Properties

Label 175.6.a.b
Level $175$
Weight $6$
Character orbit 175.a
Self dual yes
Analytic conductor $28.067$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,6,Mod(1,175)]
 
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, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

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: \(28.0671684673\)
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)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 10 q^{2} + 14 q^{3} + 68 q^{4} + 140 q^{6} + 49 q^{7} + 360 q^{8} - 47 q^{9} + 232 q^{11} + 952 q^{12} + 140 q^{13} + 490 q^{14} + 1424 q^{16} + 1722 q^{17} - 470 q^{18} - 98 q^{19} + 686 q^{21} + 2320 q^{22}+ \cdots - 10904 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
10.0000 14.0000 68.0000 0 140.000 49.0000 360.000 −47.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.6.a.b 1
5.b even 2 1 7.6.a.a 1
5.c odd 4 2 175.6.b.a 2
15.d odd 2 1 63.6.a.e 1
20.d odd 2 1 112.6.a.g 1
35.c odd 2 1 49.6.a.a 1
35.i odd 6 2 49.6.c.b 2
35.j even 6 2 49.6.c.c 2
40.e odd 2 1 448.6.a.c 1
40.f even 2 1 448.6.a.m 1
55.d odd 2 1 847.6.a.b 1
60.h even 2 1 1008.6.a.y 1
105.g even 2 1 441.6.a.k 1
140.c even 2 1 784.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 5.b even 2 1
49.6.a.a 1 35.c odd 2 1
49.6.c.b 2 35.i odd 6 2
49.6.c.c 2 35.j even 6 2
63.6.a.e 1 15.d odd 2 1
112.6.a.g 1 20.d odd 2 1
175.6.a.b 1 1.a even 1 1 trivial
175.6.b.a 2 5.c odd 4 2
441.6.a.k 1 105.g even 2 1
448.6.a.c 1 40.e odd 2 1
448.6.a.m 1 40.f even 2 1
784.6.a.c 1 140.c even 2 1
847.6.a.b 1 55.d odd 2 1
1008.6.a.y 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 10 \) Copy content Toggle raw display
$3$ \( T - 14 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 232 \) Copy content Toggle raw display
$13$ \( T - 140 \) Copy content Toggle raw display
$17$ \( T - 1722 \) Copy content Toggle raw display
$19$ \( T + 98 \) Copy content Toggle raw display
$23$ \( T + 1824 \) Copy content Toggle raw display
$29$ \( T - 3418 \) Copy content Toggle raw display
$31$ \( T + 7644 \) Copy content Toggle raw display
$37$ \( T - 10398 \) Copy content Toggle raw display
$41$ \( T + 17962 \) Copy content Toggle raw display
$43$ \( T + 10880 \) Copy content Toggle raw display
$47$ \( T + 9324 \) Copy content Toggle raw display
$53$ \( T + 2262 \) Copy content Toggle raw display
$59$ \( T + 2730 \) Copy content Toggle raw display
$61$ \( T - 25648 \) Copy content Toggle raw display
$67$ \( T - 48404 \) Copy content Toggle raw display
$71$ \( T + 58560 \) Copy content Toggle raw display
$73$ \( T + 68082 \) Copy content Toggle raw display
$79$ \( T - 31784 \) Copy content Toggle raw display
$83$ \( T - 20538 \) Copy content Toggle raw display
$89$ \( T + 50582 \) Copy content Toggle raw display
$97$ \( T - 58506 \) Copy content Toggle raw display
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