Properties

Label 175.9.d.a
Level $175$
Weight $9$
Character orbit 175.d
Self dual yes
Analytic conductor $71.291$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,9,Mod(76,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.76");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 175.d (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(71.2912567607\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 31 q^{2} + 705 q^{4} - 2401 q^{7} + 13919 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 31 q^{2} + 705 q^{4} - 2401 q^{7} + 13919 q^{8} + 6561 q^{9} + 13154 q^{11} - 74431 q^{14} + 251009 q^{16} + 203391 q^{18} + 407774 q^{22} + 20926 q^{23} - 1692705 q^{28} + 108194 q^{29} + 4218015 q^{32} + 4625505 q^{36} + 2073886 q^{37} + 6726046 q^{43} + 9273570 q^{44} + 648706 q^{46} + 5764801 q^{49} - 15377762 q^{53} - 33419519 q^{56} + 3354014 q^{58} - 15752961 q^{63} + 66500161 q^{64} + 15839326 q^{67} - 42331966 q^{71} + 91322559 q^{72} + 64290466 q^{74} - 31582754 q^{77} - 64606846 q^{79} + 43046721 q^{81} + 208507426 q^{86} + 183090526 q^{88} + 14752830 q^{92} + 178708831 q^{98} + 86303394 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0
31.0000 0 705.000 0 0 −2401.00 13919.0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.9.d.a 1
5.b even 2 1 7.9.b.a 1
5.c odd 4 2 175.9.c.a 2
7.b odd 2 1 CM 175.9.d.a 1
15.d odd 2 1 63.9.d.a 1
20.d odd 2 1 112.9.c.a 1
35.c odd 2 1 7.9.b.a 1
35.f even 4 2 175.9.c.a 2
35.i odd 6 2 49.9.d.a 2
35.j even 6 2 49.9.d.a 2
105.g even 2 1 63.9.d.a 1
140.c even 2 1 112.9.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.9.b.a 1 5.b even 2 1
7.9.b.a 1 35.c odd 2 1
49.9.d.a 2 35.i odd 6 2
49.9.d.a 2 35.j even 6 2
63.9.d.a 1 15.d odd 2 1
63.9.d.a 1 105.g even 2 1
112.9.c.a 1 20.d odd 2 1
112.9.c.a 1 140.c even 2 1
175.9.c.a 2 5.c odd 4 2
175.9.c.a 2 35.f even 4 2
175.9.d.a 1 1.a even 1 1 trivial
175.9.d.a 1 7.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 31 \) acting on \(S_{9}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 31 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2401 \) Copy content Toggle raw display
$11$ \( T - 13154 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 20926 \) Copy content Toggle raw display
$29$ \( T - 108194 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 2073886 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 6726046 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 15377762 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 15839326 \) Copy content Toggle raw display
$71$ \( T + 42331966 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 64606846 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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