Properties

Label 400.2.a.d
Level 400
Weight 2
Character orbit 400.a
Self dual yes
Analytic conductor 3.194
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.19401608085\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{7} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{7} - 2q^{9} + 3q^{11} - 4q^{13} - 3q^{17} - 5q^{19} + 2q^{21} - 6q^{23} + 5q^{27} - 2q^{31} - 3q^{33} + 2q^{37} + 4q^{39} - 3q^{41} + 4q^{43} - 12q^{47} - 3q^{49} + 3q^{51} + 6q^{53} + 5q^{57} + 2q^{61} + 4q^{63} + 13q^{67} + 6q^{69} - 12q^{71} + 11q^{73} - 6q^{77} + 10q^{79} + q^{81} + 9q^{83} + 15q^{89} + 8q^{91} + 2q^{93} + 2q^{97} - 6q^{99} + O(q^{100}) \)

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
0 −1.00000 0 0 0 −2.00000 0 −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 400.2.a.d 1
3.b odd 2 1 3600.2.a.l 1
4.b odd 2 1 50.2.a.a 1
5.b even 2 1 400.2.a.f 1
5.c odd 4 2 400.2.c.c 2
8.b even 2 1 1600.2.a.p 1
8.d odd 2 1 1600.2.a.j 1
12.b even 2 1 450.2.a.g 1
15.d odd 2 1 3600.2.a.bc 1
15.e even 4 2 3600.2.f.f 2
20.d odd 2 1 50.2.a.b yes 1
20.e even 4 2 50.2.b.a 2
28.d even 2 1 2450.2.a.g 1
40.e odd 2 1 1600.2.a.q 1
40.f even 2 1 1600.2.a.i 1
40.i odd 4 2 1600.2.c.h 2
40.k even 4 2 1600.2.c.i 2
44.c even 2 1 6050.2.a.bi 1
52.b odd 2 1 8450.2.a.v 1
60.h even 2 1 450.2.a.c 1
60.l odd 4 2 450.2.c.c 2
140.c even 2 1 2450.2.a.bd 1
140.j odd 4 2 2450.2.c.m 2
220.g even 2 1 6050.2.a.h 1
260.g odd 2 1 8450.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.a.a 1 4.b odd 2 1
50.2.a.b yes 1 20.d odd 2 1
50.2.b.a 2 20.e even 4 2
400.2.a.d 1 1.a even 1 1 trivial
400.2.a.f 1 5.b even 2 1
400.2.c.c 2 5.c odd 4 2
450.2.a.c 1 60.h even 2 1
450.2.a.g 1 12.b even 2 1
450.2.c.c 2 60.l odd 4 2
1600.2.a.i 1 40.f even 2 1
1600.2.a.j 1 8.d odd 2 1
1600.2.a.p 1 8.b even 2 1
1600.2.a.q 1 40.e odd 2 1
1600.2.c.h 2 40.i odd 4 2
1600.2.c.i 2 40.k even 4 2
2450.2.a.g 1 28.d even 2 1
2450.2.a.bd 1 140.c even 2 1
2450.2.c.m 2 140.j odd 4 2
3600.2.a.l 1 3.b odd 2 1
3600.2.a.bc 1 15.d odd 2 1
3600.2.f.f 2 15.e even 4 2
6050.2.a.h 1 220.g even 2 1
6050.2.a.bi 1 44.c even 2 1
8450.2.a.d 1 260.g odd 2 1
8450.2.a.v 1 52.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(400))\):

\( T_{3} + 1 \)
\( T_{7} + 2 \)

Hecke Characteristic Polynomials

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