LMFDB - Embedded newform 3800.1.o.g.1101.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(1101,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("3800.1101");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3800.o (of order $2$, degree $1$, 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:	$1.89644704801$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.66724352000.2

Embedding invariants

Embedding label			1101.3
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	3800.1101
Dual form			3800.1.o.g.1101.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.382683 - 0.923880i) q^{2} -1.84776 q^{3} +(-0.707107 + 0.707107i) q^{4} +(0.707107 + 1.70711i) q^{6} +(0.923880 + 0.382683i) q^{8} +2.41421 q^{9} -1.41421i q^{11} +(1.30656 - 1.30656i) q^{12} -0.765367 q^{13} -1.00000i q^{16} +(-0.923880 - 2.23044i) q^{18} +1.00000i q^{19} +(-1.30656 + 0.541196i) q^{22} +(-1.70711 - 0.707107i) q^{24} +(0.292893 + 0.707107i) q^{26} -2.61313 q^{27} +(-0.923880 + 0.382683i) q^{32} +2.61313i q^{33} +(-1.70711 + 1.70711i) q^{36} +1.84776 q^{37} +(0.923880 - 0.382683i) q^{38} +1.41421 q^{39} +(1.00000 + 1.00000i) q^{44} +1.84776i q^{48} -1.00000 q^{49} +(0.541196 - 0.541196i) q^{52} +1.84776 q^{53} +(1.00000 + 2.41421i) q^{54} -1.84776i q^{57} +1.41421i q^{61} +(0.707107 + 0.707107i) q^{64} +(2.41421 - 1.00000i) q^{66} -0.765367 q^{67} +(2.23044 + 0.923880i) q^{72} +(-0.707107 - 1.70711i) q^{74} +(-0.707107 - 0.707107i) q^{76} +(-0.541196 - 1.30656i) q^{78} +2.41421 q^{81} +(0.541196 - 1.30656i) q^{88} +(1.70711 - 0.707107i) q^{96} +0.765367i q^{97} +(0.382683 + 0.923880i) q^{98} -3.41421i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9} - 8 q^{24} + 8 q^{26} - 8 q^{36} + 8 q^{44} - 8 q^{49} + 8 q^{54} + 8 q^{66} + 8 q^{81} + 8 q^{96}+O(q^{100}) $ 8 * q + 8 * q^9 - 8 * q^24 + 8 * q^26 - 8 * q^36 + 8 * q^44 - 8 * q^49 + 8 * q^54 + 8 * q^66 + 8 * q^81 + 8 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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 $

$2$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$2$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$3$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$3$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$4$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$5$		0			0
$5$		0			0
$6$	0.707107	+	1.70711i	0.707107	+	1.70711i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	0.923880	+	0.382683i	0.923880	+	0.382683i
$9$	2.41421			2.41421
$10$		0			0
$11$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$11$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$12$	1.30656	−	1.30656i	1.30656	−	1.30656i
$13$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$13$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$14$		0			0
$15$		0			0
$16$		−	1.00000i		−	1.00000i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.923880	−	2.23044i	−0.923880	−	2.23044i
$19$			1.00000i			1.00000i
$19$			1.00000i			1.00000i
$20$		0			0
$21$		0			0
$22$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$25$		0			0
$26$	0.292893	+	0.707107i	0.292893	+	0.707107i
$27$	−2.61313			−2.61313
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$33$			2.61313i			2.61313i
$34$		0			0
$35$		0			0
$36$	−1.70711	+	1.70711i	−1.70711	+	1.70711i
$37$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$37$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$38$	0.923880	−	0.382683i	0.923880	−	0.382683i
$39$	1.41421			1.41421
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	1.00000	+	1.00000i	1.00000	+	1.00000i
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$			1.84776i			1.84776i
$49$	−1.00000			−1.00000
$50$		0			0
$51$		0			0
$52$	0.541196	−	0.541196i	0.541196	−	0.541196i
$53$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$53$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$54$	1.00000	+	2.41421i	1.00000	+	2.41421i
$55$		0			0
$56$		0			0
$57$		−	1.84776i		−	1.84776i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$61$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$		0			0
$64$	0.707107	+	0.707107i	0.707107	+	0.707107i
$65$		0			0
$66$	2.41421	−	1.00000i	2.41421	−	1.00000i
$67$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$67$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	2.23044	+	0.923880i	2.23044	+	0.923880i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$75$		0			0
$76$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$77$		0			0
$78$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	2.41421			2.41421
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.541196	−	1.30656i	0.541196	−	1.30656i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.70711	−	0.707107i	1.70711	−	0.707107i
$97$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$97$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$98$	0.382683	+	0.923880i	0.382683	+	0.923880i
$99$		−	3.41421i		−	3.41421i
$100$		0			0
$101$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$101$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$102$		0			0
$103$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$103$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$104$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$105$		0			0
$106$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$107$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$107$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$108$	1.84776	−	1.84776i	1.84776	−	1.84776i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−3.41421			−3.41421
$112$		0			0
$113$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$113$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$114$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$115$		0			0
$116$		0			0
$117$	−1.84776			−1.84776
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−1.00000
$122$	1.30656	−	0.541196i	1.30656	−	0.541196i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$127$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$128$	0.382683	−	0.923880i	0.382683	−	0.923880i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−1.84776	−	1.84776i	−1.84776	−	1.84776i
$133$		0			0
$134$	0.292893	+	0.707107i	0.292893	+	0.707107i
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$139$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.08239i			1.08239i
$144$		−	2.41421i		−	2.41421i
$145$		0			0
$146$		0			0
$147$	1.84776			1.84776
$148$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$149$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$149$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	−3.41421			−3.41421
$160$		0			0
$161$		0			0
$162$	−0.923880	−	2.23044i	−0.923880	−	2.23044i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$167$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$168$		0			0
$169$	−0.414214			−0.414214
$170$		0			0
$171$			2.41421i			2.41421i
$172$		0			0
$173$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$173$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$174$		0			0
$175$		0			0
$176$	−1.41421			−1.41421
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		−	2.61313i		−	2.61313i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$193$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$193$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$194$	0.707107	−	0.292893i	0.707107	−	0.292893i
$195$		0			0
$196$	0.707107	−	0.707107i	0.707107	−	0.707107i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−3.15432	+	1.30656i	−3.15432	+	1.30656i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	1.41421			1.41421
$202$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$207$		0			0
$208$			0.765367i			0.765367i
$209$	1.41421			1.41421
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$213$		0			0
$214$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$215$		0			0
$216$	−2.41421	−	1.00000i	−2.41421	−	1.00000i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	1.30656	+	3.15432i	1.30656	+	3.15432i
$223$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$223$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$224$		0			0
$225$		0			0
$226$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$227$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$227$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$228$	1.30656	+	1.30656i	1.30656	+	1.30656i
$229$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$229$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	0.707107	+	1.70711i	0.707107	+	1.70711i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.00000			2.00000			1.00000			$0$
$239$	2.00000			2.00000			1.00000			$0$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.382683	+	0.923880i	0.382683	+	0.923880i
$243$	−1.84776			−1.84776
$244$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$245$		0			0
$246$		0			0
$247$		−	0.765367i		−	0.765367i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$251$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$252$		0			0
$253$		0			0
$254$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$255$		0			0
$256$	−1.00000			−1.00000
$257$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$257$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	−1.00000	+	2.41421i	−1.00000	+	2.41421i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.541196	−	0.541196i	0.541196	−	0.541196i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$271$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	1.00000	−	0.414214i	1.00000	−	0.414214i
$287$		0			0
$288$	−2.23044	+	0.923880i	−2.23044	+	0.923880i
$289$	−1.00000			−1.00000
$290$		0			0
$291$		−	1.41421i		−	1.41421i
$292$		0			0
$293$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$293$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$294$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$295$		0			0
$296$	1.70711	+	0.707107i	1.70711	+	0.707107i
$297$			3.69552i			3.69552i
$298$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$			2.61313i			2.61313i
$304$	1.00000			1.00000
$305$		0			0
$306$		0			0
$307$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$307$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$308$		0			0
$309$			3.41421i			3.41421i
$310$		0			0
$311$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$311$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$312$	1.30656	+	0.541196i	1.30656	+	0.541196i
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$317$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$318$	1.30656	+	3.15432i	1.30656	+	3.15432i
$319$		0			0
$320$		0			0
$321$	−1.41421			−1.41421
$322$		0			0
$323$		0			0
$324$	−1.70711	+	1.70711i	−1.70711	+	1.70711i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	4.46088			4.46088
$334$	0.707107	−	0.292893i	0.707107	−	0.292893i
$335$		0			0
$336$		0			0
$337$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$337$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$338$	0.158513	+	0.382683i	0.158513	+	0.382683i
$339$			3.41421i			3.41421i
$340$		0			0
$341$		0			0
$342$	2.23044	−	0.923880i	2.23044	−	0.923880i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$349$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$350$		0			0
$351$	2.00000			2.00000
$352$	0.541196	+	1.30656i	0.541196	+	1.30656i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$359$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$	1.84776			1.84776
$364$		0			0
$365$		0			0
$366$	−2.41421	+	1.00000i	−2.41421	+	1.00000i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$373$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$			1.41421i			1.41421i
$382$		0			0
$383$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$383$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$384$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$385$		0			0
$386$	0.707107	−	0.292893i	0.707107	−	0.292893i
$387$		0			0
$388$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	2.41421	+	2.41421i	2.41421	+	2.41421i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$403$		0			0
$404$	1.00000	+	1.00000i	1.00000	+	1.00000i
$405$		0			0
$406$		0			0
$407$		−	2.61313i		−	2.61313i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	1.30656	+	1.30656i	1.30656	+	1.30656i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.707107	−	0.292893i	0.707107	−	0.292893i
$417$			2.61313i			2.61313i
$418$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$	1.70711	+	0.707107i	1.70711	+	0.707107i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$429$		−	2.00000i		−	2.00000i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			2.61313i			2.61313i
$433$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$433$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−2.41421			−2.41421
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	2.41421	−	2.41421i	2.41421	−	2.41421i
$445$		0			0
$446$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$447$			2.61313i			2.61313i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	1.30656	+	1.30656i	1.30656	+	1.30656i
$453$		0			0
$454$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$455$		0			0
$456$	0.707107	−	1.70711i	0.707107	−	1.70711i
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	1.30656	−	1.30656i	1.30656	−	1.30656i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.46088			4.46088
$478$	−0.765367	−	1.84776i	−0.765367	−	1.84776i
$479$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$479$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$	−1.41421			−1.41421
$482$		0			0
$483$		0			0
$484$	0.707107	−	0.707107i	0.707107	−	0.707107i
$485$		0			0
$486$	0.707107	+	1.70711i	0.707107	+	1.70711i
$487$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$487$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$488$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$489$		0			0
$490$		0			0
$491$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$491$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$499$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$500$		0			0
$501$		−	1.41421i		−	1.41421i
$502$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.765367			0.765367
$508$	0.541196	+	0.541196i	0.541196	+	0.541196i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	0.382683	+	0.923880i	0.382683	+	0.923880i
$513$		−	2.61313i		−	2.61313i
$514$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.41421			−1.41421
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$523$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	2.61313			2.61313
$529$	−1.00000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$537$		0			0
$538$		0			0
$539$			1.41421i			1.41421i
$540$		0			0
$541$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$541$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$542$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$547$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$548$		0			0
$549$			3.41421i			3.41421i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.00000	+	1.00000i	1.00000	+	1.00000i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$563$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$571$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$572$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.70711	+	1.70711i	1.70711	+	1.70711i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	0.382683	+	0.923880i	0.382683	+	0.923880i
$579$		−	1.41421i		−	1.41421i
$580$		0			0
$581$		0			0
$582$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$583$		−	2.61313i		−	2.61313i
$584$		0			0
$585$		0			0
$586$	0.707107	+	1.70711i	0.707107	+	1.70711i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		−	1.84776i		−	1.84776i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	3.41421	−	1.41421i	3.41421	−	1.41421i
$595$		0			0
$596$	1.00000	+	1.00000i	1.00000	+	1.00000i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−1.84776			−1.84776
$604$		0			0
$605$		0			0
$606$	2.41421	−	1.00000i	2.41421	−	1.00000i
$607$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$607$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$608$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	3.15432	−	1.30656i	3.15432	−	1.30656i
$619$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$619$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$623$		0			0
$624$		−	1.41421i		−	1.41421i
$625$		0			0
$626$		0			0
$627$	−2.61313			−2.61313
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$631$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$632$		0			0
$633$		0			0
$634$	0.292893	+	0.707107i	0.292893	+	0.707107i
$635$		0			0
$636$	2.41421	−	2.41421i	2.41421	−	2.41421i
$637$	0.765367			0.765367
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	0.541196	+	1.30656i	0.541196	+	1.30656i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	2.23044	+	0.923880i	2.23044	+	0.923880i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		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			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.70711	−	4.12132i	−1.70711	−	4.12132i
$667$		0			0
$668$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$669$			1.41421i			1.41421i
$670$		0			0
$671$	2.00000			2.00000
$672$		0			0
$673$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$673$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$674$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$675$		0			0
$676$	0.292893	−	0.292893i	0.292893	−	0.292893i
$677$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$677$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$678$	3.15432	−	1.30656i	3.15432	−	1.30656i
$679$		0			0
$680$		0			0
$681$	−1.41421			−1.41421
$682$		0			0
$683$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$683$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$684$	−1.70711	−	1.70711i	−1.70711	−	1.70711i
$685$		0			0
$686$		0			0
$687$			2.61313i			2.61313i
$688$		0			0
$689$	−1.41421			−1.41421
$690$		0			0
$691$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$691$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$692$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$699$		0			0
$700$		0			0
$701$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$701$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$702$	−0.765367	−	1.84776i	−0.765367	−	1.84776i
$703$			1.84776i			1.84776i
$704$	1.00000	−	1.00000i	1.00000	−	1.00000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$709$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−3.69552			−3.69552
$718$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$719$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$719$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$720$		0			0
$721$		0			0
$722$	0.382683	+	0.923880i	0.382683	+	0.923880i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	1.84776	+	1.84776i	1.84776	+	1.84776i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.08239i			1.08239i
$738$		0			0
$739$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$739$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$740$		0			0
$741$			1.41421i			1.41421i
$742$		0			0
$743$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$743$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$744$		0			0
$745$		0			0
$746$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$			3.69552i			3.69552i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$761$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$762$	1.30656	−	0.541196i	1.30656	−	0.541196i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	1.70711	−	0.707107i	1.70711	−	0.707107i
$767$		0			0
$768$	1.84776			1.84776
$769$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$769$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$			1.41421i			1.41421i
$772$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$773$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$773$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$774$		0			0
$775$		0			0
$776$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			1.00000i			1.00000i
$785$		0			0
$786$		0			0
$787$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$787$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.30656	−	3.15432i	1.30656	−	3.15432i
$793$		−	1.08239i		−	1.08239i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$797$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.541196	−	1.30656i	0.541196	−	1.30656i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$	−2.61313			−2.61313
$814$	−2.41421	+	1.00000i	−2.41421	+	1.00000i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$	0.707107	−	1.70711i	0.707107	−	1.70711i
$825$		0			0
$826$		0			0
$827$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$827$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$833$		0			0
$834$	2.41421	−	1.00000i	2.41421	−	1.00000i
$835$		0			0
$836$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		−	1.84776i		−	1.84776i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	0.707107	+	0.292893i	0.707107	+	0.292893i
$857$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$857$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$858$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$863$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$864$	2.41421	−	1.00000i	2.41421	−	1.00000i
$865$		0			0
$866$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$867$	1.84776			1.84776
$868$		0			0
$869$		0			0
$870$		0			0
$871$	0.585786			0.585786
$872$		0			0
$873$			1.84776i			1.84776i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$877$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$878$		0			0
$879$	3.41421			3.41421
$880$		0			0
$881$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$881$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$882$	0.923880	+	2.23044i	0.923880	+	2.23044i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$887$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$888$	−3.15432	−	1.30656i	−3.15432	−	1.30656i
$889$		0			0
$890$		0			0
$891$		−	3.41421i		−	3.41421i
$892$	0.541196	+	0.541196i	0.541196	+	0.541196i
$893$		0			0
$894$	2.41421	−	1.00000i	2.41421	−	1.00000i
$895$		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			0
$904$	0.707107	−	1.70711i	0.707107	−	1.70711i
$905$		0			0
$906$		0			0
$907$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$907$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$908$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$909$		−	3.41421i		−	3.41421i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−1.84776			−1.84776
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.00000	+	1.00000i	1.00000	+	1.00000i
$917$		0			0
$918$		0			0
$919$	2.00000			2.00000			1.00000			$0$
$919$	2.00000			2.00000			1.00000			$0$
$920$		0			0
$921$	−1.41421			−1.41421
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	4.46088i		−	4.46088i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		−	1.00000i		−	1.00000i
$932$		0			0
$933$	−2.61313			−2.61313
$934$		0			0
$935$		0			0
$936$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	1.41421			1.41421
$952$		0			0
$953$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$953$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$954$	−1.70711	−	4.12132i	−1.70711	−	4.12132i
$955$		0			0
$956$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$957$		0			0
$958$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$	0.541196	+	1.30656i	0.541196	+	1.30656i
$963$	1.84776			1.84776
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	1.30656	−	1.30656i	1.30656	−	1.30656i
$973$		0			0
$974$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$975$		0			0
$976$	1.41421			1.41421
$977$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$977$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1.84776	−	0.765367i	1.84776	−	0.765367i
$983$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$983$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.541196	+	0.541196i	0.541196	+	0.541196i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	1.30656	−	0.541196i	1.30656	−	0.541196i
$999$	−4.82843			−4.82843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.1.o.g.1101.3		8
5.2	odd	4		760.1.b.a.189.7	yes	8
5.3	odd	4		760.1.b.a.189.2	yes	8
5.4	even	2	inner	3800.1.o.g.1101.6		8
8.5	even	2	inner	3800.1.o.g.1101.5		8
19.18	odd	2	inner	3800.1.o.g.1101.6		8
20.3	even	4		3040.1.b.a.1329.8		8
20.7	even	4		3040.1.b.a.1329.2		8
40.3	even	4		3040.1.b.a.1329.1		8
40.13	odd	4		760.1.b.a.189.1	✓	8
40.27	even	4		3040.1.b.a.1329.7		8
40.29	even	2	inner	3800.1.o.g.1101.4		8
40.37	odd	4		760.1.b.a.189.8	yes	8
95.18	even	4		760.1.b.a.189.7	yes	8
95.37	even	4		760.1.b.a.189.2	yes	8
95.94	odd	2	CM	3800.1.o.g.1101.3		8
152.37	odd	2	inner	3800.1.o.g.1101.4		8
380.227	odd	4		3040.1.b.a.1329.8		8
380.303	odd	4		3040.1.b.a.1329.2		8
760.37	even	4		760.1.b.a.189.1	✓	8
760.189	odd	2	inner	3800.1.o.g.1101.5		8
760.227	odd	4		3040.1.b.a.1329.1		8
760.493	even	4		760.1.b.a.189.8	yes	8
760.683	odd	4		3040.1.b.a.1329.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.1.b.a.189.1	✓	8	40.13	odd	4
760.1.b.a.189.1	✓	8	760.37	even	4
760.1.b.a.189.2	yes	8	5.3	odd	4
760.1.b.a.189.2	yes	8	95.37	even	4
760.1.b.a.189.7	yes	8	5.2	odd	4
760.1.b.a.189.7	yes	8	95.18	even	4
760.1.b.a.189.8	yes	8	40.37	odd	4
760.1.b.a.189.8	yes	8	760.493	even	4
3040.1.b.a.1329.1		8	40.3	even	4
3040.1.b.a.1329.1		8	760.227	odd	4
3040.1.b.a.1329.2		8	20.7	even	4
3040.1.b.a.1329.2		8	380.303	odd	4
3040.1.b.a.1329.7		8	40.27	even	4
3040.1.b.a.1329.7		8	760.683	odd	4
3040.1.b.a.1329.8		8	20.3	even	4
3040.1.b.a.1329.8		8	380.227	odd	4
3800.1.o.g.1101.3		8	1.1	even	1	trivial
3800.1.o.g.1101.3		8	95.94	odd	2	CM
3800.1.o.g.1101.4		8	40.29	even	2	inner
3800.1.o.g.1101.4		8	152.37	odd	2	inner
3800.1.o.g.1101.5		8	8.5	even	2	inner
3800.1.o.g.1101.5		8	760.189	odd	2	inner
3800.1.o.g.1101.6		8	5.4	even	2	inner
3800.1.o.g.1101.6		8	19.18	odd	2	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3800.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.382683 - 0.923880i) q^{2} -1.84776 q^{3} +(-0.707107 + 0.707107i) q^{4} +(0.707107 + 1.70711i) q^{6} +(0.923880 + 0.382683i) q^{8} +2.41421 q^{9} -1.41421i q^{11} +(1.30656 - 1.30656i) q^{12} -0.765367 q^{13} -1.00000i q^{16} +(-0.923880 - 2.23044i) q^{18} +1.00000i q^{19} +(-1.30656 + 0.541196i) q^{22} +(-1.70711 - 0.707107i) q^{24} +(0.292893 + 0.707107i) q^{26} -2.61313 q^{27} +(-0.923880 + 0.382683i) q^{32} +2.61313i q^{33} +(-1.70711 + 1.70711i) q^{36} +1.84776 q^{37} +(0.923880 - 0.382683i) q^{38} +1.41421 q^{39} +(1.00000 + 1.00000i) q^{44} +1.84776i q^{48} -1.00000 q^{49} +(0.541196 - 0.541196i) q^{52} +1.84776 q^{53} +(1.00000 + 2.41421i) q^{54} -1.84776i q^{57} +1.41421i q^{61} +(0.707107 + 0.707107i) q^{64} +(2.41421 - 1.00000i) q^{66} -0.765367 q^{67} +(2.23044 + 0.923880i) q^{72} +(-0.707107 - 1.70711i) q^{74} +(-0.707107 - 0.707107i) q^{76} +(-0.541196 - 1.30656i) q^{78} +2.41421 q^{81} +(0.541196 - 1.30656i) q^{88} +(1.70711 - 0.707107i) q^{96} +0.765367i q^{97} +(0.382683 + 0.923880i) q^{98} -3.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9} - 8 q^{24} + 8 q^{26} - 8 q^{36} + 8 q^{44} - 8 q^{49} + 8 q^{54} + 8 q^{66} + 8 q^{81} + 8 q^{96}+O(q^{100}) \) 8 * q + 8 * q^9 - 8 * q^24 + 8 * q^26 - 8 * q^36 + 8 * q^44 - 8 * q^49 + 8 * q^54 + 8 * q^66 + 8 * q^81 + 8 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more