LMFDB - Embedded newform 228.3.s.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [228,3,Mod(5,228)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(228, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 16]))

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

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

chi := DirichletCharacter("228.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	228.s (of order $18$, degree $6$, 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:	$6.21255002741$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			17.1
Root			$-0.173648 + 0.984808i$ of defining polynomial
Character	$\chi$	$=$	228.17
Dual form			228.3.s.a.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.29813 - 1.92836i) q^{3} +(3.68732 + 6.38662i) q^{7} +(1.56283 + 8.86327i) q^{9} +O(q^{10})$	$q+(-2.29813 - 1.92836i) q^{3} +(3.68732 + 6.38662i) q^{7} +(1.56283 + 8.86327i) q^{9} +(19.4670 - 16.3348i) q^{13} +(5.50000 - 18.1865i) q^{19} +(3.84178 - 21.7878i) q^{21} +(19.1511 - 16.0697i) q^{25} +(13.5000 - 23.3827i) q^{27} +(24.4628 + 42.3707i) q^{31} +72.7443 q^{37} -76.2372 q^{39} +(-73.3478 + 26.6964i) q^{43} +(-2.69264 + 4.66379i) q^{49} +(-47.7100 + 31.1891i) q^{57} +(34.1701 + 12.4369i) q^{61} +(-50.8437 + 42.6629i) q^{63} +(10.0422 + 56.9519i) q^{67} +(-3.19122 - 2.67775i) q^{73} -75.0000 q^{75} +(-117.444 - 98.5470i) q^{79} +(-76.1151 + 27.7036i) q^{81} +(176.105 + 64.0971i) q^{91} +(25.4875 - 144.547i) q^{93} +(-29.3465 + 166.433i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 69 q^{13} + 33 q^{19} + 117 q^{21} + 81 q^{27} - 183 q^{43} - 147 q^{49} + 222 q^{61} + 54 q^{63} - 327 q^{67} - 291 q^{73} - 450 q^{75} - 426 q^{79} + 222 q^{91} + 414 q^{93}+O(q^{100}) $ 6 * q + 69 * q^13 + 33 * q^19 + 117 * q^21 + 81 * q^27 - 183 * q^43 - 147 * q^49 + 222 * q^61 + 54 * q^63 - 327 * q^67 - 291 * q^73 - 450 * q^75 - 426 * q^79 + 222 * q^91 + 414 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/228\mathbb{Z}\right)^\times$.

$n$	$77$	$97$	$115$
$\chi(n)$	$-1$	$e\left(\frac{5}{9}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.29813	−	1.92836i	−0.766044	−	0.642788i
$3$	−2.29813	−	1.92836i	−0.766044	−	0.642788i
$4$		0			0
$5$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$5$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$6$		0			0
$7$	3.68732	+	6.38662i	0.526760	+	0.912375i	0.999514	+	0.0311803i	$0.00992661\pi$
$7$	3.68732	+	6.38662i	0.526760	+	0.912375i	−0.472754	+	0.881194i	$0.656740\pi$
$8$		0			0
$9$	1.56283	+	8.86327i	0.173648	+	0.984808i
$10$		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
$13$	19.4670	−	16.3348i	1.49746	−	1.25652i	0.612849	−	0.790200i	$-0.290023\pi$
$13$	19.4670	−	16.3348i	1.49746	−	1.25652i	0.884615	−	0.466321i	$-0.154421\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$17$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$18$		0			0
$19$	5.50000	−	18.1865i	0.289474	−	0.957186i
$19$	5.50000	−	18.1865i	0.289474	−	0.957186i
$20$		0			0
$21$	3.84178	−	21.7878i	0.182942	−	1.03751i
$22$		0			0
$23$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$23$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$24$		0			0
$25$	19.1511	−	16.0697i	0.766044	−	0.642788i
$26$		0			0
$27$	13.5000	−	23.3827i	0.500000	−	0.866025i
$28$		0			0
$29$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$29$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$30$		0			0
$31$	24.4628	+	42.3707i	0.789121	+	1.36680i	0.926506	+	0.376280i	$0.122797\pi$
$31$	24.4628	+	42.3707i	0.789121	+	1.36680i	−0.137385	+	0.990518i	$0.543870\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	72.7443			1.96606			0.983032	−	0.183437i	$-0.0587222\pi$
$37$	72.7443			1.96606			0.983032	+	0.183437i	$0.0587222\pi$
$38$		0			0
$39$	−76.2372			−1.95480
$40$		0			0
$41$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$41$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$42$		0			0
$43$	−73.3478	+	26.6964i	−1.70576	+	0.620847i	−0.996461	−	0.0840574i	$-0.973212\pi$
$43$	−73.3478	+	26.6964i	−1.70576	+	0.620847i	−0.709302	+	0.704904i	$0.750990\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$47$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$48$		0			0
$49$	−2.69264	+	4.66379i	−0.0549518	+	0.0951793i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$53$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−47.7100	+	31.1891i	−0.837017	+	0.547177i
$58$		0			0
$59$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$59$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$60$		0			0
$61$	34.1701	+	12.4369i	0.560166	+	0.203884i	0.606557	−	0.795040i	$-0.292550\pi$
$61$	34.1701	+	12.4369i	0.560166	+	0.203884i	−0.0463918	+	0.998923i	$0.514772\pi$
$62$		0			0
$63$	−50.8437	+	42.6629i	−0.807043	+	0.677189i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	10.0422	+	56.9519i	0.149883	+	0.850029i	0.963316	+	0.268370i	$0.0864850\pi$
$67$	10.0422	+	56.9519i	0.149883	+	0.850029i	−0.813433	+	0.581659i	$0.802404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$71$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$72$		0			0
$73$	−3.19122	−	2.67775i	−0.0437153	−	0.0366815i	0.620668	−	0.784073i	$-0.286861\pi$
$73$	−3.19122	−	2.67775i	−0.0437153	−	0.0366815i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$	−75.0000			−1.00000
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−117.444	−	98.5470i	−1.48663	−	1.24743i	−0.898734	−	0.438494i	$-0.855512\pi$
$79$	−117.444	−	98.5470i	−1.48663	−	1.24743i	−0.587896	−	0.808937i	$-0.700043\pi$
$80$		0			0
$81$	−76.1151	+	27.7036i	−0.939693	+	0.342020i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$89$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$90$		0			0
$91$	176.105	+	64.0971i	1.93522	+	0.704363i
$92$		0			0
$93$	25.4875	−	144.547i	0.274059	−	1.55427i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−29.3465	+	166.433i	−0.302542	+	1.71580i	0.332314	+	0.943169i	$0.392170\pi$
$97$	−29.3465	+	166.433i	−0.302542	+	1.71580i	−0.634856	+	0.772630i	$0.718941\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$101$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$102$		0			0
$103$	79.3023	−	137.356i	0.769925	−	1.33355i	−0.167678	−	0.985842i	$-0.553627\pi$
$103$	79.3023	−	137.356i	0.769925	−	1.33355i	0.937603	−	0.347707i	$-0.113040\pi$
$104$		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
$109$	−66.7182	+	24.2834i	−0.612093	+	0.222784i	−0.629419	−	0.777066i	$-0.716707\pi$
$109$	−66.7182	+	24.2834i	−0.612093	+	0.222784i	0.0173256	+	0.999850i	$0.494485\pi$
$110$		0			0
$111$	−167.176	−	140.277i	−1.50609	−	1.26376i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	175.203	+	147.013i	1.49746	+	1.25652i
$118$		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
$127$	−193.809	+	162.625i	−1.52606	+	1.28051i	−0.706047	+	0.708165i	$0.749523\pi$
$127$	−193.809	+	162.625i	−1.52606	+	1.28051i	−0.820011	+	0.572348i	$0.806032\pi$
$128$		0			0
$129$	220.043	+	80.0893i	1.70576	+	0.620847i
$130$		0			0
$131$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$131$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$132$		0			0
$133$	136.431	−	31.9331i	1.02580	−	0.240099i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$137$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$138$		0			0
$139$	88.4449	−	74.2141i	0.636294	−	0.533914i	−0.266583	−	0.963812i	$-0.585895\pi$
$139$	88.4449	−	74.2141i	0.636294	−	0.533914i	0.902878	+	0.429898i	$0.141450\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	15.1815	−	5.52562i	0.103276	−	0.0375893i
$148$		0			0
$149$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$149$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$150$		0			0
$151$	227.000			1.50331			0.751656	−	0.659556i	$-0.229256\pi$
$151$	227.000			1.50331			0.751656	+	0.659556i	$0.229256\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−90.8194	+	33.0556i	−0.578468	+	0.210545i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$	−90.8194	+	33.0556i	−0.578468	+	0.210545i	0.0361820	+	0.999345i	$0.488480\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−89.9255	+	155.756i	−0.551690	+	0.955556i	0.446462	+	0.894802i	$0.352684\pi$
$163$	−89.9255	+	155.756i	−0.551690	+	0.955556i	−0.998153	+	0.0607535i	$0.980650\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$167$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$168$		0			0
$169$	82.7938	−	469.547i	0.489904	−	2.77838i
$170$		0			0
$171$	169.788	+	20.3255i	0.992911	+	0.118862i
$172$		0			0
$173$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$173$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$174$		0			0
$175$	173.247	+	63.0569i	0.989985	+	0.360325i
$176$		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
$181$	−54.3519	−	308.245i	−0.300287	−	1.70301i	−0.644901	−	0.764266i	$-0.723102\pi$
$181$	−54.3519	−	308.245i	−0.300287	−	1.70301i	0.344615	−	0.938744i	$-0.388010\pi$
$182$		0			0
$183$	−54.5446	−	94.4740i	−0.298058	−	0.516251i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	199.115			1.05352
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−9.00104	−	7.55277i	−0.0466375	−	0.0391335i	0.619171	−	0.785256i	$-0.287469\pi$
$193$	−9.00104	−	7.55277i	−0.0466375	−	0.0391335i	−0.665808	+	0.746123i	$0.731913\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	28.2264	+	160.080i	0.141841	+	0.804422i	0.969849	+	0.243706i	$0.0783631\pi$
$199$	28.2264	+	160.080i	0.141841	+	0.804422i	−0.828008	+	0.560716i	$0.810526\pi$
$200$		0			0
$201$	86.7458	−	150.248i	0.431571	−	0.747503i
$202$		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
$211$	61.3429	−	347.893i	0.290724	−	1.64878i	−0.393365	−	0.919382i	$-0.628689\pi$
$211$	61.3429	−	347.893i	0.290724	−	1.64878i	0.684089	−	0.729398i	$-0.260200\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−180.404	+	312.469i	−0.831355	+	1.43995i
$218$		0			0
$219$	2.17017	+	12.3077i	0.00990946	+	0.0561993i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−419.069	+	152.529i	−1.87923	+	0.683985i	−0.935289	+	0.353886i	$0.884860\pi$
$223$	−419.069	+	152.529i	−1.87923	+	0.683985i	−0.943946	+	0.330099i	$0.892918\pi$
$224$		0			0
$225$	172.360	+	144.627i	0.766044	+	0.642788i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	−445.800			−1.94672			−0.973362	−	0.229274i	$-0.926365\pi$
$229$	−445.800			−1.94672			−0.973362	+	0.229274i	$0.926365\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$233$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	79.8670	+	452.948i	0.336992	+	1.91117i
$238$		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
$241$	−358.874	+	301.131i	−1.48910	+	1.24951i	−0.593361	+	0.804936i	$0.702199\pi$
$241$	−358.874	+	301.131i	−1.48910	+	1.24951i	−0.895744	+	0.444571i	$0.853356\pi$
$242$		0			0
$243$	228.345	+	83.1109i	0.939693	+	0.342020i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−190.004	−	443.879i	−0.769248	−	1.79708i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$251$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$257$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$258$		0			0
$259$	268.232	+	464.591i	1.03564	+	1.79379i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$263$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$269$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$270$		0			0
$271$	−280.968	+	102.264i	−1.03678	+	0.377358i	−0.803660	−	0.595089i	$-0.797117\pi$
$271$	−280.968	+	102.264i	−1.03678	+	0.377358i	−0.233123	+	0.972447i	$0.574894\pi$
$272$		0			0
$273$	−281.111	−	486.898i	−1.02971	−	1.78351i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−203.500	+	352.472i	−0.734657	+	1.27246i	0.220217	+	0.975451i	$0.429324\pi$
$277$	−203.500	+	352.472i	−0.734657	+	1.27246i	−0.954874	+	0.297012i	$0.904010\pi$
$278$		0			0
$279$	−337.312	+	283.038i	−1.20900	+	1.01447i
$280$		0			0
$281$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$281$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$282$		0			0
$283$	10.2452	−	58.1037i	0.0362023	−	0.205313i	−0.961342	−	0.275359i	$-0.911203\pi$
$283$	10.2452	−	58.1037i	0.0362023	−	0.205313i	0.997544	+	0.0700455i	$0.0223145\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−271.571	−	98.8438i	−0.939693	−	0.342020i
$290$		0			0
$291$	388.385	−	325.893i	1.33465	−	1.11991i
$292$		0			0
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−440.957	−	370.007i	−1.46497	−	1.22926i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	468.053	+	392.743i	1.52460	+	1.27929i	0.825766	+	0.564013i	$0.190743\pi$
$307$	468.053	+	392.743i	1.52460	+	1.27929i	0.698838	+	0.715280i	$0.253701\pi$
$308$		0			0
$309$	−447.118	+	162.738i	−1.44699	+	0.526660i
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	104.015	+	589.900i	0.332317	+	1.88466i	0.452265	+	0.891884i	$0.350616\pi$
$313$	104.015	+	589.900i	0.332317	+	1.88466i	−0.119947	+	0.992780i	$0.538273\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$317$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	110.320	−	625.658i	0.339448	−	1.92510i
$326$		0			0
$327$	200.155	+	72.8503i	0.612093	+	0.222784i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	241.488	−	418.269i	0.729570	−	1.26365i	−0.227495	−	0.973779i	$-0.573054\pi$
$331$	241.488	−	418.269i	0.729570	−	1.26365i	0.957065	−	0.289873i	$-0.0936130\pi$
$332$		0			0
$333$	113.687	+	644.753i	0.341403	+	1.93619i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	577.031	−	210.022i	1.71226	−	0.623211i	0.715134	−	0.698988i	$-0.246366\pi$
$337$	577.031	−	210.022i	1.71226	−	0.623211i	0.997125	+	0.0757768i	$0.0241436\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	321.643			0.937734
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$347$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$348$		0			0
$349$	−195.217	−	338.127i	−0.559362	−	0.968844i	−0.997550	−	0.0699605i	$-0.977713\pi$
$349$	−195.217	−	338.127i	−0.559362	−	0.968844i	0.438187	−	0.898884i	$-0.355621\pi$
$350$		0			0
$351$	−119.146	−	675.711i	−0.339448	−	1.92510i
$352$		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		0.173648	−	0.984808i	$-0.444444\pi$
$359$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$360$		0			0
$361$	−300.500	−	200.052i	−0.832410	−	0.554160i
$362$		0			0
$363$	−63.0343	+	357.485i	−0.173648	+	0.984808i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	19.4770	−	16.3432i	0.0530710	−	0.0445318i	−0.615866	−	0.787851i	$-0.711194\pi$
$367$	19.4770	−	16.3432i	0.0530710	−	0.0445318i	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
$379$	547.885			1.44561			0.722803	−	0.691054i	$-0.242853\pi$
$379$	547.885			1.44561			0.722803	+	0.691054i	$0.242853\pi$
$380$		0			0
$381$	759.000			1.99213
$382$		0			0
$383$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$383$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−351.248	−	608.379i	−0.907618	−	1.57204i
$388$		0			0
$389$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$389$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−101.040	+	573.027i	−0.254509	+	1.44339i	0.542821	+	0.839848i	$0.317356\pi$
$397$	−101.040	+	573.027i	−0.254509	+	1.44339i	−0.797330	+	0.603543i	$0.793755\pi$
$398$		0			0
$399$	−375.115	−	189.702i	−0.940137	−	0.475442i
$400$		0			0
$401$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$401$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$402$		0			0
$403$	1168.33	+	425.239i	2.89909	+	1.05518i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	108.704	+	616.490i	0.265779	+	1.50731i	0.766807	+	0.641878i	$0.221845\pi$
$409$	108.704	+	616.490i	0.265779	+	1.50731i	−0.501027	+	0.865431i	$0.667044\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−346.370			−0.830623
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	642.711	+	539.299i	1.52663	+	1.28099i	0.817528	+	0.575889i	$0.195344\pi$
$421$	642.711	+	539.299i	1.52663	+	1.28099i	0.709103	+	0.705105i	$0.249100\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	46.5663	+	264.090i	0.109054	+	0.618479i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$431$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$432$		0			0
$433$	670.725	+	244.124i	1.54902	+	0.563797i	0.968187	−	0.250227i	$-0.0805053\pi$
$433$	670.725	+	244.124i	1.54902	+	0.563797i	0.580831	+	0.814024i	$0.302728\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	67.1843	−	381.021i	0.153039	−	0.867929i	−0.807517	−	0.589844i	$-0.799189\pi$
$439$	67.1843	−	381.021i	0.153039	−	0.867929i	0.960556	−	0.278085i	$-0.0896997\pi$
$440$		0			0
$441$	−45.5446	−	16.5769i	−0.103276	−	0.0375893i
$442$		0			0
$443$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$443$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−521.676	−	437.738i	−1.15160	−	0.966310i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−890.762			−1.94915			−0.974575	−	0.224060i	$-0.928069\pi$
$457$	−890.762			−1.94915			−0.974575	+	0.224060i	$0.928069\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$461$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$462$		0			0
$463$	−377.466	−	653.791i	−0.815262	−	1.41208i	−0.909140	−	0.416492i	$-0.863260\pi$
$463$	−377.466	−	653.791i	−0.815262	−	1.41208i	0.0938775	−	0.995584i	$-0.470074\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	−326.702	+	274.135i	−0.696593	+	0.584511i
$470$		0			0
$471$	272.458	+	99.1667i	0.578468	+	0.210545i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−186.921	−	436.676i	−0.393518	−	0.919317i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$479$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$480$		0			0
$481$	1416.12	−	1188.26i	2.94411	−	2.47040i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	174.500	+	302.243i	0.358316	+	0.620622i	0.987680	−	0.156489i	$-0.0500176\pi$
$487$	174.500	+	302.243i	0.358316	+	0.620622i	−0.629363	+	0.777111i	$0.716684\pi$
$488$		0			0
$489$	507.014	−	184.538i	1.03684	−	0.377378i
$490$		0			0
$491$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$491$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	343.601	−	125.060i	0.688579	−	0.250622i	0.0260521	−	0.999661i	$-0.491706\pi$
$499$	343.601	−	125.060i	0.688579	−	0.250622i	0.662527	+	0.749038i	$0.269484\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$503$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1095.73	+	919.425i	−2.16120	+	1.81346i
$508$		0			0
$509$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$509$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$510$		0			0
$511$	5.33474	−	30.2548i	0.0104398	−	0.0592071i
$512$		0			0
$513$	−351.000	−	374.123i	−0.684211	−	0.729285i
$514$		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
$523$	31.9983	+	181.471i	0.0611822	+	0.346981i	0.999997	+	0.00255837i	$0.000814355\pi$
$523$	31.9983	+	181.471i	0.0611822	+	0.346981i	−0.938815	+	0.344423i	$0.888075\pi$
$524$		0			0
$525$	−276.549	−	478.997i	−0.526760	−	0.912375i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	405.238	+	340.035i	0.766044	+	0.642788i
$530$		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
$541$	101.972	+	578.314i	0.188489	+	1.06897i	0.921391	+	0.388638i	$0.127054\pi$
$541$	101.972	+	578.314i	0.188489	+	1.06897i	−0.732902	+	0.680334i	$0.761835\pi$
$542$		0			0
$543$	−469.500	+	813.198i	−0.864641	+	1.49760i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−950.113	−	345.813i	−1.73695	−	0.632199i	−0.737867	−	0.674946i	$-0.764167\pi$
$547$	−950.113	−	345.813i	−1.73695	−	0.632199i	−0.999086	+	0.0427471i	$0.986389\pi$
$548$		0			0
$549$	−56.8294	+	322.296i	−0.103514	+	0.587059i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	196.330	−	1113.44i	0.355027	−	2.01346i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$557$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$558$		0			0
$559$	−991.784	+	1717.82i	−1.77421	+	3.07302i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−457.593	−	383.966i	−0.807043	−	0.677189i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−863.439			−1.51215			−0.756076	−	0.654484i	$-0.772886\pi$
$571$	−863.439			−1.51215			−0.756076	+	0.654484i	$0.772886\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	516.500	+	894.604i	0.895147	+	1.55044i	0.833622	+	0.552335i	$0.186263\pi$
$577$	516.500	+	894.604i	0.895147	+	1.55044i	0.0615251	+	0.998106i	$0.480404\pi$
$578$		0			0
$579$	6.12111	+	34.7145i	0.0105719	+	0.0599560i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$587$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$588$		0			0
$589$	905.122	−	211.854i	1.53671	−	0.359684i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$593$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	243.824	−	422.316i	0.408416	−	0.707397i
$598$		0			0
$599$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$599$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$600$		0			0
$601$	−62.3115	−	107.927i	−0.103680	−	0.179579i	0.809518	−	0.587095i	$-0.199728\pi$
$601$	−62.3115	−	107.927i	−0.103680	−	0.179579i	−0.913198	+	0.407516i	$0.866395\pi$
$602$		0			0
$603$	−489.086	+	178.013i	−0.811088	+	0.295212i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−44.6230			−0.0735139			−0.0367570	−	0.999324i	$-0.511703\pi$
$607$	−44.6230			−0.0735139			−0.0367570	+	0.999324i	$0.511703\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−923.718	+	336.206i	−1.50688	+	0.548460i	−0.957832	−	0.287330i	$-0.907232\pi$
$613$	−923.718	+	336.206i	−1.50688	+	0.548460i	−0.549049	+	0.835790i	$0.685010\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$617$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$618$		0			0
$619$	−309.929	+	536.813i	−0.500693	+	0.867226i	0.499306	+	0.866426i	$0.333588\pi$
$619$	−309.929	+	536.813i	−0.500693	+	0.867226i	−1.00000		0.000800838i	$0.999745\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	108.530	−	615.505i	0.173648	−	0.984808i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1097.35	−	399.402i	−1.73906	−	0.632966i	−0.739854	−	0.672768i	$-0.765105\pi$
$631$	−1097.35	−	399.402i	−1.73906	−	0.632966i	−0.999208	+	0.0398015i	$0.987327\pi$
$632$		0			0
$633$	−811.837	+	681.212i	−1.28252	+	1.07617i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	23.7643	+	134.774i	0.0373065	+	0.211576i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$641$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$642$		0			0
$643$	798.328	+	669.877i	1.24157	+	1.04180i	0.997400	+	0.0720661i	$0.0229593\pi$
$643$	798.328	+	669.877i	1.24157	+	1.04180i	0.244168	+	0.969733i	$0.421485\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1017.15	−	370.211i	1.56244	−	0.568680i
$652$		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
$657$	18.7463	−	32.4695i	0.0285332	−	0.0494209i
$658$		0			0
$659$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$659$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$660$		0			0
$661$	−114.642	−	41.7265i	−0.173438	−	0.0631263i	0.253842	−	0.967246i	$-0.418306\pi$
$661$	−114.642	−	41.7265i	−0.173438	−	0.0631263i	−0.427280	+	0.904120i	$0.640528\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1257.21	+	457.586i	1.87923	+	0.683985i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−664.676	+	1151.25i	−0.987631	+	1.71063i	−0.358027	+	0.933711i	$0.616551\pi$
$673$	−664.676	+	1151.25i	−0.987631	+	1.71063i	−0.629604	+	0.776916i	$0.716783\pi$
$674$		0			0
$675$	−117.213	−	664.745i	−0.173648	−	0.984808i
$676$		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
$679$	−1171.15	+	426.264i	−1.72482	+	0.627783i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1024.51	+	859.664i	1.49128	+	1.25133i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−509.500	−	882.480i	−0.737337	−	1.27711i	−0.953690	−	0.300790i	$-0.902750\pi$
$691$	−509.500	−	882.480i	−0.737337	−	1.27711i	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		0.173648	−	0.984808i	$-0.444444\pi$
$701$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$702$		0			0
$703$	400.094	−	1322.97i	0.569124	−	1.88189i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	951.878	−	798.721i	1.34256	−	1.12655i	0.361606	−	0.932331i	$-0.382229\pi$
$709$	951.878	−	798.721i	1.34256	−	1.12655i	0.980959	−	0.194214i	$-0.0622158\pi$
$710$		0			0
$711$	689.904	−	1194.95i	0.970329	−	1.68066i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$719$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$720$		0			0
$721$	1169.65			1.62226
$722$		0			0
$723$	1405.43			1.94389
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.2662	+	5.55646i	−0.0209990	+	0.00764300i	−0.352498	−	0.935812i	$-0.614668\pi$
$727$	−15.2662	+	5.55646i	−0.0209990	+	0.00764300i	0.331499	+	0.943455i	$0.392446\pi$
$728$		0			0
$729$	−364.500	−	631.333i	−0.500000	−	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	708.500	−	1227.16i	0.966576	−	1.67416i	0.261255	−	0.965270i	$-0.415864\pi$
$733$	708.500	−	1227.16i	0.966576	−	1.67416i	0.705321	−	0.708888i	$-0.250803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−69.7549	+	395.600i	−0.0943910	+	0.535318i	0.900541	+	0.434771i	$0.143171\pi$
$739$	−69.7549	+	395.600i	−0.0943910	+	0.535318i	−0.994932	+	0.100547i	$0.967941\pi$
$740$		0			0
$741$	−419.305	+	1386.49i	−0.565863	+	1.87111i
$742$		0			0
$743$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$743$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−249.629	−	1415.72i	−0.332396	−	1.88511i	−0.451570	−	0.892236i	$-0.649136\pi$
$751$	−249.629	−	1415.72i	−0.332396	−	1.88511i	0.119174	−	0.992873i	$-0.461975\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1062.73	−	891.735i	−1.40387	−	1.17799i	−0.959351	−	0.282216i	$-0.908931\pi$
$757$	−1062.73	−	891.735i	−1.40387	−	1.17799i	−0.444518	−	0.895770i	$-0.646625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−401.100	−	336.563i	−0.525688	−	0.441105i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	243.728	+	1382.25i	0.316942	+	1.79747i	0.561118	+	0.827736i	$0.310371\pi$
$769$	243.728	+	1382.25i	0.316942	+	1.79747i	−0.244177	+	0.969731i	$0.578518\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$773$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$774$		0			0
$775$	1149.37	+	418.338i	1.48306	+	0.539791i
$776$		0			0
$777$	279.468	−	1584.94i	0.359675	−	2.03982i
$778$		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
$787$	767.074	−	1328.61i	0.974681	−	1.68820i	0.293698	−	0.955898i	$-0.405114\pi$
$787$	767.074	−	1328.61i	0.974681	−	1.68820i	0.680983	−	0.732299i	$-0.261553\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	868.345	−	316.052i	1.09501	−	0.398552i
$794$		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
$811$	−193.809	+	162.625i	−0.238976	+	0.200524i	−0.754408	−	0.656406i	$-0.772076\pi$
$811$	−193.809	+	162.625i	−0.238976	+	0.200524i	0.515432	+	0.856930i	$0.327631\pi$
$812$		0			0
$813$	842.904	+	306.792i	1.03678	+	0.377358i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	82.1024	+	1480.77i	0.100493	+	1.81245i
$818$		0			0
$819$	−292.886	+	1661.04i	−0.357615	+	2.02813i
$820$		0			0
$821$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$821$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$822$		0			0
$823$	−1241.76	+	1041.96i	−1.50882	+	1.26605i	−0.642805	+	0.766030i	$0.722229\pi$
$823$	−1241.76	+	1041.96i	−1.50882	+	1.26605i	−0.866014	+	0.500020i	$0.833326\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$827$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$828$		0			0
$829$	336.713	+	583.203i	0.406167	+	0.703502i	0.994457	−	0.105148i	$-0.0335317\pi$
$829$	336.713	+	583.203i	0.406167	+	0.703502i	−0.588289	+	0.808650i	$0.700198\pi$
$830$		0			0
$831$	1147.36	−	417.607i	1.38070	−	0.502535i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1320.99			1.57824
$838$		0			0
$839$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$839$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$840$		0			0
$841$	−790.281	+	287.639i	−0.939693	+	0.342020i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	446.166	−	772.781i	0.526760	−	0.912375i
$848$		0			0
$849$	−135.590	+	113.773i	−0.159705	+	0.134009i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−18.7231	+	106.184i	−0.0219498	+	0.124483i	−0.993814	−	0.111059i	$-0.964576\pi$
$853$	−18.7231	+	106.184i	−0.0219498	+	0.124483i	0.971864	+	0.235543i	$0.0756867\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$857$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$858$		0			0
$859$	−1606.61	−	584.759i	−1.87033	−	0.680744i	−0.968700	−	0.248236i	$-0.920149\pi$
$859$	−1606.61	−	584.759i	−1.87033	−	0.680744i	−0.901630	−	0.432509i	$-0.857629\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	433.500	+	750.844i	0.500000	+	0.866025i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1125.79	+	944.649i	1.29252	+	1.08456i
$872$		0			0
$873$	−1521.00			−1.74227
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1162.85	−	975.748i	−1.32594	−	1.11260i	−0.985008	−	0.172511i	$-0.944812\pi$
$877$	−1162.85	−	975.748i	−1.32594	−	1.11260i	−0.340935	−	0.940087i	$-0.610744\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	−173.818	−	985.772i	−0.196850	−	1.11639i	−0.909760	−	0.415134i	$-0.863735\pi$
$883$	−173.818	−	985.772i	−0.196850	−	1.11639i	0.712911	−	0.701255i	$-0.247377\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$887$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$888$		0			0
$889$	−1753.26	−	638.135i	−1.97217	−	0.717813i
$890$		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
$903$	299.870	+	1700.65i	0.332082	+	1.88333i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1566.47	+	570.148i	−1.72709	+	0.628608i	−0.998416	−	0.0562565i	$-0.982084\pi$
$907$	−1566.47	+	570.148i	−1.72709	+	0.628608i	−0.728670	+	0.684865i	$0.759861\pi$
$908$		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
$919$	723.095	+	1252.44i	0.786828	+	1.36283i	0.927901	+	0.372827i	$0.121612\pi$
$919$	723.095	+	1252.44i	0.786828	+	1.36283i	−0.141072	+	0.989999i	$0.545055\pi$
$920$		0			0
$921$	−318.297	−	1805.15i	−0.345599	−	1.95999i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1393.13	−	1168.98i	1.50609	−	1.26376i
$926$		0			0
$927$	1341.36	+	488.213i	1.44699	+	0.526660i
$928$		0			0
$929$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$929$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$930$		0			0
$931$	70.0086	+	74.6206i	0.0751972	+	0.0801510i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1412.60	+	1185.31i	−1.50758	+	1.26501i	−0.639274	+	0.768979i	$0.720765\pi$
$937$	−1412.60	+	1185.31i	−1.50758	+	1.26501i	−0.868305	+	0.496030i	$0.834790\pi$
$938$		0			0
$939$	898.500	−	1556.25i	0.956869	−	1.65735i
$940$		0			0
$941$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$941$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$947$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$948$		0			0
$949$	−105.864			−0.111553
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$953$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−716.353	+	1240.76i	−0.745424	+	1.29111i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	155.159	−	879.950i	0.160454	−	0.909980i	−0.793175	−	0.608994i	$-0.791573\pi$
$967$	155.159	−	879.950i	0.160454	−	0.909980i	0.953629	−	0.300985i	$-0.0973156\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$971$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$972$		0			0
$973$	800.102	+	291.213i	0.822304	+	0.299294i
$974$		0			0
$975$	−1460.03	+	1225.11i	−1.49746	+	1.25652i
$976$		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
$981$	−319.500	−	553.390i	−0.325688	−	0.564108i
$982$		0			0
$983$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$983$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−486.036	−	407.833i	−0.490450	−	0.411537i	0.363737	−	0.931502i	$-0.381501\pi$
$991$	−486.036	−	407.833i	−0.490450	−	0.411537i	−0.854188	+	0.519965i	$0.825945\pi$
$992$		0			0
$993$	−1361.54	+	495.562i	−1.37114	+	0.499055i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	49.5202	+	280.843i	0.0496692	+	0.281688i	0.999519	−	0.0310190i	$-0.00987523\pi$
$997$	49.5202	+	280.843i	0.0496692	+	0.281688i	−0.949850	+	0.312707i	$0.898764\pi$
$998$		0			0
$999$	982.049	−	1700.96i	0.983032	−	1.70266i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.3.s.a.17.1	✓	6
3.2	odd	2	CM	228.3.s.a.17.1	✓	6
19.9	even	9	inner	228.3.s.a.161.1	yes	6
57.47	odd	18	inner	228.3.s.a.161.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.3.s.a.17.1	✓	6	1.1	even	1	trivial
228.3.s.a.17.1	✓	6	3.2	odd	2	CM
228.3.s.a.161.1	yes	6	19.9	even	9	inner
228.3.s.a.161.1	yes	6	57.47	odd	18	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:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	228.s (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.29813 - 1.92836i) q^{3} +(3.68732 + 6.38662i) q^{7} +(1.56283 + 8.86327i) q^{9} +O(q^{10})\)	\(q+(-2.29813 - 1.92836i) q^{3} +(3.68732 + 6.38662i) q^{7} +(1.56283 + 8.86327i) q^{9} +(19.4670 - 16.3348i) q^{13} +(5.50000 - 18.1865i) q^{19} +(3.84178 - 21.7878i) q^{21} +(19.1511 - 16.0697i) q^{25} +(13.5000 - 23.3827i) q^{27} +(24.4628 + 42.3707i) q^{31} +72.7443 q^{37} -76.2372 q^{39} +(-73.3478 + 26.6964i) q^{43} +(-2.69264 + 4.66379i) q^{49} +(-47.7100 + 31.1891i) q^{57} +(34.1701 + 12.4369i) q^{61} +(-50.8437 + 42.6629i) q^{63} +(10.0422 + 56.9519i) q^{67} +(-3.19122 - 2.67775i) q^{73} -75.0000 q^{75} +(-117.444 - 98.5470i) q^{79} +(-76.1151 + 27.7036i) q^{81} +(176.105 + 64.0971i) q^{91} +(25.4875 - 144.547i) q^{93} +(-29.3465 + 166.433i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 69 q^{13} + 33 q^{19} + 117 q^{21} + 81 q^{27} - 183 q^{43} - 147 q^{49} + 222 q^{61} + 54 q^{63} - 327 q^{67} - 291 q^{73} - 450 q^{75} - 426 q^{79} + 222 q^{91} + 414 q^{93}+O(q^{100}) \) 6 * q + 69 * q^13 + 33 * q^19 + 117 * q^21 + 81 * q^27 - 183 * q^43 - 147 * q^49 + 222 * q^61 + 54 * q^63 - 327 * q^67 - 291 * q^73 - 450 * q^75 - 426 * q^79 + 222 * q^91 + 414 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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