↑

Properties

Label 8013.2.a.c.1.16
Level $8013$
Weight $2$
Character 8013.1
Self dual yes
Analytic conductor $63.984$
Analytic rank $1$
Dimension $116$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8013,2,Mod(1,8013)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8013.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8013, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8013 = 3 \cdot 2671 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8013.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [116] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.9841271397\)
Analytic rank: \(1\)
Dimension: \(116\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 8013.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.24523 q^{2} -1.00000 q^{3} +3.04104 q^{4} -1.84042 q^{5} +2.24523 q^{6} +1.63491 q^{7} -2.33738 q^{8} +1.00000 q^{9} +4.13216 q^{10} -5.52455 q^{11} -3.04104 q^{12} +2.79664 q^{13} -3.67075 q^{14} +1.84042 q^{15} -0.834146 q^{16} -5.47709 q^{17} -2.24523 q^{18} -6.33365 q^{19} -5.59680 q^{20} -1.63491 q^{21} +12.4039 q^{22} +6.03736 q^{23} +2.33738 q^{24} -1.61285 q^{25} -6.27909 q^{26} -1.00000 q^{27} +4.97183 q^{28} -2.24543 q^{29} -4.13216 q^{30} +0.349040 q^{31} +6.54760 q^{32} +5.52455 q^{33} +12.2973 q^{34} -3.00892 q^{35} +3.04104 q^{36} +3.61606 q^{37} +14.2205 q^{38} -2.79664 q^{39} +4.30176 q^{40} +3.72173 q^{41} +3.67075 q^{42} +12.6686 q^{43} -16.8004 q^{44} -1.84042 q^{45} -13.5552 q^{46} +0.120321 q^{47} +0.834146 q^{48} -4.32707 q^{49} +3.62122 q^{50} +5.47709 q^{51} +8.50470 q^{52} +2.58473 q^{53} +2.24523 q^{54} +10.1675 q^{55} -3.82140 q^{56} +6.33365 q^{57} +5.04150 q^{58} +10.2881 q^{59} +5.59680 q^{60} +2.02333 q^{61} -0.783674 q^{62} +1.63491 q^{63} -13.0326 q^{64} -5.14699 q^{65} -12.4039 q^{66} -2.94826 q^{67} -16.6561 q^{68} -6.03736 q^{69} +6.75572 q^{70} -6.64245 q^{71} -2.33738 q^{72} +13.4702 q^{73} -8.11887 q^{74} +1.61285 q^{75} -19.2609 q^{76} -9.03215 q^{77} +6.27909 q^{78} +11.4842 q^{79} +1.53518 q^{80} +1.00000 q^{81} -8.35612 q^{82} -15.7671 q^{83} -4.97183 q^{84} +10.0801 q^{85} -28.4438 q^{86} +2.24543 q^{87} +12.9130 q^{88} -13.1816 q^{89} +4.13216 q^{90} +4.57226 q^{91} +18.3599 q^{92} -0.349040 q^{93} -0.270147 q^{94} +11.6566 q^{95} -6.54760 q^{96} +5.37299 q^{97} +9.71524 q^{98} -5.52455 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18}+ \cdots - 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.24523 −1.58761 −0.793807 − 0.608169i \(-0.791904\pi\)
−0.793807 + 0.608169i \(0.791904\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.04104 1.52052
\(5\) −1.84042 −0.823061 −0.411531 − 0.911396i \(-0.635006\pi\)
−0.411531 + 0.911396i \(0.635006\pi\)
\(6\) 2.24523 0.916610
\(7\) 1.63491 0.617938 0.308969 − 0.951072i \(-0.400016\pi\)
0.308969 + 0.951072i \(0.400016\pi\)
\(8\) −2.33738 −0.826387
\(9\) 1.00000 0.333333
\(10\) 4.13216 1.30670
\(11\) −5.52455 −1.66571 −0.832857 − 0.553488i \(-0.813297\pi\)
−0.832857 + 0.553488i \(0.813297\pi\)
\(12\) −3.04104 −0.877873
\(13\) 2.79664 0.775648 0.387824 − 0.921733i \(-0.373227\pi\)
0.387824 + 0.921733i \(0.373227\pi\)
\(14\) −3.67075 −0.981048
\(15\) 1.84042 0.475195
\(16\) −0.834146 −0.208537
\(17\) −5.47709 −1.32839 −0.664194 − 0.747560i \(-0.731225\pi\)
−0.664194 + 0.747560i \(0.731225\pi\)
\(18\) −2.24523 −0.529205
\(19\) −6.33365 −1.45304 −0.726520 − 0.687146i \(-0.758863\pi\)
−0.726520 + 0.687146i \(0.758863\pi\)
\(20\) −5.59680 −1.25148
\(21\) −1.63491 −0.356767
\(22\) 12.4039 2.64451
\(23\) 6.03736 1.25888 0.629439 − 0.777050i \(-0.283285\pi\)
0.629439 + 0.777050i \(0.283285\pi\)
\(24\) 2.33738 0.477115
\(25\) −1.61285 −0.322570
\(26\) −6.27909 −1.23143
\(27\) −1.00000 −0.192450
\(28\) 4.97183 0.939588
\(29\) −2.24543 −0.416966 −0.208483 − 0.978026i \(-0.566853\pi\)
−0.208483 + 0.978026i \(0.566853\pi\)
\(30\) −4.13216 −0.754426
\(31\) 0.349040 0.0626894 0.0313447 − 0.999509i \(-0.490021\pi\)
0.0313447 + 0.999509i \(0.490021\pi\)
\(32\) 6.54760 1.15746
\(33\) 5.52455 0.961701
\(34\) 12.2973 2.10897
\(35\) −3.00892 −0.508601
\(36\) 3.04104 0.506840
\(37\) 3.61606 0.594476 0.297238 − 0.954803i \(-0.403935\pi\)
0.297238 + 0.954803i \(0.403935\pi\)
\(38\) 14.2205 2.30687
\(39\) −2.79664 −0.447821
\(40\) 4.30176 0.680167
\(41\) 3.72173 0.581236 0.290618 − 0.956839i \(-0.406139\pi\)
0.290618 + 0.956839i \(0.406139\pi\)
\(42\) 3.67075 0.566408
\(43\) 12.6686 1.93194 0.965970 − 0.258655i \(-0.0832794\pi\)
0.965970 + 0.258655i \(0.0832794\pi\)
\(44\) −16.8004 −2.53275
\(45\) −1.84042 −0.274354
\(46\) −13.5552 −1.99861
\(47\) 0.120321 0.0175506 0.00877529 − 0.999961i \(-0.497207\pi\)
0.00877529 + 0.999961i \(0.497207\pi\)
\(48\) 0.834146 0.120399
\(49\) −4.32707 −0.618152
\(50\) 3.62122 0.512117
\(51\) 5.47709 0.766946
\(52\) 8.50470 1.17939
\(53\) 2.58473 0.355040 0.177520 − 0.984117i \(-0.443193\pi\)
0.177520 + 0.984117i \(0.443193\pi\)
\(54\) 2.24523 0.305537
\(55\) 10.1675 1.37099
\(56\) −3.82140 −0.510656
\(57\) 6.33365 0.838913
\(58\) 5.04150 0.661982
\(59\) 10.2881 1.33939 0.669696 − 0.742635i \(-0.266424\pi\)
0.669696 + 0.742635i \(0.266424\pi\)
\(60\) 5.59680 0.722544
\(61\) 2.02333 0.259060 0.129530 − 0.991575i \(-0.458653\pi\)
0.129530 + 0.991575i \(0.458653\pi\)
\(62\) −0.783674 −0.0995266
\(63\) 1.63491 0.205979
\(64\) −13.0326 −1.62907
\(65\) −5.14699 −0.638406
\(66\) −12.4039 −1.52681
\(67\) −2.94826 −0.360188 −0.180094 − 0.983649i \(-0.557640\pi\)
−0.180094 + 0.983649i \(0.557640\pi\)
\(68\) −16.6561 −2.01984
\(69\) −6.03736 −0.726813
\(70\) 6.75572 0.807463
\(71\) −6.64245 −0.788313 −0.394157 − 0.919043i \(-0.628963\pi\)
−0.394157 + 0.919043i \(0.628963\pi\)
\(72\) −2.33738 −0.275462
\(73\) 13.4702 1.57656 0.788281 − 0.615315i \(-0.210971\pi\)
0.788281 + 0.615315i \(0.210971\pi\)
\(74\) −8.11887 −0.943799
\(75\) 1.61285 0.186236
\(76\) −19.2609 −2.20938
\(77\) −9.03215 −1.02931
\(78\) 6.27909 0.710967
\(79\) 11.4842 1.29207 0.646035 − 0.763307i \(-0.276426\pi\)
0.646035 + 0.763307i \(0.276426\pi\)
\(80\) 1.53518 0.171638
\(81\) 1.00000 0.111111
\(82\) −8.35612 −0.922779
\(83\) −15.7671 −1.73066 −0.865331 − 0.501201i \(-0.832892\pi\)
−0.865331 + 0.501201i \(0.832892\pi\)
\(84\) −4.97183 −0.542472
\(85\) 10.0801 1.09335
\(86\) −28.4438 −3.06718
\(87\) 2.24543 0.240735
\(88\) 12.9130 1.37653
\(89\) −13.1816 −1.39724 −0.698621 − 0.715491i \(-0.746203\pi\)
−0.698621 + 0.715491i \(0.746203\pi\)
\(90\) 4.13216 0.435568
\(91\) 4.57226 0.479303
\(92\) 18.3599 1.91415
\(93\) −0.349040 −0.0361937
\(94\) −0.270147 −0.0278636
\(95\) 11.6566 1.19594
\(96\) −6.54760 −0.668262
\(97\) 5.37299 0.545544 0.272772 − 0.962079i \(-0.412059\pi\)
0.272772 + 0.962079i \(0.412059\pi\)
\(98\) 9.71524 0.981388
\(99\) −5.52455 −0.555238
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8013.2.a.c.1.16 ✓ 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8013.2.a.c.1.16 ✓ 116 1.1 even 1 trivial