Properties

Label 441.4.e.q
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} - \beta_{3} ) q^{2} + ( 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{5} + ( 41 + 5 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta_{1} - \beta_{3} ) q^{2} + ( 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{5} + ( 41 + 5 \beta_{2} ) q^{8} + ( -30 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{10} + ( 8 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{11} + ( 14 - 12 \beta_{2} ) q^{13} + ( -55 - 55 \beta_{1} - 27 \beta_{3} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -44 - 44 \beta_{1} + 24 \beta_{3} ) q^{19} + ( -98 - 26 \beta_{2} ) q^{20} + ( -132 - 12 \beta_{2} ) q^{22} + ( 20 + 20 \beta_{1} - 34 \beta_{3} ) q^{23} + ( -65 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{25} + ( 154 + 154 \beta_{1} + 10 \beta_{3} ) q^{26} + ( 138 - 24 \beta_{2} ) q^{29} + ( -16 \beta_{1} - 72 \beta_{2} + 72 \beta_{3} ) q^{31} + ( 105 \beta_{1} - 69 \beta_{2} + 69 \beta_{3} ) q^{32} + ( 30 + 6 \beta_{2} ) q^{34} + ( 106 + 106 \beta_{1} + 36 \beta_{3} ) q^{37} + ( -292 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{38} + ( 222 + 222 \beta_{1} + 102 \beta_{3} ) q^{40} + ( 210 + 30 \beta_{2} ) q^{41} + ( 212 - 48 \beta_{2} ) q^{43} + ( 364 + 364 \beta_{1} + 76 \beta_{3} ) q^{44} + ( 456 \beta_{1} - 48 \beta_{2} + 48 \beta_{3} ) q^{46} + ( -40 - 40 \beta_{1} + 68 \beta_{3} ) q^{47} + ( 103 - 41 \beta_{2} ) q^{50} + ( -406 \beta_{1} + 78 \beta_{2} - 78 \beta_{3} ) q^{52} + ( 554 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 264 + 24 \beta_{2} ) q^{55} + ( 198 + 198 \beta_{1} - 90 \beta_{3} ) q^{58} + ( -460 \beta_{1} - 116 \beta_{2} + 116 \beta_{3} ) q^{59} + ( 250 + 250 \beta_{1} - 72 \beta_{3} ) q^{61} + ( 992 + 128 \beta_{2} ) q^{62} + ( 631 + 27 \beta_{2} ) q^{64} + ( -308 - 308 \beta_{1} - 20 \beta_{3} ) q^{65} + ( 20 \beta_{1} - 108 \beta_{2} + 108 \beta_{3} ) q^{67} + ( -98 - 98 \beta_{1} - 26 \beta_{3} ) q^{68} + ( -492 + 30 \beta_{2} ) q^{71} + ( 530 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{73} + ( -610 \beta_{1} + 178 \beta_{2} - 178 \beta_{3} ) q^{74} + ( -700 - 108 \beta_{2} ) q^{76} + ( 232 + 232 \beta_{1} + 108 \beta_{3} ) q^{79} + ( -866 \beta_{1} + 218 \beta_{2} - 218 \beta_{3} ) q^{80} + ( -630 - 630 \beta_{1} - 270 \beta_{3} ) q^{82} + ( -924 - 96 \beta_{2} ) q^{83} + ( -60 - 12 \beta_{2} ) q^{85} + ( 460 + 460 \beta_{1} - 116 \beta_{3} ) q^{86} + ( -372 \beta_{1} + 420 \beta_{2} - 420 \beta_{3} ) q^{88} + ( 254 + 254 \beta_{1} - 142 \beta_{3} ) q^{89} + ( 1288 + 280 \beta_{2} ) q^{92} + ( -912 \beta_{1} + 96 \beta_{2} - 96 \beta_{3} ) q^{94} + ( 584 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{95} + ( 266 + 276 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} - 17q^{4} + 6q^{5} + 174q^{8} + O(q^{10}) \) \( 4q - 3q^{2} - 17q^{4} + 6q^{5} + 174q^{8} + 66q^{10} - 6q^{11} + 32q^{13} - 137q^{16} - 6q^{17} - 64q^{19} - 444q^{20} - 552q^{22} + 6q^{23} + 118q^{25} + 318q^{26} + 504q^{29} - 40q^{31} - 279q^{32} + 132q^{34} + 248q^{37} + 588q^{38} + 546q^{40} + 900q^{41} + 752q^{43} + 804q^{44} - 960q^{46} - 12q^{47} + 330q^{50} + 890q^{52} - 1104q^{53} + 1104q^{55} + 306q^{58} + 804q^{59} + 428q^{61} + 4224q^{62} + 2578q^{64} - 636q^{65} - 148q^{67} - 222q^{68} - 1908q^{71} - 1072q^{73} + 1398q^{74} - 3016q^{76} + 572q^{79} + 1950q^{80} - 1530q^{82} - 3888q^{83} - 264q^{85} + 804q^{86} + 1164q^{88} + 366q^{89} + 5712q^{92} + 1920q^{94} - 1176q^{95} + 1616q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 25 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 9 \nu + 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{3} + 2 \nu^{2} + 8 \nu - 25 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} - 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 14 \beta_{1} + 13\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 19\)\()/3\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
2.13746 0.656712i
−1.63746 + 1.52274i
2.13746 + 0.656712i
−1.63746 1.52274i
−2.63746 + 4.56821i 0 −9.91238 17.1687i 5.27492 9.13642i 0 0 62.3746 0 27.8248 + 48.1939i
226.2 1.13746 1.97014i 0 1.41238 + 2.44631i −2.27492 + 3.94027i 0 0 24.6254 0 5.17525 + 8.96379i
361.1 −2.63746 4.56821i 0 −9.91238 + 17.1687i 5.27492 + 9.13642i 0 0 62.3746 0 27.8248 48.1939i
361.2 1.13746 + 1.97014i 0 1.41238 2.44631i −2.27492 3.94027i 0 0 24.6254 0 5.17525 8.96379i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.q 4
3.b odd 2 1 147.4.e.l 4
7.b odd 2 1 441.4.e.p 4
7.c even 3 1 63.4.a.e 2
7.c even 3 1 inner 441.4.e.q 4
7.d odd 6 1 441.4.a.r 2
7.d odd 6 1 441.4.e.p 4
21.c even 2 1 147.4.e.m 4
21.g even 6 1 147.4.a.i 2
21.g even 6 1 147.4.e.m 4
21.h odd 6 1 21.4.a.c 2
21.h odd 6 1 147.4.e.l 4
28.g odd 6 1 1008.4.a.ba 2
35.j even 6 1 1575.4.a.p 2
84.j odd 6 1 2352.4.a.bz 2
84.n even 6 1 336.4.a.m 2
105.o odd 6 1 525.4.a.n 2
105.x even 12 2 525.4.d.g 4
168.s odd 6 1 1344.4.a.bg 2
168.v even 6 1 1344.4.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 21.h odd 6 1
63.4.a.e 2 7.c even 3 1
147.4.a.i 2 21.g even 6 1
147.4.e.l 4 3.b odd 2 1
147.4.e.l 4 21.h odd 6 1
147.4.e.m 4 21.c even 2 1
147.4.e.m 4 21.g even 6 1
336.4.a.m 2 84.n even 6 1
441.4.a.r 2 7.d odd 6 1
441.4.e.p 4 7.b odd 2 1
441.4.e.p 4 7.d odd 6 1
441.4.e.q 4 1.a even 1 1 trivial
441.4.e.q 4 7.c even 3 1 inner
525.4.a.n 2 105.o odd 6 1
525.4.d.g 4 105.x even 12 2
1008.4.a.ba 2 28.g odd 6 1
1344.4.a.bg 2 168.s odd 6 1
1344.4.a.bo 2 168.v even 6 1
1575.4.a.p 2 35.j even 6 1
2352.4.a.bz 2 84.j odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 3 T_{2}^{3} + 21 T_{2}^{2} - 36 T_{2} + 144 \)
\( T_{5}^{4} - 6 T_{5}^{3} + 84 T_{5}^{2} + 288 T_{5} + 2304 \)
\( T_{13}^{2} - 16 T_{13} - 1988 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 144 - 36 T + 21 T^{2} + 3 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 2304 + 288 T + 84 T^{2} - 6 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 2005056 - 8496 T + 1452 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( ( -1988 - 16 T + T^{2} )^{2} \)
$17$ \( 2304 - 288 T + 84 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( 51609856 - 459776 T + 11280 T^{2} + 64 T^{3} + T^{4} \)
$23$ \( 271063296 + 98784 T + 16500 T^{2} - 6 T^{3} + T^{4} \)
$29$ \( ( 7668 - 252 T + T^{2} )^{2} \)
$31$ \( 5398134784 - 2938880 T + 75072 T^{2} + 40 T^{3} + T^{4} \)
$37$ \( 9560464 + 766816 T + 64596 T^{2} - 248 T^{3} + T^{4} \)
$41$ \( ( 37800 - 450 T + T^{2} )^{2} \)
$43$ \( ( 2512 - 376 T + T^{2} )^{2} \)
$47$ \( 4337012736 - 790272 T + 66000 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 92705634576 + 336141504 T + 914340 T^{2} + 1104 T^{3} + T^{4} \)
$59$ \( 908660736 + 24235776 T + 676560 T^{2} - 804 T^{3} + T^{4} \)
$61$ \( 788261776 + 12016528 T + 211260 T^{2} - 428 T^{3} + T^{4} \)
$67$ \( 25836061696 - 23788928 T + 182640 T^{2} + 148 T^{3} + T^{4} \)
$71$ \( ( 214704 + 954 T + T^{2} )^{2} \)
$73$ \( 81364139536 + 305781568 T + 863940 T^{2} + 1072 T^{3} + T^{4} \)
$79$ \( 7126061056 + 48285952 T + 411600 T^{2} - 572 T^{3} + T^{4} \)
$83$ \( ( 813456 + 1944 T + T^{2} )^{2} \)
$89$ \( 64438807104 + 92908368 T + 387804 T^{2} - 366 T^{3} + T^{4} \)
$97$ \( ( -922292 - 808 T + T^{2} )^{2} \)
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