LMFDB - Embedded newform 196.4.f.a.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,4,Mod(19,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("196.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	196.f (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:	$11.5643743611$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			31.2
Root			$-0.895644 - 1.09445i$ of defining polynomial
Character	$\chi$	$=$	196.31
Dual form			196.4.f.a.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.104356 - 2.82650i) q^{2} +(-7.97822 + 0.589925i) q^{4} +(2.50000 + 22.4889i) q^{8} +(13.5000 + 23.3827i) q^{9} +O(q^{10})$	$q+(-0.104356 - 2.82650i) q^{2} +(-7.97822 + 0.589925i) q^{4} +(2.50000 + 22.4889i) q^{8} +(13.5000 + 23.3827i) q^{9} +(22.9129 + 13.2288i) q^{11} +(63.3040 - 9.41311i) q^{16} +(64.6824 - 40.5979i) q^{18} +(35.0000 - 66.1438i) q^{22} +(187.886 - 108.476i) q^{23} +(-62.5000 + 108.253i) q^{25} +166.000 q^{29} +(-33.2123 - 177.946i) q^{32} +(-121.500 - 178.588i) q^{36} +(225.000 + 389.711i) q^{37} -534.442i q^{43} +(-190.608 - 92.0250i) q^{44} +(-326.214 - 519.739i) q^{46} +(312.500 + 165.359i) q^{50} +(-295.000 + 510.955i) q^{53} +(-17.3231 - 469.199i) q^{58} +(-499.500 + 112.444i) q^{64} +(701.134 + 404.800i) q^{67} +978.928i q^{71} +(-492.101 + 362.057i) q^{72} +(1078.04 - 676.632i) q^{74} +(206.216 - 119.059i) q^{79} +(-364.500 + 631.333i) q^{81} +(-1510.60 + 55.7722i) q^{86} +(-240.218 + 548.357i) q^{88} +(-1435.00 + 976.282i) q^{92} +714.353i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} - 9 q^{4} + 10 q^{8} + 54 q^{9}+O(q^{10}) $ 4 * q - 5 * q^2 - 9 * q^4 + 10 * q^8 + 54 * q^9	$ 4 q - 5 q^{2} - 9 q^{4} + 10 q^{8} + 54 q^{9} + 47 q^{16} + 135 q^{18} + 140 q^{22} - 250 q^{25} + 664 q^{29} + 275 q^{32} - 486 q^{36} + 900 q^{37} - 350 q^{44} + 574 q^{46} + 1250 q^{50} - 1180 q^{53} - 830 q^{58} - 1998 q^{64} + 135 q^{72} + 2250 q^{74} - 1458 q^{81} - 1414 q^{86} - 1190 q^{88} - 5740 q^{92}+O(q^{100}) $ 4 * q - 5 * q^2 - 9 * q^4 + 10 * q^8 + 54 * q^9 + 47 * q^16 + 135 * q^18 + 140 * q^22 - 250 * q^25 + 664 * q^29 + 275 * q^32 - 486 * q^36 + 900 * q^37 - 350 * q^44 + 574 * q^46 + 1250 * q^50 - 1180 * q^53 - 830 * q^58 - 1998 * q^64 + 135 * q^72 + 2250 * q^74 - 1458 * q^81 - 1414 * q^86 - 1190 * q^88 - 5740 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$.

$n$	$99$	$101$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−0.104356	−	2.82650i	−0.0368954	−	0.999319i
$2$	−0.104356	−	2.82650i	−0.0368954	−	0.999319i
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$	−7.97822	+	0.589925i	−0.997277	+	0.0737406i
$5$		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
$8$	2.50000	+	22.4889i	0.110485	+	0.993878i
$9$	13.5000	+	23.3827i	0.500000	+	0.866025i
$10$		0			0
$11$	22.9129	+	13.2288i	0.628045	+	0.362602i	0.779994	−	0.625786i	$-0.215222\pi$
$11$	22.9129	+	13.2288i	0.628045	+	0.362602i	−0.151950	+	0.988388i	$0.548555\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	63.3040	−	9.41311i	0.989125	−	0.147080i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	64.6824	−	40.5979i	0.846988	−	0.531612i
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$	35.0000	−	66.1438i	0.339183	−	0.640996i
$23$	187.886	−	108.476i	1.70334	−	0.983425i	0.761012	−	0.648737i	$-0.224703\pi$
$23$	187.886	−	108.476i	1.70334	−	0.983425i	0.942329	−	0.334687i	$-0.108631\pi$
$24$		0			0
$25$	−62.5000	+	108.253i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	166.000			1.06295			0.531473	−	0.847075i	$-0.321639\pi$
$29$	166.000			1.06295			0.531473	+	0.847075i	$0.321639\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	−33.2123	−	177.946i	−0.183474	−	0.983025i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−121.500	−	178.588i	−0.562500	−	0.826797i
$37$	225.000	+	389.711i	0.999724	+	1.73157i	0.520223	+	0.854030i	$0.325849\pi$
$37$	225.000	+	389.711i	0.999724	+	1.73157i	0.479500	+	0.877542i	$0.340818\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	534.442i		−	1.89539i	−0.319183	−	0.947693i	$-0.603408\pi$
$43$		−	534.442i		−	1.89539i	0.319183	−	0.947693i	$-0.396592\pi$
$44$	−190.608	−	92.0250i	−0.653073	−	0.315302i
$45$		0			0
$46$	−326.214	−	519.739i	−1.04560	−	1.66590i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$	312.500	+	165.359i	0.883883	+	0.467707i
$51$		0			0
$52$		0			0
$53$	−295.000	+	510.955i	−0.764554	+	1.32425i	0.175928	+	0.984403i	$0.443707\pi$
$53$	−295.000	+	510.955i	−0.764554	+	1.32425i	−0.940482	+	0.339843i	$0.889626\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−17.3231	−	469.199i	−0.0392179	−	1.06222i
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−499.500	+	112.444i	−0.975586	+	0.219618i
$65$		0			0
$66$		0			0
$67$	701.134	+	404.800i	1.27847	+	0.738122i	0.976566	−	0.215218i	$-0.0690461\pi$
$67$	701.134	+	404.800i	1.27847	+	0.738122i	0.301899	+	0.953340i	$0.402379\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			978.928i			1.63630i	0.575004	+	0.818151i	$0.305000\pi$
$71$			978.928i			1.63630i	−0.575004	+	0.818151i	$0.695000\pi$
$72$	−492.101	+	362.057i	−0.805481	+	0.592622i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	1078.04	−	676.632i	1.69351	−	1.06293i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	206.216	−	119.059i	0.293685	−	0.169559i	−0.345918	−	0.938265i	$-0.612432\pi$
$79$	206.216	−	119.059i	0.293685	−	0.169559i	0.639602	+	0.768706i	$0.279099\pi$
$80$		0			0
$81$	−364.500	+	631.333i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−1510.60	+	55.7722i	−1.89410	+	0.0699311i
$87$		0			0
$88$	−240.218	+	548.357i	−0.290992	+	0.664262i
$89$		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
$92$	−1435.00	+	976.282i	−1.62619	+	1.10635i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$			714.353i			0.725204i
$100$	434.777	−	900.538i	0.434777	−	0.900538i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$	1475.00	+	780.497i	1.35155	+	0.715175i
$107$	−1342.69	+	775.205i	−1.21311	+	0.700392i	−0.963436	−	0.267937i	$-0.913658\pi$
$107$	−1342.69	+	775.205i	−1.21311	+	0.700392i	−0.249678	+	0.968329i	$0.580325\pi$
$108$		0			0
$109$	−27.0000	+	46.7654i	−0.0237260	+	0.0410946i	−0.877645	−	0.479312i	$-0.840886\pi$
$109$	−27.0000	+	46.7654i	−0.0237260	+	0.0410946i	0.853919	+	0.520407i	$0.174220\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−670.000			−0.557773			−0.278886	−	0.960324i	$-0.589965\pi$
$113$	−670.000			−0.557773			−0.278886	+	0.960324i	$0.589965\pi$
$114$		0			0
$115$		0			0
$116$	−1324.38	+	97.9276i	−1.06005	+	0.0783823i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−315.500	−	546.462i	−0.237040	−	0.410565i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2047.81i		−	1.43082i	−0.698706	−	0.715409i	$-0.746240\pi$
$127$		−	2047.81i		−	1.43082i	0.698706	−	0.715409i	$-0.253760\pi$
$128$	369.950	+	1400.10i	0.255463	+	0.966819i
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$	1071.00	−	2024.00i	0.690450	−	1.30483i
$135$		0			0
$136$		0			0
$137$	1555.00	−	2693.34i	0.969727	−	1.67962i	0.273388	−	0.961904i	$-0.411856\pi$
$137$	1555.00	−	2693.34i	0.969727	−	1.67962i	0.696339	−	0.717713i	$-0.254811\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	2766.94	−	102.157i	1.63519	−	0.0603721i
$143$		0			0
$144$	1074.71	+	1353.14i	0.621937	+	0.783067i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−2025.00	−	2976.47i	−1.12469	−	1.65314i
$149$	−407.000	−	704.945i	−0.223777	−	0.387593i	0.732175	−	0.681117i	$-0.238505\pi$
$149$	−407.000	−	704.945i	−0.223777	−	0.387593i	−0.955952	+	0.293524i	$0.905172\pi$
$150$		0			0
$151$	−1947.59	−	1124.44i	−1.04962	−	0.606000i	−0.127079	−	0.991893i	$-0.540560\pi$
$151$	−1947.59	−	1124.44i	−1.04962	−	0.606000i	−0.922544	+	0.385893i	$0.873893\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	−358.040	−	570.445i	−0.180279	−	0.287229i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1822.50	+	964.376i	0.883883	+	0.467707i
$163$	3258.21	−	1881.13i	1.56566	−	0.903935i	0.568996	−	0.822340i	$-0.307332\pi$
$163$	3258.21	−	1881.13i	1.56566	−	0.903935i	0.996666	−	0.0815946i	$-0.0260013\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	2197.00			1.00000
$170$		0			0
$171$		0			0
$172$	315.281	+	4263.89i	0.139767	+	1.89023i
$173$		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
$176$	1575.00	+	621.752i	0.674546	+	0.266286i
$177$		0			0
$178$		0			0
$179$	−3734.80	−	2156.29i	−1.55951	−	0.900383i	−0.997304	−	0.0733844i	$-0.976620\pi$
$179$	−3734.80	−	2156.29i	−1.55951	−	0.900383i	−0.562205	−	0.826998i	$-0.690047\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	2909.21	+	3954.15i	1.16560	+	1.58426i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2909.94	+	1680.05i	−1.10239	+	0.636462i	−0.936846	−	0.349741i	$-0.886270\pi$
$191$	−2909.94	+	1680.05i	−1.10239	+	0.636462i	−0.165539	+	0.986203i	$0.552936\pi$
$192$		0			0
$193$	2295.00	−	3975.06i	0.855947	−	1.48254i	−0.0198172	−	0.999804i	$-0.506308\pi$
$193$	2295.00	−	3975.06i	0.855947	−	1.48254i	0.875764	−	0.482740i	$-0.160358\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2210.00			−0.799269			−0.399634	−	0.916675i	$-0.630863\pi$
$197$	−2210.00			−0.799269			−0.399634	+	0.916675i	$0.630863\pi$
$198$	2019.12	−	74.5471i	0.724710	−	0.0267567i
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	−2590.74	−	1134.92i	−0.915966	−	0.401256i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5072.91	+	2928.85i	1.70334	+	0.983425i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			1772.65i			0.578363i	0.957274	+	0.289181i	$0.0933830\pi$
$211$			1772.65i			0.578363i	−0.957274	+	0.289181i	$0.906617\pi$
$212$	2052.15	−	4250.54i	0.664822	−	1.37702i
$213$		0			0
$214$	2331.24	+	3714.23i	0.744673	+	1.18645i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	135.000	+	71.4353i	0.0419420	+	0.0221936i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−3375.00			−1.00000
$226$	69.9186	+	1893.76i	0.0205793	+	0.557393i
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	415.000	+	3733.16i	0.117440	+	1.05644i
$233$	−2365.00	−	4096.30i	−0.664963	−	1.15175i	−0.979295	−	0.202436i	$-0.935114\pi$
$233$	−2365.00	−	4096.30i	−0.664963	−	1.15175i	0.314333	−	0.949313i	$-0.398219\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			449.778i			0.121731i	0.998146	+	0.0608655i	$0.0193861\pi$
$239$			449.778i			0.121731i	−0.998146	+	0.0608655i	$0.980614\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−1511.65	+	948.788i	−0.401540	+	0.252026i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	5740.00			1.42637
$254$	−5788.14	+	213.702i	−1.42984	+	0.0527907i
$255$		0			0
$256$	3918.79	−	1191.77i	0.956735	−	0.290960i
$257$		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
$261$	2241.00	+	3881.53i	0.531473	+	0.920538i
$262$		0			0
$263$	−3487.34	−	2013.42i	−0.817637	−	0.472063i	0.0319637	−	0.999489i	$-0.489824\pi$
$263$	−3487.34	−	2013.42i	−0.817637	−	0.472063i	−0.849601	+	0.527426i	$0.823157\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−5832.60	−	2815.97i	−1.32941	−	0.641838i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$	−7775.00	−	4114.14i	−1.71425	−	0.907097i
$275$	−2864.11	+	1653.59i	−0.628045	+	0.362602i
$276$		0			0
$277$	−3655.00	+	6330.65i	−0.792807	+	1.37318i	0.131415	+	0.991327i	$0.458048\pi$
$277$	−3655.00	+	6330.65i	−0.792807	+	1.37318i	−0.924222	+	0.381855i	$0.875285\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4342.00			−0.921786			−0.460893	−	0.887456i	$-0.652471\pi$
$281$	−4342.00			−0.921786			−0.460893	+	0.887456i	$0.652471\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	−577.494	−	7810.10i	−0.120662	−	1.63185i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	3712.50	−	3178.87i	0.759587	−	0.650405i
$289$	−2456.50	−	4254.78i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	−8201.68	+	6034.28i	−1.61052	+	1.18492i
$297$		0			0
$298$	−1950.05	+	1223.95i	−0.379072	+	0.237925i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−2975.00	+	5622.22i	−0.566861	+	1.07127i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	−1575.00	+	1071.53i	−0.280382	+	0.190754i
$317$	3485.00	+	6036.20i	0.617467	+	1.06948i	0.989946	+	0.141444i	$0.0451745\pi$
$317$	3485.00	+	6036.20i	0.617467	+	1.06948i	−0.372479	+	0.928041i	$0.621492\pi$
$318$		0			0
$319$	3803.54	+	2195.97i	0.667578	+	0.385426i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	2535.62	−	5251.94i	0.434777	−	0.900538i
$325$		0			0
$326$	−5657.03	−	9013.03i	−0.961085	−	1.53124i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4422.19	+	2553.15i	−0.734336	+	0.423969i	−0.820006	−	0.572354i	$-0.806030\pi$
$331$	−4422.19	+	2553.15i	−0.734336	+	0.423969i	0.0856702	+	0.996324i	$0.472697\pi$
$332$		0			0
$333$	−6075.00	+	10522.2i	−0.999724	+	1.73157i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3330.00			0.538269			0.269135	−	0.963103i	$-0.413262\pi$
$337$	3330.00			0.538269			0.269135	+	0.963103i	$0.413262\pi$
$338$	−229.270	−	6209.82i	−0.0368954	−	0.999319i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	12019.0	−	1336.10i	1.88378	−	0.209413i
$345$		0			0
$346$		0			0
$347$	10617.8	+	6130.21i	1.64264	+	0.948377i	0.979891	+	0.199532i	$0.0639420\pi$
$347$	10617.8	+	6130.21i	1.64264	+	0.948377i	0.662745	+	0.748845i	$0.269391\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	1593.02	−	4516.62i	0.241217	−	0.683911i
$353$		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
$358$	−5705.00	+	10781.4i	−0.842231	+	1.59167i
$359$	9463.02	−	5463.48i	1.39120	−	0.803207i	0.397747	−	0.917495i	$-0.369792\pi$
$359$	9463.02	−	5463.48i	1.39120	−	0.803207i	0.993448	+	0.114288i	$0.0364587\pi$
$360$		0			0
$361$	3429.50	−	5940.07i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$	10872.8	−	8635.54i	1.54018	−	1.22326i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6985.00	+	12098.4i	0.969624	+	1.67944i	0.696643	+	0.717418i	$0.254676\pi$
$373$	6985.00	+	12098.4i	0.969624	+	1.67944i	0.272980	+	0.962020i	$0.411991\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	8704.52i		−	1.17974i	−0.807498	−	0.589870i	$-0.799179\pi$
$379$		−	8704.52i		−	1.17974i	0.807498	−	0.589870i	$-0.200821\pi$
$380$		0			0
$381$		0			0
$382$	5052.34	+	8049.61i	0.676702	+	1.07815i
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$	−11475.0	−	6072.00i	−1.51311	−	0.800665i
$387$	12496.7	−	7214.96i	1.64145	−	0.947693i
$388$		0			0
$389$	−5263.00	+	9115.78i	−0.685976	+	1.18815i	0.287153	+	0.957885i	$0.407291\pi$
$389$	−5263.00	+	9115.78i	−0.685976	+	1.18815i	−0.973129	+	0.230261i	$0.926042\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	230.627	+	6246.57i	0.0294894	+	0.798725i
$395$		0			0
$396$	−421.415	−	5699.26i	−0.0534770	−	0.723229i
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	−2937.50	+	7441.18i	−0.367188	+	0.930147i
$401$	799.000	+	1383.91i	0.0995016	+	0.172342i	0.911478	−	0.411348i	$-0.134942\pi$
$401$	799.000	+	1383.91i	0.0995016	+	0.172342i	−0.811977	+	0.583690i	$0.801608\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			11905.9i			1.45001i
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	7749.00	−	14644.2i	0.919910	−	1.73847i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	15262.0			1.76680			0.883402	−	0.468616i	$-0.155247\pi$
$421$	15262.0			1.76680			0.883402	+	0.468616i	$0.155247\pi$
$422$	5010.41	−	184.987i	0.577969	−	0.0213389i
$423$		0			0
$424$	−12228.3	−	5356.83i	−1.40061	−	0.613563i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	10255.0	−	6976.85i	1.15816	−	0.787941i
$429$		0			0
$430$		0			0
$431$	−13587.3	−	7844.65i	−1.51851	−	0.876714i	−0.999763	−	0.0217878i	$-0.993064\pi$
$431$	−13587.3	−	7844.65i	−1.51851	−	0.876714i	−0.518750	−	0.854926i	$-0.673602\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	187.824	−	389.032i	0.0206310	−	0.0427323i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1379.36	+	796.371i	−0.147935	+	0.0854102i	−0.572140	−	0.820156i	$-0.693887\pi$
$443$	−1379.36	+	796.371i	−0.147935	+	0.0854102i	0.424205	+	0.905566i	$0.360553\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2686.00			−0.282317			−0.141158	−	0.989987i	$-0.545083\pi$
$449$	−2686.00			−0.282317			−0.141158	+	0.989987i	$0.545083\pi$
$450$	352.202	+	9539.44i	0.0368954	+	0.999319i
$451$		0			0
$452$	5345.41	−	395.250i	0.556254	−	0.0411305i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4005.00	−	6936.86i	−0.409947	−	0.710050i	0.584936	−	0.811079i	$-0.301120\pi$
$457$	−4005.00	−	6936.86i	−0.409947	−	0.710050i	−0.994883	+	0.101030i	$0.967786\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		−	18049.3i		−	1.81171i	−0.423585	−	0.905856i	$-0.639229\pi$
$463$		−	18049.3i		−	1.81171i	0.423585	−	0.905856i	$-0.360771\pi$
$464$	10508.5	−	1562.58i	1.05139	−	0.156338i
$465$		0			0
$466$	−11331.4	+	7112.15i	−1.12643	+	0.707004i
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7070.00	−	12245.6i	0.687270	−	1.19039i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−15930.0			−1.52911
$478$	1271.30	−	46.9370i	0.121648	−	0.00449132i
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2839.50	+	4173.67i	0.266670	+	0.391968i
$485$		0			0
$486$		0			0
$487$	2854.94	+	1648.30i	0.265647	+	0.153371i	0.626908	−	0.779094i	$-0.284320\pi$
$487$	2854.94	+	1648.30i	0.265647	+	0.153371i	−0.361261	+	0.932465i	$0.617654\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7646.22i		−	0.702788i	−0.936228	−	0.351394i	$-0.885708\pi$
$491$		−	7646.22i		−	0.702788i	0.936228	−	0.351394i	$-0.114292\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−18261.6	+	10543.3i	−1.63828	+	0.945859i	−0.656850	+	0.754022i	$0.728111\pi$
$499$	−18261.6	+	10543.3i	−1.63828	+	0.945859i	−0.981427	+	0.191838i	$0.938555\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$	−599.004	−	16224.1i	−0.0526264	−	1.42540i
$507$		0			0
$508$	1208.06	+	16337.9i	0.105509	+	1.42692i
$509$		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
$512$	−3777.50	−	10952.1i	−0.326062	−	0.945349i
$513$		0			0
$514$		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	$0.666667\pi$
$522$	10737.3	−	6739.25i	0.900302	−	0.565075i
$523$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$	−5327.00	+	10067.1i	−0.441575	+	0.834498i
$527$		0			0
$528$		0			0
$529$	17450.5	−	30225.2i	1.43425	−	2.48419i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−7350.66	+	16779.7i	−0.592351	+	1.35219i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7939.00	−	13750.8i	−0.630914	−	1.09277i	−0.987365	−	0.158461i	$-0.949347\pi$
$541$	−7939.00	−	13750.8i	−0.630914	−	1.09277i	0.356452	−	0.934314i	$-0.383986\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			22049.7i			1.72354i	0.507299	+	0.861770i	$0.330644\pi$
$547$			22049.7i			1.72354i	−0.507299	+	0.861770i	$0.669356\pi$
$548$	−10817.3	+	22405.4i	−0.843231	+	1.74655i
$549$		0			0
$550$	4972.77	+	7922.85i	0.385527	+	0.614239i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	18275.0	+	9670.22i	1.40150	+	0.741603i
$555$		0			0
$556$		0			0
$557$	−10235.0	+	17727.5i	−0.778583	+	1.34855i	0.154175	+	0.988044i	$0.450728\pi$
$557$	−10235.0	+	17727.5i	−0.778583	+	1.34855i	−0.932758	+	0.360502i	$0.882605\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	453.114	+	12272.7i	0.0340097	+	0.921159i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−22015.0	+	2447.32i	−1.62628	+	0.180787i
$569$	−13453.0	−	23301.3i	−0.991176	−	1.71677i	−0.610382	−	0.792107i	$-0.708984\pi$
$569$	−13453.0	−	23301.3i	−0.991176	−	1.71677i	−0.380794	−	0.924660i	$-0.624349\pi$
$570$		0			0
$571$	−22890.0	−	13215.5i	−1.67761	−	0.968569i	−0.963178	−	0.268863i	$-0.913352\pi$
$571$	−22890.0	−	13215.5i	−1.67761	−	0.968569i	−0.714431	−	0.699705i	$-0.753315\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			27119.0i			1.96685i
$576$	−9372.50	−	10161.7i	−0.677988	−	0.735073i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−11769.8	+	7387.31i	−0.846988	+	0.531612i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−13518.6	+	7804.97i	−0.960348	+	0.554457i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	17911.8	+	22552.3i	1.24353	+	1.56570i
$593$		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
$596$	3663.00	+	5384.10i	0.251749	+	0.370036i
$597$		0			0
$598$		0			0
$599$	13633.2	+	7871.11i	0.929943	+	0.536903i	0.886794	−	0.462166i	$-0.152927\pi$
$599$	13633.2	+	7871.11i	0.929943	+	0.536903i	0.0431495	+	0.999069i	$0.486261\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			21859.2i			1.47624i
$604$	16201.7	+	7822.13i	1.09145	+	0.526950i
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	7505.00	−	12999.0i	0.494493	−	0.856487i	−0.505487	−	0.862834i	$-0.668687\pi$
$613$	7505.00	−	12999.0i	0.494493	−	0.856487i	0.999980	+	0.00634752i	$0.00202049\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−30550.0			−1.99335			−0.996675	−	0.0814823i	$-0.974035\pi$
$617$	−30550.0			−1.99335			−0.996675	+	0.0814823i	$0.974035\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	17858.8i		−	1.12670i	−0.826218	−	0.563351i	$-0.809512\pi$
$631$		−	17858.8i		−	1.12670i	0.826218	−	0.563351i	$-0.190488\pi$
$632$	3193.04	+	4339.92i	0.200969	+	0.273153i
$633$		0			0
$634$	16697.6	−	10480.3i	1.04597	−	0.656506i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	5810.00	−	10979.9i	0.360533	−	0.681343i
$639$	−22890.0	+	13215.5i	−1.41708	+	0.818151i
$640$		0			0
$641$	4439.00	−	7688.57i	0.273526	−	0.473760i	−0.696236	−	0.717813i	$-0.745143\pi$
$641$	4439.00	−	7688.57i	0.273526	−	0.473760i	0.969762	+	0.244052i	$0.0784768\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	−15109.2	−	6618.87i	−0.915966	−	0.401256i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−24885.0	+	16930.2i	−1.49474	+	1.01693i
$653$	13525.0	+	23426.0i	0.810527	+	1.40387i	0.912496	+	0.409086i	$0.134153\pi$
$653$	13525.0	+	23426.0i	0.810527	+	1.40387i	−0.101969	+	0.994788i	$0.532514\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	33786.2i		−	1.99716i	−0.0533186	−	0.998578i	$-0.516980\pi$
$659$		−	33786.2i		−	1.99716i	0.0533186	−	0.998578i	$-0.483020\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	7677.96	+	12232.9i	0.450774	+	0.718194i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	30375.0	+	16072.9i	1.76728	+	0.935156i
$667$	31189.0	−	18007.0i	1.81056	−	1.04533i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	33570.0			1.92278			0.961388	−	0.275196i	$-0.0887428\pi$
$673$	33570.0			1.92278			0.961388	+	0.275196i	$0.0887428\pi$
$674$	−347.506	−	9412.25i	−0.0198597	−	0.537903i
$675$		0			0
$676$	−17528.1	+	1296.07i	−0.997277	+	0.0737406i
$677$		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
$683$	9261.39	+	5347.06i	0.518854	+	0.299560i	0.736466	−	0.676475i	$-0.236493\pi$
$683$	9261.39	+	5347.06i	0.518854	+	0.299560i	−0.217612	+	0.976035i	$0.569827\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−5030.76	−	33832.3i	−0.278773	−	1.87477i
$689$		0			0
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		0			0
$694$	16219.0	−	30651.0i	0.887125	−	1.67651i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4198.00			0.226186			0.113093	−	0.993584i	$-0.463924\pi$
$701$	4198.00			0.226186			0.113093	+	0.993584i	$0.463924\pi$
$702$		0			0
$703$		0			0
$704$	−12932.5	−	4031.34i	−0.692346	−	0.215819i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6273.00	+	10865.2i	0.332281	+	0.575528i	0.982959	−	0.183826i	$-0.0588483\pi$
$709$	6273.00	+	10865.2i	0.332281	+	0.575528i	−0.650677	+	0.759354i	$0.725515\pi$
$710$		0			0
$711$	5567.83	+	3214.59i	0.293685	+	0.169559i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	31069.1	+	15000.1i	1.62166	+	0.782932i
$717$		0			0
$718$	−16430.0	−	26177.1i	−0.853989	−	1.36061i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$	−17147.5	−	9073.60i	−0.883883	−	0.467707i
$723$		0			0
$724$		0			0
$725$	−10375.0	+	17970.0i	−0.531473	+	0.920538i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−19683.0			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	−25543.0	−	29830.8i	−1.27925	−	1.49399i
$737$	10710.0	+	18550.3i	0.535289	+	0.927148i
$738$		0			0
$739$	−27014.3	−	15596.7i	−1.34470	−	0.776365i	−0.357211	−	0.934024i	$-0.616272\pi$
$739$	−27014.3	−	15596.7i	−1.34470	−	0.776365i	−0.987494	+	0.157658i	$0.949605\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	31743.7i		−	1.56738i	−0.621151	−	0.783691i	$-0.713335\pi$
$743$		−	31743.7i		−	1.56738i	0.621151	−	0.783691i	$-0.286665\pi$
$744$		0			0
$745$		0			0
$746$	33467.1	−	21005.7i	1.64252	−	1.03093i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	35583.7	−	20544.3i	1.72898	−	0.998230i	0.834756	−	0.550620i	$-0.185609\pi$
$751$	35583.7	−	20544.3i	1.72898	−	0.998230i	0.894229	−	0.447610i	$-0.147725\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	34830.0			1.67228			0.836141	−	0.548514i	$-0.184806\pi$
$757$	34830.0			1.67228			0.836141	+	0.548514i	$0.184806\pi$
$758$	−24603.3	+	908.370i	−1.17894	+	0.0435270i
$759$		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
$764$	22225.0	−	15120.5i	1.05245	−	0.716020i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−15965.0	+	33067.8i	−0.744293	+	1.54162i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$	−21697.2	−	34569.0i	−1.00761	−	1.60537i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	26315.0	+	13924.6i	1.21265	+	0.641672i
$779$		0			0
$780$		0			0
$781$	−12950.0	+	22430.1i	−0.593326	+	1.02767i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$	17631.9	−	1303.73i	0.797093	−	0.0589386i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−16065.0	+	1785.88i	−0.720764	+	0.0801244i
$793$		0			0
$794$		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
$800$	21339.0	+	7526.32i	0.943061	+	0.332619i
$801$		0			0
$802$	3828.24	−	2402.79i	0.168553	−	0.105793i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18677.0	+	32349.5i	−0.811679	+	1.40587i	0.100009	+	0.994987i	$0.468113\pi$
$809$	−18677.0	+	32349.5i	−0.811679	+	1.40587i	−0.911688	+	0.410883i	$0.865221\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	33652.0	−	1242.45i	1.44902	−	0.0534986i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21769.0	+	37705.0i	0.925388	+	1.60282i	0.790936	+	0.611898i	$0.209594\pi$
$821$	21769.0	+	37705.0i	0.925388	+	1.60282i	0.134451	+	0.990920i	$0.457073\pi$
$822$		0			0
$823$	−8289.88	−	4786.16i	−0.351114	−	0.202716i	0.314062	−	0.949403i	$-0.398310\pi$
$823$	−8289.88	−	4786.16i	−0.351114	−	0.202716i	−0.665176	+	0.746687i	$0.731643\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			41077.9i			1.72723i	0.504151	+	0.863615i	$0.331805\pi$
$827$			41077.9i			1.72723i	−0.504151	+	0.863615i	$0.668195\pi$
$828$	−42200.6	−	20374.3i	−1.77122	−	0.855142i
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	3167.00			0.129854
$842$	−1592.68	−	43138.1i	−0.0651870	−	1.76560i
$843$		0			0
$844$	−1045.73	−	14142.6i	−0.0426488	−	0.576788i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−13865.0	+	35122.3i	−0.561469	+	1.42230i
$849$		0			0
$850$		0			0
$851$	84548.5	+	48814.1i	3.40574	+	1.96631i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−20790.2	−	28257.7i	−0.830135	−	1.12830i
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$	−20755.0	+	39223.3i	−0.820091	+	1.54983i
$863$	40276.3	−	23253.5i	1.58867	−	0.917217i	0.595140	−	0.803622i	$-0.297097\pi$
$863$	40276.3	−	23253.5i	1.58867	−	0.917217i	0.993527	−	0.113595i	$-0.0362368\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6300.00			0.245930
$870$		0			0
$871$		0			0
$872$	−1119.20	−	490.286i	−0.0434644	−	0.0190404i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−3275.00	−	5672.47i	−0.126099	−	0.218410i	0.796063	−	0.605214i	$-0.206912\pi$
$877$	−3275.00	−	5672.47i	−0.126099	−	0.218410i	−0.922162	+	0.386804i	$0.873579\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		−	43014.6i		−	1.63936i	−0.572820	−	0.819681i	$-0.694150\pi$
$883$		−	43014.6i		−	1.63936i	0.572820	−	0.819681i	$-0.305850\pi$
$884$		0			0
$885$		0			0
$886$	2394.89	+	3815.64i	0.0908102	+	0.144683i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−16703.5	+	9643.76i	−0.628045	+	0.362602i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	280.300	+	7591.98i	0.0104162	+	0.282124i
$899$		0			0
$900$	26926.5	−	1991.00i	0.997277	−	0.0737406i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1675.00	−	15067.6i	−0.0616257	−	0.554358i
$905$		0			0
$906$		0			0
$907$	12340.9	+	7125.01i	0.451788	+	0.260840i	0.708585	−	0.705625i	$-0.249334\pi$
$907$	12340.9	+	7125.01i	0.451788	+	0.260840i	−0.256797	+	0.966465i	$0.582667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			38125.3i			1.38655i	0.720673	+	0.693275i	$0.243833\pi$
$911$			38125.3i			1.38655i	−0.720673	+	0.693275i	$0.756167\pi$
$912$		0			0
$913$		0			0
$914$	−19189.1	+	12044.0i	−0.694441	+	0.435866i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−44428.1	+	25650.6i	−1.59472	+	0.920711i	−0.602238	+	0.798317i	$0.705724\pi$
$919$	−44428.1	+	25650.6i	−1.59472	+	0.920711i	−0.992482	+	0.122395i	$0.960943\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−56250.0			−1.99945
$926$	−51016.4	+	1883.56i	−1.81048	+	0.0668439i
$927$		0			0
$928$	−5513.24	−	29539.1i	−0.195023	−	1.04490i
$929$		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
$932$	21285.0	+	31286.0i	0.748083	+	1.09958i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		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
$946$	−35350.0	−	18705.5i	−1.21493	−	0.642883i
$947$	−27573.4	+	15919.5i	−0.946160	+	0.546266i	−0.891886	−	0.452260i	$-0.850618\pi$
$947$	−27573.4	+	15919.5i	−0.946160	+	0.546266i	−0.0542742	+	0.998526i	$0.517285\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29290.0			0.995589			0.497794	−	0.867295i	$-0.334143\pi$
$953$	29290.0			0.995589			0.497794	+	0.867295i	$0.334143\pi$
$954$	1662.39	+	45026.2i	0.0564171	+	1.52807i
$955$		0			0
$956$	−265.335	−	3588.43i	−0.00897652	−	0.121400i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	14895.5	+	25799.8i	0.500000	+	0.866025i
$962$		0			0
$963$	−36252.8	−	20930.5i	−1.21311	−	0.700392i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			30145.7i			1.00250i	0.865302	+	0.501251i	$0.167127\pi$
$967$			30145.7i			1.00250i	−0.865302	+	0.501251i	$0.832873\pi$
$968$	11500.6	−	8461.40i	0.381862	−	0.280950i
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$	4361.00	−	8241.52i	0.143466	−	0.271124i
$975$		0			0
$976$		0			0
$977$	−18745.0	+	32467.3i	−0.613824	+	1.06317i	0.376766	+	0.926308i	$0.377036\pi$
$977$	−18745.0	+	32467.3i	−0.613824	+	1.06317i	−0.990590	+	0.136865i	$0.956297\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1458.00			−0.0474519
$982$	−21612.1	+	797.930i	−0.702310	+	0.0259297i
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−57974.0	−	100414.i	−1.86397	−	3.22849i
$990$		0			0
$991$	−20919.5	−	12077.9i	−0.670564	−	0.387150i	0.125727	−	0.992065i	$-0.459874\pi$
$991$	−20919.5	−	12077.9i	−0.670564	−	0.387150i	−0.796290	+	0.604915i	$0.793207\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	31706.4	+	50516.1i	1.00566	+	1.60226i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.4.f.a.31.2		4
4.3	odd	2	inner	196.4.f.a.31.1		4
7.2	even	3	inner	196.4.f.a.19.1		4
7.3	odd	6		28.4.d.a.27.2	yes	2
7.4	even	3		28.4.d.a.27.2	yes	2
7.5	odd	6	inner	196.4.f.a.19.1		4
7.6	odd	2	CM	196.4.f.a.31.2		4
21.11	odd	6		252.4.b.a.55.1		2
21.17	even	6		252.4.b.a.55.1		2
28.3	even	6		28.4.d.a.27.1	✓	2
28.11	odd	6		28.4.d.a.27.1	✓	2
28.19	even	6	inner	196.4.f.a.19.2		4
28.23	odd	6	inner	196.4.f.a.19.2		4
28.27	even	2	inner	196.4.f.a.31.1		4
56.3	even	6		448.4.f.a.447.2		2
56.11	odd	6		448.4.f.a.447.2		2
56.45	odd	6		448.4.f.a.447.1		2
56.53	even	6		448.4.f.a.447.1		2
84.11	even	6		252.4.b.a.55.2		2
84.59	odd	6		252.4.b.a.55.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.d.a.27.1	✓	2	28.3	even	6
28.4.d.a.27.1	✓	2	28.11	odd	6
28.4.d.a.27.2	yes	2	7.3	odd	6
28.4.d.a.27.2	yes	2	7.4	even	3
196.4.f.a.19.1		4	7.2	even	3	inner
196.4.f.a.19.1		4	7.5	odd	6	inner
196.4.f.a.19.2		4	28.19	even	6	inner
196.4.f.a.19.2		4	28.23	odd	6	inner
196.4.f.a.31.1		4	4.3	odd	2	inner
196.4.f.a.31.1		4	28.27	even	2	inner
196.4.f.a.31.2		4	1.1	even	1	trivial
196.4.f.a.31.2		4	7.6	odd	2	CM
252.4.b.a.55.1		2	21.11	odd	6
252.4.b.a.55.1		2	21.17	even	6
252.4.b.a.55.2		2	84.11	even	6
252.4.b.a.55.2		2	84.59	odd	6
448.4.f.a.447.1		2	56.45	odd	6
448.4.f.a.447.1		2	56.53	even	6
448.4.f.a.447.2		2	56.3	even	6
448.4.f.a.447.2		2	56.11	odd	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:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	196.f (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.104356 - 2.82650i) q^{2} +(-7.97822 + 0.589925i) q^{4} +(2.50000 + 22.4889i) q^{8} +(13.5000 + 23.3827i) q^{9} +O(q^{10})\)	\(q+(-0.104356 - 2.82650i) q^{2} +(-7.97822 + 0.589925i) q^{4} +(2.50000 + 22.4889i) q^{8} +(13.5000 + 23.3827i) q^{9} +(22.9129 + 13.2288i) q^{11} +(63.3040 - 9.41311i) q^{16} +(64.6824 - 40.5979i) q^{18} +(35.0000 - 66.1438i) q^{22} +(187.886 - 108.476i) q^{23} +(-62.5000 + 108.253i) q^{25} +166.000 q^{29} +(-33.2123 - 177.946i) q^{32} +(-121.500 - 178.588i) q^{36} +(225.000 + 389.711i) q^{37} -534.442i q^{43} +(-190.608 - 92.0250i) q^{44} +(-326.214 - 519.739i) q^{46} +(312.500 + 165.359i) q^{50} +(-295.000 + 510.955i) q^{53} +(-17.3231 - 469.199i) q^{58} +(-499.500 + 112.444i) q^{64} +(701.134 + 404.800i) q^{67} +978.928i q^{71} +(-492.101 + 362.057i) q^{72} +(1078.04 - 676.632i) q^{74} +(206.216 - 119.059i) q^{79} +(-364.500 + 631.333i) q^{81} +(-1510.60 + 55.7722i) q^{86} +(-240.218 + 548.357i) q^{88} +(-1435.00 + 976.282i) q^{92} +714.353i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} - 9 q^{4} + 10 q^{8} + 54 q^{9}+O(q^{10}) \) 4 * q - 5 * q^2 - 9 * q^4 + 10 * q^8 + 54 * q^9	\( 4 q - 5 q^{2} - 9 q^{4} + 10 q^{8} + 54 q^{9} + 47 q^{16} + 135 q^{18} + 140 q^{22} - 250 q^{25} + 664 q^{29} + 275 q^{32} - 486 q^{36} + 900 q^{37} - 350 q^{44} + 574 q^{46} + 1250 q^{50} - 1180 q^{53} - 830 q^{58} - 1998 q^{64} + 135 q^{72} + 2250 q^{74} - 1458 q^{81} - 1414 q^{86} - 1190 q^{88} - 5740 q^{92}+O(q^{100}) \) 4 * q - 5 * q^2 - 9 * q^4 + 10 * q^8 + 54 * q^9 + 47 * q^16 + 135 * q^18 + 140 * q^22 - 250 * q^25 + 664 * q^29 + 275 * q^32 - 486 * q^36 + 900 * q^37 - 350 * q^44 + 574 * q^46 + 1250 * q^50 - 1180 * q^53 - 830 * q^58 - 1998 * q^64 + 135 * q^72 + 2250 * q^74 - 1458 * q^81 - 1414 * q^86 - 1190 * q^88 - 5740 * q^92

Introduction

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