Properties

Label 400.6.c.h
Level 400
Weight 6
Character orbit 400.c
Analytic conductor 64.154
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 i q^{3} -108 i q^{7} + 179 q^{9} +O(q^{10})\) \( q + 8 i q^{3} -108 i q^{7} + 179 q^{9} + 604 q^{11} -306 i q^{13} -930 i q^{17} -1324 q^{19} + 864 q^{21} + 852 i q^{23} + 3376 i q^{27} -5902 q^{29} + 3320 q^{31} + 4832 i q^{33} -10774 i q^{37} + 2448 q^{39} -17958 q^{41} -9264 i q^{43} -9796 i q^{47} + 5143 q^{49} + 7440 q^{51} -31434 i q^{53} -10592 i q^{57} + 33228 q^{59} -40210 q^{61} -19332 i q^{63} + 58864 i q^{67} -6816 q^{69} + 55312 q^{71} + 27258 i q^{73} -65232 i q^{77} + 31456 q^{79} + 16489 q^{81} -24552 i q^{83} -47216 i q^{87} + 90854 q^{89} -33048 q^{91} + 26560 i q^{93} -154706 i q^{97} + 108116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 358q^{9} + O(q^{10}) \) \( 2q + 358q^{9} + 1208q^{11} - 2648q^{19} + 1728q^{21} - 11804q^{29} + 6640q^{31} + 4896q^{39} - 35916q^{41} + 10286q^{49} + 14880q^{51} + 66456q^{59} - 80420q^{61} - 13632q^{69} + 110624q^{71} + 62912q^{79} + 32978q^{81} + 181708q^{89} - 66096q^{91} + 216232q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 8.00000i 0 0 0 108.000i 0 179.000 0
49.2 0 8.00000i 0 0 0 108.000i 0 179.000 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.c.h 2
4.b odd 2 1 200.6.c.c 2
5.b even 2 1 inner 400.6.c.h 2
5.c odd 4 1 80.6.a.f 1
5.c odd 4 1 400.6.a.f 1
15.e even 4 1 720.6.a.h 1
20.d odd 2 1 200.6.c.c 2
20.e even 4 1 40.6.a.b 1
20.e even 4 1 200.6.a.c 1
40.i odd 4 1 320.6.a.e 1
40.k even 4 1 320.6.a.l 1
60.l odd 4 1 360.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.b 1 20.e even 4 1
80.6.a.f 1 5.c odd 4 1
200.6.a.c 1 20.e even 4 1
200.6.c.c 2 4.b odd 2 1
200.6.c.c 2 20.d odd 2 1
320.6.a.e 1 40.i odd 4 1
320.6.a.l 1 40.k even 4 1
360.6.a.b 1 60.l odd 4 1
400.6.a.f 1 5.c odd 4 1
400.6.c.h 2 1.a even 1 1 trivial
400.6.c.h 2 5.b even 2 1 inner
720.6.a.h 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 64 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 422 T^{2} + 59049 T^{4} \)
$5$ 1
$7$ \( 1 - 21950 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 - 604 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 648950 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 - 1974814 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 1324 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 12146782 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 5902 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 - 3320 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 22608838 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 17958 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 208195190 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 362728398 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 + 151705370 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 - 33228 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 40210 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 + 764720282 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 - 55312 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 3403144622 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 - 31456 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 7275280582 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 - 90854 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 + 6759265922 T^{2} + 73742412689492826049 T^{4} \)
show more
show less