Properties

Label 400.6.c.b
Level 400
Weight 6
Character orbit 400.c
Analytic conductor 64.154
Analytic rank 0
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.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 10)
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.

\(f(q)\) \(=\) \( q + 24 i q^{3} + 172 i q^{7} -333 q^{9} +O(q^{10})\) \( q + 24 i q^{3} + 172 i q^{7} -333 q^{9} -132 q^{11} + 946 i q^{13} -222 i q^{17} + 500 q^{19} -4128 q^{21} + 3564 i q^{23} -2160 i q^{27} -2190 q^{29} -2312 q^{31} -3168 i q^{33} -11242 i q^{37} -22704 q^{39} + 1242 q^{41} + 20624 i q^{43} -6588 i q^{47} -12777 q^{49} + 5328 q^{51} + 21066 i q^{53} + 12000 i q^{57} + 7980 q^{59} + 16622 q^{61} -57276 i q^{63} -1808 i q^{67} -85536 q^{69} + 24528 q^{71} -20474 i q^{73} -22704 i q^{77} -46240 q^{79} -29079 q^{81} -51576 i q^{83} -52560 i q^{87} + 110310 q^{89} -162712 q^{91} -55488 i q^{93} -78382 i q^{97} + 43956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 666q^{9} + O(q^{10}) \) \( 2q - 666q^{9} - 264q^{11} + 1000q^{19} - 8256q^{21} - 4380q^{29} - 4624q^{31} - 45408q^{39} + 2484q^{41} - 25554q^{49} + 10656q^{51} + 15960q^{59} + 33244q^{61} - 171072q^{69} + 49056q^{71} - 92480q^{79} - 58158q^{81} + 220620q^{89} - 325424q^{91} + 87912q^{99} + O(q^{100}) \)

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 24.0000i 0 0 0 172.000i 0 −333.000 0
49.2 0 24.0000i 0 0 0 172.000i 0 −333.000 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.b 2
4.b odd 2 1 50.6.b.a 2
5.b even 2 1 inner 400.6.c.b 2
5.c odd 4 1 80.6.a.a 1
5.c odd 4 1 400.6.a.n 1
12.b even 2 1 450.6.c.h 2
15.e even 4 1 720.6.a.j 1
20.d odd 2 1 50.6.b.a 2
20.e even 4 1 10.6.a.b 1
20.e even 4 1 50.6.a.d 1
40.i odd 4 1 320.6.a.o 1
40.k even 4 1 320.6.a.b 1
60.h even 2 1 450.6.c.h 2
60.l odd 4 1 90.6.a.d 1
60.l odd 4 1 450.6.a.l 1
140.j odd 4 1 490.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 20.e even 4 1
50.6.a.d 1 20.e even 4 1
50.6.b.a 2 4.b odd 2 1
50.6.b.a 2 20.d odd 2 1
80.6.a.a 1 5.c odd 4 1
90.6.a.d 1 60.l odd 4 1
320.6.a.b 1 40.k even 4 1
320.6.a.o 1 40.i odd 4 1
400.6.a.n 1 5.c odd 4 1
400.6.c.b 2 1.a even 1 1 trivial
400.6.c.b 2 5.b even 2 1 inner
450.6.a.l 1 60.l odd 4 1
450.6.c.h 2 12.b even 2 1
450.6.c.h 2 60.h even 2 1
490.6.a.a 1 140.j odd 4 1
720.6.a.j 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 576 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 90 T^{2} + 59049 T^{4} \)
$5$ 1
$7$ \( 1 - 4030 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 + 132 T + 161051 T^{2} )^{2} \)
$13$ \( 1 + 152330 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 - 2790430 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 - 500 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 170590 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 2190 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 + 2312 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 12305350 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 - 1242 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 + 131332490 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 415288270 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 - 392614630 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 - 7980 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 - 16622 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 2696981350 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 - 24528 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 3726958510 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 + 46240 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 5217997510 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 - 110310 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 - 11030942590 T^{2} + 73742412689492826049 T^{4} \)
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