Properties

Label 175.4.a.f
Level $175$
Weight $4$
Character orbit 175.a
Self dual yes
Analytic conductor $10.325$
Analytic rank $0$
Dimension $3$
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] = [175,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.14360.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 17x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + (3 \beta_{2} - 4 \beta_1 + 7) q^{6} - 7 q^{7} + (3 \beta_{2} - \beta_1 + 4) q^{8} + ( - 3 \beta_{2} - 9 \beta_1 + 28) q^{9}+ \cdots + (106 \beta_{2} + 198 \beta_1 - 1198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 13 q^{4} + 24 q^{6} - 21 q^{7} + 15 q^{8} + 81 q^{9} - 74 q^{11} + 152 q^{12} - 44 q^{13} - 21 q^{14} - 79 q^{16} + 52 q^{17} + 411 q^{18} + 168 q^{19} + 14 q^{21} - 184 q^{22}+ \cdots - 3488 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 17x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) 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
4.48565
−0.861086
−3.62456
−3.48565 −0.850238 4.14976 0 2.96363 −7.00000 13.4206 −26.2771 0
1.2 1.86109 −9.53636 −4.53636 0 −17.7480 −7.00000 −23.3312 63.9421 0
1.3 4.62456 8.38660 13.3866 0 38.7844 −7.00000 24.9107 43.3350 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 175.4.a.f 3
3.b odd 2 1 1575.4.a.ba 3
5.b even 2 1 35.4.a.c 3
5.c odd 4 2 175.4.b.e 6
7.b odd 2 1 1225.4.a.y 3
15.d odd 2 1 315.4.a.p 3
20.d odd 2 1 560.4.a.u 3
35.c odd 2 1 245.4.a.l 3
35.i odd 6 2 245.4.e.n 6
35.j even 6 2 245.4.e.m 6
40.e odd 2 1 2240.4.a.bv 3
40.f even 2 1 2240.4.a.bt 3
105.g even 2 1 2205.4.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.c 3 5.b even 2 1
175.4.a.f 3 1.a even 1 1 trivial
175.4.b.e 6 5.c odd 4 2
245.4.a.l 3 35.c odd 2 1
245.4.e.m 6 35.j even 6 2
245.4.e.n 6 35.i odd 6 2
315.4.a.p 3 15.d odd 2 1
560.4.a.u 3 20.d odd 2 1
1225.4.a.y 3 7.b odd 2 1
1575.4.a.ba 3 3.b odd 2 1
2205.4.a.bm 3 105.g even 2 1
2240.4.a.bt 3 40.f even 2 1
2240.4.a.bv 3 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2}^{2} - 14T_{2} + 30 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3 T^{2} + \cdots + 30 \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} + \cdots - 68 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 74 T^{2} + \cdots + 7692 \) Copy content Toggle raw display
$13$ \( T^{3} + 44 T^{2} + \cdots + 44870 \) Copy content Toggle raw display
$17$ \( T^{3} - 52 T^{2} + \cdots + 56706 \) Copy content Toggle raw display
$19$ \( T^{3} - 168 T^{2} + \cdots - 28720 \) Copy content Toggle raw display
$23$ \( T^{3} - 124 T^{2} + \cdots + 94368 \) Copy content Toggle raw display
$29$ \( T^{3} - 332 T^{2} + \cdots + 2565450 \) Copy content Toggle raw display
$31$ \( T^{3} - 320 T^{2} + \cdots + 50176 \) Copy content Toggle raw display
$37$ \( T^{3} - 54 T^{2} + \cdots - 25736 \) Copy content Toggle raw display
$41$ \( T^{3} - 362 T^{2} + \cdots - 1536192 \) Copy content Toggle raw display
$43$ \( T^{3} - 16 T^{2} + \cdots + 1524560 \) Copy content Toggle raw display
$47$ \( T^{3} - 730 T^{2} + \cdots - 4968912 \) Copy content Toggle raw display
$53$ \( T^{3} + 110 T^{2} + \cdots - 90318336 \) Copy content Toggle raw display
$59$ \( T^{3} + 180 T^{2} + \cdots - 202459200 \) Copy content Toggle raw display
$61$ \( T^{3} - 1222 T^{2} + \cdots - 38393792 \) Copy content Toggle raw display
$67$ \( T^{3} + 204 T^{2} + \cdots - 324944128 \) Copy content Toggle raw display
$71$ \( T^{3} + 136 T^{2} + \cdots + 15575040 \) Copy content Toggle raw display
$73$ \( T^{3} + 310 T^{2} + \cdots - 48718616 \) Copy content Toggle raw display
$79$ \( T^{3} + 1034 T^{2} + \cdots - 343615600 \) Copy content Toggle raw display
$83$ \( T^{3} - 1660 T^{2} + \cdots + 42727104 \) Copy content Toggle raw display
$89$ \( T^{3} - 242 T^{2} + \cdots - 6359520 \) Copy content Toggle raw display
$97$ \( T^{3} + 100 T^{2} + \cdots + 1978018 \) Copy content Toggle raw display
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