LMFDB - Embedded newform 2144.1.bu.a.495.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2144,1,Mod(47,2144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2144, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 33, 50]))

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

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

chi := DirichletCharacter("2144.47");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2144 = 2^{5} \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2144.bu (of order $66$, degree $20$, not 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:	$1.06999538709$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 536)
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

Embedding invariants

Embedding label			495.1
Root			$0.580057 - 0.814576i$ of defining polynomial
Character	$\chi$	$=$	2144.495
Dual form			2144.1.bu.a.719.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.21769 + 0.782560i) q^{3} +(0.454947 - 0.996196i) q^{9} +O(q^{10})$	$q+(-1.21769 + 0.782560i) q^{3} +(0.454947 - 0.996196i) q^{9} +(1.78153 - 0.713215i) q^{11} +(-0.642315 - 1.85585i) q^{17} +(-1.56499 - 0.149438i) q^{19} +(-0.959493 - 0.281733i) q^{25} +(0.0196034 + 0.136345i) q^{27} +(-1.61121 + 2.26262i) q^{33} +(0.815816 + 0.157236i) q^{41} +(1.21590 - 1.40323i) q^{43} +(0.981929 - 0.189251i) q^{49} +(2.23445 + 1.75719i) q^{51} +(2.02261 - 1.04273i) q^{57} +(0.452418 - 0.132842i) q^{59} +(0.142315 - 0.989821i) q^{67} +(-0.607279 - 0.243118i) q^{73} +(1.38884 - 0.407799i) q^{75} +(0.586611 + 0.676985i) q^{81} +(-0.223734 - 0.175946i) q^{83} +(0.975950 + 0.627205i) q^{89} +(0.327068 - 0.566498i) q^{97} +(0.0999983 - 2.09922i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3}+O(q^{10}) $ 20 * q - 2 * q^3	$ 20 q - 2 q^{3} - 2 q^{11} - 12 q^{17} + q^{19} - 2 q^{25} + 2 q^{27} - 2 q^{33} + 2 q^{41} - 2 q^{43} + q^{49} - q^{51} + q^{57} + 9 q^{59} + 2 q^{67} - q^{73} + 9 q^{75} + 2 q^{81} + 9 q^{83} + 2 q^{89} - q^{97}+O(q^{100}) $ 20 * q - 2 * q^3 - 2 * q^11 - 12 * q^17 + q^19 - 2 * q^25 + 2 * q^27 - 2 * q^33 + 2 * q^41 - 2 * q^43 + q^49 - q^51 + q^57 + 9 * q^59 + 2 * q^67 - q^73 + 9 * q^75 + 2 * q^81 + 9 * q^83 + 2 * q^89 - q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2144\mathbb{Z}\right)^\times$.

$n$	$671$	$805$	$1409$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{10}{33}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.21769	+	0.782560i	−1.21769	+	0.782560i	−0.981929	−	0.189251i	$-0.939394\pi$
$3$	−1.21769	+	0.782560i	−1.21769	+	0.782560i	−0.235759	+	0.971812i	$0.575758\pi$
$4$		0			0
$5$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$5$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$6$		0			0
$7$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$7$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$8$		0			0
$9$	0.454947	−	0.996196i	0.454947	−	0.996196i
$10$		0			0
$11$	1.78153	−	0.713215i	1.78153	−	0.713215i	0.786053	−	0.618159i	$-0.212121\pi$
$11$	1.78153	−	0.713215i	1.78153	−	0.713215i	0.995472	−	0.0950560i	$-0.0303030\pi$
$12$		0			0
$13$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$13$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.642315	−	1.85585i	−0.642315	−	1.85585i	−0.500000	−	0.866025i	$-0.666667\pi$
$17$	−0.642315	−	1.85585i	−0.642315	−	1.85585i	−0.142315	−	0.989821i	$-0.545455\pi$
$18$		0			0
$19$	−1.56499	−	0.149438i	−1.56499	−	0.149438i	−0.723734	−	0.690079i	$-0.757576\pi$
$19$	−1.56499	−	0.149438i	−1.56499	−	0.149438i	−0.841254	+	0.540641i	$0.818182\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$23$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$24$		0			0
$25$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$26$		0			0
$27$	0.0196034	+	0.136345i	0.0196034	+	0.136345i
$28$		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
$31$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$31$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$32$		0			0
$33$	−1.61121	+	2.26262i	−1.61121	+	2.26262i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.815816	+	0.157236i	0.815816	+	0.157236i	0.580057	−	0.814576i	$-0.303030\pi$
$41$	0.815816	+	0.157236i	0.815816	+	0.157236i	0.235759	+	0.971812i	$0.424242\pi$
$42$		0			0
$43$	1.21590	−	1.40323i	1.21590	−	1.40323i	0.327068	−	0.945001i	$-0.393939\pi$
$43$	1.21590	−	1.40323i	1.21590	−	1.40323i	0.888835	−	0.458227i	$-0.151515\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$47$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$48$		0			0
$49$	0.981929	−	0.189251i	0.981929	−	0.189251i
$50$		0			0
$51$	2.23445	+	1.75719i	2.23445	+	1.75719i
$52$		0			0
$53$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$53$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.02261	−	1.04273i	2.02261	−	1.04273i
$58$		0			0
$59$	0.452418	−	0.132842i	0.452418	−	0.132842i	−0.0475819	−	0.998867i	$-0.515152\pi$
$59$	0.452418	−	0.132842i	0.452418	−	0.132842i	0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$61$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.142315	−	0.989821i	0.142315	−	0.989821i
$67$	0.142315	−	0.989821i	0.142315	−	0.989821i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$71$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$72$		0			0
$73$	−0.607279	−	0.243118i	−0.607279	−	0.243118i	0.0475819	−	0.998867i	$-0.484848\pi$
$73$	−0.607279	−	0.243118i	−0.607279	−	0.243118i	−0.654861	+	0.755750i	$0.727273\pi$
$74$		0			0
$75$	1.38884	−	0.407799i	1.38884	−	0.407799i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$79$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$80$		0			0
$81$	0.586611	+	0.676985i	0.586611	+	0.676985i
$82$		0			0
$83$	−0.223734	−	0.175946i	−0.223734	−	0.175946i	0.500000	−	0.866025i	$-0.333333\pi$
$83$	−0.223734	−	0.175946i	−0.223734	−	0.175946i	−0.723734	+	0.690079i	$0.757576\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.975950	+	0.627205i	0.975950	+	0.627205i	0.928368	−	0.371662i	$-0.121212\pi$
$89$	0.975950	+	0.627205i	0.975950	+	0.627205i	0.0475819	+	0.998867i	$0.484848\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		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
$99$	0.0999983	−	2.09922i	0.0999983	−	2.09922i
$100$		0			0
$101$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$101$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$102$		0			0
$103$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$103$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.239446	+	1.66538i	0.239446	+	1.66538i	0.654861	+	0.755750i	$0.272727\pi$
$107$	0.239446	+	1.66538i	0.239446	+	1.66538i	−0.415415	+	0.909632i	$0.636364\pi$
$108$		0			0
$109$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$109$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.370638	+	0.291473i	−0.370638	+	0.291473i	−0.786053	−	0.618159i	$-0.787879\pi$
$113$	−0.370638	+	0.291473i	−0.370638	+	0.291473i	0.415415	+	0.909632i	$0.363636\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.94142	−	1.85114i	1.94142	−	1.85114i
$122$		0			0
$123$	−1.11646	+	0.446961i	−1.11646	+	0.446961i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$127$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$128$		0			0
$129$	−0.382481	+	2.66021i	−0.382481	+	2.66021i
$130$		0			0
$131$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$131$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.49547	+	0.961081i	−1.49547	+	0.961081i	−0.500000	+	0.866025i	$0.666667\pi$
$137$	−1.49547	+	0.961081i	−1.49547	+	0.961081i	−0.995472	+	0.0950560i	$0.969697\pi$
$138$		0			0
$139$	−0.142315	+	0.989821i	−0.142315	+	0.989821i	0.786053	+	0.618159i	$0.212121\pi$
$139$	−0.142315	+	0.989821i	−0.142315	+	0.989821i	−0.928368	+	0.371662i	$0.878788\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.04758	+	0.998867i	−1.04758	+	0.998867i
$148$		0			0
$149$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$149$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$150$		0			0
$151$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$151$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$152$		0			0
$153$	−2.14101	−	0.204441i	−2.14101	−	0.204441i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$157$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.995472	−	1.72421i	−0.995472	−	1.72421i	−0.580057	−	0.814576i	$-0.696970\pi$
$163$	−0.995472	−	1.72421i	−0.995472	−	1.72421i	−0.415415	−	0.909632i	$-0.636364\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$167$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$168$		0			0
$169$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$170$		0			0
$171$	−0.860857	+	1.49105i	−0.860857	+	1.49105i
$172$		0			0
$173$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$173$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.446947	+	0.515804i	−0.446947	+	0.515804i
$178$		0			0
$179$	0.239446	+	0.153882i	0.239446	+	0.153882i	0.654861	−	0.755750i	$-0.272727\pi$
$179$	0.239446	+	0.153882i	0.239446	+	0.153882i	−0.415415	+	0.909632i	$0.636364\pi$
$180$		0			0
$181$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$181$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.46792	−	2.84813i	−2.46792	−	2.84813i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$191$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$192$		0			0
$193$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$193$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$197$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$198$		0			0
$199$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$199$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$200$		0			0
$201$	0.601300	+	1.31666i	0.601300	+	1.31666i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.89465	+	0.849945i	−2.89465	+	0.849945i
$210$		0			0
$211$	1.03115	−	0.531595i	1.03115	−	0.531595i	0.142315	−	0.989821i	$-0.454545\pi$
$211$	1.03115	−	0.531595i	1.03115	−	0.531595i	0.888835	+	0.458227i	$0.151515\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	0.929730	−	0.179191i	0.929730	−	0.179191i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$223$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$224$		0			0
$225$	−0.717180	+	0.827670i	−0.717180	+	0.827670i
$226$		0			0
$227$	1.95496	+	0.376789i	1.95496	+	0.376789i	0.995472	+	0.0950560i	$0.0303030\pi$
$227$	1.95496	+	0.376789i	1.95496	+	0.376789i	0.959493	+	0.281733i	$0.0909091\pi$
$228$		0			0
$229$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$229$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.0475819	+	0.998867i	−0.0475819	+	0.998867i	0.841254	+	0.540641i	$0.181818\pi$
$233$	−0.0475819	+	0.998867i	−0.0475819	+	0.998867i	−0.888835	+	0.458227i	$0.848485\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		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
$241$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$241$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$242$		0			0
$243$	−1.37626	−	0.404106i	−1.37626	−	0.404106i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.410127	+	0.0391624i	0.410127	+	0.0391624i
$250$		0			0
$251$	−0.651174	−	1.88144i	−0.651174	−	1.88144i	−0.415415	−	0.909632i	$-0.636364\pi$
$251$	−0.651174	−	1.88144i	−0.651174	−	1.88144i	−0.235759	−	0.971812i	$-0.575758\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.21590	+	0.486774i	−1.21590	+	0.486774i	−0.888835	−	0.458227i	$-0.848485\pi$
$257$	−1.21590	+	0.486774i	−1.21590	+	0.486774i	−0.327068	+	0.945001i	$0.606061\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$263$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.67923			−1.67923
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$271$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.91030	+	0.182411i	−1.91030	+	0.182411i
$276$		0			0
$277$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$277$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.42131	−	1.35522i	1.42131	−	1.35522i	0.580057	−	0.814576i	$-0.303030\pi$
$281$	1.42131	−	1.35522i	1.42131	−	1.35522i	0.841254	−	0.540641i	$-0.181818\pi$
$282$		0			0
$283$	0.415415	+	0.909632i	0.415415	+	0.909632i	0.995472	+	0.0950560i	$0.0303030\pi$
$283$	0.415415	+	0.909632i	0.415415	+	0.909632i	−0.580057	+	0.814576i	$0.696970\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.24555	+	1.76592i	−2.24555	+	1.76592i
$290$		0			0
$291$	0.0450525	+	0.945768i	0.0450525	+	0.945768i
$292$		0			0
$293$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$293$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	0.132167	+	0.228920i	0.132167	+	0.228920i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.437742	+	1.80440i	−0.437742	+	1.80440i	0.142315	+	0.989821i	$0.454545\pi$
$307$	−0.437742	+	1.80440i	−0.437742	+	1.80440i	−0.580057	+	0.814576i	$0.696970\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$311$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$312$		0			0
$313$	−1.32254	−	0.849945i	−1.32254	−	0.849945i	−0.327068	−	0.945001i	$-0.606061\pi$
$313$	−1.32254	−	0.849945i	−1.32254	−	0.849945i	−0.995472	+	0.0950560i	$0.969697\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$317$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−1.59483	−	1.84053i	−1.59483	−	1.84053i
$322$		0			0
$323$	0.727880	+	3.00036i	0.727880	+	3.00036i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.514186	+	1.48564i	−0.514186	+	1.48564i	0.327068	+	0.945001i	$0.393939\pi$
$331$	−0.514186	+	1.48564i	−0.514186	+	1.48564i	−0.841254	+	0.540641i	$0.818182\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.759713	−	1.06687i	−0.759713	−	1.06687i	−0.995472	−	0.0950560i	$-0.969697\pi$
$337$	−0.759713	−	1.06687i	−0.759713	−	1.06687i	0.235759	−	0.971812i	$-0.424242\pi$
$338$		0			0
$339$	0.223226	−	0.644970i	0.223226	−	0.644970i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.235759	+	0.971812i	0.235759	+	0.971812i	0.959493	+	0.281733i	$0.0909091\pi$
$347$	0.235759	+	0.971812i	0.235759	+	0.971812i	−0.723734	+	0.690079i	$0.757576\pi$
$348$		0			0
$349$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$349$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.65210	−	0.318417i	1.65210	−	0.318417i	0.723734	−	0.690079i	$-0.242424\pi$
$353$	1.65210	−	0.318417i	1.65210	−	0.318417i	0.928368	+	0.371662i	$0.121212\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$359$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$360$		0			0
$361$	1.44493	+	0.278487i	1.44493	+	0.278487i
$362$		0			0
$363$	−0.915415	+	3.77339i	−0.915415	+	3.77339i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$367$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$368$		0			0
$369$	0.527791	−	0.741179i	0.527791	−	0.741179i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0395325	−	0.829889i	−0.0395325	−	0.829889i	−0.928368	−	0.371662i	$-0.878788\pi$
$379$	−0.0395325	−	0.829889i	−0.0395325	−	0.829889i	0.888835	−	0.458227i	$-0.151515\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$383$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.844717	−	1.84967i	−0.844717	−	1.84967i
$388$		0			0
$389$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$389$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	0.499578	−	1.09392i	0.499578	−	1.09392i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$397$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$401$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.67489	+	0.159932i	−1.67489	+	0.159932i	−0.888835	−	0.458227i	$-0.848485\pi$
$409$	−1.67489	+	0.159932i	−1.67489	+	0.159932i	−0.786053	+	0.618159i	$0.787879\pi$
$410$		0			0
$411$	1.06891	−	2.34059i	1.06891	−	2.34059i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.601300	−	1.31666i	−0.601300	−	1.31666i
$418$		0			0
$419$	−0.213947	−	0.618159i	−0.213947	−	0.618159i	0.786053	−	0.618159i	$-0.212121\pi$
$419$	−0.213947	−	0.618159i	−0.213947	−	0.618159i	−1.00000			$\pi$
$420$		0			0
$421$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$421$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.0934441	+	1.96163i	0.0934441	+	1.96163i
$426$		0			0
$427$		0			0
$428$		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
$433$	−1.28656	−	1.22673i	−1.28656	−	1.22673i	−0.959493	−	0.281733i	$-0.909091\pi$
$433$	−1.28656	−	1.22673i	−1.28656	−	1.22673i	−0.327068	−	0.945001i	$-0.606061\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	0.258195	−	1.06429i	0.258195	−	1.06429i
$442$		0			0
$443$	−1.92837	−	0.371662i	−1.92837	−	0.371662i	−0.928368	−	0.371662i	$-0.878788\pi$
$443$	−1.92837	−	0.371662i	−1.92837	−	0.371662i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.70566	+	0.879330i	1.70566	+	0.879330i	0.981929	+	0.189251i	$0.0606061\pi$
$449$	1.70566	+	0.879330i	1.70566	+	0.879330i	0.723734	+	0.690079i	$0.242424\pi$
$450$		0			0
$451$	1.56554	−	0.301733i	1.56554	−	0.301733i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.273507	+	1.12741i	0.273507	+	1.12741i	0.928368	+	0.371662i	$0.121212\pi$
$457$	0.273507	+	1.12741i	0.273507	+	1.12741i	−0.654861	+	0.755750i	$0.727273\pi$
$458$		0			0
$459$	0.240443	−	0.123957i	0.240443	−	0.123957i
$460$		0			0
$461$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$461$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$462$		0			0
$463$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$463$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.11312	+	1.56316i	1.11312	+	1.56316i	0.786053	+	0.618159i	$0.212121\pi$
$467$	1.11312	+	1.56316i	1.11312	+	1.56316i	0.327068	+	0.945001i	$0.393939\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.16536	−	3.36709i	1.16536	−	3.36709i
$474$		0			0
$475$	1.45949	+	0.584293i	1.45949	+	0.584293i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$479$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$487$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$488$		0			0
$489$	2.56147	+	1.32053i	2.56147	+	1.32053i
$490$		0			0
$491$	0.841254	+	0.540641i	0.841254	+	0.540641i	0.888835	−	0.458227i	$-0.151515\pi$
$491$	0.841254	+	0.540641i	0.841254	+	0.540641i	−0.0475819	+	0.998867i	$0.515152\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.0475819	−	0.0824143i	0.0475819	−	0.0824143i	−0.841254	−	0.540641i	$-0.818182\pi$
$499$	0.0475819	−	0.0824143i	0.0475819	−	0.0824143i	0.888835	+	0.458227i	$0.151515\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$503$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.723734	+	1.25354i	0.723734	+	1.25354i
$508$		0			0
$509$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$509$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−0.0103040	−	0.216307i	−0.0103040	−	0.216307i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i	0.928368	−	0.371662i	$-0.121212\pi$
$521$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i	−0.888835	+	0.458227i	$0.848485\pi$
$522$		0			0
$523$	1.28656	−	1.22673i	1.28656	−	1.22673i	0.327068	−	0.945001i	$-0.393939\pi$
$523$	1.28656	−	1.22673i	1.28656	−	1.22673i	0.959493	−	0.281733i	$-0.0909091\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$530$		0			0
$531$	0.0734899	−	0.511133i	0.0734899	−	0.511133i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.411992			−0.411992
$538$		0			0
$539$	1.61435	−	1.03748i	1.61435	−	1.03748i
$540$		0			0
$541$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$541$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.264241	−	0.105786i	0.264241	−	0.105786i	−0.235759	−	0.971812i	$-0.575758\pi$
$547$	0.264241	−	0.105786i	0.264241	−	0.105786i	0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$557$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	5.23399	+	1.53684i	5.23399	+	1.53684i
$562$		0			0
$563$	0.0135432	+	0.0941952i	0.0135432	+	0.0941952i	0.995472	−	0.0950560i	$-0.0303030\pi$
$563$	0.0135432	+	0.0941952i	0.0135432	+	0.0941952i	−0.981929	+	0.189251i	$0.939394\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.379436	+	0.532843i	−0.379436	+	0.532843i	−0.959493	−	0.281733i	$-0.909091\pi$
$569$	−0.379436	+	0.532843i	−0.379436	+	0.532843i	0.580057	+	0.814576i	$0.303030\pi$
$570$		0			0
$571$	−0.0934441	+	1.96163i	−0.0934441	+	1.96163i	0.142315	+	0.989821i	$0.454545\pi$
$571$	−0.0934441	+	1.96163i	−0.0934441	+	1.96163i	−0.235759	+	0.971812i	$0.575758\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.13915	+	0.219553i	1.13915	+	0.219553i	0.723734	−	0.690079i	$-0.242424\pi$
$577$	1.13915	+	0.219553i	1.13915	+	0.219553i	0.415415	+	0.909632i	$0.363636\pi$
$578$		0			0
$579$	−1.24147	+	1.43273i	−1.24147	+	1.43273i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.514186	−	0.404360i	−0.514186	−	0.404360i	0.327068	−	0.945001i	$-0.393939\pi$
$587$	−0.514186	−	0.404360i	−0.514186	−	0.404360i	−0.841254	+	0.540641i	$0.818182\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.581419	−	0.299742i	0.581419	−	0.299742i	−0.142315	−	0.989821i	$-0.545455\pi$
$593$	0.581419	−	0.299742i	0.581419	−	0.299742i	0.723734	+	0.690079i	$0.242424\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$599$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$600$		0			0
$601$	−1.15486	−	1.62177i	−1.15486	−	1.62177i	−0.654861	−	0.755750i	$-0.727273\pi$
$601$	−1.15486	−	1.62177i	−1.15486	−	1.62177i	−0.500000	−	0.866025i	$-0.666667\pi$
$602$		0			0
$603$	−0.921310	−	0.592090i	−0.921310	−	0.592090i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$607$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$613$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$617$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$618$		0			0
$619$	−1.39734	−	1.09888i	−1.39734	−	1.09888i	−0.981929	−	0.189251i	$-0.939394\pi$
$619$	−1.39734	−	1.09888i	−1.39734	−	1.09888i	−0.415415	−	0.909632i	$-0.636364\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.841254	+	0.540641i	0.841254	+	0.540641i
$626$		0			0
$627$	2.85964	−	3.30020i	2.85964	−	3.30020i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$631$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$632$		0			0
$633$	−0.839614	+	1.45425i	−0.839614	+	1.45425i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$641$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$642$		0			0
$643$	0.0671040	+	0.466718i	0.0671040	+	0.466718i	0.995472	+	0.0950560i	$0.0303030\pi$
$643$	0.0671040	+	0.466718i	0.0671040	+	0.466718i	−0.928368	+	0.371662i	$0.878788\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$647$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$648$		0			0
$649$	0.711249	−	0.559333i	0.711249	−	0.559333i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$653$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.518473	+	0.494363i	−0.518473	+	0.494363i
$658$		0			0
$659$	0.928368	−	0.371662i	0.928368	−	0.371662i	0.142315	−	0.989821i	$-0.454545\pi$
$659$	0.928368	−	0.371662i	0.928368	−	0.371662i	0.786053	+	0.618159i	$0.212121\pi$
$660$		0			0
$661$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$661$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i	0.580057	+	0.814576i	$0.303030\pi$
$674$		0			0
$675$	0.0196034	−	0.136345i	0.0196034	−	0.136345i
$676$		0			0
$677$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$677$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−2.67540	+	1.07107i	−2.67540	+	1.07107i
$682$		0			0
$683$	1.28656	−	1.22673i	1.28656	−	1.22673i	0.327068	−	0.945001i	$-0.393939\pi$
$683$	1.28656	−	1.22673i	1.28656	−	1.22673i	0.959493	−	0.281733i	$-0.0909091\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.32254	−	1.04006i	1.32254	−	1.04006i	0.327068	−	0.945001i	$-0.393939\pi$
$691$	1.32254	−	1.04006i	1.32254	−	1.04006i	0.995472	−	0.0950560i	$-0.0303030\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.232205	−	1.61502i	−0.232205	−	1.61502i
$698$		0			0
$699$	−0.723734	−	1.25354i	−0.723734	−	1.25354i
$700$		0			0
$701$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$701$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$709$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$719$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−0.269798	−	0.311363i	−0.269798	−	0.311363i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$727$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$728$		0			0
$729$	1.13259	−	0.332560i	1.13259	−	0.332560i
$730$		0			0
$731$	−3.38517	−	1.35522i	−3.38517	−	1.35522i
$732$		0			0
$733$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$733$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.452418	−	1.86489i	−0.452418	−	1.86489i
$738$		0			0
$739$	−0.273507	−	0.384087i	−0.273507	−	0.384087i	0.654861	−	0.755750i	$-0.272727\pi$
$739$	−0.273507	−	0.384087i	−0.273507	−	0.384087i	−0.928368	+	0.371662i	$0.878788\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$743$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.277064	+	0.142837i	−0.277064	+	0.142837i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$751$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$752$		0			0
$753$	2.26527	+	1.78143i	2.26527	+	1.78143i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$757$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$761$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.959493	+	0.281733i	$0.909091\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.0947329	+	1.98869i	−0.0947329	+	1.98869i	0.0475819	+	0.998867i	$0.484848\pi$
$769$	−0.0947329	+	1.98869i	−0.0947329	+	1.98869i	−0.142315	+	0.989821i	$0.545455\pi$
$770$		0			0
$771$	1.09966	−	1.54426i	1.09966	−	1.54426i
$772$		0			0
$773$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$773$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.25324	−	0.367986i	−1.25324	−	0.367986i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.550294	+	1.58997i	0.550294	+	1.58997i	0.786053	+	0.618159i	$0.212121\pi$
$787$	0.550294	+	1.58997i	0.550294	+	1.58997i	−0.235759	+	0.971812i	$0.575758\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$797$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	1.06882	−	0.686892i	1.06882	−	0.686892i
$802$		0			0
$803$	−1.25528			−1.25528
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$809$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$810$		0			0
$811$	−0.283341	+	0.0270558i	−0.283341	+	0.0270558i	−0.235759	−	0.971812i	$-0.575758\pi$
$811$	−0.283341	+	0.0270558i	−0.283341	+	0.0270558i	−0.0475819	+	0.998867i	$0.515152\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.11257	+	2.01433i	−2.11257	+	2.01433i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$821$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$822$		0			0
$823$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$823$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$824$		0			0
$825$	2.18340	−	1.71704i	2.18340	−	1.71704i
$826$		0			0
$827$	0.0913090	+	1.91681i	0.0913090	+	1.91681i	0.327068	+	0.945001i	$0.393939\pi$
$827$	0.0913090	+	1.91681i	0.0913090	+	1.91681i	−0.235759	+	0.971812i	$0.575758\pi$
$828$		0			0
$829$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$829$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.981929	−	1.70075i	−0.981929	−	1.70075i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$839$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$840$		0			0
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$		0			0
$843$	−0.670173	+	2.76249i	−0.670173	+	2.76249i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.21769	−	0.782560i	−1.21769	−	0.782560i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$853$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$857$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$858$		0			0
$859$	−0.462997	−	1.90850i	−0.462997	−	1.90850i	−0.415415	−	0.909632i	$-0.636364\pi$
$859$	−0.462997	−	1.90850i	−0.462997	−	1.90850i	−0.0475819	−	0.998867i	$-0.515152\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$863$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.35244	−	3.90761i	1.35244	−	3.90761i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.415545	−	0.583551i	−0.415545	−	0.583551i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$877$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.70566	−	0.879330i	1.70566	−	0.879330i	0.723734	−	0.690079i	$-0.242424\pi$
$881$	1.70566	−	0.879330i	1.70566	−	0.879330i	0.981929	−	0.189251i	$-0.0606061\pi$
$882$		0			0
$883$	−0.195876	−	0.807410i	−0.195876	−	0.807410i	−0.981929	−	0.189251i	$-0.939394\pi$
$883$	−0.195876	−	0.807410i	−0.195876	−	0.807410i	0.786053	−	0.618159i	$-0.212121\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$887$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.52790	+	0.787686i	1.52790	+	0.787686i
$892$		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
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.947890	+	0.903811i	0.947890	+	0.903811i	0.995472	−	0.0950560i	$-0.0303030\pi$
$907$	0.947890	+	0.903811i	0.947890	+	0.903811i	−0.0475819	+	0.998867i	$0.515152\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$911$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$912$		0			0
$913$	−0.524075	−	0.153882i	−0.524075	−	0.153882i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$919$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$920$		0			0
$921$	−0.879017	−	2.53975i	−0.879017	−	2.53975i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	−0.888835	−	0.458227i	$-0.848485\pi$
$929$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	0.928368	+	0.371662i	$0.121212\pi$
$930$		0			0
$931$	−1.56499	+	0.149438i	−1.56499	+	0.149438i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$937$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$938$		0			0
$939$	2.27557			2.27557
$940$		0			0
$941$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$941$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.481929	+	1.05528i	−0.481929	+	1.05528i	0.500000	+	0.866025i	$0.333333\pi$
$947$	−0.481929	+	1.05528i	−0.481929	+	1.05528i	−0.981929	+	0.189251i	$0.939394\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.771316	+	1.68895i	0.771316	+	1.68895i	0.723734	+	0.690079i	$0.242424\pi$
$953$	0.771316	+	1.68895i	0.771316	+	1.68895i	0.0475819	+	0.998867i	$0.484848\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$962$		0			0
$963$	1.76798	+	0.519126i	1.76798	+	0.519126i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$	−3.23430	−	3.08390i	−3.23430	−	3.08390i
$970$		0			0
$971$	−1.07701	+	1.51245i	−1.07701	+	1.51245i	−0.235759	+	0.971812i	$0.575758\pi$
$971$	−1.07701	+	1.51245i	−1.07701	+	1.51245i	−0.841254	+	0.540641i	$0.818182\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.0671040	+	0.276606i	−0.0671040	+	0.276606i	−0.995472	−	0.0950560i	$-0.969697\pi$
$977$	−0.0671040	+	0.276606i	−0.0671040	+	0.276606i	0.928368	+	0.371662i	$0.121212\pi$
$978$		0			0
$979$	2.18601	+	0.421319i	2.18601	+	0.421319i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$983$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$991$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$992$		0			0
$993$	−0.536487	−	2.21143i	−0.536487	−	2.21143i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$997$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2144.1.bu.a.495.1		20
4.3	odd	2		536.1.ba.a.227.1	✓	20
8.3	odd	2	CM	2144.1.bu.a.495.1		20
8.5	even	2		536.1.ba.a.227.1	✓	20
67.49	even	33	inner	2144.1.bu.a.719.1		20
268.183	odd	66		536.1.ba.a.451.1	yes	20
536.317	even	66		536.1.ba.a.451.1	yes	20
536.451	odd	66	inner	2144.1.bu.a.719.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.1.ba.a.227.1	✓	20	4.3	odd	2
536.1.ba.a.227.1	✓	20	8.5	even	2
536.1.ba.a.451.1	yes	20	268.183	odd	66
536.1.ba.a.451.1	yes	20	536.317	even	66
2144.1.bu.a.495.1		20	1.1	even	1	trivial
2144.1.bu.a.495.1		20	8.3	odd	2	CM
2144.1.bu.a.719.1		20	67.49	even	33	inner
2144.1.bu.a.719.1		20	536.451	odd	66	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:	\( N \)	\(=\)	\( 2144 = 2^{5} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2144.bu (of order \(66\), degree \(20\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.21769 + 0.782560i) q^{3} +(0.454947 - 0.996196i) q^{9} +O(q^{10})\)	\(q+(-1.21769 + 0.782560i) q^{3} +(0.454947 - 0.996196i) q^{9} +(1.78153 - 0.713215i) q^{11} +(-0.642315 - 1.85585i) q^{17} +(-1.56499 - 0.149438i) q^{19} +(-0.959493 - 0.281733i) q^{25} +(0.0196034 + 0.136345i) q^{27} +(-1.61121 + 2.26262i) q^{33} +(0.815816 + 0.157236i) q^{41} +(1.21590 - 1.40323i) q^{43} +(0.981929 - 0.189251i) q^{49} +(2.23445 + 1.75719i) q^{51} +(2.02261 - 1.04273i) q^{57} +(0.452418 - 0.132842i) q^{59} +(0.142315 - 0.989821i) q^{67} +(-0.607279 - 0.243118i) q^{73} +(1.38884 - 0.407799i) q^{75} +(0.586611 + 0.676985i) q^{81} +(-0.223734 - 0.175946i) q^{83} +(0.975950 + 0.627205i) q^{89} +(0.327068 - 0.566498i) q^{97} +(0.0999983 - 2.09922i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3}+O(q^{10}) \) 20 * q - 2 * q^3	\( 20 q - 2 q^{3} - 2 q^{11} - 12 q^{17} + q^{19} - 2 q^{25} + 2 q^{27} - 2 q^{33} + 2 q^{41} - 2 q^{43} + q^{49} - q^{51} + q^{57} + 9 q^{59} + 2 q^{67} - q^{73} + 9 q^{75} + 2 q^{81} + 9 q^{83} + 2 q^{89} - q^{97}+O(q^{100}) \) 20 * q - 2 * q^3 - 2 * q^11 - 12 * q^17 + q^19 - 2 * q^25 + 2 * q^27 - 2 * q^33 + 2 * q^41 - 2 * q^43 + q^49 - q^51 + q^57 + 9 * q^59 + 2 * q^67 - q^73 + 9 * q^75 + 2 * q^81 + 9 * q^83 + 2 * q^89 - q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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