LMFDB - Embedded newform 297.3.c.b.109.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [297,3,Mod(109,297)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("297.109");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	297.c (of order $2$, degree $1$, 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:	$8.09266385150$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 420x^{12} + 1908x^{10} + 20196x^{8} - 91800x^{6} + 597348x^{4} - 4428432x^{2} + 8714304 $ x^16 + 24x^14 + 420x^12 + 1908x^10 + 20196x^8 - 91800x^6 + 597348x^4 - 4428432*x^2 + 8714304
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.11
Root			$-1.73205 + 1.64800i$ of defining polynomial
Character	$\chi$	$=$	297.109
Dual form			297.3.c.b.109.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.64800i q^{2} +1.28411 q^{4} -8.56484 q^{5} -4.92814i q^{7} +8.70819i q^{8} +O(q^{10})$	\(q+1.64800i q^{2} +1.28411 q^{4} -8.56484 q^{5} -4.92814i q^{7} +8.70819i q^{8} -14.1148i q^{10} +(-3.31180 - 10.4896i) q^{11} +8.08116i q^{13} +8.12157 q^{14} -9.21465 q^{16} -23.1444i q^{17} -32.6757i q^{19} -10.9982 q^{20} +(17.2868 - 5.45784i) q^{22} -1.91954 q^{23} +48.3566 q^{25} -13.3177 q^{26} -6.32826i q^{28} -25.9232i q^{29} -52.7189 q^{31} +19.6471i q^{32} +38.1419 q^{34} +42.2088i q^{35} -38.9360 q^{37} +53.8495 q^{38} -74.5843i q^{40} +35.7622i q^{41} -17.7961i q^{43} +(-4.25271 - 13.4698i) q^{44} -3.16339i q^{46} -7.22407 q^{47} +24.7134 q^{49} +79.6914i q^{50} +10.3771i q^{52} +80.3440 q^{53} +(28.3651 + 89.8419i) q^{55} +42.9152 q^{56} +42.7214 q^{58} -40.5905 q^{59} -33.9019i q^{61} -86.8805i q^{62} -69.2369 q^{64} -69.2138i q^{65} +3.57697 q^{67} -29.7199i q^{68} -69.5599 q^{70} -96.5057 q^{71} +39.0605i q^{73} -64.1664i q^{74} -41.9591i q^{76} +(-51.6943 + 16.3210i) q^{77} +118.280i q^{79} +78.9220 q^{80} -58.9360 q^{82} -5.50116i q^{83} +198.228i q^{85} +29.3279 q^{86} +(91.3455 - 28.8398i) q^{88} -7.57067 q^{89} +39.8251 q^{91} -2.46489 q^{92} -11.9052i q^{94} +279.863i q^{95} -97.0354 q^{97} +40.7276i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4}+O(q^{10}) $ 16 * q - 32 * q^4	$ 16 q - 32 q^{4} + 160 q^{16} + 102 q^{22} + 12 q^{25} + 68 q^{31} + 156 q^{34} - 124 q^{37} - 272 q^{49} + 182 q^{55} + 492 q^{58} - 680 q^{64} - 400 q^{67} + 324 q^{70} - 444 q^{82} - 510 q^{88} + 120 q^{91} + 152 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 160 * q^16 + 102 * q^22 + 12 * q^25 + 68 * q^31 + 156 * q^34 - 124 * q^37 - 272 * q^49 + 182 * q^55 + 492 * q^58 - 680 * q^64 - 400 * q^67 + 324 * q^70 - 444 * q^82 - 510 * q^88 + 120 * q^91 + 152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.64800i			0.823998i	0.911184	+	0.411999i	$0.135169\pi$
$2$			1.64800i			0.823998i	−0.911184	+	0.411999i	$0.864831\pi$
$3$		0			0
$3$		0			0
$4$	1.28411			0.321027
$5$	−8.56484			−1.71297			−0.856484	−	0.516173i	$-0.827356\pi$
$5$	−8.56484			−1.71297			−0.856484	+	0.516173i	$0.827356\pi$
$6$		0			0
$7$		−	4.92814i		−	0.704021i	−0.935996	−	0.352010i	$-0.885498\pi$
$7$		−	4.92814i		−	0.704021i	0.935996	−	0.352010i	$-0.114502\pi$
$8$			8.70819i			1.08852i
$9$		0			0
$10$		−	14.1148i		−	1.41148i
$11$	−3.31180	−	10.4896i	−0.301073	−	0.953601i
$11$	−3.31180	−	10.4896i	−0.301073	−	0.953601i
$12$		0			0
$13$			8.08116i			0.621627i	0.950471	+	0.310814i	$0.100602\pi$
$13$			8.08116i			0.621627i	−0.950471	+	0.310814i	$0.899398\pi$
$14$	8.12157			0.580112
$15$		0			0
$16$	−9.21465			−0.575915
$17$		−	23.1444i		−	1.36144i	−0.732546	−	0.680718i	$-0.761668\pi$
$17$		−	23.1444i		−	1.36144i	0.732546	−	0.680718i	$-0.238332\pi$
$18$		0			0
$19$		−	32.6757i		−	1.71978i	−0.510483	−	0.859888i	$-0.670534\pi$
$19$		−	32.6757i		−	1.71978i	0.510483	−	0.859888i	$-0.329466\pi$
$20$	−10.9982			−0.549909
$21$		0			0
$22$	17.2868	−	5.45784i	0.785766	−	0.248084i
$23$	−1.91954			−0.0834582			−0.0417291	−	0.999129i	$-0.513287\pi$
$23$	−1.91954			−0.0834582			−0.0417291	+	0.999129i	$0.513287\pi$
$24$		0			0
$25$	48.3566			1.93426
$26$	−13.3177			−0.512220
$27$		0			0
$28$		−	6.32826i		−	0.226009i
$29$		−	25.9232i		−	0.893904i	−0.894558	−	0.446952i	$-0.852509\pi$
$29$		−	25.9232i		−	0.893904i	0.894558	−	0.446952i	$-0.147491\pi$
$30$		0			0
$31$	−52.7189			−1.70061			−0.850305	−	0.526291i	$-0.823582\pi$
$31$	−52.7189			−1.70061			−0.850305	+	0.526291i	$0.823582\pi$
$32$			19.6471i			0.613970i
$33$		0			0
$34$	38.1419			1.12182
$35$			42.2088i			1.20597i
$36$		0			0
$37$	−38.9360			−1.05232			−0.526162	−	0.850384i	$-0.676370\pi$
$37$	−38.9360			−1.05232			−0.526162	+	0.850384i	$0.676370\pi$
$38$	53.8495			1.41709
$39$		0			0
$40$		−	74.5843i		−	1.86461i
$41$			35.7622i			0.872249i	0.899886	+	0.436125i	$0.143649\pi$
$41$			35.7622i			0.872249i	−0.899886	+	0.436125i	$0.856351\pi$
$42$		0			0
$43$		−	17.7961i		−	0.413862i	−0.978356	−	0.206931i	$-0.933652\pi$
$43$		−	17.7961i		−	0.413862i	0.978356	−	0.206931i	$-0.0663476\pi$
$44$	−4.25271	−	13.4698i	−0.0966524	−	0.306131i
$45$		0			0
$46$		−	3.16339i		−	0.0687694i
$47$	−7.22407			−0.153704			−0.0768518	−	0.997043i	$-0.524487\pi$
$47$	−7.22407			−0.153704			−0.0768518	+	0.997043i	$0.524487\pi$
$48$		0			0
$49$	24.7134			0.504355
$50$			79.6914i			1.59383i
$51$		0			0
$52$			10.3771i			0.199559i
$53$	80.3440			1.51593			0.757963	−	0.652298i	$-0.226195\pi$
$53$	80.3440			1.51593			0.757963	+	0.652298i	$0.226195\pi$
$54$		0			0
$55$	28.3651	+	89.8419i	0.515729	+	1.63349i
$56$	42.9152			0.766343
$57$		0			0
$58$	42.7214			0.736575
$59$	−40.5905			−0.687974			−0.343987	−	0.938974i	$-0.611778\pi$
$59$	−40.5905			−0.687974			−0.343987	+	0.938974i	$0.611778\pi$
$60$		0			0
$61$		−	33.9019i		−	0.555768i	−0.960615	−	0.277884i	$-0.910367\pi$
$61$		−	33.9019i		−	0.555768i	0.960615	−	0.277884i	$-0.0896331\pi$
$62$		−	86.8805i		−	1.40130i
$63$		0			0
$64$	−69.2369			−1.08183
$65$		−	69.2138i		−	1.06483i
$66$		0			0
$67$	3.57697			0.0533876			0.0266938	−	0.999644i	$-0.491502\pi$
$67$	3.57697			0.0533876			0.0266938	+	0.999644i	$0.491502\pi$
$68$		−	29.7199i		−	0.437057i
$69$		0			0
$70$	−69.5599			−0.993713
$71$	−96.5057			−1.35924			−0.679618	−	0.733566i	$-0.737854\pi$
$71$	−96.5057			−1.35924			−0.679618	+	0.733566i	$0.737854\pi$
$72$		0			0
$73$			39.0605i			0.535076i	0.963547	+	0.267538i	$0.0862100\pi$
$73$			39.0605i			0.535076i	−0.963547	+	0.267538i	$0.913790\pi$
$74$		−	64.1664i		−	0.867114i
$75$		0			0
$76$		−	41.9591i		−	0.552094i
$77$	−51.6943	+	16.3210i	−0.671355	+	0.211962i
$78$		0			0
$79$			118.280i			1.49721i	0.663014	+	0.748607i	$0.269277\pi$
$79$			118.280i			1.49721i	−0.663014	+	0.748607i	$0.730723\pi$
$80$	78.9220			0.986525
$81$		0			0
$82$	−58.9360			−0.718732
$83$		−	5.50116i		−	0.0662791i	−0.999451	−	0.0331395i	$-0.989449\pi$
$83$		−	5.50116i		−	0.0662791i	0.999451	−	0.0331395i	$-0.0105506\pi$
$84$		0			0
$85$			198.228i			2.33210i
$86$	29.3279			0.341022
$87$		0			0
$88$	91.3455	−	28.8398i	1.03802	−	0.327725i
$89$	−7.57067			−0.0850637			−0.0425319	−	0.999095i	$-0.513542\pi$
$89$	−7.57067			−0.0850637			−0.0425319	+	0.999095i	$0.513542\pi$
$90$		0			0
$91$	39.8251			0.437638
$92$	−2.46489			−0.0267923
$93$		0			0
$94$		−	11.9052i		−	0.126651i
$95$			279.863i			2.94592i
$96$		0			0
$97$	−97.0354			−1.00037			−0.500183	−	0.865920i	$-0.666734\pi$
$97$	−97.0354			−1.00037			−0.500183	+	0.865920i	$0.666734\pi$
$98$			40.7276i			0.415588i
$99$		0			0
$100$	62.0950			0.620950
$101$		−	135.569i		−	1.34227i	−0.741337	−	0.671133i	$-0.765808\pi$
$101$		−	135.569i		−	1.34227i	0.741337	−	0.671133i	$-0.234192\pi$
$102$		0			0
$103$	−101.646			−0.986856			−0.493428	−	0.869787i	$-0.664256\pi$
$103$	−101.646			−0.986856			−0.493428	+	0.869787i	$0.664256\pi$
$104$	−70.3722			−0.676656
$105$		0			0
$106$			132.407i			1.24912i
$107$		−	2.51196i		−	0.0234763i	−0.999931	−	0.0117381i	$-0.996264\pi$
$107$		−	2.51196i		−	0.0234763i	0.999931	−	0.0117381i	$-0.00373645\pi$
$108$		0			0
$109$			90.5560i			0.830789i	0.909641	+	0.415394i	$0.136356\pi$
$109$			90.5560i			0.830789i	−0.909641	+	0.415394i	$0.863644\pi$
$110$	−148.059	+	46.7455i	−1.34599	+	0.424959i
$111$		0			0
$112$			45.4111i			0.405456i
$113$	−3.71435			−0.0328704			−0.0164352	−	0.999865i	$-0.505232\pi$
$113$	−3.71435			−0.0328704			−0.0164352	+	0.999865i	$0.505232\pi$
$114$		0			0
$115$	16.4405			0.142961
$116$		−	33.2882i		−	0.286967i
$117$		0			0
$118$		−	66.8929i		−	0.566889i
$119$	−114.059			−0.958479
$120$		0			0
$121$	−99.0639	+	69.4790i	−0.818710	+	0.574207i
$122$	55.8702			0.457952
$123$		0			0
$124$	−67.6967			−0.545941
$125$	−200.045			−1.60036
$126$		0			0
$127$		−	231.179i		−	1.82031i	−0.414267	−	0.910155i	$-0.635962\pi$
$127$		−	231.179i		−	1.82031i	0.414267	−	0.910155i	$-0.364038\pi$
$128$		−	35.5139i		−	0.277452i
$129$		0			0
$130$	114.064			0.877417
$131$		−	99.2565i		−	0.757683i	−0.925462	−	0.378842i	$-0.876323\pi$
$131$		−	99.2565i		−	0.757683i	0.925462	−	0.378842i	$-0.123677\pi$
$132$		0			0
$133$	−161.031			−1.21076
$134$			5.89484i			0.0439913i
$135$		0			0
$136$	201.546			1.48196
$137$	225.206			1.64384			0.821919	−	0.569604i	$-0.192904\pi$
$137$	225.206			1.64384			0.821919	+	0.569604i	$0.192904\pi$
$138$		0			0
$139$			86.3839i			0.621467i	0.950497	+	0.310733i	$0.100575\pi$
$139$			86.3839i			0.621467i	−0.950497	+	0.310733i	$0.899425\pi$
$140$			54.2006i			0.387147i
$141$		0			0
$142$		−	159.041i		−	1.12001i
$143$	84.7682	−	26.7632i	0.592785	−	0.187155i
$144$		0			0
$145$			222.028i			1.53123i
$146$	−64.3716			−0.440901
$147$		0			0
$148$	−49.9980			−0.337824
$149$		−	155.258i		−	1.04200i	−0.853557	−	0.521000i	$-0.825559\pi$
$149$		−	155.258i		−	1.04200i	0.853557	−	0.521000i	$-0.174441\pi$
$150$		0			0
$151$			63.4603i			0.420267i	0.977673	+	0.210134i	$0.0673899\pi$
$151$			63.4603i			0.420267i	−0.977673	+	0.210134i	$0.932610\pi$
$152$	284.546			1.87202
$153$		0			0
$154$	−26.8970	−	85.1921i	−0.174656	−	0.553195i
$155$	451.529			2.91309
$156$		0			0
$157$	56.2874			0.358518			0.179259	−	0.983802i	$-0.442630\pi$
$157$	56.2874			0.358518			0.179259	+	0.983802i	$0.442630\pi$
$158$	−194.925			−1.23370
$159$		0			0
$160$		−	168.274i		−	1.05171i
$161$			9.45976i			0.0587563i
$162$		0			0
$163$	246.748			1.51379			0.756896	−	0.653535i	$-0.226715\pi$
$163$	246.748			1.51379			0.756896	+	0.653535i	$0.226715\pi$
$164$			45.9225i			0.280015i
$165$		0			0
$166$	9.06590			0.0546138
$167$			10.4392i			0.0625101i	0.999511	+	0.0312551i	$0.00995041\pi$
$167$			10.4392i			0.0625101i	−0.999511	+	0.0312551i	$0.990050\pi$
$168$		0			0
$169$	103.695			0.613579
$170$	−326.680			−1.92164
$171$		0			0
$172$		−	22.8521i		−	0.132861i
$173$		−	111.487i		−	0.644431i	−0.946666	−	0.322215i	$-0.895572\pi$
$173$		−	111.487i		−	0.644431i	0.946666	−	0.322215i	$-0.104428\pi$
$174$		0			0
$175$		−	238.308i		−	1.36176i
$176$	30.5171	+	96.6581i	0.173393	+	0.549194i
$177$		0			0
$178$		−	12.4764i		−	0.0700924i
$179$	137.962			0.770737			0.385369	−	0.922763i	$-0.374074\pi$
$179$	137.962			0.770737			0.385369	+	0.922763i	$0.374074\pi$
$180$		0			0
$181$	124.512			0.687914			0.343957	−	0.938985i	$-0.388233\pi$
$181$	124.512			0.687914			0.343957	+	0.938985i	$0.388233\pi$
$182$			65.6316i			0.360613i
$183$		0			0
$184$		−	16.7157i		−	0.0908462i
$185$	333.481			1.80260
$186$		0			0
$187$	−242.776	+	76.6497i	−1.29827	+	0.409891i
$188$	−9.27647			−0.0493429
$189$		0			0
$190$	−461.213			−2.42743
$191$	−160.750			−0.841625			−0.420813	−	0.907148i	$-0.638255\pi$
$191$	−160.750			−0.841625			−0.420813	+	0.907148i	$0.638255\pi$
$192$		0			0
$193$		−	203.277i		−	1.05325i	−0.850098	−	0.526624i	$-0.823457\pi$
$193$		−	203.277i		−	1.05325i	0.850098	−	0.526624i	$-0.176543\pi$
$194$		−	159.914i		−	0.824299i
$195$		0			0
$196$	31.7346			0.161911
$197$			306.038i			1.55349i	0.629815	+	0.776745i	$0.283131\pi$
$197$			306.038i			1.55349i	−0.629815	+	0.776745i	$0.716869\pi$
$198$		0			0
$199$	−28.8638			−0.145044			−0.0725221	−	0.997367i	$-0.523105\pi$
$199$	−28.8638			−0.145044			−0.0725221	+	0.997367i	$0.523105\pi$
$200$			421.098i			2.10549i
$201$		0			0
$202$	223.417			1.10603
$203$	−127.753			−0.629327
$204$		0			0
$205$		−	306.298i		−	1.49414i
$206$		−	167.513i		−	0.813168i
$207$		0			0
$208$		−	74.4650i		−	0.358005i
$209$	−342.756	+	108.216i	−1.63998	+	0.517778i
$210$		0			0
$211$		−	309.244i		−	1.46561i	−0.680439	−	0.732805i	$-0.738211\pi$
$211$		−	309.244i		−	1.46561i	0.680439	−	0.732805i	$-0.261789\pi$
$212$	103.170			0.486652
$213$		0			0
$214$	4.13970			0.0193444
$215$			152.421i			0.708933i
$216$		0			0
$217$			259.806i			1.19726i
$218$	−149.236			−0.684568
$219$		0			0
$220$	36.4238	+	115.367i	0.165563	+	0.524393i
$221$	187.034			0.846306
$222$		0			0
$223$	122.622			0.549874			0.274937	−	0.961462i	$-0.411343\pi$
$223$	122.622			0.549874			0.274937	+	0.961462i	$0.411343\pi$
$224$	96.8235			0.432248
$225$		0			0
$226$		−	6.12124i		−	0.0270851i
$227$		−	286.184i		−	1.26072i	−0.776302	−	0.630362i	$-0.782907\pi$
$227$		−	286.184i		−	1.26072i	0.776302	−	0.630362i	$-0.217093\pi$
$228$		0			0
$229$	254.856			1.11291			0.556455	−	0.830878i	$-0.312161\pi$
$229$	254.856			1.11291			0.556455	+	0.830878i	$0.312161\pi$
$230$			27.0940i			0.117800i
$231$		0			0
$232$	225.744			0.973036
$233$		−	46.8199i		−	0.200944i	−0.994940	−	0.100472i	$-0.967965\pi$
$233$		−	46.8199i		−	0.200944i	0.994940	−	0.100472i	$-0.0320353\pi$
$234$		0			0
$235$	61.8730			0.263289
$236$	−52.1225			−0.220858
$237$		0			0
$238$		−	187.969i		−	0.789785i
$239$			82.4962i			0.345173i	0.984994	+	0.172586i	$0.0552124\pi$
$239$			82.4962i			0.345173i	−0.984994	+	0.172586i	$0.944788\pi$
$240$		0			0
$241$		−	10.6986i		−	0.0443926i	−0.999754	−	0.0221963i	$-0.992934\pi$
$241$		−	10.6986i		−	0.0443926i	0.999754	−	0.0221963i	$-0.00706588\pi$
$242$	−114.501	−	163.257i	−0.473146	−	0.674616i
$243$		0			0
$244$		−	43.5336i		−	0.178416i
$245$	−211.666			−0.863944
$246$		0			0
$247$	264.058			1.06906
$248$		−	459.086i		−	1.85115i
$249$		0			0
$250$		−	329.674i		−	1.31870i
$251$	229.681			0.915065			0.457532	−	0.889193i	$-0.348733\pi$
$251$	229.681			0.915065			0.457532	+	0.889193i	$0.348733\pi$
$252$		0			0
$253$	6.35713	+	20.1352i	0.0251270	+	0.0795858i
$254$	380.983			1.49993
$255$		0			0
$256$	−218.421			−0.853206
$257$	−104.402			−0.406232			−0.203116	−	0.979155i	$-0.565107\pi$
$257$	−104.402			−0.406232			−0.203116	+	0.979155i	$0.565107\pi$
$258$		0			0
$259$			191.882i			0.740858i
$260$		−	88.8779i		−	0.341838i
$261$		0			0
$262$	163.574			0.624330
$263$			13.5248i			0.0514251i	0.999669	+	0.0257125i	$0.00818545\pi$
$263$			13.5248i			0.0514251i	−0.999669	+	0.0257125i	$0.991815\pi$
$264$		0			0
$265$	−688.134			−2.59673
$266$		−	265.378i		−	0.997662i
$267$		0			0
$268$	4.59321			0.0171389
$269$	−254.840			−0.947362			−0.473681	−	0.880697i	$-0.657075\pi$
$269$	−254.840			−0.947362			−0.473681	+	0.880697i	$0.657075\pi$
$270$		0			0
$271$		−	302.301i		−	1.11550i	−0.830008	−	0.557751i	$-0.811664\pi$
$271$		−	302.301i		−	1.11550i	0.830008	−	0.557751i	$-0.188336\pi$
$272$			213.267i			0.784072i
$273$		0			0
$274$			371.138i			1.35452i
$275$	−160.147	−	507.242i	−0.582354	−	1.84451i
$276$		0			0
$277$			257.195i			0.928501i	0.885704	+	0.464251i	$0.153676\pi$
$277$			257.195i			0.928501i	−0.885704	+	0.464251i	$0.846324\pi$
$278$	−142.360			−0.512088
$279$		0			0
$280$	−367.562			−1.31272
$281$			452.278i			1.60953i	0.593593	+	0.804765i	$0.297709\pi$
$281$			452.278i			1.60953i	−0.593593	+	0.804765i	$0.702291\pi$
$282$		0			0
$283$		−	278.537i		−	0.984228i	−0.870531	−	0.492114i	$-0.836224\pi$
$283$		−	278.537i		−	0.984228i	0.870531	−	0.492114i	$-0.163776\pi$
$284$	−123.924			−0.436351
$285$		0			0
$286$	44.1056	+	139.698i	0.154216	+	0.488453i
$287$	176.241			0.614081
$288$		0			0
$289$	−246.664			−0.853507
$290$	−365.902			−1.26173
$291$		0			0
$292$			50.1579i			0.171773i
$293$			179.504i			0.612642i	0.951928	+	0.306321i	$0.0990982\pi$
$293$			179.504i			0.612642i	−0.951928	+	0.306321i	$0.900902\pi$
$294$		0			0
$295$	347.651			1.17848
$296$		−	339.062i		−	1.14548i
$297$		0			0
$298$	255.865			0.858606
$299$		−	15.5121i		−	0.0518799i
$300$		0			0
$301$	−87.7017			−0.291368
$302$	−104.582			−0.346300
$303$		0			0
$304$			301.095i			0.990445i
$305$			290.364i			0.952014i
$306$		0			0
$307$		−	487.850i		−	1.58909i	−0.607206	−	0.794544i	$-0.707710\pi$
$307$		−	487.850i		−	1.58909i	0.607206	−	0.794544i	$-0.292290\pi$
$308$	−66.3810	+	20.9579i	−0.215523	+	0.0680453i
$309$		0			0
$310$			744.118i			2.40038i
$311$	−68.5229			−0.220331			−0.110166	−	0.993913i	$-0.535138\pi$
$311$	−68.5229			−0.220331			−0.110166	+	0.993913i	$0.535138\pi$
$312$		0			0
$313$	137.744			0.440075			0.220038	−	0.975491i	$-0.429382\pi$
$313$	137.744			0.440075			0.220038	+	0.975491i	$0.429382\pi$
$314$			92.7614i			0.295419i
$315$		0			0
$316$			151.884i			0.480645i
$317$	415.365			1.31030			0.655150	−	0.755498i	$-0.272605\pi$
$317$	415.365			1.31030			0.655150	+	0.755498i	$0.272605\pi$
$318$		0			0
$319$	−271.924	+	85.8526i	−0.852428	+	0.269130i
$320$	593.003			1.85313
$321$		0			0
$322$	−15.5897			−0.0484151
$323$	−756.260			−2.34136
$324$		0			0
$325$			390.777i			1.20239i
$326$			406.640i			1.24736i
$327$		0			0
$328$	−311.424			−0.949464
$329$			35.6012i			0.108210i
$330$		0			0
$331$	27.5573			0.0832547			0.0416274	−	0.999133i	$-0.486746\pi$
$331$	27.5573			0.0832547			0.0416274	+	0.999133i	$0.486746\pi$
$332$		−	7.06408i		−	0.0212773i
$333$		0			0
$334$	−17.2038			−0.0515082
$335$	−30.6362			−0.0914514
$336$		0			0
$337$			278.758i			0.827176i	0.910464	+	0.413588i	$0.135725\pi$
$337$			278.758i			0.827176i	−0.910464	+	0.413588i	$0.864275\pi$
$338$			170.889i			0.505589i
$339$		0			0
$340$			254.546i			0.748665i
$341$	174.594	+	553.001i	0.512007	+	1.62170i
$342$		0			0
$343$		−	363.270i		−	1.05910i
$344$	154.972			0.450499
$345$		0			0
$346$	183.729			0.531010
$347$		−	270.725i		−	0.780187i	−0.920775	−	0.390094i	$-0.872443\pi$
$347$		−	270.725i		−	0.780187i	0.920775	−	0.390094i	$-0.127557\pi$
$348$		0			0
$349$			253.323i			0.725854i	0.931818	+	0.362927i	$0.118223\pi$
$349$			253.323i			0.725854i	−0.931818	+	0.362927i	$0.881777\pi$
$350$	392.731			1.12209
$351$		0			0
$352$	206.090	−	65.0672i	0.585483	−	0.184850i
$353$	39.0032			0.110491			0.0552453	−	0.998473i	$-0.482406\pi$
$353$	39.0032			0.110491			0.0552453	+	0.998473i	$0.482406\pi$
$354$		0			0
$355$	826.556			2.32833
$356$	−9.72155			−0.0273077
$357$		0			0
$358$			227.361i			0.635086i
$359$			220.611i			0.614515i	0.951626	+	0.307258i	$0.0994113\pi$
$359$			220.611i			0.614515i	−0.951626	+	0.307258i	$0.900589\pi$
$360$		0			0
$361$	−706.703			−1.95763
$362$			205.196i			0.566840i
$363$		0			0
$364$	51.1397			0.140494
$365$		−	334.547i		−	0.916568i
$366$		0			0
$367$	−423.229			−1.15321			−0.576606	−	0.817022i	$-0.695623\pi$
$367$	−423.229			−1.15321			−0.576606	+	0.817022i	$0.695623\pi$
$368$	17.6879			0.0480649
$369$		0			0
$370$			549.576i			1.48534i
$371$		−	395.947i		−	1.06724i
$372$		0			0
$373$			534.925i			1.43412i	0.697013	+	0.717058i	$0.254512\pi$
$373$			534.925i			1.43412i	−0.697013	+	0.717058i	$0.745488\pi$
$374$	−126.318	−	400.094i	−0.337750	−	1.06977i
$375$		0			0
$376$		−	62.9085i		−	0.167310i
$377$	209.490			0.555675
$378$		0			0
$379$	−438.061			−1.15583			−0.577917	−	0.816096i	$-0.696134\pi$
$379$	−438.061			−1.15583			−0.577917	+	0.816096i	$0.696134\pi$
$380$			359.373i			0.945719i
$381$		0			0
$382$		−	264.916i		−	0.693498i
$383$	462.855			1.20850			0.604249	−	0.796795i	$-0.293473\pi$
$383$	462.855			1.20850			0.604249	+	0.796795i	$0.293473\pi$
$384$		0			0
$385$	442.754	−	139.787i	1.15001	−	0.363083i
$386$	335.000			0.867875
$387$		0			0
$388$	−124.604			−0.321144
$389$	−213.149			−0.547941			−0.273970	−	0.961738i	$-0.588337\pi$
$389$	−213.149			−0.547941			−0.273970	+	0.961738i	$0.588337\pi$
$390$		0			0
$391$			44.4266i			0.113623i
$392$			215.209i			0.549002i
$393$		0			0
$394$	−504.349			−1.28007
$395$		−	1013.05i		−	2.56468i
$396$		0			0
$397$	442.513			1.11464			0.557321	−	0.830297i	$-0.311829\pi$
$397$	442.513			1.11464			0.557321	+	0.830297i	$0.311829\pi$
$398$		−	47.5674i		−	0.119516i
$399$		0			0
$400$	−445.589			−1.11397
$401$	−45.0547			−0.112356			−0.0561780	−	0.998421i	$-0.517891\pi$
$401$	−45.0547			−0.112356			−0.0561780	+	0.998421i	$0.517891\pi$
$402$		0			0
$403$		−	426.029i		−	1.05715i
$404$		−	174.085i		−	0.430903i
$405$		0			0
$406$		−	210.537i		−	0.518564i
$407$	128.948	+	408.424i	0.316827	+	1.00350i
$408$		0			0
$409$		−	368.397i		−	0.900726i	−0.892845	−	0.450363i	$-0.851295\pi$
$409$		−	368.397i		−	0.900726i	0.892845	−	0.450363i	$-0.148705\pi$
$410$	504.778			1.23117
$411$		0			0
$412$	−130.524			−0.316807
$413$			200.036i			0.484348i
$414$		0			0
$415$			47.1166i			0.113534i
$416$	−158.771			−0.381661
$417$		0			0
$418$	−178.339	−	564.860i	−0.426648	−	1.35134i
$419$	−42.7918			−0.102128			−0.0510642	−	0.998695i	$-0.516261\pi$
$419$	−42.7918			−0.102128			−0.0510642	+	0.998695i	$0.516261\pi$
$420$		0			0
$421$	365.151			0.867343			0.433671	−	0.901071i	$-0.357218\pi$
$421$	365.151			0.867343			0.433671	+	0.901071i	$0.357218\pi$
$422$	509.633			1.20766
$423$		0			0
$424$			699.651i			1.65012i
$425$		−	1119.18i		−	2.63337i
$426$		0			0
$427$	−167.073			−0.391272
$428$		−	3.22562i		−	0.00753650i
$429$		0			0
$430$	−251.189			−0.584160
$431$			347.675i			0.806670i	0.915052	+	0.403335i	$0.132149\pi$
$431$			347.675i			0.806670i	−0.915052	+	0.403335i	$0.867851\pi$
$432$		0			0
$433$	462.406			1.06791			0.533956	−	0.845512i	$-0.320705\pi$
$433$	462.406			1.06791			0.533956	+	0.845512i	$0.320705\pi$
$434$	−428.160			−0.986543
$435$		0			0
$436$			116.283i			0.266705i
$437$			62.7223i			0.143529i
$438$		0			0
$439$			480.449i			1.09442i	0.836996	+	0.547209i	$0.184310\pi$
$439$			480.449i			1.09442i	−0.836996	+	0.547209i	$0.815690\pi$
$440$	−782.360	+	247.008i	−1.77809	+	0.561383i
$441$		0			0
$442$			308.231i			0.697354i
$443$	1.28783			0.00290707			0.00145354	−	0.999999i	$-0.499537\pi$
$443$	1.28783			0.00290707			0.00145354	+	0.999999i	$0.499537\pi$
$444$		0			0
$445$	64.8416			0.145712
$446$			202.080i			0.453095i
$447$		0			0
$448$			341.209i			0.761628i
$449$	−788.038			−1.75510			−0.877548	−	0.479490i	$-0.840822\pi$
$449$	−788.038			−1.75510			−0.877548	+	0.479490i	$0.840822\pi$
$450$		0			0
$451$	375.132	−	118.437i	0.831778	−	0.262611i
$452$	−4.76962			−0.0105523
$453$		0			0
$454$	471.631			1.03883
$455$	−341.096			−0.749661
$456$		0			0
$457$		−	475.774i		−	1.04108i	−0.853837	−	0.520540i	$-0.825730\pi$
$457$		−	475.774i		−	1.04108i	0.853837	−	0.520540i	$-0.174270\pi$
$458$			420.002i			0.917036i
$459$		0			0
$460$	21.1114			0.0458944
$461$		−	194.131i		−	0.421109i	−0.977582	−	0.210555i	$-0.932473\pi$
$461$		−	194.131i		−	0.421109i	0.977582	−	0.210555i	$-0.0675270\pi$
$462$		0			0
$463$	595.221			1.28557			0.642787	−	0.766045i	$-0.277778\pi$
$463$	595.221			1.28557			0.642787	+	0.766045i	$0.277778\pi$
$464$			238.873i			0.514813i
$465$		0			0
$466$	77.1591			0.165577
$467$	294.200			0.629978			0.314989	−	0.949095i	$-0.397999\pi$
$467$	294.200			0.629978			0.314989	+	0.949095i	$0.397999\pi$
$468$		0			0
$469$		−	17.6278i		−	0.0375860i
$470$			101.966i			0.216950i
$471$		0			0
$472$		−	353.469i		−	0.748876i
$473$	−186.674	+	58.9371i	−0.394660	+	0.124603i
$474$		0			0
$475$		−	1580.09i		−	3.32650i
$476$	−146.464			−0.307697
$477$		0			0
$478$	−135.954			−0.284422
$479$		−	107.253i		−	0.223909i	−0.993713	−	0.111955i	$-0.964289\pi$
$479$		−	107.253i		−	0.223909i	0.993713	−	0.111955i	$-0.0357111\pi$
$480$		0			0
$481$		−	314.648i		−	0.654154i
$482$	17.6313			0.0365794
$483$		0			0
$484$	−127.209	+	89.2185i	−0.262828	+	0.184336i
$485$	831.093			1.71359
$486$		0			0
$487$	−257.840			−0.529445			−0.264723	−	0.964325i	$-0.585280\pi$
$487$	−257.840			−0.529445			−0.264723	+	0.964325i	$0.585280\pi$
$488$	295.224			0.604967
$489$		0			0
$490$		−	348.826i		−	0.711889i
$491$			262.711i			0.535053i	0.963551	+	0.267526i	$0.0862063\pi$
$491$			262.711i			0.535053i	−0.963551	+	0.267526i	$0.913794\pi$
$492$		0			0
$493$	−599.977			−1.21699
$494$			435.166i			0.880903i
$495$		0			0
$496$	485.786			0.979407
$497$			475.594i			0.956930i
$498$		0			0
$499$	−708.627			−1.42010			−0.710048	−	0.704154i	$-0.751327\pi$
$499$	−708.627			−1.42010			−0.710048	+	0.704154i	$0.751327\pi$
$500$	−256.879			−0.513759
$501$		0			0
$502$			378.514i			0.754012i
$503$		−	87.6796i		−	0.174313i	−0.996195	−	0.0871567i	$-0.972222\pi$
$503$		−	87.6796i		−	0.174313i	0.996195	−	0.0871567i	$-0.0277781\pi$
$504$		0			0
$505$			1161.13i			2.29926i
$506$	−33.1828	+	10.4765i	−0.0655786	+	0.0207046i
$507$		0			0
$508$		−	296.859i		−	0.584368i
$509$	−913.193			−1.79409			−0.897046	−	0.441936i	$-0.854292\pi$
$509$	−913.193			−1.79409			−0.897046	+	0.441936i	$0.854292\pi$
$510$		0			0
$511$	192.496			0.376704
$512$		−	502.012i		−	0.980493i
$513$		0			0
$514$		−	172.053i		−	0.334734i
$515$	870.583			1.69045
$516$		0			0
$517$	23.9247	+	75.7776i	0.0462760	+	0.146572i
$518$	−316.221			−0.610466
$519$		0			0
$520$	602.727			1.15909
$521$	−832.282			−1.59747			−0.798735	−	0.601683i	$-0.794497\pi$
$521$	−832.282			−1.59747			−0.798735	+	0.601683i	$0.794497\pi$
$522$		0			0
$523$		−	782.157i		−	1.49552i	−0.663969	−	0.747760i	$-0.731129\pi$
$523$		−	782.157i		−	1.49552i	0.663969	−	0.747760i	$-0.268871\pi$
$524$		−	127.456i		−	0.243236i
$525$		0			0
$526$	−22.2888			−0.0423742
$527$			1220.15i			2.31527i
$528$		0			0
$529$	−525.315			−0.993035
$530$		−	1134.04i		−	2.13970i
$531$		0			0
$532$	−206.781			−0.388685
$533$	−289.000			−0.542214
$534$		0			0
$535$			21.5145i			0.0402141i
$536$			31.1490i			0.0581137i
$537$		0			0
$538$		−	419.976i		−	0.780625i
$539$	−81.8459	−	259.234i	−0.151848	−	0.480954i
$540$		0			0
$541$		−	906.866i		−	1.67628i	−0.545457	−	0.838139i	$-0.683644\pi$
$541$		−	906.866i		−	1.67628i	0.545457	−	0.838139i	$-0.316356\pi$
$542$	498.191			0.919172
$543$		0			0
$544$	454.719			0.835881
$545$		−	775.598i		−	1.42311i
$546$		0			0
$547$			110.826i			0.202606i	0.994856	+	0.101303i	$0.0323012\pi$
$547$			110.826i			0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$548$	289.188			0.527716
$549$		0			0
$550$	835.932	−	263.922i	1.51988	−	0.479859i
$551$	−847.060			−1.53731
$552$		0			0
$553$	582.900			1.05407
$554$	−423.856			−0.765084
$555$		0			0
$556$			110.926i			0.199507i
$557$		−	442.180i		−	0.793860i	−0.917849	−	0.396930i	$-0.870076\pi$
$557$		−	442.180i		−	0.793860i	0.917849	−	0.396930i	$-0.129924\pi$
$558$		0			0
$559$	143.813			0.257268
$560$		−	388.939i		−	0.694534i
$561$		0			0
$562$	−745.353			−1.32625
$563$			366.838i			0.651576i	0.945443	+	0.325788i	$0.105630\pi$
$563$			366.838i			0.651576i	−0.945443	+	0.325788i	$0.894370\pi$
$564$		0			0
$565$	31.8129			0.0563059
$566$	459.027			0.811003
$567$		0			0
$568$		−	840.390i		−	1.47956i
$569$		−	1060.07i		−	1.86305i	−0.363679	−	0.931524i	$-0.618479\pi$
$569$		−	1060.07i		−	1.86305i	0.363679	−	0.931524i	$-0.381521\pi$
$570$		0			0
$571$		−	660.459i		−	1.15667i	−0.815799	−	0.578335i	$-0.803703\pi$
$571$		−	660.459i		−	1.15667i	0.815799	−	0.578335i	$-0.196297\pi$
$572$	108.851	−	34.3668i	0.190300	−	0.0600818i
$573$		0			0
$574$			290.445i			0.506002i
$575$	−92.8223			−0.161430
$576$		0			0
$577$	−225.359			−0.390570			−0.195285	−	0.980747i	$-0.562563\pi$
$577$	−225.359			−0.390570			−0.195285	+	0.980747i	$0.562563\pi$
$578$		−	406.501i		−	0.703289i
$579$		0			0
$580$			285.108i			0.491565i
$581$	−27.1105			−0.0466618
$582$		0			0
$583$	−266.084	−	842.778i	−0.456404	−	1.44559i
$584$	−340.146			−0.582443
$585$		0			0
$586$	−295.822			−0.504816
$587$	−302.947			−0.516094			−0.258047	−	0.966132i	$-0.583079\pi$
$587$	−302.947			−0.516094			−0.258047	+	0.966132i	$0.583079\pi$
$588$		0			0
$589$			1722.63i			2.92467i
$590$			572.928i			0.971064i
$591$		0			0
$592$	358.782			0.606050
$593$			901.267i			1.51984i	0.650015	+	0.759921i	$0.274763\pi$
$593$			901.267i			1.51984i	−0.650015	+	0.759921i	$0.725237\pi$
$594$		0			0
$595$	976.897			1.64184
$596$		−	199.368i		−	0.334510i
$597$		0			0
$598$	25.5639			0.0427489
$599$	985.061			1.64451			0.822254	−	0.569120i	$-0.192716\pi$
$599$	985.061			1.64451			0.822254	+	0.569120i	$0.192716\pi$
$600$		0			0
$601$			213.400i			0.355075i	0.984114	+	0.177537i	$0.0568130\pi$
$601$			213.400i			0.355075i	−0.984114	+	0.177537i	$0.943187\pi$
$602$		−	144.532i		−	0.240086i
$603$		0			0
$604$			81.4898i			0.134917i
$605$	848.467	−	595.077i	1.40243	−	0.983599i
$606$		0			0
$607$		−	91.4622i		−	0.150679i	−0.997158	−	0.0753395i	$-0.975996\pi$
$607$		−	91.4622i		−	0.150679i	0.997158	−	0.0753395i	$-0.0240041\pi$
$608$	641.982			1.05589
$609$		0			0
$610$	−478.519			−0.784458
$611$		−	58.3788i		−	0.0955463i
$612$		0			0
$613$		−	1040.49i		−	1.69738i	−0.528890	−	0.848691i	$-0.677391\pi$
$613$		−	1040.49i		−	1.69738i	0.528890	−	0.848691i	$-0.322609\pi$
$614$	803.976			1.30941
$615$		0			0
$616$	−142.127	−	450.164i	−0.230725	−	0.730786i
$617$	383.447			0.621470			0.310735	−	0.950497i	$-0.399425\pi$
$617$	383.447			0.621470			0.310735	+	0.950497i	$0.399425\pi$
$618$		0			0
$619$	−694.045			−1.12124			−0.560618	−	0.828075i	$-0.689436\pi$
$619$	−694.045			−1.12124			−0.560618	+	0.828075i	$0.689436\pi$
$620$	579.811			0.935179
$621$		0			0
$622$		−	112.926i		−	0.181552i
$623$			37.3094i			0.0598866i
$624$		0			0
$625$	504.443			0.807108
$626$			227.001i			0.362621i
$627$		0			0
$628$	72.2790			0.115094
$629$			901.151i			1.43267i
$630$		0			0
$631$	188.500			0.298732			0.149366	−	0.988782i	$-0.452277\pi$
$631$	188.500			0.298732			0.149366	+	0.988782i	$0.452277\pi$
$632$	−1030.00			−1.62975
$633$		0			0
$634$			684.521i			1.07969i
$635$			1980.02i			3.11814i
$636$		0			0
$637$			199.713i			0.313521i
$638$	−141.485	−	448.131i	−0.221763	−	0.702399i
$639$		0			0
$640$			304.171i			0.475267i
$641$	763.956			1.19182			0.595910	−	0.803052i	$-0.296792\pi$
$641$	763.956			1.19182			0.595910	+	0.803052i	$0.296792\pi$
$642$		0			0
$643$	418.480			0.650825			0.325412	−	0.945572i	$-0.394497\pi$
$643$	418.480			0.650825			0.325412	+	0.945572i	$0.394497\pi$
$644$			12.1473i			0.0188623i
$645$		0			0
$646$		−	1246.31i		−	1.92928i
$647$	−757.593			−1.17093			−0.585466	−	0.810697i	$-0.699089\pi$
$647$	−757.593			−1.17093			−0.585466	+	0.810697i	$0.699089\pi$
$648$		0			0
$649$	134.428	+	425.778i	0.207130	+	0.656053i
$650$	−643.999			−0.990768
$651$		0			0
$652$	316.851			0.485968
$653$	603.653			0.924431			0.462215	−	0.886768i	$-0.347055\pi$
$653$	603.653			0.924431			0.462215	+	0.886768i	$0.347055\pi$
$654$		0			0
$655$			850.116i			1.29789i
$656$		−	329.536i		−	0.502342i
$657$		0			0
$658$	−58.6707			−0.0891652
$659$			409.015i			0.620661i	0.950629	+	0.310330i	$0.100440\pi$
$659$			409.015i			0.620661i	−0.950629	+	0.310330i	$0.899560\pi$
$660$		0			0
$661$	−1190.63			−1.80125			−0.900624	−	0.434599i	$-0.856890\pi$
$661$	−1190.63			−1.80125			−0.900624	+	0.434599i	$0.856890\pi$
$662$			45.4144i			0.0686018i
$663$		0			0
$664$	47.9052			0.0721463
$665$	1379.20			2.07399
$666$		0			0
$667$			49.7606i			0.0746036i
$668$			13.4050i			0.0200674i
$669$		0			0
$670$		−	50.4884i		−	0.0753558i
$671$	−355.617	+	112.276i	−0.529981	+	0.167327i
$672$		0			0
$673$			806.927i			1.19900i	0.800375	+	0.599500i	$0.204634\pi$
$673$			806.927i			1.19900i	−0.800375	+	0.599500i	$0.795366\pi$
$674$	−459.393			−0.681592
$675$		0			0
$676$	133.155			0.196975
$677$			844.174i			1.24693i	0.781850	+	0.623467i	$0.214276\pi$
$677$			844.174i			1.24693i	−0.781850	+	0.623467i	$0.785724\pi$
$678$		0			0
$679$			478.204i			0.704278i
$680$	−1726.21			−2.53854
$681$		0			0
$682$	−911.343	+	287.731i	−1.33628	+	0.421893i
$683$	374.479			0.548286			0.274143	−	0.961689i	$-0.411606\pi$
$683$	374.479			0.548286			0.274143	+	0.961689i	$0.411606\pi$
$684$		0			0
$685$	−1928.85			−2.81584
$686$	598.668			0.872694
$687$		0			0
$688$			163.985i			0.238350i
$689$			649.273i			0.942341i
$690$		0			0
$691$	885.553			1.28155			0.640776	−	0.767728i	$-0.278613\pi$
$691$	885.553			1.28155			0.640776	+	0.767728i	$0.278613\pi$
$692$		−	143.161i		−	0.206879i
$693$		0			0
$694$	446.154			0.642873
$695$		−	739.865i		−	1.06455i
$696$		0			0
$697$	827.695			1.18751
$698$	−417.475			−0.598102
$699$		0			0
$700$		−	306.013i		−	0.437161i
$701$		−	934.279i		−	1.33278i	−0.745603	−	0.666390i	$-0.767838\pi$
$701$		−	934.279i		−	1.33278i	0.745603	−	0.666390i	$-0.232162\pi$
$702$		0			0
$703$			1272.26i			1.80976i
$704$	229.299	+	726.268i	0.325709	+	1.03163i
$705$		0			0
$706$			64.2772i			0.0910442i
$707$	−668.103			−0.944983
$708$		0			0
$709$	472.473			0.666394			0.333197	−	0.942857i	$-0.391873\pi$
$709$	472.473			0.666394			0.333197	+	0.942857i	$0.391873\pi$
$710$			1362.16i			1.91854i
$711$		0			0
$712$		−	65.9269i		−	0.0925939i
$713$	101.196			0.141930
$714$		0			0
$715$	−726.026	+	229.223i	−1.01542	+	0.320591i
$716$	177.158			0.247427
$717$		0			0
$718$	−363.566			−0.506360
$719$	816.636			1.13579			0.567897	−	0.823100i	$-0.307757\pi$
$719$	816.636			1.13579			0.567897	+	0.823100i	$0.307757\pi$
$720$		0			0
$721$			500.927i			0.694767i
$722$		−	1164.64i		−	1.61308i
$723$		0			0
$724$	159.887			0.220839
$725$		−	1253.56i		−	1.72904i
$726$		0			0
$727$	829.338			1.14077			0.570384	−	0.821378i	$-0.306794\pi$
$727$	829.338			1.14077			0.570384	+	0.821378i	$0.306794\pi$
$728$			346.805i			0.476380i
$729$		0			0
$730$	551.333			0.755250
$731$	−411.880			−0.563447
$732$		0			0
$733$		−	999.783i		−	1.36396i	−0.731371	−	0.681980i	$-0.761119\pi$
$733$		−	999.783i		−	1.36396i	0.731371	−	0.681980i	$-0.238881\pi$
$734$		−	697.480i		−	0.950245i
$735$		0			0
$736$		−	37.7133i		−	0.0512409i
$737$	−11.8462	−	37.5210i	−0.0160736	−	0.0509105i
$738$		0			0
$739$			746.352i			1.00995i	0.863135	+	0.504974i	$0.168498\pi$
$739$			746.352i			1.00995i	−0.863135	+	0.504974i	$0.831502\pi$
$740$	428.225			0.578682
$741$		0			0
$742$	652.519			0.879406
$743$		−	52.0602i		−	0.0700676i	−0.999386	−	0.0350338i	$-0.988846\pi$
$743$		−	52.0602i		−	0.0700676i	0.999386	−	0.0350338i	$-0.0111539\pi$
$744$		0			0
$745$			1329.76i			1.78491i
$746$	−881.555			−1.18171
$747$		0			0
$748$	−311.750	+	98.4264i	−0.416778	+	0.131586i
$749$	−12.3793			−0.0165278
$750$		0			0
$751$	−264.446			−0.352125			−0.176063	−	0.984379i	$-0.556336\pi$
$751$	−264.446			−0.352125			−0.176063	+	0.984379i	$0.556336\pi$
$752$	66.5672			0.0885202
$753$		0			0
$754$			345.238i			0.457875i
$755$		−	543.528i		−	0.719905i
$756$		0			0
$757$	−607.325			−0.802278			−0.401139	−	0.916017i	$-0.631386\pi$
$757$	−607.325			−0.802278			−0.401139	+	0.916017i	$0.631386\pi$
$758$		−	721.923i		−	0.952405i
$759$		0			0
$760$	−2437.10			−3.20671
$761$		−	501.227i		−	0.658643i	−0.944218	−	0.329322i	$-0.893180\pi$
$761$		−	501.227i		−	0.658643i	0.944218	−	0.329322i	$-0.106820\pi$
$762$		0			0
$763$	446.273			0.584892
$764$	−206.421			−0.270184
$765$		0			0
$766$			762.783i			0.995801i
$767$		−	328.018i		−	0.427663i
$768$		0			0
$769$		−	39.9821i		−	0.0519923i	−0.999662	−	0.0259962i	$-0.991724\pi$
$769$		−	39.9821i		−	0.0519923i	0.999662	−	0.0259962i	$-0.00827577\pi$
$770$	230.369	+	729.657i	0.299180	+	0.947606i
$771$		0			0
$772$		−	261.029i		−	0.338121i
$773$	−57.3059			−0.0741344			−0.0370672	−	0.999313i	$-0.511802\pi$
$773$	−57.3059			−0.0741344			−0.0370672	+	0.999313i	$0.511802\pi$
$774$		0			0
$775$	−2549.30			−3.28942
$776$		−	845.003i		−	1.08892i
$777$		0			0
$778$		−	351.269i		−	0.451502i
$779$	1168.56			1.50007
$780$		0			0
$781$	319.608	+	1012.31i	0.409229	+	1.29617i
$782$	−73.2149			−0.0936251
$783$		0			0
$784$	−227.725			−0.290466
$785$	−482.093			−0.614131
$786$		0			0
$787$		−	324.329i		−	0.412108i	−0.978541	−	0.206054i	$-0.933938\pi$
$787$		−	324.329i		−	0.412108i	0.978541	−	0.206054i	$-0.0660623\pi$
$788$			392.985i			0.498712i
$789$		0			0
$790$	1669.50			2.11329
$791$			18.3049i			0.0231414i
$792$		0			0
$793$	273.966			0.345481
$794$			729.260i			0.918464i
$795$		0			0
$796$	−37.0642			−0.0465630
$797$	428.384			0.537495			0.268748	−	0.963211i	$-0.413390\pi$
$797$	428.384			0.537495			0.268748	+	0.963211i	$0.413390\pi$
$798$		0			0
$799$			167.197i			0.209257i
$800$			950.064i			1.18758i
$801$		0			0
$802$		−	74.2501i		−	0.0925811i
$803$	409.730	−	129.361i	0.510249	−	0.161097i
$804$		0			0
$805$		−	81.0214i		−	0.100648i
$806$	702.095			0.871086
$807$		0			0
$808$	1180.56			1.46109
$809$		−	306.627i		−	0.379019i	−0.981879	−	0.189510i	$-0.939310\pi$
$809$		−	306.627i		−	0.379019i	0.981879	−	0.189510i	$-0.0606898\pi$
$810$		0			0
$811$			129.013i			0.159079i	0.996832	+	0.0795395i	$0.0253450\pi$
$811$			129.013i			0.159079i	−0.996832	+	0.0795395i	$0.974655\pi$
$812$	−164.049			−0.202031
$813$		0			0
$814$	−673.081	+	212.507i	−0.826881	+	0.261065i
$815$	−2113.36			−2.59308
$816$		0			0
$817$	−581.500			−0.711750
$818$	607.117			0.742197
$819$		0			0
$820$		−	393.319i		−	0.479657i
$821$			1015.96i			1.23747i	0.785599	+	0.618736i	$0.212355\pi$
$821$			1015.96i			1.23747i	−0.785599	+	0.618736i	$0.787645\pi$
$822$		0			0
$823$	570.816			0.693579			0.346789	−	0.937943i	$-0.387272\pi$
$823$	570.816			0.693579			0.346789	+	0.937943i	$0.387272\pi$
$824$		−	885.154i		−	1.07422i
$825$		0			0
$826$	−329.658			−0.399102
$827$		−	1223.56i		−	1.47951i	−0.672875	−	0.739756i	$-0.734941\pi$
$827$		−	1223.56i		−	1.47951i	0.672875	−	0.739756i	$-0.265059\pi$
$828$		0			0
$829$	1076.92			1.29905			0.649527	−	0.760339i	$-0.274967\pi$
$829$	1076.92			1.29905			0.649527	+	0.760339i	$0.274967\pi$
$830$	−77.6480			−0.0935518
$831$		0			0
$832$		−	559.514i		−	0.672493i
$833$		−	571.977i		−	0.686647i
$834$		0			0
$835$		−	89.4100i		−	0.107078i
$836$	−440.135	+	138.960i	−0.526477	+	0.166220i
$837$		0			0
$838$		−	70.5208i		−	0.0841536i
$839$	−1303.84			−1.55404			−0.777018	−	0.629479i	$-0.783268\pi$
$839$	−1303.84			−1.55404			−0.777018	+	0.629479i	$0.783268\pi$
$840$		0			0
$841$	168.987			0.200936
$842$			601.768i			0.714689i
$843$		0			0
$844$		−	397.102i		−	0.470500i
$845$	−888.131			−1.05104
$846$		0			0
$847$	342.403	+	488.201i	0.404253	+	0.576389i
$848$	−740.342			−0.873045
$849$		0			0
$850$	1844.41			2.16990
$851$	74.7392			0.0878251
$852$		0			0
$853$			146.628i			0.171897i	0.996300	+	0.0859487i	$0.0273921\pi$
$853$			146.628i			0.171897i	−0.996300	+	0.0859487i	$0.972608\pi$
$854$		−	275.336i		−	0.322408i
$855$		0			0
$856$	21.8746			0.0255545
$857$		−	548.280i		−	0.639767i	−0.947457	−	0.319883i	$-0.896356\pi$
$857$		−	548.280i		−	0.639767i	0.947457	−	0.319883i	$-0.103644\pi$
$858$		0			0
$859$	1153.84			1.34324			0.671618	−	0.740897i	$-0.265599\pi$
$859$	1153.84			1.34324			0.671618	+	0.740897i	$0.265599\pi$
$860$			195.724i			0.227586i
$861$		0			0
$862$	−572.967			−0.664695
$863$	291.136			0.337354			0.168677	−	0.985671i	$-0.446051\pi$
$863$	291.136			0.337354			0.168677	+	0.985671i	$0.446051\pi$
$864$		0			0
$865$			954.865i			1.10389i
$866$			762.043i			0.879957i
$867$		0			0
$868$			333.619i			0.384353i
$869$	1240.71	−	391.719i	1.42774	−	0.450770i
$870$		0			0
$871$			28.9061i			0.0331872i
$872$	−788.579			−0.904333
$873$		0			0
$874$	−103.366			−0.118268
$875$			985.852i			1.12669i
$876$		0			0
$877$			1378.69i			1.57205i	0.618192	+	0.786027i	$0.287865\pi$
$877$			1378.69i			1.57205i	−0.618192	+	0.786027i	$0.712135\pi$
$878$	−791.779			−0.901799
$879$		0			0
$880$	−261.374	−	827.861i	−0.297016	−	0.940751i
$881$	−1423.93			−1.61626			−0.808131	−	0.589003i	$-0.799521\pi$
$881$	−1423.93			−1.61626			−0.808131	+	0.589003i	$0.799521\pi$
$882$		0			0
$883$	−1524.38			−1.72637			−0.863185	−	0.504888i	$-0.831534\pi$
$883$	−1524.38			−1.72637			−0.863185	+	0.504888i	$0.831534\pi$
$884$	240.171			0.271687
$885$		0			0
$886$			2.12234i			0.00239542i
$887$			598.912i			0.675211i	0.941288	+	0.337606i	$0.109617\pi$
$887$			598.912i			0.675211i	−0.941288	+	0.337606i	$0.890383\pi$
$888$		0			0
$889$	−1139.29			−1.28154
$890$			106.859i			0.120066i
$891$		0			0
$892$	157.459			0.176524
$893$			236.052i			0.264335i
$894$		0			0
$895$	−1181.62			−1.32025
$896$	−175.018			−0.195332
$897$		0			0
$898$		−	1298.68i		−	1.44620i
$899$			1366.64i			1.52018i
$900$		0			0
$901$		−	1859.52i		−	2.06383i
$902$	195.184	+	618.216i	0.216391	+	0.685384i
$903$		0			0
$904$		−	32.3453i		−	0.0357802i
$905$	−1066.43			−1.17838
$906$		0			0
$907$	−36.6324			−0.0403886			−0.0201943	−	0.999796i	$-0.506428\pi$
$907$	−36.6324			−0.0403886			−0.0201943	+	0.999796i	$0.506428\pi$
$908$		−	367.491i		−	0.404726i
$909$		0			0
$910$		−	562.125i		−	0.617719i
$911$	400.009			0.439088			0.219544	−	0.975603i	$-0.429543\pi$
$911$	400.009			0.439088			0.219544	+	0.975603i	$0.429543\pi$
$912$		0			0
$913$	−57.7051	+	18.2188i	−0.0632038	+	0.0199548i
$914$	784.074			0.857849
$915$		0			0
$916$	327.263			0.357274
$917$	−489.150			−0.533425
$918$		0			0
$919$			144.292i			0.157009i	0.996914	+	0.0785047i	$0.0250145\pi$
$919$			144.292i			0.157009i	−0.996914	+	0.0785047i	$0.974985\pi$
$920$			143.167i			0.155617i
$921$		0			0
$922$	319.928			0.346993
$923$		−	779.878i		−	0.844938i
$924$		0			0
$925$	−1882.81			−2.03547
$926$			980.922i			1.05931i
$927$		0			0
$928$	509.315			0.548831
$929$	1535.32			1.65266			0.826328	−	0.563189i	$-0.190426\pi$
$929$	1535.32			1.65266			0.826328	+	0.563189i	$0.190426\pi$
$930$		0			0
$931$		−	807.528i		−	0.867377i
$932$		−	60.1217i		−	0.0645083i
$933$		0			0
$934$			484.840i			0.519101i
$935$	2079.34	−	656.493i	2.22389	−	0.702131i
$936$		0			0
$937$		−	1670.95i		−	1.78329i	−0.452732	−	0.891646i	$-0.649551\pi$
$937$		−	1670.95i		−	1.78329i	0.452732	−	0.891646i	$-0.350449\pi$
$938$	29.0506			0.0309708
$939$		0			0
$940$	79.4515			0.0845229
$941$			1.54670i			0.00164367i	1.00000		0.000821837i	$0.000261599\pi$
$941$			1.54670i			0.00164367i	−1.00000		0.000821837i	$0.999738\pi$
$942$		0			0
$943$		−	68.6469i		−	0.0727963i
$944$	374.027			0.396215
$945$		0			0
$946$	−97.1282	−	307.638i	−0.102672	−	0.325199i
$947$	−778.298			−0.821856			−0.410928	−	0.911668i	$-0.634795\pi$
$947$	−778.298			−0.821856			−0.410928	+	0.911668i	$0.634795\pi$
$948$		0			0
$949$	−315.654			−0.332618
$950$	2603.98			2.74103
$951$		0			0
$952$		−	993.247i		−	1.04333i
$953$			677.960i			0.711395i	0.934601	+	0.355698i	$0.115757\pi$
$953$			677.960i			0.711395i	−0.934601	+	0.355698i	$0.884243\pi$
$954$		0			0
$955$	1376.80			1.44168
$956$			105.934i			0.110810i
$957$		0			0
$958$	176.752			0.184501
$959$		−	1109.85i		−	1.15730i
$960$		0			0
$961$	1818.28			1.89207
$962$	518.539			0.539022
$963$		0			0
$964$		−	13.7382i		−	0.0142512i
$965$			1741.04i			1.80418i
$966$		0			0
$967$			1123.90i			1.16226i	0.813812	+	0.581128i	$0.197388\pi$
$967$			1123.90i			1.16226i	−0.813812	+	0.581128i	$0.802612\pi$
$968$	−605.037	−	862.668i	−0.625038	−	0.891186i
$969$		0			0
$970$			1369.64i			1.41200i
$971$	709.561			0.730753			0.365376	−	0.930860i	$-0.380940\pi$
$971$	709.561			0.730753			0.365376	+	0.930860i	$0.380940\pi$
$972$		0			0
$973$	425.712			0.437526
$974$		−	424.919i		−	0.436262i
$975$		0			0
$976$			312.394i			0.320075i
$977$	−167.748			−0.171697			−0.0858487	−	0.996308i	$-0.527360\pi$
$977$	−167.748			−0.171697			−0.0858487	+	0.996308i	$0.527360\pi$
$978$		0			0
$979$	25.0726	+	79.4134i	0.0256104	+	0.0811169i
$980$	−271.802			−0.277349
$981$		0			0
$982$	−432.947			−0.440883
$983$	862.956			0.877880			0.438940	−	0.898516i	$-0.355354\pi$
$983$	862.956			0.877880			0.438940	+	0.898516i	$0.355354\pi$
$984$		0			0
$985$		−	2621.16i		−	2.66108i
$986$		−	988.761i		−	1.00280i
$987$		0			0
$988$	339.078			0.343196
$989$			34.1603i			0.0345402i
$990$		0			0
$991$	1594.53			1.60901			0.804507	−	0.593943i	$-0.202430\pi$
$991$	1594.53			1.60901			0.804507	+	0.593943i	$0.202430\pi$
$992$		−	1035.77i		−	1.04412i
$993$		0			0
$994$	−783.778			−0.788509
$995$	247.214			0.248456
$996$		0			0
$997$			798.471i			0.800874i	0.916324	+	0.400437i	$0.131142\pi$
$997$			798.471i			0.800874i	−0.916324	+	0.400437i	$0.868858\pi$
$998$		−	1167.82i		−	1.17016i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.3.c.b.109.11	yes	16
3.2	odd	2	inner	297.3.c.b.109.6	yes	16
11.10	odd	2	inner	297.3.c.b.109.5	✓	16
33.32	even	2	inner	297.3.c.b.109.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.3.c.b.109.5	✓	16	11.10	odd	2	inner
297.3.c.b.109.6	yes	16	3.2	odd	2	inner
297.3.c.b.109.11	yes	16	1.1	even	1	trivial
297.3.c.b.109.12	yes	16	33.32	even	2	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:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	297.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.64800i q^{2} +1.28411 q^{4} -8.56484 q^{5} -4.92814i q^{7} +8.70819i q^{8} +O(q^{10})\)	\(q+1.64800i q^{2} +1.28411 q^{4} -8.56484 q^{5} -4.92814i q^{7} +8.70819i q^{8} -14.1148i q^{10} +(-3.31180 - 10.4896i) q^{11} +8.08116i q^{13} +8.12157 q^{14} -9.21465 q^{16} -23.1444i q^{17} -32.6757i q^{19} -10.9982 q^{20} +(17.2868 - 5.45784i) q^{22} -1.91954 q^{23} +48.3566 q^{25} -13.3177 q^{26} -6.32826i q^{28} -25.9232i q^{29} -52.7189 q^{31} +19.6471i q^{32} +38.1419 q^{34} +42.2088i q^{35} -38.9360 q^{37} +53.8495 q^{38} -74.5843i q^{40} +35.7622i q^{41} -17.7961i q^{43} +(-4.25271 - 13.4698i) q^{44} -3.16339i q^{46} -7.22407 q^{47} +24.7134 q^{49} +79.6914i q^{50} +10.3771i q^{52} +80.3440 q^{53} +(28.3651 + 89.8419i) q^{55} +42.9152 q^{56} +42.7214 q^{58} -40.5905 q^{59} -33.9019i q^{61} -86.8805i q^{62} -69.2369 q^{64} -69.2138i q^{65} +3.57697 q^{67} -29.7199i q^{68} -69.5599 q^{70} -96.5057 q^{71} +39.0605i q^{73} -64.1664i q^{74} -41.9591i q^{76} +(-51.6943 + 16.3210i) q^{77} +118.280i q^{79} +78.9220 q^{80} -58.9360 q^{82} -5.50116i q^{83} +198.228i q^{85} +29.3279 q^{86} +(91.3455 - 28.8398i) q^{88} -7.57067 q^{89} +39.8251 q^{91} -2.46489 q^{92} -11.9052i q^{94} +279.863i q^{95} -97.0354 q^{97} +40.7276i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4}+O(q^{10}) \) 16 * q - 32 * q^4	\( 16 q - 32 q^{4} + 160 q^{16} + 102 q^{22} + 12 q^{25} + 68 q^{31} + 156 q^{34} - 124 q^{37} - 272 q^{49} + 182 q^{55} + 492 q^{58} - 680 q^{64} - 400 q^{67} + 324 q^{70} - 444 q^{82} - 510 q^{88} + 120 q^{91} + 152 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 160 * q^16 + 102 * q^22 + 12 * q^25 + 68 * q^31 + 156 * q^34 - 124 * q^37 - 272 * q^49 + 182 * q^55 + 492 * q^58 - 680 * q^64 - 400 * q^67 + 324 * q^70 - 444 * q^82 - 510 * q^88 + 120 * q^91 + 152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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