Properties

Label 75.3.f.a
Level $75$
Weight $3$
Character orbit 75.f
Analytic conductor $2.044$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.04360198270\)
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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} -\beta_{2} q^{4} + 3 q^{6} + 6 \beta_{1} q^{7} -5 \beta_{3} q^{8} -3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} -\beta_{2} q^{4} + 3 q^{6} + 6 \beta_{1} q^{7} -5 \beta_{3} q^{8} -3 \beta_{2} q^{9} -18 q^{11} -\beta_{1} q^{12} + 6 \beta_{3} q^{13} + 18 \beta_{2} q^{14} + 11 q^{16} -4 \beta_{1} q^{17} -3 \beta_{3} q^{18} -10 \beta_{2} q^{19} + 18 q^{21} -18 \beta_{1} q^{22} + 16 \beta_{3} q^{23} -15 \beta_{2} q^{24} -18 q^{26} -3 \beta_{1} q^{27} -6 \beta_{3} q^{28} + 22 q^{31} -9 \beta_{1} q^{32} + 18 \beta_{3} q^{33} -12 \beta_{2} q^{34} -3 q^{36} + 6 \beta_{1} q^{37} -10 \beta_{3} q^{38} + 18 \beta_{2} q^{39} -18 q^{41} + 18 \beta_{1} q^{42} -24 \beta_{3} q^{43} + 18 \beta_{2} q^{44} -48 q^{46} + 36 \beta_{1} q^{47} -11 \beta_{3} q^{48} + 59 \beta_{2} q^{49} -12 q^{51} + 6 \beta_{1} q^{52} -4 \beta_{3} q^{53} -9 \beta_{2} q^{54} + 90 q^{56} -10 \beta_{1} q^{57} -90 \beta_{2} q^{59} + 2 q^{61} + 22 \beta_{1} q^{62} -18 \beta_{3} q^{63} -71 \beta_{2} q^{64} -54 q^{66} + 36 \beta_{1} q^{67} + 4 \beta_{3} q^{68} + 48 \beta_{2} q^{69} + 72 q^{71} -15 \beta_{1} q^{72} + 36 \beta_{3} q^{73} + 18 \beta_{2} q^{74} -10 q^{76} -108 \beta_{1} q^{77} + 18 \beta_{3} q^{78} + 70 \beta_{2} q^{79} -9 q^{81} -18 \beta_{1} q^{82} -44 \beta_{3} q^{83} -18 \beta_{2} q^{84} + 72 q^{86} + 90 \beta_{3} q^{88} -90 \beta_{2} q^{89} -108 q^{91} + 16 \beta_{1} q^{92} -22 \beta_{3} q^{93} + 108 \beta_{2} q^{94} -27 q^{96} -84 \beta_{1} q^{97} + 59 \beta_{3} q^{98} + 54 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{6} + O(q^{10}) \) \( 4q + 12q^{6} - 72q^{11} + 44q^{16} + 72q^{21} - 72q^{26} + 88q^{31} - 12q^{36} - 72q^{41} - 192q^{46} - 48q^{51} + 360q^{56} + 8q^{61} - 216q^{66} + 288q^{71} - 40q^{76} - 36q^{81} + 288q^{86} - 432q^{91} - 108q^{96} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i −1.22474 + 1.22474i 1.00000i 0 3.00000 −7.34847 7.34847i −6.12372 + 6.12372i 3.00000i 0
7.2 1.22474 + 1.22474i 1.22474 1.22474i 1.00000i 0 3.00000 7.34847 + 7.34847i 6.12372 6.12372i 3.00000i 0
43.1 −1.22474 + 1.22474i −1.22474 1.22474i 1.00000i 0 3.00000 −7.34847 + 7.34847i −6.12372 6.12372i 3.00000i 0
43.2 1.22474 1.22474i 1.22474 + 1.22474i 1.00000i 0 3.00000 7.34847 7.34847i 6.12372 + 6.12372i 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
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.f.a 4
3.b odd 2 1 225.3.g.f 4
4.b odd 2 1 1200.3.bg.j 4
5.b even 2 1 inner 75.3.f.a 4
5.c odd 4 2 inner 75.3.f.a 4
15.d odd 2 1 225.3.g.f 4
15.e even 4 2 225.3.g.f 4
20.d odd 2 1 1200.3.bg.j 4
20.e even 4 2 1200.3.bg.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.f.a 4 1.a even 1 1 trivial
75.3.f.a 4 5.b even 2 1 inner
75.3.f.a 4 5.c odd 4 2 inner
225.3.g.f 4 3.b odd 2 1
225.3.g.f 4 15.d odd 2 1
225.3.g.f 4 15.e even 4 2
1200.3.bg.j 4 4.b odd 2 1
1200.3.bg.j 4 20.d odd 2 1
1200.3.bg.j 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 9 \) acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 11664 + T^{4} \)
$11$ \( ( 18 + T )^{4} \)
$13$ \( 11664 + T^{4} \)
$17$ \( 2304 + T^{4} \)
$19$ \( ( 100 + T^{2} )^{2} \)
$23$ \( 589824 + T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -22 + T )^{4} \)
$37$ \( 11664 + T^{4} \)
$41$ \( ( 18 + T )^{4} \)
$43$ \( 2985984 + T^{4} \)
$47$ \( 15116544 + T^{4} \)
$53$ \( 2304 + T^{4} \)
$59$ \( ( 8100 + T^{2} )^{2} \)
$61$ \( ( -2 + T )^{4} \)
$67$ \( 15116544 + T^{4} \)
$71$ \( ( -72 + T )^{4} \)
$73$ \( 15116544 + T^{4} \)
$79$ \( ( 4900 + T^{2} )^{2} \)
$83$ \( 33732864 + T^{4} \)
$89$ \( ( 8100 + T^{2} )^{2} \)
$97$ \( 448084224 + T^{4} \)
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