Properties

Label 175.10.a.b
Level $175$
Weight $10$
Character orbit 175.a
Self dual yes
Analytic conductor $90.131$
Analytic rank $0$
Dimension $2$
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] = [175,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(90.1312713287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{2} + ( - 11 \beta + 43) q^{3} + (6 \beta - 310) q^{4} + (10 \beta - 1994) q^{6} + 2401 q^{7} + ( - 804 \beta - 1308) q^{8} + ( - 946 \beta + 5519) q^{9} + ( - 3326 \beta + 17658) q^{11}+ \cdots + ( - 35060662 \beta + 704708930) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 86 q^{3} - 620 q^{4} - 3988 q^{6} + 4802 q^{7} - 2616 q^{8} + 11038 q^{9} + 35316 q^{11} - 52136 q^{12} + 26530 q^{13} + 14406 q^{14} - 752 q^{16} + 463920 q^{17} - 332042 q^{18} - 925426 q^{19}+ \cdots + 1409417860 q^{99}+O(q^{100}) \) 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
−6.44622
7.44622
−10.8924 195.817 −393.355 0 −2132.92 2401.00 9861.52 18661.3 0
1.2 16.8924 −109.817 −226.645 0 −1855.08 2401.00 −12477.5 −7623.25 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 175.10.a.b 2
5.b even 2 1 7.10.a.a 2
5.c odd 4 2 175.10.b.b 4
15.d odd 2 1 63.10.a.d 2
20.d odd 2 1 112.10.a.e 2
35.c odd 2 1 49.10.a.b 2
35.i odd 6 2 49.10.c.b 4
35.j even 6 2 49.10.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.a 2 5.b even 2 1
49.10.a.b 2 35.c odd 2 1
49.10.c.b 4 35.i odd 6 2
49.10.c.c 4 35.j even 6 2
63.10.a.d 2 15.d odd 2 1
112.10.a.e 2 20.d odd 2 1
175.10.a.b 2 1.a even 1 1 trivial
175.10.b.b 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 6T_{2} - 184 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T - 184 \) Copy content Toggle raw display
$3$ \( T^{2} - 86T - 21504 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 1823214304 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 22750162568 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 36657492732 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 213416091952 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 128613482496 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 23287739754332 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 3507668488800 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 224285819284476 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 51779041048756 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 207953886197312 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 14\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 34\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 459497424927744 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 33\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 60\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 70118242258304 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 40\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
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