Properties

Label 75.22.a.h
Level $75$
Weight $22$
Character orbit 75.a
Self dual yes
Analytic conductor $209.608$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 224) q^{2} + 59049 q^{3} + (\beta_{2} + 294 \beta_1 - 290854) q^{4} + (59049 \beta_1 + 13226976) q^{6} + (11 \beta_{3} + 908 \beta_{2} + \cdots + 58725955) q^{7}+ \cdots + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 224) q^{2} + 59049 q^{3} + (\beta_{2} + 294 \beta_1 - 290854) q^{4} + (59049 \beta_1 + 13226976) q^{6} + (11 \beta_{3} + 908 \beta_{2} + \cdots + 58725955) q^{7}+ \cdots + (6506339692266 \beta_{3} + \cdots + 27\!\cdots\!22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9} + 31491830256 q^{11} - 68681250027 q^{12} + 27017977768 q^{13} - 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17} + 3127645607697 q^{18} - 24270353300752 q^{19} + 13851567033696 q^{21} + 56303932793676 q^{22} - 10350924920928 q^{23} - 4538248036821 q^{24} + 474751622871378 q^{26} + 823564528378596 q^{27} + 18\!\cdots\!68 q^{28}+ \cdots + 10\!\cdots\!56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 154\nu - 1756122 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 711\nu^{2} - 2080768\nu - 978048758 ) / 22 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 154\beta _1 + 1756122 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 22\beta_{3} - 711\beta_{2} + 2190262\beta _1 - 270553984 \) 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
−1711.97
−849.272
1069.36
1492.88
−1487.97 59049.0 116903. 0 −8.78631e7 1.26833e9 2.94655e9 3.48678e9 0
1.2 −625.272 59049.0 −1.70619e6 0 −3.69217e7 −3.78633e8 2.37812e9 3.48678e9 0
1.3 1293.36 59049.0 −424379. 0 7.63714e7 −1.27960e9 −3.26124e9 3.48678e9 0
1.4 1716.88 59049.0 850541. 0 1.01380e8 6.24477e8 −2.14029e9 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.h 4
5.b even 2 1 15.22.a.e 4
5.c odd 4 2 75.22.b.h 8
15.d odd 2 1 45.22.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.e 4 5.b even 2 1
45.22.a.g 4 15.d odd 2 1
75.22.a.h 4 1.a even 1 1 trivial
75.22.b.h 8 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 897T_{2}^{3} - 3210438T_{2}^{2} + 1891862624T_{2} + 2065963121664 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + \cdots + 2065963121664 \) Copy content Toggle raw display
$3$ \( (T - 59049)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 28\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 97\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 70\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 16\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 21\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 17\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 79\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 16\!\cdots\!24 \) Copy content Toggle raw display
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