Properties

Label 75.4.e.a
Level $75$
Weight $4$
Character orbit 75.e
Analytic conductor $4.425$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,4,Mod(32,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.32");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 8 \beta_{2} q^{4} - 6 \beta_{3} q^{7} + 27 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 8 \beta_{2} q^{4} - 6 \beta_{3} q^{7} + 27 \beta_{2} q^{9} - 8 \beta_{3} q^{12} + 12 \beta_1 q^{13} - 64 q^{16} - 56 \beta_{2} q^{19} + 162 q^{21} + 27 \beta_{3} q^{27} - 48 \beta_1 q^{28} - 308 q^{31} + 216 q^{36} + 84 \beta_{3} q^{37} + 324 \beta_{2} q^{39} + 42 \beta_1 q^{43} - 64 \beta_1 q^{48} - 629 \beta_{2} q^{49} - 96 \beta_{3} q^{52} - 56 \beta_{3} q^{57} + 182 q^{61} + 162 \beta_1 q^{63} + 512 \beta_{2} q^{64} - 126 \beta_{3} q^{67} + 72 \beta_1 q^{73} - 448 q^{76} + 884 \beta_{2} q^{79} - 729 q^{81} - 1296 \beta_{2} q^{84} + 1944 q^{91} - 308 \beta_1 q^{93} + 264 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 256 q^{16} + 648 q^{21} - 1232 q^{31} + 864 q^{36} + 728 q^{61} - 1792 q^{76} - 2916 q^{81} + 7776 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 −3.67423 + 3.67423i 8.00000i 0 0 −22.0454 22.0454i 0 27.0000i 0
32.2 0 3.67423 3.67423i 8.00000i 0 0 22.0454 + 22.0454i 0 27.0000i 0
68.1 0 −3.67423 3.67423i 8.00000i 0 0 −22.0454 + 22.0454i 0 27.0000i 0
68.2 0 3.67423 + 3.67423i 8.00000i 0 0 22.0454 22.0454i 0 27.0000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.e.a 4
3.b odd 2 1 CM 75.4.e.a 4
5.b even 2 1 inner 75.4.e.a 4
5.c odd 4 2 inner 75.4.e.a 4
15.d odd 2 1 inner 75.4.e.a 4
15.e even 4 2 inner 75.4.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.e.a 4 1.a even 1 1 trivial
75.4.e.a 4 3.b odd 2 1 CM
75.4.e.a 4 5.b even 2 1 inner
75.4.e.a 4 5.c odd 4 2 inner
75.4.e.a 4 15.d odd 2 1 inner
75.4.e.a 4 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 729 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 944784 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 15116544 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3136)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 308)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 36294822144 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 2268426384 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T - 182)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 183742537104 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 19591041024 \) Copy content Toggle raw display
$79$ \( (T^{2} + 781456)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 3541141131264 \) Copy content Toggle raw display
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