LMFDB - Embedded newform 2144.1.bu.a.1807.1

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			1807.1
Root			$0.0475819 + 0.998867i$ of defining polynomial
Character	$\chi$	$=$	2144.1807
Dual form			2144.1.bu.a.1775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50842 - 0.442913i) q^{3} +(1.23792 + 0.795563i) q^{9} +O(q^{10})$	$q+(-1.50842 - 0.442913i) q^{3} +(1.23792 + 0.795563i) q^{9} +(-0.0930932 - 0.268975i) q^{11} +(-1.15486 + 0.110276i) q^{17} +(1.74555 + 0.899892i) q^{19} +(-0.142315 - 0.989821i) q^{25} +(-0.485433 - 0.560219i) q^{27} +(0.0212914 + 0.446961i) q^{33} +(0.975950 + 1.37053i) q^{41} +(0.271738 - 0.595023i) q^{43} +(0.580057 - 0.814576i) q^{49} +(1.79086 + 0.345161i) q^{51} +(-2.23445 - 2.13054i) q^{57} +(0.264241 - 1.83784i) q^{59} +(0.654861 - 0.755750i) q^{67} +(0.651174 - 1.88144i) q^{73} +(-0.223734 + 1.55610i) q^{75} +(-0.127181 - 0.278487i) q^{81} +(1.28605 + 0.247866i) q^{83} +(-0.0913090 + 0.0268107i) q^{89} +(0.995472 + 1.72421i) q^{97} +(0.0987447 - 0.407031i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2144\mathbb{Z}\right)^\times$.

$n$	$671$	$805$	$1409$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{17}{33}\right)$

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
$3$	−1.50842	−	0.442913i	−1.50842	−	0.442913i	−0.580057	−	0.814576i	$-0.696970\pi$
$3$	−1.50842	−	0.442913i	−1.50842	−	0.442913i	−0.928368	+	0.371662i	$0.878788\pi$
$4$		0			0
$5$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$5$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$6$		0			0
$7$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$7$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$8$		0			0
$9$	1.23792	+	0.795563i	1.23792	+	0.795563i
$10$		0			0
$11$	−0.0930932	−	0.268975i	−0.0930932	−	0.268975i	0.888835	−	0.458227i	$-0.151515\pi$
$11$	−0.0930932	−	0.268975i	−0.0930932	−	0.268975i	−0.981929	+	0.189251i	$0.939394\pi$
$12$		0			0
$13$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$13$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.15486	+	0.110276i	−1.15486	+	0.110276i	−0.654861	−	0.755750i	$-0.727273\pi$
$17$	−1.15486	+	0.110276i	−1.15486	+	0.110276i	−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	1.74555	+	0.899892i	1.74555	+	0.899892i	0.959493	+	0.281733i	$0.0909091\pi$
$19$	1.74555	+	0.899892i	1.74555	+	0.899892i	0.786053	+	0.618159i	$0.212121\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$23$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$24$		0			0
$25$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$26$		0			0
$27$	−0.485433	−	0.560219i	−0.485433	−	0.560219i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$31$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$32$		0			0
$33$	0.0212914	+	0.446961i	0.0212914	+	0.446961i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.975950	+	1.37053i	0.975950	+	1.37053i	0.928368	+	0.371662i	$0.121212\pi$
$41$	0.975950	+	1.37053i	0.975950	+	1.37053i	0.0475819	+	0.998867i	$0.484848\pi$
$42$		0			0
$43$	0.271738	−	0.595023i	0.271738	−	0.595023i	−0.723734	−	0.690079i	$-0.757576\pi$
$43$	0.271738	−	0.595023i	0.271738	−	0.595023i	0.995472	+	0.0950560i	$0.0303030\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$47$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$48$		0			0
$49$	0.580057	−	0.814576i	0.580057	−	0.814576i
$50$		0			0
$51$	1.79086	+	0.345161i	1.79086	+	0.345161i
$52$		0			0
$53$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$53$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.23445	−	2.13054i	−2.23445	−	2.13054i
$58$		0			0
$59$	0.264241	−	1.83784i	0.264241	−	1.83784i	−0.235759	−	0.971812i	$-0.575758\pi$
$59$	0.264241	−	1.83784i	0.264241	−	1.83784i	0.500000	−	0.866025i	$-0.333333\pi$
$60$		0			0
$61$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$61$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.654861	−	0.755750i	0.654861	−	0.755750i
$67$	0.654861	−	0.755750i	0.654861	−	0.755750i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$71$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$72$		0			0
$73$	0.651174	−	1.88144i	0.651174	−	1.88144i	0.235759	−	0.971812i	$-0.424242\pi$
$73$	0.651174	−	1.88144i	0.651174	−	1.88144i	0.415415	−	0.909632i	$-0.363636\pi$
$74$		0			0
$75$	−0.223734	+	1.55610i	−0.223734	+	1.55610i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$79$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$80$		0			0
$81$	−0.127181	−	0.278487i	−0.127181	−	0.278487i
$82$		0			0
$83$	1.28605	+	0.247866i	1.28605	+	0.247866i	0.786053	−	0.618159i	$-0.212121\pi$
$83$	1.28605	+	0.247866i	1.28605	+	0.247866i	0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.0913090	+	0.0268107i	−0.0913090	+	0.0268107i	−0.327068	−	0.945001i	$-0.606061\pi$
$89$	−0.0913090	+	0.0268107i	−0.0913090	+	0.0268107i	0.235759	+	0.971812i	$0.424242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$97$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$98$		0			0
$99$	0.0987447	−	0.407031i	0.0987447	−	0.407031i
$100$		0			0
$101$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$101$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$102$		0			0
$103$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$103$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.25667	−	1.45027i	−1.25667	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$107$	−1.25667	−	1.45027i	−1.25667	−	1.45027i	−0.415415	−	0.909632i	$-0.636364\pi$
$108$		0			0
$109$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$109$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.82318	−	0.351390i	1.82318	−	0.351390i	0.841254	−	0.540641i	$-0.181818\pi$
$113$	1.82318	−	0.351390i	1.82318	−	0.351390i	0.981929	+	0.189251i	$0.0606061\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.722372	−	0.568079i	0.722372	−	0.568079i
$122$		0			0
$123$	−0.865121	−	2.49960i	−0.865121	−	2.49960i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$127$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$128$		0			0
$129$	−0.673440	+	0.777191i	−0.673440	+	0.777191i
$130$		0			0
$131$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$131$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.38884	−	0.407799i	−1.38884	−	0.407799i	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−1.38884	−	0.407799i	−1.38884	−	0.407799i	−0.888835	+	0.458227i	$0.848485\pi$
$138$		0			0
$139$	−0.654861	+	0.755750i	−0.654861	+	0.755750i	−0.981929	−	0.189251i	$-0.939394\pi$
$139$	−0.654861	+	0.755750i	−0.654861	+	0.755750i	0.327068	+	0.945001i	$0.393939\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.23576	+	0.971812i	−1.23576	+	0.971812i
$148$		0			0
$149$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$149$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$150$		0			0
$151$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$151$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$152$		0			0
$153$	−1.51736	−	0.782251i	−1.51736	−	0.782251i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$157$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.888835	+	1.53951i	−0.888835	+	1.53951i	−0.0475819	+	0.998867i	$0.515152\pi$
$163$	−0.888835	+	1.53951i	−0.888835	+	1.53951i	−0.841254	+	0.540641i	$0.818182\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$167$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$168$		0			0
$169$	0.235759	−	0.971812i	0.235759	−	0.971812i
$170$		0			0
$171$	1.44493	+	2.50268i	1.44493	+	2.50268i
$172$		0			0
$173$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$173$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.21259	+	2.65520i	−1.21259	+	2.65520i
$178$		0			0
$179$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.841254	−	0.540641i	$-0.818182\pi$
$179$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.415415	+	0.909632i	$0.636364\pi$
$180$		0			0
$181$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$181$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.137171	+	0.300363i	0.137171	+	0.300363i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$191$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$192$		0			0
$193$	−0.118239	+	0.822373i	−0.118239	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$193$	−0.118239	+	0.822373i	−0.118239	+	0.822373i	−0.959493	+	0.281733i	$0.909091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$197$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$198$		0			0
$199$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$199$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$200$		0			0
$201$	−1.32254	+	0.849945i	−1.32254	+	0.849945i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.0795500	−	0.553283i	0.0795500	−	0.553283i
$210$		0			0
$211$	−0.0688733	−	0.0656706i	−0.0688733	−	0.0656706i	0.654861	−	0.755750i	$-0.272727\pi$
$211$	−0.0688733	−	0.0656706i	−0.0688733	−	0.0656706i	−0.723734	+	0.690079i	$0.757576\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−1.81556	+	2.54960i	−1.81556	+	2.54960i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$223$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$224$		0			0
$225$	0.611291	−	1.33854i	0.611291	−	1.33854i
$226$		0			0
$227$	1.03115	+	1.44805i	1.03115	+	1.44805i	0.888835	+	0.458227i	$0.151515\pi$
$227$	1.03115	+	1.44805i	1.03115	+	1.44805i	0.142315	+	0.989821i	$0.454545\pi$
$228$		0			0
$229$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$229$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.235759	+	0.971812i	−0.235759	+	0.971812i	0.723734	+	0.690079i	$0.242424\pi$
$233$	−0.235759	+	0.971812i	−0.235759	+	0.971812i	−0.959493	+	0.281733i	$0.909091\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$241$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$242$		0			0
$243$	0.173991	+	1.21014i	0.173991	+	1.21014i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−1.83013	−	0.943498i	−1.83013	−	0.943498i
$250$		0			0
$251$	−1.76962	+	0.168978i	−1.76962	+	0.168978i	−0.928368	−	0.371662i	$-0.878788\pi$
$251$	−1.76962	+	0.168978i	−1.76962	+	0.168978i	−0.841254	+	0.540641i	$0.818182\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.271738	−	0.785135i	−0.271738	−	0.785135i	−0.995472	−	0.0950560i	$-0.969697\pi$
$257$	−0.271738	−	0.785135i	−0.271738	−	0.785135i	0.723734	−	0.690079i	$-0.242424\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$263$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.149608			0.149608
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$271$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.252989	+	0.130425i	−0.252989	+	0.130425i
$276$		0			0
$277$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$277$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.911911	+	0.717135i	−0.911911	+	0.717135i	−0.959493	−	0.281733i	$-0.909091\pi$
$281$	−0.911911	+	0.717135i	−0.911911	+	0.717135i	0.0475819	+	0.998867i	$0.484848\pi$
$282$		0			0
$283$	0.841254	−	0.540641i	0.841254	−	0.540641i	−0.0475819	−	0.998867i	$-0.515152\pi$
$283$	0.841254	−	0.540641i	0.841254	−	0.540641i	0.888835	+	0.458227i	$0.151515\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.339614	−	0.0654552i	0.339614	−	0.0654552i
$290$		0			0
$291$	−0.737920	−	3.04175i	−0.737920	−	3.04175i
$292$		0			0
$293$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$293$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.105495	+	0.182722i	−0.105495	+	0.182722i
$298$		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
$307$	0.607279	−	0.243118i	0.607279	−	0.243118i	−0.0475819	−	0.998867i	$-0.515152\pi$
$307$	0.607279	−	0.243118i	0.607279	−	0.243118i	0.654861	+	0.755750i	$0.272727\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$311$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$312$		0			0
$313$	−1.88431	+	0.553283i	−1.88431	+	0.553283i	−0.888835	+	0.458227i	$0.848485\pi$
$313$	−1.88431	+	0.553283i	−1.88431	+	0.553283i	−0.995472	+	0.0950560i	$0.969697\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$317$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	1.25324	+	2.74422i	1.25324	+	2.74422i
$322$		0			0
$323$	−2.11510	−	0.846758i	−2.11510	−	0.846758i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.95496	+	0.186677i	1.95496	+	0.186677i	0.995472	−	0.0950560i	$-0.0303030\pi$
$331$	1.95496	+	0.186677i	1.95496	+	0.186677i	0.959493	+	0.281733i	$0.0909091\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.0395325	−	0.829889i	0.0395325	−	0.829889i	−0.888835	−	0.458227i	$-0.848485\pi$
$337$	0.0395325	−	0.829889i	0.0395325	−	0.829889i	0.928368	−	0.371662i	$-0.121212\pi$
$338$		0			0
$339$	−2.90577	−	0.277467i	−2.90577	−	0.277467i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.928368	+	0.371662i	0.928368	+	0.371662i	0.786053	−	0.618159i	$-0.212121\pi$
$347$	0.928368	+	0.371662i	0.928368	+	0.371662i	0.142315	+	0.989821i	$0.454545\pi$
$348$		0			0
$349$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$349$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.11312	+	1.56316i	−1.11312	+	1.56316i	−0.327068	+	0.945001i	$0.606061\pi$
$353$	−1.11312	+	1.56316i	−1.11312	+	1.56316i	−0.786053	+	0.618159i	$0.787879\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$359$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$360$		0			0
$361$	1.65707	+	2.32703i	1.65707	+	2.32703i
$362$		0			0
$363$	−1.34125	+	0.536957i	−1.34125	+	0.536957i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$367$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$368$		0			0
$369$	0.117805	+	2.47303i	0.117805	+	2.47303i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.396666	−	1.63508i	−0.396666	−	1.63508i	−0.723734	−	0.690079i	$-0.757576\pi$
$379$	−0.396666	−	1.63508i	−0.396666	−	1.63508i	0.327068	−	0.945001i	$-0.393939\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$383$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.809768	−	0.520406i	0.809768	−	0.520406i
$388$		0			0
$389$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$389$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−2.22518	−	1.43004i	−2.22518	−	1.43004i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$397$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$401$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.70566	−	0.879330i	1.70566	−	0.879330i	0.723734	−	0.690079i	$-0.242424\pi$
$409$	1.70566	−	0.879330i	1.70566	−	0.879330i	0.981929	−	0.189251i	$-0.0606061\pi$
$410$		0			0
$411$	1.91433	+	1.23027i	1.91433	+	1.23027i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.32254	−	0.849945i	1.32254	−	0.849945i
$418$		0			0
$419$	−1.98193	+	0.189251i	−1.98193	+	0.189251i	−0.981929	+	0.189251i	$0.939394\pi$
$419$	−1.98193	+	0.189251i	−1.98193	+	0.189251i	−1.00000			$\pi$
$420$		0			0
$421$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$421$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.273507	+	1.12741i	0.273507	+	1.12741i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−1.13779	−	0.894765i	−1.13779	−	0.894765i	−0.142315	−	0.989821i	$-0.545455\pi$
$433$	−1.13779	−	0.894765i	−1.13779	−	0.894765i	−0.995472	+	0.0950560i	$0.969697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	1.36611	−	0.546908i	1.36611	−	0.546908i
$442$		0			0
$443$	−0.672932	−	0.945001i	−0.672932	−	0.945001i	0.327068	−	0.945001i	$-0.393939\pi$
$443$	−0.672932	−	0.945001i	−0.672932	−	0.945001i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.205996	+	0.196417i	−0.205996	+	0.196417i	−0.786053	−	0.618159i	$-0.787879\pi$
$449$	−0.205996	+	0.196417i	−0.205996	+	0.196417i	0.580057	+	0.814576i	$0.303030\pi$
$450$		0			0
$451$	0.277784	−	0.390093i	0.277784	−	0.390093i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.0883470	+	0.0353688i	0.0883470	+	0.0353688i	0.415415	−	0.909632i	$-0.363636\pi$
$457$	0.0883470	+	0.0353688i	0.0883470	+	0.0353688i	−0.327068	+	0.945001i	$0.606061\pi$
$458$		0			0
$459$	0.622386	+	0.593444i	0.622386	+	0.593444i
$460$		0			0
$461$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$461$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$462$		0			0
$463$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$463$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.0135432	−	0.284307i	0.0135432	−	0.284307i	−0.981929	−	0.189251i	$-0.939394\pi$
$467$	0.0135432	−	0.284307i	0.0135432	−	0.284307i	0.995472	−	0.0950560i	$-0.0303030\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.185343	−	0.0176982i	−0.185343	−	0.0176982i
$474$		0			0
$475$	0.642315	−	1.85585i	0.642315	−	1.85585i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$479$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$487$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$488$		0			0
$489$	2.02261	−	1.92856i	2.02261	−	1.92856i
$490$		0			0
$491$	−0.959493	+	0.281733i	−0.959493	+	0.281733i	−0.723734	−	0.690079i	$-0.757576\pi$
$491$	−0.959493	+	0.281733i	−0.959493	+	0.281733i	−0.235759	+	0.971812i	$0.575758\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.235759	+	0.408346i	0.235759	+	0.408346i	0.959493	−	0.281733i	$-0.0909091\pi$
$499$	0.235759	+	0.408346i	0.235759	+	0.408346i	−0.723734	+	0.690079i	$0.757576\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$503$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.786053	+	1.36148i	−0.786053	+	1.36148i
$508$		0			0
$509$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$509$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−0.343209	−	1.41473i	−0.343209	−	1.41473i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.396666	−	0.254922i	0.396666	−	0.254922i	−0.327068	−	0.945001i	$-0.606061\pi$
$521$	0.396666	−	0.254922i	0.396666	−	0.254922i	0.723734	+	0.690079i	$0.242424\pi$
$522$		0			0
$523$	1.13779	−	0.894765i	1.13779	−	0.894765i	0.142315	−	0.989821i	$-0.454545\pi$
$523$	1.13779	−	0.894765i	1.13779	−	0.894765i	0.995472	+	0.0950560i	$0.0303030\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.888835	+	0.458227i	−0.888835	+	0.458227i
$530$		0			0
$531$	1.78922	−	2.06487i	1.78922	−	2.06487i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	2.05902			2.05902
$538$		0			0
$539$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i
$540$		0			0
$541$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$541$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.428368	−	1.23769i	−0.428368	−	1.23769i	−0.928368	−	0.371662i	$-0.878788\pi$
$547$	−0.428368	−	1.23769i	−0.428368	−	1.23769i	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		0.981929	−	0.189251i	$-0.0606061\pi$
$557$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.0738776	−	0.513830i	−0.0738776	−	0.513830i
$562$		0			0
$563$	0.308779	+	0.356349i	0.308779	+	0.356349i	0.888835	−	0.458227i	$-0.151515\pi$
$563$	0.308779	+	0.356349i	0.308779	+	0.356349i	−0.580057	+	0.814576i	$0.696970\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.0947329	−	1.98869i	−0.0947329	−	1.98869i	−0.142315	−	0.989821i	$-0.545455\pi$
$569$	−0.0947329	−	1.98869i	−0.0947329	−	1.98869i	0.0475819	−	0.998867i	$-0.484848\pi$
$570$		0			0
$571$	−0.273507	+	1.12741i	−0.273507	+	1.12741i	0.654861	+	0.755750i	$0.272727\pi$
$571$	−0.273507	+	1.12741i	−0.273507	+	1.12741i	−0.928368	+	0.371662i	$0.878788\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.0552004	+	0.0775182i	0.0552004	+	0.0775182i	0.841254	−	0.540641i	$-0.181818\pi$
$577$	0.0552004	+	0.0775182i	0.0552004	+	0.0775182i	−0.786053	+	0.618159i	$0.787879\pi$
$578$		0			0
$579$	0.542596	−	1.18812i	0.542596	−	1.18812i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.95496	+	0.376789i	1.95496	+	0.376789i	0.995472	+	0.0950560i	$0.0303030\pi$
$587$	1.95496	+	0.376789i	1.95496	+	0.376789i	0.959493	+	0.281733i	$0.0909091\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.44091	−	1.37391i	−1.44091	−	1.37391i	−0.786053	−	0.618159i	$-0.787879\pi$
$593$	−1.44091	−	1.37391i	−1.44091	−	1.37391i	−0.654861	−	0.755750i	$-0.727273\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$599$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$600$		0			0
$601$	−0.0845850	+	1.77566i	−0.0845850	+	1.77566i	0.415415	+	0.909632i	$0.363636\pi$
$601$	−0.0845850	+	1.77566i	−0.0845850	+	1.77566i	−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	1.41191	−	0.414574i	1.41191	−	0.414574i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$607$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$613$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$617$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$618$		0			0
$619$	−1.42131	−	0.273935i	−1.42131	−	0.273935i	−0.580057	−	0.814576i	$-0.696970\pi$
$619$	−1.42131	−	0.273935i	−1.42131	−	0.273935i	−0.841254	+	0.540641i	$0.818182\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$626$		0			0
$627$	−0.365052	+	0.799351i	−0.365052	+	0.799351i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$631$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$632$		0			0
$633$	0.0748038	+	0.129564i	0.0748038	+	0.129564i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$641$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$642$		0			0
$643$	1.21590	+	1.40323i	1.21590	+	1.40323i	0.888835	+	0.458227i	$0.151515\pi$
$643$	1.21590	+	1.40323i	1.21590	+	1.40323i	0.327068	+	0.945001i	$0.393939\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$647$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$648$		0			0
$649$	−0.518932	+	0.100016i	−0.518932	+	0.100016i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$653$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	2.30291	−	1.81103i	2.30291	−	1.81103i
$658$		0			0
$659$	−0.327068	−	0.945001i	−0.327068	−	0.945001i	−0.981929	−	0.189251i	$-0.939394\pi$
$659$	−0.327068	−	0.945001i	−0.327068	−	0.945001i	0.654861	−	0.755750i	$-0.272727\pi$
$660$		0			0
$661$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$661$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$662$		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
$673$	−0.452418	−	0.132842i	−0.452418	−	0.132842i	0.0475819	−	0.998867i	$-0.484848\pi$
$673$	−0.452418	−	0.132842i	−0.452418	−	0.132842i	−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$	−0.485433	+	0.560219i	−0.485433	+	0.560219i
$676$		0			0
$677$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$677$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−0.914053	−	2.64098i	−0.914053	−	2.64098i
$682$		0			0
$683$	1.13779	−	0.894765i	1.13779	−	0.894765i	0.142315	−	0.989821i	$-0.454545\pi$
$683$	1.13779	−	0.894765i	1.13779	−	0.894765i	0.995472	+	0.0950560i	$0.0303030\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.88431	−	0.363170i	1.88431	−	0.363170i	0.888835	−	0.458227i	$-0.151515\pi$
$691$	1.88431	−	0.363170i	1.88431	−	0.363170i	0.995472	+	0.0950560i	$0.0303030\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.27822	−	1.47515i	−1.27822	−	1.47515i
$698$		0			0
$699$	0.786053	−	1.36148i	0.786053	−	1.36148i
$700$		0			0
$701$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$701$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$709$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$710$		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		0.580057	−	0.814576i	$-0.303030\pi$
$719$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−0.855348	−	1.87295i	−0.855348	−	1.87295i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$727$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$728$		0			0
$729$	0.229963	−	1.59943i	0.229963	−	1.59943i
$730$		0			0
$731$	−0.248203	+	0.717135i	−0.248203	+	0.717135i
$732$		0			0
$733$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$733$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.264241	−	0.105786i	−0.264241	−	0.105786i
$738$		0			0
$739$	−0.0883470	+	1.85463i	−0.0883470	+	1.85463i	0.327068	+	0.945001i	$0.393939\pi$
$739$	−0.0883470	+	1.85463i	−0.0883470	+	1.85463i	−0.415415	+	0.909632i	$0.636364\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$743$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.39484	+	1.32997i	1.39484	+	1.32997i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$751$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$752$		0			0
$753$	2.74418	+	0.528898i	2.74418	+	0.528898i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$757$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$761$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.419102	+	1.72756i	−0.419102	+	1.72756i	0.235759	+	0.971812i	$0.424242\pi$
$769$	−0.419102	+	1.72756i	−0.419102	+	1.72756i	−0.654861	+	0.755750i	$0.727273\pi$
$770$		0			0
$771$	0.0621493	+	1.30467i	0.0621493	+	1.30467i
$772$		0			0
$773$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$773$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.470237	+	3.27057i	0.470237	+	3.27057i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.91030	+	0.182411i	−1.91030	+	0.182411i	−0.981929	−	0.189251i	$-0.939394\pi$
$787$	−1.91030	+	0.182411i	−1.91030	+	0.182411i	−0.928368	+	0.371662i	$0.878788\pi$
$788$		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		0.888835	−	0.458227i	$-0.151515\pi$
$797$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.134363	−	0.0394525i	−0.134363	−	0.0394525i
$802$		0			0
$803$	−0.566682			−0.566682
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.857685	−	0.989821i	0.857685	−	0.989821i	−0.142315	−	0.989821i	$-0.545455\pi$
$809$	0.857685	−	0.989821i	0.857685	−	0.989821i	1.00000			$0$
$810$		0			0
$811$	−1.16413	+	0.600149i	−1.16413	+	0.600149i	−0.928368	−	0.371662i	$-0.878788\pi$
$811$	−1.16413	+	0.600149i	−1.16413	+	0.600149i	−0.235759	+	0.971812i	$0.575758\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.00979	−	0.794105i	1.00979	−	0.794105i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$821$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$822$		0			0
$823$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$823$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$824$		0			0
$825$	0.439382	−	0.0846839i	0.439382	−	0.0846839i
$826$		0			0
$827$	0.0671040	+	0.276606i	0.0671040	+	0.276606i	0.995472	−	0.0950560i	$-0.0303030\pi$
$827$	0.0671040	+	0.276606i	0.0671040	+	0.276606i	−0.928368	+	0.371662i	$0.878788\pi$
$828$		0			0
$829$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$829$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.580057	+	1.00469i	−0.580057	+	1.00469i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$839$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$	1.69318	−	0.677846i	1.69318	−	0.677846i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.50842	+	0.442913i	−1.50842	+	0.442913i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$853$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.345139	+	0.755750i	0.345139	+	0.755750i	1.00000			$0$
$857$	0.345139	+	0.755750i	0.345139	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$858$		0			0
$859$	−1.07701	−	0.431171i	−1.07701	−	0.431171i	−0.235759	−	0.971812i	$-0.575758\pi$
$859$	−1.07701	−	0.431171i	−1.07701	−	0.431171i	−0.841254	+	0.540641i	$0.818182\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$863$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.541273	−	0.0516853i	−0.541273	−	0.0516853i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.139401	+	2.92639i	−0.139401	+	2.92639i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$877$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.205996	−	0.196417i	−0.205996	−	0.196417i	0.580057	−	0.814576i	$-0.303030\pi$
$881$	−0.205996	−	0.196417i	−0.205996	−	0.196417i	−0.786053	+	0.618159i	$0.787879\pi$
$882$		0			0
$883$	−1.56199	−	0.625325i	−1.56199	−	0.625325i	−0.580057	−	0.814576i	$-0.696970\pi$
$883$	−1.56199	−	0.625325i	−1.56199	−	0.625325i	−0.981929	+	0.189251i	$0.939394\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$887$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.0630664	+	0.0601336i	−0.0630664	+	0.0601336i
$892$		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
$907$	0.653077	+	0.513585i	0.653077	+	0.513585i	0.888835	−	0.458227i	$-0.151515\pi$
$907$	0.653077	+	0.513585i	0.653077	+	0.513585i	−0.235759	+	0.971812i	$0.575758\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$911$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$912$		0			0
$913$	−0.0530529	−	0.368991i	−0.0530529	−	0.368991i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$919$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$920$		0			0
$921$	−1.02371	+	0.0977529i	−1.02371	+	0.0977529i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.396666	+	0.254922i	0.396666	+	0.254922i	0.723734	−	0.690079i	$-0.242424\pi$
$929$	0.396666	+	0.254922i	0.396666	+	0.254922i	−0.327068	+	0.945001i	$0.606061\pi$
$930$		0			0
$931$	1.74555	−	0.899892i	1.74555	−	0.899892i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$937$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$938$		0			0
$939$	3.08739			3.08739
$940$		0			0
$941$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$941$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.0800569	−	0.0514495i	−0.0800569	−	0.0514495i	0.500000	−	0.866025i	$-0.333333\pi$
$947$	−0.0800569	−	0.0514495i	−0.0800569	−	0.0514495i	−0.580057	+	0.814576i	$0.696970\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.550294	+	0.353653i	−0.550294	+	0.353653i	−0.786053	−	0.618159i	$-0.787879\pi$
$953$	−0.550294	+	0.353653i	−0.550294	+	0.353653i	0.235759	+	0.971812i	$0.424242\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.235759	+	0.971812i	0.235759	+	0.971812i
$962$		0			0
$963$	−0.401872	−	2.79508i	−0.401872	−	2.79508i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$		0			0
$969$	2.81543	+	2.21408i	2.81543	+	2.21408i
$970$		0			0
$971$	0.0311250	+	0.653395i	0.0311250	+	0.653395i	0.959493	+	0.281733i	$0.0909091\pi$
$971$	0.0311250	+	0.653395i	0.0311250	+	0.653395i	−0.928368	+	0.371662i	$0.878788\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.21590	+	0.486774i	−1.21590	+	0.486774i	−0.888835	−	0.458227i	$-0.848485\pi$
$977$	−1.21590	+	0.486774i	−1.21590	+	0.486774i	−0.327068	+	0.945001i	$0.606061\pi$
$978$		0			0
$979$	0.0157117	+	0.0220640i	0.0157117	+	0.0220640i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$983$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$991$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$992$		0			0
$993$	−2.86624	−	1.14747i	−2.86624	−	1.14747i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$997$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$998$		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.1807.1		20
4.3	odd	2		536.1.ba.a.467.1	yes	20
8.3	odd	2	CM	2144.1.bu.a.1807.1		20
8.5	even	2		536.1.ba.a.467.1	yes	20
67.33	even	33	inner	2144.1.bu.a.1775.1		20
268.167	odd	66		536.1.ba.a.435.1	✓	20
536.301	even	66		536.1.ba.a.435.1	✓	20
536.435	odd	66	inner	2144.1.bu.a.1775.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.1.ba.a.435.1	✓	20	268.167	odd	66
536.1.ba.a.435.1	✓	20	536.301	even	66
536.1.ba.a.467.1	yes	20	4.3	odd	2
536.1.ba.a.467.1	yes	20	8.5	even	2
2144.1.bu.a.1775.1		20	67.33	even	33	inner
2144.1.bu.a.1775.1		20	536.435	odd	66	inner
2144.1.bu.a.1807.1		20	1.1	even	1	trivial
2144.1.bu.a.1807.1		20	8.3	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2144 = 2^{5} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2144.bu (of order \(66\), degree \(20\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50842 - 0.442913i) q^{3} +(1.23792 + 0.795563i) q^{9} +O(q^{10})\)	\(q+(-1.50842 - 0.442913i) q^{3} +(1.23792 + 0.795563i) q^{9} +(-0.0930932 - 0.268975i) q^{11} +(-1.15486 + 0.110276i) q^{17} +(1.74555 + 0.899892i) q^{19} +(-0.142315 - 0.989821i) q^{25} +(-0.485433 - 0.560219i) q^{27} +(0.0212914 + 0.446961i) q^{33} +(0.975950 + 1.37053i) q^{41} +(0.271738 - 0.595023i) q^{43} +(0.580057 - 0.814576i) q^{49} +(1.79086 + 0.345161i) q^{51} +(-2.23445 - 2.13054i) q^{57} +(0.264241 - 1.83784i) q^{59} +(0.654861 - 0.755750i) q^{67} +(0.651174 - 1.88144i) q^{73} +(-0.223734 + 1.55610i) q^{75} +(-0.127181 - 0.278487i) q^{81} +(1.28605 + 0.247866i) q^{83} +(-0.0913090 + 0.0268107i) q^{89} +(0.995472 + 1.72421i) q^{97} +(0.0987447 - 0.407031i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3}+O(q^{10}) \) 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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