LMFDB - Embedded newform 1341.2.a.h.1.5

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:	$12$
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} - 12 x^{10} + 38 x^{9} + 46 x^{8} - 162 x^{7} - 59 x^{6} + 280 x^{5} - 14 x^{4} + \cdots - 3 $ x^12 - 3x^11 - 12x^10 + 38x^9 + 46x^8 - 162x^7 - 59x^6 + 280x^5 - 14x^4 - 173x^3 + 49x^2 + 11*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.257804$ 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)

$f(q)$	$=$	$q-0.257804 q^{2} -1.93354 q^{4} -1.57497 q^{5} -2.15740 q^{7} +1.01408 q^{8} +O(q^{10})$	\(q-0.257804 q^{2} -1.93354 q^{4} -1.57497 q^{5} -2.15740 q^{7} +1.01408 q^{8} +0.406034 q^{10} -2.70433 q^{11} -2.63298 q^{13} +0.556186 q^{14} +3.60564 q^{16} -2.60842 q^{17} -3.24343 q^{19} +3.04527 q^{20} +0.697185 q^{22} +5.16911 q^{23} -2.51946 q^{25} +0.678792 q^{26} +4.17142 q^{28} +6.39128 q^{29} +8.78713 q^{31} -2.95771 q^{32} +0.672460 q^{34} +3.39785 q^{35} -5.86905 q^{37} +0.836168 q^{38} -1.59715 q^{40} +7.44313 q^{41} -6.35145 q^{43} +5.22892 q^{44} -1.33262 q^{46} -10.1452 q^{47} -2.34562 q^{49} +0.649525 q^{50} +5.09097 q^{52} +11.8085 q^{53} +4.25924 q^{55} -2.18778 q^{56} -1.64769 q^{58} +13.4540 q^{59} +3.98935 q^{61} -2.26535 q^{62} -6.44877 q^{64} +4.14688 q^{65} -0.833579 q^{67} +5.04347 q^{68} -0.875979 q^{70} +14.2573 q^{71} +8.95110 q^{73} +1.51306 q^{74} +6.27129 q^{76} +5.83432 q^{77} -7.65360 q^{79} -5.67879 q^{80} -1.91887 q^{82} -1.13916 q^{83} +4.10819 q^{85} +1.63743 q^{86} -2.74240 q^{88} -7.82574 q^{89} +5.68040 q^{91} -9.99467 q^{92} +2.61548 q^{94} +5.10832 q^{95} +9.97234 q^{97} +0.604709 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) $ 12 * q + 3 * q^2 + 9 * q^4 + 8 * q^5 - 2 * q^7 + 9 * q^8	$ 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8} + 22 q^{11} - 2 q^{13} + 12 q^{14} + 11 q^{16} + 8 q^{17} - 6 q^{19} + 24 q^{20} + 4 q^{22} + 12 q^{23} + 8 q^{25} + 26 q^{26} - 20 q^{28} + 16 q^{29} - 2 q^{31} + 21 q^{32} - 6 q^{34} + 32 q^{35} - 4 q^{37} + 15 q^{38} + 6 q^{40} + 24 q^{41} + 12 q^{43} + 37 q^{44} - 22 q^{46} + 26 q^{47} + 24 q^{49} - 5 q^{50} - 10 q^{52} + 20 q^{53} + 6 q^{55} - 13 q^{56} + 10 q^{58} + 72 q^{59} - 8 q^{61} - 2 q^{62} + q^{64} - 4 q^{65} - 14 q^{67} - 14 q^{68} + 18 q^{70} + 38 q^{71} - 27 q^{74} - 2 q^{76} + 6 q^{77} + 10 q^{79} + 56 q^{80} + 6 q^{82} + 42 q^{83} - 32 q^{85} - 14 q^{86} + 22 q^{88} + 44 q^{89} - 50 q^{92} + 2 q^{94} + 4 q^{95} - 22 q^{97} + 20 q^{98}+O(q^{100}) $ 12 * q + 3 * q^2 + 9 * q^4 + 8 * q^5 - 2 * q^7 + 9 * q^8 + 22 * q^11 - 2 * q^13 + 12 * q^14 + 11 * q^16 + 8 * q^17 - 6 * q^19 + 24 * q^20 + 4 * q^22 + 12 * q^23 + 8 * q^25 + 26 * q^26 - 20 * q^28 + 16 * q^29 - 2 * q^31 + 21 * q^32 - 6 * q^34 + 32 * q^35 - 4 * q^37 + 15 * q^38 + 6 * q^40 + 24 * q^41 + 12 * q^43 + 37 * q^44 - 22 * q^46 + 26 * q^47 + 24 * q^49 - 5 * q^50 - 10 * q^52 + 20 * q^53 + 6 * q^55 - 13 * q^56 + 10 * q^58 + 72 * q^59 - 8 * q^61 - 2 * q^62 + q^64 - 4 * q^65 - 14 * q^67 - 14 * q^68 + 18 * q^70 + 38 * q^71 - 27 * q^74 - 2 * q^76 + 6 * q^77 + 10 * q^79 + 56 * q^80 + 6 * q^82 + 42 * q^83 - 32 * q^85 - 14 * q^86 + 22 * q^88 + 44 * q^89 - 50 * q^92 + 2 * q^94 + 4 * q^95 - 22 * q^97 + 20 * 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 $

$2$	−0.257804	−0.182295	−0.0911474	−	0.995837i	$-0.529053\pi$
$2$	−0.257804	−0.182295	−0.0911474	+	0.995837i	$0.529053\pi$
$3$	0	0
$3$	0	0
$4$	−1.93354	−0.966769
$5$	−1.57497	−0.704350	−0.352175	−	0.935934i	$-0.614558\pi$
$5$	−1.57497	−0.704350	−0.352175	+	0.935934i	$0.614558\pi$
$6$	0	0
$7$	−2.15740	−0.815421	−0.407711	−	0.913111i	$-0.633673\pi$
$7$	−2.15740	−0.815421	−0.407711	+	0.913111i	$0.633673\pi$
$8$	1.01408	0.358532
$9$	0	0
$10$	0.406034	0.128399
$11$	−2.70433	−0.815385	−0.407693	−	0.913119i	$-0.633666\pi$
$11$	−2.70433	−0.815385	−0.407693	+	0.913119i	$0.633666\pi$
$12$	0	0
$13$	−2.63298	−0.730258	−0.365129	−	0.930957i	$-0.618975\pi$
$13$	−2.63298	−0.730258	−0.365129	+	0.930957i	$0.618975\pi$
$14$	0.556186	0.148647
$15$	0	0
$16$	3.60564	0.901410
$17$	−2.60842	−0.632634	−0.316317	−	0.948653i	$-0.602446\pi$
$17$	−2.60842	−0.632634	−0.316317	+	0.948653i	$0.602446\pi$
$18$	0	0
$19$	−3.24343	−0.744094	−0.372047	−	0.928214i	$-0.621344\pi$
$19$	−3.24343	−0.744094	−0.372047	+	0.928214i	$0.621344\pi$
$20$	3.04527	0.680943
$21$	0	0
$22$	0.697185	0.148640
$23$	5.16911	1.07783	0.538917	−	0.842359i	$-0.318833\pi$
$23$	5.16911	1.07783	0.538917	+	0.842359i	$0.318833\pi$
$24$	0	0
$25$	−2.51946	−0.503891
$26$	0.678792	0.133122
$27$	0	0
$28$	4.17142	0.788324
$29$	6.39128	1.18683	0.593415	−	0.804897i	$-0.297779\pi$
$29$	6.39128	1.18683	0.593415	+	0.804897i	$0.297779\pi$
$30$	0	0
$31$	8.78713	1.57821	0.789107	−	0.614255i	$-0.210544\pi$
$31$	8.78713	1.57821	0.789107	+	0.614255i	$0.210544\pi$
$32$	−2.95771	−0.522854
$33$	0	0
$34$	0.672460	0.115326
$35$	3.39785	0.574342
$36$	0	0
$37$	−5.86905	−0.964867	−0.482433	−	0.875933i	$-0.660247\pi$
$37$	−5.86905	−0.964867	−0.482433	+	0.875933i	$0.660247\pi$
$38$	0.836168	0.135644
$39$	0	0
$40$	−1.59715	−0.252532
$41$	7.44313	1.16242	0.581211	−	0.813753i	$-0.302579\pi$
$41$	7.44313	1.16242	0.581211	+	0.813753i	$0.302579\pi$
$42$	0	0
$43$	−6.35145	−0.968587	−0.484293	−	0.874906i	$-0.660923\pi$
$43$	−6.35145	−0.968587	−0.484293	+	0.874906i	$0.660923\pi$
$44$	5.22892	0.788289
$45$	0	0
$46$	−1.33262	−0.196484
$47$	−10.1452	−1.47983	−0.739917	−	0.672698i	$-0.765135\pi$
$47$	−10.1452	−1.47983	−0.739917	+	0.672698i	$0.765135\pi$
$48$	0	0
$49$	−2.34562	−0.335088
$50$	0.649525	0.0918567
$51$	0	0
$52$	5.09097	0.705990
$53$	11.8085	1.62203	0.811013	−	0.585028i	$-0.198917\pi$
$53$	11.8085	1.62203	0.811013	+	0.585028i	$0.198917\pi$
$54$	0	0
$55$	4.25924	0.574316
$56$	−2.18778	−0.292354
$57$	0	0
$58$	−1.64769	−0.216353
$59$	13.4540	1.75156	0.875782	−	0.482707i	$-0.160346\pi$
$59$	13.4540	1.75156	0.875782	+	0.482707i	$0.160346\pi$
$60$	0	0
$61$	3.98935	0.510784	0.255392	−	0.966838i	$-0.417796\pi$
$61$	3.98935	0.510784	0.255392	+	0.966838i	$0.417796\pi$
$62$	−2.26535	−0.287700
$63$	0	0
$64$	−6.44877	−0.806097
$65$	4.14688	0.514357
$66$	0	0
$67$	−0.833579	−0.101838	−0.0509189	−	0.998703i	$-0.516215\pi$
$67$	−0.833579	−0.101838	−0.0509189	+	0.998703i	$0.516215\pi$
$68$	5.04347	0.611611
$69$	0	0
$70$	−0.875979	−0.104699
$71$	14.2573	1.69203	0.846015	−	0.533159i	$-0.178995\pi$
$71$	14.2573	1.69203	0.846015	+	0.533159i	$0.178995\pi$
$72$	0	0
$73$	8.95110	1.04765	0.523823	−	0.851827i	$-0.324505\pi$
$73$	8.95110	1.04765	0.523823	+	0.851827i	$0.324505\pi$
$74$	1.51306	0.175890
$75$	0	0
$76$	6.27129	0.719367
$77$	5.83432	0.664882
$78$	0	0
$79$	−7.65360	−0.861097	−0.430549	−	0.902567i	$-0.641680\pi$
$79$	−7.65360	−0.861097	−0.430549	+	0.902567i	$0.641680\pi$
$80$	−5.67879	−0.634908
$81$	0	0
$82$	−1.91887	−0.211903
$83$	−1.13916	−0.125039	−0.0625194	−	0.998044i	$-0.519914\pi$
$83$	−1.13916	−0.125039	−0.0625194	+	0.998044i	$0.519914\pi$
$84$	0	0
$85$	4.10819	0.445596
$86$	1.63743	0.176568
$87$	0	0
$88$	−2.74240	−0.292341
$89$	−7.82574	−0.829527	−0.414763	−	0.909929i	$-0.636136\pi$
$89$	−7.82574	−0.829527	−0.414763	+	0.909929i	$0.636136\pi$
$90$	0	0
$91$	5.68040	0.595468
$92$	−9.99467	−1.04202
$93$	0	0
$94$	2.61548	0.269766
$95$	5.10832	0.524103
$96$	0	0
$97$	9.97234	1.01254	0.506269	−	0.862376i	$-0.331024\pi$
$97$	9.97234	1.01254	0.506269	+	0.862376i	$0.331024\pi$
$98$	0.604709	0.0610848
$99$	0	0
$100$	4.87146	0.487146
$101$	−7.64528	−0.760734	−0.380367	−	0.924836i	$-0.624202\pi$
$101$	−7.64528	−0.760734	−0.380367	+	0.924836i	$0.624202\pi$
$102$	0	0
$103$	−8.10727	−0.798833	−0.399416	−	0.916770i	$-0.630787\pi$
$103$	−8.10727	−0.798833	−0.399416	+	0.916770i	$0.630787\pi$
$104$	−2.67005	−0.261820
$105$	0	0
$106$	−3.04428	−0.295687
$107$	19.5042	1.88555	0.942773	−	0.333435i	$-0.108208\pi$
$107$	19.5042	1.88555	0.942773	+	0.333435i	$0.108208\pi$
$108$	0	0
$109$	13.4511	1.28838	0.644189	−	0.764866i	$-0.277195\pi$
$109$	13.4511	1.28838	0.644189	+	0.764866i	$0.277195\pi$
$110$	−1.09805	−0.104695
$111$	0	0
$112$	−7.77882	−0.735029
$113$	−5.82288	−0.547770	−0.273885	−	0.961762i	$-0.588309\pi$
$113$	−5.82288	−0.547770	−0.273885	+	0.961762i	$0.588309\pi$
$114$	0	0
$115$	−8.14122	−0.759173
$116$	−12.3578	−1.14739
$117$	0	0
$118$	−3.46850	−0.319301
$119$	5.62741	0.515864
$120$	0	0
$121$	−3.68662	−0.335147
$122$	−1.02847	−0.0931131
$123$	0	0
$124$	−16.9902	−1.52577
$125$	11.8429	1.05927
$126$	0	0
$127$	13.1098	1.16331	0.581655	−	0.813435i	$-0.302405\pi$
$127$	13.1098	1.16331	0.581655	+	0.813435i	$0.302405\pi$
$128$	7.57793	0.669801
$129$	0	0
$130$	−1.06908	−0.0937645
$131$	16.0871	1.40553	0.702766	−	0.711421i	$-0.251948\pi$
$131$	16.0871	1.40553	0.702766	+	0.711421i	$0.251948\pi$
$132$	0	0
$133$	6.99738	0.606750
$134$	0.214900	0.0185645
$135$	0	0
$136$	−2.64515	−0.226819
$137$	−15.2881	−1.30615	−0.653075	−	0.757293i	$-0.726521\pi$
$137$	−15.2881	−1.30615	−0.653075	+	0.757293i	$0.726521\pi$
$138$	0	0
$139$	−1.58265	−0.134238	−0.0671192	−	0.997745i	$-0.521381\pi$
$139$	−1.58265	−0.134238	−0.0671192	+	0.997745i	$0.521381\pi$
$140$	−6.56987	−0.555256
$141$	0	0
$142$	−3.67558	−0.308448
$143$	7.12044	0.595441
$144$	0	0
$145$	−10.0661	−0.835944
$146$	−2.30763	−0.190980
$147$	0	0
$148$	11.3480	0.932803
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−18.3857	−1.49621	−0.748103	−	0.663582i	$-0.769035\pi$
$151$	−18.3857	−1.49621	−0.748103	+	0.663582i	$0.769035\pi$
$152$	−3.28910	−0.266781
$153$	0	0
$154$	−1.50411	−0.121205
$155$	−13.8395	−1.11162
$156$	0	0
$157$	7.81966	0.624077	0.312039	−	0.950069i	$-0.398988\pi$
$157$	7.81966	0.624077	0.312039	+	0.950069i	$0.398988\pi$
$158$	1.97313	0.156973
$159$	0	0
$160$	4.65831	0.368272
$161$	−11.1519	−0.878889
$162$	0	0
$163$	2.32246	0.181909	0.0909547	−	0.995855i	$-0.471008\pi$
$163$	2.32246	0.181909	0.0909547	+	0.995855i	$0.471008\pi$
$164$	−14.3916	−1.12379
$165$	0	0
$166$	0.293679	0.0227939
$167$	−15.2290	−1.17846	−0.589229	−	0.807966i	$-0.700568\pi$
$167$	−15.2290	−1.17846	−0.589229	+	0.807966i	$0.700568\pi$
$168$	0	0
$169$	−6.06741	−0.466724
$170$	−1.05911	−0.0812298
$171$	0	0
$172$	12.2808	0.936400
$173$	−14.4407	−1.09790	−0.548951	−	0.835854i	$-0.684973\pi$
$173$	−14.4407	−1.09790	−0.548951	+	0.835854i	$0.684973\pi$
$174$	0	0
$175$	5.43548	0.410884
$176$	−9.75083	−0.734996
$177$	0	0
$178$	2.01750	0.151218
$179$	12.4863	0.933270	0.466635	−	0.884450i	$-0.345466\pi$
$179$	12.4863	0.933270	0.466635	+	0.884450i	$0.345466\pi$
$180$	0	0
$181$	−21.5812	−1.60412	−0.802061	−	0.597243i	$-0.796263\pi$
$181$	−21.5812	−1.60412	−0.802061	+	0.597243i	$0.796263\pi$
$182$	−1.46443	−0.108551
$183$	0	0
$184$	5.24190	0.386438
$185$	9.24361	0.679604
$186$	0	0
$187$	7.05402	0.515841
$188$	19.6162	1.43066
$189$	0	0
$190$	−1.31694	−0.0955411
$191$	2.91617	0.211006	0.105503	−	0.994419i	$-0.466355\pi$
$191$	2.91617	0.211006	0.105503	+	0.994419i	$0.466355\pi$
$192$	0	0
$193$	5.61063	0.403862	0.201931	−	0.979400i	$-0.435278\pi$
$193$	5.61063	0.403862	0.201931	+	0.979400i	$0.435278\pi$
$194$	−2.57091	−0.184580
$195$	0	0
$196$	4.53534	0.323953
$197$	−18.8742	−1.34473	−0.672367	−	0.740218i	$-0.734722\pi$
$197$	−18.8742	−1.34473	−0.672367	+	0.740218i	$0.734722\pi$
$198$	0	0
$199$	8.66323	0.614120	0.307060	−	0.951690i	$-0.400655\pi$
$199$	8.66323	0.614120	0.307060	+	0.951690i	$0.400655\pi$
$200$	−2.55493	−0.180661
$201$	0	0
$202$	1.97098	0.138678
$203$	−13.7885	−0.967766
$204$	0	0
$205$	−11.7227	−0.818752
$206$	2.09008	0.145623
$207$	0	0
$208$	−9.49359	−0.658262
$209$	8.77130	0.606723
$210$	0	0
$211$	−13.2286	−0.910696	−0.455348	−	0.890314i	$-0.650485\pi$
$211$	−13.2286	−0.910696	−0.455348	+	0.890314i	$0.650485\pi$
$212$	−22.8322	−1.56812
$213$	0	0
$214$	−5.02827	−0.343725
$215$	10.0034	0.682224
$216$	0	0
$217$	−18.9574	−1.28691
$218$	−3.46773	−0.234865
$219$	0	0
$220$	−8.23541	−0.555231
$221$	6.86792	0.461986
$222$	0	0
$223$	9.43359	0.631720	0.315860	−	0.948806i	$-0.397707\pi$
$223$	9.43359	0.631720	0.315860	+	0.948806i	$0.397707\pi$
$224$	6.38096	0.426346
$225$	0	0
$226$	1.50116	0.0998556
$227$	21.1680	1.40497	0.702484	−	0.711700i	$-0.252074\pi$
$227$	21.1680	1.40497	0.702484	+	0.711700i	$0.252074\pi$
$228$	0	0
$229$	22.7538	1.50362	0.751808	−	0.659382i	$-0.229182\pi$
$229$	22.7538	1.50362	0.751808	+	0.659382i	$0.229182\pi$
$230$	2.09884	0.138393
$231$	0	0
$232$	6.48127	0.425516
$233$	−12.5897	−0.824780	−0.412390	−	0.911007i	$-0.635306\pi$
$233$	−12.5897	−0.824780	−0.412390	+	0.911007i	$0.635306\pi$
$234$	0	0
$235$	15.9785	1.04232
$236$	−26.0138	−1.69336
$237$	0	0
$238$	−1.45077	−0.0940392
$239$	−15.0656	−0.974512	−0.487256	−	0.873259i	$-0.662002\pi$
$239$	−15.0656	−0.974512	−0.487256	+	0.873259i	$0.662002\pi$
$240$	0	0
$241$	26.2947	1.69379	0.846896	−	0.531758i	$-0.178469\pi$
$241$	26.2947	1.69379	0.846896	+	0.531758i	$0.178469\pi$
$242$	0.950424	0.0610956
$243$	0	0
$244$	−7.71355	−0.493810
$245$	3.69429	0.236019
$246$	0	0
$247$	8.53989	0.543380
$248$	8.91085	0.565840
$249$	0	0
$250$	−3.05316	−0.193099
$251$	6.40889	0.404526	0.202263	−	0.979331i	$-0.435170\pi$
$251$	6.40889	0.404526	0.202263	+	0.979331i	$0.435170\pi$
$252$	0	0
$253$	−13.9790	−0.878850
$254$	−3.37977	−0.212065
$255$	0	0
$256$	10.9439	0.683996
$257$	−1.41350	−0.0881719	−0.0440860	−	0.999028i	$-0.514038\pi$
$257$	−1.41350	−0.0881719	−0.0440860	+	0.999028i	$0.514038\pi$
$258$	0	0
$259$	12.6619	0.786773
$260$	−8.01814	−0.497264
$261$	0	0
$262$	−4.14730	−0.256221
$263$	−1.75694	−0.108338	−0.0541688	−	0.998532i	$-0.517251\pi$
$263$	−1.75694	−0.108338	−0.0541688	+	0.998532i	$0.517251\pi$
$264$	0	0
$265$	−18.5981	−1.14247
$266$	−1.80395	−0.110607
$267$	0	0
$268$	1.61176	0.0984536
$269$	5.74791	0.350456	0.175228	−	0.984528i	$-0.443934\pi$
$269$	5.74791	0.350456	0.175228	+	0.984528i	$0.443934\pi$
$270$	0	0
$271$	−3.40759	−0.206996	−0.103498	−	0.994630i	$-0.533004\pi$
$271$	−3.40759	−0.206996	−0.103498	+	0.994630i	$0.533004\pi$
$272$	−9.40502	−0.570263
$273$	0	0
$274$	3.94133	0.238104
$275$	6.81343	0.410865
$276$	0	0
$277$	27.8119	1.67106	0.835529	−	0.549447i	$-0.185161\pi$
$277$	27.8119	1.67106	0.835529	+	0.549447i	$0.185161\pi$
$278$	0.408012	0.0244710
$279$	0	0
$280$	3.44569	0.205920
$281$	−21.8178	−1.30154	−0.650771	−	0.759274i	$-0.725554\pi$
$281$	−21.8178	−1.30154	−0.650771	+	0.759274i	$0.725554\pi$
$282$	0	0
$283$	22.0915	1.31321	0.656603	−	0.754237i	$-0.271993\pi$
$283$	22.0915	1.31321	0.656603	+	0.754237i	$0.271993\pi$
$284$	−27.5670	−1.63580
$285$	0	0
$286$	−1.83568	−0.108546
$287$	−16.0578	−0.947864
$288$	0	0
$289$	−10.1962	−0.599774
$290$	2.59508	0.152388
$291$	0	0
$292$	−17.3073	−1.01283
$293$	15.1199	0.883314	0.441657	−	0.897184i	$-0.354391\pi$
$293$	15.1199	0.883314	0.441657	+	0.897184i	$0.354391\pi$
$294$	0	0
$295$	−21.1897	−1.23371
$296$	−5.95169	−0.345935
$297$	0	0
$298$	−0.257804	−0.0149342
$299$	−13.6102	−0.787097
$300$	0	0
$301$	13.7026	0.789806
$302$	4.73990	0.272751
$303$	0	0
$304$	−11.6946	−0.670734
$305$	−6.28312	−0.359770
$306$	0	0
$307$	−16.3063	−0.930649	−0.465325	−	0.885140i	$-0.654062\pi$
$307$	−16.3063	−0.930649	−0.465325	+	0.885140i	$0.654062\pi$
$308$	−11.2809	−0.642787
$309$	0	0
$310$	3.56787	0.202642
$311$	13.6224	0.772457	0.386229	−	0.922403i	$-0.373778\pi$
$311$	13.6224	0.772457	0.386229	+	0.922403i	$0.373778\pi$
$312$	0	0
$313$	−0.318432	−0.0179989	−0.00899943	−	0.999960i	$-0.502865\pi$
$313$	−0.318432	−0.0179989	−0.00899943	+	0.999960i	$0.502865\pi$
$314$	−2.01594	−0.113766
$315$	0	0
$316$	14.7985	0.832482
$317$	−13.8673	−0.778864	−0.389432	−	0.921055i	$-0.627329\pi$
$317$	−13.8673	−0.778864	−0.389432	+	0.921055i	$0.627329\pi$
$318$	0	0
$319$	−17.2841	−0.967724
$320$	10.1567	0.567774
$321$	0	0
$322$	2.87499	0.160217
$323$	8.46023	0.470740
$324$	0	0
$325$	6.63368	0.367971
$326$	−0.598739	−0.0331611
$327$	0	0
$328$	7.54794	0.416765
$329$	21.8873	1.20669
$330$	0	0
$331$	18.1523	0.997739	0.498869	−	0.866677i	$-0.333749\pi$
$331$	18.1523	0.997739	0.498869	+	0.866677i	$0.333749\pi$
$332$	2.20260	0.120884
$333$	0	0
$334$	3.92610	0.214827
$335$	1.31286	0.0717295
$336$	0	0
$337$	−21.8681	−1.19123	−0.595616	−	0.803269i	$-0.703092\pi$
$337$	−21.8681	−1.19123	−0.595616	+	0.803269i	$0.703092\pi$
$338$	1.56420	0.0850813
$339$	0	0
$340$	−7.94334	−0.430788
$341$	−23.7633	−1.28685
$342$	0	0
$343$	20.1623	1.08866
$344$	−6.44088	−0.347269
$345$	0	0
$346$	3.72285	0.200142
$347$	0.481718	0.0258600	0.0129300	−	0.999916i	$-0.495884\pi$
$347$	0.481718	0.0258600	0.0129300	+	0.999916i	$0.495884\pi$
$348$	0	0
$349$	17.7168	0.948360	0.474180	−	0.880428i	$-0.342745\pi$
$349$	17.7168	0.948360	0.474180	+	0.880428i	$0.342745\pi$
$350$	−1.40129	−0.0749019
$351$	0	0
$352$	7.99861	0.426327
$353$	5.62875	0.299588	0.149794	−	0.988717i	$-0.452139\pi$
$353$	5.62875	0.299588	0.149794	+	0.988717i	$0.452139\pi$
$354$	0	0
$355$	−22.4549	−1.19178
$356$	15.1314	0.801961
$357$	0	0
$358$	−3.21901	−0.170130
$359$	−19.9091	−1.05076	−0.525380	−	0.850868i	$-0.676077\pi$
$359$	−19.9091	−1.05076	−0.525380	+	0.850868i	$0.676077\pi$
$360$	0	0
$361$	−8.48016	−0.446324
$362$	5.56372	0.292423
$363$	0	0
$364$	−10.9833	−0.575679
$365$	−14.0977	−0.737910
$366$	0	0
$367$	−16.3552	−0.853733	−0.426866	−	0.904315i	$-0.640383\pi$
$367$	−16.3552	−0.853733	−0.426866	+	0.904315i	$0.640383\pi$
$368$	18.6380	0.971571
$369$	0	0
$370$	−2.38304	−0.123888
$371$	−25.4757	−1.32263
$372$	0	0
$373$	−33.6500	−1.74233	−0.871165	−	0.490990i	$-0.836635\pi$
$373$	−33.6500	−1.74233	−0.871165	+	0.490990i	$0.836635\pi$
$374$	−1.81855	−0.0940350
$375$	0	0
$376$	−10.2881	−0.530567
$377$	−16.8281	−0.866692
$378$	0	0
$379$	−10.7002	−0.549632	−0.274816	−	0.961497i	$-0.588617\pi$
$379$	−10.7002	−0.549632	−0.274816	+	0.961497i	$0.588617\pi$
$380$	−9.87713	−0.506686
$381$	0	0
$382$	−0.751798	−0.0384654
$383$	18.6494	0.952938	0.476469	−	0.879191i	$-0.341916\pi$
$383$	18.6494	0.952938	0.476469	+	0.879191i	$0.341916\pi$
$384$	0	0
$385$	−9.18890	−0.468310
$386$	−1.44644	−0.0736219
$387$	0	0
$388$	−19.2819	−0.978890
$389$	4.36216	0.221170	0.110585	−	0.993867i	$-0.464727\pi$
$389$	4.36216	0.221170	0.110585	+	0.993867i	$0.464727\pi$
$390$	0	0
$391$	−13.4832	−0.681875
$392$	−2.37865	−0.120140
$393$	0	0
$394$	4.86585	0.245138
$395$	12.0542	0.606514
$396$	0	0
$397$	10.5278	0.528377	0.264189	−	0.964471i	$-0.414896\pi$
$397$	10.5278	0.528377	0.264189	+	0.964471i	$0.414896\pi$
$398$	−2.23341	−0.111951
$399$	0	0
$400$	−9.08426	−0.454213
$401$	26.9461	1.34562	0.672811	−	0.739814i	$-0.265087\pi$
$401$	26.9461	1.34562	0.672811	+	0.739814i	$0.265087\pi$
$402$	0	0
$403$	−23.1363	−1.15250
$404$	14.7824	0.735454
$405$	0	0
$406$	3.55474	0.176419
$407$	15.8718	0.786738
$408$	0	0
$409$	4.05965	0.200737	0.100368	−	0.994950i	$-0.467998\pi$
$409$	4.05965	0.200737	0.100368	+	0.994950i	$0.467998\pi$
$410$	3.02217	0.149254
$411$	0	0
$412$	15.6757	0.772286
$413$	−29.0257	−1.42826
$414$	0	0
$415$	1.79414	0.0880710
$416$	7.78759	0.381818
$417$	0	0
$418$	−2.26127	−0.110602
$419$	7.93492	0.387646	0.193823	−	0.981037i	$-0.437911\pi$
$419$	7.93492	0.387646	0.193823	+	0.981037i	$0.437911\pi$
$420$	0	0
$421$	21.1009	1.02839	0.514197	−	0.857672i	$-0.328090\pi$
$421$	21.1009	1.02839	0.514197	+	0.857672i	$0.328090\pi$
$422$	3.41039	0.166015
$423$	0	0
$424$	11.9748	0.581547
$425$	6.57180	0.318779
$426$	0	0
$427$	−8.60662	−0.416504
$428$	−37.7122	−1.82289
$429$	0	0
$430$	−2.57890	−0.124366
$431$	24.7017	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$431$	24.7017	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$432$	0	0
$433$	12.6643	0.608608	0.304304	−	0.952575i	$-0.401576\pi$
$433$	12.6643	0.608608	0.304304	+	0.952575i	$0.401576\pi$
$434$	4.88728	0.234597
$435$	0	0
$436$	−26.0081	−1.24556
$437$	−16.7657	−0.802011
$438$	0	0
$439$	12.2475	0.584539	0.292270	−	0.956336i	$-0.405590\pi$
$439$	12.2475	0.584539	0.292270	+	0.956336i	$0.405590\pi$
$440$	4.31922	0.205911
$441$	0	0
$442$	−1.77057	−0.0842176
$443$	1.88527	0.0895718	0.0447859	−	0.998997i	$-0.485739\pi$
$443$	1.88527	0.0895718	0.0447859	+	0.998997i	$0.485739\pi$
$444$	0	0
$445$	12.3253	0.584277
$446$	−2.43201	−0.115159
$447$	0	0
$448$	13.9126	0.657308
$449$	40.4212	1.90759	0.953796	−	0.300454i	$-0.0971383\pi$
$449$	40.4212	1.90759	0.953796	+	0.300454i	$0.0971383\pi$
$450$	0	0
$451$	−20.1287	−0.947822
$452$	11.2587	0.529567
$453$	0	0
$454$	−5.45718	−0.256118
$455$	−8.94648	−0.419417
$456$	0	0
$457$	−1.70422	−0.0797200	−0.0398600	−	0.999205i	$-0.512691\pi$
$457$	−1.70422	−0.0797200	−0.0398600	+	0.999205i	$0.512691\pi$
$458$	−5.86602	−0.274101
$459$	0	0
$460$	15.7414	0.733944
$461$	−10.3204	−0.480669	−0.240335	−	0.970690i	$-0.577257\pi$
$461$	−10.3204	−0.480669	−0.240335	+	0.970690i	$0.577257\pi$
$462$	0	0
$463$	35.4560	1.64778	0.823889	−	0.566751i	$-0.191800\pi$
$463$	35.4560	1.64778	0.823889	+	0.566751i	$0.191800\pi$
$464$	23.0446	1.06982
$465$	0	0
$466$	3.24568	0.150353
$467$	−14.4318	−0.667826	−0.333913	−	0.942604i	$-0.608369\pi$
$467$	−14.4318	−0.667826	−0.333913	+	0.942604i	$0.608369\pi$
$468$	0	0
$469$	1.79836	0.0830407
$470$	−4.11931	−0.190010
$471$	0	0
$472$	13.6435	0.627991
$473$	17.1764	0.789771
$474$	0	0
$475$	8.17168	0.374943
$476$	−10.8808	−0.498721
$477$	0	0
$478$	3.88396	0.177648
$479$	−16.2831	−0.743995	−0.371998	−	0.928234i	$-0.621327\pi$
$479$	−16.2831	−0.743995	−0.371998	+	0.928234i	$0.621327\pi$
$480$	0	0
$481$	15.4531	0.704601
$482$	−6.77888	−0.308769
$483$	0	0
$484$	7.12822	0.324010
$485$	−15.7062	−0.713181
$486$	0	0
$487$	−24.9035	−1.12849	−0.564243	−	0.825609i	$-0.690832\pi$
$487$	−24.9035	−1.12849	−0.564243	+	0.825609i	$0.690832\pi$
$488$	4.04552	0.183132
$489$	0	0
$490$	−0.952401	−0.0430251
$491$	23.9612	1.08136	0.540678	−	0.841230i	$-0.318168\pi$
$491$	23.9612	1.08136	0.540678	+	0.841230i	$0.318168\pi$
$492$	0	0
$493$	−16.6711	−0.750830
$494$	−2.20162	−0.0990554
$495$	0	0
$496$	31.6832	1.42262
$497$	−30.7587	−1.37972
$498$	0	0
$499$	7.82546	0.350316	0.175158	−	0.984540i	$-0.443956\pi$
$499$	7.82546	0.350316	0.175158	+	0.984540i	$0.443956\pi$
$500$	−22.8988	−1.02406
$501$	0	0
$502$	−1.65224	−0.0737429
$503$	7.01992	0.313003	0.156501	−	0.987678i	$-0.449978\pi$
$503$	7.01992	0.313003	0.156501	+	0.987678i	$0.449978\pi$
$504$	0	0
$505$	12.0411	0.535823
$506$	3.60383	0.160210
$507$	0	0
$508$	−25.3484	−1.12465
$509$	−22.1122	−0.980105	−0.490053	−	0.871693i	$-0.663022\pi$
$509$	−22.1122	−0.980105	−0.490053	+	0.871693i	$0.663022\pi$
$510$	0	0
$511$	−19.3111	−0.854273
$512$	−17.9773	−0.794490
$513$	0	0
$514$	0.364406	0.0160733
$515$	12.7687	0.562658
$516$	0	0
$517$	27.4360	1.20664
$518$	−3.26429	−0.143424
$519$	0	0
$520$	4.20527	0.184413
$521$	13.3065	0.582970	0.291485	−	0.956575i	$-0.405851\pi$
$521$	13.3065	0.582970	0.291485	+	0.956575i	$0.405851\pi$
$522$	0	0
$523$	−1.54366	−0.0674994	−0.0337497	−	0.999430i	$-0.510745\pi$
$523$	−1.54366	−0.0674994	−0.0337497	+	0.999430i	$0.510745\pi$
$524$	−31.1049	−1.35883
$525$	0	0
$526$	0.452946	0.0197494
$527$	−22.9205	−0.998433
$528$	0	0
$529$	3.71974	0.161728
$530$	4.79466	0.208267
$531$	0	0
$532$	−13.5297	−0.586587
$533$	−19.5976	−0.848868
$534$	0	0
$535$	−30.7187	−1.32808
$536$	−0.845316	−0.0365121
$537$	0	0
$538$	−1.48183	−0.0638863
$539$	6.34332	0.273226
$540$	0	0
$541$	2.14727	0.0923183	0.0461592	−	0.998934i	$-0.485302\pi$
$541$	2.14727	0.0923183	0.0461592	+	0.998934i	$0.485302\pi$
$542$	0.878489	0.0377343
$543$	0	0
$544$	7.71494	0.330775
$545$	−21.1851	−0.907469
$546$	0	0
$547$	16.3032	0.697075	0.348538	−	0.937295i	$-0.386678\pi$
$547$	16.3032	0.697075	0.348538	+	0.937295i	$0.386678\pi$
$548$	29.5601	1.26275
$549$	0	0
$550$	−1.75653	−0.0748986
$551$	−20.7297	−0.883113
$552$	0	0
$553$	16.5119	0.702157
$554$	−7.17002	−0.304625
$555$	0	0
$556$	3.06011	0.129777
$557$	6.87933	0.291486	0.145743	−	0.989322i	$-0.453443\pi$
$557$	6.87933	0.291486	0.145743	+	0.989322i	$0.453443\pi$
$558$	0	0
$559$	16.7232	0.707318
$560$	12.2514	0.517718
$561$	0	0
$562$	5.62471	0.237264
$563$	27.3957	1.15459	0.577295	−	0.816536i	$-0.304108\pi$
$563$	27.3957	1.15459	0.577295	+	0.816536i	$0.304108\pi$
$564$	0	0
$565$	9.17088	0.385822
$566$	−5.69528	−0.239390
$567$	0	0
$568$	14.4580	0.606646
$569$	−3.18604	−0.133566	−0.0667829	−	0.997768i	$-0.521274\pi$
$569$	−3.18604	−0.133566	−0.0667829	+	0.997768i	$0.521274\pi$
$570$	0	0
$571$	−12.4627	−0.521547	−0.260773	−	0.965400i	$-0.583978\pi$
$571$	−12.4627	−0.521547	−0.260773	+	0.965400i	$0.583978\pi$
$572$	−13.7676	−0.575654
$573$	0	0
$574$	4.13977	0.172791
$575$	−13.0234	−0.543112
$576$	0	0
$577$	−34.9606	−1.45543	−0.727714	−	0.685881i	$-0.759417\pi$
$577$	−34.9606	−1.45543	−0.727714	+	0.685881i	$0.759417\pi$
$578$	2.62861	0.109336
$579$	0	0
$580$	19.4632	0.808164
$581$	2.45762	0.101959
$582$	0	0
$583$	−31.9341	−1.32258
$584$	9.07713	0.375614
$585$	0	0
$586$	−3.89796	−0.161023
$587$	34.7712	1.43516	0.717580	−	0.696476i	$-0.245250\pi$
$587$	34.7712	1.43516	0.717580	+	0.696476i	$0.245250\pi$
$588$	0	0
$589$	−28.5004	−1.17434
$590$	5.46279	0.224900
$591$	0	0
$592$	−21.1617	−0.869741
$593$	20.7540	0.852265	0.426133	−	0.904661i	$-0.359876\pi$
$593$	20.7540	0.852265	0.426133	+	0.904661i	$0.359876\pi$
$594$	0	0
$595$	−8.86302	−0.363348
$596$	−1.93354	−0.0792008
$597$	0	0
$598$	3.50875	0.143484
$599$	28.4666	1.16311	0.581557	−	0.813506i	$-0.302444\pi$
$599$	28.4666	1.16311	0.581557	+	0.813506i	$0.302444\pi$
$600$	0	0
$601$	−15.0455	−0.613719	−0.306860	−	0.951755i	$-0.599278\pi$
$601$	−15.0455	−0.613719	−0.306860	+	0.951755i	$0.599278\pi$
$602$	−3.53259	−0.143978
$603$	0	0
$604$	35.5494	1.44649
$605$	5.80633	0.236061
$606$	0	0
$607$	−8.14452	−0.330576	−0.165288	−	0.986245i	$-0.552855\pi$
$607$	−8.14452	−0.330576	−0.165288	+	0.986245i	$0.552855\pi$
$608$	9.59312	0.389052
$609$	0	0
$610$	1.61981	0.0655842
$611$	26.7122	1.08066
$612$	0	0
$613$	28.9347	1.16866	0.584331	−	0.811515i	$-0.301357\pi$
$613$	28.9347	1.16866	0.584331	+	0.811515i	$0.301357\pi$
$614$	4.20382	0.169652
$615$	0	0
$616$	5.91647	0.238381
$617$	23.9426	0.963895	0.481947	−	0.876200i	$-0.339930\pi$
$617$	23.9426	0.963895	0.481947	+	0.876200i	$0.339930\pi$
$618$	0	0
$619$	33.9702	1.36538	0.682689	−	0.730709i	$-0.260810\pi$
$619$	33.9702	1.36538	0.682689	+	0.730709i	$0.260810\pi$
$620$	26.7592	1.07467
$621$	0	0
$622$	−3.51191	−0.140815
$623$	16.8833	0.676414
$624$	0	0
$625$	−6.05506	−0.242202
$626$	0.0820930	0.00328110
$627$	0	0
$628$	−15.1196	−0.603338
$629$	15.3090	0.610408
$630$	0	0
$631$	3.50993	0.139728	0.0698640	−	0.997557i	$-0.477743\pi$
$631$	3.50993	0.139728	0.0698640	+	0.997557i	$0.477743\pi$
$632$	−7.76136	−0.308730
$633$	0	0
$634$	3.57503	0.141983
$635$	−20.6477	−0.819378
$636$	0	0
$637$	6.17597	0.244701
$638$	4.45590	0.176411
$639$	0	0
$640$	−11.9350	−0.471774
$641$	3.78037	0.149316	0.0746578	−	0.997209i	$-0.476214\pi$
$641$	3.78037	0.149316	0.0746578	+	0.997209i	$0.476214\pi$
$642$	0	0
$643$	34.4632	1.35910	0.679549	−	0.733630i	$-0.262176\pi$
$643$	34.4632	1.35910	0.679549	+	0.733630i	$0.262176\pi$
$644$	21.5625	0.849683
$645$	0	0
$646$	−2.18108	−0.0858133
$647$	−2.44801	−0.0962411	−0.0481205	−	0.998842i	$-0.515323\pi$
$647$	−2.44801	−0.0962411	−0.0481205	+	0.998842i	$0.515323\pi$
$648$	0	0
$649$	−36.3841	−1.42820
$650$	−1.71019	−0.0670791
$651$	0	0
$652$	−4.49057	−0.175864
$653$	30.8194	1.20606	0.603029	−	0.797719i	$-0.293960\pi$
$653$	30.8194	1.20606	0.603029	+	0.797719i	$0.293960\pi$
$654$	0	0
$655$	−25.3367	−0.989987
$656$	26.8373	1.04782
$657$	0	0
$658$	−5.64264	−0.219973
$659$	12.9266	0.503550	0.251775	−	0.967786i	$-0.418986\pi$
$659$	12.9266	0.503550	0.251775	+	0.967786i	$0.418986\pi$
$660$	0	0
$661$	−26.6915	−1.03818	−0.519089	−	0.854720i	$-0.673729\pi$
$661$	−26.6915	−1.03818	−0.519089	+	0.854720i	$0.673729\pi$
$662$	−4.67972	−0.181882
$663$	0	0
$664$	−1.15520	−0.0448303
$665$	−11.0207	−0.427364
$666$	0	0
$667$	33.0372	1.27921
$668$	29.4459	1.13930
$669$	0	0
$670$	−0.338461	−0.0130759
$671$	−10.7885	−0.416485
$672$	0	0
$673$	47.1551	1.81769	0.908847	−	0.417130i	$-0.136964\pi$
$673$	47.1551	1.81769	0.908847	+	0.417130i	$0.136964\pi$
$674$	5.63768	0.217155
$675$	0	0
$676$	11.7316	0.451214
$677$	−48.4902	−1.86363	−0.931816	−	0.362932i	$-0.881776\pi$
$677$	−48.4902	−1.86363	−0.931816	+	0.362932i	$0.881776\pi$
$678$	0	0
$679$	−21.5143	−0.825645
$680$	4.16604	0.159760
$681$	0	0
$682$	6.12625	0.234586
$683$	48.7203	1.86423	0.932115	−	0.362163i	$-0.117962\pi$
$683$	48.7203	1.86423	0.932115	+	0.362163i	$0.117962\pi$
$684$	0	0
$685$	24.0784	0.919987
$686$	−5.19790	−0.198457
$687$	0	0
$688$	−22.9010	−0.873094
$689$	−31.0916	−1.18450
$690$	0	0
$691$	33.8251	1.28677	0.643384	−	0.765544i	$-0.277530\pi$
$691$	33.8251	1.28677	0.643384	+	0.765544i	$0.277530\pi$
$692$	27.9215	1.06142
$693$	0	0
$694$	−0.124189	−0.00471413
$695$	2.49263	0.0945508
$696$	0	0
$697$	−19.4148	−0.735388
$698$	−4.56746	−0.172881
$699$	0	0
$700$	−10.5097	−0.397229
$701$	−4.73714	−0.178919	−0.0894596	−	0.995990i	$-0.528514\pi$
$701$	−4.73714	−0.178919	−0.0894596	+	0.995990i	$0.528514\pi$
$702$	0	0
$703$	19.0359	0.717952
$704$	17.4396	0.657279
$705$	0	0
$706$	−1.45111	−0.0546134
$707$	16.4939	0.620319
$708$	0	0
$709$	−7.74555	−0.290890	−0.145445	−	0.989366i	$-0.546461\pi$
$709$	−7.74555	−0.290890	−0.145445	+	0.989366i	$0.546461\pi$
$710$	5.78895	0.217255
$711$	0	0
$712$	−7.93593	−0.297412
$713$	45.4217	1.70105
$714$	0	0
$715$	−11.2145	−0.419399
$716$	−24.1427	−0.902256
$717$	0	0
$718$	5.13263	0.191548
$719$	52.8263	1.97009	0.985045	−	0.172296i	$-0.0551187\pi$
$719$	52.8263	1.97009	0.985045	+	0.172296i	$0.0551187\pi$
$720$	0	0
$721$	17.4906	0.651385
$722$	2.18621	0.0813625
$723$	0	0
$724$	41.7281	1.55081
$725$	−16.1025	−0.598033
$726$	0	0
$727$	−7.27171	−0.269693	−0.134846	−	0.990867i	$-0.543054\pi$
$727$	−7.27171	−0.269693	−0.134846	+	0.990867i	$0.543054\pi$
$728$	5.76038	0.213494
$729$	0	0
$730$	3.63445	0.134517
$731$	16.5672	0.612762
$732$	0	0
$733$	−10.6200	−0.392260	−0.196130	−	0.980578i	$-0.562837\pi$
$733$	−10.6200	−0.392260	−0.196130	+	0.980578i	$0.562837\pi$
$734$	4.21642	0.155631
$735$	0	0
$736$	−15.2887	−0.563550
$737$	2.25427	0.0830371
$738$	0	0
$739$	12.0392	0.442870	0.221435	−	0.975175i	$-0.428926\pi$
$739$	12.0392	0.442870	0.221435	+	0.975175i	$0.428926\pi$
$740$	−17.8729	−0.657020
$741$	0	0
$742$	6.56774	0.241109
$743$	3.48680	0.127918	0.0639591	−	0.997953i	$-0.479627\pi$
$743$	3.48680	0.127918	0.0639591	+	0.997953i	$0.479627\pi$
$744$	0	0
$745$	−1.57497	−0.0577026
$746$	8.67509	0.317618
$747$	0	0
$748$	−13.6392	−0.498699
$749$	−42.0785	−1.53751
$750$	0	0
$751$	45.7147	1.66815	0.834077	−	0.551649i	$-0.186001\pi$
$751$	45.7147	1.66815	0.834077	+	0.551649i	$0.186001\pi$
$752$	−36.5801	−1.33394
$753$	0	0
$754$	4.33835	0.157993
$755$	28.9570	1.05385
$756$	0	0
$757$	−30.9290	−1.12413	−0.562066	−	0.827092i	$-0.689993\pi$
$757$	−30.9290	−1.12413	−0.562066	+	0.827092i	$0.689993\pi$
$758$	2.75855	0.100195
$759$	0	0
$760$	5.18025	0.187907
$761$	24.1898	0.876880	0.438440	−	0.898761i	$-0.355531\pi$
$761$	24.1898	0.876880	0.438440	+	0.898761i	$0.355531\pi$
$762$	0	0
$763$	−29.0193	−1.05057
$764$	−5.63852	−0.203994
$765$	0	0
$766$	−4.80787	−0.173716
$767$	−35.4242	−1.27909
$768$	0	0
$769$	−24.9222	−0.898719	−0.449359	−	0.893351i	$-0.648348\pi$
$769$	−24.9222	−0.898719	−0.449359	+	0.893351i	$0.648348\pi$
$770$	2.36893	0.0853704
$771$	0	0
$772$	−10.8484	−0.390441
$773$	41.1988	1.48182	0.740909	−	0.671605i	$-0.234395\pi$
$773$	41.1988	1.48182	0.740909	+	0.671605i	$0.234395\pi$
$774$	0	0
$775$	−22.1388	−0.795249
$776$	10.1128	0.363027
$777$	0	0
$778$	−1.12458	−0.0403182
$779$	−24.1413	−0.864952
$780$	0	0
$781$	−38.5564	−1.37966
$782$	3.47602	0.124302
$783$	0	0
$784$	−8.45746	−0.302052
$785$	−12.3158	−0.439569
$786$	0	0
$787$	−0.965166	−0.0344045	−0.0172022	−	0.999852i	$-0.505476\pi$
$787$	−0.965166	−0.0344045	−0.0172022	+	0.999852i	$0.505476\pi$
$788$	36.4941	1.30005
$789$	0	0
$790$	−3.10762	−0.110564
$791$	12.5623	0.446663
$792$	0	0
$793$	−10.5039	−0.373004
$794$	−2.71412	−0.0963204
$795$	0	0
$796$	−16.7507	−0.593712
$797$	−8.87385	−0.314328	−0.157164	−	0.987573i	$-0.550235\pi$
$797$	−8.87385	−0.314328	−0.157164	+	0.987573i	$0.550235\pi$
$798$	0	0
$799$	26.4630	0.936194
$800$	7.45182	0.263462
$801$	0	0
$802$	−6.94679	−0.245300
$803$	−24.2067	−0.854236
$804$	0	0
$805$	17.5639	0.619046
$806$	5.96463	0.210095
$807$	0	0
$808$	−7.75293	−0.272747
$809$	39.5607	1.39088	0.695440	−	0.718584i	$-0.255210\pi$
$809$	39.5607	1.39088	0.695440	+	0.718584i	$0.255210\pi$
$810$	0	0
$811$	33.3474	1.17099	0.585493	−	0.810677i	$-0.300901\pi$
$811$	33.3474	1.17099	0.585493	+	0.810677i	$0.300901\pi$
$812$	26.6607	0.935606
$813$	0	0
$814$	−4.09182	−0.143418
$815$	−3.65782	−0.128128
$816$	0	0
$817$	20.6005	0.720720
$818$	−1.04659	−0.0365932
$819$	0	0
$820$	22.6664	0.791544
$821$	3.98485	0.139072	0.0695361	−	0.997579i	$-0.477848\pi$
$821$	3.98485	0.139072	0.0695361	+	0.997579i	$0.477848\pi$
$822$	0	0
$823$	−28.4304	−0.991023	−0.495511	−	0.868601i	$-0.665019\pi$
$823$	−28.4304	−0.991023	−0.495511	+	0.868601i	$0.665019\pi$
$824$	−8.22142	−0.286407
$825$	0	0
$826$	7.48294	0.260365
$827$	25.8975	0.900543	0.450271	−	0.892892i	$-0.351327\pi$
$827$	25.8975	0.900543	0.450271	+	0.892892i	$0.351327\pi$
$828$	0	0
$829$	−27.6407	−0.960002	−0.480001	−	0.877268i	$-0.659364\pi$
$829$	−27.6407	−0.960002	−0.480001	+	0.877268i	$0.659364\pi$
$830$	−0.462537	−0.0160549
$831$	0	0
$832$	16.9795	0.588658
$833$	6.11836	0.211988
$834$	0	0
$835$	23.9853	0.830047
$836$	−16.9596	−0.586561
$837$	0	0
$838$	−2.04565	−0.0706658
$839$	45.9366	1.58591	0.792955	−	0.609281i	$-0.208542\pi$
$839$	45.9366	1.58591	0.792955	+	0.609281i	$0.208542\pi$
$840$	0	0
$841$	11.8484	0.408566
$842$	−5.43988	−0.187471
$843$	0	0
$844$	25.5780	0.880432
$845$	9.55601	0.328737
$846$	0	0
$847$	7.95352	0.273286
$848$	42.5773	1.46211
$849$	0	0
$850$	−1.69423	−0.0581117
$851$	−30.3378	−1.03997
$852$	0	0
$853$	−38.2424	−1.30939	−0.654697	−	0.755891i	$-0.727204\pi$
$853$	−38.2424	−1.30939	−0.654697	+	0.755891i	$0.727204\pi$
$854$	2.21882	0.0759264
$855$	0	0
$856$	19.7789	0.676028
$857$	9.27507	0.316830	0.158415	−	0.987373i	$-0.449362\pi$
$857$	9.27507	0.316830	0.158415	+	0.987373i	$0.449362\pi$
$858$	0	0
$859$	−40.5538	−1.38368	−0.691839	−	0.722052i	$-0.743199\pi$
$859$	−40.5538	−1.38368	−0.691839	+	0.722052i	$0.743199\pi$
$860$	−19.3419	−0.659553
$861$	0	0
$862$	−6.36819	−0.216902
$863$	−38.8883	−1.32377	−0.661886	−	0.749604i	$-0.730244\pi$
$863$	−38.8883	−1.32377	−0.661886	+	0.749604i	$0.730244\pi$
$864$	0	0
$865$	22.7437	0.773307
$866$	−3.26490	−0.110946
$867$	0	0
$868$	36.6548	1.24414
$869$	20.6978	0.702126
$870$	0	0
$871$	2.19480	0.0743679
$872$	13.6405	0.461924
$873$	0	0
$874$	4.32225	0.146202
$875$	−25.5500	−0.863748
$876$	0	0
$877$	−5.87241	−0.198297	−0.0991485	−	0.995073i	$-0.531612\pi$
$877$	−5.87241	−0.198297	−0.0991485	+	0.995073i	$0.531612\pi$
$878$	−3.15744	−0.106558
$879$	0	0
$880$	15.3573	0.517695
$881$	−43.2046	−1.45560	−0.727800	−	0.685790i	$-0.759457\pi$
$881$	−43.2046	−1.45560	−0.727800	+	0.685790i	$0.759457\pi$
$882$	0	0
$883$	26.3709	0.887453	0.443727	−	0.896162i	$-0.353656\pi$
$883$	26.3709	0.887453	0.443727	+	0.896162i	$0.353656\pi$
$884$	−13.2794	−0.446634
$885$	0	0
$886$	−0.486029	−0.0163285
$887$	15.3419	0.515129	0.257565	−	0.966261i	$-0.417080\pi$
$887$	15.3419	0.515129	0.257565	+	0.966261i	$0.417080\pi$
$888$	0	0
$889$	−28.2832	−0.948588
$890$	−3.17752	−0.106511
$891$	0	0
$892$	−18.2402	−0.610727
$893$	32.9054	1.10114
$894$	0	0
$895$	−19.6656	−0.657349
$896$	−16.3486	−0.546170
$897$	0	0
$898$	−10.4207	−0.347744
$899$	56.1610	1.87307
$900$	0	0
$901$	−30.8016	−1.02615
$902$	5.18924	0.172783
$903$	0	0
$904$	−5.90486	−0.196393
$905$	33.9899	1.12986
$906$	0	0
$907$	−27.1781	−0.902433	−0.451217	−	0.892414i	$-0.649010\pi$
$907$	−27.1781	−0.902433	−0.451217	+	0.892414i	$0.649010\pi$
$908$	−40.9290	−1.35828
$909$	0	0
$910$	2.30644	0.0764576
$911$	−32.6908	−1.08309	−0.541547	−	0.840671i	$-0.682161\pi$
$911$	−32.6908	−1.08309	−0.541547	+	0.840671i	$0.682161\pi$
$912$	0	0
$913$	3.08065	0.101955
$914$	0.439354	0.0145325
$915$	0	0
$916$	−43.9954	−1.45365
$917$	−34.7062	−1.14610
$918$	0	0
$919$	−36.4818	−1.20342	−0.601712	−	0.798713i	$-0.705514\pi$
$919$	−36.4818	−1.20342	−0.601712	+	0.798713i	$0.705514\pi$
$920$	−8.25585	−0.272187
$921$	0	0
$922$	2.66064	0.0876235
$923$	−37.5392	−1.23562
$924$	0	0
$925$	14.7868	0.486188
$926$	−9.14067	−0.300381
$927$	0	0
$928$	−18.9035	−0.620539
$929$	−12.6412	−0.414744	−0.207372	−	0.978262i	$-0.566491\pi$
$929$	−12.6412	−0.414744	−0.207372	+	0.978262i	$0.566491\pi$
$930$	0	0
$931$	7.60785	0.249337
$932$	24.3427	0.797372
$933$	0	0
$934$	3.72058	0.121741
$935$	−11.1099	−0.363332
$936$	0	0
$937$	15.0849	0.492801	0.246401	−	0.969168i	$-0.420752\pi$
$937$	15.0849	0.492801	0.246401	+	0.969168i	$0.420752\pi$
$938$	−0.463625	−0.0151379
$939$	0	0
$940$	−30.8950	−1.00768
$941$	−33.2657	−1.08443	−0.542215	−	0.840240i	$-0.682414\pi$
$941$	−33.2657	−1.08443	−0.542215	+	0.840240i	$0.682414\pi$
$942$	0	0
$943$	38.4744	1.25290
$944$	48.5104	1.57888
$945$	0	0
$946$	−4.42814	−0.143971
$947$	20.8665	0.678070	0.339035	−	0.940774i	$-0.389900\pi$
$947$	20.8665	0.678070	0.339035	+	0.940774i	$0.389900\pi$
$948$	0	0
$949$	−23.5681	−0.765052
$950$	−2.10669	−0.0683500
$951$	0	0
$952$	5.70664	0.184953
$953$	−58.7016	−1.90153	−0.950766	−	0.309910i	$-0.899701\pi$
$953$	−58.7016	−1.90153	−0.950766	+	0.309910i	$0.899701\pi$
$954$	0	0
$955$	−4.59289	−0.148622
$956$	29.1299	0.942127
$957$	0	0
$958$	4.19785	0.135626
$959$	32.9826	1.06506
$960$	0	0
$961$	46.2136	1.49076
$962$	−3.98387	−0.128445
$963$	0	0
$964$	−50.8418	−1.63751
$965$	−8.83660	−0.284460
$966$	0	0
$967$	30.5730	0.983161	0.491581	−	0.870832i	$-0.336419\pi$
$967$	30.5730	0.983161	0.491581	+	0.870832i	$0.336419\pi$
$968$	−3.73853	−0.120161
$969$	0	0
$970$	4.04911	0.130009
$971$	−38.9576	−1.25021	−0.625104	−	0.780541i	$-0.714943\pi$
$971$	−38.9576	−1.25021	−0.625104	+	0.780541i	$0.714943\pi$
$972$	0	0
$973$	3.41441	0.109461
$974$	6.42022	0.205717
$975$	0	0
$976$	14.3842	0.460426
$977$	−22.1280	−0.707938	−0.353969	−	0.935257i	$-0.615168\pi$
$977$	−22.1280	−0.707938	−0.353969	+	0.935257i	$0.615168\pi$
$978$	0	0
$979$	21.1634	0.676384
$980$	−7.14304	−0.228176
$981$	0	0
$982$	−6.17729	−0.197125
$983$	−8.91769	−0.284430	−0.142215	−	0.989836i	$-0.545422\pi$
$983$	−8.91769	−0.284430	−0.142215	+	0.989836i	$0.545422\pi$
$984$	0	0
$985$	29.7265	0.947164
$986$	4.29788	0.136872
$987$	0	0
$988$	−16.5122	−0.525323
$989$	−32.8314	−1.04398
$990$	0	0
$991$	−13.7306	−0.436167	−0.218084	−	0.975930i	$-0.569981\pi$
$991$	−13.7306	−0.436167	−0.218084	+	0.975930i	$0.569981\pi$
$992$	−25.9898	−0.825176
$993$	0	0
$994$	7.92971	0.251515
$995$	−13.6444	−0.432555
$996$	0	0
$997$	14.3691	0.455075	0.227537	−	0.973769i	$-0.426933\pi$
$997$	14.3691	0.455075	0.227537	+	0.973769i	$0.426933\pi$
$998$	−2.01743	−0.0638607
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.h.1.5	yes	12
3.2	odd	2		1341.2.a.g.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1341.2.a.g.1.8	✓	12	3.2	odd	2
1341.2.a.h.1.5	yes	12	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-0.257804 q^{2} -1.93354 q^{4} -1.57497 q^{5} -2.15740 q^{7} +1.01408 q^{8} +O(q^{10})\)	\(q-0.257804 q^{2} -1.93354 q^{4} -1.57497 q^{5} -2.15740 q^{7} +1.01408 q^{8} +0.406034 q^{10} -2.70433 q^{11} -2.63298 q^{13} +0.556186 q^{14} +3.60564 q^{16} -2.60842 q^{17} -3.24343 q^{19} +3.04527 q^{20} +0.697185 q^{22} +5.16911 q^{23} -2.51946 q^{25} +0.678792 q^{26} +4.17142 q^{28} +6.39128 q^{29} +8.78713 q^{31} -2.95771 q^{32} +0.672460 q^{34} +3.39785 q^{35} -5.86905 q^{37} +0.836168 q^{38} -1.59715 q^{40} +7.44313 q^{41} -6.35145 q^{43} +5.22892 q^{44} -1.33262 q^{46} -10.1452 q^{47} -2.34562 q^{49} +0.649525 q^{50} +5.09097 q^{52} +11.8085 q^{53} +4.25924 q^{55} -2.18778 q^{56} -1.64769 q^{58} +13.4540 q^{59} +3.98935 q^{61} -2.26535 q^{62} -6.44877 q^{64} +4.14688 q^{65} -0.833579 q^{67} +5.04347 q^{68} -0.875979 q^{70} +14.2573 q^{71} +8.95110 q^{73} +1.51306 q^{74} +6.27129 q^{76} +5.83432 q^{77} -7.65360 q^{79} -5.67879 q^{80} -1.91887 q^{82} -1.13916 q^{83} +4.10819 q^{85} +1.63743 q^{86} -2.74240 q^{88} -7.82574 q^{89} +5.68040 q^{91} -9.99467 q^{92} +2.61548 q^{94} +5.10832 q^{95} +9.97234 q^{97} +0.604709 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) 12 * q + 3 * q^2 + 9 * q^4 + 8 * q^5 - 2 * q^7 + 9 * q^8	\( 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8} + 22 q^{11} - 2 q^{13} + 12 q^{14} + 11 q^{16} + 8 q^{17} - 6 q^{19} + 24 q^{20} + 4 q^{22} + 12 q^{23} + 8 q^{25} + 26 q^{26} - 20 q^{28} + 16 q^{29} - 2 q^{31} + 21 q^{32} - 6 q^{34} + 32 q^{35} - 4 q^{37} + 15 q^{38} + 6 q^{40} + 24 q^{41} + 12 q^{43} + 37 q^{44} - 22 q^{46} + 26 q^{47} + 24 q^{49} - 5 q^{50} - 10 q^{52} + 20 q^{53} + 6 q^{55} - 13 q^{56} + 10 q^{58} + 72 q^{59} - 8 q^{61} - 2 q^{62} + q^{64} - 4 q^{65} - 14 q^{67} - 14 q^{68} + 18 q^{70} + 38 q^{71} - 27 q^{74} - 2 q^{76} + 6 q^{77} + 10 q^{79} + 56 q^{80} + 6 q^{82} + 42 q^{83} - 32 q^{85} - 14 q^{86} + 22 q^{88} + 44 q^{89} - 50 q^{92} + 2 q^{94} + 4 q^{95} - 22 q^{97} + 20 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 + 9 * q^4 + 8 * q^5 - 2 * q^7 + 9 * q^8 + 22 * q^11 - 2 * q^13 + 12 * q^14 + 11 * q^16 + 8 * q^17 - 6 * q^19 + 24 * q^20 + 4 * q^22 + 12 * q^23 + 8 * q^25 + 26 * q^26 - 20 * q^28 + 16 * q^29 - 2 * q^31 + 21 * q^32 - 6 * q^34 + 32 * q^35 - 4 * q^37 + 15 * q^38 + 6 * q^40 + 24 * q^41 + 12 * q^43 + 37 * q^44 - 22 * q^46 + 26 * q^47 + 24 * q^49 - 5 * q^50 - 10 * q^52 + 20 * q^53 + 6 * q^55 - 13 * q^56 + 10 * q^58 + 72 * q^59 - 8 * q^61 - 2 * q^62 + q^64 - 4 * q^65 - 14 * q^67 - 14 * q^68 + 18 * q^70 + 38 * q^71 - 27 * q^74 - 2 * q^76 + 6 * q^77 + 10 * q^79 + 56 * q^80 + 6 * q^82 + 42 * q^83 - 32 * q^85 - 14 * q^86 + 22 * q^88 + 44 * q^89 - 50 * q^92 + 2 * q^94 + 4 * q^95 - 22 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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