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Properties

Label 3936.2.j.h.3361.1
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 3361.1
Character \(\chi\) \(=\) 3936.3361
Dual form 3936.2.j.h.3361.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -3.38755 q^{5} -4.17077i q^{7} -1.00000 q^{9} +1.81617i q^{11} -0.200740i q^{13} +3.38755i q^{15} +2.34794i q^{17} +3.22881i q^{19} -4.17077 q^{21} +8.52069 q^{23} +6.47546 q^{25} +1.00000i q^{27} +5.91394i q^{29} +5.25238 q^{31} +1.81617 q^{33} +14.1287i q^{35} -0.0290050 q^{37} -0.200740 q^{39} +(-4.84808 + 4.18284i) q^{41} -7.25557 q^{43} +3.38755 q^{45} +3.14543i q^{47} -10.3954 q^{49} +2.34794 q^{51} -8.60894i q^{53} -6.15236i q^{55} +3.22881 q^{57} +9.78265 q^{59} +5.96215 q^{61} +4.17077i q^{63} +0.680014i q^{65} +3.61305i q^{67} -8.52069i q^{69} -0.0964833i q^{71} +6.01658 q^{73} -6.47546i q^{75} +7.57484 q^{77} -6.31675i q^{79} +1.00000 q^{81} +0.863801 q^{83} -7.95376i q^{85} +5.91394 q^{87} -9.64957i q^{89} -0.837239 q^{91} -5.25238i q^{93} -10.9377i q^{95} -11.8342i q^{97} -1.81617i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 4 q^{5} - 22 q^{9} + 4 q^{21} - 8 q^{23} + 30 q^{25} - 16 q^{31} - 4 q^{33} + 8 q^{37} - 12 q^{39} + 2 q^{41} + 8 q^{43} + 4 q^{45} - 30 q^{49} + 20 q^{51} - 4 q^{57} + 12 q^{59} + 16 q^{61} - 4 q^{73}+ \cdots + 24 q^{91}+O(q^{100}) \) 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
\(5\) −3.38755 −1.51496 −0.757478 − 0.652861i \(-0.773569\pi\)
−0.757478 + 0.652861i \(0.773569\pi\)
\(6\) 0 0
\(7\) − 4.17077i − 1.57640i −0.615416 − 0.788202i \(-0.711012\pi\)
0.615416 − 0.788202i \(-0.288988\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.81617i 0.547596i 0.961787 + 0.273798i \(0.0882801\pi\)
−0.961787 + 0.273798i \(0.911720\pi\)
\(12\) 0 0
\(13\) − 0.200740i − 0.0556751i −0.999612 − 0.0278376i \(-0.991138\pi\)
0.999612 − 0.0278376i \(-0.00886212\pi\)
\(14\) 0 0
\(15\) 3.38755i 0.874660i
\(16\) 0 0
\(17\) 2.34794i 0.569460i 0.958608 + 0.284730i \(0.0919039\pi\)
−0.958608 + 0.284730i \(0.908096\pi\)
\(18\) 0 0
\(19\) 3.22881i 0.740739i 0.928884 + 0.370370i \(0.120769\pi\)
−0.928884 + 0.370370i \(0.879231\pi\)
\(20\) 0 0
\(21\) −4.17077 −0.910138
\(22\) 0 0
\(23\) 8.52069 1.77669 0.888343 − 0.459181i \(-0.151857\pi\)
0.888343 + 0.459181i \(0.151857\pi\)
\(24\) 0 0
\(25\) 6.47546 1.29509
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 5.91394i 1.09819i 0.835759 + 0.549096i \(0.185028\pi\)
−0.835759 + 0.549096i \(0.814972\pi\)
\(30\) 0 0
\(31\) 5.25238 0.943355 0.471678 − 0.881771i \(-0.343649\pi\)
0.471678 + 0.881771i \(0.343649\pi\)
\(32\) 0 0
\(33\) 1.81617 0.316155
\(34\) 0 0
\(35\) 14.1287i 2.38818i
\(36\) 0 0
\(37\) −0.0290050 −0.00476840 −0.00238420 − 0.999997i \(-0.500759\pi\)
−0.00238420 + 0.999997i \(0.500759\pi\)
\(38\) 0 0
\(39\) −0.200740 −0.0321440
\(40\) 0 0
\(41\) −4.84808 + 4.18284i −0.757143 + 0.653249i
\(42\) 0 0
\(43\) −7.25557 −1.10646 −0.553232 − 0.833027i \(-0.686606\pi\)
−0.553232 + 0.833027i \(0.686606\pi\)
\(44\) 0 0
\(45\) 3.38755 0.504985
\(46\) 0 0
\(47\) 3.14543i 0.458808i 0.973331 + 0.229404i \(0.0736777\pi\)
−0.973331 + 0.229404i \(0.926322\pi\)
\(48\) 0 0
\(49\) −10.3954 −1.48505
\(50\) 0 0
\(51\) 2.34794 0.328778
\(52\) 0 0
\(53\) − 8.60894i − 1.18253i −0.806478 − 0.591264i \(-0.798629\pi\)
0.806478 − 0.591264i \(-0.201371\pi\)
\(54\) 0 0
\(55\) − 6.15236i − 0.829585i
\(56\) 0 0
\(57\) 3.22881 0.427666
\(58\) 0 0
\(59\) 9.78265 1.27359 0.636796 − 0.771032i \(-0.280259\pi\)
0.636796 + 0.771032i \(0.280259\pi\)
\(60\) 0 0
\(61\) 5.96215 0.763375 0.381687 − 0.924291i \(-0.375343\pi\)
0.381687 + 0.924291i \(0.375343\pi\)
\(62\) 0 0
\(63\) 4.17077i 0.525468i
\(64\) 0 0
\(65\) 0.680014i 0.0843454i
\(66\) 0 0
\(67\) 3.61305i 0.441405i 0.975341 + 0.220702i \(0.0708350\pi\)
−0.975341 + 0.220702i \(0.929165\pi\)
\(68\) 0 0
\(69\) − 8.52069i − 1.02577i
\(70\) 0 0
\(71\) − 0.0964833i − 0.0114505i −0.999984 − 0.00572523i \(-0.998178\pi\)
0.999984 − 0.00572523i \(-0.00182241\pi\)
\(72\) 0 0
\(73\) 6.01658 0.704187 0.352093 − 0.935965i \(-0.385470\pi\)
0.352093 + 0.935965i \(0.385470\pi\)
\(74\) 0 0
\(75\) − 6.47546i − 0.747722i
\(76\) 0 0
\(77\) 7.57484 0.863234
\(78\) 0 0
\(79\) − 6.31675i − 0.710690i −0.934735 − 0.355345i \(-0.884363\pi\)
0.934735 − 0.355345i \(-0.115637\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.863801 0.0948145 0.0474072 − 0.998876i \(-0.484904\pi\)
0.0474072 + 0.998876i \(0.484904\pi\)
\(84\) 0 0
\(85\) − 7.95376i − 0.862707i
\(86\) 0 0
\(87\) 5.91394 0.634041
\(88\) 0 0
\(89\) − 9.64957i − 1.02285i −0.859327 − 0.511426i \(-0.829117\pi\)
0.859327 − 0.511426i \(-0.170883\pi\)
\(90\) 0 0
\(91\) −0.837239 −0.0877665
\(92\) 0 0
\(93\) − 5.25238i − 0.544646i
\(94\) 0 0
\(95\) − 10.9377i − 1.12219i
\(96\) 0 0
\(97\) − 11.8342i − 1.20158i −0.799406 − 0.600791i \(-0.794852\pi\)
0.799406 − 0.600791i \(-0.205148\pi\)
\(98\) 0 0
\(99\) − 1.81617i − 0.182532i
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.3361.1 ✓ 22
4.3 odd 2 3936.2.j.i.3361.2 yes 22
41.40 even 2 inner 3936.2.j.h.3361.2 yes 22
164.163 odd 2 3936.2.j.i.3361.1 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3936.2.j.h.3361.1 ✓ 22 1.1 even 1 trivial
3936.2.j.h.3361.2 yes 22 41.40 even 2 inner
3936.2.j.i.3361.1 yes 22 164.163 odd 2
3936.2.j.i.3361.2 yes 22 4.3 odd 2