Properties

Label 75.22.a.f
Level $75$
Weight $22$
Character orbit 75.a
Self dual yes
Analytic conductor $209.608$
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] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 157936x - 9799664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 234) q^{2} + 59049 q^{3} + (2 \beta_{2} + 1030 \beta_1 + 1748076) q^{4} + ( - 59049 \beta_1 - 13817466) q^{6} + (403 \beta_{2} + 167580 \beta_1 + 690806068) q^{7} + ( - 1404 \beta_{2} - 2050196 \beta_1 - 3820355944) q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 234) q^{2} + 59049 q^{3} + (2 \beta_{2} + 1030 \beta_1 + 1748076) q^{4} + ( - 59049 \beta_1 - 13817466) q^{6} + (403 \beta_{2} + 167580 \beta_1 + 690806068) q^{7} + ( - 1404 \beta_{2} - 2050196 \beta_1 - 3820355944) q^{8} + 3486784401 q^{9} + ( - 54496 \beta_{2} + \cdots - 2822533480) q^{11}+ \cdots + ( - 190015802716896 \beta_{2} + \cdots - 98\!\cdots\!80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 702 q^{2} + 177147 q^{3} + 5244228 q^{4} - 41452398 q^{6} + 2072418204 q^{7} - 11461067832 q^{8} + 10460353203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 702 q^{2} + 177147 q^{3} + 5244228 q^{4} - 41452398 q^{6} + 2072418204 q^{7} - 11461067832 q^{8} + 10460353203 q^{9} - 8467600440 q^{11} + 309666419172 q^{12} + 469447548570 q^{13} - 2389270234872 q^{14} + 14993057845776 q^{16} + 6865096035486 q^{17} - 2447722649502 q^{18} + 33215593555044 q^{19} + 122374222527996 q^{21} - 301288671211728 q^{22} - 48760125154728 q^{23} - 676764594411768 q^{24} - 21\!\cdots\!24 q^{26}+ \cdots - 29\!\cdots\!40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 157936x - 9799664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 18\nu^{2} - 1698\nu - 1894672 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 283\beta _1 + 1895238 ) / 18 \) 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
425.878
−63.7115
−361.167
−2787.27 59049.0 5.67172e6 0 −1.64586e8 1.37938e9 −9.96329e9 3.48678e9 0
1.2 150.269 59049.0 −2.07457e6 0 8.87323e6 −6.41000e7 −6.26880e8 3.48678e9 0
1.3 1935.00 59049.0 1.64708e6 0 1.14260e8 7.57139e8 −8.70895e8 3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)

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 75.22.a.f 3
5.b even 2 1 15.22.a.c 3
5.c odd 4 2 75.22.b.f 6
15.d odd 2 1 45.22.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.c 3 5.b even 2 1
45.22.a.c 3 15.d odd 2 1
75.22.a.f 3 1.a even 1 1 trivial
75.22.b.f 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 702T_{2}^{2} - 5521440T_{2} + 810456064 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 702 T^{2} + \cdots + 810456064 \) Copy content Toggle raw display
$3$ \( (T - 59049)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 66\!\cdots\!20 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 86\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 19\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 64\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 91\!\cdots\!60 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 79\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 26\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 18\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 20\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 97\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 14\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 62\!\cdots\!12 \) Copy content Toggle raw display
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