LMFDB - Embedded newform 1511.2.a.b.1.80

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1511,2,Mod(1,1511)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1511, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1511.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1511.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:	$12.0653957454$
Analytic rank:	$0$
Dimension:	$87$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.80
Character	$\chi$	$=$	1511.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.57611 q^{2} -0.138176 q^{3} +4.63637 q^{4} +2.90494 q^{5} -0.355956 q^{6} +4.21629 q^{7} +6.79159 q^{8} -2.98091 q^{9} +O(q^{10})$	\(q+2.57611 q^{2} -0.138176 q^{3} +4.63637 q^{4} +2.90494 q^{5} -0.355956 q^{6} +4.21629 q^{7} +6.79159 q^{8} -2.98091 q^{9} +7.48346 q^{10} -5.78037 q^{11} -0.640633 q^{12} +2.39484 q^{13} +10.8616 q^{14} -0.401392 q^{15} +8.22317 q^{16} -1.83124 q^{17} -7.67916 q^{18} +2.36435 q^{19} +13.4684 q^{20} -0.582588 q^{21} -14.8909 q^{22} -8.99973 q^{23} -0.938432 q^{24} +3.43868 q^{25} +6.16938 q^{26} +0.826416 q^{27} +19.5483 q^{28} -3.01538 q^{29} -1.03403 q^{30} +2.24494 q^{31} +7.60067 q^{32} +0.798706 q^{33} -4.71748 q^{34} +12.2481 q^{35} -13.8206 q^{36} -3.86770 q^{37} +6.09083 q^{38} -0.330908 q^{39} +19.7292 q^{40} +9.80185 q^{41} -1.50081 q^{42} -10.2310 q^{43} -26.7999 q^{44} -8.65936 q^{45} -23.1843 q^{46} -13.3608 q^{47} -1.13624 q^{48} +10.7771 q^{49} +8.85844 q^{50} +0.253033 q^{51} +11.1034 q^{52} -8.94411 q^{53} +2.12894 q^{54} -16.7916 q^{55} +28.6353 q^{56} -0.326695 q^{57} -7.76796 q^{58} +3.29384 q^{59} -1.86100 q^{60} +11.6301 q^{61} +5.78323 q^{62} -12.5684 q^{63} +3.13384 q^{64} +6.95687 q^{65} +2.05756 q^{66} +6.52897 q^{67} -8.49030 q^{68} +1.24354 q^{69} +31.5524 q^{70} +5.85505 q^{71} -20.2451 q^{72} +12.0498 q^{73} -9.96365 q^{74} -0.475142 q^{75} +10.9620 q^{76} -24.3717 q^{77} -0.852458 q^{78} +9.58991 q^{79} +23.8878 q^{80} +8.82853 q^{81} +25.2507 q^{82} +10.1833 q^{83} -2.70109 q^{84} -5.31964 q^{85} -26.3563 q^{86} +0.416652 q^{87} -39.2579 q^{88} -8.38870 q^{89} -22.3075 q^{90} +10.0973 q^{91} -41.7261 q^{92} -0.310196 q^{93} -34.4190 q^{94} +6.86828 q^{95} -1.05023 q^{96} +1.39657 q^{97} +27.7630 q^{98} +17.2307 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 87 q + 6 q^{2} + 4 q^{3} + 110 q^{4} + 10 q^{5} + 8 q^{6} + 21 q^{7} + 15 q^{8} + 133 q^{9}+O(q^{10}) $ 87 * q + 6 * q^2 + 4 * q^3 + 110 * q^4 + 10 * q^5 + 8 * q^6 + 21 * q^7 + 15 * q^8 + 133 * q^9	\( 87 q + 6 q^{2} + 4 q^{3} + 110 q^{4} + 10 q^{5} + 8 q^{6} + 21 q^{7} + 15 q^{8} + 133 q^{9} + 25 q^{10} + 17 q^{11} + 34 q^{13} + 12 q^{14} + 16 q^{15} + 152 q^{16} + 32 q^{17} + 14 q^{18} + 56 q^{19} + 3 q^{20} + 38 q^{21} + 32 q^{22} + 8 q^{23} + 8 q^{24} + 179 q^{25} + 11 q^{26} - 2 q^{27} + 45 q^{28} + 40 q^{29} + 24 q^{30} + 31 q^{31} + 26 q^{32} + 31 q^{33} + 31 q^{34} + 22 q^{35} + 180 q^{36} + 35 q^{37} - 15 q^{38} + 59 q^{39} + 42 q^{40} + 45 q^{41} - 30 q^{42} + 82 q^{43} + 25 q^{44} + 20 q^{45} + 69 q^{46} - 7 q^{47} - 39 q^{48} + 222 q^{49} + 17 q^{50} + 53 q^{51} + 54 q^{52} + 16 q^{53} - 7 q^{54} + 49 q^{55} + 12 q^{56} + 52 q^{57} + 17 q^{58} - 7 q^{59} - 6 q^{60} + 131 q^{61} - 8 q^{62} + 19 q^{63} + 213 q^{64} + 57 q^{65} + 17 q^{66} + 38 q^{67} + 13 q^{68} + 45 q^{69} - 5 q^{71} + 4 q^{72} + 91 q^{73} + q^{74} - 44 q^{75} + 150 q^{76} + 5 q^{77} - 87 q^{78} + 120 q^{79} - 41 q^{80} + 247 q^{81} + 20 q^{82} - 33 q^{83} - 16 q^{84} + 110 q^{85} - 22 q^{86} - 13 q^{87} + 78 q^{88} + 53 q^{89} - 33 q^{90} + 32 q^{91} - 31 q^{92} + 13 q^{93} + 79 q^{94} - 25 q^{95} - 51 q^{96} + 92 q^{97} - 36 q^{98} + 35 q^{99}+O(q^{100}) \) 87 * q + 6 * q^2 + 4 * q^3 + 110 * q^4 + 10 * q^5 + 8 * q^6 + 21 * q^7 + 15 * q^8 + 133 * q^9 + 25 * q^10 + 17 * q^11 + 34 * q^13 + 12 * q^14 + 16 * q^15 + 152 * q^16 + 32 * q^17 + 14 * q^18 + 56 * q^19 + 3 * q^20 + 38 * q^21 + 32 * q^22 + 8 * q^23 + 8 * q^24 + 179 * q^25 + 11 * q^26 - 2 * q^27 + 45 * q^28 + 40 * q^29 + 24 * q^30 + 31 * q^31 + 26 * q^32 + 31 * q^33 + 31 * q^34 + 22 * q^35 + 180 * q^36 + 35 * q^37 - 15 * q^38 + 59 * q^39 + 42 * q^40 + 45 * q^41 - 30 * q^42 + 82 * q^43 + 25 * q^44 + 20 * q^45 + 69 * q^46 - 7 * q^47 - 39 * q^48 + 222 * q^49 + 17 * q^50 + 53 * q^51 + 54 * q^52 + 16 * q^53 - 7 * q^54 + 49 * q^55 + 12 * q^56 + 52 * q^57 + 17 * q^58 - 7 * q^59 - 6 * q^60 + 131 * q^61 - 8 * q^62 + 19 * q^63 + 213 * q^64 + 57 * q^65 + 17 * q^66 + 38 * q^67 + 13 * q^68 + 45 * q^69 - 5 * q^71 + 4 * q^72 + 91 * q^73 + q^74 - 44 * q^75 + 150 * q^76 + 5 * q^77 - 87 * q^78 + 120 * q^79 - 41 * q^80 + 247 * q^81 + 20 * q^82 - 33 * q^83 - 16 * q^84 + 110 * q^85 - 22 * q^86 - 13 * q^87 + 78 * q^88 + 53 * q^89 - 33 * q^90 + 32 * q^91 - 31 * q^92 + 13 * q^93 + 79 * q^94 - 25 * q^95 - 51 * q^96 + 92 * q^97 - 36 * q^98 + 35 * q^99

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 $

$2$	2.57611	1.82159	0.910794	−	0.412861i	$-0.135470\pi$
$2$	2.57611	1.82159	0.910794	+	0.412861i	$0.135470\pi$
$3$	−0.138176	−0.0797757	−0.0398879	−	0.999204i	$-0.512700\pi$
$3$	−0.138176	−0.0797757	−0.0398879	+	0.999204i	$0.512700\pi$
$4$	4.63637	2.31818
$5$	2.90494	1.29913	0.649565	−	0.760306i	$-0.274951\pi$
$5$	2.90494	1.29913	0.649565	+	0.760306i	$0.274951\pi$
$6$	−0.355956	−0.145319
$7$	4.21629	1.59361	0.796804	−	0.604238i	$-0.206522\pi$
$7$	4.21629	1.59361	0.796804	+	0.604238i	$0.206522\pi$
$8$	6.79159	2.40119
$9$	−2.98091	−0.993636
$10$	7.48346	2.36648
$11$	−5.78037	−1.74285	−0.871423	−	0.490532i	$-0.836802\pi$
$11$	−5.78037	−1.74285	−0.871423	+	0.490532i	$0.836802\pi$
$12$	−0.640633	−0.184935
$13$	2.39484	0.664209	0.332104	−	0.943243i	$-0.392241\pi$
$13$	2.39484	0.664209	0.332104	+	0.943243i	$0.392241\pi$
$14$	10.8616	2.90290
$15$	−0.401392	−0.103639
$16$	8.22317	2.05579
$17$	−1.83124	−0.444141	−0.222070	−	0.975031i	$-0.571281\pi$
$17$	−1.83124	−0.444141	−0.222070	+	0.975031i	$0.571281\pi$
$18$	−7.67916	−1.81000
$19$	2.36435	0.542418	0.271209	−	0.962520i	$-0.412577\pi$
$19$	2.36435	0.542418	0.271209	+	0.962520i	$0.412577\pi$
$20$	13.4684	3.01162
$21$	−0.582588	−0.127131
$22$	−14.8909	−3.17475
$23$	−8.99973	−1.87657	−0.938287	−	0.345858i	$-0.887588\pi$
$23$	−8.99973	−1.87657	−0.938287	+	0.345858i	$0.887588\pi$
$24$	−0.938432	−0.191557
$25$	3.43868	0.687736
$26$	6.16938	1.20992
$27$	0.826416	0.159044
$28$	19.5483	3.69428
$29$	−3.01538	−0.559941	−0.279971	−	0.960009i	$-0.590325\pi$
$29$	−3.01538	−0.559941	−0.279971	+	0.960009i	$0.590325\pi$
$30$	−1.03403	−0.188788
$31$	2.24494	0.403204	0.201602	−	0.979468i	$-0.435385\pi$
$31$	2.24494	0.403204	0.201602	+	0.979468i	$0.435385\pi$
$32$	7.60067	1.34362
$33$	0.798706	0.139037
$34$	−4.71748	−0.809042
$35$	12.2481	2.07030
$36$	−13.8206	−2.30343
$37$	−3.86770	−0.635847	−0.317923	−	0.948116i	$-0.602985\pi$
$37$	−3.86770	−0.635847	−0.317923	+	0.948116i	$0.602985\pi$
$38$	6.09083	0.988062
$39$	−0.330908	−0.0529878
$40$	19.7292	3.11945
$41$	9.80185	1.53079	0.765396	−	0.643560i	$-0.222543\pi$
$41$	9.80185	1.53079	0.765396	+	0.643560i	$0.222543\pi$
$42$	−1.50081	−0.231581
$43$	−10.2310	−1.56021	−0.780107	−	0.625646i	$-0.784836\pi$
$43$	−10.2310	−1.56021	−0.780107	+	0.625646i	$0.784836\pi$
$44$	−26.7999	−4.04024
$45$	−8.65936	−1.29086
$46$	−23.1843	−3.41834
$47$	−13.3608	−1.94887	−0.974437	−	0.224662i	$-0.927872\pi$
$47$	−13.3608	−1.94887	−0.974437	+	0.224662i	$0.927872\pi$
$48$	−1.13624	−0.164002
$49$	10.7771	1.53959
$50$	8.85844	1.25277
$51$	0.253033	0.0354317
$52$	11.1034	1.53976
$53$	−8.94411	−1.22857	−0.614284	−	0.789085i	$-0.710555\pi$
$53$	−8.94411	−1.22857	−0.614284	+	0.789085i	$0.710555\pi$
$54$	2.12894	0.289712
$55$	−16.7916	−2.26418
$56$	28.6353	3.82655
$57$	−0.326695	−0.0432718
$58$	−7.76796	−1.01998
$59$	3.29384	0.428821	0.214410	−	0.976744i	$-0.431217\pi$
$59$	3.29384	0.428821	0.214410	+	0.976744i	$0.431217\pi$
$60$	−1.86100	−0.240254
$61$	11.6301	1.48908	0.744540	−	0.667578i	$-0.232669\pi$
$61$	11.6301	1.48908	0.744540	+	0.667578i	$0.232669\pi$
$62$	5.78323	0.734471
$63$	−12.5684	−1.58347
$64$	3.13384	0.391730
$65$	6.95687	0.862893
$66$	2.05756	0.253268
$67$	6.52897	0.797641	0.398821	−	0.917029i	$-0.369420\pi$
$67$	6.52897	0.797641	0.398821	+	0.917029i	$0.369420\pi$
$68$	−8.49030	−1.02960
$69$	1.24354	0.149705
$70$	31.5524	3.77124
$71$	5.85505	0.694866	0.347433	−	0.937705i	$-0.387053\pi$
$71$	5.85505	0.694866	0.347433	+	0.937705i	$0.387053\pi$
$72$	−20.2451	−2.38591
$73$	12.0498	1.41032	0.705159	−	0.709049i	$-0.250875\pi$
$73$	12.0498	1.41032	0.705159	+	0.709049i	$0.250875\pi$
$74$	−9.96365	−1.15825
$75$	−0.475142	−0.0548647
$76$	10.9620	1.25742
$77$	−24.3717	−2.77741
$78$	−0.852458	−0.0965219
$79$	9.58991	1.07895	0.539475	−	0.842002i	$-0.318623\pi$
$79$	9.58991	1.07895	0.539475	+	0.842002i	$0.318623\pi$
$80$	23.8878	2.67074
$81$	8.82853	0.980948
$82$	25.2507	2.78847
$83$	10.1833	1.11776	0.558879	−	0.829249i	$-0.311232\pi$
$83$	10.1833	1.11776	0.558879	+	0.829249i	$0.311232\pi$
$84$	−2.70109	−0.294714
$85$	−5.31964	−0.576996
$86$	−26.3563	−2.84207
$87$	0.416652	0.0446697
$88$	−39.2579	−4.18490
$89$	−8.38870	−0.889201	−0.444600	−	0.895729i	$-0.646654\pi$
$89$	−8.38870	−0.889201	−0.444600	+	0.895729i	$0.646654\pi$
$90$	−22.3075	−2.35142
$91$	10.0973	1.05849
$92$	−41.7261	−4.35024
$93$	−0.310196	−0.0321659
$94$	−34.4190	−3.55005
$95$	6.86828	0.704671
$96$	−1.05023	−0.107188
$97$	1.39657	0.141800	0.0709000	−	0.997483i	$-0.477413\pi$
$97$	1.39657	0.141800	0.0709000	+	0.997483i	$0.477413\pi$
$98$	27.7630	2.80449
$99$	17.2307	1.73175
$100$	15.9430	1.59430
$101$	2.89326	0.287890	0.143945	−	0.989586i	$-0.454021\pi$
$101$	2.89326	0.287890	0.143945	+	0.989586i	$0.454021\pi$
$102$	0.651841	0.0645419
$103$	6.63679	0.653942	0.326971	−	0.945034i	$-0.393972\pi$
$103$	6.63679	0.653942	0.326971	+	0.945034i	$0.393972\pi$
$104$	16.2648	1.59489
$105$	−1.69239	−0.165160
$106$	−23.0411	−2.23795
$107$	5.31240	0.513569	0.256785	−	0.966469i	$-0.417337\pi$
$107$	5.31240	0.513569	0.256785	+	0.966469i	$0.417337\pi$
$108$	3.83157	0.368693
$109$	−4.18516	−0.400865	−0.200433	−	0.979707i	$-0.564235\pi$
$109$	−4.18516	−0.400865	−0.200433	+	0.979707i	$0.564235\pi$
$110$	−43.2572	−4.12441
$111$	0.534422	0.0507251
$112$	34.6713	3.27613
$113$	−6.29190	−0.591892	−0.295946	−	0.955205i	$-0.595635\pi$
$113$	−6.29190	−0.591892	−0.295946	+	0.955205i	$0.595635\pi$
$114$	−0.841604	−0.0788234
$115$	−26.1437	−2.43791
$116$	−13.9804	−1.29805
$117$	−7.13879	−0.659982
$118$	8.48530	0.781135
$119$	−7.72104	−0.707786
$120$	−2.72609	−0.248857
$121$	22.4126	2.03751
$122$	29.9604	2.71249
$123$	−1.35438	−0.122120
$124$	10.4084	0.934700
$125$	−4.53554	−0.405671
$126$	−32.3776	−2.88442
$127$	2.98393	0.264781	0.132390	−	0.991198i	$-0.457735\pi$
$127$	2.98393	0.264781	0.132390	+	0.991198i	$0.457735\pi$
$128$	−7.12819	−0.630049
$129$	1.41368	0.124467
$130$	17.9217	1.57184
$131$	−4.32868	−0.378199	−0.189099	−	0.981958i	$-0.560557\pi$
$131$	−4.32868	−0.378199	−0.189099	+	0.981958i	$0.560557\pi$
$132$	3.70309	0.322313
$133$	9.96877	0.864402
$134$	16.8194	1.45297
$135$	2.40069	0.206618
$136$	−12.4370	−1.06647
$137$	19.5671	1.67173	0.835864	−	0.548937i	$-0.184967\pi$
$137$	19.5671	1.67173	0.835864	+	0.548937i	$0.184967\pi$
$138$	3.20351	0.272701
$139$	0.203114	0.0172279	0.00861395	−	0.999963i	$-0.497258\pi$
$139$	0.203114	0.0172279	0.00861395	+	0.999963i	$0.497258\pi$
$140$	56.7866	4.79934
$141$	1.84614	0.155473
$142$	15.0833	1.26576
$143$	−13.8430	−1.15761
$144$	−24.5125	−2.04271
$145$	−8.75949	−0.727436
$146$	31.0416	2.56902
$147$	−1.48913	−0.122822
$148$	−17.9321	−1.47401
$149$	−12.3740	−1.01372	−0.506858	−	0.862030i	$-0.669193\pi$
$149$	−12.3740	−1.01372	−0.506858	+	0.862030i	$0.669193\pi$
$150$	−1.22402	−0.0999409
$151$	9.89285	0.805069	0.402534	−	0.915405i	$-0.368129\pi$
$151$	9.89285	0.805069	0.402534	+	0.915405i	$0.368129\pi$
$152$	16.0577	1.30245
$153$	5.45876	0.441314
$154$	−62.7843	−5.05930
$155$	6.52143	0.523814
$156$	−1.53421	−0.122835
$157$	16.6076	1.32543	0.662715	−	0.748872i	$-0.269404\pi$
$157$	16.6076	1.32543	0.662715	+	0.748872i	$0.269404\pi$
$158$	24.7047	1.96540
$159$	1.23586	0.0980100
$160$	22.0795	1.74554
$161$	−37.9455	−2.99052
$162$	22.7433	1.78688
$163$	−13.9835	−1.09527	−0.547636	−	0.836717i	$-0.684472\pi$
$163$	−13.9835	−1.09527	−0.547636	+	0.836717i	$0.684472\pi$
$164$	45.4450	3.54866
$165$	2.32019	0.180627
$166$	26.2332	2.03609
$167$	−7.77386	−0.601559	−0.300780	−	0.953694i	$-0.597247\pi$
$167$	−7.77386	−0.601559	−0.300780	+	0.953694i	$0.597247\pi$
$168$	−3.95670	−0.305266
$169$	−7.26474	−0.558826
$170$	−13.7040	−1.05105
$171$	−7.04790	−0.538966
$172$	−47.4347	−3.61687
$173$	−2.49164	−0.189436	−0.0947181	−	0.995504i	$-0.530195\pi$
$173$	−2.49164	−0.189436	−0.0947181	+	0.995504i	$0.530195\pi$
$174$	1.07334	0.0813699
$175$	14.4985	1.09598
$176$	−47.5330	−3.58293
$177$	−0.455128	−0.0342095
$178$	−21.6103	−1.61976
$179$	13.4325	1.00399	0.501994	−	0.864871i	$-0.332600\pi$
$179$	13.4325	1.00399	0.501994	+	0.864871i	$0.332600\pi$
$180$	−40.1480	−2.99245
$181$	16.5089	1.22710	0.613548	−	0.789657i	$-0.289742\pi$
$181$	16.5089	1.22710	0.613548	+	0.789657i	$0.289742\pi$
$182$	26.0119	1.92813
$183$	−1.60699	−0.118792
$184$	−61.1225	−4.50601
$185$	−11.2355	−0.826047
$186$	−0.799101	−0.0585930
$187$	10.5852	0.774069
$188$	−61.9456	−4.51785
$189$	3.48441	0.253453
$190$	17.6935	1.28362
$191$	−1.04756	−0.0757988	−0.0378994	−	0.999282i	$-0.512067\pi$
$191$	−1.04756	−0.0757988	−0.0378994	+	0.999282i	$0.512067\pi$
$192$	−0.433021	−0.0312506
$193$	−11.5562	−0.831835	−0.415918	−	0.909402i	$-0.636540\pi$
$193$	−11.5562	−0.831835	−0.415918	+	0.909402i	$0.636540\pi$
$194$	3.59772	0.258301
$195$	−0.961270	−0.0688379
$196$	49.9666	3.56904
$197$	−8.27467	−0.589546	−0.294773	−	0.955567i	$-0.595244\pi$
$197$	−8.27467	−0.589546	−0.294773	+	0.955567i	$0.595244\pi$
$198$	44.3884	3.15454
$199$	−2.93201	−0.207844	−0.103922	−	0.994585i	$-0.533139\pi$
$199$	−2.93201	−0.207844	−0.103922	+	0.994585i	$0.533139\pi$
$200$	23.3541	1.65139
$201$	−0.902145	−0.0636324
$202$	7.45338	0.524418
$203$	−12.7137	−0.892327
$204$	1.17315	0.0821371
$205$	28.4738	1.98870
$206$	17.0971	1.19121
$207$	26.8274	1.86463
$208$	19.6932	1.36548
$209$	−13.6668	−0.945351
$210$	−4.35978	−0.300853
$211$	14.9653	1.03025	0.515127	−	0.857114i	$-0.327745\pi$
$211$	14.9653	1.03025	0.515127	+	0.857114i	$0.327745\pi$
$212$	−41.4682	−2.84805
$213$	−0.809025	−0.0554335
$214$	13.6854	0.935512
$215$	−29.7205	−2.02692
$216$	5.61267	0.381894
$217$	9.46533	0.642548
$218$	−10.7814	−0.730212
$219$	−1.66498	−0.112509
$220$	−77.8522	−5.24879
$221$	−4.38553	−0.295002
$222$	1.37673	0.0924003
$223$	−25.6703	−1.71901	−0.859505	−	0.511127i	$-0.829228\pi$
$223$	−25.6703	−1.71901	−0.859505	+	0.511127i	$0.829228\pi$
$224$	32.0466	2.14120
$225$	−10.2504	−0.683360
$226$	−16.2087	−1.07818
$227$	−18.5721	−1.23267	−0.616335	−	0.787484i	$-0.711383\pi$
$227$	−18.5721	−1.23267	−0.616335	+	0.787484i	$0.711383\pi$
$228$	−1.51468	−0.100312
$229$	−15.3137	−1.01196	−0.505979	−	0.862546i	$-0.668869\pi$
$229$	−15.3137	−1.01196	−0.505979	+	0.862546i	$0.668869\pi$
$230$	−67.3491	−4.44087
$231$	3.36758	0.221570
$232$	−20.4792	−1.34453
$233$	−2.24420	−0.147023	−0.0735113	−	0.997294i	$-0.523420\pi$
$233$	−2.24420	−0.147023	−0.0735113	+	0.997294i	$0.523420\pi$
$234$	−18.3904	−1.20222
$235$	−38.8123	−2.53184
$236$	15.2714	0.994086
$237$	−1.32509	−0.0860740
$238$	−19.8903	−1.28930
$239$	−18.1907	−1.17666	−0.588330	−	0.808621i	$-0.700214\pi$
$239$	−18.1907	−1.17666	−0.588330	+	0.808621i	$0.700214\pi$
$240$	−3.30072	−0.213060
$241$	4.52412	0.291424	0.145712	−	0.989327i	$-0.453453\pi$
$241$	4.52412	0.291424	0.145712	+	0.989327i	$0.453453\pi$
$242$	57.7375	3.71151
$243$	−3.69913	−0.237300
$244$	53.9214	3.45196
$245$	31.3068	2.00012
$246$	−3.48903	−0.222452
$247$	5.66223	0.360279
$248$	15.2467	0.968168
$249$	−1.40708	−0.0891700
$250$	−11.6841	−0.738965
$251$	13.1324	0.828908	0.414454	−	0.910070i	$-0.363973\pi$
$251$	13.1324	0.828908	0.414454	+	0.910070i	$0.363973\pi$
$252$	−58.2716	−3.67077
$253$	52.0217	3.27058
$254$	7.68694	0.482321
$255$	0.735045	0.0460303
$256$	−24.6307	−1.53942
$257$	12.8042	0.798705	0.399353	−	0.916797i	$-0.369235\pi$
$257$	12.8042	0.798705	0.399353	+	0.916797i	$0.369235\pi$
$258$	3.64179	0.226728
$259$	−16.3074	−1.01329
$260$	32.2546	2.00035
$261$	8.98856	0.556378
$262$	−11.1512	−0.688923
$263$	−16.9614	−1.04589	−0.522943	−	0.852368i	$-0.675166\pi$
$263$	−16.9614	−1.04589	−0.522943	+	0.852368i	$0.675166\pi$
$264$	5.42448	0.333854
$265$	−25.9821	−1.59607
$266$	25.6807	1.57458
$267$	1.15911	0.0709367
$268$	30.2707	1.84908
$269$	29.5344	1.80074	0.900371	−	0.435123i	$-0.143295\pi$
$269$	29.5344	1.80074	0.900371	+	0.435123i	$0.143295\pi$
$270$	6.18445	0.376374
$271$	−11.4390	−0.694872	−0.347436	−	0.937704i	$-0.612948\pi$
$271$	−11.4390	−0.694872	−0.347436	+	0.937704i	$0.612948\pi$
$272$	−15.0586	−0.913062
$273$	−1.39521	−0.0844417
$274$	50.4070	3.04520
$275$	−19.8768	−1.19862
$276$	5.76552	0.347044
$277$	−25.5392	−1.53450	−0.767252	−	0.641346i	$-0.778377\pi$
$277$	−25.5392	−1.53450	−0.767252	+	0.641346i	$0.778377\pi$
$278$	0.523245	0.0313821
$279$	−6.69197	−0.400638
$280$	83.1839	4.97119
$281$	21.3362	1.27281	0.636404	−	0.771356i	$-0.280421\pi$
$281$	21.3362	1.27281	0.636404	+	0.771356i	$0.280421\pi$
$282$	4.75586	0.283207
$283$	7.24867	0.430889	0.215444	−	0.976516i	$-0.430880\pi$
$283$	7.24867	0.430889	0.215444	+	0.976516i	$0.430880\pi$
$284$	27.1462	1.61083
$285$	−0.949030	−0.0562157
$286$	−35.6613	−2.10870
$287$	41.3274	2.43948
$288$	−22.6569	−1.33507
$289$	−13.6466	−0.802739
$290$	−22.5655	−1.32509
$291$	−0.192972	−0.0113122
$292$	55.8672	3.26938
$293$	12.2196	0.713877	0.356939	−	0.934128i	$-0.383821\pi$
$293$	12.2196	0.713877	0.356939	+	0.934128i	$0.383821\pi$
$294$	−3.83618	−0.223730
$295$	9.56840	0.557094
$296$	−26.2679	−1.52679
$297$	−4.77699	−0.277189
$298$	−31.8768	−1.84657
$299$	−21.5529	−1.24644
$300$	−2.20293	−0.127186
$301$	−43.1369	−2.48637
$302$	25.4851	1.46650
$303$	−0.399778	−0.0229667
$304$	19.4424	1.11510
$305$	33.7847	1.93451
$306$	14.0624	0.803893
$307$	−25.3574	−1.44722	−0.723612	−	0.690207i	$-0.757520\pi$
$307$	−25.3574	−1.44722	−0.723612	+	0.690207i	$0.757520\pi$
$308$	−112.996	−6.43855
$309$	−0.917043	−0.0521687
$310$	16.7999	0.954173
$311$	−6.66181	−0.377757	−0.188878	−	0.982001i	$-0.560485\pi$
$311$	−6.66181	−0.377757	−0.188878	+	0.982001i	$0.560485\pi$
$312$	−2.24739	−0.127234
$313$	6.97271	0.394121	0.197060	−	0.980391i	$-0.436860\pi$
$313$	6.97271	0.394121	0.197060	+	0.980391i	$0.436860\pi$
$314$	42.7830	2.41439
$315$	−36.5104	−2.05713
$316$	44.4624	2.50120
$317$	25.9650	1.45834	0.729169	−	0.684334i	$-0.239907\pi$
$317$	25.9650	1.45834	0.729169	+	0.684334i	$0.239907\pi$
$318$	3.18371	0.178534
$319$	17.4300	0.975892
$320$	9.10363	0.508908
$321$	−0.734045	−0.0409704
$322$	−97.7519	−5.44750
$323$	−4.32968	−0.240910
$324$	40.9323	2.27402
$325$	8.23509	0.456801
$326$	−36.0231	−1.99513
$327$	0.578287	0.0319793
$328$	66.5701	3.67572
$329$	−56.3330	−3.10574
$330$	5.97708	0.329028
$331$	1.10765	0.0608822	0.0304411	−	0.999537i	$-0.490309\pi$
$331$	1.10765	0.0608822	0.0304411	+	0.999537i	$0.490309\pi$
$332$	47.2133	2.59117
$333$	11.5293	0.631800
$334$	−20.0264	−1.09579
$335$	18.9663	1.03624
$336$	−4.79073	−0.261356
$337$	−31.0200	−1.68977	−0.844883	−	0.534952i	$-0.820330\pi$
$337$	−31.0200	−1.68977	−0.844883	+	0.534952i	$0.820330\pi$
$338$	−18.7148	−1.01795
$339$	0.869387	0.0472186
$340$	−24.6638	−1.33758
$341$	−12.9766	−0.702722
$342$	−18.1562	−0.981774
$343$	15.9253	0.859888
$344$	−69.4848	−3.74637
$345$	3.61242	0.194486
$346$	−6.41876	−0.345075
$347$	−17.2446	−0.925739	−0.462870	−	0.886426i	$-0.653180\pi$
$347$	−17.2446	−0.925739	−0.462870	+	0.886426i	$0.653180\pi$
$348$	1.93175	0.103553
$349$	8.97990	0.480683	0.240342	−	0.970688i	$-0.422741\pi$
$349$	8.97990	0.480683	0.240342	+	0.970688i	$0.422741\pi$
$350$	37.3498	1.99643
$351$	1.97913	0.105638
$352$	−43.9346	−2.34172
$353$	16.7055	0.889146	0.444573	−	0.895743i	$-0.353355\pi$
$353$	16.7055	0.889146	0.444573	+	0.895743i	$0.353355\pi$
$354$	−1.17246	−0.0623156
$355$	17.0086	0.902721
$356$	−38.8931	−2.06133
$357$	1.06686	0.0564642
$358$	34.6035	1.82885
$359$	3.63293	0.191739	0.0958693	−	0.995394i	$-0.469437\pi$
$359$	3.63293	0.191739	0.0958693	+	0.995394i	$0.469437\pi$
$360$	−58.8108	−3.09960
$361$	−13.4099	−0.705783
$362$	42.5288	2.23526
$363$	−3.09688	−0.162544
$364$	46.8150	2.45377
$365$	35.0039	1.83219
$366$	−4.13980	−0.216391
$367$	10.8105	0.564303	0.282152	−	0.959370i	$-0.408952\pi$
$367$	10.8105	0.564303	0.282152	+	0.959370i	$0.408952\pi$
$368$	−74.0063	−3.85785
$369$	−29.2184	−1.52105
$370$	−28.9438	−1.50472
$371$	−37.7110	−1.95786
$372$	−1.43818	−0.0745664
$373$	−25.4319	−1.31681	−0.658407	−	0.752662i	$-0.728770\pi$
$373$	−25.4319	−1.31681	−0.658407	+	0.752662i	$0.728770\pi$
$374$	27.2688	1.41004
$375$	0.626700	0.0323627
$376$	−90.7411	−4.67961
$377$	−7.22134	−0.371918
$378$	8.97624	0.461688
$379$	33.2190	1.70635	0.853173	−	0.521628i	$-0.174675\pi$
$379$	33.2190	1.70635	0.853173	+	0.521628i	$0.174675\pi$
$380$	31.8439	1.63356
$381$	−0.412306	−0.0211231
$382$	−2.69864	−0.138074
$383$	−15.2746	−0.780497	−0.390249	−	0.920710i	$-0.627611\pi$
$383$	−15.2746	−0.780497	−0.390249	+	0.920710i	$0.627611\pi$
$384$	0.984943	0.0502626
$385$	−70.7984	−3.60822
$386$	−29.7702	−1.51526
$387$	30.4977	1.55029
$388$	6.47500	0.328718
$389$	−18.1625	−0.920878	−0.460439	−	0.887691i	$-0.652308\pi$
$389$	−18.1625	−0.920878	−0.460439	+	0.887691i	$0.652308\pi$
$390$	−2.47634	−0.125394
$391$	16.4807	0.833463
$392$	73.1936	3.69684
$393$	0.598119	0.0301711
$394$	−21.3165	−1.07391
$395$	27.8581	1.40169
$396$	79.8880	4.01453
$397$	12.6468	0.634726	0.317363	−	0.948304i	$-0.397203\pi$
$397$	12.6468	0.634726	0.317363	+	0.948304i	$0.397203\pi$
$398$	−7.55319	−0.378607
$399$	−1.37744	−0.0689583
$400$	28.2769	1.41384
$401$	11.1564	0.557123	0.278561	−	0.960418i	$-0.410142\pi$
$401$	11.1564	0.557123	0.278561	+	0.960418i	$0.410142\pi$
$402$	−2.32403	−0.115912
$403$	5.37628	0.267811
$404$	13.4142	0.667383
$405$	25.6464	1.27438
$406$	−32.7520	−1.62545
$407$	22.3567	1.10818
$408$	1.71849	0.0850781
$409$	7.93371	0.392297	0.196148	−	0.980574i	$-0.437157\pi$
$409$	7.93371	0.392297	0.196148	+	0.980574i	$0.437157\pi$
$410$	73.3518	3.62259
$411$	−2.70369	−0.133363
$412$	30.7706	1.51596
$413$	13.8878	0.683372
$414$	69.1104	3.39659
$415$	29.5818	1.45211
$416$	18.2024	0.892445
$417$	−0.0280654	−0.00137437
$418$	−35.2072	−1.72204
$419$	−28.7052	−1.40234	−0.701170	−	0.712994i	$-0.747339\pi$
$419$	−28.7052	−1.40234	−0.701170	+	0.712994i	$0.747339\pi$
$420$	−7.84652	−0.382871
$421$	−2.67489	−0.130366	−0.0651831	−	0.997873i	$-0.520763\pi$
$421$	−2.67489	−0.130366	−0.0651831	+	0.997873i	$0.520763\pi$
$422$	38.5523	1.87670
$423$	39.8273	1.93647
$424$	−60.7447	−2.95003
$425$	−6.29705	−0.305452
$426$	−2.08414	−0.100977
$427$	49.0358	2.37301
$428$	24.6303	1.19055
$429$	1.91277	0.0923495
$430$	−76.5634	−3.69221
$431$	22.5164	1.08457	0.542287	−	0.840193i	$-0.317558\pi$
$431$	22.5164	1.08457	0.542287	+	0.840193i	$0.317558\pi$
$432$	6.79576	0.326961
$433$	21.0314	1.01070	0.505352	−	0.862913i	$-0.331363\pi$
$433$	21.0314	1.01070	0.505352	+	0.862913i	$0.331363\pi$
$434$	24.3838	1.17046
$435$	1.21035	0.0580318
$436$	−19.4039	−0.929280
$437$	−21.2785	−1.01789
$438$	−4.28919	−0.204945
$439$	−14.8487	−0.708690	−0.354345	−	0.935115i	$-0.615296\pi$
$439$	−14.8487	−0.708690	−0.354345	+	0.935115i	$0.615296\pi$
$440$	−114.042	−5.43673
$441$	−32.1255	−1.52979
$442$	−11.2976	−0.537373
$443$	27.0028	1.28294	0.641471	−	0.767147i	$-0.278324\pi$
$443$	27.0028	1.28294	0.641471	+	0.767147i	$0.278324\pi$
$444$	2.47778	0.117590
$445$	−24.3687	−1.15519
$446$	−66.1296	−3.13133
$447$	1.70978	0.0808699
$448$	13.2132	0.624265
$449$	−7.69748	−0.363266	−0.181633	−	0.983366i	$-0.558138\pi$
$449$	−7.69748	−0.363266	−0.181633	+	0.983366i	$0.558138\pi$
$450$	−26.4062	−1.24480
$451$	−56.6583	−2.66793
$452$	−29.1716	−1.37211
$453$	−1.36695	−0.0642249
$454$	−47.8438	−2.24542
$455$	29.3322	1.37511
$456$	−2.21878	−0.103904
$457$	16.3008	0.762519	0.381259	−	0.924468i	$-0.375490\pi$
$457$	16.3008	0.762519	0.381259	+	0.924468i	$0.375490\pi$
$458$	−39.4499	−1.84337
$459$	−1.51337	−0.0706378
$460$	−121.212	−5.65153
$461$	35.2925	1.64373	0.821867	−	0.569679i	$-0.192933\pi$
$461$	35.2925	1.64373	0.821867	+	0.569679i	$0.192933\pi$
$462$	8.67526	0.403610
$463$	9.20688	0.427880	0.213940	−	0.976847i	$-0.431370\pi$
$463$	9.20688	0.427880	0.213940	+	0.976847i	$0.431370\pi$
$464$	−24.7960	−1.15112
$465$	−0.901102	−0.0417876
$466$	−5.78132	−0.267815
$467$	18.5605	0.858878	0.429439	−	0.903096i	$-0.358711\pi$
$467$	18.5605	0.858878	0.429439	+	0.903096i	$0.358711\pi$
$468$	−33.0981	−1.52996
$469$	27.5280	1.27113
$470$	−99.9851	−4.61197
$471$	−2.29476	−0.105737
$472$	22.3704	1.02968
$473$	59.1390	2.71921
$474$	−3.41359	−0.156791
$475$	8.13023	0.373041
$476$	−35.7976	−1.64078
$477$	26.6616	1.22075
$478$	−46.8614	−2.14339
$479$	−28.5036	−1.30236	−0.651182	−	0.758922i	$-0.725727\pi$
$479$	−28.5036	−1.30236	−0.651182	+	0.758922i	$0.725727\pi$
$480$	−3.05085	−0.139251
$481$	−9.26253	−0.422335
$482$	11.6547	0.530855
$483$	5.24314	0.238571
$484$	103.913	4.72333
$485$	4.05695	0.184216
$486$	−9.52940	−0.432262
$487$	30.8626	1.39852	0.699259	−	0.714868i	$-0.253513\pi$
$487$	30.8626	1.39852	0.699259	+	0.714868i	$0.253513\pi$
$488$	78.9867	3.57556
$489$	1.93218	0.0873761
$490$	80.6500	3.64340
$491$	−26.0297	−1.17471	−0.587353	−	0.809331i	$-0.699830\pi$
$491$	−26.0297	−1.17471	−0.587353	+	0.809331i	$0.699830\pi$
$492$	−6.27939	−0.283097
$493$	5.52188	0.248693
$494$	14.5865	0.656280
$495$	50.0543	2.24977
$496$	18.4606	0.828903
$497$	24.6866	1.10734
$498$	−3.62479	−0.162431
$499$	−23.8061	−1.06571	−0.532854	−	0.846207i	$-0.678880\pi$
$499$	−23.8061	−1.06571	−0.532854	+	0.846207i	$0.678880\pi$
$500$	−21.0284	−0.940419
$501$	1.07416	0.0479899
$502$	33.8305	1.50993
$503$	1.02256	0.0455938	0.0227969	−	0.999740i	$-0.492743\pi$
$503$	1.02256	0.0455938	0.0227969	+	0.999740i	$0.492743\pi$
$504$	−85.3592	−3.80220
$505$	8.40476	0.374007
$506$	134.014	5.95765
$507$	1.00381	0.0445808
$508$	13.8346	0.613810
$509$	18.2649	0.809577	0.404789	−	0.914410i	$-0.367345\pi$
$509$	18.2649	0.809577	0.404789	+	0.914410i	$0.367345\pi$
$510$	1.89356	0.0838483
$511$	50.8053	2.24749
$512$	−49.1952	−2.17414
$513$	1.95393	0.0862682
$514$	32.9851	1.45491
$515$	19.2795	0.849556
$516$	6.55433	0.288538
$517$	77.2303	3.39659
$518$	−42.0096	−1.84580
$519$	0.344284	0.0151124
$520$	47.2482	2.07197
$521$	19.4937	0.854035	0.427018	−	0.904243i	$-0.359564\pi$
$521$	19.4937	0.854035	0.427018	+	0.904243i	$0.359564\pi$
$522$	23.1556	1.01349
$523$	−19.4820	−0.851888	−0.425944	−	0.904750i	$-0.640058\pi$
$523$	−19.4820	−0.851888	−0.425944	+	0.904750i	$0.640058\pi$
$524$	−20.0694	−0.876735
$525$	−2.00334	−0.0874328
$526$	−43.6946	−1.90517
$527$	−4.11103	−0.179079
$528$	6.56790	0.285831
$529$	57.9951	2.52153
$530$	−66.9329	−2.90738
$531$	−9.81862	−0.426092
$532$	46.2189	2.00384
$533$	23.4739	1.01677
$534$	2.98601	0.129217
$535$	15.4322	0.667193
$536$	44.3421	1.91529
$537$	−1.85604	−0.0800939
$538$	76.0839	3.28021
$539$	−62.2956	−2.68326
$540$	11.1305	0.478979
$541$	17.7155	0.761649	0.380824	−	0.924647i	$-0.375640\pi$
$541$	17.7155	0.761649	0.380824	+	0.924647i	$0.375640\pi$
$542$	−29.4683	−1.26577
$543$	−2.28113	−0.0978925
$544$	−13.9186	−0.596757
$545$	−12.1576	−0.520776
$546$	−3.59421	−0.153818
$547$	−31.4507	−1.34474	−0.672368	−	0.740217i	$-0.734723\pi$
$547$	−31.4507	−1.34474	−0.672368	+	0.740217i	$0.734723\pi$
$548$	90.7201	3.87537
$549$	−34.6682	−1.47960
$550$	−51.2050	−2.18339
$551$	−7.12939	−0.303722
$552$	8.44563	0.359470
$553$	40.4338	1.71942
$554$	−65.7920	−2.79524
$555$	1.55247	0.0658985
$556$	0.941711	0.0399374
$557$	−42.6481	−1.80706	−0.903530	−	0.428525i	$-0.859033\pi$
$557$	−42.6481	−1.80706	−0.903530	+	0.428525i	$0.859033\pi$
$558$	−17.2393	−0.729797
$559$	−24.5016	−1.03631
$560$	100.718	4.25611
$561$	−1.46262	−0.0617519
$562$	54.9644	2.31853
$563$	28.0911	1.18390	0.591949	−	0.805976i	$-0.298359\pi$
$563$	28.0911	1.18390	0.591949	+	0.805976i	$0.298359\pi$
$564$	8.55937	0.360415
$565$	−18.2776	−0.768944
$566$	18.6734	0.784902
$567$	37.2236	1.56325
$568$	39.7651	1.66851
$569$	38.1499	1.59933	0.799663	−	0.600449i	$-0.205012\pi$
$569$	38.1499	1.59933	0.799663	+	0.600449i	$0.205012\pi$
$570$	−2.44481	−0.102402
$571$	−35.1612	−1.47145	−0.735725	−	0.677280i	$-0.763159\pi$
$571$	−35.1612	−1.47145	−0.735725	+	0.677280i	$0.763159\pi$
$572$	−64.1815	−2.68356
$573$	0.144747	0.00604691
$574$	106.464	4.44373
$575$	−30.9472	−1.29059
$576$	−9.34170	−0.389237
$577$	1.58747	0.0660874	0.0330437	−	0.999454i	$-0.489480\pi$
$577$	1.58747	0.0660874	0.0330437	+	0.999454i	$0.489480\pi$
$578$	−35.1551	−1.46226
$579$	1.59679	0.0663603
$580$	−40.6122	−1.68633
$581$	42.9356	1.78127
$582$	−0.497117	−0.0206062
$583$	51.7003	2.14121
$584$	81.8371	3.38644
$585$	−20.7378	−0.857402
$586$	31.4791	1.30039
$587$	34.5962	1.42794	0.713970	−	0.700177i	$-0.246895\pi$
$587$	34.5962	1.42794	0.713970	+	0.700177i	$0.246895\pi$
$588$	−6.90417	−0.284723
$589$	5.30782	0.218705
$590$	24.6493	1.01480
$591$	1.14336	0.0470315
$592$	−31.8048	−1.30717
$593$	−1.33225	−0.0547089	−0.0273545	−	0.999626i	$-0.508708\pi$
$593$	−1.33225	−0.0547089	−0.0273545	+	0.999626i	$0.508708\pi$
$594$	−12.3061	−0.504924
$595$	−22.4292	−0.919506
$596$	−57.3703	−2.34998
$597$	0.405132	0.0165809
$598$	−55.5228	−2.27049
$599$	−21.5190	−0.879243	−0.439621	−	0.898183i	$-0.644887\pi$
$599$	−21.5190	−0.879243	−0.439621	+	0.898183i	$0.644887\pi$
$600$	−3.22697	−0.131740
$601$	44.0581	1.79717	0.898584	−	0.438803i	$-0.144597\pi$
$601$	44.0581	1.79717	0.898584	+	0.438803i	$0.144597\pi$
$602$	−111.126	−4.52914
$603$	−19.4623	−0.792565
$604$	45.8669	1.86630
$605$	65.1074	2.64699
$606$	−1.02988	−0.0418358
$607$	−22.0221	−0.893848	−0.446924	−	0.894572i	$-0.647481\pi$
$607$	−22.0221	−0.893848	−0.446924	+	0.894572i	$0.647481\pi$
$608$	17.9706	0.728804
$609$	1.75672	0.0711860
$610$	87.0333	3.52388
$611$	−31.9970	−1.29446
$612$	25.3088	1.02305
$613$	−39.8568	−1.60980	−0.804900	−	0.593410i	$-0.797781\pi$
$613$	−39.8568	−1.60980	−0.804900	+	0.593410i	$0.797781\pi$
$614$	−65.3236	−2.63625
$615$	−3.93438	−0.158650
$616$	−165.523	−6.66909
$617$	26.4154	1.06344	0.531721	−	0.846919i	$-0.321545\pi$
$617$	26.4154	1.06344	0.531721	+	0.846919i	$0.321545\pi$
$618$	−2.36241	−0.0950300
$619$	8.64942	0.347650	0.173825	−	0.984777i	$-0.444387\pi$
$619$	8.64942	0.347650	0.173825	+	0.984777i	$0.444387\pi$
$620$	30.2357	1.21430
$621$	−7.43752	−0.298457
$622$	−17.1616	−0.688117
$623$	−35.3692	−1.41704
$624$	−2.72112	−0.108932
$625$	−30.3689	−1.21476
$626$	17.9625	0.717926
$627$	1.88842	0.0754161
$628$	76.9989	3.07259
$629$	7.08269	0.282406
$630$	−94.0549	−3.74724
$631$	−5.53218	−0.220233	−0.110116	−	0.993919i	$-0.535122\pi$
$631$	−5.53218	−0.220233	−0.110116	+	0.993919i	$0.535122\pi$
$632$	65.1307	2.59076
$633$	−2.06784	−0.0821892
$634$	66.8888	2.65649
$635$	8.66813	0.343984
$636$	5.72990	0.227205
$637$	25.8094	1.02261
$638$	44.9016	1.77767
$639$	−17.4534	−0.690444
$640$	−20.7070	−0.818515
$641$	−10.6549	−0.420843	−0.210421	−	0.977611i	$-0.567484\pi$
$641$	−10.6549	−0.420843	−0.210421	+	0.977611i	$0.567484\pi$
$642$	−1.89098	−0.0746312
$643$	−32.8430	−1.29520	−0.647602	−	0.761979i	$-0.724228\pi$
$643$	−32.8430	−1.29520	−0.647602	+	0.761979i	$0.724228\pi$
$644$	−175.929	−6.93258
$645$	4.10665	0.161699
$646$	−11.1538	−0.438839
$647$	25.7526	1.01244	0.506219	−	0.862405i	$-0.331043\pi$
$647$	25.7526	1.01244	0.506219	+	0.862405i	$0.331043\pi$
$648$	59.9598	2.35544
$649$	−19.0396	−0.747369
$650$	21.2145	0.832103
$651$	−1.30788	−0.0512598
$652$	−64.8326	−2.53904
$653$	−35.9795	−1.40799	−0.703993	−	0.710206i	$-0.748602\pi$
$653$	−35.9795	−1.40799	−0.703993	+	0.710206i	$0.748602\pi$
$654$	1.48973	0.0582532
$655$	−12.5746	−0.491329
$656$	80.6023	3.14699
$657$	−35.9192	−1.40134
$658$	−145.120	−5.65738
$659$	36.2244	1.41110	0.705551	−	0.708659i	$-0.250700\pi$
$659$	36.2244	1.41110	0.705551	+	0.708659i	$0.250700\pi$
$660$	10.7573	0.418726
$661$	−12.4979	−0.486110	−0.243055	−	0.970012i	$-0.578150\pi$
$661$	−12.4979	−0.486110	−0.243055	+	0.970012i	$0.578150\pi$
$662$	2.85344	0.110902
$663$	0.605973	0.0235340
$664$	69.1605	2.68395
$665$	28.9587	1.12297
$666$	29.7007	1.15088
$667$	27.1376	1.05077
$668$	−36.0425	−1.39453
$669$	3.54701	0.137135
$670$	48.8593	1.88760
$671$	−67.2262	−2.59524
$672$	−4.42806	−0.170816
$673$	−15.2082	−0.586233	−0.293116	−	0.956077i	$-0.594692\pi$
$673$	−15.2082	−0.586233	−0.293116	+	0.956077i	$0.594692\pi$
$674$	−79.9110	−3.07806
$675$	2.84178	0.109380
$676$	−33.6820	−1.29546
$677$	27.0864	1.04101	0.520507	−	0.853857i	$-0.325743\pi$
$677$	27.0864	1.04101	0.520507	+	0.853857i	$0.325743\pi$
$678$	2.23964	0.0860129
$679$	5.88833	0.225974
$680$	−36.1288	−1.38548
$681$	2.56621	0.0983372
$682$	−33.4292	−1.28007
$683$	−0.169165	−0.00647293	−0.00323647	−	0.999995i	$-0.501030\pi$
$683$	−0.169165	−0.00647293	−0.00323647	+	0.999995i	$0.501030\pi$
$684$	−32.6766	−1.24942
$685$	56.8412	2.17179
$686$	41.0255	1.56636
$687$	2.11598	0.0807297
$688$	−84.1314	−3.20748
$689$	−21.4197	−0.816026
$690$	9.30601	0.354274
$691$	−28.6276	−1.08904	−0.544522	−	0.838746i	$-0.683289\pi$
$691$	−28.6276	−1.08904	−0.544522	+	0.838746i	$0.683289\pi$
$692$	−11.5522	−0.439148
$693$	72.6498	2.75974
$694$	−44.4241	−1.68632
$695$	0.590034	0.0223813
$696$	2.82973	0.107260
$697$	−17.9495	−0.679887
$698$	23.1333	0.875607
$699$	0.310094	0.0117288
$700$	67.2203	2.54069
$701$	11.1182	0.419928	0.209964	−	0.977709i	$-0.432665\pi$
$701$	11.1182	0.419928	0.209964	+	0.977709i	$0.432665\pi$
$702$	5.09847	0.192429
$703$	−9.14459	−0.344895
$704$	−18.1148	−0.682726
$705$	5.36292	0.201979
$706$	43.0354	1.61966
$707$	12.1988	0.458784
$708$	−2.11014	−0.0793039
$709$	23.3720	0.877755	0.438877	−	0.898547i	$-0.355376\pi$
$709$	23.3720	0.877755	0.438877	+	0.898547i	$0.355376\pi$
$710$	43.8160	1.64439
$711$	−28.5866	−1.07208
$712$	−56.9726	−2.13514
$713$	−20.2039	−0.756641
$714$	2.74835	0.102854
$715$	−40.2132	−1.50389
$716$	62.2778	2.32743
$717$	2.51352	0.0938689
$718$	9.35884	0.349269
$719$	−15.2767	−0.569724	−0.284862	−	0.958569i	$-0.591948\pi$
$719$	−15.2767	−0.569724	−0.284862	+	0.958569i	$0.591948\pi$
$720$	−71.2074	−2.65374
$721$	27.9826	1.04213
$722$	−34.5454	−1.28565
$723$	−0.625124	−0.0232486
$724$	76.5413	2.84463
$725$	−10.3689	−0.385092
$726$	−7.97792	−0.296088
$727$	−38.2461	−1.41847	−0.709235	−	0.704972i	$-0.750959\pi$
$727$	−38.2461	−1.41847	−0.709235	+	0.704972i	$0.750959\pi$
$728$	68.5769	2.54163
$729$	−25.9745	−0.962017
$730$	90.1740	3.33749
$731$	18.7354	0.692955
$732$	−7.45062	−0.275383
$733$	−19.4280	−0.717590	−0.358795	−	0.933416i	$-0.616812\pi$
$733$	−19.4280	−0.717590	−0.358795	+	0.933416i	$0.616812\pi$
$734$	27.8491	1.02793
$735$	−4.32584	−0.159561
$736$	−68.4039	−2.52140
$737$	−37.7399	−1.39017
$738$	−75.2700	−2.77073
$739$	1.96445	0.0722633	0.0361316	−	0.999347i	$-0.488496\pi$
$739$	1.96445	0.0722633	0.0361316	+	0.999347i	$0.488496\pi$
$740$	−52.0917	−1.91493
$741$	−0.782382	−0.0287415
$742$	−97.1478	−3.56641
$743$	6.24149	0.228978	0.114489	−	0.993425i	$-0.463477\pi$
$743$	6.24149	0.228978	0.114489	+	0.993425i	$0.463477\pi$
$744$	−2.10673	−0.0772363
$745$	−35.9457	−1.31695
$746$	−65.5155	−2.39869
$747$	−30.3554	−1.11064
$748$	49.0771	1.79444
$749$	22.3986	0.818428
$750$	1.61445	0.0589515
$751$	11.9271	0.435226	0.217613	−	0.976035i	$-0.430173\pi$
$751$	11.9271	0.435226	0.217613	+	0.976035i	$0.430173\pi$
$752$	−109.868	−4.00648
$753$	−1.81457	−0.0661267
$754$	−18.6030	−0.677482
$755$	28.7381	1.04589
$756$	16.1550	0.587552
$757$	−40.0027	−1.45392	−0.726962	−	0.686678i	$-0.759068\pi$
$757$	−40.0027	−1.45392	−0.726962	+	0.686678i	$0.759068\pi$
$758$	85.5760	3.10826
$759$	−7.18814	−0.260913
$760$	46.6466	1.69205
$761$	11.7197	0.424839	0.212420	−	0.977179i	$-0.431866\pi$
$761$	11.7197	0.424839	0.212420	+	0.977179i	$0.431866\pi$
$762$	−1.06215	−0.0384775
$763$	−17.6458	−0.638822
$764$	−4.85688	−0.175716
$765$	15.8574	0.573324
$766$	−39.3492	−1.42174
$767$	7.88821	0.284827
$768$	3.40337	0.122808
$769$	−49.1782	−1.77341	−0.886706	−	0.462335i	$-0.847012\pi$
$769$	−49.1782	−1.77341	−0.886706	+	0.462335i	$0.847012\pi$
$770$	−182.385	−6.57269
$771$	−1.76923	−0.0637173
$772$	−53.5789	−1.92835
$773$	41.6392	1.49766	0.748829	−	0.662763i	$-0.230616\pi$
$773$	41.6392	1.49766	0.748829	+	0.662763i	$0.230616\pi$
$774$	78.5656	2.82398
$775$	7.71964	0.277298
$776$	9.48491	0.340488
$777$	2.25328	0.0808360
$778$	−46.7888	−1.67746
$779$	23.1750	0.830329
$780$	−4.45680	−0.159579
$781$	−33.8443	−1.21105
$782$	42.4561	1.51823
$783$	−2.49195	−0.0890552
$784$	88.6220	3.16507
$785$	48.2441	1.72190
$786$	1.54082	0.0549593
$787$	16.5783	0.590952	0.295476	−	0.955350i	$-0.404522\pi$
$787$	16.5783	0.590952	0.295476	+	0.955350i	$0.404522\pi$
$788$	−38.3644	−1.36668
$789$	2.34365	0.0834363
$790$	71.7657	2.55331
$791$	−26.5285	−0.943244
$792$	117.024	4.15827
$793$	27.8522	0.989060
$794$	32.5797	1.15621
$795$	3.59010	0.127328
$796$	−13.5939	−0.481822
$797$	33.0248	1.16980	0.584899	−	0.811106i	$-0.301134\pi$
$797$	33.0248	1.16980	0.584899	+	0.811106i	$0.301134\pi$
$798$	−3.54845	−0.125614
$799$	24.4668	0.865574
$800$	26.1363	0.924057
$801$	25.0060	0.883542
$802$	28.7401	1.01485
$803$	−69.6521	−2.45797
$804$	−4.18268	−0.147512
$805$	−110.229	−3.88507
$806$	13.8499	0.487842
$807$	−4.08093	−0.143656
$808$	19.6499	0.691279
$809$	22.7658	0.800402	0.400201	−	0.916427i	$-0.368940\pi$
$809$	22.7658	0.800402	0.400201	+	0.916427i	$0.368940\pi$
$810$	66.0680	2.32139
$811$	27.6178	0.969792	0.484896	−	0.874572i	$-0.338857\pi$
$811$	27.6178	0.969792	0.484896	+	0.874572i	$0.338857\pi$
$812$	−58.9454	−2.06858
$813$	1.58060	0.0554339
$814$	57.5936	2.01865
$815$	−40.6212	−1.42290
$816$	2.08073	0.0728402
$817$	−24.1896	−0.846289
$818$	20.4382	0.714603
$819$	−30.0992	−1.05175
$820$	132.015	4.61016
$821$	−20.8279	−0.726897	−0.363449	−	0.931614i	$-0.618401\pi$
$821$	−20.8279	−0.726897	−0.363449	+	0.931614i	$0.618401\pi$
$822$	−6.96502	−0.242933
$823$	2.49785	0.0870695	0.0435347	−	0.999052i	$-0.486138\pi$
$823$	2.49785	0.0870695	0.0435347	+	0.999052i	$0.486138\pi$
$824$	45.0743	1.57024
$825$	2.74650	0.0956207
$826$	35.7765	1.24482
$827$	−15.5908	−0.542144	−0.271072	−	0.962559i	$-0.587378\pi$
$827$	−15.5908	−0.542144	−0.271072	+	0.962559i	$0.587378\pi$
$828$	124.382	4.32256
$829$	41.9310	1.45632	0.728162	−	0.685405i	$-0.240375\pi$
$829$	41.9310	1.45632	0.728162	+	0.685405i	$0.240375\pi$
$830$	76.2060	2.64515
$831$	3.52890	0.122416
$832$	7.50505	0.260191
$833$	−19.7355	−0.683793
$834$	−0.0722997	−0.00250353
$835$	−22.5826	−0.781503
$836$	−63.3642	−2.19150
$837$	1.85526	0.0641270
$838$	−73.9479	−2.55449
$839$	6.60371	0.227985	0.113993	−	0.993482i	$-0.463636\pi$
$839$	6.60371	0.227985	0.113993	+	0.993482i	$0.463636\pi$
$840$	−11.4940	−0.396580
$841$	−19.9075	−0.686466
$842$	−6.89083	−0.237474
$843$	−2.94814	−0.101539
$844$	69.3846	2.38832
$845$	−21.1037	−0.725988
$846$	102.600	3.52745
$847$	94.4982	3.24700
$848$	−73.5490	−2.52568
$849$	−1.00159	−0.0343745
$850$	−16.2219	−0.556408
$851$	34.8083	1.19321
$852$	−3.75094	−0.128505
$853$	−3.29165	−0.112704	−0.0563520	−	0.998411i	$-0.517947\pi$
$853$	−3.29165	−0.112704	−0.0563520	+	0.998411i	$0.517947\pi$
$854$	126.322	4.32265
$855$	−20.4737	−0.700186
$856$	36.0797	1.23318
$857$	29.7850	1.01744	0.508718	−	0.860933i	$-0.330120\pi$
$857$	29.7850	1.01744	0.508718	+	0.860933i	$0.330120\pi$
$858$	4.92752	0.168223
$859$	−36.0056	−1.22849	−0.614247	−	0.789113i	$-0.710540\pi$
$859$	−36.0056	−1.22849	−0.614247	+	0.789113i	$0.710540\pi$
$860$	−137.795	−4.69877
$861$	−5.71044	−0.194611
$862$	58.0047	1.97565
$863$	−34.1539	−1.16261	−0.581305	−	0.813685i	$-0.697458\pi$
$863$	−34.1539	−1.16261	−0.581305	+	0.813685i	$0.697458\pi$
$864$	6.28131	0.213695
$865$	−7.23808	−0.246102
$866$	54.1793	1.84109
$867$	1.88562	0.0640391
$868$	43.8847	1.48955
$869$	−55.4332	−1.88044
$870$	3.11800	0.105710
$871$	15.6358	0.529800
$872$	−28.4239	−0.962554
$873$	−4.16304	−0.140898
$874$	−54.8158	−1.85417
$875$	−19.1231	−0.646480
$876$	−7.71948	−0.260817
$877$	16.4832	0.556599	0.278299	−	0.960494i	$-0.410229\pi$
$877$	16.4832	0.556599	0.278299	+	0.960494i	$0.410229\pi$
$878$	−38.2520	−1.29094
$879$	−1.68845	−0.0569501
$880$	−138.080	−4.65469
$881$	−23.4154	−0.788885	−0.394442	−	0.918921i	$-0.629062\pi$
$881$	−23.4154	−0.788885	−0.394442	+	0.918921i	$0.629062\pi$
$882$	−82.7591	−2.78664
$883$	2.42886	0.0817377	0.0408689	−	0.999165i	$-0.486987\pi$
$883$	2.42886	0.0817377	0.0408689	+	0.999165i	$0.486987\pi$
$884$	−20.3329	−0.683870
$885$	−1.32212	−0.0444426
$886$	69.5623	2.33699
$887$	34.9260	1.17270	0.586350	−	0.810058i	$-0.300564\pi$
$887$	34.9260	1.17270	0.586350	+	0.810058i	$0.300564\pi$
$888$	3.62958	0.121801
$889$	12.5811	0.421957
$890$	−62.7765	−2.10427
$891$	−51.0322	−1.70964
$892$	−119.017	−3.98498
$893$	−31.5896	−1.05710
$894$	4.40459	0.147312
$895$	39.0205	1.30431
$896$	−30.0545	−1.00405
$897$	2.97809	0.0994354
$898$	−19.8296	−0.661722
$899$	−6.76935	−0.225770
$900$	−47.5246	−1.58415
$901$	16.3788	0.545658
$902$	−145.958	−4.85988
$903$	5.96047	0.198352
$904$	−42.7320	−1.42124
$905$	47.9573	1.59416
$906$	−3.52142	−0.116991
$907$	12.3049	0.408578	0.204289	−	0.978911i	$-0.434512\pi$
$907$	12.3049	0.408578	0.204289	+	0.978911i	$0.434512\pi$
$908$	−86.1069	−2.85756
$909$	−8.62455	−0.286058
$910$	75.5630	2.50489
$911$	−20.0540	−0.664418	−0.332209	−	0.943206i	$-0.607794\pi$
$911$	−20.0540	−0.664418	−0.332209	+	0.943206i	$0.607794\pi$
$912$	−2.68647	−0.0889579
$913$	−58.8630	−1.94808
$914$	41.9927	1.38900
$915$	−4.66822	−0.154327
$916$	−71.0000	−2.34591
$917$	−18.2510	−0.602701
$918$	−3.89860	−0.128673
$919$	30.5317	1.00715	0.503574	−	0.863952i	$-0.332018\pi$
$919$	30.5317	1.00715	0.503574	+	0.863952i	$0.332018\pi$
$920$	−177.557	−5.85389
$921$	3.50378	0.115453
$922$	90.9174	2.99421
$923$	14.0219	0.461536
$924$	15.6133	0.513640
$925$	−13.2998	−0.437295
$926$	23.7180	0.779421
$927$	−19.7837	−0.649781
$928$	−22.9189	−0.752349
$929$	39.4029	1.29277	0.646384	−	0.763012i	$-0.276280\pi$
$929$	39.4029	1.29277	0.646384	+	0.763012i	$0.276280\pi$
$930$	−2.32134	−0.0761198
$931$	25.4808	0.835099
$932$	−10.4049	−0.340825
$933$	0.920500	0.0301358
$934$	47.8140	1.56452
$935$	30.7495	1.00562
$936$	−48.4838	−1.58474
$937$	−7.96646	−0.260253	−0.130127	−	0.991497i	$-0.541538\pi$
$937$	−7.96646	−0.260253	−0.130127	+	0.991497i	$0.541538\pi$
$938$	70.9154	2.31547
$939$	−0.963459	−0.0314413
$940$	−179.948	−5.86927
$941$	22.9729	0.748895	0.374447	−	0.927248i	$-0.377832\pi$
$941$	22.9729	0.748895	0.374447	+	0.927248i	$0.377832\pi$
$942$	−5.91157	−0.192610
$943$	−88.2140	−2.87264
$944$	27.0858	0.881567
$945$	10.1220	0.329269
$946$	152.349	4.95329
$947$	−48.6515	−1.58096	−0.790481	−	0.612487i	$-0.790169\pi$
$947$	−48.6515	−1.58096	−0.790481	+	0.612487i	$0.790169\pi$
$948$	−6.14361	−0.199535
$949$	28.8573	0.936746
$950$	20.9444	0.679527
$951$	−3.58773	−0.116340
$952$	−52.4381	−1.69953
$953$	26.7316	0.865920	0.432960	−	0.901413i	$-0.357469\pi$
$953$	26.7316	0.865920	0.432960	+	0.901413i	$0.357469\pi$
$954$	68.6833	2.22370
$955$	−3.04310	−0.0984725
$956$	−84.3389	−2.72772
$957$	−2.40840	−0.0778525
$958$	−73.4286	−2.37237
$959$	82.5004	2.66408
$960$	−1.25790	−0.0405985
$961$	−25.9602	−0.837427
$962$	−23.8613	−0.769321
$963$	−15.8358	−0.510301
$964$	20.9755	0.675575
$965$	−33.5702	−1.08066
$966$	13.5069	0.434578
$967$	−7.87216	−0.253152	−0.126576	−	0.991957i	$-0.540399\pi$
$967$	−7.87216	−0.253152	−0.126576	+	0.991957i	$0.540399\pi$
$968$	152.217	4.89245
$969$	0.598257	0.0192188
$970$	10.4512	0.335567
$971$	−52.0685	−1.67096	−0.835479	−	0.549522i	$-0.814810\pi$
$971$	−52.0685	−1.67096	−0.835479	+	0.549522i	$0.814810\pi$
$972$	−17.1506	−0.550104
$973$	0.856387	0.0274545
$974$	79.5056	2.54752
$975$	−1.13789	−0.0364416
$976$	95.6362	3.06124
$977$	41.7763	1.33654	0.668271	−	0.743918i	$-0.267035\pi$
$977$	41.7763	1.33654	0.668271	+	0.743918i	$0.267035\pi$
$978$	4.97751	0.159163
$979$	48.4898	1.54974
$980$	145.150	4.63665
$981$	12.4756	0.398314
$982$	−67.0556	−2.13983
$983$	−22.2321	−0.709095	−0.354547	−	0.935038i	$-0.615365\pi$
$983$	−22.2321	−0.709095	−0.354547	+	0.935038i	$0.615365\pi$
$984$	−9.19837	−0.293233
$985$	−24.0374	−0.765896
$986$	14.2250	0.453016
$987$	7.78385	0.247763
$988$	26.2522	0.835193
$989$	92.0763	2.92786
$990$	128.946	4.09816
$991$	29.2647	0.929624	0.464812	−	0.885409i	$-0.346122\pi$
$991$	29.2647	0.929624	0.464812	+	0.885409i	$0.346122\pi$
$992$	17.0631	0.541753
$993$	−0.153051	−0.00485692
$994$	63.5955	2.01713
$995$	−8.51731	−0.270017
$996$	−6.52373	−0.206712
$997$	14.4368	0.457219	0.228610	−	0.973518i	$-0.426582\pi$
$997$	14.4368	0.457219	0.228610	+	0.973518i	$0.426582\pi$
$998$	−61.3273	−1.94128
$999$	−3.19633	−0.101127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1511.2.a.b.1.80	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1511.2.a.b.1.80	✓	87	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1511.a (trivial)

\(f(q)\)	\(=\)	\(q+2.57611 q^{2} -0.138176 q^{3} +4.63637 q^{4} +2.90494 q^{5} -0.355956 q^{6} +4.21629 q^{7} +6.79159 q^{8} -2.98091 q^{9} +O(q^{10})\)	\(q+2.57611 q^{2} -0.138176 q^{3} +4.63637 q^{4} +2.90494 q^{5} -0.355956 q^{6} +4.21629 q^{7} +6.79159 q^{8} -2.98091 q^{9} +7.48346 q^{10} -5.78037 q^{11} -0.640633 q^{12} +2.39484 q^{13} +10.8616 q^{14} -0.401392 q^{15} +8.22317 q^{16} -1.83124 q^{17} -7.67916 q^{18} +2.36435 q^{19} +13.4684 q^{20} -0.582588 q^{21} -14.8909 q^{22} -8.99973 q^{23} -0.938432 q^{24} +3.43868 q^{25} +6.16938 q^{26} +0.826416 q^{27} +19.5483 q^{28} -3.01538 q^{29} -1.03403 q^{30} +2.24494 q^{31} +7.60067 q^{32} +0.798706 q^{33} -4.71748 q^{34} +12.2481 q^{35} -13.8206 q^{36} -3.86770 q^{37} +6.09083 q^{38} -0.330908 q^{39} +19.7292 q^{40} +9.80185 q^{41} -1.50081 q^{42} -10.2310 q^{43} -26.7999 q^{44} -8.65936 q^{45} -23.1843 q^{46} -13.3608 q^{47} -1.13624 q^{48} +10.7771 q^{49} +8.85844 q^{50} +0.253033 q^{51} +11.1034 q^{52} -8.94411 q^{53} +2.12894 q^{54} -16.7916 q^{55} +28.6353 q^{56} -0.326695 q^{57} -7.76796 q^{58} +3.29384 q^{59} -1.86100 q^{60} +11.6301 q^{61} +5.78323 q^{62} -12.5684 q^{63} +3.13384 q^{64} +6.95687 q^{65} +2.05756 q^{66} +6.52897 q^{67} -8.49030 q^{68} +1.24354 q^{69} +31.5524 q^{70} +5.85505 q^{71} -20.2451 q^{72} +12.0498 q^{73} -9.96365 q^{74} -0.475142 q^{75} +10.9620 q^{76} -24.3717 q^{77} -0.852458 q^{78} +9.58991 q^{79} +23.8878 q^{80} +8.82853 q^{81} +25.2507 q^{82} +10.1833 q^{83} -2.70109 q^{84} -5.31964 q^{85} -26.3563 q^{86} +0.416652 q^{87} -39.2579 q^{88} -8.38870 q^{89} -22.3075 q^{90} +10.0973 q^{91} -41.7261 q^{92} -0.310196 q^{93} -34.4190 q^{94} +6.86828 q^{95} -1.05023 q^{96} +1.39657 q^{97} +27.7630 q^{98} +17.2307 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 6 q^{2} + 4 q^{3} + 110 q^{4} + 10 q^{5} + 8 q^{6} + 21 q^{7} + 15 q^{8} + 133 q^{9}+O(q^{10}) \) 87 * q + 6 * q^2 + 4 * q^3 + 110 * q^4 + 10 * q^5 + 8 * q^6 + 21 * q^7 + 15 * q^8 + 133 * q^9	\( 87 q + 6 q^{2} + 4 q^{3} + 110 q^{4} + 10 q^{5} + 8 q^{6} + 21 q^{7} + 15 q^{8} + 133 q^{9} + 25 q^{10} + 17 q^{11} + 34 q^{13} + 12 q^{14} + 16 q^{15} + 152 q^{16} + 32 q^{17} + 14 q^{18} + 56 q^{19} + 3 q^{20} + 38 q^{21} + 32 q^{22} + 8 q^{23} + 8 q^{24} + 179 q^{25} + 11 q^{26} - 2 q^{27} + 45 q^{28} + 40 q^{29} + 24 q^{30} + 31 q^{31} + 26 q^{32} + 31 q^{33} + 31 q^{34} + 22 q^{35} + 180 q^{36} + 35 q^{37} - 15 q^{38} + 59 q^{39} + 42 q^{40} + 45 q^{41} - 30 q^{42} + 82 q^{43} + 25 q^{44} + 20 q^{45} + 69 q^{46} - 7 q^{47} - 39 q^{48} + 222 q^{49} + 17 q^{50} + 53 q^{51} + 54 q^{52} + 16 q^{53} - 7 q^{54} + 49 q^{55} + 12 q^{56} + 52 q^{57} + 17 q^{58} - 7 q^{59} - 6 q^{60} + 131 q^{61} - 8 q^{62} + 19 q^{63} + 213 q^{64} + 57 q^{65} + 17 q^{66} + 38 q^{67} + 13 q^{68} + 45 q^{69} - 5 q^{71} + 4 q^{72} + 91 q^{73} + q^{74} - 44 q^{75} + 150 q^{76} + 5 q^{77} - 87 q^{78} + 120 q^{79} - 41 q^{80} + 247 q^{81} + 20 q^{82} - 33 q^{83} - 16 q^{84} + 110 q^{85} - 22 q^{86} - 13 q^{87} + 78 q^{88} + 53 q^{89} - 33 q^{90} + 32 q^{91} - 31 q^{92} + 13 q^{93} + 79 q^{94} - 25 q^{95} - 51 q^{96} + 92 q^{97} - 36 q^{98} + 35 q^{99}+O(q^{100}) \) 87 * q + 6 * q^2 + 4 * q^3 + 110 * q^4 + 10 * q^5 + 8 * q^6 + 21 * q^7 + 15 * q^8 + 133 * q^9 + 25 * q^10 + 17 * q^11 + 34 * q^13 + 12 * q^14 + 16 * q^15 + 152 * q^16 + 32 * q^17 + 14 * q^18 + 56 * q^19 + 3 * q^20 + 38 * q^21 + 32 * q^22 + 8 * q^23 + 8 * q^24 + 179 * q^25 + 11 * q^26 - 2 * q^27 + 45 * q^28 + 40 * q^29 + 24 * q^30 + 31 * q^31 + 26 * q^32 + 31 * q^33 + 31 * q^34 + 22 * q^35 + 180 * q^36 + 35 * q^37 - 15 * q^38 + 59 * q^39 + 42 * q^40 + 45 * q^41 - 30 * q^42 + 82 * q^43 + 25 * q^44 + 20 * q^45 + 69 * q^46 - 7 * q^47 - 39 * q^48 + 222 * q^49 + 17 * q^50 + 53 * q^51 + 54 * q^52 + 16 * q^53 - 7 * q^54 + 49 * q^55 + 12 * q^56 + 52 * q^57 + 17 * q^58 - 7 * q^59 - 6 * q^60 + 131 * q^61 - 8 * q^62 + 19 * q^63 + 213 * q^64 + 57 * q^65 + 17 * q^66 + 38 * q^67 + 13 * q^68 + 45 * q^69 - 5 * q^71 + 4 * q^72 + 91 * q^73 + q^74 - 44 * q^75 + 150 * q^76 + 5 * q^77 - 87 * q^78 + 120 * q^79 - 41 * q^80 + 247 * q^81 + 20 * q^82 - 33 * q^83 - 16 * q^84 + 110 * q^85 - 22 * q^86 - 13 * q^87 + 78 * q^88 + 53 * q^89 - 33 * q^90 + 32 * q^91 - 31 * q^92 + 13 * q^93 + 79 * q^94 - 25 * q^95 - 51 * q^96 + 92 * q^97 - 36 * q^98 + 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more