Properties

Label 441.4.e.v
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{2} + ( 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{4} + ( 10 + 7 \beta_{1} + 10 \beta_{2} ) q^{5} + ( 9 - 11 \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{2} + ( 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{4} + ( 10 + 7 \beta_{1} + 10 \beta_{2} ) q^{5} + ( 9 - 11 \beta_{3} ) q^{8} + ( 17 \beta_{1} + 24 \beta_{2} + 17 \beta_{3} ) q^{10} + ( -24 \beta_{1} + 10 \beta_{2} - 24 \beta_{3} ) q^{11} + ( 52 - 25 \beta_{3} ) q^{13} + ( -9 + 36 \beta_{1} - 9 \beta_{2} ) q^{16} + ( 45 \beta_{1} - 58 \beta_{2} + 45 \beta_{3} ) q^{17} + ( -96 + 22 \beta_{1} - 96 \beta_{2} ) q^{19} + ( 22 - 15 \beta_{3} ) q^{20} + ( 38 - 14 \beta_{3} ) q^{22} + ( 14 - 28 \beta_{1} + 14 \beta_{2} ) q^{23} + ( 140 \beta_{1} + 73 \beta_{2} + 140 \beta_{3} ) q^{25} + ( 102 + 77 \beta_{1} + 102 \beta_{2} ) q^{26} + ( -148 - 62 \beta_{3} ) q^{29} + ( -50 \beta_{1} - 52 \beta_{2} - 50 \beta_{3} ) q^{31} + ( -61 \beta_{1} - 9 \beta_{2} - 61 \beta_{3} ) q^{32} + ( -32 - 13 \beta_{3} ) q^{34} + ( 124 - 48 \beta_{1} + 124 \beta_{2} ) q^{37} + ( -74 \beta_{1} - 52 \beta_{2} - 74 \beta_{3} ) q^{38} + ( 244 + 173 \beta_{1} + 244 \beta_{2} ) q^{40} + ( -10 - 219 \beta_{3} ) q^{41} + ( -360 + 100 \beta_{3} ) q^{43} + ( 146 - 140 \beta_{1} + 146 \beta_{2} ) q^{44} + ( -14 \beta_{1} - 42 \beta_{2} - 14 \beta_{3} ) q^{46} + ( -48 + 250 \beta_{1} - 48 \beta_{2} ) q^{47} + ( -353 + 213 \beta_{3} ) q^{50} + ( -21 \beta_{1} - 160 \beta_{2} - 21 \beta_{3} ) q^{52} + ( 360 \beta_{1} - 134 \beta_{2} + 360 \beta_{3} ) q^{53} + ( 236 - 170 \beta_{3} ) q^{55} + ( -24 - 86 \beta_{1} - 24 \beta_{2} ) q^{58} + ( 226 \beta_{1} + 308 \beta_{2} + 226 \beta_{3} ) q^{59} + ( -8 + 3 \beta_{1} - 8 \beta_{2} ) q^{61} + ( 152 - 102 \beta_{3} ) q^{62} + ( 59 - 358 \beta_{3} ) q^{64} + ( 870 + 614 \beta_{1} + 870 \beta_{2} ) q^{65} + ( -524 \beta_{1} - 72 \beta_{2} - 524 \beta_{3} ) q^{67} + ( -470 + 341 \beta_{1} - 470 \beta_{2} ) q^{68} + ( -494 - 232 \beta_{3} ) q^{71} + ( 401 \beta_{1} + 52 \beta_{2} + 401 \beta_{3} ) q^{73} + ( 76 \beta_{1} + 28 \beta_{2} + 76 \beta_{3} ) q^{74} + ( -568 - 302 \beta_{3} ) q^{76} + ( 472 - 236 \beta_{1} + 472 \beta_{2} ) q^{79} + ( 297 \beta_{1} + 414 \beta_{2} + 297 \beta_{3} ) q^{80} + ( 428 + 209 \beta_{1} + 428 \beta_{2} ) q^{82} + ( -508 + 80 \beta_{3} ) q^{83} + ( -50 + 44 \beta_{3} ) q^{85} + ( -560 - 460 \beta_{1} - 560 \beta_{2} ) q^{86} + ( -106 \beta_{1} - 438 \beta_{2} - 106 \beta_{3} ) q^{88} + ( -194 + 339 \beta_{1} - 194 \beta_{2} ) q^{89} + ( 182 + 168 \beta_{3} ) q^{92} + ( 202 \beta_{1} + 452 \beta_{2} + 202 \beta_{3} ) q^{94} + ( -452 \beta_{1} - 652 \beta_{2} - 452 \beta_{3} ) q^{95} + ( 244 + 599 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 10q^{4} + 20q^{5} + 36q^{8} + O(q^{10}) \) \( 4q + 2q^{2} + 10q^{4} + 20q^{5} + 36q^{8} - 48q^{10} - 20q^{11} + 208q^{13} - 18q^{16} + 116q^{17} - 192q^{19} + 88q^{20} + 152q^{22} + 28q^{23} - 146q^{25} + 204q^{26} - 592q^{29} + 104q^{31} + 18q^{32} - 128q^{34} + 248q^{37} + 104q^{38} + 488q^{40} - 40q^{41} - 1440q^{43} + 292q^{44} + 84q^{46} - 96q^{47} - 1412q^{50} + 320q^{52} + 268q^{53} + 944q^{55} - 48q^{58} - 616q^{59} - 16q^{61} + 608q^{62} + 236q^{64} + 1740q^{65} + 144q^{67} - 940q^{68} - 1976q^{71} - 104q^{73} - 56q^{74} - 2272q^{76} + 944q^{79} - 828q^{80} + 856q^{82} - 2032q^{83} - 200q^{85} - 1120q^{86} + 876q^{88} - 388q^{89} + 728q^{92} - 904q^{94} + 1304q^{95} + 976q^{97} + O(q^{100}) \)

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

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

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.207107 + 0.358719i 0 3.91421 + 6.77962i 0.0502525 0.0870399i 0 0 −6.55635 0 0.0208153 + 0.0360531i
226.2 1.20711 2.09077i 0 1.08579 + 1.88064i 9.94975 17.2335i 0 0 24.5563 0 −24.0208 41.6053i
361.1 −0.207107 0.358719i 0 3.91421 6.77962i 0.0502525 + 0.0870399i 0 0 −6.55635 0 0.0208153 0.0360531i
361.2 1.20711 + 2.09077i 0 1.08579 1.88064i 9.94975 + 17.2335i 0 0 24.5563 0 −24.0208 + 41.6053i
\(n\): 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.v 4
3.b odd 2 1 147.4.e.k 4
7.b odd 2 1 441.4.e.u 4
7.c even 3 1 441.4.a.n 2
7.c even 3 1 inner 441.4.e.v 4
7.d odd 6 1 441.4.a.o 2
7.d odd 6 1 441.4.e.u 4
21.c even 2 1 147.4.e.j 4
21.g even 6 1 147.4.a.k yes 2
21.g even 6 1 147.4.e.j 4
21.h odd 6 1 147.4.a.j 2
21.h odd 6 1 147.4.e.k 4
84.j odd 6 1 2352.4.a.bl 2
84.n even 6 1 2352.4.a.cf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.j 2 21.h odd 6 1
147.4.a.k yes 2 21.g even 6 1
147.4.e.j 4 21.c even 2 1
147.4.e.j 4 21.g even 6 1
147.4.e.k 4 3.b odd 2 1
147.4.e.k 4 21.h odd 6 1
441.4.a.n 2 7.c even 3 1
441.4.a.o 2 7.d odd 6 1
441.4.e.u 4 7.b odd 2 1
441.4.e.u 4 7.d odd 6 1
441.4.e.v 4 1.a even 1 1 trivial
441.4.e.v 4 7.c even 3 1 inner
2352.4.a.bl 2 84.j odd 6 1
2352.4.a.cf 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 \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{4} - 2 T_{2}^{3} + 5 T_{2}^{2} + 2 T_{2} + 1 \)
\( T_{5}^{4} - 20 T_{5}^{3} + 398 T_{5}^{2} - 40 T_{5} + 4 \)
\( T_{13}^{2} - 104 T_{13} + 1454 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 4 - 40 T + 398 T^{2} - 20 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 1106704 - 21040 T + 1452 T^{2} + 20 T^{3} + T^{4} \)
$13$ \( ( 1454 - 104 T + T^{2} )^{2} \)
$17$ \( 470596 + 79576 T + 14142 T^{2} - 116 T^{3} + T^{4} \)
$19$ \( 68029504 + 1583616 T + 28616 T^{2} + 192 T^{3} + T^{4} \)
$23$ \( 1882384 + 38416 T + 2156 T^{2} - 28 T^{3} + T^{4} \)
$29$ \( ( 14216 + 296 T + T^{2} )^{2} \)
$31$ \( 5271616 + 238784 T + 13112 T^{2} - 104 T^{3} + T^{4} \)
$37$ \( 115949824 - 2670464 T + 50736 T^{2} - 248 T^{3} + T^{4} \)
$41$ \( ( -95822 + 20 T + T^{2} )^{2} \)
$43$ \( ( 109600 + 720 T + T^{2} )^{2} \)
$47$ \( 15054308416 - 11778816 T + 131912 T^{2} + 96 T^{3} + T^{4} \)
$53$ \( 58198667536 + 64653392 T + 313068 T^{2} - 268 T^{3} + T^{4} \)
$59$ \( 53114944 - 4489408 T + 386744 T^{2} + 616 T^{3} + T^{4} \)
$61$ \( 2116 + 736 T + 210 T^{2} + 16 T^{3} + T^{4} \)
$67$ \( 295901185024 + 78331392 T + 564704 T^{2} - 144 T^{3} + T^{4} \)
$71$ \( ( 136388 + 988 T + T^{2} )^{2} \)
$73$ \( 101695934404 - 33165392 T + 329714 T^{2} + 104 T^{3} + T^{4} \)
$79$ \( 12408177664 - 105154048 T + 779744 T^{2} - 944 T^{3} + T^{4} \)
$83$ \( ( 245264 + 1016 T + T^{2} )^{2} \)
$89$ \( 36943146436 - 74575928 T + 342750 T^{2} + 388 T^{3} + T^{4} \)
$97$ \( ( -658066 - 488 T + T^{2} )^{2} \)
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