.809017i	$0.300000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$449$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$450$		0			0
$451$		0			0
$452$	0.786610	−	0.0826761i	0.786610	−	0.0826761i
$453$		0			0
$454$	−1.30902	−	0.278240i	−1.30902	−	0.278240i
$455$		0			0
$456$		0			0
$457$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$457$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$458$	−1.05269	+	1.16913i	−1.05269	+	1.16913i
$459$		0			0
$460$	−1.30290	−	0.752232i	−1.30290	−	0.752232i
$461$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$461$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$462$		0			0
$463$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$463$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$464$		0			0
$465$		0			0
$466$	0.279773			0.279773
$467$	0.951057	−	1.30902i	0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$467$	0.951057	−	1.30902i	0.951057	−	1.30902i	0.951057	−	0.309017i	$-0.100000\pi$
$468$		0			0
$469$		0			0
$470$	−2.26728	−	1.30902i	−2.26728	−	1.30902i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.66913	+	0.743145i	−1.66913	+	0.743145i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$479$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$480$		0			0
$481$		0			0
$482$	−1.06485	+	0.226341i	−1.06485	+	0.226341i
$483$		0			0
$484$	0.0826761	−	0.786610i	0.0826761	−	0.786610i
$485$		0			0
$486$		0			0
$487$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$487$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$488$	−0.171962	+	0.529244i	−0.171962	+	0.529244i
$489$		0			0
$490$	−0.669131	−	1.15897i	−0.669131	−	1.15897i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.16535			−1.16535
$497$		0			0
$498$		0			0
$499$	−0.478148	+	1.07394i	−0.478148	+	1.07394i	0.500000	+	0.866025i	$0.333333\pi$
$499$	−0.478148	+	1.07394i	−0.478148	+	1.07394i	−0.978148	+	0.207912i	$0.933333\pi$
$500$	0.752232	−	0.244415i	0.752232	−	0.244415i
$501$		0			0
$502$		0			0
$503$	−1.20243	−	1.08268i	−1.20243	−	1.08268i	−0.994522	−	0.104528i	$-0.966667\pi$
$503$	−1.20243	−	1.08268i	−1.20243	−	1.08268i	−0.207912	−	0.978148i	$-0.566667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$509$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$510$		0			0
$511$		0			0
$512$	−0.684977	−	0.942790i	−0.684977	−	0.942790i
$513$		0			0
$514$	0.255585	+	2.43173i	0.255585	+	2.43173i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$523$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$524$		0			0
$525$		0			0
$526$			1.08864i			1.08864i
$527$	1.58231	+	0.704489i	1.58231	+	0.704489i
$528$		0			0
$529$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$530$	−1.43721	−	0.639886i	−1.43721	−	0.639886i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.30902	+	0.278240i	−1.30902	+	0.278240i	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	−1.30902	+	0.278240i	−1.30902	+	0.278240i	−0.500000	+	0.866025i	$0.666667\pi$
$542$	2.15349	−	1.56460i	2.15349	−	1.56460i
$543$		0			0
$544$	0.460864	+	2.16819i	0.460864	+	2.16819i
$545$	−1.60917	+	0.169131i	−1.60917	+	0.169131i
$546$		0			0
$547$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$547$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$548$	1.07394	−	0.478148i	1.07394	−	0.478148i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.193318	+	0.266080i	−0.193318	+	0.266080i
$557$	−1.98904			−1.98904			−0.994522	−	0.104528i	$-0.966667\pi$
$557$	−1.98904			−1.98904			−0.994522	+	0.104528i	$0.966667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.181049	+	0.104528i	0.181049	+	0.104528i	0.587785	−	0.809017i	$-0.300000\pi$
$563$	0.181049	+	0.104528i	0.181049	+	0.104528i	−0.406737	+	0.913545i	$0.633333\pi$
$564$		0			0
$565$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$569$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$570$		0			0
$571$	1.16913	−	0.122881i	1.16913	−	0.122881i	0.500000	−	0.866025i	$-0.333333\pi$
$571$	1.16913	−	0.122881i	1.16913	−	0.122881i	0.669131	+	0.743145i	$0.266667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.86055	−	0.395472i	1.86055	−	0.395472i
$576$		0			0
$577$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$577$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$578$	0.556480	−	2.61803i	0.556480	−	2.61803i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.255585	−	0.113794i	−0.255585	−	0.113794i
$587$	0.658114	+	0.478148i	0.658114	+	0.478148i	0.866025	−	0.500000i	$-0.166667\pi$
$587$	0.658114	+	0.478148i	0.658114	+	0.478148i	−0.207912	+	0.978148i	$0.566667\pi$
$588$		0			0
$589$	1.22256	−	1.35779i	1.22256	−	1.35779i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.86055	−	0.604528i	1.86055	−	0.604528i	0.866025	−	0.500000i	$-0.166667\pi$
$593$	1.86055	−	0.604528i	1.86055	−	0.604528i	0.994522	−	0.104528i	$-0.0333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$599$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$600$		0			0
$601$	−1.47815	−	0.155360i	−1.47815	−	0.155360i	−0.669131	−	0.743145i	$-0.733333\pi$
$601$	−1.47815	−	0.155360i	−1.47815	−	0.155360i	−0.809017	+	0.587785i	$0.800000\pi$
$602$		0			0
$603$		0			0
$604$	−0.546528	−	0.752232i	−0.546528	−	0.752232i
$605$	0.587785	+	0.809017i	0.587785	+	0.809017i
$606$		0			0
$607$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$607$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$608$	2.32545	+	0.244415i	2.32545	+	0.244415i
$609$		0			0
$610$	1.08268	+	2.43173i	1.08268	+	2.43173i
$611$		0			0
$612$		0			0
$613$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$613$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.658114	−	1.47815i	0.658114	−	1.47815i	−0.207912	−	0.978148i	$-0.566667\pi$
$617$	0.658114	−	1.47815i	0.658114	−	1.47815i	0.866025	−	0.500000i	$-0.166667\pi$
$618$		0			0
$619$		−	1.48629i		−	1.48629i	−0.669131	−	0.743145i	$-0.733333\pi$
$619$		−	1.48629i		−	1.48629i	0.669131	−	0.743145i	$-0.266667\pi$
$620$	−0.587785	+	0.529244i	−0.587785	+	0.529244i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.309017	−	1.45381i	0.309017	−	1.45381i	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	0.309017	−	1.45381i	0.309017	−	1.45381i	0.809017	−	0.587785i	$-0.200000\pi$
$632$	0.0237894	−	0.226341i	0.0237894	−	0.226341i
$633$		0			0
$634$	2.56082	−	0.544320i	2.56082	−	0.544320i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.535358	+	0.113794i	0.535358	+	0.113794i
$641$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$641$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$642$		0			0
$643$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$643$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$644$		0			0
$645$		0			0
$646$	−3.66769	−	2.11754i	−3.66769	−	2.11754i
$647$	0.614648	+	1.89169i	0.614648	+	1.89169i	0.406737	+	0.913545i	$0.366667\pi$
$647$	0.614648	+	1.89169i	0.614648	+	1.89169i	0.207912	+	0.978148i	$0.433333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.994522	−	0.104528i	$-0.966667\pi$
$653$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	0.406737	+	0.913545i	$0.366667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$659$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$660$		0			0
$661$	1.78716	−	0.795697i	1.78716	−	0.795697i	0.809017	−	0.587785i	$-0.200000\pi$
$661$	1.78716	−	0.795697i	1.78716	−	0.795697i	0.978148	−	0.207912i	$-0.0666667\pi$
$662$	−2.60369	−	0.553432i	−2.60369	−	0.553432i
$663$		0			0
$664$	−0.115699	+	0.0121604i	−0.115699	+	0.0121604i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.14988	+	0.244415i	−1.14988	+	0.244415i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$673$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$674$		0			0
$675$		0			0
$676$	−0.395472	−	0.684977i	−0.395472	−	0.684977i
$677$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$677$	0.951057	−	1.64728i	0.951057	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$678$		0			0
$679$		0			0
$680$	−0.392034	−	0.284829i	−0.392034	−	0.284829i
$681$		0			0
$682$		0			0
$683$		−	1.82709i		−	1.82709i	−0.406737	−	0.913545i	$-0.633333\pi$
$683$		−	1.82709i		−	1.82709i	0.406737	−	0.913545i	$-0.366667\pi$
$684$		0			0
$685$	−0.604528	+	1.35779i	−0.604528	+	1.35779i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.895472	+	0.994522i	0.895472	+	0.994522i	1.00000			$0$
$691$	0.895472	+	0.994522i	0.895472	+	0.994522i	−0.104528	+	0.994522i	$0.533333\pi$
$692$	−0.198825	−	0.446568i	−0.198825	−	0.446568i
$693$		0			0
$694$	1.97815	+	0.207912i	1.97815	+	0.207912i
$695$	−0.0434654	−	0.413545i	−0.0434654	−	0.413545i
$696$		0			0
$697$		0			0
$698$	−1.53884	−	2.11803i	−1.53884	−	2.11803i
$699$		0			0
$700$		0			0
$701$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$701$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.413545	−	0.372358i	−0.413545	−	0.372358i
$707$		0			0
$708$		0			0
$709$	1.41355	−	0.459289i	1.41355	−	0.459289i	0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.41355	−	0.459289i	1.41355	−	0.459289i	0.913545	+	0.406737i	$0.133333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	−2.32545	+	2.09385i	−2.32545	+	2.09385i
$723$		0			0
$724$	0.312795	−	1.47159i	0.312795	−	1.47159i
$725$		0			0
$726$		0			0
$727$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$727$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$733$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$734$		0			0
$735$		0			0
$736$	−2.31513	−	0.752232i	−2.31513	−	0.752232i
$737$		0			0
$738$		0			0
$739$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$739$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$743$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.895472	−	0.994522i	0.895472	−	0.994522i	−0.104528	−	0.994522i	$-0.533333\pi$
$751$	0.895472	−	0.994522i	0.895472	−	0.994522i	1.00000			$0$
$752$	−2.16819	−	0.704489i	−2.16819	−	0.704489i
$753$		0			0
$754$		0			0
$755$	1.14988	+	0.244415i	1.14988	+	0.244415i
$756$		0			0
$757$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$757$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$758$	0.450202	+	2.11803i	0.450202	+	2.11803i
$759$		0			0
$760$	−0.413545	+	0.300458i	−0.413545	+	0.300458i
$761$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$761$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	1.97815	−	1.78113i	1.97815	−	1.78113i
$767$		0			0
$768$		0			0
$769$	0.913545	+	1.58231i	0.913545	+	1.58231i	0.809017	+	0.587785i	$0.200000\pi$
$769$	0.913545	+	1.58231i	0.913545	+	1.58231i	0.104528	+	0.994522i	$0.466667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$773$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$774$		0			0
$775$	0.104528	−	0.994522i	0.104528	−	0.994522i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	3.27651	+	2.95018i	3.27651	+	2.95018i
$783$		0			0
$784$	−0.779773	−	0.866025i	−0.779773	−	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$787$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$788$	−0.0343786	−	0.327091i	−0.0343786	−	0.327091i
$789$		0			0
$790$	−0.639886	−	0.880728i	−0.639886	−	0.880728i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.557210	+	1.25151i	0.557210	+	1.25151i
$797$	0.994522	+	1.10453i	0.994522	+	1.10453i	0.994522	+	0.104528i	$0.0333333\pi$
$797$	0.994522	+	1.10453i	0.994522	+	1.10453i			1.00000i	$0.5\pi$
$798$		0			0
$799$	2.51807	+	2.26728i	2.51807	+	2.26728i
$800$	1.10832	−	0.639886i	1.10832	−	0.639886i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$809$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$810$		0			0
$811$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$811$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.0581680	−	0.553432i	0.0581680	−	0.553432i
$819$		0			0
$820$		0			0
$821$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$823$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.81708	+	0.809017i	−1.81708	+	0.809017i	−0.866025	+	0.500000i	$0.833333\pi$
$827$	−1.81708	+	0.809017i	−1.81708	+	0.809017i	−0.951057	+	0.309017i	$0.900000\pi$
$828$		0			0
$829$	−0.773659	−	0.251377i	−0.773659	−	0.251377i	−0.104528	−	0.994522i	$-0.533333\pi$
$829$	−0.773659	−	0.251377i	−0.773659	−	0.251377i	−0.669131	+	0.743145i	$0.733333\pi$
$830$	−0.372358	+	0.413545i	−0.372358	+	0.413545i
$831$		0			0
$832$		0			0
$833$	0.535233	+	1.64728i	0.535233	+	1.64728i
$834$		0			0
$835$	0.873619	−	1.20243i	0.873619	−	1.20243i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$839$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$840$		0			0
$841$	0.309017	+	0.951057i	0.309017	+	0.951057i
$842$	−1.15897	−	0.669131i	−1.15897	−	0.669131i
$843$		0			0
$844$	0.327091	−	0.363271i	0.327091	−	0.363271i
$845$	0.951057	+	0.309017i	0.951057	+	0.309017i
$846$		0			0
$847$		0			0
$848$	−1.34002	−	0.284829i	−1.34002	−	0.284829i
$849$		0			0
$850$	−2.30524	+	0.242290i	−2.30524	+	0.242290i
$851$		0			0
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$		0			0
$856$	−0.0180739	+	0.171962i	−0.0180739	+	0.171962i
$857$	0.128496	−	0.604528i	0.128496	−	0.604528i	−0.866025	−	0.500000i	$-0.833333\pi$
$857$	0.128496	−	0.604528i	0.128496	−	0.604528i	0.994522	−	0.104528i	$-0.0333333\pi$
$858$		0			0
$859$	−0.604528	+	0.544320i	−0.604528	+	0.544320i	−0.913545	−	0.406737i	$-0.866667\pi$
$859$	−0.604528	+	0.544320i	−0.604528	+	0.544320i	0.309017	+	0.951057i	$0.400000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$863$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$	0.564602	+	0.251377i	0.564602	+	0.251377i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.430526	+	0.139886i	−0.430526	+	0.139886i
$873$		0			0
$874$	4.02780	−	2.32545i	4.02780	−	2.32545i
$875$		0			0
$876$		0			0
$877$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$877$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$878$	−0.336408	−	0.755585i	−0.336408	−	0.755585i
$879$		0			0
$880$		0			0
$881$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$881$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$882$		0			0
$883$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$883$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$884$		0			0
$885$		0			0
$886$	0.0292442	+	0.278240i	0.0292442	+	0.278240i
$887$	−0.614648	−	0.0646021i	−0.614648	−	0.0646021i	−0.207912	−	0.978148i	$-0.566667\pi$
$887$	−0.614648	−	0.0646021i	−0.614648	−	0.0646021i	−0.406737	+	0.913545i	$0.633333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.09546	−	1.78716i	3.09546	−	1.78716i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	1.64728	+	1.19682i	1.64728	+	1.19682i
$902$		0			0
$903$		0			0
$904$	−0.139886	+	0.242290i	−0.139886	+	0.242290i
$905$	0.951057	+	1.64728i	0.951057	+	1.64728i
$906$		0			0
$907$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$907$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$908$	−0.587785	+	0.529244i	−0.587785	+	0.529244i
$909$		0			0
$910$		0			0
$911$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$911$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.193318	+	0.909491i	0.193318	+	0.909491i
$917$		0			0
$918$		0			0
$919$	0.204489	+	0.0434654i	0.204489	+	0.0434654i	0.309017	−	0.951057i	$-0.400000\pi$
$919$	0.204489	+	0.0434654i	0.204489	+	0.0434654i	−0.104528	+	0.994522i	$0.533333\pi$
$920$	0.486152	−	0.216449i	0.486152	−	0.216449i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	1.82709			1.82709
$932$	0.0971915	−	0.133773i	0.0971915	−	0.133773i
$933$		0			0
$934$	−0.669131	−	2.05937i	−0.669131	−	2.05937i
$935$		0			0
$936$		0			0
$937$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$937$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$938$		0			0
$939$		0			0
$940$	−1.41355	+	0.629351i	−1.41355	+	0.629351i
$941$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$941$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.45381	+	0.309017i	−1.45381	+	0.309017i	−0.866025	−	0.500000i	$-0.833333\pi$
$947$	−1.45381	+	0.309017i	−1.45381	+	0.309017i	−0.587785	+	0.809017i	$0.700000\pi$
$948$		0			0
$949$		0			0
$950$	−0.508370	+	2.39169i	−0.508370	+	2.39169i
$951$		0			0
$952$		0			0
$953$	−0.614648	+	1.89169i	−0.614648	+	1.89169i	−0.207912	+	0.978148i	$0.566667\pi$
$953$	−0.614648	+	1.89169i	−0.614648	+	1.89169i	−0.406737	+	0.913545i	$0.633333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.309017	+	0.951057i	0.309017	+	0.951057i
$962$		0			0
$963$		0			0
$964$	−0.261699	+	0.587785i	−0.261699	+	0.587785i
$965$		0			0
$966$		0			0
$967$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$967$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$968$	0.207912	+	0.187205i	0.207912	+	0.187205i
$969$		0			0
$970$		0			0
$971$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$971$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.36245	+	1.87525i	1.36245	+	1.87525i
$977$	−1.07394	−	1.47815i	−1.07394	−	1.47815i	−0.866025	−	0.500000i	$-0.833333\pi$
$977$	−1.07394	−	1.47815i	−1.07394	−	1.47815i	−0.207912	−	0.978148i	$-0.566667\pi$
$978$		0			0
$979$		0			0
$980$	−0.786610	−	0.0826761i	−0.786610	−	0.0826761i
$981$		0			0
$982$		0			0
$983$	0.544320	+	0.604528i	0.544320	+	0.604528i	0.951057	−	0.309017i	$-0.100000\pi$
$983$	0.544320	+	0.604528i	0.544320	+	0.604528i	−0.406737	+	0.913545i	$0.633333\pi$
$984$		0			0
$985$	0.309017	+	0.278240i	0.309017	+	0.278240i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	0.415823i		−	0.415823i	−0.978148	−	0.207912i	$-0.933333\pi$
$991$		−	0.415823i		−	0.415823i	0.978148	−	0.207912i	$-0.0666667\pi$
$992$	−0.752232	+	1.03536i	−0.752232	+	1.03536i
$993$		0			0
$994$		0			0
$995$	−1.58231	−	0.704489i	−1.58231	−	0.704489i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	0.786610	+	1.36245i	0.786610	+	1.36245i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1395.1.dw.a.694.2	yes	16
3.2	odd	2	inner	1395.1.dw.a.694.1	yes	16
5.4	even	2	inner	1395.1.dw.a.694.1	yes	16
15.14	odd	2	CM	1395.1.dw.a.694.2	yes	16
31.13	odd	30	inner	1395.1.dw.a.199.1	✓	16
93.44	even	30	inner	1395.1.dw.a.199.2	yes	16
155.44	odd	30	inner	1395.1.dw.a.199.2	yes	16
465.44	even	30	inner	1395.1.dw.a.199.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.1.dw.a.199.1	✓	16	31.13	odd	30	inner
1395.1.dw.a.199.1	✓	16	465.44	even	30	inner
1395.1.dw.a.199.2	yes	16	93.44	even	30	inner
1395.1.dw.a.199.2	yes	16	155.44	odd	30	inner
1395.1.dw.a.694.1	yes	16	3.2	odd	2	inner
1395.1.dw.a.694.1	yes	16	5.4	even	2	inner
1395.1.dw.a.694.2	yes	16	1.1	even	1	trivial
1395.1.dw.a.694.2	yes	16	15.14	odd	2	CM

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:	\( N \)	\(=\)	\( 1395 = 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1395.dw (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.786610 - 1.08268i) q^{2} +(-0.244415 - 0.752232i) q^{4} +(0.866025 + 0.500000i) q^{5} +(0.266080 + 0.0864545i) q^{8} +O(q^{10})\)	\(q+(0.786610 - 1.08268i) q^{2} +(-0.244415 - 0.752232i) q^{4} +(0.866025 + 0.500000i) q^{5} +(0.266080 + 0.0864545i) q^{8} +(1.22256 - 0.544320i) q^{10} +(0.942790 - 0.684977i) q^{16} +(-1.69420 + 0.360114i) q^{17} +(-0.190983 + 1.81708i) q^{19} +(0.164446 - 0.773659i) q^{20} +(0.587785 - 1.80902i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-0.809017 - 0.587785i) q^{31} -1.27977i q^{32} +(-0.942790 + 2.11754i) q^{34} +(1.81708 + 1.63611i) q^{38} +(0.187205 + 0.207912i) q^{40} +(-1.49622 - 2.05937i) q^{46} +(-1.14988 - 1.58268i) q^{47} +(-0.104528 - 0.994522i) q^{49} +(1.33093 + 0.139886i) q^{50} +(-0.786610 - 0.873619i) q^{53} +1.98904i q^{61} +(-1.27276 + 0.413545i) q^{62} +(-0.442790 - 0.321706i) q^{64} +(0.684977 + 1.18641i) q^{68} +(1.41355 - 0.300458i) q^{76} +(-0.169131 - 0.795697i) q^{79} +(1.15897 - 0.121812i) q^{80} +(-0.379874 + 0.169131i) q^{83} +(-1.64728 - 0.535233i) q^{85} -1.50446 q^{92} -2.61803 q^{94} +(-1.07394 + 1.47815i) q^{95} +(-1.15897 - 0.669131i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 2 q^{10} - 6 q^{16} - 12 q^{19} + 8 q^{25} - 4 q^{31} + 6 q^{34} - 22 q^{40} + 10 q^{46} + 2 q^{49} + 14 q^{64} + 10 q^{76} + 6 q^{79} - 24 q^{94}+O(q^{100}) \) 16 * q - 2 * q^10 - 6 * q^16 - 12 * q^19 + 8 * q^25 - 4 * q^31 + 6 * q^34 - 22 * q^40 + 10 * q^46 + 2 * q^49 + 14 * q^64 + 10 * q^76 + 6 * q^79 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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