Properties

Label 400.6.c.f
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 4)
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.

\(f(q)\) \(=\) \( q + 12 i q^{3} -88 i q^{7} + 99 q^{9} +O(q^{10})\) \( q + 12 i q^{3} -88 i q^{7} + 99 q^{9} -540 q^{11} -418 i q^{13} -594 i q^{17} + 836 q^{19} + 1056 q^{21} + 4104 i q^{23} + 4104 i q^{27} + 594 q^{29} -4256 q^{31} -6480 i q^{33} + 298 i q^{37} + 5016 q^{39} + 17226 q^{41} + 12100 i q^{43} -1296 i q^{47} + 9063 q^{49} + 7128 q^{51} + 19494 i q^{53} + 10032 i q^{57} -7668 q^{59} -34738 q^{61} -8712 i q^{63} + 21812 i q^{67} -49248 q^{69} + 46872 q^{71} + 67562 i q^{73} + 47520 i q^{77} -76912 q^{79} -25191 q^{81} -67716 i q^{83} + 7128 i q^{87} -29754 q^{89} -36784 q^{91} -51072 i q^{93} + 122398 i q^{97} -53460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 198q^{9} + O(q^{10}) \) \( 2q + 198q^{9} - 1080q^{11} + 1672q^{19} + 2112q^{21} + 1188q^{29} - 8512q^{31} + 10032q^{39} + 34452q^{41} + 18126q^{49} + 14256q^{51} - 15336q^{59} - 69476q^{61} - 98496q^{69} + 93744q^{71} - 153824q^{79} - 50382q^{81} - 59508q^{89} - 73568q^{91} - 106920q^{99} + O(q^{100}) \)

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 12.0000i 0 0 0 88.0000i 0 99.0000 0
49.2 0 12.0000i 0 0 0 88.0000i 0 99.0000 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.f 2
4.b odd 2 1 100.6.c.b 2
5.b even 2 1 inner 400.6.c.f 2
5.c odd 4 1 16.6.a.b 1
5.c odd 4 1 400.6.a.d 1
12.b even 2 1 900.6.d.a 2
15.e even 4 1 144.6.a.c 1
20.d odd 2 1 100.6.c.b 2
20.e even 4 1 4.6.a.a 1
20.e even 4 1 100.6.a.b 1
35.f even 4 1 784.6.a.d 1
40.i odd 4 1 64.6.a.b 1
40.k even 4 1 64.6.a.f 1
60.h even 2 1 900.6.d.a 2
60.l odd 4 1 36.6.a.a 1
60.l odd 4 1 900.6.a.h 1
80.i odd 4 1 256.6.b.c 2
80.j even 4 1 256.6.b.g 2
80.s even 4 1 256.6.b.g 2
80.t odd 4 1 256.6.b.c 2
120.q odd 4 1 576.6.a.bc 1
120.w even 4 1 576.6.a.bd 1
140.j odd 4 1 196.6.a.e 1
140.w even 12 2 196.6.e.g 2
140.x odd 12 2 196.6.e.d 2
180.v odd 12 2 324.6.e.d 2
180.x even 12 2 324.6.e.a 2
220.i odd 4 1 484.6.a.a 1
260.l odd 4 1 676.6.d.a 2
260.p even 4 1 676.6.a.a 1
260.s odd 4 1 676.6.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 20.e even 4 1
16.6.a.b 1 5.c odd 4 1
36.6.a.a 1 60.l odd 4 1
64.6.a.b 1 40.i odd 4 1
64.6.a.f 1 40.k even 4 1
100.6.a.b 1 20.e even 4 1
100.6.c.b 2 4.b odd 2 1
100.6.c.b 2 20.d odd 2 1
144.6.a.c 1 15.e even 4 1
196.6.a.e 1 140.j odd 4 1
196.6.e.d 2 140.x odd 12 2
196.6.e.g 2 140.w even 12 2
256.6.b.c 2 80.i odd 4 1
256.6.b.c 2 80.t odd 4 1
256.6.b.g 2 80.j even 4 1
256.6.b.g 2 80.s even 4 1
324.6.e.a 2 180.x even 12 2
324.6.e.d 2 180.v odd 12 2
400.6.a.d 1 5.c odd 4 1
400.6.c.f 2 1.a even 1 1 trivial
400.6.c.f 2 5.b even 2 1 inner
484.6.a.a 1 220.i odd 4 1
576.6.a.bc 1 120.q odd 4 1
576.6.a.bd 1 120.w even 4 1
676.6.a.a 1 260.p even 4 1
676.6.d.a 2 260.l odd 4 1
676.6.d.a 2 260.s odd 4 1
784.6.a.d 1 35.f even 4 1
900.6.a.h 1 60.l odd 4 1
900.6.d.a 2 12.b even 2 1
900.6.d.a 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 144 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 144 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7744 + T^{2} \)
$11$ \( ( 540 + T )^{2} \)
$13$ \( 174724 + T^{2} \)
$17$ \( 352836 + T^{2} \)
$19$ \( ( -836 + T )^{2} \)
$23$ \( 16842816 + T^{2} \)
$29$ \( ( -594 + T )^{2} \)
$31$ \( ( 4256 + T )^{2} \)
$37$ \( 88804 + T^{2} \)
$41$ \( ( -17226 + T )^{2} \)
$43$ \( 146410000 + T^{2} \)
$47$ \( 1679616 + T^{2} \)
$53$ \( 380016036 + T^{2} \)
$59$ \( ( 7668 + T )^{2} \)
$61$ \( ( 34738 + T )^{2} \)
$67$ \( 475763344 + T^{2} \)
$71$ \( ( -46872 + T )^{2} \)
$73$ \( 4564623844 + T^{2} \)
$79$ \( ( 76912 + T )^{2} \)
$83$ \( 4585456656 + T^{2} \)
$89$ \( ( 29754 + T )^{2} \)
$97$ \( 14981270404 + T^{2} \)
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