Properties

Label 150.2.a.a
Level $150$
Weight $2$
Character orbit 150.a
Self dual yes
Analytic conductor $1.198$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} + 2 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + q^{6} + 2 q^{7} - q^{8} + q^{9} + 2 q^{11} - q^{12} + 6 q^{13} - 2 q^{14} + q^{16} + 2 q^{17} - q^{18} - 2 q^{21} - 2 q^{22} - 4 q^{23} + q^{24} - 6 q^{26} - q^{27} + 2 q^{28} - 8 q^{31} - q^{32} - 2 q^{33} - 2 q^{34} + q^{36} + 2 q^{37} - 6 q^{39} + 2 q^{41} + 2 q^{42} - 4 q^{43} + 2 q^{44} + 4 q^{46} - 8 q^{47} - q^{48} - 3 q^{49} - 2 q^{51} + 6 q^{52} + 6 q^{53} + q^{54} - 2 q^{56} + 10 q^{59} + 2 q^{61} + 8 q^{62} + 2 q^{63} + q^{64} + 2 q^{66} - 8 q^{67} + 2 q^{68} + 4 q^{69} + 12 q^{71} - q^{72} - 4 q^{73} - 2 q^{74} + 4 q^{77} + 6 q^{78} + q^{81} - 2 q^{82} - 4 q^{83} - 2 q^{84} + 4 q^{86} - 2 q^{88} - 10 q^{89} + 12 q^{91} - 4 q^{92} + 8 q^{93} + 8 q^{94} + q^{96} - 8 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100}) \) 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
−1.00000 −1.00000 1.00000 0 1.00000 2.00000 −1.00000 1.00000 0
\(n\): 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 150.2.a.a 1
3.b odd 2 1 450.2.a.f 1
4.b odd 2 1 1200.2.a.m 1
5.b even 2 1 150.2.a.c 1
5.c odd 4 2 30.2.c.a 2
7.b odd 2 1 7350.2.a.bg 1
8.b even 2 1 4800.2.a.cg 1
8.d odd 2 1 4800.2.a.m 1
12.b even 2 1 3600.2.a.o 1
15.d odd 2 1 450.2.a.b 1
15.e even 4 2 90.2.c.a 2
20.d odd 2 1 1200.2.a.g 1
20.e even 4 2 240.2.f.a 2
35.c odd 2 1 7350.2.a.cc 1
35.f even 4 2 1470.2.g.g 2
35.k even 12 4 1470.2.n.a 4
35.l odd 12 4 1470.2.n.h 4
40.e odd 2 1 4800.2.a.cj 1
40.f even 2 1 4800.2.a.l 1
40.i odd 4 2 960.2.f.h 2
40.k even 4 2 960.2.f.i 2
45.k odd 12 4 810.2.i.e 4
45.l even 12 4 810.2.i.b 4
60.h even 2 1 3600.2.a.bg 1
60.l odd 4 2 720.2.f.f 2
80.i odd 4 2 3840.2.d.y 2
80.j even 4 2 3840.2.d.x 2
80.s even 4 2 3840.2.d.j 2
80.t odd 4 2 3840.2.d.g 2
120.q odd 4 2 2880.2.f.c 2
120.w even 4 2 2880.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 5.c odd 4 2
90.2.c.a 2 15.e even 4 2
150.2.a.a 1 1.a even 1 1 trivial
150.2.a.c 1 5.b even 2 1
240.2.f.a 2 20.e even 4 2
450.2.a.b 1 15.d odd 2 1
450.2.a.f 1 3.b odd 2 1
720.2.f.f 2 60.l odd 4 2
810.2.i.b 4 45.l even 12 4
810.2.i.e 4 45.k odd 12 4
960.2.f.h 2 40.i odd 4 2
960.2.f.i 2 40.k even 4 2
1200.2.a.g 1 20.d odd 2 1
1200.2.a.m 1 4.b odd 2 1
1470.2.g.g 2 35.f even 4 2
1470.2.n.a 4 35.k even 12 4
1470.2.n.h 4 35.l odd 12 4
2880.2.f.c 2 120.q odd 4 2
2880.2.f.e 2 120.w even 4 2
3600.2.a.o 1 12.b even 2 1
3600.2.a.bg 1 60.h even 2 1
3840.2.d.g 2 80.t odd 4 2
3840.2.d.j 2 80.s even 4 2
3840.2.d.x 2 80.j even 4 2
3840.2.d.y 2 80.i odd 4 2
4800.2.a.l 1 40.f even 2 1
4800.2.a.m 1 8.d odd 2 1
4800.2.a.cg 1 8.b even 2 1
4800.2.a.cj 1 40.e odd 2 1
7350.2.a.bg 1 7.b odd 2 1
7350.2.a.cc 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(150))\). Copy content Toggle raw display

Hecke characteristic polynomials

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