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Properties

Label 2352.4.a.ci.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [2352,4,Mod(1,2352)] 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("2352.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,9,0,11,0,0,0,27,0,-35] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 24x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.30829\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +5.56140 q^{5} +9.00000 q^{9} +13.9174 q^{11} +38.6718 q^{13} +16.6842 q^{15} +43.4788 q^{17} +109.028 q^{19} +74.8778 q^{23} -94.0708 q^{25} +27.0000 q^{27} -72.3589 q^{29} -64.0431 q^{31} +41.7521 q^{33} +188.727 q^{37} +116.015 q^{39} -24.7923 q^{41} +243.881 q^{43} +50.0526 q^{45} +620.549 q^{47} +130.436 q^{51} -287.839 q^{53} +77.4001 q^{55} +327.083 q^{57} -525.051 q^{59} -383.436 q^{61} +215.069 q^{65} -198.117 q^{67} +224.634 q^{69} -785.432 q^{71} -331.141 q^{73} -282.213 q^{75} -437.647 q^{79} +81.0000 q^{81} -241.241 q^{83} +241.803 q^{85} -217.077 q^{87} +1585.54 q^{89} -192.129 q^{93} +606.347 q^{95} +79.2754 q^{97} +125.256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{3} + 11 q^{5} + 27 q^{9} - 35 q^{11} + 62 q^{13} + 33 q^{15} + 48 q^{17} + 202 q^{19} - 216 q^{23} + 130 q^{25} + 81 q^{27} + 53 q^{29} + 95 q^{31} - 105 q^{33} + 262 q^{37} + 186 q^{39} + 244 q^{41}+ \cdots - 315 q^{99}+O(q^{100}) \) 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 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 5.56140 0.497427 0.248713 − 0.968577i \(-0.419992\pi\)
0.248713 + 0.968577i \(0.419992\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 13.9174 0.381477 0.190738 − 0.981641i \(-0.438912\pi\)
0.190738 + 0.981641i \(0.438912\pi\)
\(12\) 0 0
\(13\) 38.6718 0.825048 0.412524 − 0.910947i \(-0.364647\pi\)
0.412524 + 0.910947i \(0.364647\pi\)
\(14\) 0 0
\(15\) 16.6842 0.287189
\(16\) 0 0
\(17\) 43.4788 0.620303 0.310152 − 0.950687i \(-0.399620\pi\)
0.310152 + 0.950687i \(0.399620\pi\)
\(18\) 0 0
\(19\) 109.028 1.31646 0.658228 − 0.752818i \(-0.271306\pi\)
0.658228 + 0.752818i \(0.271306\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 74.8778 0.678831 0.339415 − 0.940637i \(-0.389771\pi\)
0.339415 + 0.940637i \(0.389771\pi\)
\(24\) 0 0
\(25\) −94.0708 −0.752567
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −72.3589 −0.463335 −0.231667 − 0.972795i \(-0.574418\pi\)
−0.231667 + 0.972795i \(0.574418\pi\)
\(30\) 0 0
\(31\) −64.0431 −0.371048 −0.185524 − 0.982640i \(-0.559398\pi\)
−0.185524 + 0.982640i \(0.559398\pi\)
\(32\) 0 0
\(33\) 41.7521 0.220246
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 188.727 0.838556 0.419278 − 0.907858i \(-0.362283\pi\)
0.419278 + 0.907858i \(0.362283\pi\)
\(38\) 0 0
\(39\) 116.015 0.476342
\(40\) 0 0
\(41\) −24.7923 −0.0944367 −0.0472184 − 0.998885i \(-0.515036\pi\)
−0.0472184 + 0.998885i \(0.515036\pi\)
\(42\) 0 0
\(43\) 243.881 0.864920 0.432460 − 0.901653i \(-0.357646\pi\)
0.432460 + 0.901653i \(0.357646\pi\)
\(44\) 0 0
\(45\) 50.0526 0.165809
\(46\) 0 0
\(47\) 620.549 1.92588 0.962940 − 0.269717i \(-0.0869303\pi\)
0.962940 + 0.269717i \(0.0869303\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 130.436 0.358132
\(52\) 0 0
\(53\) −287.839 −0.745995 −0.372997 − 0.927832i \(-0.621670\pi\)
−0.372997 + 0.927832i \(0.621670\pi\)
\(54\) 0 0
\(55\) 77.4001 0.189757
\(56\) 0 0
\(57\) 327.083 0.760057
\(58\) 0 0
\(59\) −525.051 −1.15857 −0.579287 − 0.815124i \(-0.696669\pi\)
−0.579287 + 0.815124i \(0.696669\pi\)
\(60\) 0 0
\(61\) −383.436 −0.804818 −0.402409 − 0.915460i \(-0.631827\pi\)
−0.402409 + 0.915460i \(0.631827\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 215.069 0.410401
\(66\) 0 0
\(67\) −198.117 −0.361251 −0.180625 − 0.983552i \(-0.557812\pi\)
−0.180625 + 0.983552i \(0.557812\pi\)
\(68\) 0 0
\(69\) 224.634 0.391923
\(70\) 0 0
\(71\) −785.432 −1.31287 −0.656434 − 0.754384i \(-0.727936\pi\)
−0.656434 + 0.754384i \(0.727936\pi\)
\(72\) 0 0
\(73\) −331.141 −0.530919 −0.265459 − 0.964122i \(-0.585524\pi\)
−0.265459 + 0.964122i \(0.585524\pi\)
\(74\) 0 0
\(75\) −282.213 −0.434495
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −437.647 −0.623280 −0.311640 − 0.950200i \(-0.600878\pi\)
−0.311640 + 0.950200i \(0.600878\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −241.241 −0.319032 −0.159516 − 0.987195i \(-0.550993\pi\)
−0.159516 + 0.987195i \(0.550993\pi\)
\(84\) 0 0
\(85\) 241.803 0.308555
\(86\) 0 0
\(87\) −217.077 −0.267506
\(88\) 0 0
\(89\) 1585.54 1.88840 0.944198 − 0.329378i \(-0.106839\pi\)
0.944198 + 0.329378i \(0.106839\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −192.129 −0.214224
\(94\) 0 0
\(95\) 606.347 0.654841
\(96\) 0 0
\(97\) 79.2754 0.0829814 0.0414907 − 0.999139i \(-0.486789\pi\)
0.0414907 + 0.999139i \(0.486789\pi\)
\(98\) 0 0
\(99\) 125.256 0.127159
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 2352.4.a.ci.1.2 3
4.3 odd 2 147.4.a.l.1.3 3
7.2 even 3 336.4.q.k.193.2 6
7.4 even 3 336.4.q.k.289.2 6
7.6 odd 2 2352.4.a.cg.1.2 3
12.11 even 2 441.4.a.s.1.1 3
28.3 even 6 147.4.e.n.79.1 6
28.11 odd 6 21.4.e.b.16.1 yes 6
28.19 even 6 147.4.e.n.67.1 6
28.23 odd 6 21.4.e.b.4.1 ✓ 6
28.27 even 2 147.4.a.m.1.3 3
84.11 even 6 63.4.e.c.37.3 6
84.23 even 6 63.4.e.c.46.3 6
84.47 odd 6 441.4.e.w.361.3 6
84.59 odd 6 441.4.e.w.226.3 6
84.83 odd 2 441.4.a.t.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.1 ✓ 6 28.23 odd 6
21.4.e.b.16.1 yes 6 28.11 odd 6
63.4.e.c.37.3 6 84.11 even 6
63.4.e.c.46.3 6 84.23 even 6
147.4.a.l.1.3 3 4.3 odd 2
147.4.a.m.1.3 3 28.27 even 2
147.4.e.n.67.1 6 28.19 even 6
147.4.e.n.79.1 6 28.3 even 6
336.4.q.k.193.2 6 7.2 even 3
336.4.q.k.289.2 6 7.4 even 3
441.4.a.s.1.1 3 12.11 even 2
441.4.a.t.1.1 3 84.83 odd 2
441.4.e.w.226.3 6 84.59 odd 6
441.4.e.w.361.3 6 84.47 odd 6
2352.4.a.cg.1.2 3 7.6 odd 2
2352.4.a.ci.1.2 3 1.1 even 1 trivial