LMFDB - Embedded newform 588.3.p.c.557.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,3,Mod(557,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("588.557");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	588.p (of order $6$, degree $2$, 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:	$16.0218395444$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			557.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	588.557
Dual form			588.3.p.c.569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 2.59808i) q^{3} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})$	$q+(1.50000 - 2.59808i) q^{3} +(-4.50000 - 7.79423i) q^{9} -22.0000 q^{13} +(-13.0000 - 22.5167i) q^{19} +(-12.5000 + 21.6506i) q^{25} -27.0000 q^{27} +(23.0000 - 39.8372i) q^{31} +(-13.0000 - 22.5167i) q^{37} +(-33.0000 + 57.1577i) q^{39} -22.0000 q^{43} -78.0000 q^{57} +(-37.0000 - 64.0859i) q^{61} +(-61.0000 + 105.655i) q^{67} +(23.0000 - 39.8372i) q^{73} +(37.5000 + 64.9519i) q^{75} +(71.0000 + 122.976i) q^{79} +(-40.5000 + 70.1481i) q^{81} +(-69.0000 - 119.512i) q^{93} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 9 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - 9 * q^9	$ 2 q + 3 q^{3} - 9 q^{9} - 44 q^{13} - 26 q^{19} - 25 q^{25} - 54 q^{27} + 46 q^{31} - 26 q^{37} - 66 q^{39} - 44 q^{43} - 156 q^{57} - 74 q^{61} - 122 q^{67} + 46 q^{73} + 75 q^{75} + 142 q^{79} - 81 q^{81} - 138 q^{93} + 4 q^{97}+O(q^{100}) $ 2 * q + 3 * q^3 - 9 * q^9 - 44 * q^13 - 26 * q^19 - 25 * q^25 - 54 * q^27 + 46 * q^31 - 26 * q^37 - 66 * q^39 - 44 * q^43 - 156 * q^57 - 74 * q^61 - 122 * q^67 + 46 * q^73 + 75 * q^75 + 142 * q^79 - 81 * q^81 - 138 * q^93 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	1.50000	−	2.59808i	0.500000	−	0.866025i
$3$	1.50000	−	2.59808i	0.500000	−	0.866025i
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	−22.0000			−1.69231			−0.846154	−	0.532939i	$-0.821088\pi$
$13$	−22.0000			−1.69231			−0.846154	+	0.532939i	$0.821088\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−13.0000	−	22.5167i	−0.684211	−	1.18509i	−0.973684	−	0.227901i	$-0.926814\pi$
$19$	−13.0000	−	22.5167i	−0.684211	−	1.18509i	0.289474	−	0.957186i	$-0.406520\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$26$		0			0
$27$	−27.0000			−1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	23.0000	−	39.8372i	0.741935	−	1.28507i	−0.209677	−	0.977771i	$-0.567241\pi$
$31$	23.0000	−	39.8372i	0.741935	−	1.28507i	0.951613	−	0.307299i	$-0.0994253\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−13.0000	−	22.5167i	−0.351351	−	0.608558i	0.635135	−	0.772401i	$-0.280944\pi$
$37$	−13.0000	−	22.5167i	−0.351351	−	0.608558i	−0.986486	+	0.163843i	$0.947611\pi$
$38$		0			0
$39$	−33.0000	+	57.1577i	−0.846154	+	1.46558i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−22.0000			−0.511628			−0.255814	−	0.966726i	$-0.582343\pi$
$43$	−22.0000			−0.511628			−0.255814	+	0.966726i	$0.582343\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−78.0000			−1.36842
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	−37.0000	−	64.0859i	−0.606557	−	1.05059i	−0.991803	−	0.127774i	$-0.959217\pi$
$61$	−37.0000	−	64.0859i	−0.606557	−	1.05059i	0.385246	−	0.922814i	$-0.374117\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−61.0000	+	105.655i	−0.910448	+	1.57694i	−0.0970149	+	0.995283i	$0.530929\pi$
$67$	−61.0000	+	105.655i	−0.910448	+	1.57694i	−0.813433	+	0.581659i	$0.802404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	23.0000	−	39.8372i	0.315068	−	0.545715i	−0.664384	−	0.747392i	$-0.731306\pi$
$73$	23.0000	−	39.8372i	0.315068	−	0.545715i	0.979452	+	0.201677i	$0.0646392\pi$
$74$		0			0
$75$	37.5000	+	64.9519i	0.500000	+	0.866025i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	71.0000	+	122.976i	0.898734	+	1.55665i	0.829114	+	0.559080i	$0.188845\pi$
$79$	71.0000	+	122.976i	0.898734	+	1.55665i	0.0696203	+	0.997574i	$0.477821\pi$
$80$		0			0
$81$	−40.5000	+	70.1481i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−69.0000	−	119.512i	−0.741935	−	1.28507i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.00000			0.0206186			0.0103093	−	0.999947i	$-0.496718\pi$
$97$	2.00000			0.0206186			0.0103093	+	0.999947i	$0.496718\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−97.0000	−	168.009i	−0.941748	−	1.63115i	−0.762136	−	0.647417i	$-0.775849\pi$
$103$	−97.0000	−	168.009i	−0.941748	−	1.63115i	−0.179612	−	0.983738i	$-0.557484\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	107.000	−	185.329i	0.981651	−	1.70027i	0.325688	−	0.945477i	$-0.394404\pi$
$109$	107.000	−	185.329i	0.981651	−	1.70027i	0.655963	−	0.754793i	$-0.272263\pi$
$110$		0			0
$111$	−78.0000			−0.702703
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	99.0000	+	171.473i	0.846154	+	1.46558i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	146.000			1.14961			0.574803	−	0.818292i	$-0.305079\pi$
$127$	146.000			1.14961			0.574803	+	0.818292i	$0.305079\pi$
$128$		0			0
$129$	−33.0000	+	57.1577i	−0.255814	+	0.443083i
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−22.0000			−0.158273			−0.0791367	−	0.996864i	$-0.525216\pi$
$139$	−22.0000			−0.158273			−0.0791367	+	0.996864i	$0.525216\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	143.000	−	247.683i	0.947020	−	1.64029i	0.195364	−	0.980731i	$-0.437411\pi$
$151$	143.000	−	247.683i	0.947020	−	1.64029i	0.751656	−	0.659556i	$-0.229256\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	0.990446	+	0.137902i	$0.0440359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	131.000	+	226.899i	0.803681	+	1.39202i	0.917178	+	0.398478i	$0.130461\pi$
$163$	131.000	+	226.899i	0.803681	+	1.39202i	−0.113497	+	0.993538i	$0.536205\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	315.000			1.86391
$170$		0			0
$171$	−117.000	+	202.650i	−0.684211	+	1.18509i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	314.000			1.73481			0.867403	−	0.497606i	$-0.165787\pi$
$181$	314.000			1.73481			0.867403	+	0.497606i	$0.165787\pi$
$182$		0			0
$183$	−222.000			−1.21311
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	191.000	−	330.822i	0.989637	−	1.71410i	0.370466	−	0.928846i	$-0.379198\pi$
$193$	191.000	−	330.822i	0.989637	−	1.71410i	0.619171	−	0.785256i	$-0.287469\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−193.000	+	334.286i	−0.969849	+	1.67983i	−0.273869	+	0.961767i	$0.588304\pi$
$199$	−193.000	+	334.286i	−0.969849	+	1.67983i	−0.695980	+	0.718061i	$0.745030\pi$
$200$		0			0
$201$	183.000	+	316.965i	0.910448	+	1.57694i
$202$		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
$211$	−166.000			−0.786730			−0.393365	−	0.919382i	$-0.628689\pi$
$211$	−166.000			−0.786730			−0.393365	+	0.919382i	$0.628689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−69.0000	−	119.512i	−0.315068	−	0.545715i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	338.000			1.51570			0.757848	−	0.652432i	$-0.226251\pi$
$223$	338.000			1.51570			0.757848	+	0.652432i	$0.226251\pi$
$224$		0			0
$225$	225.000			1.00000
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−13.0000	−	22.5167i	−0.0567686	−	0.0983260i	0.836245	−	0.548357i	$-0.184746\pi$
$229$	−13.0000	−	22.5167i	−0.0567686	−	0.0983260i	−0.893013	+	0.450031i	$0.851413\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	426.000			1.79747
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	143.000	−	247.683i	0.593361	−	1.02773i	−0.400415	−	0.916334i	$-0.631134\pi$
$241$	143.000	−	247.683i	0.593361	−	1.02773i	0.993776	−	0.111397i	$-0.0355327\pi$
$242$		0			0
$243$	121.500	+	210.444i	0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	286.000	+	495.367i	1.15789	+	2.00553i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−121.000	−	209.578i	−0.446494	−	0.773351i	0.551661	−	0.834069i	$-0.313994\pi$
$271$	−121.000	−	209.578i	−0.446494	−	0.773351i	−0.998155	+	0.0607176i	$0.980661\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−61.0000	+	105.655i	−0.220217	+	0.381426i	−0.954874	−	0.297012i	$-0.904010\pi$
$277$	−61.0000	+	105.655i	−0.220217	+	0.381426i	0.734657	+	0.678439i	$0.237343\pi$
$278$		0			0
$279$	−414.000			−1.48387
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−229.000	+	396.640i	−0.809187	+	1.40155i	0.104240	+	0.994552i	$0.466759\pi$
$283$	−229.000	+	396.640i	−0.809187	+	1.40155i	−0.913428	+	0.407001i	$0.866574\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−144.500	−	250.281i	−0.500000	−	0.866025i
$290$		0			0
$291$	3.00000	−	5.19615i	0.0103093	−	0.0178562i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		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
$307$	−358.000			−1.16612			−0.583062	−	0.812428i	$-0.698145\pi$
$307$	−358.000			−1.16612			−0.583062	+	0.812428i	$0.698145\pi$
$308$		0			0
$309$	−582.000			−1.88350
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	71.0000	+	122.976i	0.226837	+	0.392893i	0.956869	−	0.290520i	$-0.0938282\pi$
$313$	71.0000	+	122.976i	0.226837	+	0.392893i	−0.730032	+	0.683413i	$0.760495\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	275.000	−	476.314i	0.846154	−	1.46558i
$326$		0			0
$327$	−321.000	−	555.988i	−0.981651	−	1.70027i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−181.000	−	313.501i	−0.546828	−	0.947134i	−0.998489	−	0.0549442i	$-0.982502\pi$
$331$	−181.000	−	313.501i	−0.546828	−	0.947134i	0.451662	−	0.892189i	$-0.350831\pi$
$332$		0			0
$333$	−117.000	+	202.650i	−0.351351	+	0.608558i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	482.000			1.43027			0.715134	−	0.698988i	$-0.246366\pi$
$337$	482.000			1.43027			0.715134	+	0.698988i	$0.246366\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−502.000			−1.43840			−0.719198	−	0.694805i	$-0.755490\pi$
$349$	−502.000			−1.43840			−0.719198	+	0.694805i	$0.755490\pi$
$350$		0			0
$351$	594.000			1.69231
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−157.500	+	272.798i	−0.436288	+	0.755673i
$362$		0			0
$363$	−363.000			−1.00000
$364$		0			0
$365$		0			0
$366$		0			0
$367$	359.000	−	621.806i	0.978202	−	1.69429i	0.309264	−	0.950976i	$-0.399917\pi$
$367$	359.000	−	621.806i	0.978202	−	1.69429i	0.668937	−	0.743319i	$-0.266749\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−349.000	−	604.486i	−0.935657	−	1.62061i	−0.773458	−	0.633847i	$-0.781475\pi$
$373$	−349.000	−	604.486i	−0.935657	−	1.62061i	−0.162198	−	0.986758i	$-0.551858\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−694.000			−1.83113			−0.915567	−	0.402165i	$-0.868258\pi$
$379$	−694.000			−1.83113			−0.915567	+	0.402165i	$0.868258\pi$
$380$		0			0
$381$	219.000	−	379.319i	0.574803	−	0.995588i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	99.0000	+	171.473i	0.255814	+	0.443083i
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−181.000	−	313.501i	−0.455919	−	0.789676i	0.542821	−	0.839848i	$-0.317356\pi$
$397$	−181.000	−	313.501i	−0.455919	−	0.789676i	−0.998741	+	0.0501728i	$0.984023\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−506.000	+	876.418i	−1.25558	+	2.17473i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−313.000	+	542.132i	−0.765281	+	1.32551i	0.174817	+	0.984601i	$0.444067\pi$
$409$	−313.000	+	542.132i	−0.765281	+	1.32551i	−0.940098	+	0.340905i	$0.889267\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−33.0000	+	57.1577i	−0.0791367	+	0.137069i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−358.000			−0.850356			−0.425178	−	0.905110i	$-0.639789\pi$
$421$	−358.000			−0.850356			−0.425178	+	0.905110i	$0.639789\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		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
$433$	−862.000			−1.99076			−0.995381	−	0.0960028i	$-0.969394\pi$
$433$	−862.000			−1.99076			−0.995381	+	0.0960028i	$0.969394\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	47.0000	+	81.4064i	0.107062	+	0.185436i	0.914579	−	0.404408i	$-0.132522\pi$
$439$	47.0000	+	81.4064i	0.107062	+	0.185436i	−0.807517	+	0.589844i	$0.799189\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−429.000	−	743.050i	−0.947020	−	1.64029i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	407.000	+	704.945i	0.890591	+	1.54255i	0.839168	+	0.543872i	$0.183042\pi$
$457$	407.000	+	704.945i	0.890591	+	1.54255i	0.0514223	+	0.998677i	$0.483625\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−526.000			−1.13607			−0.568035	−	0.823005i	$-0.692296\pi$
$463$	−526.000			−1.13607			−0.568035	+	0.823005i	$0.692296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−177.000	−	306.573i	−0.375796	−	0.650898i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	650.000			1.36842
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	286.000	+	495.367i	0.594595	+	1.02987i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−481.000	+	833.116i	−0.987680	+	1.71071i	−0.358316	+	0.933600i	$0.616649\pi$
$487$	−481.000	+	833.116i	−0.987680	+	1.71071i	−0.629363	+	0.777111i	$0.716684\pi$
$488$		0			0
$489$	786.000			1.60736
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.0000	−	22.5167i	−0.0260521	−	0.0451236i	0.852705	−	0.522392i	$-0.174960\pi$
$499$	−13.0000	−	22.5167i	−0.0260521	−	0.0451236i	−0.878758	+	0.477269i	$0.841627\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	472.500	−	818.394i	0.931953	−	1.61419i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	351.000	+	607.950i	0.684211	+	1.18509i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	491.000	+	850.437i	0.938815	+	1.62607i	0.767686	+	0.640826i	$0.221408\pi$
$523$	491.000	+	850.437i	0.938815	+	1.62607i	0.171128	+	0.985249i	$0.445259\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	+	458.127i	−0.500000	+	0.866025i
$530$		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
$541$	−517.000	−	895.470i	−0.955638	−	1.65521i	−0.732902	−	0.680334i	$-0.761835\pi$
$541$	−517.000	−	895.470i	−0.955638	−	1.65521i	−0.222736	−	0.974879i	$-0.571499\pi$
$542$		0			0
$543$	471.000	−	815.796i	0.867403	−	1.50239i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	506.000			0.925046			0.462523	−	0.886607i	$-0.346944\pi$
$547$	506.000			0.925046			0.462523	+	0.886607i	$0.346944\pi$
$548$		0			0
$549$	−333.000	+	576.773i	−0.606557	+	1.05059i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	484.000			0.865832
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	443.000	−	767.299i	0.775832	−	1.34378i	−0.158494	−	0.987360i	$-0.550664\pi$
$571$	443.000	−	767.299i	0.775832	−	1.34378i	0.934326	−	0.356420i	$-0.116003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−481.000	+	833.116i	−0.833622	+	1.44388i	0.0615251	+	0.998106i	$0.480404\pi$
$577$	−481.000	+	833.116i	−0.833622	+	1.44388i	−0.895147	+	0.445770i	$0.852930\pi$
$578$		0			0
$579$	−573.000	−	992.465i	−0.989637	−	1.71410i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	−1196.00			−2.03056
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	579.000	+	1002.86i	0.969849	+	1.67983i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	−526.000			−0.875208			−0.437604	−	0.899168i	$-0.644173\pi$
$601$	−526.000			−0.875208			−0.437604	+	0.899168i	$0.644173\pi$
$602$		0			0
$603$	1098.00			1.82090
$604$		0			0
$605$		0			0
$606$		0			0
$607$	407.000	+	704.945i	0.670511	+	1.16136i	0.977759	+	0.209729i	$0.0672583\pi$
$607$	407.000	+	704.945i	0.670511	+	1.16136i	−0.307249	+	0.951629i	$0.599408\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	563.000	−	975.145i	0.918434	−	1.59077i	0.116639	−	0.993174i	$-0.462788\pi$
$613$	563.000	−	975.145i	0.918434	−	1.59077i	0.801794	−	0.597600i	$-0.203879\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	107.000	−	185.329i	0.172859	−	0.299401i	−0.766559	−	0.642174i	$-0.778033\pi$
$619$	107.000	−	185.329i	0.172859	−	0.299401i	0.939418	+	0.342773i	$0.111366\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	−	541.266i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	674.000			1.06815			0.534073	−	0.845438i	$-0.320661\pi$
$631$	674.000			1.06815			0.534073	+	0.845438i	$0.320661\pi$
$632$		0			0
$633$	−249.000	+	431.281i	−0.393365	+	0.681328i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	314.000			0.488336			0.244168	−	0.969733i	$-0.421485\pi$
$643$	314.000			0.488336			0.244168	+	0.969733i	$0.421485\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−414.000			−0.630137
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−61.0000	+	105.655i	−0.0922844	+	0.159841i	−0.908472	−	0.417946i	$-0.862750\pi$
$661$	−61.0000	+	105.655i	−0.0922844	+	0.159841i	0.816188	+	0.577787i	$0.196084\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	507.000	−	878.150i	0.757848	−	1.31263i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1154.00			1.71471			0.857355	−	0.514725i	$-0.172106\pi$
$673$	1154.00			1.71471			0.857355	+	0.514725i	$0.172106\pi$
$674$		0			0
$675$	337.500	−	584.567i	0.500000	−	0.866025i
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−78.0000			−0.113537
$688$		0			0
$689$		0			0
$690$		0			0
$691$	659.000	+	1141.42i	0.953690	+	1.65184i	0.737337	+	0.675525i	$0.236083\pi$
$691$	659.000	+	1141.42i	0.953690	+	1.65184i	0.216353	+	0.976315i	$0.430584\pi$
$692$		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		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−338.000	+	585.433i	−0.480797	+	0.832764i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	467.000	+	808.868i	0.658674	+	1.14086i	0.980959	+	0.194214i	$0.0622158\pi$
$709$	467.000	+	808.868i	0.658674	+	1.14086i	−0.322285	+	0.946643i	$0.604451\pi$
$710$		0			0
$711$	639.000	−	1106.78i	0.898734	−	1.55665i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−429.000	−	743.050i	−0.593361	−	1.02773i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	482.000			0.662999			0.331499	−	0.943455i	$-0.392446\pi$
$727$	482.000			0.662999			0.331499	+	0.943455i	$0.392446\pi$
$728$		0			0
$729$	729.000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−517.000	−	895.470i	−0.705321	−	1.22165i	−0.966576	−	0.256381i	$-0.917470\pi$
$733$	−517.000	−	895.470i	−0.705321	−	1.22165i	0.261255	−	0.965270i	$-0.415864\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	611.000	−	1058.28i	0.826793	−	1.43205i	−0.0737483	−	0.997277i	$-0.523496\pi$
$739$	611.000	−	1058.28i	0.826793	−	1.43205i	0.900541	−	0.434771i	$-0.143171\pi$
$740$		0			0
$741$	1716.00			2.31579
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−601.000	−	1040.96i	−0.800266	−	1.38610i	−0.919441	−	0.393229i	$-0.871358\pi$
$751$	−601.000	−	1040.96i	−0.800266	−	1.38610i	0.119174	−	0.992873i	$-0.461975\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−838.000			−1.10700			−0.553501	−	0.832849i	$-0.686708\pi$
$757$	−838.000			−1.10700			−0.553501	+	0.832849i	$0.686708\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1534.00			−1.99480			−0.997399	−	0.0720749i	$-0.977038\pi$
$769$	−1534.00			−1.99480			−0.997399	+	0.0720749i	$0.977038\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	575.000	+	995.929i	0.741935	+	1.28507i
$776$		0			0
$777$		0			0
$778$		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
$787$	−781.000	+	1352.73i	−0.992376	+	1.71885i	−0.389454	+	0.921046i	$0.627336\pi$
$787$	−781.000	+	1352.73i	−0.992376	+	1.71885i	−0.602922	+	0.797800i	$0.705997\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	814.000	+	1409.89i	1.02648	+	1.77792i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		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		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	1514.00			1.86683			0.933416	−	0.358797i	$-0.116813\pi$
$811$	1514.00			1.86683			0.933416	+	0.358797i	$0.116813\pi$
$812$		0			0
$813$	−726.000			−0.892989
$814$		0			0
$815$		0			0
$816$		0			0
$817$	286.000	+	495.367i	0.350061	+	0.606324i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−529.000	+	916.255i	−0.642770	+	1.11331i	0.342041	+	0.939685i	$0.388882\pi$
$823$	−529.000	+	916.255i	−0.642770	+	1.11331i	−0.984812	+	0.173626i	$0.944452\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−229.000	+	396.640i	−0.276236	+	0.478456i	−0.970446	−	0.241317i	$-0.922421\pi$
$829$	−229.000	+	396.640i	−0.276236	+	0.478456i	0.694210	+	0.719773i	$0.255754\pi$
$830$		0			0
$831$	183.000	+	316.965i	0.220217	+	0.381426i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−621.000	+	1075.60i	−0.741935	+	1.28507i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	687.000	+	1189.92i	0.809187	+	1.40155i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1658.00			1.94373			0.971864	−	0.235543i	$-0.0756867\pi$
$853$	1658.00			1.94373			0.971864	+	0.235543i	$0.0756867\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	−709.000	−	1228.02i	−0.825378	−	1.42960i	−0.901630	−	0.432509i	$-0.857629\pi$
$859$	−709.000	−	1228.02i	−0.825378	−	1.42960i	0.0762515	−	0.997089i	$-0.475705\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−867.000			−1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1342.00	−	2324.41i	1.54076	−	2.66867i
$872$		0			0
$873$	−9.00000	−	15.5885i	−0.0103093	−	0.0178562i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	299.000	+	517.883i	0.340935	+	0.590517i	0.984607	−	0.174785i	$-0.0559231\pi$
$877$	299.000	+	517.883i	0.340935	+	0.590517i	−0.643672	+	0.765302i	$0.722590\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−1702.00			−1.92752			−0.963760	−	0.266771i	$-0.914043\pi$
$883$	−1702.00			−1.92752			−0.963760	+	0.266771i	$0.914043\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		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
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	107.000	−	185.329i	0.117971	−	0.204332i	−0.800992	−	0.598675i	$-0.795694\pi$
$907$	107.000	−	185.329i	0.117971	−	0.204332i	0.918964	+	0.394342i	$0.129028\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−433.000	−	749.978i	−0.471164	−	0.816081i	0.528292	−	0.849063i	$-0.322833\pi$
$919$	−433.000	−	749.978i	−0.471164	−	0.816081i	−0.999456	+	0.0329825i	$0.989499\pi$
$920$		0			0
$921$	−537.000	+	930.111i	−0.583062	+	1.00989i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	650.000			0.702703
$926$		0			0
$927$	−873.000	+	1512.08i	−0.941748	+	1.63115i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1198.00			−1.27855			−0.639274	−	0.768979i	$-0.720765\pi$
$937$	−1198.00			−1.27855			−0.639274	+	0.768979i	$0.720765\pi$
$938$		0			0
$939$	426.000			0.453674
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−506.000	+	876.418i	−0.533193	+	0.923517i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−577.500	−	1000.26i	−0.600937	−	1.04085i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1534.00			−1.58635			−0.793175	−	0.608994i	$-0.791573\pi$
$967$	−1534.00			−1.58635			−0.793175	+	0.608994i	$0.791573\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−825.000	−	1428.94i	−0.846154	−	1.46558i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1926.00			−1.96330
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	23.0000	−	39.8372i	0.0232089	−	0.0401990i	−0.854188	−	0.519965i	$-0.825945\pi$
$991$	23.0000	−	39.8372i	0.0232089	−	0.0401990i	0.877397	+	0.479766i	$0.159278\pi$
$992$		0			0
$993$	−1086.00			−1.09366
$994$		0			0
$995$		0			0
$996$		0			0
$997$	947.000	−	1640.25i	0.949850	−	1.64519i	0.204112	−	0.978947i	$-0.434569\pi$
$997$	947.000	−	1640.25i	0.949850	−	1.64519i	0.745737	−	0.666240i	$-0.232097\pi$
$998$		0			0
$999$	351.000	+	607.950i	0.351351	+	0.608558i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.3.p.c.557.1		2
3.2	odd	2	CM	588.3.p.c.557.1		2
7.2	even	3	inner	588.3.p.c.569.1		2
7.3	odd	6		588.3.c.c.197.1		1
7.4	even	3		12.3.c.a.5.1	✓	1
7.5	odd	6		588.3.p.b.569.1		2
7.6	odd	2		588.3.p.b.557.1		2
21.2	odd	6	inner	588.3.p.c.569.1		2
21.5	even	6		588.3.p.b.569.1		2
21.11	odd	6		12.3.c.a.5.1	✓	1
21.17	even	6		588.3.c.c.197.1		1
21.20	even	2		588.3.p.b.557.1		2
28.11	odd	6		48.3.e.a.17.1		1
35.4	even	6		300.3.g.b.101.1		1
35.18	odd	12		300.3.b.a.149.1		2
35.32	odd	12		300.3.b.a.149.2		2
56.11	odd	6		192.3.e.a.65.1		1
56.53	even	6		192.3.e.b.65.1		1
63.4	even	3		324.3.g.b.269.1		2
63.11	odd	6		324.3.g.b.53.1		2
63.25	even	3		324.3.g.b.53.1		2
63.32	odd	6		324.3.g.b.269.1		2
77.32	odd	6		1452.3.e.b.485.1		1
84.11	even	6		48.3.e.a.17.1		1
105.32	even	12		300.3.b.a.149.2		2
105.53	even	12		300.3.b.a.149.1		2
105.74	odd	6		300.3.g.b.101.1		1
112.11	odd	12		768.3.h.b.641.1		2
112.53	even	12		768.3.h.a.641.2		2
112.67	odd	12		768.3.h.b.641.2		2
112.109	even	12		768.3.h.a.641.1		2
140.39	odd	6		1200.3.l.b.401.1		1
140.67	even	12		1200.3.c.c.449.1		2
140.123	even	12		1200.3.c.c.449.2		2
168.11	even	6		192.3.e.a.65.1		1
168.53	odd	6		192.3.e.b.65.1		1
231.32	even	6		1452.3.e.b.485.1		1
252.11	even	6		1296.3.q.b.1025.1		2
252.67	odd	6		1296.3.q.b.593.1		2
252.95	even	6		1296.3.q.b.593.1		2
252.151	odd	6		1296.3.q.b.1025.1		2
336.11	even	12		768.3.h.b.641.1		2
336.53	odd	12		768.3.h.a.641.2		2
336.179	even	12		768.3.h.b.641.2		2
336.221	odd	12		768.3.h.a.641.1		2
420.179	even	6		1200.3.l.b.401.1		1
420.263	odd	12		1200.3.c.c.449.2		2
420.347	odd	12		1200.3.c.c.449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.3.c.a.5.1	✓	1	7.4	even	3
12.3.c.a.5.1	✓	1	21.11	odd	6
48.3.e.a.17.1		1	28.11	odd	6
48.3.e.a.17.1		1	84.11	even	6
192.3.e.a.65.1		1	56.11	odd	6
192.3.e.a.65.1		1	168.11	even	6
192.3.e.b.65.1		1	56.53	even	6
192.3.e.b.65.1		1	168.53	odd	6
300.3.b.a.149.1		2	35.18	odd	12
300.3.b.a.149.1		2	105.53	even	12
300.3.b.a.149.2		2	35.32	odd	12
300.3.b.a.149.2		2	105.32	even	12
300.3.g.b.101.1		1	35.4	even	6
300.3.g.b.101.1		1	105.74	odd	6
324.3.g.b.53.1		2	63.11	odd	6
324.3.g.b.53.1		2	63.25	even	3
324.3.g.b.269.1		2	63.4	even	3
324.3.g.b.269.1		2	63.32	odd	6
588.3.c.c.197.1		1	7.3	odd	6
588.3.c.c.197.1		1	21.17	even	6
588.3.p.b.557.1		2	7.6	odd	2
588.3.p.b.557.1		2	21.20	even	2
588.3.p.b.569.1		2	7.5	odd	6
588.3.p.b.569.1		2	21.5	even	6
588.3.p.c.557.1		2	1.1	even	1	trivial
588.3.p.c.557.1		2	3.2	odd	2	CM
588.3.p.c.569.1		2	7.2	even	3	inner
588.3.p.c.569.1		2	21.2	odd	6	inner
768.3.h.a.641.1		2	112.109	even	12
768.3.h.a.641.1		2	336.221	odd	12
768.3.h.a.641.2		2	112.53	even	12
768.3.h.a.641.2		2	336.53	odd	12
768.3.h.b.641.1		2	112.11	odd	12
768.3.h.b.641.1		2	336.11	even	12
768.3.h.b.641.2		2	112.67	odd	12
768.3.h.b.641.2		2	336.179	even	12
1200.3.c.c.449.1		2	140.67	even	12
1200.3.c.c.449.1		2	420.347	odd	12
1200.3.c.c.449.2		2	140.123	even	12
1200.3.c.c.449.2		2	420.263	odd	12
1200.3.l.b.401.1		1	140.39	odd	6
1200.3.l.b.401.1		1	420.179	even	6
1296.3.q.b.593.1		2	252.67	odd	6
1296.3.q.b.593.1		2	252.95	even	6
1296.3.q.b.1025.1		2	252.11	even	6
1296.3.q.b.1025.1		2	252.151	odd	6
1452.3.e.b.485.1		1	77.32	odd	6
1452.3.e.b.485.1		1	231.32	even	6

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	588.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 2.59808i) q^{3} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)	\(q+(1.50000 - 2.59808i) q^{3} +(-4.50000 - 7.79423i) q^{9} -22.0000 q^{13} +(-13.0000 - 22.5167i) q^{19} +(-12.5000 + 21.6506i) q^{25} -27.0000 q^{27} +(23.0000 - 39.8372i) q^{31} +(-13.0000 - 22.5167i) q^{37} +(-33.0000 + 57.1577i) q^{39} -22.0000 q^{43} -78.0000 q^{57} +(-37.0000 - 64.0859i) q^{61} +(-61.0000 + 105.655i) q^{67} +(23.0000 - 39.8372i) q^{73} +(37.5000 + 64.9519i) q^{75} +(71.0000 + 122.976i) q^{79} +(-40.5000 + 70.1481i) q^{81} +(-69.0000 - 119.512i) q^{93} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 9 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - 9 * q^9	\( 2 q + 3 q^{3} - 9 q^{9} - 44 q^{13} - 26 q^{19} - 25 q^{25} - 54 q^{27} + 46 q^{31} - 26 q^{37} - 66 q^{39} - 44 q^{43} - 156 q^{57} - 74 q^{61} - 122 q^{67} + 46 q^{73} + 75 q^{75} + 142 q^{79} - 81 q^{81} - 138 q^{93} + 4 q^{97}+O(q^{100}) \) 2 * q + 3 * q^3 - 9 * q^9 - 44 * q^13 - 26 * q^19 - 25 * q^25 - 54 * q^27 + 46 * q^31 - 26 * q^37 - 66 * q^39 - 44 * q^43 - 156 * q^57 - 74 * q^61 - 122 * q^67 + 46 * q^73 + 75 * q^75 + 142 * q^79 - 81 * q^81 - 138 * q^93 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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