LMFDB - Embedded newform 2151.2.b.a.2150.16

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			2150.16
Character	$\chi$	$=$	2151.2150
Dual form			2151.2.b.a.2150.65

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83893i q^{2} -1.38165 q^{4} -0.488895i q^{5} +1.94727i q^{7} -1.13709i q^{8} +O(q^{10})$	\(q-1.83893i q^{2} -1.38165 q^{4} -0.488895i q^{5} +1.94727i q^{7} -1.13709i q^{8} -0.899043 q^{10} +4.14123i q^{11} -2.39267i q^{13} +3.58089 q^{14} -4.85434 q^{16} +6.47857i q^{17} -6.20139i q^{19} +0.675484i q^{20} +7.61542 q^{22} +7.33885 q^{23} +4.76098 q^{25} -4.39994 q^{26} -2.69045i q^{28} +4.65358i q^{29} -1.04908 q^{31} +6.65259i q^{32} +11.9136 q^{34} +0.952011 q^{35} +0.902591i q^{37} -11.4039 q^{38} -0.555920 q^{40} -5.83617 q^{41} -5.46817i q^{43} -5.72174i q^{44} -13.4956i q^{46} +9.41703 q^{47} +3.20814 q^{49} -8.75510i q^{50} +3.30584i q^{52} +7.35874 q^{53} +2.02463 q^{55} +2.21423 q^{56} +8.55760 q^{58} +12.8740 q^{59} +2.45943 q^{61} +1.92919i q^{62} +2.52495 q^{64} -1.16976 q^{65} -2.55500 q^{67} -8.95114i q^{68} -1.75068i q^{70} -1.82546i q^{71} -3.90000i q^{73} +1.65980 q^{74} +8.56818i q^{76} -8.06408 q^{77} +2.77679i q^{79} +2.37326i q^{80} +10.7323i q^{82} +6.67645i q^{83} +3.16734 q^{85} -10.0556 q^{86} +4.70896 q^{88} +4.44903 q^{89} +4.65916 q^{91} -10.1397 q^{92} -17.3172i q^{94} -3.03183 q^{95} +1.32217i q^{97} -5.89954i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.83893i		−	1.30032i	−0.759798	−	0.650159i	$-0.774702\pi$
$2$		−	1.83893i		−	1.30032i	0.759798	−	0.650159i	$-0.225298\pi$
$3$		0			0
$3$		0			0
$4$	−1.38165			−0.690827
$5$		−	0.488895i		−	0.218641i	−0.994007	−	0.109320i	$-0.965133\pi$
$5$		−	0.488895i		−	0.218641i	0.994007	−	0.109320i	$-0.0348674\pi$
$6$		0			0
$7$			1.94727i			0.735998i	0.929826	+	0.367999i	$0.119957\pi$
$7$			1.94727i			0.735998i	−0.929826	+	0.367999i	$0.880043\pi$
$8$		−	1.13709i		−	0.402023i
$9$		0			0
$10$	−0.899043			−0.284302
$11$			4.14123i			1.24863i	0.781174	+	0.624314i	$0.214621\pi$
$11$			4.14123i			1.24863i	−0.781174	+	0.624314i	$0.785379\pi$
$12$		0			0
$13$		−	2.39267i		−	0.663606i	−0.943349	−	0.331803i	$-0.892343\pi$
$13$		−	2.39267i		−	0.663606i	0.943349	−	0.331803i	$-0.107657\pi$
$14$	3.58089			0.957032
$15$		0			0
$16$	−4.85434			−1.21359
$17$			6.47857i			1.57128i	0.618681	+	0.785642i	$0.287667\pi$
$17$			6.47857i			1.57128i	−0.618681	+	0.785642i	$0.712333\pi$
$18$		0			0
$19$		−	6.20139i		−	1.42270i	−0.702839	−	0.711349i	$-0.748085\pi$
$19$		−	6.20139i		−	1.42270i	0.702839	−	0.711349i	$-0.251915\pi$
$20$			0.675484i			0.151043i
$21$		0			0
$22$	7.61542			1.62361
$23$	7.33885			1.53026			0.765128	−	0.643879i	$-0.222676\pi$
$23$	7.33885			1.53026			0.765128	+	0.643879i	$0.222676\pi$
$24$		0			0
$25$	4.76098			0.952196
$26$	−4.39994			−0.862899
$27$		0			0
$28$		−	2.69045i		−	0.508448i
$29$			4.65358i			0.864149i	0.901838	+	0.432074i	$0.142218\pi$
$29$			4.65358i			0.864149i	−0.901838	+	0.432074i	$0.857782\pi$
$30$		0			0
$31$	−1.04908			−0.188421			−0.0942104	−	0.995552i	$-0.530033\pi$
$31$	−1.04908			−0.188421			−0.0942104	+	0.995552i	$0.530033\pi$
$32$			6.65259i			1.17602i
$33$		0			0
$34$	11.9136			2.04317
$35$	0.952011			0.160919
$36$		0			0
$37$			0.902591i			0.148385i	0.997244	+	0.0741925i	$0.0236379\pi$
$37$			0.902591i			0.148385i	−0.997244	+	0.0741925i	$0.976362\pi$
$38$	−11.4039			−1.84996
$39$		0			0
$40$	−0.555920			−0.0878986
$41$	−5.83617			−0.911456			−0.455728	−	0.890119i	$-0.650621\pi$
$41$	−5.83617			−0.911456			−0.455728	+	0.890119i	$0.650621\pi$
$42$		0			0
$43$		−	5.46817i		−	0.833889i	−0.908932	−	0.416944i	$-0.863101\pi$
$43$		−	5.46817i		−	0.833889i	0.908932	−	0.416944i	$-0.136899\pi$
$44$		−	5.72174i		−	0.862585i
$45$		0			0
$46$		−	13.4956i		−	1.98982i
$47$	9.41703			1.37361			0.686807	−	0.726839i	$-0.259012\pi$
$47$	9.41703			1.37361			0.686807	+	0.726839i	$0.259012\pi$
$48$		0			0
$49$	3.20814			0.458306
$50$		−	8.75510i		−	1.23816i
$51$		0			0
$52$			3.30584i			0.458437i
$53$	7.35874			1.01080			0.505400	−	0.862885i	$-0.331345\pi$
$53$	7.35874			1.01080			0.505400	+	0.862885i	$0.331345\pi$
$54$		0			0
$55$	2.02463			0.273001
$56$	2.21423			0.295889
$57$		0			0
$58$	8.55760			1.12367
$59$	12.8740			1.67606			0.838029	−	0.545626i	$-0.183708\pi$
$59$	12.8740			1.67606			0.838029	+	0.545626i	$0.183708\pi$
$60$		0			0
$61$	2.45943			0.314898			0.157449	−	0.987527i	$-0.449673\pi$
$61$	2.45943			0.314898			0.157449	+	0.987527i	$0.449673\pi$
$62$			1.92919i			0.245007i
$63$		0			0
$64$	2.52495			0.315619
$65$	−1.16976			−0.145091
$66$		0			0
$67$	−2.55500			−0.312143			−0.156071	−	0.987746i	$-0.549883\pi$
$67$	−2.55500			−0.312143			−0.156071	+	0.987746i	$0.549883\pi$
$68$		−	8.95114i		−	1.08549i
$69$		0			0
$70$		−	1.75068i		−	0.209246i
$71$		−	1.82546i		−	0.216642i	−0.994116	−	0.108321i	$-0.965453\pi$
$71$		−	1.82546i		−	0.216642i	0.994116	−	0.108321i	$-0.0345474\pi$
$72$		0			0
$73$		−	3.90000i		−	0.456461i	−0.973607	−	0.228230i	$-0.926706\pi$
$73$		−	3.90000i		−	0.456461i	0.973607	−	0.228230i	$-0.0732939\pi$
$74$	1.65980			0.192948
$75$		0			0
$76$			8.56818i			0.982838i
$77$	−8.06408			−0.918988
$78$		0			0
$79$			2.77679i			0.312414i	0.987724	+	0.156207i	$0.0499267\pi$
$79$			2.77679i			0.312414i	−0.987724	+	0.156207i	$0.950073\pi$
$80$			2.37326i			0.265339i
$81$		0			0
$82$			10.7323i			1.18518i
$83$			6.67645i			0.732836i	0.930450	+	0.366418i	$0.119416\pi$
$83$			6.67645i			0.732836i	−0.930450	+	0.366418i	$0.880584\pi$
$84$		0			0
$85$	3.16734			0.343547
$86$	−10.0556			−1.08432
$87$		0			0
$88$	4.70896			0.501977
$89$	4.44903			0.471596			0.235798	−	0.971802i	$-0.424230\pi$
$89$	4.44903			0.471596			0.235798	+	0.971802i	$0.424230\pi$
$90$		0			0
$91$	4.65916			0.488413
$92$	−10.1397			−1.05714
$93$		0			0
$94$		−	17.3172i		−	1.78614i
$95$	−3.03183			−0.311059
$96$		0			0
$97$			1.32217i			0.134246i	0.997745	+	0.0671228i	$0.0213819\pi$
$97$			1.32217i			0.134246i	−0.997745	+	0.0671228i	$0.978618\pi$
$98$		−	5.89954i		−	0.595944i
$99$		0			0
$100$	−6.57803			−0.657803
$101$		−	4.05044i		−	0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$101$		−	4.05044i		−	0.403034i	0.979485	−	0.201517i	$-0.0645871\pi$
$102$		0			0
$103$			13.8035i			1.36010i	0.733168	+	0.680048i	$0.238041\pi$
$103$			13.8035i			1.36010i	−0.733168	+	0.680048i	$0.761959\pi$
$104$	−2.72068			−0.266785
$105$		0			0
$106$		−	13.5322i		−	1.31436i
$107$	0.214562			0.0207425			0.0103712	−	0.999946i	$-0.496699\pi$
$107$	0.214562			0.0207425			0.0103712	+	0.999946i	$0.496699\pi$
$108$		0			0
$109$	4.45269			0.426490			0.213245	−	0.976999i	$-0.431597\pi$
$109$	4.45269			0.426490			0.213245	+	0.976999i	$0.431597\pi$
$110$		−	3.72314i		−	0.354988i
$111$		0			0
$112$		−	9.45271i		−	0.893197i
$113$		−	11.8408i		−	1.11389i	−0.830551	−	0.556943i	$-0.811974\pi$
$113$		−	11.8408i		−	1.11389i	0.830551	−	0.556943i	$-0.188026\pi$
$114$		0			0
$115$		−	3.58793i		−	0.334576i
$116$		−	6.42964i		−	0.596977i
$117$		0			0
$118$		−	23.6744i		−	2.17941i
$119$	−12.6155			−1.15646
$120$		0			0
$121$	−6.14977			−0.559070
$122$		−	4.52271i		−	0.409467i
$123$		0			0
$124$	1.44947			0.130166
$125$		−	4.77210i		−	0.426830i
$126$		0			0
$127$	14.3811			1.27611			0.638057	−	0.769989i	$-0.279738\pi$
$127$	14.3811			1.27611			0.638057	+	0.769989i	$0.279738\pi$
$128$			8.66198i			0.765618i
$129$		0			0
$130$			2.15111i			0.188665i
$131$	−14.2138			−1.24187			−0.620934	−	0.783863i	$-0.713247\pi$
$131$	−14.2138			−1.24187			−0.620934	+	0.783863i	$0.713247\pi$
$132$		0			0
$133$	12.0758			1.04710
$134$			4.69845i			0.405885i
$135$		0			0
$136$	7.36674			0.631693
$137$	20.8231			1.77904			0.889520	−	0.456897i	$-0.151039\pi$
$137$	20.8231			1.77904			0.889520	+	0.456897i	$0.151039\pi$
$138$		0			0
$139$		−	6.17483i		−	0.523742i	−0.965103	−	0.261871i	$-0.915660\pi$
$139$		−	6.17483i		−	0.523742i	0.965103	−	0.261871i	$-0.0843396\pi$
$140$	−1.31535			−0.111167
$141$		0			0
$142$	−3.35688			−0.281703
$143$	9.90857			0.828597
$144$		0			0
$145$	2.27512			0.188938
$146$	−7.17182			−0.593544
$147$		0			0
$148$		−	1.24707i		−	0.102508i
$149$	−5.00820			−0.410288			−0.205144	−	0.978732i	$-0.565766\pi$
$149$	−5.00820			−0.410288			−0.205144	+	0.978732i	$0.565766\pi$
$150$		0			0
$151$		−	11.7974i		−	0.960059i	−0.877252	−	0.480030i	$-0.840626\pi$
$151$		−	11.7974i		−	0.960059i	0.877252	−	0.480030i	$-0.159374\pi$
$152$	−7.05157			−0.571957
$153$		0			0
$154$			14.8293i			1.19498i
$155$			0.512892i			0.0411965i
$156$		0			0
$157$	−2.03923			−0.162748			−0.0813740	−	0.996684i	$-0.525931\pi$
$157$	−2.03923			−0.162748			−0.0813740	+	0.996684i	$0.525931\pi$
$158$	5.10632			0.406237
$159$		0			0
$160$	3.25242			0.257127
$161$			14.2907i			1.12627i
$162$		0			0
$163$	−6.08984			−0.476993			−0.238497	−	0.971143i	$-0.576655\pi$
$163$	−6.08984			−0.476993			−0.238497	+	0.971143i	$0.576655\pi$
$164$	8.06356			0.629658
$165$		0			0
$166$	12.2775			0.952919
$167$	−17.4413			−1.34965			−0.674825	−	0.737978i	$-0.735781\pi$
$167$	−17.4413			−1.34965			−0.674825	+	0.737978i	$0.735781\pi$
$168$		0			0
$169$	7.27515			0.559627
$170$		−	5.82451i		−	0.446720i
$171$		0			0
$172$			7.55512i			0.576073i
$173$	−24.5070			−1.86323			−0.931615	−	0.363447i	$-0.881600\pi$
$173$	−24.5070			−1.86323			−0.931615	+	0.363447i	$0.881600\pi$
$174$		0			0
$175$			9.27091i			0.700815i
$176$		−	20.1029i		−	1.51532i
$177$		0			0
$178$		−	8.18145i		−	0.613225i
$179$	4.71389			0.352332			0.176166	−	0.984360i	$-0.443630\pi$
$179$	4.71389			0.352332			0.176166	+	0.984360i	$0.443630\pi$
$180$		0			0
$181$		−	9.71270i		−	0.721939i	−0.932578	−	0.360970i	$-0.882446\pi$
$181$		−	9.71270i		−	0.721939i	0.932578	−	0.360970i	$-0.117554\pi$
$182$		−	8.56786i		−	0.635092i
$183$		0			0
$184$		−	8.34495i		−	0.615198i
$185$	0.441272			0.0324430
$186$		0			0
$187$	−26.8292			−1.96195
$188$	−13.0111			−0.948930
$189$		0			0
$190$			5.57532i			0.404476i
$191$	−16.3758			−1.18491			−0.592455	−	0.805603i	$-0.701841\pi$
$191$	−16.3758			−1.18491			−0.592455	+	0.805603i	$0.701841\pi$
$192$		0			0
$193$	−19.1214			−1.37639			−0.688193	−	0.725528i	$-0.741596\pi$
$193$	−19.1214			−1.37639			−0.688193	+	0.725528i	$0.741596\pi$
$194$	2.43137			0.174562
$195$		0			0
$196$	−4.43255			−0.316610
$197$			3.49678i			0.249136i	0.992211	+	0.124568i	$0.0397544\pi$
$197$			3.49678i			0.249136i	−0.992211	+	0.124568i	$0.960246\pi$
$198$		0			0
$199$			14.7722i			1.04717i	0.851973	+	0.523586i	$0.175406\pi$
$199$			14.7722i			1.04717i	−0.851973	+	0.523586i	$0.824594\pi$
$200$		−	5.41368i		−	0.382805i
$201$		0			0
$202$	−7.44846			−0.524072
$203$	−9.06178			−0.636012
$204$		0			0
$205$			2.85327i			0.199281i
$206$	25.3836			1.76856
$207$		0			0
$208$			11.6148i			0.805342i
$209$	25.6814			1.77642
$210$		0			0
$211$	−8.32039			−0.572799			−0.286400	−	0.958110i	$-0.592458\pi$
$211$	−8.32039			−0.572799			−0.286400	+	0.958110i	$0.592458\pi$
$212$	−10.1672			−0.698288
$213$		0			0
$214$		−	0.394563i		−	0.0269718i
$215$	−2.67337			−0.182322
$216$		0			0
$217$		−	2.04285i		−	0.138677i
$218$		−	8.18817i		−	0.554573i
$219$		0			0
$220$	−2.79733			−0.188596
$221$	15.5011			1.04271
$222$		0			0
$223$			4.02136i			0.269290i	0.990894	+	0.134645i	$0.0429895\pi$
$223$			4.02136i			0.269290i	−0.990894	+	0.134645i	$0.957011\pi$
$224$	−12.9544			−0.865551
$225$		0			0
$226$	−21.7743			−1.44841
$227$	−2.27179			−0.150784			−0.0753919	−	0.997154i	$-0.524021\pi$
$227$	−2.27179			−0.150784			−0.0753919	+	0.997154i	$0.524021\pi$
$228$		0			0
$229$			6.79380i			0.448947i	0.974480	+	0.224473i	$0.0720662\pi$
$229$			6.79380i			0.448947i	−0.974480	+	0.224473i	$0.927934\pi$
$230$	−6.59794			−0.435055
$231$		0			0
$232$	5.29156			0.347408
$233$	27.0573			1.77258			0.886292	−	0.463127i	$-0.153273\pi$
$233$	27.0573			1.77258			0.886292	+	0.463127i	$0.153273\pi$
$234$		0			0
$235$		−	4.60394i		−	0.300328i
$236$	−17.7875			−1.15787
$237$		0			0
$238$			23.1990i			1.50377i
$239$	10.0043	−	11.7861i	0.647127	−	0.762382i
$239$	10.0043	−	11.7861i	0.647127	−	0.762382i
$240$		0			0
$241$	23.3260			1.50256			0.751280	−	0.659983i	$-0.229437\pi$
$241$	23.3260			1.50256			0.751280	+	0.659983i	$0.229437\pi$
$242$			11.3090i			0.726969i
$243$		0			0
$244$	−3.39808			−0.217540
$245$		−	1.56845i		−	0.100204i
$246$		0			0
$247$	−14.8379			−0.944111
$248$			1.19291i			0.0757496i
$249$		0			0
$250$	−8.77554			−0.555014
$251$			28.1938i			1.77957i	0.456376	+	0.889787i	$0.349147\pi$
$251$			28.1938i			1.77957i	−0.456376	+	0.889787i	$0.650853\pi$
$252$		0			0
$253$			30.3918i			1.91072i
$254$		−	26.4457i		−	1.65935i
$255$		0			0
$256$	20.9787			1.31117
$257$		−	12.1086i		−	0.755312i	−0.925946	−	0.377656i	$-0.876730\pi$
$257$		−	12.1086i		−	0.755312i	0.925946	−	0.377656i	$-0.123270\pi$
$258$		0			0
$259$	−1.75759			−0.109211
$260$	1.61621			0.100233
$261$		0			0
$262$			26.1382i			1.61482i
$263$			18.4508i			1.13772i	0.822433	+	0.568862i	$0.192616\pi$
$263$			18.4508i			1.13772i	−0.822433	+	0.568862i	$0.807384\pi$
$264$		0			0
$265$		−	3.59765i		−	0.221002i
$266$		−	22.2065i		−	1.36157i
$267$		0			0
$268$	3.53012			0.215637
$269$			15.6532i			0.954390i	0.878797	+	0.477195i	$0.158347\pi$
$269$			15.6532i			0.954390i	−0.878797	+	0.477195i	$0.841653\pi$
$270$		0			0
$271$	0.770632			0.0468126			0.0234063	−	0.999726i	$-0.492549\pi$
$271$	0.770632			0.0468126			0.0234063	+	0.999726i	$0.492549\pi$
$272$		−	31.4492i		−	1.90689i
$273$		0			0
$274$		−	38.2922i		−	2.31332i
$275$			19.7163i			1.18894i
$276$		0			0
$277$		−	22.8190i		−	1.37106i	−0.728043	−	0.685531i	$-0.759570\pi$
$277$		−	22.8190i		−	1.37106i	0.728043	−	0.685531i	$-0.240430\pi$
$278$	−11.3551			−0.681032
$279$		0			0
$280$		−	1.08253i		−	0.0646933i
$281$	−27.5804			−1.64531			−0.822653	−	0.568544i	$-0.807507\pi$
$281$	−27.5804			−1.64531			−0.822653	+	0.568544i	$0.807507\pi$
$282$		0			0
$283$	−20.1539			−1.19803			−0.599013	−	0.800739i	$-0.704440\pi$
$283$	−20.1539			−1.19803			−0.599013	+	0.800739i	$0.704440\pi$
$284$			2.52215i			0.149662i
$285$		0			0
$286$		−	18.2211i		−	1.07744i
$287$		−	11.3646i		−	0.670830i
$288$		0			0
$289$	−24.9719			−1.46893
$290$		−	4.18377i		−	0.245680i
$291$		0			0
$292$			5.38845i			0.315335i
$293$			5.12838i			0.299603i	0.988716	+	0.149802i	$0.0478635\pi$
$293$			5.12838i			0.299603i	−0.988716	+	0.149802i	$0.952136\pi$
$294$		0			0
$295$		−	6.29406i		−	0.366454i
$296$	1.02633			0.0596542
$297$		0			0
$298$			9.20972i			0.533505i
$299$		−	17.5594i		−	1.01549i
$300$		0			0
$301$	10.6480			0.613741
$302$	−21.6946			−1.24838
$303$		0			0
$304$			30.1037i			1.72656i
$305$		−	1.20240i		−	0.0688495i
$306$		0			0
$307$	−1.42885			−0.0815487			−0.0407744	−	0.999168i	$-0.512982\pi$
$307$	−1.42885			−0.0815487			−0.0407744	+	0.999168i	$0.512982\pi$
$308$	11.1418			0.634861
$309$		0			0
$310$	0.943171			0.0535685
$311$			4.15310i			0.235501i	0.993043	+	0.117750i	$0.0375683\pi$
$311$			4.15310i			0.235501i	−0.993043	+	0.117750i	$0.962432\pi$
$312$		0			0
$313$			28.4813i			1.60986i	0.593372	+	0.804929i	$0.297796\pi$
$313$			28.4813i			1.60986i	−0.593372	+	0.804929i	$0.702204\pi$
$314$			3.74999i			0.211624i
$315$		0			0
$316$		−	3.83657i		−	0.215824i
$317$	8.22446			0.461932			0.230966	−	0.972962i	$-0.425811\pi$
$317$	8.22446			0.461932			0.230966	+	0.972962i	$0.425811\pi$
$318$		0			0
$319$	−19.2716			−1.07900
$320$		−	1.23444i		−	0.0690072i
$321$		0			0
$322$	26.2796			1.46450
$323$	40.1762			2.23546
$324$		0			0
$325$		−	11.3914i		−	0.631883i
$326$			11.1988i			0.620243i
$327$		0			0
$328$			6.63627i			0.366427i
$329$			18.3375i			1.01098i
$330$		0			0
$331$		−	21.1089i		−	1.16025i	−0.814528	−	0.580124i	$-0.803004\pi$
$331$		−	21.1089i		−	1.16025i	0.814528	−	0.580124i	$-0.196996\pi$
$332$		−	9.22454i		−	0.506263i
$333$		0			0
$334$			32.0733i			1.75497i
$335$			1.24913i			0.0682471i
$336$		0			0
$337$	−6.74368			−0.367352			−0.183676	−	0.982987i	$-0.558800\pi$
$337$	−6.74368			−0.367352			−0.183676	+	0.982987i	$0.558800\pi$
$338$		−	13.3785i		−	0.727693i
$339$		0			0
$340$	−4.37617			−0.237331
$341$		−	4.34449i		−	0.235267i
$342$		0			0
$343$			19.8780i			1.07331i
$344$	−6.21783			−0.335243
$345$		0			0
$346$			45.0665i			2.42279i
$347$		−	27.5805i		−	1.48060i	−0.672276	−	0.740300i	$-0.734683\pi$
$347$		−	27.5805i		−	1.48060i	0.672276	−	0.740300i	$-0.265317\pi$
$348$		0			0
$349$	22.6721			1.21361			0.606805	−	0.794851i	$-0.292451\pi$
$349$	22.6721			1.21361			0.606805	+	0.794851i	$0.292451\pi$
$350$	17.0485			0.911282
$351$		0			0
$352$	−27.5499			−1.46841
$353$	19.5022			1.03800			0.518999	−	0.854775i	$-0.326305\pi$
$353$	19.5022			1.03800			0.518999	+	0.854775i	$0.326305\pi$
$354$		0			0
$355$	−0.892457			−0.0473667
$356$	−6.14702			−0.325792
$357$		0			0
$358$		−	8.66849i		−	0.458144i
$359$			4.77398i			0.251961i	0.992033	+	0.125981i	$0.0402077\pi$
$359$			4.77398i			0.251961i	−0.992033	+	0.125981i	$0.959792\pi$
$360$		0			0
$361$	−19.4573			−1.02407
$362$	−17.8609			−0.938751
$363$		0			0
$364$	−6.43735			−0.337409
$365$	−1.90669			−0.0998009
$366$		0			0
$367$	−15.0489			−0.785544			−0.392772	−	0.919636i	$-0.628484\pi$
$367$	−15.0489			−0.785544			−0.392772	+	0.919636i	$0.628484\pi$
$368$	−35.6253			−1.85709
$369$		0			0
$370$		−	0.811468i		−	0.0421862i
$371$			14.3294i			0.743948i
$372$		0			0
$373$	−36.6470			−1.89751			−0.948756	−	0.316011i	$-0.897656\pi$
$373$	−36.6470			−1.89751			−0.948756	+	0.316011i	$0.897656\pi$
$374$			49.3370i			2.55116i
$375$		0			0
$376$		−	10.7080i		−	0.552225i
$377$	11.1345			0.573454
$378$		0			0
$379$			5.15699i			0.264897i	0.991190	+	0.132448i	$0.0422839\pi$
$379$			5.15699i			0.264897i	−0.991190	+	0.132448i	$0.957716\pi$
$380$	4.18894			0.214888
$381$		0			0
$382$			30.1139i			1.54076i
$383$			3.81829i			0.195105i	0.995230	+	0.0975527i	$0.0311015\pi$
$383$			3.81829i			0.195105i	−0.995230	+	0.0975527i	$0.968899\pi$
$384$		0			0
$385$			3.94249i			0.200928i
$386$			35.1628i			1.78974i
$387$		0			0
$388$		−	1.82678i		−	0.0927405i
$389$		−	24.3507i		−	1.23463i	−0.786716	−	0.617315i	$-0.788221\pi$
$389$		−	24.3507i		−	1.23463i	0.786716	−	0.617315i	$-0.211779\pi$
$390$		0			0
$391$			47.5452i			2.40447i
$392$		−	3.64796i		−	0.184250i
$393$		0			0
$394$	6.43033			0.323955
$395$	1.35756			0.0683064
$396$		0			0
$397$		−	26.3192i		−	1.32092i	−0.750861	−	0.660460i	$-0.770361\pi$
$397$		−	26.3192i		−	1.32092i	0.750861	−	0.660460i	$-0.229639\pi$
$398$	27.1649			1.36166
$399$		0			0
$400$	−23.1114			−1.15557
$401$			36.1240i			1.80395i	0.431792	+	0.901973i	$0.357881\pi$
$401$			36.1240i			1.80395i	−0.431792	+	0.901973i	$0.642119\pi$
$402$		0			0
$403$			2.51010i			0.125037i
$404$			5.59631i			0.278427i
$405$		0			0
$406$			16.6640i			0.827018i
$407$	−3.73783			−0.185278
$408$		0			0
$409$	33.7461			1.66864			0.834319	−	0.551282i	$-0.185861\pi$
$409$	33.7461			1.66864			0.834319	+	0.551282i	$0.185861\pi$
$410$	5.24696			0.259129
$411$		0			0
$412$		−	19.0716i		−	0.939591i
$413$			25.0692i			1.23358i
$414$		0			0
$415$	3.26409			0.160228
$416$	15.9174			0.780416
$417$		0			0
$418$		−	47.2262i		−	2.30991i
$419$		−	3.59885i		−	0.175815i	−0.996129	−	0.0879077i	$-0.971982\pi$
$419$		−	3.59885i		−	0.175815i	0.996129	−	0.0879077i	$-0.0280180\pi$
$420$		0			0
$421$	5.80299			0.282821			0.141410	−	0.989951i	$-0.454836\pi$
$421$	5.80299			0.282821			0.141410	+	0.989951i	$0.454836\pi$
$422$			15.3006i			0.744821i
$423$		0			0
$424$		−	8.36758i		−	0.406365i
$425$			30.8444i			1.49617i
$426$		0			0
$427$			4.78917i			0.231764i
$428$	−0.296450			−0.0143294
$429$		0			0
$430$			4.91612i			0.237077i
$431$			12.5877i			0.606326i	0.952939	+	0.303163i	$0.0980427\pi$
$431$			12.5877i			0.606326i	−0.952939	+	0.303163i	$0.901957\pi$
$432$		0			0
$433$		−	13.8107i		−	0.663698i	−0.943333	−	0.331849i	$-0.892328\pi$
$433$		−	13.8107i		−	0.663698i	0.943333	−	0.331849i	$-0.107672\pi$
$434$	−3.75665			−0.180325
$435$		0			0
$436$	−6.15208			−0.294631
$437$		−	45.5111i		−	2.17709i
$438$		0			0
$439$	−9.04406			−0.431649			−0.215825	−	0.976432i	$-0.569244\pi$
$439$	−9.04406			−0.431649			−0.215825	+	0.976432i	$0.569244\pi$
$440$		−	2.30219i		−	0.109753i
$441$		0			0
$442$		−	28.5053i		−	1.35586i
$443$			23.7036i			1.12619i	0.826392	+	0.563095i	$0.190389\pi$
$443$			23.7036i			1.12619i	−0.826392	+	0.563095i	$0.809611\pi$
$444$		0			0
$445$		−	2.17511i		−	0.103110i
$446$	7.39499			0.350163
$447$		0			0
$448$			4.91676i			0.232295i
$449$	9.20499			0.434410			0.217205	−	0.976126i	$-0.430306\pi$
$449$	9.20499			0.434410			0.217205	+	0.976126i	$0.430306\pi$
$450$		0			0
$451$		−	24.1689i		−	1.13807i
$452$			16.3598i			0.769502i
$453$		0			0
$454$			4.17765i			0.196067i
$455$		−	2.27784i		−	0.106787i
$456$		0			0
$457$	23.6202			1.10491			0.552453	−	0.833544i	$-0.313692\pi$
$457$	23.6202			1.10491			0.552453	+	0.833544i	$0.313692\pi$
$458$	12.4933			0.583774
$459$		0			0
$460$			4.95727i			0.231134i
$461$	12.4625			0.580437			0.290218	−	0.956960i	$-0.406272\pi$
$461$	12.4625			0.580437			0.290218	+	0.956960i	$0.406272\pi$
$462$		0			0
$463$			7.38980i			0.343433i	0.985146	+	0.171717i	$0.0549313\pi$
$463$			7.38980i			0.343433i	−0.985146	+	0.171717i	$0.945069\pi$
$464$		−	22.5901i		−	1.04872i
$465$		0			0
$466$		−	49.7564i		−	2.30492i
$467$	−13.5619			−0.627570			−0.313785	−	0.949494i	$-0.601597\pi$
$467$	−13.5619			−0.627570			−0.313785	+	0.949494i	$0.601597\pi$
$468$		0			0
$469$		−	4.97527i		−	0.229736i
$470$	−8.46631			−0.390522
$471$		0			0
$472$		−	14.6390i		−	0.673814i
$473$	22.6450			1.04122
$474$		0			0
$475$		−	29.5247i		−	1.35469i
$476$	17.4303			0.798916
$477$		0			0
$478$	−21.6739	−	18.3973i	−0.991339	−	0.841471i
$479$			40.2336i			1.83832i	0.393882	+	0.919161i	$0.371132\pi$
$479$			40.2336i			1.83832i	−0.393882	+	0.919161i	$0.628868\pi$
$480$		0			0
$481$	2.15960			0.0984692
$482$		−	42.8948i		−	1.95381i
$483$		0			0
$484$	8.49686			0.386221
$485$	0.646401			0.0293515
$486$		0			0
$487$	35.6679			1.61627			0.808133	−	0.589000i	$-0.200478\pi$
$487$	35.6679			1.61627			0.808133	+	0.589000i	$0.200478\pi$
$488$		−	2.79660i		−	0.126596i
$489$		0			0
$490$	−2.88426			−0.130298
$491$	−0.519007			−0.0234225			−0.0117112	−	0.999931i	$-0.503728\pi$
$491$	−0.519007			−0.0234225			−0.0117112	+	0.999931i	$0.503728\pi$
$492$		0			0
$493$	−30.1486			−1.35782
$494$			27.2857i			1.22764i
$495$		0			0
$496$	5.09261			0.228665
$497$	3.55465			0.159448
$498$		0			0
$499$			34.4551i			1.54242i	0.636578	+	0.771212i	$0.280349\pi$
$499$			34.4551i			1.54242i	−0.636578	+	0.771212i	$0.719651\pi$
$500$			6.59339i			0.294865i
$501$		0			0
$502$	51.8463			2.31401
$503$		−	33.1063i		−	1.47614i	−0.674726	−	0.738068i	$-0.735738\pi$
$503$		−	33.1063i		−	1.47614i	0.674726	−	0.738068i	$-0.264262\pi$
$504$		0			0
$505$	−1.98024			−0.0881196
$506$	55.8884			2.48454
$507$		0			0
$508$	−19.8697			−0.881574
$509$		−	19.2548i		−	0.853456i	−0.904380	−	0.426728i	$-0.859666\pi$
$509$		−	19.2548i		−	0.853456i	0.904380	−	0.426728i	$-0.140334\pi$
$510$		0			0
$511$	7.59435			0.335954
$512$		−	21.2543i		−	0.939315i
$513$		0			0
$514$	−22.2668			−0.982146
$515$	6.74845			0.297372
$516$		0			0
$517$			38.9981i			1.71513i
$518$			3.23207i			0.142009i
$519$		0			0
$520$			1.33013i			0.0583301i
$521$	−6.25534			−0.274051			−0.137026	−	0.990567i	$-0.543754\pi$
$521$	−6.25534			−0.274051			−0.137026	+	0.990567i	$0.543754\pi$
$522$		0			0
$523$	−12.9905			−0.568037			−0.284018	−	0.958819i	$-0.591668\pi$
$523$	−12.9905			−0.568037			−0.284018	+	0.958819i	$0.591668\pi$
$524$	19.6386			0.857916
$525$		0			0
$526$	33.9297			1.47940
$527$		−	6.79656i		−	0.296063i
$528$		0			0
$529$	30.8586			1.34168
$530$	−6.61582			−0.287373
$531$		0			0
$532$	−16.6845			−0.723367
$533$			13.9640i			0.604848i
$534$		0			0
$535$		−	0.104898i		−	0.00453514i
$536$			2.90527i			0.125489i
$537$		0			0
$538$	28.7850			1.24101
$539$			13.2857i			0.572254i
$540$		0			0
$541$		−	25.5388i		−	1.09800i	−0.835823	−	0.548999i	$-0.815009\pi$
$541$		−	25.5388i		−	1.09800i	0.835823	−	0.548999i	$-0.184991\pi$
$542$		−	1.41714i		−	0.0608712i
$543$		0			0
$544$	−43.0993			−1.84787
$545$		−	2.17690i		−	0.0932481i
$546$		0			0
$547$			19.5619i			0.836408i	0.908353	+	0.418204i	$0.137340\pi$
$547$			19.5619i			0.836408i	−0.908353	+	0.418204i	$0.862660\pi$
$548$	−28.7703			−1.22901
$549$		0			0
$550$	36.2569			1.54600
$551$	28.8587			1.22942
$552$		0			0
$553$	−5.40717			−0.229936
$554$	−41.9625			−1.78282
$555$		0			0
$556$			8.53148i			0.361815i
$557$	−16.0002			−0.677952			−0.338976	−	0.940795i	$-0.610081\pi$
$557$	−16.0002			−0.677952			−0.338976	+	0.940795i	$0.610081\pi$
$558$		0			0
$559$	−13.0835			−0.553374
$560$	−4.62138			−0.195289
$561$		0			0
$562$			50.7183i			2.13942i
$563$			41.3543i			1.74288i	0.490505	+	0.871438i	$0.336812\pi$
$563$			41.3543i			1.74288i	−0.490505	+	0.871438i	$0.663188\pi$
$564$		0			0
$565$	−5.78890			−0.243541
$566$			37.0616i			1.55781i
$567$		0			0
$568$	−2.07571			−0.0870951
$569$			12.0592i			0.505547i	0.967526	+	0.252773i	$0.0813427\pi$
$569$			12.0592i			0.505547i	−0.967526	+	0.252773i	$0.918657\pi$
$570$		0			0
$571$	−20.8156			−0.871104			−0.435552	−	0.900164i	$-0.643447\pi$
$571$	−20.8156			−0.871104			−0.435552	+	0.900164i	$0.643447\pi$
$572$	−13.6902			−0.572417
$573$		0			0
$574$	−20.8986			−0.872292
$575$	34.9401			1.45710
$576$		0			0
$577$	15.9704			0.664856			0.332428	−	0.943129i	$-0.392132\pi$
$577$	15.9704			0.664856			0.332428	+	0.943129i	$0.392132\pi$
$578$			45.9215i			1.91008i
$579$		0			0
$580$	−3.14342			−0.130524
$581$	−13.0008			−0.539366
$582$		0			0
$583$			30.4742i			1.26211i
$584$	−4.43467			−0.183508
$585$		0			0
$586$	9.43072			0.389580
$587$			21.7500i			0.897718i	0.893603	+	0.448859i	$0.148170\pi$
$587$			21.7500i			0.897718i	−0.893603	+	0.448859i	$0.851830\pi$
$588$		0			0
$589$			6.50578i			0.268066i
$590$	−11.5743			−0.476507
$591$		0			0
$592$		−	4.38148i		−	0.180078i
$593$	−30.1967			−1.24003			−0.620015	−	0.784590i	$-0.712873\pi$
$593$	−30.1967			−1.24003			−0.620015	+	0.784590i	$0.712873\pi$
$594$		0			0
$595$			6.16767i			0.252850i
$596$	6.91960			0.283438
$597$		0			0
$598$	−32.2905			−1.32046
$599$		−	4.45634i		−	0.182081i	−0.995847	−	0.0910406i	$-0.970981\pi$
$599$		−	4.45634i		−	0.182081i	0.995847	−	0.0910406i	$-0.0290193\pi$
$600$		0			0
$601$			39.4164i			1.60783i	0.594746	+	0.803914i	$0.297253\pi$
$601$			39.4164i			1.60783i	−0.594746	+	0.803914i	$0.702747\pi$
$602$		−	19.5809i		−	0.798058i
$603$		0			0
$604$			16.2999i			0.663235i
$605$			3.00660i			0.122235i
$606$		0			0
$607$		−	22.0326i		−	0.894274i	−0.894465	−	0.447137i	$-0.852444\pi$
$607$		−	22.0326i		−	0.894274i	0.894465	−	0.447137i	$-0.147556\pi$
$608$	41.2553			1.67313
$609$		0			0
$610$	−2.21113			−0.0895262
$611$		−	22.5318i		−	0.911539i
$612$		0			0
$613$	19.0297			0.768601			0.384300	−	0.923208i	$-0.374443\pi$
$613$	19.0297			0.768601			0.384300	+	0.923208i	$0.374443\pi$
$614$			2.62755i			0.106039i
$615$		0			0
$616$			9.16962i			0.369454i
$617$	−20.6059			−0.829561			−0.414781	−	0.909921i	$-0.636142\pi$
$617$	−20.6059			−0.829561			−0.414781	+	0.909921i	$0.636142\pi$
$618$		0			0
$619$		−	19.8000i		−	0.795831i	−0.917422	−	0.397915i	$-0.869734\pi$
$619$		−	19.8000i		−	0.795831i	0.917422	−	0.397915i	$-0.130266\pi$
$620$		−	0.708639i		−	0.0284596i
$621$		0			0
$622$	7.63726			0.306226
$623$			8.66346i			0.347094i
$624$		0			0
$625$	21.4718			0.858874
$626$	52.3750			2.09333
$627$		0			0
$628$	2.81751			0.112431
$629$	−5.84750			−0.233155
$630$		0			0
$631$	−11.1945			−0.445644			−0.222822	−	0.974859i	$-0.571527\pi$
$631$	−11.1945			−0.445644			−0.222822	+	0.974859i	$0.571527\pi$
$632$	3.15748			0.125598
$633$		0			0
$634$		−	15.1242i		−	0.600658i
$635$		−	7.03084i		−	0.279010i
$636$		0			0
$637$		−	7.67602i		−	0.304135i
$638$			35.4390i			1.40304i
$639$		0			0
$640$	4.23480			0.167395
$641$		−	23.9514i		−	0.946024i	−0.881056	−	0.473012i	$-0.843167\pi$
$641$		−	23.9514i		−	0.946024i	0.881056	−	0.473012i	$-0.156833\pi$
$642$		0			0
$643$	−16.7338			−0.659915			−0.329958	−	0.943996i	$-0.607034\pi$
$643$	−16.7338			−0.659915			−0.329958	+	0.943996i	$0.607034\pi$
$644$		−	19.7448i		−	0.778054i
$645$		0			0
$646$		−	73.8811i		−	2.90681i
$647$		−	19.4085i		−	0.763029i	−0.924363	−	0.381514i	$-0.875403\pi$
$647$		−	19.4085i		−	0.763029i	0.924363	−	0.381514i	$-0.124597\pi$
$648$		0			0
$649$			53.3143i			2.09277i
$650$	−20.9480			−0.821649
$651$		0			0
$652$	8.41406			0.329520
$653$	−21.0442			−0.823523			−0.411762	−	0.911292i	$-0.635086\pi$
$653$	−21.0442			−0.823523			−0.411762	+	0.911292i	$0.635086\pi$
$654$		0			0
$655$			6.94908i			0.271523i
$656$	28.3307			1.10613
$657$		0			0
$658$	33.7213			1.31459
$659$	2.30634			0.0898424			0.0449212	−	0.998991i	$-0.485696\pi$
$659$	2.30634			0.0898424			0.0449212	+	0.998991i	$0.485696\pi$
$660$		0			0
$661$	−17.0752			−0.664147			−0.332074	−	0.943254i	$-0.607748\pi$
$661$	−17.0752			−0.664147			−0.332074	+	0.943254i	$0.607748\pi$
$662$	−38.8177			−1.50869
$663$		0			0
$664$	7.59175			0.294617
$665$		−	5.90379i		−	0.228939i
$666$		0			0
$667$			34.1519i			1.32237i
$668$	24.0979			0.932374
$669$		0			0
$670$	2.29705			0.0887429
$671$			10.1851i			0.393190i
$672$		0			0
$673$		−	47.1755i		−	1.81848i	−0.416272	−	0.909240i	$-0.636664\pi$
$673$		−	47.1755i		−	1.81848i	0.416272	−	0.909240i	$-0.363336\pi$
$674$			12.4011i			0.477674i
$675$		0			0
$676$	−10.0517			−0.386605
$677$	−5.63750			−0.216667			−0.108333	−	0.994115i	$-0.534551\pi$
$677$	−5.63750			−0.216667			−0.108333	+	0.994115i	$0.534551\pi$
$678$		0			0
$679$	−2.57461			−0.0988045
$680$		−	3.60157i		−	0.138114i
$681$		0			0
$682$	−7.98920			−0.305922
$683$	33.7921			1.29302			0.646509	−	0.762907i	$-0.276228\pi$
$683$	33.7921			1.29302			0.646509	+	0.762907i	$0.276228\pi$
$684$		0			0
$685$		−	10.1803i		−	0.388970i
$686$	36.5542			1.39565
$687$		0			0
$688$			26.5444i			1.01199i
$689$		−	17.6070i		−	0.670773i
$690$		0			0
$691$	0.398249			0.0151501			0.00757505	−	0.999971i	$-0.497589\pi$
$691$	0.398249			0.0151501			0.00757505	+	0.999971i	$0.497589\pi$
$692$	33.8601			1.28717
$693$		0			0
$694$	−50.7186			−1.92525
$695$	−3.01885			−0.114511
$696$		0			0
$697$		−	37.8100i		−	1.43216i
$698$		−	41.6923i		−	1.57808i
$699$		0			0
$700$		−	12.8092i		−	0.484142i
$701$	13.6963			0.517304			0.258652	−	0.965971i	$-0.416722\pi$
$701$	13.6963			0.517304			0.258652	+	0.965971i	$0.416722\pi$
$702$		0			0
$703$	5.59732			0.211107
$704$			10.4564i			0.394091i
$705$		0			0
$706$		−	35.8631i		−	1.34973i
$707$	7.88729			0.296632
$708$		0			0
$709$		−	5.75043i		−	0.215962i	−0.994153	−	0.107981i	$-0.965561\pi$
$709$		−	5.75043i		−	0.215962i	0.994153	−	0.107981i	$-0.0344385\pi$
$710$			1.64116i			0.0615918i
$711$		0			0
$712$		−	5.05897i		−	0.189593i
$713$	−7.69906			−0.288332
$714$		0			0
$715$		−	4.84426i		−	0.181165i
$716$	−6.51296			−0.243401
$717$		0			0
$718$	8.77900			0.327629
$719$			20.2931i			0.756806i	0.925641	+	0.378403i	$0.123527\pi$
$719$			20.2931i			0.756806i	−0.925641	+	0.378403i	$0.876473\pi$
$720$		0			0
$721$	−26.8791			−1.00103
$722$			35.7805i			1.33161i
$723$		0			0
$724$			13.4196i			0.498735i
$725$			22.1556i			0.822839i
$726$		0			0
$727$	−33.3882			−1.23830			−0.619150	−	0.785273i	$-0.712523\pi$
$727$	−33.3882			−1.23830			−0.619150	+	0.785273i	$0.712523\pi$
$728$		−	5.29790i		−	0.196353i
$729$		0			0
$730$			3.50627i			0.129773i
$731$	35.4259			1.31028
$732$		0			0
$733$	−23.0228			−0.850365			−0.425183	−	0.905108i	$-0.639790\pi$
$733$	−23.0228			−0.850365			−0.425183	+	0.905108i	$0.639790\pi$
$734$			27.6738i			1.02146i
$735$		0			0
$736$			48.8223i			1.79962i
$737$		−	10.5808i		−	0.389750i
$738$		0			0
$739$	43.3706			1.59541			0.797706	−	0.603046i	$-0.206046\pi$
$739$	43.3706			1.59541			0.797706	+	0.603046i	$0.206046\pi$
$740$	−0.609686			−0.0224125
$741$		0			0
$742$	26.3508			0.967369
$743$	−32.3767			−1.18779			−0.593893	−	0.804544i	$-0.702410\pi$
$743$	−32.3767			−1.18779			−0.593893	+	0.804544i	$0.702410\pi$
$744$		0			0
$745$			2.44849i			0.0897056i
$746$			67.3912i			2.46737i
$747$		0			0
$748$	37.0687			1.35537
$749$			0.417809i			0.0152664i
$750$		0			0
$751$	20.1318			0.734621			0.367310	−	0.930098i	$-0.380279\pi$
$751$	20.1318			0.734621			0.367310	+	0.930098i	$0.380279\pi$
$752$	−45.7135			−1.66700
$753$		0			0
$754$		−	20.4755i		−	0.745673i
$755$	−5.76770			−0.209908
$756$		0			0
$757$	−21.9291			−0.797026			−0.398513	−	0.917163i	$-0.630474\pi$
$757$	−21.9291			−0.797026			−0.398513	+	0.917163i	$0.630474\pi$
$758$	9.48333			0.344450
$759$		0			0
$760$			3.44748i			0.125053i
$761$		−	32.9216i		−	1.19341i	−0.802462	−	0.596703i	$-0.796477\pi$
$761$		−	32.9216i		−	1.19341i	0.802462	−	0.596703i	$-0.203523\pi$
$762$		0			0
$763$			8.67058i			0.313896i
$764$	22.6257			0.818568
$765$		0			0
$766$	7.02156			0.253699
$767$		−	30.8033i		−	1.11224i
$768$		0			0
$769$		−	27.7852i		−	1.00196i	−0.865459	−	0.500980i	$-0.832973\pi$
$769$		−	27.7852i		−	1.00196i	0.865459	−	0.500980i	$-0.167027\pi$
$770$	7.24996			0.261270
$771$		0			0
$772$	26.4191			0.950845
$773$	−54.6049			−1.96400			−0.982001	−	0.188878i	$-0.939515\pi$
$773$	−54.6049			−1.96400			−0.982001	+	0.188878i	$0.939515\pi$
$774$		0			0
$775$	−4.99466			−0.179414
$776$	1.50343			0.0539698
$777$		0			0
$778$	−44.7792			−1.60541
$779$			36.1924i			1.29673i
$780$		0			0
$781$	7.55963			0.270505
$782$	87.4322			3.12657
$783$		0			0
$784$	−15.5734			−0.556194
$785$			0.996969i			0.0355834i
$786$		0			0
$787$			0.825260i			0.0294173i	0.999892	+	0.0147087i	$0.00468208\pi$
$787$			0.825260i			0.0294173i	−0.999892	+	0.0147087i	$0.995318\pi$
$788$		−	4.83134i		−	0.172110i
$789$		0			0
$790$		−	2.49646i		−	0.0888200i
$791$	23.0572			0.819818
$792$		0			0
$793$		−	5.88459i		−	0.208968i
$794$	−48.3990			−1.71762
$795$		0			0
$796$		−	20.4100i		−	0.723414i
$797$		−	50.2896i		−	1.78135i	−0.454640	−	0.890675i	$-0.650232\pi$
$797$		−	50.2896i		−	1.78135i	0.454640	−	0.890675i	$-0.349768\pi$
$798$		0			0
$799$			61.0089i			2.15834i
$800$			31.6729i			1.11980i
$801$		0			0
$802$	66.4294			2.34570
$803$	16.1508			0.569949
$804$		0			0
$805$	6.98666			0.246247
$806$	4.61590			0.162588
$807$		0			0
$808$	−4.60573			−0.162029
$809$	−16.0151			−0.563060			−0.281530	−	0.959552i	$-0.590842\pi$
$809$	−16.0151			−0.563060			−0.281530	+	0.959552i	$0.590842\pi$
$810$		0			0
$811$		−	14.4262i		−	0.506572i	−0.967391	−	0.253286i	$-0.918489\pi$
$811$		−	14.4262i		−	0.506572i	0.967391	−	0.253286i	$-0.0815114\pi$
$812$	12.5202			0.439374
$813$		0			0
$814$			6.87361i			0.240920i
$815$			2.97730i			0.104290i
$816$		0			0
$817$	−33.9103			−1.18637
$818$		−	62.0567i		−	2.16976i
$819$		0			0
$820$		−	3.94224i		−	0.137669i
$821$	34.1562			1.19206			0.596031	−	0.802962i	$-0.296744\pi$
$821$	34.1562			1.19206			0.596031	+	0.802962i	$0.296744\pi$
$822$		0			0
$823$			14.6203i			0.509632i	0.966990	+	0.254816i	$0.0820149\pi$
$823$			14.6203i			0.509632i	−0.966990	+	0.254816i	$0.917985\pi$
$824$	15.6958			0.546790
$825$		0			0
$826$	46.1005			1.60404
$827$			34.8131i			1.21057i	0.796009	+	0.605285i	$0.206941\pi$
$827$			34.8131i			1.21057i	−0.796009	+	0.605285i	$0.793059\pi$
$828$		0			0
$829$		−	7.01522i		−	0.243649i	−0.992552	−	0.121824i	$-0.961126\pi$
$829$		−	7.01522i		−	0.243649i	0.992552	−	0.121824i	$-0.0388745\pi$
$830$		−	6.00242i		−	0.208347i
$831$		0			0
$832$		−	6.04137i		−	0.209447i
$833$			20.7842i			0.720129i
$834$		0			0
$835$			8.52698i			0.295088i
$836$	−35.4828			−1.22720
$837$		0			0
$838$	−6.61802			−0.228616
$839$		−	47.6004i		−	1.64335i	−0.569957	−	0.821675i	$-0.693040\pi$
$839$		−	47.6004i		−	1.64335i	0.569957	−	0.821675i	$-0.306960\pi$
$840$		0			0
$841$	7.34415			0.253247
$842$		−	10.6713i		−	0.367757i
$843$		0			0
$844$	11.4959			0.395705
$845$		−	3.55679i		−	0.122357i
$846$		0			0
$847$		−	11.9753i		−	0.411475i
$848$	−35.7218			−1.22669
$849$		0			0
$850$	56.7205			1.94550
$851$			6.62397i			0.227067i
$852$		0			0
$853$	19.5911			0.670786			0.335393	−	0.942078i	$-0.391131\pi$
$853$	19.5911			0.670786			0.335393	+	0.942078i	$0.391131\pi$
$854$	8.80694			0.301367
$855$		0			0
$856$		−	0.243977i		−	0.00833895i
$857$	4.88591			0.166900			0.0834498	−	0.996512i	$-0.473406\pi$
$857$	4.88591			0.166900			0.0834498	+	0.996512i	$0.473406\pi$
$858$		0			0
$859$	−35.5629			−1.21339			−0.606695	−	0.794935i	$-0.707505\pi$
$859$	−35.5629			−1.21339			−0.606695	+	0.794935i	$0.707505\pi$
$860$	3.69367			0.125953
$861$		0			0
$862$	23.1478			0.788417
$863$	7.34275			0.249950			0.124975	−	0.992160i	$-0.460115\pi$
$863$	7.34275			0.249950			0.124975	+	0.992160i	$0.460115\pi$
$864$		0			0
$865$			11.9813i			0.407378i
$866$	−25.3968			−0.863019
$867$		0			0
$868$			2.82251i			0.0958021i
$869$	−11.4993			−0.390088
$870$		0			0
$871$			6.11325i			0.207140i
$872$		−	5.06312i		−	0.171459i
$873$		0			0
$874$	−83.6916			−2.83091
$875$	9.29256			0.314146
$876$		0			0
$877$	18.2889			0.617571			0.308786	−	0.951132i	$-0.400077\pi$
$877$	18.2889			0.617571			0.308786	+	0.951132i	$0.400077\pi$
$878$			16.6314i			0.561281i
$879$		0			0
$880$	−9.82823			−0.331310
$881$	−46.3106			−1.56024			−0.780122	−	0.625627i	$-0.784843\pi$
$881$	−46.3106			−1.56024			−0.780122	+	0.625627i	$0.784843\pi$
$882$		0			0
$883$	8.65568			0.291287			0.145643	−	0.989337i	$-0.453475\pi$
$883$	8.65568			0.291287			0.145643	+	0.989337i	$0.453475\pi$
$884$	−21.4171			−0.720335
$885$		0			0
$886$	43.5891			1.46440
$887$		−	25.8063i		−	0.866489i	−0.901276	−	0.433245i	$-0.857369\pi$
$887$		−	25.8063i		−	0.866489i	0.901276	−	0.433245i	$-0.142631\pi$
$888$		0			0
$889$			28.0038i			0.939218i
$890$	−3.99987			−0.134076
$891$		0			0
$892$		−	5.55613i		−	0.186033i
$893$		−	58.3987i		−	1.95424i
$894$		0			0
$895$		−	2.30460i		−	0.0770342i
$896$	−16.8672			−0.563494
$897$		0			0
$898$		−	16.9273i		−	0.564872i
$899$		−	4.88200i		−	0.162824i
$900$		0			0
$901$			47.6741i			1.58825i
$902$	−44.4448			−1.47985
$903$		0			0
$904$	−13.4641			−0.447808
$905$	−4.74849			−0.157845
$906$		0			0
$907$		−	23.7846i		−	0.789756i	−0.918734	−	0.394878i	$-0.870787\pi$
$907$		−	23.7846i		−	0.789756i	0.918734	−	0.394878i	$-0.129213\pi$
$908$	3.13882			0.104166
$909$		0			0
$910$	−4.18879			−0.138857
$911$	26.5576			0.879893			0.439947	−	0.898024i	$-0.354997\pi$
$911$	26.5576			0.879893			0.439947	+	0.898024i	$0.354997\pi$
$912$		0			0
$913$	−27.6487			−0.915039
$914$		−	43.4358i		−	1.43673i
$915$		0			0
$916$		−	9.38668i		−	0.310145i
$917$		−	27.6781i		−	0.914013i
$918$		0			0
$919$	−22.1678			−0.731249			−0.365625	−	0.930762i	$-0.619145\pi$
$919$	−22.1678			−0.731249			−0.365625	+	0.930762i	$0.619145\pi$
$920$	−4.07981			−0.134507
$921$		0			0
$922$		−	22.9176i		−	0.754752i
$923$	−4.36771			−0.143765
$924$		0			0
$925$			4.29722i			0.141292i
$926$	13.5893			0.446572
$927$		0			0
$928$	−30.9584			−1.01626
$929$	6.65602			0.218377			0.109189	−	0.994021i	$-0.465175\pi$
$929$	6.65602			0.218377			0.109189	+	0.994021i	$0.465175\pi$
$930$		0			0
$931$		−	19.8950i		−	0.652031i
$932$	−37.3838			−1.22455
$933$		0			0
$934$			24.9393i			0.816040i
$935$			13.1167i			0.428962i
$936$		0			0
$937$	−15.7406			−0.514221			−0.257111	−	0.966382i	$-0.582771\pi$
$937$	−15.7406			−0.514221			−0.257111	+	0.966382i	$0.582771\pi$
$938$	−9.14915			−0.298730
$939$		0			0
$940$			6.36105i			0.207475i
$941$	−8.12923			−0.265005			−0.132503	−	0.991183i	$-0.542301\pi$
$941$	−8.12923			−0.265005			−0.132503	+	0.991183i	$0.542301\pi$
$942$		0			0
$943$	−42.8307			−1.39476
$944$	−62.4950			−2.03404
$945$		0			0
$946$		−	41.6424i		−	1.35391i
$947$	−53.3737			−1.73441			−0.867206	−	0.497949i	$-0.834087\pi$
$947$	−53.3737			−1.73441			−0.867206	+	0.497949i	$0.834087\pi$
$948$		0			0
$949$	−9.33140			−0.302910
$950$	−54.2938			−1.76152
$951$		0			0
$952$			14.3450i			0.464925i
$953$	20.5499			0.665678			0.332839	−	0.942984i	$-0.391993\pi$
$953$	20.5499			0.665678			0.332839	+	0.942984i	$0.391993\pi$
$954$		0			0
$955$			8.00605i			0.259070i
$956$	−13.8225	+	16.2844i	−0.447053	+	0.526674i
$957$		0			0
$958$	73.9867			2.39040
$959$			40.5482i			1.30937i
$960$		0			0
$961$	−29.8994			−0.964498
$962$		−	3.97134i		−	0.128041i
$963$		0			0
$964$	−32.2285			−1.03801
$965$			9.34835i			0.300934i
$966$		0			0
$967$	−48.7684			−1.56829			−0.784143	−	0.620580i	$-0.786897\pi$
$967$	−48.7684			−1.56829			−0.784143	+	0.620580i	$0.786897\pi$
$968$			6.99287i			0.224759i
$969$		0			0
$970$		−	1.18868i		−	0.0381663i
$971$		−	45.8586i		−	1.47167i	−0.677159	−	0.735836i	$-0.736789\pi$
$971$		−	45.8586i		−	1.47167i	0.677159	−	0.735836i	$-0.263211\pi$
$972$		0			0
$973$	12.0241			0.385474
$974$		−	65.5906i		−	2.10166i
$975$		0			0
$976$	−11.9389			−0.382155
$977$	−45.2019			−1.44614			−0.723069	−	0.690776i	$-0.757269\pi$
$977$	−45.2019			−1.44614			−0.723069	+	0.690776i	$0.757269\pi$
$978$		0			0
$979$			18.4245i			0.588848i
$980$			2.16705i			0.0692239i
$981$		0			0
$982$			0.954417i			0.0304567i
$983$		−	45.9605i		−	1.46591i	−0.680276	−	0.732956i	$-0.738140\pi$
$983$		−	45.9605i		−	1.46591i	0.680276	−	0.732956i	$-0.261860\pi$
$984$		0			0
$985$	1.70956			0.0544712
$986$			55.4410i			1.76560i
$987$		0			0
$988$	20.5008			0.652217
$989$		−	40.1301i		−	1.27606i
$990$		0			0
$991$		−	19.0315i		−	0.604557i	−0.953220	−	0.302278i	$-0.902253\pi$
$991$		−	19.0315i		−	0.604557i	0.953220	−	0.302278i	$-0.0977472\pi$
$992$		−	6.97912i		−	0.221587i
$993$		0			0
$994$		−	6.53675i		−	0.207333i
$995$	7.22205			0.228954
$996$		0			0
$997$		−	55.8685i		−	1.76937i	−0.466188	−	0.884686i	$-0.654373\pi$
$997$		−	55.8685i		−	1.76937i	0.466188	−	0.884686i	$-0.345627\pi$
$998$	63.3605			2.00564
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.b.a.2150.16	yes	80
3.2	odd	2	inner	2151.2.b.a.2150.66	yes	80
239.238	odd	2	inner	2151.2.b.a.2150.15	✓	80
717.716	even	2	inner	2151.2.b.a.2150.65	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.b.a.2150.15	✓	80	239.238	odd	2	inner
2151.2.b.a.2150.16	yes	80	1.1	even	1	trivial
2151.2.b.a.2150.65	yes	80	717.716	even	2	inner
2151.2.b.a.2150.66	yes	80	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-1.83893i q^{2} -1.38165 q^{4} -0.488895i q^{5} +1.94727i q^{7} -1.13709i q^{8} +O(q^{10})\)	\(q-1.83893i q^{2} -1.38165 q^{4} -0.488895i q^{5} +1.94727i q^{7} -1.13709i q^{8} -0.899043 q^{10} +4.14123i q^{11} -2.39267i q^{13} +3.58089 q^{14} -4.85434 q^{16} +6.47857i q^{17} -6.20139i q^{19} +0.675484i q^{20} +7.61542 q^{22} +7.33885 q^{23} +4.76098 q^{25} -4.39994 q^{26} -2.69045i q^{28} +4.65358i q^{29} -1.04908 q^{31} +6.65259i q^{32} +11.9136 q^{34} +0.952011 q^{35} +0.902591i q^{37} -11.4039 q^{38} -0.555920 q^{40} -5.83617 q^{41} -5.46817i q^{43} -5.72174i q^{44} -13.4956i q^{46} +9.41703 q^{47} +3.20814 q^{49} -8.75510i q^{50} +3.30584i q^{52} +7.35874 q^{53} +2.02463 q^{55} +2.21423 q^{56} +8.55760 q^{58} +12.8740 q^{59} +2.45943 q^{61} +1.92919i q^{62} +2.52495 q^{64} -1.16976 q^{65} -2.55500 q^{67} -8.95114i q^{68} -1.75068i q^{70} -1.82546i q^{71} -3.90000i q^{73} +1.65980 q^{74} +8.56818i q^{76} -8.06408 q^{77} +2.77679i q^{79} +2.37326i q^{80} +10.7323i q^{82} +6.67645i q^{83} +3.16734 q^{85} -10.0556 q^{86} +4.70896 q^{88} +4.44903 q^{89} +4.65916 q^{91} -10.1397 q^{92} -17.3172i q^{94} -3.03183 q^{95} +1.32217i q^{97} -5.89954i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 80 q^{4}+O(q^{10}) \) 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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