Properties

Label 350.2.a.d
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

Related objects

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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

\(f(q)\) \(=\) \( q + q^{2} - 3q^{3} + q^{4} - 3q^{6} - q^{7} + q^{8} + 6q^{9} + O(q^{10}) \) \( q + q^{2} - 3q^{3} + q^{4} - 3q^{6} - q^{7} + q^{8} + 6q^{9} - 5q^{11} - 3q^{12} - 6q^{13} - q^{14} + q^{16} - q^{17} + 6q^{18} - 3q^{19} + 3q^{21} - 5q^{22} - 3q^{24} - 6q^{26} - 9q^{27} - q^{28} - 6q^{29} - 4q^{31} + q^{32} + 15q^{33} - q^{34} + 6q^{36} + 8q^{37} - 3q^{38} + 18q^{39} + 11q^{41} + 3q^{42} - 8q^{43} - 5q^{44} + 2q^{47} - 3q^{48} + q^{49} + 3q^{51} - 6q^{52} + 4q^{53} - 9q^{54} - q^{56} + 9q^{57} - 6q^{58} + 4q^{59} - 2q^{61} - 4q^{62} - 6q^{63} + q^{64} + 15q^{66} + 9q^{67} - q^{68} - 10q^{71} + 6q^{72} - 7q^{73} + 8q^{74} - 3q^{76} + 5q^{77} + 18q^{78} - 2q^{79} + 9q^{81} + 11q^{82} + 11q^{83} + 3q^{84} - 8q^{86} + 18q^{87} - 5q^{88} - 11q^{89} + 6q^{91} + 12q^{93} + 2q^{94} - 3q^{96} - 10q^{97} + q^{98} - 30q^{99} + O(q^{100}) \)

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 −3.00000 1.00000 0 −3.00000 −1.00000 1.00000 6.00000 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.d yes 1
3.b odd 2 1 3150.2.a.j 1
4.b odd 2 1 2800.2.a.bg 1
5.b even 2 1 350.2.a.c 1
5.c odd 4 2 350.2.c.a 2
7.b odd 2 1 2450.2.a.bg 1
15.d odd 2 1 3150.2.a.bq 1
15.e even 4 2 3150.2.g.v 2
20.d odd 2 1 2800.2.a.b 1
20.e even 4 2 2800.2.g.a 2
35.c odd 2 1 2450.2.a.a 1
35.f even 4 2 2450.2.c.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 5.b even 2 1
350.2.a.d yes 1 1.a even 1 1 trivial
350.2.c.a 2 5.c odd 4 2
2450.2.a.a 1 35.c odd 2 1
2450.2.a.bg 1 7.b odd 2 1
2450.2.c.r 2 35.f even 4 2
2800.2.a.b 1 20.d odd 2 1
2800.2.a.bg 1 4.b odd 2 1
2800.2.g.a 2 20.e even 4 2
3150.2.a.j 1 3.b odd 2 1
3150.2.a.bq 1 15.d odd 2 1
3150.2.g.v 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3} + 3 \)
\( T_{13} + 6 \)

Hecke Characteristic Polynomials

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