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Properties

Label 2352.2.bl.q.607.4
Level $2352$
Weight $2$
Character 2352.607
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
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] = [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,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,24] 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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 607.4
Root \(-1.60021 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 2352.607
Dual form 2352.2.bl.q.31.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(2.78415 + 1.60743i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-1.18394 + 0.683549i) q^{11} +2.93015i q^{13} -3.21486i q^{15} +(-5.98456 + 3.45519i) q^{17} +(3.67725 - 6.36918i) q^{19} +(3.14171 + 1.81387i) q^{23} +(2.66765 + 4.62051i) q^{25} +1.00000 q^{27} -1.11185 q^{29} +(4.35159 + 7.53718i) q^{31} +(1.18394 + 0.683549i) q^{33} +(-3.81896 + 6.61463i) q^{37} +(2.53759 - 1.46508i) q^{39} +0.833147i q^{41} +4.82362i q^{43} +(-2.78415 + 1.60743i) q^{45} +(1.47683 - 2.55795i) q^{47} +(5.98456 + 3.45519i) q^{51} +(2.28897 + 3.96462i) q^{53} -4.39502 q^{55} -7.35449 q^{57} +(-7.04923 - 12.2096i) q^{59} +(9.57981 + 5.53091i) q^{61} +(-4.71001 + 8.15797i) q^{65} +(-10.5261 + 6.07723i) q^{67} -3.62774i q^{69} +12.6249i q^{71} +(5.62060 - 3.24505i) q^{73} +(2.66765 - 4.62051i) q^{75} +(-6.74952 - 3.89684i) q^{79} +(-0.500000 - 0.866025i) q^{81} -8.87502 q^{83} -22.2159 q^{85} +(0.555927 + 0.962893i) q^{87} +(10.9246 + 6.30731i) q^{89} +(4.35159 - 7.53718i) q^{93} +(20.4760 - 11.8218i) q^{95} -13.8811i q^{97} -1.36710i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{9} + 24 q^{23} + 12 q^{25} + 8 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{47} - 8 q^{53} - 64 q^{55} - 24 q^{59} + 48 q^{61} + 8 q^{65} - 48 q^{67} + 48 q^{73} + 12 q^{75} - 24 q^{79} - 4 q^{81}+ \cdots + 72 q^{95}+O(q^{100}) \) 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
\(5\) 2.78415 + 1.60743i 1.24511 + 0.718864i 0.970130 − 0.242586i \(-0.0779958\pi\)
0.274979 + 0.961450i \(0.411329\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −1.18394 + 0.683549i −0.356972 + 0.206098i −0.667752 − 0.744384i \(-0.732743\pi\)
0.310780 + 0.950482i \(0.399410\pi\)
\(12\) 0 0
\(13\) 2.93015i 0.812678i 0.913722 + 0.406339i \(0.133195\pi\)
−0.913722 + 0.406339i \(0.866805\pi\)
\(14\) 0 0
\(15\) − 3.21486i − 0.830072i
\(16\) 0 0
\(17\) −5.98456 + 3.45519i −1.45147 + 0.838006i −0.998565 − 0.0535532i \(-0.982945\pi\)
−0.452904 + 0.891559i \(0.649612\pi\)
\(18\) 0 0
\(19\) 3.67725 − 6.36918i 0.843618 − 1.46119i −0.0431974 − 0.999067i \(-0.513754\pi\)
0.886816 − 0.462123i \(-0.152912\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.14171 + 1.81387i 0.655092 + 0.378218i 0.790405 − 0.612585i \(-0.209870\pi\)
−0.135312 + 0.990803i \(0.543204\pi\)
\(24\) 0 0
\(25\) 2.66765 + 4.62051i 0.533530 + 0.924102i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.11185 −0.206466 −0.103233 − 0.994657i \(-0.532919\pi\)
−0.103233 + 0.994657i \(0.532919\pi\)
\(30\) 0 0
\(31\) 4.35159 + 7.53718i 0.781569 + 1.35372i 0.931027 + 0.364949i \(0.118914\pi\)
−0.149458 + 0.988768i \(0.547753\pi\)
\(32\) 0 0
\(33\) 1.18394 + 0.683549i 0.206098 + 0.118991i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.81896 + 6.61463i −0.627833 + 1.08744i 0.360152 + 0.932893i \(0.382725\pi\)
−0.987986 + 0.154546i \(0.950609\pi\)
\(38\) 0 0
\(39\) 2.53759 − 1.46508i 0.406339 − 0.234600i
\(40\) 0 0
\(41\) 0.833147i 0.130116i 0.997881 + 0.0650579i \(0.0207232\pi\)
−0.997881 + 0.0650579i \(0.979277\pi\)
\(42\) 0 0
\(43\) 4.82362i 0.735595i 0.929906 + 0.367797i \(0.119888\pi\)
−0.929906 + 0.367797i \(0.880112\pi\)
\(44\) 0 0
\(45\) −2.78415 + 1.60743i −0.415036 + 0.239621i
\(46\) 0 0
\(47\) 1.47683 − 2.55795i 0.215418 − 0.373116i −0.737983 − 0.674819i \(-0.764222\pi\)
0.953402 + 0.301703i \(0.0975551\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.98456 + 3.45519i 0.838006 + 0.483823i
\(52\) 0 0
\(53\) 2.28897 + 3.96462i 0.314414 + 0.544582i 0.979313 − 0.202352i \(-0.0648585\pi\)
−0.664898 + 0.746934i \(0.731525\pi\)
\(54\) 0 0
\(55\) −4.39502 −0.592625
\(56\) 0 0
\(57\) −7.35449 −0.974127
\(58\) 0 0
\(59\) −7.04923 − 12.2096i −0.917732 − 1.58956i −0.802852 − 0.596179i \(-0.796685\pi\)
−0.114880 − 0.993379i \(-0.536648\pi\)
\(60\) 0 0
\(61\) 9.57981 + 5.53091i 1.22657 + 0.708160i 0.966311 − 0.257379i \(-0.0828588\pi\)
0.260259 + 0.965539i \(0.416192\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.71001 + 8.15797i −0.584205 + 1.01187i
\(66\) 0 0
\(67\) −10.5261 + 6.07723i −1.28596 + 0.742452i −0.977932 − 0.208924i \(-0.933004\pi\)
−0.308032 + 0.951376i \(0.599670\pi\)
\(68\) 0 0
\(69\) − 3.62774i − 0.436728i
\(70\) 0 0
\(71\) 12.6249i 1.49830i 0.662402 + 0.749148i \(0.269537\pi\)
−0.662402 + 0.749148i \(0.730463\pi\)
\(72\) 0 0
\(73\) 5.62060 − 3.24505i 0.657841 − 0.379805i −0.133613 − 0.991034i \(-0.542658\pi\)
0.791454 + 0.611229i \(0.209324\pi\)
\(74\) 0 0
\(75\) 2.66765 − 4.62051i 0.308034 − 0.533530i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.74952 − 3.89684i −0.759380 − 0.438428i 0.0696931 − 0.997568i \(-0.477798\pi\)
−0.829073 + 0.559140i \(0.811131\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −8.87502 −0.974159 −0.487080 − 0.873358i \(-0.661938\pi\)
−0.487080 + 0.873358i \(0.661938\pi\)
\(84\) 0 0
\(85\) −22.2159 −2.40965
\(86\) 0 0
\(87\) 0.555927 + 0.962893i 0.0596016 + 0.103233i
\(88\) 0 0
\(89\) 10.9246 + 6.30731i 1.15800 + 0.668574i 0.950825 − 0.309729i \(-0.100238\pi\)
0.207179 + 0.978303i \(0.433572\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.35159 − 7.53718i 0.451239 − 0.781569i
\(94\) 0 0
\(95\) 20.4760 − 11.8218i 2.10079 − 1.21289i
\(96\) 0 0
\(97\) − 13.8811i − 1.40942i −0.709497 − 0.704708i \(-0.751078\pi\)
0.709497 − 0.704708i \(-0.248922\pi\)
\(98\) 0 0
\(99\) − 1.36710i − 0.137398i
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.q.607.4 8
4.3 odd 2 2352.2.bl.r.607.4 8
7.2 even 3 2352.2.b.l.1567.2 yes 8
7.3 odd 6 2352.2.bl.r.31.4 8
7.4 even 3 2352.2.bl.o.31.1 8
7.5 odd 6 2352.2.b.k.1567.7 yes 8
7.6 odd 2 2352.2.bl.t.607.1 8
21.2 odd 6 7056.2.b.w.1567.7 8
21.5 even 6 7056.2.b.x.1567.2 8
28.3 even 6 inner 2352.2.bl.q.31.4 8
28.11 odd 6 2352.2.bl.t.31.1 8
28.19 even 6 2352.2.b.l.1567.7 yes 8
28.23 odd 6 2352.2.b.k.1567.2 ✓ 8
28.27 even 2 2352.2.bl.o.607.1 8
84.23 even 6 7056.2.b.x.1567.7 8
84.47 odd 6 7056.2.b.w.1567.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.2 ✓ 8 28.23 odd 6
2352.2.b.k.1567.7 yes 8 7.5 odd 6
2352.2.b.l.1567.2 yes 8 7.2 even 3
2352.2.b.l.1567.7 yes 8 28.19 even 6
2352.2.bl.o.31.1 8 7.4 even 3
2352.2.bl.o.607.1 8 28.27 even 2
2352.2.bl.q.31.4 8 28.3 even 6 inner
2352.2.bl.q.607.4 8 1.1 even 1 trivial
2352.2.bl.r.31.4 8 7.3 odd 6
2352.2.bl.r.607.4 8 4.3 odd 2
2352.2.bl.t.31.1 8 28.11 odd 6
2352.2.bl.t.607.1 8 7.6 odd 2
7056.2.b.w.1567.2 8 84.47 odd 6
7056.2.b.w.1567.7 8 21.2 odd 6
7056.2.b.x.1567.2 8 21.5 even 6
7056.2.b.x.1567.7 8 84.23 even 6