Properties

Label 75.4.e.b
Level $75$
Weight $4$
Character orbit 75.e
Analytic conductor $4.425$
Analytic rank $0$
Dimension $4$
CM discriminant -15
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + 19 \beta_{2} q^{4} + 27 q^{6} + 11 \beta_{3} q^{8} -27 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + 19 \beta_{2} q^{4} + 27 q^{6} + 11 \beta_{3} q^{8} -27 \beta_{2} q^{9} + 19 \beta_{1} q^{12} -145 q^{16} -4 \beta_{1} q^{17} -27 \beta_{3} q^{18} -164 \beta_{2} q^{19} + 38 \beta_{3} q^{23} + 297 \beta_{2} q^{24} -27 \beta_{1} q^{27} + 232 q^{31} -57 \beta_{1} q^{32} -108 \beta_{2} q^{34} + 513 q^{36} -164 \beta_{3} q^{38} -1026 q^{46} + 66 \beta_{1} q^{47} + 145 \beta_{3} q^{48} + 343 \beta_{2} q^{49} -108 q^{51} + 88 \beta_{3} q^{53} -729 \beta_{2} q^{54} -164 \beta_{1} q^{57} -358 q^{61} + 232 \beta_{1} q^{62} -379 \beta_{2} q^{64} -76 \beta_{3} q^{68} + 1026 \beta_{2} q^{69} + 297 \beta_{1} q^{72} + 3116 q^{76} -304 \beta_{2} q^{79} -729 q^{81} + 158 \beta_{3} q^{83} -722 \beta_{1} q^{92} -232 \beta_{3} q^{93} + 1782 \beta_{2} q^{94} -1539 q^{96} + 343 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 108q^{6} + O(q^{10}) \) \( 4q + 108q^{6} - 580q^{16} + 928q^{31} + 2052q^{36} - 4104q^{46} - 432q^{51} - 1432q^{61} + 12464q^{76} - 2916q^{81} - 6156q^{96} + O(q^{100}) \)

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
−3.67423 + 3.67423i −3.67423 3.67423i 19.0000i 0 27.0000 0 40.4166 + 40.4166i 27.0000i 0
32.2 3.67423 3.67423i 3.67423 + 3.67423i 19.0000i 0 27.0000 0 −40.4166 40.4166i 27.0000i 0
68.1 −3.67423 3.67423i −3.67423 + 3.67423i 19.0000i 0 27.0000 0 40.4166 40.4166i 27.0000i 0
68.2 3.67423 + 3.67423i 3.67423 3.67423i 19.0000i 0 27.0000 0 −40.4166 + 40.4166i 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 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.b 4
3.b odd 2 1 inner 75.4.e.b 4
5.b even 2 1 inner 75.4.e.b 4
5.c odd 4 2 inner 75.4.e.b 4
15.d odd 2 1 CM 75.4.e.b 4
15.e even 4 2 inner 75.4.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.e.b 4 1.a even 1 1 trivial
75.4.e.b 4 3.b odd 2 1 inner
75.4.e.b 4 5.b even 2 1 inner
75.4.e.b 4 5.c odd 4 2 inner
75.4.e.b 4 15.d odd 2 1 CM
75.4.e.b 4 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 729 \) acting on \(S_{4}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 729 + T^{4} \)
$3$ \( 729 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( 186624 + T^{4} \)
$19$ \( ( 26896 + T^{2} )^{2} \)
$23$ \( 1520064144 + T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -232 + T )^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( 13832582544 + T^{4} \)
$53$ \( 43717791744 + T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 358 + T )^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 92416 + T^{2} )^{2} \)
$83$ \( 454313744784 + T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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