Properties

Label 210.2.a.c
Level $210$
Weight $2$
Character orbit 210.a
Self dual yes
Analytic conductor $1.677$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.67685844245\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{14} - q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + q^{20} - q^{21} + 4q^{22} - 8q^{23} - q^{24} + q^{25} - 2q^{26} - q^{27} + q^{28} + 6q^{29} - q^{30} - 8q^{31} + q^{32} - 4q^{33} + 2q^{34} + q^{35} + q^{36} - 2q^{37} - 4q^{38} + 2q^{39} + q^{40} + 2q^{41} - q^{42} - 12q^{43} + 4q^{44} + q^{45} - 8q^{46} - 8q^{47} - q^{48} + q^{49} + q^{50} - 2q^{51} - 2q^{52} + 6q^{53} - q^{54} + 4q^{55} + q^{56} + 4q^{57} + 6q^{58} + 4q^{59} - q^{60} - 2q^{61} - 8q^{62} + q^{63} + q^{64} - 2q^{65} - 4q^{66} + 12q^{67} + 2q^{68} + 8q^{69} + q^{70} + 8q^{71} + q^{72} - 14q^{73} - 2q^{74} - q^{75} - 4q^{76} + 4q^{77} + 2q^{78} + q^{80} + q^{81} + 2q^{82} + 12q^{83} - q^{84} + 2q^{85} - 12q^{86} - 6q^{87} + 4q^{88} + 2q^{89} + q^{90} - 2q^{91} - 8q^{92} + 8q^{93} - 8q^{94} - 4q^{95} - q^{96} + 10q^{97} + q^{98} + 4q^{99} + O(q^{100}) \)

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
1.00000 −1.00000 1.00000 1.00000 −1.00000 1.00000 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 210.2.a.c 1
3.b odd 2 1 630.2.a.b 1
4.b odd 2 1 1680.2.a.q 1
5.b even 2 1 1050.2.a.h 1
5.c odd 4 2 1050.2.g.d 2
7.b odd 2 1 1470.2.a.q 1
7.c even 3 2 1470.2.i.f 2
7.d odd 6 2 1470.2.i.b 2
8.b even 2 1 6720.2.a.bp 1
8.d odd 2 1 6720.2.a.k 1
12.b even 2 1 5040.2.a.i 1
15.d odd 2 1 3150.2.a.w 1
15.e even 4 2 3150.2.g.e 2
20.d odd 2 1 8400.2.a.p 1
21.c even 2 1 4410.2.a.l 1
35.c odd 2 1 7350.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.c 1 1.a even 1 1 trivial
630.2.a.b 1 3.b odd 2 1
1050.2.a.h 1 5.b even 2 1
1050.2.g.d 2 5.c odd 4 2
1470.2.a.q 1 7.b odd 2 1
1470.2.i.b 2 7.d odd 6 2
1470.2.i.f 2 7.c even 3 2
1680.2.a.q 1 4.b odd 2 1
3150.2.a.w 1 15.d odd 2 1
3150.2.g.e 2 15.e even 4 2
4410.2.a.l 1 21.c even 2 1
5040.2.a.i 1 12.b even 2 1
6720.2.a.k 1 8.d odd 2 1
6720.2.a.bp 1 8.b even 2 1
7350.2.a.p 1 35.c odd 2 1
8400.2.a.p 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(210))\):

\( T_{11} - 4 \)
\( T_{17} - 2 \)
\( T_{19} + 4 \)