LMFDB - Embedded newform 3042.2.b.i.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not 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:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.i.1351.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.73205i q^{5} +2.73205i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -3.73205i q^{5} +2.73205i q^{7} +1.00000i q^{8} -3.73205 q^{10} +1.26795i q^{11} +2.73205 q^{14} +1.00000 q^{16} -5.73205 q^{17} +4.73205i q^{19} +3.73205i q^{20} +1.26795 q^{22} +4.19615 q^{23} -8.92820 q^{25} -2.73205i q^{28} +4.46410 q^{29} +1.46410i q^{31} -1.00000i q^{32} +5.73205i q^{34} +10.1962 q^{35} -3.53590i q^{37} +4.73205 q^{38} +3.73205 q^{40} +9.39230i q^{41} +9.66025 q^{43} -1.26795i q^{44} -4.19615i q^{46} +2.19615i q^{47} -0.464102 q^{49} +8.92820i q^{50} +6.46410 q^{53} +4.73205 q^{55} -2.73205 q^{56} -4.46410i q^{58} +8.00000i q^{59} -9.19615 q^{61} +1.46410 q^{62} -1.00000 q^{64} +13.1244i q^{67} +5.73205 q^{68} -10.1962i q^{70} -4.73205i q^{71} +6.26795i q^{73} -3.53590 q^{74} -4.73205i q^{76} -3.46410 q^{77} -2.53590 q^{79} -3.73205i q^{80} +9.39230 q^{82} +0.196152i q^{83} +21.3923i q^{85} -9.66025i q^{86} -1.26795 q^{88} -9.46410i q^{89} -4.19615 q^{92} +2.19615 q^{94} +17.6603 q^{95} -6.00000i q^{97} +0.464102i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 8 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} - 4 q^{23} - 8 q^{25} + 4 q^{29} + 20 q^{35} + 12 q^{38} + 8 q^{40} + 4 q^{43} + 12 q^{49} + 12 q^{53} + 12 q^{55} - 4 q^{56} - 16 q^{61} - 8 q^{62} - 4 q^{64} + 16 q^{68} - 28 q^{74} - 24 q^{79} - 4 q^{82} - 12 q^{88} + 4 q^{92} - 12 q^{94} + 36 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^10 + 4 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 - 4 * q^23 - 8 * q^25 + 4 * q^29 + 20 * q^35 + 12 * q^38 + 8 * q^40 + 4 * q^43 + 12 * q^49 + 12 * q^53 + 12 * q^55 - 4 * q^56 - 16 * q^61 - 8 * q^62 - 4 * q^64 + 16 * q^68 - 28 * q^74 - 24 * q^79 - 4 * q^82 - 12 * q^88 + 4 * q^92 - 12 * q^94 + 36 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 3.73205i	− 1.66902i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	− 3.73205i	− 1.66902i	0.550990	−	0.834512i	$-0.314250\pi$
$6$	0	0
$7$	2.73205i	1.03262i	0.856402	+	0.516309i	$0.172694\pi$
$7$	2.73205i	1.03262i	−0.856402	+	0.516309i	$0.827306\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	−3.73205	−1.18018
$11$	1.26795i	0.382301i	0.981561	+	0.191151i	$0.0612219\pi$
$11$	1.26795i	0.382301i	−0.981561	+	0.191151i	$0.938778\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.73205	0.730171
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.73205	−1.39023	−0.695113	−	0.718900i	$-0.744646\pi$
$17$	−5.73205	−1.39023	−0.695113	+	0.718900i	$0.744646\pi$
$18$	0	0
$19$	4.73205i	1.08561i	0.839860	+	0.542803i	$0.182637\pi$
$19$	4.73205i	1.08561i	−0.839860	+	0.542803i	$0.817363\pi$
$20$	3.73205i	0.834512i
$21$	0	0
$22$	1.26795	0.270328
$23$	4.19615	0.874958	0.437479	−	0.899229i	$-0.355871\pi$
$23$	4.19615	0.874958	0.437479	+	0.899229i	$0.355871\pi$
$24$	0	0
$25$	−8.92820	−1.78564
$26$	0	0
$27$	0	0
$28$	− 2.73205i	− 0.516309i
$29$	4.46410	0.828963	0.414481	−	0.910058i	$-0.363963\pi$
$29$	4.46410	0.828963	0.414481	+	0.910058i	$0.363963\pi$
$30$	0	0
$31$	1.46410i	0.262960i	0.991319	+	0.131480i	$0.0419730\pi$
$31$	1.46410i	0.262960i	−0.991319	+	0.131480i	$0.958027\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	5.73205i	0.983039i
$35$	10.1962	1.72346
$36$	0	0
$37$	− 3.53590i	− 0.581298i	−0.956830	−	0.290649i	$-0.906129\pi$
$37$	− 3.53590i	− 0.581298i	0.956830	−	0.290649i	$-0.0938712\pi$
$38$	4.73205	0.767640
$39$	0	0
$40$	3.73205	0.590089
$41$	9.39230i	1.46683i	0.679780	+	0.733416i	$0.262075\pi$
$41$	9.39230i	1.46683i	−0.679780	+	0.733416i	$0.737925\pi$
$42$	0	0
$43$	9.66025	1.47317	0.736587	−	0.676342i	$-0.236436\pi$
$43$	9.66025	1.47317	0.736587	+	0.676342i	$0.236436\pi$
$44$	− 1.26795i	− 0.191151i
$45$	0	0
$46$	− 4.19615i	− 0.618689i
$47$	2.19615i	0.320342i	0.987089	+	0.160171i	$0.0512045\pi$
$47$	2.19615i	0.320342i	−0.987089	+	0.160171i	$0.948795\pi$
$48$	0	0
$49$	−0.464102	−0.0663002
$50$	8.92820i	1.26264i
$51$	0	0
$52$	0	0
$53$	6.46410	0.887913	0.443956	−	0.896048i	$-0.353575\pi$
$53$	6.46410	0.887913	0.443956	+	0.896048i	$0.353575\pi$
$54$	0	0
$55$	4.73205	0.638070
$56$	−2.73205	−0.365086
$57$	0	0
$58$	− 4.46410i	− 0.586165i
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	−9.19615	−1.17745	−0.588723	−	0.808335i	$-0.700369\pi$
$61$	−9.19615	−1.17745	−0.588723	+	0.808335i	$0.700369\pi$
$62$	1.46410	0.185941
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	13.1244i	1.60340i	0.597730	+	0.801698i	$0.296070\pi$
$67$	13.1244i	1.60340i	−0.597730	+	0.801698i	$0.703930\pi$
$68$	5.73205	0.695113
$69$	0	0
$70$	− 10.1962i	− 1.21867i
$71$	− 4.73205i	− 0.561591i	−0.959768	−	0.280796i	$-0.909402\pi$
$71$	− 4.73205i	− 0.561591i	0.959768	−	0.280796i	$-0.0905983\pi$
$72$	0	0
$73$	6.26795i	0.733608i	0.930298	+	0.366804i	$0.119548\pi$
$73$	6.26795i	0.733608i	−0.930298	+	0.366804i	$0.880452\pi$
$74$	−3.53590	−0.411040
$75$	0	0
$76$	− 4.73205i	− 0.542803i
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−2.53590	−0.285311	−0.142655	−	0.989772i	$-0.545564\pi$
$79$	−2.53590	−0.285311	−0.142655	+	0.989772i	$0.545564\pi$
$80$	− 3.73205i	− 0.417256i
$81$	0	0
$82$	9.39230	1.03721
$83$	0.196152i	0.0215305i	0.999942	+	0.0107653i	$0.00342676\pi$
$83$	0.196152i	0.0215305i	−0.999942	+	0.0107653i	$0.996573\pi$
$84$	0	0
$85$	21.3923i	2.32032i
$86$	− 9.66025i	− 1.04169i
$87$	0	0
$88$	−1.26795	−0.135164
$89$	− 9.46410i	− 1.00319i	−0.865102	−	0.501596i	$-0.832746\pi$
$89$	− 9.46410i	− 1.00319i	0.865102	−	0.501596i	$-0.167254\pi$
$90$	0	0
$91$	0	0
$92$	−4.19615	−0.437479
$93$	0	0
$94$	2.19615	0.226516
$95$	17.6603	1.81190
$96$	0	0
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	0.464102i	0.0468813i
$99$	0	0
$100$	8.92820	0.892820
$101$	1.92820	0.191863	0.0959317	−	0.995388i	$-0.469417\pi$
$101$	1.92820	0.191863	0.0959317	+	0.995388i	$0.469417\pi$
$102$	0	0
$103$	15.2679	1.50440	0.752198	−	0.658937i	$-0.228994\pi$
$103$	15.2679	1.50440	0.752198	+	0.658937i	$0.228994\pi$
$104$	0	0
$105$	0	0
$106$	− 6.46410i	− 0.627849i
$107$	−10.1962	−0.985699	−0.492850	−	0.870114i	$-0.664045\pi$
$107$	−10.1962	−0.985699	−0.492850	+	0.870114i	$0.664045\pi$
$108$	0	0
$109$	1.46410i	0.140236i	0.997539	+	0.0701178i	$0.0223375\pi$
$109$	1.46410i	0.140236i	−0.997539	+	0.0701178i	$0.977662\pi$
$110$	− 4.73205i	− 0.451183i
$111$	0	0
$112$	2.73205i	0.258155i
$113$	1.33975	0.126033	0.0630163	−	0.998012i	$-0.479928\pi$
$113$	1.33975	0.126033	0.0630163	+	0.998012i	$0.479928\pi$
$114$	0	0
$115$	− 15.6603i	− 1.46033i
$116$	−4.46410	−0.414481
$117$	0	0
$118$	8.00000	0.736460
$119$	− 15.6603i	− 1.43557i
$120$	0	0
$121$	9.39230	0.853846
$122$	9.19615i	0.832581i
$123$	0	0
$124$	− 1.46410i	− 0.131480i
$125$	14.6603i	1.31125i
$126$	0	0
$127$	9.85641	0.874615	0.437307	−	0.899312i	$-0.355932\pi$
$127$	9.85641	0.874615	0.437307	+	0.899312i	$0.355932\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−6.53590	−0.571044	−0.285522	−	0.958372i	$-0.592167\pi$
$131$	−6.53590	−0.571044	−0.285522	+	0.958372i	$0.592167\pi$
$132$	0	0
$133$	−12.9282	−1.12102
$134$	13.1244	1.13377
$135$	0	0
$136$	− 5.73205i	− 0.491519i
$137$	11.9282i	1.01910i	0.860442	+	0.509548i	$0.170187\pi$
$137$	11.9282i	1.01910i	−0.860442	+	0.509548i	$0.829813\pi$
$138$	0	0
$139$	17.8564	1.51456	0.757280	−	0.653090i	$-0.226528\pi$
$139$	17.8564	1.51456	0.757280	+	0.653090i	$0.226528\pi$
$140$	−10.1962	−0.861732
$141$	0	0
$142$	−4.73205	−0.397105
$143$	0	0
$144$	0	0
$145$	− 16.6603i	− 1.38356i
$146$	6.26795	0.518739
$147$	0	0
$148$	3.53590i	0.290649i
$149$	− 13.1962i	− 1.08107i	−0.841321	−	0.540535i	$-0.818222\pi$
$149$	− 13.1962i	− 1.08107i	0.841321	−	0.540535i	$-0.181778\pi$
$150$	0	0
$151$	− 6.73205i	− 0.547847i	−0.961752	−	0.273923i	$-0.911679\pi$
$151$	− 6.73205i	− 0.547847i	0.961752	−	0.273923i	$-0.0883214\pi$
$152$	−4.73205	−0.383820
$153$	0	0
$154$	3.46410i	0.279145i
$155$	5.46410	0.438887
$156$	0	0
$157$	7.58846	0.605625	0.302812	−	0.953050i	$-0.402074\pi$
$157$	7.58846	0.605625	0.302812	+	0.953050i	$0.402074\pi$
$158$	2.53590i	0.201745i
$159$	0	0
$160$	−3.73205	−0.295045
$161$	11.4641i	0.903498i
$162$	0	0
$163$	13.4641i	1.05459i	0.849682	+	0.527295i	$0.176794\pi$
$163$	13.4641i	1.05459i	−0.849682	+	0.527295i	$0.823206\pi$
$164$	− 9.39230i	− 0.733416i
$165$	0	0
$166$	0.196152	0.0152244
$167$	− 9.46410i	− 0.732354i	−0.930545	−	0.366177i	$-0.880666\pi$
$167$	− 9.46410i	− 0.732354i	0.930545	−	0.366177i	$-0.119334\pi$
$168$	0	0
$169$	0	0
$170$	21.3923	1.64071
$171$	0	0
$172$	−9.66025	−0.736587
$173$	4.39230	0.333941	0.166970	−	0.985962i	$-0.446602\pi$
$173$	4.39230	0.333941	0.166970	+	0.985962i	$0.446602\pi$
$174$	0	0
$175$	− 24.3923i	− 1.84388i
$176$	1.26795i	0.0955753i
$177$	0	0
$178$	−9.46410	−0.709364
$179$	−16.0526	−1.19982	−0.599912	−	0.800066i	$-0.704798\pi$
$179$	−16.0526	−1.19982	−0.599912	+	0.800066i	$0.704798\pi$
$180$	0	0
$181$	−19.1962	−1.42684	−0.713419	−	0.700737i	$-0.752855\pi$
$181$	−19.1962	−1.42684	−0.713419	+	0.700737i	$0.752855\pi$
$182$	0	0
$183$	0	0
$184$	4.19615i	0.309344i
$185$	−13.1962	−0.970200
$186$	0	0
$187$	− 7.26795i	− 0.531485i
$188$	− 2.19615i	− 0.160171i
$189$	0	0
$190$	− 17.6603i	− 1.28121i
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	0	0
$193$	11.7321i	0.844491i	0.906481	+	0.422246i	$0.138758\pi$
$193$	11.7321i	0.844491i	−0.906481	+	0.422246i	$0.861242\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	0.464102	0.0331501
$197$	17.8564i	1.27222i	0.771600	+	0.636108i	$0.219457\pi$
$197$	17.8564i	1.27222i	−0.771600	+	0.636108i	$0.780543\pi$
$198$	0	0
$199$	14.1962	1.00634	0.503169	−	0.864188i	$-0.332167\pi$
$199$	14.1962	1.00634	0.503169	+	0.864188i	$0.332167\pi$
$200$	− 8.92820i	− 0.631319i
$201$	0	0
$202$	− 1.92820i	− 0.135668i
$203$	12.1962i	0.856002i
$204$	0	0
$205$	35.0526	2.44818
$206$	− 15.2679i	− 1.06377i
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	16.3923	1.12849	0.564246	−	0.825606i	$-0.309167\pi$
$211$	16.3923	1.12849	0.564246	+	0.825606i	$0.309167\pi$
$212$	−6.46410	−0.443956
$213$	0	0
$214$	10.1962i	0.696995i
$215$	− 36.0526i	− 2.45876i
$216$	0	0
$217$	−4.00000	−0.271538
$218$	1.46410	0.0991615
$219$	0	0
$220$	−4.73205	−0.319035
$221$	0	0
$222$	0	0
$223$	− 26.9282i	− 1.80325i	−0.432523	−	0.901623i	$-0.642377\pi$
$223$	− 26.9282i	− 1.80325i	0.432523	−	0.901623i	$-0.357623\pi$
$224$	2.73205	0.182543
$225$	0	0
$226$	− 1.33975i	− 0.0891186i
$227$	12.1962i	0.809487i	0.914430	+	0.404744i	$0.132639\pi$
$227$	12.1962i	0.809487i	−0.914430	+	0.404744i	$0.867361\pi$
$228$	0	0
$229$	11.8564i	0.783493i	0.920073	+	0.391747i	$0.128129\pi$
$229$	11.8564i	0.783493i	−0.920073	+	0.391747i	$0.871871\pi$
$230$	−15.6603	−1.03261
$231$	0	0
$232$	4.46410i	0.293083i
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	0	0
$235$	8.19615	0.534658
$236$	− 8.00000i	− 0.520756i
$237$	0	0
$238$	−15.6603	−1.01510
$239$	− 7.66025i	− 0.495501i	−0.968824	−	0.247750i	$-0.920309\pi$
$239$	− 7.66025i	− 0.495501i	0.968824	−	0.247750i	$-0.0796913\pi$
$240$	0	0
$241$	− 13.5885i	− 0.875309i	−0.899143	−	0.437655i	$-0.855809\pi$
$241$	− 13.5885i	− 0.875309i	0.899143	−	0.437655i	$-0.144191\pi$
$242$	− 9.39230i	− 0.603760i
$243$	0	0
$244$	9.19615	0.588723
$245$	1.73205i	0.110657i
$246$	0	0
$247$	0	0
$248$	−1.46410	−0.0929705
$249$	0	0
$250$	14.6603	0.927196
$251$	13.4641	0.849847	0.424923	−	0.905229i	$-0.360301\pi$
$251$	13.4641	0.849847	0.424923	+	0.905229i	$0.360301\pi$
$252$	0	0
$253$	5.32051i	0.334497i
$254$	− 9.85641i	− 0.618446i
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.33975	0.582597	0.291299	−	0.956632i	$-0.405913\pi$
$257$	9.33975	0.582597	0.291299	+	0.956632i	$0.405913\pi$
$258$	0	0
$259$	9.66025	0.600259
$260$	0	0
$261$	0	0
$262$	6.53590i	0.403789i
$263$	10.0526	0.619867	0.309934	−	0.950758i	$-0.399693\pi$
$263$	10.0526	0.619867	0.309934	+	0.950758i	$0.399693\pi$
$264$	0	0
$265$	− 24.1244i	− 1.48195i
$266$	12.9282i	0.792679i
$267$	0	0
$268$	− 13.1244i	− 0.801698i
$269$	−5.46410	−0.333152	−0.166576	−	0.986029i	$-0.553271\pi$
$269$	−5.46410	−0.333152	−0.166576	+	0.986029i	$0.553271\pi$
$270$	0	0
$271$	21.8564i	1.32768i	0.747874	+	0.663841i	$0.231075\pi$
$271$	21.8564i	1.32768i	−0.747874	+	0.663841i	$0.768925\pi$
$272$	−5.73205	−0.347557
$273$	0	0
$274$	11.9282	0.720609
$275$	− 11.3205i	− 0.682652i
$276$	0	0
$277$	5.73205	0.344406	0.172203	−	0.985062i	$-0.444912\pi$
$277$	5.73205	0.344406	0.172203	+	0.985062i	$0.444912\pi$
$278$	− 17.8564i	− 1.07096i
$279$	0	0
$280$	10.1962i	0.609337i
$281$	12.3205i	0.734980i	0.930027	+	0.367490i	$0.119783\pi$
$281$	12.3205i	0.734980i	−0.930027	+	0.367490i	$0.880217\pi$
$282$	0	0
$283$	−25.6603	−1.52534	−0.762672	−	0.646786i	$-0.776113\pi$
$283$	−25.6603	−1.52534	−0.762672	+	0.646786i	$0.776113\pi$
$284$	4.73205i	0.280796i
$285$	0	0
$286$	0	0
$287$	−25.6603	−1.51468
$288$	0	0
$289$	15.8564	0.932730
$290$	−16.6603	−0.978324
$291$	0	0
$292$	− 6.26795i	− 0.366804i
$293$	30.5167i	1.78280i	0.453215	+	0.891401i	$0.350277\pi$
$293$	30.5167i	1.78280i	−0.453215	+	0.891401i	$0.649723\pi$
$294$	0	0
$295$	29.8564	1.73831
$296$	3.53590	0.205520
$297$	0	0
$298$	−13.1962	−0.764433
$299$	0	0
$300$	0	0
$301$	26.3923i	1.52123i
$302$	−6.73205	−0.387386
$303$	0	0
$304$	4.73205i	0.271402i
$305$	34.3205i	1.96519i
$306$	0	0
$307$	22.5885i	1.28919i	0.764524	+	0.644596i	$0.222974\pi$
$307$	22.5885i	1.28919i	−0.764524	+	0.644596i	$0.777026\pi$
$308$	3.46410	0.197386
$309$	0	0
$310$	− 5.46410i	− 0.310340i
$311$	−1.66025	−0.0941444	−0.0470722	−	0.998891i	$-0.514989\pi$
$311$	−1.66025	−0.0941444	−0.0470722	+	0.998891i	$0.514989\pi$
$312$	0	0
$313$	6.53590	0.369431	0.184715	−	0.982792i	$-0.440864\pi$
$313$	6.53590	0.369431	0.184715	+	0.982792i	$0.440864\pi$
$314$	− 7.58846i	− 0.428241i
$315$	0	0
$316$	2.53590	0.142655
$317$	− 20.6603i	− 1.16040i	−0.814476	−	0.580198i	$-0.802975\pi$
$317$	− 20.6603i	− 1.16040i	0.814476	−	0.580198i	$-0.197025\pi$
$318$	0	0
$319$	5.66025i	0.316913i
$320$	3.73205i	0.208628i
$321$	0	0
$322$	11.4641	0.638869
$323$	− 27.1244i	− 1.50924i
$324$	0	0
$325$	0	0
$326$	13.4641	0.745708
$327$	0	0
$328$	−9.39230	−0.518603
$329$	−6.00000	−0.330791
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	− 0.196152i	− 0.0107653i
$333$	0	0
$334$	−9.46410	−0.517853
$335$	48.9808	2.67610
$336$	0	0
$337$	20.8564	1.13612	0.568060	−	0.822987i	$-0.307694\pi$
$337$	20.8564	1.13612	0.568060	+	0.822987i	$0.307694\pi$
$338$	0	0
$339$	0	0
$340$	− 21.3923i	− 1.16016i
$341$	−1.85641	−0.100530
$342$	0	0
$343$	17.8564i	0.964155i
$344$	9.66025i	0.520846i
$345$	0	0
$346$	− 4.39230i	− 0.236132i
$347$	−33.1244	−1.77821	−0.889104	−	0.457705i	$-0.848672\pi$
$347$	−33.1244	−1.77821	−0.889104	+	0.457705i	$0.848672\pi$
$348$	0	0
$349$	− 15.3205i	− 0.820088i	−0.912066	−	0.410044i	$-0.865513\pi$
$349$	− 15.3205i	− 0.820088i	0.912066	−	0.410044i	$-0.134487\pi$
$350$	−24.3923	−1.30382
$351$	0	0
$352$	1.26795	0.0675819
$353$	21.7846i	1.15948i	0.814802	+	0.579739i	$0.196845\pi$
$353$	21.7846i	1.15948i	−0.814802	+	0.579739i	$0.803155\pi$
$354$	0	0
$355$	−17.6603	−0.937309
$356$	9.46410i	0.501596i
$357$	0	0
$358$	16.0526i	0.848404i
$359$	− 1.12436i	− 0.0593412i	−0.999560	−	0.0296706i	$-0.990554\pi$
$359$	− 1.12436i	− 0.0593412i	0.999560	−	0.0296706i	$-0.00944584\pi$
$360$	0	0
$361$	−3.39230	−0.178542
$362$	19.1962i	1.00893i
$363$	0	0
$364$	0	0
$365$	23.3923	1.22441
$366$	0	0
$367$	−11.2679	−0.588182	−0.294091	−	0.955777i	$-0.595017\pi$
$367$	−11.2679	−0.588182	−0.294091	+	0.955777i	$0.595017\pi$
$368$	4.19615	0.218740
$369$	0	0
$370$	13.1962i	0.686035i
$371$	17.6603i	0.916875i
$372$	0	0
$373$	−13.7321	−0.711019	−0.355509	−	0.934673i	$-0.615693\pi$
$373$	−13.7321	−0.711019	−0.355509	+	0.934673i	$0.615693\pi$
$374$	−7.26795	−0.375817
$375$	0	0
$376$	−2.19615	−0.113258
$377$	0	0
$378$	0	0
$379$	− 5.46410i	− 0.280672i	−0.990104	−	0.140336i	$-0.955182\pi$
$379$	− 5.46410i	− 0.280672i	0.990104	−	0.140336i	$-0.0448183\pi$
$380$	−17.6603	−0.905952
$381$	0	0
$382$	− 6.92820i	− 0.354478i
$383$	1.46410i	0.0748121i	0.999300	+	0.0374060i	$0.0119095\pi$
$383$	1.46410i	0.0748121i	−0.999300	+	0.0374060i	$0.988091\pi$
$384$	0	0
$385$	12.9282i	0.658882i
$386$	11.7321	0.597146
$387$	0	0
$388$	6.00000i	0.304604i
$389$	11.7846	0.597503	0.298752	−	0.954331i	$-0.403430\pi$
$389$	11.7846	0.597503	0.298752	+	0.954331i	$0.403430\pi$
$390$	0	0
$391$	−24.0526	−1.21639
$392$	− 0.464102i	− 0.0234407i
$393$	0	0
$394$	17.8564	0.899593
$395$	9.46410i	0.476191i
$396$	0	0
$397$	− 20.3923i	− 1.02346i	−0.859146	−	0.511730i	$-0.829005\pi$
$397$	− 20.3923i	− 1.02346i	0.859146	−	0.511730i	$-0.170995\pi$
$398$	− 14.1962i	− 0.711589i
$399$	0	0
$400$	−8.92820	−0.446410
$401$	8.07180i	0.403086i	0.979480	+	0.201543i	$0.0645956\pi$
$401$	8.07180i	0.403086i	−0.979480	+	0.201543i	$0.935404\pi$
$402$	0	0
$403$	0	0
$404$	−1.92820	−0.0959317
$405$	0	0
$406$	12.1962	0.605285
$407$	4.48334	0.222231
$408$	0	0
$409$	17.7321i	0.876793i	0.898782	+	0.438397i	$0.144454\pi$
$409$	17.7321i	0.876793i	−0.898782	+	0.438397i	$0.855546\pi$
$410$	− 35.0526i	− 1.73112i
$411$	0	0
$412$	−15.2679	−0.752198
$413$	−21.8564	−1.07548
$414$	0	0
$415$	0.732051	0.0359350
$416$	0	0
$417$	0	0
$418$	6.00000i	0.293470i
$419$	17.4641	0.853177	0.426589	−	0.904446i	$-0.359715\pi$
$419$	17.4641	0.853177	0.426589	+	0.904446i	$0.359715\pi$
$420$	0	0
$421$	− 22.7128i	− 1.10695i	−0.832864	−	0.553477i	$-0.813301\pi$
$421$	− 22.7128i	− 1.10695i	0.832864	−	0.553477i	$-0.186699\pi$
$422$	− 16.3923i	− 0.797965i
$423$	0	0
$424$	6.46410i	0.313925i
$425$	51.1769	2.48244
$426$	0	0
$427$	− 25.1244i	− 1.21585i
$428$	10.1962	0.492850
$429$	0	0
$430$	−36.0526	−1.73861
$431$	− 13.1244i	− 0.632178i	−0.948730	−	0.316089i	$-0.897630\pi$
$431$	− 13.1244i	− 0.632178i	0.948730	−	0.316089i	$-0.102370\pi$
$432$	0	0
$433$	−12.8564	−0.617839	−0.308920	−	0.951088i	$-0.599967\pi$
$433$	−12.8564	−0.617839	−0.308920	+	0.951088i	$0.599967\pi$
$434$	4.00000i	0.192006i
$435$	0	0
$436$	− 1.46410i	− 0.0701178i
$437$	19.8564i	0.949861i
$438$	0	0
$439$	0.339746	0.0162152	0.00810760	−	0.999967i	$-0.497419\pi$
$439$	0.339746	0.0162152	0.00810760	+	0.999967i	$0.497419\pi$
$440$	4.73205i	0.225592i
$441$	0	0
$442$	0	0
$443$	−15.6077	−0.741544	−0.370772	−	0.928724i	$-0.620907\pi$
$443$	−15.6077	−0.741544	−0.370772	+	0.928724i	$0.620907\pi$
$444$	0	0
$445$	−35.3205	−1.67435
$446$	−26.9282	−1.27509
$447$	0	0
$448$	− 2.73205i	− 0.129077i
$449$	11.3205i	0.534248i	0.963662	+	0.267124i	$0.0860733\pi$
$449$	11.3205i	0.534248i	−0.963662	+	0.267124i	$0.913927\pi$
$450$	0	0
$451$	−11.9090	−0.560771
$452$	−1.33975	−0.0630163
$453$	0	0
$454$	12.1962	0.572394
$455$	0	0
$456$	0	0
$457$	1.33975i	0.0626707i	0.999509	+	0.0313353i	$0.00997598\pi$
$457$	1.33975i	0.0626707i	−0.999509	+	0.0313353i	$0.990024\pi$
$458$	11.8564	0.554013
$459$	0	0
$460$	15.6603i	0.730163i
$461$	− 22.2679i	− 1.03712i	−0.855041	−	0.518561i	$-0.826468\pi$
$461$	− 22.2679i	− 1.03712i	0.855041	−	0.518561i	$-0.173532\pi$
$462$	0	0
$463$	− 10.0526i	− 0.467182i	−0.972335	−	0.233591i	$-0.924952\pi$
$463$	− 10.0526i	− 0.467182i	0.972335	−	0.233591i	$-0.0750477\pi$
$464$	4.46410	0.207241
$465$	0	0
$466$	7.85641i	0.363941i
$467$	18.5885	0.860171	0.430086	−	0.902788i	$-0.358483\pi$
$467$	18.5885	0.860171	0.430086	+	0.902788i	$0.358483\pi$
$468$	0	0
$469$	−35.8564	−1.65570
$470$	− 8.19615i	− 0.378060i
$471$	0	0
$472$	−8.00000	−0.368230
$473$	12.2487i	0.563196i
$474$	0	0
$475$	− 42.2487i	− 1.93850i
$476$	15.6603i	0.717787i
$477$	0	0
$478$	−7.66025	−0.350372
$479$	33.4641i	1.52901i	0.644616	+	0.764507i	$0.277017\pi$
$479$	33.4641i	1.52901i	−0.644616	+	0.764507i	$0.722983\pi$
$480$	0	0
$481$	0	0
$482$	−13.5885	−0.618937
$483$	0	0
$484$	−9.39230	−0.426923
$485$	−22.3923	−1.01678
$486$	0	0
$487$	− 3.12436i	− 0.141578i	−0.997491	−	0.0707890i	$-0.977448\pi$
$487$	− 3.12436i	− 0.141578i	0.997491	−	0.0707890i	$-0.0225517\pi$
$488$	− 9.19615i	− 0.416290i
$489$	0	0
$490$	1.73205	0.0782461
$491$	8.73205	0.394072	0.197036	−	0.980396i	$-0.436868\pi$
$491$	8.73205	0.394072	0.197036	+	0.980396i	$0.436868\pi$
$492$	0	0
$493$	−25.5885	−1.15245
$494$	0	0
$495$	0	0
$496$	1.46410i	0.0657401i
$497$	12.9282	0.579909
$498$	0	0
$499$	32.0000i	1.43252i	0.697835	+	0.716258i	$0.254147\pi$
$499$	32.0000i	1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$	− 14.6603i	− 0.655626i
$501$	0	0
$502$	− 13.4641i	− 0.600932i
$503$	−40.9808	−1.82724	−0.913621	−	0.406567i	$-0.866726\pi$
$503$	−40.9808	−1.82724	−0.913621	+	0.406567i	$0.866726\pi$
$504$	0	0
$505$	− 7.19615i	− 0.320225i
$506$	5.32051	0.236525
$507$	0	0
$508$	−9.85641	−0.437307
$509$	− 13.7321i	− 0.608662i	−0.952566	−	0.304331i	$-0.901567\pi$
$509$	− 13.7321i	− 0.608662i	0.952566	−	0.304331i	$-0.0984330\pi$
$510$	0	0
$511$	−17.1244	−0.757537
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 9.33975i	− 0.411959i
$515$	− 56.9808i	− 2.51087i
$516$	0	0
$517$	−2.78461	−0.122467
$518$	− 9.66025i	− 0.424447i
$519$	0	0
$520$	0	0
$521$	−41.4449	−1.81573	−0.907866	−	0.419260i	$-0.862290\pi$
$521$	−41.4449	−1.81573	−0.907866	+	0.419260i	$0.862290\pi$
$522$	0	0
$523$	−22.4449	−0.981445	−0.490723	−	0.871316i	$-0.663267\pi$
$523$	−22.4449	−0.981445	−0.490723	+	0.871316i	$0.663267\pi$
$524$	6.53590	0.285522
$525$	0	0
$526$	− 10.0526i	− 0.438312i
$527$	− 8.39230i	− 0.365575i
$528$	0	0
$529$	−5.39230	−0.234448
$530$	−24.1244	−1.04790
$531$	0	0
$532$	12.9282	0.560509
$533$	0	0
$534$	0	0
$535$	38.0526i	1.64516i
$536$	−13.1244	−0.566886
$537$	0	0
$538$	5.46410i	0.235574i
$539$	− 0.588457i	− 0.0253466i
$540$	0	0
$541$	− 5.67949i	− 0.244180i	−0.992519	−	0.122090i	$-0.961040\pi$
$541$	− 5.67949i	− 0.244180i	0.992519	−	0.122090i	$-0.0389597\pi$
$542$	21.8564	0.938813
$543$	0	0
$544$	5.73205i	0.245760i
$545$	5.46410	0.234056
$546$	0	0
$547$	−4.19615	−0.179415	−0.0897073	−	0.995968i	$-0.528593\pi$
$547$	−4.19615	−0.179415	−0.0897073	+	0.995968i	$0.528593\pi$
$548$	− 11.9282i	− 0.509548i
$549$	0	0
$550$	−11.3205	−0.482708
$551$	21.1244i	0.899928i
$552$	0	0
$553$	− 6.92820i	− 0.294617i
$554$	− 5.73205i	− 0.243532i
$555$	0	0
$556$	−17.8564	−0.757280
$557$	42.3731i	1.79540i	0.440603	+	0.897702i	$0.354765\pi$
$557$	42.3731i	1.79540i	−0.440603	+	0.897702i	$0.645235\pi$
$558$	0	0
$559$	0	0
$560$	10.1962	0.430866
$561$	0	0
$562$	12.3205	0.519709
$563$	34.9282	1.47205	0.736024	−	0.676955i	$-0.236701\pi$
$563$	34.9282	1.47205	0.736024	+	0.676955i	$0.236701\pi$
$564$	0	0
$565$	− 5.00000i	− 0.210352i
$566$	25.6603i	1.07858i
$567$	0	0
$568$	4.73205	0.198552
$569$	30.6410	1.28454	0.642269	−	0.766479i	$-0.277993\pi$
$569$	30.6410	1.28454	0.642269	+	0.766479i	$0.277993\pi$
$570$	0	0
$571$	−14.0526	−0.588081	−0.294041	−	0.955793i	$-0.595000\pi$
$571$	−14.0526	−0.588081	−0.294041	+	0.955793i	$0.595000\pi$
$572$	0	0
$573$	0	0
$574$	25.6603i	1.07104i
$575$	−37.4641	−1.56236
$576$	0	0
$577$	3.73205i	0.155367i	0.996978	+	0.0776837i	$0.0247524\pi$
$577$	3.73205i	0.155367i	−0.996978	+	0.0776837i	$0.975248\pi$
$578$	− 15.8564i	− 0.659540i
$579$	0	0
$580$	16.6603i	0.691779i
$581$	−0.535898	−0.0222328
$582$	0	0
$583$	8.19615i	0.339450i
$584$	−6.26795	−0.259370
$585$	0	0
$586$	30.5167	1.26063
$587$	− 16.0000i	− 0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$	− 16.0000i	− 0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$	0	0
$589$	−6.92820	−0.285472
$590$	− 29.8564i	− 1.22917i
$591$	0	0
$592$	− 3.53590i	− 0.145325i
$593$	− 9.14359i	− 0.375482i	−0.982219	−	0.187741i	$-0.939883\pi$
$593$	− 9.14359i	− 0.375482i	0.982219	−	0.187741i	$-0.0601166\pi$
$594$	0	0
$595$	−58.4449	−2.39601
$596$	13.1962i	0.540535i
$597$	0	0
$598$	0	0
$599$	2.53590	0.103614	0.0518070	−	0.998657i	$-0.483502\pi$
$599$	2.53590	0.103614	0.0518070	+	0.998657i	$0.483502\pi$
$600$	0	0
$601$	7.92820	0.323398	0.161699	−	0.986840i	$-0.448303\pi$
$601$	7.92820	0.323398	0.161699	+	0.986840i	$0.448303\pi$
$602$	26.3923	1.07567
$603$	0	0
$604$	6.73205i	0.273923i
$605$	− 35.0526i	− 1.42509i
$606$	0	0
$607$	−40.7846	−1.65540	−0.827698	−	0.561174i	$-0.810350\pi$
$607$	−40.7846	−1.65540	−0.827698	+	0.561174i	$0.810350\pi$
$608$	4.73205	0.191910
$609$	0	0
$610$	34.3205	1.38960
$611$	0	0
$612$	0	0
$613$	− 9.39230i	− 0.379352i	−0.981847	−	0.189676i	$-0.939256\pi$
$613$	− 9.39230i	− 0.379352i	0.981847	−	0.189676i	$-0.0607437\pi$
$614$	22.5885	0.911596
$615$	0	0
$616$	− 3.46410i	− 0.139573i
$617$	13.2487i	0.533373i	0.963783	+	0.266687i	$0.0859288\pi$
$617$	13.2487i	0.533373i	−0.963783	+	0.266687i	$0.914071\pi$
$618$	0	0
$619$	− 17.4641i	− 0.701942i	−0.936386	−	0.350971i	$-0.885852\pi$
$619$	− 17.4641i	− 0.701942i	0.936386	−	0.350971i	$-0.114148\pi$
$620$	−5.46410	−0.219444
$621$	0	0
$622$	1.66025i	0.0665701i
$623$	25.8564	1.03592
$624$	0	0
$625$	10.0718	0.402872
$626$	− 6.53590i	− 0.261227i
$627$	0	0
$628$	−7.58846	−0.302812
$629$	20.2679i	0.808136i
$630$	0	0
$631$	7.71281i	0.307042i	0.988145	+	0.153521i	$0.0490613\pi$
$631$	7.71281i	0.307042i	−0.988145	+	0.153521i	$0.950939\pi$
$632$	− 2.53590i	− 0.100873i
$633$	0	0
$634$	−20.6603	−0.820524
$635$	− 36.7846i	− 1.45975i
$636$	0	0
$637$	0	0
$638$	5.66025	0.224092
$639$	0	0
$640$	3.73205	0.147522
$641$	−25.9808	−1.02618	−0.513089	−	0.858335i	$-0.671499\pi$
$641$	−25.9808	−1.02618	−0.513089	+	0.858335i	$0.671499\pi$
$642$	0	0
$643$	− 13.8564i	− 0.546443i	−0.961951	−	0.273222i	$-0.911911\pi$
$643$	− 13.8564i	− 0.546443i	0.961951	−	0.273222i	$-0.0880892\pi$
$644$	− 11.4641i	− 0.451749i
$645$	0	0
$646$	−27.1244	−1.06719
$647$	−22.2487	−0.874687	−0.437344	−	0.899295i	$-0.644081\pi$
$647$	−22.2487	−0.874687	−0.437344	+	0.899295i	$0.644081\pi$
$648$	0	0
$649$	−10.1436	−0.398171
$650$	0	0
$651$	0	0
$652$	− 13.4641i	− 0.527295i
$653$	17.4641	0.683423	0.341712	−	0.939805i	$-0.388993\pi$
$653$	17.4641	0.683423	0.341712	+	0.939805i	$0.388993\pi$
$654$	0	0
$655$	24.3923i	0.953086i
$656$	9.39230i	0.366708i
$657$	0	0
$658$	6.00000i	0.233904i
$659$	10.2487	0.399233	0.199617	−	0.979874i	$-0.436030\pi$
$659$	10.2487	0.399233	0.199617	+	0.979874i	$0.436030\pi$
$660$	0	0
$661$	− 11.3923i	− 0.443109i	−0.975148	−	0.221555i	$-0.928887\pi$
$661$	− 11.3923i	− 0.443109i	0.975148	−	0.221555i	$-0.0711131\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−0.196152	−0.00761219
$665$	48.2487i	1.87100i
$666$	0	0
$667$	18.7321	0.725308
$668$	9.46410i	0.366177i
$669$	0	0
$670$	− 48.9808i	− 1.89229i
$671$	− 11.6603i	− 0.450139i
$672$	0	0
$673$	−27.9282	−1.07655	−0.538277	−	0.842768i	$-0.680924\pi$
$673$	−27.9282	−1.07655	−0.538277	+	0.842768i	$0.680924\pi$
$674$	− 20.8564i	− 0.803359i
$675$	0	0
$676$	0	0
$677$	45.4641	1.74733	0.873664	−	0.486530i	$-0.161738\pi$
$677$	45.4641	1.74733	0.873664	+	0.486530i	$0.161738\pi$
$678$	0	0
$679$	16.3923	0.629079
$680$	−21.3923	−0.820357
$681$	0	0
$682$	1.85641i	0.0710855i
$683$	− 10.1436i	− 0.388134i	−0.980988	−	0.194067i	$-0.937832\pi$
$683$	− 10.1436i	− 0.388134i	0.980988	−	0.194067i	$-0.0621679\pi$
$684$	0	0
$685$	44.5167	1.70089
$686$	17.8564	0.681761
$687$	0	0
$688$	9.66025	0.368294
$689$	0	0
$690$	0	0
$691$	43.6603i	1.66091i	0.557082	+	0.830457i	$0.311921\pi$
$691$	43.6603i	1.66091i	−0.557082	+	0.830457i	$0.688079\pi$
$692$	−4.39230	−0.166970
$693$	0	0
$694$	33.1244i	1.25738i
$695$	− 66.6410i	− 2.52784i
$696$	0	0
$697$	− 53.8372i	− 2.03923i
$698$	−15.3205	−0.579890
$699$	0	0
$700$	24.3923i	0.921942i
$701$	−3.32051	−0.125414	−0.0627069	−	0.998032i	$-0.519973\pi$
$701$	−3.32051	−0.125414	−0.0627069	+	0.998032i	$0.519973\pi$
$702$	0	0
$703$	16.7321	0.631061
$704$	− 1.26795i	− 0.0477876i
$705$	0	0
$706$	21.7846	0.819875
$707$	5.26795i	0.198122i
$708$	0	0
$709$	− 13.1436i	− 0.493618i	−0.969064	−	0.246809i	$-0.920618\pi$
$709$	− 13.1436i	− 0.493618i	0.969064	−	0.246809i	$-0.0793820\pi$
$710$	17.6603i	0.662778i
$711$	0	0
$712$	9.46410	0.354682
$713$	6.14359i	0.230079i
$714$	0	0
$715$	0	0
$716$	16.0526	0.599912
$717$	0	0
$718$	−1.12436	−0.0419606
$719$	−29.4641	−1.09883	−0.549413	−	0.835551i	$-0.685149\pi$
$719$	−29.4641	−1.09883	−0.549413	+	0.835551i	$0.685149\pi$
$720$	0	0
$721$	41.7128i	1.55347i
$722$	3.39230i	0.126249i
$723$	0	0
$724$	19.1962	0.713419
$725$	−39.8564	−1.48023
$726$	0	0
$727$	30.9808	1.14901	0.574506	−	0.818500i	$-0.305194\pi$
$727$	30.9808	1.14901	0.574506	+	0.818500i	$0.305194\pi$
$728$	0	0
$729$	0	0
$730$	− 23.3923i	− 0.865788i
$731$	−55.3731	−2.04805
$732$	0	0
$733$	19.0000i	0.701781i	0.936416	+	0.350891i	$0.114121\pi$
$733$	19.0000i	0.701781i	−0.936416	+	0.350891i	$0.885879\pi$
$734$	11.2679i	0.415908i
$735$	0	0
$736$	− 4.19615i	− 0.154672i
$737$	−16.6410	−0.612980
$738$	0	0
$739$	− 2.92820i	− 0.107716i	−0.998549	−	0.0538578i	$-0.982848\pi$
$739$	− 2.92820i	− 0.107716i	0.998549	−	0.0538578i	$-0.0171518\pi$
$740$	13.1962	0.485100
$741$	0	0
$742$	17.6603	0.648328
$743$	48.3923i	1.77534i	0.460479	+	0.887671i	$0.347678\pi$
$743$	48.3923i	1.77534i	−0.460479	+	0.887671i	$0.652322\pi$
$744$	0	0
$745$	−49.2487	−1.80433
$746$	13.7321i	0.502766i
$747$	0	0
$748$	7.26795i	0.265743i
$749$	− 27.8564i	− 1.01785i
$750$	0	0
$751$	49.9090	1.82120	0.910602	−	0.413284i	$-0.135618\pi$
$751$	49.9090	1.82120	0.910602	+	0.413284i	$0.135618\pi$
$752$	2.19615i	0.0800854i
$753$	0	0
$754$	0	0
$755$	−25.1244	−0.914369
$756$	0	0
$757$	20.9282	0.760648	0.380324	−	0.924853i	$-0.375812\pi$
$757$	20.9282	0.760648	0.380324	+	0.924853i	$0.375812\pi$
$758$	−5.46410	−0.198465
$759$	0	0
$760$	17.6603i	0.640605i
$761$	11.3205i	0.410368i	0.978723	+	0.205184i	$0.0657793\pi$
$761$	11.3205i	0.410368i	−0.978723	+	0.205184i	$0.934221\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	−6.92820	−0.250654
$765$	0	0
$766$	1.46410	0.0529001
$767$	0	0
$768$	0	0
$769$	− 43.8564i	− 1.58150i	−0.612138	−	0.790751i	$-0.709690\pi$
$769$	− 43.8564i	− 1.58150i	0.612138	−	0.790751i	$-0.290310\pi$
$770$	12.9282	0.465900
$771$	0	0
$772$	− 11.7321i	− 0.422246i
$773$	48.9282i	1.75983i	0.475136	+	0.879913i	$0.342399\pi$
$773$	48.9282i	1.75983i	−0.475136	+	0.879913i	$0.657601\pi$
$774$	0	0
$775$	− 13.0718i	− 0.469553i
$776$	6.00000	0.215387
$777$	0	0
$778$	− 11.7846i	− 0.422499i
$779$	−44.4449	−1.59240
$780$	0	0
$781$	6.00000	0.214697
$782$	24.0526i	0.860118i
$783$	0	0
$784$	−0.464102	−0.0165751
$785$	− 28.3205i	− 1.01080i
$786$	0	0
$787$	− 4.67949i	− 0.166806i	−0.996516	−	0.0834029i	$-0.973421\pi$
$787$	− 4.67949i	− 0.166806i	0.996516	−	0.0834029i	$-0.0265789\pi$
$788$	− 17.8564i	− 0.636108i
$789$	0	0
$790$	9.46410	0.336718
$791$	3.66025i	0.130144i
$792$	0	0
$793$	0	0
$794$	−20.3923	−0.723696
$795$	0	0
$796$	−14.1962	−0.503169
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	− 12.5885i	− 0.445348i
$800$	8.92820i	0.315660i
$801$	0	0
$802$	8.07180	0.285025
$803$	−7.94744	−0.280459
$804$	0	0
$805$	42.7846	1.50796
$806$	0	0
$807$	0	0
$808$	1.92820i	0.0678340i
$809$	53.5885	1.88407	0.942035	−	0.335515i	$-0.108910\pi$
$809$	53.5885	1.88407	0.942035	+	0.335515i	$0.108910\pi$
$810$	0	0
$811$	17.1769i	0.603163i	0.953440	+	0.301582i	$0.0975145\pi$
$811$	17.1769i	0.603163i	−0.953440	+	0.301582i	$0.902485\pi$
$812$	− 12.1962i	− 0.428001i
$813$	0	0
$814$	− 4.48334i	− 0.157141i
$815$	50.2487	1.76014
$816$	0	0
$817$	45.7128i	1.59929i
$818$	17.7321	0.619987
$819$	0	0
$820$	−35.0526	−1.22409
$821$	0.928203i	0.0323945i	0.999869	+	0.0161973i	$0.00515597\pi$
$821$	0.928203i	0.0323945i	−0.999869	+	0.0161973i	$0.994844\pi$
$822$	0	0
$823$	−41.5692	−1.44901	−0.724506	−	0.689269i	$-0.757932\pi$
$823$	−41.5692	−1.44901	−0.724506	+	0.689269i	$0.757932\pi$
$824$	15.2679i	0.531884i
$825$	0	0
$826$	21.8564i	0.760482i
$827$	26.5359i	0.922744i	0.887207	+	0.461372i	$0.152643\pi$
$827$	26.5359i	0.922744i	−0.887207	+	0.461372i	$0.847357\pi$
$828$	0	0
$829$	12.1244	0.421096	0.210548	−	0.977583i	$-0.432475\pi$
$829$	12.1244	0.421096	0.210548	+	0.977583i	$0.432475\pi$
$830$	− 0.732051i	− 0.0254099i
$831$	0	0
$832$	0	0
$833$	2.66025	0.0921723
$834$	0	0
$835$	−35.3205	−1.22232
$836$	6.00000	0.207514
$837$	0	0
$838$	− 17.4641i	− 0.603287i
$839$	41.8564i	1.44504i	0.691348	+	0.722522i	$0.257017\pi$
$839$	41.8564i	1.44504i	−0.691348	+	0.722522i	$0.742983\pi$
$840$	0	0
$841$	−9.07180	−0.312821
$842$	−22.7128	−0.782735
$843$	0	0
$844$	−16.3923	−0.564246
$845$	0	0
$846$	0	0
$847$	25.6603i	0.881697i
$848$	6.46410	0.221978
$849$	0	0
$850$	− 51.1769i	− 1.75535i
$851$	− 14.8372i	− 0.508612i
$852$	0	0
$853$	54.1769i	1.85498i	0.373845	+	0.927491i	$0.378039\pi$
$853$	54.1769i	1.85498i	−0.373845	+	0.927491i	$0.621961\pi$
$854$	−25.1244	−0.859738
$855$	0	0
$856$	− 10.1962i	− 0.348497i
$857$	39.4449	1.34741	0.673705	−	0.739000i	$-0.264702\pi$
$857$	39.4449	1.34741	0.673705	+	0.739000i	$0.264702\pi$
$858$	0	0
$859$	−47.1244	−1.60786	−0.803931	−	0.594722i	$-0.797262\pi$
$859$	−47.1244	−1.60786	−0.803931	+	0.594722i	$0.797262\pi$
$860$	36.0526i	1.22938i
$861$	0	0
$862$	−13.1244	−0.447017
$863$	17.1244i	0.582920i	0.956583	+	0.291460i	$0.0941410\pi$
$863$	17.1244i	0.582920i	−0.956583	+	0.291460i	$0.905859\pi$
$864$	0	0
$865$	− 16.3923i	− 0.557355i
$866$	12.8564i	0.436878i
$867$	0	0
$868$	4.00000	0.135769
$869$	− 3.21539i	− 0.109075i
$870$	0	0
$871$	0	0
$872$	−1.46410	−0.0495807
$873$	0	0
$874$	19.8564	0.671653
$875$	−40.0526	−1.35402
$876$	0	0
$877$	23.9282i	0.807998i	0.914759	+	0.403999i	$0.132380\pi$
$877$	23.9282i	0.807998i	−0.914759	+	0.403999i	$0.867620\pi$
$878$	− 0.339746i	− 0.0114659i
$879$	0	0
$880$	4.73205	0.159517
$881$	−27.8372	−0.937858	−0.468929	−	0.883236i	$-0.655360\pi$
$881$	−27.8372	−0.937858	−0.468929	+	0.883236i	$0.655360\pi$
$882$	0	0
$883$	−42.9282	−1.44465	−0.722325	−	0.691554i	$-0.756926\pi$
$883$	−42.9282	−1.44465	−0.722325	+	0.691554i	$0.756926\pi$
$884$	0	0
$885$	0	0
$886$	15.6077i	0.524351i
$887$	−37.8564	−1.27109	−0.635547	−	0.772062i	$-0.719225\pi$
$887$	−37.8564	−1.27109	−0.635547	+	0.772062i	$0.719225\pi$
$888$	0	0
$889$	26.9282i	0.903143i
$890$	35.3205i	1.18395i
$891$	0	0
$892$	26.9282i	0.901623i
$893$	−10.3923	−0.347765
$894$	0	0
$895$	59.9090i	2.00254i
$896$	−2.73205	−0.0912714
$897$	0	0
$898$	11.3205	0.377770
$899$	6.53590i	0.217984i
$900$	0	0
$901$	−37.0526	−1.23440
$902$	11.9090i	0.396525i
$903$	0	0
$904$	1.33975i	0.0445593i
$905$	71.6410i	2.38143i
$906$	0	0
$907$	−36.3923	−1.20839	−0.604193	−	0.796838i	$-0.706505\pi$
$907$	−36.3923	−1.20839	−0.604193	+	0.796838i	$0.706505\pi$
$908$	− 12.1962i	− 0.404744i
$909$	0	0
$910$	0	0
$911$	2.53590	0.0840181	0.0420090	−	0.999117i	$-0.486624\pi$
$911$	2.53590	0.0840181	0.0420090	+	0.999117i	$0.486624\pi$
$912$	0	0
$913$	−0.248711	−0.00823114
$914$	1.33975	0.0443149
$915$	0	0
$916$	− 11.8564i	− 0.391747i
$917$	− 17.8564i	− 0.589670i
$918$	0	0
$919$	45.9615	1.51613	0.758065	−	0.652179i	$-0.226145\pi$
$919$	45.9615	1.51613	0.758065	+	0.652179i	$0.226145\pi$
$920$	15.6603	0.516303
$921$	0	0
$922$	−22.2679	−0.733356
$923$	0	0
$924$	0	0
$925$	31.5692i	1.03799i
$926$	−10.0526	−0.330348
$927$	0	0
$928$	− 4.46410i	− 0.146541i
$929$	39.2487i	1.28771i	0.765148	+	0.643854i	$0.222666\pi$
$929$	39.2487i	1.28771i	−0.765148	+	0.643854i	$0.777334\pi$
$930$	0	0
$931$	− 2.19615i	− 0.0719760i
$932$	7.85641	0.257345
$933$	0	0
$934$	− 18.5885i	− 0.608233i
$935$	−27.1244	−0.887061
$936$	0	0
$937$	−5.24871	−0.171468	−0.0857340	−	0.996318i	$-0.527324\pi$
$937$	−5.24871	−0.171468	−0.0857340	+	0.996318i	$0.527324\pi$
$938$	35.8564i	1.17075i
$939$	0	0
$940$	−8.19615	−0.267329
$941$	− 12.6410i	− 0.412085i	−0.978543	−	0.206043i	$-0.933941\pi$
$941$	− 12.6410i	− 0.412085i	0.978543	−	0.206043i	$-0.0660586\pi$
$942$	0	0
$943$	39.4115i	1.28342i
$944$	8.00000i	0.260378i
$945$	0	0
$946$	12.2487	0.398240
$947$	21.0718i	0.684741i	0.939565	+	0.342371i	$0.111230\pi$
$947$	21.0718i	0.684741i	−0.939565	+	0.342371i	$0.888770\pi$
$948$	0	0
$949$	0	0
$950$	−42.2487	−1.37073
$951$	0	0
$952$	15.6603	0.507552
$953$	41.5692	1.34656	0.673280	−	0.739388i	$-0.264885\pi$
$953$	41.5692	1.34656	0.673280	+	0.739388i	$0.264885\pi$
$954$	0	0
$955$	− 25.8564i	− 0.836694i
$956$	7.66025i	0.247750i
$957$	0	0
$958$	33.4641	1.08118
$959$	−32.5885	−1.05234
$960$	0	0
$961$	28.8564	0.930852
$962$	0	0
$963$	0	0
$964$	13.5885i	0.437655i
$965$	43.7846	1.40948
$966$	0	0
$967$	− 43.1244i	− 1.38679i	−0.720560	−	0.693393i	$-0.756115\pi$
$967$	− 43.1244i	− 1.38679i	0.720560	−	0.693393i	$-0.243885\pi$
$968$	9.39230i	0.301880i
$969$	0	0
$970$	22.3923i	0.718974i
$971$	−30.2487	−0.970727	−0.485364	−	0.874312i	$-0.661313\pi$
$971$	−30.2487	−0.970727	−0.485364	+	0.874312i	$0.661313\pi$
$972$	0	0
$973$	48.7846i	1.56396i
$974$	−3.12436	−0.100111
$975$	0	0
$976$	−9.19615	−0.294362
$977$	− 45.9282i	− 1.46937i	−0.678407	−	0.734687i	$-0.737329\pi$
$977$	− 45.9282i	− 1.46937i	0.678407	−	0.734687i	$-0.262671\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	− 1.73205i	− 0.0553283i
$981$	0	0
$982$	− 8.73205i	− 0.278651i
$983$	20.7846i	0.662926i	0.943468	+	0.331463i	$0.107542\pi$
$983$	20.7846i	0.662926i	−0.943468	+	0.331463i	$0.892458\pi$
$984$	0	0
$985$	66.6410	2.12336
$986$	25.5885i	0.814902i
$987$	0	0
$988$	0	0
$989$	40.5359	1.28897
$990$	0	0
$991$	22.5885	0.717546	0.358773	−	0.933425i	$-0.383195\pi$
$991$	22.5885	0.717546	0.358773	+	0.933425i	$0.383195\pi$
$992$	1.46410	0.0464853
$993$	0	0
$994$	− 12.9282i	− 0.410058i
$995$	− 52.9808i	− 1.67960i
$996$	0	0
$997$	21.3397	0.675837	0.337918	−	0.941175i	$-0.390277\pi$
$997$	21.3397	0.675837	0.337918	+	0.941175i	$0.390277\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.i.1351.1		4
3.2	odd	2		1014.2.b.e.337.4		4
13.3	even	3		234.2.l.c.199.2		4
13.4	even	6		234.2.l.c.127.2		4
13.5	odd	4		3042.2.a.p.1.1		2
13.8	odd	4		3042.2.a.y.1.2		2
13.12	even	2	inner	3042.2.b.i.1351.4		4
39.2	even	12		1014.2.e.g.529.2		4
39.5	even	4		1014.2.a.k.1.2		2
39.8	even	4		1014.2.a.i.1.1		2
39.11	even	12		1014.2.e.i.529.1		4
39.17	odd	6		78.2.i.a.49.1	yes	4
39.20	even	12		1014.2.e.i.991.1		4
39.23	odd	6		1014.2.i.a.823.2		4
39.29	odd	6		78.2.i.a.43.1	✓	4
39.32	even	12		1014.2.e.g.991.2		4
39.35	odd	6		1014.2.i.a.361.2		4
39.38	odd	2		1014.2.b.e.337.1		4
52.3	odd	6		1872.2.by.h.433.1		4
52.43	odd	6		1872.2.by.h.1297.2		4
156.47	odd	4		8112.2.a.bj.1.1		2
156.83	odd	4		8112.2.a.bp.1.2		2
156.95	even	6		624.2.bv.e.49.1		4
156.107	even	6		624.2.bv.e.433.2		4
195.17	even	12		1950.2.y.b.49.1		4
195.29	odd	6		1950.2.bc.d.901.2		4
195.68	even	12		1950.2.y.b.199.1		4
195.107	even	12		1950.2.y.g.199.2		4
195.134	odd	6		1950.2.bc.d.751.2		4
195.173	even	12		1950.2.y.g.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.a.43.1	✓	4	39.29	odd	6
78.2.i.a.49.1	yes	4	39.17	odd	6
234.2.l.c.127.2		4	13.4	even	6
234.2.l.c.199.2		4	13.3	even	3
624.2.bv.e.49.1		4	156.95	even	6
624.2.bv.e.433.2		4	156.107	even	6
1014.2.a.i.1.1		2	39.8	even	4
1014.2.a.k.1.2		2	39.5	even	4
1014.2.b.e.337.1		4	39.38	odd	2
1014.2.b.e.337.4		4	3.2	odd	2
1014.2.e.g.529.2		4	39.2	even	12
1014.2.e.g.991.2		4	39.32	even	12
1014.2.e.i.529.1		4	39.11	even	12
1014.2.e.i.991.1		4	39.20	even	12
1014.2.i.a.361.2		4	39.35	odd	6
1014.2.i.a.823.2		4	39.23	odd	6
1872.2.by.h.433.1		4	52.3	odd	6
1872.2.by.h.1297.2		4	52.43	odd	6
1950.2.y.b.49.1		4	195.17	even	12
1950.2.y.b.199.1		4	195.68	even	12
1950.2.y.g.49.2		4	195.173	even	12
1950.2.y.g.199.2		4	195.107	even	12
1950.2.bc.d.751.2		4	195.134	odd	6
1950.2.bc.d.901.2		4	195.29	odd	6
3042.2.a.p.1.1		2	13.5	odd	4
3042.2.a.y.1.2		2	13.8	odd	4
3042.2.b.i.1351.1		4	1.1	even	1	trivial
3042.2.b.i.1351.4		4	13.12	even	2	inner
8112.2.a.bj.1.1		2	156.47	odd	4
8112.2.a.bp.1.2		2	156.83	odd	4

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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.73205i q^{5} +2.73205i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.73205i q^{5} +2.73205i q^{7} +1.00000i q^{8} -3.73205 q^{10} +1.26795i q^{11} +2.73205 q^{14} +1.00000 q^{16} -5.73205 q^{17} +4.73205i q^{19} +3.73205i q^{20} +1.26795 q^{22} +4.19615 q^{23} -8.92820 q^{25} -2.73205i q^{28} +4.46410 q^{29} +1.46410i q^{31} -1.00000i q^{32} +5.73205i q^{34} +10.1962 q^{35} -3.53590i q^{37} +4.73205 q^{38} +3.73205 q^{40} +9.39230i q^{41} +9.66025 q^{43} -1.26795i q^{44} -4.19615i q^{46} +2.19615i q^{47} -0.464102 q^{49} +8.92820i q^{50} +6.46410 q^{53} +4.73205 q^{55} -2.73205 q^{56} -4.46410i q^{58} +8.00000i q^{59} -9.19615 q^{61} +1.46410 q^{62} -1.00000 q^{64} +13.1244i q^{67} +5.73205 q^{68} -10.1962i q^{70} -4.73205i q^{71} +6.26795i q^{73} -3.53590 q^{74} -4.73205i q^{76} -3.46410 q^{77} -2.53590 q^{79} -3.73205i q^{80} +9.39230 q^{82} +0.196152i q^{83} +21.3923i q^{85} -9.66025i q^{86} -1.26795 q^{88} -9.46410i q^{89} -4.19615 q^{92} +2.19615 q^{94} +17.6603 q^{95} -6.00000i q^{97} +0.464102i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 8 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} - 4 q^{23} - 8 q^{25} + 4 q^{29} + 20 q^{35} + 12 q^{38} + 8 q^{40} + 4 q^{43} + 12 q^{49} + 12 q^{53} + 12 q^{55} - 4 q^{56} - 16 q^{61} - 8 q^{62} - 4 q^{64} + 16 q^{68} - 28 q^{74} - 24 q^{79} - 4 q^{82} - 12 q^{88} + 4 q^{92} - 12 q^{94} + 36 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^10 + 4 * q^14 + 4 * q^16 - 16 * q^17 + 12 * q^22 - 4 * q^23 - 8 * q^25 + 4 * q^29 + 20 * q^35 + 12 * q^38 + 8 * q^40 + 4 * q^43 + 12 * q^49 + 12 * q^53 + 12 * q^55 - 4 * q^56 - 16 * q^61 - 8 * q^62 - 4 * q^64 + 16 * q^68 - 28 * q^74 - 24 * q^79 - 4 * q^82 - 12 * q^88 + 4 * q^92 - 12 * q^94 + 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more