Properties

Label 8670.2.a.g
Level $8670$
Weight $2$
Character orbit 8670.a
Self dual yes
Analytic conductor $69.230$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.2302985525\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 4q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 4q^{7} - q^{8} + q^{9} - q^{10} - q^{12} + 2q^{13} - 4q^{14} - q^{15} + q^{16} - q^{18} - 4q^{19} + q^{20} - 4q^{21} + q^{24} + q^{25} - 2q^{26} - q^{27} + 4q^{28} + 6q^{29} + q^{30} - 8q^{31} - q^{32} + 4q^{35} + q^{36} - 2q^{37} + 4q^{38} - 2q^{39} - q^{40} + 6q^{41} + 4q^{42} - 4q^{43} + q^{45} - q^{48} + 9q^{49} - q^{50} + 2q^{52} - 6q^{53} + q^{54} - 4q^{56} + 4q^{57} - 6q^{58} - q^{60} + 10q^{61} + 8q^{62} + 4q^{63} + q^{64} + 2q^{65} - 4q^{67} - 4q^{70} - q^{72} - 2q^{73} + 2q^{74} - q^{75} - 4q^{76} + 2q^{78} - 8q^{79} + q^{80} + q^{81} - 6q^{82} + 12q^{83} - 4q^{84} + 4q^{86} - 6q^{87} + 18q^{89} - q^{90} + 8q^{91} + 8q^{93} - 4q^{95} + q^{96} - 2q^{97} - 9q^{98} + O(q^{100}) \)

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
−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
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(17\) \(1\)

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 8670.2.a.g 1
17.b even 2 1 30.2.a.a 1
51.c odd 2 1 90.2.a.c 1
68.d odd 2 1 240.2.a.b 1
85.c even 2 1 150.2.a.b 1
85.g odd 4 2 150.2.c.a 2
119.d odd 2 1 1470.2.a.d 1
119.h odd 6 2 1470.2.i.q 2
119.j even 6 2 1470.2.i.o 2
136.e odd 2 1 960.2.a.p 1
136.h even 2 1 960.2.a.e 1
153.h even 6 2 810.2.e.l 2
153.i odd 6 2 810.2.e.b 2
187.b odd 2 1 3630.2.a.w 1
204.h even 2 1 720.2.a.j 1
221.b even 2 1 5070.2.a.w 1
221.g odd 4 2 5070.2.b.k 2
255.h odd 2 1 450.2.a.d 1
255.o even 4 2 450.2.c.b 2
272.k odd 4 2 3840.2.k.f 2
272.r even 4 2 3840.2.k.y 2
340.d odd 2 1 1200.2.a.k 1
340.r even 4 2 1200.2.f.e 2
357.c even 2 1 4410.2.a.z 1
408.b odd 2 1 2880.2.a.a 1
408.h even 2 1 2880.2.a.q 1
595.b odd 2 1 7350.2.a.ct 1
680.h even 2 1 4800.2.a.cq 1
680.k odd 2 1 4800.2.a.d 1
680.u even 4 2 4800.2.f.w 2
680.bi odd 4 2 4800.2.f.p 2
1020.b even 2 1 3600.2.a.f 1
1020.x 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 17.b even 2 1
90.2.a.c 1 51.c odd 2 1
150.2.a.b 1 85.c even 2 1
150.2.c.a 2 85.g odd 4 2
240.2.a.b 1 68.d odd 2 1
450.2.a.d 1 255.h odd 2 1
450.2.c.b 2 255.o even 4 2
720.2.a.j 1 204.h even 2 1
810.2.e.b 2 153.i odd 6 2
810.2.e.l 2 153.h even 6 2
960.2.a.e 1 136.h even 2 1
960.2.a.p 1 136.e odd 2 1
1200.2.a.k 1 340.d odd 2 1
1200.2.f.e 2 340.r even 4 2
1470.2.a.d 1 119.d odd 2 1
1470.2.i.o 2 119.j even 6 2
1470.2.i.q 2 119.h odd 6 2
2880.2.a.a 1 408.b odd 2 1
2880.2.a.q 1 408.h even 2 1
3600.2.a.f 1 1020.b even 2 1
3600.2.f.i 2 1020.x odd 4 2
3630.2.a.w 1 187.b odd 2 1
3840.2.k.f 2 272.k odd 4 2
3840.2.k.y 2 272.r even 4 2
4410.2.a.z 1 357.c even 2 1
4800.2.a.d 1 680.k odd 2 1
4800.2.a.cq 1 680.h even 2 1
4800.2.f.p 2 680.bi odd 4 2
4800.2.f.w 2 680.u even 4 2
5070.2.a.w 1 221.b even 2 1
5070.2.b.k 2 221.g odd 4 2
7350.2.a.ct 1 595.b odd 2 1
8670.2.a.g 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8670))\):

\( T_{7} - 4 \)
\( T_{11} \)
\( T_{13} - 2 \)
\( T_{23} \)

Hecke characteristic polynomials

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