LMFDB - Embedded newform 2151.2.b.a.2150.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(2150,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2151.2150");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.b (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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2150.11
Character	$\chi$	$=$	2151.2150
Dual form			2151.2.b.a.2150.70

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14709i q^{2} -2.61000 q^{4} +2.81441i q^{5} -1.25298i q^{7} +1.30973i q^{8} +O(q^{10})$	\(q-2.14709i q^{2} -2.61000 q^{4} +2.81441i q^{5} -1.25298i q^{7} +1.30973i q^{8} +6.04279 q^{10} +0.892556i q^{11} +3.41689i q^{13} -2.69025 q^{14} -2.40789 q^{16} -1.69865i q^{17} +0.501134i q^{19} -7.34561i q^{20} +1.91640 q^{22} -5.03781 q^{23} -2.92088 q^{25} +7.33637 q^{26} +3.27027i q^{28} +2.92096i q^{29} -7.40471 q^{31} +7.78943i q^{32} -3.64716 q^{34} +3.52638 q^{35} -9.92211i q^{37} +1.07598 q^{38} -3.68612 q^{40} -5.83967 q^{41} -12.0059i q^{43} -2.32957i q^{44} +10.8166i q^{46} -4.49607 q^{47} +5.43005 q^{49} +6.27140i q^{50} -8.91808i q^{52} -10.7539 q^{53} -2.51201 q^{55} +1.64106 q^{56} +6.27156 q^{58} -12.0371 q^{59} -9.20925 q^{61} +15.8986i q^{62} +11.9088 q^{64} -9.61650 q^{65} -1.56742 q^{67} +4.43349i q^{68} -7.57146i q^{70} +3.30270i q^{71} -8.58126i q^{73} -21.3037 q^{74} -1.30796i q^{76} +1.11835 q^{77} +8.26175i q^{79} -6.77678i q^{80} +12.5383i q^{82} +10.7499i q^{83} +4.78069 q^{85} -25.7778 q^{86} -1.16901 q^{88} +13.3771 q^{89} +4.28127 q^{91} +13.1487 q^{92} +9.65348i q^{94} -1.41039 q^{95} +8.14234i q^{97} -11.6588i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 80 q^{4}+O(q^{10}) $ 80 * q - 80 * q^4	$ 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) $ 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$.

$n$	$479$	$1441$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.14709i		−	1.51822i	−0.650961	−	0.759112i	$-0.725634\pi$
$2$		−	2.14709i		−	1.51822i	0.650961	−	0.759112i	$-0.274366\pi$
$3$		0			0
$3$		0			0
$4$	−2.61000			−1.30500
$5$			2.81441i			1.25864i	0.777146	+	0.629320i	$0.216667\pi$
$5$			2.81441i			1.25864i	−0.777146	+	0.629320i	$0.783333\pi$
$6$		0			0
$7$		−	1.25298i		−	0.473580i	−0.971561	−	0.236790i	$-0.923905\pi$
$7$		−	1.25298i		−	0.473580i	0.971561	−	0.236790i	$-0.0760953\pi$
$8$			1.30973i			0.463060i
$9$		0			0
$10$	6.04279			1.91090
$11$			0.892556i			0.269116i	0.990906	+	0.134558i	$0.0429614\pi$
$11$			0.892556i			0.269116i	−0.990906	+	0.134558i	$0.957039\pi$
$12$		0			0
$13$			3.41689i			0.947674i	0.880613	+	0.473837i	$0.157131\pi$
$13$			3.41689i			0.947674i	−0.880613	+	0.473837i	$0.842869\pi$
$14$	−2.69025			−0.719000
$15$		0			0
$16$	−2.40789			−0.601973
$17$		−	1.69865i		−	0.411984i	−0.978554	−	0.205992i	$-0.933958\pi$
$17$		−	1.69865i		−	0.411984i	0.978554	−	0.205992i	$-0.0660420\pi$
$18$		0			0
$19$			0.501134i			0.114968i	0.998346	+	0.0574840i	$0.0183078\pi$
$19$			0.501134i			0.114968i	−0.998346	+	0.0574840i	$0.981692\pi$
$20$		−	7.34561i		−	1.64253i
$21$		0			0
$22$	1.91640			0.408578
$23$	−5.03781			−1.05046			−0.525228	−	0.850961i	$-0.676020\pi$
$23$	−5.03781			−1.05046			−0.525228	+	0.850961i	$0.676020\pi$
$24$		0			0
$25$	−2.92088			−0.584176
$26$	7.33637			1.43878
$27$		0			0
$28$			3.27027i			0.618023i
$29$			2.92096i			0.542408i	0.962522	+	0.271204i	$0.0874218\pi$
$29$			2.92096i			0.542408i	−0.962522	+	0.271204i	$0.912578\pi$
$30$		0			0
$31$	−7.40471			−1.32993			−0.664963	−	0.746877i	$-0.731553\pi$
$31$	−7.40471			−1.32993			−0.664963	+	0.746877i	$0.731553\pi$
$32$			7.78943i			1.37699i
$33$		0			0
$34$	−3.64716			−0.625483
$35$	3.52638			0.596067
$36$		0			0
$37$		−	9.92211i		−	1.63118i	−0.578627	−	0.815592i	$-0.696411\pi$
$37$		−	9.92211i		−	1.63118i	0.578627	−	0.815592i	$-0.303589\pi$
$38$	1.07598			0.174547
$39$		0			0
$40$	−3.68612			−0.582826
$41$	−5.83967			−0.912003			−0.456001	−	0.889979i	$-0.650719\pi$
$41$	−5.83967			−0.912003			−0.456001	+	0.889979i	$0.650719\pi$
$42$		0			0
$43$		−	12.0059i		−	1.83088i	−0.402452	−	0.915441i	$-0.631842\pi$
$43$		−	12.0059i		−	1.83088i	0.402452	−	0.915441i	$-0.368158\pi$
$44$		−	2.32957i		−	0.351196i
$45$		0			0
$46$			10.8166i			1.59483i
$47$	−4.49607			−0.655820			−0.327910	−	0.944709i	$-0.606344\pi$
$47$	−4.49607			−0.655820			−0.327910	+	0.944709i	$0.606344\pi$
$48$		0			0
$49$	5.43005			0.775722
$50$			6.27140i			0.886909i
$51$		0			0
$52$		−	8.91808i		−	1.23672i
$53$	−10.7539			−1.47717			−0.738583	−	0.674163i	$-0.764505\pi$
$53$	−10.7539			−1.47717			−0.738583	+	0.674163i	$0.764505\pi$
$54$		0			0
$55$	−2.51201			−0.338720
$56$	1.64106			0.219296
$57$		0			0
$58$	6.27156			0.823496
$59$	−12.0371			−1.56709			−0.783547	−	0.621333i	$-0.786591\pi$
$59$	−12.0371			−1.56709			−0.783547	+	0.621333i	$0.786591\pi$
$60$		0			0
$61$	−9.20925			−1.17912			−0.589562	−	0.807723i	$-0.700700\pi$
$61$	−9.20925			−1.17912			−0.589562	+	0.807723i	$0.700700\pi$
$62$			15.8986i			2.01912i
$63$		0			0
$64$	11.9088			1.48860
$65$	−9.61650			−1.19278
$66$		0			0
$67$	−1.56742			−0.191491			−0.0957456	−	0.995406i	$-0.530524\pi$
$67$	−1.56742			−0.191491			−0.0957456	+	0.995406i	$0.530524\pi$
$68$			4.43349i			0.537639i
$69$		0			0
$70$		−	7.57146i		−	0.904963i
$71$			3.30270i			0.391958i	0.980608	+	0.195979i	$0.0627884\pi$
$71$			3.30270i			0.391958i	−0.980608	+	0.195979i	$0.937212\pi$
$72$		0			0
$73$		−	8.58126i		−	1.00436i	−0.864763	−	0.502180i	$-0.832531\pi$
$73$		−	8.58126i		−	1.00436i	0.864763	−	0.502180i	$-0.167469\pi$
$74$	−21.3037			−2.47650
$75$		0			0
$76$		−	1.30796i		−	0.150033i
$77$	1.11835			0.127448
$78$		0			0
$79$			8.26175i			0.929520i	0.885437	+	0.464760i	$0.153859\pi$
$79$			8.26175i			0.929520i	−0.885437	+	0.464760i	$0.846141\pi$
$80$		−	6.77678i		−	0.757667i
$81$		0			0
$82$			12.5383i			1.38462i
$83$			10.7499i			1.17996i	0.807418	+	0.589980i	$0.200864\pi$
$83$			10.7499i			1.17996i	−0.807418	+	0.589980i	$0.799136\pi$
$84$		0			0
$85$	4.78069			0.518539
$86$	−25.7778			−2.77969
$87$		0			0
$88$	−1.16901			−0.124617
$89$	13.3771			1.41797			0.708985	−	0.705224i	$-0.249153\pi$
$89$	13.3771			1.41797			0.708985	+	0.705224i	$0.249153\pi$
$90$		0			0
$91$	4.28127			0.448799
$92$	13.1487			1.37085
$93$		0			0
$94$			9.65348i			0.995681i
$95$	−1.41039			−0.144703
$96$		0			0
$97$			8.14234i			0.826730i	0.910566	+	0.413365i	$0.135647\pi$
$97$			8.14234i			0.826730i	−0.910566	+	0.413365i	$0.864353\pi$
$98$		−	11.6588i		−	1.17772i
$99$		0			0
$100$	7.62350			0.762350
$101$			17.3700i			1.72838i	0.503164	+	0.864191i	$0.332169\pi$
$101$			17.3700i			1.72838i	−0.503164	+	0.864191i	$0.667831\pi$
$102$		0			0
$103$		−	3.41714i		−	0.336701i	−0.985727	−	0.168351i	$-0.946156\pi$
$103$		−	3.41714i		−	0.336701i	0.985727	−	0.168351i	$-0.0538440\pi$
$104$	−4.47521			−0.438830
$105$		0			0
$106$			23.0897i			2.24267i
$107$	1.91002			0.184648			0.0923241	−	0.995729i	$-0.470570\pi$
$107$	1.91002			0.184648			0.0923241	+	0.995729i	$0.470570\pi$
$108$		0			0
$109$	10.1035			0.967741			0.483871	−	0.875139i	$-0.339230\pi$
$109$	10.1035			0.967741			0.483871	+	0.875139i	$0.339230\pi$
$110$			5.39352i			0.514252i
$111$		0			0
$112$			3.01703i			0.285082i
$113$			3.96549i			0.373042i	0.982451	+	0.186521i	$0.0597213\pi$
$113$			3.96549i			0.373042i	−0.982451	+	0.186521i	$0.940279\pi$
$114$		0			0
$115$		−	14.1784i		−	1.32215i
$116$		−	7.62370i		−	0.707843i
$117$		0			0
$118$			25.8447i			2.37920i
$119$	−2.12837			−0.195107
$120$		0			0
$121$	10.2033			0.927577
$122$			19.7731i			1.79017i
$123$		0			0
$124$	19.3263			1.73555
$125$			5.85149i			0.523373i
$126$		0			0
$127$	−7.84643			−0.696258			−0.348129	−	0.937447i	$-0.613183\pi$
$127$	−7.84643			−0.696258			−0.348129	+	0.937447i	$0.613183\pi$
$128$		−	9.99050i		−	0.883044i
$129$		0			0
$130$			20.6475i			1.81091i
$131$	−18.2977			−1.59868			−0.799338	−	0.600882i	$-0.794816\pi$
$131$	−18.2977			−1.59868			−0.799338	+	0.600882i	$0.794816\pi$
$132$		0			0
$133$	0.627908			0.0544465
$134$			3.36540i			0.290726i
$135$		0			0
$136$	2.22478			0.190773
$137$	8.77744			0.749907			0.374954	−	0.927044i	$-0.377659\pi$
$137$	8.77744			0.749907			0.374954	+	0.927044i	$0.377659\pi$
$138$		0			0
$139$		−	7.25476i		−	0.615341i	−0.951493	−	0.307671i	$-0.900451\pi$
$139$		−	7.25476i		−	0.615341i	0.951493	−	0.307671i	$-0.0995495\pi$
$140$	−9.20386			−0.777868
$141$		0			0
$142$	7.09119			0.595080
$143$	−3.04976			−0.255034
$144$		0			0
$145$	−8.22076			−0.682697
$146$	−18.4247			−1.52484
$147$		0			0
$148$			25.8967i			2.12870i
$149$	−5.77848			−0.473392			−0.236696	−	0.971584i	$-0.576065\pi$
$149$	−5.77848			−0.473392			−0.236696	+	0.971584i	$0.576065\pi$
$150$		0			0
$151$		−	1.84578i		−	0.150208i	−0.997176	−	0.0751039i	$-0.976071\pi$
$151$		−	1.84578i		−	0.150208i	0.997176	−	0.0751039i	$-0.0239288\pi$
$152$	−0.656351			−0.0532371
$153$		0			0
$154$		−	2.40120i		−	0.193494i
$155$		−	20.8399i		−	1.67390i
$156$		0			0
$157$	−0.0255164			−0.00203643			−0.00101821	−	0.999999i	$-0.500324\pi$
$157$	−0.0255164			−0.00203643			−0.00101821	+	0.999999i	$0.500324\pi$
$158$	17.7387			1.41122
$159$		0			0
$160$	−21.9226			−1.73313
$161$			6.31225i			0.497475i
$162$		0			0
$163$	−19.0147			−1.48935			−0.744674	−	0.667429i	$-0.767395\pi$
$163$	−19.0147			−1.48935			−0.744674	+	0.667429i	$0.767395\pi$
$164$	15.2415			1.19016
$165$		0			0
$166$	23.0811			1.79144
$167$	−20.0669			−1.55282			−0.776411	−	0.630227i	$-0.782962\pi$
$167$	−20.0669			−1.55282			−0.776411	+	0.630227i	$0.782962\pi$
$168$		0			0
$169$	1.32489			0.101914
$170$		−	10.2646i		−	0.787258i
$171$		0			0
$172$			31.3354i			2.38930i
$173$	−10.3270			−0.785145			−0.392572	−	0.919721i	$-0.628415\pi$
$173$	−10.3270			−0.785145			−0.392572	+	0.919721i	$0.628415\pi$
$174$		0			0
$175$			3.65979i			0.276654i
$176$		−	2.14918i		−	0.162000i
$177$		0			0
$178$		−	28.7219i		−	2.15279i
$179$	21.6417			1.61757			0.808787	−	0.588101i	$-0.200124\pi$
$179$	21.6417			1.61757			0.808787	+	0.588101i	$0.200124\pi$
$180$		0			0
$181$			4.63625i			0.344610i	0.985044	+	0.172305i	$0.0551215\pi$
$181$			4.63625i			0.344610i	−0.985044	+	0.172305i	$0.944879\pi$
$182$		−	9.19229i		−	0.681378i
$183$		0			0
$184$		−	6.59818i		−	0.486424i
$185$	27.9248			2.05307
$186$		0			0
$187$	1.51614			0.110871
$188$	11.7348			0.855846
$189$		0			0
$190$			3.02824i			0.219692i
$191$	−5.71009			−0.413168			−0.206584	−	0.978429i	$-0.566235\pi$
$191$	−5.71009			−0.413168			−0.206584	+	0.978429i	$0.566235\pi$
$192$		0			0
$193$	−13.6733			−0.984226			−0.492113	−	0.870531i	$-0.663775\pi$
$193$	−13.6733			−0.984226			−0.492113	+	0.870531i	$0.663775\pi$
$194$	17.4824			1.25516
$195$		0			0
$196$	−14.1725			−1.01232
$197$		−	24.6990i		−	1.75973i	−0.475224	−	0.879865i	$-0.657633\pi$
$197$		−	24.6990i		−	1.75973i	0.475224	−	0.879865i	$-0.342367\pi$
$198$		0			0
$199$			2.56719i			0.181983i	0.995852	+	0.0909917i	$0.0290037\pi$
$199$			2.56719i			0.181983i	−0.995852	+	0.0909917i	$0.970996\pi$
$200$		−	3.82557i		−	0.270509i
$201$		0			0
$202$	37.2950			2.62407
$203$	3.65989			0.256874
$204$		0			0
$205$		−	16.4352i		−	1.14788i
$206$	−7.33692			−0.511187
$207$		0			0
$208$		−	8.22749i		−	0.570474i
$209$	−0.447290			−0.0309397
$210$		0			0
$211$	−5.86014			−0.403429			−0.201714	−	0.979444i	$-0.564651\pi$
$211$	−5.86014			−0.403429			−0.201714	+	0.979444i	$0.564651\pi$
$212$	28.0678			1.92770
$213$		0			0
$214$		−	4.10098i		−	0.280337i
$215$	33.7895			2.30442
$216$		0			0
$217$			9.27792i			0.629826i
$218$		−	21.6932i		−	1.46925i
$219$		0			0
$220$	6.55636			0.442030
$221$	5.80410			0.390426
$222$		0			0
$223$			6.48435i			0.434224i	0.976147	+	0.217112i	$0.0696637\pi$
$223$			6.48435i			0.434224i	−0.976147	+	0.217112i	$0.930336\pi$
$224$	9.75996			0.652115
$225$		0			0
$226$	8.51428			0.566361
$227$	−5.17846			−0.343707			−0.171853	−	0.985123i	$-0.554976\pi$
$227$	−5.17846			−0.343707			−0.171853	+	0.985123i	$0.554976\pi$
$228$		0			0
$229$			2.91509i			0.192635i	0.995351	+	0.0963174i	$0.0307064\pi$
$229$			2.91509i			0.192635i	−0.995351	+	0.0963174i	$0.969294\pi$
$230$	−30.4424			−2.00731
$231$		0			0
$232$	−3.82567			−0.251168
$233$	22.0843			1.44679			0.723394	−	0.690435i	$-0.242581\pi$
$233$	22.0843			1.44679			0.723394	+	0.690435i	$0.242581\pi$
$234$		0			0
$235$		−	12.6538i		−	0.825441i
$236$	31.4168			2.04506
$237$		0			0
$238$			4.56980i			0.296216i
$239$	13.1444	+	8.13780i	0.850243	+	0.526391i
$239$	13.1444	+	8.13780i	0.850243	+	0.526391i
$240$		0			0
$241$	−14.9933			−0.965806			−0.482903	−	0.875674i	$-0.660418\pi$
$241$	−14.9933			−0.965806			−0.482903	+	0.875674i	$0.660418\pi$
$242$		−	21.9075i		−	1.40827i
$243$		0			0
$244$	24.0362			1.53876
$245$			15.2824i			0.976355i
$246$		0			0
$247$	−1.71232			−0.108952
$248$		−	9.69819i		−	0.615835i
$249$		0			0
$250$	12.5637			0.794597
$251$			1.32004i			0.0833200i	0.999132	+	0.0416600i	$0.0132646\pi$
$251$			1.32004i			0.0833200i	−0.999132	+	0.0416600i	$0.986735\pi$
$252$		0			0
$253$		−	4.49653i		−	0.282694i
$254$			16.8470i			1.05708i
$255$		0			0
$256$	2.36714			0.147946
$257$		−	7.66843i		−	0.478343i	−0.970977	−	0.239172i	$-0.923124\pi$
$257$		−	7.66843i		−	0.478343i	0.970977	−	0.239172i	$-0.0768759\pi$
$258$		0			0
$259$	−12.4322			−0.772496
$260$	25.0991			1.55658
$261$		0			0
$262$			39.2868i			2.42715i
$263$		−	13.4753i		−	0.830921i	−0.909611	−	0.415461i	$-0.863620\pi$
$263$		−	13.4753i		−	0.830921i	0.909611	−	0.415461i	$-0.136380\pi$
$264$		0			0
$265$		−	30.2659i		−	1.85922i
$266$		−	1.34818i		−	0.0826620i
$267$		0			0
$268$	4.09098			0.249896
$269$		−	17.4523i		−	1.06409i	−0.846717	−	0.532043i	$-0.821425\pi$
$269$		−	17.4523i		−	1.06409i	0.846717	−	0.532043i	$-0.178575\pi$
$270$		0			0
$271$	0.889735			0.0540476			0.0270238	−	0.999635i	$-0.491397\pi$
$271$	0.889735			0.0540476			0.0270238	+	0.999635i	$0.491397\pi$
$272$			4.09017i			0.248003i
$273$		0			0
$274$		−	18.8460i		−	1.13853i
$275$		−	2.60705i		−	0.157211i
$276$		0			0
$277$		−	4.09808i		−	0.246230i	−0.992392	−	0.123115i	$-0.960712\pi$
$277$		−	4.09808i		−	0.246230i	0.992392	−	0.123115i	$-0.0392884\pi$
$278$	−15.5766			−0.934225
$279$		0			0
$280$			4.61861i			0.276015i
$281$	−14.2532			−0.850273			−0.425137	−	0.905129i	$-0.639774\pi$
$281$	−14.2532			−0.850273			−0.425137	+	0.905129i	$0.639774\pi$
$282$		0			0
$283$	3.71649			0.220922			0.110461	−	0.993880i	$-0.464767\pi$
$283$	3.71649			0.220922			0.110461	+	0.993880i	$0.464767\pi$
$284$		−	8.62005i		−	0.511506i
$285$		0			0
$286$			6.54812i			0.387198i
$287$			7.31696i			0.431906i
$288$		0			0
$289$	14.1146			0.830270
$290$			17.6507i			1.03649i
$291$		0			0
$292$			22.3971i			1.31069i
$293$		−	4.18223i		−	0.244329i	−0.992510	−	0.122164i	$-0.961017\pi$
$293$		−	4.18223i		−	0.244329i	0.992510	−	0.122164i	$-0.0389835\pi$
$294$		0			0
$295$		−	33.8772i		−	1.97241i
$296$	12.9953			0.755336
$297$		0			0
$298$			12.4069i			0.718714i
$299$		−	17.2136i		−	0.995490i
$300$		0			0
$301$	−15.0431			−0.867069
$302$	−3.96307			−0.228049
$303$		0			0
$304$		−	1.20668i		−	0.0692076i
$305$		−	25.9186i		−	1.48409i
$306$		0			0
$307$	1.86109			0.106218			0.0531091	−	0.998589i	$-0.483087\pi$
$307$	1.86109			0.106218			0.0531091	+	0.998589i	$0.483087\pi$
$308$	−2.91890			−0.166320
$309$		0			0
$310$	−44.7451			−2.54135
$311$			5.02427i			0.284900i	0.989802	+	0.142450i	$0.0454981\pi$
$311$			5.02427i			0.284900i	−0.989802	+	0.142450i	$0.954502\pi$
$312$		0			0
$313$			21.7173i			1.22754i	0.789487	+	0.613768i	$0.210347\pi$
$313$			21.7173i			1.22754i	−0.789487	+	0.613768i	$0.789653\pi$
$314$			0.0547860i			0.00309175i
$315$		0			0
$316$		−	21.5632i		−	1.21302i
$317$	0.956250			0.0537084			0.0268542	−	0.999639i	$-0.491451\pi$
$317$	0.956250			0.0537084			0.0268542	+	0.999639i	$0.491451\pi$
$318$		0			0
$319$	−2.60712			−0.145970
$320$			33.5163i			1.87362i
$321$		0			0
$322$	13.5530			0.755278
$323$	0.851252			0.0473649
$324$		0			0
$325$		−	9.98031i		−	0.553608i
$326$			40.8263i			2.26116i
$327$		0			0
$328$		−	7.64840i		−	0.422312i
$329$			5.63347i			0.310583i
$330$		0			0
$331$		−	7.37466i		−	0.405348i	−0.979246	−	0.202674i	$-0.935037\pi$
$331$		−	7.37466i		−	0.405348i	0.979246	−	0.202674i	$-0.0649632\pi$
$332$		−	28.0574i		−	1.53985i
$333$		0			0
$334$			43.0854i			2.35753i
$335$		−	4.41136i		−	0.241019i
$336$		0			0
$337$	22.0353			1.20034			0.600169	−	0.799873i	$-0.295100\pi$
$337$	22.0353			1.20034			0.600169	+	0.799873i	$0.295100\pi$
$338$		−	2.84466i		−	0.154729i
$339$		0			0
$340$	−12.4776			−0.676694
$341$		−	6.60912i		−	0.357904i
$342$		0			0
$343$		−	15.5745i		−	0.840947i
$344$	15.7245			0.847809
$345$		0			0
$346$			22.1729i			1.19203i
$347$		−	22.8440i		−	1.22633i	−0.789955	−	0.613165i	$-0.789896\pi$
$347$		−	22.8440i		−	1.22633i	0.789955	−	0.613165i	$-0.210104\pi$
$348$		0			0
$349$	20.7100			1.10858			0.554290	−	0.832324i	$-0.312990\pi$
$349$	20.7100			1.10858			0.554290	+	0.832324i	$0.312990\pi$
$350$	7.85790			0.420023
$351$		0			0
$352$	−6.95250			−0.370569
$353$	−0.870860			−0.0463512			−0.0231756	−	0.999731i	$-0.507378\pi$
$353$	−0.870860			−0.0463512			−0.0231756	+	0.999731i	$0.507378\pi$
$354$		0			0
$355$	−9.29513			−0.493334
$356$	−34.9143			−1.85045
$357$		0			0
$358$		−	46.4666i		−	2.45584i
$359$			24.6128i			1.29902i	0.760355	+	0.649508i	$0.225025\pi$
$359$			24.6128i			1.29902i	−0.760355	+	0.649508i	$0.774975\pi$
$360$		0			0
$361$	18.7489			0.986782
$362$	9.95446			0.523195
$363$		0			0
$364$	−11.1741			−0.585684
$365$	24.1511			1.26413
$366$		0			0
$367$	−25.2759			−1.31939			−0.659695	−	0.751534i	$-0.729314\pi$
$367$	−25.2759			−1.31939			−0.659695	+	0.751534i	$0.729314\pi$
$368$	12.1305			0.632346
$369$		0			0
$370$		−	59.9572i		−	3.11702i
$371$			13.4744i			0.699556i
$372$		0			0
$373$	2.32448			0.120357			0.0601786	−	0.998188i	$-0.480833\pi$
$373$	2.32448			0.120357			0.0601786	+	0.998188i	$0.480833\pi$
$374$		−	3.25529i		−	0.168327i
$375$		0			0
$376$		−	5.88865i		−	0.303684i
$377$	−9.98057			−0.514026
$378$		0			0
$379$			18.9667i			0.974255i	0.873331	+	0.487128i	$0.161955\pi$
$379$			18.9667i			0.974255i	−0.873331	+	0.487128i	$0.838045\pi$
$380$	3.68113			0.188838
$381$		0			0
$382$			12.2601i			0.627280i
$383$		−	5.66284i		−	0.289357i	−0.989479	−	0.144679i	$-0.953785\pi$
$383$		−	5.66284i		−	0.289357i	0.989479	−	0.144679i	$-0.0462149\pi$
$384$		0			0
$385$			3.14749i			0.160411i
$386$			29.3578i			1.49427i
$387$		0			0
$388$		−	21.2515i		−	1.07888i
$389$			0.298679i			0.0151436i	0.999971	+	0.00757182i	$0.00241021\pi$
$389$			0.298679i			0.0151436i	−0.999971	+	0.00757182i	$0.997590\pi$
$390$		0			0
$391$			8.55749i			0.432771i
$392$			7.11191i			0.359206i
$393$		0			0
$394$	−53.0310			−2.67166
$395$	−23.2519			−1.16993
$396$		0			0
$397$			12.4547i			0.625083i	0.949904	+	0.312542i	$0.101180\pi$
$397$			12.4547i			0.625083i	−0.949904	+	0.312542i	$0.898820\pi$
$398$	5.51199			0.276291
$399$		0			0
$400$	7.03316			0.351658
$401$		−	20.7745i		−	1.03743i	−0.854948	−	0.518714i	$-0.826411\pi$
$401$		−	20.7745i		−	1.03743i	0.854948	−	0.518714i	$-0.173589\pi$
$402$		0			0
$403$		−	25.3011i		−	1.26034i
$404$		−	45.3358i		−	2.25554i
$405$		0			0
$406$		−	7.85811i		−	0.389991i
$407$	8.85603			0.438977
$408$		0			0
$409$	−9.44100			−0.466827			−0.233414	−	0.972378i	$-0.574990\pi$
$409$	−9.44100			−0.466827			−0.233414	+	0.972378i	$0.574990\pi$
$410$	−35.2879			−1.74274
$411$		0			0
$412$			8.91875i			0.439395i
$413$			15.0822i			0.742144i
$414$		0			0
$415$	−30.2547			−1.48515
$416$	−26.6156			−1.30494
$417$		0			0
$418$			0.960372i			0.0469733i
$419$		−	8.35648i		−	0.408241i	−0.978946	−	0.204120i	$-0.934567\pi$
$419$		−	8.35648i		−	0.408241i	0.978946	−	0.204120i	$-0.0654334\pi$
$420$		0			0
$421$	−20.6460			−1.00622			−0.503112	−	0.864221i	$-0.667812\pi$
$421$	−20.6460			−1.00622			−0.503112	+	0.864221i	$0.667812\pi$
$422$			12.5823i			0.612495i
$423$		0			0
$424$		−	14.0848i		−	0.684017i
$425$			4.96156i			0.240671i
$426$		0			0
$427$			11.5390i			0.558409i
$428$	−4.98515			−0.240966
$429$		0			0
$430$		−	72.5491i		−	3.49863i
$431$		−	5.69747i		−	0.274438i	−0.990541	−	0.137219i	$-0.956184\pi$
$431$		−	5.69747i		−	0.274438i	0.990541	−	0.137219i	$-0.0438163\pi$
$432$		0			0
$433$			28.4058i			1.36510i	0.730840	+	0.682549i	$0.239129\pi$
$433$			28.4058i			1.36510i	−0.730840	+	0.682549i	$0.760871\pi$
$434$	19.9205			0.956217
$435$		0			0
$436$	−26.3702			−1.26290
$437$		−	2.52462i		−	0.120769i
$438$		0			0
$439$	−8.74981			−0.417606			−0.208803	−	0.977958i	$-0.566957\pi$
$439$	−8.74981			−0.417606			−0.208803	+	0.977958i	$0.566957\pi$
$440$		−	3.29006i		−	0.156848i
$441$		0			0
$442$		−	12.4619i		−	0.592754i
$443$			8.80587i			0.418379i	0.977875	+	0.209190i	$0.0670826\pi$
$443$			8.80587i			0.418379i	−0.977875	+	0.209190i	$0.932917\pi$
$444$		0			0
$445$			37.6486i			1.78471i
$446$	13.9225			0.659249
$447$		0			0
$448$		−	14.9215i		−	0.704973i
$449$	2.90960			0.137312			0.0686562	−	0.997640i	$-0.478129\pi$
$449$	2.90960			0.137312			0.0686562	+	0.997640i	$0.478129\pi$
$450$		0			0
$451$		−	5.21223i		−	0.245434i
$452$		−	10.3499i		−	0.486821i
$453$		0			0
$454$			11.1186i			0.521824i
$455$			12.0492i			0.564877i
$456$		0			0
$457$	34.2391			1.60164			0.800820	−	0.598905i	$-0.204397\pi$
$457$	34.2391			1.60164			0.800820	+	0.598905i	$0.204397\pi$
$458$	6.25897			0.292463
$459$		0			0
$460$			37.0058i			1.72540i
$461$	−14.1242			−0.657831			−0.328915	−	0.944359i	$-0.606683\pi$
$461$	−14.1242			−0.657831			−0.328915	+	0.944359i	$0.606683\pi$
$462$		0			0
$463$		−	14.9849i		−	0.696409i	−0.937419	−	0.348204i	$-0.886792\pi$
$463$		−	14.9849i		−	0.696409i	0.937419	−	0.348204i	$-0.113208\pi$
$464$		−	7.03334i		−	0.326515i
$465$		0			0
$466$		−	47.4169i		−	2.19655i
$467$	6.44771			0.298364			0.149182	−	0.988810i	$-0.452336\pi$
$467$	6.44771			0.298364			0.149182	+	0.988810i	$0.452336\pi$
$468$		0			0
$469$			1.96394i			0.0906864i
$470$	−27.1688			−1.25320
$471$		0			0
$472$		−	15.7653i		−	0.725658i
$473$	10.7159			0.492719
$474$		0			0
$475$		−	1.46375i		−	0.0671615i
$476$	5.55505			0.254615
$477$		0			0
$478$	17.4726	−	28.2223i	0.799178	−	1.29086i
$479$		−	6.47890i		−	0.296028i	−0.988985	−	0.148014i	$-0.952712\pi$
$479$		−	6.47890i		−	0.296028i	0.988985	−	0.148014i	$-0.0472881\pi$
$480$		0			0
$481$	33.9027			1.54583
$482$			32.1921i			1.46631i
$483$		0			0
$484$	−26.6308			−1.21049
$485$	−22.9159			−1.04056
$486$		0			0
$487$	13.1781			0.597156			0.298578	−	0.954385i	$-0.403488\pi$
$487$	13.1781			0.597156			0.298578	+	0.954385i	$0.403488\pi$
$488$		−	12.0616i		−	0.546005i
$489$		0			0
$490$	32.8127			1.48232
$491$	−15.6595			−0.706703			−0.353352	−	0.935491i	$-0.614958\pi$
$491$	−15.6595			−0.706703			−0.353352	+	0.935491i	$0.614958\pi$
$492$		0			0
$493$	4.96169			0.223463
$494$			3.67650i			0.165414i
$495$		0			0
$496$	17.8297			0.800579
$497$	4.13820			0.185624
$498$		0			0
$499$		−	41.4724i		−	1.85656i	−0.371882	−	0.928280i	$-0.621287\pi$
$499$		−	41.4724i		−	1.85656i	0.371882	−	0.928280i	$-0.378713\pi$
$500$		−	15.2724i		−	0.683003i
$501$		0			0
$502$	2.83424			0.126498
$503$			16.8753i			0.752430i	0.926532	+	0.376215i	$0.122775\pi$
$503$			16.8753i			0.752430i	−0.926532	+	0.376215i	$0.877225\pi$
$504$		0			0
$505$	−48.8863			−2.17541
$506$	−9.65445			−0.429193
$507$		0			0
$508$	20.4792			0.908618
$509$			12.6322i			0.559914i	0.960013	+	0.279957i	$0.0903202\pi$
$509$			12.6322i			0.559914i	−0.960013	+	0.279957i	$0.909680\pi$
$510$		0			0
$511$	−10.7521			−0.475645
$512$		−	25.0635i		−	1.10766i
$513$		0			0
$514$	−16.4648			−0.726232
$515$	9.61722			0.423786
$516$		0			0
$517$		−	4.01300i		−	0.176491i
$518$			26.6930i			1.17282i
$519$		0			0
$520$		−	12.5950i		−	0.552329i
$521$	−21.4191			−0.938387			−0.469194	−	0.883095i	$-0.655455\pi$
$521$	−21.4191			−0.938387			−0.469194	+	0.883095i	$0.655455\pi$
$522$		0			0
$523$	−40.8932			−1.78813			−0.894067	−	0.447933i	$-0.852160\pi$
$523$	−40.8932			−1.78813			−0.894067	+	0.447933i	$0.852160\pi$
$524$	47.7570			2.08627
$525$		0			0
$526$	−28.9327			−1.26152
$527$			12.5780i			0.547907i
$528$		0			0
$529$	2.37954			0.103458
$530$	−64.9837			−2.82271
$531$		0			0
$532$	−1.63884			−0.0710528
$533$		−	19.9535i		−	0.864281i
$534$		0			0
$535$			5.37556i			0.232406i
$536$		−	2.05290i		−	0.0886719i
$537$		0			0
$538$	−37.4717			−1.61552
$539$			4.84662i			0.208759i
$540$		0			0
$541$		−	25.4365i		−	1.09360i	−0.837263	−	0.546800i	$-0.815846\pi$
$541$		−	25.4365i		−	1.09360i	0.837263	−	0.546800i	$-0.184154\pi$
$542$		−	1.91034i		−	0.0820562i
$543$		0			0
$544$	13.2315			0.567297
$545$			28.4354i			1.21804i
$546$		0			0
$547$		−	5.97227i		−	0.255356i	−0.991816	−	0.127678i	$-0.959248\pi$
$547$		−	5.97227i		−	0.255356i	0.991816	−	0.127678i	$-0.0407524\pi$
$548$	−22.9091			−0.978630
$549$		0			0
$550$	−5.59757			−0.238681
$551$	−1.46379			−0.0623595
$552$		0			0
$553$	10.3518			0.440202
$554$	−8.79896			−0.373832
$555$		0			0
$556$			18.9350i			0.803021i
$557$	−22.9110			−0.970770			−0.485385	−	0.874300i	$-0.661321\pi$
$557$	−22.9110			−0.970770			−0.485385	+	0.874300i	$0.661321\pi$
$558$		0			0
$559$	41.0228			1.73508
$560$	−8.49114			−0.358816
$561$		0			0
$562$			30.6029i			1.29090i
$563$		−	13.1756i		−	0.555286i	−0.960684	−	0.277643i	$-0.910447\pi$
$563$		−	13.1756i		−	0.555286i	0.960684	−	0.277643i	$-0.0895532\pi$
$564$		0			0
$565$	−11.1605			−0.469526
$566$		−	7.97964i		−	0.335409i
$567$		0			0
$568$	−4.32565			−0.181500
$569$			12.4016i			0.519901i	0.965622	+	0.259950i	$0.0837062\pi$
$569$			12.4016i			0.519901i	−0.965622	+	0.259950i	$0.916294\pi$
$570$		0			0
$571$	−1.49424			−0.0625320			−0.0312660	−	0.999511i	$-0.509954\pi$
$571$	−1.49424			−0.0625320			−0.0312660	+	0.999511i	$0.509954\pi$
$572$	7.95988			0.332819
$573$		0			0
$574$	15.7102			0.655730
$575$	14.7148			0.613651
$576$		0			0
$577$	36.4101			1.51577			0.757885	−	0.652388i	$-0.226233\pi$
$577$	36.4101			1.51577			0.757885	+	0.652388i	$0.226233\pi$
$578$		−	30.3053i		−	1.26053i
$579$		0			0
$580$	21.4562			0.890920
$581$	13.4694			0.558806
$582$		0			0
$583$		−	9.59848i		−	0.397528i
$584$	11.2391			0.465079
$585$		0			0
$586$	−8.97964			−0.370945
$587$			35.8436i			1.47942i	0.672924	+	0.739712i	$0.265038\pi$
$587$			35.8436i			1.47942i	−0.672924	+	0.739712i	$0.734962\pi$
$588$		0			0
$589$		−	3.71075i		−	0.152899i
$590$	−72.7375			−2.99455
$591$		0			0
$592$			23.8913i			0.981928i
$593$	44.4771			1.82646			0.913228	−	0.407449i	$-0.133582\pi$
$593$	44.4771			1.82646			0.913228	+	0.407449i	$0.133582\pi$
$594$		0			0
$595$		−	5.99009i		−	0.245570i
$596$	15.0819			0.617777
$597$		0			0
$598$	−36.9592			−1.51138
$599$			41.2103i			1.68381i	0.539627	+	0.841904i	$0.318565\pi$
$599$			41.2103i			1.68381i	−0.539627	+	0.841904i	$0.681435\pi$
$600$		0			0
$601$			44.4286i			1.81228i	0.422979	+	0.906140i	$0.360985\pi$
$601$			44.4286i			1.81228i	−0.422979	+	0.906140i	$0.639015\pi$
$602$			32.2989i			1.31640i
$603$		0			0
$604$			4.81750i			0.196021i
$605$			28.7164i			1.16749i
$606$		0			0
$607$		−	35.1506i		−	1.42672i	−0.700798	−	0.713359i	$-0.747173\pi$
$607$		−	35.1506i		−	1.42672i	0.700798	−	0.713359i	$-0.252827\pi$
$608$	−3.90354			−0.158310
$609$		0			0
$610$	−55.6495			−2.25318
$611$		−	15.3626i		−	0.621503i
$612$		0			0
$613$	−30.1121			−1.21622			−0.608109	−	0.793853i	$-0.708072\pi$
$613$	−30.1121			−1.21622			−0.608109	+	0.793853i	$0.708072\pi$
$614$		−	3.99594i		−	0.161263i
$615$		0			0
$616$			1.46474i			0.0590160i
$617$	17.7832			0.715924			0.357962	−	0.933736i	$-0.383472\pi$
$617$	17.7832			0.715924			0.357962	+	0.933736i	$0.383472\pi$
$618$		0			0
$619$			32.6299i			1.31151i	0.754975	+	0.655754i	$0.227649\pi$
$619$			32.6299i			1.31151i	−0.754975	+	0.655754i	$0.772351\pi$
$620$			54.3921i			2.18444i
$621$		0			0
$622$	10.7876			0.432542
$623$		−	16.7612i		−	0.671522i
$624$		0			0
$625$	−31.0729			−1.24291
$626$	46.6291			1.86367
$627$		0			0
$628$	0.0665978			0.00265754
$629$	−16.8542			−0.672021
$630$		0			0
$631$	17.6253			0.701653			0.350827	−	0.936440i	$-0.385901\pi$
$631$	17.6253			0.701653			0.350827	+	0.936440i	$0.385901\pi$
$632$	−10.8207			−0.430424
$633$		0			0
$634$		−	2.05316i		−	0.0815413i
$635$		−	22.0830i		−	0.876339i
$636$		0			0
$637$			18.5539i			0.735131i
$638$			5.59772i			0.221616i
$639$		0			0
$640$	28.1173			1.11143
$641$		−	34.0833i		−	1.34621i	−0.739548	−	0.673104i	$-0.764961\pi$
$641$		−	34.0833i		−	1.34621i	0.739548	−	0.673104i	$-0.235039\pi$
$642$		0			0
$643$	30.1707			1.18982			0.594909	−	0.803793i	$-0.297188\pi$
$643$	30.1707			1.18982			0.594909	+	0.803793i	$0.297188\pi$
$644$		−	16.4750i		−	0.649206i
$645$		0			0
$646$		−	1.82772i		−	0.0719105i
$647$			36.1007i			1.41927i	0.704571	+	0.709633i	$0.251139\pi$
$647$			36.1007i			1.41927i	−0.704571	+	0.709633i	$0.748861\pi$
$648$		0			0
$649$		−	10.7438i		−	0.421729i
$650$	−21.4286			−0.840501
$651$		0			0
$652$	49.6285			1.94360
$653$	−39.0920			−1.52979			−0.764894	−	0.644156i	$-0.777209\pi$
$653$	−39.0920			−1.52979			−0.764894	+	0.644156i	$0.777209\pi$
$654$		0			0
$655$		−	51.4971i		−	2.01216i
$656$	14.0613			0.549001
$657$		0			0
$658$	12.0956			0.471535
$659$	5.43700			0.211795			0.105898	−	0.994377i	$-0.466228\pi$
$659$	5.43700			0.211795			0.105898	+	0.994377i	$0.466228\pi$
$660$		0			0
$661$	−13.5903			−0.528603			−0.264301	−	0.964440i	$-0.585141\pi$
$661$	−13.5903			−0.528603			−0.264301	+	0.964440i	$0.585141\pi$
$662$	−15.8341			−0.615408
$663$		0			0
$664$	−14.0796			−0.546393
$665$			1.76719i			0.0685286i
$666$		0			0
$667$		−	14.7152i		−	0.569776i
$668$	52.3746			2.02644
$669$		0			0
$670$	−9.47160			−0.365920
$671$		−	8.21977i		−	0.317321i
$672$		0			0
$673$			11.5222i			0.444148i	0.975030	+	0.222074i	$0.0712827\pi$
$673$			11.5222i			0.444148i	−0.975030	+	0.222074i	$0.928717\pi$
$674$		−	47.3117i		−	1.82238i
$675$		0			0
$676$	−3.45796			−0.132999
$677$	−1.99304			−0.0765987			−0.0382993	−	0.999266i	$-0.512194\pi$
$677$	−1.99304			−0.0765987			−0.0382993	+	0.999266i	$0.512194\pi$
$678$		0			0
$679$	10.2022			0.391523
$680$			6.26143i			0.240115i
$681$		0			0
$682$	−14.1904			−0.543378
$683$	11.9271			0.456377			0.228188	−	0.973617i	$-0.426720\pi$
$683$	11.9271			0.456377			0.228188	+	0.973617i	$0.426720\pi$
$684$		0			0
$685$			24.7033i			0.943863i
$686$	−33.4400			−1.27674
$687$		0			0
$688$			28.9089i			1.10214i
$689$		−	36.7450i		−	1.39987i
$690$		0			0
$691$	0.971197			0.0369461			0.0184730	−	0.999829i	$-0.494120\pi$
$691$	0.971197			0.0369461			0.0184730	+	0.999829i	$0.494120\pi$
$692$	26.9534			1.02462
$693$		0			0
$694$	−49.0481			−1.86184
$695$	20.4179			0.774493
$696$		0			0
$697$			9.91956i			0.375730i
$698$		−	44.4662i		−	1.68307i
$699$		0			0
$700$		−	9.55206i		−	0.361034i
$701$	2.34775			0.0886733			0.0443367	−	0.999017i	$-0.485883\pi$
$701$	2.34775			0.0886733			0.0443367	+	0.999017i	$0.485883\pi$
$702$		0			0
$703$	4.97230			0.187534
$704$			10.6293i			0.400607i
$705$		0			0
$706$			1.86982i			0.0703715i
$707$	21.7642			0.818527
$708$		0			0
$709$		−	10.1437i		−	0.380953i	−0.981692	−	0.190477i	$-0.938997\pi$
$709$		−	10.1437i		−	0.380953i	0.981692	−	0.190477i	$-0.0610034\pi$
$710$			19.9575i			0.748991i
$711$		0			0
$712$			17.5204i			0.656605i
$713$	37.3035			1.39703
$714$		0			0
$715$		−	8.58326i		−	0.320996i
$716$	−56.4848			−2.11094
$717$		0			0
$718$	52.8460			1.97220
$719$		−	46.2070i		−	1.72323i	−0.507561	−	0.861616i	$-0.669453\pi$
$719$		−	46.2070i		−	1.72323i	0.507561	−	0.861616i	$-0.330547\pi$
$720$		0			0
$721$	−4.28159			−0.159455
$722$		−	40.2555i		−	1.49816i
$723$		0			0
$724$		−	12.1006i		−	0.449717i
$725$		−	8.53176i		−	0.316862i
$726$		0			0
$727$	34.9319			1.29555			0.647776	−	0.761831i	$-0.275699\pi$
$727$	34.9319			1.29555			0.647776	+	0.761831i	$0.275699\pi$
$728$			5.60732i			0.207821i
$729$		0			0
$730$		−	51.8547i		−	1.91923i
$731$	−20.3938			−0.754293
$732$		0			0
$733$	−37.2177			−1.37467			−0.687334	−	0.726341i	$-0.741219\pi$
$733$	−37.2177			−1.37467			−0.687334	+	0.726341i	$0.741219\pi$
$734$			54.2696i			2.00313i
$735$		0			0
$736$		−	39.2417i		−	1.44647i
$737$		−	1.39901i		−	0.0515333i
$738$		0			0
$739$	−28.4005			−1.04473			−0.522365	−	0.852722i	$-0.674950\pi$
$739$	−28.4005			−1.04473			−0.522365	+	0.852722i	$0.674950\pi$
$740$	−72.8839			−2.67926
$741$		0			0
$742$	28.9308			1.06208
$743$	31.6891			1.16256			0.581281	−	0.813703i	$-0.302552\pi$
$743$	31.6891			1.16256			0.581281	+	0.813703i	$0.302552\pi$
$744$		0			0
$745$		−	16.2630i		−	0.595830i
$746$		−	4.99088i		−	0.182729i
$747$		0			0
$748$	−3.95713			−0.144687
$749$		−	2.39320i		−	0.0874458i
$750$		0			0
$751$	−17.1743			−0.626701			−0.313350	−	0.949638i	$-0.601451\pi$
$751$	−17.1743			−0.626701			−0.313350	+	0.949638i	$0.601451\pi$
$752$	10.8261			0.394786
$753$		0			0
$754$			21.4292i			0.780406i
$755$	5.19478			0.189058
$756$		0			0
$757$	48.2662			1.75426			0.877132	−	0.480249i	$-0.159454\pi$
$757$	48.2662			1.75426			0.877132	+	0.480249i	$0.159454\pi$
$758$	40.7233			1.47914
$759$		0			0
$760$		−	1.84724i		−	0.0670063i
$761$			25.1757i			0.912619i	0.889821	+	0.456309i	$0.150829\pi$
$761$			25.1757i			0.912619i	−0.889821	+	0.456309i	$0.849171\pi$
$762$		0			0
$763$		−	12.6595i		−	0.458303i
$764$	14.9033			0.539184
$765$		0			0
$766$	−12.1586			−0.439309
$767$		−	41.1293i		−	1.48509i
$768$		0			0
$769$			8.83633i			0.318646i	0.987227	+	0.159323i	$0.0509311\pi$
$769$			8.83633i			0.318646i	−0.987227	+	0.159323i	$0.949069\pi$
$770$	6.75795			0.243540
$771$		0			0
$772$	35.6874			1.28442
$773$	19.4407			0.699234			0.349617	−	0.936893i	$-0.386312\pi$
$773$	19.4407			0.699234			0.349617	+	0.936893i	$0.386312\pi$
$774$		0			0
$775$	21.6283			0.776910
$776$	−10.6643			−0.382826
$777$		0			0
$778$	0.641292			0.0229914
$779$		−	2.92645i		−	0.104851i
$780$		0			0
$781$	−2.94784			−0.105482
$782$	18.3737			0.657042
$783$		0			0
$784$	−13.0750			−0.466963
$785$		−	0.0718134i		−	0.00256313i
$786$		0			0
$787$			3.73920i			0.133288i	0.997777	+	0.0666441i	$0.0212292\pi$
$787$			3.73920i			0.133288i	−0.997777	+	0.0666441i	$0.978771\pi$
$788$			64.4644i			2.29645i
$789$		0			0
$790$			49.9240i			1.77622i
$791$	4.96866			0.176665
$792$		0			0
$793$		−	31.4670i		−	1.11742i
$794$	26.7414			0.949015
$795$		0			0
$796$		−	6.70037i		−	0.237488i
$797$		−	31.9996i		−	1.13348i	−0.823895	−	0.566742i	$-0.808204\pi$
$797$		−	31.9996i		−	1.13348i	0.823895	−	0.566742i	$-0.191796\pi$
$798$		0			0
$799$			7.63726i			0.270187i
$800$		−	22.7520i		−	0.804404i
$801$		0			0
$802$	−44.6047			−1.57505
$803$	7.65925			0.270289
$804$		0			0
$805$	−17.7652			−0.626142
$806$	−54.3237			−1.91347
$807$		0			0
$808$	−22.7501			−0.800345
$809$	10.9555			0.385175			0.192588	−	0.981280i	$-0.438312\pi$
$809$	10.9555			0.385175			0.192588	+	0.981280i	$0.438312\pi$
$810$		0			0
$811$			28.9623i			1.01700i	0.861061	+	0.508501i	$0.169800\pi$
$811$			28.9623i			1.01700i	−0.861061	+	0.508501i	$0.830200\pi$
$812$	−9.55231			−0.335220
$813$		0			0
$814$		−	19.0147i		−	0.666465i
$815$		−	53.5151i		−	1.87455i
$816$		0			0
$817$	6.01656			0.210493
$818$			20.2707i			0.708748i
$819$		0			0
$820$			42.8959i			1.49799i
$821$	−17.0318			−0.594413			−0.297206	−	0.954813i	$-0.596055\pi$
$821$	−17.0318			−0.594413			−0.297206	+	0.954813i	$0.596055\pi$
$822$		0			0
$823$		−	48.9831i		−	1.70744i	−0.520730	−	0.853722i	$-0.674340\pi$
$823$		−	48.9831i		−	1.70744i	0.520730	−	0.853722i	$-0.325660\pi$
$824$	4.47554			0.155913
$825$		0			0
$826$	32.3828			1.12674
$827$		−	14.2350i		−	0.494999i	−0.968888	−	0.247499i	$-0.920391\pi$
$827$		−	14.2350i		−	0.494999i	0.968888	−	0.247499i	$-0.0796088\pi$
$828$		0			0
$829$		−	29.7202i		−	1.03223i	−0.856520	−	0.516113i	$-0.827378\pi$
$829$		−	29.7202i		−	1.03223i	0.856520	−	0.516113i	$-0.172622\pi$
$830$			64.9596i			2.25478i
$831$		0			0
$832$			40.6911i			1.41071i
$833$		−	9.22377i		−	0.319585i
$834$		0			0
$835$		−	56.4764i		−	1.95444i
$836$	1.16743			0.0403763
$837$		0			0
$838$	−17.9421			−0.619800
$839$		−	5.31575i		−	0.183520i	−0.995781	−	0.0917600i	$-0.970751\pi$
$839$		−	5.31575i		−	0.183520i	0.995781	−	0.0917600i	$-0.0292493\pi$
$840$		0			0
$841$	20.4680			0.705794
$842$			44.3289i			1.52767i
$843$		0			0
$844$	15.2950			0.526475
$845$			3.72877i			0.128274i
$846$		0			0
$847$		−	12.7845i		−	0.439282i
$848$	25.8943			0.889214
$849$		0			0
$850$	10.6529			0.365392
$851$			49.9857i			1.71349i
$852$		0			0
$853$	−33.7395			−1.15522			−0.577609	−	0.816314i	$-0.696014\pi$
$853$	−33.7395			−1.15522			−0.577609	+	0.816314i	$0.696014\pi$
$854$	24.7752			0.847790
$855$		0			0
$856$			2.50161i			0.0855033i
$857$	46.0372			1.57260			0.786300	−	0.617844i	$-0.211994\pi$
$857$	46.0372			1.57260			0.786300	+	0.617844i	$0.211994\pi$
$858$		0			0
$859$	−39.8720			−1.36042			−0.680208	−	0.733019i	$-0.738111\pi$
$859$	−39.8720			−1.36042			−0.680208	+	0.733019i	$0.738111\pi$
$860$	−88.1906			−3.00727
$861$		0			0
$862$	−12.2330			−0.416658
$863$	14.7023			0.500471			0.250235	−	0.968185i	$-0.419492\pi$
$863$	14.7023			0.500471			0.250235	+	0.968185i	$0.419492\pi$
$864$		0			0
$865$		−	29.0643i		−	0.988215i
$866$	60.9900			2.07252
$867$		0			0
$868$		−	24.2154i		−	0.821924i
$869$	−7.37407			−0.250148
$870$		0			0
$871$		−	5.35570i		−	0.181471i
$872$			13.2329i			0.448123i
$873$		0			0
$874$	−5.42058			−0.183354
$875$	7.33177			0.247859
$876$		0			0
$877$	−20.6823			−0.698390			−0.349195	−	0.937050i	$-0.613545\pi$
$877$	−20.6823			−0.698390			−0.349195	+	0.937050i	$0.613545\pi$
$878$			18.7867i			0.634019i
$879$		0			0
$880$	6.04865			0.203900
$881$	−42.0848			−1.41787			−0.708936	−	0.705273i	$-0.750825\pi$
$881$	−42.0848			−1.41787			−0.708936	+	0.705273i	$0.750825\pi$
$882$		0			0
$883$	9.01791			0.303477			0.151738	−	0.988421i	$-0.451513\pi$
$883$	9.01791			0.303477			0.151738	+	0.988421i	$0.451513\pi$
$884$	−15.1487			−0.509506
$885$		0			0
$886$	18.9070			0.635193
$887$		−	22.5426i		−	0.756908i	−0.925620	−	0.378454i	$-0.876456\pi$
$887$		−	22.5426i		−	0.756908i	0.925620	−	0.378454i	$-0.123544\pi$
$888$		0			0
$889$			9.83139i			0.329734i
$890$	80.8350			2.70959
$891$		0			0
$892$		−	16.9242i		−	0.566663i
$893$		−	2.25313i		−	0.0753983i
$894$		0			0
$895$			60.9084i			2.03594i
$896$	−12.5178			−0.418192
$897$		0			0
$898$		−	6.24717i		−	0.208471i
$899$		−	21.6288i		−	0.721362i
$900$		0			0
$901$			18.2672i			0.608568i
$902$	−11.1911			−0.372624
$903$		0			0
$904$	−5.19373			−0.172741
$905$	−13.0483			−0.433740
$906$		0			0
$907$			53.2653i			1.76865i	0.466875	+	0.884323i	$0.345380\pi$
$907$			53.2653i			1.76865i	−0.466875	+	0.884323i	$0.654620\pi$
$908$	13.5158			0.448538
$909$		0			0
$910$	25.8708			0.857609
$911$	33.6080			1.11348			0.556742	−	0.830686i	$-0.312051\pi$
$911$	33.6080			1.11348			0.556742	+	0.830686i	$0.312051\pi$
$912$		0			0
$913$	−9.59493			−0.317546
$914$		−	73.5146i		−	2.43165i
$915$		0			0
$916$		−	7.60840i		−	0.251389i
$917$			22.9265i			0.757101i
$918$		0			0
$919$	−57.1500			−1.88521			−0.942603	−	0.333916i	$-0.891630\pi$
$919$	−57.1500			−1.88521			−0.942603	+	0.333916i	$0.891630\pi$
$920$	18.5700			0.612233
$921$		0			0
$922$			30.3260i			0.998734i
$923$	−11.2849			−0.371448
$924$		0			0
$925$			28.9813i			0.952898i
$926$	−32.1740			−1.05730
$927$		0			0
$928$	−22.7526			−0.746890
$929$	25.0435			0.821651			0.410825	−	0.911714i	$-0.365241\pi$
$929$	25.0435			0.821651			0.410825	+	0.911714i	$0.365241\pi$
$930$		0			0
$931$			2.72118i			0.0891832i
$932$	−57.6400			−1.88806
$933$		0			0
$934$		−	13.8438i		−	0.452984i
$935$			4.26704i			0.139547i
$936$		0			0
$937$	−22.5850			−0.737819			−0.368909	−	0.929465i	$-0.620269\pi$
$937$	−22.5850			−0.737819			−0.368909	+	0.929465i	$0.620269\pi$
$938$	4.21676			0.137682
$939$		0			0
$940$			33.0264i			1.07720i
$941$	−10.4839			−0.341766			−0.170883	−	0.985291i	$-0.554662\pi$
$941$	−10.4839			−0.341766			−0.170883	+	0.985291i	$0.554662\pi$
$942$		0			0
$943$	29.4191			0.958019
$944$	28.9840			0.943347
$945$		0			0
$946$		−	23.0081i		−	0.748057i
$947$	39.4112			1.28069			0.640345	−	0.768087i	$-0.278791\pi$
$947$	39.4112			1.28069			0.640345	+	0.768087i	$0.278791\pi$
$948$		0			0
$949$	29.3212			0.951806
$950$	−3.14281			−0.101966
$951$		0			0
$952$		−	2.78759i		−	0.0903464i
$953$	56.9455			1.84465			0.922323	−	0.386419i	$-0.126288\pi$
$953$	56.9455			1.84465			0.922323	+	0.386419i	$0.126288\pi$
$954$		0			0
$955$		−	16.0705i		−	0.520029i
$956$	−34.3070	−	21.2397i	−1.10957	−	0.686941i
$957$		0			0
$958$	−13.9108			−0.449437
$959$		−	10.9979i		−	0.355141i
$960$		0			0
$961$	23.8298			0.768702
$962$		−	72.7922i		−	2.34692i
$963$		0			0
$964$	39.1327			1.26038
$965$		−	38.4822i		−	1.23879i
$966$		0			0
$967$	20.9438			0.673507			0.336754	−	0.941593i	$-0.390671\pi$
$967$	20.9438			0.673507			0.336754	+	0.941593i	$0.390671\pi$
$968$			13.3636i			0.429524i
$969$		0			0
$970$			49.2024i			1.57980i
$971$			19.7919i			0.635151i	0.948233	+	0.317576i	$0.102869\pi$
$971$			19.7919i			0.635151i	−0.948233	+	0.317576i	$0.897131\pi$
$972$		0			0
$973$	−9.09004			−0.291413
$974$		−	28.2946i		−	0.906617i
$975$		0			0
$976$	22.1749			0.709800
$977$	−50.1381			−1.60406			−0.802030	−	0.597283i	$-0.796247\pi$
$977$	−50.1381			−1.60406			−0.802030	+	0.597283i	$0.796247\pi$
$978$		0			0
$979$			11.9398i			0.381598i
$980$		−	39.8870i		−	1.27414i
$981$		0			0
$982$			33.6224i			1.07293i
$983$		−	56.5609i		−	1.80401i	−0.431723	−	0.902006i	$-0.642094\pi$
$983$		−	56.5609i		−	1.80401i	0.431723	−	0.902006i	$-0.357906\pi$
$984$		0			0
$985$	69.5129			2.21487
$986$		−	10.6532i		−	0.339267i
$987$		0			0
$988$	4.46915			0.142183
$989$			60.4835i			1.92326i
$990$		0			0
$991$		−	31.6148i		−	1.00428i	−0.864787	−	0.502139i	$-0.832547\pi$
$991$		−	31.6148i		−	1.00428i	0.864787	−	0.502139i	$-0.167453\pi$
$992$		−	57.6785i		−	1.83129i
$993$		0			0
$994$		−	8.88509i		−	0.281818i
$995$	−7.22512			−0.229052
$996$		0			0
$997$			38.7314i			1.22664i	0.789836	+	0.613318i	$0.210166\pi$
$997$			38.7314i			1.22664i	−0.789836	+	0.613318i	$0.789834\pi$
$998$	−89.0451			−2.81867
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.b.a.2150.11	✓	80
3.2	odd	2	inner	2151.2.b.a.2150.69	yes	80
239.238	odd	2	inner	2151.2.b.a.2150.12	yes	80
717.716	even	2	inner	2151.2.b.a.2150.70	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.b.a.2150.11	✓	80	1.1	even	1	trivial
2151.2.b.a.2150.12	yes	80	239.238	odd	2	inner
2151.2.b.a.2150.69	yes	80	3.2	odd	2	inner
2151.2.b.a.2150.70	yes	80	717.716	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.14709i q^{2} -2.61000 q^{4} +2.81441i q^{5} -1.25298i q^{7} +1.30973i q^{8} +O(q^{10})\)	\(q-2.14709i q^{2} -2.61000 q^{4} +2.81441i q^{5} -1.25298i q^{7} +1.30973i q^{8} +6.04279 q^{10} +0.892556i q^{11} +3.41689i q^{13} -2.69025 q^{14} -2.40789 q^{16} -1.69865i q^{17} +0.501134i q^{19} -7.34561i q^{20} +1.91640 q^{22} -5.03781 q^{23} -2.92088 q^{25} +7.33637 q^{26} +3.27027i q^{28} +2.92096i q^{29} -7.40471 q^{31} +7.78943i q^{32} -3.64716 q^{34} +3.52638 q^{35} -9.92211i q^{37} +1.07598 q^{38} -3.68612 q^{40} -5.83967 q^{41} -12.0059i q^{43} -2.32957i q^{44} +10.8166i q^{46} -4.49607 q^{47} +5.43005 q^{49} +6.27140i q^{50} -8.91808i q^{52} -10.7539 q^{53} -2.51201 q^{55} +1.64106 q^{56} +6.27156 q^{58} -12.0371 q^{59} -9.20925 q^{61} +15.8986i q^{62} +11.9088 q^{64} -9.61650 q^{65} -1.56742 q^{67} +4.43349i q^{68} -7.57146i q^{70} +3.30270i q^{71} -8.58126i q^{73} -21.3037 q^{74} -1.30796i q^{76} +1.11835 q^{77} +8.26175i q^{79} -6.77678i q^{80} +12.5383i q^{82} +10.7499i q^{83} +4.78069 q^{85} -25.7778 q^{86} -1.16901 q^{88} +13.3771 q^{89} +4.28127 q^{91} +13.1487 q^{92} +9.65348i q^{94} -1.41039 q^{95} +8.14234i q^{97} -11.6588i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 80 q^{4}+O(q^{10}) \) 80 * q - 80 * q^4	\( 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) \) 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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