Properties

Label 400.6.a.s
Level 400
Weight 6
Character orbit 400.a
Self dual yes
Analytic conductor 64.154
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1535279252\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
Defining polynomial: \(x^{2} - 31\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -11 \beta q^{7} -119 q^{9} +O(q^{10})\) \( q + \beta q^{3} -11 \beta q^{7} -119 q^{9} + 100 q^{11} + 66 \beta q^{13} -88 \beta q^{17} + 2244 q^{19} -1364 q^{21} + 307 \beta q^{23} -362 \beta q^{27} -7854 q^{29} + 2144 q^{31} + 100 \beta q^{33} -934 \beta q^{37} + 8184 q^{39} -7414 q^{41} -1595 \beta q^{43} -847 \beta q^{47} -1803 q^{49} -10912 q^{51} -2178 \beta q^{53} + 2244 \beta q^{57} -25972 q^{59} -3058 q^{61} + 1309 \beta q^{63} + 5279 \beta q^{67} + 38068 q^{69} -37608 q^{71} + 2156 \beta q^{73} -1100 \beta q^{77} -79728 q^{79} -15971 q^{81} -1463 \beta q^{83} -7854 \beta q^{87} -826 q^{89} -90024 q^{91} + 2144 \beta q^{93} + 3376 \beta q^{97} -11900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 238q^{9} + O(q^{10}) \) \( 2q - 238q^{9} + 200q^{11} + 4488q^{19} - 2728q^{21} - 15708q^{29} + 4288q^{31} + 16368q^{39} - 14828q^{41} - 3606q^{49} - 21824q^{51} - 51944q^{59} - 6116q^{61} + 76136q^{69} - 75216q^{71} - 159456q^{79} - 31942q^{81} - 1652q^{89} - 180048q^{91} - 23800q^{99} + O(q^{100}) \)

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
−5.56776
5.56776
0 −11.1355 0 0 0 122.491 0 −119.000 0
1.2 0 11.1355 0 0 0 −122.491 0 −119.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.a.s 2
4.b odd 2 1 100.6.a.d 2
5.b even 2 1 inner 400.6.a.s 2
5.c odd 4 2 80.6.c.b 2
12.b even 2 1 900.6.a.q 2
15.e even 4 2 720.6.f.d 2
20.d odd 2 1 100.6.a.d 2
20.e even 4 2 20.6.c.a 2
40.i odd 4 2 320.6.c.d 2
40.k even 4 2 320.6.c.e 2
60.h even 2 1 900.6.a.q 2
60.l odd 4 2 180.6.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.c.a 2 20.e even 4 2
80.6.c.b 2 5.c odd 4 2
100.6.a.d 2 4.b odd 2 1
100.6.a.d 2 20.d odd 2 1
180.6.d.b 2 60.l odd 4 2
320.6.c.d 2 40.i odd 4 2
320.6.c.e 2 40.k even 4 2
400.6.a.s 2 1.a even 1 1 trivial
400.6.a.s 2 5.b even 2 1 inner
720.6.f.d 2 15.e even 4 2
900.6.a.q 2 12.b even 2 1
900.6.a.q 2 60.h even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 124 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 362 T^{2} + 59049 T^{4} \)
$5$ 1
$7$ \( 1 + 18610 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 - 100 T + 161051 T^{2} )^{2} \)
$13$ \( 1 + 202442 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 + 1879458 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 - 2244 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 + 1185810 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 7854 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 - 2144 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 + 30515770 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 7414 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 21442214 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 + 369731298 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 + 248174170 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 25972 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 3058 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 755362070 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 37608 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 + 3569749522 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 + 79728 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 + 7612675530 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 826 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 + 15761405890 T^{2} + 73742412689492826049 T^{4} \)
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