LMFDB - Embedded newform 804.1.be.a.725.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,1,Mod(17,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([0, 33, 64]))

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

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

chi := DirichletCharacter("804.17");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	804.be (of order $66$, degree $20$, 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:	$0.401248270157$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

Embedding invariants

Embedding label			725.1
Root			$0.723734 - 0.690079i$ of defining polynomial
Character	$\chi$	$=$	804.725
Dual form			804.1.be.a.173.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.654861 + 0.755750i) q^{3} +(1.50842 - 1.18624i) q^{7} +(-0.142315 - 0.989821i) q^{9} +O(q^{10})$	$q+(-0.654861 + 0.755750i) q^{3} +(1.50842 - 1.18624i) q^{7} +(-0.142315 - 0.989821i) q^{9} +(-0.911911 - 1.28060i) q^{13} +(1.02951 + 0.809616i) q^{19} +(-0.0913090 + 1.91681i) q^{21} +(0.415415 - 0.909632i) q^{25} +(0.841254 + 0.540641i) q^{27} +(-1.03115 + 1.44805i) q^{31} +(-0.415415 + 0.719520i) q^{37} +(1.56499 + 0.149438i) q^{39} +(1.91030 + 0.560914i) q^{43} +(0.632425 - 2.60689i) q^{49} +(-1.28605 + 0.247866i) q^{57} +(-1.74555 + 0.899892i) q^{61} +(-1.38884 - 1.32425i) q^{63} +(-0.995472 + 0.0950560i) q^{67} +(-0.0845850 + 0.0436066i) q^{73} +(0.415415 + 0.909632i) q^{75} +(-1.15486 + 0.110276i) q^{79} +(-0.959493 + 0.281733i) q^{81} +(-2.89465 - 0.849945i) q^{91} +(-0.419102 - 1.72756i) q^{93} +(0.327068 - 0.566498i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3} + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 20 * q - 2 * q^3 + 2 * q^7 - 2 * q^9	$ 20 q - 2 q^{3} + 2 q^{7} - 2 q^{9} - q^{13} + 2 q^{19} + 2 q^{21} - 2 q^{25} - 2 q^{27} - q^{31} + 2 q^{37} - q^{39} + 2 q^{43} + 3 q^{49} - 9 q^{57} - q^{61} - 9 q^{63} + q^{67} - 12 q^{73} - 2 q^{75} - 12 q^{79} - 2 q^{81} + 4 q^{91} - q^{93} - q^{97}+O(q^{100}) $ 20 * q - 2 * q^3 + 2 * q^7 - 2 * q^9 - q^13 + 2 * q^19 + 2 * q^21 - 2 * q^25 - 2 * q^27 - q^31 + 2 * q^37 - q^39 + 2 * q^43 + 3 * q^49 - 9 * q^57 - q^61 - 9 * q^63 + q^67 - 12 * q^73 - 2 * q^75 - 12 * q^79 - 2 * q^81 + 4 * q^91 - q^93 - q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{4}{33}\right)$	$1$

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			0
$2$		0			0
$3$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$3$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$4$		0			0
$5$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$5$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$6$		0			0
$7$	1.50842	−	1.18624i	1.50842	−	1.18624i	0.580057	−	0.814576i	$-0.303030\pi$
$7$	1.50842	−	1.18624i	1.50842	−	1.18624i	0.928368	−	0.371662i	$-0.121212\pi$
$8$		0			0
$9$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$10$		0			0
$11$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$11$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$12$		0			0
$13$	−0.911911	−	1.28060i	−0.911911	−	1.28060i	−0.959493	−	0.281733i	$-0.909091\pi$
$13$	−0.911911	−	1.28060i	−0.911911	−	1.28060i	0.0475819	−	0.998867i	$-0.484848\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$17$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$18$		0			0
$19$	1.02951	+	0.809616i	1.02951	+	0.809616i	0.981929	−	0.189251i	$-0.0606061\pi$
$19$	1.02951	+	0.809616i	1.02951	+	0.809616i	0.0475819	+	0.998867i	$0.484848\pi$
$20$		0			0
$21$	−0.0913090	+	1.91681i	−0.0913090	+	1.91681i
$22$		0			0
$23$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$23$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$24$		0			0
$25$	0.415415	−	0.909632i	0.415415	−	0.909632i
$26$		0			0
$27$	0.841254	+	0.540641i	0.841254	+	0.540641i
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−1.03115	+	1.44805i	−1.03115	+	1.44805i	−0.142315	+	0.989821i	$0.545455\pi$
$31$	−1.03115	+	1.44805i	−1.03115	+	1.44805i	−0.888835	+	0.458227i	$0.848485\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$37$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$38$		0			0
$39$	1.56499	+	0.149438i	1.56499	+	0.149438i
$40$		0			0
$41$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$41$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$42$		0			0
$43$	1.91030	+	0.560914i	1.91030	+	0.560914i	0.981929	+	0.189251i	$0.0606061\pi$
$43$	1.91030	+	0.560914i	1.91030	+	0.560914i	0.928368	+	0.371662i	$0.121212\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$47$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$48$		0			0
$49$	0.632425	−	2.60689i	0.632425	−	2.60689i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$53$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.28605	+	0.247866i	−1.28605	+	0.247866i
$58$		0			0
$59$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$59$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$60$		0			0
$61$	−1.74555	+	0.899892i	−1.74555	+	0.899892i	−0.786053	+	0.618159i	$0.787879\pi$
$61$	−1.74555	+	0.899892i	−1.74555	+	0.899892i	−0.959493	+	0.281733i	$0.909091\pi$
$62$		0			0
$63$	−1.38884	−	1.32425i	−1.38884	−	1.32425i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$67$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$71$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$72$		0			0
$73$	−0.0845850	+	0.0436066i	−0.0845850	+	0.0436066i	−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−0.0845850	+	0.0436066i	−0.0845850	+	0.0436066i	0.415415	+	0.909632i	$0.363636\pi$
$74$		0			0
$75$	0.415415	+	0.909632i	0.415415	+	0.909632i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.15486	+	0.110276i	−1.15486	+	0.110276i	−0.654861	−	0.755750i	$-0.727273\pi$
$79$	−1.15486	+	0.110276i	−1.15486	+	0.110276i	−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$82$		0			0
$83$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$83$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$89$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$90$		0			0
$91$	−2.89465	−	0.849945i	−2.89465	−	0.849945i
$92$		0			0
$93$	−0.419102	−	1.72756i	−0.419102	−	1.72756i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$97$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$101$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$102$		0			0
$103$	−0.165101	+	0.231852i	−0.165101	+	0.231852i	−0.888835	−	0.458227i	$-0.848485\pi$
$103$	−0.165101	+	0.231852i	−0.165101	+	0.231852i	0.723734	+	0.690079i	$0.242424\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$107$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$108$		0			0
$109$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	−0.888835	−	0.458227i	$-0.848485\pi$
$109$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	0.928368	+	0.371662i	$0.121212\pi$
$110$		0			0
$111$	−0.271738	−	0.785135i	−0.271738	−	0.785135i
$112$		0			0
$113$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$113$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.13779	+	1.08488i	−1.13779	+	1.08488i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.580057	+	0.814576i	0.580057	+	0.814576i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.56499	−	1.23072i	1.56499	−	1.23072i	0.723734	−	0.690079i	$-0.242424\pi$
$127$	1.56499	−	1.23072i	1.56499	−	1.23072i	0.841254	−	0.540641i	$-0.181818\pi$
$128$		0			0
$129$	−1.67489	+	1.07639i	−1.67489	+	1.07639i
$130$		0			0
$131$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$131$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$132$		0			0
$133$	2.51334			2.51334
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$137$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$138$		0			0
$139$	−0.550294	+	0.353653i	−0.550294	+	0.353653i	−0.786053	−	0.618159i	$-0.787879\pi$
$139$	−0.550294	+	0.353653i	−0.550294	+	0.353653i	0.235759	+	0.971812i	$0.424242\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.55601	+	2.18511i	1.55601	+	2.18511i
$148$		0			0
$149$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$149$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$150$		0			0
$151$	−1.28656	+	1.22673i	−1.28656	+	1.22673i	−0.327068	+	0.945001i	$0.606061\pi$
$151$	−1.28656	+	1.22673i	−1.28656	+	1.22673i	−0.959493	+	0.281733i	$0.909091\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.0930932	+	0.268975i	0.0930932	+	0.268975i	0.981929	−	0.189251i	$-0.0606061\pi$
$157$	0.0930932	+	0.268975i	0.0930932	+	0.268975i	−0.888835	+	0.458227i	$0.848485\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$163$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$167$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$168$		0			0
$169$	−0.481286	+	1.39059i	−0.481286	+	1.39059i
$170$		0			0
$171$	0.654861	−	1.13425i	0.654861	−	1.13425i
$172$		0			0
$173$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$173$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$174$		0			0
$175$	−0.452418	−	1.86489i	−0.452418	−	1.86489i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$179$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$180$		0			0
$181$	−0.642315	−	0.123796i	−0.642315	−	0.123796i	−0.142315	−	0.989821i	$-0.545455\pi$
$181$	−0.642315	−	0.123796i	−0.642315	−	0.123796i	−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$	0.462997	−	1.90850i	0.462997	−	1.90850i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.91030	−	0.182411i	1.91030	−	0.182411i
$190$		0			0
$191$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$191$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$192$		0			0
$193$	0.771316	+	1.68895i	0.771316	+	1.68895i	0.723734	+	0.690079i	$0.242424\pi$
$193$	0.771316	+	1.68895i	0.771316	+	1.68895i	0.0475819	+	0.998867i	$0.484848\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$197$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$198$		0			0
$199$	0.437742	+	0.175245i	0.437742	+	0.175245i	0.580057	−	0.814576i	$-0.303030\pi$
$199$	0.437742	+	0.175245i	0.437742	+	0.175245i	−0.142315	+	0.989821i	$0.545455\pi$
$200$		0			0
$201$	0.580057	−	0.814576i	0.580057	−	0.814576i
$202$		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
$211$	0.0934441	−	0.0180099i	0.0934441	−	0.0180099i	−0.142315	−	0.989821i	$-0.545455\pi$
$211$	0.0934441	−	0.0180099i	0.0934441	−	0.0180099i	0.235759	+	0.971812i	$0.424242\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.162317	+	3.40746i	0.162317	+	3.40746i
$218$		0			0
$219$	0.0224357	−	0.0924813i	0.0224357	−	0.0924813i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.428368	+	0.494363i	0.428368	+	0.494363i	0.928368	−	0.371662i	$-0.121212\pi$
$223$	0.428368	+	0.494363i	0.428368	+	0.494363i	−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$226$		0			0
$227$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$227$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$228$		0			0
$229$	−1.15486	−	0.110276i	−1.15486	−	0.110276i	−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.15486	−	0.110276i	−1.15486	−	0.110276i	−0.654861	+	0.755750i	$0.727273\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$233$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.672932	−	0.945001i	0.672932	−	0.945001i
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	0.975950	+	0.627205i	0.975950	+	0.627205i	0.928368	−	0.371662i	$-0.121212\pi$
$241$	0.975950	+	0.627205i	0.975950	+	0.627205i	0.0475819	+	0.998867i	$0.484848\pi$
$242$		0			0
$243$	0.415415	−	0.909632i	0.415415	−	0.909632i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.0979722	−	2.05669i	0.0979722	−	2.05669i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$251$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$257$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$258$		0			0
$259$	0.226900	+	1.57812i	0.226900	+	1.57812i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$263$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−1.21590	+	1.40323i	−1.21590	+	1.40323i	−0.327068	+	0.945001i	$0.606061\pi$
$271$	−1.21590	+	1.40323i	−1.21590	+	1.40323i	−0.888835	+	0.458227i	$0.848485\pi$
$272$		0			0
$273$	2.53794	−	1.63103i	2.53794	−	1.63103i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	−0.786053	−	0.618159i	$-0.787879\pi$
$277$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	0.580057	−	0.814576i	$-0.303030\pi$
$278$		0			0
$279$	1.58006	+	0.814576i	1.58006	+	0.814576i
$280$		0			0
$281$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$281$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$282$		0			0
$283$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$283$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$290$		0			0
$291$	0.213947	+	0.618159i	0.213947	+	0.618159i
$292$		0			0
$293$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$293$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	3.54692	−	1.41997i	3.54692	−	1.41997i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.469383	−	0.0448206i	−0.469383	−	0.0448206i	−0.142315	−	0.989821i	$-0.545455\pi$
$307$	−0.469383	−	0.0448206i	−0.469383	−	0.0448206i	−0.327068	+	0.945001i	$0.606061\pi$
$308$		0			0
$309$	−0.0671040	−	0.276606i	−0.0671040	−	0.276606i
$310$		0			0
$311$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$311$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$312$		0			0
$313$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$313$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$317$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.54370	+	0.297523i	−1.54370	+	0.297523i
$326$		0			0
$327$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.42131	+	1.35522i	1.42131	+	1.35522i	0.841254	+	0.540641i	$0.181818\pi$
$331$	1.42131	+	1.35522i	1.42131	+	1.35522i	0.580057	+	0.814576i	$0.303030\pi$
$332$		0			0
$333$	0.771316	+	0.308788i	0.771316	+	0.308788i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.84833	−	0.739959i	−1.84833	−	0.739959i	−0.959493	−	0.281733i	$-0.909091\pi$
$337$	−1.84833	−	0.739959i	−1.84833	−	0.739959i	−0.888835	−	0.458227i	$-0.848485\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.34125	−	2.93694i	−1.34125	−	2.93694i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$347$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$348$		0			0
$349$	1.50842	−	0.442913i	1.50842	−	0.442913i	0.580057	−	0.814576i	$-0.303030\pi$
$349$	1.50842	−	0.442913i	1.50842	−	0.442913i	0.928368	+	0.371662i	$0.121212\pi$
$350$		0			0
$351$	−0.0748038	−	1.57033i	−0.0748038	−	1.57033i
$352$		0			0
$353$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$353$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$359$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$360$		0			0
$361$	0.168655	+	0.695205i	0.168655	+	0.695205i
$362$		0			0
$363$	−0.995472	−	0.0950560i	−0.995472	−	0.0950560i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.550294	+	1.58997i	−0.550294	+	1.58997i	0.235759	+	0.971812i	$0.424242\pi$
$367$	−0.550294	+	1.58997i	−0.550294	+	1.58997i	−0.786053	+	0.618159i	$0.787879\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$373$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.607279	−	1.75462i	−0.607279	−	1.75462i	−0.654861	−	0.755750i	$-0.727273\pi$
$379$	−0.607279	−	1.75462i	−0.607279	−	1.75462i	0.0475819	−	0.998867i	$-0.484848\pi$
$380$		0			0
$381$	−0.0947329	+	1.98869i	−0.0947329	+	1.98869i
$382$		0			0
$383$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$383$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.283341	−	1.97068i	0.283341	−	1.97068i
$388$		0			0
$389$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$389$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.396666	−	0.254922i	0.396666	−	0.254922i	−0.327068	−	0.945001i	$-0.606061\pi$
$397$	0.396666	−	0.254922i	0.396666	−	0.254922i	0.723734	+	0.690079i	$0.242424\pi$
$398$		0			0
$399$	−1.64589	+	1.89945i	−1.64589	+	1.89945i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	2.79469			2.79469
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.32254	+	1.04006i	−1.32254	+	1.04006i	−0.327068	+	0.945001i	$0.606061\pi$
$409$	−1.32254	+	1.04006i	−1.32254	+	1.04006i	−0.995472	+	0.0950560i	$0.969697\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.0930932	−	0.647478i	0.0930932	−	0.647478i
$418$		0			0
$419$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$419$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$420$		0			0
$421$	1.56499	+	1.23072i	1.56499	+	1.23072i	0.841254	+	0.540641i	$0.181818\pi$
$421$	1.56499	+	1.23072i	1.56499	+	1.23072i	0.723734	+	0.690079i	$0.242424\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.56554	+	3.42805i	−1.56554	+	3.42805i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	0.481929	−	0.676774i	0.481929	−	0.676774i	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	0.481929	−	0.676774i	0.481929	−	0.676774i	0.981929	+	0.189251i	$0.0606061\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$439$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$440$		0			0
$441$	−2.67036	−	0.254989i	−2.67036	−	0.254989i
$442$		0			0
$443$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$443$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$449$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.0845850	−	1.77566i	−0.0845850	−	1.77566i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.56499	−	0.149438i	1.56499	−	0.149438i	0.723734	−	0.690079i	$-0.242424\pi$
$457$	1.56499	−	0.149438i	1.56499	−	0.149438i	0.841254	+	0.540641i	$0.181818\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$461$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$462$		0			0
$463$	0.252989	−	0.130425i	0.252989	−	0.130425i	−0.327068	−	0.945001i	$-0.606061\pi$
$463$	0.252989	−	0.130425i	0.252989	−	0.130425i	0.580057	+	0.814576i	$0.303030\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$467$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$468$		0			0
$469$	−1.38884	+	1.32425i	−1.38884	+	1.32425i
$470$		0			0
$471$	−0.264241	−	0.105786i	−0.264241	−	0.105786i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.16413	−	0.600149i	1.16413	−	0.600149i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$479$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$480$		0			0
$481$	1.30024	−	0.124158i	1.30024	−	0.124158i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.341254	−	1.40667i	0.341254	−	1.40667i	−0.500000	−	0.866025i	$-0.666667\pi$
$487$	0.341254	−	1.40667i	0.341254	−	1.40667i	0.841254	−	0.540641i	$-0.181818\pi$
$488$		0			0
$489$	1.42131	+	0.273935i	1.42131	+	0.273935i
$490$		0			0
$491$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$491$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$499$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$503$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.735759	−	1.27437i	−0.735759	−	1.27437i
$508$		0			0
$509$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$509$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$510$		0			0
$511$	−0.0758623	+	0.166115i	−0.0758623	+	0.166115i
$512$		0			0
$513$	0.428368	+	1.23769i	0.428368	+	1.23769i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$521$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$522$		0			0
$523$	−0.165101	−	0.231852i	−0.165101	−	0.231852i	0.723734	−	0.690079i	$-0.242424\pi$
$523$	−0.165101	−	0.231852i	−0.165101	−	0.231852i	−0.888835	+	0.458227i	$0.848485\pi$
$524$		0			0
$525$	1.70566	+	0.879330i	1.70566	+	0.879330i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.786053	+	0.618159i	−0.786053	+	0.618159i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.21769	−	0.782560i	1.21769	−	0.782560i	0.235759	−	0.971812i	$-0.424242\pi$
$541$	1.21769	−	0.782560i	1.21769	−	0.782560i	0.981929	+	0.189251i	$0.0606061\pi$
$542$		0			0
$543$	0.514186	−	0.404360i	0.514186	−	0.404360i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.28656	−	0.663268i	−1.28656	−	0.663268i	−0.327068	−	0.945001i	$-0.606061\pi$
$547$	−1.28656	−	0.663268i	−1.28656	−	0.663268i	−0.959493	+	0.281733i	$0.909091\pi$
$548$		0			0
$549$	1.13915	+	1.59971i	1.13915	+	1.59971i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.61121	+	1.53628i	−1.61121	+	1.53628i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$557$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$558$		0			0
$559$	−1.02371	−	2.95783i	−1.02371	−	2.95783i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$563$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.11312	+	1.56316i	−1.11312	+	1.56316i
$568$		0			0
$569$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$569$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$570$		0			0
$571$	−0.642315	+	1.85585i	−0.642315	+	1.85585i	−0.142315	+	0.989821i	$0.545455\pi$
$571$	−0.642315	+	1.85585i	−0.642315	+	1.85585i	−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.0224357	+	0.0924813i	0.0224357	+	0.0924813i	0.981929	−	0.189251i	$-0.0606061\pi$
$577$	0.0224357	+	0.0924813i	0.0224357	+	0.0924813i	−0.959493	+	0.281733i	$0.909091\pi$
$578$		0			0
$579$	−1.78153	−	0.523103i	−1.78153	−	0.523103i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$587$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$588$		0			0
$589$	−2.23394	+	0.655945i	−2.23394	+	0.655945i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$593$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.419102	+	0.216062i	−0.419102	+	0.216062i
$598$		0			0
$599$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$599$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$600$		0			0
$601$	−1.65033	−	0.660694i	−1.65033	−	0.660694i	−0.654861	−	0.755750i	$-0.727273\pi$
$601$	−1.65033	−	0.660694i	−1.65033	−	0.660694i	−0.995472	+	0.0950560i	$0.969697\pi$
$602$		0			0
$603$	0.235759	+	0.971812i	0.235759	+	0.971812i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.34378	+	1.28129i	1.34378	+	1.28129i	0.928368	+	0.371662i	$0.121212\pi$
$607$	1.34378	+	1.28129i	1.34378	+	1.28129i	0.415415	+	0.909632i	$0.363636\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.462997	−	0.0892353i	0.462997	−	0.0892353i	0.0475819	−	0.998867i	$-0.484848\pi$
$613$	0.462997	−	0.0892353i	0.462997	−	0.0892353i	0.415415	+	0.909632i	$0.363636\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$617$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$618$		0			0
$619$	−0.0845850	−	1.77566i	−0.0845850	−	1.77566i	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−0.0845850	−	1.77566i	−0.0845850	−	1.77566i	0.415415	−	0.909632i	$-0.363636\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.30379	+	0.124497i	1.30379	+	0.124497i	0.723734	−	0.690079i	$-0.242424\pi$
$631$	1.30379	+	0.124497i	1.30379	+	0.124497i	0.580057	+	0.814576i	$0.303030\pi$
$632$		0			0
$633$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.91510	+	1.56737i	−3.91510	+	1.56737i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$643$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$647$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.68148	−	2.10874i	−2.68148	−	2.10874i
$652$		0			0
$653$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$653$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.0552004	+	0.0775182i	0.0552004	+	0.0775182i
$658$		0			0
$659$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$659$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$660$		0			0
$661$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	−0.786053	−	0.618159i	$-0.787879\pi$
$661$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	0.580057	−	0.814576i	$-0.303030\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.654136			−0.654136
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.16413	−	1.34347i	1.16413	−	1.34347i	0.235759	−	0.971812i	$-0.424242\pi$
$673$	1.16413	−	1.34347i	1.16413	−	1.34347i	0.928368	−	0.371662i	$-0.121212\pi$
$674$		0			0
$675$	0.841254	−	0.540641i	0.841254	−	0.540641i
$676$		0			0
$677$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$677$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$678$		0			0
$679$	−0.178645	−	1.24250i	−0.178645	−	1.24250i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$683$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	0.839614	−	0.800570i	0.839614	−	0.800570i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.0395325	−	0.829889i	0.0395325	−	0.829889i	−0.888835	−	0.458227i	$-0.848485\pi$
$691$	0.0395325	−	0.829889i	0.0395325	−	0.829889i	0.928368	−	0.371662i	$-0.121212\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$701$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$702$		0			0
$703$	−1.01021	+	0.404427i	−1.01021	+	0.404427i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.30379	+	0.124497i	1.30379	+	0.124497i	0.723734	−	0.690079i	$-0.242424\pi$
$709$	1.30379	+	0.124497i	1.30379	+	0.124497i	0.580057	+	0.814576i	$0.303030\pi$
$710$		0			0
$711$	0.273507	+	1.12741i	0.273507	+	1.12741i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$719$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$720$		0			0
$721$	0.0259893	+	0.545582i	0.0259893	+	0.545582i
$722$		0			0
$723$	−1.11312	+	0.326842i	−1.11312	+	0.326842i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.462997	−	0.0892353i	0.462997	−	0.0892353i	0.0475819	−	0.998867i	$-0.484848\pi$
$727$	0.462997	−	0.0892353i	0.462997	−	0.0892353i	0.415415	+	0.909632i	$0.363636\pi$
$728$		0			0
$729$	0.415415	+	0.909632i	0.415415	+	0.909632i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.205996	−	0.196417i	−0.205996	−	0.196417i	0.580057	−	0.814576i	$-0.303030\pi$
$733$	−0.205996	−	0.196417i	−0.205996	−	0.196417i	−0.786053	+	0.618159i	$0.787879\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.437742	+	0.175245i	0.437742	+	0.175245i	0.580057	−	0.814576i	$-0.303030\pi$
$739$	0.437742	+	0.175245i	0.437742	+	0.175245i	−0.142315	+	0.989821i	$0.545455\pi$
$740$		0			0
$741$	1.49018	+	1.42089i	1.49018	+	1.42089i
$742$		0			0
$743$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$743$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$751$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.74555	−	0.336426i	−1.74555	−	0.336426i	−0.786053	−	0.618159i	$-0.787879\pi$
$757$	−1.74555	−	0.336426i	−1.74555	−	0.336426i	−0.959493	+	0.281733i	$0.909091\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$761$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$762$		0			0
$763$	−0.0430538	−	0.177470i	−0.0430538	−	0.177470i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.651174	−	1.88144i	0.651174	−	1.88144i	0.235759	−	0.971812i	$-0.424242\pi$
$769$	0.651174	−	1.88144i	0.651174	−	1.88144i	0.415415	−	0.909632i	$-0.363636\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$773$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$774$		0			0
$775$	0.888835	+	1.53951i	0.888835	+	1.53951i
$776$		0			0
$777$	−1.34125	−	0.861971i	−1.34125	−	0.861971i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.601300	−	0.573338i	0.601300	−	0.573338i	−0.327068	−	0.945001i	$-0.606061\pi$
$787$	0.601300	−	0.573338i	0.601300	−	0.573338i	0.928368	+	0.371662i	$0.121212\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.74418	+	1.41473i	2.74418	+	1.41473i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$797$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		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		0.841254	−	0.540641i	$-0.181818\pi$
$809$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$810$		0			0
$811$	−0.911911	+	0.717135i	−0.911911	+	0.717135i	−0.959493	−	0.281733i	$-0.909091\pi$
$811$	−0.911911	+	0.717135i	−0.911911	+	0.717135i	0.0475819	+	0.998867i	$0.484848\pi$
$812$		0			0
$813$	−0.264241	−	1.83784i	−0.264241	−	1.83784i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.51255	+	2.12407i	1.51255	+	2.12407i
$818$		0			0
$819$	−0.429342	+	2.98614i	−0.429342	+	2.98614i
$820$		0			0
$821$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$821$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$822$		0			0
$823$	−1.45949	−	1.14776i	−1.45949	−	1.14776i	−0.959493	−	0.281733i	$-0.909091\pi$
$823$	−1.45949	−	1.14776i	−1.45949	−	1.14776i	−0.500000	−	0.866025i	$-0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$827$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$828$		0			0
$829$	0.601300	−	1.31666i	0.601300	−	1.31666i	−0.327068	−	0.945001i	$-0.606061\pi$
$829$	0.601300	−	1.31666i	0.601300	−	1.31666i	0.928368	−	0.371662i	$-0.121212\pi$
$830$		0			0
$831$	1.21769	+	0.782560i	1.21769	+	0.782560i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.65033	+	0.660694i	−1.65033	+	0.660694i
$838$		0			0
$839$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$839$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$840$		0			0
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.84125	+	0.540641i	1.84125	+	0.540641i
$848$		0			0
$849$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.469383	+	1.93482i	−0.469383	+	1.93482i	−0.142315	+	0.989821i	$0.545455\pi$
$853$	−0.469383	+	1.93482i	−0.469383	+	1.93482i	−0.327068	+	0.945001i	$0.606061\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$857$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$858$		0			0
$859$	1.98193	−	0.189251i	1.98193	−	0.189251i	0.981929	−	0.189251i	$-0.0606061\pi$
$859$	1.98193	−	0.189251i	1.98193	−	0.189251i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$863$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.723734	+	0.690079i	0.723734	+	0.690079i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1.02951	+	1.18812i	1.02951	+	1.18812i
$872$		0			0
$873$	−0.607279	−	0.243118i	−0.607279	−	0.243118i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.39734	−	0.720381i	1.39734	−	0.720381i	0.415415	−	0.909632i	$-0.363636\pi$
$877$	1.39734	−	0.720381i	1.39734	−	0.720381i	0.981929	+	0.189251i	$0.0606061\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$881$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$882$		0			0
$883$	1.30379	−	0.124497i	1.30379	−	0.124497i	0.580057	−	0.814576i	$-0.303030\pi$
$883$	1.30379	−	0.124497i	1.30379	−	0.124497i	0.723734	+	0.690079i	$0.242424\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$887$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$888$		0			0
$889$	0.900739	−	3.71290i	0.900739	−	3.71290i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−1.24959	+	3.61046i	−1.24959	+	3.61046i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.975950	−	1.37053i	0.975950	−	1.37053i	0.0475819	−	0.998867i	$-0.484848\pi$
$907$	0.975950	−	1.37053i	0.975950	−	1.37053i	0.928368	−	0.371662i	$-0.121212\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$911$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.32254	−	1.04006i	−1.32254	−	1.04006i	−0.995472	−	0.0950560i	$-0.969697\pi$
$919$	−1.32254	−	1.04006i	−1.32254	−	1.04006i	−0.327068	−	0.945001i	$-0.606061\pi$
$920$		0			0
$921$	0.341254	−	0.325385i	0.341254	−	0.325385i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.481929	+	0.676774i	0.481929	+	0.676774i
$926$		0			0
$927$	0.252989	+	0.130425i	0.252989	+	0.130425i
$928$		0			0
$929$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$929$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$930$		0			0
$931$	2.76167	−	2.17180i	2.76167	−	2.17180i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$937$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$938$		0			0
$939$	−0.284630			−0.284630
$940$		0			0
$941$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$941$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$947$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$948$		0			0
$949$	0.132977	+	0.0685542i	0.132977	+	0.0685542i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$953$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.706504	−	2.04131i	−0.706504	−	2.04131i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$967$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$971$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$972$		0			0
$973$	−0.410561	+	1.18624i	−0.410561	+	1.18624i
$974$		0			0
$975$	0.786053	−	1.36148i	0.786053	−	1.36148i
$976$		0			0
$977$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$977$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.0913090	−	0.0268107i	−0.0913090	−	0.0268107i
$982$		0			0
$983$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$983$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$991$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$992$		0			0
$993$	−1.95496	+	0.186677i	−1.95496	+	0.186677i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$997$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$998$		0			0
$999$	−0.738471	+	0.380708i	−0.738471	+	0.380708i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.1.be.a.725.1	yes	20
3.2	odd	2	CM	804.1.be.a.725.1	yes	20
4.3	odd	2		3216.1.dh.a.3137.1		20
12.11	even	2		3216.1.dh.a.3137.1		20
67.39	even	33	inner	804.1.be.a.173.1	✓	20
201.173	odd	66	inner	804.1.be.a.173.1	✓	20
268.39	odd	66		3216.1.dh.a.977.1		20
804.575	even	66		3216.1.dh.a.977.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.1.be.a.173.1	✓	20	67.39	even	33	inner
804.1.be.a.173.1	✓	20	201.173	odd	66	inner
804.1.be.a.725.1	yes	20	1.1	even	1	trivial
804.1.be.a.725.1	yes	20	3.2	odd	2	CM
3216.1.dh.a.977.1		20	268.39	odd	66
3216.1.dh.a.977.1		20	804.575	even	66
3216.1.dh.a.3137.1		20	4.3	odd	2
3216.1.dh.a.3137.1		20	12.11	even	2

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:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	804.be (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.654861 + 0.755750i) q^{3} +(1.50842 - 1.18624i) q^{7} +(-0.142315 - 0.989821i) q^{9} +O(q^{10})\)	\(q+(-0.654861 + 0.755750i) q^{3} +(1.50842 - 1.18624i) q^{7} +(-0.142315 - 0.989821i) q^{9} +(-0.911911 - 1.28060i) q^{13} +(1.02951 + 0.809616i) q^{19} +(-0.0913090 + 1.91681i) q^{21} +(0.415415 - 0.909632i) q^{25} +(0.841254 + 0.540641i) q^{27} +(-1.03115 + 1.44805i) q^{31} +(-0.415415 + 0.719520i) q^{37} +(1.56499 + 0.149438i) q^{39} +(1.91030 + 0.560914i) q^{43} +(0.632425 - 2.60689i) q^{49} +(-1.28605 + 0.247866i) q^{57} +(-1.74555 + 0.899892i) q^{61} +(-1.38884 - 1.32425i) q^{63} +(-0.995472 + 0.0950560i) q^{67} +(-0.0845850 + 0.0436066i) q^{73} +(0.415415 + 0.909632i) q^{75} +(-1.15486 + 0.110276i) q^{79} +(-0.959493 + 0.281733i) q^{81} +(-2.89465 - 0.849945i) q^{91} +(-0.419102 - 1.72756i) q^{93} +(0.327068 - 0.566498i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 20 * q - 2 * q^3 + 2 * q^7 - 2 * q^9	\( 20 q - 2 q^{3} + 2 q^{7} - 2 q^{9} - q^{13} + 2 q^{19} + 2 q^{21} - 2 q^{25} - 2 q^{27} - q^{31} + 2 q^{37} - q^{39} + 2 q^{43} + 3 q^{49} - 9 q^{57} - q^{61} - 9 q^{63} + q^{67} - 12 q^{73} - 2 q^{75} - 12 q^{79} - 2 q^{81} + 4 q^{91} - q^{93} - q^{97}+O(q^{100}) \) 20 * q - 2 * q^3 + 2 * q^7 - 2 * q^9 - q^13 + 2 * q^19 + 2 * q^21 - 2 * q^25 - 2 * q^27 - q^31 + 2 * q^37 - q^39 + 2 * q^43 + 3 * q^49 - 9 * q^57 - q^61 - 9 * q^63 + q^67 - 12 * q^73 - 2 * q^75 - 12 * q^79 - 2 * q^81 + 4 * q^91 - q^93 - q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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