Properties

Label 400.6.c.o
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: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 200)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 4 \beta_1) q^{3} + ( - 2 \beta_{2} - 4 \beta_1) q^{7} + ( - 8 \beta_{3} - 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 4 \beta_1) q^{3} + ( - 2 \beta_{2} - 4 \beta_1) q^{7} + ( - 8 \beta_{3} - 14) q^{9} + (21 \beta_{3} + 100) q^{11} + ( - 52 \beta_{2} - 296 \beta_1) q^{13} + (16 \beta_{2} - 139 \beta_1) q^{17} + (59 \beta_{3} - 420) q^{19} + ( - 12 \beta_{3} - 498) q^{21} + (98 \beta_{2} + 976 \beta_1) q^{23} + ( - 197 \beta_{2} + 1012 \beta_1) q^{27} + ( - 148 \beta_{3} + 2340) q^{29} + (354 \beta_{3} + 2504) q^{31} + ( - 184 \beta_{2} - 5461 \beta_1) q^{33} + ( - 484 \beta_{2} - 6250 \beta_1) q^{37} + ( - 504 \beta_{3} - 13716) q^{39} + (936 \beta_{3} - 2667) q^{41} + ( - 1076 \beta_{2} - 112 \beta_1) q^{43} + 13036 \beta_1 q^{47} + ( - 16 \beta_{3} + 15827) q^{49} + ( - 75 \beta_{3} + 3300) q^{51} + (360 \beta_{2} - 23406 \beta_1) q^{53} + (184 \beta_{2} - 12539 \beta_1) q^{57} + ( - 416 \beta_{3} - 40888) q^{59} + (44 \beta_{3} - 23466) q^{61} + (60 \beta_{2} + 3912 \beta_1) q^{63} + (323 \beta_{2} + 34404 \beta_1) q^{67} + (1368 \beta_{3} + 27522) q^{69} + (344 \beta_{3} - 3724) q^{71} + (72 \beta_{2} - 54411 \beta_1) q^{73} + ( - 284 \beta_{2} - 10522 \beta_1) q^{77} + (3062 \beta_{3} - 54052) q^{79} + ( - 1720 \beta_{3} - 46831) q^{81} + (3841 \beta_{2} - 13612 \beta_1) q^{83} + ( - 1748 \beta_{2} + 26308 \beta_1) q^{87} + ( - 5024 \beta_{3} - 35495) q^{89} + ( - 800 \beta_{3} - 26248) q^{91} + ( - 3920 \beta_{2} - 95330 \beta_1) q^{93} + (3800 \beta_{2} + 48426 \beta_1) q^{97} + ( - 1094 \beta_{3} - 41888) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 56 q^{9} + 400 q^{11} - 1680 q^{19} - 1992 q^{21} + 9360 q^{29} + 10016 q^{31} - 54864 q^{39} - 10668 q^{41} + 63308 q^{49} + 13200 q^{51} - 163552 q^{59} - 93864 q^{61} + 110088 q^{69} - 14896 q^{71} - 216208 q^{79} - 187324 q^{81} - 141980 q^{89} - 104992 q^{91} - 167552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 61\nu ) / 60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 181\nu ) / 60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 121 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 121 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -61\beta_{2} + 181\beta_1 ) / 2 \) Copy content Toggle raw display

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 19.5242i 0 0 0 35.0483i 0 −138.193 0
49.2 0 11.5242i 0 0 0 27.0483i 0 110.193 0
49.3 0 11.5242i 0 0 0 27.0483i 0 110.193 0
49.4 0 19.5242i 0 0 0 35.0483i 0 −138.193 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.o 4
4.b odd 2 1 200.6.c.f 4
5.b even 2 1 inner 400.6.c.o 4
5.c odd 4 1 400.6.a.r 2
5.c odd 4 1 400.6.a.u 2
20.d odd 2 1 200.6.c.f 4
20.e even 4 1 200.6.a.e 2
20.e even 4 1 200.6.a.f yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.a.e 2 20.e even 4 1
200.6.a.f yes 2 20.e even 4 1
200.6.c.f 4 4.b odd 2 1
200.6.c.f 4 20.d odd 2 1
400.6.a.r 2 5.c odd 4 1
400.6.a.u 2 5.c odd 4 1
400.6.c.o 4 1.a even 1 1 trivial
400.6.c.o 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 514T_{3}^{2} + 50625 \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 514 T^{2} + 50625 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1960 T^{2} + 898704 \) Copy content Toggle raw display
$11$ \( (T^{2} - 200 T - 96281)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 1478560 T^{2} + \cdots + 318150146304 \) Copy content Toggle raw display
$17$ \( T^{4} + 162034 T^{2} + \cdots + 1795640625 \) Copy content Toggle raw display
$19$ \( (T^{2} + 840 T - 662521)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 6534280 T^{2} + \cdots + 1855011312144 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4680 T + 196736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 5008 T - 23931140)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 302523267094416 \) Copy content Toggle raw display
$41$ \( (T^{2} + 5334 T - 204026247)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 558073120 T^{2} + \cdots + 77\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{2} + 169937296)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 1158148872 T^{2} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{2} + 81776 T + 1630122048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 46932 T + 550186580)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 2417557010 T^{2} + \cdots + 13\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( (T^{2} + 7448 T - 14650800)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 5923612530 T^{2} + \cdots + 87\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( (T^{2} + 108104 T + 662040300)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 7481654530 T^{2} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( (T^{2} + 70990 T - 4823083791)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 11650234952 T^{2} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
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