Properties

Label 75.22.a.e
Level $75$
Weight $22$
Character orbit 75.a
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
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: \(1\)
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: \( x^{3} - x^{2} - 6395796x - 2792983104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 268) q^{2} - 59049 q^{3} + (\beta_{2} + 120 \beta_1 + 2238318) q^{4} + ( - 59049 \beta_1 + 15825132) q^{6} + ( - 264 \beta_{2} + 153760 \beta_1 + 525814764) q^{7} + ( - 803 \beta_{2} + 1890196 \beta_1 + 474100006) q^{8} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 268) q^{2} - 59049 q^{3} + (\beta_{2} + 120 \beta_1 + 2238318) q^{4} + ( - 59049 \beta_1 + 15825132) q^{6} + ( - 264 \beta_{2} + 153760 \beta_1 + 525814764) q^{7} + ( - 803 \beta_{2} + 1890196 \beta_1 + 474100006) q^{8} + 3486784401 q^{9} + ( - 7808 \beta_{2} + \cdots + 28162718864) q^{11}+ \cdots + ( - 27224812603008 \beta_{2} + \cdots + 98\!\cdots\!64) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9} + 84497282000 q^{11} - 396518345577 q^{12} - 1065489966310 q^{13} + 1543881561348 q^{14} + 9712801855841 q^{16} - 13851876239906 q^{17} - 2799887874003 q^{18} + 26858848298644 q^{19} - 93155602961484 q^{21} + 93991312008688 q^{22} - 75776598293952 q^{23} - 84097055362833 q^{24} + 16\!\cdots\!86 q^{26}+ \cdots + 29\!\cdots\!00 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6395796x - 2792983104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 656\nu - 4263646 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 656\beta _1 + 4263646 \) 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
−2272.56
−451.072
2724.63
−2540.56 −59049.0 4.35728e6 0 1.50017e8 −4.55015e8 −5.74199e9 3.48678e9 0
1.2 −719.072 −59049.0 −1.58009e6 0 4.24605e7 1.45023e9 2.64420e9 3.48678e9 0
1.3 2456.63 −59049.0 3.93788e6 0 −1.45062e8 5.82386e8 4.52198e9 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.e 3
5.b even 2 1 15.22.a.d 3
5.c odd 4 2 75.22.b.e 6
15.d odd 2 1 45.22.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.d 3 5.b even 2 1
45.22.a.b 3 15.d odd 2 1
75.22.a.e 3 1.a even 1 1 trivial
75.22.b.e 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 803T_{2}^{2} - 6180860T_{2} - 4487879424 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + \cdots - 4487879424 \) Copy content Toggle raw display
$3$ \( (T + 59049)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 38\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 12\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 18\!\cdots\!60 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 62\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 96\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 67\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 29\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 61\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
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