Properties

Label 400.6.c.n
Level $400$
Weight $6$
Character orbit 400.c
Analytic conductor $64.154$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Defining polynomial: \(x^{4} + 121 x^{2} + 3600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - 2 \beta_{2} ) q^{3} + ( -2 \beta_{1} + 20 \beta_{2} ) q^{7} + ( -98 - 4 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - 2 \beta_{2} ) q^{3} + ( -2 \beta_{1} + 20 \beta_{2} ) q^{7} + ( -98 - 4 \beta_{3} ) q^{9} + ( 98 - 5 \beta_{3} ) q^{11} + ( -16 \beta_{1} - 36 \beta_{2} ) q^{13} + ( -68 \beta_{1} - 149 \beta_{2} ) q^{17} + ( -1590 - 7 \beta_{3} ) q^{19} + ( 1482 + 24 \beta_{3} ) q^{21} + ( 6 \beta_{1} - 156 \beta_{2} ) q^{23} + ( -55 \beta_{1} + 674 \beta_{2} ) q^{27} + ( 1960 + 8 \beta_{3} ) q^{29} + ( 548 + 110 \beta_{3} ) q^{31} + ( -152 \beta_{1} + 1009 \beta_{2} ) q^{33} + ( 192 \beta_{1} - 202 \beta_{2} ) q^{37} + ( 2056 - 4 \beta_{3} ) q^{39} + ( 13877 + 40 \beta_{3} ) q^{41} + ( -1064 \beta_{1} + 300 \beta_{2} ) q^{43} + ( -772 \beta_{1} + 2576 \beta_{2} ) q^{47} + ( 5843 - 80 \beta_{3} ) q^{49} + ( 8938 - 13 \beta_{3} ) q^{51} + ( -376 \beta_{1} - 2698 \beta_{2} ) q^{53} + ( -1940 \beta_{1} + 4867 \beta_{2} ) q^{57} + ( 5980 - 196 \beta_{3} ) q^{59} + ( -12198 + 200 \beta_{3} ) q^{61} + ( 2196 \beta_{1} - 3888 \beta_{2} ) q^{63} + ( 793 \beta_{1} - 4006 \beta_{2} ) q^{67} + ( -9246 - 168 \beta_{3} ) q^{69} + ( 43648 - 100 \beta_{3} ) q^{71} + ( -556 \beta_{1} - 7029 \beta_{2} ) q^{73} + ( 2304 \beta_{1} - 450 \beta_{2} ) q^{77} + ( 32740 + 502 \beta_{3} ) q^{79} + ( 23141 - 188 \beta_{3} ) q^{81} + ( -429 \beta_{1} - 9258 \beta_{2} ) q^{83} + ( 2360 \beta_{1} - 5848 \beta_{2} ) q^{87} + ( 36405 + 1044 \beta_{3} ) q^{89} + ( 10288 - 248 \beta_{3} ) q^{91} + ( 6048 \beta_{1} - 27606 \beta_{2} ) q^{93} + ( 5472 \beta_{1} + 12614 \beta_{2} ) q^{97} + ( 110896 + 98 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 392q^{9} + O(q^{10}) \) \( 4q - 392q^{9} + 392q^{11} - 6360q^{19} + 5928q^{21} + 7840q^{29} + 2192q^{31} + 8224q^{39} + 55508q^{41} + 23372q^{49} + 35752q^{51} + 23920q^{59} - 48792q^{61} - 36984q^{69} + 174592q^{71} + 130960q^{79} + 92564q^{81} + 145620q^{89} + 41152q^{91} + 443584q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 121 x^{2} + 3600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 181 \nu \)\()/60\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 61 \nu \)\()/12\)
\(\beta_{3}\)\(=\)\( 10 \nu^{2} + 605 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 5 \beta_{1}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 605\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-181 \beta_{2} - 305 \beta_{1}\)\()/10\)

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
7.26209i
8.26209i
8.26209i
7.26209i
0 25.5242i 0 0 0 131.048i 0 −408.483 0
49.2 0 5.52417i 0 0 0 68.9517i 0 212.483 0
49.3 0 5.52417i 0 0 0 68.9517i 0 212.483 0
49.4 0 25.5242i 0 0 0 131.048i 0 −408.483 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.n 4
4.b odd 2 1 25.6.b.b 4
5.b even 2 1 inner 400.6.c.n 4
5.c odd 4 1 400.6.a.o 2
5.c odd 4 1 400.6.a.w 2
12.b even 2 1 225.6.b.i 4
20.d odd 2 1 25.6.b.b 4
20.e even 4 1 25.6.a.b 2
20.e even 4 1 25.6.a.d yes 2
60.h even 2 1 225.6.b.i 4
60.l odd 4 1 225.6.a.l 2
60.l odd 4 1 225.6.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 20.e even 4 1
25.6.a.d yes 2 20.e even 4 1
25.6.b.b 4 4.b odd 2 1
25.6.b.b 4 20.d odd 2 1
225.6.a.l 2 60.l odd 4 1
225.6.a.s 2 60.l odd 4 1
225.6.b.i 4 12.b even 2 1
225.6.b.i 4 60.h even 2 1
400.6.a.o 2 5.c odd 4 1
400.6.a.w 2 5.c odd 4 1
400.6.c.n 4 1.a even 1 1 trivial
400.6.c.n 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 682 T_{3}^{2} + 19881 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 19881 + 682 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 81649296 + 21928 T^{2} + T^{4} \)
$11$ \( ( -141021 - 196 T + T^{2} )^{2} \)
$13$ \( 858255616 + 188192 T^{2} + T^{4} \)
$17$ \( 312882490881 + 3338818 T^{2} + T^{4} \)
$19$ \( ( 2232875 + 3180 T + T^{2} )^{2} \)
$23$ \( 359668876176 + 1234152 T^{2} + T^{4} \)
$29$ \( ( 3456000 - 3920 T + T^{2} )^{2} \)
$31$ \( ( -72602196 - 1096 T + T^{2} )^{2} \)
$37$ \( 61844446287376 + 19808648 T^{2} + T^{4} \)
$41$ \( ( 182931129 - 27754 T + T^{2} )^{2} \)
$43$ \( 73216315824138496 + 550170272 T^{2} + T^{4} \)
$47$ \( 495608042209536 + 619053088 T^{2} + T^{4} \)
$53$ \( 21876919639178256 + 432103432 T^{2} + T^{4} \)
$59$ \( ( -195696000 - 11960 T + T^{2} )^{2} \)
$61$ \( ( -92208796 + 24396 T + T^{2} )^{2} \)
$67$ \( 62324269199220681 + 1105507018 T^{2} + T^{4} \)
$71$ \( ( 1844897904 - 87296 T + T^{2} )^{2} \)
$73$ \( 1347153105574224001 + 2619345602 T^{2} + T^{4} \)
$79$ \( ( -446416500 - 65480 T + T^{2} )^{2} \)
$83$ \( 4403325447203627961 + 4374235962 T^{2} + T^{4} \)
$89$ \( ( -5241540375 - 72810 T + T^{2} )^{2} \)
$97$ \( 10487144169973969936 + 22388071688 T^{2} + T^{4} \)
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