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Properties

Label 2376.2.b.d
Level $2376$
Weight $2$
Character orbit 2376.b
Analytic conductor $18.972$
Analytic rank $0$
Dimension $12$
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] = [2376,2,Mod(593,2376)] 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("2376.593"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2376.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,5] 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: no
Analytic conductor: \(18.9724555203\)
Analytic rank: \(0\)
Dimension: \(12\)
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: \( x^{12} + 35x^{10} + 432x^{8} + 2280x^{6} + 4784x^{4} + 2224x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{4} q^{7} - \beta_{7} q^{11} + \beta_{11} q^{13} + ( - \beta_{9} + 1) q^{17} + ( - \beta_{10} + \beta_{3} - \beta_1) q^{19} + (\beta_{11} - \beta_{10}) q^{23} + (\beta_{9} - \beta_{8} + \beta_{2}) q^{25}+ \cdots + ( - 2 \beta_{9} + \beta_{8} + 3 \beta_{7} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{11} + 10 q^{17} - 10 q^{25} - 2 q^{29} + 6 q^{31} + 10 q^{35} - 6 q^{37} - 2 q^{41} - 16 q^{49} - q^{55} + 32 q^{65} + 4 q^{67} - 7 q^{77} + 10 q^{83} + 24 q^{91} + 44 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 35x^{10} + 432x^{8} + 2280x^{6} + 4784x^{4} + 2224x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 29\nu^{8} + 258\nu^{6} + 732\nu^{4} + 520\nu^{2} + 896 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 31\nu^{9} + 312\nu^{7} + 1156\nu^{5} + 1376\nu^{3} + 736\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 41\nu^{9} + 614\nu^{7} + 4012\nu^{5} + 10136\nu^{3} + 4416\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{10} - 37\nu^{8} - 490\nu^{6} - 2764\nu^{4} - 5704\nu^{2} - 1152 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} - 6 \nu^{9} + 25 \nu^{8} - 178 \nu^{7} + 134 \nu^{6} - 1640 \nu^{5} - 484 \nu^{4} + \cdots - 896 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 6 \nu^{9} + 25 \nu^{8} + 178 \nu^{7} + 134 \nu^{6} + 1640 \nu^{5} - 484 \nu^{4} + \cdots - 896 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{10} - 157\nu^{8} - 1614\nu^{6} - 6172\nu^{4} - 7192\nu^{2} - 1280 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{10} - 93\nu^{8} - 936\nu^{6} - 3452\nu^{4} - 3792\nu^{2} - 768 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{11} + 65\nu^{9} + 713\nu^{7} + 3116\nu^{5} + 4852\nu^{3} + 1040\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} + 34\nu^{9} + 401\nu^{7} + 1968\nu^{5} + 3644\nu^{3} + 1056\nu ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - 2\beta_{10} - \beta_{7} + \beta_{6} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -14\beta_{9} + 12\beta_{8} - 3\beta_{7} - 3\beta_{6} + 6\beta_{5} - 12\beta_{2} + 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -38\beta_{11} + 34\beta_{10} + 21\beta_{7} - 21\beta_{6} + 16\beta_{3} + 116\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 180\beta_{9} - 130\beta_{8} + 67\beta_{7} + 67\beta_{6} - 142\beta_{5} + 154\beta_{2} - 558 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 594\beta_{11} - 502\beta_{10} - 363\beta_{7} + 363\beta_{6} + 48\beta_{4} - 392\beta_{3} - 1488\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2312\beta_{9} + 1370\beta_{8} - 1181\beta_{7} - 1181\beta_{6} + 2578\beta_{5} - 2082\beta_{2} + 7198 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -8838\beta_{11} + 7202\beta_{10} + 5841\beta_{7} - 5841\beta_{6} - 1424\beta_{4} + 7256\beta_{3} + 20200\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 30336\beta_{9} - 14454\beta_{8} + 19159\beta_{7} + 19159\beta_{6} - 42518\beta_{5} + 29038\beta_{2} - 98210 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 129826 \beta_{11} - 103190 \beta_{10} - 90715 \beta_{7} + 90715 \beta_{6} + 29168 \beta_{4} + \cdots - 283016 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(353\) \(1189\) \(1729\) \(1783\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
− 3.83827i
− 3.11093i
− 2.26309i
− 2.21040i
− 0.636690i
− 0.420721i
0.420721i
0.636690i
2.21040i
2.26309i
3.11093i
3.83827i
0 0 0 − 3.83827i 0 − 0.451228i 0 0 0
593.2 0 0 0 − 3.11093i 0 3.71703i 0 0 0
593.3 0 0 0 − 2.26309i 0 − 0.524655i 0 0 0
593.4 0 0 0 − 2.21040i 0 − 3.17087i 0 0 0
593.5 0 0 0 − 0.636690i 0 2.30454i 0 0 0
593.6 0 0 0 − 0.420721i 0 4.50993i 0 0 0
593.7 0 0 0 0.420721i 0 − 4.50993i 0 0 0
593.8 0 0 0 0.636690i 0 − 2.30454i 0 0 0
593.9 0 0 0 2.21040i 0 3.17087i 0 0 0
593.10 0 0 0 2.26309i 0 0.524655i 0 0 0
593.11 0 0 0 3.11093i 0 − 3.71703i 0 0 0
593.12 0 0 0 3.83827i 0 0.451228i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2376.2.b.d yes 12
3.b odd 2 1 2376.2.b.a ✓ 12
4.b odd 2 1 4752.2.b.i 12
11.b odd 2 1 2376.2.b.a ✓ 12
12.b even 2 1 4752.2.b.l 12
33.d even 2 1 inner 2376.2.b.d yes 12
44.c even 2 1 4752.2.b.l 12
132.d odd 2 1 4752.2.b.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2376.2.b.a ✓ 12 3.b odd 2 1
2376.2.b.a ✓ 12 11.b odd 2 1
2376.2.b.d yes 12 1.a even 1 1 trivial
2376.2.b.d yes 12 33.d even 2 1 inner
4752.2.b.i 12 4.b odd 2 1
4752.2.b.i 12 132.d odd 2 1
4752.2.b.l 12 12.b even 2 1
4752.2.b.l 12 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2376, [\chi])\):

\( T_{5}^{12} + 35T_{5}^{10} + 432T_{5}^{8} + 2280T_{5}^{6} + 4784T_{5}^{4} + 2224T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{17}^{6} - 5T_{17}^{5} - 34T_{17}^{4} + 222T_{17}^{3} - 9T_{17}^{2} - 1663T_{17} + 2320 \) 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
$5$ \( T^{12} + 35 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{12} + 50 T^{10} + \cdots + 841 \) Copy content Toggle raw display
$11$ \( T^{12} - 5 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 104 T^{10} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( (T^{6} - 5 T^{5} + \cdots + 2320)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 96 T^{10} + \cdots + 12544 \) Copy content Toggle raw display
$23$ \( T^{12} + 131 T^{10} + \cdots + 8988004 \) Copy content Toggle raw display
$29$ \( (T^{6} + T^{5} - 60 T^{4} + \cdots - 14)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} - 92 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 3 T^{5} + \cdots - 7394)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + T^{5} + \cdots - 54110)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 235 T^{10} + \cdots + 33339076 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 524318404 \) Copy content Toggle raw display
$53$ \( T^{12} + 371 T^{10} + \cdots + 5017600 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 505800100 \) Copy content Toggle raw display
$61$ \( T^{12} + 120 T^{10} + \cdots + 23658496 \) Copy content Toggle raw display
$67$ \( (T^{6} - 2 T^{5} + \cdots - 476288)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1677721600 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 443355136 \) Copy content Toggle raw display
$79$ \( T^{12} + 227 T^{10} + \cdots + 114244 \) Copy content Toggle raw display
$83$ \( (T^{6} - 5 T^{5} + \cdots - 1917776)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 107794022400 \) Copy content Toggle raw display
$97$ \( (T^{6} - 4 T^{5} + \cdots + 80815)^{2} \) Copy content Toggle raw display
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