Properties

Label 441.4.e.r
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
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: \( k \) \(=\) \( 4 \)
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: \(\Q(\sqrt{-3}, \sqrt{19})\)
Defining polynomial: \(x^{4} + 19 x^{2} + 361\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 11 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + 3 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 11 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + 3 \beta_{3} q^{8} + 38 \beta_{2} q^{10} + ( 10 \beta_{1} + 10 \beta_{3} ) q^{11} + 82 q^{13} + ( 31 + 31 \beta_{2} ) q^{16} + ( 18 \beta_{1} + 18 \beta_{3} ) q^{17} + ( 20 + 20 \beta_{2} ) q^{19} + 22 \beta_{3} q^{20} -190 q^{22} -30 \beta_{1} q^{23} -49 \beta_{2} q^{25} + 82 \beta_{1} q^{26} + 56 \beta_{3} q^{29} + 156 \beta_{2} q^{31} + ( 55 \beta_{1} + 55 \beta_{3} ) q^{32} -342 q^{34} + ( -186 - 186 \beta_{2} ) q^{37} + ( 20 \beta_{1} + 20 \beta_{3} ) q^{38} + ( -114 - 114 \beta_{2} ) q^{40} -38 \beta_{3} q^{41} + 164 q^{43} -110 \beta_{1} q^{44} -570 \beta_{2} q^{46} + 108 \beta_{1} q^{47} -49 \beta_{3} q^{50} + 902 \beta_{2} q^{52} + ( 36 \beta_{1} + 36 \beta_{3} ) q^{53} -380 q^{55} + ( -1064 - 1064 \beta_{2} ) q^{58} + ( -36 \beta_{1} - 36 \beta_{3} ) q^{59} + ( -790 - 790 \beta_{2} ) q^{61} + 156 \beta_{3} q^{62} -797 q^{64} + 164 \beta_{1} q^{65} -44 \beta_{2} q^{67} -198 \beta_{1} q^{68} + 102 \beta_{3} q^{71} + 126 \beta_{2} q^{73} + ( -186 \beta_{1} - 186 \beta_{3} ) q^{74} -220 q^{76} + ( 712 + 712 \beta_{2} ) q^{79} + ( 62 \beta_{1} + 62 \beta_{3} ) q^{80} + ( 722 + 722 \beta_{2} ) q^{82} -336 \beta_{3} q^{83} -684 q^{85} + 164 \beta_{1} q^{86} -570 \beta_{2} q^{88} -334 \beta_{1} q^{89} -330 \beta_{3} q^{92} + 2052 \beta_{2} q^{94} + ( 40 \beta_{1} + 40 \beta_{3} ) q^{95} + 798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 22q^{4} + O(q^{10}) \) \( 4q - 22q^{4} - 76q^{10} + 328q^{13} + 62q^{16} + 40q^{19} - 760q^{22} + 98q^{25} - 312q^{31} - 1368q^{34} - 372q^{37} - 228q^{40} + 656q^{43} + 1140q^{46} - 1804q^{52} - 1520q^{55} - 2128q^{58} - 1580q^{61} - 3188q^{64} + 88q^{67} - 252q^{73} - 880q^{76} + 1424q^{79} + 1444q^{82} - 2736q^{85} + 1140q^{88} - 4104q^{94} + 3192q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 19 x^{2} + 361\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/19\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(19 \beta_{2}\)
\(\nu^{3}\)\(=\)\(19 \beta_{3}\)

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
−2.17945 + 3.77492i
2.17945 3.77492i
−2.17945 3.77492i
2.17945 + 3.77492i
−2.17945 + 3.77492i 0 −5.50000 9.52628i −4.35890 + 7.54983i 0 0 13.0767 0 −19.0000 32.9090i
226.2 2.17945 3.77492i 0 −5.50000 9.52628i 4.35890 7.54983i 0 0 −13.0767 0 −19.0000 32.9090i
361.1 −2.17945 3.77492i 0 −5.50000 + 9.52628i −4.35890 7.54983i 0 0 13.0767 0 −19.0000 + 32.9090i
361.2 2.17945 + 3.77492i 0 −5.50000 + 9.52628i 4.35890 + 7.54983i 0 0 −13.0767 0 −19.0000 + 32.9090i
\(n\): 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.c even 3 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.r 4
3.b odd 2 1 inner 441.4.e.r 4
7.b odd 2 1 441.4.e.s 4
7.c even 3 1 63.4.a.d 2
7.c even 3 1 inner 441.4.e.r 4
7.d odd 6 1 441.4.a.q 2
7.d odd 6 1 441.4.e.s 4
21.c even 2 1 441.4.e.s 4
21.g even 6 1 441.4.a.q 2
21.g even 6 1 441.4.e.s 4
21.h odd 6 1 63.4.a.d 2
21.h odd 6 1 inner 441.4.e.r 4
28.g odd 6 1 1008.4.a.be 2
35.j even 6 1 1575.4.a.t 2
84.n even 6 1 1008.4.a.be 2
105.o odd 6 1 1575.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 7.c even 3 1
63.4.a.d 2 21.h odd 6 1
441.4.a.q 2 7.d odd 6 1
441.4.a.q 2 21.g even 6 1
441.4.e.r 4 1.a even 1 1 trivial
441.4.e.r 4 3.b odd 2 1 inner
441.4.e.r 4 7.c even 3 1 inner
441.4.e.r 4 21.h odd 6 1 inner
441.4.e.s 4 7.b odd 2 1
441.4.e.s 4 7.d odd 6 1
441.4.e.s 4 21.c even 2 1
441.4.e.s 4 21.g even 6 1
1008.4.a.be 2 28.g odd 6 1
1008.4.a.be 2 84.n even 6 1
1575.4.a.t 2 35.j even 6 1
1575.4.a.t 2 105.o odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 19 T_{2}^{2} + 361 \)
\( T_{5}^{4} + 76 T_{5}^{2} + 5776 \)
\( T_{13} - 82 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 361 + 19 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 5776 + 76 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 3610000 + 1900 T^{2} + T^{4} \)
$13$ \( ( -82 + T )^{4} \)
$17$ \( 37896336 + 6156 T^{2} + T^{4} \)
$19$ \( ( 400 - 20 T + T^{2} )^{2} \)
$23$ \( 292410000 + 17100 T^{2} + T^{4} \)
$29$ \( ( -59584 + T^{2} )^{2} \)
$31$ \( ( 24336 + 156 T + T^{2} )^{2} \)
$37$ \( ( 34596 + 186 T + T^{2} )^{2} \)
$41$ \( ( -27436 + T^{2} )^{2} \)
$43$ \( ( -164 + T )^{4} \)
$47$ \( 49113651456 + 221616 T^{2} + T^{4} \)
$53$ \( 606341376 + 24624 T^{2} + T^{4} \)
$59$ \( 606341376 + 24624 T^{2} + T^{4} \)
$61$ \( ( 624100 + 790 T + T^{2} )^{2} \)
$67$ \( ( 1936 - 44 T + T^{2} )^{2} \)
$71$ \( ( -197676 + T^{2} )^{2} \)
$73$ \( ( 15876 + 126 T + T^{2} )^{2} \)
$79$ \( ( 506944 - 712 T + T^{2} )^{2} \)
$83$ \( ( -2145024 + T^{2} )^{2} \)
$89$ \( 4492551550096 + 2119564 T^{2} + T^{4} \)
$97$ \( ( -798 + T )^{4} \)
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