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 $4$

Related objects

Downloads

Learn more

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: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} + 24 q^{4} - 3 \beta_{3} q^{5} - 112 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} + 24 q^{4} - 3 \beta_{3} q^{5} - 112 \beta_1 q^{8} + 12 \beta_{2} q^{10} + 409 \beta_1 q^{11} + 95 \beta_{2} q^{13} + 320 q^{16} - 22 \beta_{3} q^{17} - 143 \beta_{2} q^{19} - 72 \beta_{3} q^{20} + 1636 q^{22} - 3260 \beta_1 q^{23} - 479 q^{25} + 190 \beta_{3} q^{26} - 3254 \beta_1 q^{29} + 187 \beta_{2} q^{31} - 4224 \beta_1 q^{32} + 88 \beta_{2} q^{34} + 203 q^{37} - 286 \beta_{3} q^{38} + 672 \beta_{2} q^{40} + 5 \beta_{3} q^{41} + 4697 q^{43} + 9816 \beta_1 q^{44} - 13040 q^{46} - 1399 \beta_{3} q^{47} + 958 \beta_1 q^{50} + 2280 \beta_{2} q^{52} - 5246 \beta_1 q^{53} - 2454 \beta_{2} q^{55} - 13016 q^{58} - 290 \beta_{3} q^{59} - 1438 \beta_{2} q^{61} + 374 \beta_{3} q^{62} - 6656 q^{64} - 41895 \beta_1 q^{65} + 30197 q^{67} - 528 \beta_{3} q^{68} - 2801 \beta_1 q^{71} - 4399 \beta_{2} q^{73} - 406 \beta_1 q^{74} - 3432 \beta_{2} q^{76} - 39385 q^{79} - 960 \beta_{3} q^{80} - 20 \beta_{2} q^{82} - 3707 \beta_{3} q^{83} + 19404 q^{85} - 9394 \beta_1 q^{86} + 91616 q^{88} - 6146 \beta_{3} q^{89} - 78240 \beta_1 q^{92} + 5596 \beta_{2} q^{94} + 63063 \beta_1 q^{95} - 170 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 96 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 96 q^{4} + 1280 q^{16} + 6544 q^{22} - 1916 q^{25} + 812 q^{37} + 18788 q^{43} - 52160 q^{46} - 52064 q^{58} - 26624 q^{64} + 120788 q^{67} - 157540 q^{79} + 77616 q^{85} + 366464 q^{88}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 7\nu^{2} - 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{3} + 28\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 7\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 7 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

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 \(S_{6}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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