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Properties

Label 2352.2.bl.s.607.1
Level $2352$
Weight $2$
Character 2352.607
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [2352,2,Mod(31,2352)] 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("2352.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,4,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 607.1
Root \(0.662827 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 2352.607
Dual form 2352.2.bl.s.31.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-1.60021 - 0.923880i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(4.52607 - 2.61313i) q^{11} -4.46088i q^{13} -1.84776i q^{15} +(-2.53759 + 1.46508i) q^{17} +(-2.82843 + 4.89898i) q^{19} +(-1.87476 - 1.08239i) q^{23} +(-0.792893 - 1.37333i) q^{25} -1.00000 q^{27} -5.41421 q^{29} +(-4.82843 - 8.36308i) q^{31} +(4.52607 + 2.61313i) q^{33} +(-0.707107 + 1.22474i) q^{37} +(3.86324 - 2.23044i) q^{39} +4.01254i q^{41} +3.06147i q^{43} +(1.60021 - 0.923880i) q^{45} +(-0.828427 + 1.43488i) q^{47} +(-2.53759 - 1.46508i) q^{51} +(4.82843 + 8.36308i) q^{53} -9.65685 q^{55} -5.65685 q^{57} +(-2.82843 - 4.89898i) q^{59} +(-1.21193 - 0.699709i) q^{61} +(-4.12132 + 7.13834i) q^{65} +(11.7034 - 6.75699i) q^{67} -2.16478i q^{69} -6.49435i q^{71} +(-4.25151 + 2.45461i) q^{73} +(0.792893 - 1.37333i) q^{75} +(-10.1503 - 5.86030i) q^{79} +(-0.500000 - 0.866025i) q^{81} -17.6569 q^{83} +5.41421 q^{85} +(-2.70711 - 4.68885i) q^{87} +(-7.84020 - 4.52654i) q^{89} +(4.82843 - 8.36308i) q^{93} +(9.05213 - 5.22625i) q^{95} -9.23880i q^{97} +5.22625i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{9} - 12 q^{25} - 8 q^{27} - 32 q^{29} - 16 q^{31} + 16 q^{47} + 16 q^{53} - 32 q^{55} - 16 q^{65} + 12 q^{75} - 4 q^{81} - 96 q^{83} + 32 q^{85} - 16 q^{87} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −1.60021 − 0.923880i −0.715634 − 0.413171i 0.0975096 − 0.995235i \(-0.468912\pi\)
−0.813144 + 0.582063i \(0.802246\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 4.52607 − 2.61313i 1.36466 − 0.787887i 0.374420 − 0.927259i \(-0.377842\pi\)
0.990240 + 0.139372i \(0.0445084\pi\)
\(12\) 0 0
\(13\) − 4.46088i − 1.23723i −0.785695 − 0.618613i \(-0.787695\pi\)
0.785695 − 0.618613i \(-0.212305\pi\)
\(14\) 0 0
\(15\) − 1.84776i − 0.477089i
\(16\) 0 0
\(17\) −2.53759 + 1.46508i −0.615455 + 0.355333i −0.775097 − 0.631842i \(-0.782299\pi\)
0.159642 + 0.987175i \(0.448966\pi\)
\(18\) 0 0
\(19\) −2.82843 + 4.89898i −0.648886 + 1.12390i 0.334504 + 0.942394i \(0.391431\pi\)
−0.983389 + 0.181509i \(0.941902\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.87476 − 1.08239i −0.390914 − 0.225694i 0.291642 − 0.956528i \(-0.405798\pi\)
−0.682556 + 0.730833i \(0.739132\pi\)
\(24\) 0 0
\(25\) −0.792893 − 1.37333i −0.158579 − 0.274666i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.41421 −1.00539 −0.502697 − 0.864463i \(-0.667659\pi\)
−0.502697 + 0.864463i \(0.667659\pi\)
\(30\) 0 0
\(31\) −4.82843 − 8.36308i −0.867211 − 1.50205i −0.864835 − 0.502057i \(-0.832577\pi\)
−0.00237631 − 0.999997i \(-0.500756\pi\)
\(32\) 0 0
\(33\) 4.52607 + 2.61313i 0.787887 + 0.454887i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.707107 + 1.22474i −0.116248 + 0.201347i −0.918278 − 0.395937i \(-0.870420\pi\)
0.802030 + 0.597284i \(0.203753\pi\)
\(38\) 0 0
\(39\) 3.86324 − 2.23044i 0.618613 − 0.357157i
\(40\) 0 0
\(41\) 4.01254i 0.626654i 0.949645 + 0.313327i \(0.101444\pi\)
−0.949645 + 0.313327i \(0.898556\pi\)
\(42\) 0 0
\(43\) 3.06147i 0.466869i 0.972372 + 0.233435i \(0.0749965\pi\)
−0.972372 + 0.233435i \(0.925003\pi\)
\(44\) 0 0
\(45\) 1.60021 − 0.923880i 0.238545 − 0.137724i
\(46\) 0 0
\(47\) −0.828427 + 1.43488i −0.120839 + 0.209298i −0.920099 − 0.391687i \(-0.871892\pi\)
0.799260 + 0.600985i \(0.205225\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.53759 − 1.46508i −0.355333 − 0.205152i
\(52\) 0 0
\(53\) 4.82843 + 8.36308i 0.663235 + 1.14876i 0.979760 + 0.200173i \(0.0641505\pi\)
−0.316525 + 0.948584i \(0.602516\pi\)
\(54\) 0 0
\(55\) −9.65685 −1.30213
\(56\) 0 0
\(57\) −5.65685 −0.749269
\(58\) 0 0
\(59\) −2.82843 − 4.89898i −0.368230 − 0.637793i 0.621059 − 0.783764i \(-0.286703\pi\)
−0.989289 + 0.145971i \(0.953369\pi\)
\(60\) 0 0
\(61\) −1.21193 − 0.699709i −0.155172 − 0.0895885i 0.420403 − 0.907337i \(-0.361889\pi\)
−0.575575 + 0.817749i \(0.695222\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.12132 + 7.13834i −0.511187 + 0.885402i
\(66\) 0 0
\(67\) 11.7034 − 6.75699i 1.42980 − 0.825497i 0.432698 − 0.901539i \(-0.357562\pi\)
0.997105 + 0.0760416i \(0.0242282\pi\)
\(68\) 0 0
\(69\) − 2.16478i − 0.260609i
\(70\) 0 0
\(71\) − 6.49435i − 0.770738i −0.922763 − 0.385369i \(-0.874074\pi\)
0.922763 − 0.385369i \(-0.125926\pi\)
\(72\) 0 0
\(73\) −4.25151 + 2.45461i −0.497602 + 0.287291i −0.727723 − 0.685871i \(-0.759421\pi\)
0.230121 + 0.973162i \(0.426088\pi\)
\(74\) 0 0
\(75\) 0.792893 − 1.37333i 0.0915554 − 0.158579i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.1503 − 5.86030i −1.14200 − 0.659336i −0.195077 − 0.980788i \(-0.562496\pi\)
−0.946926 + 0.321452i \(0.895829\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −17.6569 −1.93809 −0.969046 − 0.246881i \(-0.920594\pi\)
−0.969046 + 0.246881i \(0.920594\pi\)
\(84\) 0 0
\(85\) 5.41421 0.587254
\(86\) 0 0
\(87\) −2.70711 − 4.68885i −0.290232 − 0.502697i
\(88\) 0 0
\(89\) −7.84020 − 4.52654i −0.831060 − 0.479813i 0.0231556 − 0.999732i \(-0.492629\pi\)
−0.854215 + 0.519919i \(0.825962\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.82843 − 8.36308i 0.500685 − 0.867211i
\(94\) 0 0
\(95\) 9.05213 − 5.22625i 0.928729 − 0.536202i
\(96\) 0 0
\(97\) − 9.23880i − 0.938058i −0.883183 − 0.469029i \(-0.844604\pi\)
0.883183 − 0.469029i \(-0.155396\pi\)
\(98\) 0 0
\(99\) 5.22625i 0.525258i
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 2352.2.bl.s.607.1 8
4.3 odd 2 2352.2.bl.p.607.1 8
7.2 even 3 2352.2.b.i.1567.4 yes 4
7.3 odd 6 2352.2.bl.p.31.1 8
7.4 even 3 inner 2352.2.bl.s.31.4 8
7.5 odd 6 2352.2.b.j.1567.1 yes 4
7.6 odd 2 2352.2.bl.p.607.4 8
21.2 odd 6 7056.2.b.u.1567.1 4
21.5 even 6 7056.2.b.t.1567.4 4
28.3 even 6 inner 2352.2.bl.s.31.1 8
28.11 odd 6 2352.2.bl.p.31.4 8
28.19 even 6 2352.2.b.i.1567.1 ✓ 4
28.23 odd 6 2352.2.b.j.1567.4 yes 4
28.27 even 2 inner 2352.2.bl.s.607.4 8
84.23 even 6 7056.2.b.t.1567.1 4
84.47 odd 6 7056.2.b.u.1567.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.i.1567.1 ✓ 4 28.19 even 6
2352.2.b.i.1567.4 yes 4 7.2 even 3
2352.2.b.j.1567.1 yes 4 7.5 odd 6
2352.2.b.j.1567.4 yes 4 28.23 odd 6
2352.2.bl.p.31.1 8 7.3 odd 6
2352.2.bl.p.31.4 8 28.11 odd 6
2352.2.bl.p.607.1 8 4.3 odd 2
2352.2.bl.p.607.4 8 7.6 odd 2
2352.2.bl.s.31.1 8 28.3 even 6 inner
2352.2.bl.s.31.4 8 7.4 even 3 inner
2352.2.bl.s.607.1 8 1.1 even 1 trivial
2352.2.bl.s.607.4 8 28.27 even 2 inner
7056.2.b.t.1567.1 4 84.23 even 6
7056.2.b.t.1567.4 4 21.5 even 6
7056.2.b.u.1567.1 4 21.2 odd 6
7056.2.b.u.1567.4 4 84.47 odd 6