Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7935,2,Mod(1,7935)] 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("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [25,1,-25,31,-25,-1,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3612940039\)
Analytic rank: \(0\)
Dimension: \(25\)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 7935.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.137935 q^{2} -1.00000 q^{3} -1.98097 q^{4} -1.00000 q^{5} -0.137935 q^{6} -4.46257 q^{7} -0.549115 q^{8} +1.00000 q^{9} -0.137935 q^{10} +2.77203 q^{11} +1.98097 q^{12} +1.01349 q^{13} -0.615543 q^{14} +1.00000 q^{15} +3.88621 q^{16} +5.40876 q^{17} +0.137935 q^{18} +2.91494 q^{19} +1.98097 q^{20} +4.46257 q^{21} +0.382360 q^{22} +0.549115 q^{24} +1.00000 q^{25} +0.139796 q^{26} -1.00000 q^{27} +8.84023 q^{28} -4.86108 q^{29} +0.137935 q^{30} -10.5262 q^{31} +1.63427 q^{32} -2.77203 q^{33} +0.746056 q^{34} +4.46257 q^{35} -1.98097 q^{36} -9.12003 q^{37} +0.402072 q^{38} -1.01349 q^{39} +0.549115 q^{40} +6.38318 q^{41} +0.615543 q^{42} -4.91952 q^{43} -5.49133 q^{44} -1.00000 q^{45} +9.95133 q^{47} -3.88621 q^{48} +12.9145 q^{49} +0.137935 q^{50} -5.40876 q^{51} -2.00770 q^{52} +2.17396 q^{53} -0.137935 q^{54} -2.77203 q^{55} +2.45046 q^{56} -2.91494 q^{57} -0.670512 q^{58} -11.9951 q^{59} -1.98097 q^{60} -10.1527 q^{61} -1.45192 q^{62} -4.46257 q^{63} -7.54699 q^{64} -1.01349 q^{65} -0.382360 q^{66} -1.27158 q^{67} -10.7146 q^{68} +0.615543 q^{70} -1.98303 q^{71} -0.549115 q^{72} +5.37766 q^{73} -1.25797 q^{74} -1.00000 q^{75} -5.77443 q^{76} -12.3704 q^{77} -0.139796 q^{78} -2.74641 q^{79} -3.88621 q^{80} +1.00000 q^{81} +0.880463 q^{82} +2.13769 q^{83} -8.84023 q^{84} -5.40876 q^{85} -0.678572 q^{86} +4.86108 q^{87} -1.52216 q^{88} +6.25296 q^{89} -0.137935 q^{90} -4.52278 q^{91} +10.5262 q^{93} +1.37263 q^{94} -2.91494 q^{95} -1.63427 q^{96} -4.61045 q^{97} +1.78136 q^{98} +2.77203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q + q^{2} - 25 q^{3} + 31 q^{4} - 25 q^{5} - q^{6} + 15 q^{7} + 3 q^{8} + 25 q^{9} - q^{10} - 15 q^{11} - 31 q^{12} + 24 q^{13} - 5 q^{14} + 25 q^{15} + 39 q^{16} + 6 q^{17} + q^{18} - 13 q^{19}+ \cdots - 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.137935 0.0975346 0.0487673 − 0.998810i \(-0.484471\pi\)
0.0487673 + 0.998810i \(0.484471\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.98097 −0.990487
\(5\) −1.00000 −0.447214
\(6\) −0.137935 −0.0563116
\(7\) −4.46257 −1.68669 −0.843346 − 0.537371i \(-0.819418\pi\)
−0.843346 + 0.537371i \(0.819418\pi\)
\(8\) −0.549115 −0.194141
\(9\) 1.00000 0.333333
\(10\) −0.137935 −0.0436188
\(11\) 2.77203 0.835799 0.417900 − 0.908493i \(-0.362766\pi\)
0.417900 + 0.908493i \(0.362766\pi\)
\(12\) 1.98097 0.571858
\(13\) 1.01349 0.281092 0.140546 − 0.990074i \(-0.455114\pi\)
0.140546 + 0.990074i \(0.455114\pi\)
\(14\) −0.615543 −0.164511
\(15\) 1.00000 0.258199
\(16\) 3.88621 0.971551
\(17\) 5.40876 1.31182 0.655909 − 0.754840i \(-0.272286\pi\)
0.655909 + 0.754840i \(0.272286\pi\)
\(18\) 0.137935 0.0325115
\(19\) 2.91494 0.668734 0.334367 − 0.942443i \(-0.391477\pi\)
0.334367 + 0.942443i \(0.391477\pi\)
\(20\) 1.98097 0.442959
\(21\) 4.46257 0.973812
\(22\) 0.382360 0.0815194
\(23\) 0 0
\(24\) 0.549115 0.112088
\(25\) 1.00000 0.200000
\(26\) 0.139796 0.0274162
\(27\) −1.00000 −0.192450
\(28\) 8.84023 1.67065
\(29\) −4.86108 −0.902680 −0.451340 − 0.892352i \(-0.649054\pi\)
−0.451340 + 0.892352i \(0.649054\pi\)
\(30\) 0.137935 0.0251833
\(31\) −10.5262 −1.89055 −0.945277 − 0.326269i \(-0.894208\pi\)
−0.945277 + 0.326269i \(0.894208\pi\)
\(32\) 1.63427 0.288901
\(33\) −2.77203 −0.482549
\(34\) 0.746056 0.127948
\(35\) 4.46257 0.754312
\(36\) −1.98097 −0.330162
\(37\) −9.12003 −1.49932 −0.749662 − 0.661821i \(-0.769784\pi\)
−0.749662 + 0.661821i \(0.769784\pi\)
\(38\) 0.402072 0.0652247
\(39\) −1.01349 −0.162289
\(40\) 0.549115 0.0868227
\(41\) 6.38318 0.996885 0.498443 − 0.866923i \(-0.333905\pi\)
0.498443 + 0.866923i \(0.333905\pi\)
\(42\) 0.615543 0.0949804
\(43\) −4.91952 −0.750219 −0.375110 − 0.926980i \(-0.622395\pi\)
−0.375110 + 0.926980i \(0.622395\pi\)
\(44\) −5.49133 −0.827848
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 9.95133 1.45155 0.725775 − 0.687932i \(-0.241481\pi\)
0.725775 + 0.687932i \(0.241481\pi\)
\(48\) −3.88621 −0.560926
\(49\) 12.9145 1.84493
\(50\) 0.137935 0.0195069
\(51\) −5.40876 −0.757378
\(52\) −2.00770 −0.278418
\(53\) 2.17396 0.298616 0.149308 − 0.988791i \(-0.452295\pi\)
0.149308 + 0.988791i \(0.452295\pi\)
\(54\) −0.137935 −0.0187705
\(55\) −2.77203 −0.373781
\(56\) 2.45046 0.327457
\(57\) −2.91494 −0.386094
\(58\) −0.670512 −0.0880426
\(59\) −11.9951 −1.56162 −0.780811 − 0.624767i \(-0.785194\pi\)
−0.780811 + 0.624767i \(0.785194\pi\)
\(60\) −1.98097 −0.255743
\(61\) −10.1527 −1.29992 −0.649959 − 0.759969i \(-0.725214\pi\)
−0.649959 + 0.759969i \(0.725214\pi\)
\(62\) −1.45192 −0.184394
\(63\) −4.46257 −0.562231
\(64\) −7.54699 −0.943374
\(65\) −1.01349 −0.125708
\(66\) −0.382360 −0.0470652
\(67\) −1.27158 −0.155349 −0.0776744 − 0.996979i \(-0.524749\pi\)
−0.0776744 + 0.996979i \(0.524749\pi\)
\(68\) −10.7146 −1.29934
\(69\) 0 0
\(70\) 0.615543 0.0735715
\(71\) −1.98303 −0.235342 −0.117671 − 0.993053i \(-0.537543\pi\)
−0.117671 + 0.993053i \(0.537543\pi\)
\(72\) −0.549115 −0.0647138
\(73\) 5.37766 0.629408 0.314704 − 0.949190i \(-0.398095\pi\)
0.314704 + 0.949190i \(0.398095\pi\)
\(74\) −1.25797 −0.146236
\(75\) −1.00000 −0.115470
\(76\) −5.77443 −0.662372
\(77\) −12.3704 −1.40974
\(78\) −0.139796 −0.0158288
\(79\) −2.74641 −0.308995 −0.154497 − 0.987993i \(-0.549376\pi\)
−0.154497 + 0.987993i \(0.549376\pi\)
\(80\) −3.88621 −0.434491
\(81\) 1.00000 0.111111
\(82\) 0.880463 0.0972308
\(83\) 2.13769 0.234642 0.117321 − 0.993094i \(-0.462569\pi\)
0.117321 + 0.993094i \(0.462569\pi\)
\(84\) −8.84023 −0.964548
\(85\) −5.40876 −0.586663
\(86\) −0.678572 −0.0731723
\(87\) 4.86108 0.521163
\(88\) −1.52216 −0.162263
\(89\) 6.25296 0.662812 0.331406 − 0.943488i \(-0.392477\pi\)
0.331406 + 0.943488i \(0.392477\pi\)
\(90\) −0.137935 −0.0145396
\(91\) −4.52278 −0.474116
\(92\) 0 0
\(93\) 10.5262 1.09151
\(94\) 1.37263 0.141576
\(95\) −2.91494 −0.299067
\(96\) −1.63427 −0.166797
\(97\) −4.61045 −0.468120 −0.234060 − 0.972222i \(-0.575201\pi\)
−0.234060 + 0.972222i \(0.575201\pi\)
\(98\) 1.78136 0.179945
\(99\) 2.77203 0.278600
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 7935.2.a.bt.1.13 25
23.5 odd 22 345.2.m.d.301.3 yes 50
23.14 odd 22 345.2.m.d.196.3 ✓ 50
23.22 odd 2 7935.2.a.bu.1.13 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.m.d.196.3 ✓ 50 23.14 odd 22
345.2.m.d.301.3 yes 50 23.5 odd 22
7935.2.a.bt.1.13 25 1.1 even 1 trivial
7935.2.a.bu.1.13 25 23.22 odd 2