Properties

Label 75.3.f.b
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.

\(f(q)\) \(=\) \( q + 2 \beta_{1} q^{2} -\beta_{3} q^{3} + 8 \beta_{2} q^{4} + 6 q^{6} -5 \beta_{1} q^{7} + 8 \beta_{3} q^{8} -3 \beta_{2} q^{9} +O(q^{10})\) \( q + 2 \beta_{1} q^{2} -\beta_{3} q^{3} + 8 \beta_{2} q^{4} + 6 q^{6} -5 \beta_{1} q^{7} + 8 \beta_{3} q^{8} -3 \beta_{2} q^{9} + 6 q^{11} + 8 \beta_{1} q^{12} + 3 \beta_{3} q^{13} -30 \beta_{2} q^{14} -16 q^{16} -14 \beta_{1} q^{17} -6 \beta_{3} q^{18} + 23 \beta_{2} q^{19} -15 q^{21} + 12 \beta_{1} q^{22} -10 \beta_{3} q^{23} + 24 \beta_{2} q^{24} -18 q^{26} -3 \beta_{1} q^{27} -40 \beta_{3} q^{28} + 6 \beta_{2} q^{29} + 25 q^{31} -6 \beta_{3} q^{33} -84 \beta_{2} q^{34} + 24 q^{36} + 20 \beta_{1} q^{37} + 46 \beta_{3} q^{38} + 9 \beta_{2} q^{39} -60 q^{41} -30 \beta_{1} q^{42} + 49 \beta_{3} q^{43} + 48 \beta_{2} q^{44} + 60 q^{46} -6 \beta_{1} q^{47} + 16 \beta_{3} q^{48} + 26 \beta_{2} q^{49} -42 q^{51} -24 \beta_{1} q^{52} -20 \beta_{3} q^{53} -18 \beta_{2} q^{54} + 120 q^{56} + 23 \beta_{1} q^{57} + 12 \beta_{3} q^{58} -18 \beta_{2} q^{59} -37 q^{61} + 50 \beta_{1} q^{62} + 15 \beta_{3} q^{63} + 64 \beta_{2} q^{64} + 36 q^{66} + 21 \beta_{1} q^{67} -112 \beta_{3} q^{68} -30 \beta_{2} q^{69} + 132 q^{71} + 24 \beta_{1} q^{72} -20 \beta_{3} q^{73} + 120 \beta_{2} q^{74} -184 q^{76} -30 \beta_{1} q^{77} + 18 \beta_{3} q^{78} + 10 \beta_{2} q^{79} -9 q^{81} -120 \beta_{1} q^{82} + 2 \beta_{3} q^{83} -120 \beta_{2} q^{84} -294 q^{86} + 6 \beta_{1} q^{87} + 48 \beta_{3} q^{88} -132 \beta_{2} q^{89} + 45 q^{91} + 80 \beta_{1} q^{92} -25 \beta_{3} q^{93} -36 \beta_{2} q^{94} -19 \beta_{1} q^{97} + 52 \beta_{3} q^{98} -18 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 24q^{6} + O(q^{10}) \) \( 4q + 24q^{6} + 24q^{11} - 64q^{16} - 60q^{21} - 72q^{26} + 100q^{31} + 96q^{36} - 240q^{41} + 240q^{46} - 168q^{51} + 480q^{56} - 148q^{61} + 144q^{66} + 528q^{71} - 736q^{76} - 36q^{81} - 1176q^{86} + 180q^{91} + O(q^{100}) \)

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
−2.44949 2.44949i −1.22474 + 1.22474i 8.00000i 0 6.00000 6.12372 + 6.12372i 9.79796 9.79796i 3.00000i 0
7.2 2.44949 + 2.44949i 1.22474 1.22474i 8.00000i 0 6.00000 −6.12372 6.12372i −9.79796 + 9.79796i 3.00000i 0
43.1 −2.44949 + 2.44949i −1.22474 1.22474i 8.00000i 0 6.00000 6.12372 6.12372i 9.79796 + 9.79796i 3.00000i 0
43.2 2.44949 2.44949i 1.22474 + 1.22474i 8.00000i 0 6.00000 −6.12372 + 6.12372i −9.79796 9.79796i 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.b 4
3.b odd 2 1 225.3.g.b 4
4.b odd 2 1 1200.3.bg.g 4
5.b even 2 1 inner 75.3.f.b 4
5.c odd 4 2 inner 75.3.f.b 4
15.d odd 2 1 225.3.g.b 4
15.e even 4 2 225.3.g.b 4
20.d odd 2 1 1200.3.bg.g 4
20.e even 4 2 1200.3.bg.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.f.b 4 1.a even 1 1 trivial
75.3.f.b 4 5.b even 2 1 inner
75.3.f.b 4 5.c odd 4 2 inner
225.3.g.b 4 3.b odd 2 1
225.3.g.b 4 15.d odd 2 1
225.3.g.b 4 15.e even 4 2
1200.3.bg.g 4 4.b odd 2 1
1200.3.bg.g 4 20.d odd 2 1
1200.3.bg.g 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 144 \) acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 144 + T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 5625 + T^{4} \)
$11$ \( ( -6 + T )^{4} \)
$13$ \( 729 + T^{4} \)
$17$ \( 345744 + T^{4} \)
$19$ \( ( 529 + T^{2} )^{2} \)
$23$ \( 90000 + T^{4} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( -25 + T )^{4} \)
$37$ \( 1440000 + T^{4} \)
$41$ \( ( 60 + T )^{4} \)
$43$ \( 51883209 + T^{4} \)
$47$ \( 11664 + T^{4} \)
$53$ \( 1440000 + T^{4} \)
$59$ \( ( 324 + T^{2} )^{2} \)
$61$ \( ( 37 + T )^{4} \)
$67$ \( 1750329 + T^{4} \)
$71$ \( ( -132 + T )^{4} \)
$73$ \( 1440000 + T^{4} \)
$79$ \( ( 100 + T^{2} )^{2} \)
$83$ \( 144 + T^{4} \)
$89$ \( ( 17424 + T^{2} )^{2} \)
$97$ \( 1172889 + T^{4} \)
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