Properties

Label 30.2.a.a
Level $30$
Weight $2$
Character orbit 30.a
Self dual yes
Analytic conductor $0.240$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.239551206064\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} + 4 q^{14} - q^{15} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} - q^{20} - 4 q^{21} - q^{24} + q^{25} - 2 q^{26} + q^{27} - 4 q^{28} - 6 q^{29} + q^{30} + 8 q^{31} - q^{32} - 6 q^{34} + 4 q^{35} + q^{36} + 2 q^{37} + 4 q^{38} + 2 q^{39} + q^{40} - 6 q^{41} + 4 q^{42} - 4 q^{43} - q^{45} + q^{48} + 9 q^{49} - q^{50} + 6 q^{51} + 2 q^{52} - 6 q^{53} - q^{54} + 4 q^{56} - 4 q^{57} + 6 q^{58} - q^{60} - 10 q^{61} - 8 q^{62} - 4 q^{63} + q^{64} - 2 q^{65} - 4 q^{67} + 6 q^{68} - 4 q^{70} - q^{72} + 2 q^{73} - 2 q^{74} + q^{75} - 4 q^{76} - 2 q^{78} + 8 q^{79} - q^{80} + q^{81} + 6 q^{82} + 12 q^{83} - 4 q^{84} - 6 q^{85} + 4 q^{86} - 6 q^{87} + 18 q^{89} + q^{90} - 8 q^{91} + 8 q^{93} + 4 q^{95} - q^{96} + 2 q^{97} - 9 q^{98}+O(q^{100}) \) 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
−1.00000 1.00000 1.00000 −1.00000 −1.00000 −4.00000 −1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)

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 30.2.a.a 1
3.b odd 2 1 90.2.a.c 1
4.b odd 2 1 240.2.a.b 1
5.b even 2 1 150.2.a.b 1
5.c odd 4 2 150.2.c.a 2
7.b odd 2 1 1470.2.a.d 1
7.c even 3 2 1470.2.i.o 2
7.d odd 6 2 1470.2.i.q 2
8.b even 2 1 960.2.a.e 1
8.d odd 2 1 960.2.a.p 1
9.c even 3 2 810.2.e.l 2
9.d odd 6 2 810.2.e.b 2
11.b odd 2 1 3630.2.a.w 1
12.b even 2 1 720.2.a.j 1
13.b even 2 1 5070.2.a.w 1
13.d odd 4 2 5070.2.b.k 2
15.d odd 2 1 450.2.a.d 1
15.e even 4 2 450.2.c.b 2
16.e even 4 2 3840.2.k.y 2
16.f odd 4 2 3840.2.k.f 2
17.b even 2 1 8670.2.a.g 1
20.d odd 2 1 1200.2.a.k 1
20.e even 4 2 1200.2.f.e 2
21.c even 2 1 4410.2.a.z 1
24.f even 2 1 2880.2.a.q 1
24.h odd 2 1 2880.2.a.a 1
35.c odd 2 1 7350.2.a.ct 1
40.e odd 2 1 4800.2.a.d 1
40.f even 2 1 4800.2.a.cq 1
40.i odd 4 2 4800.2.f.p 2
40.k even 4 2 4800.2.f.w 2
60.h even 2 1 3600.2.a.f 1
60.l odd 4 2 3600.2.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 1.a even 1 1 trivial
90.2.a.c 1 3.b odd 2 1
150.2.a.b 1 5.b even 2 1
150.2.c.a 2 5.c odd 4 2
240.2.a.b 1 4.b odd 2 1
450.2.a.d 1 15.d odd 2 1
450.2.c.b 2 15.e even 4 2
720.2.a.j 1 12.b even 2 1
810.2.e.b 2 9.d odd 6 2
810.2.e.l 2 9.c even 3 2
960.2.a.e 1 8.b even 2 1
960.2.a.p 1 8.d odd 2 1
1200.2.a.k 1 20.d odd 2 1
1200.2.f.e 2 20.e even 4 2
1470.2.a.d 1 7.b odd 2 1
1470.2.i.o 2 7.c even 3 2
1470.2.i.q 2 7.d odd 6 2
2880.2.a.a 1 24.h odd 2 1
2880.2.a.q 1 24.f even 2 1
3600.2.a.f 1 60.h even 2 1
3600.2.f.i 2 60.l odd 4 2
3630.2.a.w 1 11.b odd 2 1
3840.2.k.f 2 16.f odd 4 2
3840.2.k.y 2 16.e even 4 2
4410.2.a.z 1 21.c even 2 1
4800.2.a.d 1 40.e odd 2 1
4800.2.a.cq 1 40.f even 2 1
4800.2.f.p 2 40.i odd 4 2
4800.2.f.w 2 40.k even 4 2
5070.2.a.w 1 13.b even 2 1
5070.2.b.k 2 13.d odd 4 2
7350.2.a.ct 1 35.c odd 2 1
8670.2.a.g 1 17.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(30))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 18 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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