LMFDB - Embedded newform 99.3.c.b.10.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,3,Mod(10,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, 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("99.10");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	99.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:	$2.69755461717$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.39744.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 12x^{2} + 4x + 22 $ x^4 - 2x^3 + 12x^2 + 4*x + 22
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			10.2
Root			$1.36603 + 3.21405i$ of defining polynomial
Character	$\chi$	$=$	99.10
Dual form			99.3.c.b.10.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35285i q^{2} -1.53590 q^{4} +4.19615 q^{5} -6.42810i q^{7} -5.79766i q^{8} +O(q^{10})$	\(q-2.35285i q^{2} -1.53590 q^{4} +4.19615 q^{5} -6.42810i q^{7} -5.79766i q^{8} -9.87291i q^{10} +(-3.26795 + 10.5034i) q^{11} -7.68899i q^{13} -15.1244 q^{14} -19.7846 q^{16} -8.15051i q^{17} +30.4181i q^{19} -6.44486 q^{20} +(24.7128 + 7.68899i) q^{22} +31.6603 q^{23} -7.39230 q^{25} -18.0910 q^{26} +9.87291i q^{28} +6.88962i q^{29} +51.1769 q^{31} +23.3596i q^{32} -19.1769 q^{34} -26.9733i q^{35} -19.4641 q^{37} +71.5692 q^{38} -24.3279i q^{40} +47.9800i q^{41} +81.8429i q^{43} +(5.01924 - 16.1321i) q^{44} -74.4918i q^{46} -30.1962 q^{47} +7.67949 q^{49} +17.3930i q^{50} +11.8095i q^{52} -26.0526 q^{53} +(-13.7128 + 44.0737i) q^{55} -37.2679 q^{56} +16.2102 q^{58} -82.7461 q^{59} -75.4148i q^{61} -120.412i q^{62} -24.1769 q^{64} -32.2642i q^{65} -34.0000 q^{67} +12.5184i q^{68} -63.4641 q^{70} +72.7321 q^{71} -54.8696i q^{73} +45.7961i q^{74} -46.7191i q^{76} +(67.5167 + 21.0067i) q^{77} -24.3279i q^{79} -83.0192 q^{80} +112.890 q^{82} +0.923034i q^{83} -34.2008i q^{85} +192.564 q^{86} +(60.8949 + 18.9465i) q^{88} -44.8231 q^{89} -49.4256 q^{91} -48.6269 q^{92} +71.0470i q^{94} +127.639i q^{95} -21.2539 q^{97} -18.0687i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{4} - 4 q^{5}+O(q^{10}) $ 4 * q - 20 * q^4 - 4 * q^5	$ 4 q - 20 q^{4} - 4 q^{5} - 20 q^{11} - 12 q^{14} + 4 q^{16} + 92 q^{20} - 12 q^{22} + 92 q^{23} + 12 q^{25} - 204 q^{26} + 80 q^{31} + 48 q^{34} - 64 q^{37} + 120 q^{38} + 124 q^{44} - 100 q^{47} + 100 q^{49} - 28 q^{53} + 56 q^{55} - 156 q^{56} - 240 q^{58} - 40 q^{59} + 28 q^{64} - 136 q^{67} - 240 q^{70} + 284 q^{71} + 180 q^{77} - 436 q^{80} + 216 q^{82} + 216 q^{86} + 396 q^{88} - 304 q^{89} + 24 q^{91} - 340 q^{92} - 376 q^{97}+O(q^{100}) $ 4 * q - 20 * q^4 - 4 * q^5 - 20 * q^11 - 12 * q^14 + 4 * q^16 + 92 * q^20 - 12 * q^22 + 92 * q^23 + 12 * q^25 - 204 * q^26 + 80 * q^31 + 48 * q^34 - 64 * q^37 + 120 * q^38 + 124 * q^44 - 100 * q^47 + 100 * q^49 - 28 * q^53 + 56 * q^55 - 156 * q^56 - 240 * q^58 - 40 * q^59 + 28 * q^64 - 136 * q^67 - 240 * q^70 + 284 * q^71 + 180 * q^77 - 436 * q^80 + 216 * q^82 + 216 * q^86 + 396 * q^88 - 304 * q^89 + 24 * q^91 - 340 * q^92 - 376 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	2.35285i		−	1.17642i	−0.808707	−	0.588212i	$-0.799832\pi$
$2$		−	2.35285i		−	1.17642i	0.808707	−	0.588212i	$-0.200168\pi$
$3$		0			0
$3$		0			0
$4$	−1.53590			−0.383975
$5$	4.19615			0.839230			0.419615	−	0.907702i	$-0.362165\pi$
$5$	4.19615			0.839230			0.419615	+	0.907702i	$0.362165\pi$
$6$		0			0
$7$		−	6.42810i		−	0.918300i	−0.888359	−	0.459150i	$-0.848154\pi$
$7$		−	6.42810i		−	0.918300i	0.888359	−	0.459150i	$-0.151846\pi$
$8$		−	5.79766i		−	0.724707i
$9$		0			0
$10$		−	9.87291i		−	0.987291i
$11$	−3.26795	+	10.5034i	−0.297086	+	0.954851i
$11$	−3.26795	+	10.5034i	−0.297086	+	0.954851i
$12$		0			0
$13$		−	7.68899i		−	0.591461i	−0.955271	−	0.295730i	$-0.904437\pi$
$13$		−	7.68899i		−	0.591461i	0.955271	−	0.295730i	$-0.0955630\pi$
$14$	−15.1244			−1.08031
$15$		0			0
$16$	−19.7846			−1.23654
$17$		−	8.15051i		−	0.479442i	−0.970842	−	0.239721i	$-0.922944\pi$
$17$		−	8.15051i		−	0.479442i	0.970842	−	0.239721i	$-0.0770559\pi$
$18$		0			0
$19$			30.4181i			1.60095i	0.599364	+	0.800477i	$0.295420\pi$
$19$			30.4181i			1.60095i	−0.599364	+	0.800477i	$0.704580\pi$
$20$	−6.44486			−0.322243
$21$		0			0
$22$	24.7128	+	7.68899i	1.12331	+	0.349500i
$23$	31.6603			1.37653			0.688266	−	0.725458i	$-0.258372\pi$
$23$	31.6603			1.37653			0.688266	+	0.725458i	$0.258372\pi$
$24$		0			0
$25$	−7.39230			−0.295692
$26$	−18.0910			−0.695809
$27$		0			0
$28$			9.87291i			0.352604i
$29$			6.88962i			0.237573i	0.992920	+	0.118787i	$0.0379004\pi$
$29$			6.88962i			0.237573i	−0.992920	+	0.118787i	$0.962100\pi$
$30$		0			0
$31$	51.1769			1.65087			0.825434	−	0.564498i	$-0.190930\pi$
$31$	51.1769			1.65087			0.825434	+	0.564498i	$0.190930\pi$
$32$			23.3596i			0.729986i
$33$		0			0
$34$	−19.1769			−0.564027
$35$		−	26.9733i		−	0.770666i
$36$		0			0
$37$	−19.4641			−0.526057			−0.263028	−	0.964788i	$-0.584721\pi$
$37$	−19.4641			−0.526057			−0.263028	+	0.964788i	$0.584721\pi$
$38$	71.5692			1.88340
$39$		0			0
$40$		−	24.3279i		−	0.608197i
$41$			47.9800i			1.17024i	0.810945	+	0.585122i	$0.198953\pi$
$41$			47.9800i			1.17024i	−0.810945	+	0.585122i	$0.801047\pi$
$42$		0			0
$43$			81.8429i			1.90332i	0.307147	+	0.951662i	$0.400626\pi$
$43$			81.8429i			1.90332i	−0.307147	+	0.951662i	$0.599374\pi$
$44$	5.01924	−	16.1321i	0.114074	−	0.366638i
$45$		0			0
$46$		−	74.4918i		−	1.61939i
$47$	−30.1962			−0.642471			−0.321236	−	0.946999i	$-0.604098\pi$
$47$	−30.1962			−0.642471			−0.321236	+	0.946999i	$0.604098\pi$
$48$		0			0
$49$	7.67949			0.156724
$50$			17.3930i			0.347860i
$51$		0			0
$52$			11.8095i			0.227106i
$53$	−26.0526			−0.491558			−0.245779	−	0.969326i	$-0.579044\pi$
$53$	−26.0526			−0.491558			−0.245779	+	0.969326i	$0.579044\pi$
$54$		0			0
$55$	−13.7128	+	44.0737i	−0.249324	+	0.801340i
$56$	−37.2679			−0.665499
$57$		0			0
$58$	16.2102			0.279487
$59$	−82.7461			−1.40248			−0.701238	−	0.712927i	$-0.747369\pi$
$59$	−82.7461			−1.40248			−0.701238	+	0.712927i	$0.747369\pi$
$60$		0			0
$61$		−	75.4148i		−	1.23631i	−0.786057	−	0.618154i	$-0.787881\pi$
$61$		−	75.4148i		−	1.23631i	0.786057	−	0.618154i	$-0.212119\pi$
$62$		−	120.412i		−	1.94212i
$63$		0			0
$64$	−24.1769			−0.377764
$65$		−	32.2642i		−	0.496372i
$66$		0			0
$67$	−34.0000			−0.507463			−0.253731	−	0.967275i	$-0.581658\pi$
$67$	−34.0000			−0.507463			−0.253731	+	0.967275i	$0.581658\pi$
$68$			12.5184i			0.184093i
$69$		0			0
$70$	−63.4641			−0.906630
$71$	72.7321			1.02440			0.512198	−	0.858868i	$-0.328832\pi$
$71$	72.7321			1.02440			0.512198	+	0.858868i	$0.328832\pi$
$72$		0			0
$73$		−	54.8696i		−	0.751639i	−0.926693	−	0.375819i	$-0.877361\pi$
$73$		−	54.8696i		−	0.751639i	0.926693	−	0.375819i	$-0.122639\pi$
$74$			45.7961i			0.618866i
$75$		0			0
$76$		−	46.7191i		−	0.614725i
$77$	67.5167	+	21.0067i	0.876840	+	0.272814i
$78$		0			0
$79$		−	24.3279i		−	0.307948i	−0.988075	−	0.153974i	$-0.950793\pi$
$79$		−	24.3279i		−	0.307948i	0.988075	−	0.153974i	$-0.0492071\pi$
$80$	−83.0192			−1.03774
$81$		0			0
$82$	112.890			1.37670
$83$			0.923034i			0.0111209i	0.999985	+	0.00556045i	$0.00176995\pi$
$83$			0.923034i			0.0111209i	−0.999985	+	0.00556045i	$0.998230\pi$
$84$		0			0
$85$		−	34.2008i		−	0.402362i
$86$	192.564			2.23912
$87$		0			0
$88$	60.8949	+	18.9465i	0.691987	+	0.215301i
$89$	−44.8231			−0.503630			−0.251815	−	0.967775i	$-0.581027\pi$
$89$	−44.8231			−0.503630			−0.251815	+	0.967775i	$0.581027\pi$
$90$		0			0
$91$	−49.4256			−0.543139
$92$	−48.6269			−0.528554
$93$		0			0
$94$			71.0470i			0.755819i
$95$			127.639i			1.34357i
$96$		0			0
$97$	−21.2539			−0.219112			−0.109556	−	0.993981i	$-0.534943\pi$
$97$	−21.2539			−0.219112			−0.109556	+	0.993981i	$0.534943\pi$
$98$		−	18.0687i		−	0.184374i
$99$		0			0
$100$	11.3538			0.113538
$101$		−	52.3479i		−	0.518296i	−0.965838	−	0.259148i	$-0.916558\pi$
$101$		−	52.3479i		−	0.518296i	0.965838	−	0.259148i	$-0.0834417\pi$
$102$		0			0
$103$	−4.74613			−0.0460790			−0.0230395	−	0.999735i	$-0.507334\pi$
$103$	−4.74613			−0.0460790			−0.0230395	+	0.999735i	$0.507334\pi$
$104$	−44.5781			−0.428636
$105$		0			0
$106$			61.2977i			0.578281i
$107$		−	147.723i		−	1.38059i	−0.723530	−	0.690293i	$-0.757482\pi$
$107$		−	147.723i		−	1.38059i	0.723530	−	0.690293i	$-0.242518\pi$
$108$		0			0
$109$		−	3.56847i		−	0.0327383i	−0.999866	−	0.0163691i	$-0.994789\pi$
$109$		−	3.56847i		−	0.0327383i	0.999866	−	0.0163691i	$-0.00521069\pi$
$110$	103.699	+	32.2642i	0.942716	+	0.293311i
$111$		0			0
$112$			127.178i			1.13551i
$113$	62.3154			0.551463			0.275732	−	0.961235i	$-0.411080\pi$
$113$	62.3154			0.551463			0.275732	+	0.961235i	$0.411080\pi$
$114$		0			0
$115$	132.851			1.15523
$116$		−	10.5818i		−	0.0912220i
$117$		0			0
$118$			194.689i			1.64991i
$119$	−52.3923			−0.440271
$120$		0			0
$121$	−99.6410	−	68.6489i	−0.823479	−	0.567346i
$122$	−177.440			−1.45442
$123$		0			0
$124$	−78.6025			−0.633891
$125$	−135.923			−1.08738
$126$		0			0
$127$			28.0200i			0.220630i	0.993897	+	0.110315i	$0.0351860\pi$
$127$			28.0200i			0.220630i	−0.993897	+	0.110315i	$0.964814\pi$
$128$			150.323i			1.17440i
$129$		0			0
$130$	−75.9127			−0.583944
$131$			113.184i			0.864001i	0.901873	+	0.432000i	$0.142192\pi$
$131$			113.184i			0.864001i	−0.901873	+	0.432000i	$0.857808\pi$
$132$		0			0
$133$	195.531			1.47016
$134$			79.9969i			0.596992i
$135$		0			0
$136$	−47.2539			−0.347455
$137$	70.7846			0.516676			0.258338	−	0.966055i	$-0.416825\pi$
$137$	70.7846			0.516676			0.258338	+	0.966055i	$0.416825\pi$
$138$		0			0
$139$			130.499i			0.938839i	0.882975	+	0.469420i	$0.155537\pi$
$139$			130.499i			0.938839i	−0.882975	+	0.469420i	$0.844463\pi$
$140$			41.4282i			0.295916i
$141$		0			0
$142$		−	171.128i		−	1.20512i
$143$	80.7602	+	25.1272i	0.564757	+	0.175715i
$144$		0			0
$145$			28.9099i			0.199379i
$146$	−129.100			−0.884246
$147$		0			0
$148$	29.8949			0.201992
$149$		−	125.793i		−	0.844248i	−0.906538	−	0.422124i	$-0.861285\pi$
$149$		−	125.793i		−	0.844248i	0.906538	−	0.422124i	$-0.138715\pi$
$150$		0			0
$151$		−	275.238i		−	1.82277i	−0.411556	−	0.911384i	$-0.635015\pi$
$151$		−	275.238i		−	1.82277i	0.411556	−	0.911384i	$-0.364985\pi$
$152$	176.354			1.16022
$153$		0			0
$154$	49.4256	−	158.857i	0.320946	−	1.03154i
$155$	214.746			1.38546
$156$		0			0
$157$	−273.138			−1.73974			−0.869868	−	0.493285i	$-0.835796\pi$
$157$	−273.138			−1.73974			−0.869868	+	0.493285i	$0.835796\pi$
$158$	−57.2398			−0.362277
$159$		0			0
$160$			98.0203i			0.612627i
$161$		−	203.515i		−	1.26407i
$162$		0			0
$163$	62.0666			0.380777			0.190388	−	0.981709i	$-0.439025\pi$
$163$	62.0666			0.380777			0.190388	+	0.981709i	$0.439025\pi$
$164$		−	73.6924i		−	0.449344i
$165$		0			0
$166$	2.17176			0.0130829
$167$			47.9800i			0.287305i	0.989628	+	0.143653i	$0.0458848\pi$
$167$			47.9800i			0.287305i	−0.989628	+	0.143653i	$0.954115\pi$
$168$		0			0
$169$	109.879			0.650174
$170$	−80.4693			−0.473349
$171$		0			0
$172$		−	125.702i		−	0.730828i
$173$		−	143.017i		−	0.826688i	−0.910575	−	0.413344i	$-0.864361\pi$
$173$		−	143.017i		−	0.826688i	0.910575	−	0.413344i	$-0.135639\pi$
$174$		0			0
$175$			47.5185i			0.271534i
$176$	64.6551	−	207.805i	0.367359	−	1.18071i
$177$		0			0
$178$			105.462i			0.592483i
$179$	−36.6025			−0.204483			−0.102242	−	0.994760i	$-0.532602\pi$
$179$	−36.6025			−0.204483			−0.102242	+	0.994760i	$0.532602\pi$
$180$		0			0
$181$	−84.1718			−0.465037			−0.232519	−	0.972592i	$-0.574697\pi$
$181$	−84.1718			−0.465037			−0.232519	+	0.972592i	$0.574697\pi$
$182$			116.291i			0.638962i
$183$		0			0
$184$		−	183.555i		−	0.997583i
$185$	−81.6743			−0.441483
$186$		0			0
$187$	85.6077	+	26.6354i	0.457795	+	0.142436i
$188$	46.3782			0.246693
$189$		0			0
$190$	300.315			1.58061
$191$	−44.2346			−0.231595			−0.115797	−	0.993273i	$-0.536942\pi$
$191$	−44.2346			−0.231595			−0.115797	+	0.993273i	$0.536942\pi$
$192$		0			0
$193$		−	136.127i		−	0.705323i	−0.935751	−	0.352662i	$-0.885277\pi$
$193$		−	136.127i		−	0.705323i	0.935751	−	0.352662i	$-0.114723\pi$
$194$			50.0071i			0.257769i
$195$		0			0
$196$	−11.7949			−0.0601782
$197$		−	5.96659i		−	0.0302872i	−0.999885	−	0.0151436i	$-0.995179\pi$
$197$		−	5.96659i		−	0.0302872i	0.999885	−	0.0151436i	$-0.00482055\pi$
$198$		0			0
$199$	−234.056			−1.17616			−0.588081	−	0.808802i	$-0.700116\pi$
$199$	−234.056			−1.17616			−0.588081	+	0.808802i	$0.700116\pi$
$200$			42.8581i			0.214290i
$201$		0			0
$202$	−123.167			−0.609736
$203$	44.2872			0.218163
$204$		0			0
$205$			201.331i			0.982105i
$206$			11.1669i			0.0542084i
$207$		0			0
$208$			152.124i			0.731364i
$209$	−319.492	−	99.4048i	−1.52867	−	0.475621i
$210$		0			0
$211$			167.469i			0.793690i	0.917886	+	0.396845i	$0.129895\pi$
$211$			167.469i			0.793690i	−0.917886	+	0.396845i	$0.870105\pi$
$212$	40.0141			0.188746
$213$		0			0
$214$	−347.569			−1.62416
$215$			343.425i			1.59733i
$216$		0			0
$217$		−	328.970i		−	1.51599i
$218$	−8.39608			−0.0385141
$219$		0			0
$220$	21.0615	−	67.6927i	0.0957340	−	0.307694i
$221$	−62.6692			−0.283571
$222$		0			0
$223$	82.2872			0.369001			0.184500	−	0.982832i	$-0.440933\pi$
$223$	82.2872			0.369001			0.184500	+	0.982832i	$0.440933\pi$
$224$	150.158			0.670347
$225$		0			0
$226$		−	146.619i		−	0.648755i
$227$		−	40.9999i		−	0.180616i	−0.995914	−	0.0903081i	$-0.971215\pi$
$227$		−	40.9999i		−	0.180616i	0.995914	−	0.0903081i	$-0.0287852\pi$
$228$		0			0
$229$	160.718			0.701825			0.350913	−	0.936408i	$-0.385871\pi$
$229$	160.718			0.701825			0.350913	+	0.936408i	$0.385871\pi$
$230$		−	312.579i		−	1.35904i
$231$		0			0
$232$	39.9437			0.172171
$233$			407.954i			1.75087i	0.483332	+	0.875437i	$0.339427\pi$
$233$			407.954i			1.75087i	−0.483332	+	0.875437i	$0.660573\pi$
$234$		0			0
$235$	−126.708			−0.539182
$236$	127.090			0.538515
$237$		0			0
$238$			123.271i			0.517946i
$239$			202.254i			0.846253i	0.906071	+	0.423127i	$0.139067\pi$
$239$			202.254i			0.846253i	−0.906071	+	0.423127i	$0.860933\pi$
$240$		0			0
$241$		−	26.8828i		−	0.111547i	−0.998443	−	0.0557734i	$-0.982238\pi$
$241$		−	26.8828i		−	0.111547i	0.998443	−	0.0557734i	$-0.0177624\pi$
$242$	−161.520	+	234.440i	−0.667440	+	0.968761i
$243$		0			0
$244$			115.830i			0.474711i
$245$	32.2243			0.131528
$246$		0			0
$247$	233.885			0.946901
$248$		−	296.706i		−	1.19640i
$249$		0			0
$250$			319.806i			1.27923i
$251$	−398.277			−1.58676			−0.793380	−	0.608726i	$-0.791681\pi$
$251$	−398.277			−1.58676			−0.793380	+	0.608726i	$0.791681\pi$
$252$		0			0
$253$	−103.464	+	332.539i	−0.408949	+	1.31438i
$254$	65.9268			0.259554
$255$		0			0
$256$	256.979			1.00383
$257$	337.818			1.31447			0.657233	−	0.753687i	$-0.271727\pi$
$257$	337.818			1.31447			0.657233	+	0.753687i	$0.271727\pi$
$258$		0			0
$259$			125.117i			0.483078i
$260$			49.5545i			0.190594i
$261$		0			0
$262$	266.305			1.01643
$263$			134.529i			0.511516i	0.966741	+	0.255758i	$0.0823250\pi$
$263$			134.529i			0.511516i	−0.966741	+	0.255758i	$0.917675\pi$
$264$		0			0
$265$	−109.321			−0.412530
$266$		−	460.054i		−	1.72953i
$267$		0			0
$268$	52.2205			0.194853
$269$	341.870			1.27089			0.635447	−	0.772144i	$-0.280816\pi$
$269$	341.870			1.27089			0.635447	+	0.772144i	$0.280816\pi$
$270$		0			0
$271$			213.298i			0.787077i	0.919308	+	0.393538i	$0.128749\pi$
$271$			213.298i			0.787077i	−0.919308	+	0.393538i	$0.871251\pi$
$272$			161.255i			0.592848i
$273$		0			0
$274$		−	166.545i		−	0.607830i
$275$	24.1577	−	77.6440i	0.0878461	−	0.282342i
$276$		0			0
$277$		−	84.5789i		−	0.305339i	−0.988277	−	0.152669i	$-0.951213\pi$
$277$		−	84.5789i		−	0.305339i	0.988277	−	0.152669i	$-0.0487870\pi$
$278$	307.044			1.10447
$279$		0			0
$280$	−156.382			−0.558507
$281$		−	351.148i		−	1.24964i	−0.780771	−	0.624818i	$-0.785173\pi$
$281$		−	351.148i		−	1.24964i	0.780771	−	0.624818i	$-0.214827\pi$
$282$		0			0
$283$			285.943i			1.01040i	0.863002	+	0.505201i	$0.168581\pi$
$283$			285.943i			1.01040i	−0.863002	+	0.505201i	$0.831419\pi$
$284$	−111.709			−0.393342
$285$		0			0
$286$	59.1206	−	190.017i	0.206715	−	0.664394i
$287$	308.420			1.07464
$288$		0			0
$289$	222.569			0.770136
$290$	68.0206			0.234554
$291$		0			0
$292$			84.2742i			0.288610i
$293$			393.927i			1.34446i	0.740342	+	0.672231i	$0.234664\pi$
$293$			393.927i			1.34446i	−0.740342	+	0.672231i	$0.765336\pi$
$294$		0			0
$295$	−347.215			−1.17700
$296$			112.846i			0.381237i
$297$		0			0
$298$	−295.972			−0.993194
$299$		−	243.435i		−	0.814165i
$300$		0			0
$301$	526.095			1.74782
$302$	−647.594			−2.14435
$303$		0			0
$304$		−	601.810i		−	1.97964i
$305$		−	316.452i		−	1.03755i
$306$		0			0
$307$		−	58.2239i		−	0.189654i	−0.995494	−	0.0948272i	$-0.969770\pi$
$307$		−	58.2239i		−	0.189654i	0.995494	−	0.0948272i	$-0.0302299\pi$
$308$	−103.699	−	32.2642i	−0.336684	−	0.104754i
$309$		0			0
$310$		−	505.265i		−	1.62989i
$311$	−207.611			−0.667561			−0.333780	−	0.942651i	$-0.608324\pi$
$311$	−207.611			−0.667561			−0.333780	+	0.942651i	$0.608324\pi$
$312$		0			0
$313$	−376.697			−1.20351			−0.601753	−	0.798682i	$-0.705531\pi$
$313$	−376.697			−1.20351			−0.601753	+	0.798682i	$0.705531\pi$
$314$			642.654i			2.04667i
$315$		0			0
$316$			37.3651i			0.118244i
$317$	−246.809			−0.778577			−0.389289	−	0.921116i	$-0.627279\pi$
$317$	−246.809			−0.778577			−0.389289	+	0.921116i	$0.627279\pi$
$318$		0			0
$319$	−72.3641	−	22.5149i	−0.226847	−	0.0705797i
$320$	−101.450			−0.317031
$321$		0			0
$322$	−478.841			−1.48708
$323$	247.923			0.767564
$324$		0			0
$325$			56.8394i			0.174890i
$326$		−	146.033i		−	0.447955i
$327$		0			0
$328$	278.172			0.848085
$329$			194.104i			0.589982i
$330$		0			0
$331$	146.431			0.442389			0.221195	−	0.975230i	$-0.429004\pi$
$331$	146.431			0.442389			0.221195	+	0.975230i	$0.429004\pi$
$332$		−	1.41769i		−	0.00427014i
$333$		0			0
$334$	112.890			0.337993
$335$	−142.669			−0.425878
$336$		0			0
$337$		−	354.007i		−	1.05047i	−0.850958	−	0.525233i	$-0.823978\pi$
$337$		−	354.007i		−	1.05047i	0.850958	−	0.525233i	$-0.176022\pi$
$338$		−	258.530i		−	0.764881i
$339$		0			0
$340$			52.5289i			0.154497i
$341$	−167.244	+	537.529i	−0.490450	+	1.57633i
$342$		0			0
$343$		−	364.342i		−	1.06222i
$344$	474.497			1.37935
$345$		0			0
$346$	−336.497			−0.972536
$347$		−	529.807i		−	1.52682i	−0.645913	−	0.763411i	$-0.723523\pi$
$347$		−	529.807i		−	1.52682i	0.645913	−	0.763411i	$-0.276477\pi$
$348$		0			0
$349$			342.873i			0.982445i	0.871034	+	0.491223i	$0.163450\pi$
$349$			342.873i			0.982445i	−0.871034	+	0.491223i	$0.836550\pi$
$350$	111.804			0.319440
$351$		0			0
$352$	−245.354	−	76.3379i	−0.697028	−	0.216869i
$353$	−191.990			−0.543880			−0.271940	−	0.962314i	$-0.587665\pi$
$353$	−191.990			−0.543880			−0.271940	+	0.962314i	$0.587665\pi$
$354$		0			0
$355$	305.195			0.859704
$356$	68.8437			0.193381
$357$		0			0
$358$			86.1203i			0.240559i
$359$		−	601.654i		−	1.67592i	−0.545735	−	0.837958i	$-0.683750\pi$
$359$		−	601.654i		−	1.67592i	0.545735	−	0.837958i	$-0.316250\pi$
$360$		0			0
$361$	−564.261			−1.56305
$362$			198.043i			0.547081i
$363$		0			0
$364$	75.9127			0.208551
$365$		−	230.241i		−	0.630798i
$366$		0			0
$367$	−139.415			−0.379878			−0.189939	−	0.981796i	$-0.560829\pi$
$367$	−139.415			−0.379878			−0.189939	+	0.981796i	$0.560829\pi$
$368$	−626.386			−1.70214
$369$		0			0
$370$			192.167i			0.519371i
$371$			167.469i			0.451398i
$372$		0			0
$373$			303.629i			0.814019i	0.913424	+	0.407009i	$0.133428\pi$
$373$			303.629i			0.814019i	−0.913424	+	0.407009i	$0.866572\pi$
$374$	62.6692	−	201.422i	0.167565	−	0.538561i
$375$		0			0
$376$			175.067i			0.465604i
$377$	52.9742			0.140515
$378$		0			0
$379$	690.046			1.82070			0.910351	−	0.413837i	$-0.135812\pi$
$379$	690.046			1.82070			0.910351	+	0.413837i	$0.135812\pi$
$380$		−	196.041i		−	0.515896i
$381$		0			0
$382$			104.077i			0.272454i
$383$	53.2576			0.139054			0.0695269	−	0.997580i	$-0.477851\pi$
$383$	53.2576			0.139054			0.0695269	+	0.997580i	$0.477851\pi$
$384$		0			0
$385$	283.310	+	88.1474i	0.735871	+	0.228954i
$386$	−320.287			−0.829760
$387$		0			0
$388$	32.6438			0.0841334
$389$	587.027			1.50907			0.754533	−	0.656262i	$-0.227863\pi$
$389$	587.027			1.50907			0.754533	+	0.656262i	$0.227863\pi$
$390$		0			0
$391$		−	258.047i		−	0.659967i
$392$		−	44.5231i		−	0.113579i
$393$		0			0
$394$	−14.0385			−0.0356306
$395$		−	102.083i		−	0.258439i
$396$		0			0
$397$	485.797			1.22367			0.611835	−	0.790985i	$-0.290431\pi$
$397$	485.797			1.22367			0.611835	+	0.790985i	$0.290431\pi$
$398$			550.699i			1.38367i
$399$		0			0
$400$	146.254			0.365635
$401$	166.469			0.415135			0.207568	−	0.978221i	$-0.433445\pi$
$401$	166.469			0.415135			0.207568	+	0.978221i	$0.433445\pi$
$402$		0			0
$403$		−	393.499i		−	0.976424i
$404$			80.4010i			0.199012i
$405$		0			0
$406$		−	104.201i		−	0.256653i
$407$	63.6077	−	204.438i	0.156284	−	0.502306i
$408$		0			0
$409$		−	270.161i		−	0.660541i	−0.943886	−	0.330271i	$-0.892860\pi$
$409$		−	270.161i		−	0.660541i	0.943886	−	0.330271i	$-0.107140\pi$
$410$	473.703			1.15537
$411$		0			0
$412$	7.28958			0.0176932
$413$			531.901i			1.28790i
$414$		0			0
$415$			3.87319i			0.00933299i
$416$	179.611			0.431758
$417$		0			0
$418$	−233.885	+	751.717i	−0.559532	+	1.79837i
$419$	356.631			0.851147			0.425574	−	0.904924i	$-0.360072\pi$
$419$	356.631			0.851147			0.425574	+	0.904924i	$0.360072\pi$
$420$		0			0
$421$	−462.238			−1.09795			−0.548977	−	0.835838i	$-0.684982\pi$
$421$	−462.238			−1.09795			−0.548977	+	0.835838i	$0.684982\pi$
$422$	394.028			0.933716
$423$		0			0
$424$			151.044i			0.356236i
$425$			60.2510i			0.141767i
$426$		0			0
$427$	−484.774			−1.13530
$428$			226.887i			0.530110i
$429$		0			0
$430$	808.028			1.87914
$431$		−	199.552i		−	0.462997i	−0.972835	−	0.231498i	$-0.925637\pi$
$431$		−	199.552i		−	0.462997i	0.972835	−	0.231498i	$-0.0743628\pi$
$432$		0			0
$433$	−45.3693			−0.104779			−0.0523895	−	0.998627i	$-0.516684\pi$
$433$	−45.3693			−0.104779			−0.0523895	+	0.998627i	$0.516684\pi$
$434$	−774.018			−1.78345
$435$		0			0
$436$			5.48081i			0.0125707i
$437$			963.045i			2.20376i
$438$		0			0
$439$			484.630i			1.10394i	0.833864	+	0.551970i	$0.186124\pi$
$439$			484.630i			1.10394i	−0.833864	+	0.551970i	$0.813876\pi$
$440$	255.524	+	79.5022i	0.580737	+	0.180687i
$441$		0			0
$442$			147.451i			0.333600i
$443$	−375.538			−0.847716			−0.423858	−	0.905729i	$-0.639324\pi$
$443$	−375.538			−0.847716			−0.423858	+	0.905729i	$0.639324\pi$
$444$		0			0
$445$	−188.084			−0.422662
$446$		−	193.609i		−	0.434102i
$447$		0			0
$448$			155.412i			0.346901i
$449$	385.636			0.858877			0.429439	−	0.903096i	$-0.358711\pi$
$449$	385.636			0.858877			0.429439	+	0.903096i	$0.358711\pi$
$450$		0			0
$451$	−503.951	−	156.796i	−1.11741	−	0.347664i
$452$	−95.7101			−0.211748
$453$		0			0
$454$	−96.4665			−0.212481
$455$	−207.397			−0.455819
$456$		0			0
$457$		−	378.368i		−	0.827939i	−0.910291	−	0.413970i	$-0.864142\pi$
$457$		−	378.368i		−	0.827939i	0.910291	−	0.413970i	$-0.135858\pi$
$458$		−	378.145i		−	0.825644i
$459$		0			0
$460$	−204.046			−0.443578
$461$		−	224.522i		−	0.487033i	−0.969897	−	0.243516i	$-0.921699\pi$
$461$		−	224.522i		−	0.487033i	0.969897	−	0.243516i	$-0.0783010\pi$
$462$		0			0
$463$	−52.3820			−0.113136			−0.0565680	−	0.998399i	$-0.518016\pi$
$463$	−52.3820			−0.113136			−0.0565680	+	0.998399i	$0.518016\pi$
$464$		−	136.308i		−	0.293768i
$465$		0			0
$466$	959.854			2.05977
$467$	−513.387			−1.09933			−0.549665	−	0.835385i	$-0.685245\pi$
$467$	−513.387			−1.09933			−0.549665	+	0.835385i	$0.685245\pi$
$468$		0			0
$469$			218.556i			0.466003i
$470$			298.124i			0.634306i
$471$		0			0
$472$			479.734i			1.01639i
$473$	−859.626	−	267.459i	−1.81739	−	0.565451i
$474$		0			0
$475$		−	224.860i		−	0.473389i
$476$	80.4693			0.169053
$477$		0			0
$478$	475.874			0.995553
$479$			238.730i			0.498392i	0.968453	+	0.249196i	$0.0801663\pi$
$479$			238.730i			0.498392i	−0.968453	+	0.249196i	$0.919834\pi$
$480$		0			0
$481$			149.659i			0.311142i
$482$	−63.2511			−0.131226
$483$		0			0
$484$	153.038	+	105.438i	0.316195	+	0.217846i
$485$	−89.1845			−0.183885
$486$		0			0
$487$	593.251			1.21817			0.609087	−	0.793103i	$-0.291536\pi$
$487$	593.251			1.21817			0.609087	+	0.793103i	$0.291536\pi$
$488$	−437.229			−0.895962
$489$		0			0
$490$		−	75.8190i		−	0.154733i
$491$			858.935i			1.74936i	0.484703	+	0.874679i	$0.338928\pi$
$491$			858.935i			1.74936i	−0.484703	+	0.874679i	$0.661072\pi$
$492$		0			0
$493$	56.1539			0.113902
$494$		−	550.295i		−	1.11396i
$495$		0			0
$496$	−1012.52			−2.04136
$497$		−	467.529i		−	0.940702i
$498$		0			0
$499$	−803.692			−1.61061			−0.805303	−	0.592864i	$-0.797997\pi$
$499$	−803.692			−1.61061			−0.805303	+	0.592864i	$0.797997\pi$
$500$	208.764			0.417528
$501$		0			0
$502$			937.085i			1.86670i
$503$		−	752.302i		−	1.49563i	−0.663907	−	0.747815i	$-0.731103\pi$
$503$		−	752.302i		−	1.49563i	0.663907	−	0.747815i	$-0.268897\pi$
$504$		0			0
$505$		−	219.660i		−	0.434969i
$506$	782.414	+	243.435i	1.54627	+	0.481098i
$507$		0			0
$508$		−	43.0359i		−	0.0847163i
$509$	249.268			0.489721			0.244860	−	0.969558i	$-0.421258\pi$
$509$	249.268			0.489721			0.244860	+	0.969558i	$0.421258\pi$
$510$		0			0
$511$	−352.708			−0.690230
$512$		−	3.34215i		−	0.00652764i
$513$		0			0
$514$		−	794.835i		−	1.54637i
$515$	−19.9155			−0.0386709
$516$		0			0
$517$	98.6795	−	317.161i	0.190869	−	0.613464i
$518$	294.382			0.568305
$519$		0			0
$520$	−187.057			−0.359724
$521$	−475.864			−0.913367			−0.456683	−	0.889629i	$-0.650963\pi$
$521$	−475.864			−0.913367			−0.456683	+	0.889629i	$0.650963\pi$
$522$		0			0
$523$		−	362.833i		−	0.693754i	−0.937911	−	0.346877i	$-0.887242\pi$
$523$		−	362.833i		−	0.693754i	0.937911	−	0.346877i	$-0.112758\pi$
$524$		−	173.839i		−	0.331754i
$525$		0			0
$526$	316.526			0.601760
$527$		−	417.118i		−	0.791495i
$528$		0			0
$529$	473.372			0.894843
$530$			257.215i			0.485311i
$531$		0			0
$532$	−300.315			−0.564503
$533$	368.918			0.692154
$534$		0			0
$535$		−	619.867i		−	1.15863i
$536$			197.120i			0.367762i
$537$		0			0
$538$		−	804.370i		−	1.49511i
$539$	−25.0962	+	80.6604i	−0.0465606	+	0.149648i
$540$		0			0
$541$			830.667i			1.53543i	0.640792	+	0.767715i	$0.278606\pi$
$541$			830.667i			1.53543i	−0.640792	+	0.767715i	$0.721394\pi$
$542$	501.857			0.925936
$543$		0			0
$544$	190.392			0.349986
$545$		−	14.9739i		−	0.0274750i
$546$		0			0
$547$		−	456.091i		−	0.833804i	−0.908952	−	0.416902i	$-0.863116\pi$
$547$		−	456.091i		−	0.833804i	0.908952	−	0.416902i	$-0.136884\pi$
$548$	−108.718			−0.198390
$549$		0			0
$550$	−182.685	−	56.8394i	−0.332154	−	0.103344i
$551$	−209.569			−0.380343
$552$		0			0
$553$	−156.382			−0.282788
$554$	−199.001			−0.359208
$555$		0			0
$556$		−	200.433i		−	0.360490i
$557$		−	338.810i		−	0.608277i	−0.952628	−	0.304138i	$-0.901631\pi$
$557$		−	338.810i		−	0.608277i	0.952628	−	0.304138i	$-0.0983686\pi$
$558$		0			0
$559$	629.290			1.12574
$560$			533.656i			0.952958i
$561$		0			0
$562$	−826.197			−1.47010
$563$			447.041i			0.794034i	0.917811	+	0.397017i	$0.129955\pi$
$563$			447.041i			0.794034i	−0.917811	+	0.397017i	$0.870045\pi$
$564$		0			0
$565$	261.485			0.462805
$566$	672.782			1.18866
$567$		0			0
$568$		−	421.676i		−	0.742387i
$569$		−	522.893i		−	0.918969i	−0.888186	−	0.459485i	$-0.848034\pi$
$569$		−	522.893i		−	0.918969i	0.888186	−	0.459485i	$-0.151966\pi$
$570$		0			0
$571$		−	904.574i		−	1.58419i	−0.610396	−	0.792096i	$-0.708990\pi$
$571$		−	904.574i		−	1.58419i	0.610396	−	0.792096i	$-0.291010\pi$
$572$	−124.039	−	38.5929i	−0.216852	−	0.0674701i
$573$		0			0
$574$		−	725.667i		−	1.26423i
$575$	−234.042			−0.407030
$576$		0			0
$577$	237.674			0.411914			0.205957	−	0.978561i	$-0.433969\pi$
$577$	237.674			0.411914			0.205957	+	0.978561i	$0.433969\pi$
$578$		−	523.672i		−	0.906007i
$579$		0			0
$580$		−	44.4027i		−	0.0765563i
$581$	5.93336			0.0102123
$582$		0			0
$583$	85.1384	−	273.639i	0.146035	−	0.469364i
$584$	−318.115			−0.544718
$585$		0			0
$586$	926.851			1.58166
$587$	578.515			0.985546			0.492773	−	0.870158i	$-0.335983\pi$
$587$	578.515			0.985546			0.492773	+	0.870158i	$0.335983\pi$
$588$		0			0
$589$			1556.71i			2.64296i
$590$			816.945i			1.38465i
$591$		0			0
$592$	385.090			0.650489
$593$		−	333.857i		−	0.562997i	−0.959562	−	0.281499i	$-0.909169\pi$
$593$		−	333.857i		−	0.562997i	0.959562	−	0.281499i	$-0.0908314\pi$
$594$		0			0
$595$	−219.846			−0.369489
$596$			193.205i			0.324170i
$597$		0			0
$598$	−572.767			−0.957804
$599$	846.483			1.41316			0.706580	−	0.707633i	$-0.250237\pi$
$599$	846.483			1.41316			0.706580	+	0.707633i	$0.250237\pi$
$600$		0			0
$601$			455.011i			0.757089i	0.925583	+	0.378545i	$0.123575\pi$
$601$			455.011i			0.757089i	−0.925583	+	0.378545i	$0.876425\pi$
$602$		−	1237.82i		−	2.05618i
$603$		0			0
$604$			422.738i			0.699897i
$605$	−418.109	−	288.061i	−0.691089	−	0.476134i
$606$		0			0
$607$			726.557i			1.19696i	0.801137	+	0.598482i	$0.204229\pi$
$607$			726.557i			1.19696i	−0.801137	+	0.598482i	$0.795771\pi$
$608$	−710.554			−1.16867
$609$		0			0
$610$	−744.564			−1.22060
$611$			232.178i			0.379997i
$612$		0			0
$613$			574.532i			0.937247i	0.883398	+	0.468624i	$0.155250\pi$
$613$			574.532i			0.937247i	−0.883398	+	0.468624i	$0.844750\pi$
$614$	−136.992			−0.223114
$615$		0			0
$616$	121.790	−	391.439i	0.197711	−	0.635452i
$617$	278.946			0.452101			0.226050	−	0.974116i	$-0.427419\pi$
$617$	278.946			0.452101			0.226050	+	0.974116i	$0.427419\pi$
$618$		0			0
$619$	−593.061			−0.958096			−0.479048	−	0.877789i	$-0.659018\pi$
$619$	−593.061			−0.958096			−0.479048	+	0.877789i	$0.659018\pi$
$620$	−329.828			−0.531981
$621$		0			0
$622$			488.478i			0.785335i
$623$			288.127i			0.462484i
$624$		0			0
$625$	−385.546			−0.616874
$626$			886.312i			1.41583i
$627$		0			0
$628$	419.513			0.668014
$629$			158.642i			0.252214i
$630$		0			0
$631$	−361.664			−0.573160			−0.286580	−	0.958056i	$-0.592518\pi$
$631$	−361.664			−0.573160			−0.286580	+	0.958056i	$0.592518\pi$
$632$	−141.045			−0.223172
$633$		0			0
$634$			580.704i			0.915937i
$635$			117.576i			0.185159i
$636$		0			0
$637$		−	59.0475i		−	0.0926963i
$638$	−52.9742	+	170.262i	−0.0830317	+	0.266868i
$639$		0			0
$640$			630.778i			0.985590i
$641$	−784.382			−1.22368			−0.611842	−	0.790980i	$-0.709571\pi$
$641$	−784.382			−1.22368			−0.611842	+	0.790980i	$0.709571\pi$
$642$		0			0
$643$	362.764			0.564174			0.282087	−	0.959389i	$-0.408973\pi$
$643$	362.764			0.564174			0.282087	+	0.959389i	$0.408973\pi$
$644$			312.579i			0.485371i
$645$		0			0
$646$		−	583.325i		−	0.902981i
$647$	−921.727			−1.42462			−0.712308	−	0.701867i	$-0.752350\pi$
$647$	−921.727			−1.42462			−0.712308	+	0.701867i	$0.752350\pi$
$648$		0			0
$649$	270.410	−	869.112i	0.416657	−	1.33916i
$650$	133.734			0.205745
$651$		0			0
$652$	−95.3281			−0.146209
$653$	−26.8653			−0.0411414			−0.0205707	−	0.999788i	$-0.506548\pi$
$653$	−26.8653			−0.0411414			−0.0205707	+	0.999788i	$0.506548\pi$
$654$		0			0
$655$			474.938i			0.725096i
$656$		−	949.266i		−	1.44705i
$657$		0			0
$658$	456.697			0.694069
$659$			320.820i			0.486828i	0.969922	+	0.243414i	$0.0782674\pi$
$659$			320.820i			0.486828i	−0.969922	+	0.243414i	$0.921733\pi$
$660$		0			0
$661$	331.969			0.502222			0.251111	−	0.967958i	$-0.419204\pi$
$661$	331.969			0.502222			0.251111	+	0.967958i	$0.419204\pi$
$662$		−	344.530i		−	0.520437i
$663$		0			0
$664$	5.35144			0.00805939
$665$	820.477			1.23380
$666$		0			0
$667$			218.127i			0.327027i
$668$		−	73.6924i		−	0.110318i
$669$		0			0
$670$			335.679i			0.501013i
$671$	792.109	+	246.452i	1.18049	+	0.367290i
$672$		0			0
$673$			797.019i			1.18428i	0.805836	+	0.592139i	$0.201716\pi$
$673$			797.019i			1.18428i	−0.805836	+	0.592139i	$0.798284\pi$
$674$	−832.925			−1.23579
$675$		0			0
$676$	−168.764			−0.249650
$677$			534.175i			0.789033i	0.918889	+	0.394516i	$0.129088\pi$
$677$			534.175i			0.789033i	−0.918889	+	0.394516i	$0.870912\pi$
$678$		0			0
$679$			136.622i			0.201211i
$680$	−198.284			−0.291595
$681$		0			0
$682$	1264.73	+	393.499i	1.85444	+	0.576978i
$683$	843.097			1.23440			0.617201	−	0.786805i	$-0.288266\pi$
$683$	843.097			1.23440			0.617201	+	0.786805i	$0.288266\pi$
$684$		0			0
$685$	297.023			0.433610
$686$	−857.241			−1.24962
$687$		0			0
$688$		−	1619.23i		−	2.35353i
$689$			200.318i			0.290737i
$690$		0			0
$691$	−193.615			−0.280196			−0.140098	−	0.990138i	$-0.544742\pi$
$691$	−193.615			−0.280196			−0.140098	+	0.990138i	$0.544742\pi$
$692$			219.660i			0.317427i
$693$		0			0
$694$	−1246.56			−1.79619
$695$			547.592i			0.787903i
$696$		0			0
$697$	391.061			0.561064
$698$	806.729			1.15577
$699$		0			0
$700$		−	72.9836i		−	0.104262i
$701$			448.863i			0.640318i	0.947364	+	0.320159i	$0.103736\pi$
$701$			448.863i			0.640318i	−0.947364	+	0.320159i	$0.896264\pi$
$702$		0			0
$703$		−	592.061i		−	0.842192i
$704$	79.0089	−	253.939i	0.112229	−	0.360708i
$705$		0			0
$706$			451.723i			0.639834i
$707$	−336.497			−0.475951
$708$		0			0
$709$	−311.031			−0.438689			−0.219345	−	0.975647i	$-0.570392\pi$
$709$	−311.031			−0.438689			−0.219345	+	0.975647i	$0.570392\pi$
$710$		−	718.077i		−	1.01138i
$711$		0			0
$712$			259.869i			0.364985i
$713$	1620.27			2.27247
$714$		0			0
$715$	338.882	+	105.438i	0.473961	+	0.147465i
$716$	56.2178			0.0785165
$717$		0			0
$718$	−1415.60			−1.97159
$719$	−486.014			−0.675958			−0.337979	−	0.941154i	$-0.609743\pi$
$719$	−486.014			−0.675958			−0.337979	+	0.941154i	$0.609743\pi$
$720$		0			0
$721$			30.5086i			0.0423143i
$722$			1327.62i			1.83881i
$723$		0			0
$724$	129.279			0.178563
$725$		−	50.9302i		−	0.0702485i
$726$		0			0
$727$	38.8616			0.0534547			0.0267273	−	0.999643i	$-0.491491\pi$
$727$	38.8616			0.0534547			0.0267273	+	0.999643i	$0.491491\pi$
$728$			286.553i			0.393617i
$729$		0			0
$730$	−541.723			−0.742086
$731$	667.061			0.912533
$732$		0			0
$733$			175.562i			0.239511i	0.992803	+	0.119756i	$0.0382111\pi$
$733$			175.562i			0.239511i	−0.992803	+	0.119756i	$0.961789\pi$
$734$			328.023i			0.446898i
$735$		0			0
$736$			739.570i			1.00485i
$737$	111.110	−	357.114i	0.150760	−	0.484551i
$738$		0			0
$739$		−	494.527i		−	0.669184i	−0.942363	−	0.334592i	$-0.891402\pi$
$739$		−	494.527i		−	0.669184i	0.942363	−	0.334592i	$-0.108598\pi$
$740$	125.443			0.169518
$741$		0			0
$742$	394.028			0.531035
$743$		−	1150.84i		−	1.54892i	−0.632625	−	0.774458i	$-0.718023\pi$
$743$		−	1150.84i		−	1.54892i	0.632625	−	0.774458i	$-0.281977\pi$
$744$		0			0
$745$		−	527.846i		−	0.708519i
$746$	714.393			0.957632
$747$		0			0
$748$	−131.485	−	40.9093i	−0.175782	−	0.0546916i
$749$	−949.577			−1.26779
$750$		0			0
$751$	1343.73			1.78926			0.894628	−	0.446813i	$-0.147441\pi$
$751$	1343.73			1.78926			0.894628	+	0.446813i	$0.147441\pi$
$752$	597.419			0.794440
$753$		0			0
$754$		−	124.640i		−	0.165306i
$755$		−	1154.94i		−	1.52972i
$756$		0			0
$757$	1034.58			1.36669			0.683343	−	0.730097i	$-0.260525\pi$
$757$	1034.58			1.36669			0.683343	+	0.730097i	$0.260525\pi$
$758$		−	1623.57i		−	2.14192i
$759$		0			0
$760$	740.008			0.973694
$761$			798.527i			1.04931i	0.851314	+	0.524656i	$0.175806\pi$
$761$			798.527i			1.04931i	−0.851314	+	0.524656i	$0.824194\pi$
$762$		0			0
$763$	−22.9385			−0.0300636
$764$	67.9399			0.0889266
$765$		0			0
$766$		−	125.307i		−	0.163586i
$767$			636.234i			0.829510i
$768$		0			0
$769$		−	261.311i		−	0.339806i	−0.985461	−	0.169903i	$-0.945655\pi$
$769$		−	261.311i		−	0.339806i	0.985461	−	0.169903i	$-0.0543455\pi$
$770$	207.397	−	666.586i	0.269347	−	0.865696i
$771$		0			0
$772$			209.078i			0.270826i
$773$	327.734			0.423977			0.211989	−	0.977272i	$-0.432006\pi$
$773$	327.734			0.423977			0.211989	+	0.977272i	$0.432006\pi$
$774$		0			0
$775$	−378.315			−0.488149
$776$			123.223i			0.158792i
$777$		0			0
$778$		−	1381.19i		−	1.77530i
$779$	−1459.46			−1.87351
$780$		0			0
$781$	−237.685	+	763.931i	−0.304334	+	0.978144i
$782$	−607.146			−0.776402
$783$		0			0
$784$	−151.936			−0.193796
$785$	−1146.13			−1.46004
$786$		0			0
$787$		−	289.388i		−	0.367711i	−0.982953	−	0.183855i	$-0.941142\pi$
$787$		−	289.388i		−	0.367711i	0.982953	−	0.183855i	$-0.0588578\pi$
$788$			9.16407i			0.0116295i
$789$		0			0
$790$	−240.187			−0.304034
$791$		−	400.570i		−	0.506409i
$792$		0			0
$793$	−579.864			−0.731228
$794$		−	1143.01i		−	1.43956i
$795$		0			0
$796$	359.487			0.451617
$797$	−113.681			−0.142636			−0.0713180	−	0.997454i	$-0.522721\pi$
$797$	−113.681			−0.142636			−0.0713180	+	0.997454i	$0.522721\pi$
$798$		0			0
$799$			246.114i			0.308028i
$800$		−	172.681i		−	0.215851i
$801$		0			0
$802$		−	391.677i		−	0.488375i
$803$	576.315	+	179.311i	0.717703	+	0.223302i
$804$		0			0
$805$		−	853.982i		−	1.06085i
$806$	−925.843			−1.14869
$807$		0			0
$808$	−303.495			−0.375613
$809$			854.163i			1.05583i	0.849299	+	0.527913i	$0.177025\pi$
$809$			854.163i			1.05583i	−0.849299	+	0.527913i	$0.822975\pi$
$810$		0			0
$811$			486.533i			0.599917i	0.953952	+	0.299959i	$0.0969729\pi$
$811$			486.533i			0.599917i	−0.953952	+	0.299959i	$0.903027\pi$
$812$	−68.0206			−0.0837692
$813$		0			0
$814$	−481.013	−	149.659i	−0.590925	−	0.183857i
$815$	260.441			0.319560
$816$		0			0
$817$	−2489.51			−3.04713
$818$	−635.649			−0.777077
$819$		0			0
$820$		−	309.225i		−	0.377103i
$821$		−	527.104i		−	0.642027i	−0.947075	−	0.321014i	$-0.895976\pi$
$821$		−	527.104i		−	0.642027i	0.947075	−	0.321014i	$-0.104024\pi$
$822$		0			0
$823$	−314.805			−0.382509			−0.191255	−	0.981540i	$-0.561256\pi$
$823$	−314.805			−0.382509			−0.191255	+	0.981540i	$0.561256\pi$
$824$			27.5165i			0.0333938i
$825$		0			0
$826$	1251.48			1.51511
$827$			111.652i			0.135008i	0.997719	+	0.0675040i	$0.0215035\pi$
$827$			111.652i			0.135008i	−0.997719	+	0.0675040i	$0.978496\pi$
$828$		0			0
$829$	−695.395			−0.838836			−0.419418	−	0.907793i	$-0.637766\pi$
$829$	−695.395			−0.838836			−0.419418	+	0.907793i	$0.637766\pi$
$830$	9.11303			0.0109796
$831$		0			0
$832$			185.896i			0.223433i
$833$		−	62.5918i		−	0.0751402i
$834$		0			0
$835$			201.331i			0.241116i
$836$	490.708	+	152.676i	0.586971	+	0.182626i
$837$		0			0
$838$		−	839.098i		−	1.00131i
$839$	383.968			0.457650			0.228825	−	0.973468i	$-0.426512\pi$
$839$	383.968			0.457650			0.228825	+	0.973468i	$0.426512\pi$
$840$		0			0
$841$	793.533			0.943559
$842$			1087.58i			1.29166i
$843$		0			0
$844$		−	257.215i		−	0.304757i
$845$	461.071			0.545646
$846$		0			0
$847$	−441.282	+	640.503i	−0.520994	+	0.756202i
$848$	515.440			0.607830
$849$		0			0
$850$	141.762			0.166778
$851$	−616.238			−0.724134
$852$		0			0
$853$		−	570.527i		−	0.668847i	−0.942423	−	0.334424i	$-0.891458\pi$
$853$		−	570.527i		−	0.668847i	0.942423	−	0.334424i	$-0.108542\pi$
$854$			1140.60i			1.33560i
$855$		0			0
$856$	−856.446			−1.00052
$857$			315.553i			0.368207i	0.982907	+	0.184103i	$0.0589382\pi$
$857$			315.553i			0.368207i	−0.982907	+	0.184103i	$0.941062\pi$
$858$		0			0
$859$	−107.923			−0.125638			−0.0628190	−	0.998025i	$-0.520009\pi$
$859$	−107.923			−0.125638			−0.0628190	+	0.998025i	$0.520009\pi$
$860$		−	527.467i		−	0.613333i
$861$		0			0
$862$	−469.515			−0.544681
$863$	−1431.55			−1.65881			−0.829403	−	0.558651i	$-0.811319\pi$
$863$	−1431.55			−1.65881			−0.829403	+	0.558651i	$0.811319\pi$
$864$		0			0
$865$		−	600.121i		−	0.693782i
$866$			106.747i			0.123265i
$867$		0			0
$868$			505.265i			0.582103i
$869$	255.524	+	79.5022i	0.294044	+	0.0914870i
$870$		0			0
$871$			261.426i			0.300144i
$872$	−20.6888			−0.0237257
$873$		0			0
$874$	2265.90			2.59256
$875$			873.727i			0.998546i
$876$		0			0
$877$			567.758i			0.647386i	0.946162	+	0.323693i	$0.104925\pi$
$877$			567.758i			0.647386i	−0.946162	+	0.323693i	$0.895075\pi$
$878$	1140.26			1.29870
$879$		0			0
$880$	271.303	−	871.981i	0.308298	−	0.990887i
$881$	−488.641			−0.554644			−0.277322	−	0.960777i	$-0.589447\pi$
$881$	−488.641			−0.554644			−0.277322	+	0.960777i	$0.589447\pi$
$882$		0			0
$883$	840.144			0.951465			0.475732	−	0.879590i	$-0.342183\pi$
$883$	840.144			0.951465			0.475732	+	0.879590i	$0.342183\pi$
$884$	96.2535			0.108884
$885$		0			0
$886$			883.585i			0.997274i
$887$		−	981.439i		−	1.10647i	−0.833025	−	0.553235i	$-0.813393\pi$
$887$		−	981.439i		−	1.10647i	0.833025	−	0.553235i	$-0.186607\pi$
$888$		0			0
$889$	180.115			0.202605
$890$			442.534i			0.497230i
$891$		0			0
$892$	−126.385			−0.141687
$893$		−	918.510i		−	1.02857i
$894$		0			0
$895$	−153.590			−0.171609
$896$	966.291			1.07845
$897$		0			0
$898$		−	907.343i		−	1.01040i
$899$			352.589i			0.392202i
$900$		0			0
$901$			212.342i			0.235673i
$902$	−368.918	+	1185.72i	−0.409000	+	1.31455i
$903$		0			0
$904$		−	361.283i		−	0.399650i
$905$	−353.198			−0.390274
$906$		0			0
$907$	438.610			0.483583			0.241792	−	0.970328i	$-0.422265\pi$
$907$	438.610			0.483583			0.241792	+	0.970328i	$0.422265\pi$
$908$			62.9716i			0.0693520i
$909$		0			0
$910$			487.975i			0.536236i
$911$	170.042			0.186654			0.0933272	−	0.995635i	$-0.470250\pi$
$911$	170.042			0.186654			0.0933272	+	0.995635i	$0.470250\pi$
$912$		0			0
$913$	−9.69496	−	3.01643i	−0.0106188	−	0.00330386i
$914$	−890.243			−0.974008
$915$		0			0
$916$	−246.846			−0.269483
$917$	727.559			0.793412
$918$		0			0
$919$			1089.73i			1.18578i	0.805285	+	0.592888i	$0.202012\pi$
$919$			1089.73i			1.18578i	−0.805285	+	0.592888i	$0.797988\pi$
$920$		−	770.226i		−	0.837202i
$921$		0			0
$922$	−528.267			−0.572957
$923$		−	559.236i		−	0.605890i
$924$		0			0
$925$	143.885			0.155551
$926$			123.247i			0.133096i
$927$		0			0
$928$	−160.939			−0.173425
$929$	−769.446			−0.828252			−0.414126	−	0.910220i	$-0.635913\pi$
$929$	−769.446			−0.828252			−0.414126	+	0.910220i	$0.635913\pi$
$930$		0			0
$931$			233.596i			0.250908i
$932$		−	626.576i		−	0.672291i
$933$		0			0
$934$			1207.92i			1.29328i
$935$	359.223	+	111.766i	0.384196	+	0.119536i
$936$		0			0
$937$		−	1606.78i		−	1.71481i	−0.514641	−	0.857406i	$-0.672075\pi$
$937$		−	1606.78i		−	1.71481i	0.514641	−	0.857406i	$-0.327925\pi$
$938$	514.228			0.548218
$939$		0			0
$940$	194.610			0.207032
$941$		−	1279.34i		−	1.35955i	−0.733419	−	0.679777i	$-0.762077\pi$
$941$		−	1279.34i		−	1.35955i	0.733419	−	0.679777i	$-0.237923\pi$
$942$		0			0
$943$			1519.06i			1.61088i
$944$	1637.10			1.73422
$945$		0			0
$946$	−629.290	+	2022.57i	−0.665211	+	2.13802i
$947$	985.800			1.04097			0.520486	−	0.853870i	$-0.325751\pi$
$947$	985.800			1.04097			0.520486	+	0.853870i	$0.325751\pi$
$948$		0			0
$949$	−421.892			−0.444565
$950$	−529.061			−0.556907
$951$		0			0
$952$			303.753i			0.319068i
$953$			761.038i			0.798571i	0.916827	+	0.399285i	$0.130742\pi$
$953$			761.038i			0.798571i	−0.916827	+	0.399285i	$0.869258\pi$
$954$		0			0
$955$	−185.615			−0.194362
$956$		−	310.642i		−	0.324940i
$957$		0			0
$958$	561.695			0.586320
$959$		−	455.011i		−	0.474464i
$960$		0			0
$961$	1658.08			1.72537
$962$	352.126			0.366035
$963$		0			0
$964$			41.2892i			0.0428311i
$965$		−	571.211i		−	0.591929i
$966$		0			0
$967$		−	702.938i		−	0.726926i	−0.931609	−	0.363463i	$-0.881594\pi$
$967$		−	702.938i		−	0.726926i	0.931609	−	0.363463i	$-0.118406\pi$
$968$	−398.003	+	577.685i	−0.411160	+	0.596782i
$969$		0			0
$970$			209.838i			0.216327i
$971$	1289.42			1.32793			0.663963	−	0.747766i	$-0.268873\pi$
$971$	1289.42			1.32793			0.663963	+	0.747766i	$0.268873\pi$
$972$		0			0
$973$	838.859			0.862136
$974$		−	1395.83i		−	1.43309i
$975$		0			0
$976$			1492.05i			1.52874i
$977$	690.526			0.706782			0.353391	−	0.935476i	$-0.385029\pi$
$977$	690.526			0.706782			0.353391	+	0.935476i	$0.385029\pi$
$978$		0			0
$979$	146.480	−	470.793i	0.149622	−	0.480892i
$980$	−49.4933			−0.0505033
$981$		0			0
$982$	2020.94			2.05799
$983$	−1080.40			−1.09908			−0.549540	−	0.835467i	$-0.685197\pi$
$983$	−1080.40			−1.09908			−0.549540	+	0.835467i	$0.685197\pi$
$984$		0			0
$985$		−	25.0367i		−	0.0254180i
$986$		−	132.122i		−	0.133998i
$987$		0			0
$988$	−359.223			−0.363586
$989$			2591.17i			2.61999i
$990$		0			0
$991$	−686.561			−0.692796			−0.346398	−	0.938088i	$-0.612595\pi$
$991$	−686.561			−0.692796			−0.346398	+	0.938088i	$0.612595\pi$
$992$			1195.47i			1.20511i
$993$		0			0
$994$	−1100.03			−1.10667
$995$	−982.136			−0.987071
$996$		0			0
$997$		−	1413.50i		−	1.41775i	−0.705333	−	0.708876i	$-0.749203\pi$
$997$		−	1413.50i		−	1.41775i	0.705333	−	0.708876i	$-0.250797\pi$
$998$			1890.97i			1.89476i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.3.c.b.10.2		4
3.2	odd	2		33.3.c.a.10.3	yes	4
4.3	odd	2		1584.3.j.f.1297.4		4
11.10	odd	2	inner	99.3.c.b.10.3		4
12.11	even	2		528.3.j.c.241.2		4
15.2	even	4		825.3.h.a.274.3		8
15.8	even	4		825.3.h.a.274.6		8
15.14	odd	2		825.3.b.a.76.2		4
24.5	odd	2		2112.3.j.a.769.1		4
24.11	even	2		2112.3.j.d.769.4		4
33.2	even	10		363.3.g.e.40.2		16
33.5	odd	10		363.3.g.e.118.2		16
33.8	even	10		363.3.g.e.112.2		16
33.14	odd	10		363.3.g.e.112.3		16
33.17	even	10		363.3.g.e.118.3		16
33.20	odd	10		363.3.g.e.40.3		16
33.26	odd	10		363.3.g.e.94.2		16
33.29	even	10		363.3.g.e.94.3		16
33.32	even	2		33.3.c.a.10.2	✓	4
44.43	even	2		1584.3.j.f.1297.3		4
132.131	odd	2		528.3.j.c.241.1		4
165.32	odd	4		825.3.h.a.274.5		8
165.98	odd	4		825.3.h.a.274.4		8
165.164	even	2		825.3.b.a.76.3		4
264.131	odd	2		2112.3.j.d.769.3		4
264.197	even	2		2112.3.j.a.769.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.c.a.10.2	✓	4	33.32	even	2
33.3.c.a.10.3	yes	4	3.2	odd	2
99.3.c.b.10.2		4	1.1	even	1	trivial
99.3.c.b.10.3		4	11.10	odd	2	inner
363.3.g.e.40.2		16	33.2	even	10
363.3.g.e.40.3		16	33.20	odd	10
363.3.g.e.94.2		16	33.26	odd	10
363.3.g.e.94.3		16	33.29	even	10
363.3.g.e.112.2		16	33.8	even	10
363.3.g.e.112.3		16	33.14	odd	10
363.3.g.e.118.2		16	33.5	odd	10
363.3.g.e.118.3		16	33.17	even	10
528.3.j.c.241.1		4	132.131	odd	2
528.3.j.c.241.2		4	12.11	even	2
825.3.b.a.76.2		4	15.14	odd	2
825.3.b.a.76.3		4	165.164	even	2
825.3.h.a.274.3		8	15.2	even	4
825.3.h.a.274.4		8	165.98	odd	4
825.3.h.a.274.5		8	165.32	odd	4
825.3.h.a.274.6		8	15.8	even	4
1584.3.j.f.1297.3		4	44.43	even	2
1584.3.j.f.1297.4		4	4.3	odd	2
2112.3.j.a.769.1		4	24.5	odd	2
2112.3.j.a.769.2		4	264.197	even	2
2112.3.j.d.769.3		4	264.131	odd	2
2112.3.j.d.769.4		4	24.11	even	2

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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	99.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.35285i q^{2} -1.53590 q^{4} +4.19615 q^{5} -6.42810i q^{7} -5.79766i q^{8} +O(q^{10})\)	\(q-2.35285i q^{2} -1.53590 q^{4} +4.19615 q^{5} -6.42810i q^{7} -5.79766i q^{8} -9.87291i q^{10} +(-3.26795 + 10.5034i) q^{11} -7.68899i q^{13} -15.1244 q^{14} -19.7846 q^{16} -8.15051i q^{17} +30.4181i q^{19} -6.44486 q^{20} +(24.7128 + 7.68899i) q^{22} +31.6603 q^{23} -7.39230 q^{25} -18.0910 q^{26} +9.87291i q^{28} +6.88962i q^{29} +51.1769 q^{31} +23.3596i q^{32} -19.1769 q^{34} -26.9733i q^{35} -19.4641 q^{37} +71.5692 q^{38} -24.3279i q^{40} +47.9800i q^{41} +81.8429i q^{43} +(5.01924 - 16.1321i) q^{44} -74.4918i q^{46} -30.1962 q^{47} +7.67949 q^{49} +17.3930i q^{50} +11.8095i q^{52} -26.0526 q^{53} +(-13.7128 + 44.0737i) q^{55} -37.2679 q^{56} +16.2102 q^{58} -82.7461 q^{59} -75.4148i q^{61} -120.412i q^{62} -24.1769 q^{64} -32.2642i q^{65} -34.0000 q^{67} +12.5184i q^{68} -63.4641 q^{70} +72.7321 q^{71} -54.8696i q^{73} +45.7961i q^{74} -46.7191i q^{76} +(67.5167 + 21.0067i) q^{77} -24.3279i q^{79} -83.0192 q^{80} +112.890 q^{82} +0.923034i q^{83} -34.2008i q^{85} +192.564 q^{86} +(60.8949 + 18.9465i) q^{88} -44.8231 q^{89} -49.4256 q^{91} -48.6269 q^{92} +71.0470i q^{94} +127.639i q^{95} -21.2539 q^{97} -18.0687i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{4} - 4 q^{5}+O(q^{10}) \) 4 * q - 20 * q^4 - 4 * q^5	\( 4 q - 20 q^{4} - 4 q^{5} - 20 q^{11} - 12 q^{14} + 4 q^{16} + 92 q^{20} - 12 q^{22} + 92 q^{23} + 12 q^{25} - 204 q^{26} + 80 q^{31} + 48 q^{34} - 64 q^{37} + 120 q^{38} + 124 q^{44} - 100 q^{47} + 100 q^{49} - 28 q^{53} + 56 q^{55} - 156 q^{56} - 240 q^{58} - 40 q^{59} + 28 q^{64} - 136 q^{67} - 240 q^{70} + 284 q^{71} + 180 q^{77} - 436 q^{80} + 216 q^{82} + 216 q^{86} + 396 q^{88} - 304 q^{89} + 24 q^{91} - 340 q^{92} - 376 q^{97}+O(q^{100}) \) 4 * q - 20 * q^4 - 4 * q^5 - 20 * q^11 - 12 * q^14 + 4 * q^16 + 92 * q^20 - 12 * q^22 + 92 * q^23 + 12 * q^25 - 204 * q^26 + 80 * q^31 + 48 * q^34 - 64 * q^37 + 120 * q^38 + 124 * q^44 - 100 * q^47 + 100 * q^49 - 28 * q^53 + 56 * q^55 - 156 * q^56 - 240 * q^58 - 40 * q^59 + 28 * q^64 - 136 * q^67 - 240 * q^70 + 284 * q^71 + 180 * q^77 - 436 * q^80 + 216 * q^82 + 216 * q^86 + 396 * q^88 - 304 * q^89 + 24 * q^91 - 340 * q^92 - 376 * q^97

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