Properties

Label 441.2.e.h
Level 441
Weight 2
Character orbit 441.e
Analytic conductor 3.521
Analytic rank 0
Dimension 4
CM discriminant -7
Inner twists 8

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 5 \beta_{2} q^{4} + 3 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 5 \beta_{2} q^{4} + 3 \beta_{3} q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -11 - 11 \beta_{2} ) q^{16} -14 q^{22} + 2 \beta_{1} q^{23} -5 \beta_{2} q^{25} -4 \beta_{3} q^{29} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{32} + ( -6 - 6 \beta_{2} ) q^{37} + 12 q^{43} -10 \beta_{1} q^{44} + 14 \beta_{2} q^{46} -5 \beta_{3} q^{50} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{53} + ( 28 + 28 \beta_{2} ) q^{58} + 13 q^{64} + 4 \beta_{2} q^{67} -2 \beta_{3} q^{71} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{74} + ( -8 - 8 \beta_{2} ) q^{79} + 12 \beta_{1} q^{86} -42 \beta_{2} q^{88} + 10 \beta_{3} q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 10q^{4} + O(q^{10}) \) \( 4q - 10q^{4} - 22q^{16} - 56q^{22} + 10q^{25} - 12q^{37} + 48q^{43} - 28q^{46} + 56q^{58} + 52q^{64} - 8q^{67} - 16q^{79} + 84q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

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

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i 0 −2.50000 4.33013i 0 0 0 7.93725 0 0
226.2 1.32288 2.29129i 0 −2.50000 4.33013i 0 0 0 −7.93725 0 0
361.1 −1.32288 2.29129i 0 −2.50000 + 4.33013i 0 0 0 7.93725 0 0
361.2 1.32288 + 2.29129i 0 −2.50000 + 4.33013i 0 0 0 −7.93725 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.h 4
3.b odd 2 1 inner 441.2.e.h 4
7.b odd 2 1 CM 441.2.e.h 4
7.c even 3 1 441.2.a.h 2
7.c even 3 1 inner 441.2.e.h 4
7.d odd 6 1 441.2.a.h 2
7.d odd 6 1 inner 441.2.e.h 4
21.c even 2 1 inner 441.2.e.h 4
21.g even 6 1 441.2.a.h 2
21.g even 6 1 inner 441.2.e.h 4
21.h odd 6 1 441.2.a.h 2
21.h odd 6 1 inner 441.2.e.h 4
28.f even 6 1 7056.2.a.co 2
28.g odd 6 1 7056.2.a.co 2
84.j odd 6 1 7056.2.a.co 2
84.n even 6 1 7056.2.a.co 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 7.c even 3 1
441.2.a.h 2 7.d odd 6 1
441.2.a.h 2 21.g even 6 1
441.2.a.h 2 21.h odd 6 1
441.2.e.h 4 1.a even 1 1 trivial
441.2.e.h 4 3.b odd 2 1 inner
441.2.e.h 4 7.b odd 2 1 CM
441.2.e.h 4 7.c even 3 1 inner
441.2.e.h 4 7.d odd 6 1 inner
441.2.e.h 4 21.c even 2 1 inner
441.2.e.h 4 21.g even 6 1 inner
441.2.e.h 4 21.h odd 6 1 inner
7056.2.a.co 2 28.f even 6 1
7056.2.a.co 2 28.g odd 6 1
7056.2.a.co 2 84.j odd 6 1
7056.2.a.co 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{4} + 7 T_{2}^{2} + 49 \)
\( T_{5} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \)
$3$ 1
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ 1
$11$ \( 1 + 6 T^{2} - 85 T^{4} + 726 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \)
$23$ \( 1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 54 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( 1 + 6 T^{2} - 2773 T^{4} + 16854 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 114 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 97 T^{2} )^{4} \)
show more
show less