Properties

Label 441.4.e.t
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{2} q^{4} -9 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{2} q^{4} -9 \beta_{3} q^{8} + ( -10 \beta_{1} - 10 \beta_{3} ) q^{11} + ( 55 + 55 \beta_{2} ) q^{16} + 70 q^{22} -82 \beta_{1} q^{23} -125 \beta_{2} q^{25} -100 \beta_{3} q^{29} + ( -17 \beta_{1} - 17 \beta_{3} ) q^{32} + ( 450 + 450 \beta_{2} ) q^{37} + 180 q^{43} -10 \beta_{1} q^{44} -574 \beta_{2} q^{46} -125 \beta_{3} q^{50} + ( -188 \beta_{1} - 188 \beta_{3} ) q^{53} + ( 700 + 700 \beta_{2} ) q^{58} + 559 q^{64} -740 \beta_{2} q^{67} + 370 \beta_{3} q^{71} + ( 450 \beta_{1} + 450 \beta_{3} ) q^{74} + ( 1384 + 1384 \beta_{2} ) q^{79} + 180 \beta_{1} q^{86} -630 \beta_{2} q^{88} + 82 \beta_{3} q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} + 110q^{16} + 280q^{22} + 250q^{25} + 900q^{37} + 720q^{43} + 1148q^{46} + 1400q^{58} + 2236q^{64} + 1480q^{67} + 2768q^{79} + 1260q^{88} + O(q^{100}) \)

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 0.500000 + 0.866025i 0 0 0 −23.8118 0 0
226.2 1.32288 2.29129i 0 0.500000 + 0.866025i 0 0 0 23.8118 0 0
361.1 −1.32288 2.29129i 0 0.500000 0.866025i 0 0 0 −23.8118 0 0
361.2 1.32288 + 2.29129i 0 0.500000 0.866025i 0 0 0 23.8118 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 441.4.e.t 4
3.b odd 2 1 inner 441.4.e.t 4
7.b odd 2 1 CM 441.4.e.t 4
7.c even 3 1 441.4.a.p 2
7.c even 3 1 inner 441.4.e.t 4
7.d odd 6 1 441.4.a.p 2
7.d odd 6 1 inner 441.4.e.t 4
21.c even 2 1 inner 441.4.e.t 4
21.g even 6 1 441.4.a.p 2
21.g even 6 1 inner 441.4.e.t 4
21.h odd 6 1 441.4.a.p 2
21.h odd 6 1 inner 441.4.e.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.p 2 7.c even 3 1
441.4.a.p 2 7.d odd 6 1
441.4.a.p 2 21.g even 6 1
441.4.a.p 2 21.h odd 6 1
441.4.e.t 4 1.a even 1 1 trivial
441.4.e.t 4 3.b odd 2 1 inner
441.4.e.t 4 7.b odd 2 1 CM
441.4.e.t 4 7.c even 3 1 inner
441.4.e.t 4 7.d odd 6 1 inner
441.4.e.t 4 21.c even 2 1 inner
441.4.e.t 4 21.g even 6 1 inner
441.4.e.t 4 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 49 + 7 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 490000 + 700 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 2215396624 + 47068 T^{2} + T^{4} \)
$29$ \( ( -70000 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 202500 - 450 T + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -180 + T )^{4} \)
$47$ \( T^{4} \)
$53$ \( 61210718464 + 247408 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( 547600 - 740 T + T^{2} )^{2} \)
$71$ \( ( -958300 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( 1915456 - 1384 T + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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