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Properties

Label 3751.1.t.f
Level $3751$
Weight $1$
Character orbit 3751.t
Analytic conductor $1.872$
Analytic rank $0$
Dimension $12$
Projective image $D_{9}$
CM discriminant -31
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3751,1,Mod(2138,3751)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3751, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3751.2138");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3751 = 11^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3751.t (of order \(10\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.87199286239\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{10})\)
Coefficient field: 12.0.84075626953125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3x^{10} - x^{9} + 9x^{8} + 9x^{7} + 28x^{6} + 18x^{5} + 75x^{4} + 26x^{3} + 9x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.1636073786281.1
Artin image: $C_5\times D_9$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{45} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{8} + \beta_{7} + \beta_{6}) q^{4} + (\beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{5}+ \cdots + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_{8} + \beta_{7} + \beta_{6}) q^{4} + (\beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - \beta_{2} + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{4} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{4} + 3 q^{8} - 3 q^{9} + 24 q^{10} + 3 q^{14} - 3 q^{16} + 3 q^{20} - 3 q^{25} + 3 q^{28} - 3 q^{31} - 12 q^{32} + 3 q^{35} - 3 q^{36} - 6 q^{38} - 6 q^{40} + 3 q^{47} - 3 q^{49} + 3 q^{50} - 12 q^{56} - 12 q^{67} + 3 q^{70} + 3 q^{72} - 12 q^{76} + 6 q^{80} - 3 q^{81} + 3 q^{82} - 6 q^{90} - 6 q^{95} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3x^{10} - x^{9} + 9x^{8} + 9x^{7} + 28x^{6} + 18x^{5} + 75x^{4} + 26x^{3} + 9x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 90\nu^{5} + 243 ) / 701 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{10} - 199\nu^{5} + 1075 ) / 701 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{11} - 109\nu^{6} + 2019\nu ) / 701 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\nu^{11} + 308\nu^{6} - 3795\nu ) / 701 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 27 \nu^{10} + 9 \nu^{9} - 81 \nu^{8} + 28 \nu^{7} - 252 \nu^{6} - 162 \nu^{5} - 675 \nu^{4} + \cdots - 9 ) / 701 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 84 \nu^{10} - 28 \nu^{9} + 252 \nu^{8} - 165 \nu^{7} + 784 \nu^{6} + 504 \nu^{5} + 2100 \nu^{4} + \cdots + 28 ) / 701 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 168 \nu^{10} + 56 \nu^{9} - 504 \nu^{8} + 330 \nu^{7} - 1568 \nu^{6} - 1008 \nu^{5} - 4200 \nu^{4} + \cdots - 56 ) / 701 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 84 \nu^{11} + 28 \nu^{10} - 252 \nu^{9} + 165 \nu^{8} - 784 \nu^{7} - 504 \nu^{6} - 2100 \nu^{5} + \cdots - 28 \nu ) / 701 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 252 \nu^{11} + 84 \nu^{10} - 756 \nu^{9} + 495 \nu^{8} - 2352 \nu^{7} - 1512 \nu^{6} + \cdots - 84 \nu ) / 701 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 477 \nu^{11} - 159 \nu^{10} + 1431 \nu^{9} - 962 \nu^{8} + 4452 \nu^{7} + 2862 \nu^{6} + \cdots + 159 \nu ) / 701 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 2\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 3\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{11} + 6 \beta_{10} - \beta_{9} - 3 \beta_{8} - 6 \beta_{7} - \beta_{6} - 3 \beta_{5} - 6 \beta_{4} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{3} + 10\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{5} + 19\beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -6\beta_{8} - 21\beta_{7} - 28\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{11} - 62\beta_{10} + 27\beta_{9} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 27 \beta_{11} - 82 \beta_{10} + 90 \beta_{9} + 27 \beta_{8} + 82 \beta_{7} + 90 \beta_{6} + 27 \beta_{5} + \cdots + 82 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -90\beta_{3} - 199\beta_{2} + 207 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -109\beta_{5} - 308\beta_{4} + 297\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times\).

\(n\) \(2421\) \(2543\)
\(\chi(n)\) \(-1\) \(\beta_{7}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2138.1
1.52045 + 1.10467i
−1.23949 − 0.900539i
−0.280969 − 0.204136i
1.52045 − 1.10467i
−1.23949 + 0.900539i
−0.280969 + 0.204136i
0.107320 − 0.330298i
0.473442 − 1.45710i
−0.580762 + 1.78740i
0.107320 + 0.330298i
0.473442 + 1.45710i
−0.580762 − 1.78740i
−1.23949 − 0.900539i 0 0.416337 + 1.28135i −1.23949 + 0.900539i 0 −0.580762 − 1.78740i 0.164425 − 0.506047i −0.809017 − 0.587785i 2.34730
2138.2 −0.280969 − 0.204136i 0 −0.271745 − 0.836345i −0.280969 + 0.204136i 0 0.473442 + 1.45710i −0.201697 + 0.620758i −0.809017 − 0.587785i 0.120615
2138.3 1.52045 + 1.10467i 0 0.782458 + 2.40816i 1.52045 − 1.10467i 0 0.107320 + 0.330298i −0.889779 + 2.73846i −0.809017 − 0.587785i 3.53209
2665.1 −1.23949 + 0.900539i 0 0.416337 − 1.28135i −1.23949 − 0.900539i 0 −0.580762 + 1.78740i 0.164425 + 0.506047i −0.809017 + 0.587785i 2.34730
2665.2 −0.280969 + 0.204136i 0 −0.271745 + 0.836345i −0.280969 − 0.204136i 0 0.473442 − 1.45710i −0.201697 − 0.620758i −0.809017 + 0.587785i 0.120615
2665.3 1.52045 − 1.10467i 0 0.782458 − 2.40816i 1.52045 + 1.10467i 0 0.107320 − 0.330298i −0.889779 − 2.73846i −0.809017 + 0.587785i 3.53209
2913.1 −0.580762 + 1.78740i 0 −2.04850 − 1.48832i −0.580762 − 1.78740i 0 −0.280969 − 0.204136i 2.32947 − 1.69246i 0.309017 − 0.951057i 3.53209
2913.2 0.107320 − 0.330298i 0 0.711438 + 0.516890i 0.107320 + 0.330298i 0 −1.23949 − 0.900539i 0.528048 − 0.383650i 0.309017 − 0.951057i 0.120615
2913.3 0.473442 − 1.45710i 0 −1.08999 − 0.791921i 0.473442 + 1.45710i 0 1.52045 + 1.10467i −0.430469 + 0.312754i 0.309017 − 0.951057i 2.34730
3657.1 −0.580762 − 1.78740i 0 −2.04850 + 1.48832i −0.580762 + 1.78740i 0 −0.280969 + 0.204136i 2.32947 + 1.69246i 0.309017 + 0.951057i 3.53209
3657.2 0.107320 + 0.330298i 0 0.711438 − 0.516890i 0.107320 − 0.330298i 0 −1.23949 + 0.900539i 0.528048 + 0.383650i 0.309017 + 0.951057i 0.120615
3657.3 0.473442 + 1.45710i 0 −1.08999 + 0.791921i 0.473442 − 1.45710i 0 1.52045 − 1.10467i −0.430469 − 0.312754i 0.309017 + 0.951057i 2.34730
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2138.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
11.c even 5 3 inner
341.t odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3751.1.t.f 12
11.b odd 2 1 3751.1.t.e 12
11.c even 5 1 3751.1.d.d ✓ 3
11.c even 5 3 inner 3751.1.t.f 12
11.d odd 10 1 3751.1.d.e yes 3
11.d odd 10 3 3751.1.t.e 12
31.b odd 2 1 CM 3751.1.t.f 12
341.b even 2 1 3751.1.t.e 12
341.t odd 10 1 3751.1.d.d ✓ 3
341.t odd 10 3 inner 3751.1.t.f 12
341.ba even 10 1 3751.1.d.e yes 3
341.ba even 10 3 3751.1.t.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3751.1.d.d ✓ 3 11.c even 5 1
3751.1.d.d ✓ 3 341.t odd 10 1
3751.1.d.e yes 3 11.d odd 10 1
3751.1.d.e yes 3 341.ba even 10 1
3751.1.t.e 12 11.b odd 2 1
3751.1.t.e 12 11.d odd 10 3
3751.1.t.e 12 341.b even 2 1
3751.1.t.e 12 341.ba even 10 3
3751.1.t.f 12 1.a even 1 1 trivial
3751.1.t.f 12 11.c even 5 3 inner
3751.1.t.f 12 31.b odd 2 1 CM
3751.1.t.f 12 341.t odd 10 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 3T_{2}^{10} - T_{2}^{9} + 9T_{2}^{8} + 9T_{2}^{7} + 28T_{2}^{6} + 18T_{2}^{5} + 75T_{2}^{4} + 26T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3751, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( (T + 1)^{12} \) Copy content Toggle raw display
$71$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
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