Properties

Label 350.4.a.m
Level $350$
Weight $4$
Character orbit 350.a
Self dual yes
Analytic conductor $20.651$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.6506685020\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} - 7 q^{3} + 4 q^{4} - 14 q^{6} + 7 q^{7} + 8 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 7 q^{3} + 4 q^{4} - 14 q^{6} + 7 q^{7} + 8 q^{8} + 22 q^{9} - 37 q^{11} - 28 q^{12} + 51 q^{13} + 14 q^{14} + 16 q^{16} + 41 q^{17} + 44 q^{18} - 108 q^{19} - 49 q^{21} - 74 q^{22} - 70 q^{23} - 56 q^{24} + 102 q^{26} + 35 q^{27} + 28 q^{28} - 249 q^{29} - 134 q^{31} + 32 q^{32} + 259 q^{33} + 82 q^{34} + 88 q^{36} - 334 q^{37} - 216 q^{38} - 357 q^{39} + 206 q^{41} - 98 q^{42} - 376 q^{43} - 148 q^{44} - 140 q^{46} - 287 q^{47} - 112 q^{48} + 49 q^{49} - 287 q^{51} + 204 q^{52} - 6 q^{53} + 70 q^{54} + 56 q^{56} + 756 q^{57} - 498 q^{58} - 2 q^{59} - 940 q^{61} - 268 q^{62} + 154 q^{63} + 64 q^{64} + 518 q^{66} + 106 q^{67} + 164 q^{68} + 490 q^{69} + 456 q^{71} + 176 q^{72} + 650 q^{73} - 668 q^{74} - 432 q^{76} - 259 q^{77} - 714 q^{78} - 1239 q^{79} - 839 q^{81} + 412 q^{82} + 428 q^{83} - 196 q^{84} - 752 q^{86} + 1743 q^{87} - 296 q^{88} - 220 q^{89} + 357 q^{91} - 280 q^{92} + 938 q^{93} - 574 q^{94} - 224 q^{96} - 1055 q^{97} + 98 q^{98} - 814 q^{99}+O(q^{100}) \) 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
2.00000 −7.00000 4.00000 0 −14.0000 7.00000 8.00000 22.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 350.4.a.m 1
5.b even 2 1 350.4.a.i 1
5.c odd 4 2 70.4.c.a 2
7.b odd 2 1 2450.4.a.bn 1
15.e even 4 2 630.4.g.a 2
20.e even 4 2 560.4.g.c 2
35.c odd 2 1 2450.4.a.c 1
35.f even 4 2 490.4.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.c.a 2 5.c odd 4 2
350.4.a.i 1 5.b even 2 1
350.4.a.m 1 1.a even 1 1 trivial
490.4.c.a 2 35.f even 4 2
560.4.g.c 2 20.e even 4 2
630.4.g.a 2 15.e even 4 2
2450.4.a.c 1 35.c odd 2 1
2450.4.a.bn 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3} + 7 \) Copy content Toggle raw display
\( T_{11} + 37 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 7 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 37 \) Copy content Toggle raw display
$13$ \( T - 51 \) Copy content Toggle raw display
$17$ \( T - 41 \) Copy content Toggle raw display
$19$ \( T + 108 \) Copy content Toggle raw display
$23$ \( T + 70 \) Copy content Toggle raw display
$29$ \( T + 249 \) Copy content Toggle raw display
$31$ \( T + 134 \) Copy content Toggle raw display
$37$ \( T + 334 \) Copy content Toggle raw display
$41$ \( T - 206 \) Copy content Toggle raw display
$43$ \( T + 376 \) Copy content Toggle raw display
$47$ \( T + 287 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 2 \) Copy content Toggle raw display
$61$ \( T + 940 \) Copy content Toggle raw display
$67$ \( T - 106 \) Copy content Toggle raw display
$71$ \( T - 456 \) Copy content Toggle raw display
$73$ \( T - 650 \) Copy content Toggle raw display
$79$ \( T + 1239 \) Copy content Toggle raw display
$83$ \( T - 428 \) Copy content Toggle raw display
$89$ \( T + 220 \) Copy content Toggle raw display
$97$ \( T + 1055 \) Copy content Toggle raw display
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