LMFDB - Embedded newform 1925.1.dn.c.1707.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,1,Mod(118,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([15, 10, 6]))

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

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

chi := DirichletCharacter("1925.118");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1925.dn (of order $20$, 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:	$0.960700149319$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$4$ over $\Q(\zeta_{20})$
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:	$\Z[a_1, \ldots, a_{11}]$
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			1707.3
Root			$0.933580 - 0.358368i$ of defining polynomial
Character	$\chi$	$=$	1925.1707
Dual form			1925.1.dn.c.468.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.0949099 + 0.186271i) q^{2} +(0.562096 - 0.773659i) q^{4} +(0.987688 - 0.156434i) q^{7} +(0.403942 + 0.0639781i) q^{8} +(-0.951057 - 0.309017i) q^{9} +(0.669131 - 0.743145i) q^{11} +(0.122881 + 0.169131i) q^{14} +(-0.269091 - 0.828176i) q^{16} +(-0.0327037 - 0.206483i) q^{18} +(0.201933 + 0.0541079i) q^{22} +(-0.575212 + 0.575212i) q^{23} +(0.434149 - 0.852065i) q^{28} +(-1.20243 - 0.873619i) q^{29} +(0.417916 - 0.417916i) q^{32} +(-0.773659 + 0.562096i) q^{36} +(0.311155 + 1.96456i) q^{37} +(0.946294 + 0.946294i) q^{43} +(-0.198825 - 0.935398i) q^{44} +(-0.161739 - 0.0525521i) q^{46} +(0.951057 - 0.309017i) q^{49} +(-0.863541 - 1.69480i) q^{53} +0.408977 q^{56} +(0.0486072 - 0.306894i) q^{58} +(-0.987688 - 0.156434i) q^{63} +(-0.710666 - 0.230909i) q^{64} +(0.294032 + 0.294032i) q^{67} +(0.413545 + 1.27276i) q^{71} +(-0.364401 - 0.185672i) q^{72} +(-0.336408 + 0.244415i) q^{74} +(0.544639 - 0.838671i) q^{77} +(-0.128496 + 0.395472i) q^{79} +(0.809017 + 0.587785i) q^{81} +(-0.0864545 + 0.266080i) q^{86} +(0.317835 - 0.257378i) q^{88} +(0.121694 + 0.768343i) q^{92} +(0.147826 + 0.147826i) q^{98} +(-0.866025 + 0.500000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{11} + 12 q^{16} - 40 q^{46} - 8 q^{56} - 12 q^{71} + 8 q^{81} - 28 q^{86}+O(q^{100}) $ 32 * q + 4 * q^11 + 12 * q^16 - 40 * q^46 - 8 * q^56 - 12 * q^71 + 8 * q^81 - 28 * q^86

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$.

$n$	$276$	$1002$	$1751$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{1}{10}\right)$

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 $

$2$	0.0949099	+	0.186271i	0.0949099	+	0.186271i	0.933580	−	0.358368i	$-0.116667\pi$
$2$	0.0949099	+	0.186271i	0.0949099	+	0.186271i	−0.838671	+	0.544639i	$0.816667\pi$
$3$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$3$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$4$	0.562096	−	0.773659i	0.562096	−	0.773659i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.987688	−	0.156434i	0.987688	−	0.156434i
$7$	0.987688	−	0.156434i	0.987688	−	0.156434i
$8$	0.403942	+	0.0639781i	0.403942	+	0.0639781i
$9$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$10$		0			0
$11$	0.669131	−	0.743145i	0.669131	−	0.743145i
$11$	0.669131	−	0.743145i	0.669131	−	0.743145i
$12$		0			0
$13$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$13$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$14$	0.122881	+	0.169131i	0.122881	+	0.169131i
$15$		0			0
$16$	−0.269091	−	0.828176i	−0.269091	−	0.828176i
$17$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$17$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$18$	−0.0327037	−	0.206483i	−0.0327037	−	0.206483i
$19$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$19$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$20$		0			0
$21$		0			0
$22$	0.201933	+	0.0541079i	0.201933	+	0.0541079i
$23$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	−0.933580	−	0.358368i	$-0.883333\pi$
$23$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	0.358368	+	0.933580i	$0.383333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	0.434149	−	0.852065i	0.434149	−	0.852065i
$29$	−1.20243	−	0.873619i	−1.20243	−	0.873619i	−0.207912	−	0.978148i	$-0.566667\pi$
$29$	−1.20243	−	0.873619i	−1.20243	−	0.873619i	−0.994522	+	0.104528i	$0.966667\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$	0.417916	−	0.417916i	0.417916	−	0.417916i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.773659	+	0.562096i	−0.773659	+	0.562096i
$37$	0.311155	+	1.96456i	0.311155	+	1.96456i	0.258819	+	0.965926i	$0.416667\pi$
$37$	0.311155	+	1.96456i	0.311155	+	1.96456i	0.0523360	+	0.998630i	$0.483333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$41$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$42$		0			0
$43$	0.946294	+	0.946294i	0.946294	+	0.946294i	0.998630	−	0.0523360i	$-0.0166667\pi$
$43$	0.946294	+	0.946294i	0.946294	+	0.946294i	−0.0523360	+	0.998630i	$0.516667\pi$
$44$	−0.198825	−	0.935398i	−0.198825	−	0.935398i
$45$		0			0
$46$	−0.161739	−	0.0525521i	−0.161739	−	0.0525521i
$47$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$47$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$48$		0			0
$49$	0.951057	−	0.309017i	0.951057	−	0.309017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.863541	−	1.69480i	−0.863541	−	1.69480i	−0.707107	−	0.707107i	$-0.750000\pi$
$53$	−0.863541	−	1.69480i	−0.863541	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$54$		0			0
$55$		0			0
$56$	0.408977			0.408977
$57$		0			0
$58$	0.0486072	−	0.306894i	0.0486072	−	0.306894i
$59$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$59$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$60$		0			0
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$64$	−0.710666	−	0.230909i	−0.710666	−	0.230909i
$65$		0			0
$66$		0			0
$67$	0.294032	+	0.294032i	0.294032	+	0.294032i	0.838671	−	0.544639i	$-0.183333\pi$
$67$	0.294032	+	0.294032i	0.294032	+	0.294032i	−0.544639	+	0.838671i	$0.683333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.413545	+	1.27276i	0.413545	+	1.27276i	0.913545	+	0.406737i	$0.133333\pi$
$71$	0.413545	+	1.27276i	0.413545	+	1.27276i	−0.500000	+	0.866025i	$0.666667\pi$
$72$	−0.364401	−	0.185672i	−0.364401	−	0.185672i
$73$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$73$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$74$	−0.336408	+	0.244415i	−0.336408	+	0.244415i
$75$		0			0
$76$		0			0
$77$	0.544639	−	0.838671i	0.544639	−	0.838671i
$78$		0			0
$79$	−0.128496	+	0.395472i	−0.128496	+	0.395472i	−0.994522	−	0.104528i	$-0.966667\pi$
$79$	−0.128496	+	0.395472i	−0.128496	+	0.395472i	0.866025	+	0.500000i	$0.166667\pi$
$80$		0			0
$81$	0.809017	+	0.587785i	0.809017	+	0.587785i
$82$		0			0
$83$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$83$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$84$		0			0
$85$		0			0
$86$	−0.0864545	+	0.266080i	−0.0864545	+	0.266080i
$87$		0			0
$88$	0.317835	−	0.257378i	0.317835	−	0.257378i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	0.121694	+	0.768343i	0.121694	+	0.768343i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$97$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$98$	0.147826	+	0.147826i	0.147826	+	0.147826i
$99$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$100$		0			0
$101$		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		0.987688	−	0.156434i	$-0.0500000\pi$
$103$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$104$		0			0
$105$		0			0
$106$	0.233733	−	0.321706i	0.233733	−	0.321706i
$107$	0.253116	−	1.59811i	0.253116	−	1.59811i	−0.453990	−	0.891007i	$-0.650000\pi$
$107$	0.253116	−	1.59811i	0.253116	−	1.59811i	0.707107	−	0.707107i	$-0.250000\pi$
$108$		0			0
$109$	1.98904			1.98904			0.994522	−	0.104528i	$-0.0333333\pi$
$109$	1.98904			1.98904			0.994522	+	0.104528i	$0.0333333\pi$
$110$		0			0
$111$		0			0
$112$	−0.395333	−	0.775885i	−0.395333	−	0.775885i
$113$	−0.232507	+	1.46799i	−0.232507	+	1.46799i	0.544639	+	0.838671i	$0.316667\pi$
$113$	−0.232507	+	1.46799i	−0.232507	+	1.46799i	−0.777146	+	0.629320i	$0.783333\pi$
$114$		0			0
$115$		0			0
$116$	−1.35177	+	0.439216i	−1.35177	+	0.439216i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.0646021	−	0.198825i	−0.0646021	−	0.198825i
$127$	−1.62795	−	0.829482i	−1.62795	−	0.829482i	−0.998630	−	0.0523360i	$-0.983333\pi$
$127$	−1.62795	−	0.829482i	−1.62795	−	0.829482i	−0.629320	−	0.777146i	$-0.716667\pi$
$128$	−0.116894	−	0.738039i	−0.116894	−	0.738039i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	−0.0268631	+	0.0826761i	−0.0268631	+	0.0826761i
$135$		0			0
$136$		0			0
$137$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	0.453990	+	0.891007i	$0.350000\pi$
$137$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	−0.987688	+	0.156434i	$0.950000\pi$
$138$		0			0
$139$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$139$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$140$		0			0
$141$		0			0
$142$	−0.197829	+	0.197829i	−0.197829	+	0.197829i
$143$		0			0
$144$			0.870796i			0.870796i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.69480	+	0.863541i	1.69480	+	0.863541i
$149$	0.535233	+	1.64728i	0.535233	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$149$	0.535233	+	1.64728i	0.535233	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$150$		0			0
$151$	−0.478148	−	0.658114i	−0.478148	−	0.658114i	0.500000	−	0.866025i	$-0.333333\pi$
$151$	−0.478148	−	0.658114i	−0.478148	−	0.658114i	−0.978148	+	0.207912i	$0.933333\pi$
$152$		0			0
$153$		0			0
$154$	0.207912	+	0.0218524i	0.207912	+	0.0218524i
$155$		0			0
$156$		0			0
$157$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$157$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$158$	−0.0858605	+	0.0135990i	−0.0858605	+	0.0135990i
$159$		0			0
$160$		0			0
$161$	−0.478148	+	0.658114i	−0.478148	+	0.658114i
$162$	−0.0327037	+	0.206483i	−0.0327037	+	0.206483i
$163$	0.533698	+	1.04744i	0.533698	+	1.04744i	0.987688	+	0.156434i	$0.0500000\pi$
$163$	0.533698	+	1.04744i	0.533698	+	1.04744i	−0.453990	+	0.891007i	$0.650000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$167$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$168$		0			0
$169$	0.587785	−	0.809017i	0.587785	−	0.809017i
$170$		0			0
$171$		0			0
$172$	1.26402	−	0.200201i	1.26402	−	0.200201i
$173$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$173$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$174$		0			0
$175$		0			0
$176$	−0.795511	−	0.354185i	−0.795511	−	0.354185i
$177$		0			0
$178$		0			0
$179$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$179$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$181$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$	−0.269153	+	0.195551i	−0.269153	+	0.195551i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$191$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$192$		0			0
$193$	−0.888139	+	1.74307i	−0.888139	+	1.74307i	−0.258819	+	0.965926i	$0.583333\pi$
$193$	−0.888139	+	1.74307i	−0.888139	+	1.74307i	−0.629320	+	0.777146i	$0.716667\pi$
$194$		0			0
$195$		0			0
$196$	0.295511	−	0.909491i	0.295511	−	0.909491i
$197$	−1.29195	+	1.29195i	−1.29195	+	1.29195i	−0.358368	+	0.933580i	$0.616667\pi$
$197$	−1.29195	+	1.29195i	−1.29195	+	1.29195i	−0.933580	+	0.358368i	$0.883333\pi$
$198$	−0.175330	−	0.113861i	−0.175330	−	0.113861i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.32429	−	0.674761i	−1.32429	−	0.674761i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.724810	−	0.369309i	0.724810	−	0.369309i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$211$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$212$	−1.79659	−	0.284551i	−1.79659	−	0.284551i
$213$		0			0
$214$	0.321706	−	0.104528i	0.321706	−	0.104528i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.188780	+	0.370501i	0.188780	+	0.370501i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$223$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$224$	0.347395	−	0.478148i	0.347395	−	0.478148i
$225$		0			0
$226$	−0.295511	+	0.0960175i	−0.295511	+	0.0960175i
$227$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$227$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$228$		0			0
$229$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$229$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$230$		0			0
$231$		0			0
$232$	−0.429821	−	0.429821i	−0.429821	−	0.429821i
$233$	0.891007	−	0.453990i	0.891007	−	0.453990i	0.0523360	−	0.998630i	$-0.483333\pi$
$233$	0.891007	−	0.453990i	0.891007	−	0.453990i	0.838671	+	0.544639i	$0.183333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$239$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	0.175330	−	0.113861i	0.175330	−	0.113861i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$251$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$252$	−0.676203	+	0.676203i	−0.676203	+	0.676203i
$253$	0.0425739	+	0.812358i	0.0425739	+	0.812358i
$254$		−	0.381966i		−	0.381966i
$255$		0			0
$256$	−0.478148	+	0.347395i	−0.478148	+	0.347395i
$257$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$257$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$258$		0			0
$259$	0.614648	+	1.89169i	0.614648	+	1.89169i
$260$		0			0
$261$	0.873619	+	1.20243i	0.873619	+	1.20243i
$262$		0			0
$263$	−1.38331	−	1.38331i	−1.38331	−	1.38331i	−0.838671	−	0.544639i	$-0.816667\pi$
$263$	−1.38331	−	1.38331i	−1.38331	−	1.38331i	−0.544639	−	0.838671i	$-0.683333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.392754	−	0.0622062i	0.392754	−	0.0622062i
$269$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$269$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$		0			0
$273$		0			0
$274$	−0.245761			−0.245761
$275$		0			0
$276$		0			0
$277$	−0.734572	−	1.44168i	−0.734572	−	1.44168i	−0.891007	−	0.453990i	$-0.850000\pi$
$277$	−0.734572	−	1.44168i	−0.734572	−	1.44168i	0.156434	−	0.987688i	$-0.450000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.395472	+	0.128496i	−0.395472	+	0.128496i	−0.500000	−	0.866025i	$-0.666667\pi$
$281$	−0.395472	+	0.128496i	−0.395472	+	0.128496i	0.104528	+	0.994522i	$0.466667\pi$
$282$		0			0
$283$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$283$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$284$	1.21714	+	0.395472i	1.21714	+	0.395472i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.526605	+	0.268319i	−0.526605	+	0.268319i
$289$	0.587785	+	0.809017i	0.587785	+	0.809017i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$293$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$294$		0			0
$295$		0			0
$296$			0.813473i			0.813473i
$297$		0			0
$298$	−0.256041	+	0.256041i	−0.256041	+	0.256041i
$299$		0			0
$300$		0			0
$301$	1.08268	+	0.786610i	1.08268	+	0.786610i
$302$	0.0772066	−	0.151527i	0.0772066	−	0.151527i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$	−0.342706	−	0.892778i	−0.342706	−	0.892778i
$309$		0			0
$310$		0			0
$311$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$311$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$313$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$314$		0			0
$315$		0			0
$316$	0.233733	+	0.321706i	0.233733	+	0.321706i
$317$	0.370501	−	0.188780i	0.370501	−	0.188780i	−0.258819	−	0.965926i	$-0.583333\pi$
$317$	0.370501	−	0.188780i	0.370501	−	0.188780i	0.629320	+	0.777146i	$0.283333\pi$
$318$		0			0
$319$	−1.45381	+	0.309017i	−1.45381	+	0.309017i
$320$		0			0
$321$		0			0
$322$	−0.167968	−	0.0266036i	−0.167968	−	0.0266036i
$323$		0			0
$324$	0.909491	−	0.295511i	0.909491	−	0.295511i
$325$		0			0
$326$	−0.144455	+	0.198825i	−0.144455	+	0.198825i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.82709			−1.82709			−0.913545	−	0.406737i	$-0.866667\pi$
$331$	−1.82709			−1.82709			−0.913545	+	0.406737i	$0.866667\pi$
$332$		0			0
$333$	0.311155	−	1.96456i	0.311155	−	1.96456i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.891007	−	0.453990i	$-0.850000\pi$
$337$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.707107	+	0.707107i	$0.750000\pi$
$338$	0.206483	+	0.0327037i	0.206483	+	0.0327037i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.891007	−	0.453990i	0.891007	−	0.453990i
$344$	0.321706	+	0.442790i	0.321706	+	0.442790i
$345$		0			0
$346$		0			0
$347$	−1.74307	−	0.888139i	−1.74307	−	0.888139i	−0.965926	−	0.258819i	$-0.916667\pi$
$347$	−1.74307	−	0.888139i	−1.74307	−	0.888139i	−0.777146	−	0.629320i	$-0.783333\pi$
$348$		0			0
$349$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$349$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$350$		0			0
$351$		0			0
$352$	−0.0309318	−	0.590213i	−0.0309318	−	0.590213i
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.153567	+	0.301393i	−0.153567	+	0.301393i
$359$	−1.60917	−	1.16913i	−1.60917	−	1.16913i	−0.866025	−	0.500000i	$-0.833333\pi$
$359$	−1.60917	−	1.16913i	−1.60917	−	1.16913i	−0.743145	−	0.669131i	$-0.766667\pi$
$360$		0			0
$361$	0.309017	−	0.951057i	0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$367$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$368$	0.631161	+	0.321593i	0.631161	+	0.321593i
$369$		0			0
$370$		0			0
$371$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$372$		0			0
$373$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$373$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.198825	+	0.0646021i	−0.198825	+	0.0646021i	−0.406737	−	0.913545i	$-0.633333\pi$
$379$	−0.198825	+	0.0646021i	−0.198825	+	0.0646021i	0.207912	+	0.978148i	$0.433333\pi$
$380$		0			0
$381$		0			0
$382$	−0.0202120	+	0.127614i	−0.0202120	+	0.127614i
$383$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$383$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$384$		0			0
$385$		0			0
$386$	−0.408977			−0.408977
$387$	−0.607558	−	1.19240i	−0.607558	−	1.19240i
$388$		0			0
$389$	1.07394	−	1.47815i	1.07394	−	1.47815i	0.207912	−	0.978148i	$-0.433333\pi$
$389$	1.07394	−	1.47815i	1.07394	−	1.47815i	0.866025	−	0.500000i	$-0.166667\pi$
$390$		0			0
$391$		0			0
$392$	0.403942	−	0.0639781i	0.403942	−	0.0639781i
$393$		0			0
$394$	−0.363271	−	0.118034i	−0.363271	−	0.118034i
$395$		0			0
$396$	−0.0999601	+	0.951057i	−0.0999601	+	0.951057i
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.564602	+	1.73767i	0.564602	+	1.73767i	0.669131	+	0.743145i	$0.266667\pi$
$401$	0.564602	+	1.73767i	0.564602	+	1.73767i	−0.104528	+	0.994522i	$0.533333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		−	0.310719i		−	0.310719i
$407$	1.66815	+	1.08331i	1.66815	+	1.08331i
$408$		0			0
$409$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$409$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.137583	+	0.0999601i	0.137583	+	0.0999601i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−0.169131	+	0.122881i	−0.169131	+	0.122881i	−0.669131	−	0.743145i	$-0.733333\pi$
$421$	−0.169131	+	0.122881i	−0.169131	+	0.122881i	0.500000	+	0.866025i	$0.333333\pi$
$422$	−0.0384455	−	0.242735i	−0.0384455	−	0.242735i
$423$		0			0
$424$	−0.240391	−	0.739846i	−0.240391	−	0.739846i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.09412	−	1.09412i	−1.09412	−	1.09412i
$429$		0			0
$430$		0			0
$431$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$431$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$432$		0			0
$433$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$433$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$434$		0			0
$435$		0			0
$436$	1.11803	−	1.53884i	1.11803	−	1.53884i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$		0			0
$443$	−0.297556	+	1.87869i	−0.297556	+	1.87869i	0.156434	+	0.987688i	$0.450000\pi$
$443$	−0.297556	+	1.87869i	−0.297556	+	1.87869i	−0.453990	+	0.891007i	$0.650000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.738039	−	0.116894i	−0.738039	−	0.116894i
$449$	−0.198825	−	0.0646021i	−0.198825	−	0.0646021i	0.207912	−	0.978148i	$-0.433333\pi$
$449$	−0.198825	−	0.0646021i	−0.198825	−	0.0646021i	−0.406737	+	0.913545i	$0.633333\pi$
$450$		0			0
$451$		0			0
$452$	1.00503	+	1.00503i	1.00503	+	1.00503i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.62795	+	0.829482i	1.62795	+	0.829482i	0.998630	+	0.0523360i	$0.0166667\pi$
$457$	1.62795	+	0.829482i	1.62795	+	0.829482i	0.629320	+	0.777146i	$0.283333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$463$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$464$	−0.399947	+	1.23091i	−0.399947	+	1.23091i
$465$		0			0
$466$	0.169131	+	0.122881i	0.169131	+	0.122881i
$467$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$467$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$468$		0			0
$469$	0.336408	+	0.244415i	0.336408	+	0.244415i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.33643	−	0.0700392i	1.33643	−	0.0700392i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.297556	+	1.87869i	0.297556	+	1.87869i
$478$	−0.354309	−	0.180529i	−0.354309	−	0.180529i
$479$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$479$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.828176	−	0.478148i	−0.828176	−	0.478148i
$485$		0			0
$486$		0			0
$487$	−1.46799	−	0.232507i	−1.46799	−	0.232507i	−0.629320	−	0.777146i	$-0.716667\pi$
$487$	−1.46799	−	0.232507i	−1.46799	−	0.232507i	−0.838671	+	0.544639i	$0.816667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.01807	−	1.40126i	1.01807	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$491$	1.01807	−	1.40126i	1.01807	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.607558	+	1.19240i	0.607558	+	1.19240i
$498$		0			0
$499$	0.363271	−	0.500000i	0.363271	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$499$	0.363271	−	0.500000i	0.363271	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$503$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$504$	−0.388960	−	0.126381i	−0.388960	−	0.126381i
$505$		0			0
$506$	−0.147278	+	0.0850311i	−0.147278	+	0.0850311i
$507$		0			0
$508$	−1.55680	+	0.793230i	−1.55680	+	0.793230i
$509$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$509$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$510$		0			0
$511$		0			0
$512$	−0.775885	−	0.395333i	−0.775885	−	0.395333i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.294032	+	0.294032i	−0.294032	+	0.294032i
$519$		0			0
$520$		0			0
$521$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$521$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$522$	−0.141064	+	0.276853i	−0.141064	+	0.276853i
$523$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$523$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$524$		0			0
$525$		0			0
$526$	0.126381	−	0.388960i	0.126381	−	0.388960i
$527$		0			0
$528$		0			0
$529$			0.338261i			0.338261i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.0999601	+	0.137583i	0.0999601	+	0.137583i
$537$		0			0
$538$		0			0
$539$	0.406737	−	0.913545i	0.406737	−	0.913545i
$540$		0			0
$541$	−0.773659	−	0.251377i	−0.773659	−	0.251377i	−0.104528	−	0.994522i	$-0.533333\pi$
$541$	−0.773659	−	0.251377i	−0.773659	−	0.251377i	−0.669131	+	0.743145i	$0.733333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.156434	−	0.987688i	0.156434	−	0.987688i	−0.777146	−	0.629320i	$-0.783333\pi$
$547$	0.156434	−	0.987688i	0.156434	−	0.987688i	0.933580	−	0.358368i	$-0.116667\pi$
$548$	0.510373	+	1.00166i	0.510373	+	1.00166i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.0650491	+	0.410704i	−0.0650491	+	0.410704i
$554$	0.198825	−	0.273659i	0.198825	−	0.273659i
$555$		0			0
$556$		0			0
$557$	1.93221	−	0.306032i	1.93221	−	0.306032i	0.933580	−	0.358368i	$-0.116667\pi$
$557$	1.93221	−	0.306032i	1.93221	−	0.306032i	0.998630	+	0.0523360i	$0.0166667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.0614693	−	0.0614693i	−0.0614693	−	0.0614693i
$563$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$563$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.891007	+	0.453990i	0.891007	+	0.453990i
$568$	0.0856194	+	0.540580i	0.0856194	+	0.540580i
$569$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$569$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.994522	−	0.104528i	$-0.0333333\pi$
$570$		0			0
$571$		−	0.813473i		−	0.813473i	−0.913545	−	0.406737i	$-0.866667\pi$
$571$		−	0.813473i		−	0.813473i	0.913545	−	0.406737i	$-0.133333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.604528	+	0.439216i	0.604528	+	0.439216i
$577$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$577$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$578$	−0.0949099	+	0.186271i	−0.0949099	+	0.186271i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.83730	−	0.492303i	−1.83730	−	0.492303i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$587$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.54327	−	0.786335i	1.54327	−	0.786335i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$	1.57528	+	0.511841i	1.57528	+	0.511841i
$597$		0			0
$598$		0			0
$599$	−1.73767	+	0.564602i	−1.73767	+	0.564602i	−0.994522	−	0.104528i	$-0.966667\pi$
$599$	−1.73767	+	0.564602i	−1.73767	+	0.564602i	−0.743145	+	0.669131i	$0.766667\pi$
$600$		0			0
$601$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$601$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$602$	−0.0437661	+	0.276328i	−0.0437661	+	0.276328i
$603$	−0.188780	−	0.370501i	−0.188780	−	0.370501i
$604$	−0.777921			−0.777921
$605$		0			0
$606$		0			0
$607$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$607$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.32178	+	0.209350i	1.32178	+	0.209350i	0.777146	−	0.629320i	$-0.216667\pi$
$613$	1.32178	+	0.209350i	1.32178	+	0.209350i	0.544639	+	0.838671i	$0.316667\pi$
$614$		0			0
$615$		0			0
$616$	0.273659	−	0.303929i	0.273659	−	0.303929i
$617$	−0.294032	−	0.294032i	−0.294032	−	0.294032i	0.544639	−	0.838671i	$-0.316667\pi$
$617$	−0.294032	−	0.294032i	−0.294032	−	0.294032i	−0.838671	+	0.544639i	$0.816667\pi$
$618$		0			0
$619$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$619$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.169131	−	0.122881i	−0.169131	−	0.122881i	0.500000	−	0.866025i	$-0.333333\pi$
$631$	−0.169131	−	0.122881i	−0.169131	−	0.122881i	−0.669131	+	0.743145i	$0.733333\pi$
$632$	−0.0772066	+	0.151527i	−0.0772066	+	0.151527i
$633$		0			0
$634$	0.0703285	+	0.0510966i	0.0703285	+	0.0510966i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.195542	−	0.241474i	−0.195542	−	0.241474i
$639$		−	1.33826i		−	1.33826i
$640$		0			0
$641$	1.08268	−	0.786610i	1.08268	−	0.786610i	0.104528	−	0.994522i	$-0.466667\pi$
$641$	1.08268	−	0.786610i	1.08268	−	0.786610i	0.978148	+	0.207912i	$0.0666667\pi$
$642$		0			0
$643$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$643$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$644$	0.240391	+	0.739846i	0.240391	+	0.739846i
$645$		0			0
$646$		0			0
$647$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$647$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$648$	0.289190	+	0.289190i	0.289190	+	0.289190i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.11035	+	0.175862i	1.11035	+	0.175862i
$653$	1.16110	−	0.183900i	1.16110	−	0.183900i	0.453990	−	0.891007i	$-0.350000\pi$
$653$	1.16110	−	0.183900i	1.16110	−	0.183900i	0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−0.173409	−	0.340334i	−0.173409	−	0.340334i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.395472	−	0.128496i	0.395472	−	0.128496i
$667$	1.19417	−	0.189138i	1.19417	−	0.189138i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.550672	+	0.280582i	−0.550672	+	0.280582i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	−0.550672	+	0.280582i	−0.550672	+	0.280582i	0.156434	+	0.987688i	$0.450000\pi$
$674$	−0.198825	−	0.273659i	−0.198825	−	0.273659i
$675$		0			0
$676$	−0.295511	−	0.909491i	−0.295511	−	0.909491i
$677$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$677$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.629320	−	0.777146i	$-0.283333\pi$
$683$	1.40647	−	1.40647i	1.40647	−	1.40647i	0.777146	−	0.629320i	$-0.216667\pi$
$684$		0			0
$685$		0			0
$686$	0.169131	+	0.122881i	0.169131	+	0.122881i
$687$		0			0
$688$	0.529059	−	1.03834i	0.529059	−	1.03834i
$689$		0			0
$690$		0			0
$691$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$691$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$692$		0			0
$693$	−0.777146	+	0.629320i	−0.777146	+	0.629320i
$694$		−	0.408977i		−	0.408977i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$701$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−0.647127	+	0.373619i	−0.647127	+	0.373619i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.587785	−	0.190983i	0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$709$	0.587785	−	0.190983i	0.587785	−	0.190983i	0.587785	+	0.809017i	$0.300000\pi$
$710$		0			0
$711$	0.244415	−	0.336408i	0.244415	−	0.336408i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.54732			1.54732
$717$		0			0
$718$	0.0650491	−	0.410704i	0.0650491	−	0.410704i
$719$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$719$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$720$		0			0
$721$		0			0
$722$	0.206483	−	0.0327037i	0.206483	−	0.0327037i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$733$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$734$		0			0
$735$		0			0
$736$			0.480781i			0.480781i
$737$	0.415254	−	0.0217625i	0.415254	−	0.0217625i
$738$		0			0
$739$	0.459289	−	1.41355i	0.459289	−	1.41355i	−0.406737	−	0.913545i	$-0.633333\pi$
$739$	0.459289	−	1.41355i	0.459289	−	1.41355i	0.866025	−	0.500000i	$-0.166667\pi$
$740$		0			0
$741$		0			0
$742$	0.180529	−	0.354309i	0.180529	−	0.354309i
$743$	−0.280582	+	0.550672i	−0.280582	+	0.550672i	−0.987688	−	0.156434i	$-0.950000\pi$
$743$	−0.280582	+	0.550672i	−0.280582	+	0.550672i	0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i
$747$		0			0
$748$		0			0
$749$		−	1.61803i		−	1.61803i
$750$		0			0
$751$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$751$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.32429	−	0.674761i	1.32429	−	0.674761i	0.358368	−	0.933580i	$-0.383333\pi$
$757$	1.32429	−	0.674761i	1.32429	−	0.674761i	0.965926	+	0.258819i	$0.0833333\pi$
$758$	−0.0309040	−	0.0309040i	−0.0309040	−	0.0309040i
$759$		0			0
$760$		0			0
$761$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$761$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$762$		0			0
$763$	1.96456	−	0.311155i	1.96456	−	0.311155i
$764$	0.562096	−	0.182636i	0.562096	−	0.182636i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	0.849323	+	1.66689i	0.849323	+	1.66689i
$773$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$773$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$774$	0.164446	−	0.226341i	0.164446	−	0.226341i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.377263	+	0.0597526i	0.377263	+	0.0597526i
$779$		0			0
$780$		0			0
$781$	1.22256	+	0.544320i	1.22256	+	0.544320i
$782$		0			0
$783$		0			0
$784$	−0.511841	−	0.704489i	−0.511841	−	0.704489i
$785$		0			0
$786$		0			0
$787$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$787$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$788$	0.273328	+	1.72573i	0.273328	+	1.72573i
$789$		0			0
$790$		0			0
$791$			1.48629i			1.48629i
$792$	−0.381813	+	0.146564i	−0.381813	+	0.146564i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$797$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.614648	−	1.89169i	−0.614648	−	1.89169i	−0.406737	−	0.913545i	$-0.633333\pi$
$809$	−0.614648	−	1.89169i	−0.614648	−	1.89169i	−0.207912	−	0.978148i	$-0.566667\pi$
$810$		0			0
$811$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$811$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$812$	−1.26642	+	0.645271i	−1.26642	+	0.645271i
$813$		0			0
$814$	−0.0434654	+	0.413545i	−0.0434654	+	0.413545i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$821$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$822$		0			0
$823$	−0.674761	−	1.32429i	−0.674761	−	1.32429i	−0.933580	−	0.358368i	$-0.883333\pi$
$823$	−0.674761	−	1.32429i	−0.674761	−	1.32429i	0.258819	−	0.965926i	$-0.416667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.453990	+	0.891007i	0.453990	+	0.891007i	0.998630	+	0.0523360i	$0.0166667\pi$
$827$	0.453990	+	0.891007i	0.453990	+	0.891007i	−0.544639	+	0.838671i	$0.683333\pi$
$828$	0.121694	−	0.768343i	0.121694	−	0.768343i
$829$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$829$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$839$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$840$		0			0
$841$	0.373619	+	1.14988i	0.373619	+	1.14988i
$842$	−0.0389413	−	0.0198416i	−0.0389413	−	0.0198416i
$843$		0			0
$844$	−0.909491	+	0.660784i	−0.909491	+	0.660784i
$845$		0			0
$846$		0			0
$847$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$848$	−1.17122	+	1.17122i	−1.17122	+	1.17122i
$849$		0			0
$850$		0			0
$851$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$852$		0			0
$853$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$853$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$854$		0			0
$855$		0			0
$856$	0.204489	−	0.629351i	0.204489	−	0.629351i
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	0.0384455	+	0.242735i	0.0384455	+	0.242735i
$863$	−1.54327	−	0.786335i	−1.54327	−	0.786335i	−0.544639	−	0.838671i	$-0.683333\pi$
$863$	−1.54327	−	0.786335i	−1.54327	−	0.786335i	−0.998630	+	0.0523360i	$0.983333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.207912	+	0.360114i	0.207912	+	0.360114i
$870$		0			0
$871$		0			0
$872$	0.803458	+	0.127255i	0.803458	+	0.127255i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i	0.891007	+	0.453990i	$0.150000\pi$
$877$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i	−0.987688	+	0.156434i	$0.950000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.0949099	−	0.186271i	−0.0949099	−	0.186271i
$883$	0.270952	−	1.71073i	0.270952	−	1.71073i	−0.358368	−	0.933580i	$-0.616667\pi$
$883$	0.270952	−	1.71073i	0.270952	−	1.71073i	0.629320	−	0.777146i	$-0.283333\pi$
$884$		0			0
$885$		0			0
$886$	−0.378188	+	0.122881i	−0.378188	+	0.122881i
$887$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$887$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$888$		0			0
$889$	−1.73767	−	0.564602i	−1.73767	−	0.564602i
$890$		0			0
$891$	0.978148	−	0.207912i	0.978148	−	0.207912i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.230909	−	0.710666i	−0.230909	−	0.710666i
$897$		0			0
$898$	−0.00683694	−	0.0431667i	−0.00683694	−	0.0431667i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.187839	+	0.578108i	−0.187839	+	0.578108i
$905$		0			0
$906$		0			0
$907$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$907$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$911$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$912$		0			0
$913$		0			0
$914$			0.381966i			0.381966i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.459289	+	1.41355i	0.459289	+	1.41355i	0.866025	+	0.500000i	$0.166667\pi$
$919$	0.459289	+	1.41355i	0.459289	+	1.41355i	−0.406737	+	0.913545i	$0.633333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	0.378188	+	0.122881i	0.378188	+	0.122881i
$927$		0			0
$928$	−0.867616	+	0.137417i	−0.867616	+	0.137417i
$929$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$929$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$930$		0			0
$931$		0			0
$932$	0.149598	−	0.944522i	0.149598	−	0.944522i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$937$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$938$	−0.0135990	+	0.0858605i	−0.0135990	+	0.0858605i
$939$		0			0
$940$		0			0
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.139886	+	0.242290i	0.139886	+	0.242290i
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.285820	−	1.80460i	−0.285820	−	1.80460i	−0.544639	−	0.838671i	$-0.683333\pi$
$953$	−0.285820	−	1.80460i	−0.285820	−	1.80460i	0.258819	−	0.965926i	$-0.416667\pi$
$954$	−0.321706	+	0.233733i	−0.321706	+	0.233733i
$955$		0			0
$956$			1.81898i			1.81898i
$957$		0			0
$958$		0			0
$959$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$960$		0			0
$961$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$962$		0			0
$963$	−0.734572	+	1.44168i	−0.734572	+	1.44168i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$967$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$968$	0.0214042	−	0.408417i	0.0214042	−	0.408417i
$969$		0			0
$970$		0			0
$971$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$971$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$972$		0			0
$973$		0			0
$974$	−0.0960175	−	0.295511i	−0.0960175	−	0.295511i
$975$		0			0
$976$		0			0
$977$	−0.724810	+	0.369309i	−0.724810	+	0.369309i	−0.777146	−	0.629320i	$-0.783333\pi$
$977$	−0.724810	+	0.369309i	−0.724810	+	0.369309i	0.0523360	+	0.998630i	$0.483333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.89169	−	0.614648i	−1.89169	−	0.614648i
$982$	0.357639	+	0.0566445i	0.357639	+	0.0566445i
$983$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$983$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.08864			−1.08864
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$	−0.164446	+	0.226341i	−0.164446	+	0.226341i
$995$		0			0
$996$		0			0
$997$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$997$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$998$	0.127614	+	0.0202120i	0.127614	+	0.0202120i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.1.dn.c.1707.3	yes	32
5.2	odd	4	inner	1925.1.dn.c.1168.2	yes	32
5.3	odd	4	inner	1925.1.dn.c.1168.3	yes	32
5.4	even	2	inner	1925.1.dn.c.1707.2	yes	32
7.6	odd	2	CM	1925.1.dn.c.1707.3	yes	32
11.6	odd	10	inner	1925.1.dn.c.1007.3	yes	32
35.13	even	4	inner	1925.1.dn.c.1168.3	yes	32
35.27	even	4	inner	1925.1.dn.c.1168.2	yes	32
35.34	odd	2	inner	1925.1.dn.c.1707.2	yes	32
55.17	even	20	inner	1925.1.dn.c.468.2	✓	32
55.28	even	20	inner	1925.1.dn.c.468.3	yes	32
55.39	odd	10	inner	1925.1.dn.c.1007.2	yes	32
77.6	even	10	inner	1925.1.dn.c.1007.3	yes	32
385.83	odd	20	inner	1925.1.dn.c.468.3	yes	32
385.237	odd	20	inner	1925.1.dn.c.468.2	✓	32
385.314	even	10	inner	1925.1.dn.c.1007.2	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.1.dn.c.468.2	✓	32	55.17	even	20	inner
1925.1.dn.c.468.2	✓	32	385.237	odd	20	inner
1925.1.dn.c.468.3	yes	32	55.28	even	20	inner
1925.1.dn.c.468.3	yes	32	385.83	odd	20	inner
1925.1.dn.c.1007.2	yes	32	55.39	odd	10	inner
1925.1.dn.c.1007.2	yes	32	385.314	even	10	inner
1925.1.dn.c.1007.3	yes	32	11.6	odd	10	inner
1925.1.dn.c.1007.3	yes	32	77.6	even	10	inner
1925.1.dn.c.1168.2	yes	32	5.2	odd	4	inner
1925.1.dn.c.1168.2	yes	32	35.27	even	4	inner
1925.1.dn.c.1168.3	yes	32	5.3	odd	4	inner
1925.1.dn.c.1168.3	yes	32	35.13	even	4	inner
1925.1.dn.c.1707.2	yes	32	5.4	even	2	inner
1925.1.dn.c.1707.2	yes	32	35.34	odd	2	inner
1925.1.dn.c.1707.3	yes	32	1.1	even	1	trivial
1925.1.dn.c.1707.3	yes	32	7.6	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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1925.dn (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0949099 + 0.186271i) q^{2} +(0.562096 - 0.773659i) q^{4} +(0.987688 - 0.156434i) q^{7} +(0.403942 + 0.0639781i) q^{8} +(-0.951057 - 0.309017i) q^{9} +(0.669131 - 0.743145i) q^{11} +(0.122881 + 0.169131i) q^{14} +(-0.269091 - 0.828176i) q^{16} +(-0.0327037 - 0.206483i) q^{18} +(0.201933 + 0.0541079i) q^{22} +(-0.575212 + 0.575212i) q^{23} +(0.434149 - 0.852065i) q^{28} +(-1.20243 - 0.873619i) q^{29} +(0.417916 - 0.417916i) q^{32} +(-0.773659 + 0.562096i) q^{36} +(0.311155 + 1.96456i) q^{37} +(0.946294 + 0.946294i) q^{43} +(-0.198825 - 0.935398i) q^{44} +(-0.161739 - 0.0525521i) q^{46} +(0.951057 - 0.309017i) q^{49} +(-0.863541 - 1.69480i) q^{53} +0.408977 q^{56} +(0.0486072 - 0.306894i) q^{58} +(-0.987688 - 0.156434i) q^{63} +(-0.710666 - 0.230909i) q^{64} +(0.294032 + 0.294032i) q^{67} +(0.413545 + 1.27276i) q^{71} +(-0.364401 - 0.185672i) q^{72} +(-0.336408 + 0.244415i) q^{74} +(0.544639 - 0.838671i) q^{77} +(-0.128496 + 0.395472i) q^{79} +(0.809017 + 0.587785i) q^{81} +(-0.0864545 + 0.266080i) q^{86} +(0.317835 - 0.257378i) q^{88} +(0.121694 + 0.768343i) q^{92} +(0.147826 + 0.147826i) q^{98} +(-0.866025 + 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{11} + 12 q^{16} - 40 q^{46} - 8 q^{56} - 12 q^{71} + 8 q^{81} - 28 q^{86}+O(q^{100}) \) 32 * q + 4 * q^11 + 12 * q^16 - 40 * q^46 - 8 * q^56 - 12 * q^71 + 8 * q^81 - 28 * q^86

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