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Properties

Label 2376.2.q.f
Level $2376$
Weight $2$
Character orbit 2376.q
Analytic conductor $18.972$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2376,2,Mod(793,2376)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2376.793"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2376.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.9724555203\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 7x^{10} - 2x^{9} + 39x^{8} - 9x^{7} + 67x^{6} - 18x^{5} + 88x^{4} - 16x^{3} + 24x^{2} + 4x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 792)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{5} + 1) q^{5} + ( - \beta_{5} - \beta_1) q^{7} + \beta_{5} q^{11} + ( - \beta_{11} + \beta_{8} + \cdots + \beta_{6}) q^{13} + (\beta_{9} + \beta_{4} - 1) q^{17} + (\beta_{10} + \beta_{2} - \beta_1 + 1) q^{19}+ \cdots + (2 \beta_{9} + 4 \beta_{8} + \cdots - 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{5} - 5 q^{7} + 6 q^{11} - 3 q^{13} - 14 q^{17} + 10 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{29} - 4 q^{31} - 16 q^{35} + 6 q^{37} - 5 q^{43} + 17 q^{47} - 3 q^{49} - 28 q^{53} + 8 q^{55} + 4 q^{59}+ \cdots + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 7x^{10} - 2x^{9} + 39x^{8} - 9x^{7} + 67x^{6} - 18x^{5} + 88x^{4} - 16x^{3} + 24x^{2} + 4x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6647 \nu^{11} + 42545 \nu^{10} - 15881 \nu^{9} + 277191 \nu^{8} - 264513 \nu^{7} + 1665582 \nu^{6} + \cdots + 740800 ) / 721692 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4540 \nu^{11} - 4391 \nu^{10} + 30185 \nu^{9} - 32247 \nu^{8} + 175701 \nu^{7} - 167199 \nu^{6} + \cdots + 653222 ) / 360846 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10401 \nu^{11} + 12475 \nu^{10} - 74827 \nu^{9} + 104517 \nu^{8} - 438279 \nu^{7} + \cdots + 236300 ) / 240564 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35953 \nu^{11} - 122905 \nu^{10} - 315011 \nu^{9} - 777771 \nu^{8} - 1544367 \nu^{7} + \cdots - 559256 ) / 721692 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39985 \nu^{11} + 34235 \nu^{10} - 270815 \nu^{9} + 310833 \nu^{8} - 1567515 \nu^{7} + \cdots + 672580 ) / 721692 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25645 \nu^{11} - 18776 \nu^{10} + 170969 \nu^{9} - 148572 \nu^{8} + 984729 \nu^{7} - 706605 \nu^{6} + \cdots + 1050128 ) / 360846 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39985 \nu^{11} - 34235 \nu^{10} + 270815 \nu^{9} - 310833 \nu^{8} + 1567515 \nu^{7} - 1630536 \nu^{6} + \cdots + 49112 ) / 360846 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26365 \nu^{11} - 21062 \nu^{10} + 180260 \nu^{9} - 214092 \nu^{8} + 1040412 \nu^{7} - 1128939 \nu^{6} + \cdots - 467170 ) / 180423 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 37369 \nu^{11} + 13901 \nu^{10} - 235053 \nu^{9} + 193059 \nu^{8} - 1299045 \nu^{7} + \cdots + 375960 ) / 240564 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41411 \nu^{11} + 5685 \nu^{10} - 305421 \nu^{9} + 119847 \nu^{8} - 1711377 \nu^{7} + \cdots - 156984 ) / 240564 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 64079 \nu^{11} - 16565 \nu^{10} - 438559 \nu^{9} + 17559 \nu^{8} - 2415003 \nu^{7} + \cdots - 301720 ) / 360846 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 2\beta_{5} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 2\beta_{10} - 2\beta_{9} + 2\beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 6\beta_{7} - 8\beta_{5} + 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{11} - 9\beta_{10} - 10\beta_{8} + \beta_{7} - 10\beta_{6} + \beta_{5} + 10\beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} + 8\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 32\beta_{2} - \beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 51\beta_{9} + 52\beta_{8} - 11\beta_{7} - 43\beta_{5} + 32\beta_{3} + 11\beta_{2} + 44\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -11\beta_{11} - 12\beta_{10} - 52\beta_{8} + 167\beta_{7} - 52\beta_{6} + 191\beta_{5} + 13\beta_{4} - 191 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 167 \beta_{11} + 223 \beta_{10} - 262 \beta_{9} + 274 \beta_{6} + 167 \beta_{5} - 262 \beta_{4} + \cdots + 89 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 112\beta_{9} + 318\beta_{8} - 870\beta_{7} - 1069\beta_{5} + 87\beta_{3} + 870\beta_{2} + 100\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 870 \beta_{11} - 1149 \beta_{10} - 1454 \beta_{8} + 604 \beta_{7} - 1454 \beta_{6} + 630 \beta_{5} + \cdots - 630 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2376\mathbb{Z}\right)^\times\).

\(n\) \(353\) \(1189\) \(1729\) \(1783\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
793.1
−1.17557 − 2.03615i
1.08357 + 1.87680i
−0.654157 − 1.13303i
0.613168 + 1.06204i
0.322589 + 0.558741i
−0.189597 − 0.328392i
−1.17557 + 2.03615i
1.08357 − 1.87680i
−0.654157 + 1.13303i
0.613168 − 1.06204i
0.322589 − 0.558741i
−0.189597 + 0.328392i
0 0 0 −1.26394 + 2.18921i 0 0.129686 + 0.224623i 0 0 0
793.2 0 0 0 −0.848245 + 1.46920i 0 −0.951006 − 1.64719i 0 0 0
793.3 0 0 0 0.644157 − 1.11571i 0 2.04089 + 3.53492i 0 0 0
793.4 0 0 0 0.748050 − 1.29566i 0 −1.94143 − 3.36265i 0 0 0
793.5 0 0 0 1.29187 − 2.23759i 0 −0.251298 − 0.435260i 0 0 0
793.6 0 0 0 1.42811 − 2.47355i 0 −1.52684 − 2.64457i 0 0 0
1585.1 0 0 0 −1.26394 − 2.18921i 0 0.129686 − 0.224623i 0 0 0
1585.2 0 0 0 −0.848245 − 1.46920i 0 −0.951006 + 1.64719i 0 0 0
1585.3 0 0 0 0.644157 + 1.11571i 0 2.04089 − 3.53492i 0 0 0
1585.4 0 0 0 0.748050 + 1.29566i 0 −1.94143 + 3.36265i 0 0 0
1585.5 0 0 0 1.29187 + 2.23759i 0 −0.251298 + 0.435260i 0 0 0
1585.6 0 0 0 1.42811 + 2.47355i 0 −1.52684 + 2.64457i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 793.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2376.2.q.f 12
3.b odd 2 1 792.2.q.f ✓ 12
9.c even 3 1 inner 2376.2.q.f 12
9.c even 3 1 7128.2.a.w 6
9.d odd 6 1 792.2.q.f ✓ 12
9.d odd 6 1 7128.2.a.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
792.2.q.f ✓ 12 3.b odd 2 1
792.2.q.f ✓ 12 9.d odd 6 1
2376.2.q.f 12 1.a even 1 1 trivial
2376.2.q.f 12 9.c even 3 1 inner
7128.2.a.w 6 9.c even 3 1
7128.2.a.ba 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 4 T_{5}^{11} + 22 T_{5}^{10} - 50 T_{5}^{9} + 196 T_{5}^{8} - 391 T_{5}^{7} + 1127 T_{5}^{6} + \cdots + 3721 \) acting on \(S_{2}^{\mathrm{new}}(2376, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots + 3721 \) Copy content Toggle raw display
$7$ \( T^{12} + 5 T^{11} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 4624 \) Copy content Toggle raw display
$17$ \( (T^{6} + 7 T^{5} + 6 T^{4} + \cdots + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 5 T^{5} - 18 T^{4} + \cdots - 72)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 8 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 169312144 \) Copy content Toggle raw display
$31$ \( T^{12} + 4 T^{11} + \cdots + 2601 \) Copy content Toggle raw display
$37$ \( (T^{6} - 3 T^{5} + \cdots + 19819)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 91 T^{10} + \cdots + 1354896 \) Copy content Toggle raw display
$43$ \( T^{12} + 5 T^{11} + \cdots + 394384 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 244078129 \) Copy content Toggle raw display
$53$ \( (T^{6} + 14 T^{5} + \cdots - 36261)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2306016441 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 205520896 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10217368561 \) Copy content Toggle raw display
$71$ \( (T^{6} + 7 T^{5} + \cdots + 1161)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 12 T^{5} + \cdots + 146764)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 24150403216 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 9910600704 \) Copy content Toggle raw display
$89$ \( (T^{6} + 22 T^{5} + \cdots + 106432)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2035182769 \) Copy content Toggle raw display
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