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Properties

Label 3775.1.c.c
Level $3775$
Weight $1$
Character orbit 3775.c
Analytic conductor $1.884$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -151
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] = [3775,1,Mod(3774,3775)] 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("3775.3774"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3775 = 5^{2} \cdot 151 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3775.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.88397042269\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 151)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.3442951.1
Artin image: $C_4\times D_7$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_1) q^{8} - q^{9} - \beta_{2} q^{11} + \beta_{4} q^{16} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{17} + \beta_1 q^{18} + \beta_{4} q^{19}+ \cdots + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} - 6 q^{9} - 2 q^{11} + 2 q^{16} + 2 q^{19} + 2 q^{29} - 2 q^{31} + 4 q^{34} + 4 q^{36} - 8 q^{44} - 6 q^{49} + 2 q^{59} + 4 q^{74} - 6 q^{76} + 6 q^{81} - 4 q^{86} - 10 q^{94} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3775\mathbb{Z}\right)^\times\).

\(n\) \(152\) \(3026\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3774.1
1.80194i
1.24698i
0.445042i
− 0.445042i
− 1.24698i
− 1.80194i
− 1.80194i 0 −2.24698 0 0 0 2.24698i −1.00000 0
3774.2 − 1.24698i 0 −0.554958 0 0 0 − 0.554958i −1.00000 0
3774.3 − 0.445042i 0 0.801938 0 0 0 − 0.801938i −1.00000 0
3774.4 0.445042i 0 0.801938 0 0 0 0.801938i −1.00000 0
3774.5 1.24698i 0 −0.554958 0 0 0 0.554958i −1.00000 0
3774.6 1.80194i 0 −2.24698 0 0 0 − 2.24698i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3774.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
151.b odd 2 1 CM by \(\Q(\sqrt{-151}) \)
5.b even 2 1 inner
755.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3775.1.c.c 6
5.b even 2 1 inner 3775.1.c.c 6
5.c odd 4 1 151.1.b.a ✓ 3
5.c odd 4 1 3775.1.d.d 3
15.e even 4 1 1359.1.d.b 3
20.e even 4 1 2416.1.e.a 3
151.b odd 2 1 CM 3775.1.c.c 6
755.c odd 2 1 inner 3775.1.c.c 6
755.f even 4 1 151.1.b.a ✓ 3
755.f even 4 1 3775.1.d.d 3
2265.j odd 4 1 1359.1.d.b 3
3020.j odd 4 1 2416.1.e.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
151.1.b.a ✓ 3 5.c odd 4 1
151.1.b.a ✓ 3 755.f even 4 1
1359.1.d.b 3 15.e even 4 1
1359.1.d.b 3 2265.j odd 4 1
2416.1.e.a 3 20.e even 4 1
2416.1.e.a 3 3020.j odd 4 1
3775.1.c.c 6 1.a even 1 1 trivial
3775.1.c.c 6 5.b even 2 1 inner
3775.1.c.c 6 151.b odd 2 1 CM
3775.1.c.c 6 755.c odd 2 1 inner
3775.1.d.d 3 5.c odd 4 1
3775.1.d.d 3 755.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3775, [\chi])\):

\( T_{2}^{6} + 5T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{3} - T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( (T^{3} - T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( (T^{3} - T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
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