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Properties

Label 3936.2.j.h.3361.15
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 3361.15
Character \(\chi\) \(=\) 3936.3361
Dual form 3936.2.j.h.3361.16

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -2.14706 q^{5} +1.11744i q^{7} -1.00000 q^{9} +2.16266i q^{11} +2.31296i q^{13} +2.14706i q^{15} -5.16871i q^{17} -6.83492i q^{19} +1.11744 q^{21} +2.59321 q^{23} -0.390132 q^{25} +1.00000i q^{27} -0.866182i q^{29} -5.09661 q^{31} +2.16266 q^{33} -2.39921i q^{35} -7.90806 q^{37} +2.31296 q^{39} +(4.70909 + 4.33872i) q^{41} +6.47687 q^{43} +2.14706 q^{45} +7.81398i q^{47} +5.75133 q^{49} -5.16871 q^{51} +0.106576i q^{53} -4.64335i q^{55} -6.83492 q^{57} -6.81749 q^{59} +11.5665 q^{61} -1.11744i q^{63} -4.96607i q^{65} +3.90138i q^{67} -2.59321i q^{69} +11.2356i q^{71} +1.01229 q^{73} +0.390132i q^{75} -2.41663 q^{77} -6.28644i q^{79} +1.00000 q^{81} -13.2372 q^{83} +11.0975i q^{85} -0.866182 q^{87} +17.8485i q^{89} -2.58459 q^{91} +5.09661i q^{93} +14.6750i q^{95} +6.39864i q^{97} -2.16266i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 4 q^{5} - 22 q^{9} + 4 q^{21} - 8 q^{23} + 30 q^{25} - 16 q^{31} - 4 q^{33} + 8 q^{37} - 12 q^{39} + 2 q^{41} + 8 q^{43} + 4 q^{45} - 30 q^{49} + 20 q^{51} - 4 q^{57} + 12 q^{59} + 16 q^{61} - 4 q^{73}+ \cdots + 24 q^{91}+O(q^{100}) \) 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
\(5\) −2.14706 −0.960195 −0.480097 − 0.877215i \(-0.659399\pi\)
−0.480097 + 0.877215i \(0.659399\pi\)
\(6\) 0 0
\(7\) 1.11744i 0.422352i 0.977448 + 0.211176i \(0.0677293\pi\)
−0.977448 + 0.211176i \(0.932271\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.16266i 0.652065i 0.945359 + 0.326033i \(0.105712\pi\)
−0.945359 + 0.326033i \(0.894288\pi\)
\(12\) 0 0
\(13\) 2.31296i 0.641500i 0.947164 + 0.320750i \(0.103935\pi\)
−0.947164 + 0.320750i \(0.896065\pi\)
\(14\) 0 0
\(15\) 2.14706i 0.554369i
\(16\) 0 0
\(17\) − 5.16871i − 1.25360i −0.779182 − 0.626798i \(-0.784365\pi\)
0.779182 − 0.626798i \(-0.215635\pi\)
\(18\) 0 0
\(19\) − 6.83492i − 1.56804i −0.620737 − 0.784019i \(-0.713166\pi\)
0.620737 − 0.784019i \(-0.286834\pi\)
\(20\) 0 0
\(21\) 1.11744 0.243845
\(22\) 0 0
\(23\) 2.59321 0.540722 0.270361 − 0.962759i \(-0.412857\pi\)
0.270361 + 0.962759i \(0.412857\pi\)
\(24\) 0 0
\(25\) −0.390132 −0.0780264
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 0.866182i − 0.160846i −0.996761 − 0.0804230i \(-0.974373\pi\)
0.996761 − 0.0804230i \(-0.0256271\pi\)
\(30\) 0 0
\(31\) −5.09661 −0.915379 −0.457689 − 0.889112i \(-0.651323\pi\)
−0.457689 + 0.889112i \(0.651323\pi\)
\(32\) 0 0
\(33\) 2.16266 0.376470
\(34\) 0 0
\(35\) − 2.39921i − 0.405540i
\(36\) 0 0
\(37\) −7.90806 −1.30008 −0.650038 − 0.759901i \(-0.725247\pi\)
−0.650038 + 0.759901i \(0.725247\pi\)
\(38\) 0 0
\(39\) 2.31296 0.370370
\(40\) 0 0
\(41\) 4.70909 + 4.33872i 0.735436 + 0.677594i
\(42\) 0 0
\(43\) 6.47687 0.987714 0.493857 − 0.869543i \(-0.335587\pi\)
0.493857 + 0.869543i \(0.335587\pi\)
\(44\) 0 0
\(45\) 2.14706 0.320065
\(46\) 0 0
\(47\) 7.81398i 1.13979i 0.821719 + 0.569893i \(0.193015\pi\)
−0.821719 + 0.569893i \(0.806985\pi\)
\(48\) 0 0
\(49\) 5.75133 0.821619
\(50\) 0 0
\(51\) −5.16871 −0.723764
\(52\) 0 0
\(53\) 0.106576i 0.0146393i 0.999973 + 0.00731965i \(0.00232994\pi\)
−0.999973 + 0.00731965i \(0.997670\pi\)
\(54\) 0 0
\(55\) − 4.64335i − 0.626110i
\(56\) 0 0
\(57\) −6.83492 −0.905307
\(58\) 0 0
\(59\) −6.81749 −0.887562 −0.443781 − 0.896135i \(-0.646363\pi\)
−0.443781 + 0.896135i \(0.646363\pi\)
\(60\) 0 0
\(61\) 11.5665 1.48094 0.740471 − 0.672088i \(-0.234602\pi\)
0.740471 + 0.672088i \(0.234602\pi\)
\(62\) 0 0
\(63\) − 1.11744i − 0.140784i
\(64\) 0 0
\(65\) − 4.96607i − 0.615965i
\(66\) 0 0
\(67\) 3.90138i 0.476629i 0.971188 + 0.238315i \(0.0765949\pi\)
−0.971188 + 0.238315i \(0.923405\pi\)
\(68\) 0 0
\(69\) − 2.59321i − 0.312186i
\(70\) 0 0
\(71\) 11.2356i 1.33342i 0.745318 + 0.666709i \(0.232297\pi\)
−0.745318 + 0.666709i \(0.767703\pi\)
\(72\) 0 0
\(73\) 1.01229 0.118480 0.0592398 − 0.998244i \(-0.481132\pi\)
0.0592398 + 0.998244i \(0.481132\pi\)
\(74\) 0 0
\(75\) 0.390132i 0.0450485i
\(76\) 0 0
\(77\) −2.41663 −0.275401
\(78\) 0 0
\(79\) − 6.28644i − 0.707279i −0.935382 − 0.353640i \(-0.884944\pi\)
0.935382 − 0.353640i \(-0.115056\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.2372 −1.45297 −0.726485 − 0.687182i \(-0.758847\pi\)
−0.726485 + 0.687182i \(0.758847\pi\)
\(84\) 0 0
\(85\) 11.0975i 1.20370i
\(86\) 0 0
\(87\) −0.866182 −0.0928644
\(88\) 0 0
\(89\) 17.8485i 1.89194i 0.324251 + 0.945971i \(0.394888\pi\)
−0.324251 + 0.945971i \(0.605112\pi\)
\(90\) 0 0
\(91\) −2.58459 −0.270939
\(92\) 0 0
\(93\) 5.09661i 0.528494i
\(94\) 0 0
\(95\) 14.6750i 1.50562i
\(96\) 0 0
\(97\) 6.39864i 0.649684i 0.945768 + 0.324842i \(0.105311\pi\)
−0.945768 + 0.324842i \(0.894689\pi\)
\(98\) 0 0
\(99\) − 2.16266i − 0.217355i
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.3361.15 ✓ 22
4.3 odd 2 3936.2.j.i.3361.16 yes 22
41.40 even 2 inner 3936.2.j.h.3361.16 yes 22
164.163 odd 2 3936.2.j.i.3361.15 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3936.2.j.h.3361.15 ✓ 22 1.1 even 1 trivial
3936.2.j.h.3361.16 yes 22 41.40 even 2 inner
3936.2.j.i.3361.15 yes 22 164.163 odd 2
3936.2.j.i.3361.16 yes 22 4.3 odd 2