LMFDB - Embedded newform 8046.2.a.o.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.a (trivial)

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:	yes
Analytic conductor:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} + \cdots + 423 $ x^12 - 3x^11 - 29x^10 + 76x^9 + 320x^8 - 724x^7 - 1643x^6 + 3265x^5 + 3921x^4 - 6927x^3 - 3639x^2 + 5508*x + 423
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.0737297$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.0737297 q^{5} +2.82077 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.0737297 q^{5} +2.82077 q^{7} +1.00000 q^{8} -0.0737297 q^{10} +0.846877 q^{11} +6.61317 q^{13} +2.82077 q^{14} +1.00000 q^{16} -2.99097 q^{17} -4.47691 q^{19} -0.0737297 q^{20} +0.846877 q^{22} +3.02516 q^{23} -4.99456 q^{25} +6.61317 q^{26} +2.82077 q^{28} +7.80750 q^{29} +8.54086 q^{31} +1.00000 q^{32} -2.99097 q^{34} -0.207975 q^{35} -4.28035 q^{37} -4.47691 q^{38} -0.0737297 q^{40} +4.12319 q^{41} -10.9934 q^{43} +0.846877 q^{44} +3.02516 q^{46} +3.07244 q^{47} +0.956763 q^{49} -4.99456 q^{50} +6.61317 q^{52} +5.55611 q^{53} -0.0624400 q^{55} +2.82077 q^{56} +7.80750 q^{58} +13.5281 q^{59} -2.57187 q^{61} +8.54086 q^{62} +1.00000 q^{64} -0.487586 q^{65} -0.662125 q^{67} -2.99097 q^{68} -0.207975 q^{70} +10.6685 q^{71} +5.88447 q^{73} -4.28035 q^{74} -4.47691 q^{76} +2.38885 q^{77} -7.96274 q^{79} -0.0737297 q^{80} +4.12319 q^{82} +1.58303 q^{83} +0.220523 q^{85} -10.9934 q^{86} +0.846877 q^{88} +5.09852 q^{89} +18.6542 q^{91} +3.02516 q^{92} +3.07244 q^{94} +0.330081 q^{95} -11.6060 q^{97} +0.956763 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8 + 3 * q^10 + 10 * q^11 + 5 * q^13 + 6 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^19 + 3 * q^20 + 10 * q^22 + 9 * q^23 + 7 * q^25 + 5 * q^26 + 6 * q^28 + 19 * q^29 + 10 * q^31 + 12 * q^32 + 8 * q^34 + 20 * q^35 + 11 * q^37 + 2 * q^38 + 3 * q^40 + 8 * q^41 + 13 * q^43 + 10 * q^44 + 9 * q^46 + 11 * q^47 + 2 * q^49 + 7 * q^50 + 5 * q^52 + 24 * q^53 + 3 * q^55 + 6 * q^56 + 19 * q^58 + 10 * q^59 + 10 * q^62 + 12 * q^64 + 28 * q^65 + 21 * q^67 + 8 * q^68 + 20 * q^70 + 37 * q^71 - 2 * q^73 + 11 * q^74 + 2 * q^76 + 2 * q^77 + 7 * q^79 + 3 * q^80 + 8 * q^82 + 22 * q^83 + 15 * q^85 + 13 * q^86 + 10 * q^88 + 40 * q^89 + q^91 + 9 * q^92 + 11 * q^94 + 11 * q^95 + 7 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.0737297	−0.0329729	−0.0164865	−	0.999864i	$-0.505248\pi$
$5$	−0.0737297	−0.0329729	−0.0164865	+	0.999864i	$0.505248\pi$
$6$	0	0
$7$	2.82077	1.06615	0.533076	−	0.846067i	$-0.321036\pi$
$7$	2.82077	1.06615	0.533076	+	0.846067i	$0.321036\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.0737297	−0.0233154
$11$	0.846877	0.255343	0.127672	−	0.991817i	$-0.459250\pi$
$11$	0.846877	0.255343	0.127672	+	0.991817i	$0.459250\pi$
$12$	0	0
$13$	6.61317	1.83416	0.917081	−	0.398701i	$-0.130539\pi$
$13$	6.61317	1.83416	0.917081	+	0.398701i	$0.130539\pi$
$14$	2.82077	0.753883
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.99097	−0.725418	−0.362709	−	0.931903i	$-0.618148\pi$
$17$	−2.99097	−0.725418	−0.362709	+	0.931903i	$0.618148\pi$
$18$	0	0
$19$	−4.47691	−1.02707	−0.513537	−	0.858067i	$-0.671665\pi$
$19$	−4.47691	−1.02707	−0.513537	+	0.858067i	$0.671665\pi$
$20$	−0.0737297	−0.0164865
$21$	0	0
$22$	0.846877	0.180555
$23$	3.02516	0.630789	0.315394	−	0.948961i	$-0.397863\pi$
$23$	3.02516	0.630789	0.315394	+	0.948961i	$0.397863\pi$
$24$	0	0
$25$	−4.99456	−0.998913
$26$	6.61317	1.29695
$27$	0	0
$28$	2.82077	0.533076
$29$	7.80750	1.44982	0.724908	−	0.688846i	$-0.241882\pi$
$29$	7.80750	1.44982	0.724908	+	0.688846i	$0.241882\pi$
$30$	0	0
$31$	8.54086	1.53398	0.766992	−	0.641657i	$-0.221753\pi$
$31$	8.54086	1.53398	0.766992	+	0.641657i	$0.221753\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.99097	−0.512948
$35$	−0.207975	−0.0351541
$36$	0	0
$37$	−4.28035	−0.703685	−0.351842	−	0.936059i	$-0.614445\pi$
$37$	−4.28035	−0.703685	−0.351842	+	0.936059i	$0.614445\pi$
$38$	−4.47691	−0.726251
$39$	0	0
$40$	−0.0737297	−0.0116577
$41$	4.12319	0.643933	0.321967	−	0.946751i	$-0.395656\pi$
$41$	4.12319	0.643933	0.321967	+	0.946751i	$0.395656\pi$
$42$	0	0
$43$	−10.9934	−1.67648	−0.838242	−	0.545298i	$-0.816417\pi$
$43$	−10.9934	−1.67648	−0.838242	+	0.545298i	$0.816417\pi$
$44$	0.846877	0.127672
$45$	0	0
$46$	3.02516	0.446035
$47$	3.07244	0.448162	0.224081	−	0.974571i	$-0.428062\pi$
$47$	3.07244	0.448162	0.224081	+	0.974571i	$0.428062\pi$
$48$	0	0
$49$	0.956763	0.136680
$50$	−4.99456	−0.706338
$51$	0	0
$52$	6.61317	0.917081
$53$	5.55611	0.763190	0.381595	−	0.924330i	$-0.375375\pi$
$53$	5.55611	0.763190	0.381595	+	0.924330i	$0.375375\pi$
$54$	0	0
$55$	−0.0624400	−0.00841940
$56$	2.82077	0.376942
$57$	0	0
$58$	7.80750	1.02517
$59$	13.5281	1.76120	0.880602	−	0.473858i	$-0.157139\pi$
$59$	13.5281	1.76120	0.880602	+	0.473858i	$0.157139\pi$
$60$	0	0
$61$	−2.57187	−0.329294	−0.164647	−	0.986353i	$-0.552649\pi$
$61$	−2.57187	−0.329294	−0.164647	+	0.986353i	$0.552649\pi$
$62$	8.54086	1.08469
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.487586	−0.0604777
$66$	0	0
$67$	−0.662125	−0.0808914	−0.0404457	−	0.999182i	$-0.512878\pi$
$67$	−0.662125	−0.0808914	−0.0404457	+	0.999182i	$0.512878\pi$
$68$	−2.99097	−0.362709
$69$	0	0
$70$	−0.207975	−0.0248577
$71$	10.6685	1.26612	0.633062	−	0.774101i	$-0.281798\pi$
$71$	10.6685	1.26612	0.633062	+	0.774101i	$0.281798\pi$
$72$	0	0
$73$	5.88447	0.688725	0.344363	−	0.938837i	$-0.388095\pi$
$73$	5.88447	0.688725	0.344363	+	0.938837i	$0.388095\pi$
$74$	−4.28035	−0.497580
$75$	0	0
$76$	−4.47691	−0.513537
$77$	2.38885	0.272235
$78$	0	0
$79$	−7.96274	−0.895878	−0.447939	−	0.894064i	$-0.647842\pi$
$79$	−7.96274	−0.895878	−0.447939	+	0.894064i	$0.647842\pi$
$80$	−0.0737297	−0.00824323
$81$	0	0
$82$	4.12319	0.455330
$83$	1.58303	0.173761	0.0868803	−	0.996219i	$-0.472310\pi$
$83$	1.58303	0.173761	0.0868803	+	0.996219i	$0.472310\pi$
$84$	0	0
$85$	0.220523	0.0239191
$86$	−10.9934	−1.18545
$87$	0	0
$88$	0.846877	0.0902774
$89$	5.09852	0.540442	0.270221	−	0.962798i	$-0.412903\pi$
$89$	5.09852	0.540442	0.270221	+	0.962798i	$0.412903\pi$
$90$	0	0
$91$	18.6542	1.95550
$92$	3.02516	0.315394
$93$	0	0
$94$	3.07244	0.316898
$95$	0.330081	0.0338656
$96$	0	0
$97$	−11.6060	−1.17841	−0.589206	−	0.807983i	$-0.700560\pi$
$97$	−11.6060	−1.17841	−0.589206	+	0.807983i	$0.700560\pi$
$98$	0.956763	0.0966477
$99$	0	0
$100$	−4.99456	−0.499456
$101$	0.530840	0.0528205	0.0264103	−	0.999651i	$-0.491592\pi$
$101$	0.530840	0.0528205	0.0264103	+	0.999651i	$0.491592\pi$
$102$	0	0
$103$	−10.3632	−1.02111	−0.510556	−	0.859844i	$-0.670560\pi$
$103$	−10.3632	−1.02111	−0.510556	+	0.859844i	$0.670560\pi$
$104$	6.61317	0.648474
$105$	0	0
$106$	5.55611	0.539657
$107$	−17.9925	−1.73940	−0.869701	−	0.493579i	$-0.835689\pi$
$107$	−17.9925	−1.73940	−0.869701	+	0.493579i	$0.835689\pi$
$108$	0	0
$109$	−2.56786	−0.245957	−0.122978	−	0.992409i	$-0.539245\pi$
$109$	−2.56786	−0.245957	−0.122978	+	0.992409i	$0.539245\pi$
$110$	−0.0624400	−0.00595342
$111$	0	0
$112$	2.82077	0.266538
$113$	18.5242	1.74261	0.871305	−	0.490741i	$-0.163274\pi$
$113$	18.5242	1.74261	0.871305	+	0.490741i	$0.163274\pi$
$114$	0	0
$115$	−0.223044	−0.0207989
$116$	7.80750	0.724908
$117$	0	0
$118$	13.5281	1.24536
$119$	−8.43686	−0.773405
$120$	0	0
$121$	−10.2828	−0.934800
$122$	−2.57187	−0.232846
$123$	0	0
$124$	8.54086	0.766992
$125$	0.736896	0.0659100
$126$	0	0
$127$	−9.61735	−0.853402	−0.426701	−	0.904393i	$-0.640324\pi$
$127$	−9.61735	−0.853402	−0.426701	+	0.904393i	$0.640324\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.487586	−0.0427642
$131$	−11.3291	−0.989826	−0.494913	−	0.868943i	$-0.664800\pi$
$131$	−11.3291	−0.989826	−0.494913	+	0.868943i	$0.664800\pi$
$132$	0	0
$133$	−12.6284	−1.09502
$134$	−0.662125	−0.0571989
$135$	0	0
$136$	−2.99097	−0.256474
$137$	−11.1716	−0.954455	−0.477227	−	0.878780i	$-0.658358\pi$
$137$	−11.1716	−0.954455	−0.477227	+	0.878780i	$0.658358\pi$
$138$	0	0
$139$	22.5964	1.91660	0.958300	−	0.285764i	$-0.0922475\pi$
$139$	22.5964	1.91660	0.958300	+	0.285764i	$0.0922475\pi$
$140$	−0.207975	−0.0175771
$141$	0	0
$142$	10.6685	0.895285
$143$	5.60054	0.468341
$144$	0	0
$145$	−0.575644	−0.0478046
$146$	5.88447	0.487002
$147$	0	0
$148$	−4.28035	−0.351842
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	21.1615	1.72210	0.861051	−	0.508519i	$-0.169807\pi$
$151$	21.1615	1.72210	0.861051	+	0.508519i	$0.169807\pi$
$152$	−4.47691	−0.363126
$153$	0	0
$154$	2.38885	0.192499
$155$	−0.629715	−0.0505799
$156$	0	0
$157$	−3.31978	−0.264947	−0.132474	−	0.991187i	$-0.542292\pi$
$157$	−3.31978	−0.264947	−0.132474	+	0.991187i	$0.542292\pi$
$158$	−7.96274	−0.633482
$159$	0	0
$160$	−0.0737297	−0.00582884
$161$	8.53328	0.672517
$162$	0	0
$163$	−10.5398	−0.825544	−0.412772	−	0.910834i	$-0.635439\pi$
$163$	−10.5398	−0.825544	−0.412772	+	0.910834i	$0.635439\pi$
$164$	4.12319	0.321967
$165$	0	0
$166$	1.58303	0.122867
$167$	−5.76356	−0.445997	−0.222999	−	0.974819i	$-0.571585\pi$
$167$	−5.76356	−0.445997	−0.222999	+	0.974819i	$0.571585\pi$
$168$	0	0
$169$	30.7340	2.36415
$170$	0.220523	0.0169134
$171$	0	0
$172$	−10.9934	−0.838242
$173$	14.7159	1.11883	0.559416	−	0.828887i	$-0.311025\pi$
$173$	14.7159	1.11883	0.559416	+	0.828887i	$0.311025\pi$
$174$	0	0
$175$	−14.0885	−1.06499
$176$	0.846877	0.0638358
$177$	0	0
$178$	5.09852	0.382150
$179$	13.1718	0.984507	0.492253	−	0.870452i	$-0.336173\pi$
$179$	13.1718	0.984507	0.492253	+	0.870452i	$0.336173\pi$
$180$	0	0
$181$	−3.62996	−0.269813	−0.134906	−	0.990858i	$-0.543073\pi$
$181$	−3.62996	−0.269813	−0.134906	+	0.990858i	$0.543073\pi$
$182$	18.6542	1.38274
$183$	0	0
$184$	3.02516	0.223018
$185$	0.315589	0.0232025
$186$	0	0
$187$	−2.53299	−0.185230
$188$	3.07244	0.224081
$189$	0	0
$190$	0.330081	0.0239466
$191$	−8.47963	−0.613564	−0.306782	−	0.951780i	$-0.599252\pi$
$191$	−8.47963	−0.613564	−0.306782	+	0.951780i	$0.599252\pi$
$192$	0	0
$193$	−17.8965	−1.28822	−0.644108	−	0.764934i	$-0.722771\pi$
$193$	−17.8965	−1.28822	−0.644108	+	0.764934i	$0.722771\pi$
$194$	−11.6060	−0.833264
$195$	0	0
$196$	0.956763	0.0683402
$197$	−20.6524	−1.47142	−0.735710	−	0.677297i	$-0.763151\pi$
$197$	−20.6524	−1.47142	−0.735710	+	0.677297i	$0.763151\pi$
$198$	0	0
$199$	21.3635	1.51442	0.757210	−	0.653171i	$-0.226562\pi$
$199$	21.3635	1.51442	0.757210	+	0.653171i	$0.226562\pi$
$200$	−4.99456	−0.353169
$201$	0	0
$202$	0.530840	0.0373497
$203$	22.0232	1.54572
$204$	0	0
$205$	−0.304001	−0.0212324
$206$	−10.3632	−0.722036
$207$	0	0
$208$	6.61317	0.458541
$209$	−3.79140	−0.262256
$210$	0	0
$211$	23.8145	1.63946	0.819729	−	0.572752i	$-0.194124\pi$
$211$	23.8145	1.63946	0.819729	+	0.572752i	$0.194124\pi$
$212$	5.55611	0.381595
$213$	0	0
$214$	−17.9925	−1.22994
$215$	0.810543	0.0552786
$216$	0	0
$217$	24.0918	1.63546
$218$	−2.56786	−0.173918
$219$	0	0
$220$	−0.0624400	−0.00420970
$221$	−19.7798	−1.33053
$222$	0	0
$223$	−12.4209	−0.831765	−0.415883	−	0.909418i	$-0.636527\pi$
$223$	−12.4209	−0.831765	−0.415883	+	0.909418i	$0.636527\pi$
$224$	2.82077	0.188471
$225$	0	0
$226$	18.5242	1.23221
$227$	14.9424	0.991759	0.495879	−	0.868391i	$-0.334846\pi$
$227$	14.9424	0.991759	0.495879	+	0.868391i	$0.334846\pi$
$228$	0	0
$229$	6.02872	0.398389	0.199194	−	0.979960i	$-0.436167\pi$
$229$	6.02872	0.398389	0.199194	+	0.979960i	$0.436167\pi$
$230$	−0.223044	−0.0147071
$231$	0	0
$232$	7.80750	0.512587
$233$	15.1492	0.992460	0.496230	−	0.868191i	$-0.334717\pi$
$233$	15.1492	0.992460	0.496230	+	0.868191i	$0.334717\pi$
$234$	0	0
$235$	−0.226530	−0.0147772
$236$	13.5281	0.880602
$237$	0	0
$238$	−8.43686	−0.546880
$239$	−20.3706	−1.31767	−0.658833	−	0.752289i	$-0.728950\pi$
$239$	−20.3706	−1.31767	−0.658833	+	0.752289i	$0.728950\pi$
$240$	0	0
$241$	−7.15656	−0.460994	−0.230497	−	0.973073i	$-0.574035\pi$
$241$	−7.15656	−0.460994	−0.230497	+	0.973073i	$0.574035\pi$
$242$	−10.2828	−0.661003
$243$	0	0
$244$	−2.57187	−0.164647
$245$	−0.0705418	−0.00450675
$246$	0	0
$247$	−29.6066	−1.88382
$248$	8.54086	0.542345
$249$	0	0
$250$	0.736896	0.0466054
$251$	13.4244	0.847343	0.423671	−	0.905816i	$-0.360741\pi$
$251$	13.4244	0.847343	0.423671	+	0.905816i	$0.360741\pi$
$252$	0	0
$253$	2.56194	0.161068
$254$	−9.61735	−0.603446
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.8316	1.17468	0.587340	−	0.809340i	$-0.300175\pi$
$257$	18.8316	1.17468	0.587340	+	0.809340i	$0.300175\pi$
$258$	0	0
$259$	−12.0739	−0.750235
$260$	−0.487586	−0.0302388
$261$	0	0
$262$	−11.3291	−0.699912
$263$	2.43969	0.150437	0.0752187	−	0.997167i	$-0.476034\pi$
$263$	2.43969	0.150437	0.0752187	+	0.997167i	$0.476034\pi$
$264$	0	0
$265$	−0.409650	−0.0251646
$266$	−12.6284	−0.774295
$267$	0	0
$268$	−0.662125	−0.0404457
$269$	−19.1312	−1.16645	−0.583225	−	0.812311i	$-0.698209\pi$
$269$	−19.1312	−1.16645	−0.583225	+	0.812311i	$0.698209\pi$
$270$	0	0
$271$	−28.8732	−1.75392	−0.876962	−	0.480560i	$-0.840433\pi$
$271$	−28.8732	−1.75392	−0.876962	+	0.480560i	$0.840433\pi$
$272$	−2.99097	−0.181354
$273$	0	0
$274$	−11.1716	−0.674902
$275$	−4.22978	−0.255065
$276$	0	0
$277$	19.8577	1.19313	0.596567	−	0.802563i	$-0.296531\pi$
$277$	19.8577	1.19313	0.596567	+	0.802563i	$0.296531\pi$
$278$	22.5964	1.35524
$279$	0	0
$280$	−0.207975	−0.0124289
$281$	16.1625	0.964174	0.482087	−	0.876123i	$-0.339879\pi$
$281$	16.1625	0.964174	0.482087	+	0.876123i	$0.339879\pi$
$282$	0	0
$283$	14.3573	0.853453	0.426727	−	0.904381i	$-0.359667\pi$
$283$	14.3573	0.853453	0.426727	+	0.904381i	$0.359667\pi$
$284$	10.6685	0.633062
$285$	0	0
$286$	5.60054	0.331167
$287$	11.6306	0.686531
$288$	0	0
$289$	−8.05408	−0.473769
$290$	−0.575644	−0.0338030
$291$	0	0
$292$	5.88447	0.344363
$293$	−8.57843	−0.501157	−0.250579	−	0.968096i	$-0.580621\pi$
$293$	−8.57843	−0.501157	−0.250579	+	0.968096i	$0.580621\pi$
$294$	0	0
$295$	−0.997419	−0.0580720
$296$	−4.28035	−0.248790
$297$	0	0
$298$	1.00000	0.0579284
$299$	20.0059	1.15697
$300$	0	0
$301$	−31.0100	−1.78739
$302$	21.1615	1.21771
$303$	0	0
$304$	−4.47691	−0.256769
$305$	0.189623	0.0108578
$306$	0	0
$307$	−6.96009	−0.397233	−0.198617	−	0.980077i	$-0.563645\pi$
$307$	−6.96009	−0.397233	−0.198617	+	0.980077i	$0.563645\pi$
$308$	2.38885	0.136117
$309$	0	0
$310$	−0.629715	−0.0357654
$311$	−27.5601	−1.56279	−0.781395	−	0.624036i	$-0.785492\pi$
$311$	−27.5601	−1.56279	−0.781395	+	0.624036i	$0.785492\pi$
$312$	0	0
$313$	−12.0236	−0.679615	−0.339807	−	0.940495i	$-0.610362\pi$
$313$	−12.0236	−0.679615	−0.339807	+	0.940495i	$0.610362\pi$
$314$	−3.31978	−0.187346
$315$	0	0
$316$	−7.96274	−0.447939
$317$	−26.5999	−1.49400	−0.747001	−	0.664823i	$-0.768507\pi$
$317$	−26.5999	−1.49400	−0.747001	+	0.664823i	$0.768507\pi$
$318$	0	0
$319$	6.61199	0.370200
$320$	−0.0737297	−0.00412161
$321$	0	0
$322$	8.53328	0.475541
$323$	13.3903	0.745058
$324$	0	0
$325$	−33.0299	−1.83217
$326$	−10.5398	−0.583748
$327$	0	0
$328$	4.12319	0.227665
$329$	8.66667	0.477809
$330$	0	0
$331$	24.9022	1.36875	0.684373	−	0.729132i	$-0.260076\pi$
$331$	24.9022	1.36875	0.684373	+	0.729132i	$0.260076\pi$
$332$	1.58303	0.0868803
$333$	0	0
$334$	−5.76356	−0.315368
$335$	0.0488182	0.00266723
$336$	0	0
$337$	18.9366	1.03154	0.515771	−	0.856727i	$-0.327506\pi$
$337$	18.9366	1.03154	0.515771	+	0.856727i	$0.327506\pi$
$338$	30.7340	1.67171
$339$	0	0
$340$	0.220523	0.0119596
$341$	7.23306	0.391692
$342$	0	0
$343$	−17.0466	−0.920430
$344$	−10.9934	−0.592727
$345$	0	0
$346$	14.7159	0.791134
$347$	−23.5737	−1.26550	−0.632751	−	0.774356i	$-0.718074\pi$
$347$	−23.5737	−1.26550	−0.632751	+	0.774356i	$0.718074\pi$
$348$	0	0
$349$	25.1375	1.34558	0.672790	−	0.739833i	$-0.265096\pi$
$349$	25.1375	1.34558	0.672790	+	0.739833i	$0.265096\pi$
$350$	−14.0885	−0.753064
$351$	0	0
$352$	0.846877	0.0451387
$353$	−9.49147	−0.505180	−0.252590	−	0.967573i	$-0.581282\pi$
$353$	−9.49147	−0.505180	−0.252590	+	0.967573i	$0.581282\pi$
$354$	0	0
$355$	−0.786588	−0.0417478
$356$	5.09852	0.270221
$357$	0	0
$358$	13.1718	0.696152
$359$	16.0817	0.848761	0.424381	−	0.905484i	$-0.360492\pi$
$359$	16.0817	0.848761	0.424381	+	0.905484i	$0.360492\pi$
$360$	0	0
$361$	1.04277	0.0548824
$362$	−3.62996	−0.190787
$363$	0	0
$364$	18.6542	0.977748
$365$	−0.433860	−0.0227093
$366$	0	0
$367$	28.0566	1.46454	0.732271	−	0.681013i	$-0.238460\pi$
$367$	28.0566	1.46454	0.732271	+	0.681013i	$0.238460\pi$
$368$	3.02516	0.157697
$369$	0	0
$370$	0.315589	0.0164067
$371$	15.6725	0.813677
$372$	0	0
$373$	7.06289	0.365703	0.182851	−	0.983141i	$-0.441467\pi$
$373$	7.06289	0.365703	0.182851	+	0.983141i	$0.441467\pi$
$374$	−2.53299	−0.130978
$375$	0	0
$376$	3.07244	0.158449
$377$	51.6323	2.65920
$378$	0	0
$379$	28.1458	1.44575	0.722876	−	0.690978i	$-0.242820\pi$
$379$	28.1458	1.44575	0.722876	+	0.690978i	$0.242820\pi$
$380$	0.330081	0.0169328
$381$	0	0
$382$	−8.47963	−0.433855
$383$	37.3687	1.90945	0.954726	−	0.297486i	$-0.0961481\pi$
$383$	37.3687	1.90945	0.954726	+	0.297486i	$0.0961481\pi$
$384$	0	0
$385$	−0.176129	−0.00897637
$386$	−17.8965	−0.910907
$387$	0	0
$388$	−11.6060	−0.589206
$389$	35.1791	1.78365	0.891826	−	0.452379i	$-0.149424\pi$
$389$	35.1791	1.78365	0.891826	+	0.452379i	$0.149424\pi$
$390$	0	0
$391$	−9.04816	−0.457585
$392$	0.956763	0.0483238
$393$	0	0
$394$	−20.6524	−1.04045
$395$	0.587090	0.0295397
$396$	0	0
$397$	−18.0007	−0.903430	−0.451715	−	0.892162i	$-0.649188\pi$
$397$	−18.0007	−0.903430	−0.451715	+	0.892162i	$0.649188\pi$
$398$	21.3635	1.07086
$399$	0	0
$400$	−4.99456	−0.249728
$401$	−7.81326	−0.390175	−0.195088	−	0.980786i	$-0.562499\pi$
$401$	−7.81326	−0.390175	−0.195088	+	0.980786i	$0.562499\pi$
$402$	0	0
$403$	56.4821	2.81357
$404$	0.530840	0.0264103
$405$	0	0
$406$	22.0232	1.09299
$407$	−3.62493	−0.179681
$408$	0	0
$409$	5.56848	0.275344	0.137672	−	0.990478i	$-0.456038\pi$
$409$	5.56848	0.275344	0.137672	+	0.990478i	$0.456038\pi$
$410$	−0.304001	−0.0150135
$411$	0	0
$412$	−10.3632	−0.510556
$413$	38.1596	1.87771
$414$	0	0
$415$	−0.116717	−0.00572939
$416$	6.61317	0.324237
$417$	0	0
$418$	−3.79140	−0.185443
$419$	−16.2733	−0.795003	−0.397502	−	0.917601i	$-0.630123\pi$
$419$	−16.2733	−0.795003	−0.397502	+	0.917601i	$0.630123\pi$
$420$	0	0
$421$	−8.69362	−0.423701	−0.211851	−	0.977302i	$-0.567949\pi$
$421$	−8.69362	−0.423701	−0.211851	+	0.977302i	$0.567949\pi$
$422$	23.8145	1.15927
$423$	0	0
$424$	5.55611	0.269829
$425$	14.9386	0.724629
$426$	0	0
$427$	−7.25467	−0.351078
$428$	−17.9925	−0.869701
$429$	0	0
$430$	0.810543	0.0390879
$431$	17.6996	0.852561	0.426281	−	0.904591i	$-0.359824\pi$
$431$	17.6996	0.852561	0.426281	+	0.904591i	$0.359824\pi$
$432$	0	0
$433$	18.4256	0.885478	0.442739	−	0.896650i	$-0.354007\pi$
$433$	18.4256	0.885478	0.442739	+	0.896650i	$0.354007\pi$
$434$	24.0918	1.15645
$435$	0	0
$436$	−2.56786	−0.122978
$437$	−13.5434	−0.647867
$438$	0	0
$439$	−8.35941	−0.398973	−0.199486	−	0.979901i	$-0.563927\pi$
$439$	−8.35941	−0.398973	−0.199486	+	0.979901i	$0.563927\pi$
$440$	−0.0624400	−0.00297671
$441$	0	0
$442$	−19.7798	−0.940829
$443$	−32.2101	−1.53035	−0.765173	−	0.643824i	$-0.777347\pi$
$443$	−32.2101	−1.53035	−0.765173	+	0.643824i	$0.777347\pi$
$444$	0	0
$445$	−0.375912	−0.0178200
$446$	−12.4209	−0.588147
$447$	0	0
$448$	2.82077	0.133269
$449$	−16.0870	−0.759191	−0.379595	−	0.925153i	$-0.623937\pi$
$449$	−16.0870	−0.759191	−0.379595	+	0.925153i	$0.623937\pi$
$450$	0	0
$451$	3.49183	0.164424
$452$	18.5242	0.871305
$453$	0	0
$454$	14.9424	0.701279
$455$	−1.37537	−0.0644784
$456$	0	0
$457$	31.2568	1.46213	0.731065	−	0.682308i	$-0.239024\pi$
$457$	31.2568	1.46213	0.731065	+	0.682308i	$0.239024\pi$
$458$	6.02872	0.281704
$459$	0	0
$460$	−0.223044	−0.0103995
$461$	19.0181	0.885763	0.442881	−	0.896580i	$-0.353956\pi$
$461$	19.0181	0.885763	0.442881	+	0.896580i	$0.353956\pi$
$462$	0	0
$463$	11.0685	0.514395	0.257198	−	0.966359i	$-0.417201\pi$
$463$	11.0685	0.514395	0.257198	+	0.966359i	$0.417201\pi$
$464$	7.80750	0.362454
$465$	0	0
$466$	15.1492	0.701775
$467$	−19.8084	−0.916624	−0.458312	−	0.888791i	$-0.651546\pi$
$467$	−19.8084	−0.916624	−0.458312	+	0.888791i	$0.651546\pi$
$468$	0	0
$469$	−1.86770	−0.0862426
$470$	−0.226530	−0.0104491
$471$	0	0
$472$	13.5281	0.622679
$473$	−9.31010	−0.428079
$474$	0	0
$475$	22.3602	1.02596
$476$	−8.43686	−0.386703
$477$	0	0
$478$	−20.3706	−0.931731
$479$	−5.02223	−0.229472	−0.114736	−	0.993396i	$-0.536602\pi$
$479$	−5.02223	−0.229472	−0.114736	+	0.993396i	$0.536602\pi$
$480$	0	0
$481$	−28.3066	−1.29067
$482$	−7.15656	−0.325972
$483$	0	0
$484$	−10.2828	−0.467400
$485$	0.855708	0.0388557
$486$	0	0
$487$	−19.9240	−0.902842	−0.451421	−	0.892311i	$-0.649083\pi$
$487$	−19.9240	−0.902842	−0.451421	+	0.892311i	$0.649083\pi$
$488$	−2.57187	−0.116423
$489$	0	0
$490$	−0.0705418	−0.00318675
$491$	13.0304	0.588054	0.294027	−	0.955797i	$-0.405004\pi$
$491$	13.0304	0.588054	0.294027	+	0.955797i	$0.405004\pi$
$492$	0	0
$493$	−23.3520	−1.05172
$494$	−29.6066	−1.33206
$495$	0	0
$496$	8.54086	0.383496
$497$	30.0936	1.34988
$498$	0	0
$499$	27.3213	1.22307	0.611535	−	0.791218i	$-0.290552\pi$
$499$	27.3213	1.22307	0.611535	+	0.791218i	$0.290552\pi$
$500$	0.736896	0.0329550
$501$	0	0
$502$	13.4244	0.599162
$503$	39.0248	1.74003	0.870016	−	0.493024i	$-0.164108\pi$
$503$	39.0248	1.74003	0.870016	+	0.493024i	$0.164108\pi$
$504$	0	0
$505$	−0.0391386	−0.00174165
$506$	2.56194	0.113892
$507$	0	0
$508$	−9.61735	−0.426701
$509$	30.4449	1.34945	0.674724	−	0.738070i	$-0.264263\pi$
$509$	30.4449	1.34945	0.674724	+	0.738070i	$0.264263\pi$
$510$	0	0
$511$	16.5988	0.734286
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.8316	0.830625
$515$	0.764072	0.0336690
$516$	0	0
$517$	2.60198	0.114435
$518$	−12.0739	−0.530496
$519$	0	0
$520$	−0.487586	−0.0213821
$521$	−26.0464	−1.14111	−0.570557	−	0.821258i	$-0.693272\pi$
$521$	−26.0464	−1.14111	−0.570557	+	0.821258i	$0.693272\pi$
$522$	0	0
$523$	−2.50636	−0.109596	−0.0547978	−	0.998497i	$-0.517451\pi$
$523$	−2.50636	−0.109596	−0.0547978	+	0.998497i	$0.517451\pi$
$524$	−11.3291	−0.494913
$525$	0	0
$526$	2.43969	0.106375
$527$	−25.5455	−1.11278
$528$	0	0
$529$	−13.8484	−0.602105
$530$	−0.409650	−0.0177941
$531$	0	0
$532$	−12.6284	−0.547509
$533$	27.2673	1.18108
$534$	0	0
$535$	1.32658	0.0573532
$536$	−0.662125	−0.0285994
$537$	0	0
$538$	−19.1312	−0.824805
$539$	0.810261	0.0349004
$540$	0	0
$541$	−13.1124	−0.563746	−0.281873	−	0.959452i	$-0.590956\pi$
$541$	−13.1124	−0.563746	−0.281873	+	0.959452i	$0.590956\pi$
$542$	−28.8732	−1.24021
$543$	0	0
$544$	−2.99097	−0.128237
$545$	0.189328	0.00810991
$546$	0	0
$547$	−33.1933	−1.41924	−0.709622	−	0.704583i	$-0.751134\pi$
$547$	−33.1933	−1.41924	−0.709622	+	0.704583i	$0.751134\pi$
$548$	−11.1716	−0.477227
$549$	0	0
$550$	−4.22978	−0.180359
$551$	−34.9535	−1.48907
$552$	0	0
$553$	−22.4611	−0.955143
$554$	19.8577	0.843673
$555$	0	0
$556$	22.5964	0.958300
$557$	−3.66313	−0.155212	−0.0776060	−	0.996984i	$-0.524728\pi$
$557$	−3.66313	−0.155212	−0.0776060	+	0.996984i	$0.524728\pi$
$558$	0	0
$559$	−72.7015	−3.07495
$560$	−0.207975	−0.00878853
$561$	0	0
$562$	16.1625	0.681774
$563$	−44.6936	−1.88361	−0.941805	−	0.336160i	$-0.890872\pi$
$563$	−44.6936	−1.88361	−0.941805	+	0.336160i	$0.890872\pi$
$564$	0	0
$565$	−1.36578	−0.0574589
$566$	14.3573	0.603482
$567$	0	0
$568$	10.6685	0.447642
$569$	−28.3965	−1.19044	−0.595222	−	0.803561i	$-0.702936\pi$
$569$	−28.3965	−1.19044	−0.595222	+	0.803561i	$0.702936\pi$
$570$	0	0
$571$	14.1958	0.594075	0.297038	−	0.954866i	$-0.404001\pi$
$571$	14.1958	0.594075	0.297038	+	0.954866i	$0.404001\pi$
$572$	5.60054	0.234170
$573$	0	0
$574$	11.6306	0.485451
$575$	−15.1093	−0.630103
$576$	0	0
$577$	−6.95543	−0.289558	−0.144779	−	0.989464i	$-0.546247\pi$
$577$	−6.95543	−0.289558	−0.144779	+	0.989464i	$0.546247\pi$
$578$	−8.05408	−0.335006
$579$	0	0
$580$	−0.575644	−0.0239023
$581$	4.46538	0.185255
$582$	0	0
$583$	4.70534	0.194875
$584$	5.88447	0.243501
$585$	0	0
$586$	−8.57843	−0.354372
$587$	−20.2969	−0.837744	−0.418872	−	0.908045i	$-0.637574\pi$
$587$	−20.2969	−0.837744	−0.418872	+	0.908045i	$0.637574\pi$
$588$	0	0
$589$	−38.2367	−1.57552
$590$	−0.997419	−0.0410631
$591$	0	0
$592$	−4.28035	−0.175921
$593$	−37.4732	−1.53884	−0.769419	−	0.638744i	$-0.779454\pi$
$593$	−37.4732	−1.53884	−0.769419	+	0.638744i	$0.779454\pi$
$594$	0	0
$595$	0.622047	0.0255014
$596$	1.00000	0.0409616
$597$	0	0
$598$	20.0059	0.818101
$599$	10.4893	0.428580	0.214290	−	0.976770i	$-0.431256\pi$
$599$	10.4893	0.428580	0.214290	+	0.976770i	$0.431256\pi$
$600$	0	0
$601$	0.305383	0.0124568	0.00622841	−	0.999981i	$-0.498017\pi$
$601$	0.305383	0.0124568	0.00622841	+	0.999981i	$0.498017\pi$
$602$	−31.0100	−1.26387
$603$	0	0
$604$	21.1615	0.861051
$605$	0.758147	0.0308231
$606$	0	0
$607$	−10.6928	−0.434006	−0.217003	−	0.976171i	$-0.569628\pi$
$607$	−10.6928	−0.434006	−0.217003	+	0.976171i	$0.569628\pi$
$608$	−4.47691	−0.181563
$609$	0	0
$610$	0.189623	0.00767762
$611$	20.3186	0.822002
$612$	0	0
$613$	0.270765	0.0109361	0.00546805	−	0.999985i	$-0.498259\pi$
$613$	0.270765	0.0109361	0.00546805	+	0.999985i	$0.498259\pi$
$614$	−6.96009	−0.280886
$615$	0	0
$616$	2.38885	0.0962494
$617$	17.7655	0.715213	0.357606	−	0.933872i	$-0.383593\pi$
$617$	17.7655	0.715213	0.357606	+	0.933872i	$0.383593\pi$
$618$	0	0
$619$	5.99701	0.241040	0.120520	−	0.992711i	$-0.461544\pi$
$619$	5.99701	0.241040	0.120520	+	0.992711i	$0.461544\pi$
$620$	−0.629715	−0.0252900
$621$	0	0
$622$	−27.5601	−1.10506
$623$	14.3818	0.576194
$624$	0	0
$625$	24.9185	0.996740
$626$	−12.0236	−0.480560
$627$	0	0
$628$	−3.31978	−0.132474
$629$	12.8024	0.510465
$630$	0	0
$631$	−40.9808	−1.63142	−0.815710	−	0.578461i	$-0.803654\pi$
$631$	−40.9808	−1.63142	−0.815710	+	0.578461i	$0.803654\pi$
$632$	−7.96274	−0.316741
$633$	0	0
$634$	−26.5999	−1.05642
$635$	0.709084	0.0281392
$636$	0	0
$637$	6.32723	0.250694
$638$	6.61199	0.261771
$639$	0	0
$640$	−0.0737297	−0.00291442
$641$	29.3792	1.16041	0.580204	−	0.814472i	$-0.302973\pi$
$641$	29.3792	1.16041	0.580204	+	0.814472i	$0.302973\pi$
$642$	0	0
$643$	−15.8828	−0.626357	−0.313179	−	0.949694i	$-0.601394\pi$
$643$	−15.8828	−0.626357	−0.313179	+	0.949694i	$0.601394\pi$
$644$	8.53328	0.336258
$645$	0	0
$646$	13.3903	0.526836
$647$	30.4711	1.19794	0.598972	−	0.800770i	$-0.295576\pi$
$647$	30.4711	1.19794	0.598972	+	0.800770i	$0.295576\pi$
$648$	0	0
$649$	11.4566	0.449711
$650$	−33.0299	−1.29554
$651$	0	0
$652$	−10.5398	−0.412772
$653$	−24.3418	−0.952566	−0.476283	−	0.879292i	$-0.658016\pi$
$653$	−24.3418	−0.952566	−0.476283	+	0.879292i	$0.658016\pi$
$654$	0	0
$655$	0.835289	0.0326374
$656$	4.12319	0.160983
$657$	0	0
$658$	8.66667	0.337862
$659$	−6.33161	−0.246645	−0.123322	−	0.992367i	$-0.539355\pi$
$659$	−6.33161	−0.246645	−0.123322	+	0.992367i	$0.539355\pi$
$660$	0	0
$661$	−25.5389	−0.993348	−0.496674	−	0.867937i	$-0.665445\pi$
$661$	−25.5389	−0.993348	−0.496674	+	0.867937i	$0.665445\pi$
$662$	24.9022	0.967850
$663$	0	0
$664$	1.58303	0.0614337
$665$	0.931085	0.0361059
$666$	0	0
$667$	23.6189	0.914528
$668$	−5.76356	−0.222999
$669$	0	0
$670$	0.0488182	0.00188601
$671$	−2.17806	−0.0840830
$672$	0	0
$673$	−27.9146	−1.07603	−0.538014	−	0.842936i	$-0.680825\pi$
$673$	−27.9146	−1.07603	−0.538014	+	0.842936i	$0.680825\pi$
$674$	18.9366	0.729410
$675$	0	0
$676$	30.7340	1.18208
$677$	−45.8343	−1.76155	−0.880777	−	0.473531i	$-0.842979\pi$
$677$	−45.8343	−1.76155	−0.880777	+	0.473531i	$0.842979\pi$
$678$	0	0
$679$	−32.7380	−1.25637
$680$	0.220523	0.00845669
$681$	0	0
$682$	7.23306	0.276968
$683$	3.67086	0.140461	0.0702307	−	0.997531i	$-0.477626\pi$
$683$	3.67086	0.140461	0.0702307	+	0.997531i	$0.477626\pi$
$684$	0	0
$685$	0.823679	0.0314712
$686$	−17.0466	−0.650842
$687$	0	0
$688$	−10.9934	−0.419121
$689$	36.7435	1.39981
$690$	0	0
$691$	−2.86087	−0.108833	−0.0544164	−	0.998518i	$-0.517330\pi$
$691$	−2.86087	−0.108833	−0.0544164	+	0.998518i	$0.517330\pi$
$692$	14.7159	0.559416
$693$	0	0
$694$	−23.5737	−0.894844
$695$	−1.66602	−0.0631959
$696$	0	0
$697$	−12.3323	−0.467121
$698$	25.1375	0.951469
$699$	0	0
$700$	−14.0885	−0.532497
$701$	14.6876	0.554745	0.277372	−	0.960762i	$-0.410536\pi$
$701$	14.6876	0.554745	0.277372	+	0.960762i	$0.410536\pi$
$702$	0	0
$703$	19.1627	0.722737
$704$	0.846877	0.0319179
$705$	0	0
$706$	−9.49147	−0.357216
$707$	1.49738	0.0563147
$708$	0	0
$709$	−23.9566	−0.899708	−0.449854	−	0.893102i	$-0.648524\pi$
$709$	−23.9566	−0.899708	−0.449854	+	0.893102i	$0.648524\pi$
$710$	−0.786588	−0.0295201
$711$	0	0
$712$	5.09852	0.191075
$713$	25.8374	0.967620
$714$	0	0
$715$	−0.412926	−0.0154425
$716$	13.1718	0.492253
$717$	0	0
$718$	16.0817	0.600165
$719$	41.3366	1.54159	0.770797	−	0.637081i	$-0.219858\pi$
$719$	41.3366	1.54159	0.770797	+	0.637081i	$0.219858\pi$
$720$	0	0
$721$	−29.2321	−1.08866
$722$	1.04277	0.0388077
$723$	0	0
$724$	−3.62996	−0.134906
$725$	−38.9950	−1.44824
$726$	0	0
$727$	−48.7090	−1.80652	−0.903258	−	0.429098i	$-0.858832\pi$
$727$	−48.7090	−1.80652	−0.903258	+	0.429098i	$0.858832\pi$
$728$	18.6542	0.691372
$729$	0	0
$730$	−0.433860	−0.0160579
$731$	32.8811	1.21615
$732$	0	0
$733$	12.7973	0.472680	0.236340	−	0.971670i	$-0.424052\pi$
$733$	12.7973	0.472680	0.236340	+	0.971670i	$0.424052\pi$
$734$	28.0566	1.03559
$735$	0	0
$736$	3.02516	0.111509
$737$	−0.560738	−0.0206551
$738$	0	0
$739$	12.8730	0.473541	0.236770	−	0.971566i	$-0.423911\pi$
$739$	12.8730	0.473541	0.236770	+	0.971566i	$0.423911\pi$
$740$	0.315589	0.0116013
$741$	0	0
$742$	15.6725	0.575357
$743$	−39.5383	−1.45052	−0.725259	−	0.688476i	$-0.758280\pi$
$743$	−39.5383	−1.45052	−0.725259	+	0.688476i	$0.758280\pi$
$744$	0	0
$745$	−0.0737297	−0.00270125
$746$	7.06289	0.258591
$747$	0	0
$748$	−2.53299	−0.0926152
$749$	−50.7528	−1.85447
$750$	0	0
$751$	8.92207	0.325571	0.162786	−	0.986661i	$-0.447952\pi$
$751$	8.92207	0.325571	0.162786	+	0.986661i	$0.447952\pi$
$752$	3.07244	0.112040
$753$	0	0
$754$	51.6323	1.88034
$755$	−1.56023	−0.0567827
$756$	0	0
$757$	−39.3899	−1.43165	−0.715824	−	0.698280i	$-0.753949\pi$
$757$	−39.3899	−1.43165	−0.715824	+	0.698280i	$0.753949\pi$
$758$	28.1458	1.02230
$759$	0	0
$760$	0.330081	0.0119733
$761$	38.3367	1.38970	0.694852	−	0.719153i	$-0.255470\pi$
$761$	38.3367	1.38970	0.694852	+	0.719153i	$0.255470\pi$
$762$	0	0
$763$	−7.24336	−0.262227
$764$	−8.47963	−0.306782
$765$	0	0
$766$	37.3687	1.35019
$767$	89.4633	3.23033
$768$	0	0
$769$	−8.25167	−0.297563	−0.148781	−	0.988870i	$-0.547535\pi$
$769$	−8.25167	−0.297563	−0.148781	+	0.988870i	$0.547535\pi$
$770$	−0.176129	−0.00634725
$771$	0	0
$772$	−17.8965	−0.644108
$773$	0.822376	0.0295788	0.0147894	−	0.999891i	$-0.495292\pi$
$773$	0.822376	0.0295788	0.0147894	+	0.999891i	$0.495292\pi$
$774$	0	0
$775$	−42.6579	−1.53232
$776$	−11.6060	−0.416632
$777$	0	0
$778$	35.1791	1.26123
$779$	−18.4592	−0.661368
$780$	0	0
$781$	9.03495	0.323296
$782$	−9.04816	−0.323562
$783$	0	0
$784$	0.956763	0.0341701
$785$	0.244766	0.00873608
$786$	0	0
$787$	−21.6214	−0.770718	−0.385359	−	0.922767i	$-0.625922\pi$
$787$	−21.6214	−0.770718	−0.385359	+	0.922767i	$0.625922\pi$
$788$	−20.6524	−0.735710
$789$	0	0
$790$	0.587090	0.0208877
$791$	52.2526	1.85789
$792$	0	0
$793$	−17.0082	−0.603979
$794$	−18.0007	−0.638822
$795$	0	0
$796$	21.3635	0.757210
$797$	3.98465	0.141144	0.0705718	−	0.997507i	$-0.477518\pi$
$797$	3.98465	0.141144	0.0705718	+	0.997507i	$0.477518\pi$
$798$	0	0
$799$	−9.18960	−0.325105
$800$	−4.99456	−0.176585
$801$	0	0
$802$	−7.81326	−0.275896
$803$	4.98342	0.175861
$804$	0	0
$805$	−0.629156	−0.0221748
$806$	56.4821	1.98950
$807$	0	0
$808$	0.530840	0.0186749
$809$	−15.7371	−0.553287	−0.276643	−	0.960973i	$-0.589222\pi$
$809$	−15.7371	−0.553287	−0.276643	+	0.960973i	$0.589222\pi$
$810$	0	0
$811$	1.23071	0.0432160	0.0216080	−	0.999767i	$-0.493121\pi$
$811$	1.23071	0.0432160	0.0216080	+	0.999767i	$0.493121\pi$
$812$	22.0232	0.772862
$813$	0	0
$814$	−3.62493	−0.127054
$815$	0.777099	0.0272206
$816$	0	0
$817$	49.2167	1.72188
$818$	5.56848	0.194697
$819$	0	0
$820$	−0.304001	−0.0106162
$821$	−17.9576	−0.626724	−0.313362	−	0.949634i	$-0.601455\pi$
$821$	−17.9576	−0.626724	−0.313362	+	0.949634i	$0.601455\pi$
$822$	0	0
$823$	−16.1046	−0.561370	−0.280685	−	0.959800i	$-0.590562\pi$
$823$	−16.1046	−0.561370	−0.280685	+	0.959800i	$0.590562\pi$
$824$	−10.3632	−0.361018
$825$	0	0
$826$	38.1596	1.32774
$827$	36.7883	1.27926	0.639628	−	0.768685i	$-0.279088\pi$
$827$	36.7883	1.27926	0.639628	+	0.768685i	$0.279088\pi$
$828$	0	0
$829$	−54.8626	−1.90546	−0.952729	−	0.303821i	$-0.901738\pi$
$829$	−54.8626	−1.90546	−0.952729	+	0.303821i	$0.901738\pi$
$830$	−0.116717	−0.00405129
$831$	0	0
$832$	6.61317	0.229270
$833$	−2.86165	−0.0991504
$834$	0	0
$835$	0.424945	0.0147058
$836$	−3.79140	−0.131128
$837$	0	0
$838$	−16.2733	−0.562152
$839$	−42.0659	−1.45228	−0.726138	−	0.687549i	$-0.758687\pi$
$839$	−42.0659	−1.45228	−0.726138	+	0.687549i	$0.758687\pi$
$840$	0	0
$841$	31.9570	1.10197
$842$	−8.69362	−0.299602
$843$	0	0
$844$	23.8145	0.819729
$845$	−2.26600	−0.0779529
$846$	0	0
$847$	−29.0054	−0.996639
$848$	5.55611	0.190798
$849$	0	0
$850$	14.9386	0.512390
$851$	−12.9487	−0.443876
$852$	0	0
$853$	−23.1030	−0.791032	−0.395516	−	0.918459i	$-0.629434\pi$
$853$	−23.1030	−0.791032	−0.395516	+	0.918459i	$0.629434\pi$
$854$	−7.25467	−0.248250
$855$	0	0
$856$	−17.9925	−0.614972
$857$	13.1296	0.448498	0.224249	−	0.974532i	$-0.428007\pi$
$857$	13.1296	0.448498	0.224249	+	0.974532i	$0.428007\pi$
$858$	0	0
$859$	−27.7488	−0.946776	−0.473388	−	0.880854i	$-0.656969\pi$
$859$	−27.7488	−0.946776	−0.473388	+	0.880854i	$0.656969\pi$
$860$	0.810543	0.0276393
$861$	0	0
$862$	17.6996	0.602852
$863$	13.4467	0.457731	0.228866	−	0.973458i	$-0.426498\pi$
$863$	13.4467	0.457731	0.228866	+	0.973458i	$0.426498\pi$
$864$	0	0
$865$	−1.08500	−0.0368911
$866$	18.4256	0.626128
$867$	0	0
$868$	24.0918	0.817730
$869$	−6.74346	−0.228756
$870$	0	0
$871$	−4.37874	−0.148368
$872$	−2.56786	−0.0869588
$873$	0	0
$874$	−13.5434	−0.458111
$875$	2.07862	0.0702701
$876$	0	0
$877$	52.0581	1.75788	0.878939	−	0.476933i	$-0.158252\pi$
$877$	52.0581	1.75788	0.878939	+	0.476933i	$0.158252\pi$
$878$	−8.35941	−0.282116
$879$	0	0
$880$	−0.0624400	−0.00210485
$881$	−32.5420	−1.09637	−0.548184	−	0.836358i	$-0.684681\pi$
$881$	−32.5420	−1.09637	−0.548184	+	0.836358i	$0.684681\pi$
$882$	0	0
$883$	4.00628	0.134822	0.0674111	−	0.997725i	$-0.478526\pi$
$883$	4.00628	0.134822	0.0674111	+	0.997725i	$0.478526\pi$
$884$	−19.7798	−0.665267
$885$	0	0
$886$	−32.2101	−1.08212
$887$	−34.9539	−1.17364	−0.586818	−	0.809719i	$-0.699620\pi$
$887$	−34.9539	−1.17364	−0.586818	+	0.809719i	$0.699620\pi$
$888$	0	0
$889$	−27.1284	−0.909857
$890$	−0.375912	−0.0126006
$891$	0	0
$892$	−12.4209	−0.415883
$893$	−13.7551	−0.460296
$894$	0	0
$895$	−0.971153	−0.0324621
$896$	2.82077	0.0942354
$897$	0	0
$898$	−16.0870	−0.536829
$899$	66.6827	2.22399
$900$	0	0
$901$	−16.6182	−0.553632
$902$	3.49183	0.116265
$903$	0	0
$904$	18.5242	0.616106
$905$	0.267636	0.00889652
$906$	0	0
$907$	11.9878	0.398049	0.199024	−	0.979995i	$-0.436223\pi$
$907$	11.9878	0.398049	0.199024	+	0.979995i	$0.436223\pi$
$908$	14.9424	0.495879
$909$	0	0
$910$	−1.37537	−0.0455931
$911$	−27.0943	−0.897675	−0.448838	−	0.893613i	$-0.648162\pi$
$911$	−27.0943	−0.897675	−0.448838	+	0.893613i	$0.648162\pi$
$912$	0	0
$913$	1.34064	0.0443686
$914$	31.2568	1.03388
$915$	0	0
$916$	6.02872	0.199194
$917$	−31.9567	−1.05530
$918$	0	0
$919$	35.1911	1.16085	0.580424	−	0.814314i	$-0.302887\pi$
$919$	35.1911	1.16085	0.580424	+	0.814314i	$0.302887\pi$
$920$	−0.223044	−0.00735354
$921$	0	0
$922$	19.0181	0.626329
$923$	70.5529	2.32228
$924$	0	0
$925$	21.3785	0.702920
$926$	11.0685	0.363732
$927$	0	0
$928$	7.80750	0.256294
$929$	7.64764	0.250911	0.125455	−	0.992099i	$-0.459961\pi$
$929$	7.64764	0.250911	0.125455	+	0.992099i	$0.459961\pi$
$930$	0	0
$931$	−4.28335	−0.140381
$932$	15.1492	0.496230
$933$	0	0
$934$	−19.8084	−0.648151
$935$	0.186756	0.00610758
$936$	0	0
$937$	−1.21281	−0.0396209	−0.0198104	−	0.999804i	$-0.506306\pi$
$937$	−1.21281	−0.0396209	−0.0198104	+	0.999804i	$0.506306\pi$
$938$	−1.86770	−0.0609827
$939$	0	0
$940$	−0.226530	−0.00738860
$941$	12.2619	0.399726	0.199863	−	0.979824i	$-0.435950\pi$
$941$	12.2619	0.399726	0.199863	+	0.979824i	$0.435950\pi$
$942$	0	0
$943$	12.4733	0.406186
$944$	13.5281	0.440301
$945$	0	0
$946$	−9.31010	−0.302697
$947$	17.5657	0.570808	0.285404	−	0.958407i	$-0.407872\pi$
$947$	17.5657	0.570808	0.285404	+	0.958407i	$0.407872\pi$
$948$	0	0
$949$	38.9150	1.26323
$950$	22.3602	0.725462
$951$	0	0
$952$	−8.43686	−0.273440
$953$	−26.6383	−0.862898	−0.431449	−	0.902137i	$-0.641998\pi$
$953$	−26.6383	−0.862898	−0.431449	+	0.902137i	$0.641998\pi$
$954$	0	0
$955$	0.625200	0.0202310
$956$	−20.3706	−0.658833
$957$	0	0
$958$	−5.02223	−0.162261
$959$	−31.5126	−1.01759
$960$	0	0
$961$	41.9463	1.35311
$962$	−28.3066	−0.912643
$963$	0	0
$964$	−7.15656	−0.230497
$965$	1.31950	0.0424763
$966$	0	0
$967$	−33.0542	−1.06295	−0.531475	−	0.847074i	$-0.678362\pi$
$967$	−33.0542	−1.06295	−0.531475	+	0.847074i	$0.678362\pi$
$968$	−10.2828	−0.330502
$969$	0	0
$970$	0.855708	0.0274751
$971$	−26.2117	−0.841172	−0.420586	−	0.907253i	$-0.638176\pi$
$971$	−26.2117	−0.841172	−0.420586	+	0.907253i	$0.638176\pi$
$972$	0	0
$973$	63.7393	2.04339
$974$	−19.9240	−0.638406
$975$	0	0
$976$	−2.57187	−0.0823236
$977$	4.11029	0.131500	0.0657498	−	0.997836i	$-0.479056\pi$
$977$	4.11029	0.131500	0.0657498	+	0.997836i	$0.479056\pi$
$978$	0	0
$979$	4.31782	0.137998
$980$	−0.0705418	−0.00225338
$981$	0	0
$982$	13.0304	0.415817
$983$	2.85827	0.0911647	0.0455823	−	0.998961i	$-0.485486\pi$
$983$	2.85827	0.0911647	0.0455823	+	0.998961i	$0.485486\pi$
$984$	0	0
$985$	1.52269	0.0485170
$986$	−23.3520	−0.743680
$987$	0	0
$988$	−29.6066	−0.941911
$989$	−33.2569	−1.05751
$990$	0	0
$991$	−6.01338	−0.191021	−0.0955106	−	0.995428i	$-0.530448\pi$
$991$	−6.01338	−0.191021	−0.0955106	+	0.995428i	$0.530448\pi$
$992$	8.54086	0.271173
$993$	0	0
$994$	30.0936	0.954510
$995$	−1.57513	−0.0499349
$996$	0	0
$997$	24.1427	0.764608	0.382304	−	0.924037i	$-0.375131\pi$
$997$	24.1427	0.764608	0.382304	+	0.924037i	$0.375131\pi$
$998$	27.3213	0.864841
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.o.1.6	yes	12
3.2	odd	2		8046.2.a.j.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.7	✓	12	3.2	odd	2
8046.2.a.o.1.6	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.0737297 q^{5} +2.82077 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.0737297 q^{5} +2.82077 q^{7} +1.00000 q^{8} -0.0737297 q^{10} +0.846877 q^{11} +6.61317 q^{13} +2.82077 q^{14} +1.00000 q^{16} -2.99097 q^{17} -4.47691 q^{19} -0.0737297 q^{20} +0.846877 q^{22} +3.02516 q^{23} -4.99456 q^{25} +6.61317 q^{26} +2.82077 q^{28} +7.80750 q^{29} +8.54086 q^{31} +1.00000 q^{32} -2.99097 q^{34} -0.207975 q^{35} -4.28035 q^{37} -4.47691 q^{38} -0.0737297 q^{40} +4.12319 q^{41} -10.9934 q^{43} +0.846877 q^{44} +3.02516 q^{46} +3.07244 q^{47} +0.956763 q^{49} -4.99456 q^{50} +6.61317 q^{52} +5.55611 q^{53} -0.0624400 q^{55} +2.82077 q^{56} +7.80750 q^{58} +13.5281 q^{59} -2.57187 q^{61} +8.54086 q^{62} +1.00000 q^{64} -0.487586 q^{65} -0.662125 q^{67} -2.99097 q^{68} -0.207975 q^{70} +10.6685 q^{71} +5.88447 q^{73} -4.28035 q^{74} -4.47691 q^{76} +2.38885 q^{77} -7.96274 q^{79} -0.0737297 q^{80} +4.12319 q^{82} +1.58303 q^{83} +0.220523 q^{85} -10.9934 q^{86} +0.846877 q^{88} +5.09852 q^{89} +18.6542 q^{91} +3.02516 q^{92} +3.07244 q^{94} +0.330081 q^{95} -11.6060 q^{97} +0.956763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8 + 3 * q^10 + 10 * q^11 + 5 * q^13 + 6 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^19 + 3 * q^20 + 10 * q^22 + 9 * q^23 + 7 * q^25 + 5 * q^26 + 6 * q^28 + 19 * q^29 + 10 * q^31 + 12 * q^32 + 8 * q^34 + 20 * q^35 + 11 * q^37 + 2 * q^38 + 3 * q^40 + 8 * q^41 + 13 * q^43 + 10 * q^44 + 9 * q^46 + 11 * q^47 + 2 * q^49 + 7 * q^50 + 5 * q^52 + 24 * q^53 + 3 * q^55 + 6 * q^56 + 19 * q^58 + 10 * q^59 + 10 * q^62 + 12 * q^64 + 28 * q^65 + 21 * q^67 + 8 * q^68 + 20 * q^70 + 37 * q^71 - 2 * q^73 + 11 * q^74 + 2 * q^76 + 2 * q^77 + 7 * q^79 + 3 * q^80 + 8 * q^82 + 22 * q^83 + 15 * q^85 + 13 * q^86 + 10 * q^88 + 40 * q^89 + q^91 + 9 * q^92 + 11 * q^94 + 11 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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