Properties

Label 75.4.a.a
Level $75$
Weight $4$
Character orbit 75.a
Self dual yes
Analytic conductor $4.425$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.42514325043\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{2} + 3 q^{3} + q^{4} - 9 q^{6} - 20 q^{7} + 21 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} + 3 q^{3} + q^{4} - 9 q^{6} - 20 q^{7} + 21 q^{8} + 9 q^{9} - 24 q^{11} + 3 q^{12} - 74 q^{13} + 60 q^{14} - 71 q^{16} - 54 q^{17} - 27 q^{18} - 124 q^{19} - 60 q^{21} + 72 q^{22} + 120 q^{23} + 63 q^{24} + 222 q^{26} + 27 q^{27} - 20 q^{28} - 78 q^{29} + 200 q^{31} + 45 q^{32} - 72 q^{33} + 162 q^{34} + 9 q^{36} + 70 q^{37} + 372 q^{38} - 222 q^{39} + 330 q^{41} + 180 q^{42} - 92 q^{43} - 24 q^{44} - 360 q^{46} + 24 q^{47} - 213 q^{48} + 57 q^{49} - 162 q^{51} - 74 q^{52} - 450 q^{53} - 81 q^{54} - 420 q^{56} - 372 q^{57} + 234 q^{58} + 24 q^{59} - 322 q^{61} - 600 q^{62} - 180 q^{63} + 433 q^{64} + 216 q^{66} + 196 q^{67} - 54 q^{68} + 360 q^{69} - 288 q^{71} + 189 q^{72} + 430 q^{73} - 210 q^{74} - 124 q^{76} + 480 q^{77} + 666 q^{78} - 520 q^{79} + 81 q^{81} - 990 q^{82} - 156 q^{83} - 60 q^{84} + 276 q^{86} - 234 q^{87} - 504 q^{88} + 1026 q^{89} + 1480 q^{91} + 120 q^{92} + 600 q^{93} - 72 q^{94} + 135 q^{96} + 286 q^{97} - 171 q^{98} - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−3.00000 3.00000 1.00000 0 −9.00000 −20.0000 21.0000 9.00000 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 75.4.a.a 1
3.b odd 2 1 225.4.a.g 1
4.b odd 2 1 1200.4.a.o 1
5.b even 2 1 15.4.a.b 1
5.c odd 4 2 75.4.b.a 2
15.d odd 2 1 45.4.a.b 1
15.e even 4 2 225.4.b.d 2
20.d odd 2 1 240.4.a.f 1
20.e even 4 2 1200.4.f.m 2
35.c odd 2 1 735.4.a.i 1
40.e odd 2 1 960.4.a.l 1
40.f even 2 1 960.4.a.bi 1
45.h odd 6 2 405.4.e.k 2
45.j even 6 2 405.4.e.d 2
55.d odd 2 1 1815.4.a.a 1
60.h even 2 1 720.4.a.r 1
105.g even 2 1 2205.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 5.b even 2 1
45.4.a.b 1 15.d odd 2 1
75.4.a.a 1 1.a even 1 1 trivial
75.4.b.a 2 5.c odd 4 2
225.4.a.g 1 3.b odd 2 1
225.4.b.d 2 15.e even 4 2
240.4.a.f 1 20.d odd 2 1
405.4.e.d 2 45.j even 6 2
405.4.e.k 2 45.h odd 6 2
720.4.a.r 1 60.h even 2 1
735.4.a.i 1 35.c odd 2 1
960.4.a.l 1 40.e odd 2 1
960.4.a.bi 1 40.f even 2 1
1200.4.a.o 1 4.b odd 2 1
1200.4.f.m 2 20.e even 4 2
1815.4.a.a 1 55.d odd 2 1
2205.4.a.c 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T + 24 \) Copy content Toggle raw display
$13$ \( T + 74 \) Copy content Toggle raw display
$17$ \( T + 54 \) Copy content Toggle raw display
$19$ \( T + 124 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T + 78 \) Copy content Toggle raw display
$31$ \( T - 200 \) Copy content Toggle raw display
$37$ \( T - 70 \) Copy content Toggle raw display
$41$ \( T - 330 \) Copy content Toggle raw display
$43$ \( T + 92 \) Copy content Toggle raw display
$47$ \( T - 24 \) Copy content Toggle raw display
$53$ \( T + 450 \) Copy content Toggle raw display
$59$ \( T - 24 \) Copy content Toggle raw display
$61$ \( T + 322 \) Copy content Toggle raw display
$67$ \( T - 196 \) Copy content Toggle raw display
$71$ \( T + 288 \) Copy content Toggle raw display
$73$ \( T - 430 \) Copy content Toggle raw display
$79$ \( T + 520 \) Copy content Toggle raw display
$83$ \( T + 156 \) Copy content Toggle raw display
$89$ \( T - 1026 \) Copy content Toggle raw display
$97$ \( T - 286 \) Copy content Toggle raw display
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