LMFDB - Embedded newform 2169.2.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$1$
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} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.0822506$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0822506 q^{2} -1.99323 q^{4} -4.31963 q^{5} +0.690569 q^{7} +0.328446 q^{8} +O(q^{10})$	\(q-0.0822506 q^{2} -1.99323 q^{4} -4.31963 q^{5} +0.690569 q^{7} +0.328446 q^{8} +0.355292 q^{10} +2.95833 q^{11} -1.93470 q^{13} -0.0567997 q^{14} +3.95945 q^{16} +2.07477 q^{17} +3.74689 q^{19} +8.61004 q^{20} -0.243324 q^{22} -4.34832 q^{23} +13.6592 q^{25} +0.159131 q^{26} -1.37647 q^{28} +8.10772 q^{29} -2.80197 q^{31} -0.982559 q^{32} -0.170651 q^{34} -2.98300 q^{35} -9.72312 q^{37} -0.308183 q^{38} -1.41876 q^{40} +4.09401 q^{41} +3.02779 q^{43} -5.89665 q^{44} +0.357652 q^{46} -6.71240 q^{47} -6.52312 q^{49} -1.12348 q^{50} +3.85632 q^{52} +0.0484656 q^{53} -12.7789 q^{55} +0.226814 q^{56} -0.666864 q^{58} -4.50676 q^{59} -9.62232 q^{61} +0.230464 q^{62} -7.83809 q^{64} +8.35721 q^{65} -0.964160 q^{67} -4.13551 q^{68} +0.245353 q^{70} -7.76289 q^{71} +16.4250 q^{73} +0.799732 q^{74} -7.46842 q^{76} +2.04293 q^{77} -6.83310 q^{79} -17.1034 q^{80} -0.336735 q^{82} +9.15477 q^{83} -8.96225 q^{85} -0.249037 q^{86} +0.971651 q^{88} +2.36597 q^{89} -1.33605 q^{91} +8.66723 q^{92} +0.552099 q^{94} -16.1852 q^{95} -5.25335 q^{97} +0.536530 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * 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 $

$2$	−0.0822506	−0.0581599	−0.0290800	−	0.999577i	$-0.509258\pi$
$2$	−0.0822506	−0.0581599	−0.0290800	+	0.999577i	$0.509258\pi$
$3$	0	0
$3$	0	0
$4$	−1.99323	−0.996617
$5$	−4.31963	−1.93180	−0.965899	−	0.258920i	$-0.916633\pi$
$5$	−4.31963	−1.93180	−0.965899	+	0.258920i	$0.916633\pi$
$6$	0	0
$7$	0.690569	0.261010	0.130505	−	0.991448i	$-0.458340\pi$
$7$	0.690569	0.261010	0.130505	+	0.991448i	$0.458340\pi$
$8$	0.328446	0.116123
$9$	0	0
$10$	0.355292	0.112353
$11$	2.95833	0.891970	0.445985	−	0.895040i	$-0.352853\pi$
$11$	2.95833	0.891970	0.445985	+	0.895040i	$0.352853\pi$
$12$	0	0
$13$	−1.93470	−0.536591	−0.268295	−	0.963337i	$-0.586460\pi$
$13$	−1.93470	−0.536591	−0.268295	+	0.963337i	$0.586460\pi$
$14$	−0.0567997	−0.0151803
$15$	0	0
$16$	3.95945	0.989864
$17$	2.07477	0.503206	0.251603	−	0.967830i	$-0.419042\pi$
$17$	2.07477	0.503206	0.251603	+	0.967830i	$0.419042\pi$
$18$	0	0
$19$	3.74689	0.859595	0.429797	−	0.902925i	$-0.358585\pi$
$19$	3.74689	0.859595	0.429797	+	0.902925i	$0.358585\pi$
$20$	8.61004	1.92526
$21$	0	0
$22$	−0.243324	−0.0518769
$23$	−4.34832	−0.906688	−0.453344	−	0.891336i	$-0.649769\pi$
$23$	−4.34832	−0.906688	−0.453344	+	0.891336i	$0.649769\pi$
$24$	0	0
$25$	13.6592	2.73184
$26$	0.159131	0.0312081
$27$	0	0
$28$	−1.37647	−0.260127
$29$	8.10772	1.50557	0.752783	−	0.658269i	$-0.228711\pi$
$29$	8.10772	1.50557	0.752783	+	0.658269i	$0.228711\pi$
$30$	0	0
$31$	−2.80197	−0.503249	−0.251624	−	0.967825i	$-0.580965\pi$
$31$	−2.80197	−0.503249	−0.251624	+	0.967825i	$0.580965\pi$
$32$	−0.982559	−0.173694
$33$	0	0
$34$	−0.170651	−0.0292664
$35$	−2.98300	−0.504219
$36$	0	0
$37$	−9.72312	−1.59847	−0.799236	−	0.601018i	$-0.794762\pi$
$37$	−9.72312	−1.59847	−0.799236	+	0.601018i	$0.794762\pi$
$38$	−0.308183	−0.0499940
$39$	0	0
$40$	−1.41876	−0.224326
$41$	4.09401	0.639378	0.319689	−	0.947523i	$-0.396422\pi$
$41$	4.09401	0.639378	0.319689	+	0.947523i	$0.396422\pi$
$42$	0	0
$43$	3.02779	0.461733	0.230867	−	0.972985i	$-0.425844\pi$
$43$	3.02779	0.461733	0.230867	+	0.972985i	$0.425844\pi$
$44$	−5.89665	−0.888953
$45$	0	0
$46$	0.357652	0.0527329
$47$	−6.71240	−0.979104	−0.489552	−	0.871974i	$-0.662840\pi$
$47$	−6.71240	−0.979104	−0.489552	+	0.871974i	$0.662840\pi$
$48$	0	0
$49$	−6.52312	−0.931874
$50$	−1.12348	−0.158884
$51$	0	0
$52$	3.85632	0.534776
$53$	0.0484656	0.00665726	0.00332863	−	0.999994i	$-0.498940\pi$
$53$	0.0484656	0.00665726	0.00332863	+	0.999994i	$0.498940\pi$
$54$	0	0
$55$	−12.7789	−1.72311
$56$	0.226814	0.0303093
$57$	0	0
$58$	−0.666864	−0.0875636
$59$	−4.50676	−0.586730	−0.293365	−	0.956000i	$-0.594775\pi$
$59$	−4.50676	−0.586730	−0.293365	+	0.956000i	$0.594775\pi$
$60$	0	0
$61$	−9.62232	−1.23201	−0.616006	−	0.787742i	$-0.711250\pi$
$61$	−9.62232	−1.23201	−0.616006	+	0.787742i	$0.711250\pi$
$62$	0.230464	0.0292689
$63$	0	0
$64$	−7.83809	−0.979762
$65$	8.35721	1.03658
$66$	0	0
$67$	−0.964160	−0.117791	−0.0588954	−	0.998264i	$-0.518758\pi$
$67$	−0.964160	−0.117791	−0.0588954	+	0.998264i	$0.518758\pi$
$68$	−4.13551	−0.501504
$69$	0	0
$70$	0.245353	0.0293254
$71$	−7.76289	−0.921286	−0.460643	−	0.887586i	$-0.652381\pi$
$71$	−7.76289	−0.921286	−0.460643	+	0.887586i	$0.652381\pi$
$72$	0	0
$73$	16.4250	1.92241	0.961203	−	0.275841i	$-0.0889563\pi$
$73$	16.4250	1.92241	0.961203	+	0.275841i	$0.0889563\pi$
$74$	0.799732	0.0929670
$75$	0	0
$76$	−7.46842	−0.856687
$77$	2.04293	0.232813
$78$	0	0
$79$	−6.83310	−0.768783	−0.384392	−	0.923170i	$-0.625589\pi$
$79$	−6.83310	−0.768783	−0.384392	+	0.923170i	$0.625589\pi$
$80$	−17.1034	−1.91222
$81$	0	0
$82$	−0.336735	−0.0371862
$83$	9.15477	1.00487	0.502433	−	0.864616i	$-0.332438\pi$
$83$	9.15477	1.00487	0.502433	+	0.864616i	$0.332438\pi$
$84$	0	0
$85$	−8.96225	−0.972092
$86$	−0.249037	−0.0268544
$87$	0	0
$88$	0.971651	0.103578
$89$	2.36597	0.250793	0.125396	−	0.992107i	$-0.459980\pi$
$89$	2.36597	0.250793	0.125396	+	0.992107i	$0.459980\pi$
$90$	0	0
$91$	−1.33605	−0.140056
$92$	8.66723	0.903621
$93$	0	0
$94$	0.552099	0.0569447
$95$	−16.1852	−1.66056
$96$	0	0
$97$	−5.25335	−0.533396	−0.266698	−	0.963780i	$-0.585933\pi$
$97$	−5.25335	−0.533396	−0.266698	+	0.963780i	$0.585933\pi$
$98$	0.536530	0.0541977
$99$	0	0
$100$	−27.2260	−2.72260
$101$	−9.93584	−0.988653	−0.494327	−	0.869276i	$-0.664585\pi$
$101$	−9.93584	−0.988653	−0.494327	+	0.869276i	$0.664585\pi$
$102$	0	0
$103$	19.2644	1.89818	0.949090	−	0.315005i	$-0.102006\pi$
$103$	19.2644	1.89818	0.949090	+	0.315005i	$0.102006\pi$
$104$	−0.635446	−0.0623106
$105$	0	0
$106$	−0.00398632	−0.000387186 0
$107$	7.88030	0.761817	0.380909	−	0.924613i	$-0.375611\pi$
$107$	7.88030	0.761817	0.380909	+	0.924613i	$0.375611\pi$
$108$	0	0
$109$	−1.72728	−0.165443	−0.0827215	−	0.996573i	$-0.526361\pi$
$109$	−1.72728	−0.165443	−0.0827215	+	0.996573i	$0.526361\pi$
$110$	1.05107	0.100216
$111$	0	0
$112$	2.73427	0.258365
$113$	−14.0431	−1.32107	−0.660533	−	0.750797i	$-0.729670\pi$
$113$	−14.0431	−1.32107	−0.660533	+	0.750797i	$0.729670\pi$
$114$	0	0
$115$	18.7831	1.75154
$116$	−16.1606	−1.50047
$117$	0	0
$118$	0.370684	0.0341242
$119$	1.43277	0.131342
$120$	0	0
$121$	−2.24829	−0.204390
$122$	0.791441	0.0716537
$123$	0	0
$124$	5.58499	0.501547
$125$	−37.4046	−3.34557
$126$	0	0
$127$	−1.65498	−0.146856	−0.0734278	−	0.997301i	$-0.523394\pi$
$127$	−1.65498	−0.146856	−0.0734278	+	0.997301i	$0.523394\pi$
$128$	2.60981	0.230676
$129$	0	0
$130$	−0.687385	−0.0602877
$131$	−17.1541	−1.49876	−0.749381	−	0.662139i	$-0.769649\pi$
$131$	−17.1541	−1.49876	−0.749381	+	0.662139i	$0.769649\pi$
$132$	0	0
$133$	2.58748	0.224363
$134$	0.0793027	0.00685071
$135$	0	0
$136$	0.681450	0.0584339
$137$	−14.7651	−1.26147	−0.630733	−	0.776000i	$-0.717245\pi$
$137$	−14.7651	−1.26147	−0.630733	+	0.776000i	$0.717245\pi$
$138$	0	0
$139$	20.3570	1.72666	0.863329	−	0.504641i	$-0.168375\pi$
$139$	20.3570	1.72666	0.863329	+	0.504641i	$0.168375\pi$
$140$	5.94582	0.502514
$141$	0	0
$142$	0.638502	0.0535819
$143$	−5.72349	−0.478623
$144$	0	0
$145$	−35.0223	−2.90845
$146$	−1.35097	−0.111807
$147$	0	0
$148$	19.3805	1.59306
$149$	−8.37820	−0.686369	−0.343185	−	0.939268i	$-0.611506\pi$
$149$	−8.37820	−0.686369	−0.343185	+	0.939268i	$0.611506\pi$
$150$	0	0
$151$	−22.0074	−1.79093	−0.895467	−	0.445128i	$-0.853158\pi$
$151$	−22.0074	−1.79093	−0.895467	+	0.445128i	$0.853158\pi$
$152$	1.23065	0.0998188
$153$	0	0
$154$	−0.168032	−0.0135404
$155$	12.1035	0.972175
$156$	0	0
$157$	−0.208187	−0.0166151	−0.00830757	−	0.999965i	$-0.502644\pi$
$157$	−0.208187	−0.0166151	−0.00830757	+	0.999965i	$0.502644\pi$
$158$	0.562026	0.0447124
$159$	0	0
$160$	4.24429	0.335541
$161$	−3.00281	−0.236655
$162$	0	0
$163$	7.85060	0.614906	0.307453	−	0.951563i	$-0.400523\pi$
$163$	7.85060	0.614906	0.307453	+	0.951563i	$0.400523\pi$
$164$	−8.16033	−0.637215
$165$	0	0
$166$	−0.752985	−0.0584430
$167$	13.8766	1.07380	0.536900	−	0.843646i	$-0.319595\pi$
$167$	13.8766	1.07380	0.536900	+	0.843646i	$0.319595\pi$
$168$	0	0
$169$	−9.25692	−0.712071
$170$	0.737150	0.0565368
$171$	0	0
$172$	−6.03509	−0.460171
$173$	−19.3500	−1.47115	−0.735577	−	0.677441i	$-0.763089\pi$
$173$	−19.3500	−1.47115	−0.735577	+	0.677441i	$0.763089\pi$
$174$	0	0
$175$	9.43262	0.713039
$176$	11.7134	0.882929
$177$	0	0
$178$	−0.194603	−0.0145861
$179$	7.02479	0.525057	0.262529	−	0.964924i	$-0.415444\pi$
$179$	7.02479	0.525057	0.262529	+	0.964924i	$0.415444\pi$
$180$	0	0
$181$	7.22067	0.536708	0.268354	−	0.963320i	$-0.413520\pi$
$181$	7.22067	0.536708	0.268354	+	0.963320i	$0.413520\pi$
$182$	0.109891	0.00814563
$183$	0	0
$184$	−1.42819	−0.105287
$185$	42.0003	3.08792
$186$	0	0
$187$	6.13786	0.448845
$188$	13.3794	0.975793
$189$	0	0
$190$	1.33124	0.0965782
$191$	−20.7694	−1.50282	−0.751412	−	0.659834i	$-0.770627\pi$
$191$	−20.7694	−1.50282	−0.751412	+	0.659834i	$0.770627\pi$
$192$	0	0
$193$	−19.7206	−1.41952	−0.709762	−	0.704442i	$-0.751197\pi$
$193$	−19.7206	−1.41952	−0.709762	+	0.704442i	$0.751197\pi$
$194$	0.432091	0.0310223
$195$	0	0
$196$	13.0021	0.928721
$197$	16.2977	1.16116	0.580582	−	0.814201i	$-0.302825\pi$
$197$	16.2977	1.16116	0.580582	+	0.814201i	$0.302825\pi$
$198$	0	0
$199$	4.79014	0.339564	0.169782	−	0.985482i	$-0.445694\pi$
$199$	4.79014	0.339564	0.169782	+	0.985482i	$0.445694\pi$
$200$	4.48631	0.317230
$201$	0	0
$202$	0.817229	0.0575000
$203$	5.59893	0.392968
$204$	0	0
$205$	−17.6846	−1.23515
$206$	−1.58451	−0.110398
$207$	0	0
$208$	−7.66038	−0.531152
$209$	11.0845	0.766733
$210$	0	0
$211$	−4.30473	−0.296350	−0.148175	−	0.988961i	$-0.547340\pi$
$211$	−4.30473	−0.296350	−0.148175	+	0.988961i	$0.547340\pi$
$212$	−0.0966033	−0.00663474
$213$	0	0
$214$	−0.648159	−0.0443072
$215$	−13.0789	−0.891975
$216$	0	0
$217$	−1.93495	−0.131353
$218$	0.142069	0.00962215
$219$	0	0
$220$	25.4713	1.71728
$221$	−4.01407	−0.270016
$222$	0	0
$223$	14.8068	0.991540	0.495770	−	0.868454i	$-0.334886\pi$
$223$	14.8068	0.991540	0.495770	+	0.868454i	$0.334886\pi$
$224$	−0.678524	−0.0453358
$225$	0	0
$226$	1.15505	0.0768331
$227$	11.9026	0.790000	0.395000	−	0.918681i	$-0.370745\pi$
$227$	11.9026	0.790000	0.395000	+	0.918681i	$0.370745\pi$
$228$	0	0
$229$	−5.11465	−0.337985	−0.168993	−	0.985617i	$-0.554051\pi$
$229$	−5.11465	−0.337985	−0.168993	+	0.985617i	$0.554051\pi$
$230$	−1.54492	−0.101869
$231$	0	0
$232$	2.66295	0.174831
$233$	14.9300	0.978097	0.489048	−	0.872257i	$-0.337344\pi$
$233$	14.9300	0.978097	0.489048	+	0.872257i	$0.337344\pi$
$234$	0	0
$235$	28.9951	1.89143
$236$	8.98303	0.584746
$237$	0	0
$238$	−0.117846	−0.00763884
$239$	−29.8624	−1.93164	−0.965820	−	0.259215i	$-0.916536\pi$
$239$	−29.8624	−1.93164	−0.965820	+	0.259215i	$0.916536\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0.184923	0.0118873
$243$	0	0
$244$	19.1795	1.22784
$245$	28.1774	1.80019
$246$	0	0
$247$	−7.24912	−0.461250
$248$	−0.920296	−0.0584388
$249$	0	0
$250$	3.07655	0.194578
$251$	−13.0620	−0.824464	−0.412232	−	0.911079i	$-0.635251\pi$
$251$	−13.0620	−0.824464	−0.412232	+	0.911079i	$0.635251\pi$
$252$	0	0
$253$	−12.8638	−0.808738
$254$	0.136123	0.00854111
$255$	0	0
$256$	15.4615	0.966346
$257$	−6.35830	−0.396620	−0.198310	−	0.980139i	$-0.563545\pi$
$257$	−6.35830	−0.396620	−0.198310	+	0.980139i	$0.563545\pi$
$258$	0	0
$259$	−6.71448	−0.417218
$260$	−16.6579	−1.03308
$261$	0	0
$262$	1.41094	0.0871679
$263$	−4.66189	−0.287465	−0.143732	−	0.989617i	$-0.545910\pi$
$263$	−4.66189	−0.287465	−0.143732	+	0.989617i	$0.545910\pi$
$264$	0	0
$265$	−0.209353	−0.0128605
$266$	−0.212822	−0.0130489
$267$	0	0
$268$	1.92180	0.117392
$269$	10.5251	0.641728	0.320864	−	0.947125i	$-0.396027\pi$
$269$	10.5251	0.641728	0.320864	+	0.947125i	$0.396027\pi$
$270$	0	0
$271$	10.3212	0.626968	0.313484	−	0.949593i	$-0.398504\pi$
$271$	10.3212	0.626968	0.313484	+	0.949593i	$0.398504\pi$
$272$	8.21497	0.498106
$273$	0	0
$274$	1.21444	0.0733667
$275$	40.4084	2.43672
$276$	0	0
$277$	22.1023	1.32800	0.663999	−	0.747734i	$-0.268858\pi$
$277$	22.1023	1.32800	0.663999	+	0.747734i	$0.268858\pi$
$278$	−1.67437	−0.100422
$279$	0	0
$280$	−0.979754	−0.0585515
$281$	−4.22803	−0.252223	−0.126111	−	0.992016i	$-0.540250\pi$
$281$	−4.22803	−0.252223	−0.126111	+	0.992016i	$0.540250\pi$
$282$	0	0
$283$	−3.06769	−0.182355	−0.0911776	−	0.995835i	$-0.529063\pi$
$283$	−3.06769	−0.182355	−0.0911776	+	0.995835i	$0.529063\pi$
$284$	15.4733	0.918169
$285$	0	0
$286$	0.470761	0.0278367
$287$	2.82720	0.166884
$288$	0	0
$289$	−12.6953	−0.746783
$290$	2.88061	0.169155
$291$	0	0
$292$	−32.7390	−1.91590
$293$	15.4694	0.903735	0.451867	−	0.892085i	$-0.350758\pi$
$293$	15.4694	0.903735	0.451867	+	0.892085i	$0.350758\pi$
$294$	0	0
$295$	19.4675	1.13344
$296$	−3.19352	−0.185619
$297$	0	0
$298$	0.689112	0.0399192
$299$	8.41272	0.486520
$300$	0	0
$301$	2.09089	0.120517
$302$	1.81012	0.104161
$303$	0	0
$304$	14.8356	0.850881
$305$	41.5648	2.38000
$306$	0	0
$307$	−20.2412	−1.15523	−0.577614	−	0.816310i	$-0.696016\pi$
$307$	−20.2412	−1.15523	−0.577614	+	0.816310i	$0.696016\pi$
$308$	−4.07204	−0.232026
$309$	0	0
$310$	−0.995518	−0.0565416
$311$	−1.49434	−0.0847363	−0.0423681	−	0.999102i	$-0.513490\pi$
$311$	−1.49434	−0.0847363	−0.0423681	+	0.999102i	$0.513490\pi$
$312$	0	0
$313$	−21.6470	−1.22356	−0.611780	−	0.791028i	$-0.709546\pi$
$313$	−21.6470	−1.22356	−0.611780	+	0.791028i	$0.709546\pi$
$314$	0.0171235	0.000966335 0
$315$	0	0
$316$	13.6200	0.766183
$317$	−28.9329	−1.62504	−0.812518	−	0.582936i	$-0.801904\pi$
$317$	−28.9329	−1.62504	−0.812518	+	0.582936i	$0.801904\pi$
$318$	0	0
$319$	23.9853	1.34292
$320$	33.8577	1.89270
$321$	0	0
$322$	0.246983	0.0137638
$323$	7.77394	0.432553
$324$	0	0
$325$	−26.4265	−1.46588
$326$	−0.645716	−0.0357629
$327$	0	0
$328$	1.34466	0.0742465
$329$	−4.63537	−0.255556
$330$	0	0
$331$	−27.5701	−1.51539	−0.757695	−	0.652609i	$-0.773675\pi$
$331$	−27.5701	−1.51539	−0.757695	+	0.652609i	$0.773675\pi$
$332$	−18.2476	−1.00147
$333$	0	0
$334$	−1.14135	−0.0624521
$335$	4.16481	0.227548
$336$	0	0
$337$	−8.25450	−0.449651	−0.224826	−	0.974399i	$-0.572181\pi$
$337$	−8.25450	−0.449651	−0.224826	+	0.974399i	$0.572181\pi$
$338$	0.761387	0.0414140
$339$	0	0
$340$	17.8639	0.968804
$341$	−8.28915	−0.448883
$342$	0	0
$343$	−9.33864	−0.504239
$344$	0.994464	0.0536179
$345$	0	0
$346$	1.59155	0.0855622
$347$	−21.2018	−1.13817	−0.569085	−	0.822278i	$-0.692703\pi$
$347$	−21.2018	−1.13817	−0.569085	+	0.822278i	$0.692703\pi$
$348$	0	0
$349$	21.3257	1.14154	0.570770	−	0.821110i	$-0.306645\pi$
$349$	21.3257	1.14154	0.570770	+	0.821110i	$0.306645\pi$
$350$	−0.775838	−0.0414703
$351$	0	0
$352$	−2.90673	−0.154929
$353$	−2.80184	−0.149127	−0.0745635	−	0.997216i	$-0.523756\pi$
$353$	−2.80184	−0.149127	−0.0745635	+	0.997216i	$0.523756\pi$
$354$	0	0
$355$	33.5328	1.77974
$356$	−4.71594	−0.249944
$357$	0	0
$358$	−0.577793	−0.0305373
$359$	−4.62788	−0.244250	−0.122125	−	0.992515i	$-0.538971\pi$
$359$	−4.62788	−0.244250	−0.122125	+	0.992515i	$0.538971\pi$
$360$	0	0
$361$	−4.96085	−0.261097
$362$	−0.593904	−0.0312149
$363$	0	0
$364$	2.66305	0.139582
$365$	−70.9501	−3.71370
$366$	0	0
$367$	−0.106183	−0.00554270	−0.00277135	−	0.999996i	$-0.500882\pi$
$367$	−0.106183	−0.00554270	−0.00277135	+	0.999996i	$0.500882\pi$
$368$	−17.2170	−0.897497
$369$	0	0
$370$	−3.45455	−0.179593
$371$	0.0334688	0.00173761
$372$	0	0
$373$	−16.6703	−0.863157	−0.431579	−	0.902075i	$-0.642043\pi$
$373$	−16.6703	−0.863157	−0.431579	+	0.902075i	$0.642043\pi$
$374$	−0.504843	−0.0261048
$375$	0	0
$376$	−2.20466	−0.113697
$377$	−15.6860	−0.807872
$378$	0	0
$379$	−16.5660	−0.850939	−0.425469	−	0.904973i	$-0.639891\pi$
$379$	−16.5660	−0.850939	−0.425469	+	0.904973i	$0.639891\pi$
$380$	32.2608	1.65495
$381$	0	0
$382$	1.70830	0.0874041
$383$	25.0198	1.27845	0.639226	−	0.769019i	$-0.279255\pi$
$383$	25.0198	1.27845	0.639226	+	0.769019i	$0.279255\pi$
$384$	0	0
$385$	−8.82470	−0.449748
$386$	1.62203	0.0825594
$387$	0	0
$388$	10.4712	0.531592
$389$	32.6696	1.65641	0.828207	−	0.560423i	$-0.189361\pi$
$389$	32.6696	1.65641	0.828207	+	0.560423i	$0.189361\pi$
$390$	0	0
$391$	−9.02178	−0.456251
$392$	−2.14249	−0.108212
$393$	0	0
$394$	−1.34050	−0.0675333
$395$	29.5165	1.48513
$396$	0	0
$397$	−21.9103	−1.09964	−0.549822	−	0.835282i	$-0.685305\pi$
$397$	−21.9103	−1.09964	−0.549822	+	0.835282i	$0.685305\pi$
$398$	−0.393992	−0.0197490
$399$	0	0
$400$	54.0830	2.70415
$401$	−10.2229	−0.510508	−0.255254	−	0.966874i	$-0.582159\pi$
$401$	−10.2229	−0.510508	−0.255254	+	0.966874i	$0.582159\pi$
$402$	0	0
$403$	5.42099	0.270039
$404$	19.8045	0.985309
$405$	0	0
$406$	−0.460516	−0.0228550
$407$	−28.7642	−1.42579
$408$	0	0
$409$	−16.7739	−0.829417	−0.414709	−	0.909954i	$-0.636117\pi$
$409$	−16.7739	−0.829417	−0.414709	+	0.909954i	$0.636117\pi$
$410$	1.45457	0.0718361
$411$	0	0
$412$	−38.3985	−1.89176
$413$	−3.11223	−0.153143
$414$	0	0
$415$	−39.5452	−1.94120
$416$	1.90096	0.0932023
$417$	0	0
$418$	−0.911708	−0.0445931
$419$	−32.5790	−1.59159	−0.795793	−	0.605569i	$-0.792946\pi$
$419$	−32.5790	−1.59159	−0.795793	+	0.605569i	$0.792946\pi$
$420$	0	0
$421$	11.5346	0.562163	0.281082	−	0.959684i	$-0.409307\pi$
$421$	11.5346	0.562163	0.281082	+	0.959684i	$0.409307\pi$
$422$	0.354066	0.0172357
$423$	0	0
$424$	0.0159183	0.000773062 0
$425$	28.3397	1.37468
$426$	0	0
$427$	−6.64487	−0.321568
$428$	−15.7073	−0.759240
$429$	0	0
$430$	1.07575	0.0518772
$431$	−15.2713	−0.735590	−0.367795	−	0.929907i	$-0.619887\pi$
$431$	−15.2713	−0.735590	−0.367795	+	0.929907i	$0.619887\pi$
$432$	0	0
$433$	8.69941	0.418067	0.209034	−	0.977908i	$-0.432968\pi$
$433$	8.69941	0.418067	0.209034	+	0.977908i	$0.432968\pi$
$434$	0.159151	0.00763949
$435$	0	0
$436$	3.44287	0.164883
$437$	−16.2927	−0.779384
$438$	0	0
$439$	29.4040	1.40338	0.701689	−	0.712484i	$-0.252430\pi$
$439$	29.4040	1.40338	0.701689	+	0.712484i	$0.252430\pi$
$440$	−4.19717	−0.200092
$441$	0	0
$442$	0.330160	0.0157041
$443$	35.7878	1.70033	0.850164	−	0.526518i	$-0.176503\pi$
$443$	35.7878	1.70033	0.850164	+	0.526518i	$0.176503\pi$
$444$	0	0
$445$	−10.2201	−0.484481
$446$	−1.21787	−0.0576679
$447$	0	0
$448$	−5.41274	−0.255728
$449$	7.50777	0.354314	0.177157	−	0.984183i	$-0.443310\pi$
$449$	7.50777	0.354314	0.177157	+	0.984183i	$0.443310\pi$
$450$	0	0
$451$	12.1114	0.570306
$452$	27.9912	1.31660
$453$	0	0
$454$	−0.978992	−0.0459464
$455$	5.77123	0.270559
$456$	0	0
$457$	−31.7902	−1.48708	−0.743540	−	0.668691i	$-0.766855\pi$
$457$	−31.7902	−1.48708	−0.743540	+	0.668691i	$0.766855\pi$
$458$	0.420683	0.0196572
$459$	0	0
$460$	−37.4392	−1.74561
$461$	12.8715	0.599487	0.299743	−	0.954020i	$-0.403099\pi$
$461$	12.8715	0.599487	0.299743	+	0.954020i	$0.403099\pi$
$462$	0	0
$463$	−11.8687	−0.551587	−0.275794	−	0.961217i	$-0.588941\pi$
$463$	−11.8687	−0.551587	−0.275794	+	0.961217i	$0.588941\pi$
$464$	32.1021	1.49030
$465$	0	0
$466$	−1.22800	−0.0568860
$467$	18.9191	0.875473	0.437736	−	0.899103i	$-0.355780\pi$
$467$	18.9191	0.875473	0.437736	+	0.899103i	$0.355780\pi$
$468$	0	0
$469$	−0.665818	−0.0307446
$470$	−2.38486	−0.110006
$471$	0	0
$472$	−1.48023	−0.0681329
$473$	8.95719	0.411852
$474$	0	0
$475$	51.1795	2.34828
$476$	−2.85585	−0.130898
$477$	0	0
$478$	2.45620	0.112344
$479$	−23.7911	−1.08704	−0.543522	−	0.839395i	$-0.682909\pi$
$479$	−23.7911	−1.08704	−0.543522	+	0.839395i	$0.682909\pi$
$480$	0	0
$481$	18.8114	0.857725
$482$	−0.0822506	−0.00374641
$483$	0	0
$484$	4.48136	0.203698
$485$	22.6925	1.03041
$486$	0	0
$487$	1.59589	0.0723165	0.0361583	−	0.999346i	$-0.488488\pi$
$487$	1.59589	0.0723165	0.0361583	+	0.999346i	$0.488488\pi$
$488$	−3.16041	−0.143065
$489$	0	0
$490$	−2.31761	−0.104699
$491$	31.8604	1.43784	0.718920	−	0.695093i	$-0.244637\pi$
$491$	31.8604	1.43784	0.718920	+	0.695093i	$0.244637\pi$
$492$	0	0
$493$	16.8217	0.757610
$494$	0.596244	0.0268263
$495$	0	0
$496$	−11.0943	−0.498148
$497$	−5.36081	−0.240465
$498$	0	0
$499$	17.3389	0.776194	0.388097	−	0.921619i	$-0.373133\pi$
$499$	17.3389	0.776194	0.388097	+	0.921619i	$0.373133\pi$
$500$	74.5561	3.33425
$501$	0	0
$502$	1.07435	0.0479508
$503$	13.8066	0.615607	0.307803	−	0.951450i	$-0.400406\pi$
$503$	13.8066	0.615607	0.307803	+	0.951450i	$0.400406\pi$
$504$	0	0
$505$	42.9192	1.90988
$506$	1.05805	0.0470362
$507$	0	0
$508$	3.29876	0.146359
$509$	−37.2876	−1.65274	−0.826372	−	0.563124i	$-0.809599\pi$
$509$	−37.2876	−1.65274	−0.826372	+	0.563124i	$0.809599\pi$
$510$	0	0
$511$	11.3426	0.501768
$512$	−6.49133	−0.286879
$513$	0	0
$514$	0.522974	0.0230674
$515$	−83.2152	−3.66690
$516$	0	0
$517$	−19.8575	−0.873332
$518$	0.552270	0.0242653
$519$	0	0
$520$	2.74489	0.120371
$521$	−35.4359	−1.55248	−0.776238	−	0.630439i	$-0.782875\pi$
$521$	−35.4359	−1.55248	−0.776238	+	0.630439i	$0.782875\pi$
$522$	0	0
$523$	−22.2983	−0.975036	−0.487518	−	0.873113i	$-0.662098\pi$
$523$	−22.2983	−0.975036	−0.487518	+	0.873113i	$0.662098\pi$
$524$	34.1922	1.49369
$525$	0	0
$526$	0.383443	0.0167189
$527$	−5.81345	−0.253238
$528$	0	0
$529$	−4.09210	−0.177917
$530$	0.0172194	0.000747964 0
$531$	0	0
$532$	−5.15746	−0.223604
$533$	−7.92071	−0.343084
$534$	0	0
$535$	−34.0400	−1.47168
$536$	−0.316674	−0.0136782
$537$	0	0
$538$	−0.865698	−0.0373229
$539$	−19.2975	−0.831203
$540$	0	0
$541$	−28.0367	−1.20539	−0.602697	−	0.797970i	$-0.705907\pi$
$541$	−28.0367	−1.20539	−0.602697	+	0.797970i	$0.705907\pi$
$542$	−0.848925	−0.0364644
$543$	0	0
$544$	−2.03859	−0.0874037
$545$	7.46119	0.319602
$546$	0	0
$547$	−13.1556	−0.562491	−0.281245	−	0.959636i	$-0.590747\pi$
$547$	−13.1556	−0.562491	−0.281245	+	0.959636i	$0.590747\pi$
$548$	29.4303	1.25720
$549$	0	0
$550$	−3.32362	−0.141719
$551$	30.3787	1.29418
$552$	0	0
$553$	−4.71872	−0.200660
$554$	−1.81793	−0.0772363
$555$	0	0
$556$	−40.5763	−1.72082
$557$	20.1003	0.851678	0.425839	−	0.904799i	$-0.359979\pi$
$557$	20.1003	0.851678	0.425839	+	0.904799i	$0.359979\pi$
$558$	0	0
$559$	−5.85787	−0.247762
$560$	−11.8111	−0.499108
$561$	0	0
$562$	0.347757	0.0146693
$563$	1.97240	0.0831268	0.0415634	−	0.999136i	$-0.486766\pi$
$563$	1.97240	0.0831268	0.0415634	+	0.999136i	$0.486766\pi$
$564$	0	0
$565$	60.6611	2.55203
$566$	0.252319	0.0106058
$567$	0	0
$568$	−2.54969	−0.106983
$569$	−19.0863	−0.800139	−0.400070	−	0.916485i	$-0.631014\pi$
$569$	−19.0863	−0.800139	−0.400070	+	0.916485i	$0.631014\pi$
$570$	0	0
$571$	10.1484	0.424697	0.212348	−	0.977194i	$-0.431889\pi$
$571$	10.1484	0.424697	0.212348	+	0.977194i	$0.431889\pi$
$572$	11.4083	0.477004
$573$	0	0
$574$	−0.232539	−0.00970597
$575$	−59.3946	−2.47693
$576$	0	0
$577$	11.2993	0.470395	0.235198	−	0.971948i	$-0.424426\pi$
$577$	11.2993	0.470395	0.235198	+	0.971948i	$0.424426\pi$
$578$	1.04420	0.0434329
$579$	0	0
$580$	69.8078	2.89861
$581$	6.32200	0.262281
$582$	0	0
$583$	0.143377	0.00593807
$584$	5.39474	0.223236
$585$	0	0
$586$	−1.27237	−0.0525611
$587$	−36.1850	−1.49352	−0.746758	−	0.665095i	$-0.768391\pi$
$587$	−36.1850	−1.49352	−0.746758	+	0.665095i	$0.768391\pi$
$588$	0	0
$589$	−10.4987	−0.432590
$590$	−1.60122	−0.0659210
$591$	0	0
$592$	−38.4983	−1.58227
$593$	28.2897	1.16172	0.580858	−	0.814005i	$-0.302717\pi$
$593$	28.2897	1.16172	0.580858	+	0.814005i	$0.302717\pi$
$594$	0	0
$595$	−6.18905	−0.253726
$596$	16.6997	0.684047
$597$	0	0
$598$	−0.691951	−0.0282960
$599$	10.9989	0.449404	0.224702	−	0.974427i	$-0.427859\pi$
$599$	10.9989	0.449404	0.224702	+	0.974427i	$0.427859\pi$
$600$	0	0
$601$	−10.3224	−0.421058	−0.210529	−	0.977588i	$-0.567519\pi$
$601$	−10.3224	−0.421058	−0.210529	+	0.977588i	$0.567519\pi$
$602$	−0.171977	−0.00700927
$603$	0	0
$604$	43.8658	1.78488
$605$	9.71176	0.394839
$606$	0	0
$607$	−26.4559	−1.07381	−0.536906	−	0.843642i	$-0.680407\pi$
$607$	−26.4559	−1.07381	−0.536906	+	0.843642i	$0.680407\pi$
$608$	−3.68154	−0.149306
$609$	0	0
$610$	−3.41873	−0.138420
$611$	12.9865	0.525378
$612$	0	0
$613$	33.8484	1.36712	0.683561	−	0.729893i	$-0.260430\pi$
$613$	33.8484	1.36712	0.683561	+	0.729893i	$0.260430\pi$
$614$	1.66485	0.0671879
$615$	0	0
$616$	0.670992	0.0270350
$617$	0.659865	0.0265651	0.0132826	−	0.999912i	$-0.495772\pi$
$617$	0.659865	0.0265651	0.0132826	+	0.999912i	$0.495772\pi$
$618$	0	0
$619$	−34.8866	−1.40221	−0.701106	−	0.713057i	$-0.747310\pi$
$619$	−34.8866	−1.40221	−0.701106	+	0.713057i	$0.747310\pi$
$620$	−24.1251	−0.968886
$621$	0	0
$622$	0.122910	0.00492826
$623$	1.63387	0.0654595
$624$	0	0
$625$	93.2779	3.73111
$626$	1.78048	0.0711622
$627$	0	0
$628$	0.414966	0.0165589
$629$	−20.1733	−0.804361
$630$	0	0
$631$	−6.39474	−0.254571	−0.127285	−	0.991866i	$-0.540626\pi$
$631$	−6.39474	−0.254571	−0.127285	+	0.991866i	$0.540626\pi$
$632$	−2.24430	−0.0892735
$633$	0	0
$634$	2.37975	0.0945120
$635$	7.14890	0.283695
$636$	0	0
$637$	12.6203	0.500035
$638$	−1.97280	−0.0781041
$639$	0	0
$640$	−11.2734	−0.445620
$641$	−16.9466	−0.669352	−0.334676	−	0.942333i	$-0.608627\pi$
$641$	−16.9466	−0.669352	−0.334676	+	0.942333i	$0.608627\pi$
$642$	0	0
$643$	14.3993	0.567852	0.283926	−	0.958846i	$-0.408363\pi$
$643$	14.3993	0.567852	0.283926	+	0.958846i	$0.408363\pi$
$644$	5.98531	0.235854
$645$	0	0
$646$	−0.639411	−0.0251573
$647$	−4.38744	−0.172488	−0.0862441	−	0.996274i	$-0.527486\pi$
$647$	−4.38744	−0.172488	−0.0862441	+	0.996274i	$0.527486\pi$
$648$	0	0
$649$	−13.3325	−0.523346
$650$	2.17360	0.0852555
$651$	0	0
$652$	−15.6481	−0.612826
$653$	4.56970	0.178826	0.0894132	−	0.995995i	$-0.471501\pi$
$653$	4.56970	0.178826	0.0894132	+	0.995995i	$0.471501\pi$
$654$	0	0
$655$	74.0994	2.89530
$656$	16.2101	0.632897
$657$	0	0
$658$	0.381262	0.0148631
$659$	13.6506	0.531751	0.265875	−	0.964007i	$-0.414339\pi$
$659$	13.6506	0.531751	0.265875	+	0.964007i	$0.414339\pi$
$660$	0	0
$661$	32.4151	1.26080	0.630401	−	0.776270i	$-0.282891\pi$
$661$	32.4151	1.26080	0.630401	+	0.776270i	$0.282891\pi$
$662$	2.26766	0.0881350
$663$	0	0
$664$	3.00685	0.116688
$665$	−11.1770	−0.433424
$666$	0	0
$667$	−35.2550	−1.36508
$668$	−27.6592	−1.07017
$669$	0	0
$670$	−0.342558	−0.0132342
$671$	−28.4660	−1.09892
$672$	0	0
$673$	0.237992	0.00917391	0.00458696	−	0.999989i	$-0.498540\pi$
$673$	0.237992	0.00917391	0.00458696	+	0.999989i	$0.498540\pi$
$674$	0.678938	0.0261517
$675$	0	0
$676$	18.4512	0.709662
$677$	30.9534	1.18964	0.594818	−	0.803860i	$-0.297224\pi$
$677$	30.9534	1.18964	0.594818	+	0.803860i	$0.297224\pi$
$678$	0	0
$679$	−3.62780	−0.139222
$680$	−2.94361	−0.112882
$681$	0	0
$682$	0.681788	0.0261070
$683$	28.5801	1.09359	0.546794	−	0.837267i	$-0.315848\pi$
$683$	28.5801	1.09359	0.546794	+	0.837267i	$0.315848\pi$
$684$	0	0
$685$	63.7796	2.43689
$686$	0.768108	0.0293265
$687$	0	0
$688$	11.9884	0.457053
$689$	−0.0937666	−0.00357222
$690$	0	0
$691$	−6.89241	−0.262200	−0.131100	−	0.991369i	$-0.541851\pi$
$691$	−6.89241	−0.262200	−0.131100	+	0.991369i	$0.541851\pi$
$692$	38.5691	1.46618
$693$	0	0
$694$	1.74386	0.0661959
$695$	−87.9347	−3.33555
$696$	0	0
$697$	8.49415	0.321739
$698$	−1.75405	−0.0663919
$699$	0	0
$700$	−18.8014	−0.710627
$701$	29.3972	1.11032	0.555158	−	0.831745i	$-0.312658\pi$
$701$	29.3972	1.11032	0.555158	+	0.831745i	$0.312658\pi$
$702$	0	0
$703$	−36.4314	−1.37404
$704$	−23.1877	−0.873918
$705$	0	0
$706$	0.230453	0.00867321
$707$	−6.86138	−0.258049
$708$	0	0
$709$	−10.8690	−0.408195	−0.204098	−	0.978951i	$-0.565426\pi$
$709$	−10.8690	−0.408195	−0.204098	+	0.978951i	$0.565426\pi$
$710$	−2.75809	−0.103509
$711$	0	0
$712$	0.777094	0.0291228
$713$	12.1839	0.456290
$714$	0	0
$715$	24.7234	0.924602
$716$	−14.0021	−0.523281
$717$	0	0
$718$	0.380646	0.0142056
$719$	30.8283	1.14970	0.574850	−	0.818259i	$-0.305060\pi$
$719$	30.8283	1.14970	0.574850	+	0.818259i	$0.305060\pi$
$720$	0	0
$721$	13.3034	0.495445
$722$	0.408032	0.0151854
$723$	0	0
$724$	−14.3925	−0.534892
$725$	110.745	4.11297
$726$	0	0
$727$	28.0322	1.03966	0.519828	−	0.854271i	$-0.325996\pi$
$727$	28.0322	1.03966	0.519828	+	0.854271i	$0.325996\pi$
$728$	−0.438819	−0.0162637
$729$	0	0
$730$	5.83569	0.215989
$731$	6.28197	0.232347
$732$	0	0
$733$	25.8442	0.954578	0.477289	−	0.878746i	$-0.341619\pi$
$733$	25.8442	0.954578	0.477289	+	0.878746i	$0.341619\pi$
$734$	0.00873360	0.000322363 0
$735$	0	0
$736$	4.27248	0.157486
$737$	−2.85230	−0.105066
$738$	0	0
$739$	−43.0752	−1.58455	−0.792273	−	0.610166i	$-0.791103\pi$
$739$	−43.0752	−1.58455	−0.792273	+	0.610166i	$0.791103\pi$
$740$	−83.7164	−3.07748
$741$	0	0
$742$	−0.00275283	−0.000101059 0
$743$	−13.1212	−0.481372	−0.240686	−	0.970603i	$-0.577372\pi$
$743$	−13.1212	−0.481372	−0.240686	+	0.970603i	$0.577372\pi$
$744$	0	0
$745$	36.1907	1.32593
$746$	1.37114	0.0502012
$747$	0	0
$748$	−12.2342	−0.447327
$749$	5.44189	0.198842
$750$	0	0
$751$	33.6870	1.22926	0.614629	−	0.788817i	$-0.289306\pi$
$751$	33.6870	1.22926	0.614629	+	0.788817i	$0.289306\pi$
$752$	−26.5775	−0.969180
$753$	0	0
$754$	1.29019	0.0469858
$755$	95.0637	3.45972
$756$	0	0
$757$	−46.0911	−1.67521	−0.837604	−	0.546278i	$-0.816044\pi$
$757$	−46.0911	−1.67521	−0.837604	+	0.546278i	$0.816044\pi$
$758$	1.36256	0.0494905
$759$	0	0
$760$	−5.31595	−0.192830
$761$	−14.2727	−0.517387	−0.258693	−	0.965960i	$-0.583292\pi$
$761$	−14.2727	−0.517387	−0.258693	+	0.965960i	$0.583292\pi$
$762$	0	0
$763$	−1.19280	−0.0431823
$764$	41.3983	1.49774
$765$	0	0
$766$	−2.05789	−0.0743547
$767$	8.71925	0.314834
$768$	0	0
$769$	13.0154	0.469349	0.234674	−	0.972074i	$-0.424598\pi$
$769$	13.0154	0.469349	0.234674	+	0.972074i	$0.424598\pi$
$770$	0.725837	0.0261573
$771$	0	0
$772$	39.3079	1.41472
$773$	29.9438	1.07700	0.538502	−	0.842624i	$-0.318990\pi$
$773$	29.9438	1.07700	0.538502	+	0.842624i	$0.318990\pi$
$774$	0	0
$775$	−38.2727	−1.37480
$776$	−1.72544	−0.0619397
$777$	0	0
$778$	−2.68709	−0.0963369
$779$	15.3398	0.549606
$780$	0	0
$781$	−22.9652	−0.821759
$782$	0.742046	0.0265355
$783$	0	0
$784$	−25.8280	−0.922428
$785$	0.899291	0.0320971
$786$	0	0
$787$	13.7932	0.491676	0.245838	−	0.969311i	$-0.420937\pi$
$787$	13.7932	0.491676	0.245838	+	0.969311i	$0.420937\pi$
$788$	−32.4852	−1.15724
$789$	0	0
$790$	−2.42774	−0.0863753
$791$	−9.69774	−0.344812
$792$	0	0
$793$	18.6163	0.661086
$794$	1.80213	0.0639553
$795$	0	0
$796$	−9.54788	−0.338416
$797$	−18.4152	−0.652300	−0.326150	−	0.945318i	$-0.605751\pi$
$797$	−18.4152	−0.652300	−0.326150	+	0.945318i	$0.605751\pi$
$798$	0	0
$799$	−13.9267	−0.492691
$800$	−13.4210	−0.474503
$801$	0	0
$802$	0.840841	0.0296911
$803$	48.5907	1.71473
$804$	0	0
$805$	12.9710	0.457169
$806$	−0.445879	−0.0157054
$807$	0	0
$808$	−3.26339	−0.114806
$809$	14.7007	0.516849	0.258425	−	0.966031i	$-0.416797\pi$
$809$	14.7007	0.516849	0.258425	+	0.966031i	$0.416797\pi$
$810$	0	0
$811$	−46.3555	−1.62776	−0.813882	−	0.581031i	$-0.802650\pi$
$811$	−46.3555	−1.62776	−0.813882	+	0.581031i	$0.802650\pi$
$812$	−11.1600	−0.391639
$813$	0	0
$814$	2.36587	0.0829238
$815$	−33.9117	−1.18787
$816$	0	0
$817$	11.3448	0.396903
$818$	1.37967	0.0482389
$819$	0	0
$820$	35.2496	1.23097
$821$	−39.2626	−1.37027	−0.685137	−	0.728414i	$-0.740258\pi$
$821$	−39.2626	−1.37027	−0.685137	+	0.728414i	$0.740258\pi$
$822$	0	0
$823$	−29.0804	−1.01368	−0.506839	−	0.862041i	$-0.669186\pi$
$823$	−29.0804	−1.01368	−0.506839	+	0.862041i	$0.669186\pi$
$824$	6.32732	0.220423
$825$	0	0
$826$	0.255982	0.00890677
$827$	−30.7270	−1.06848	−0.534242	−	0.845332i	$-0.679403\pi$
$827$	−30.7270	−1.06848	−0.534242	+	0.845332i	$0.679403\pi$
$828$	0	0
$829$	−16.4599	−0.571675	−0.285837	−	0.958278i	$-0.592272\pi$
$829$	−16.4599	−0.571675	−0.285837	+	0.958278i	$0.592272\pi$
$830$	3.25262	0.112900
$831$	0	0
$832$	15.1644	0.525731
$833$	−13.5340	−0.468925
$834$	0	0
$835$	−59.9416	−2.07436
$836$	−22.0941	−0.764139
$837$	0	0
$838$	2.67964	0.0925665
$839$	−43.2285	−1.49241	−0.746206	−	0.665715i	$-0.768127\pi$
$839$	−43.2285	−1.49241	−0.746206	+	0.665715i	$0.768127\pi$
$840$	0	0
$841$	36.7351	1.26673
$842$	−0.948729	−0.0326954
$843$	0	0
$844$	8.58034	0.295347
$845$	39.9865	1.37558
$846$	0	0
$847$	−1.55260	−0.0533478
$848$	0.191897	0.00658978
$849$	0	0
$850$	−2.33096	−0.0799513
$851$	42.2793	1.44931
$852$	0	0
$853$	8.36996	0.286582	0.143291	−	0.989681i	$-0.454232\pi$
$853$	8.36996	0.286582	0.143291	+	0.989681i	$0.454232\pi$
$854$	0.546544	0.0187024
$855$	0	0
$856$	2.58825	0.0884646
$857$	−18.3641	−0.627305	−0.313653	−	0.949538i	$-0.601553\pi$
$857$	−18.3641	−0.627305	−0.313653	+	0.949538i	$0.601553\pi$
$858$	0	0
$859$	2.47499	0.0844457	0.0422228	−	0.999108i	$-0.486556\pi$
$859$	2.47499	0.0844457	0.0422228	+	0.999108i	$0.486556\pi$
$860$	26.0694	0.888958
$861$	0	0
$862$	1.25607	0.0427819
$863$	10.0917	0.343525	0.171762	−	0.985138i	$-0.445054\pi$
$863$	10.0917	0.343525	0.171762	+	0.985138i	$0.445054\pi$
$864$	0	0
$865$	83.5849	2.84197
$866$	−0.715532	−0.0243148
$867$	0	0
$868$	3.85682	0.130909
$869$	−20.2146	−0.685732
$870$	0	0
$871$	1.86536	0.0632055
$872$	−0.567316	−0.0192118
$873$	0	0
$874$	1.34008	0.0453289
$875$	−25.8304	−0.873227
$876$	0	0
$877$	53.6453	1.81147	0.905736	−	0.423843i	$-0.139319\pi$
$877$	53.6453	1.81147	0.905736	+	0.423843i	$0.139319\pi$
$878$	−2.41850	−0.0816203
$879$	0	0
$880$	−50.5974	−1.70564
$881$	−30.6985	−1.03426	−0.517130	−	0.855907i	$-0.672999\pi$
$881$	−30.6985	−1.03426	−0.517130	+	0.855907i	$0.672999\pi$
$882$	0	0
$883$	−21.8831	−0.736424	−0.368212	−	0.929742i	$-0.620030\pi$
$883$	−21.8831	−0.736424	−0.368212	+	0.929742i	$0.620030\pi$
$884$	8.00099	0.269102
$885$	0	0
$886$	−2.94356	−0.0988910
$887$	23.3921	0.785431	0.392715	−	0.919660i	$-0.371536\pi$
$887$	23.3921	0.785431	0.392715	+	0.919660i	$0.371536\pi$
$888$	0	0
$889$	−1.14288	−0.0383308
$890$	0.840612	0.0281774
$891$	0	0
$892$	−29.5135	−0.988186
$893$	−25.1506	−0.841633
$894$	0	0
$895$	−30.3445	−1.01430
$896$	1.80225	0.0602089
$897$	0	0
$898$	−0.617518	−0.0206069
$899$	−22.7176	−0.757674
$900$	0	0
$901$	0.100555	0.00334997
$902$	−0.996173	−0.0331689
$903$	0	0
$904$	−4.61240	−0.153406
$905$	−31.1906	−1.03681
$906$	0	0
$907$	−2.98723	−0.0991893	−0.0495947	−	0.998769i	$-0.515793\pi$
$907$	−2.98723	−0.0991893	−0.0495947	+	0.998769i	$0.515793\pi$
$908$	−23.7246	−0.787328
$909$	0	0
$910$	−0.474687	−0.0157357
$911$	−42.1883	−1.39776	−0.698880	−	0.715239i	$-0.746318\pi$
$911$	−42.1883	−1.39776	−0.698880	+	0.715239i	$0.746318\pi$
$912$	0	0
$913$	27.0828	0.896311
$914$	2.61476	0.0864885
$915$	0	0
$916$	10.1947	0.336842
$917$	−11.8461	−0.391192
$918$	0	0
$919$	−53.2428	−1.75632	−0.878160	−	0.478367i	$-0.841229\pi$
$919$	−53.2428	−1.75632	−0.878160	+	0.478367i	$0.841229\pi$
$920$	6.16924	0.203394
$921$	0	0
$922$	−1.05869	−0.0348661
$923$	15.0189	0.494353
$924$	0	0
$925$	−132.810	−4.36677
$926$	0.976211	0.0320803
$927$	0	0
$928$	−7.96631	−0.261507
$929$	27.2213	0.893103	0.446552	−	0.894758i	$-0.352652\pi$
$929$	27.2213	0.893103	0.446552	+	0.894758i	$0.352652\pi$
$930$	0	0
$931$	−24.4414	−0.801033
$932$	−29.7590	−0.974788
$933$	0	0
$934$	−1.55611	−0.0509174
$935$	−26.5133	−0.867077
$936$	0	0
$937$	7.54293	0.246417	0.123208	−	0.992381i	$-0.460682\pi$
$937$	7.54293	0.246417	0.123208	+	0.992381i	$0.460682\pi$
$938$	0.0547639	0.00178811
$939$	0	0
$940$	−57.7940	−1.88503
$941$	−35.4553	−1.15581	−0.577905	−	0.816104i	$-0.696130\pi$
$941$	−35.4553	−1.15581	−0.577905	+	0.816104i	$0.696130\pi$
$942$	0	0
$943$	−17.8021	−0.579716
$944$	−17.8443	−0.580783
$945$	0	0
$946$	−0.736734	−0.0239533
$947$	−33.0630	−1.07440	−0.537202	−	0.843454i	$-0.680519\pi$
$947$	−33.0630	−1.07440	−0.537202	+	0.843454i	$0.680519\pi$
$948$	0	0
$949$	−31.7776	−1.03155
$950$	−4.20954	−0.136576
$951$	0	0
$952$	0.470588	0.0152518
$953$	5.14501	0.166663	0.0833316	−	0.996522i	$-0.473444\pi$
$953$	5.14501	0.166663	0.0833316	+	0.996522i	$0.473444\pi$
$954$	0	0
$955$	89.7162	2.90315
$956$	59.5228	1.92511
$957$	0	0
$958$	1.95683	0.0632223
$959$	−10.1963	−0.329255
$960$	0	0
$961$	−23.1490	−0.746741
$962$	−1.54725	−0.0498852
$963$	0	0
$964$	−1.99323	−0.0641978
$965$	85.1859	2.74223
$966$	0	0
$967$	2.82789	0.0909390	0.0454695	−	0.998966i	$-0.485522\pi$
$967$	2.82789	0.0909390	0.0454695	+	0.998966i	$0.485522\pi$
$968$	−0.738440	−0.0237344
$969$	0	0
$970$	−1.86647	−0.0599288
$971$	43.0152	1.38042	0.690211	−	0.723608i	$-0.257518\pi$
$971$	43.0152	1.38042	0.690211	+	0.723608i	$0.257518\pi$
$972$	0	0
$973$	14.0579	0.450676
$974$	−0.131263	−0.00420592
$975$	0	0
$976$	−38.0991	−1.21952
$977$	5.45876	0.174641	0.0873207	−	0.996180i	$-0.472170\pi$
$977$	5.45876	0.174641	0.0873207	+	0.996180i	$0.472170\pi$
$978$	0	0
$979$	6.99933	0.223700
$980$	−56.1643	−1.79410
$981$	0	0
$982$	−2.62054	−0.0836246
$983$	−16.8518	−0.537489	−0.268745	−	0.963211i	$-0.586609\pi$
$983$	−16.8518	−0.537489	−0.268745	+	0.963211i	$0.586609\pi$
$984$	0	0
$985$	−70.4002	−2.24314
$986$	−1.38359	−0.0440625
$987$	0	0
$988$	14.4492	0.459690
$989$	−13.1658	−0.418648
$990$	0	0
$991$	57.3144	1.82065	0.910326	−	0.413892i	$-0.135831\pi$
$991$	57.3144	1.82065	0.910326	+	0.413892i	$0.135831\pi$
$992$	2.75310	0.0874111
$993$	0	0
$994$	0.440929	0.0139854
$995$	−20.6916	−0.655969
$996$	0	0
$997$	−39.3642	−1.24668	−0.623339	−	0.781952i	$-0.714224\pi$
$997$	−39.3642	−1.24668	−0.623339	+	0.781952i	$0.714224\pi$
$998$	−1.42613	−0.0451434
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.7		12
3.2	odd	2		241.2.a.b.1.6	✓	12
12.11	even	2		3856.2.a.n.1.4		12
15.14	odd	2		6025.2.a.h.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.6	✓	12	3.2	odd	2
2169.2.a.h.1.7		12	1.1	even	1	trivial
3856.2.a.n.1.4		12	12.11	even	2
6025.2.a.h.1.7		12	15.14	odd	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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0822506 q^{2} -1.99323 q^{4} -4.31963 q^{5} +0.690569 q^{7} +0.328446 q^{8} +O(q^{10})\)	\(q-0.0822506 q^{2} -1.99323 q^{4} -4.31963 q^{5} +0.690569 q^{7} +0.328446 q^{8} +0.355292 q^{10} +2.95833 q^{11} -1.93470 q^{13} -0.0567997 q^{14} +3.95945 q^{16} +2.07477 q^{17} +3.74689 q^{19} +8.61004 q^{20} -0.243324 q^{22} -4.34832 q^{23} +13.6592 q^{25} +0.159131 q^{26} -1.37647 q^{28} +8.10772 q^{29} -2.80197 q^{31} -0.982559 q^{32} -0.170651 q^{34} -2.98300 q^{35} -9.72312 q^{37} -0.308183 q^{38} -1.41876 q^{40} +4.09401 q^{41} +3.02779 q^{43} -5.89665 q^{44} +0.357652 q^{46} -6.71240 q^{47} -6.52312 q^{49} -1.12348 q^{50} +3.85632 q^{52} +0.0484656 q^{53} -12.7789 q^{55} +0.226814 q^{56} -0.666864 q^{58} -4.50676 q^{59} -9.62232 q^{61} +0.230464 q^{62} -7.83809 q^{64} +8.35721 q^{65} -0.964160 q^{67} -4.13551 q^{68} +0.245353 q^{70} -7.76289 q^{71} +16.4250 q^{73} +0.799732 q^{74} -7.46842 q^{76} +2.04293 q^{77} -6.83310 q^{79} -17.1034 q^{80} -0.336735 q^{82} +9.15477 q^{83} -8.96225 q^{85} -0.249037 q^{86} +0.971651 q^{88} +2.36597 q^{89} -1.33605 q^{91} +8.66723 q^{92} +0.552099 q^{94} -16.1852 q^{95} -5.25335 q^{97} +0.536530 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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