LMFDB - Embedded newform 1341.2.a.d.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:	$1$
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} - 4x^{8} - 6x^{7} + 37x^{6} - 3x^{5} - 101x^{4} + 49x^{3} + 72x^{2} - 21x - 13 $ x^9 - 4x^8 - 6x^7 + 37x^6 - 3x^5 - 101x^4 + 49x^3 + 72x^2 - 21x - 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 447)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.720171$ 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)

$f(q)$	$=$	$q+0.720171 q^{2} -1.48135 q^{4} +3.36155 q^{5} +0.451528 q^{7} -2.50717 q^{8} +O(q^{10})$	\(q+0.720171 q^{2} -1.48135 q^{4} +3.36155 q^{5} +0.451528 q^{7} -2.50717 q^{8} +2.42089 q^{10} -5.21771 q^{11} -5.16313 q^{13} +0.325177 q^{14} +1.15712 q^{16} -7.12598 q^{17} -0.740583 q^{19} -4.97965 q^{20} -3.75764 q^{22} -7.26909 q^{23} +6.30002 q^{25} -3.71833 q^{26} -0.668873 q^{28} -0.897164 q^{29} +6.36414 q^{31} +5.84766 q^{32} -5.13192 q^{34} +1.51783 q^{35} +0.200318 q^{37} -0.533346 q^{38} -8.42797 q^{40} +5.33009 q^{41} +6.16561 q^{43} +7.72928 q^{44} -5.23499 q^{46} -8.69209 q^{47} -6.79612 q^{49} +4.53709 q^{50} +7.64842 q^{52} +12.3472 q^{53} -17.5396 q^{55} -1.13206 q^{56} -0.646111 q^{58} -6.95828 q^{59} -2.55490 q^{61} +4.58327 q^{62} +1.89708 q^{64} -17.3561 q^{65} +2.74020 q^{67} +10.5561 q^{68} +1.09310 q^{70} -8.88925 q^{71} +3.10321 q^{73} +0.144263 q^{74} +1.09707 q^{76} -2.35594 q^{77} -0.613299 q^{79} +3.88971 q^{80} +3.83857 q^{82} -0.114313 q^{83} -23.9543 q^{85} +4.44029 q^{86} +13.0817 q^{88} -7.68351 q^{89} -2.33130 q^{91} +10.7681 q^{92} -6.25979 q^{94} -2.48951 q^{95} -6.73173 q^{97} -4.89437 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 4 q^{2} + 10 q^{4} - 8 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 4 * q^2 + 10 * q^4 - 8 * q^5 + q^7 - 9 * q^8	$ 9 q - 4 q^{2} + 10 q^{4} - 8 q^{5} + q^{7} - 9 q^{8} - 4 q^{10} - 13 q^{11} - 2 q^{13} + 5 q^{14} - 10 q^{17} - 11 q^{19} - 6 q^{20} - 16 q^{22} - 11 q^{23} + 15 q^{25} - 6 q^{26} - 14 q^{28} - 13 q^{31} - 7 q^{32} - 28 q^{34} - 6 q^{35} + q^{37} + 5 q^{38} - 32 q^{40} - 17 q^{41} - 8 q^{43} - 3 q^{44} + 5 q^{46} - 10 q^{47} - 4 q^{49} + 8 q^{50} - 4 q^{52} - 2 q^{53} - 12 q^{55} + 2 q^{56} + 10 q^{58} - 25 q^{59} - 13 q^{61} + 18 q^{62} - 23 q^{64} - 16 q^{65} + q^{67} + 18 q^{68} - 32 q^{70} - 13 q^{71} + 5 q^{73} + 17 q^{74} - 43 q^{76} - 20 q^{77} - 22 q^{79} + 70 q^{80} + 22 q^{82} - q^{83} - 16 q^{85} - 10 q^{86} + 3 q^{88} - 23 q^{89} - 62 q^{91} - 8 q^{92} - 28 q^{94} + 18 q^{95} + 16 q^{97} + 61 q^{98}+O(q^{100}) $ 9 * q - 4 * q^2 + 10 * q^4 - 8 * q^5 + q^7 - 9 * q^8 - 4 * q^10 - 13 * q^11 - 2 * q^13 + 5 * q^14 - 10 * q^17 - 11 * q^19 - 6 * q^20 - 16 * q^22 - 11 * q^23 + 15 * q^25 - 6 * q^26 - 14 * q^28 - 13 * q^31 - 7 * q^32 - 28 * q^34 - 6 * q^35 + q^37 + 5 * q^38 - 32 * q^40 - 17 * q^41 - 8 * q^43 - 3 * q^44 + 5 * q^46 - 10 * q^47 - 4 * q^49 + 8 * q^50 - 4 * q^52 - 2 * q^53 - 12 * q^55 + 2 * q^56 + 10 * q^58 - 25 * q^59 - 13 * q^61 + 18 * q^62 - 23 * q^64 - 16 * q^65 + q^67 + 18 * q^68 - 32 * q^70 - 13 * q^71 + 5 * q^73 + 17 * q^74 - 43 * q^76 - 20 * q^77 - 22 * q^79 + 70 * q^80 + 22 * q^82 - q^83 - 16 * q^85 - 10 * q^86 + 3 * q^88 - 23 * q^89 - 62 * q^91 - 8 * q^92 - 28 * q^94 + 18 * q^95 + 16 * q^97 + 61 * 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 $

$2$	0.720171	0.509238	0.254619	−	0.967042i	$-0.418050\pi$
$2$	0.720171	0.509238	0.254619	+	0.967042i	$0.418050\pi$
$3$	0	0
$3$	0	0
$4$	−1.48135	−0.740677
$5$	3.36155	1.50333	0.751665	−	0.659545i	$-0.229251\pi$
$5$	3.36155	1.50333	0.751665	+	0.659545i	$0.229251\pi$
$6$	0	0
$7$	0.451528	0.170662	0.0853308	−	0.996353i	$-0.472805\pi$
$7$	0.451528	0.170662	0.0853308	+	0.996353i	$0.472805\pi$
$8$	−2.50717	−0.886418
$9$	0	0
$10$	2.42089	0.765552
$11$	−5.21771	−1.57320	−0.786599	−	0.617464i	$-0.788160\pi$
$11$	−5.21771	−1.57320	−0.786599	+	0.617464i	$0.788160\pi$
$12$	0	0
$13$	−5.16313	−1.43199	−0.715997	−	0.698103i	$-0.754028\pi$
$13$	−5.16313	−1.43199	−0.715997	+	0.698103i	$0.754028\pi$
$14$	0.325177	0.0869072
$15$	0	0
$16$	1.15712	0.289280
$17$	−7.12598	−1.72830	−0.864152	−	0.503230i	$-0.832145\pi$
$17$	−7.12598	−1.72830	−0.864152	+	0.503230i	$0.832145\pi$
$18$	0	0
$19$	−0.740583	−0.169901	−0.0849507	−	0.996385i	$-0.527073\pi$
$19$	−0.740583	−0.169901	−0.0849507	+	0.996385i	$0.527073\pi$
$20$	−4.97965	−1.11348
$21$	0	0
$22$	−3.75764	−0.801132
$23$	−7.26909	−1.51571	−0.757855	−	0.652423i	$-0.773753\pi$
$23$	−7.26909	−1.51571	−0.757855	+	0.652423i	$0.773753\pi$
$24$	0	0
$25$	6.30002	1.26000
$26$	−3.71833	−0.729225
$27$	0	0
$28$	−0.668873	−0.126405
$29$	−0.897164	−0.166599	−0.0832996	−	0.996525i	$-0.526546\pi$
$29$	−0.897164	−0.166599	−0.0832996	+	0.996525i	$0.526546\pi$
$30$	0	0
$31$	6.36414	1.14303	0.571517	−	0.820590i	$-0.306355\pi$
$31$	6.36414	1.14303	0.571517	+	0.820590i	$0.306355\pi$
$32$	5.84766	1.03373
$33$	0	0
$34$	−5.13192	−0.880118
$35$	1.51783	0.256561
$36$	0	0
$37$	0.200318	0.0329321	0.0164661	−	0.999864i	$-0.494758\pi$
$37$	0.200318	0.0329321	0.0164661	+	0.999864i	$0.494758\pi$
$38$	−0.533346	−0.0865202
$39$	0	0
$40$	−8.42797	−1.33258
$41$	5.33009	0.832420	0.416210	−	0.909269i	$-0.363358\pi$
$41$	5.33009	0.832420	0.416210	+	0.909269i	$0.363358\pi$
$42$	0	0
$43$	6.16561	0.940246	0.470123	−	0.882601i	$-0.344210\pi$
$43$	6.16561	0.940246	0.470123	+	0.882601i	$0.344210\pi$
$44$	7.72928	1.16523
$45$	0	0
$46$	−5.23499	−0.771857
$47$	−8.69209	−1.26787	−0.633936	−	0.773386i	$-0.718562\pi$
$47$	−8.69209	−1.26787	−0.633936	+	0.773386i	$0.718562\pi$
$48$	0	0
$49$	−6.79612	−0.970875
$50$	4.53709	0.641641
$51$	0	0
$52$	7.64842	1.06065
$53$	12.3472	1.69602	0.848010	−	0.529981i	$-0.177801\pi$
$53$	12.3472	1.69602	0.848010	+	0.529981i	$0.177801\pi$
$54$	0	0
$55$	−17.5396	−2.36504
$56$	−1.13206	−0.151277
$57$	0	0
$58$	−0.646111	−0.0848386
$59$	−6.95828	−0.905890	−0.452945	−	0.891538i	$-0.649627\pi$
$59$	−6.95828	−0.905890	−0.452945	+	0.891538i	$0.649627\pi$
$60$	0	0
$61$	−2.55490	−0.327122	−0.163561	−	0.986533i	$-0.552298\pi$
$61$	−2.55490	−0.327122	−0.163561	+	0.986533i	$0.552298\pi$
$62$	4.58327	0.582076
$63$	0	0
$64$	1.89708	0.237134
$65$	−17.3561	−2.15276
$66$	0	0
$67$	2.74020	0.334769	0.167384	−	0.985892i	$-0.446468\pi$
$67$	2.74020	0.334769	0.167384	+	0.985892i	$0.446468\pi$
$68$	10.5561	1.28012
$69$	0	0
$70$	1.09310	0.130650
$71$	−8.88925	−1.05496	−0.527480	−	0.849567i	$-0.676863\pi$
$71$	−8.88925	−1.05496	−0.527480	+	0.849567i	$0.676863\pi$
$72$	0	0
$73$	3.10321	0.363203	0.181602	−	0.983372i	$-0.441872\pi$
$73$	3.10321	0.363203	0.181602	+	0.983372i	$0.441872\pi$
$74$	0.144263	0.0167703
$75$	0	0
$76$	1.09707	0.125842
$77$	−2.35594	−0.268484
$78$	0	0
$79$	−0.613299	−0.0690016	−0.0345008	−	0.999405i	$-0.510984\pi$
$79$	−0.613299	−0.0690016	−0.0345008	+	0.999405i	$0.510984\pi$
$80$	3.88971	0.434883
$81$	0	0
$82$	3.83857	0.423899
$83$	−0.114313	−0.0125475	−0.00627376	−	0.999980i	$-0.501997\pi$
$83$	−0.114313	−0.0125475	−0.00627376	+	0.999980i	$0.501997\pi$
$84$	0	0
$85$	−23.9543	−2.59821
$86$	4.44029	0.478809
$87$	0	0
$88$	13.0817	1.39451
$89$	−7.68351	−0.814451	−0.407225	−	0.913328i	$-0.633504\pi$
$89$	−7.68351	−0.814451	−0.407225	+	0.913328i	$0.633504\pi$
$90$	0	0
$91$	−2.33130	−0.244386
$92$	10.7681	1.12265
$93$	0	0
$94$	−6.25979	−0.645648
$95$	−2.48951	−0.255418
$96$	0	0
$97$	−6.73173	−0.683504	−0.341752	−	0.939790i	$-0.611020\pi$
$97$	−6.73173	−0.683504	−0.341752	+	0.939790i	$0.611020\pi$
$98$	−4.89437	−0.494406
$99$	0	0
$100$	−9.33256	−0.933256
$101$	13.9525	1.38833	0.694164	−	0.719817i	$-0.255774\pi$
$101$	13.9525	1.38833	0.694164	+	0.719817i	$0.255774\pi$
$102$	0	0
$103$	−1.55565	−0.153283	−0.0766413	−	0.997059i	$-0.524420\pi$
$103$	−1.55565	−0.153283	−0.0766413	+	0.997059i	$0.524420\pi$
$104$	12.9448	1.26935
$105$	0	0
$106$	8.89210	0.863677
$107$	5.70887	0.551897	0.275949	−	0.961172i	$-0.411008\pi$
$107$	5.70887	0.551897	0.275949	+	0.961172i	$0.411008\pi$
$108$	0	0
$109$	−0.602118	−0.0576725	−0.0288362	−	0.999584i	$-0.509180\pi$
$109$	−0.602118	−0.0576725	−0.0288362	+	0.999584i	$0.509180\pi$
$110$	−12.6315	−1.20437
$111$	0	0
$112$	0.522472	0.0493689
$113$	8.04301	0.756622	0.378311	−	0.925678i	$-0.376505\pi$
$113$	8.04301	0.756622	0.378311	+	0.925678i	$0.376505\pi$
$114$	0	0
$115$	−24.4354	−2.27861
$116$	1.32902	0.123396
$117$	0	0
$118$	−5.01115	−0.461313
$119$	−3.21758	−0.294955
$120$	0	0
$121$	16.2245	1.47495
$122$	−1.83997	−0.166583
$123$	0	0
$124$	−9.42755	−0.846619
$125$	4.37007	0.390871
$126$	0	0
$127$	16.0237	1.42187	0.710937	−	0.703256i	$-0.248271\pi$
$127$	16.0237	1.42187	0.710937	+	0.703256i	$0.248271\pi$
$128$	−10.3291	−0.912972
$129$	0	0
$130$	−12.4994	−1.09627
$131$	−20.5714	−1.79733	−0.898664	−	0.438637i	$-0.855461\pi$
$131$	−20.5714	−1.79733	−0.898664	+	0.438637i	$0.855461\pi$
$132$	0	0
$133$	−0.334394	−0.0289956
$134$	1.97341	0.170477
$135$	0	0
$136$	17.8660	1.53200
$137$	−14.8831	−1.27155	−0.635773	−	0.771876i	$-0.719318\pi$
$137$	−14.8831	−1.27155	−0.635773	+	0.771876i	$0.719318\pi$
$138$	0	0
$139$	−19.8984	−1.68776	−0.843881	−	0.536531i	$-0.819734\pi$
$139$	−19.8984	−1.68776	−0.843881	+	0.536531i	$0.819734\pi$
$140$	−2.24845	−0.190029
$141$	0	0
$142$	−6.40178	−0.537225
$143$	26.9397	2.25281
$144$	0	0
$145$	−3.01586	−0.250454
$146$	2.23484	0.184957
$147$	0	0
$148$	−0.296743	−0.0243921
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−11.4528	−0.932019	−0.466009	−	0.884780i	$-0.654309\pi$
$151$	−11.4528	−0.932019	−0.466009	+	0.884780i	$0.654309\pi$
$152$	1.85677	0.150604
$153$	0	0
$154$	−1.69668	−0.136722
$155$	21.3934	1.71836
$156$	0	0
$157$	9.01945	0.719830	0.359915	−	0.932985i	$-0.382806\pi$
$157$	9.01945	0.719830	0.359915	+	0.932985i	$0.382806\pi$
$158$	−0.441680	−0.0351382
$159$	0	0
$160$	19.6572	1.55404
$161$	−3.28220	−0.258673
$162$	0	0
$163$	−2.75071	−0.215452	−0.107726	−	0.994181i	$-0.534357\pi$
$163$	−2.75071	−0.215452	−0.107726	+	0.994181i	$0.534357\pi$
$164$	−7.89575	−0.616554
$165$	0	0
$166$	−0.0823251	−0.00638967
$167$	11.4623	0.886976	0.443488	−	0.896280i	$-0.353741\pi$
$167$	11.4623	0.886976	0.443488	+	0.896280i	$0.353741\pi$
$168$	0	0
$169$	13.6579	1.05061
$170$	−17.2512	−1.32311
$171$	0	0
$172$	−9.13345	−0.696419
$173$	23.8734	1.81506	0.907530	−	0.419987i	$-0.137965\pi$
$173$	23.8734	1.81506	0.907530	+	0.419987i	$0.137965\pi$
$174$	0	0
$175$	2.84463	0.215034
$176$	−6.03751	−0.455095
$177$	0	0
$178$	−5.53344	−0.414749
$179$	1.33357	0.0996754	0.0498377	−	0.998757i	$-0.484130\pi$
$179$	1.33357	0.0996754	0.0498377	+	0.998757i	$0.484130\pi$
$180$	0	0
$181$	11.3268	0.841916	0.420958	−	0.907080i	$-0.361694\pi$
$181$	11.3268	0.841916	0.420958	+	0.907080i	$0.361694\pi$
$182$	−1.67893	−0.124451
$183$	0	0
$184$	18.2248	1.34355
$185$	0.673380	0.0495079
$186$	0	0
$187$	37.1813	2.71897
$188$	12.8761	0.939083
$189$	0	0
$190$	−1.79287	−0.130068
$191$	−16.8660	−1.22038	−0.610192	−	0.792254i	$-0.708908\pi$
$191$	−16.8660	−1.22038	−0.610192	+	0.792254i	$0.708908\pi$
$192$	0	0
$193$	12.7723	0.919372	0.459686	−	0.888081i	$-0.347962\pi$
$193$	12.7723	0.919372	0.459686	+	0.888081i	$0.347962\pi$
$194$	−4.84800	−0.348066
$195$	0	0
$196$	10.0675	0.719105
$197$	−19.9874	−1.42404	−0.712022	−	0.702157i	$-0.752221\pi$
$197$	−19.9874	−1.42404	−0.712022	+	0.702157i	$0.752221\pi$
$198$	0	0
$199$	−23.2196	−1.64599	−0.822997	−	0.568046i	$-0.807699\pi$
$199$	−23.2196	−1.64599	−0.822997	+	0.568046i	$0.807699\pi$
$200$	−15.7952	−1.11689
$201$	0	0
$202$	10.0482	0.706989
$203$	−0.405095	−0.0284321
$204$	0	0
$205$	17.9174	1.25140
$206$	−1.12033	−0.0780573
$207$	0	0
$208$	−5.97435	−0.414247
$209$	3.86415	0.267289
$210$	0	0
$211$	19.5749	1.34759	0.673795	−	0.738919i	$-0.264663\pi$
$211$	19.5749	1.34759	0.673795	+	0.738919i	$0.264663\pi$
$212$	−18.2906	−1.25620
$213$	0	0
$214$	4.11136	0.281047
$215$	20.7260	1.41350
$216$	0	0
$217$	2.87359	0.195072
$218$	−0.433628	−0.0293690
$219$	0	0
$220$	25.9824	1.75173
$221$	36.7924	2.47492
$222$	0	0
$223$	−16.2663	−1.08927	−0.544636	−	0.838673i	$-0.683332\pi$
$223$	−16.2663	−1.08927	−0.544636	+	0.838673i	$0.683332\pi$
$224$	2.64038	0.176418
$225$	0	0
$226$	5.79234	0.385300
$227$	−12.1842	−0.808692	−0.404346	−	0.914606i	$-0.632501\pi$
$227$	−12.1842	−0.808692	−0.404346	+	0.914606i	$0.632501\pi$
$228$	0	0
$229$	−11.6463	−0.769610	−0.384805	−	0.922998i	$-0.625731\pi$
$229$	−11.6463	−0.769610	−0.384805	+	0.922998i	$0.625731\pi$
$230$	−17.5977	−1.16036
$231$	0	0
$232$	2.24934	0.147677
$233$	−7.59598	−0.497629	−0.248815	−	0.968551i	$-0.580041\pi$
$233$	−7.59598	−0.497629	−0.248815	+	0.968551i	$0.580041\pi$
$234$	0	0
$235$	−29.2189	−1.90603
$236$	10.3077	0.670972
$237$	0	0
$238$	−2.31721	−0.150202
$239$	7.68084	0.496832	0.248416	−	0.968653i	$-0.420090\pi$
$239$	7.68084	0.496832	0.248416	+	0.968653i	$0.420090\pi$
$240$	0	0
$241$	6.20603	0.399766	0.199883	−	0.979820i	$-0.435944\pi$
$241$	6.20603	0.399766	0.199883	+	0.979820i	$0.435944\pi$
$242$	11.6844	0.751102
$243$	0	0
$244$	3.78472	0.242292
$245$	−22.8455	−1.45955
$246$	0	0
$247$	3.82372	0.243298
$248$	−15.9560	−1.01321
$249$	0	0
$250$	3.14720	0.199046
$251$	−4.16240	−0.262729	−0.131364	−	0.991334i	$-0.541936\pi$
$251$	−4.16240	−0.262729	−0.131364	+	0.991334i	$0.541936\pi$
$252$	0	0
$253$	37.9280	2.38451
$254$	11.5398	0.724071
$255$	0	0
$256$	−11.2329	−0.702054
$257$	29.0645	1.81300	0.906498	−	0.422210i	$-0.138745\pi$
$257$	29.0645	1.81300	0.906498	+	0.422210i	$0.138745\pi$
$258$	0	0
$259$	0.0904494	0.00562025
$260$	25.7105	1.59450
$261$	0	0
$262$	−14.8149	−0.915267
$263$	−19.4771	−1.20101	−0.600504	−	0.799622i	$-0.705033\pi$
$263$	−19.4771	−1.20101	−0.600504	+	0.799622i	$0.705033\pi$
$264$	0	0
$265$	41.5058	2.54968
$266$	−0.240821	−0.0147657
$267$	0	0
$268$	−4.05921	−0.247956
$269$	21.4119	1.30550	0.652752	−	0.757571i	$-0.273614\pi$
$269$	21.4119	1.30550	0.652752	+	0.757571i	$0.273614\pi$
$270$	0	0
$271$	−5.02455	−0.305220	−0.152610	−	0.988287i	$-0.548768\pi$
$271$	−5.02455	−0.305220	−0.152610	+	0.988287i	$0.548768\pi$
$272$	−8.24561	−0.499964
$273$	0	0
$274$	−10.7183	−0.647518
$275$	−32.8717	−1.98224
$276$	0	0
$277$	−31.2845	−1.87970	−0.939850	−	0.341586i	$-0.889036\pi$
$277$	−31.2845	−1.87970	−0.939850	+	0.341586i	$0.889036\pi$
$278$	−14.3303	−0.859471
$279$	0	0
$280$	−3.80547	−0.227420
$281$	24.0698	1.43589	0.717943	−	0.696102i	$-0.245084\pi$
$281$	24.0698	1.43589	0.717943	+	0.696102i	$0.245084\pi$
$282$	0	0
$283$	10.2762	0.610856	0.305428	−	0.952215i	$-0.401200\pi$
$283$	10.2762	0.610856	0.305428	+	0.952215i	$0.401200\pi$
$284$	13.1681	0.781385
$285$	0	0
$286$	19.4012	1.14722
$287$	2.40668	0.142062
$288$	0	0
$289$	33.7796	1.98704
$290$	−2.17193	−0.127540
$291$	0	0
$292$	−4.59695	−0.269016
$293$	−22.0220	−1.28654	−0.643270	−	0.765640i	$-0.722423\pi$
$293$	−22.0220	−1.28654	−0.643270	+	0.765640i	$0.722423\pi$
$294$	0	0
$295$	−23.3906	−1.36185
$296$	−0.502232	−0.0291917
$297$	0	0
$298$	0.720171	0.0417184
$299$	37.5313	2.17049
$300$	0	0
$301$	2.78394	0.160464
$302$	−8.24800	−0.474619
$303$	0	0
$304$	−0.856943	−0.0491490
$305$	−8.58843	−0.491772
$306$	0	0
$307$	19.8201	1.13119	0.565597	−	0.824682i	$-0.308646\pi$
$307$	19.8201	1.13119	0.565597	+	0.824682i	$0.308646\pi$
$308$	3.48998	0.198860
$309$	0	0
$310$	15.4069	0.875052
$311$	9.75657	0.553244	0.276622	−	0.960979i	$-0.410785\pi$
$311$	9.75657	0.553244	0.276622	+	0.960979i	$0.410785\pi$
$312$	0	0
$313$	6.21602	0.351350	0.175675	−	0.984448i	$-0.443789\pi$
$313$	6.21602	0.351350	0.175675	+	0.984448i	$0.443789\pi$
$314$	6.49554	0.366564
$315$	0	0
$316$	0.908513	0.0511079
$317$	−23.5747	−1.32409	−0.662043	−	0.749466i	$-0.730310\pi$
$317$	−23.5747	−1.32409	−0.662043	+	0.749466i	$0.730310\pi$
$318$	0	0
$319$	4.68114	0.262094
$320$	6.37711	0.356492
$321$	0	0
$322$	−2.36374	−0.131726
$323$	5.27738	0.293641
$324$	0	0
$325$	−32.5278	−1.80432
$326$	−1.98098	−0.109716
$327$	0	0
$328$	−13.3634	−0.737872
$329$	−3.92472	−0.216377
$330$	0	0
$331$	−32.6167	−1.79278	−0.896389	−	0.443268i	$-0.853819\pi$
$331$	−32.6167	−1.79278	−0.896389	+	0.443268i	$0.853819\pi$
$332$	0.169339	0.00929366
$333$	0	0
$334$	8.25478	0.451681
$335$	9.21132	0.503268
$336$	0	0
$337$	14.2181	0.774511	0.387256	−	0.921972i	$-0.373423\pi$
$337$	14.2181	0.774511	0.387256	+	0.921972i	$0.373423\pi$
$338$	9.83601	0.535008
$339$	0	0
$340$	35.4849	1.92444
$341$	−33.2062	−1.79822
$342$	0	0
$343$	−6.22933	−0.336352
$344$	−15.4582	−0.833451
$345$	0	0
$346$	17.1929	0.924297
$347$	13.4582	0.722476	0.361238	−	0.932474i	$-0.382354\pi$
$347$	13.4582	0.722476	0.361238	+	0.932474i	$0.382354\pi$
$348$	0	0
$349$	−9.78661	−0.523865	−0.261933	−	0.965086i	$-0.584360\pi$
$349$	−9.78661	−0.523865	−0.261933	+	0.965086i	$0.584360\pi$
$350$	2.04862	0.109503
$351$	0	0
$352$	−30.5114	−1.62626
$353$	11.2306	0.597747	0.298873	−	0.954293i	$-0.403389\pi$
$353$	11.2306	0.597747	0.298873	+	0.954293i	$0.403389\pi$
$354$	0	0
$355$	−29.8817	−1.58595
$356$	11.3820	0.603245
$357$	0	0
$358$	0.960395	0.0507585
$359$	−12.6694	−0.668663	−0.334331	−	0.942456i	$-0.608510\pi$
$359$	−12.6694	−0.668663	−0.334331	+	0.942456i	$0.608510\pi$
$360$	0	0
$361$	−18.4515	−0.971134
$362$	8.15724	0.428735
$363$	0	0
$364$	3.45348	0.181011
$365$	10.4316	0.546015
$366$	0	0
$367$	−23.5529	−1.22945	−0.614726	−	0.788741i	$-0.710733\pi$
$367$	−23.5529	−1.22945	−0.614726	+	0.788741i	$0.710733\pi$
$368$	−8.41121	−0.438464
$369$	0	0
$370$	0.484949	0.0252113
$371$	5.57511	0.289445
$372$	0	0
$373$	−15.7862	−0.817379	−0.408690	−	0.912673i	$-0.634014\pi$
$373$	−15.7862	−0.817379	−0.408690	+	0.912673i	$0.634014\pi$
$374$	26.7769	1.38460
$375$	0	0
$376$	21.7925	1.12386
$377$	4.63217	0.238569
$378$	0	0
$379$	−7.59277	−0.390014	−0.195007	−	0.980802i	$-0.562473\pi$
$379$	−7.59277	−0.390014	−0.195007	+	0.980802i	$0.562473\pi$
$380$	3.68784	0.189182
$381$	0	0
$382$	−12.1464	−0.621465
$383$	−16.6497	−0.850762	−0.425381	−	0.905014i	$-0.639860\pi$
$383$	−16.6497	−0.850762	−0.425381	+	0.905014i	$0.639860\pi$
$384$	0	0
$385$	−7.91962	−0.403621
$386$	9.19826	0.468179
$387$	0	0
$388$	9.97208	0.506256
$389$	−30.0408	−1.52313	−0.761564	−	0.648090i	$-0.775568\pi$
$389$	−30.0408	−1.52313	−0.761564	+	0.648090i	$0.775568\pi$
$390$	0	0
$391$	51.7994	2.61961
$392$	17.0390	0.860601
$393$	0	0
$394$	−14.3944	−0.725177
$395$	−2.06164	−0.103732
$396$	0	0
$397$	15.5365	0.779753	0.389877	−	0.920867i	$-0.372518\pi$
$397$	15.5365	0.779753	0.389877	+	0.920867i	$0.372518\pi$
$398$	−16.7221	−0.838202
$399$	0	0
$400$	7.28987	0.364494
$401$	29.5840	1.47736	0.738678	−	0.674058i	$-0.235450\pi$
$401$	29.5840	1.47736	0.738678	+	0.674058i	$0.235450\pi$
$402$	0	0
$403$	−32.8589	−1.63682
$404$	−20.6686	−1.02830
$405$	0	0
$406$	−0.291737	−0.0144787
$407$	−1.04520	−0.0518088
$408$	0	0
$409$	−14.5908	−0.721470	−0.360735	−	0.932668i	$-0.617474\pi$
$409$	−14.5908	−0.721470	−0.360735	+	0.932668i	$0.617474\pi$
$410$	12.9036	0.637261
$411$	0	0
$412$	2.30447	0.113533
$413$	−3.14186	−0.154601
$414$	0	0
$415$	−0.384270	−0.0188631
$416$	−30.1922	−1.48030
$417$	0	0
$418$	2.78285	0.136113
$419$	−9.01986	−0.440649	−0.220325	−	0.975427i	$-0.570712\pi$
$419$	−9.01986	−0.440649	−0.220325	+	0.975427i	$0.570712\pi$
$420$	0	0
$421$	−6.78157	−0.330514	−0.165257	−	0.986251i	$-0.552845\pi$
$421$	−6.78157	−0.330514	−0.165257	+	0.986251i	$0.552845\pi$
$422$	14.0972	0.686243
$423$	0	0
$424$	−30.9565	−1.50338
$425$	−44.8938	−2.17767
$426$	0	0
$427$	−1.15361	−0.0558271
$428$	−8.45686	−0.408778
$429$	0	0
$430$	14.9263	0.719808
$431$	0.0835960	0.00402668	0.00201334	−	0.999998i	$-0.499359\pi$
$431$	0.0835960	0.00402668	0.00201334	+	0.999998i	$0.499359\pi$
$432$	0	0
$433$	−10.5942	−0.509127	−0.254564	−	0.967056i	$-0.581932\pi$
$433$	−10.5942	−0.509127	−0.254564	+	0.967056i	$0.581932\pi$
$434$	2.06947	0.0993379
$435$	0	0
$436$	0.891950	0.0427167
$437$	5.38337	0.257521
$438$	0	0
$439$	−25.1030	−1.19810	−0.599051	−	0.800711i	$-0.704455\pi$
$439$	−25.1030	−1.19810	−0.599051	+	0.800711i	$0.704455\pi$
$440$	43.9747	2.09641
$441$	0	0
$442$	26.4968	1.26032
$443$	−37.0991	−1.76263	−0.881315	−	0.472529i	$-0.843341\pi$
$443$	−37.0991	−1.76263	−0.881315	+	0.472529i	$0.843341\pi$
$444$	0	0
$445$	−25.8285	−1.22439
$446$	−11.7145	−0.554698
$447$	0	0
$448$	0.856583	0.0404697
$449$	20.7464	0.979083	0.489541	−	0.871980i	$-0.337164\pi$
$449$	20.7464	0.979083	0.489541	+	0.871980i	$0.337164\pi$
$450$	0	0
$451$	−27.8109	−1.30956
$452$	−11.9145	−0.560413
$453$	0	0
$454$	−8.77468	−0.411816
$455$	−7.83677	−0.367393
$456$	0	0
$457$	30.8934	1.44513	0.722566	−	0.691302i	$-0.242962\pi$
$457$	30.8934	1.44513	0.722566	+	0.691302i	$0.242962\pi$
$458$	−8.38733	−0.391914
$459$	0	0
$460$	36.1975	1.68772
$461$	11.5978	0.540165	0.270082	−	0.962837i	$-0.412949\pi$
$461$	11.5978	0.540165	0.270082	+	0.962837i	$0.412949\pi$
$462$	0	0
$463$	29.4892	1.37048	0.685239	−	0.728319i	$-0.259698\pi$
$463$	29.4892	1.37048	0.685239	+	0.728319i	$0.259698\pi$
$464$	−1.03813	−0.0481938
$465$	0	0
$466$	−5.47040	−0.253411
$467$	30.8781	1.42887	0.714433	−	0.699704i	$-0.246685\pi$
$467$	30.8781	1.42887	0.714433	+	0.699704i	$0.246685\pi$
$468$	0	0
$469$	1.23728	0.0571322
$470$	−21.0426	−0.970622
$471$	0	0
$472$	17.4456	0.802998
$473$	−32.1703	−1.47919
$474$	0	0
$475$	−4.66569	−0.214076
$476$	4.76638	0.218466
$477$	0	0
$478$	5.53152	0.253006
$479$	2.60033	0.118812	0.0594060	−	0.998234i	$-0.481079\pi$
$479$	2.60033	0.118812	0.0594060	+	0.998234i	$0.481079\pi$
$480$	0	0
$481$	−1.03427	−0.0471586
$482$	4.46940	0.203576
$483$	0	0
$484$	−24.0342	−1.09247
$485$	−22.6291	−1.02753
$486$	0	0
$487$	−1.06798	−0.0483949	−0.0241975	−	0.999707i	$-0.507703\pi$
$487$	−1.06798	−0.0483949	−0.0241975	+	0.999707i	$0.507703\pi$
$488$	6.40557	0.289967
$489$	0	0
$490$	−16.4527	−0.743255
$491$	39.9095	1.80109	0.900545	−	0.434763i	$-0.143168\pi$
$491$	39.9095	1.80109	0.900545	+	0.434763i	$0.143168\pi$
$492$	0	0
$493$	6.39318	0.287934
$494$	2.75373	0.123896
$495$	0	0
$496$	7.36407	0.330656
$497$	−4.01375	−0.180041
$498$	0	0
$499$	−33.3923	−1.49484	−0.747422	−	0.664349i	$-0.768709\pi$
$499$	−33.3923	−1.49484	−0.747422	+	0.664349i	$0.768709\pi$
$500$	−6.47363	−0.289509
$501$	0	0
$502$	−2.99764	−0.133791
$503$	31.9976	1.42670	0.713352	−	0.700806i	$-0.247176\pi$
$503$	31.9976	1.42670	0.713352	+	0.700806i	$0.247176\pi$
$504$	0	0
$505$	46.9021	2.08712
$506$	27.3146	1.21428
$507$	0	0
$508$	−23.7368	−1.05315
$509$	3.52569	0.156274	0.0781368	−	0.996943i	$-0.475103\pi$
$509$	3.52569	0.156274	0.0781368	+	0.996943i	$0.475103\pi$
$510$	0	0
$511$	1.40119	0.0619848
$512$	12.5686	0.555460
$513$	0	0
$514$	20.9314	0.923246
$515$	−5.22939	−0.230435
$516$	0	0
$517$	45.3528	1.99461
$518$	0.0651390	0.00286204
$519$	0	0
$520$	43.5147	1.90825
$521$	32.4565	1.42194	0.710972	−	0.703221i	$-0.248255\pi$
$521$	32.4565	1.42194	0.710972	+	0.703221i	$0.248255\pi$
$522$	0	0
$523$	−2.10090	−0.0918658	−0.0459329	−	0.998945i	$-0.514626\pi$
$523$	−2.10090	−0.0918658	−0.0459329	+	0.998945i	$0.514626\pi$
$524$	30.4735	1.33124
$525$	0	0
$526$	−14.0268	−0.611599
$527$	−45.3508	−1.97551
$528$	0	0
$529$	29.8397	1.29738
$530$	29.8912	1.29839
$531$	0	0
$532$	0.495356	0.0214764
$533$	−27.5199	−1.19202
$534$	0	0
$535$	19.1906	0.829684
$536$	−6.87015	−0.296745
$537$	0	0
$538$	15.4202	0.664812
$539$	35.4602	1.52738
$540$	0	0
$541$	−31.8044	−1.36738	−0.683689	−	0.729773i	$-0.739626\pi$
$541$	−31.8044	−1.36738	−0.683689	+	0.729773i	$0.739626\pi$
$542$	−3.61853	−0.155429
$543$	0	0
$544$	−41.6703	−1.78660
$545$	−2.02405	−0.0867008
$546$	0	0
$547$	−33.4548	−1.43042	−0.715212	−	0.698908i	$-0.753670\pi$
$547$	−33.4548	−1.43042	−0.715212	+	0.698908i	$0.753670\pi$
$548$	22.0471	0.941804
$549$	0	0
$550$	−23.6732	−1.00943
$551$	0.664424	0.0283054
$552$	0	0
$553$	−0.276922	−0.0117759
$554$	−22.5301	−0.957214
$555$	0	0
$556$	29.4766	1.25009
$557$	16.7773	0.710877	0.355439	−	0.934700i	$-0.384331\pi$
$557$	16.7773	0.710877	0.355439	+	0.934700i	$0.384331\pi$
$558$	0	0
$559$	−31.8338	−1.34643
$560$	1.75631	0.0742178
$561$	0	0
$562$	17.3344	0.731207
$563$	−9.47252	−0.399219	−0.199609	−	0.979876i	$-0.563967\pi$
$563$	−9.47252	−0.399219	−0.199609	+	0.979876i	$0.563967\pi$
$564$	0	0
$565$	27.0370	1.13745
$566$	7.40061	0.311071
$567$	0	0
$568$	22.2869	0.935136
$569$	−7.93748	−0.332756	−0.166378	−	0.986062i	$-0.553207\pi$
$569$	−7.93748	−0.332756	−0.166378	+	0.986062i	$0.553207\pi$
$570$	0	0
$571$	7.16768	0.299958	0.149979	−	0.988689i	$-0.452079\pi$
$571$	7.16768	0.299958	0.149979	+	0.988689i	$0.452079\pi$
$572$	−39.9072	−1.66861
$573$	0	0
$574$	1.73322	0.0723433
$575$	−45.7954	−1.90980
$576$	0	0
$577$	0.222271	0.00925327	0.00462664	−	0.999989i	$-0.498527\pi$
$577$	0.222271	0.00925327	0.00462664	+	0.999989i	$0.498527\pi$
$578$	24.3271	1.01187
$579$	0	0
$580$	4.46756	0.185505
$581$	−0.0516157	−0.00214138
$582$	0	0
$583$	−64.4242	−2.66818
$584$	−7.78027	−0.321950
$585$	0	0
$586$	−15.8596	−0.655154
$587$	−13.0374	−0.538113	−0.269056	−	0.963124i	$-0.586712\pi$
$587$	−13.0374	−0.538113	−0.269056	+	0.963124i	$0.586712\pi$
$588$	0	0
$589$	−4.71317	−0.194203
$590$	−16.8452	−0.693507
$591$	0	0
$592$	0.231792	0.00952660
$593$	−19.5281	−0.801921	−0.400960	−	0.916095i	$-0.631324\pi$
$593$	−19.5281	−0.801921	−0.400960	+	0.916095i	$0.631324\pi$
$594$	0	0
$595$	−10.8161	−0.443415
$596$	−1.48135	−0.0606786
$597$	0	0
$598$	27.0289	1.10529
$599$	15.5924	0.637088	0.318544	−	0.947908i	$-0.396806\pi$
$599$	15.5924	0.637088	0.318544	+	0.947908i	$0.396806\pi$
$600$	0	0
$601$	−14.3793	−0.586544	−0.293272	−	0.956029i	$-0.594744\pi$
$601$	−14.3793	−0.586544	−0.293272	+	0.956029i	$0.594744\pi$
$602$	2.00491	0.0817142
$603$	0	0
$604$	16.9657	0.690325
$605$	54.5395	2.21734
$606$	0	0
$607$	37.3377	1.51549	0.757745	−	0.652550i	$-0.226301\pi$
$607$	37.3377	1.51549	0.757745	+	0.652550i	$0.226301\pi$
$608$	−4.33068	−0.175632
$609$	0	0
$610$	−6.18514	−0.250429
$611$	44.8784	1.81558
$612$	0	0
$613$	43.2650	1.74746	0.873728	−	0.486415i	$-0.161696\pi$
$613$	43.2650	1.74746	0.873728	+	0.486415i	$0.161696\pi$
$614$	14.2739	0.576047
$615$	0	0
$616$	5.90674	0.237990
$617$	34.0698	1.37160	0.685799	−	0.727791i	$-0.259453\pi$
$617$	34.0698	1.37160	0.685799	+	0.727791i	$0.259453\pi$
$618$	0	0
$619$	−35.8451	−1.44074	−0.720368	−	0.693592i	$-0.756027\pi$
$619$	−35.8451	−1.44074	−0.720368	+	0.693592i	$0.756027\pi$
$620$	−31.6912	−1.27275
$621$	0	0
$622$	7.02640	0.281733
$623$	−3.46932	−0.138995
$624$	0	0
$625$	−16.8099	−0.672395
$626$	4.47660	0.178921
$627$	0	0
$628$	−13.3610	−0.533162
$629$	−1.42747	−0.0569168
$630$	0	0
$631$	−7.20445	−0.286805	−0.143402	−	0.989664i	$-0.545804\pi$
$631$	−7.20445	−0.286805	−0.143402	+	0.989664i	$0.545804\pi$
$632$	1.53764	0.0611642
$633$	0	0
$634$	−16.9778	−0.674274
$635$	53.8645	2.13755
$636$	0	0
$637$	35.0892	1.39029
$638$	3.37122	0.133468
$639$	0	0
$640$	−34.7218	−1.37250
$641$	−29.1170	−1.15005	−0.575027	−	0.818135i	$-0.695008\pi$
$641$	−29.1170	−1.15005	−0.575027	+	0.818135i	$0.695008\pi$
$642$	0	0
$643$	−6.88797	−0.271635	−0.135818	−	0.990734i	$-0.543366\pi$
$643$	−6.88797	−0.271635	−0.135818	+	0.990734i	$0.543366\pi$
$644$	4.86210	0.191594
$645$	0	0
$646$	3.80061	0.149533
$647$	−2.96107	−0.116412	−0.0582058	−	0.998305i	$-0.518538\pi$
$647$	−2.96107	−0.116412	−0.0582058	+	0.998305i	$0.518538\pi$
$648$	0	0
$649$	36.3063	1.42515
$650$	−23.4256	−0.918826
$651$	0	0
$652$	4.07477	0.159580
$653$	−31.2447	−1.22270	−0.611350	−	0.791360i	$-0.709373\pi$
$653$	−31.2447	−1.22270	−0.611350	+	0.791360i	$0.709373\pi$
$654$	0	0
$655$	−69.1517	−2.70198
$656$	6.16755	0.240802
$657$	0	0
$658$	−2.82647	−0.110187
$659$	17.4557	0.679977	0.339989	−	0.940430i	$-0.389577\pi$
$659$	17.4557	0.679977	0.339989	+	0.940430i	$0.389577\pi$
$660$	0	0
$661$	−12.8896	−0.501349	−0.250674	−	0.968071i	$-0.580652\pi$
$661$	−12.8896	−0.501349	−0.250674	+	0.968071i	$0.580652\pi$
$662$	−23.4896	−0.912950
$663$	0	0
$664$	0.286603	0.0111223
$665$	−1.12408	−0.0435900
$666$	0	0
$667$	6.52157	0.252516
$668$	−16.9797	−0.656963
$669$	0	0
$670$	6.63372	0.256283
$671$	13.3307	0.514627
$672$	0	0
$673$	−45.0072	−1.73490	−0.867450	−	0.497525i	$-0.834242\pi$
$673$	−45.0072	−1.73490	−0.867450	+	0.497525i	$0.834242\pi$
$674$	10.2395	0.394410
$675$	0	0
$676$	−20.2322	−0.778160
$677$	24.0367	0.923806	0.461903	−	0.886931i	$-0.347167\pi$
$677$	24.0367	0.923806	0.461903	+	0.886931i	$0.347167\pi$
$678$	0	0
$679$	−3.03957	−0.116648
$680$	60.0576	2.30310
$681$	0	0
$682$	−23.9142	−0.915721
$683$	−5.10700	−0.195414	−0.0977070	−	0.995215i	$-0.531151\pi$
$683$	−5.10700	−0.195414	−0.0977070	+	0.995215i	$0.531151\pi$
$684$	0	0
$685$	−50.0301	−1.91155
$686$	−4.48618	−0.171283
$687$	0	0
$688$	7.13434	0.271994
$689$	−63.7502	−2.42869
$690$	0	0
$691$	40.3702	1.53575	0.767877	−	0.640598i	$-0.221313\pi$
$691$	40.3702	1.53575	0.767877	+	0.640598i	$0.221313\pi$
$692$	−35.3649	−1.34437
$693$	0	0
$694$	9.69222	0.367912
$695$	−66.8895	−2.53726
$696$	0	0
$697$	−37.9821	−1.43868
$698$	−7.04803	−0.266772
$699$	0	0
$700$	−4.21391	−0.159271
$701$	−5.21867	−0.197106	−0.0985532	−	0.995132i	$-0.531421\pi$
$701$	−5.21867	−0.197106	−0.0985532	+	0.995132i	$0.531421\pi$
$702$	0	0
$703$	−0.148352	−0.00559522
$704$	−9.89839	−0.373060
$705$	0	0
$706$	8.08798	0.304395
$707$	6.29996	0.236934
$708$	0	0
$709$	−23.1327	−0.868765	−0.434383	−	0.900728i	$-0.643033\pi$
$709$	−23.1327	−0.868765	−0.434383	+	0.900728i	$0.643033\pi$
$710$	−21.5199	−0.807627
$711$	0	0
$712$	19.2639	0.721944
$713$	−46.2615	−1.73251
$714$	0	0
$715$	90.5592	3.38672
$716$	−1.97548	−0.0738273
$717$	0	0
$718$	−9.12409	−0.340508
$719$	−34.3541	−1.28119	−0.640597	−	0.767878i	$-0.721313\pi$
$719$	−34.3541	−1.28119	−0.640597	+	0.767878i	$0.721313\pi$
$720$	0	0
$721$	−0.702419	−0.0261594
$722$	−13.2883	−0.494538
$723$	0	0
$724$	−16.7790	−0.623588
$725$	−5.65215	−0.209916
$726$	0	0
$727$	36.9315	1.36971	0.684857	−	0.728677i	$-0.259865\pi$
$727$	36.9315	1.36971	0.684857	+	0.728677i	$0.259865\pi$
$728$	5.84495	0.216628
$729$	0	0
$730$	7.51253	0.278051
$731$	−43.9360	−1.62503
$732$	0	0
$733$	16.3451	0.603718	0.301859	−	0.953353i	$-0.402393\pi$
$733$	16.3451	0.603718	0.301859	+	0.953353i	$0.402393\pi$
$734$	−16.9621	−0.626083
$735$	0	0
$736$	−42.5072	−1.56684
$737$	−14.2976	−0.526658
$738$	0	0
$739$	15.5073	0.570446	0.285223	−	0.958461i	$-0.407932\pi$
$739$	15.5073	0.570446	0.285223	+	0.958461i	$0.407932\pi$
$740$	−0.997515	−0.0366694
$741$	0	0
$742$	4.01503	0.147396
$743$	27.5884	1.01212	0.506059	−	0.862499i	$-0.331102\pi$
$743$	27.5884	1.01212	0.506059	+	0.862499i	$0.331102\pi$
$744$	0	0
$745$	3.36155	0.123158
$746$	−11.3688	−0.416240
$747$	0	0
$748$	−55.0787	−2.01388
$749$	2.57771	0.0941876
$750$	0	0
$751$	2.00340	0.0731050	0.0365525	−	0.999332i	$-0.488362\pi$
$751$	2.00340	0.0731050	0.0365525	+	0.999332i	$0.488362\pi$
$752$	−10.0578	−0.366770
$753$	0	0
$754$	3.33595	0.121488
$755$	−38.4993	−1.40113
$756$	0	0
$757$	−7.30957	−0.265671	−0.132835	−	0.991138i	$-0.542408\pi$
$757$	−7.30957	−0.265671	−0.132835	+	0.991138i	$0.542408\pi$
$758$	−5.46809	−0.198610
$759$	0	0
$760$	6.24161	0.226407
$761$	−35.0825	−1.27174	−0.635870	−	0.771796i	$-0.719359\pi$
$761$	−35.0825	−1.27174	−0.635870	+	0.771796i	$0.719359\pi$
$762$	0	0
$763$	−0.271873	−0.00984247
$764$	24.9846	0.903911
$765$	0	0
$766$	−11.9907	−0.433240
$767$	35.9265	1.29723
$768$	0	0
$769$	−44.9301	−1.62022	−0.810111	−	0.586277i	$-0.800593\pi$
$769$	−44.9301	−1.62022	−0.810111	+	0.586277i	$0.800593\pi$
$770$	−5.70347	−0.205539
$771$	0	0
$772$	−18.9203	−0.680958
$773$	−7.86001	−0.282705	−0.141353	−	0.989959i	$-0.545145\pi$
$773$	−7.86001	−0.282705	−0.141353	+	0.989959i	$0.545145\pi$
$774$	0	0
$775$	40.0942	1.44023
$776$	16.8776	0.605870
$777$	0	0
$778$	−21.6345	−0.775634
$779$	−3.94737	−0.141429
$780$	0	0
$781$	46.3815	1.65966
$782$	37.3044	1.33400
$783$	0	0
$784$	−7.86392	−0.280854
$785$	30.3193	1.08214
$786$	0	0
$787$	−23.7998	−0.848373	−0.424186	−	0.905575i	$-0.639440\pi$
$787$	−23.7998	−0.848373	−0.424186	+	0.905575i	$0.639440\pi$
$788$	29.6085	1.05476
$789$	0	0
$790$	−1.48473	−0.0528243
$791$	3.63164	0.129126
$792$	0	0
$793$	13.1913	0.468436
$794$	11.1889	0.397080
$795$	0	0
$796$	34.3964	1.21915
$797$	−25.2752	−0.895295	−0.447647	−	0.894210i	$-0.647738\pi$
$797$	−25.2752	−0.895295	−0.447647	+	0.894210i	$0.647738\pi$
$798$	0	0
$799$	61.9397	2.19127
$800$	36.8404	1.30250
$801$	0	0
$802$	21.3056	0.752325
$803$	−16.1916	−0.571391
$804$	0	0
$805$	−11.0333	−0.388872
$806$	−23.6640	−0.833529
$807$	0	0
$808$	−34.9813	−1.23064
$809$	−23.0247	−0.809505	−0.404752	−	0.914426i	$-0.632642\pi$
$809$	−23.0247	−0.809505	−0.404752	+	0.914426i	$0.632642\pi$
$810$	0	0
$811$	12.4750	0.438058	0.219029	−	0.975718i	$-0.429711\pi$
$811$	12.4750	0.438058	0.219029	+	0.975718i	$0.429711\pi$
$812$	0.600089	0.0210590
$813$	0	0
$814$	−0.752725	−0.0263830
$815$	−9.24664	−0.323896
$816$	0	0
$817$	−4.56614	−0.159749
$818$	−10.5079	−0.367400
$819$	0	0
$820$	−26.5420	−0.926885
$821$	6.26577	0.218677	0.109338	−	0.994005i	$-0.465127\pi$
$821$	6.26577	0.218677	0.109338	+	0.994005i	$0.465127\pi$
$822$	0	0
$823$	−9.06515	−0.315991	−0.157996	−	0.987440i	$-0.550503\pi$
$823$	−9.06515	−0.315991	−0.157996	+	0.987440i	$0.550503\pi$
$824$	3.90027	0.135873
$825$	0	0
$826$	−2.26267	−0.0787284
$827$	10.2760	0.357331	0.178666	−	0.983910i	$-0.442822\pi$
$827$	10.2760	0.357331	0.178666	+	0.983910i	$0.442822\pi$
$828$	0	0
$829$	−21.4179	−0.743874	−0.371937	−	0.928258i	$-0.621306\pi$
$829$	−21.4179	−0.743874	−0.371937	+	0.928258i	$0.621306\pi$
$830$	−0.276740	−0.00960578
$831$	0	0
$832$	−9.79484	−0.339575
$833$	48.4291	1.67797
$834$	0	0
$835$	38.5309	1.33342
$836$	−5.72417	−0.197975
$837$	0	0
$838$	−6.49584	−0.224395
$839$	−14.4573	−0.499120	−0.249560	−	0.968359i	$-0.580286\pi$
$839$	−14.4573	−0.499120	−0.249560	+	0.968359i	$0.580286\pi$
$840$	0	0
$841$	−28.1951	−0.972245
$842$	−4.88389	−0.168310
$843$	0	0
$844$	−28.9973	−0.998128
$845$	45.9117	1.57941
$846$	0	0
$847$	7.32581	0.251718
$848$	14.2872	0.490624
$849$	0	0
$850$	−32.3312	−1.10895
$851$	−1.45613	−0.0499156
$852$	0	0
$853$	−16.9262	−0.579544	−0.289772	−	0.957096i	$-0.593579\pi$
$853$	−16.9262	−0.579544	−0.289772	+	0.957096i	$0.593579\pi$
$854$	−0.830796	−0.0284292
$855$	0	0
$856$	−14.3131	−0.489212
$857$	27.3260	0.933440	0.466720	−	0.884405i	$-0.345436\pi$
$857$	27.3260	0.933440	0.466720	+	0.884405i	$0.345436\pi$
$858$	0	0
$859$	2.63516	0.0899106	0.0449553	−	0.998989i	$-0.485685\pi$
$859$	2.63516	0.0899106	0.0449553	+	0.998989i	$0.485685\pi$
$860$	−30.7025	−1.04695
$861$	0	0
$862$	0.0602034	0.00205054
$863$	−34.5485	−1.17604	−0.588022	−	0.808845i	$-0.700093\pi$
$863$	−34.5485	−1.17604	−0.588022	+	0.808845i	$0.700093\pi$
$864$	0	0
$865$	80.2516	2.72864
$866$	−7.62967	−0.259267
$867$	0	0
$868$	−4.25680	−0.144485
$869$	3.20002	0.108553
$870$	0	0
$871$	−14.1480	−0.479387
$872$	1.50961	0.0511219
$873$	0	0
$874$	3.87694	0.131140
$875$	1.97321	0.0667067
$876$	0	0
$877$	42.3766	1.43096	0.715479	−	0.698635i	$-0.246209\pi$
$877$	42.3766	1.43096	0.715479	+	0.698635i	$0.246209\pi$
$878$	−18.0785	−0.610119
$879$	0	0
$880$	−20.2954	−0.684158
$881$	−0.414494	−0.0139646	−0.00698232	−	0.999976i	$-0.502223\pi$
$881$	−0.414494	−0.0139646	−0.00698232	+	0.999976i	$0.502223\pi$
$882$	0	0
$883$	27.4216	0.922810	0.461405	−	0.887190i	$-0.347346\pi$
$883$	27.4216	0.922810	0.461405	+	0.887190i	$0.347346\pi$
$884$	−54.5025	−1.83312
$885$	0	0
$886$	−26.7177	−0.897597
$887$	32.6899	1.09762	0.548809	−	0.835948i	$-0.315081\pi$
$887$	32.6899	1.09762	0.548809	+	0.835948i	$0.315081\pi$
$888$	0	0
$889$	7.23515	0.242659
$890$	−18.6009	−0.623505
$891$	0	0
$892$	24.0961	0.806799
$893$	6.43721	0.215413
$894$	0	0
$895$	4.48285	0.149845
$896$	−4.66388	−0.155809
$897$	0	0
$898$	14.9409	0.498586
$899$	−5.70968	−0.190428
$900$	0	0
$901$	−87.9860	−2.93124
$902$	−20.0286	−0.666878
$903$	0	0
$904$	−20.1652	−0.670684
$905$	38.0757	1.26568
$906$	0	0
$907$	21.5627	0.715977	0.357989	−	0.933726i	$-0.383463\pi$
$907$	21.5627	0.715977	0.357989	+	0.933726i	$0.383463\pi$
$908$	18.0491	0.598980
$909$	0	0
$910$	−5.64381	−0.187090
$911$	−38.6529	−1.28063	−0.640314	−	0.768114i	$-0.721196\pi$
$911$	−38.6529	−1.28063	−0.640314	+	0.768114i	$0.721196\pi$
$912$	0	0
$913$	0.596454	0.0197397
$914$	22.2485	0.735916
$915$	0	0
$916$	17.2523	0.570033
$917$	−9.28854	−0.306735
$918$	0	0
$919$	−19.2187	−0.633968	−0.316984	−	0.948431i	$-0.602670\pi$
$919$	−19.2187	−0.633968	−0.316984	+	0.948431i	$0.602670\pi$
$920$	61.2637	2.01981
$921$	0	0
$922$	8.35242	0.275072
$923$	45.8963	1.51070
$924$	0	0
$925$	1.26201	0.0414946
$926$	21.2372	0.697898
$927$	0	0
$928$	−5.24631	−0.172219
$929$	−28.3944	−0.931591	−0.465795	−	0.884892i	$-0.654232\pi$
$929$	−28.3944	−0.931591	−0.465795	+	0.884892i	$0.654232\pi$
$930$	0	0
$931$	5.03309	0.164953
$932$	11.2523	0.368582
$933$	0	0
$934$	22.2375	0.727632
$935$	124.987	4.08751
$936$	0	0
$937$	−54.3272	−1.77479	−0.887396	−	0.461007i	$-0.847488\pi$
$937$	−54.3272	−1.77479	−0.887396	+	0.461007i	$0.847488\pi$
$938$	0.891051	0.0290938
$939$	0	0
$940$	43.2835	1.41175
$941$	17.4042	0.567359	0.283680	−	0.958919i	$-0.408445\pi$
$941$	17.4042	0.567359	0.283680	+	0.958919i	$0.408445\pi$
$942$	0	0
$943$	−38.7449	−1.26171
$944$	−8.05155	−0.262056
$945$	0	0
$946$	−23.1681	−0.753261
$947$	11.9576	0.388571	0.194285	−	0.980945i	$-0.437761\pi$
$947$	11.9576	0.388571	0.194285	+	0.980945i	$0.437761\pi$
$948$	0	0
$949$	−16.0223	−0.520105
$950$	−3.36009	−0.109016
$951$	0	0
$952$	8.06702	0.261454
$953$	35.3150	1.14396	0.571982	−	0.820266i	$-0.306175\pi$
$953$	35.3150	1.14396	0.571982	+	0.820266i	$0.306175\pi$
$954$	0	0
$955$	−56.6960	−1.83464
$956$	−11.3781	−0.367992
$957$	0	0
$958$	1.87268	0.0605035
$959$	−6.72011	−0.217004
$960$	0	0
$961$	9.50229	0.306526
$962$	−0.744851	−0.0240149
$963$	0	0
$964$	−9.19333	−0.296097
$965$	42.9348	1.38212
$966$	0	0
$967$	−46.8426	−1.50635	−0.753177	−	0.657817i	$-0.771480\pi$
$967$	−46.8426	−1.50635	−0.753177	+	0.657817i	$0.771480\pi$
$968$	−40.6776	−1.30743
$969$	0	0
$970$	−16.2968	−0.523258
$971$	−40.1836	−1.28955	−0.644776	−	0.764371i	$-0.723050\pi$
$971$	−40.1836	−1.28955	−0.644776	+	0.764371i	$0.723050\pi$
$972$	0	0
$973$	−8.98469	−0.288036
$974$	−0.769130	−0.0246445
$975$	0	0
$976$	−2.95633	−0.0946297
$977$	−45.5671	−1.45782	−0.728911	−	0.684609i	$-0.759973\pi$
$977$	−45.5671	−1.45782	−0.728911	+	0.684609i	$0.759973\pi$
$978$	0	0
$979$	40.0903	1.28129
$980$	33.8423	1.08105
$981$	0	0
$982$	28.7416	0.917182
$983$	−60.9266	−1.94326	−0.971628	−	0.236514i	$-0.923995\pi$
$983$	−60.9266	−1.94326	−0.971628	+	0.236514i	$0.923995\pi$
$984$	0	0
$985$	−67.1887	−2.14081
$986$	4.60418	0.146627
$987$	0	0
$988$	−5.66429	−0.180205
$989$	−44.8184	−1.42514
$990$	0	0
$991$	18.5041	0.587803	0.293902	−	0.955836i	$-0.405046\pi$
$991$	18.5041	0.587803	0.293902	+	0.955836i	$0.405046\pi$
$992$	37.2153	1.18159
$993$	0	0
$994$	−2.89058	−0.0916837
$995$	−78.0538	−2.47447
$996$	0	0
$997$	−35.3876	−1.12074	−0.560369	−	0.828243i	$-0.689341\pi$
$997$	−35.3876	−1.12074	−0.560369	+	0.828243i	$0.689341\pi$
$998$	−24.0481	−0.761231
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.d.1.7		9
3.2	odd	2		447.2.a.c.1.3	✓	9
12.11	even	2		7152.2.a.z.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
447.2.a.c.1.3	✓	9	3.2	odd	2
1341.2.a.d.1.7		9	1.1	even	1	trivial
7152.2.a.z.1.1		9	12.11	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)

\(f(q)\)	\(=\)	\(q+0.720171 q^{2} -1.48135 q^{4} +3.36155 q^{5} +0.451528 q^{7} -2.50717 q^{8} +O(q^{10})\)	\(q+0.720171 q^{2} -1.48135 q^{4} +3.36155 q^{5} +0.451528 q^{7} -2.50717 q^{8} +2.42089 q^{10} -5.21771 q^{11} -5.16313 q^{13} +0.325177 q^{14} +1.15712 q^{16} -7.12598 q^{17} -0.740583 q^{19} -4.97965 q^{20} -3.75764 q^{22} -7.26909 q^{23} +6.30002 q^{25} -3.71833 q^{26} -0.668873 q^{28} -0.897164 q^{29} +6.36414 q^{31} +5.84766 q^{32} -5.13192 q^{34} +1.51783 q^{35} +0.200318 q^{37} -0.533346 q^{38} -8.42797 q^{40} +5.33009 q^{41} +6.16561 q^{43} +7.72928 q^{44} -5.23499 q^{46} -8.69209 q^{47} -6.79612 q^{49} +4.53709 q^{50} +7.64842 q^{52} +12.3472 q^{53} -17.5396 q^{55} -1.13206 q^{56} -0.646111 q^{58} -6.95828 q^{59} -2.55490 q^{61} +4.58327 q^{62} +1.89708 q^{64} -17.3561 q^{65} +2.74020 q^{67} +10.5561 q^{68} +1.09310 q^{70} -8.88925 q^{71} +3.10321 q^{73} +0.144263 q^{74} +1.09707 q^{76} -2.35594 q^{77} -0.613299 q^{79} +3.88971 q^{80} +3.83857 q^{82} -0.114313 q^{83} -23.9543 q^{85} +4.44029 q^{86} +13.0817 q^{88} -7.68351 q^{89} -2.33130 q^{91} +10.7681 q^{92} -6.25979 q^{94} -2.48951 q^{95} -6.73173 q^{97} -4.89437 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 4 q^{2} + 10 q^{4} - 8 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 4 * q^2 + 10 * q^4 - 8 * q^5 + q^7 - 9 * q^8	\( 9 q - 4 q^{2} + 10 q^{4} - 8 q^{5} + q^{7} - 9 q^{8} - 4 q^{10} - 13 q^{11} - 2 q^{13} + 5 q^{14} - 10 q^{17} - 11 q^{19} - 6 q^{20} - 16 q^{22} - 11 q^{23} + 15 q^{25} - 6 q^{26} - 14 q^{28} - 13 q^{31} - 7 q^{32} - 28 q^{34} - 6 q^{35} + q^{37} + 5 q^{38} - 32 q^{40} - 17 q^{41} - 8 q^{43} - 3 q^{44} + 5 q^{46} - 10 q^{47} - 4 q^{49} + 8 q^{50} - 4 q^{52} - 2 q^{53} - 12 q^{55} + 2 q^{56} + 10 q^{58} - 25 q^{59} - 13 q^{61} + 18 q^{62} - 23 q^{64} - 16 q^{65} + q^{67} + 18 q^{68} - 32 q^{70} - 13 q^{71} + 5 q^{73} + 17 q^{74} - 43 q^{76} - 20 q^{77} - 22 q^{79} + 70 q^{80} + 22 q^{82} - q^{83} - 16 q^{85} - 10 q^{86} + 3 q^{88} - 23 q^{89} - 62 q^{91} - 8 q^{92} - 28 q^{94} + 18 q^{95} + 16 q^{97} + 61 q^{98}+O(q^{100}) \) 9 * q - 4 * q^2 + 10 * q^4 - 8 * q^5 + q^7 - 9 * q^8 - 4 * q^10 - 13 * q^11 - 2 * q^13 + 5 * q^14 - 10 * q^17 - 11 * q^19 - 6 * q^20 - 16 * q^22 - 11 * q^23 + 15 * q^25 - 6 * q^26 - 14 * q^28 - 13 * q^31 - 7 * q^32 - 28 * q^34 - 6 * q^35 + q^37 + 5 * q^38 - 32 * q^40 - 17 * q^41 - 8 * q^43 - 3 * q^44 + 5 * q^46 - 10 * q^47 - 4 * q^49 + 8 * q^50 - 4 * q^52 - 2 * q^53 - 12 * q^55 + 2 * q^56 + 10 * q^58 - 25 * q^59 - 13 * q^61 + 18 * q^62 - 23 * q^64 - 16 * q^65 + q^67 + 18 * q^68 - 32 * q^70 - 13 * q^71 + 5 * q^73 + 17 * q^74 - 43 * q^76 - 20 * q^77 - 22 * q^79 + 70 * q^80 + 22 * q^82 - q^83 - 16 * q^85 - 10 * q^86 + 3 * q^88 - 23 * q^89 - 62 * q^91 - 8 * q^92 - 28 * q^94 + 18 * q^95 + 16 * q^97 + 61 * q^98

Introduction

L-functions

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