Properties

Label 350.2.a.a
Level 350
Weight 2
Character orbit 350.a
Self dual yes
Analytic conductor 2.795
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.79476407074\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} - 2q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} - 2q^{9} + 3q^{11} - q^{12} - 2q^{13} + q^{14} + q^{16} - 3q^{17} + 2q^{18} - 7q^{19} + q^{21} - 3q^{22} + q^{24} + 2q^{26} + 5q^{27} - q^{28} - 6q^{29} - 4q^{31} - q^{32} - 3q^{33} + 3q^{34} - 2q^{36} - 8q^{37} + 7q^{38} + 2q^{39} - 9q^{41} - q^{42} - 8q^{43} + 3q^{44} + 6q^{47} - q^{48} + q^{49} + 3q^{51} - 2q^{52} + 12q^{53} - 5q^{54} + q^{56} + 7q^{57} + 6q^{58} + 12q^{59} - 10q^{61} + 4q^{62} + 2q^{63} + q^{64} + 3q^{66} + 7q^{67} - 3q^{68} + 6q^{71} + 2q^{72} - 5q^{73} + 8q^{74} - 7q^{76} - 3q^{77} - 2q^{78} + 14q^{79} + q^{81} + 9q^{82} + 9q^{83} + q^{84} + 8q^{86} + 6q^{87} - 3q^{88} - 15q^{89} + 2q^{91} + 4q^{93} - 6q^{94} + q^{96} + 10q^{97} - q^{98} - 6q^{99} + O(q^{100}) \)

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 0 1.00000 −1.00000 −1.00000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 350.2.a.a 1
3.b odd 2 1 3150.2.a.x 1
4.b odd 2 1 2800.2.a.x 1
5.b even 2 1 350.2.a.e yes 1
5.c odd 4 2 350.2.c.c 2
7.b odd 2 1 2450.2.a.m 1
15.d odd 2 1 3150.2.a.m 1
15.e even 4 2 3150.2.g.f 2
20.d odd 2 1 2800.2.a.h 1
20.e even 4 2 2800.2.g.i 2
35.c odd 2 1 2450.2.a.x 1
35.f even 4 2 2450.2.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 1.a even 1 1 trivial
350.2.a.e yes 1 5.b even 2 1
350.2.c.c 2 5.c odd 4 2
2450.2.a.m 1 7.b odd 2 1
2450.2.a.x 1 35.c odd 2 1
2450.2.c.h 2 35.f even 4 2
2800.2.a.h 1 20.d odd 2 1
2800.2.a.x 1 4.b odd 2 1
2800.2.g.i 2 20.e even 4 2
3150.2.a.m 1 15.d odd 2 1
3150.2.a.x 1 3.b odd 2 1
3150.2.g.f 2 15.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3} + 1 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T + 3 T^{2} \)
$5$ 1
$7$ \( 1 + T \)
$11$ \( 1 - 3 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 + 7 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 + 8 T + 37 T^{2} \)
$41$ \( 1 + 9 T + 41 T^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( 1 - 6 T + 47 T^{2} \)
$53$ \( 1 - 12 T + 53 T^{2} \)
$59$ \( 1 - 12 T + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 - 7 T + 67 T^{2} \)
$71$ \( 1 - 6 T + 71 T^{2} \)
$73$ \( 1 + 5 T + 73 T^{2} \)
$79$ \( 1 - 14 T + 79 T^{2} \)
$83$ \( 1 - 9 T + 83 T^{2} \)
$89$ \( 1 + 15 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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