Properties

Label 75.2.a.a
Level $75$
Weight $2$
Character orbit 75.a
Self dual yes
Analytic conductor $0.599$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + 3 q^{7} + q^{9} + O(q^{10}) \) \( q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + 3 q^{7} + q^{9} + 2 q^{11} + 2 q^{12} - q^{13} - 6 q^{14} - 4 q^{16} - 2 q^{17} - 2 q^{18} - 5 q^{19} + 3 q^{21} - 4 q^{22} - 6 q^{23} + 2 q^{26} + q^{27} + 6 q^{28} + 10 q^{29} - 3 q^{31} + 8 q^{32} + 2 q^{33} + 4 q^{34} + 2 q^{36} - 2 q^{37} + 10 q^{38} - q^{39} - 8 q^{41} - 6 q^{42} - q^{43} + 4 q^{44} + 12 q^{46} - 2 q^{47} - 4 q^{48} + 2 q^{49} - 2 q^{51} - 2 q^{52} + 4 q^{53} - 2 q^{54} - 5 q^{57} - 20 q^{58} - 10 q^{59} + 7 q^{61} + 6 q^{62} + 3 q^{63} - 8 q^{64} - 4 q^{66} + 3 q^{67} - 4 q^{68} - 6 q^{69} - 8 q^{71} + 14 q^{73} + 4 q^{74} - 10 q^{76} + 6 q^{77} + 2 q^{78} + q^{81} + 16 q^{82} - 6 q^{83} + 6 q^{84} + 2 q^{86} + 10 q^{87} - 3 q^{91} - 12 q^{92} - 3 q^{93} + 4 q^{94} + 8 q^{96} - 17 q^{97} - 4 q^{98} + 2 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 1.00000 2.00000 0 −2.00000 3.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.2.a.a 1
3.b odd 2 1 225.2.a.e 1
4.b odd 2 1 1200.2.a.c 1
5.b even 2 1 75.2.a.c yes 1
5.c odd 4 2 75.2.b.a 2
7.b odd 2 1 3675.2.a.b 1
8.b even 2 1 4800.2.a.bb 1
8.d odd 2 1 4800.2.a.br 1
11.b odd 2 1 9075.2.a.s 1
12.b even 2 1 3600.2.a.j 1
15.d odd 2 1 225.2.a.a 1
15.e even 4 2 225.2.b.a 2
20.d odd 2 1 1200.2.a.p 1
20.e even 4 2 1200.2.f.d 2
35.c odd 2 1 3675.2.a.q 1
40.e odd 2 1 4800.2.a.be 1
40.f even 2 1 4800.2.a.bq 1
40.i odd 4 2 4800.2.f.l 2
40.k even 4 2 4800.2.f.y 2
55.d odd 2 1 9075.2.a.a 1
60.h even 2 1 3600.2.a.bk 1
60.l odd 4 2 3600.2.f.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 1.a even 1 1 trivial
75.2.a.c yes 1 5.b even 2 1
75.2.b.a 2 5.c odd 4 2
225.2.a.a 1 15.d odd 2 1
225.2.a.e 1 3.b odd 2 1
225.2.b.a 2 15.e even 4 2
1200.2.a.c 1 4.b odd 2 1
1200.2.a.p 1 20.d odd 2 1
1200.2.f.d 2 20.e even 4 2
3600.2.a.j 1 12.b even 2 1
3600.2.a.bk 1 60.h even 2 1
3600.2.f.p 2 60.l odd 4 2
3675.2.a.b 1 7.b odd 2 1
3675.2.a.q 1 35.c odd 2 1
4800.2.a.bb 1 8.b even 2 1
4800.2.a.be 1 40.e odd 2 1
4800.2.a.bq 1 40.f even 2 1
4800.2.a.br 1 8.d odd 2 1
4800.2.f.l 2 40.i odd 4 2
4800.2.f.y 2 40.k even 4 2
9075.2.a.a 1 55.d odd 2 1
9075.2.a.s 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(75))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -3 + T \)
$11$ \( -2 + T \)
$13$ \( 1 + T \)
$17$ \( 2 + T \)
$19$ \( 5 + T \)
$23$ \( 6 + T \)
$29$ \( -10 + T \)
$31$ \( 3 + T \)
$37$ \( 2 + T \)
$41$ \( 8 + T \)
$43$ \( 1 + T \)
$47$ \( 2 + T \)
$53$ \( -4 + T \)
$59$ \( 10 + T \)
$61$ \( -7 + T \)
$67$ \( -3 + T \)
$71$ \( 8 + T \)
$73$ \( -14 + T \)
$79$ \( T \)
$83$ \( 6 + T \)
$89$ \( T \)
$97$ \( 17 + T \)
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