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Properties

Label 3936.2.j.h.3361.4
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 3361.4
Character \(\chi\) \(=\) 3936.3361
Dual form 3936.2.j.h.3361.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +2.18458 q^{5} -4.41839i q^{7} -1.00000 q^{9} -0.447067i q^{11} +6.30215i q^{13} +2.18458i q^{15} +4.49774i q^{17} +7.27555i q^{19} +4.41839 q^{21} -1.25935 q^{23} -0.227623 q^{25} -1.00000i q^{27} -3.20275i q^{29} -0.894052 q^{31} +0.447067 q^{33} -9.65232i q^{35} +7.00344 q^{37} -6.30215 q^{39} +(-6.38451 + 0.487874i) q^{41} -1.27240 q^{43} -2.18458 q^{45} +3.86589i q^{47} -12.5222 q^{49} -4.49774 q^{51} +13.5896i q^{53} -0.976653i q^{55} -7.27555 q^{57} +4.35209 q^{59} -2.86248 q^{61} +4.41839i q^{63} +13.7675i q^{65} +7.02949i q^{67} -1.25935i q^{69} -9.20874i q^{71} -5.14275 q^{73} -0.227623i q^{75} -1.97532 q^{77} -9.72794i q^{79} +1.00000 q^{81} -5.68836 q^{83} +9.82567i q^{85} +3.20275 q^{87} +17.3309i q^{89} +27.8454 q^{91} -0.894052i q^{93} +15.8940i q^{95} -9.74439i q^{97} +0.447067i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 4 q^{5} - 22 q^{9} + 4 q^{21} - 8 q^{23} + 30 q^{25} - 16 q^{31} - 4 q^{33} + 8 q^{37} - 12 q^{39} + 2 q^{41} + 8 q^{43} + 4 q^{45} - 30 q^{49} + 20 q^{51} - 4 q^{57} + 12 q^{59} + 16 q^{61} - 4 q^{73}+ \cdots + 24 q^{91}+O(q^{100}) \) 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
\(5\) 2.18458 0.976973 0.488486 − 0.872572i \(-0.337549\pi\)
0.488486 + 0.872572i \(0.337549\pi\)
\(6\) 0 0
\(7\) − 4.41839i − 1.67000i −0.550253 − 0.834998i \(-0.685469\pi\)
0.550253 − 0.834998i \(-0.314531\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 0.447067i − 0.134796i −0.997726 − 0.0673979i \(-0.978530\pi\)
0.997726 − 0.0673979i \(-0.0214697\pi\)
\(12\) 0 0
\(13\) 6.30215i 1.74790i 0.486015 + 0.873951i \(0.338450\pi\)
−0.486015 + 0.873951i \(0.661550\pi\)
\(14\) 0 0
\(15\) 2.18458i 0.564055i
\(16\) 0 0
\(17\) 4.49774i 1.09086i 0.838155 + 0.545431i \(0.183634\pi\)
−0.838155 + 0.545431i \(0.816366\pi\)
\(18\) 0 0
\(19\) 7.27555i 1.66913i 0.550913 + 0.834563i \(0.314280\pi\)
−0.550913 + 0.834563i \(0.685720\pi\)
\(20\) 0 0
\(21\) 4.41839 0.964173
\(22\) 0 0
\(23\) −1.25935 −0.262592 −0.131296 − 0.991343i \(-0.541914\pi\)
−0.131296 + 0.991343i \(0.541914\pi\)
\(24\) 0 0
\(25\) −0.227623 −0.0455245
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 3.20275i − 0.594736i −0.954763 − 0.297368i \(-0.903891\pi\)
0.954763 − 0.297368i \(-0.0961088\pi\)
\(30\) 0 0
\(31\) −0.894052 −0.160577 −0.0802883 − 0.996772i \(-0.525584\pi\)
−0.0802883 + 0.996772i \(0.525584\pi\)
\(32\) 0 0
\(33\) 0.447067 0.0778244
\(34\) 0 0
\(35\) − 9.65232i − 1.63154i
\(36\) 0 0
\(37\) 7.00344 1.15136 0.575679 − 0.817676i \(-0.304738\pi\)
0.575679 + 0.817676i \(0.304738\pi\)
\(38\) 0 0
\(39\) −6.30215 −1.00915
\(40\) 0 0
\(41\) −6.38451 + 0.487874i −0.997093 + 0.0761931i
\(42\) 0 0
\(43\) −1.27240 −0.194039 −0.0970193 − 0.995282i \(-0.530931\pi\)
−0.0970193 + 0.995282i \(0.530931\pi\)
\(44\) 0 0
\(45\) −2.18458 −0.325658
\(46\) 0 0
\(47\) 3.86589i 0.563898i 0.959429 + 0.281949i \(0.0909809\pi\)
−0.959429 + 0.281949i \(0.909019\pi\)
\(48\) 0 0
\(49\) −12.5222 −1.78889
\(50\) 0 0
\(51\) −4.49774 −0.629810
\(52\) 0 0
\(53\) 13.5896i 1.86667i 0.359005 + 0.933336i \(0.383116\pi\)
−0.359005 + 0.933336i \(0.616884\pi\)
\(54\) 0 0
\(55\) − 0.976653i − 0.131692i
\(56\) 0 0
\(57\) −7.27555 −0.963670
\(58\) 0 0
\(59\) 4.35209 0.566594 0.283297 − 0.959032i \(-0.408572\pi\)
0.283297 + 0.959032i \(0.408572\pi\)
\(60\) 0 0
\(61\) −2.86248 −0.366503 −0.183252 − 0.983066i \(-0.558662\pi\)
−0.183252 + 0.983066i \(0.558662\pi\)
\(62\) 0 0
\(63\) 4.41839i 0.556665i
\(64\) 0 0
\(65\) 13.7675i 1.70765i
\(66\) 0 0
\(67\) 7.02949i 0.858789i 0.903117 + 0.429394i \(0.141273\pi\)
−0.903117 + 0.429394i \(0.858727\pi\)
\(68\) 0 0
\(69\) − 1.25935i − 0.151608i
\(70\) 0 0
\(71\) − 9.20874i − 1.09288i −0.837500 − 0.546438i \(-0.815983\pi\)
0.837500 − 0.546438i \(-0.184017\pi\)
\(72\) 0 0
\(73\) −5.14275 −0.601914 −0.300957 − 0.953638i \(-0.597306\pi\)
−0.300957 + 0.953638i \(0.597306\pi\)
\(74\) 0 0
\(75\) − 0.227623i − 0.0262836i
\(76\) 0 0
\(77\) −1.97532 −0.225109
\(78\) 0 0
\(79\) − 9.72794i − 1.09448i −0.836976 − 0.547240i \(-0.815679\pi\)
0.836976 − 0.547240i \(-0.184321\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.68836 −0.624379 −0.312189 − 0.950020i \(-0.601062\pi\)
−0.312189 + 0.950020i \(0.601062\pi\)
\(84\) 0 0
\(85\) 9.82567i 1.06574i
\(86\) 0 0
\(87\) 3.20275 0.343371
\(88\) 0 0
\(89\) 17.3309i 1.83707i 0.395341 + 0.918535i \(0.370627\pi\)
−0.395341 + 0.918535i \(0.629373\pi\)
\(90\) 0 0
\(91\) 27.8454 2.91899
\(92\) 0 0
\(93\) − 0.894052i − 0.0927089i
\(94\) 0 0
\(95\) 15.8940i 1.63069i
\(96\) 0 0
\(97\) − 9.74439i − 0.989393i −0.869066 − 0.494697i \(-0.835279\pi\)
0.869066 − 0.494697i \(-0.164721\pi\)
\(98\) 0 0
\(99\) 0.447067i 0.0449319i
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.3361.4 yes 22
4.3 odd 2 3936.2.j.i.3361.3 yes 22
41.40 even 2 inner 3936.2.j.h.3361.3 ✓ 22
164.163 odd 2 3936.2.j.i.3361.4 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3936.2.j.h.3361.3 ✓ 22 41.40 even 2 inner
3936.2.j.h.3361.4 yes 22 1.1 even 1 trivial
3936.2.j.i.3361.3 yes 22 4.3 odd 2
3936.2.j.i.3361.4 yes 22 164.163 odd 2