Properties

Label 400.2.a.c
Level 400
Weight 2
Character orbit 400.a
Self dual yes
Analytic conductor 3.194
Analytic rank 0
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} + 2q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{3} + 2q^{7} + q^{9} - 2q^{13} + 6q^{17} + 4q^{19} - 4q^{21} + 6q^{23} + 4q^{27} + 6q^{29} + 4q^{31} - 2q^{37} + 4q^{39} + 6q^{41} - 10q^{43} - 6q^{47} - 3q^{49} - 12q^{51} + 6q^{53} - 8q^{57} - 12q^{59} + 2q^{61} + 2q^{63} + 2q^{67} - 12q^{69} + 12q^{71} - 2q^{73} - 8q^{79} - 11q^{81} + 6q^{83} - 12q^{87} - 6q^{89} - 4q^{91} - 8q^{93} - 2q^{97} + O(q^{100}) \)

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 −2.00000 0 0 0 2.00000 0 1.00000 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.c 1
3.b odd 2 1 3600.2.a.be 1
4.b odd 2 1 100.2.a.a 1
5.b even 2 1 80.2.a.b 1
5.c odd 4 2 400.2.c.b 2
8.b even 2 1 1600.2.a.w 1
8.d odd 2 1 1600.2.a.c 1
12.b even 2 1 900.2.a.b 1
15.d odd 2 1 720.2.a.h 1
15.e even 4 2 3600.2.f.j 2
20.d odd 2 1 20.2.a.a 1
20.e even 4 2 100.2.c.a 2
28.d even 2 1 4900.2.a.e 1
35.c odd 2 1 3920.2.a.h 1
40.e odd 2 1 320.2.a.f 1
40.f even 2 1 320.2.a.a 1
40.i odd 4 2 1600.2.c.e 2
40.k even 4 2 1600.2.c.d 2
55.d odd 2 1 9680.2.a.ba 1
60.h even 2 1 180.2.a.a 1
60.l odd 4 2 900.2.d.c 2
80.k odd 4 2 1280.2.d.c 2
80.q even 4 2 1280.2.d.g 2
120.i odd 2 1 2880.2.a.f 1
120.m even 2 1 2880.2.a.m 1
140.c even 2 1 980.2.a.h 1
140.j odd 4 2 4900.2.e.f 2
140.p odd 6 2 980.2.i.i 2
140.s even 6 2 980.2.i.c 2
180.n even 6 2 1620.2.i.b 2
180.p odd 6 2 1620.2.i.h 2
220.g even 2 1 2420.2.a.a 1
260.g odd 2 1 3380.2.a.c 1
260.u even 4 2 3380.2.f.b 2
340.d odd 2 1 5780.2.a.f 1
340.n odd 4 2 5780.2.c.a 2
380.d even 2 1 7220.2.a.f 1
420.o odd 2 1 8820.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 20.d odd 2 1
80.2.a.b 1 5.b even 2 1
100.2.a.a 1 4.b odd 2 1
100.2.c.a 2 20.e even 4 2
180.2.a.a 1 60.h even 2 1
320.2.a.a 1 40.f even 2 1
320.2.a.f 1 40.e odd 2 1
400.2.a.c 1 1.a even 1 1 trivial
400.2.c.b 2 5.c odd 4 2
720.2.a.h 1 15.d odd 2 1
900.2.a.b 1 12.b even 2 1
900.2.d.c 2 60.l odd 4 2
980.2.a.h 1 140.c even 2 1
980.2.i.c 2 140.s even 6 2
980.2.i.i 2 140.p odd 6 2
1280.2.d.c 2 80.k odd 4 2
1280.2.d.g 2 80.q even 4 2
1600.2.a.c 1 8.d odd 2 1
1600.2.a.w 1 8.b even 2 1
1600.2.c.d 2 40.k even 4 2
1600.2.c.e 2 40.i odd 4 2
1620.2.i.b 2 180.n even 6 2
1620.2.i.h 2 180.p odd 6 2
2420.2.a.a 1 220.g even 2 1
2880.2.a.f 1 120.i odd 2 1
2880.2.a.m 1 120.m even 2 1
3380.2.a.c 1 260.g odd 2 1
3380.2.f.b 2 260.u even 4 2
3600.2.a.be 1 3.b odd 2 1
3600.2.f.j 2 15.e even 4 2
3920.2.a.h 1 35.c odd 2 1
4900.2.a.e 1 28.d even 2 1
4900.2.e.f 2 140.j odd 4 2
5780.2.a.f 1 340.d odd 2 1
5780.2.c.a 2 340.n odd 4 2
7220.2.a.f 1 380.d even 2 1
8820.2.a.g 1 420.o odd 2 1
9680.2.a.ba 1 55.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(400))\):

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

Hecke characteristic polynomials

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