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Properties

Label 3936.2.j.h.3361.7
Level $3936$
Weight $2$
Character 3936.3361
Analytic conductor $31.429$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3936,2,Mod(3361,3936)] 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("3936.3361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3936 = 2^{5} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3936.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,0,-4,0,0,0,-22,0,0,0,0,0,0,0,0,0,0,0,4,0,-8,0,30,0,0, 0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(31.4291182356\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3361.7
Character \(\chi\) \(=\) 3936.3361
Dual form 3936.2.j.h.3361.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +3.19623 q^{5} -1.38035i q^{7} -1.00000 q^{9} +5.51809i q^{11} -2.80416i q^{13} -3.19623i q^{15} +4.81721i q^{17} +1.88227i q^{19} -1.38035 q^{21} -2.47763 q^{23} +5.21587 q^{25} +1.00000i q^{27} -6.99971i q^{29} +1.24956 q^{31} +5.51809 q^{33} -4.41190i q^{35} +8.79280 q^{37} -2.80416 q^{39} +(-0.746440 + 6.35947i) q^{41} +9.72539 q^{43} -3.19623 q^{45} +0.964314i q^{47} +5.09465 q^{49} +4.81721 q^{51} -2.30373i q^{53} +17.6371i q^{55} +1.88227 q^{57} -10.1465 q^{59} +11.8309 q^{61} +1.38035i q^{63} -8.96272i q^{65} +12.0650i q^{67} +2.47763i q^{69} +2.46025i q^{71} +12.9530 q^{73} -5.21587i q^{75} +7.61687 q^{77} -5.69316i q^{79} +1.00000 q^{81} +15.3590 q^{83} +15.3969i q^{85} -6.99971 q^{87} -0.581063i q^{89} -3.87070 q^{91} -1.24956i q^{93} +6.01617i q^{95} -17.7917i q^{97} -5.51809i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 4 q^{5} - 22 q^{9} + 4 q^{21} - 8 q^{23} + 30 q^{25} - 16 q^{31} - 4 q^{33} + 8 q^{37} - 12 q^{39} + 2 q^{41} + 8 q^{43} + 4 q^{45} - 30 q^{49} + 20 q^{51} - 4 q^{57} + 12 q^{59} + 16 q^{61} - 4 q^{73}+ \cdots + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3936\mathbb{Z}\right)^\times\).

\(n\) \(1313\) \(1441\) \(1477\) \(3199\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 3.19623 1.42940 0.714698 − 0.699433i \(-0.246564\pi\)
0.714698 + 0.699433i \(0.246564\pi\)
\(6\) 0 0
\(7\) − 1.38035i − 0.521721i −0.965376 − 0.260861i \(-0.915994\pi\)
0.965376 − 0.260861i \(-0.0840063\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.51809i 1.66377i 0.554950 + 0.831884i \(0.312737\pi\)
−0.554950 + 0.831884i \(0.687263\pi\)
\(12\) 0 0
\(13\) − 2.80416i − 0.777733i −0.921294 − 0.388866i \(-0.872867\pi\)
0.921294 − 0.388866i \(-0.127133\pi\)
\(14\) 0 0
\(15\) − 3.19623i − 0.825262i
\(16\) 0 0
\(17\) 4.81721i 1.16834i 0.811630 + 0.584172i \(0.198581\pi\)
−0.811630 + 0.584172i \(0.801419\pi\)
\(18\) 0 0
\(19\) 1.88227i 0.431822i 0.976413 + 0.215911i \(0.0692722\pi\)
−0.976413 + 0.215911i \(0.930728\pi\)
\(20\) 0 0
\(21\) −1.38035 −0.301216
\(22\) 0 0
\(23\) −2.47763 −0.516622 −0.258311 − 0.966062i \(-0.583166\pi\)
−0.258311 + 0.966062i \(0.583166\pi\)
\(24\) 0 0
\(25\) 5.21587 1.04317
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 6.99971i − 1.29981i −0.760014 − 0.649907i \(-0.774808\pi\)
0.760014 − 0.649907i \(-0.225192\pi\)
\(30\) 0 0
\(31\) 1.24956 0.224428 0.112214 − 0.993684i \(-0.464206\pi\)
0.112214 + 0.993684i \(0.464206\pi\)
\(32\) 0 0
\(33\) 5.51809 0.960577
\(34\) 0 0
\(35\) − 4.41190i − 0.745747i
\(36\) 0 0
\(37\) 8.79280 1.44553 0.722764 − 0.691095i \(-0.242871\pi\)
0.722764 + 0.691095i \(0.242871\pi\)
\(38\) 0 0
\(39\) −2.80416 −0.449024
\(40\) 0 0
\(41\) −0.746440 + 6.35947i −0.116574 + 0.993182i
\(42\) 0 0
\(43\) 9.72539 1.48311 0.741554 − 0.670893i \(-0.234089\pi\)
0.741554 + 0.670893i \(0.234089\pi\)
\(44\) 0 0
\(45\) −3.19623 −0.476466
\(46\) 0 0
\(47\) 0.964314i 0.140660i 0.997524 + 0.0703298i \(0.0224052\pi\)
−0.997524 + 0.0703298i \(0.977595\pi\)
\(48\) 0 0
\(49\) 5.09465 0.727807
\(50\) 0 0
\(51\) 4.81721 0.674544
\(52\) 0 0
\(53\) − 2.30373i − 0.316442i −0.987404 − 0.158221i \(-0.949424\pi\)
0.987404 − 0.158221i \(-0.0505758\pi\)
\(54\) 0 0
\(55\) 17.6371i 2.37818i
\(56\) 0 0
\(57\) 1.88227 0.249313
\(58\) 0 0
\(59\) −10.1465 −1.32096 −0.660481 − 0.750843i \(-0.729648\pi\)
−0.660481 + 0.750843i \(0.729648\pi\)
\(60\) 0 0
\(61\) 11.8309 1.51479 0.757396 − 0.652955i \(-0.226471\pi\)
0.757396 + 0.652955i \(0.226471\pi\)
\(62\) 0 0
\(63\) 1.38035i 0.173907i
\(64\) 0 0
\(65\) − 8.96272i − 1.11169i
\(66\) 0 0
\(67\) 12.0650i 1.47397i 0.675909 + 0.736985i \(0.263751\pi\)
−0.675909 + 0.736985i \(0.736249\pi\)
\(68\) 0 0
\(69\) 2.47763i 0.298272i
\(70\) 0 0
\(71\) 2.46025i 0.291978i 0.989286 + 0.145989i \(0.0466364\pi\)
−0.989286 + 0.145989i \(0.953364\pi\)
\(72\) 0 0
\(73\) 12.9530 1.51603 0.758016 − 0.652236i \(-0.226169\pi\)
0.758016 + 0.652236i \(0.226169\pi\)
\(74\) 0 0
\(75\) − 5.21587i − 0.602277i
\(76\) 0 0
\(77\) 7.61687 0.868023
\(78\) 0 0
\(79\) − 5.69316i − 0.640531i −0.947328 − 0.320265i \(-0.896228\pi\)
0.947328 − 0.320265i \(-0.103772\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.3590 1.68587 0.842933 − 0.538018i \(-0.180827\pi\)
0.842933 + 0.538018i \(0.180827\pi\)
\(84\) 0 0
\(85\) 15.3969i 1.67003i
\(86\) 0 0
\(87\) −6.99971 −0.750448
\(88\) 0 0
\(89\) − 0.581063i − 0.0615926i −0.999526 − 0.0307963i \(-0.990196\pi\)
0.999526 − 0.0307963i \(-0.00980431\pi\)
\(90\) 0 0
\(91\) −3.87070 −0.405760
\(92\) 0 0
\(93\) − 1.24956i − 0.129574i
\(94\) 0 0
\(95\) 6.01617i 0.617246i
\(96\) 0 0
\(97\) − 17.7917i − 1.80647i −0.429144 − 0.903236i \(-0.641185\pi\)
0.429144 − 0.903236i \(-0.358815\pi\)
\(98\) 0 0
\(99\) − 5.51809i − 0.554589i
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 3936.2.j.h.3361.7 ✓ 22
4.3 odd 2 3936.2.j.i.3361.8 yes 22
41.40 even 2 inner 3936.2.j.h.3361.8 yes 22
164.163 odd 2 3936.2.j.i.3361.7 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3936.2.j.h.3361.7 ✓ 22 1.1 even 1 trivial
3936.2.j.h.3361.8 yes 22 41.40 even 2 inner
3936.2.j.i.3361.7 yes 22 164.163 odd 2
3936.2.j.i.3361.8 yes 22 4.3 odd 2