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

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

Newform invariants

Self dual: no
Analytic conductor: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
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

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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -8 \beta_{2} q^{4} -6 \beta_{3} q^{7} + 27 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -8 \beta_{2} q^{4} -6 \beta_{3} q^{7} + 27 \beta_{2} q^{9} -8 \beta_{3} q^{12} + 12 \beta_{1} q^{13} -64 q^{16} -56 \beta_{2} q^{19} + 162 q^{21} + 27 \beta_{3} q^{27} -48 \beta_{1} q^{28} -308 q^{31} + 216 q^{36} + 84 \beta_{3} q^{37} + 324 \beta_{2} q^{39} + 42 \beta_{1} q^{43} -64 \beta_{1} q^{48} -629 \beta_{2} q^{49} -96 \beta_{3} q^{52} -56 \beta_{3} q^{57} + 182 q^{61} + 162 \beta_{1} q^{63} + 512 \beta_{2} q^{64} -126 \beta_{3} q^{67} + 72 \beta_{1} q^{73} -448 q^{76} + 884 \beta_{2} q^{79} -729 q^{81} -1296 \beta_{2} q^{84} + 1944 q^{91} -308 \beta_{1} q^{93} + 264 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 256q^{16} + 648q^{21} - 1232q^{31} + 864q^{36} + 728q^{61} - 1792q^{76} - 2916q^{81} + 7776q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/3\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)

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 −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 \(S_{4}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

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