LMFDB - Embedded newform 2151.2.b.a.2150.4

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			2150.4
Character	$\chi$	$=$	2151.2150
Dual form			2151.2.b.a.2150.77

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56329i q^{2} -4.57046 q^{4} -2.12793i q^{5} +2.62545i q^{7} +6.58884i q^{8} +O(q^{10})$	\(q-2.56329i q^{2} -4.57046 q^{4} -2.12793i q^{5} +2.62545i q^{7} +6.58884i q^{8} -5.45450 q^{10} +0.0947060i q^{11} +3.19243i q^{13} +6.72979 q^{14} +7.74819 q^{16} -0.618466i q^{17} +2.41430i q^{19} +9.72561i q^{20} +0.242759 q^{22} +6.02141 q^{23} +0.471925 q^{25} +8.18312 q^{26} -11.9995i q^{28} -2.84177i q^{29} -1.87532 q^{31} -6.68319i q^{32} -1.58531 q^{34} +5.58676 q^{35} -2.46645i q^{37} +6.18855 q^{38} +14.0206 q^{40} +9.58680 q^{41} +2.85199i q^{43} -0.432850i q^{44} -15.4346i q^{46} -2.13934 q^{47} +0.107026 q^{49} -1.20968i q^{50} -14.5909i q^{52} +3.79749 q^{53} +0.201528 q^{55} -17.2987 q^{56} -7.28430 q^{58} -6.69642 q^{59} +6.16339 q^{61} +4.80698i q^{62} -1.63458 q^{64} +6.79325 q^{65} +2.62094 q^{67} +2.82667i q^{68} -14.3205i q^{70} +1.76688i q^{71} -8.65314i q^{73} -6.32223 q^{74} -11.0344i q^{76} -0.248646 q^{77} -0.727776i q^{79} -16.4876i q^{80} -24.5737i q^{82} -8.15116i q^{83} -1.31605 q^{85} +7.31047 q^{86} -0.624003 q^{88} +7.54271 q^{89} -8.38155 q^{91} -27.5206 q^{92} +5.48376i q^{94} +5.13745 q^{95} -2.58921i q^{97} -0.274338i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.56329i		−	1.81252i	−0.422720	−	0.906260i	$-0.638925\pi$
$2$		−	2.56329i		−	1.81252i	0.422720	−	0.906260i	$-0.361075\pi$
$3$		0			0
$3$		0			0
$4$	−4.57046			−2.28523
$5$		−	2.12793i		−	0.951638i	−0.879543	−	0.475819i	$-0.842152\pi$
$5$		−	2.12793i		−	0.951638i	0.879543	−	0.475819i	$-0.157848\pi$
$6$		0			0
$7$			2.62545i			0.992326i	0.868229	+	0.496163i	$0.165258\pi$
$7$			2.62545i			0.992326i	−0.868229	+	0.496163i	$0.834742\pi$
$8$			6.58884i			2.32951i
$9$		0			0
$10$	−5.45450			−1.72486
$11$			0.0947060i			0.0285549i	0.999898	+	0.0142775i	$0.00454481\pi$
$11$			0.0947060i			0.0285549i	−0.999898	+	0.0142775i	$0.995455\pi$
$12$		0			0
$13$			3.19243i			0.885420i	0.896665	+	0.442710i	$0.145983\pi$
$13$			3.19243i			0.885420i	−0.896665	+	0.442710i	$0.854017\pi$
$14$	6.72979			1.79861
$15$		0			0
$16$	7.74819			1.93705
$17$		−	0.618466i		−	0.150000i	−0.997184	−	0.0750000i	$-0.976104\pi$
$17$		−	0.618466i		−	0.150000i	0.997184	−	0.0750000i	$-0.0238957\pi$
$18$		0			0
$19$			2.41430i			0.553878i	0.960888	+	0.276939i	$0.0893199\pi$
$19$			2.41430i			0.553878i	−0.960888	+	0.276939i	$0.910680\pi$
$20$			9.72561i			2.17471i
$21$		0			0
$22$	0.242759			0.0517564
$23$	6.02141			1.25555			0.627775	−	0.778395i	$-0.283966\pi$
$23$	6.02141			1.25555			0.627775	+	0.778395i	$0.283966\pi$
$24$		0			0
$25$	0.471925			0.0943851
$26$	8.18312			1.60484
$27$		0			0
$28$		−	11.9995i		−	2.26769i
$29$		−	2.84177i		−	0.527704i	−0.964563	−	0.263852i	$-0.915007\pi$
$29$		−	2.84177i		−	0.527704i	0.964563	−	0.263852i	$-0.0849931\pi$
$30$		0			0
$31$	−1.87532			−0.336817			−0.168408	−	0.985717i	$-0.553863\pi$
$31$	−1.87532			−0.336817			−0.168408	+	0.985717i	$0.553863\pi$
$32$		−	6.68319i		−	1.18143i
$33$		0			0
$34$	−1.58531			−0.271878
$35$	5.58676			0.944335
$36$		0			0
$37$		−	2.46645i		−	0.405482i	−0.979232	−	0.202741i	$-0.935015\pi$
$37$		−	2.46645i		−	0.405482i	0.979232	−	0.202741i	$-0.0649849\pi$
$38$	6.18855			1.00391
$39$		0			0
$40$	14.0206			2.21685
$41$	9.58680			1.49721			0.748603	−	0.663018i	$-0.230725\pi$
$41$	9.58680			1.49721			0.748603	+	0.663018i	$0.230725\pi$
$42$		0			0
$43$			2.85199i			0.434924i	0.976069	+	0.217462i	$0.0697778\pi$
$43$			2.85199i			0.434924i	−0.976069	+	0.217462i	$0.930222\pi$
$44$		−	0.432850i		−	0.0652546i
$45$		0			0
$46$		−	15.4346i		−	2.27571i
$47$	−2.13934			−0.312055			−0.156028	−	0.987753i	$-0.549869\pi$
$47$	−2.13934			−0.312055			−0.156028	+	0.987753i	$0.549869\pi$
$48$		0			0
$49$	0.107026			0.0152894
$50$		−	1.20968i		−	0.171075i
$51$		0			0
$52$		−	14.5909i		−	2.02339i
$53$	3.79749			0.521626			0.260813	−	0.965389i	$-0.416009\pi$
$53$	3.79749			0.521626			0.260813	+	0.965389i	$0.416009\pi$
$54$		0			0
$55$	0.201528			0.0271740
$56$	−17.2987			−2.31163
$57$		0			0
$58$	−7.28430			−0.956475
$59$	−6.69642			−0.871800			−0.435900	−	0.899995i	$-0.643570\pi$
$59$	−6.69642			−0.871800			−0.435900	+	0.899995i	$0.643570\pi$
$60$		0			0
$61$	6.16339			0.789141			0.394570	−	0.918866i	$-0.370893\pi$
$61$	6.16339			0.789141			0.394570	+	0.918866i	$0.370893\pi$
$62$			4.80698i			0.610487i
$63$		0			0
$64$	−1.63458			−0.204322
$65$	6.79325			0.842599
$66$		0			0
$67$	2.62094			0.320199			0.160099	−	0.987101i	$-0.448819\pi$
$67$	2.62094			0.320199			0.160099	+	0.987101i	$0.448819\pi$
$68$			2.82667i			0.342785i
$69$		0			0
$70$		−	14.3205i		−	1.71163i
$71$			1.76688i			0.209690i	0.994489	+	0.104845i	$0.0334347\pi$
$71$			1.76688i			0.209690i	−0.994489	+	0.104845i	$0.966565\pi$
$72$		0			0
$73$		−	8.65314i		−	1.01277i	−0.862306	−	0.506387i	$-0.830981\pi$
$73$		−	8.65314i		−	1.01277i	0.862306	−	0.506387i	$-0.169019\pi$
$74$	−6.32223			−0.734944
$75$		0			0
$76$		−	11.0344i		−	1.26574i
$77$	−0.248646			−0.0283358
$78$		0			0
$79$		−	0.727776i		−	0.0818812i	−0.999162	−	0.0409406i	$-0.986965\pi$
$79$		−	0.727776i		−	0.0818812i	0.999162	−	0.0409406i	$-0.0130354\pi$
$80$		−	16.4876i		−	1.84337i
$81$		0			0
$82$		−	24.5737i		−	2.71372i
$83$		−	8.15116i		−	0.894706i	−0.894358	−	0.447353i	$-0.852367\pi$
$83$		−	8.15116i		−	0.894706i	0.894358	−	0.447353i	$-0.147633\pi$
$84$		0			0
$85$	−1.31605			−0.142746
$86$	7.31047			0.788309
$87$		0			0
$88$	−0.624003			−0.0665189
$89$	7.54271			0.799526			0.399763	−	0.916619i	$-0.369092\pi$
$89$	7.54271			0.799526			0.399763	+	0.916619i	$0.369092\pi$
$90$		0			0
$91$	−8.38155			−0.878625
$92$	−27.5206			−2.86922
$93$		0			0
$94$			5.48376i			0.565607i
$95$	5.13745			0.527091
$96$		0			0
$97$		−	2.58921i		−	0.262894i	−0.991323	−	0.131447i	$-0.958038\pi$
$97$		−	2.58921i		−	0.262894i	0.991323	−	0.131447i	$-0.0419624\pi$
$98$		−	0.274338i		−	0.0277124i
$99$		0			0
$100$	−2.15692			−0.215692
$101$			8.59648i			0.855382i	0.903925	+	0.427691i	$0.140673\pi$
$101$			8.59648i			0.855382i	−0.903925	+	0.427691i	$0.859327\pi$
$102$		0			0
$103$			6.28634i			0.619411i	0.950832	+	0.309706i	$0.100231\pi$
$103$			6.28634i			0.619411i	−0.950832	+	0.309706i	$0.899769\pi$
$104$	−21.0344			−2.06259
$105$		0			0
$106$		−	9.73408i		−	0.945457i
$107$	9.78523			0.945974			0.472987	−	0.881069i	$-0.343176\pi$
$107$	9.78523			0.945974			0.472987	+	0.881069i	$0.343176\pi$
$108$		0			0
$109$	9.15599			0.876985			0.438492	−	0.898735i	$-0.355513\pi$
$109$	9.15599			0.876985			0.438492	+	0.898735i	$0.355513\pi$
$110$		−	0.516574i		−	0.0492534i
$111$		0			0
$112$			20.3425i			1.92218i
$113$			18.3525i			1.72646i	0.504814	+	0.863228i	$0.331561\pi$
$113$			18.3525i			1.72646i	−0.504814	+	0.863228i	$0.668439\pi$
$114$		0			0
$115$		−	12.8131i		−	1.19483i
$116$			12.9882i			1.20593i
$117$		0			0
$118$			17.1649i			1.58016i
$119$	1.62375			0.148849
$120$		0			0
$121$	10.9910			0.999185
$122$		−	15.7986i		−	1.43033i
$123$		0			0
$124$	8.57105			0.769703
$125$		−	11.6439i		−	1.04146i
$126$		0			0
$127$	0.0983942			0.00873107			0.00436554	−	0.999990i	$-0.498610\pi$
$127$	0.0983942			0.00873107			0.00436554	+	0.999990i	$0.498610\pi$
$128$		−	9.17648i		−	0.811094i
$129$		0			0
$130$		−	17.4131i		−	1.52723i
$131$	−6.93657			−0.606051			−0.303025	−	0.952983i	$-0.597997\pi$
$131$	−6.93657			−0.606051			−0.303025	+	0.952983i	$0.597997\pi$
$132$		0			0
$133$	−6.33861			−0.549627
$134$		−	6.71823i		−	0.580367i
$135$		0			0
$136$	4.07497			0.349426
$137$	−16.2996			−1.39257			−0.696285	−	0.717765i	$-0.745165\pi$
$137$	−16.2996			−1.39257			−0.696285	+	0.717765i	$0.745165\pi$
$138$		0			0
$139$			20.5358i			1.74182i	0.491440	+	0.870911i	$0.336471\pi$
$139$			20.5358i			1.74182i	−0.491440	+	0.870911i	$0.663529\pi$
$140$	−25.5341			−2.15802
$141$		0			0
$142$	4.52903			0.380068
$143$	−0.302342			−0.0252831
$144$		0			0
$145$	−6.04709			−0.502183
$146$	−22.1805			−1.83567
$147$		0			0
$148$			11.2728i			0.926620i
$149$	−2.43014			−0.199085			−0.0995425	−	0.995033i	$-0.531738\pi$
$149$	−2.43014			−0.199085			−0.0995425	+	0.995033i	$0.531738\pi$
$150$		0			0
$151$		−	10.5452i		−	0.858158i	−0.903267	−	0.429079i	$-0.858838\pi$
$151$		−	10.5452i		−	0.858158i	0.903267	−	0.429079i	$-0.141162\pi$
$152$	−15.9074			−1.29026
$153$		0			0
$154$			0.637351i			0.0513592i
$155$			3.99053i			0.320527i
$156$		0			0
$157$	−5.28303			−0.421632			−0.210816	−	0.977526i	$-0.567612\pi$
$157$	−5.28303			−0.421632			−0.210816	+	0.977526i	$0.567612\pi$
$158$	−1.86550			−0.148411
$159$		0			0
$160$	−14.2213			−1.12430
$161$			15.8089i			1.24591i
$162$		0			0
$163$	−5.27627			−0.413269			−0.206635	−	0.978418i	$-0.566251\pi$
$163$	−5.27627			−0.413269			−0.206635	+	0.978418i	$0.566251\pi$
$164$	−43.8161			−3.42146
$165$		0			0
$166$	−20.8938			−1.62167
$167$	−17.1068			−1.32377			−0.661884	−	0.749607i	$-0.730243\pi$
$167$	−17.1068			−1.32377			−0.661884	+	0.749607i	$0.730243\pi$
$168$		0			0
$169$	2.80841			0.216031
$170$			3.37342i			0.258730i
$171$		0			0
$172$		−	13.0349i		−	0.993902i
$173$	18.2963			1.39105			0.695523	−	0.718504i	$-0.255173\pi$
$173$	18.2963			1.39105			0.695523	+	0.718504i	$0.255173\pi$
$174$		0			0
$175$			1.23902i			0.0936608i
$176$			0.733800i			0.0553123i
$177$		0			0
$178$		−	19.3342i		−	1.44916i
$179$	−3.66974			−0.274290			−0.137145	−	0.990551i	$-0.543793\pi$
$179$	−3.66974			−0.274290			−0.137145	+	0.990551i	$0.543793\pi$
$180$		0			0
$181$		−	7.85640i		−	0.583962i	−0.956424	−	0.291981i	$-0.905686\pi$
$181$		−	7.85640i		−	0.583962i	0.956424	−	0.291981i	$-0.0943143\pi$
$182$			21.4844i			1.59253i
$183$		0			0
$184$			39.6741i			2.92481i
$185$	−5.24843			−0.385872
$186$		0			0
$187$	0.0585724			0.00428324
$188$	9.77779			0.713118
$189$		0			0
$190$		−	13.1688i		−	0.955363i
$191$	5.23126			0.378521			0.189260	−	0.981927i	$-0.439391\pi$
$191$	5.23126			0.378521			0.189260	+	0.981927i	$0.439391\pi$
$192$		0			0
$193$	11.6300			0.837147			0.418574	−	0.908183i	$-0.362530\pi$
$193$	11.6300			0.837147			0.418574	+	0.908183i	$0.362530\pi$
$194$	−6.63689			−0.476501
$195$		0			0
$196$	−0.489157			−0.0349398
$197$		−	7.81077i		−	0.556494i	−0.960510	−	0.278247i	$-0.910247\pi$
$197$		−	7.81077i		−	0.556494i	0.960510	−	0.278247i	$-0.0897534\pi$
$198$		0			0
$199$		−	6.61163i		−	0.468686i	−0.972154	−	0.234343i	$-0.924706\pi$
$199$		−	6.61163i		−	0.468686i	0.972154	−	0.234343i	$-0.0752939\pi$
$200$			3.10944i			0.219871i
$201$		0			0
$202$	22.0353			1.55040
$203$	7.46093			0.523655
$204$		0			0
$205$		−	20.4000i		−	1.42480i
$206$	16.1137			1.12270
$207$		0			0
$208$			24.7355i			1.71510i
$209$	−0.228648			−0.0158159
$210$		0			0
$211$	−12.9710			−0.892958			−0.446479	−	0.894794i	$-0.647322\pi$
$211$	−12.9710			−0.892958			−0.446479	+	0.894794i	$0.647322\pi$
$212$	−17.3563			−1.19204
$213$		0			0
$214$		−	25.0824i		−	1.71460i
$215$	6.06882			0.413890
$216$		0			0
$217$		−	4.92354i		−	0.334232i
$218$		−	23.4695i		−	1.58955i
$219$		0			0
$220$	−0.921074			−0.0620988
$221$	1.97441			0.132813
$222$		0			0
$223$			10.3898i			0.695751i	0.937541	+	0.347875i	$0.113097\pi$
$223$			10.3898i			0.695751i	−0.937541	+	0.347875i	$0.886903\pi$
$224$	17.5464			1.17237
$225$		0			0
$226$	47.0427			3.12924
$227$	21.1323			1.40260			0.701299	−	0.712868i	$-0.252604\pi$
$227$	21.1323			1.40260			0.701299	+	0.712868i	$0.252604\pi$
$228$		0			0
$229$			23.9522i			1.58281i	0.611295	+	0.791403i	$0.290649\pi$
$229$			23.9522i			1.58281i	−0.611295	+	0.791403i	$0.709351\pi$
$230$	−32.8437			−2.16565
$231$		0			0
$232$	18.7240			1.22929
$233$	16.6084			1.08805			0.544026	−	0.839068i	$-0.316899\pi$
$233$	16.6084			1.08805			0.544026	+	0.839068i	$0.316899\pi$
$234$		0			0
$235$			4.55237i			0.296964i
$236$	30.6057			1.99226
$237$		0			0
$238$		−	4.16214i		−	0.269792i
$239$	15.3811	−	1.55641i	0.994919	−	0.100676i
$239$	15.3811	−	1.55641i	0.994919	−	0.100676i
$240$		0			0
$241$	−21.3445			−1.37492			−0.687462	−	0.726221i	$-0.741275\pi$
$241$	−21.3445			−1.37492			−0.687462	+	0.726221i	$0.741275\pi$
$242$		−	28.1732i		−	1.81104i
$243$		0			0
$244$	−28.1695			−1.80337
$245$		−	0.227743i		−	0.0145500i
$246$		0			0
$247$	−7.70747			−0.490414
$248$		−	12.3562i		−	0.784616i
$249$		0			0
$250$	−29.8466			−1.88766
$251$			3.41444i			0.215517i	0.994177	+	0.107759i	$0.0343674\pi$
$251$			3.41444i			0.215517i	−0.994177	+	0.107759i	$0.965633\pi$
$252$		0			0
$253$			0.570263i			0.0358522i
$254$		−	0.252213i		−	0.0158252i
$255$		0			0
$256$	−26.7911			−1.67445
$257$		−	12.7112i		−	0.792901i	−0.918056	−	0.396450i	$-0.870242\pi$
$257$		−	12.7112i		−	0.792901i	0.918056	−	0.396450i	$-0.129758\pi$
$258$		0			0
$259$	6.47554			0.402370
$260$	−31.0483			−1.92553
$261$		0			0
$262$			17.7804i			1.09848i
$263$			4.08627i			0.251970i	0.992032	+	0.125985i	$0.0402091\pi$
$263$			4.08627i			0.251970i	−0.992032	+	0.125985i	$0.959791\pi$
$264$		0			0
$265$		−	8.08079i		−	0.496399i
$266$			16.2477i			0.996210i
$267$		0			0
$268$	−11.9789			−0.731728
$269$			1.77041i			0.107944i	0.998542	+	0.0539719i	$0.0171882\pi$
$269$			1.77041i			0.107944i	−0.998542	+	0.0539719i	$0.982812\pi$
$270$		0			0
$271$	19.0825			1.15918			0.579591	−	0.814908i	$-0.303212\pi$
$271$	19.0825			1.15918			0.579591	+	0.814908i	$0.303212\pi$
$272$		−	4.79199i		−	0.290557i
$273$		0			0
$274$			41.7807i			2.52406i
$275$			0.0446942i			0.00269516i
$276$		0			0
$277$		−	10.1545i		−	0.610125i	−0.952332	−	0.305063i	$-0.901323\pi$
$277$		−	10.1545i		−	0.610125i	0.952332	−	0.305063i	$-0.0986774\pi$
$278$	52.6392			3.15709
$279$		0			0
$280$			36.8103i			2.19983i
$281$	23.4831			1.40088			0.700441	−	0.713710i	$-0.252987\pi$
$281$	23.4831			1.40088			0.700441	+	0.713710i	$0.252987\pi$
$282$		0			0
$283$	6.58740			0.391581			0.195790	−	0.980646i	$-0.437273\pi$
$283$	6.58740			0.391581			0.195790	+	0.980646i	$0.437273\pi$
$284$		−	8.07546i		−	0.479190i
$285$		0			0
$286$			0.774991i			0.0458262i
$287$			25.1696i			1.48572i
$288$		0			0
$289$	16.6175			0.977500
$290$			15.5005i			0.910218i
$291$		0			0
$292$			39.5489i			2.31442i
$293$		−	29.3334i		−	1.71367i	−0.515587	−	0.856837i	$-0.672426\pi$
$293$		−	29.3334i		−	1.71367i	0.515587	−	0.856837i	$-0.327574\pi$
$294$		0			0
$295$			14.2495i			0.829638i
$296$	16.2510			0.944573
$297$		0			0
$298$			6.22916i			0.360846i
$299$			19.2229i			1.11169i
$300$		0			0
$301$	−7.48774			−0.431586
$302$	−27.0305			−1.55543
$303$		0			0
$304$			18.7064i			1.07289i
$305$		−	13.1152i		−	0.750976i
$306$		0			0
$307$	−8.59250			−0.490400			−0.245200	−	0.969473i	$-0.578854\pi$
$307$	−8.59250			−0.490400			−0.245200	+	0.969473i	$0.578854\pi$
$308$	1.13643			0.0647538
$309$		0			0
$310$	10.2289			0.580963
$311$			3.63242i			0.205976i	0.994683	+	0.102988i	$0.0328403\pi$
$311$			3.63242i			0.205976i	−0.994683	+	0.102988i	$0.967160\pi$
$312$		0			0
$313$			11.7421i			0.663701i	0.943332	+	0.331850i	$0.107673\pi$
$313$			11.7421i			0.663701i	−0.943332	+	0.331850i	$0.892327\pi$
$314$			13.5419i			0.764216i
$315$		0			0
$316$			3.32627i			0.187117i
$317$	18.1138			1.01737			0.508686	−	0.860952i	$-0.330131\pi$
$317$	18.1138			1.01737			0.508686	+	0.860952i	$0.330131\pi$
$318$		0			0
$319$	0.269133			0.0150686
$320$			3.47826i			0.194441i
$321$		0			0
$322$	40.5228			2.25825
$323$	1.49316			0.0830817
$324$		0			0
$325$			1.50659i			0.0835704i
$326$			13.5246i			0.749059i
$327$		0			0
$328$			63.1659i			3.48775i
$329$		−	5.61674i		−	0.309661i
$330$		0			0
$331$			22.4535i			1.23416i	0.786902	+	0.617078i	$0.211684\pi$
$331$			22.4535i			1.23416i	−0.786902	+	0.617078i	$0.788316\pi$
$332$			37.2546i			2.04461i
$333$		0			0
$334$			43.8498i			2.39935i
$335$		−	5.57717i		−	0.304713i
$336$		0			0
$337$	−30.7860			−1.67702			−0.838509	−	0.544887i	$-0.816573\pi$
$337$	−30.7860			−1.67702			−0.838509	+	0.544887i	$0.816573\pi$
$338$		−	7.19877i		−	0.391561i
$339$		0			0
$340$	6.01496			0.326207
$341$		−	0.177604i		−	0.00961778i
$342$		0			0
$343$			18.6591i			1.00750i
$344$	−18.7913			−1.01316
$345$		0			0
$346$		−	46.8988i		−	2.52130i
$347$		−	16.7604i		−	0.899747i	−0.893092	−	0.449873i	$-0.851469\pi$
$347$		−	16.7604i		−	0.899747i	0.893092	−	0.449873i	$-0.148531\pi$
$348$		0			0
$349$	−1.51410			−0.0810478			−0.0405239	−	0.999179i	$-0.512903\pi$
$349$	−1.51410			−0.0810478			−0.0405239	+	0.999179i	$0.512903\pi$
$350$	3.17596			0.169762
$351$		0			0
$352$	0.632938			0.0337357
$353$	31.0981			1.65519			0.827593	−	0.561329i	$-0.189710\pi$
$353$	31.0981			1.65519			0.827593	+	0.561329i	$0.189710\pi$
$354$		0			0
$355$	3.75979			0.199549
$356$	−34.4737			−1.82710
$357$		0			0
$358$			9.40662i			0.497155i
$359$			8.89194i			0.469299i	0.972080	+	0.234649i	$0.0753942\pi$
$359$			8.89194i			0.469299i	−0.972080	+	0.234649i	$0.924606\pi$
$360$		0			0
$361$	13.1712			0.693220
$362$	−20.1382			−1.05844
$363$		0			0
$364$	38.3075			2.00786
$365$	−18.4133			−0.963794
$366$		0			0
$367$	3.36080			0.175432			0.0877162	−	0.996146i	$-0.472043\pi$
$367$	3.36080			0.175432			0.0877162	+	0.996146i	$0.472043\pi$
$368$	46.6550			2.43206
$369$		0			0
$370$			13.4532i			0.699401i
$371$			9.97012i			0.517623i
$372$		0			0
$373$	2.68531			0.139040			0.0695201	−	0.997581i	$-0.477853\pi$
$373$	2.68531			0.139040			0.0695201	+	0.997581i	$0.477853\pi$
$374$		−	0.150138i		−	0.00776346i
$375$		0			0
$376$		−	14.0958i		−	0.726935i
$377$	9.07216			0.467240
$378$		0			0
$379$		−	10.7130i		−	0.550289i	−0.961403	−	0.275144i	$-0.911274\pi$
$379$		−	10.7130i		−	0.550289i	0.961403	−	0.275144i	$-0.0887256\pi$
$380$	−23.4805			−1.20452
$381$		0			0
$382$		−	13.4092i		−	0.686077i
$383$			7.37011i			0.376595i	0.982112	+	0.188297i	$0.0602969\pi$
$383$			7.37011i			0.376595i	−0.982112	+	0.188297i	$0.939703\pi$
$384$		0			0
$385$			0.529100i			0.0269654i
$386$		−	29.8111i		−	1.51735i
$387$		0			0
$388$			11.8339i			0.600774i
$389$			28.5565i			1.44787i	0.689867	+	0.723936i	$0.257669\pi$
$389$			28.5565i			1.44787i	−0.689867	+	0.723936i	$0.742331\pi$
$390$		0			0
$391$		−	3.72403i		−	0.188332i
$392$			0.705176i			0.0356168i
$393$		0			0
$394$	−20.0213			−1.00866
$395$	−1.54865			−0.0779213
$396$		0			0
$397$		−	4.98495i		−	0.250188i	−0.992145	−	0.125094i	$-0.960077\pi$
$397$		−	4.98495i		−	0.250188i	0.992145	−	0.125094i	$-0.0399232\pi$
$398$	−16.9475			−0.849503
$399$		0			0
$400$	3.65657			0.182828
$401$		−	13.2904i		−	0.663693i	−0.943333	−	0.331846i	$-0.892328\pi$
$401$		−	13.2904i		−	0.663693i	0.943333	−	0.331846i	$-0.107672\pi$
$402$		0			0
$403$		−	5.98681i		−	0.298224i
$404$		−	39.2899i		−	1.95474i
$405$		0			0
$406$		−	19.1245i		−	0.949135i
$407$	0.233588			0.0115785
$408$		0			0
$409$	−22.5885			−1.11693			−0.558465	−	0.829528i	$-0.688609\pi$
$409$	−22.5885			−1.11693			−0.558465	+	0.829528i	$0.688609\pi$
$410$	−52.2912			−2.58248
$411$		0			0
$412$		−	28.7315i		−	1.41550i
$413$		−	17.5811i		−	0.865110i
$414$		0			0
$415$	−17.3451			−0.851436
$416$	21.3356			1.04606
$417$		0			0
$418$			0.586093i			0.0286667i
$419$			14.4459i			0.705731i	0.935674	+	0.352865i	$0.114793\pi$
$419$			14.4459i			0.705731i	−0.935674	+	0.352865i	$0.885207\pi$
$420$		0			0
$421$	28.1177			1.37037			0.685187	−	0.728367i	$-0.259720\pi$
$421$	28.1177			1.37037			0.685187	+	0.728367i	$0.259720\pi$
$422$			33.2484i			1.61851i
$423$		0			0
$424$			25.0211i			1.21513i
$425$		−	0.291870i		−	0.0141578i
$426$		0			0
$427$			16.1816i			0.783085i
$428$	−44.7230			−2.16177
$429$		0			0
$430$		−	15.5562i		−	0.750185i
$431$		−	30.8092i		−	1.48403i	−0.670384	−	0.742014i	$-0.733871\pi$
$431$		−	30.8092i		−	1.48403i	0.670384	−	0.742014i	$-0.266129\pi$
$432$		0			0
$433$		−	17.4595i		−	0.839049i	−0.907744	−	0.419525i	$-0.862197\pi$
$433$		−	17.4595i		−	0.839049i	0.907744	−	0.419525i	$-0.137803\pi$
$434$	−12.6205			−0.605802
$435$		0			0
$436$	−41.8471			−2.00411
$437$			14.5375i			0.695421i
$438$		0			0
$439$	−36.9043			−1.76135			−0.880673	−	0.473725i	$-0.842909\pi$
$439$	−36.9043			−1.76135			−0.880673	+	0.473725i	$0.842909\pi$
$440$			1.32783i			0.0633019i
$441$		0			0
$442$		−	5.06098i		−	0.240726i
$443$		−	10.0227i		−	0.476193i	−0.971242	−	0.238096i	$-0.923477\pi$
$443$		−	10.0227i		−	0.476193i	0.971242	−	0.238096i	$-0.0765234\pi$
$444$		0			0
$445$		−	16.0503i		−	0.760859i
$446$	26.6320			1.26106
$447$		0			0
$448$		−	4.29150i		−	0.202754i
$449$	−1.38244			−0.0652412			−0.0326206	−	0.999468i	$-0.510385\pi$
$449$	−1.38244			−0.0652412			−0.0326206	+	0.999468i	$0.510385\pi$
$450$		0			0
$451$			0.907927i			0.0427526i
$452$		−	83.8793i		−	3.94535i
$453$		0			0
$454$		−	54.1681i		−	2.54224i
$455$			17.8353i			0.836133i
$456$		0			0
$457$	−12.8653			−0.601815			−0.300907	−	0.953653i	$-0.597290\pi$
$457$	−12.8653			−0.601815			−0.300907	+	0.953653i	$0.597290\pi$
$458$	61.3964			2.86887
$459$		0			0
$460$			58.5618i			2.73046i
$461$	−27.7622			−1.29301			−0.646507	−	0.762908i	$-0.723771\pi$
$461$	−27.7622			−1.29301			−0.646507	+	0.762908i	$0.723771\pi$
$462$		0			0
$463$			29.6863i			1.37964i	0.723982	+	0.689819i	$0.242310\pi$
$463$			29.6863i			1.37964i	−0.723982	+	0.689819i	$0.757690\pi$
$464$		−	22.0186i		−	1.02219i
$465$		0			0
$466$		−	42.5721i		−	1.97212i
$467$	30.4205			1.40769			0.703846	−	0.710352i	$-0.251464\pi$
$467$	30.4205			1.40769			0.703846	+	0.710352i	$0.251464\pi$
$468$		0			0
$469$			6.88114i			0.317742i
$470$	11.6690			0.538253
$471$		0			0
$472$		−	44.1216i		−	2.03086i
$473$	−0.270100			−0.0124192
$474$		0			0
$475$			1.13937i			0.0522778i
$476$	−7.42128			−0.340154
$477$		0			0
$478$	−3.98953	−	39.4262i	−0.182477	−	1.80331i
$479$		−	42.2355i		−	1.92979i	−0.262635	−	0.964895i	$-0.584591\pi$
$479$		−	42.2355i		−	1.92979i	0.262635	−	0.964895i	$-0.415409\pi$
$480$		0			0
$481$	7.87396			0.359022
$482$			54.7123i			2.49208i
$483$		0			0
$484$	−50.2341			−2.28337
$485$	−5.50965			−0.250180
$486$		0			0
$487$	22.1272			1.00268			0.501339	−	0.865251i	$-0.332841\pi$
$487$	22.1272			1.00268			0.501339	+	0.865251i	$0.332841\pi$
$488$			40.6096i			1.83831i
$489$		0			0
$490$	−0.583772			−0.0263721
$491$	8.82156			0.398112			0.199056	−	0.979988i	$-0.436213\pi$
$491$	8.82156			0.398112			0.199056	+	0.979988i	$0.436213\pi$
$492$		0			0
$493$	−1.75754			−0.0791556
$494$			19.7565i			0.888886i
$495$		0			0
$496$	−14.5303			−0.652430
$497$	−4.63885			−0.208081
$498$		0			0
$499$		−	12.6921i		−	0.568175i	−0.958798	−	0.284088i	$-0.908309\pi$
$499$		−	12.6921i		−	0.568175i	0.958798	−	0.284088i	$-0.0916907\pi$
$500$			53.2178i			2.37997i
$501$		0			0
$502$	8.75219			0.390629
$503$			14.5144i			0.647165i	0.946200	+	0.323583i	$0.104887\pi$
$503$			14.5144i			0.647165i	−0.946200	+	0.323583i	$0.895113\pi$
$504$		0			0
$505$	18.2927			0.814014
$506$	1.46175			0.0649828
$507$		0			0
$508$	−0.449707			−0.0199525
$509$		−	0.152513i		−	0.00676003i	−0.999994	−	0.00338002i	$-0.998924\pi$
$509$		−	0.152513i		−	0.00676003i	0.999994	−	0.00338002i	$-0.00107589\pi$
$510$		0			0
$511$	22.7184			1.00500
$512$			50.3205i			2.22387i
$513$		0			0
$514$	−32.5824			−1.43715
$515$	13.3769			0.589455
$516$		0			0
$517$		−	0.202609i		−	0.00891072i
$518$		−	16.5987i		−	0.729304i
$519$		0			0
$520$			44.7597i			1.96284i
$521$	27.7906			1.21753			0.608765	−	0.793351i	$-0.291665\pi$
$521$	27.7906			1.21753			0.608765	+	0.793351i	$0.291665\pi$
$522$		0			0
$523$	−15.2289			−0.665915			−0.332958	−	0.942942i	$-0.608047\pi$
$523$	−15.2289			−0.665915			−0.332958	+	0.942942i	$0.608047\pi$
$524$	31.7033			1.38497
$525$		0			0
$526$	10.4743			0.456701
$527$			1.15982i			0.0505225i
$528$		0			0
$529$	13.2573			0.576406
$530$	−20.7134			−0.899733
$531$		0			0
$532$	28.9704			1.25602
$533$			30.6051i			1.32566i
$534$		0			0
$535$		−	20.8223i		−	0.900225i
$536$			17.2690i			0.745905i
$537$		0			0
$538$	4.53808			0.195651
$539$			0.0101360i			0.000436588i
$540$		0			0
$541$		−	14.3791i		−	0.618208i	−0.951028	−	0.309104i	$-0.899971\pi$
$541$		−	14.3791i		−	0.618208i	0.951028	−	0.309104i	$-0.100029\pi$
$542$		−	48.9141i		−	2.10104i
$543$		0			0
$544$	−4.13332			−0.177215
$545$		−	19.4833i		−	0.834572i
$546$		0			0
$547$			36.8515i			1.57566i	0.615894	+	0.787829i	$0.288795\pi$
$547$			36.8515i			1.57566i	−0.615894	+	0.787829i	$0.711205\pi$
$548$	74.4968			3.18234
$549$		0			0
$550$	0.114564			0.00488503
$551$	6.86089			0.292284
$552$		0			0
$553$	1.91074			0.0812528
$554$	−26.0289			−1.10586
$555$		0			0
$556$		−	93.8580i		−	3.98047i
$557$	−39.6539			−1.68019			−0.840094	−	0.542441i	$-0.817500\pi$
$557$	−39.6539			−1.68019			−0.840094	+	0.542441i	$0.817500\pi$
$558$		0			0
$559$	−9.10476			−0.385090
$560$	43.2873			1.82922
$561$		0			0
$562$		−	60.1939i		−	2.53913i
$563$			17.5891i			0.741290i	0.928775	+	0.370645i	$0.120863\pi$
$563$			17.5891i			0.741290i	−0.928775	+	0.370645i	$0.879137\pi$
$564$		0			0
$565$	39.0527			1.64296
$566$		−	16.8854i		−	0.709748i
$567$		0			0
$568$	−11.6417			−0.488474
$569$			4.04551i			0.169596i	0.996398	+	0.0847982i	$0.0270246\pi$
$569$			4.04551i			0.169596i	−0.996398	+	0.0847982i	$0.972975\pi$
$570$		0			0
$571$	−14.6453			−0.612887			−0.306444	−	0.951889i	$-0.599139\pi$
$571$	−14.6453			−0.612887			−0.306444	+	0.951889i	$0.599139\pi$
$572$	1.38184			0.0577777
$573$		0			0
$574$	64.5171			2.69289
$575$	2.84165			0.118505
$576$		0			0
$577$	10.6776			0.444515			0.222257	−	0.974988i	$-0.428658\pi$
$577$	10.6776			0.444515			0.222257	+	0.974988i	$0.428658\pi$
$578$		−	42.5955i		−	1.77174i
$579$		0			0
$580$	27.6380			1.14761
$581$	21.4004			0.887840
$582$		0			0
$583$			0.359645i			0.0148950i
$584$	57.0142			2.35926
$585$		0			0
$586$	−75.1900			−3.10607
$587$		−	12.2891i		−	0.507226i	−0.967306	−	0.253613i	$-0.918381\pi$
$587$		−	12.2891i		−	0.507226i	0.967306	−	0.253613i	$-0.0816190\pi$
$588$		0			0
$589$		−	4.52757i		−	0.186555i
$590$	36.5256			1.50374
$591$		0			0
$592$		−	19.1105i		−	0.785438i
$593$	15.6855			0.644125			0.322062	−	0.946718i	$-0.395624\pi$
$593$	15.6855			0.644125			0.322062	+	0.946718i	$0.395624\pi$
$594$		0			0
$595$		−	3.45522i		−	0.141650i
$596$	11.1069			0.454955
$597$		0			0
$598$	49.2739			2.01496
$599$			28.9214i			1.18170i	0.806782	+	0.590849i	$0.201207\pi$
$599$			28.9214i			1.18170i	−0.806782	+	0.590849i	$0.798793\pi$
$600$		0			0
$601$		−	35.5479i		−	1.45003i	−0.688733	−	0.725015i	$-0.741833\pi$
$601$		−	35.5479i		−	1.45003i	0.688733	−	0.725015i	$-0.258167\pi$
$602$			19.1933i			0.782259i
$603$		0			0
$604$			48.1965i			1.96109i
$605$		−	23.3881i		−	0.950862i
$606$		0			0
$607$			30.4290i			1.23508i	0.786541	+	0.617538i	$0.211870\pi$
$607$			30.4290i			1.23508i	−0.786541	+	0.617538i	$0.788130\pi$
$608$	16.1352			0.654369
$609$		0			0
$610$	−33.6182			−1.36116
$611$		−	6.82970i		−	0.276300i
$612$		0			0
$613$	0.790399			0.0319239			0.0159620	−	0.999873i	$-0.494919\pi$
$613$	0.790399			0.0319239			0.0159620	+	0.999873i	$0.494919\pi$
$614$			22.0251i			0.888860i
$615$		0			0
$616$		−	1.63829i		−	0.0660084i
$617$	−3.88374			−0.156354			−0.0781768	−	0.996940i	$-0.524910\pi$
$617$	−3.88374			−0.156354			−0.0781768	+	0.996940i	$0.524910\pi$
$618$		0			0
$619$			0.624752i			0.0251109i	0.999921	+	0.0125554i	$0.00399663\pi$
$619$			0.624752i			0.0251109i	−0.999921	+	0.0125554i	$0.996003\pi$
$620$		−	18.2386i		−	0.732479i
$621$		0			0
$622$	9.31096			0.373335
$623$			19.8030i			0.793390i
$624$		0			0
$625$	−22.4177			−0.896706
$626$	30.0983			1.20297
$627$		0			0
$628$	24.1459			0.963525
$629$	−1.52542			−0.0608223
$630$		0			0
$631$	47.6735			1.89785			0.948926	−	0.315500i	$-0.102172\pi$
$631$	47.6735			1.89785			0.948926	+	0.315500i	$0.102172\pi$
$632$	4.79520			0.190743
$633$		0			0
$634$		−	46.4309i		−	1.84401i
$635$		−	0.209376i		−	0.00830882i
$636$		0			0
$637$			0.341672i			0.0135375i
$638$		−	0.689867i		−	0.0273121i
$639$		0			0
$640$	−19.5269			−0.771868
$641$			36.4882i			1.44120i	0.693353	+	0.720598i	$0.256133\pi$
$641$			36.4882i			1.44120i	−0.693353	+	0.720598i	$0.743867\pi$
$642$		0			0
$643$	−3.86938			−0.152593			−0.0762967	−	0.997085i	$-0.524310\pi$
$643$	−3.86938			−0.152593			−0.0762967	+	0.997085i	$0.524310\pi$
$644$		−	72.2539i		−	2.84720i
$645$		0			0
$646$		−	3.82740i		−	0.150587i
$647$		−	12.3506i		−	0.485550i	−0.970083	−	0.242775i	$-0.921942\pi$
$647$		−	12.3506i		−	0.485550i	0.970083	−	0.242775i	$-0.0780577\pi$
$648$		0			0
$649$		−	0.634192i		−	0.0248942i
$650$	3.86182			0.151473
$651$		0			0
$652$	24.1150			0.944415
$653$	2.65890			0.104051			0.0520254	−	0.998646i	$-0.483432\pi$
$653$	2.65890			0.104051			0.0520254	+	0.998646i	$0.483432\pi$
$654$		0			0
$655$			14.7605i			0.576741i
$656$	74.2803			2.90016
$657$		0			0
$658$	−14.3973			−0.561266
$659$	−28.1528			−1.09668			−0.548338	−	0.836257i	$-0.684739\pi$
$659$	−28.1528			−1.09668			−0.548338	+	0.836257i	$0.684739\pi$
$660$		0			0
$661$	31.6591			1.23140			0.615698	−	0.787982i	$-0.288874\pi$
$661$	31.6591			1.23140			0.615698	+	0.787982i	$0.288874\pi$
$662$	57.5549			2.23693
$663$		0			0
$664$	53.7067			2.08422
$665$			13.4881i			0.523046i
$666$		0			0
$667$		−	17.1115i		−	0.662559i
$668$	78.1861			3.02511
$669$		0			0
$670$	−14.2959			−0.552299
$671$			0.583710i			0.0225339i
$672$		0			0
$673$			16.3388i			0.629814i	0.949123	+	0.314907i	$0.101973\pi$
$673$			16.3388i			0.629814i	−0.949123	+	0.314907i	$0.898027\pi$
$674$			78.9134i			3.03963i
$675$		0			0
$676$	−12.8357			−0.493682
$677$	−6.23511			−0.239635			−0.119817	−	0.992796i	$-0.538231\pi$
$677$	−6.23511			−0.239635			−0.119817	+	0.992796i	$0.538231\pi$
$678$		0			0
$679$	6.79783			0.260877
$680$		−	8.67124i		−	0.332527i
$681$		0			0
$682$	−0.455250			−0.0174324
$683$	−16.4566			−0.629694			−0.314847	−	0.949142i	$-0.601953\pi$
$683$	−16.4566			−0.629694			−0.314847	+	0.949142i	$0.601953\pi$
$684$		0			0
$685$			34.6844i			1.32522i
$686$	47.8288			1.82611
$687$		0			0
$688$			22.0977i			0.842469i
$689$			12.1232i			0.461858i
$690$		0			0
$691$	−26.6851			−1.01515			−0.507575	−	0.861607i	$-0.669458\pi$
$691$	−26.6851			−1.01515			−0.507575	+	0.861607i	$0.669458\pi$
$692$	−83.6227			−3.17886
$693$		0			0
$694$	−42.9618			−1.63081
$695$	43.6987			1.65758
$696$		0			0
$697$		−	5.92911i		−	0.224581i
$698$			3.88107i			0.146901i
$699$		0			0
$700$		−	5.66287i		−	0.214036i
$701$	−31.8205			−1.20185			−0.600923	−	0.799307i	$-0.705200\pi$
$701$	−31.8205			−1.20185			−0.600923	+	0.799307i	$0.705200\pi$
$702$		0			0
$703$	5.95474			0.224587
$704$		−	0.154804i		−	0.00583441i
$705$		0			0
$706$		−	79.7135i		−	3.00006i
$707$	−22.5696			−0.848817
$708$		0			0
$709$		−	6.17962i		−	0.232080i	−0.993245	−	0.116040i	$-0.962980\pi$
$709$		−	6.17962i		−	0.232080i	0.993245	−	0.116040i	$-0.0370201\pi$
$710$		−	9.63744i		−	0.361687i
$711$		0			0
$712$			49.6977i			1.86250i
$713$	−11.2920			−0.422890
$714$		0			0
$715$			0.643362i			0.0240604i
$716$	16.7724			0.626815
$717$		0			0
$718$	22.7926			0.850613
$719$		−	44.8374i		−	1.67215i	−0.548612	−	0.836077i	$-0.684844\pi$
$719$		−	44.8374i		−	1.67215i	0.548612	−	0.836077i	$-0.315156\pi$
$720$		0			0
$721$	−16.5045			−0.614658
$722$		−	33.7615i		−	1.25647i
$723$		0			0
$724$			35.9074i			1.33449i
$725$		−	1.34111i		−	0.0498074i
$726$		0			0
$727$	−5.42701			−0.201277			−0.100638	−	0.994923i	$-0.532089\pi$
$727$	−5.42701			−0.201277			−0.100638	+	0.994923i	$0.532089\pi$
$728$		−	55.2247i		−	2.04676i
$729$		0			0
$730$			47.1985i			1.74690i
$731$	1.76386			0.0652386
$732$		0			0
$733$	−28.5590			−1.05485			−0.527425	−	0.849602i	$-0.676842\pi$
$733$	−28.5590			−1.05485			−0.527425	+	0.849602i	$0.676842\pi$
$734$		−	8.61471i		−	0.317975i
$735$		0			0
$736$		−	40.2422i		−	1.48335i
$737$			0.248219i			0.00914326i
$738$		0			0
$739$	−39.2828			−1.44504			−0.722521	−	0.691349i	$-0.757017\pi$
$739$	−39.2828			−1.44504			−0.722521	+	0.691349i	$0.757017\pi$
$740$	23.9877			0.881807
$741$		0			0
$742$	25.5563			0.938202
$743$	3.00507			0.110245			0.0551227	−	0.998480i	$-0.482445\pi$
$743$	3.00507			0.110245			0.0551227	+	0.998480i	$0.482445\pi$
$744$		0			0
$745$			5.17117i			0.189457i
$746$		−	6.88324i		−	0.252013i
$747$		0			0
$748$	−0.267703			−0.00978819
$749$			25.6906i			0.938714i
$750$		0			0
$751$	−17.9955			−0.656667			−0.328333	−	0.944562i	$-0.606487\pi$
$751$	−17.9955			−0.656667			−0.328333	+	0.944562i	$0.606487\pi$
$752$	−16.5760			−0.604466
$753$		0			0
$754$		−	23.2546i		−	0.846882i
$755$	−22.4395			−0.816656
$756$		0			0
$757$	8.74112			0.317701			0.158851	−	0.987303i	$-0.449221\pi$
$757$	8.74112			0.317701			0.158851	+	0.987303i	$0.449221\pi$
$758$	−27.4605			−0.997409
$759$		0			0
$760$			33.8498i			1.22786i
$761$		−	3.84418i		−	0.139352i	−0.997570	−	0.0696758i	$-0.977804\pi$
$761$		−	3.84418i		−	0.139352i	0.997570	−	0.0696758i	$-0.0221965\pi$
$762$		0			0
$763$			24.0386i			0.870254i
$764$	−23.9093			−0.865008
$765$		0			0
$766$	18.8917			0.682586
$767$		−	21.3778i		−	0.771909i
$768$		0			0
$769$			36.0152i			1.29874i	0.760471	+	0.649372i	$0.224968\pi$
$769$			36.0152i			1.29874i	−0.760471	+	0.649372i	$0.775032\pi$
$770$	1.35624			0.0488754
$771$		0			0
$772$	−53.1545			−1.91307
$773$	−23.1702			−0.833373			−0.416686	−	0.909050i	$-0.636809\pi$
$773$	−23.1702			−0.833373			−0.416686	+	0.909050i	$0.636809\pi$
$774$		0			0
$775$	−0.885009			−0.0317905
$776$	17.0599			0.612414
$777$		0			0
$778$	73.1986			2.62430
$779$			23.1454i			0.829269i
$780$		0			0
$781$	−0.167334			−0.00598769
$782$	−9.54578			−0.341357
$783$		0			0
$784$	0.829256			0.0296163
$785$			11.2419i			0.401241i
$786$		0			0
$787$			49.3478i			1.75906i	0.475844	+	0.879530i	$0.342143\pi$
$787$			49.3478i			1.75906i	−0.475844	+	0.879530i	$0.657857\pi$
$788$			35.6988i			1.27172i
$789$		0			0
$790$			3.96965i			0.141234i
$791$	−48.1835			−1.71321
$792$		0			0
$793$			19.6762i			0.698721i
$794$	−12.7779			−0.453470
$795$		0			0
$796$			30.2182i			1.07106i
$797$		−	47.2893i		−	1.67507i	−0.546380	−	0.837537i	$-0.683995\pi$
$797$		−	47.2893i		−	1.67507i	0.546380	−	0.837537i	$-0.316005\pi$
$798$		0			0
$799$			1.32311i			0.0468083i
$800$		−	3.15397i		−	0.111510i
$801$		0			0
$802$	−34.0673			−1.20296
$803$	0.819505			0.0289197
$804$		0			0
$805$	33.6402			1.18566
$806$	−15.3459			−0.540537
$807$		0			0
$808$	−56.6408			−1.99262
$809$	−22.5981			−0.794507			−0.397253	−	0.917709i	$-0.630037\pi$
$809$	−22.5981			−0.794507			−0.397253	+	0.917709i	$0.630037\pi$
$810$		0			0
$811$		−	20.0728i		−	0.704853i	−0.935840	−	0.352426i	$-0.885357\pi$
$811$		−	20.0728i		−	0.704853i	0.935840	−	0.352426i	$-0.114643\pi$
$812$	−34.0999			−1.19667
$813$		0			0
$814$		−	0.598753i		−	0.0209863i
$815$			11.2275i			0.393282i
$816$		0			0
$817$	−6.88555			−0.240895
$818$			57.9009i			2.02446i
$819$		0			0
$820$			93.2374i			3.25599i
$821$	−30.4957			−1.06431			−0.532154	−	0.846648i	$-0.678617\pi$
$821$	−30.4957			−1.06431			−0.532154	+	0.846648i	$0.678617\pi$
$822$		0			0
$823$			53.1213i			1.85169i	0.377899	+	0.925847i	$0.376647\pi$
$823$			53.1213i			1.85169i	−0.377899	+	0.925847i	$0.623353\pi$
$824$	−41.4197			−1.44292
$825$		0			0
$826$	−45.0655			−1.56803
$827$		−	20.5301i		−	0.713901i	−0.934123	−	0.356950i	$-0.883817\pi$
$827$		−	20.5301i		−	0.713901i	0.934123	−	0.356950i	$-0.116183\pi$
$828$		0			0
$829$		−	30.9861i		−	1.07619i	−0.842884	−	0.538095i	$-0.819144\pi$
$829$		−	30.9861i		−	1.07619i	0.842884	−	0.538095i	$-0.180856\pi$
$830$			44.4605i			1.54325i
$831$		0			0
$832$		−	5.21827i		−	0.180911i
$833$		−	0.0661918i		−	0.00229341i
$834$		0			0
$835$			36.4021i			1.25975i
$836$	1.04503			0.0361431
$837$		0			0
$838$	37.0292			1.27915
$839$		−	52.6372i		−	1.81724i	−0.417627	−	0.908619i	$-0.637138\pi$
$839$		−	52.6372i		−	1.81724i	0.417627	−	0.908619i	$-0.362862\pi$
$840$		0			0
$841$	20.9243			0.721528
$842$		−	72.0739i		−	2.48383i
$843$		0			0
$844$	59.2833			2.04062
$845$		−	5.97609i		−	0.205584i
$846$		0			0
$847$			28.8564i			0.991517i
$848$	29.4237			1.01041
$849$		0			0
$850$	−0.748147			−0.0256612
$851$		−	14.8515i		−	0.509103i
$852$		0			0
$853$	4.86592			0.166606			0.0833030	−	0.996524i	$-0.473453\pi$
$853$	4.86592			0.166606			0.0833030	+	0.996524i	$0.473453\pi$
$854$	41.4783			1.41936
$855$		0			0
$856$			64.4733i			2.20365i
$857$	−14.8802			−0.508297			−0.254148	−	0.967165i	$-0.581795\pi$
$857$	−14.8802			−0.508297			−0.254148	+	0.967165i	$0.581795\pi$
$858$		0			0
$859$	39.0568			1.33260			0.666300	−	0.745684i	$-0.267877\pi$
$859$	39.0568			1.33260			0.666300	+	0.745684i	$0.267877\pi$
$860$	−27.7373			−0.945835
$861$		0			0
$862$	−78.9730			−2.68983
$863$	−16.7253			−0.569334			−0.284667	−	0.958626i	$-0.591883\pi$
$863$	−16.7253			−0.569334			−0.284667	+	0.958626i	$0.591883\pi$
$864$		0			0
$865$		−	38.9333i		−	1.32377i
$866$	−44.7537			−1.52079
$867$		0			0
$868$			22.5029i			0.763797i
$869$	0.0689248			0.00233811
$870$		0			0
$871$			8.36716i			0.283510i
$872$			60.3273i			2.04294i
$873$		0			0
$874$	37.2637			1.26047
$875$	30.5703			1.03347
$876$		0			0
$877$	−40.8967			−1.38098			−0.690492	−	0.723340i	$-0.742606\pi$
$877$	−40.8967			−1.38098			−0.690492	+	0.723340i	$0.742606\pi$
$878$			94.5965i			3.19248i
$879$		0			0
$880$	1.56147			0.0526373
$881$	−20.5090			−0.690966			−0.345483	−	0.938425i	$-0.612285\pi$
$881$	−20.5090			−0.690966			−0.345483	+	0.938425i	$0.612285\pi$
$882$		0			0
$883$	−49.6797			−1.67185			−0.835927	−	0.548841i	$-0.815069\pi$
$883$	−49.6797			−1.67185			−0.835927	+	0.548841i	$0.815069\pi$
$884$	−9.02395			−0.303508
$885$		0			0
$886$	−25.6911			−0.863109
$887$			4.12591i			0.138535i	0.997598	+	0.0692673i	$0.0220661\pi$
$887$			4.12591i			0.138535i	−0.997598	+	0.0692673i	$0.977934\pi$
$888$		0			0
$889$			0.258329i			0.00866407i
$890$	−41.1417			−1.37907
$891$		0			0
$892$		−	47.4861i		−	1.58995i
$893$		−	5.16501i		−	0.172841i
$894$		0			0
$895$			7.80895i			0.261024i
$896$	24.0924			0.804869
$897$		0			0
$898$			3.54359i			0.118251i
$899$			5.32922i			0.177740i
$900$		0			0
$901$		−	2.34862i		−	0.0782439i
$902$	2.32728			0.0774900
$903$		0			0
$904$	−120.922			−4.02179
$905$	−16.7178			−0.555720
$906$		0			0
$907$		−	0.994052i		−	0.0330069i	−0.999864	−	0.0165035i	$-0.994747\pi$
$907$		−	0.994052i		−	0.0330069i	0.999864	−	0.0165035i	$-0.00525346\pi$
$908$	−96.5842			−3.20526
$909$		0			0
$910$	45.7171			1.51551
$911$	17.8199			0.590399			0.295200	−	0.955436i	$-0.404614\pi$
$911$	17.8199			0.590399			0.295200	+	0.955436i	$0.404614\pi$
$912$		0			0
$913$	0.771964			0.0255483
$914$			32.9776i			1.09080i
$915$		0			0
$916$		−	109.473i		−	3.61708i
$917$		−	18.2116i		−	0.601400i
$918$		0			0
$919$	45.3813			1.49699			0.748496	−	0.663139i	$-0.230776\pi$
$919$	45.3813			1.49699			0.748496	+	0.663139i	$0.230776\pi$
$920$	84.4236			2.78336
$921$		0			0
$922$			71.1626i			2.34362i
$923$	−5.64064			−0.185664
$924$		0			0
$925$		−	1.16398i		−	0.0382715i
$926$	76.0946			2.50062
$927$		0			0
$928$	−18.9921			−0.623447
$929$	−34.7162			−1.13900			−0.569500	−	0.821991i	$-0.692863\pi$
$929$	−34.7162			−1.13900			−0.569500	+	0.821991i	$0.692863\pi$
$930$		0			0
$931$			0.258392i			0.00846846i
$932$	−75.9080			−2.48645
$933$		0			0
$934$		−	77.9766i		−	2.55147i
$935$		−	0.124638i		−	0.00407610i
$936$		0			0
$937$	−36.6777			−1.19821			−0.599104	−	0.800671i	$-0.704477\pi$
$937$	−36.6777			−1.19821			−0.599104	+	0.800671i	$0.704477\pi$
$938$	17.6384			0.575913
$939$		0			0
$940$		−	20.8064i		−	0.678631i
$941$	8.78024			0.286228			0.143114	−	0.989706i	$-0.454288\pi$
$941$	8.78024			0.286228			0.143114	+	0.989706i	$0.454288\pi$
$942$		0			0
$943$	57.7260			1.87982
$944$	−51.8852			−1.68872
$945$		0			0
$946$			0.692346i			0.0225101i
$947$	−10.8844			−0.353696			−0.176848	−	0.984238i	$-0.556590\pi$
$947$	−10.8844			−0.353696			−0.176848	+	0.984238i	$0.556590\pi$
$948$		0			0
$949$	27.6245			0.896730
$950$	2.92053			0.0947546
$951$		0			0
$952$			10.6986i			0.346744i
$953$	−7.05059			−0.228391			−0.114196	−	0.993458i	$-0.536429\pi$
$953$	−7.05059			−0.228391			−0.114196	+	0.993458i	$0.536429\pi$
$954$		0			0
$955$		−	11.1317i		−	0.360215i
$956$	−70.2986	+	7.11351i	−2.27362	+	0.230068i
$957$		0			0
$958$	−108.262			−3.49779
$959$		−	42.7938i		−	1.38188i
$960$		0			0
$961$	−27.4832			−0.886555
$962$		−	20.1833i		−	0.650734i
$963$		0			0
$964$	97.5544			3.14202
$965$		−	24.7478i		−	0.796661i
$966$		0			0
$967$	−32.2883			−1.03832			−0.519160	−	0.854677i	$-0.673755\pi$
$967$	−32.2883			−1.03832			−0.519160	+	0.854677i	$0.673755\pi$
$968$			72.4181i			2.32761i
$969$		0			0
$970$			14.1228i			0.453457i
$971$			23.3122i			0.748123i	0.927404	+	0.374062i	$0.122035\pi$
$971$			23.3122i			0.748123i	−0.927404	+	0.374062i	$0.877965\pi$
$972$		0			0
$973$	−53.9156			−1.72846
$974$		−	56.7184i		−	1.81738i
$975$		0			0
$976$	47.7551			1.52860
$977$	−5.36417			−0.171615			−0.0858075	−	0.996312i	$-0.527347\pi$
$977$	−5.36417			−0.171615			−0.0858075	+	0.996312i	$0.527347\pi$
$978$		0			0
$979$			0.714340i			0.0228304i
$980$			1.04089i			0.0332500i
$981$		0			0
$982$		−	22.6122i		−	0.721585i
$983$			38.9930i			1.24368i	0.783143	+	0.621841i	$0.213615\pi$
$983$			38.9930i			1.24368i	−0.783143	+	0.621841i	$0.786385\pi$
$984$		0			0
$985$	−16.6207			−0.529581
$986$			4.50509i			0.143471i
$987$		0			0
$988$	35.2267			1.12071
$989$			17.1730i			0.546069i
$990$		0			0
$991$		−	45.9628i		−	1.46006i	−0.683417	−	0.730028i	$-0.739507\pi$
$991$		−	45.9628i		−	1.46006i	0.683417	−	0.730028i	$-0.260493\pi$
$992$			12.5331i			0.397926i
$993$		0			0
$994$			11.8907i			0.377151i
$995$	−14.0691			−0.446020
$996$		0			0
$997$		−	4.14507i		−	0.131276i	−0.997844	−	0.0656379i	$-0.979092\pi$
$997$		−	4.14507i		−	0.131276i	0.997844	−	0.0656379i	$-0.0209082\pi$
$998$	−32.5335			−1.02983
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.b.a.2150.4	yes	80
3.2	odd	2	inner	2151.2.b.a.2150.78	yes	80
239.238	odd	2	inner	2151.2.b.a.2150.3	✓	80
717.716	even	2	inner	2151.2.b.a.2150.77	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.b.a.2150.3	✓	80	239.238	odd	2	inner
2151.2.b.a.2150.4	yes	80	1.1	even	1	trivial
2151.2.b.a.2150.77	yes	80	717.716	even	2	inner
2151.2.b.a.2150.78	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)

\(f(q)\)	\(=\)	\(q-2.56329i q^{2} -4.57046 q^{4} -2.12793i q^{5} +2.62545i q^{7} +6.58884i q^{8} +O(q^{10})\)	\(q-2.56329i q^{2} -4.57046 q^{4} -2.12793i q^{5} +2.62545i q^{7} +6.58884i q^{8} -5.45450 q^{10} +0.0947060i q^{11} +3.19243i q^{13} +6.72979 q^{14} +7.74819 q^{16} -0.618466i q^{17} +2.41430i q^{19} +9.72561i q^{20} +0.242759 q^{22} +6.02141 q^{23} +0.471925 q^{25} +8.18312 q^{26} -11.9995i q^{28} -2.84177i q^{29} -1.87532 q^{31} -6.68319i q^{32} -1.58531 q^{34} +5.58676 q^{35} -2.46645i q^{37} +6.18855 q^{38} +14.0206 q^{40} +9.58680 q^{41} +2.85199i q^{43} -0.432850i q^{44} -15.4346i q^{46} -2.13934 q^{47} +0.107026 q^{49} -1.20968i q^{50} -14.5909i q^{52} +3.79749 q^{53} +0.201528 q^{55} -17.2987 q^{56} -7.28430 q^{58} -6.69642 q^{59} +6.16339 q^{61} +4.80698i q^{62} -1.63458 q^{64} +6.79325 q^{65} +2.62094 q^{67} +2.82667i q^{68} -14.3205i q^{70} +1.76688i q^{71} -8.65314i q^{73} -6.32223 q^{74} -11.0344i q^{76} -0.248646 q^{77} -0.727776i q^{79} -16.4876i q^{80} -24.5737i q^{82} -8.15116i q^{83} -1.31605 q^{85} +7.31047 q^{86} -0.624003 q^{88} +7.54271 q^{89} -8.38155 q^{91} -27.5206 q^{92} +5.48376i q^{94} +5.13745 q^{95} -2.58921i q^{97} -0.274338i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 80 q^{4}+O(q^{10}) \) 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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