LMFDB - Embedded newform 1341.2.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1341,2,Mod(1,1341)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1341, 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("1341.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1341 = 3^{2} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1341.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:	$10.7079389111$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 15x^{7} + 12x^{6} + 75x^{5} - 48x^{4} - 137x^{3} + 76x^{2} + 68x - 39 $ x^9 - x^8 - 15x^7 + 12x^6 + 75x^5 - 48x^4 - 137x^3 + 76x^2 + 68*x - 39
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 149)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.60510$ of defining polynomial
Character	$\chi$	$=$	1341.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60510 q^{2} +0.576343 q^{4} +1.61810 q^{5} +4.58029 q^{7} -2.28511 q^{8} +O(q^{10})$	\(q+1.60510 q^{2} +0.576343 q^{4} +1.61810 q^{5} +4.58029 q^{7} -2.28511 q^{8} +2.59720 q^{10} +2.18841 q^{11} +4.15663 q^{13} +7.35182 q^{14} -4.82051 q^{16} -4.56008 q^{17} +1.14926 q^{19} +0.932579 q^{20} +3.51262 q^{22} -0.565875 q^{23} -2.38176 q^{25} +6.67180 q^{26} +2.63982 q^{28} -0.935532 q^{29} -6.36311 q^{31} -3.16718 q^{32} -7.31938 q^{34} +7.41135 q^{35} +1.90519 q^{37} +1.84467 q^{38} -3.69753 q^{40} +10.5238 q^{41} -3.11444 q^{43} +1.26128 q^{44} -0.908286 q^{46} +12.6142 q^{47} +13.9790 q^{49} -3.82297 q^{50} +2.39565 q^{52} +5.73780 q^{53} +3.54106 q^{55} -10.4665 q^{56} -1.50162 q^{58} -11.8479 q^{59} -8.30754 q^{61} -10.2134 q^{62} +4.55739 q^{64} +6.72583 q^{65} +15.6061 q^{67} -2.62817 q^{68} +11.8959 q^{70} +0.851833 q^{71} +3.90582 q^{73} +3.05802 q^{74} +0.662366 q^{76} +10.0236 q^{77} -12.2344 q^{79} -7.80006 q^{80} +16.8917 q^{82} -13.9499 q^{83} -7.37864 q^{85} -4.99899 q^{86} -5.00077 q^{88} -1.15067 q^{89} +19.0386 q^{91} -0.326138 q^{92} +20.2470 q^{94} +1.85961 q^{95} -7.77105 q^{97} +22.4377 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8	$ 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) $ 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 6 * q^10 - 5 * q^11 + 7 * q^13 + 8 * q^14 + 13 * q^16 + 5 * q^17 + 30 * q^19 + 10 * q^20 - 7 * q^22 + 4 * q^23 + 6 * q^25 + 15 * q^26 - 12 * q^28 + 16 * q^29 + 22 * q^31 + 38 * q^32 + 9 * q^34 + 11 * q^35 - 7 * q^37 + 18 * q^38 - 7 * q^40 - 6 * q^41 + 4 * q^43 - 6 * q^44 + q^46 + 6 * q^47 + 14 * q^49 + 16 * q^50 + 50 * q^52 + 2 * q^53 - 2 * q^55 - 7 * q^56 + 2 * q^58 - 43 * q^59 + q^61 - 33 * q^62 + 18 * q^64 + 20 * q^65 + 33 * q^67 + 16 * q^68 - 3 * q^70 - 15 * q^71 - 11 * q^73 - 33 * q^74 + 59 * q^76 + 30 * q^77 + q^79 - 65 * q^80 + 5 * q^82 + 4 * q^83 - 34 * q^85 + 7 * q^86 - 37 * q^88 + 19 * q^89 + 62 * q^91 - 17 * q^92 + 17 * q^94 + 21 * q^95 - q^97 - 36 * 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 $

$2$	1.60510	1.13498	0.567488	−	0.823381i	$-0.307915\pi$
$2$	1.60510	1.13498	0.567488	+	0.823381i	$0.307915\pi$
$3$	0	0
$3$	0	0
$4$	0.576343	0.288172
$5$	1.61810	0.723635	0.361817	−	0.932249i	$-0.382156\pi$
$5$	1.61810	0.723635	0.361817	+	0.932249i	$0.382156\pi$
$6$	0	0
$7$	4.58029	1.73119	0.865593	−	0.500748i	$-0.166942\pi$
$7$	4.58029	1.73119	0.865593	+	0.500748i	$0.166942\pi$
$8$	−2.28511	−0.807908
$9$	0	0
$10$	2.59720	0.821308
$11$	2.18841	0.659831	0.329916	−	0.944010i	$-0.392980\pi$
$11$	2.18841	0.659831	0.329916	+	0.944010i	$0.392980\pi$
$12$	0	0
$13$	4.15663	1.15284	0.576421	−	0.817153i	$-0.304449\pi$
$13$	4.15663	1.15284	0.576421	+	0.817153i	$0.304449\pi$
$14$	7.35182	1.96486
$15$	0	0
$16$	−4.82051	−1.20513
$17$	−4.56008	−1.10598	−0.552991	−	0.833187i	$-0.686513\pi$
$17$	−4.56008	−1.10598	−0.552991	+	0.833187i	$0.686513\pi$
$18$	0	0
$19$	1.14926	0.263657	0.131829	−	0.991273i	$-0.457915\pi$
$19$	1.14926	0.263657	0.131829	+	0.991273i	$0.457915\pi$
$20$	0.932579	0.208531
$21$	0	0
$22$	3.51262	0.748893
$23$	−0.565875	−0.117993	−0.0589966	−	0.998258i	$-0.518790\pi$
$23$	−0.565875	−0.117993	−0.0589966	+	0.998258i	$0.518790\pi$
$24$	0	0
$25$	−2.38176	−0.476353
$26$	6.67180	1.30845
$27$	0	0
$28$	2.63982	0.498879
$29$	−0.935532	−0.173724	−0.0868620	−	0.996220i	$-0.527684\pi$
$29$	−0.935532	−0.173724	−0.0868620	+	0.996220i	$0.527684\pi$
$30$	0	0
$31$	−6.36311	−1.14285	−0.571425	−	0.820655i	$-0.693609\pi$
$31$	−6.36311	−1.14285	−0.571425	+	0.820655i	$0.693609\pi$
$32$	−3.16718	−0.559884
$33$	0	0
$34$	−7.31938	−1.25526
$35$	7.41135	1.25275
$36$	0	0
$37$	1.90519	0.313212	0.156606	−	0.987661i	$-0.449945\pi$
$37$	1.90519	0.313212	0.156606	+	0.987661i	$0.449945\pi$
$38$	1.84467	0.299245
$39$	0	0
$40$	−3.69753	−0.584631
$41$	10.5238	1.64354	0.821769	−	0.569821i	$-0.192987\pi$
$41$	10.5238	1.64354	0.821769	+	0.569821i	$0.192987\pi$
$42$	0	0
$43$	−3.11444	−0.474948	−0.237474	−	0.971394i	$-0.576319\pi$
$43$	−3.11444	−0.474948	−0.237474	+	0.971394i	$0.576319\pi$
$44$	1.26128	0.190145
$45$	0	0
$46$	−0.908286	−0.133919
$47$	12.6142	1.83997	0.919984	−	0.391955i	$-0.128201\pi$
$47$	12.6142	1.83997	0.919984	+	0.391955i	$0.128201\pi$
$48$	0	0
$49$	13.9790	1.99700
$50$	−3.82297	−0.540649
$51$	0	0
$52$	2.39565	0.332216
$53$	5.73780	0.788148	0.394074	−	0.919079i	$-0.371065\pi$
$53$	5.73780	0.788148	0.394074	+	0.919079i	$0.371065\pi$
$54$	0	0
$55$	3.54106	0.477477
$56$	−10.4665	−1.39864
$57$	0	0
$58$	−1.50162	−0.197173
$59$	−11.8479	−1.54246	−0.771232	−	0.636555i	$-0.780359\pi$
$59$	−11.8479	−1.54246	−0.771232	+	0.636555i	$0.780359\pi$
$60$	0	0
$61$	−8.30754	−1.06367	−0.531836	−	0.846848i	$-0.678498\pi$
$61$	−8.30754	−1.06367	−0.531836	+	0.846848i	$0.678498\pi$
$62$	−10.2134	−1.29711
$63$	0	0
$64$	4.55739	0.569673
$65$	6.72583	0.834236
$66$	0	0
$67$	15.6061	1.90658	0.953291	−	0.302052i	$-0.0976716\pi$
$67$	15.6061	1.90658	0.953291	+	0.302052i	$0.0976716\pi$
$68$	−2.62817	−0.318712
$69$	0	0
$70$	11.8959	1.42184
$71$	0.851833	0.101094	0.0505470	−	0.998722i	$-0.483904\pi$
$71$	0.851833	0.101094	0.0505470	+	0.998722i	$0.483904\pi$
$72$	0	0
$73$	3.90582	0.457142	0.228571	−	0.973527i	$-0.426595\pi$
$73$	3.90582	0.457142	0.228571	+	0.973527i	$0.426595\pi$
$74$	3.05802	0.355488
$75$	0	0
$76$	0.662366	0.0759786
$77$	10.0236	1.14229
$78$	0	0
$79$	−12.2344	−1.37648	−0.688241	−	0.725482i	$-0.741617\pi$
$79$	−12.2344	−1.37648	−0.688241	+	0.725482i	$0.741617\pi$
$80$	−7.80006	−0.872073
$81$	0	0
$82$	16.8917	1.86538
$83$	−13.9499	−1.53120	−0.765602	−	0.643315i	$-0.777559\pi$
$83$	−13.9499	−1.53120	−0.765602	+	0.643315i	$0.777559\pi$
$84$	0	0
$85$	−7.37864	−0.800326
$86$	−4.99899	−0.539055
$87$	0	0
$88$	−5.00077	−0.533083
$89$	−1.15067	−0.121971	−0.0609853	−	0.998139i	$-0.519424\pi$
$89$	−1.15067	−0.121971	−0.0609853	+	0.998139i	$0.519424\pi$
$90$	0	0
$91$	19.0386	1.99578
$92$	−0.326138	−0.0340023
$93$	0	0
$94$	20.2470	2.08832
$95$	1.85961	0.190792
$96$	0	0
$97$	−7.77105	−0.789030	−0.394515	−	0.918889i	$-0.629087\pi$
$97$	−7.77105	−0.789030	−0.394515	+	0.918889i	$0.629087\pi$
$98$	22.4377	2.26655
$99$	0	0
$100$	−1.37271	−0.137271
$101$	−2.59676	−0.258387	−0.129194	−	0.991619i	$-0.541239\pi$
$101$	−2.59676	−0.258387	−0.129194	+	0.991619i	$0.541239\pi$
$102$	0	0
$103$	−16.3525	−1.61126	−0.805630	−	0.592419i	$-0.798173\pi$
$103$	−16.3525	−1.61126	−0.805630	+	0.592419i	$0.798173\pi$
$104$	−9.49836	−0.931391
$105$	0	0
$106$	9.20974	0.894529
$107$	−13.5767	−1.31251	−0.656257	−	0.754538i	$-0.727861\pi$
$107$	−13.5767	−1.31251	−0.656257	+	0.754538i	$0.727861\pi$
$108$	0	0
$109$	1.94406	0.186207	0.0931037	−	0.995656i	$-0.470321\pi$
$109$	1.94406	0.186207	0.0931037	+	0.995656i	$0.470321\pi$
$110$	5.68376	0.541925
$111$	0	0
$112$	−22.0793	−2.08630
$113$	−5.94772	−0.559514	−0.279757	−	0.960071i	$-0.590254\pi$
$113$	−5.94772	−0.559514	−0.279757	+	0.960071i	$0.590254\pi$
$114$	0	0
$115$	−0.915640	−0.0853839
$116$	−0.539188	−0.0500623
$117$	0	0
$118$	−19.0170	−1.75066
$119$	−20.8865	−1.91466
$120$	0	0
$121$	−6.21085	−0.564622
$122$	−13.3344	−1.20724
$123$	0	0
$124$	−3.66734	−0.329337
$125$	−11.9444	−1.06834
$126$	0	0
$127$	1.14813	0.101880	0.0509398	−	0.998702i	$-0.483778\pi$
$127$	1.14813	0.101880	0.0509398	+	0.998702i	$0.483778\pi$
$128$	13.6494	1.20645
$129$	0	0
$130$	10.7956	0.946839
$131$	5.62287	0.491272	0.245636	−	0.969362i	$-0.421003\pi$
$131$	5.62287	0.491272	0.245636	+	0.969362i	$0.421003\pi$
$132$	0	0
$133$	5.26392	0.456440
$134$	25.0493	2.16393
$135$	0	0
$136$	10.4203	0.893532
$137$	−0.803810	−0.0686741	−0.0343371	−	0.999410i	$-0.510932\pi$
$137$	−0.803810	−0.0686741	−0.0343371	+	0.999410i	$0.510932\pi$
$138$	0	0
$139$	0.931286	0.0789906	0.0394953	−	0.999220i	$-0.487425\pi$
$139$	0.931286	0.0789906	0.0394953	+	0.999220i	$0.487425\pi$
$140$	4.27148	0.361006
$141$	0	0
$142$	1.36728	0.114739
$143$	9.09643	0.760681
$144$	0	0
$145$	−1.51378	−0.125713
$146$	6.26923	0.518845
$147$	0	0
$148$	1.09805	0.0902588
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	19.2396	1.56569	0.782847	−	0.622215i	$-0.213767\pi$
$151$	19.2396	1.56569	0.782847	+	0.622215i	$0.213767\pi$
$152$	−2.62618	−0.213011
$153$	0	0
$154$	16.0888	1.29647
$155$	−10.2961	−0.827005
$156$	0	0
$157$	2.58762	0.206515	0.103257	−	0.994655i	$-0.467073\pi$
$157$	2.58762	0.206515	0.103257	+	0.994655i	$0.467073\pi$
$158$	−19.6375	−1.56228
$159$	0	0
$160$	−5.12481	−0.405152
$161$	−2.59187	−0.204268
$162$	0	0
$163$	−8.64759	−0.677332	−0.338666	−	0.940907i	$-0.609976\pi$
$163$	−8.64759	−0.677332	−0.338666	+	0.940907i	$0.609976\pi$
$164$	6.06531	0.473621
$165$	0	0
$166$	−22.3910	−1.73788
$167$	9.71609	0.751854	0.375927	−	0.926649i	$-0.377324\pi$
$167$	9.71609	0.751854	0.375927	+	0.926649i	$0.377324\pi$
$168$	0	0
$169$	4.27758	0.329045
$170$	−11.8435	−0.908351
$171$	0	0
$172$	−1.79499	−0.136867
$173$	6.60313	0.502027	0.251014	−	0.967984i	$-0.419236\pi$
$173$	6.60313	0.502027	0.251014	+	0.967984i	$0.419236\pi$
$174$	0	0
$175$	−10.9092	−0.824656
$176$	−10.5493	−0.795182
$177$	0	0
$178$	−1.84694	−0.138434
$179$	6.85957	0.512708	0.256354	−	0.966583i	$-0.417479\pi$
$179$	6.85957	0.512708	0.256354	+	0.966583i	$0.417479\pi$
$180$	0	0
$181$	−22.9083	−1.70276	−0.851381	−	0.524547i	$-0.824235\pi$
$181$	−22.9083	−1.70276	−0.851381	+	0.524547i	$0.824235\pi$
$182$	30.5588	2.26517
$183$	0	0
$184$	1.29309	0.0953276
$185$	3.08279	0.226651
$186$	0	0
$187$	−9.97933	−0.729761
$188$	7.27010	0.530227
$189$	0	0
$190$	2.98485	0.216544
$191$	7.15017	0.517368	0.258684	−	0.965962i	$-0.416711\pi$
$191$	7.15017	0.517368	0.258684	+	0.965962i	$0.416711\pi$
$192$	0	0
$193$	−7.96445	−0.573293	−0.286647	−	0.958036i	$-0.592541\pi$
$193$	−7.96445	−0.573293	−0.286647	+	0.958036i	$0.592541\pi$
$194$	−12.4733	−0.895531
$195$	0	0
$196$	8.05672	0.575480
$197$	4.57914	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$197$	4.57914	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$198$	0	0
$199$	11.9473	0.846918	0.423459	−	0.905915i	$-0.360816\pi$
$199$	11.9473	0.846918	0.423459	+	0.905915i	$0.360816\pi$
$200$	5.44259	0.384850
$201$	0	0
$202$	−4.16806	−0.293264
$203$	−4.28501	−0.300748
$204$	0	0
$205$	17.0285	1.18932
$206$	−26.2474	−1.82874
$207$	0	0
$208$	−20.0371	−1.38932
$209$	2.51505	0.173969
$210$	0	0
$211$	−16.5224	−1.13745	−0.568724	−	0.822528i	$-0.692563\pi$
$211$	−16.5224	−1.13745	−0.568724	+	0.822528i	$0.692563\pi$
$212$	3.30694	0.227122
$213$	0	0
$214$	−21.7920	−1.48967
$215$	−5.03947	−0.343689
$216$	0	0
$217$	−29.1449	−1.97848
$218$	3.12041	0.211341
$219$	0	0
$220$	2.04087	0.137595
$221$	−18.9546	−1.27502
$222$	0	0
$223$	23.4598	1.57098	0.785491	−	0.618873i	$-0.212410\pi$
$223$	23.4598	1.57098	0.785491	+	0.618873i	$0.212410\pi$
$224$	−14.5066	−0.969264
$225$	0	0
$226$	−9.54667	−0.635035
$227$	−21.2145	−1.40806	−0.704029	−	0.710171i	$-0.748618\pi$
$227$	−21.2145	−1.40806	−0.704029	+	0.710171i	$0.748618\pi$
$228$	0	0
$229$	2.56128	0.169254	0.0846272	−	0.996413i	$-0.473030\pi$
$229$	2.56128	0.169254	0.0846272	+	0.996413i	$0.473030\pi$
$230$	−1.46969	−0.0969087
$231$	0	0
$232$	2.13779	0.140353
$233$	−29.3104	−1.92019	−0.960093	−	0.279681i	$-0.909771\pi$
$233$	−29.3104	−1.92019	−0.960093	+	0.279681i	$0.909771\pi$
$234$	0	0
$235$	20.4110	1.33147
$236$	−6.82845	−0.444494
$237$	0	0
$238$	−33.5248	−2.17309
$239$	−0.786631	−0.0508829	−0.0254415	−	0.999676i	$-0.508099\pi$
$239$	−0.786631	−0.0508829	−0.0254415	+	0.999676i	$0.508099\pi$
$240$	0	0
$241$	−25.0809	−1.61560	−0.807801	−	0.589455i	$-0.799343\pi$
$241$	−25.0809	−1.61560	−0.807801	+	0.589455i	$0.799343\pi$
$242$	−9.96902	−0.640833
$243$	0	0
$244$	−4.78799	−0.306520
$245$	22.6194	1.44510
$246$	0	0
$247$	4.77703	0.303955
$248$	14.5404	0.923317
$249$	0	0
$250$	−19.1720	−1.21254
$251$	−26.4649	−1.67045	−0.835225	−	0.549908i	$-0.814663\pi$
$251$	−26.4649	−1.67045	−0.835225	+	0.549908i	$0.814663\pi$
$252$	0	0
$253$	−1.23837	−0.0778556
$254$	1.84286	0.115631
$255$	0	0
$256$	12.7939	0.799619
$257$	18.4020	1.14788	0.573942	−	0.818896i	$-0.305414\pi$
$257$	18.4020	1.14788	0.573942	+	0.818896i	$0.305414\pi$
$258$	0	0
$259$	8.72633	0.542228
$260$	3.87639	0.240403
$261$	0	0
$262$	9.02526	0.557583
$263$	−2.50403	−0.154405	−0.0772025	−	0.997015i	$-0.524599\pi$
$263$	−2.50403	−0.154405	−0.0772025	+	0.997015i	$0.524599\pi$
$264$	0	0
$265$	9.28432	0.570331
$266$	8.44912	0.518049
$267$	0	0
$268$	8.99444	0.549423
$269$	12.5575	0.765645	0.382822	−	0.923822i	$-0.374952\pi$
$269$	12.5575	0.765645	0.382822	+	0.923822i	$0.374952\pi$
$270$	0	0
$271$	28.2606	1.71671	0.858356	−	0.513055i	$-0.171486\pi$
$271$	28.2606	1.71671	0.858356	+	0.513055i	$0.171486\pi$
$272$	21.9819	1.33285
$273$	0	0
$274$	−1.29019	−0.0779435
$275$	−5.21229	−0.314313
$276$	0	0
$277$	9.64560	0.579548	0.289774	−	0.957095i	$-0.406420\pi$
$277$	9.64560	0.579548	0.289774	+	0.957095i	$0.406420\pi$
$278$	1.49481	0.0896525
$279$	0	0
$280$	−16.9357	−1.01210
$281$	−29.9804	−1.78848	−0.894241	−	0.447586i	$-0.852284\pi$
$281$	−29.9804	−1.78848	−0.894241	+	0.447586i	$0.852284\pi$
$282$	0	0
$283$	−11.3871	−0.676893	−0.338446	−	0.940986i	$-0.609901\pi$
$283$	−11.3871	−0.676893	−0.338446	+	0.940986i	$0.609901\pi$
$284$	0.490948	0.0291324
$285$	0	0
$286$	14.6007	0.863356
$287$	48.2019	2.84527
$288$	0	0
$289$	3.79430	0.223194
$290$	−2.42977	−0.142681
$291$	0	0
$292$	2.25109	0.131735
$293$	−19.3369	−1.12967	−0.564836	−	0.825203i	$-0.691061\pi$
$293$	−19.3369	−1.12967	−0.564836	+	0.825203i	$0.691061\pi$
$294$	0	0
$295$	−19.1710	−1.11618
$296$	−4.35358	−0.253047
$297$	0	0
$298$	−1.60510	−0.0929809
$299$	−2.35213	−0.136027
$300$	0	0
$301$	−14.2650	−0.822223
$302$	30.8814	1.77702
$303$	0	0
$304$	−5.54001	−0.317741
$305$	−13.4424	−0.769709
$306$	0	0
$307$	19.9732	1.13993	0.569964	−	0.821670i	$-0.306957\pi$
$307$	19.9732	1.13993	0.569964	+	0.821670i	$0.306957\pi$
$308$	5.77701	0.329176
$309$	0	0
$310$	−16.5263	−0.938631
$311$	−21.5927	−1.22441	−0.612204	−	0.790700i	$-0.709717\pi$
$311$	−21.5927	−1.22441	−0.612204	+	0.790700i	$0.709717\pi$
$312$	0	0
$313$	28.7932	1.62749	0.813744	−	0.581223i	$-0.197426\pi$
$313$	28.7932	1.62749	0.813744	+	0.581223i	$0.197426\pi$
$314$	4.15339	0.234389
$315$	0	0
$316$	−7.05124	−0.396663
$317$	9.81392	0.551204	0.275602	−	0.961272i	$-0.411123\pi$
$317$	9.81392	0.551204	0.275602	+	0.961272i	$0.411123\pi$
$318$	0	0
$319$	−2.04733	−0.114629
$320$	7.37429	0.412235
$321$	0	0
$322$	−4.16021	−0.231839
$323$	−5.24070	−0.291600
$324$	0	0
$325$	−9.90012	−0.549160
$326$	−13.8802	−0.768756
$327$	0	0
$328$	−24.0480	−1.32783
$329$	57.7766	3.18533
$330$	0	0
$331$	10.6126	0.583320	0.291660	−	0.956522i	$-0.405792\pi$
$331$	10.6126	0.583320	0.291660	+	0.956522i	$0.405792\pi$
$332$	−8.03995	−0.441249
$333$	0	0
$334$	15.5953	0.853336
$335$	25.2521	1.37967
$336$	0	0
$337$	−17.2831	−0.941472	−0.470736	−	0.882274i	$-0.656012\pi$
$337$	−17.2831	−0.941472	−0.470736	+	0.882274i	$0.656012\pi$
$338$	6.86594	0.373458
$339$	0	0
$340$	−4.25263	−0.230631
$341$	−13.9251	−0.754088
$342$	0	0
$343$	31.9660	1.72600
$344$	7.11685	0.383715
$345$	0	0
$346$	10.5987	0.569789
$347$	26.3745	1.41586	0.707929	−	0.706284i	$-0.249630\pi$
$347$	26.3745	1.41586	0.707929	+	0.706284i	$0.249630\pi$
$348$	0	0
$349$	−19.0250	−1.01838	−0.509192	−	0.860653i	$-0.670056\pi$
$349$	−19.0250	−1.01838	−0.509192	+	0.860653i	$0.670056\pi$
$350$	−17.5103	−0.935965
$351$	0	0
$352$	−6.93111	−0.369429
$353$	15.0218	0.799530	0.399765	−	0.916618i	$-0.369092\pi$
$353$	15.0218	0.799530	0.399765	+	0.916618i	$0.369092\pi$
$354$	0	0
$355$	1.37835	0.0731551
$356$	−0.663180	−0.0351485
$357$	0	0
$358$	11.0103	0.581912
$359$	−0.226039	−0.0119299	−0.00596493	−	0.999982i	$-0.501899\pi$
$359$	−0.226039	−0.0119299	−0.00596493	+	0.999982i	$0.501899\pi$
$360$	0	0
$361$	−17.6792	−0.930485
$362$	−36.7701	−1.93260
$363$	0	0
$364$	10.9727	0.575128
$365$	6.32000	0.330804
$366$	0	0
$367$	−23.6050	−1.23217	−0.616084	−	0.787680i	$-0.711282\pi$
$367$	−23.6050	−1.23217	−0.616084	+	0.787680i	$0.711282\pi$
$368$	2.72781	0.142197
$369$	0	0
$370$	4.94818	0.257243
$371$	26.2808	1.36443
$372$	0	0
$373$	16.8003	0.869885	0.434942	−	0.900458i	$-0.356769\pi$
$373$	16.8003	0.869885	0.434942	+	0.900458i	$0.356769\pi$
$374$	−16.0178	−0.828262
$375$	0	0
$376$	−28.8248	−1.48653
$377$	−3.88866	−0.200276
$378$	0	0
$379$	32.2575	1.65695	0.828477	−	0.560023i	$-0.189208\pi$
$379$	32.2575	1.65695	0.828477	+	0.560023i	$0.189208\pi$
$380$	1.07177	0.0549807
$381$	0	0
$382$	11.4767	0.587200
$383$	1.33342	0.0681348	0.0340674	−	0.999420i	$-0.489154\pi$
$383$	1.33342	0.0681348	0.0340674	+	0.999420i	$0.489154\pi$
$384$	0	0
$385$	16.2191	0.826601
$386$	−12.7837	−0.650674
$387$	0	0
$388$	−4.47879	−0.227376
$389$	2.10792	0.106876	0.0534380	−	0.998571i	$-0.482982\pi$
$389$	2.10792	0.106876	0.0534380	+	0.998571i	$0.482982\pi$
$390$	0	0
$391$	2.58043	0.130498
$392$	−31.9436	−1.61340
$393$	0	0
$394$	7.34997	0.370286
$395$	−19.7965	−0.996071
$396$	0	0
$397$	15.3814	0.771971	0.385985	−	0.922505i	$-0.373861\pi$
$397$	15.3814	0.771971	0.385985	+	0.922505i	$0.373861\pi$
$398$	19.1765	0.961232
$399$	0	0
$400$	11.4813	0.574067
$401$	5.31766	0.265551	0.132776	−	0.991146i	$-0.457611\pi$
$401$	5.31766	0.265551	0.132776	+	0.991146i	$0.457611\pi$
$402$	0	0
$403$	−26.4491	−1.31752
$404$	−1.49663	−0.0744599
$405$	0	0
$406$	−6.87786	−0.341342
$407$	4.16935	0.206667
$408$	0	0
$409$	14.1544	0.699888	0.349944	−	0.936771i	$-0.386201\pi$
$409$	14.1544	0.699888	0.349944	+	0.936771i	$0.386201\pi$
$410$	27.3324	1.34985
$411$	0	0
$412$	−9.42465	−0.464319
$413$	−54.2667	−2.67029
$414$	0	0
$415$	−22.5723	−1.10803
$416$	−13.1648	−0.645458
$417$	0	0
$418$	4.03690	0.197451
$419$	1.90815	0.0932192	0.0466096	−	0.998913i	$-0.485158\pi$
$419$	1.90815	0.0932192	0.0466096	+	0.998913i	$0.485158\pi$
$420$	0	0
$421$	−23.0811	−1.12491	−0.562453	−	0.826829i	$-0.690142\pi$
$421$	−23.0811	−1.12491	−0.562453	+	0.826829i	$0.690142\pi$
$422$	−26.5201	−1.29098
$423$	0	0
$424$	−13.1115	−0.636751
$425$	10.8610	0.526837
$426$	0	0
$427$	−38.0509	−1.84141
$428$	−7.82486	−0.378229
$429$	0	0
$430$	−8.08885	−0.390079
$431$	21.7118	1.04582	0.522911	−	0.852387i	$-0.324846\pi$
$431$	21.7118	1.04582	0.522911	+	0.852387i	$0.324846\pi$
$432$	0	0
$433$	−1.36060	−0.0653862	−0.0326931	−	0.999465i	$-0.510408\pi$
$433$	−1.36060	−0.0653862	−0.0326931	+	0.999465i	$0.510408\pi$
$434$	−46.7804	−2.24553
$435$	0	0
$436$	1.12045	0.0536597
$437$	−0.650335	−0.0311098
$438$	0	0
$439$	27.7517	1.32452	0.662258	−	0.749275i	$-0.269598\pi$
$439$	27.7517	1.32452	0.662258	+	0.749275i	$0.269598\pi$
$440$	−8.09172	−0.385758
$441$	0	0
$442$	−30.4239	−1.44712
$443$	16.4055	0.779450	0.389725	−	0.920931i	$-0.372570\pi$
$443$	16.4055	0.779450	0.389725	+	0.920931i	$0.372570\pi$
$444$	0	0
$445$	−1.86189	−0.0882621
$446$	37.6553	1.78303
$447$	0	0
$448$	20.8741	0.986210
$449$	−18.7210	−0.883499	−0.441749	−	0.897139i	$-0.645642\pi$
$449$	−18.7210	−0.883499	−0.441749	+	0.897139i	$0.645642\pi$
$450$	0	0
$451$	23.0304	1.08446
$452$	−3.42793	−0.161236
$453$	0	0
$454$	−34.0514	−1.59811
$455$	30.8062	1.44422
$456$	0	0
$457$	26.6978	1.24887	0.624436	−	0.781076i	$-0.285329\pi$
$457$	26.6978	1.24887	0.624436	+	0.781076i	$0.285329\pi$
$458$	4.11112	0.192100
$459$	0	0
$460$	−0.527723	−0.0246052
$461$	15.2809	0.711704	0.355852	−	0.934542i	$-0.384191\pi$
$461$	15.2809	0.711704	0.355852	+	0.934542i	$0.384191\pi$
$462$	0	0
$463$	36.2380	1.68412	0.842061	−	0.539383i	$-0.181342\pi$
$463$	36.2380	1.68412	0.842061	+	0.539383i	$0.181342\pi$
$464$	4.50975	0.209360
$465$	0	0
$466$	−47.0460	−2.17937
$467$	−8.01421	−0.370853	−0.185427	−	0.982658i	$-0.559367\pi$
$467$	−8.01421	−0.370853	−0.185427	+	0.982658i	$0.559367\pi$
$468$	0	0
$469$	71.4802	3.30065
$470$	32.7616	1.51118
$471$	0	0
$472$	27.0737	1.24617
$473$	−6.81569	−0.313386
$474$	0	0
$475$	−2.73726	−0.125594
$476$	−12.0378	−0.551750
$477$	0	0
$478$	−1.26262	−0.0577509
$479$	12.1072	0.553194	0.276597	−	0.960986i	$-0.410793\pi$
$479$	12.1072	0.553194	0.276597	+	0.960986i	$0.410793\pi$
$480$	0	0
$481$	7.91919	0.361084
$482$	−40.2573	−1.83367
$483$	0	0
$484$	−3.57958	−0.162708
$485$	−12.5743	−0.570970
$486$	0	0
$487$	−13.7922	−0.624982	−0.312491	−	0.949921i	$-0.601163\pi$
$487$	−13.7922	−0.624982	−0.312491	+	0.949921i	$0.601163\pi$
$488$	18.9836	0.859349
$489$	0	0
$490$	36.3064	1.64016
$491$	−21.4486	−0.967960	−0.483980	−	0.875079i	$-0.660809\pi$
$491$	−21.4486	−0.967960	−0.483980	+	0.875079i	$0.660809\pi$
$492$	0	0
$493$	4.26610	0.192135
$494$	7.66761	0.344982
$495$	0	0
$496$	30.6735	1.37728
$497$	3.90164	0.175013
$498$	0	0
$499$	−15.3767	−0.688356	−0.344178	−	0.938904i	$-0.611842\pi$
$499$	−15.3767	−0.688356	−0.344178	+	0.938904i	$0.611842\pi$
$500$	−6.88408	−0.307865
$501$	0	0
$502$	−42.4788	−1.89592
$503$	22.7495	1.01435	0.507176	−	0.861842i	$-0.330689\pi$
$503$	22.7495	1.01435	0.507176	+	0.861842i	$0.330689\pi$
$504$	0	0
$505$	−4.20181	−0.186978
$506$	−1.98770	−0.0883642
$507$	0	0
$508$	0.661714	0.0293588
$509$	−14.5956	−0.646939	−0.323469	−	0.946239i	$-0.604849\pi$
$509$	−14.5956	−0.646939	−0.323469	+	0.946239i	$0.604849\pi$
$510$	0	0
$511$	17.8898	0.791398
$512$	−6.76336	−0.298901
$513$	0	0
$514$	29.5370	1.30282
$515$	−26.4599	−1.16596
$516$	0	0
$517$	27.6051	1.21407
$518$	14.0066	0.615416
$519$	0	0
$520$	−15.3693	−0.673987
$521$	−27.5974	−1.20906	−0.604532	−	0.796581i	$-0.706640\pi$
$521$	−27.5974	−1.20906	−0.604532	+	0.796581i	$0.706640\pi$
$522$	0	0
$523$	6.56814	0.287205	0.143602	−	0.989635i	$-0.454131\pi$
$523$	6.56814	0.287205	0.143602	+	0.989635i	$0.454131\pi$
$524$	3.24070	0.141571
$525$	0	0
$526$	−4.01921	−0.175246
$527$	29.0163	1.26397
$528$	0	0
$529$	−22.6798	−0.986078
$530$	14.9022	0.647312
$531$	0	0
$532$	3.03383	0.131533
$533$	43.7435	1.89474
$534$	0	0
$535$	−21.9685	−0.949780
$536$	−35.6616	−1.54034
$537$	0	0
$538$	20.1560	0.868989
$539$	30.5919	1.31769
$540$	0	0
$541$	1.51611	0.0651826	0.0325913	−	0.999469i	$-0.489624\pi$
$541$	1.51611	0.0651826	0.0325913	+	0.999469i	$0.489624\pi$
$542$	45.3611	1.94843
$543$	0	0
$544$	14.4426	0.619222
$545$	3.14568	0.134746
$546$	0	0
$547$	22.2054	0.949435	0.474718	−	0.880138i	$-0.342550\pi$
$547$	22.2054	0.949435	0.474718	+	0.880138i	$0.342550\pi$
$548$	−0.463271	−0.0197899
$549$	0	0
$550$	−8.36624	−0.356737
$551$	−1.07517	−0.0458036
$552$	0	0
$553$	−56.0373	−2.38295
$554$	15.4821	0.657773
$555$	0	0
$556$	0.536740	0.0227629
$557$	4.56601	0.193468	0.0967340	−	0.995310i	$-0.469160\pi$
$557$	4.56601	0.193468	0.0967340	+	0.995310i	$0.469160\pi$
$558$	0	0
$559$	−12.9456	−0.547540
$560$	−35.7265	−1.50972
$561$	0	0
$562$	−48.1215	−2.02988
$563$	−1.81784	−0.0766127	−0.0383063	−	0.999266i	$-0.512196\pi$
$563$	−1.81784	−0.0766127	−0.0383063	+	0.999266i	$0.512196\pi$
$564$	0	0
$565$	−9.62398	−0.404884
$566$	−18.2774	−0.768257
$567$	0	0
$568$	−1.94653	−0.0816747
$569$	18.2290	0.764199	0.382100	−	0.924121i	$-0.375201\pi$
$569$	18.2290	0.764199	0.382100	+	0.924121i	$0.375201\pi$
$570$	0	0
$571$	−6.71113	−0.280852	−0.140426	−	0.990091i	$-0.544847\pi$
$571$	−6.71113	−0.280852	−0.140426	+	0.990091i	$0.544847\pi$
$572$	5.24266	0.219207
$573$	0	0
$574$	77.3689	3.22932
$575$	1.34778	0.0562064
$576$	0	0
$577$	−1.54458	−0.0643019	−0.0321509	−	0.999483i	$-0.510236\pi$
$577$	−1.54458	−0.0643019	−0.0321509	+	0.999483i	$0.510236\pi$
$578$	6.09023	0.253320
$579$	0	0
$580$	−0.872457	−0.0362268
$581$	−63.8947	−2.65080
$582$	0	0
$583$	12.5567	0.520045
$584$	−8.92524	−0.369329
$585$	0	0
$586$	−31.0376	−1.28215
$587$	−6.07810	−0.250870	−0.125435	−	0.992102i	$-0.540033\pi$
$587$	−6.07810	−0.250870	−0.125435	+	0.992102i	$0.540033\pi$
$588$	0	0
$589$	−7.31285	−0.301321
$590$	−30.7714	−1.26684
$591$	0	0
$592$	−9.18401	−0.377461
$593$	−16.9776	−0.697185	−0.348592	−	0.937274i	$-0.613340\pi$
$593$	−16.9776	−0.697185	−0.348592	+	0.937274i	$0.613340\pi$
$594$	0	0
$595$	−33.7963	−1.38551
$596$	−0.576343	−0.0236079
$597$	0	0
$598$	−3.77541	−0.154388
$599$	−3.77340	−0.154177	−0.0770886	−	0.997024i	$-0.524562\pi$
$599$	−3.77340	−0.154177	−0.0770886	+	0.997024i	$0.524562\pi$
$600$	0	0
$601$	−16.2959	−0.664723	−0.332361	−	0.943152i	$-0.607845\pi$
$601$	−16.2959	−0.664723	−0.332361	+	0.943152i	$0.607845\pi$
$602$	−22.8968	−0.933204
$603$	0	0
$604$	11.0886	0.451188
$605$	−10.0497	−0.408580
$606$	0	0
$607$	27.0629	1.09845	0.549224	−	0.835675i	$-0.314923\pi$
$607$	27.0629	1.09845	0.549224	+	0.835675i	$0.314923\pi$
$608$	−3.63991	−0.147618
$609$	0	0
$610$	−21.5764	−0.873602
$611$	52.4325	2.12119
$612$	0	0
$613$	−2.03060	−0.0820150	−0.0410075	−	0.999159i	$-0.513057\pi$
$613$	−2.03060	−0.0820150	−0.0410075	+	0.999159i	$0.513057\pi$
$614$	32.0589	1.29379
$615$	0	0
$616$	−22.9049	−0.922867
$617$	−17.7024	−0.712673	−0.356336	−	0.934358i	$-0.615974\pi$
$617$	−17.7024	−0.712673	−0.356336	+	0.934358i	$0.615974\pi$
$618$	0	0
$619$	36.2600	1.45741	0.728706	−	0.684826i	$-0.240122\pi$
$619$	36.2600	1.45741	0.728706	+	0.684826i	$0.240122\pi$
$620$	−5.93411	−0.238319
$621$	0	0
$622$	−34.6584	−1.38967
$623$	−5.27039	−0.211154
$624$	0	0
$625$	−7.41837	−0.296735
$626$	46.2160	1.84716
$627$	0	0
$628$	1.49136	0.0595117
$629$	−8.68783	−0.346406
$630$	0	0
$631$	−43.3373	−1.72523	−0.862614	−	0.505862i	$-0.831175\pi$
$631$	−43.3373	−1.72523	−0.862614	+	0.505862i	$0.831175\pi$
$632$	27.9571	1.11207
$633$	0	0
$634$	15.7523	0.625604
$635$	1.85778	0.0737237
$636$	0	0
$637$	58.1057	2.30223
$638$	−3.28617	−0.130101
$639$	0	0
$640$	22.0861	0.873029
$641$	8.63702	0.341142	0.170571	−	0.985345i	$-0.445439\pi$
$641$	8.63702	0.341142	0.170571	+	0.985345i	$0.445439\pi$
$642$	0	0
$643$	0.609504	0.0240365	0.0120182	−	0.999928i	$-0.496174\pi$
$643$	0.609504	0.0240365	0.0120182	+	0.999928i	$0.496174\pi$
$644$	−1.49381	−0.0588642
$645$	0	0
$646$	−8.41184	−0.330959
$647$	−4.34556	−0.170842	−0.0854208	−	0.996345i	$-0.527223\pi$
$647$	−4.34556	−0.170842	−0.0854208	+	0.996345i	$0.527223\pi$
$648$	0	0
$649$	−25.9281	−1.01777
$650$	−15.8907	−0.623283
$651$	0	0
$652$	−4.98398	−0.195188
$653$	7.19764	0.281666	0.140833	−	0.990033i	$-0.455022\pi$
$653$	7.19764	0.281666	0.140833	+	0.990033i	$0.455022\pi$
$654$	0	0
$655$	9.09834	0.355502
$656$	−50.7300	−1.98068
$657$	0	0
$658$	92.7372	3.61527
$659$	−25.1558	−0.979930	−0.489965	−	0.871742i	$-0.662990\pi$
$659$	−25.1558	−0.979930	−0.489965	+	0.871742i	$0.662990\pi$
$660$	0	0
$661$	−11.0508	−0.429825	−0.214912	−	0.976633i	$-0.568947\pi$
$661$	−11.0508	−0.429825	−0.214912	+	0.976633i	$0.568947\pi$
$662$	17.0342	0.662055
$663$	0	0
$664$	31.8771	1.23707
$665$	8.51753	0.330296
$666$	0	0
$667$	0.529394	0.0204982
$668$	5.59980	0.216663
$669$	0	0
$670$	40.5321	1.56589
$671$	−18.1803	−0.701844
$672$	0	0
$673$	15.9152	0.613485	0.306743	−	0.951793i	$-0.400761\pi$
$673$	15.9152	0.613485	0.306743	+	0.951793i	$0.400761\pi$
$674$	−27.7411	−1.06855
$675$	0	0
$676$	2.46535	0.0948213
$677$	18.0453	0.693538	0.346769	−	0.937951i	$-0.387279\pi$
$677$	18.0453	0.693538	0.346769	+	0.937951i	$0.387279\pi$
$678$	0	0
$679$	−35.5936	−1.36596
$680$	16.8610	0.646590
$681$	0	0
$682$	−22.3512	−0.855872
$683$	−6.40694	−0.245155	−0.122577	−	0.992459i	$-0.539116\pi$
$683$	−6.40694	−0.245155	−0.122577	+	0.992459i	$0.539116\pi$
$684$	0	0
$685$	−1.30064	−0.0496950
$686$	51.3086	1.95897
$687$	0	0
$688$	15.0132	0.572374
$689$	23.8499	0.908610
$690$	0	0
$691$	47.5301	1.80813	0.904065	−	0.427395i	$-0.140569\pi$
$691$	47.5301	1.80813	0.904065	+	0.427395i	$0.140569\pi$
$692$	3.80567	0.144670
$693$	0	0
$694$	42.3337	1.60697
$695$	1.50691	0.0571604
$696$	0	0
$697$	−47.9892	−1.81772
$698$	−30.5370	−1.15584
$699$	0	0
$700$	−6.28742	−0.237642
$701$	20.6766	0.780944	0.390472	−	0.920615i	$-0.372312\pi$
$701$	20.6766	0.780944	0.390472	+	0.920615i	$0.372312\pi$
$702$	0	0
$703$	2.18955	0.0825806
$704$	9.97344	0.375888
$705$	0	0
$706$	24.1115	0.907447
$707$	−11.8939	−0.447316
$708$	0	0
$709$	−21.1294	−0.793532	−0.396766	−	0.917920i	$-0.629867\pi$
$709$	−21.1294	−0.793532	−0.396766	+	0.917920i	$0.629867\pi$
$710$	2.21239	0.0830294
$711$	0	0
$712$	2.62940	0.0985411
$713$	3.60073	0.134848
$714$	0	0
$715$	14.7189	0.550455
$716$	3.95347	0.147748
$717$	0	0
$718$	−0.362814	−0.0135401
$719$	7.53028	0.280832	0.140416	−	0.990093i	$-0.455156\pi$
$719$	7.53028	0.280832	0.140416	+	0.990093i	$0.455156\pi$
$720$	0	0
$721$	−74.8992	−2.78939
$722$	−28.3769	−1.05608
$723$	0	0
$724$	−13.2031	−0.490688
$725$	2.22822	0.0827539
$726$	0	0
$727$	28.8353	1.06944	0.534721	−	0.845029i	$-0.320417\pi$
$727$	28.8353	1.06944	0.534721	+	0.845029i	$0.320417\pi$
$728$	−43.5052	−1.61241
$729$	0	0
$730$	10.1442	0.375454
$731$	14.2021	0.525284
$732$	0	0
$733$	17.5591	0.648560	0.324280	−	0.945961i	$-0.394878\pi$
$733$	17.5591	0.648560	0.324280	+	0.945961i	$0.394878\pi$
$734$	−37.8883	−1.39848
$735$	0	0
$736$	1.79223	0.0660625
$737$	34.1525	1.25802
$738$	0	0
$739$	38.3981	1.41250	0.706248	−	0.707964i	$-0.250386\pi$
$739$	38.3981	1.41250	0.706248	+	0.707964i	$0.250386\pi$
$740$	1.77674	0.0653144
$741$	0	0
$742$	42.1833	1.54860
$743$	−23.4989	−0.862091	−0.431045	−	0.902330i	$-0.641855\pi$
$743$	−23.4989	−0.862091	−0.431045	+	0.902330i	$0.641855\pi$
$744$	0	0
$745$	−1.61810	−0.0592825
$746$	26.9661	0.987299
$747$	0	0
$748$	−5.75152	−0.210296
$749$	−62.1854	−2.27220
$750$	0	0
$751$	15.8551	0.578563	0.289281	−	0.957244i	$-0.406584\pi$
$751$	15.8551	0.578563	0.289281	+	0.957244i	$0.406584\pi$
$752$	−60.8069	−2.21740
$753$	0	0
$754$	−6.24169	−0.227309
$755$	31.1315	1.13299
$756$	0	0
$757$	2.29130	0.0832788	0.0416394	−	0.999133i	$-0.486742\pi$
$757$	2.29130	0.0832788	0.0416394	+	0.999133i	$0.486742\pi$
$758$	51.7764	1.88060
$759$	0	0
$760$	−4.24941	−0.154142
$761$	24.1144	0.874145	0.437072	−	0.899426i	$-0.356015\pi$
$761$	24.1144	0.874145	0.437072	+	0.899426i	$0.356015\pi$
$762$	0	0
$763$	8.90436	0.322359
$764$	4.12095	0.149091
$765$	0	0
$766$	2.14028	0.0773314
$767$	−49.2473	−1.77822
$768$	0	0
$769$	−50.2033	−1.81038	−0.905188	−	0.425011i	$-0.860270\pi$
$769$	−50.2033	−1.81038	−0.905188	+	0.425011i	$0.860270\pi$
$770$	26.0332	0.938173
$771$	0	0
$772$	−4.59025	−0.165207
$773$	47.2029	1.69777	0.848886	−	0.528576i	$-0.177274\pi$
$773$	47.2029	1.69777	0.848886	+	0.528576i	$0.177274\pi$
$774$	0	0
$775$	15.1554	0.544399
$776$	17.7577	0.637464
$777$	0	0
$778$	3.38343	0.121302
$779$	12.0945	0.433331
$780$	0	0
$781$	1.86416	0.0667050
$782$	4.14185	0.148112
$783$	0	0
$784$	−67.3861	−2.40665
$785$	4.18702	0.149441
$786$	0	0
$787$	−17.6905	−0.630599	−0.315299	−	0.948992i	$-0.602105\pi$
$787$	−17.6905	−0.630599	−0.315299	+	0.948992i	$0.602105\pi$
$788$	2.63916	0.0940160
$789$	0	0
$790$	−31.7754	−1.13052
$791$	−27.2423	−0.968623
$792$	0	0
$793$	−34.5314	−1.22624
$794$	24.6887	0.876169
$795$	0	0
$796$	6.88572	0.244058
$797$	−8.31270	−0.294451	−0.147225	−	0.989103i	$-0.547034\pi$
$797$	−8.31270	−0.294451	−0.147225	+	0.989103i	$0.547034\pi$
$798$	0	0
$799$	−57.5217	−2.03497
$800$	7.54349	0.266703
$801$	0	0
$802$	8.53537	0.301395
$803$	8.54756	0.301637
$804$	0	0
$805$	−4.19390	−0.147815
$806$	−42.4535	−1.49536
$807$	0	0
$808$	5.93388	0.208753
$809$	34.7175	1.22060	0.610301	−	0.792169i	$-0.291048\pi$
$809$	34.7175	1.22060	0.610301	+	0.792169i	$0.291048\pi$
$810$	0	0
$811$	48.7191	1.71076	0.855379	−	0.518002i	$-0.173324\pi$
$811$	48.7191	1.71076	0.855379	+	0.518002i	$0.173324\pi$
$812$	−2.46963	−0.0866672
$813$	0	0
$814$	6.69222	0.234562
$815$	−13.9926	−0.490141
$816$	0	0
$817$	−3.57929	−0.125224
$818$	22.7191	0.794356
$819$	0	0
$820$	9.81425	0.342729
$821$	11.1342	0.388585	0.194292	−	0.980944i	$-0.437759\pi$
$821$	11.1342	0.388585	0.194292	+	0.980944i	$0.437759\pi$
$822$	0	0
$823$	−26.0154	−0.906840	−0.453420	−	0.891297i	$-0.649796\pi$
$823$	−26.0154	−0.906840	−0.453420	+	0.891297i	$0.649796\pi$
$824$	37.3673	1.30175
$825$	0	0
$826$	−87.1035	−3.03072
$827$	37.6183	1.30812	0.654058	−	0.756444i	$-0.273065\pi$
$827$	37.6183	1.30812	0.654058	+	0.756444i	$0.273065\pi$
$828$	0	0
$829$	45.3259	1.57423	0.787116	−	0.616805i	$-0.211573\pi$
$829$	45.3259	1.57423	0.787116	+	0.616805i	$0.211573\pi$
$830$	−36.2308	−1.25759
$831$	0	0
$832$	18.9434	0.656743
$833$	−63.7455	−2.20865
$834$	0	0
$835$	15.7216	0.544067
$836$	1.44953	0.0501331
$837$	0	0
$838$	3.06277	0.105802
$839$	37.5412	1.29606	0.648032	−	0.761613i	$-0.275592\pi$
$839$	37.5412	1.29606	0.648032	+	0.761613i	$0.275592\pi$
$840$	0	0
$841$	−28.1248	−0.969820
$842$	−37.0475	−1.27674
$843$	0	0
$844$	−9.52256	−0.327780
$845$	6.92154	0.238108
$846$	0	0
$847$	−28.4475	−0.977466
$848$	−27.6592	−0.949820
$849$	0	0
$850$	17.4330	0.597948
$851$	−1.07810	−0.0369568
$852$	0	0
$853$	3.62937	0.124267	0.0621336	−	0.998068i	$-0.480210\pi$
$853$	3.62937	0.124267	0.0621336	+	0.998068i	$0.480210\pi$
$854$	−61.0755	−2.08996
$855$	0	0
$856$	31.0244	1.06039
$857$	30.1700	1.03059	0.515293	−	0.857014i	$-0.327683\pi$
$857$	30.1700	1.03059	0.515293	+	0.857014i	$0.327683\pi$
$858$	0	0
$859$	25.4804	0.869380	0.434690	−	0.900580i	$-0.356858\pi$
$859$	25.4804	0.869380	0.434690	+	0.900580i	$0.356858\pi$
$860$	−2.90446	−0.0990414
$861$	0	0
$862$	34.8497	1.18698
$863$	41.9712	1.42871	0.714357	−	0.699781i	$-0.246719\pi$
$863$	41.9712	1.42871	0.714357	+	0.699781i	$0.246719\pi$
$864$	0	0
$865$	10.6845	0.363284
$866$	−2.18390	−0.0742118
$867$	0	0
$868$	−16.7975	−0.570143
$869$	−26.7740	−0.908247
$870$	0	0
$871$	64.8686	2.19799
$872$	−4.44239	−0.150438
$873$	0	0
$874$	−1.04385	−0.0353088
$875$	−54.7088	−1.84950
$876$	0	0
$877$	−5.40315	−0.182451	−0.0912257	−	0.995830i	$-0.529078\pi$
$877$	−5.40315	−0.182451	−0.0912257	+	0.995830i	$0.529078\pi$
$878$	44.5442	1.50330
$879$	0	0
$880$	−17.0697	−0.575421
$881$	−46.6700	−1.57235	−0.786176	−	0.618003i	$-0.787942\pi$
$881$	−46.6700	−1.57235	−0.786176	+	0.618003i	$0.787942\pi$
$882$	0	0
$883$	−16.0855	−0.541321	−0.270660	−	0.962675i	$-0.587242\pi$
$883$	−16.0855	−0.541321	−0.270660	+	0.962675i	$0.587242\pi$
$884$	−10.9243	−0.367425
$885$	0	0
$886$	26.3325	0.884658
$887$	−22.9835	−0.771709	−0.385855	−	0.922560i	$-0.626093\pi$
$887$	−22.9835	−0.771709	−0.385855	+	0.922560i	$0.626093\pi$
$888$	0	0
$889$	5.25875	0.176373
$890$	−2.98852	−0.100175
$891$	0	0
$892$	13.5209	0.452713
$893$	14.4969	0.485121
$894$	0	0
$895$	11.0994	0.371014
$896$	62.5183	2.08859
$897$	0	0
$898$	−30.0491	−1.00275
$899$	5.95290	0.198540
$900$	0	0
$901$	−26.1648	−0.871677
$902$	36.9660	1.23083
$903$	0	0
$904$	13.5912	0.452036
$905$	−37.0679	−1.23218
$906$	0	0
$907$	36.6183	1.21589	0.607946	−	0.793979i	$-0.291994\pi$
$907$	36.6183	1.21589	0.607946	+	0.793979i	$0.291994\pi$
$908$	−12.2269	−0.405762
$909$	0	0
$910$	49.4471	1.63915
$911$	3.73066	0.123602	0.0618011	−	0.998088i	$-0.480316\pi$
$911$	3.73066	0.123602	0.0618011	+	0.998088i	$0.480316\pi$
$912$	0	0
$913$	−30.5282	−1.01034
$914$	42.8527	1.41744
$915$	0	0
$916$	1.47618	0.0487743
$917$	25.7544	0.850484
$918$	0	0
$919$	−18.1560	−0.598911	−0.299456	−	0.954110i	$-0.596805\pi$
$919$	−18.1560	−0.598911	−0.299456	+	0.954110i	$0.596805\pi$
$920$	2.09234	0.0689824
$921$	0	0
$922$	24.5274	0.807768
$923$	3.54076	0.116545
$924$	0	0
$925$	−4.53772	−0.149199
$926$	58.1656	1.91144
$927$	0	0
$928$	2.96300	0.0972653
$929$	−52.2751	−1.71509	−0.857545	−	0.514409i	$-0.828011\pi$
$929$	−52.2751	−1.71509	−0.857545	+	0.514409i	$0.828011\pi$
$930$	0	0
$931$	16.0655	0.526525
$932$	−16.8928	−0.553343
$933$	0	0
$934$	−12.8636	−0.420910
$935$	−16.1475	−0.528080
$936$	0	0
$937$	−37.0812	−1.21139	−0.605694	−	0.795697i	$-0.707105\pi$
$937$	−37.0812	−1.21139	−0.605694	+	0.795697i	$0.707105\pi$
$938$	114.733	3.74616
$939$	0	0
$940$	11.7637	0.383690
$941$	56.6561	1.84694	0.923468	−	0.383676i	$-0.125342\pi$
$941$	56.6561	1.84694	0.923468	+	0.383676i	$0.125342\pi$
$942$	0	0
$943$	−5.95514	−0.193926
$944$	57.1129	1.85887
$945$	0	0
$946$	−10.9399	−0.355685
$947$	5.52652	0.179588	0.0897938	−	0.995960i	$-0.471379\pi$
$947$	5.52652	0.179588	0.0897938	+	0.995960i	$0.471379\pi$
$948$	0	0
$949$	16.2351	0.527012
$950$	−4.39357	−0.142546
$951$	0	0
$952$	47.7279	1.54687
$953$	−29.2755	−0.948326	−0.474163	−	0.880437i	$-0.657249\pi$
$953$	−29.2755	−0.948326	−0.474163	+	0.880437i	$0.657249\pi$
$954$	0	0
$955$	11.5697	0.374385
$956$	−0.453370	−0.0146630
$957$	0	0
$958$	19.4333	0.627862
$959$	−3.68168	−0.118888
$960$	0	0
$961$	9.48922	0.306104
$962$	12.7111	0.409822
$963$	0	0
$964$	−14.4552	−0.465571
$965$	−12.8872	−0.414855
$966$	0	0
$967$	−48.2493	−1.55159	−0.775797	−	0.630983i	$-0.782652\pi$
$967$	−48.2493	−1.55159	−0.775797	+	0.630983i	$0.782652\pi$
$968$	14.1925	0.456163
$969$	0	0
$970$	−20.1830	−0.648037
$971$	−13.7662	−0.441777	−0.220889	−	0.975299i	$-0.570896\pi$
$971$	−13.7662	−0.441777	−0.220889	+	0.975299i	$0.570896\pi$
$972$	0	0
$973$	4.26556	0.136747
$974$	−22.1378	−0.709340
$975$	0	0
$976$	40.0466	1.28186
$977$	4.49107	0.143682	0.0718410	−	0.997416i	$-0.477113\pi$
$977$	4.49107	0.143682	0.0718410	+	0.997416i	$0.477113\pi$
$978$	0	0
$979$	−2.51814	−0.0804800
$980$	13.0366	0.416437
$981$	0	0
$982$	−34.4271	−1.09861
$983$	−21.0432	−0.671175	−0.335587	−	0.942009i	$-0.608935\pi$
$983$	−21.0432	−0.671175	−0.335587	+	0.942009i	$0.608935\pi$
$984$	0	0
$985$	7.40949	0.236086
$986$	6.84751	0.218069
$987$	0	0
$988$	2.75321	0.0875913
$989$	1.76239	0.0560406
$990$	0	0
$991$	20.2185	0.642263	0.321132	−	0.947035i	$-0.395937\pi$
$991$	20.2185	0.642263	0.321132	+	0.947035i	$0.395937\pi$
$992$	20.1532	0.639863
$993$	0	0
$994$	6.26252	0.198635
$995$	19.3318	0.612859
$996$	0	0
$997$	0.392026	0.0124156	0.00620780	−	0.999981i	$-0.498024\pi$
$997$	0.392026	0.0124156	0.00620780	+	0.999981i	$0.498024\pi$
$998$	−24.6812	−0.781268
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.e.1.7		9
3.2	odd	2		149.2.a.b.1.3	✓	9
12.11	even	2		2384.2.a.j.1.3		9
15.14	odd	2		3725.2.a.c.1.7		9
21.20	even	2		7301.2.a.j.1.3		9
24.5	odd	2		9536.2.a.v.1.3		9
24.11	even	2		9536.2.a.w.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
149.2.a.b.1.3	✓	9	3.2	odd	2
1341.2.a.e.1.7		9	1.1	even	1	trivial
2384.2.a.j.1.3		9	12.11	even	2
3725.2.a.c.1.7		9	15.14	odd	2
7301.2.a.j.1.3		9	21.20	even	2
9536.2.a.v.1.3		9	24.5	odd	2
9536.2.a.w.1.7		9	24.11	even	2

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:	\( N \)	\(=\)	\( 1341 = 3^{2} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1341.a (trivial)

\(f(q)\)	\(=\)	\(q+1.60510 q^{2} +0.576343 q^{4} +1.61810 q^{5} +4.58029 q^{7} -2.28511 q^{8} +O(q^{10})\)	\(q+1.60510 q^{2} +0.576343 q^{4} +1.61810 q^{5} +4.58029 q^{7} -2.28511 q^{8} +2.59720 q^{10} +2.18841 q^{11} +4.15663 q^{13} +7.35182 q^{14} -4.82051 q^{16} -4.56008 q^{17} +1.14926 q^{19} +0.932579 q^{20} +3.51262 q^{22} -0.565875 q^{23} -2.38176 q^{25} +6.67180 q^{26} +2.63982 q^{28} -0.935532 q^{29} -6.36311 q^{31} -3.16718 q^{32} -7.31938 q^{34} +7.41135 q^{35} +1.90519 q^{37} +1.84467 q^{38} -3.69753 q^{40} +10.5238 q^{41} -3.11444 q^{43} +1.26128 q^{44} -0.908286 q^{46} +12.6142 q^{47} +13.9790 q^{49} -3.82297 q^{50} +2.39565 q^{52} +5.73780 q^{53} +3.54106 q^{55} -10.4665 q^{56} -1.50162 q^{58} -11.8479 q^{59} -8.30754 q^{61} -10.2134 q^{62} +4.55739 q^{64} +6.72583 q^{65} +15.6061 q^{67} -2.62817 q^{68} +11.8959 q^{70} +0.851833 q^{71} +3.90582 q^{73} +3.05802 q^{74} +0.662366 q^{76} +10.0236 q^{77} -12.2344 q^{79} -7.80006 q^{80} +16.8917 q^{82} -13.9499 q^{83} -7.37864 q^{85} -4.99899 q^{86} -5.00077 q^{88} -1.15067 q^{89} +19.0386 q^{91} -0.326138 q^{92} +20.2470 q^{94} +1.85961 q^{95} -7.77105 q^{97} +22.4377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8	\( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) \) 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 6 * q^10 - 5 * q^11 + 7 * q^13 + 8 * q^14 + 13 * q^16 + 5 * q^17 + 30 * q^19 + 10 * q^20 - 7 * q^22 + 4 * q^23 + 6 * q^25 + 15 * q^26 - 12 * q^28 + 16 * q^29 + 22 * q^31 + 38 * q^32 + 9 * q^34 + 11 * q^35 - 7 * q^37 + 18 * q^38 - 7 * q^40 - 6 * q^41 + 4 * q^43 - 6 * q^44 + q^46 + 6 * q^47 + 14 * q^49 + 16 * q^50 + 50 * q^52 + 2 * q^53 - 2 * q^55 - 7 * q^56 + 2 * q^58 - 43 * q^59 + q^61 - 33 * q^62 + 18 * q^64 + 20 * q^65 + 33 * q^67 + 16 * q^68 - 3 * q^70 - 15 * q^71 - 11 * q^73 - 33 * q^74 + 59 * q^76 + 30 * q^77 + q^79 - 65 * q^80 + 5 * q^82 + 4 * q^83 - 34 * q^85 + 7 * q^86 - 37 * q^88 + 19 * q^89 + 62 * q^91 - 17 * q^92 + 17 * q^94 + 21 * q^95 - q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more