LMFDB - Embedded newform 297.3.c.b.109.9

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			109.9
Root			$1.73205 + 0.184385i$ of defining polynomial
Character	$\chi$	$=$	297.109
Dual form			297.3.c.b.109.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.184385i q^{2} +3.96600 q^{4} -4.00859 q^{5} -8.43922i q^{7} +1.46881i q^{8} +O(q^{10})$	\(q+0.184385i q^{2} +3.96600 q^{4} -4.00859 q^{5} -8.43922i q^{7} +1.46881i q^{8} -0.739122i q^{10} +(6.50908 + 8.86746i) q^{11} -19.7418i q^{13} +1.55606 q^{14} +15.5932 q^{16} -30.7073i q^{17} +8.86630i q^{19} -15.8981 q^{20} +(-1.63502 + 1.20017i) q^{22} +12.9593 q^{23} -8.93123 q^{25} +3.64009 q^{26} -33.4700i q^{28} +17.1818i q^{29} +37.4224 q^{31} +8.75037i q^{32} +5.66196 q^{34} +33.8293i q^{35} +31.7612 q^{37} -1.63481 q^{38} -5.88784i q^{40} -63.7865i q^{41} -22.6623i q^{43} +(25.8150 + 35.1684i) q^{44} +2.38949i q^{46} -80.8340 q^{47} -22.2204 q^{49} -1.64678i q^{50} -78.2962i q^{52} -2.76682 q^{53} +(-26.0922 - 35.5460i) q^{55} +12.3956 q^{56} -3.16805 q^{58} +98.8565 q^{59} +73.9334i q^{61} +6.90012i q^{62} +60.7593 q^{64} +79.1369i q^{65} -54.0844 q^{67} -121.785i q^{68} -6.23761 q^{70} -62.1321 q^{71} +81.7158i q^{73} +5.85628i q^{74} +35.1637i q^{76} +(74.8344 - 54.9315i) q^{77} +121.303i q^{79} -62.5066 q^{80} +11.7612 q^{82} +61.1145i q^{83} +123.093i q^{85} +4.17859 q^{86} +(-13.0246 + 9.56059i) q^{88} -56.6145 q^{89} -166.606 q^{91} +51.3965 q^{92} -14.9046i q^{94} -35.5413i q^{95} -113.203 q^{97} -4.09709i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			0.184385i			0.0921923i	0.998937	+	0.0460961i	$0.0146781\pi$
$2$			0.184385i			0.0921923i	−0.998937	+	0.0460961i	$0.985322\pi$
$3$		0			0
$3$		0			0
$4$	3.96600			0.991501
$5$	−4.00859			−0.801718			−0.400859	−	0.916140i	$-0.631288\pi$
$5$	−4.00859			−0.801718			−0.400859	+	0.916140i	$0.631288\pi$
$6$		0			0
$7$		−	8.43922i		−	1.20560i	−0.797891	−	0.602801i	$-0.794051\pi$
$7$		−	8.43922i		−	1.20560i	0.797891	−	0.602801i	$-0.205949\pi$
$8$			1.46881i			0.183601i
$9$		0			0
$10$		−	0.739122i		−	0.0739122i
$11$	6.50908	+	8.86746i	0.591735	+	0.806133i
$11$	6.50908	+	8.86746i	0.591735	+	0.806133i
$12$		0			0
$13$		−	19.7418i		−	1.51860i	−0.650739	−	0.759302i	$-0.725541\pi$
$13$		−	19.7418i		−	1.51860i	0.650739	−	0.759302i	$-0.274459\pi$
$14$	1.55606			0.111147
$15$		0			0
$16$	15.5932			0.974574
$17$		−	30.7073i		−	1.80631i	−0.429311	−	0.903157i	$-0.641243\pi$
$17$		−	30.7073i		−	1.80631i	0.429311	−	0.903157i	$-0.358757\pi$
$18$		0			0
$19$			8.86630i			0.466647i	0.972399	+	0.233324i	$0.0749601\pi$
$19$			8.86630i			0.466647i	−0.972399	+	0.233324i	$0.925040\pi$
$20$	−15.8981			−0.794903
$21$		0			0
$22$	−1.63502	+	1.20017i	−0.0743192	+	0.0545534i
$23$	12.9593			0.563447			0.281723	−	0.959496i	$-0.409094\pi$
$23$	12.9593			0.563447			0.281723	+	0.959496i	$0.409094\pi$
$24$		0			0
$25$	−8.93123			−0.357249
$26$	3.64009			0.140004
$27$		0			0
$28$		−	33.4700i		−	1.19536i
$29$			17.1818i			0.592475i	0.955114	+	0.296237i	$0.0957320\pi$
$29$			17.1818i			0.592475i	−0.955114	+	0.296237i	$0.904268\pi$
$30$		0			0
$31$	37.4224			1.20717			0.603587	−	0.797297i	$-0.293737\pi$
$31$	37.4224			1.20717			0.603587	+	0.797297i	$0.293737\pi$
$32$			8.75037i			0.273449i
$33$		0			0
$34$	5.66196			0.166528
$35$			33.8293i			0.966552i
$36$		0			0
$37$	31.7612			0.858412			0.429206	−	0.903207i	$-0.358793\pi$
$37$	31.7612			0.858412			0.429206	+	0.903207i	$0.358793\pi$
$38$	−1.63481			−0.0430213
$39$		0			0
$40$		−	5.88784i		−	0.147196i
$41$		−	63.7865i		−	1.55577i	−0.628408	−	0.777884i	$-0.716293\pi$
$41$		−	63.7865i		−	1.55577i	0.628408	−	0.777884i	$-0.283707\pi$
$42$		0			0
$43$		−	22.6623i		−	0.527031i	−0.964655	−	0.263516i	$-0.915118\pi$
$43$		−	22.6623i		−	0.527031i	0.964655	−	0.263516i	$-0.0848821\pi$
$44$	25.8150	+	35.1684i	0.586705	+	0.799281i
$45$		0			0
$46$			2.38949i			0.0519454i
$47$	−80.8340			−1.71987			−0.859937	−	0.510401i	$-0.829497\pi$
$47$	−80.8340			−1.71987			−0.859937	+	0.510401i	$0.829497\pi$
$48$		0			0
$49$	−22.2204			−0.453477
$50$		−	1.64678i		−	0.0329356i
$51$		0			0
$52$		−	78.2962i		−	1.50570i
$53$	−2.76682			−0.0522041			−0.0261020	−	0.999659i	$-0.508309\pi$
$53$	−2.76682			−0.0522041			−0.0261020	+	0.999659i	$0.508309\pi$
$54$		0			0
$55$	−26.0922	−	35.5460i	−0.474404	−	0.646291i
$56$	12.3956			0.221350
$57$		0			0
$58$	−3.16805			−0.0546216
$59$	98.8565			1.67553			0.837767	−	0.546028i	$-0.183861\pi$
$59$	98.8565			1.67553			0.837767	+	0.546028i	$0.183861\pi$
$60$		0			0
$61$			73.9334i			1.21202i	0.795456	+	0.606012i	$0.207232\pi$
$61$			73.9334i			1.21202i	−0.795456	+	0.606012i	$0.792768\pi$
$62$			6.90012i			0.111292i
$63$		0			0
$64$	60.7593			0.949364
$65$			79.1369i			1.21749i
$66$		0			0
$67$	−54.0844			−0.807229			−0.403615	−	0.914929i	$-0.632246\pi$
$67$	−54.0844			−0.807229			−0.403615	+	0.914929i	$0.632246\pi$
$68$		−	121.785i		−	1.79096i
$69$		0			0
$70$	−6.23761			−0.0891087
$71$	−62.1321			−0.875100			−0.437550	−	0.899194i	$-0.644154\pi$
$71$	−62.1321			−0.875100			−0.437550	+	0.899194i	$0.644154\pi$
$72$		0			0
$73$			81.7158i			1.11939i	0.828697	+	0.559697i	$0.189082\pi$
$73$			81.7158i			1.11939i	−0.828697	+	0.559697i	$0.810918\pi$
$74$			5.85628i			0.0791389i
$75$		0			0
$76$			35.1637i			0.462681i
$77$	74.8344	−	54.9315i	0.971876	−	0.713396i
$78$		0			0
$79$			121.303i			1.53548i	0.640763	+	0.767739i	$0.278618\pi$
$79$			121.303i			1.53548i	−0.640763	+	0.767739i	$0.721382\pi$
$80$	−62.5066			−0.781333
$81$		0			0
$82$	11.7612			0.143430
$83$			61.1145i			0.736319i	0.929763	+	0.368160i	$0.120012\pi$
$83$			61.1145i			0.736319i	−0.929763	+	0.368160i	$0.879988\pi$
$84$		0			0
$85$			123.093i			1.44815i
$86$	4.17859			0.0485882
$87$		0			0
$88$	−13.0246	+	9.56059i	−0.148007	+	0.108643i
$89$	−56.6145			−0.636118			−0.318059	−	0.948071i	$-0.603031\pi$
$89$	−56.6145			−0.636118			−0.318059	+	0.948071i	$0.603031\pi$
$90$		0			0
$91$	−166.606			−1.83083
$92$	51.3965			0.558658
$93$		0			0
$94$		−	14.9046i		−	0.158559i
$95$		−	35.5413i		−	0.374119i
$96$		0			0
$97$	−113.203			−1.16704			−0.583521	−	0.812098i	$-0.698326\pi$
$97$	−113.203			−1.16704			−0.583521	+	0.812098i	$0.698326\pi$
$98$		−	4.09709i		−	0.0418071i
$99$		0			0
$100$	−35.4213			−0.354213
$101$			56.1360i			0.555802i	0.960610	+	0.277901i	$0.0896387\pi$
$101$			56.1360i			0.555802i	−0.960610	+	0.277901i	$0.910361\pi$
$102$		0			0
$103$	−3.83273			−0.0372109			−0.0186055	−	0.999827i	$-0.505923\pi$
$103$	−3.83273			−0.0372109			−0.0186055	+	0.999827i	$0.505923\pi$
$104$	28.9970			0.278817
$105$		0			0
$106$		−	0.510158i		−	0.00481281i
$107$			26.8478i			0.250914i	0.992099	+	0.125457i	$0.0400397\pi$
$107$			26.8478i			0.250914i	−0.992099	+	0.125457i	$0.959960\pi$
$108$		0			0
$109$			70.7041i			0.648661i	0.945944	+	0.324331i	$0.105139\pi$
$109$			70.7041i			0.648661i	−0.945944	+	0.324331i	$0.894861\pi$
$110$	6.55413	−	4.81100i	0.0595830	−	0.0437364i
$111$		0			0
$112$		−	131.594i		−	1.17495i
$113$	−26.8369			−0.237494			−0.118747	−	0.992925i	$-0.537888\pi$
$113$	−26.8369			−0.237494			−0.118747	+	0.992925i	$0.537888\pi$
$114$		0			0
$115$	−51.9484			−0.451725
$116$			68.1429i			0.587439i
$117$		0			0
$118$			18.2276i			0.154471i
$119$	−259.146			−2.17770
$120$		0			0
$121$	−36.2638	+	115.438i	−0.299701	+	0.954033i
$122$	−13.6322			−0.111739
$123$		0			0
$124$	148.417			1.19691
$125$	136.016			1.08813
$126$		0			0
$127$			72.7110i			0.572528i	0.958151	+	0.286264i	$0.0924134\pi$
$127$			72.7110i			0.572528i	−0.958151	+	0.286264i	$0.907587\pi$
$128$			46.2046i			0.360973i
$129$		0			0
$130$	−14.5916			−0.112243
$131$		−	140.074i		−	1.06926i	−0.845085	−	0.534632i	$-0.820450\pi$
$131$		−	140.074i		−	1.06926i	0.845085	−	0.534632i	$-0.179550\pi$
$132$		0			0
$133$	74.8246			0.562591
$134$		−	9.97232i		−	0.0744203i
$135$		0			0
$136$	45.1032			0.331641
$137$	134.176			0.979384			0.489692	−	0.871896i	$-0.337109\pi$
$137$	134.176			0.979384			0.489692	+	0.871896i	$0.337109\pi$
$138$		0			0
$139$		−	144.138i		−	1.03696i	−0.855088	−	0.518482i	$-0.826497\pi$
$139$		−	144.138i		−	1.03696i	0.855088	−	0.518482i	$-0.173503\pi$
$140$			134.167i			0.958337i
$141$		0			0
$142$		−	11.4562i		−	0.0806775i
$143$	175.060	−	128.501i	1.22420	−	0.898610i
$144$		0			0
$145$		−	68.8746i		−	0.474997i
$146$	−15.0671			−0.103199
$147$		0			0
$148$	125.965			0.851116
$149$		−	95.1722i		−	0.638740i	−0.947630	−	0.319370i	$-0.896529\pi$
$149$		−	95.1722i		−	0.638740i	0.947630	−	0.319370i	$-0.103471\pi$
$150$		0			0
$151$		−	49.0897i		−	0.325097i	−0.986701	−	0.162549i	$-0.948029\pi$
$151$		−	49.0897i		−	0.325097i	0.986701	−	0.162549i	$-0.0519715\pi$
$152$	−13.0229			−0.0856769
$153$		0			0
$154$	10.1285	+	13.7983i	0.0657696	+	0.0895994i
$155$	−150.011			−0.967813
$156$		0			0
$157$	39.1517			0.249374			0.124687	−	0.992196i	$-0.460207\pi$
$157$	39.1517			0.249374			0.124687	+	0.992196i	$0.460207\pi$
$158$	−22.3664			−0.141559
$159$		0			0
$160$		−	35.0766i		−	0.219229i
$161$		−	109.366i		−	0.679293i
$162$		0			0
$163$	80.6986			0.495084			0.247542	−	0.968877i	$-0.420377\pi$
$163$	80.6986			0.495084			0.247542	+	0.968877i	$0.420377\pi$
$164$		−	252.977i		−	1.54254i
$165$		0			0
$166$	−11.2686			−0.0678830
$167$			261.825i			1.56781i	0.620880	+	0.783906i	$0.286776\pi$
$167$			261.825i			1.56781i	−0.620880	+	0.783906i	$0.713224\pi$
$168$		0			0
$169$	−220.740			−1.30616
$170$	−22.6965			−0.133509
$171$		0			0
$172$		−	89.8789i		−	0.522552i
$173$			162.346i			0.938416i	0.883088	+	0.469208i	$0.155461\pi$
$173$			162.346i			0.938416i	−0.883088	+	0.469208i	$0.844539\pi$
$174$		0			0
$175$			75.3725i			0.430700i
$176$	101.497	+	138.272i	0.576689	+	0.785636i
$177$		0			0
$178$		−	10.4388i		−	0.0586452i
$179$	145.636			0.813607			0.406804	−	0.913516i	$-0.366643\pi$
$179$	145.636			0.813607			0.406804	+	0.913516i	$0.366643\pi$
$180$		0			0
$181$	−139.130			−0.768673			−0.384336	−	0.923193i	$-0.625570\pi$
$181$	−139.130			−0.768673			−0.384336	+	0.923193i	$0.625570\pi$
$182$		−	30.7195i		−	0.168789i
$183$		0			0
$184$			19.0347i			0.103449i
$185$	−127.318			−0.688204
$186$		0			0
$187$	272.296	−	199.876i	1.45613	−	1.06886i
$188$	−320.588			−1.70526
$189$		0			0
$190$	6.55327			0.0344909
$191$	288.606			1.51103			0.755513	−	0.655133i	$-0.227388\pi$
$191$	288.606			1.51103			0.755513	+	0.655133i	$0.227388\pi$
$192$		0			0
$193$		−	54.4234i		−	0.281986i	−0.990011	−	0.140993i	$-0.954970\pi$
$193$		−	54.4234i		−	0.281986i	0.990011	−	0.140993i	$-0.0450296\pi$
$194$		−	20.8729i		−	0.107592i
$195$		0			0
$196$	−88.1260			−0.449623
$197$			90.9850i			0.461853i	0.972971	+	0.230926i	$0.0741757\pi$
$197$			90.9850i			0.461853i	−0.972971	+	0.230926i	$0.925824\pi$
$198$		0			0
$199$	181.217			0.910638			0.455319	−	0.890328i	$-0.349525\pi$
$199$	181.217			0.910638			0.455319	+	0.890328i	$0.349525\pi$
$200$		−	13.1183i		−	0.0655913i
$201$		0			0
$202$	−10.3506			−0.0512406
$203$	145.001			0.714289
$204$		0			0
$205$			255.694i			1.24729i
$206$		−	0.706696i		−	0.00343056i
$207$		0			0
$208$		−	307.838i		−	1.47999i
$209$	−78.6215	+	57.7114i	−0.376180	+	0.276131i
$210$		0			0
$211$		−	24.2157i		−	0.114766i	−0.998352	−	0.0573832i	$-0.981724\pi$
$211$		−	24.2157i		−	0.114766i	0.998352	−	0.0573832i	$-0.0182757\pi$
$212$	−10.9732			−0.0517604
$213$		0			0
$214$	−4.95032			−0.0231323
$215$			90.8440i			0.422530i
$216$		0			0
$217$		−	315.816i		−	1.45537i
$218$	−13.0367			−0.0598016
$219$		0			0
$220$	−103.482	−	140.976i	−0.470372	−	0.640798i
$221$	−606.219			−2.74307
$222$		0			0
$223$	354.013			1.58750			0.793751	−	0.608242i	$-0.208125\pi$
$223$	354.013			1.58750			0.793751	+	0.608242i	$0.208125\pi$
$224$	73.8463			0.329671
$225$		0			0
$226$		−	4.94830i		−	0.0218951i
$227$		−	242.374i		−	1.06773i	−0.845570	−	0.533864i	$-0.820739\pi$
$227$		−	242.374i		−	1.06773i	0.845570	−	0.533864i	$-0.179261\pi$
$228$		0			0
$229$	−87.9800			−0.384192			−0.192096	−	0.981376i	$-0.561528\pi$
$229$	−87.9800			−0.384192			−0.192096	+	0.981376i	$0.561528\pi$
$230$		−	9.57848i		−	0.0416456i
$231$		0			0
$232$	−25.2367			−0.108779
$233$		−	251.183i		−	1.07804i	−0.842293	−	0.539020i	$-0.818795\pi$
$233$		−	251.183i		−	1.07804i	0.842293	−	0.539020i	$-0.181205\pi$
$234$		0			0
$235$	324.030			1.37885
$236$	392.065			1.66129
$237$		0			0
$238$		−	47.7825i		−	0.200767i
$239$			30.2323i			0.126495i	0.997998	+	0.0632475i	$0.0201457\pi$
$239$			30.2323i			0.126495i	−0.997998	+	0.0632475i	$0.979854\pi$
$240$		0			0
$241$		−	102.699i		−	0.426135i	−0.977037	−	0.213068i	$-0.931655\pi$
$241$		−	102.699i		−	0.426135i	0.977037	−	0.213068i	$-0.0683455\pi$
$242$	−21.2850	−	6.68648i	−0.0879545	−	0.0276301i
$243$		0			0
$244$			293.220i			1.20172i
$245$	89.0723			0.363560
$246$		0			0
$247$	175.037			0.708652
$248$			54.9663i			0.221638i
$249$		0			0
$250$			25.0793i			0.100317i
$251$	−88.9176			−0.354253			−0.177127	−	0.984188i	$-0.556680\pi$
$251$	−88.9176			−0.354253			−0.177127	+	0.984188i	$0.556680\pi$
$252$		0			0
$253$	84.3529	+	114.916i	0.333411	+	0.454213i
$254$	−13.4068			−0.0527826
$255$		0			0
$256$	234.518			0.916085
$257$	346.967			1.35007			0.675033	−	0.737787i	$-0.264129\pi$
$257$	346.967			1.35007			0.675033	+	0.737787i	$0.264129\pi$
$258$		0			0
$259$		−	268.040i		−	1.03490i
$260$			313.857i			1.20714i
$261$		0			0
$262$	25.8274			0.0985779
$263$			248.571i			0.945137i	0.881294	+	0.472568i	$0.156673\pi$
$263$			248.571i			0.945137i	−0.881294	+	0.472568i	$0.843327\pi$
$264$		0			0
$265$	11.0910			0.0418529
$266$			13.7965i			0.0518665i
$267$		0			0
$268$	−214.499			−0.800368
$269$	229.929			0.854756			0.427378	−	0.904073i	$-0.359437\pi$
$269$	229.929			0.854756			0.427378	+	0.904073i	$0.359437\pi$
$270$		0			0
$271$		−	181.016i		−	0.667954i	−0.942581	−	0.333977i	$-0.891609\pi$
$271$		−	181.016i		−	0.667954i	0.942581	−	0.333977i	$-0.108391\pi$
$272$		−	478.825i		−	1.76039i
$273$		0			0
$274$			24.7399i			0.0902916i
$275$	−58.1341	−	79.1973i	−0.211397	−	0.287990i
$276$		0			0
$277$		−	181.207i		−	0.654176i	−0.944994	−	0.327088i	$-0.893933\pi$
$277$		−	181.207i		−	0.654176i	0.944994	−	0.327088i	$-0.106067\pi$
$278$	26.5768			0.0956001
$279$		0			0
$280$	−49.6888			−0.177460
$281$		−	151.895i		−	0.540552i	−0.962783	−	0.270276i	$-0.912885\pi$
$281$		−	151.895i		−	0.540552i	0.962783	−	0.270276i	$-0.0871149\pi$
$282$		0			0
$283$		−	182.527i		−	0.644972i	−0.946574	−	0.322486i	$-0.895481\pi$
$283$		−	182.527i		−	0.644972i	0.946574	−	0.322486i	$-0.104519\pi$
$284$	−246.416			−0.867662
$285$		0			0
$286$	23.6936	+	32.2784i	0.0828449	+	0.112861i
$287$	−538.308			−1.87564
$288$		0			0
$289$	−653.941			−2.26277
$290$	12.6994			0.0437911
$291$		0			0
$292$			324.085i			1.10988i
$293$			432.249i			1.47525i	0.675209	+	0.737627i	$0.264054\pi$
$293$			432.249i			1.47525i	−0.675209	+	0.737627i	$0.735946\pi$
$294$		0			0
$295$	−396.275			−1.34331
$296$			46.6512i			0.157605i
$297$		0			0
$298$	17.5483			0.0588869
$299$		−	255.840i		−	0.855652i
$300$		0			0
$301$	−191.252			−0.635390
$302$	9.05139			0.0299715
$303$		0			0
$304$			138.254i			0.454782i
$305$		−	296.369i		−	0.971701i
$306$		0			0
$307$			394.653i			1.28552i	0.766070	+	0.642758i	$0.222210\pi$
$307$			394.653i			1.28552i	−0.766070	+	0.642758i	$0.777790\pi$
$308$	296.794	−	217.859i	0.963615	−	0.707333i
$309$		0			0
$310$		−	27.6597i		−	0.0892249i
$311$	95.5227			0.307147			0.153574	−	0.988137i	$-0.450922\pi$
$311$	95.5227			0.307147			0.153574	+	0.988137i	$0.450922\pi$
$312$		0			0
$313$	−13.3342			−0.0426012			−0.0213006	−	0.999773i	$-0.506781\pi$
$313$	−13.3342			−0.0426012			−0.0213006	+	0.999773i	$0.506781\pi$
$314$			7.21896i			0.0229903i
$315$		0			0
$316$			481.087i			1.52243i
$317$	−313.920			−0.990284			−0.495142	−	0.868812i	$-0.664884\pi$
$317$	−313.920			−0.990284			−0.495142	+	0.868812i	$0.664884\pi$
$318$		0			0
$319$	−152.359	+	111.838i	−0.477613	+	0.350588i
$320$	−243.559			−0.761122
$321$		0			0
$322$	20.1654			0.0626255
$323$	272.260			0.842911
$324$		0			0
$325$			176.319i			0.542520i
$326$			14.8796i			0.0456429i
$327$		0			0
$328$	93.6900			0.285640
$329$			682.176i			2.07348i
$330$		0			0
$331$	88.8016			0.268283			0.134141	−	0.990962i	$-0.457172\pi$
$331$	88.8016			0.268283			0.134141	+	0.990962i	$0.457172\pi$
$332$			242.380i			0.730061i
$333$		0			0
$334$	−48.2764			−0.144540
$335$	216.802			0.647170
$336$		0			0
$337$			342.194i			1.01541i	0.861530	+	0.507707i	$0.169507\pi$
$337$			342.194i			1.01541i	−0.861530	+	0.507707i	$0.830493\pi$
$338$		−	40.7011i		−	0.120418i
$339$		0			0
$340$			488.187i			1.43585i
$341$	243.585	+	331.842i	0.714327	+	0.973143i
$342$		0			0
$343$		−	225.999i		−	0.658890i
$344$	33.2866			0.0967635
$345$		0			0
$346$	−29.9341			−0.0865147
$347$			209.419i			0.603514i	0.953385	+	0.301757i	$0.0975731\pi$
$347$			209.419i			0.603514i	−0.953385	+	0.301757i	$0.902427\pi$
$348$		0			0
$349$			629.752i			1.80445i	0.431268	+	0.902224i	$0.358066\pi$
$349$			629.752i			1.80445i	−0.431268	+	0.902224i	$0.641934\pi$
$350$	−13.8975			−0.0397072
$351$		0			0
$352$	−77.5936	+	56.9569i	−0.220436	+	0.161809i
$353$	−504.524			−1.42925			−0.714623	−	0.699510i	$-0.753402\pi$
$353$	−504.524			−1.42925			−0.714623	+	0.699510i	$0.753402\pi$
$354$		0			0
$355$	249.062			0.701583
$356$	−224.533			−0.630711
$357$		0			0
$358$			26.8530i			0.0750083i
$359$		−	423.320i		−	1.17916i	−0.807708	−	0.589582i	$-0.799292\pi$
$359$		−	423.320i		−	1.17916i	0.807708	−	0.589582i	$-0.200708\pi$
$360$		0			0
$361$	282.389			0.782240
$362$		−	25.6534i		−	0.0708657i
$363$		0			0
$364$	−660.759			−1.81527
$365$		−	327.565i		−	0.897438i
$366$		0			0
$367$	174.159			0.474549			0.237274	−	0.971443i	$-0.423746\pi$
$367$	174.159			0.474549			0.237274	+	0.971443i	$0.423746\pi$
$368$	202.076			0.549120
$369$		0			0
$370$		−	23.4754i		−	0.0634471i
$371$			23.3498i			0.0629374i
$372$		0			0
$373$		−	396.798i		−	1.06380i	−0.846807	−	0.531900i	$-0.821478\pi$
$373$		−	396.798i		−	1.06380i	0.846807	−	0.531900i	$-0.178522\pi$
$374$	36.8541	+	50.2072i	0.0985405	+	0.134244i
$375$		0			0
$376$		−	118.730i		−	0.315770i
$377$	339.200			0.899734
$378$		0			0
$379$	−361.031			−0.952588			−0.476294	−	0.879286i	$-0.658020\pi$
$379$	−361.031			−0.952588			−0.476294	+	0.879286i	$0.658020\pi$
$380$		−	140.957i		−	0.370939i
$381$		0			0
$382$			53.2145i			0.139305i
$383$	−328.677			−0.858166			−0.429083	−	0.903265i	$-0.641163\pi$
$383$	−328.677			−0.858166			−0.429083	+	0.903265i	$0.641163\pi$
$384$		0			0
$385$	−299.980	+	220.198i	−0.779170	+	0.571942i
$386$	10.0348			0.0259970
$387$		0			0
$388$	−448.964			−1.15712
$389$	477.994			1.22878			0.614389	−	0.789004i	$-0.289403\pi$
$389$	477.994			1.22878			0.614389	+	0.789004i	$0.289403\pi$
$390$		0			0
$391$		−	397.945i		−	1.01776i
$392$		−	32.6374i		−	0.0832588i
$393$		0			0
$394$	−16.7762			−0.0425793
$395$		−	486.253i		−	1.23102i
$396$		0			0
$397$	345.963			0.871442			0.435721	−	0.900082i	$-0.356493\pi$
$397$	345.963			0.871442			0.435721	+	0.900082i	$0.356493\pi$
$398$			33.4136i			0.0839537i
$399$		0			0
$400$	−139.266			−0.348166
$401$	−224.682			−0.560305			−0.280152	−	0.959956i	$-0.590385\pi$
$401$	−224.682			−0.560305			−0.280152	+	0.959956i	$0.590385\pi$
$402$		0			0
$403$		−	738.788i		−	1.83322i
$404$			222.635i			0.551078i
$405$		0			0
$406$			26.7359i			0.0658519i
$407$	206.736	+	281.642i	0.507952	+	0.691994i
$408$		0			0
$409$			770.992i			1.88507i	0.334114	+	0.942533i	$0.391563\pi$
$409$			770.992i			1.88507i	−0.334114	+	0.942533i	$0.608437\pi$
$410$	−47.1459			−0.114990
$411$		0			0
$412$	−15.2006			−0.0368947
$413$		−	834.272i		−	2.02003i
$414$		0			0
$415$		−	244.983i		−	0.590320i
$416$	172.749			0.415261
$417$		0			0
$418$	−10.6411	−	14.4966i	−0.0254572	−	0.0346809i
$419$	−26.9779			−0.0643864			−0.0321932	−	0.999482i	$-0.510249\pi$
$419$	−26.9779			−0.0643864			−0.0321932	+	0.999482i	$0.510249\pi$
$420$		0			0
$421$	−61.4181			−0.145886			−0.0729431	−	0.997336i	$-0.523239\pi$
$421$	−61.4181			−0.145886			−0.0729431	+	0.997336i	$0.523239\pi$
$422$	4.46501			0.0105806
$423$		0			0
$424$		−	4.06392i		−	0.00958472i
$425$			274.254i			0.645304i
$426$		0			0
$427$	623.940			1.46122
$428$			106.478i			0.248781i
$429$		0			0
$430$	−16.7502			−0.0389540
$431$			363.704i			0.843860i	0.906628	+	0.421930i	$0.138647\pi$
$431$			363.704i			0.843860i	−0.906628	+	0.421930i	$0.861353\pi$
$432$		0			0
$433$	410.930			0.949031			0.474515	−	0.880247i	$-0.342623\pi$
$433$	410.930			0.949031			0.474515	+	0.880247i	$0.342623\pi$
$434$	58.2316			0.134174
$435$		0			0
$436$			280.412i			0.643148i
$437$			114.901i			0.262931i
$438$		0			0
$439$		−	455.775i		−	1.03821i	−0.854710	−	0.519106i	$-0.826265\pi$
$439$		−	455.775i		−	1.03821i	0.854710	−	0.519106i	$-0.173735\pi$
$440$	52.2102	−	38.3245i	0.118660	−	0.0871010i
$441$		0			0
$442$		−	111.778i		−	0.252890i
$443$	−190.937			−0.431009			−0.215504	−	0.976503i	$-0.569140\pi$
$443$	−190.937			−0.431009			−0.215504	+	0.976503i	$0.569140\pi$
$444$		0			0
$445$	226.944			0.509987
$446$			65.2746i			0.146356i
$447$		0			0
$448$		−	512.761i		−	1.14456i
$449$	−704.187			−1.56835			−0.784173	−	0.620543i	$-0.786912\pi$
$449$	−704.187			−1.56835			−0.784173	+	0.620543i	$0.786912\pi$
$450$		0			0
$451$	565.624	−	415.191i	1.25416	−	0.920601i
$452$	−106.435			−0.235476
$453$		0			0
$454$	44.6901			0.0984363
$455$	667.854			1.46781
$456$		0			0
$457$			619.143i			1.35480i	0.735616	+	0.677399i	$0.236893\pi$
$457$			619.143i			1.35480i	−0.735616	+	0.677399i	$0.763107\pi$
$458$		−	16.2221i		−	0.0354195i
$459$		0			0
$460$	−206.027			−0.447886
$461$		−	453.625i		−	0.984003i	−0.870594	−	0.492001i	$-0.836265\pi$
$461$		−	453.625i		−	0.984003i	0.870594	−	0.492001i	$-0.163735\pi$
$462$		0			0
$463$	−569.010			−1.22896			−0.614481	−	0.788931i	$-0.710635\pi$
$463$	−569.010			−1.22896			−0.614481	+	0.788931i	$0.710635\pi$
$464$			267.919i			0.577411i
$465$		0			0
$466$	46.3144			0.0993870
$467$	−42.0007			−0.0899372			−0.0449686	−	0.998988i	$-0.514319\pi$
$467$	−42.0007			−0.0899372			−0.0449686	+	0.998988i	$0.514319\pi$
$468$		0			0
$469$			456.430i			0.973198i
$470$			59.7462i			0.127120i
$471$		0			0
$472$			145.201i			0.307630i
$473$	200.957	−	147.511i	0.424857	−	0.311863i
$474$		0			0
$475$		−	79.1869i		−	0.166709i
$476$	−1027.77			−2.15919
$477$		0			0
$478$	−5.57437			−0.0116619
$479$			618.652i			1.29155i	0.763528	+	0.645774i	$0.223465\pi$
$479$			618.652i			1.29155i	−0.763528	+	0.645774i	$0.776535\pi$
$480$		0			0
$481$		−	627.025i		−	1.30359i
$482$	18.9360			0.0392864
$483$		0			0
$484$	−143.822	+	457.828i	−0.297153	+	0.945925i
$485$	453.785			0.935639
$486$		0			0
$487$	−510.974			−1.04923			−0.524614	−	0.851340i	$-0.675790\pi$
$487$	−510.974			−1.04923			−0.524614	+	0.851340i	$0.675790\pi$
$488$	−108.594			−0.222529
$489$		0			0
$490$			16.4236i			0.0335175i
$491$		−	514.598i		−	1.04806i	−0.851699	−	0.524031i	$-0.824428\pi$
$491$		−	514.598i		−	1.04806i	0.851699	−	0.524031i	$-0.175572\pi$
$492$		0			0
$493$	527.606			1.07020
$494$			32.2741i			0.0653322i
$495$		0			0
$496$	583.535			1.17648
$497$			524.346i			1.05502i
$498$		0			0
$499$	170.143			0.340967			0.170484	−	0.985360i	$-0.445467\pi$
$499$	170.143			0.340967			0.170484	+	0.985360i	$0.445467\pi$
$500$	539.441			1.07888
$501$		0			0
$502$		−	16.3950i		−	0.0326594i
$503$			475.638i			0.945602i	0.881169	+	0.472801i	$0.156757\pi$
$503$			475.638i			0.945602i	−0.881169	+	0.472801i	$0.843243\pi$
$504$		0			0
$505$		−	225.026i		−	0.445596i
$506$	−21.1887	+	15.5534i	−0.0418749	+	0.0307379i
$507$		0			0
$508$			288.372i			0.567662i
$509$	−456.072			−0.896017			−0.448008	−	0.894029i	$-0.647867\pi$
$509$	−456.072			−0.896017			−0.448008	+	0.894029i	$0.647867\pi$
$510$		0			0
$511$	689.617			1.34954
$512$			228.060i			0.445429i
$513$		0			0
$514$			63.9754i			0.124466i
$515$	15.3638			0.0298327
$516$		0			0
$517$	−526.155	−	716.793i	−1.01771	−	1.38645i
$518$	49.4224			0.0954101
$519$		0			0
$520$	−116.237			−0.223533
$521$	693.828			1.33172			0.665862	−	0.746075i	$-0.268064\pi$
$521$	693.828			1.33172			0.665862	+	0.746075i	$0.268064\pi$
$522$		0			0
$523$		−	800.408i		−	1.53042i	−0.643783	−	0.765208i	$-0.722636\pi$
$523$		−	800.408i		−	1.53042i	0.643783	−	0.765208i	$-0.277364\pi$
$524$		−	555.532i		−	1.06018i
$525$		0			0
$526$	−45.8326			−0.0871343
$527$		−	1149.14i		−	2.18054i
$528$		0			0
$529$	−361.057			−0.682528
$530$			2.04501i			0.00385852i
$531$		0			0
$532$	296.754			0.557809
$533$	−1259.26			−2.36259
$534$		0			0
$535$		−	107.622i		−	0.201162i
$536$		−	79.4396i		−	0.148208i
$537$		0			0
$538$			42.3954i			0.0788019i
$539$	−144.634	−	197.038i	−0.268338	−	0.365563i
$540$		0			0
$541$		−	181.442i		−	0.335383i	−0.985840	−	0.167692i	$-0.946369\pi$
$541$		−	181.442i		−	0.335383i	0.985840	−	0.167692i	$-0.0536313\pi$
$542$	33.3765			0.0615802
$543$		0			0
$544$	268.701			0.493935
$545$		−	283.423i		−	0.520043i
$546$		0			0
$547$			33.6006i			0.0614270i	0.999528	+	0.0307135i	$0.00977795\pi$
$547$			33.6006i			0.0614270i	−0.999528	+	0.0307135i	$0.990222\pi$
$548$	532.141			0.971060
$549$		0			0
$550$	14.6028	−	10.7190i	0.0265505	−	0.0194891i
$551$	−152.339			−0.276477
$552$		0			0
$553$	1023.70			1.85118
$554$	33.4117			0.0603100
$555$		0			0
$556$		−	571.652i		−	1.02815i
$557$			902.783i			1.62080i	0.585880	+	0.810398i	$0.300749\pi$
$557$			902.783i			1.62080i	−0.585880	+	0.810398i	$0.699251\pi$
$558$		0			0
$559$	−447.397			−0.800352
$560$			527.507i			0.941977i
$561$		0			0
$562$	28.0071			0.0498347
$563$		−	439.514i		−	0.780664i	−0.920674	−	0.390332i	$-0.872360\pi$
$563$		−	439.514i		−	0.780664i	0.920674	−	0.390332i	$-0.127640\pi$
$564$		0			0
$565$	107.578			0.190403
$566$	33.6552			0.0594614
$567$		0			0
$568$		−	91.2601i		−	0.160669i
$569$			490.302i			0.861691i	0.902426	+	0.430846i	$0.141785\pi$
$569$			490.302i			0.861691i	−0.902426	+	0.430846i	$0.858215\pi$
$570$		0			0
$571$		−	377.347i		−	0.660853i	−0.943832	−	0.330427i	$-0.892807\pi$
$571$		−	377.347i		−	0.660853i	0.943832	−	0.330427i	$-0.107193\pi$
$572$	694.289	−	509.636i	1.21379	−	0.890972i
$573$		0			0
$574$		−	99.2556i		−	0.172919i
$575$	−115.742			−0.201291
$576$		0			0
$577$	−811.655			−1.40668			−0.703341	−	0.710853i	$-0.748309\pi$
$577$	−811.655			−1.40668			−0.703341	+	0.710853i	$0.748309\pi$
$578$		−	120.577i		−	0.208610i
$579$		0			0
$580$		−	273.157i		−	0.470960i
$581$	515.758			0.887708
$582$		0			0
$583$	−18.0094	−	24.5346i	−0.0308910	−	0.0420834i
$584$	−120.025			−0.205522
$585$		0			0
$586$	−79.7001			−0.136007
$587$	−201.416			−0.343128			−0.171564	−	0.985173i	$-0.554882\pi$
$587$	−201.416			−0.343128			−0.171564	+	0.985173i	$0.554882\pi$
$588$		0			0
$589$			331.798i			0.563325i
$590$		−	73.0670i		−	0.123842i
$591$		0			0
$592$	495.259			0.836586
$593$			122.322i			0.206276i	0.994667	+	0.103138i	$0.0328883\pi$
$593$			122.322i			0.206276i	−0.994667	+	0.103138i	$0.967112\pi$
$594$		0			0
$595$	1038.81			1.74590
$596$		−	377.453i		−	0.633311i
$597$		0			0
$598$	47.1729			0.0788845
$599$	134.364			0.224314			0.112157	−	0.993691i	$-0.464224\pi$
$599$	134.364			0.224314			0.112157	+	0.993691i	$0.464224\pi$
$600$		0			0
$601$		−	1027.75i		−	1.71006i	−0.518578	−	0.855030i	$-0.673539\pi$
$601$		−	1027.75i		−	1.71006i	0.518578	−	0.855030i	$-0.326461\pi$
$602$		−	35.2640i		−	0.0585781i
$603$		0			0
$604$		−	194.690i		−	0.322334i
$605$	145.366	−	462.743i	0.240275	−	0.764865i
$606$		0			0
$607$		−	600.866i		−	0.989895i	−0.868923	−	0.494947i	$-0.835187\pi$
$607$		−	600.866i		−	0.989895i	0.868923	−	0.494947i	$-0.164813\pi$
$608$	−77.5834			−0.127604
$609$		0			0
$610$	54.6458			0.0895833
$611$			1595.81i			2.61181i
$612$		0			0
$613$			727.464i			1.18673i	0.804934	+	0.593364i	$0.202201\pi$
$613$			727.464i			1.18673i	−0.804934	+	0.593364i	$0.797799\pi$
$614$	−72.7680			−0.118515
$615$		0			0
$616$	80.6839	+	109.917i	0.130980	+	0.178437i
$617$	890.821			1.44379			0.721897	−	0.692001i	$-0.243271\pi$
$617$	890.821			1.44379			0.721897	+	0.692001i	$0.243271\pi$
$618$		0			0
$619$	−281.563			−0.454868			−0.227434	−	0.973794i	$-0.573034\pi$
$619$	−281.563			−0.454868			−0.227434	+	0.973794i	$0.573034\pi$
$620$	−594.944			−0.959587
$621$		0			0
$622$			17.6129i			0.0283166i
$623$			477.782i			0.766905i
$624$		0			0
$625$	−321.953			−0.515124
$626$		−	2.45862i		−	0.00392750i
$627$		0			0
$628$	155.276			0.247254
$629$		−	975.303i		−	1.55056i
$630$		0			0
$631$	−30.6804			−0.0486218			−0.0243109	−	0.999704i	$-0.507739\pi$
$631$	−30.6804			−0.0486218			−0.0243109	+	0.999704i	$0.507739\pi$
$632$	−178.170			−0.281915
$633$		0			0
$634$		−	57.8820i		−	0.0912965i
$635$		−	291.468i		−	0.459006i
$636$		0			0
$637$			438.671i			0.688652i
$638$	−20.6211	−	28.0926i	−0.0323215	−	0.0440323i
$639$		0			0
$640$		−	185.215i		−	0.289399i
$641$	−0.625949			−0.000976520			−0.000488260	−	1.00000i	$-0.500155\pi$
$641$	−0.625949			−0.000976520			−0.000488260		1.00000i	$0.500155\pi$
$642$		0			0
$643$	982.428			1.52788			0.763941	−	0.645286i	$-0.223262\pi$
$643$	982.428			1.52788			0.763941	+	0.645286i	$0.223262\pi$
$644$		−	433.746i		−	0.673519i
$645$		0			0
$646$			50.2006i			0.0777099i
$647$	−1023.63			−1.58212			−0.791060	−	0.611738i	$-0.790471\pi$
$647$	−1023.63			−1.58212			−0.791060	+	0.611738i	$0.790471\pi$
$648$		0			0
$649$	643.465	+	876.606i	0.991471	+	1.35070i
$650$	−32.5105			−0.0500161
$651$		0			0
$652$	320.051			0.490876
$653$	151.716			0.232337			0.116168	−	0.993230i	$-0.462939\pi$
$653$	151.716			0.232337			0.116168	+	0.993230i	$0.462939\pi$
$654$		0			0
$655$			561.497i			0.857248i
$656$		−	994.634i		−	1.51621i
$657$		0			0
$658$	−125.783			−0.191159
$659$			279.304i			0.423830i	0.977288	+	0.211915i	$0.0679700\pi$
$659$			279.304i			0.423830i	−0.977288	+	0.211915i	$0.932030\pi$
$660$		0			0
$661$	−469.503			−0.710292			−0.355146	−	0.934811i	$-0.615569\pi$
$661$	−469.503			−0.710292			−0.355146	+	0.934811i	$0.615569\pi$
$662$			16.3736i			0.0247336i
$663$		0			0
$664$	−89.7655			−0.135189
$665$	−299.941			−0.451039
$666$		0			0
$667$			222.663i			0.333828i
$668$			1038.40i			1.55449i
$669$		0			0
$670$			39.9749i			0.0596641i
$671$	−655.602	+	481.239i	−0.977052	+	0.717196i
$672$		0			0
$673$			200.977i			0.298628i	0.988790	+	0.149314i	$0.0477065\pi$
$673$			200.977i			0.298628i	−0.988790	+	0.149314i	$0.952293\pi$
$674$	−63.0954			−0.0936133
$675$		0			0
$676$	−875.457			−1.29505
$677$			473.333i			0.699163i	0.936906	+	0.349581i	$0.113676\pi$
$677$			473.333i			0.699163i	−0.936906	+	0.349581i	$0.886324\pi$
$678$		0			0
$679$			955.346i			1.40699i
$680$	−180.800			−0.265882
$681$		0			0
$682$	−61.1865	+	44.9134i	−0.0897163	+	0.0658554i
$683$	1082.96			1.58559			0.792795	−	0.609488i	$-0.208625\pi$
$683$	1082.96			1.58559			0.792795	+	0.609488i	$0.208625\pi$
$684$		0			0
$685$	−537.855			−0.785189
$686$	41.6707			0.0607445
$687$		0			0
$688$		−	353.378i		−	0.513631i
$689$			54.6221i			0.0792773i
$690$		0			0
$691$	193.866			0.280558			0.140279	−	0.990112i	$-0.455200\pi$
$691$	193.866			0.280558			0.140279	+	0.990112i	$0.455200\pi$
$692$			643.865i			0.930440i
$693$		0			0
$694$	−38.6137			−0.0556393
$695$			577.790i			0.831353i
$696$		0			0
$697$	−1958.71			−2.81020
$698$	−116.117			−0.166356
$699$		0			0
$700$			298.928i			0.427040i
$701$		−	960.662i		−	1.37042i	−0.728347	−	0.685208i	$-0.759711\pi$
$701$		−	960.662i		−	1.37042i	0.728347	−	0.685208i	$-0.240289\pi$
$702$		0			0
$703$			281.605i			0.400575i
$704$	395.487	+	538.781i	0.561772	+	0.765314i
$705$		0			0
$706$		−	93.0264i		−	0.131766i
$707$	473.744			0.670076
$708$		0			0
$709$	−758.596			−1.06995			−0.534976	−	0.844867i	$-0.679679\pi$
$709$	−758.596			−1.06995			−0.534976	+	0.844867i	$0.679679\pi$
$710$			45.9232i			0.0646806i
$711$		0			0
$712$		−	83.1558i		−	0.116792i
$713$	484.967			0.680179
$714$		0			0
$715$	−701.744	+	515.108i	−0.981460	+	0.720431i
$716$	577.592			0.806692
$717$		0			0
$718$	78.0537			0.108710
$719$	−205.989			−0.286494			−0.143247	−	0.989687i	$-0.545754\pi$
$719$	−205.989			−0.286494			−0.143247	+	0.989687i	$0.545754\pi$
$720$		0			0
$721$			32.3452i			0.0448616i
$722$			52.0681i			0.0721165i
$723$		0			0
$724$	−551.789			−0.762140
$725$		−	153.454i		−	0.211661i
$726$		0			0
$727$	−1373.39			−1.88912			−0.944560	−	0.328339i	$-0.893511\pi$
$727$	−1373.39			−1.88912			−0.944560	+	0.328339i	$0.893511\pi$
$728$		−	244.712i		−	0.336143i
$729$		0			0
$730$	60.3979			0.0827368
$731$	−695.900			−0.951984
$732$		0			0
$733$			500.407i			0.682683i	0.939939	+	0.341341i	$0.110881\pi$
$733$			500.407i			0.682683i	−0.939939	+	0.341341i	$0.889119\pi$
$734$			32.1123i			0.0437497i
$735$		0			0
$736$			113.398i			0.154074i
$737$	−352.040	−	479.591i	−0.477666	−	0.650734i
$738$		0			0
$739$			759.480i			1.02771i	0.857876	+	0.513856i	$0.171784\pi$
$739$			759.480i			1.02771i	−0.857876	+	0.513856i	$0.828216\pi$
$740$	−504.942			−0.682354
$741$		0			0
$742$	−4.30533			−0.00580234
$743$		−	802.958i		−	1.08070i	−0.841441	−	0.540349i	$-0.818292\pi$
$743$		−	802.958i		−	1.08070i	0.841441	−	0.540349i	$-0.181708\pi$
$744$		0			0
$745$			381.506i			0.512089i
$746$	73.1634			0.0980742
$747$		0			0
$748$	1079.93	−	792.711i	1.44375	−	1.05977i
$749$	226.574			0.302502
$750$		0			0
$751$	1242.17			1.65402			0.827009	−	0.562188i	$-0.190040\pi$
$751$	1242.17			1.65402			0.827009	+	0.562188i	$0.190040\pi$
$752$	−1260.46			−1.67614
$753$		0			0
$754$			62.5432i			0.0829486i
$755$			196.780i			0.260636i
$756$		0			0
$757$	−1311.26			−1.73218			−0.866089	−	0.499890i	$-0.833374\pi$
$757$	−1311.26			−1.73218			−0.866089	+	0.499890i	$0.833374\pi$
$758$		−	66.5685i		−	0.0878213i
$759$		0			0
$760$	52.2034			0.0686886
$761$			396.593i			0.521147i	0.965454	+	0.260574i	$0.0839117\pi$
$761$			396.593i			0.521147i	−0.965454	+	0.260574i	$0.916088\pi$
$762$		0			0
$763$	596.687			0.782027
$764$	1144.61			1.49818
$765$		0			0
$766$		−	60.6031i		−	0.0791163i
$767$		−	1951.61i		−	2.54447i
$768$		0			0
$769$			1414.55i			1.83946i	0.392550	+	0.919731i	$0.371593\pi$
$769$			1414.55i			1.83946i	−0.392550	+	0.919731i	$0.628407\pi$
$770$	−40.6011	−	55.3117i	−0.0527287	−	0.0718334i
$771$		0			0
$772$		−	215.843i		−	0.279590i
$773$	727.960			0.941734			0.470867	−	0.882204i	$-0.343941\pi$
$773$	727.960			0.941734			0.470867	+	0.882204i	$0.343941\pi$
$774$		0			0
$775$	−334.228			−0.431262
$776$		−	166.274i		−	0.214270i
$777$		0			0
$778$			88.1348i			0.113284i
$779$	565.550			0.725994
$780$		0			0
$781$	−404.423	−	550.954i	−0.517827	−	0.705447i
$782$	73.3749			0.0938298
$783$		0			0
$784$	−346.486			−0.441947
$785$	−156.943			−0.199927
$786$		0			0
$787$			298.294i			0.379026i	0.981878	+	0.189513i	$0.0606910\pi$
$787$			298.294i			0.379026i	−0.981878	+	0.189513i	$0.939309\pi$
$788$			360.847i			0.457927i
$789$		0			0
$790$	89.6575			0.113491
$791$			226.482i			0.286324i
$792$		0			0
$793$	1459.58			1.84058
$794$			63.7902i			0.0803403i
$795$		0			0
$796$	718.707			0.902898
$797$	−628.735			−0.788877			−0.394439	−	0.918922i	$-0.629061\pi$
$797$	−628.735			−0.788877			−0.394439	+	0.918922i	$0.629061\pi$
$798$		0			0
$799$			2482.20i			3.10663i
$800$		−	78.1516i		−	0.0976894i
$801$		0			0
$802$		−	41.4279i		−	0.0516558i
$803$	−724.611	+	531.894i	−0.902380	+	0.662384i
$804$		0			0
$805$			438.404i			0.544601i
$806$	136.221			0.169009
$807$		0			0
$808$	−82.4530			−0.102046
$809$		−	1105.89i		−	1.36698i	−0.729959	−	0.683491i	$-0.760461\pi$
$809$		−	1105.89i		−	1.36698i	0.729959	−	0.683491i	$-0.239539\pi$
$810$		0			0
$811$		−	1351.15i		−	1.66603i	−0.553249	−	0.833016i	$-0.686612\pi$
$811$		−	1351.15i		−	1.66603i	0.553249	−	0.833016i	$-0.313388\pi$
$812$	575.073			0.708218
$813$		0			0
$814$	−51.9304	+	38.1190i	−0.0637965	+	0.0468292i
$815$	−323.487			−0.396917
$816$		0			0
$817$	200.931			0.245938
$818$	−142.159			−0.173788
$819$		0			0
$820$			1014.08i			1.23668i
$821$			411.388i			0.501082i	0.968106	+	0.250541i	$0.0806084\pi$
$821$			411.388i			0.501082i	−0.968106	+	0.250541i	$0.919392\pi$
$822$		0			0
$823$	−376.122			−0.457013			−0.228506	−	0.973542i	$-0.573384\pi$
$823$	−376.122			−0.457013			−0.228506	+	0.973542i	$0.573384\pi$
$824$		−	5.62954i		−	0.00683196i
$825$		0			0
$826$	153.827			0.186231
$827$			1181.11i			1.42818i	0.700051	+	0.714092i	$0.253160\pi$
$827$			1181.11i			1.42818i	−0.700051	+	0.714092i	$0.746840\pi$
$828$		0			0
$829$	−880.646			−1.06230			−0.531149	−	0.847278i	$-0.678240\pi$
$829$	−880.646			−1.06230			−0.531149	+	0.847278i	$0.678240\pi$
$830$	45.1711			0.0544230
$831$		0			0
$832$		−	1199.50i		−	1.44171i
$833$			682.328i			0.819122i
$834$		0			0
$835$		−	1049.55i		−	1.25694i
$836$	−311.813	+	228.884i	−0.372982	+	0.273784i
$837$		0			0
$838$		−	4.97430i		−	0.00593592i
$839$	907.784			1.08198			0.540992	−	0.841028i	$-0.318049\pi$
$839$	907.784			1.08198			0.540992	+	0.841028i	$0.318049\pi$
$840$		0			0
$841$	545.787			0.648974
$842$		−	11.3245i		−	0.0134496i
$843$		0			0
$844$		−	96.0396i		−	0.113791i
$845$	884.857			1.04717
$846$		0			0
$847$	974.206	+	306.038i	1.15018	+	0.361320i
$848$	−43.1435			−0.0508767
$849$		0			0
$850$	−50.5682			−0.0594920
$851$	411.603			0.483669
$852$		0			0
$853$		−	1217.18i		−	1.42694i	−0.700684	−	0.713472i	$-0.747122\pi$
$853$		−	1217.18i		−	1.42694i	0.700684	−	0.713472i	$-0.252878\pi$
$854$			115.045i			0.134713i
$855$		0			0
$856$	−39.4343			−0.0460681
$857$		−	486.093i		−	0.567203i	−0.958942	−	0.283601i	$-0.908471\pi$
$857$		−	486.093i		−	0.567203i	0.958942	−	0.283601i	$-0.0915292\pi$
$858$		0			0
$859$	−1315.42			−1.53134			−0.765669	−	0.643235i	$-0.777592\pi$
$859$	−1315.42			−1.53134			−0.765669	+	0.643235i	$0.777592\pi$
$860$			360.288i			0.418939i
$861$		0			0
$862$	−67.0613			−0.0777974
$863$	−545.263			−0.631823			−0.315911	−	0.948789i	$-0.602310\pi$
$863$	−545.263			−0.631823			−0.315911	+	0.948789i	$0.602310\pi$
$864$		0			0
$865$		−	650.778i		−	0.752345i
$866$			75.7692i			0.0874933i
$867$		0			0
$868$		−	1252.53i		−	1.44300i
$869$	−1075.65	+	789.569i	−1.23780	+	0.908595i
$870$		0			0
$871$			1067.73i			1.22586i
$872$	−103.851			−0.119095
$873$		0			0
$874$	−21.1859			−0.0242402
$875$		−	1147.87i		−	1.31185i
$876$		0			0
$877$		−	249.249i		−	0.284206i	−0.989852	−	0.142103i	$-0.954614\pi$
$877$		−	249.249i		−	0.284206i	0.989852	−	0.142103i	$-0.0453864\pi$
$878$	84.0379			0.0957152
$879$		0			0
$880$	−406.861	−	554.275i	−0.462342	−	0.629858i
$881$	−244.861			−0.277935			−0.138968	−	0.990297i	$-0.544378\pi$
$881$	−244.861			−0.277935			−0.138968	+	0.990297i	$0.544378\pi$
$882$		0			0
$883$	−617.166			−0.698942			−0.349471	−	0.936947i	$-0.613639\pi$
$883$	−617.166			−0.698942			−0.349471	+	0.936947i	$0.613639\pi$
$884$	−2404.27			−2.71976
$885$		0			0
$886$		−	35.2058i		−	0.0397357i
$887$		−	177.796i		−	0.200446i	−0.994965	−	0.100223i	$-0.968044\pi$
$887$		−	177.796i		−	0.200446i	0.994965	−	0.100223i	$-0.0319556\pi$
$888$		0			0
$889$	613.624			0.690241
$890$			41.8450i			0.0470169i
$891$		0			0
$892$	1404.02			1.57401
$893$		−	716.699i		−	0.802574i
$894$		0			0
$895$	−583.793			−0.652283
$896$	389.930			0.435190
$897$		0			0
$898$		−	129.841i		−	0.144589i
$899$			642.983i			0.715221i
$900$		0			0
$901$			84.9616i			0.0942970i
$902$	76.5548	+	104.292i	0.0848723	+	0.115623i
$903$		0			0
$904$		−	39.4182i		−	0.0436042i
$905$	557.714			0.616258
$906$		0			0
$907$	753.634			0.830909			0.415454	−	0.909614i	$-0.363623\pi$
$907$	753.634			0.830909			0.415454	+	0.909614i	$0.363623\pi$
$908$		−	961.257i		−	1.05865i
$909$		0			0
$910$			123.142i			0.135321i
$911$	1254.78			1.37737			0.688685	−	0.725061i	$-0.258188\pi$
$911$	1254.78			1.37737			0.688685	+	0.725061i	$0.258188\pi$
$912$		0			0
$913$	−541.931	+	397.799i	−0.593571	+	0.435706i
$914$	−114.160			−0.124902
$915$		0			0
$916$	−348.929			−0.380927
$917$	−1182.11			−1.28911
$918$		0			0
$919$			682.679i			0.742850i	0.928463	+	0.371425i	$0.121131\pi$
$919$			682.679i			0.742850i	−0.928463	+	0.371425i	$0.878869\pi$
$920$		−	76.3022i		−	0.0829372i
$921$		0			0
$922$	83.6415			0.0907175
$923$			1226.60i			1.32893i
$924$		0			0
$925$	−283.667			−0.306667
$926$		−	104.917i		−	0.113301i
$927$		0			0
$928$	−150.347			−0.162012
$929$	327.556			0.352590			0.176295	−	0.984337i	$-0.443589\pi$
$929$	327.556			0.352590			0.176295	+	0.984337i	$0.443589\pi$
$930$		0			0
$931$		−	197.012i		−	0.211614i
$932$		−	996.194i		−	1.06888i
$933$		0			0
$934$		−	7.74427i		−	0.00829151i
$935$	−1091.52	+	801.222i	−1.16740	+	0.856922i
$936$		0			0
$937$			418.748i			0.446903i	0.974715	+	0.223451i	$0.0717324\pi$
$937$			418.748i			0.446903i	−0.974715	+	0.223451i	$0.928268\pi$
$938$	−84.1586			−0.0897213
$939$		0			0
$940$	1285.11			1.36713
$941$			1117.76i			1.18784i	0.804525	+	0.593919i	$0.202420\pi$
$941$			1117.76i			1.18784i	−0.804525	+	0.593919i	$0.797580\pi$
$942$		0			0
$943$		−	826.626i		−	0.876592i
$944$	1541.49			1.63293
$945$		0			0
$946$	27.1988	+	37.0535i	0.0287513	+	0.0391686i
$947$	1623.89			1.71477			0.857384	−	0.514677i	$-0.172088\pi$
$947$	1623.89			1.71477			0.857384	+	0.514677i	$0.172088\pi$
$948$		0			0
$949$	1613.22			1.69992
$950$	14.6008			0.0153693
$951$		0			0
$952$		−	380.635i		−	0.399827i
$953$		−	176.904i		−	0.185629i	−0.995683	−	0.0928143i	$-0.970414\pi$
$953$		−	176.904i		−	0.185629i	0.995683	−	0.0928143i	$-0.0295863\pi$
$954$		0			0
$955$	−1156.90			−1.21142
$956$			119.901i			0.125420i
$957$		0			0
$958$	−114.070			−0.119071
$959$		−	1132.34i		−	1.18075i
$960$		0			0
$961$	439.437			0.457271
$962$	115.614			0.120181
$963$		0			0
$964$		−	407.303i		−	0.422513i
$965$			218.161i			0.226073i
$966$		0			0
$967$			181.993i			0.188204i	0.995563	+	0.0941021i	$0.0299980\pi$
$967$			181.993i			0.188204i	−0.995563	+	0.0941021i	$0.970002\pi$
$968$	−169.556	−	53.2645i	−0.175161	−	0.0550253i
$969$		0			0
$970$			83.6709i			0.0862587i
$971$	234.120			0.241112			0.120556	−	0.992707i	$-0.461532\pi$
$971$	234.120			0.241112			0.120556	+	0.992707i	$0.461532\pi$
$972$		0			0
$973$	−1216.41			−1.25017
$974$		−	94.2157i		−	0.0967307i
$975$		0			0
$976$			1152.86i			1.18121i
$977$	−389.869			−0.399047			−0.199524	−	0.979893i	$-0.563939\pi$
$977$	−389.869			−0.399047			−0.199524	+	0.979893i	$0.563939\pi$
$978$		0			0
$979$	−368.508	−	502.027i	−0.376413	−	0.512796i
$980$	353.261			0.360470
$981$		0			0
$982$	94.8840			0.0966232
$983$	536.559			0.545839			0.272919	−	0.962037i	$-0.412011\pi$
$983$	536.559			0.545839			0.272919	+	0.962037i	$0.412011\pi$
$984$		0			0
$985$		−	364.721i		−	0.370275i
$986$			97.2825i			0.0986638i
$987$		0			0
$988$	694.197			0.702629
$989$		−	293.688i		−	0.296954i
$990$		0			0
$991$	517.023			0.521718			0.260859	−	0.965377i	$-0.415994\pi$
$991$	517.023			0.521718			0.260859	+	0.965377i	$0.415994\pi$
$992$			327.460i			0.330101i
$993$		0			0
$994$	−96.6814			−0.0972650
$995$	−726.424			−0.730074
$996$		0			0
$997$		−	249.325i		−	0.250075i	−0.992152	−	0.125037i	$-0.960095\pi$
$997$		−	249.325i		−	0.250075i	0.992152	−	0.125037i	$-0.0399051\pi$
$998$			31.3717i			0.0314346i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.3.c.b.109.9	yes	16
3.2	odd	2	inner	297.3.c.b.109.8	yes	16
11.10	odd	2	inner	297.3.c.b.109.7	✓	16
33.32	even	2	inner	297.3.c.b.109.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.3.c.b.109.7	✓	16	11.10	odd	2	inner
297.3.c.b.109.8	yes	16	3.2	odd	2	inner
297.3.c.b.109.9	yes	16	1.1	even	1	trivial
297.3.c.b.109.10	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)

\(f(q)\)	\(=\)	\(q+0.184385i q^{2} +3.96600 q^{4} -4.00859 q^{5} -8.43922i q^{7} +1.46881i q^{8} +O(q^{10})\)	\(q+0.184385i q^{2} +3.96600 q^{4} -4.00859 q^{5} -8.43922i q^{7} +1.46881i q^{8} -0.739122i q^{10} +(6.50908 + 8.86746i) q^{11} -19.7418i q^{13} +1.55606 q^{14} +15.5932 q^{16} -30.7073i q^{17} +8.86630i q^{19} -15.8981 q^{20} +(-1.63502 + 1.20017i) q^{22} +12.9593 q^{23} -8.93123 q^{25} +3.64009 q^{26} -33.4700i q^{28} +17.1818i q^{29} +37.4224 q^{31} +8.75037i q^{32} +5.66196 q^{34} +33.8293i q^{35} +31.7612 q^{37} -1.63481 q^{38} -5.88784i q^{40} -63.7865i q^{41} -22.6623i q^{43} +(25.8150 + 35.1684i) q^{44} +2.38949i q^{46} -80.8340 q^{47} -22.2204 q^{49} -1.64678i q^{50} -78.2962i q^{52} -2.76682 q^{53} +(-26.0922 - 35.5460i) q^{55} +12.3956 q^{56} -3.16805 q^{58} +98.8565 q^{59} +73.9334i q^{61} +6.90012i q^{62} +60.7593 q^{64} +79.1369i q^{65} -54.0844 q^{67} -121.785i q^{68} -6.23761 q^{70} -62.1321 q^{71} +81.7158i q^{73} +5.85628i q^{74} +35.1637i q^{76} +(74.8344 - 54.9315i) q^{77} +121.303i q^{79} -62.5066 q^{80} +11.7612 q^{82} +61.1145i q^{83} +123.093i q^{85} +4.17859 q^{86} +(-13.0246 + 9.56059i) q^{88} -56.6145 q^{89} -166.606 q^{91} +51.3965 q^{92} -14.9046i q^{94} -35.5413i q^{95} -113.203 q^{97} -4.09709i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4}+O(q^{10}) \) 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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