Properties

Label 75.2.e.b
Level $75$
Weight $2$
Character orbit 75.e
Analytic conductor $0.599$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + \beta_{3} q^{7} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + \beta_{3} q^{7} + 3 \beta_{2} q^{9} - 2 \beta_{3} q^{12} - 3 \beta_1 q^{13} - 4 q^{16} + \beta_{2} q^{19} - 3 q^{21} + 3 \beta_{3} q^{27} + 2 \beta_1 q^{28} + 7 q^{31} + 6 q^{36} - 4 \beta_{3} q^{37} - 9 \beta_{2} q^{39} + 7 \beta_1 q^{43} - 4 \beta_1 q^{48} + 4 \beta_{2} q^{49} + 6 \beta_{3} q^{52} + \beta_{3} q^{57} - 13 q^{61} - 3 \beta_1 q^{63} + 8 \beta_{2} q^{64} - 9 \beta_{3} q^{67} - 8 \beta_1 q^{73} + 2 q^{76} - 4 \beta_{2} q^{79} - 9 q^{81} + 6 \beta_{2} q^{84} + 9 q^{91} + 7 \beta_1 q^{93} + 11 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{16} - 12 q^{21} + 28 q^{31} + 24 q^{36} - 52 q^{61} + 8 q^{76} - 36 q^{81} + 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

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

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 −1.22474 + 1.22474i 2.00000i 0 0 1.22474 + 1.22474i 0 3.00000i 0
32.2 0 1.22474 1.22474i 2.00000i 0 0 −1.22474 1.22474i 0 3.00000i 0
68.1 0 −1.22474 1.22474i 2.00000i 0 0 1.22474 1.22474i 0 3.00000i 0
68.2 0 1.22474 + 1.22474i 2.00000i 0 0 −1.22474 + 1.22474i 0 3.00000i 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 75.2.e.b 4
3.b odd 2 1 CM 75.2.e.b 4
4.b odd 2 1 1200.2.v.f 4
5.b even 2 1 inner 75.2.e.b 4
5.c odd 4 2 inner 75.2.e.b 4
12.b even 2 1 1200.2.v.f 4
15.d odd 2 1 inner 75.2.e.b 4
15.e even 4 2 inner 75.2.e.b 4
20.d odd 2 1 1200.2.v.f 4
20.e even 4 2 1200.2.v.f 4
60.h even 2 1 1200.2.v.f 4
60.l odd 4 2 1200.2.v.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.e.b 4 1.a even 1 1 trivial
75.2.e.b 4 3.b odd 2 1 CM
75.2.e.b 4 5.b even 2 1 inner
75.2.e.b 4 5.c odd 4 2 inner
75.2.e.b 4 15.d odd 2 1 inner
75.2.e.b 4 15.e even 4 2 inner
1200.2.v.f 4 4.b odd 2 1
1200.2.v.f 4 12.b even 2 1
1200.2.v.f 4 20.d odd 2 1
1200.2.v.f 4 20.e even 4 2
1200.2.v.f 4 60.h even 2 1
1200.2.v.f 4 60.l odd 4 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 9 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 729 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 7)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 2304 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 21609 \) 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
$61$ \( (T + 13)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 59049 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 36864 \) Copy content Toggle raw display
$79$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 131769 \) Copy content Toggle raw display
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