Properties

Label 400.6.c.c
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 20)
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.

\(f(q)\) \(=\) \( q + 22 i q^{3} -218 i q^{7} -241 q^{9} +O(q^{10})\) \( q + 22 i q^{3} -218 i q^{7} -241 q^{9} + 480 q^{11} + 622 i q^{13} + 186 i q^{17} -1204 q^{19} + 4796 q^{21} -3186 i q^{23} + 44 i q^{27} -5526 q^{29} -9356 q^{31} + 10560 i q^{33} + 5618 i q^{37} -13684 q^{39} -14394 q^{41} -370 i q^{43} -16146 i q^{47} -30717 q^{49} -4092 q^{51} + 4374 i q^{53} -26488 i q^{57} -11748 q^{59} + 13202 q^{61} + 52538 i q^{63} + 11542 i q^{67} + 70092 q^{69} + 29532 q^{71} -33698 i q^{73} -104640 i q^{77} + 31208 q^{79} -59531 q^{81} -38466 i q^{83} -121572 i q^{87} -119514 q^{89} + 135596 q^{91} -205832 i q^{93} + 94658 i q^{97} -115680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 482q^{9} + O(q^{10}) \) \( 2q - 482q^{9} + 960q^{11} - 2408q^{19} + 9592q^{21} - 11052q^{29} - 18712q^{31} - 27368q^{39} - 28788q^{41} - 61434q^{49} - 8184q^{51} - 23496q^{59} + 26404q^{61} + 140184q^{69} + 59064q^{71} + 62416q^{79} - 119062q^{81} - 239028q^{89} + 271192q^{91} - 231360q^{99} + O(q^{100}) \)

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 22.0000i 0 0 0 218.000i 0 −241.000 0
49.2 0 22.0000i 0 0 0 218.000i 0 −241.000 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.c 2
4.b odd 2 1 100.6.c.a 2
5.b even 2 1 inner 400.6.c.c 2
5.c odd 4 1 80.6.a.b 1
5.c odd 4 1 400.6.a.m 1
12.b even 2 1 900.6.d.h 2
15.e even 4 1 720.6.a.l 1
20.d odd 2 1 100.6.c.a 2
20.e even 4 1 20.6.a.a 1
20.e even 4 1 100.6.a.a 1
40.i odd 4 1 320.6.a.n 1
40.k even 4 1 320.6.a.c 1
60.h even 2 1 900.6.d.h 2
60.l odd 4 1 180.6.a.e 1
60.l odd 4 1 900.6.a.b 1
140.j odd 4 1 980.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 20.e even 4 1
80.6.a.b 1 5.c odd 4 1
100.6.a.a 1 20.e even 4 1
100.6.c.a 2 4.b odd 2 1
100.6.c.a 2 20.d odd 2 1
180.6.a.e 1 60.l odd 4 1
320.6.a.c 1 40.k even 4 1
320.6.a.n 1 40.i odd 4 1
400.6.a.m 1 5.c odd 4 1
400.6.c.c 2 1.a even 1 1 trivial
400.6.c.c 2 5.b even 2 1 inner
720.6.a.l 1 15.e even 4 1
900.6.a.b 1 60.l odd 4 1
900.6.d.h 2 12.b even 2 1
900.6.d.h 2 60.h even 2 1
980.6.a.b 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 484 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 59049 T^{4} \)
$5$ 1
$7$ \( 1 + 13910 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 - 480 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 355702 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 - 2805118 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 1204 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 2722090 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 5526 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 + 9356 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 107125990 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 14394 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 293879986 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 197996698 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 - 817259110 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 11748 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 - 13202 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 2567032450 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 - 29532 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 3010587982 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 - 31208 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 6398448130 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 119514 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 - 8214543550 T^{2} + 73742412689492826049 T^{4} \)
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