Properties

Label 400.6.c.e
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 18 i q^{3} + 242 i q^{7} -81 q^{9} +O(q^{10})\) \( q + 18 i q^{3} + 242 i q^{7} -81 q^{9} -656 q^{11} -206 i q^{13} -1690 i q^{17} -1364 q^{19} -4356 q^{21} -2198 i q^{23} + 2916 i q^{27} + 2218 q^{29} + 1700 q^{31} -11808 i q^{33} + 846 i q^{37} + 3708 q^{39} -1818 q^{41} -10534 i q^{43} + 12074 i q^{47} -41757 q^{49} + 30420 q^{51} + 32586 i q^{53} -24552 i q^{57} + 8668 q^{59} -34670 q^{61} -19602 i q^{63} -47566 i q^{67} + 39564 q^{69} -948 q^{71} -63102 i q^{73} -158752 i q^{77} + 46536 q^{79} -72171 q^{81} + 88778 i q^{83} + 39924 i q^{87} + 104934 q^{89} + 49852 q^{91} + 30600 i q^{93} + 36254 i q^{97} + 53136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 162q^{9} + O(q^{10}) \) \( 2q - 162q^{9} - 1312q^{11} - 2728q^{19} - 8712q^{21} + 4436q^{29} + 3400q^{31} + 7416q^{39} - 3636q^{41} - 83514q^{49} + 60840q^{51} + 17336q^{59} - 69340q^{61} + 79128q^{69} - 1896q^{71} + 93072q^{79} - 144342q^{81} + 209868q^{89} + 99704q^{91} + 106272q^{99} + O(q^{100}) \)

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 18.0000i 0 0 0 242.000i 0 −81.0000 0
49.2 0 18.0000i 0 0 0 242.000i 0 −81.0000 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.e 2
4.b odd 2 1 200.6.c.b 2
5.b even 2 1 inner 400.6.c.e 2
5.c odd 4 1 80.6.a.g 1
5.c odd 4 1 400.6.a.b 1
15.e even 4 1 720.6.a.k 1
20.d odd 2 1 200.6.c.b 2
20.e even 4 1 40.6.a.a 1
20.e even 4 1 200.6.a.d 1
40.i odd 4 1 320.6.a.d 1
40.k even 4 1 320.6.a.m 1
60.l odd 4 1 360.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.a 1 20.e even 4 1
80.6.a.g 1 5.c odd 4 1
200.6.a.d 1 20.e even 4 1
200.6.c.b 2 4.b odd 2 1
200.6.c.b 2 20.d odd 2 1
320.6.a.d 1 40.i odd 4 1
320.6.a.m 1 40.k even 4 1
360.6.a.i 1 60.l odd 4 1
400.6.a.b 1 5.c odd 4 1
400.6.c.e 2 1.a even 1 1 trivial
400.6.c.e 2 5.b even 2 1 inner
720.6.a.k 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 324 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 162 T^{2} + 59049 T^{4} \)
$5$ 1
$7$ \( 1 + 24950 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 + 656 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 700150 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 + 16386 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 1364 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 8041482 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 - 2218 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 - 1700 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 137972198 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 1818 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 183051730 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 312908538 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 + 225456410 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 - 8668 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 34670 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 437725858 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 948 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 164280782 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 - 46536 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 + 3451998 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 - 104934 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 - 15860327998 T^{2} + 73742412689492826049 T^{4} \)
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