LMFDB - Embedded newform 8014.2.a.d.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8014,2,Mod(1,8014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8014, 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("8014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8014 = 2 \cdot 4007 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8014.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:	$63.9921121799$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.73672 q^{3} +1.00000 q^{4} +0.860575 q^{5} -2.73672 q^{6} +1.51345 q^{7} +1.00000 q^{8} +4.48964 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.73672 q^{3} +1.00000 q^{4} +0.860575 q^{5} -2.73672 q^{6} +1.51345 q^{7} +1.00000 q^{8} +4.48964 q^{9} +0.860575 q^{10} -0.565196 q^{11} -2.73672 q^{12} +3.02292 q^{13} +1.51345 q^{14} -2.35515 q^{15} +1.00000 q^{16} +7.90410 q^{17} +4.48964 q^{18} +4.55582 q^{19} +0.860575 q^{20} -4.14188 q^{21} -0.565196 q^{22} -3.13092 q^{23} -2.73672 q^{24} -4.25941 q^{25} +3.02292 q^{26} -4.07672 q^{27} +1.51345 q^{28} +2.45096 q^{29} -2.35515 q^{30} +6.96205 q^{31} +1.00000 q^{32} +1.54678 q^{33} +7.90410 q^{34} +1.30243 q^{35} +4.48964 q^{36} -8.51411 q^{37} +4.55582 q^{38} -8.27290 q^{39} +0.860575 q^{40} +5.92178 q^{41} -4.14188 q^{42} +9.58102 q^{43} -0.565196 q^{44} +3.86367 q^{45} -3.13092 q^{46} +9.19350 q^{47} -2.73672 q^{48} -4.70948 q^{49} -4.25941 q^{50} -21.6313 q^{51} +3.02292 q^{52} +2.34935 q^{53} -4.07672 q^{54} -0.486394 q^{55} +1.51345 q^{56} -12.4680 q^{57} +2.45096 q^{58} +4.26552 q^{59} -2.35515 q^{60} -6.95587 q^{61} +6.96205 q^{62} +6.79482 q^{63} +1.00000 q^{64} +2.60145 q^{65} +1.54678 q^{66} +5.69331 q^{67} +7.90410 q^{68} +8.56846 q^{69} +1.30243 q^{70} -1.40899 q^{71} +4.48964 q^{72} +6.84165 q^{73} -8.51411 q^{74} +11.6568 q^{75} +4.55582 q^{76} -0.855393 q^{77} -8.27290 q^{78} -8.96033 q^{79} +0.860575 q^{80} -2.31208 q^{81} +5.92178 q^{82} -16.8624 q^{83} -4.14188 q^{84} +6.80207 q^{85} +9.58102 q^{86} -6.70759 q^{87} -0.565196 q^{88} -1.51510 q^{89} +3.86367 q^{90} +4.57503 q^{91} -3.13092 q^{92} -19.0532 q^{93} +9.19350 q^{94} +3.92062 q^{95} -2.73672 q^{96} +3.15982 q^{97} -4.70948 q^{98} -2.53752 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 88 q^{2} + 22 q^{3} + 88 q^{4} + 25 q^{5} + 22 q^{6} + 33 q^{7} + 88 q^{8} + 108 q^{9}+O(q^{10}) $ 88 * q + 88 * q^2 + 22 * q^3 + 88 * q^4 + 25 * q^5 + 22 * q^6 + 33 * q^7 + 88 * q^8 + 108 * q^9	\( 88 q + 88 q^{2} + 22 q^{3} + 88 q^{4} + 25 q^{5} + 22 q^{6} + 33 q^{7} + 88 q^{8} + 108 q^{9} + 25 q^{10} + 70 q^{11} + 22 q^{12} + 31 q^{13} + 33 q^{14} + 47 q^{15} + 88 q^{16} + 19 q^{17} + 108 q^{18} + 33 q^{19} + 25 q^{20} + 48 q^{21} + 70 q^{22} + 77 q^{23} + 22 q^{24} + 109 q^{25} + 31 q^{26} + 88 q^{27} + 33 q^{28} + 83 q^{29} + 47 q^{30} + 51 q^{31} + 88 q^{32} + 30 q^{33} + 19 q^{34} + 40 q^{35} + 108 q^{36} + 45 q^{37} + 33 q^{38} + 82 q^{39} + 25 q^{40} + 35 q^{41} + 48 q^{42} + 78 q^{43} + 70 q^{44} + 37 q^{45} + 77 q^{46} + 59 q^{47} + 22 q^{48} + 103 q^{49} + 109 q^{50} + 21 q^{51} + 31 q^{52} + 58 q^{53} + 88 q^{54} + 35 q^{55} + 33 q^{56} - 16 q^{57} + 83 q^{58} + 54 q^{59} + 47 q^{60} + 18 q^{61} + 51 q^{62} + 47 q^{63} + 88 q^{64} + 34 q^{65} + 30 q^{66} + 88 q^{67} + 19 q^{68} + 62 q^{69} + 40 q^{70} + 139 q^{71} + 108 q^{72} - 6 q^{73} + 45 q^{74} + 45 q^{75} + 33 q^{76} + 37 q^{77} + 82 q^{78} + 94 q^{79} + 25 q^{80} + 112 q^{81} + 35 q^{82} + 58 q^{83} + 48 q^{84} + 83 q^{85} + 78 q^{86} + 21 q^{87} + 70 q^{88} + 99 q^{89} + 37 q^{90} + 53 q^{91} + 77 q^{92} + 57 q^{93} + 59 q^{94} + 92 q^{95} + 22 q^{96} + 16 q^{97} + 103 q^{98} + 150 q^{99}+O(q^{100}) \) 88 * q + 88 * q^2 + 22 * q^3 + 88 * q^4 + 25 * q^5 + 22 * q^6 + 33 * q^7 + 88 * q^8 + 108 * q^9 + 25 * q^10 + 70 * q^11 + 22 * q^12 + 31 * q^13 + 33 * q^14 + 47 * q^15 + 88 * q^16 + 19 * q^17 + 108 * q^18 + 33 * q^19 + 25 * q^20 + 48 * q^21 + 70 * q^22 + 77 * q^23 + 22 * q^24 + 109 * q^25 + 31 * q^26 + 88 * q^27 + 33 * q^28 + 83 * q^29 + 47 * q^30 + 51 * q^31 + 88 * q^32 + 30 * q^33 + 19 * q^34 + 40 * q^35 + 108 * q^36 + 45 * q^37 + 33 * q^38 + 82 * q^39 + 25 * q^40 + 35 * q^41 + 48 * q^42 + 78 * q^43 + 70 * q^44 + 37 * q^45 + 77 * q^46 + 59 * q^47 + 22 * q^48 + 103 * q^49 + 109 * q^50 + 21 * q^51 + 31 * q^52 + 58 * q^53 + 88 * q^54 + 35 * q^55 + 33 * q^56 - 16 * q^57 + 83 * q^58 + 54 * q^59 + 47 * q^60 + 18 * q^61 + 51 * q^62 + 47 * q^63 + 88 * q^64 + 34 * q^65 + 30 * q^66 + 88 * q^67 + 19 * q^68 + 62 * q^69 + 40 * q^70 + 139 * q^71 + 108 * q^72 - 6 * q^73 + 45 * q^74 + 45 * q^75 + 33 * q^76 + 37 * q^77 + 82 * q^78 + 94 * q^79 + 25 * q^80 + 112 * q^81 + 35 * q^82 + 58 * q^83 + 48 * q^84 + 83 * q^85 + 78 * q^86 + 21 * q^87 + 70 * q^88 + 99 * q^89 + 37 * q^90 + 53 * q^91 + 77 * q^92 + 57 * q^93 + 59 * q^94 + 92 * q^95 + 22 * q^96 + 16 * q^97 + 103 * q^98 + 150 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.73672	−1.58005	−0.790023	−	0.613077i	$-0.789931\pi$
$3$	−2.73672	−1.58005	−0.790023	+	0.613077i	$0.789931\pi$
$4$	1.00000	0.500000
$5$	0.860575	0.384861	0.192431	−	0.981311i	$-0.438363\pi$
$5$	0.860575	0.384861	0.192431	+	0.981311i	$0.438363\pi$
$6$	−2.73672	−1.11726
$7$	1.51345	0.572029	0.286014	−	0.958225i	$-0.407670\pi$
$7$	1.51345	0.572029	0.286014	+	0.958225i	$0.407670\pi$
$8$	1.00000	0.353553
$9$	4.48964	1.49655
$10$	0.860575	0.272138
$11$	−0.565196	−0.170413	−0.0852065	−	0.996363i	$-0.527155\pi$
$11$	−0.565196	−0.170413	−0.0852065	+	0.996363i	$0.527155\pi$
$12$	−2.73672	−0.790023
$13$	3.02292	0.838408	0.419204	−	0.907892i	$-0.362309\pi$
$13$	3.02292	0.838408	0.419204	+	0.907892i	$0.362309\pi$
$14$	1.51345	0.404485
$15$	−2.35515	−0.608098
$16$	1.00000	0.250000
$17$	7.90410	1.91702	0.958512	−	0.285051i	$-0.0920106\pi$
$17$	7.90410	1.91702	0.958512	+	0.285051i	$0.0920106\pi$
$18$	4.48964	1.05822
$19$	4.55582	1.04518	0.522588	−	0.852585i	$-0.324967\pi$
$19$	4.55582	1.04518	0.522588	+	0.852585i	$0.324967\pi$
$20$	0.860575	0.192431
$21$	−4.14188	−0.903832
$22$	−0.565196	−0.120500
$23$	−3.13092	−0.652843	−0.326421	−	0.945224i	$-0.605843\pi$
$23$	−3.13092	−0.652843	−0.326421	+	0.945224i	$0.605843\pi$
$24$	−2.73672	−0.558631
$25$	−4.25941	−0.851882
$26$	3.02292	0.592844
$27$	−4.07672	−0.784565
$28$	1.51345	0.286014
$29$	2.45096	0.455132	0.227566	−	0.973763i	$-0.426923\pi$
$29$	2.45096	0.455132	0.227566	+	0.973763i	$0.426923\pi$
$30$	−2.35515	−0.429990
$31$	6.96205	1.25042	0.625210	−	0.780456i	$-0.285013\pi$
$31$	6.96205	1.25042	0.625210	+	0.780456i	$0.285013\pi$
$32$	1.00000	0.176777
$33$	1.54678	0.269260
$34$	7.90410	1.35554
$35$	1.30243	0.220152
$36$	4.48964	0.748273
$37$	−8.51411	−1.39971	−0.699856	−	0.714284i	$-0.746752\pi$
$37$	−8.51411	−1.39971	−0.699856	+	0.714284i	$0.746752\pi$
$38$	4.55582	0.739051
$39$	−8.27290	−1.32472
$40$	0.860575	0.136069
$41$	5.92178	0.924827	0.462413	−	0.886664i	$-0.346984\pi$
$41$	5.92178	0.924827	0.462413	+	0.886664i	$0.346984\pi$
$42$	−4.14188	−0.639105
$43$	9.58102	1.46109	0.730546	−	0.682864i	$-0.239266\pi$
$43$	9.58102	1.46109	0.730546	+	0.682864i	$0.239266\pi$
$44$	−0.565196	−0.0852065
$45$	3.86367	0.575962
$46$	−3.13092	−0.461630
$47$	9.19350	1.34101	0.670505	−	0.741905i	$-0.266078\pi$
$47$	9.19350	1.34101	0.670505	+	0.741905i	$0.266078\pi$
$48$	−2.73672	−0.395012
$49$	−4.70948	−0.672783
$50$	−4.25941	−0.602372
$51$	−21.6313	−3.02899
$52$	3.02292	0.419204
$53$	2.34935	0.322708	0.161354	−	0.986897i	$-0.448414\pi$
$53$	2.34935	0.322708	0.161354	+	0.986897i	$0.448414\pi$
$54$	−4.07672	−0.554771
$55$	−0.486394	−0.0655853
$56$	1.51345	0.202243
$57$	−12.4680	−1.65143
$58$	2.45096	0.321827
$59$	4.26552	0.555323	0.277661	−	0.960679i	$-0.410441\pi$
$59$	4.26552	0.555323	0.277661	+	0.960679i	$0.410441\pi$
$60$	−2.35515	−0.304049
$61$	−6.95587	−0.890608	−0.445304	−	0.895380i	$-0.646904\pi$
$61$	−6.95587	−0.890608	−0.445304	+	0.895380i	$0.646904\pi$
$62$	6.96205	0.884181
$63$	6.79482	0.856067
$64$	1.00000	0.125000
$65$	2.60145	0.322671
$66$	1.54678	0.190396
$67$	5.69331	0.695548	0.347774	−	0.937578i	$-0.386938\pi$
$67$	5.69331	0.695548	0.347774	+	0.937578i	$0.386938\pi$
$68$	7.90410	0.958512
$69$	8.56846	1.03152
$70$	1.30243	0.155671
$71$	−1.40899	−0.167217	−0.0836083	−	0.996499i	$-0.526644\pi$
$71$	−1.40899	−0.167217	−0.0836083	+	0.996499i	$0.526644\pi$
$72$	4.48964	0.529109
$73$	6.84165	0.800755	0.400377	−	0.916350i	$-0.368879\pi$
$73$	6.84165	0.800755	0.400377	+	0.916350i	$0.368879\pi$
$74$	−8.51411	−0.989745
$75$	11.6568	1.34601
$76$	4.55582	0.522588
$77$	−0.855393	−0.0974811
$78$	−8.27290	−0.936721
$79$	−8.96033	−1.00812	−0.504058	−	0.863670i	$-0.668160\pi$
$79$	−8.96033	−1.00812	−0.504058	+	0.863670i	$0.668160\pi$
$80$	0.860575	0.0962153
$81$	−2.31208	−0.256897
$82$	5.92178	0.653951
$83$	−16.8624	−1.85089	−0.925444	−	0.378885i	$-0.876308\pi$
$83$	−16.8624	−1.85089	−0.925444	+	0.378885i	$0.876308\pi$
$84$	−4.14188	−0.451916
$85$	6.80207	0.737788
$86$	9.58102	1.03315
$87$	−6.70759	−0.719129
$88$	−0.565196	−0.0602501
$89$	−1.51510	−0.160600	−0.0803001	−	0.996771i	$-0.525588\pi$
$89$	−1.51510	−0.160600	−0.0803001	+	0.996771i	$0.525588\pi$
$90$	3.86367	0.407267
$91$	4.57503	0.479593
$92$	−3.13092	−0.326421
$93$	−19.0532	−1.97572
$94$	9.19350	0.948237
$95$	3.92062	0.402247
$96$	−2.73672	−0.279315
$97$	3.15982	0.320831	0.160416	−	0.987050i	$-0.448717\pi$
$97$	3.15982	0.320831	0.160416	+	0.987050i	$0.448717\pi$
$98$	−4.70948	−0.475730
$99$	−2.53752	−0.255031
$100$	−4.25941	−0.425941
$101$	−15.5459	−1.54687	−0.773436	−	0.633874i	$-0.781464\pi$
$101$	−15.5459	−1.54687	−0.773436	+	0.633874i	$0.781464\pi$
$102$	−21.6313	−2.14182
$103$	8.06180	0.794353	0.397176	−	0.917742i	$-0.369990\pi$
$103$	8.06180	0.794353	0.397176	+	0.917742i	$0.369990\pi$
$104$	3.02292	0.296422
$105$	−3.56440	−0.347850
$106$	2.34935	0.228189
$107$	14.4035	1.39244	0.696219	−	0.717829i	$-0.254864\pi$
$107$	14.4035	1.39244	0.696219	+	0.717829i	$0.254864\pi$
$108$	−4.07672	−0.392282
$109$	−10.2850	−0.985129	−0.492565	−	0.870276i	$-0.663940\pi$
$109$	−10.2850	−0.985129	−0.492565	+	0.870276i	$0.663940\pi$
$110$	−0.486394	−0.0463758
$111$	23.3007	2.21161
$112$	1.51345	0.143007
$113$	−11.6674	−1.09758	−0.548790	−	0.835960i	$-0.684911\pi$
$113$	−11.6674	−1.09758	−0.548790	+	0.835960i	$0.684911\pi$
$114$	−12.4680	−1.16773
$115$	−2.69440	−0.251254
$116$	2.45096	0.227566
$117$	13.5718	1.25472
$118$	4.26552	0.392673
$119$	11.9624	1.09659
$120$	−2.35515	−0.214995
$121$	−10.6806	−0.970959
$122$	−6.95587	−0.629755
$123$	−16.2063	−1.46127
$124$	6.96205	0.625210
$125$	−7.96842	−0.712717
$126$	6.79482	0.605331
$127$	−4.75260	−0.421725	−0.210862	−	0.977516i	$-0.567627\pi$
$127$	−4.75260	−0.421725	−0.210862	+	0.977516i	$0.567627\pi$
$128$	1.00000	0.0883883
$129$	−26.2206	−2.30859
$130$	2.60145	0.228163
$131$	4.32109	0.377535	0.188768	−	0.982022i	$-0.439551\pi$
$131$	4.32109	0.377535	0.188768	+	0.982022i	$0.439551\pi$
$132$	1.54678	0.134630
$133$	6.89498	0.597870
$134$	5.69331	0.491827
$135$	−3.50832	−0.301948
$136$	7.90410	0.677771
$137$	−16.4756	−1.40761	−0.703804	−	0.710394i	$-0.748517\pi$
$137$	−16.4756	−1.40761	−0.703804	+	0.710394i	$0.748517\pi$
$138$	8.56846	0.729396
$139$	21.0914	1.78895	0.894476	−	0.447116i	$-0.147549\pi$
$139$	21.0914	1.78895	0.894476	+	0.447116i	$0.147549\pi$
$140$	1.30243	0.110076
$141$	−25.1600	−2.11886
$142$	−1.40899	−0.118240
$143$	−1.70854	−0.142876
$144$	4.48964	0.374136
$145$	2.10924	0.175163
$146$	6.84165	0.566219
$147$	12.8885	1.06303
$148$	−8.51411	−0.699856
$149$	−2.88618	−0.236445	−0.118223	−	0.992987i	$-0.537720\pi$
$149$	−2.88618	−0.236445	−0.118223	+	0.992987i	$0.537720\pi$
$150$	11.6568	0.951775
$151$	1.87975	0.152972	0.0764859	−	0.997071i	$-0.475630\pi$
$151$	1.87975	0.152972	0.0764859	+	0.997071i	$0.475630\pi$
$152$	4.55582	0.369525
$153$	35.4865	2.86891
$154$	−0.855393	−0.0689296
$155$	5.99137	0.481238
$156$	−8.27290	−0.662362
$157$	10.8446	0.865495	0.432747	−	0.901515i	$-0.357544\pi$
$157$	10.8446	0.865495	0.432747	+	0.901515i	$0.357544\pi$
$158$	−8.96033	−0.712845
$159$	−6.42951	−0.509893
$160$	0.860575	0.0680345
$161$	−4.73848	−0.373445
$162$	−2.31208	−0.181654
$163$	7.92617	0.620825	0.310413	−	0.950602i	$-0.399533\pi$
$163$	7.92617	0.620825	0.310413	+	0.950602i	$0.399533\pi$
$164$	5.92178	0.462413
$165$	1.33112	0.103628
$166$	−16.8624	−1.30878
$167$	−6.92920	−0.536197	−0.268099	−	0.963391i	$-0.586395\pi$
$167$	−6.92920	−0.536197	−0.268099	+	0.963391i	$0.586395\pi$
$168$	−4.14188	−0.319553
$169$	−3.86193	−0.297072
$170$	6.80207	0.521695
$171$	20.4540	1.56415
$172$	9.58102	0.730546
$173$	−3.54696	−0.269670	−0.134835	−	0.990868i	$-0.543050\pi$
$173$	−3.54696	−0.269670	−0.134835	+	0.990868i	$0.543050\pi$
$174$	−6.70759	−0.508501
$175$	−6.44638	−0.487301
$176$	−0.565196	−0.0426033
$177$	−11.6735	−0.877436
$178$	−1.51510	−0.113561
$179$	20.7005	1.54723	0.773616	−	0.633655i	$-0.218446\pi$
$179$	20.7005	1.54723	0.773616	+	0.633655i	$0.218446\pi$
$180$	3.86367	0.287981
$181$	16.1627	1.20136	0.600681	−	0.799489i	$-0.294896\pi$
$181$	16.1627	1.20136	0.600681	+	0.799489i	$0.294896\pi$
$182$	4.57503	0.339124
$183$	19.0363	1.40720
$184$	−3.13092	−0.230815
$185$	−7.32703	−0.538694
$186$	−19.0532	−1.39705
$187$	−4.46736	−0.326686
$188$	9.19350	0.670505
$189$	−6.16989	−0.448793
$190$	3.92062	0.284432
$191$	−11.6903	−0.845878	−0.422939	−	0.906158i	$-0.639001\pi$
$191$	−11.6903	−0.845878	−0.422939	+	0.906158i	$0.639001\pi$
$192$	−2.73672	−0.197506
$193$	−8.90067	−0.640684	−0.320342	−	0.947302i	$-0.603798\pi$
$193$	−8.90067	−0.640684	−0.320342	+	0.947302i	$0.603798\pi$
$194$	3.15982	0.226862
$195$	−7.11945	−0.509835
$196$	−4.70948	−0.336392
$197$	7.98979	0.569249	0.284624	−	0.958639i	$-0.408131\pi$
$197$	7.98979	0.569249	0.284624	+	0.958639i	$0.408131\pi$
$198$	−2.53752	−0.180334
$199$	20.5928	1.45978	0.729891	−	0.683564i	$-0.239571\pi$
$199$	20.5928	1.45978	0.729891	+	0.683564i	$0.239571\pi$
$200$	−4.25941	−0.301186
$201$	−15.5810	−1.09900
$202$	−15.5459	−1.09380
$203$	3.70939	0.260348
$204$	−21.6313	−1.51449
$205$	5.09614	0.355930
$206$	8.06180	0.561692
$207$	−14.0567	−0.977009
$208$	3.02292	0.209602
$209$	−2.57493	−0.178112
$210$	−3.56440	−0.245967
$211$	−3.34882	−0.230542	−0.115271	−	0.993334i	$-0.536774\pi$
$211$	−3.34882	−0.230542	−0.115271	+	0.993334i	$0.536774\pi$
$212$	2.34935	0.161354
$213$	3.85602	0.264210
$214$	14.4035	0.984603
$215$	8.24519	0.562317
$216$	−4.07672	−0.277385
$217$	10.5367	0.715276
$218$	−10.2850	−0.696591
$219$	−18.7237	−1.26523
$220$	−0.486394	−0.0327927
$221$	23.8935	1.60725
$222$	23.3007	1.56384
$223$	−17.0415	−1.14118	−0.570590	−	0.821235i	$-0.693286\pi$
$223$	−17.0415	−1.14118	−0.570590	+	0.821235i	$0.693286\pi$
$224$	1.51345	0.101121
$225$	−19.1232	−1.27488
$226$	−11.6674	−0.776106
$227$	−1.74461	−0.115794	−0.0578971	−	0.998323i	$-0.518440\pi$
$227$	−1.74461	−0.115794	−0.0578971	+	0.998323i	$0.518440\pi$
$228$	−12.4680	−0.825713
$229$	−7.25971	−0.479735	−0.239868	−	0.970806i	$-0.577104\pi$
$229$	−7.25971	−0.479735	−0.239868	+	0.970806i	$0.577104\pi$
$230$	−2.69440	−0.177663
$231$	2.34097	0.154025
$232$	2.45096	0.160913
$233$	−0.960473	−0.0629227	−0.0314613	−	0.999505i	$-0.510016\pi$
$233$	−0.960473	−0.0629227	−0.0314613	+	0.999505i	$0.510016\pi$
$234$	13.5718	0.887218
$235$	7.91170	0.516102
$236$	4.26552	0.277661
$237$	24.5219	1.59287
$238$	11.9624	0.775408
$239$	−12.5554	−0.812138	−0.406069	−	0.913842i	$-0.633101\pi$
$239$	−12.5554	−0.812138	−0.406069	+	0.913842i	$0.633101\pi$
$240$	−2.35515	−0.152025
$241$	−11.7569	−0.757327	−0.378663	−	0.925534i	$-0.623616\pi$
$241$	−11.7569	−0.757327	−0.378663	+	0.925534i	$0.623616\pi$
$242$	−10.6806	−0.686572
$243$	18.5577	1.19047
$244$	−6.95587	−0.445304
$245$	−4.05287	−0.258928
$246$	−16.2063	−1.03327
$247$	13.7719	0.876284
$248$	6.96205	0.442090
$249$	46.1476	2.92449
$250$	−7.96842	−0.503967
$251$	5.64100	0.356057	0.178028	−	0.984025i	$-0.443028\pi$
$251$	5.64100	0.356057	0.178028	+	0.984025i	$0.443028\pi$
$252$	6.79482	0.428033
$253$	1.76959	0.111253
$254$	−4.75260	−0.298205
$255$	−18.6154	−1.16574
$256$	1.00000	0.0625000
$257$	−19.0882	−1.19069	−0.595343	−	0.803471i	$-0.702984\pi$
$257$	−19.0882	−1.19069	−0.595343	+	0.803471i	$0.702984\pi$
$258$	−26.2206	−1.63242
$259$	−12.8856	−0.800675
$260$	2.60145	0.161335
$261$	11.0039	0.681125
$262$	4.32109	0.266958
$263$	−0.376581	−0.0232210	−0.0116105	−	0.999933i	$-0.503696\pi$
$263$	−0.376581	−0.0232210	−0.0116105	+	0.999933i	$0.503696\pi$
$264$	1.54678	0.0951979
$265$	2.02179	0.124198
$266$	6.89498	0.422758
$267$	4.14640	0.253756
$268$	5.69331	0.347774
$269$	15.2623	0.930561	0.465280	−	0.885163i	$-0.345953\pi$
$269$	15.2623	0.930561	0.465280	+	0.885163i	$0.345953\pi$
$270$	−3.50832	−0.213510
$271$	0.575961	0.0349871	0.0174936	−	0.999847i	$-0.494431\pi$
$271$	0.575961	0.0349871	0.0174936	+	0.999847i	$0.494431\pi$
$272$	7.90410	0.479256
$273$	−12.5206	−0.757780
$274$	−16.4756	−0.995329
$275$	2.40740	0.145172
$276$	8.56846	0.515761
$277$	−3.50440	−0.210559	−0.105280	−	0.994443i	$-0.533574\pi$
$277$	−3.50440	−0.210559	−0.105280	+	0.994443i	$0.533574\pi$
$278$	21.0914	1.26498
$279$	31.2571	1.87131
$280$	1.30243	0.0778353
$281$	−1.04794	−0.0625151	−0.0312575	−	0.999511i	$-0.509951\pi$
$281$	−1.04794	−0.0625151	−0.0312575	+	0.999511i	$0.509951\pi$
$282$	−25.1600	−1.49826
$283$	−29.3215	−1.74298	−0.871492	−	0.490410i	$-0.836847\pi$
$283$	−29.3215	−1.74298	−0.871492	+	0.490410i	$0.836847\pi$
$284$	−1.40899	−0.0836083
$285$	−10.7296	−0.635569
$286$	−1.70854	−0.101028
$287$	8.96229	0.529027
$288$	4.48964	0.264554
$289$	45.4747	2.67498
$290$	2.10924	0.123859
$291$	−8.64754	−0.506928
$292$	6.84165	0.400377
$293$	11.0528	0.645714	0.322857	−	0.946448i	$-0.395357\pi$
$293$	11.0528	0.645714	0.322857	+	0.946448i	$0.395357\pi$
$294$	12.8885	0.751675
$295$	3.67080	0.213722
$296$	−8.51411	−0.494873
$297$	2.30414	0.133700
$298$	−2.88618	−0.167192
$299$	−9.46454	−0.547349
$300$	11.6568	0.673006
$301$	14.5003	0.835786
$302$	1.87975	0.108167
$303$	42.5447	2.44413
$304$	4.55582	0.261294
$305$	−5.98605	−0.342760
$306$	35.4865	2.02863
$307$	−1.69618	−0.0968063	−0.0484032	−	0.998828i	$-0.515413\pi$
$307$	−1.69618	−0.0968063	−0.0484032	+	0.998828i	$0.515413\pi$
$308$	−0.855393	−0.0487406
$309$	−22.0629	−1.25511
$310$	5.99137	0.340287
$311$	1.77178	0.100468	0.0502341	−	0.998737i	$-0.484003\pi$
$311$	1.77178	0.100468	0.0502341	+	0.998737i	$0.484003\pi$
$312$	−8.27290	−0.468361
$313$	13.9288	0.787301	0.393650	−	0.919260i	$-0.371212\pi$
$313$	13.9288	0.787301	0.393650	+	0.919260i	$0.371212\pi$
$314$	10.8446	0.611997
$315$	5.84746	0.329467
$316$	−8.96033	−0.504058
$317$	24.9716	1.40254	0.701271	−	0.712894i	$-0.252616\pi$
$317$	24.9716	1.40254	0.701271	+	0.712894i	$0.252616\pi$
$318$	−6.42951	−0.360549
$319$	−1.38527	−0.0775604
$320$	0.860575	0.0481076
$321$	−39.4183	−2.20012
$322$	−4.73848	−0.264065
$323$	36.0096	2.00363
$324$	−2.31208	−0.128449
$325$	−12.8759	−0.714225
$326$	7.92617	0.438990
$327$	28.1473	1.55655
$328$	5.92178	0.326976
$329$	13.9139	0.767096
$330$	1.33112	0.0732760
$331$	5.36926	0.295121	0.147561	−	0.989053i	$-0.452858\pi$
$331$	5.36926	0.295121	0.147561	+	0.989053i	$0.452858\pi$
$332$	−16.8624	−0.925444
$333$	−38.2253	−2.09473
$334$	−6.92920	−0.379149
$335$	4.89952	0.267689
$336$	−4.14188	−0.225958
$337$	−2.68149	−0.146070	−0.0730350	−	0.997329i	$-0.523268\pi$
$337$	−2.68149	−0.146070	−0.0730350	+	0.997329i	$0.523268\pi$
$338$	−3.86193	−0.210061
$339$	31.9305	1.73423
$340$	6.80207	0.368894
$341$	−3.93492	−0.213088
$342$	20.4540	1.10602
$343$	−17.7217	−0.956880
$344$	9.58102	0.516574
$345$	7.37381	0.396992
$346$	−3.54696	−0.190686
$347$	31.1551	1.67249	0.836247	−	0.548353i	$-0.184745\pi$
$347$	31.1551	1.67249	0.836247	+	0.548353i	$0.184745\pi$
$348$	−6.70759	−0.359565
$349$	32.8399	1.75788	0.878941	−	0.476931i	$-0.158251\pi$
$349$	32.8399	1.75788	0.878941	+	0.476931i	$0.158251\pi$
$350$	−6.44638	−0.344574
$351$	−12.3236	−0.657785
$352$	−0.565196	−0.0301251
$353$	34.0166	1.81052	0.905262	−	0.424854i	$-0.139675\pi$
$353$	34.0166	1.81052	0.905262	+	0.424854i	$0.139675\pi$
$354$	−11.6735	−0.620441
$355$	−1.21254	−0.0643552
$356$	−1.51510	−0.0803001
$357$	−32.7378	−1.73267
$358$	20.7005	1.09406
$359$	18.4725	0.974939	0.487469	−	0.873140i	$-0.337920\pi$
$359$	18.4725	0.974939	0.487469	+	0.873140i	$0.337920\pi$
$360$	3.86367	0.203633
$361$	1.75545	0.0923923
$362$	16.1627	0.849491
$363$	29.2297	1.53416
$364$	4.57503	0.239797
$365$	5.88776	0.308179
$366$	19.0363	0.995041
$367$	−17.2080	−0.898248	−0.449124	−	0.893469i	$-0.648264\pi$
$367$	−17.2080	−0.898248	−0.449124	+	0.893469i	$0.648264\pi$
$368$	−3.13092	−0.163211
$369$	26.5866	1.38405
$370$	−7.32703	−0.380914
$371$	3.55561	0.184598
$372$	−19.0532	−0.987861
$373$	31.1554	1.61316	0.806582	−	0.591123i	$-0.201315\pi$
$373$	31.1554	1.61316	0.806582	+	0.591123i	$0.201315\pi$
$374$	−4.46736	−0.231002
$375$	21.8073	1.12613
$376$	9.19350	0.474119
$377$	7.40907	0.381586
$378$	−6.16989	−0.317345
$379$	12.1380	0.623489	0.311745	−	0.950166i	$-0.399087\pi$
$379$	12.1380	0.623489	0.311745	+	0.950166i	$0.399087\pi$
$380$	3.92062	0.201124
$381$	13.0065	0.666345
$382$	−11.6903	−0.598126
$383$	8.45074	0.431813	0.215906	−	0.976414i	$-0.430729\pi$
$383$	8.45074	0.431813	0.215906	+	0.976414i	$0.430729\pi$
$384$	−2.73672	−0.139658
$385$	−0.736131	−0.0375167
$386$	−8.90067	−0.453032
$387$	43.0153	2.18659
$388$	3.15982	0.160416
$389$	−24.0460	−1.21918	−0.609590	−	0.792717i	$-0.708666\pi$
$389$	−24.0460	−1.21918	−0.609590	+	0.792717i	$0.708666\pi$
$390$	−7.11945	−0.360507
$391$	−24.7471	−1.25152
$392$	−4.70948	−0.237865
$393$	−11.8256	−0.596523
$394$	7.98979	0.402520
$395$	−7.71104	−0.387984
$396$	−2.53752	−0.127515
$397$	−35.6374	−1.78859	−0.894296	−	0.447476i	$-0.852323\pi$
$397$	−35.6374	−1.78859	−0.894296	+	0.447476i	$0.852323\pi$
$398$	20.5928	1.03222
$399$	−18.8696	−0.944663
$400$	−4.25941	−0.212970
$401$	5.59313	0.279308	0.139654	−	0.990200i	$-0.455401\pi$
$401$	5.59313	0.279308	0.139654	+	0.990200i	$0.455401\pi$
$402$	−15.5810	−0.777109
$403$	21.0457	1.04836
$404$	−15.5459	−0.773436
$405$	−1.98972	−0.0988698
$406$	3.70939	0.184094
$407$	4.81214	0.238529
$408$	−21.6313	−1.07091
$409$	−24.4082	−1.20691	−0.603454	−	0.797398i	$-0.706209\pi$
$409$	−24.4082	−1.20691	−0.603454	+	0.797398i	$0.706209\pi$
$410$	5.09614	0.251680
$411$	45.0892	2.22409
$412$	8.06180	0.397176
$413$	6.45563	0.317661
$414$	−14.0567	−0.690850
$415$	−14.5114	−0.712335
$416$	3.02292	0.148211
$417$	−57.7213	−2.82663
$418$	−2.57493	−0.125944
$419$	10.8805	0.531549	0.265774	−	0.964035i	$-0.414372\pi$
$419$	10.8805	0.531549	0.265774	+	0.964035i	$0.414372\pi$
$420$	−3.56440	−0.173925
$421$	3.35606	0.163564	0.0817821	−	0.996650i	$-0.473939\pi$
$421$	3.35606	0.163564	0.0817821	+	0.996650i	$0.473939\pi$
$422$	−3.34882	−0.163018
$423$	41.2755	2.00688
$424$	2.34935	0.114094
$425$	−33.6668	−1.63308
$426$	3.85602	0.186825
$427$	−10.5273	−0.509453
$428$	14.4035	0.696219
$429$	4.67581	0.225750
$430$	8.24519	0.397618
$431$	−4.67278	−0.225080	−0.112540	−	0.993647i	$-0.535899\pi$
$431$	−4.67278	−0.225080	−0.112540	+	0.993647i	$0.535899\pi$
$432$	−4.07672	−0.196141
$433$	−12.5400	−0.602633	−0.301317	−	0.953524i	$-0.597426\pi$
$433$	−12.5400	−0.602633	−0.301317	+	0.953524i	$0.597426\pi$
$434$	10.5367	0.505777
$435$	−5.77239	−0.276765
$436$	−10.2850	−0.492565
$437$	−14.2639	−0.682335
$438$	−18.7237	−0.894652
$439$	18.5102	0.883446	0.441723	−	0.897152i	$-0.354367\pi$
$439$	18.5102	0.883446	0.441723	+	0.897152i	$0.354367\pi$
$440$	−0.486394	−0.0231879
$441$	−21.1439	−1.00685
$442$	23.8935	1.13650
$443$	−26.6770	−1.26746	−0.633732	−	0.773553i	$-0.718478\pi$
$443$	−26.6770	−1.26746	−0.633732	+	0.773553i	$0.718478\pi$
$444$	23.3007	1.10580
$445$	−1.30386	−0.0618087
$446$	−17.0415	−0.806937
$447$	7.89867	0.373594
$448$	1.51345	0.0715036
$449$	−10.6878	−0.504391	−0.252195	−	0.967676i	$-0.581153\pi$
$449$	−10.6878	−0.504391	−0.252195	+	0.967676i	$0.581153\pi$
$450$	−19.1232	−0.901476
$451$	−3.34697	−0.157603
$452$	−11.6674	−0.548790
$453$	−5.14435	−0.241702
$454$	−1.74461	−0.0818788
$455$	3.93716	0.184577
$456$	−12.4680	−0.583867
$457$	0.751043	0.0351323	0.0175662	−	0.999846i	$-0.494408\pi$
$457$	0.751043	0.0351323	0.0175662	+	0.999846i	$0.494408\pi$
$458$	−7.25971	−0.339224
$459$	−32.2228	−1.50403
$460$	−2.69440	−0.125627
$461$	23.3622	1.08809	0.544044	−	0.839057i	$-0.316892\pi$
$461$	23.3622	1.08809	0.544044	+	0.839057i	$0.316892\pi$
$462$	2.34097	0.108912
$463$	5.72639	0.266128	0.133064	−	0.991107i	$-0.457518\pi$
$463$	5.72639	0.266128	0.133064	+	0.991107i	$0.457518\pi$
$464$	2.45096	0.113783
$465$	−16.3967	−0.760379
$466$	−0.960473	−0.0444930
$467$	−9.21292	−0.426323	−0.213161	−	0.977017i	$-0.568376\pi$
$467$	−9.21292	−0.426323	−0.213161	+	0.977017i	$0.568376\pi$
$468$	13.5718	0.627358
$469$	8.61651	0.397873
$470$	7.91170	0.364940
$471$	−29.6787	−1.36752
$472$	4.26552	0.196336
$473$	−5.41515	−0.248989
$474$	24.5219	1.12633
$475$	−19.4051	−0.890366
$476$	11.9624	0.548296
$477$	10.5477	0.482947
$478$	−12.5554	−0.574269
$479$	−17.6585	−0.806839	−0.403419	−	0.915015i	$-0.632178\pi$
$479$	−17.6585	−0.806839	−0.403419	+	0.915015i	$0.632178\pi$
$480$	−2.35515	−0.107498
$481$	−25.7375	−1.17353
$482$	−11.7569	−0.535511
$483$	12.9679	0.590060
$484$	−10.6806	−0.485480
$485$	2.71926	0.123475
$486$	18.5577	0.841792
$487$	−2.97595	−0.134853	−0.0674266	−	0.997724i	$-0.521479\pi$
$487$	−2.97595	−0.134853	−0.0674266	+	0.997724i	$0.521479\pi$
$488$	−6.95587	−0.314877
$489$	−21.6917	−0.980932
$490$	−4.05287	−0.183090
$491$	−7.00895	−0.316310	−0.158155	−	0.987414i	$-0.550555\pi$
$491$	−7.00895	−0.316310	−0.158155	+	0.987414i	$0.550555\pi$
$492$	−16.2063	−0.730634
$493$	19.3726	0.872499
$494$	13.7719	0.619626
$495$	−2.18373	−0.0981514
$496$	6.96205	0.312605
$497$	−2.13243	−0.0956527
$498$	46.1476	2.06793
$499$	22.6541	1.01414	0.507068	−	0.861906i	$-0.330729\pi$
$499$	22.6541	1.01414	0.507068	+	0.861906i	$0.330729\pi$
$500$	−7.96842	−0.356359
$501$	18.9633	0.847216
$502$	5.64100	0.251770
$503$	27.1397	1.21010	0.605049	−	0.796189i	$-0.293154\pi$
$503$	27.1397	1.21010	0.605049	+	0.796189i	$0.293154\pi$
$504$	6.79482	0.302665
$505$	−13.3784	−0.595331
$506$	1.76959	0.0786677
$507$	10.5690	0.469387
$508$	−4.75260	−0.210862
$509$	−0.466416	−0.0206735	−0.0103368	−	0.999947i	$-0.503290\pi$
$509$	−0.466416	−0.0206735	−0.0103368	+	0.999947i	$0.503290\pi$
$510$	−18.6154	−0.824302
$511$	10.3545	0.458054
$512$	1.00000	0.0441942
$513$	−18.5728	−0.820008
$514$	−19.0882	−0.841943
$515$	6.93779	0.305715
$516$	−26.2206	−1.15430
$517$	−5.19613	−0.228526
$518$	−12.8856	−0.566163
$519$	9.70702	0.426091
$520$	2.60145	0.114081
$521$	32.4285	1.42072	0.710360	−	0.703839i	$-0.248532\pi$
$521$	32.4285	1.42072	0.710360	+	0.703839i	$0.248532\pi$
$522$	11.0039	0.481628
$523$	−39.5928	−1.73127	−0.865636	−	0.500674i	$-0.833085\pi$
$523$	−39.5928	−1.73127	−0.865636	+	0.500674i	$0.833085\pi$
$524$	4.32109	0.188768
$525$	17.6419	0.769958
$526$	−0.376581	−0.0164197
$527$	55.0287	2.39709
$528$	1.54678	0.0673151
$529$	−13.1973	−0.573796
$530$	2.02179	0.0878210
$531$	19.1506	0.831066
$532$	6.89498	0.298935
$533$	17.9011	0.775382
$534$	4.14640	0.179432
$535$	12.3953	0.535895
$536$	5.69331	0.245913
$537$	−56.6516	−2.44470
$538$	15.2623	0.658006
$539$	2.66178	0.114651
$540$	−3.50832	−0.150974
$541$	16.6344	0.715168	0.357584	−	0.933881i	$-0.383601\pi$
$541$	16.6344	0.715168	0.357584	+	0.933881i	$0.383601\pi$
$542$	0.575961	0.0247396
$543$	−44.2327	−1.89821
$544$	7.90410	0.338885
$545$	−8.85106	−0.379138
$546$	−12.5206	−0.535831
$547$	−16.2679	−0.695566	−0.347783	−	0.937575i	$-0.613065\pi$
$547$	−16.2679	−0.695566	−0.347783	+	0.937575i	$0.613065\pi$
$548$	−16.4756	−0.703804
$549$	−31.2293	−1.33283
$550$	2.40740	0.102652
$551$	11.1661	0.475693
$552$	8.56846	0.364698
$553$	−13.5610	−0.576671
$554$	−3.50440	−0.148888
$555$	20.0520	0.851162
$556$	21.0914	0.894476
$557$	38.0094	1.61051	0.805254	−	0.592930i	$-0.202029\pi$
$557$	38.0094	1.61051	0.805254	+	0.592930i	$0.202029\pi$
$558$	31.2571	1.32322
$559$	28.9627	1.22499
$560$	1.30243	0.0550379
$561$	12.2259	0.516179
$562$	−1.04794	−0.0442048
$563$	−3.74533	−0.157847	−0.0789234	−	0.996881i	$-0.525148\pi$
$563$	−3.74533	−0.157847	−0.0789234	+	0.996881i	$0.525148\pi$
$564$	−25.1600	−1.05943
$565$	−10.0407	−0.422416
$566$	−29.3215	−1.23248
$567$	−3.49920	−0.146953
$568$	−1.40899	−0.0591200
$569$	2.05013	0.0859458	0.0429729	−	0.999076i	$-0.486317\pi$
$569$	2.05013	0.0859458	0.0429729	+	0.999076i	$0.486317\pi$
$570$	−10.7296	−0.449415
$571$	−41.5450	−1.73860	−0.869302	−	0.494281i	$-0.835431\pi$
$571$	−41.5450	−1.73860	−0.869302	+	0.494281i	$0.835431\pi$
$572$	−1.70854	−0.0714378
$573$	31.9930	1.33653
$574$	8.96229	0.374079
$575$	13.3359	0.556145
$576$	4.48964	0.187068
$577$	30.4812	1.26895	0.634475	−	0.772943i	$-0.281216\pi$
$577$	30.4812	1.26895	0.634475	+	0.772943i	$0.281216\pi$
$578$	45.4747	1.89150
$579$	24.3586	1.01231
$580$	2.10924	0.0875813
$581$	−25.5203	−1.05876
$582$	−8.64754	−0.358452
$583$	−1.32784	−0.0549936
$584$	6.84165	0.283109
$585$	11.6796	0.482891
$586$	11.0528	0.456589
$587$	12.7626	0.526770	0.263385	−	0.964691i	$-0.415161\pi$
$587$	12.7626	0.526770	0.263385	+	0.964691i	$0.415161\pi$
$588$	12.8885	0.531514
$589$	31.7178	1.30691
$590$	3.67080	0.151124
$591$	−21.8658	−0.899439
$592$	−8.51411	−0.349928
$593$	18.1671	0.746033	0.373016	−	0.927825i	$-0.378324\pi$
$593$	18.1671	0.746033	0.373016	+	0.927825i	$0.378324\pi$
$594$	2.30414	0.0945402
$595$	10.2946	0.422036
$596$	−2.88618	−0.118223
$597$	−56.3566	−2.30652
$598$	−9.46454	−0.387034
$599$	28.3740	1.15933	0.579665	−	0.814855i	$-0.303183\pi$
$599$	28.3740	1.15933	0.579665	+	0.814855i	$0.303183\pi$
$600$	11.6568	0.475887
$601$	30.7312	1.25355	0.626775	−	0.779200i	$-0.284375\pi$
$601$	30.7312	1.25355	0.626775	+	0.779200i	$0.284375\pi$
$602$	14.5003	0.590990
$603$	25.5609	1.04092
$604$	1.87975	0.0764859
$605$	−9.19142	−0.373684
$606$	42.5447	1.72826
$607$	27.0028	1.09601	0.548005	−	0.836475i	$-0.315387\pi$
$607$	27.0028	1.09601	0.548005	+	0.836475i	$0.315387\pi$
$608$	4.55582	0.184763
$609$	−10.1516	−0.411363
$610$	−5.98605	−0.242368
$611$	27.7913	1.12431
$612$	35.4865	1.43446
$613$	44.0604	1.77958	0.889792	−	0.456366i	$-0.150849\pi$
$613$	44.0604	1.77958	0.889792	+	0.456366i	$0.150849\pi$
$614$	−1.69618	−0.0684524
$615$	−13.9467	−0.562385
$616$	−0.855393	−0.0344648
$617$	5.26728	0.212053	0.106026	−	0.994363i	$-0.466187\pi$
$617$	5.26728	0.212053	0.106026	+	0.994363i	$0.466187\pi$
$618$	−22.0629	−0.887500
$619$	40.5653	1.63046	0.815228	−	0.579140i	$-0.196611\pi$
$619$	40.5653	1.63046	0.815228	+	0.579140i	$0.196611\pi$
$620$	5.99137	0.240619
$621$	12.7639	0.512197
$622$	1.77178	0.0710418
$623$	−2.29302	−0.0918679
$624$	−8.27290	−0.331181
$625$	14.4396	0.577585
$626$	13.9288	0.556706
$627$	7.04686	0.281424
$628$	10.8446	0.432747
$629$	−67.2963	−2.68328
$630$	5.84746	0.232968
$631$	5.05493	0.201234	0.100617	−	0.994925i	$-0.467918\pi$
$631$	5.05493	0.201234	0.100617	+	0.994925i	$0.467918\pi$
$632$	−8.96033	−0.356423
$633$	9.16479	0.364268
$634$	24.9716	0.991747
$635$	−4.08997	−0.162305
$636$	−6.42951	−0.254947
$637$	−14.2364	−0.564067
$638$	−1.38527	−0.0548435
$639$	−6.32586	−0.250247
$640$	0.860575	0.0340172
$641$	−28.5409	−1.12730	−0.563648	−	0.826015i	$-0.690603\pi$
$641$	−28.5409	−1.12730	−0.563648	+	0.826015i	$0.690603\pi$
$642$	−39.4183	−1.55572
$643$	−27.9780	−1.10335	−0.551673	−	0.834061i	$-0.686010\pi$
$643$	−27.9780	−1.10335	−0.551673	+	0.834061i	$0.686010\pi$
$644$	−4.73848	−0.186722
$645$	−22.5648	−0.888487
$646$	36.0096	1.41678
$647$	2.00792	0.0789396	0.0394698	−	0.999221i	$-0.487433\pi$
$647$	2.00792	0.0789396	0.0394698	+	0.999221i	$0.487433\pi$
$648$	−2.31208	−0.0908269
$649$	−2.41085	−0.0946343
$650$	−12.8759	−0.505033
$651$	−28.8359	−1.13017
$652$	7.92617	0.310413
$653$	−20.1059	−0.786806	−0.393403	−	0.919366i	$-0.628702\pi$
$653$	−20.1059	−0.786806	−0.393403	+	0.919366i	$0.628702\pi$
$654$	28.1473	1.10065
$655$	3.71862	0.145299
$656$	5.92178	0.231207
$657$	30.7165	1.19837
$658$	13.9139	0.542419
$659$	−7.94205	−0.309379	−0.154689	−	0.987963i	$-0.549438\pi$
$659$	−7.94205	−0.309379	−0.154689	+	0.987963i	$0.549438\pi$
$660$	1.33112	0.0518139
$661$	−35.5213	−1.38162	−0.690809	−	0.723037i	$-0.742746\pi$
$661$	−35.5213	−1.38162	−0.690809	+	0.723037i	$0.742746\pi$
$662$	5.36926	0.208682
$663$	−65.3898	−2.53953
$664$	−16.8624	−0.654388
$665$	5.93365	0.230097
$666$	−38.2253	−1.48120
$667$	−7.67377	−0.297130
$668$	−6.92920	−0.268099
$669$	46.6377	1.80312
$670$	4.89952	0.189285
$671$	3.93143	0.151771
$672$	−4.14188	−0.159776
$673$	24.1768	0.931945	0.465973	−	0.884799i	$-0.345704\pi$
$673$	24.1768	0.931945	0.465973	+	0.884799i	$0.345704\pi$
$674$	−2.68149	−0.103287
$675$	17.3644	0.668356
$676$	−3.86193	−0.148536
$677$	−30.7618	−1.18227	−0.591136	−	0.806572i	$-0.701320\pi$
$677$	−30.7618	−1.18227	−0.591136	+	0.806572i	$0.701320\pi$
$678$	31.9305	1.22628
$679$	4.78221	0.183525
$680$	6.80207	0.260847
$681$	4.77452	0.182960
$682$	−3.93492	−0.150676
$683$	7.37698	0.282272	0.141136	−	0.989990i	$-0.454924\pi$
$683$	7.37698	0.282272	0.141136	+	0.989990i	$0.454924\pi$
$684$	20.4540	0.782076
$685$	−14.1785	−0.541733
$686$	−17.7217	−0.676616
$687$	19.8678	0.758004
$688$	9.58102	0.365273
$689$	7.10190	0.270561
$690$	7.37381	0.280716
$691$	−11.0789	−0.421461	−0.210730	−	0.977544i	$-0.567584\pi$
$691$	−11.0789	−0.421461	−0.210730	+	0.977544i	$0.567584\pi$
$692$	−3.54696	−0.134835
$693$	−3.84041	−0.145885
$694$	31.1551	1.18263
$695$	18.1508	0.688498
$696$	−6.70759	−0.254251
$697$	46.8063	1.77292
$698$	32.8399	1.24301
$699$	2.62855	0.0994207
$700$	−6.44638	−0.243650
$701$	−7.49331	−0.283018	−0.141509	−	0.989937i	$-0.545195\pi$
$701$	−7.49331	−0.283018	−0.141509	+	0.989937i	$0.545195\pi$
$702$	−12.3236	−0.465125
$703$	−38.7887	−1.46294
$704$	−0.565196	−0.0213016
$705$	−21.6521	−0.815466
$706$	34.0166	1.28023
$707$	−23.5278	−0.884855
$708$	−11.6735	−0.438718
$709$	0.160535	0.00602902	0.00301451	−	0.999995i	$-0.499040\pi$
$709$	0.160535	0.00602902	0.00301451	+	0.999995i	$0.499040\pi$
$710$	−1.21254	−0.0455060
$711$	−40.2286	−1.50869
$712$	−1.51510	−0.0567807
$713$	−21.7976	−0.816328
$714$	−32.7378	−1.22518
$715$	−1.47033	−0.0549873
$716$	20.7005	0.773616
$717$	34.3605	1.28322
$718$	18.4725	0.689386
$719$	8.29195	0.309238	0.154619	−	0.987974i	$-0.450585\pi$
$719$	8.29195	0.309238	0.154619	+	0.987974i	$0.450585\pi$
$720$	3.86367	0.143991
$721$	12.2011	0.454392
$722$	1.75545	0.0653312
$723$	32.1753	1.19661
$724$	16.1627	0.600681
$725$	−10.4396	−0.387719
$726$	29.2297	1.08482
$727$	−32.6709	−1.21170	−0.605849	−	0.795580i	$-0.707166\pi$
$727$	−32.6709	−1.21170	−0.605849	+	0.795580i	$0.707166\pi$
$728$	4.57503	0.169562
$729$	−43.8509	−1.62411
$730$	5.88776	0.217916
$731$	75.7293	2.80095
$732$	19.0363	0.703601
$733$	22.1314	0.817442	0.408721	−	0.912659i	$-0.365975\pi$
$733$	22.1314	0.817442	0.408721	+	0.912659i	$0.365975\pi$
$734$	−17.2080	−0.635158
$735$	11.0916	0.409118
$736$	−3.13092	−0.115407
$737$	−3.21783	−0.118530
$738$	26.5866	0.978668
$739$	−24.8178	−0.912937	−0.456468	−	0.889740i	$-0.650886\pi$
$739$	−24.8178	−0.912937	−0.456468	+	0.889740i	$0.650886\pi$
$740$	−7.32703	−0.269347
$741$	−37.6898	−1.38457
$742$	3.55561	0.130531
$743$	6.31589	0.231708	0.115854	−	0.993266i	$-0.463040\pi$
$743$	6.31589	0.231708	0.115854	+	0.993266i	$0.463040\pi$
$744$	−19.0532	−0.698523
$745$	−2.48378	−0.0909986
$746$	31.1554	1.14068
$747$	−75.7060	−2.76994
$748$	−4.46736	−0.163343
$749$	21.7989	0.796514
$750$	21.8073	0.796291
$751$	8.33289	0.304071	0.152036	−	0.988375i	$-0.451417\pi$
$751$	8.33289	0.304071	0.152036	+	0.988375i	$0.451417\pi$
$752$	9.19350	0.335252
$753$	−15.4378	−0.562586
$754$	7.40907	0.269822
$755$	1.61767	0.0588729
$756$	−6.16989	−0.224397
$757$	−31.6587	−1.15066	−0.575328	−	0.817923i	$-0.695126\pi$
$757$	−31.6587	−1.15066	−0.575328	+	0.817923i	$0.695126\pi$
$758$	12.1380	0.440874
$759$	−4.84286	−0.175785
$760$	3.92062	0.142216
$761$	−28.4089	−1.02982	−0.514912	−	0.857243i	$-0.672175\pi$
$761$	−28.4089	−1.02982	−0.514912	+	0.857243i	$0.672175\pi$
$762$	13.0065	0.471177
$763$	−15.5659	−0.563522
$764$	−11.6903	−0.422939
$765$	30.5388	1.10413
$766$	8.45074	0.305338
$767$	12.8943	0.465587
$768$	−2.73672	−0.0987529
$769$	−16.3372	−0.589135	−0.294568	−	0.955631i	$-0.595176\pi$
$769$	−16.3372	−0.589135	−0.294568	+	0.955631i	$0.595176\pi$
$770$	−0.736131	−0.0265283
$771$	52.2389	1.88134
$772$	−8.90067	−0.320342
$773$	−12.5508	−0.451420	−0.225710	−	0.974195i	$-0.572470\pi$
$773$	−12.5508	−0.451420	−0.225710	+	0.974195i	$0.572470\pi$
$774$	43.0153	1.54615
$775$	−29.6542	−1.06521
$776$	3.15982	0.113431
$777$	35.2644	1.26510
$778$	−24.0460	−0.862090
$779$	26.9785	0.966607
$780$	−7.11945	−0.254917
$781$	0.796357	0.0284959
$782$	−24.7471	−0.884955
$783$	−9.99187	−0.357080
$784$	−4.70948	−0.168196
$785$	9.33261	0.333095
$786$	−11.8256	−0.421806
$787$	−33.8808	−1.20772	−0.603860	−	0.797090i	$-0.706371\pi$
$787$	−33.8808	−1.20772	−0.603860	+	0.797090i	$0.706371\pi$
$788$	7.98979	0.284624
$789$	1.03060	0.0366902
$790$	−7.71104	−0.274346
$791$	−17.6580	−0.627847
$792$	−2.53752	−0.0901670
$793$	−21.0271	−0.746693
$794$	−35.6374	−1.26473
$795$	−5.53308	−0.196238
$796$	20.5928	0.729891
$797$	−12.3937	−0.439006	−0.219503	−	0.975612i	$-0.570444\pi$
$797$	−12.3937	−0.439006	−0.219503	+	0.975612i	$0.570444\pi$
$798$	−18.8696	−0.667977
$799$	72.6663	2.57075
$800$	−4.25941	−0.150593
$801$	−6.80224	−0.240345
$802$	5.59313	0.197500
$803$	−3.86687	−0.136459
$804$	−15.5810	−0.549499
$805$	−4.07782	−0.143724
$806$	21.0457	0.741305
$807$	−41.7687	−1.47033
$808$	−15.5459	−0.546902
$809$	7.67182	0.269727	0.134863	−	0.990864i	$-0.456940\pi$
$809$	7.67182	0.269727	0.134863	+	0.990864i	$0.456940\pi$
$810$	−1.98972	−0.0699115
$811$	−9.73702	−0.341913	−0.170957	−	0.985279i	$-0.554686\pi$
$811$	−9.73702	−0.341913	−0.170957	+	0.985279i	$0.554686\pi$
$812$	3.70939	0.130174
$813$	−1.57624	−0.0552813
$814$	4.81214	0.168665
$815$	6.82106	0.238931
$816$	−21.6313	−0.757247
$817$	43.6493	1.52710
$818$	−24.4082	−0.853412
$819$	20.5402	0.717733
$820$	5.09614	0.177965
$821$	12.2129	0.426234	0.213117	−	0.977027i	$-0.431639\pi$
$821$	12.2129	0.426234	0.213117	+	0.977027i	$0.431639\pi$
$822$	45.0892	1.57267
$823$	16.3618	0.570335	0.285168	−	0.958478i	$-0.407951\pi$
$823$	16.3618	0.570335	0.285168	+	0.958478i	$0.407951\pi$
$824$	8.06180	0.280846
$825$	−6.58838	−0.229378
$826$	6.45563	0.224620
$827$	−46.1562	−1.60501	−0.802504	−	0.596647i	$-0.796499\pi$
$827$	−46.1562	−1.60501	−0.802504	+	0.596647i	$0.796499\pi$
$828$	−14.0567	−0.488504
$829$	−4.07902	−0.141670	−0.0708352	−	0.997488i	$-0.522566\pi$
$829$	−4.07902	−0.141670	−0.0708352	+	0.997488i	$0.522566\pi$
$830$	−14.5114	−0.503697
$831$	9.59057	0.332693
$832$	3.02292	0.104801
$833$	−37.2242	−1.28974
$834$	−57.7213	−1.99873
$835$	−5.96310	−0.206361
$836$	−2.57493	−0.0890558
$837$	−28.3823	−0.981036
$838$	10.8805	0.375862
$839$	44.3332	1.53055	0.765276	−	0.643703i	$-0.222603\pi$
$839$	44.3332	1.53055	0.765276	+	0.643703i	$0.222603\pi$
$840$	−3.56440	−0.122983
$841$	−22.9928	−0.792855
$842$	3.35606	0.115657
$843$	2.86793	0.0987767
$844$	−3.34882	−0.115271
$845$	−3.32348	−0.114331
$846$	41.2755	1.41908
$847$	−16.1644	−0.555417
$848$	2.34935	0.0806769
$849$	80.2448	2.75399
$850$	−33.6668	−1.15476
$851$	26.6570	0.913791
$852$	3.85602	0.132105
$853$	−8.74433	−0.299400	−0.149700	−	0.988731i	$-0.547831\pi$
$853$	−8.74433	−0.299400	−0.149700	+	0.988731i	$0.547831\pi$
$854$	−10.5273	−0.360238
$855$	17.6022	0.601982
$856$	14.4035	0.492301
$857$	−14.3925	−0.491639	−0.245819	−	0.969316i	$-0.579057\pi$
$857$	−14.3925	−0.491639	−0.245819	+	0.969316i	$0.579057\pi$
$858$	4.67581	0.159629
$859$	−0.00120618	−4.11544e−5 0	−2.05772e−5	−	1.00000i	$-0.500007\pi$
$859$	−0.00120618	−4.11544e−5 0	−2.05772e−5		1.00000i	$0.500007\pi$
$860$	8.24519	0.281159
$861$	−24.5273	−0.835888
$862$	−4.67278	−0.159156
$863$	29.8214	1.01513	0.507567	−	0.861612i	$-0.330545\pi$
$863$	29.8214	1.01513	0.507567	+	0.861612i	$0.330545\pi$
$864$	−4.07672	−0.138693
$865$	−3.05242	−0.103785
$866$	−12.5400	−0.426126
$867$	−124.452	−4.22660
$868$	10.5367	0.357638
$869$	5.06434	0.171796
$870$	−5.77239	−0.195702
$871$	17.2104	0.583153
$872$	−10.2850	−0.348296
$873$	14.1864	0.480138
$874$	−14.2639	−0.482484
$875$	−12.0598	−0.407695
$876$	−18.7237	−0.632614
$877$	26.5226	0.895606	0.447803	−	0.894132i	$-0.352207\pi$
$877$	26.5226	0.895606	0.447803	+	0.894132i	$0.352207\pi$
$878$	18.5102	0.624690
$879$	−30.2485	−1.02026
$880$	−0.486394	−0.0163963
$881$	39.2478	1.32229	0.661147	−	0.750257i	$-0.270070\pi$
$881$	39.2478	1.32229	0.661147	+	0.750257i	$0.270070\pi$
$882$	−21.1439	−0.711951
$883$	−48.8056	−1.64244	−0.821220	−	0.570611i	$-0.806706\pi$
$883$	−48.8056	−1.64244	−0.821220	+	0.570611i	$0.806706\pi$
$884$	23.8935	0.803625
$885$	−10.0459	−0.337691
$886$	−26.6770	−0.896232
$887$	−27.5649	−0.925540	−0.462770	−	0.886478i	$-0.653144\pi$
$887$	−27.5649	−0.925540	−0.462770	+	0.886478i	$0.653144\pi$
$888$	23.3007	0.781922
$889$	−7.19280	−0.241239
$890$	−1.30386	−0.0437054
$891$	1.30678	0.0437786
$892$	−17.0415	−0.570590
$893$	41.8839	1.40159
$894$	7.89867	0.264171
$895$	17.8144	0.595469
$896$	1.51345	0.0505607
$897$	25.9018	0.864836
$898$	−10.6878	−0.356658
$899$	17.0637	0.569106
$900$	−19.1232	−0.637440
$901$	18.5695	0.618639
$902$	−3.34697	−0.111442
$903$	−39.6834	−1.32058
$904$	−11.6674	−0.388053
$905$	13.9092	0.462357
$906$	−5.14435	−0.170909
$907$	39.3128	1.30536	0.652680	−	0.757633i	$-0.273644\pi$
$907$	39.3128	1.30536	0.652680	+	0.757633i	$0.273644\pi$
$908$	−1.74461	−0.0578971
$909$	−69.7953	−2.31496
$910$	3.93716	0.130516
$911$	−11.1063	−0.367970	−0.183985	−	0.982929i	$-0.558900\pi$
$911$	−11.1063	−0.367970	−0.183985	+	0.982929i	$0.558900\pi$
$912$	−12.4680	−0.412856
$913$	9.53056	0.315415
$914$	0.751043	0.0248423
$915$	16.3821	0.541577
$916$	−7.25971	−0.239868
$917$	6.53973	0.215961
$918$	−32.2228	−1.06351
$919$	−14.6513	−0.483302	−0.241651	−	0.970363i	$-0.577689\pi$
$919$	−14.6513	−0.483302	−0.241651	+	0.970363i	$0.577689\pi$
$920$	−2.69440	−0.0888316
$921$	4.64198	0.152958
$922$	23.3622	0.769394
$923$	−4.25928	−0.140196
$924$	2.34097	0.0770123
$925$	36.2651	1.19239
$926$	5.72639	0.188181
$927$	36.1945	1.18878
$928$	2.45096	0.0804567
$929$	−16.2608	−0.533501	−0.266750	−	0.963766i	$-0.585950\pi$
$929$	−16.2608	−0.533501	−0.266750	+	0.963766i	$0.585950\pi$
$930$	−16.3967	−0.537669
$931$	−21.4555	−0.703177
$932$	−0.960473	−0.0314613
$933$	−4.84886	−0.158744
$934$	−9.21292	−0.301456
$935$	−3.84450	−0.125729
$936$	13.5718	0.443609
$937$	−18.2860	−0.597379	−0.298690	−	0.954350i	$-0.596550\pi$
$937$	−18.2860	−0.597379	−0.298690	+	0.954350i	$0.596550\pi$
$938$	8.61651	0.281339
$939$	−38.1192	−1.24397
$940$	7.91170	0.258051
$941$	6.75727	0.220281	0.110140	−	0.993916i	$-0.464870\pi$
$941$	6.75727	0.220281	0.110140	+	0.993916i	$0.464870\pi$
$942$	−29.6787	−0.966984
$943$	−18.5406	−0.603766
$944$	4.26552	0.138831
$945$	−5.30966	−0.172723
$946$	−5.41515	−0.176062
$947$	3.70507	0.120398	0.0601992	−	0.998186i	$-0.480826\pi$
$947$	3.70507	0.120398	0.0601992	+	0.998186i	$0.480826\pi$
$948$	24.5219	0.796434
$949$	20.6818	0.671359
$950$	−19.4051	−0.629584
$951$	−68.3402	−2.21608
$952$	11.9624	0.387704
$953$	−1.67876	−0.0543804	−0.0271902	−	0.999630i	$-0.508656\pi$
$953$	−1.67876	−0.0543804	−0.0271902	+	0.999630i	$0.508656\pi$
$954$	10.5477	0.341495
$955$	−10.0604	−0.325545
$956$	−12.5554	−0.406069
$957$	3.79110	0.122549
$958$	−17.6585	−0.570521
$959$	−24.9350	−0.805192
$960$	−2.35515	−0.0760123
$961$	17.4701	0.563552
$962$	−25.7375	−0.829811
$963$	64.6664	2.08385
$964$	−11.7569	−0.378663
$965$	−7.65970	−0.246574
$966$	12.9679	0.417235
$967$	−20.0050	−0.643318	−0.321659	−	0.946856i	$-0.604241\pi$
$967$	−20.0050	−0.643318	−0.321659	+	0.946856i	$0.604241\pi$
$968$	−10.6806	−0.343286
$969$	−98.5482	−3.16582
$970$	2.71926	0.0873103
$971$	−11.4129	−0.366257	−0.183129	−	0.983089i	$-0.558622\pi$
$971$	−11.4129	−0.366257	−0.183129	+	0.983089i	$0.558622\pi$
$972$	18.5577	0.595237
$973$	31.9207	1.02333
$974$	−2.97595	−0.0953556
$975$	35.2377	1.12851
$976$	−6.95587	−0.222652
$977$	−18.7911	−0.601181	−0.300590	−	0.953753i	$-0.597184\pi$
$977$	−18.7911	−0.601181	−0.300590	+	0.953753i	$0.597184\pi$
$978$	−21.6917	−0.693624
$979$	0.856328	0.0273684
$980$	−4.05287	−0.129464
$981$	−46.1761	−1.47429
$982$	−7.00895	−0.223665
$983$	43.6405	1.39192	0.695958	−	0.718082i	$-0.254980\pi$
$983$	43.6405	1.39192	0.695958	+	0.718082i	$0.254980\pi$
$984$	−16.2063	−0.516637
$985$	6.87582	0.219082
$986$	19.3726	0.616950
$987$	−38.0783	−1.21205
$988$	13.7719	0.438142
$989$	−29.9974	−0.953863
$990$	−2.18373	−0.0694035
$991$	40.1201	1.27446	0.637228	−	0.770675i	$-0.280081\pi$
$991$	40.1201	1.27446	0.637228	+	0.770675i	$0.280081\pi$
$992$	6.96205	0.221045
$993$	−14.6942	−0.466305
$994$	−2.13243	−0.0676367
$995$	17.7216	0.561813
$996$	46.1476	1.46224
$997$	−17.3521	−0.549546	−0.274773	−	0.961509i	$-0.588603\pi$
$997$	−17.3521	−0.549546	−0.274773	+	0.961509i	$0.588603\pi$
$998$	22.6541	0.717102
$999$	34.7096	1.09816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8014.2.a.d.1.8	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8014.2.a.d.1.8	✓	88	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:	\( N \)	\(=\)	\( 8014 = 2 \cdot 4007 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.73672 q^{3} +1.00000 q^{4} +0.860575 q^{5} -2.73672 q^{6} +1.51345 q^{7} +1.00000 q^{8} +4.48964 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.73672 q^{3} +1.00000 q^{4} +0.860575 q^{5} -2.73672 q^{6} +1.51345 q^{7} +1.00000 q^{8} +4.48964 q^{9} +0.860575 q^{10} -0.565196 q^{11} -2.73672 q^{12} +3.02292 q^{13} +1.51345 q^{14} -2.35515 q^{15} +1.00000 q^{16} +7.90410 q^{17} +4.48964 q^{18} +4.55582 q^{19} +0.860575 q^{20} -4.14188 q^{21} -0.565196 q^{22} -3.13092 q^{23} -2.73672 q^{24} -4.25941 q^{25} +3.02292 q^{26} -4.07672 q^{27} +1.51345 q^{28} +2.45096 q^{29} -2.35515 q^{30} +6.96205 q^{31} +1.00000 q^{32} +1.54678 q^{33} +7.90410 q^{34} +1.30243 q^{35} +4.48964 q^{36} -8.51411 q^{37} +4.55582 q^{38} -8.27290 q^{39} +0.860575 q^{40} +5.92178 q^{41} -4.14188 q^{42} +9.58102 q^{43} -0.565196 q^{44} +3.86367 q^{45} -3.13092 q^{46} +9.19350 q^{47} -2.73672 q^{48} -4.70948 q^{49} -4.25941 q^{50} -21.6313 q^{51} +3.02292 q^{52} +2.34935 q^{53} -4.07672 q^{54} -0.486394 q^{55} +1.51345 q^{56} -12.4680 q^{57} +2.45096 q^{58} +4.26552 q^{59} -2.35515 q^{60} -6.95587 q^{61} +6.96205 q^{62} +6.79482 q^{63} +1.00000 q^{64} +2.60145 q^{65} +1.54678 q^{66} +5.69331 q^{67} +7.90410 q^{68} +8.56846 q^{69} +1.30243 q^{70} -1.40899 q^{71} +4.48964 q^{72} +6.84165 q^{73} -8.51411 q^{74} +11.6568 q^{75} +4.55582 q^{76} -0.855393 q^{77} -8.27290 q^{78} -8.96033 q^{79} +0.860575 q^{80} -2.31208 q^{81} +5.92178 q^{82} -16.8624 q^{83} -4.14188 q^{84} +6.80207 q^{85} +9.58102 q^{86} -6.70759 q^{87} -0.565196 q^{88} -1.51510 q^{89} +3.86367 q^{90} +4.57503 q^{91} -3.13092 q^{92} -19.0532 q^{93} +9.19350 q^{94} +3.92062 q^{95} -2.73672 q^{96} +3.15982 q^{97} -4.70948 q^{98} -2.53752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 88 q^{2} + 22 q^{3} + 88 q^{4} + 25 q^{5} + 22 q^{6} + 33 q^{7} + 88 q^{8} + 108 q^{9}+O(q^{10}) \) 88 * q + 88 * q^2 + 22 * q^3 + 88 * q^4 + 25 * q^5 + 22 * q^6 + 33 * q^7 + 88 * q^8 + 108 * q^9	\( 88 q + 88 q^{2} + 22 q^{3} + 88 q^{4} + 25 q^{5} + 22 q^{6} + 33 q^{7} + 88 q^{8} + 108 q^{9} + 25 q^{10} + 70 q^{11} + 22 q^{12} + 31 q^{13} + 33 q^{14} + 47 q^{15} + 88 q^{16} + 19 q^{17} + 108 q^{18} + 33 q^{19} + 25 q^{20} + 48 q^{21} + 70 q^{22} + 77 q^{23} + 22 q^{24} + 109 q^{25} + 31 q^{26} + 88 q^{27} + 33 q^{28} + 83 q^{29} + 47 q^{30} + 51 q^{31} + 88 q^{32} + 30 q^{33} + 19 q^{34} + 40 q^{35} + 108 q^{36} + 45 q^{37} + 33 q^{38} + 82 q^{39} + 25 q^{40} + 35 q^{41} + 48 q^{42} + 78 q^{43} + 70 q^{44} + 37 q^{45} + 77 q^{46} + 59 q^{47} + 22 q^{48} + 103 q^{49} + 109 q^{50} + 21 q^{51} + 31 q^{52} + 58 q^{53} + 88 q^{54} + 35 q^{55} + 33 q^{56} - 16 q^{57} + 83 q^{58} + 54 q^{59} + 47 q^{60} + 18 q^{61} + 51 q^{62} + 47 q^{63} + 88 q^{64} + 34 q^{65} + 30 q^{66} + 88 q^{67} + 19 q^{68} + 62 q^{69} + 40 q^{70} + 139 q^{71} + 108 q^{72} - 6 q^{73} + 45 q^{74} + 45 q^{75} + 33 q^{76} + 37 q^{77} + 82 q^{78} + 94 q^{79} + 25 q^{80} + 112 q^{81} + 35 q^{82} + 58 q^{83} + 48 q^{84} + 83 q^{85} + 78 q^{86} + 21 q^{87} + 70 q^{88} + 99 q^{89} + 37 q^{90} + 53 q^{91} + 77 q^{92} + 57 q^{93} + 59 q^{94} + 92 q^{95} + 22 q^{96} + 16 q^{97} + 103 q^{98} + 150 q^{99}+O(q^{100}) \) 88 * q + 88 * q^2 + 22 * q^3 + 88 * q^4 + 25 * q^5 + 22 * q^6 + 33 * q^7 + 88 * q^8 + 108 * q^9 + 25 * q^10 + 70 * q^11 + 22 * q^12 + 31 * q^13 + 33 * q^14 + 47 * q^15 + 88 * q^16 + 19 * q^17 + 108 * q^18 + 33 * q^19 + 25 * q^20 + 48 * q^21 + 70 * q^22 + 77 * q^23 + 22 * q^24 + 109 * q^25 + 31 * q^26 + 88 * q^27 + 33 * q^28 + 83 * q^29 + 47 * q^30 + 51 * q^31 + 88 * q^32 + 30 * q^33 + 19 * q^34 + 40 * q^35 + 108 * q^36 + 45 * q^37 + 33 * q^38 + 82 * q^39 + 25 * q^40 + 35 * q^41 + 48 * q^42 + 78 * q^43 + 70 * q^44 + 37 * q^45 + 77 * q^46 + 59 * q^47 + 22 * q^48 + 103 * q^49 + 109 * q^50 + 21 * q^51 + 31 * q^52 + 58 * q^53 + 88 * q^54 + 35 * q^55 + 33 * q^56 - 16 * q^57 + 83 * q^58 + 54 * q^59 + 47 * q^60 + 18 * q^61 + 51 * q^62 + 47 * q^63 + 88 * q^64 + 34 * q^65 + 30 * q^66 + 88 * q^67 + 19 * q^68 + 62 * q^69 + 40 * q^70 + 139 * q^71 + 108 * q^72 - 6 * q^73 + 45 * q^74 + 45 * q^75 + 33 * q^76 + 37 * q^77 + 82 * q^78 + 94 * q^79 + 25 * q^80 + 112 * q^81 + 35 * q^82 + 58 * q^83 + 48 * q^84 + 83 * q^85 + 78 * q^86 + 21 * q^87 + 70 * q^88 + 99 * q^89 + 37 * q^90 + 53 * q^91 + 77 * q^92 + 57 * q^93 + 59 * q^94 + 92 * q^95 + 22 * q^96 + 16 * q^97 + 103 * q^98 + 150 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more