Properties

Label 441.6.c.a
Level $441$
Weight $6$
Character orbit 441.c
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q -2 \beta_{1} q^{2} + 24 q^{4} -3 \beta_{3} q^{5} -112 \beta_{1} q^{8} +O(q^{10})\) \( q -2 \beta_{1} q^{2} + 24 q^{4} -3 \beta_{3} q^{5} -112 \beta_{1} q^{8} + 12 \beta_{2} q^{10} + 409 \beta_{1} q^{11} + 95 \beta_{2} q^{13} + 320 q^{16} -22 \beta_{3} q^{17} -143 \beta_{2} q^{19} -72 \beta_{3} q^{20} + 1636 q^{22} -3260 \beta_{1} q^{23} -479 q^{25} + 190 \beta_{3} q^{26} -3254 \beta_{1} q^{29} + 187 \beta_{2} q^{31} -4224 \beta_{1} q^{32} + 88 \beta_{2} q^{34} + 203 q^{37} -286 \beta_{3} q^{38} + 672 \beta_{2} q^{40} + 5 \beta_{3} q^{41} + 4697 q^{43} + 9816 \beta_{1} q^{44} -13040 q^{46} -1399 \beta_{3} q^{47} + 958 \beta_{1} q^{50} + 2280 \beta_{2} q^{52} -5246 \beta_{1} q^{53} -2454 \beta_{2} q^{55} -13016 q^{58} -290 \beta_{3} q^{59} -1438 \beta_{2} q^{61} + 374 \beta_{3} q^{62} -6656 q^{64} -41895 \beta_{1} q^{65} + 30197 q^{67} -528 \beta_{3} q^{68} -2801 \beta_{1} q^{71} -4399 \beta_{2} q^{73} -406 \beta_{1} q^{74} -3432 \beta_{2} q^{76} -39385 q^{79} -960 \beta_{3} q^{80} -20 \beta_{2} q^{82} -3707 \beta_{3} q^{83} + 19404 q^{85} -9394 \beta_{1} q^{86} + 91616 q^{88} -6146 \beta_{3} q^{89} -78240 \beta_{1} q^{92} + 5596 \beta_{2} q^{94} + 63063 \beta_{1} q^{95} -170 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 96 q^{4} + O(q^{10}) \) \( 4 q + 96 q^{4} + 1280 q^{16} + 6544 q^{22} - 1916 q^{25} + 812 q^{37} + 18788 q^{43} - 52160 q^{46} - 52064 q^{58} - 26624 q^{64} + 120788 q^{67} - 157540 q^{79} + 77616 q^{85} + 366464 q^{88} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( 7 \nu^{2} - 7 \)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{3} + 28 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 7 \beta_{1}\)\()/14\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 7\)\()/7\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
440.1
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
2.82843i 0 24.0000 −51.4393 0 0 158.392i 0 145.492i
440.2 2.82843i 0 24.0000 51.4393 0 0 158.392i 0 145.492i
440.3 2.82843i 0 24.0000 −51.4393 0 0 158.392i 0 145.492i
440.4 2.82843i 0 24.0000 51.4393 0 0 158.392i 0 145.492i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.c.a 4
3.b odd 2 1 inner 441.6.c.a 4
7.b odd 2 1 inner 441.6.c.a 4
7.c even 3 1 63.6.p.a 4
7.d odd 6 1 63.6.p.a 4
21.c even 2 1 inner 441.6.c.a 4
21.g even 6 1 63.6.p.a 4
21.h odd 6 1 63.6.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.p.a 4 7.c even 3 1
63.6.p.a 4 7.d odd 6 1
63.6.p.a 4 21.g even 6 1
63.6.p.a 4 21.h odd 6 1
441.6.c.a 4 1.a even 1 1 trivial
441.6.c.a 4 3.b odd 2 1 inner
441.6.c.a 4 7.b odd 2 1 inner
441.6.c.a 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 8 \) acting on \(S_{6}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( -2646 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 334562 + T^{2} )^{2} \)
$13$ \( ( 1326675 + T^{2} )^{2} \)
$17$ \( ( -142296 + T^{2} )^{2} \)
$19$ \( ( 3006003 + T^{2} )^{2} \)
$23$ \( ( 21255200 + T^{2} )^{2} \)
$29$ \( ( 21177032 + T^{2} )^{2} \)
$31$ \( ( 5140443 + T^{2} )^{2} \)
$37$ \( ( -203 + T )^{4} \)
$41$ \( ( -7350 + T^{2} )^{2} \)
$43$ \( ( -4697 + T )^{4} \)
$47$ \( ( -575417094 + T^{2} )^{2} \)
$53$ \( ( 55041032 + T^{2} )^{2} \)
$59$ \( ( -24725400 + T^{2} )^{2} \)
$61$ \( ( 303973068 + T^{2} )^{2} \)
$67$ \( ( -30197 + T )^{4} \)
$71$ \( ( 15691202 + T^{2} )^{2} \)
$73$ \( ( 2844626547 + T^{2} )^{2} \)
$79$ \( ( 39385 + T )^{4} \)
$83$ \( ( -4040103606 + T^{2} )^{2} \)
$89$ \( ( -11105354904 + T^{2} )^{2} \)
$97$ \( ( 4248300 + T^{2} )^{2} \)
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