Properties

Label 441.4.p.b
Level $441$
Weight $4$
Character orbit 441.p
Analytic conductor $26.020$
Analytic rank $0$
Dimension $8$
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.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
Defining polynomial: \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( 8 + 5 \beta_{1} - 8 \beta_{2} + 5 \beta_{4} ) q^{4} + ( 10 \beta_{3} - 5 \beta_{5} + 10 \beta_{6} - 5 \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( 8 + 5 \beta_{1} - 8 \beta_{2} + 5 \beta_{4} ) q^{4} + ( 10 \beta_{3} - 5 \beta_{5} + 10 \beta_{6} - 5 \beta_{7} ) q^{8} + ( 13 \beta_{6} + \beta_{7} ) q^{11} + ( -111 \beta_{2} + 40 \beta_{4} ) q^{16} + ( -205 - 59 \beta_{1} ) q^{22} + ( 7 \beta_{3} - 25 \beta_{5} ) q^{23} + ( 125 - 125 \beta_{2} ) q^{25} + ( 11 \beta_{3} + 37 \beta_{5} + 11 \beta_{6} + 37 \beta_{7} ) q^{29} + 111 \beta_{6} q^{32} + 4 \beta_{4} q^{37} + 202 \beta_{1} q^{43} + ( -219 \beta_{3} + 67 \beta_{5} ) q^{44} + ( 187 + 185 \beta_{1} - 187 \beta_{2} + 185 \beta_{4} ) q^{46} + ( 125 \beta_{3} + 125 \beta_{6} ) q^{50} + ( 67 \beta_{6} + 85 \beta_{7} ) q^{53} + ( -65 \beta_{2} - 167 \beta_{4} ) q^{58} + ( -888 - 235 \beta_{1} ) q^{64} + ( -740 + 740 \beta_{2} ) q^{67} + ( 53 \beta_{3} + 141 \beta_{5} + 53 \beta_{6} + 141 \beta_{7} ) q^{71} + ( 8 \beta_{6} - 4 \beta_{7} ) q^{74} + 1384 \beta_{2} q^{79} + ( 404 \beta_{3} - 202 \beta_{5} ) q^{86} + ( -2065 - 1025 \beta_{1} + 2065 \beta_{2} - 1025 \beta_{4} ) q^{88} + ( 501 \beta_{3} + 15 \beta_{5} + 501 \beta_{6} + 15 \beta_{7} ) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 32q^{4} + O(q^{10}) \) \( 8q + 32q^{4} - 444q^{16} - 1640q^{22} + 500q^{25} + 748q^{46} - 260q^{58} - 7104q^{64} - 2960q^{67} + 5536q^{79} - 8260q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{6} - 148 \)\()/55\)
\(\beta_{2}\)\(=\)\((\)\( -8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576 \)\()/495\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{7} - 241 \nu \)\()/165\)
\(\beta_{4}\)\(=\)\((\)\( -23 \nu^{6} + 220 \nu^{4} - 1265 \nu^{2} + 1656 \)\()/495\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 368 \nu \)\()/55\)
\(\beta_{6}\)\(=\)\((\)\( -38 \nu^{7} + 385 \nu^{5} - 2090 \nu^{3} + 2736 \nu \)\()/1485\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 8 \nu^{5} - 55 \nu^{3} + 72 \nu \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} + 3 \beta_{3}\)\()/9\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 4 \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{7} + 24 \beta_{6} - 7 \beta_{5} + 24 \beta_{3}\)\()/9\)
\(\nu^{4}\)\(=\)\(8 \beta_{4} - 23 \beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(-38 \beta_{7} + 165 \beta_{6}\)\()/9\)
\(\nu^{6}\)\(=\)\(-55 \beta_{1} - 148\)
\(\nu^{7}\)\(=\)\((\)\(241 \beta_{5} - 1104 \beta_{3}\)\()/9\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
−2.23256 + 1.28897i
1.00781 0.581861i
−1.00781 + 0.581861i
2.23256 1.28897i
−2.23256 1.28897i
1.00781 + 0.581861i
−1.00781 0.581861i
2.23256 + 1.28897i
−4.68205 + 2.70318i 0 10.6144 18.3846i 0 0 0 71.5195i 0 0
80.2 −1.44168 + 0.832353i 0 −2.61438 + 4.52824i 0 0 0 22.0220i 0 0
80.3 1.44168 0.832353i 0 −2.61438 + 4.52824i 0 0 0 22.0220i 0 0
80.4 4.68205 2.70318i 0 10.6144 18.3846i 0 0 0 71.5195i 0 0
215.1 −4.68205 2.70318i 0 10.6144 + 18.3846i 0 0 0 71.5195i 0 0
215.2 −1.44168 0.832353i 0 −2.61438 4.52824i 0 0 0 22.0220i 0 0
215.3 1.44168 + 0.832353i 0 −2.61438 4.52824i 0 0 0 22.0220i 0 0
215.4 4.68205 + 2.70318i 0 10.6144 + 18.3846i 0 0 0 71.5195i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.4
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.p.b 8
3.b odd 2 1 inner 441.4.p.b 8
7.b odd 2 1 CM 441.4.p.b 8
7.c even 3 1 63.4.c.a 4
7.c even 3 1 inner 441.4.p.b 8
7.d odd 6 1 63.4.c.a 4
7.d odd 6 1 inner 441.4.p.b 8
21.c even 2 1 inner 441.4.p.b 8
21.g even 6 1 63.4.c.a 4
21.g even 6 1 inner 441.4.p.b 8
21.h odd 6 1 63.4.c.a 4
21.h odd 6 1 inner 441.4.p.b 8
28.f even 6 1 1008.4.k.a 4
28.g odd 6 1 1008.4.k.a 4
84.j odd 6 1 1008.4.k.a 4
84.n even 6 1 1008.4.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.a 4 7.c even 3 1
63.4.c.a 4 7.d odd 6 1
63.4.c.a 4 21.g even 6 1
63.4.c.a 4 21.h odd 6 1
441.4.p.b 8 1.a even 1 1 trivial
441.4.p.b 8 3.b odd 2 1 inner
441.4.p.b 8 7.b odd 2 1 CM
441.4.p.b 8 7.c even 3 1 inner
441.4.p.b 8 7.d odd 6 1 inner
441.4.p.b 8 21.c even 2 1 inner
441.4.p.b 8 21.g even 6 1 inner
441.4.p.b 8 21.h odd 6 1 inner
1008.4.k.a 4 28.f even 6 1
1008.4.k.a 4 28.g odd 6 1
1008.4.k.a 4 84.j odd 6 1
1008.4.k.a 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 32 T_{2}^{6} + 943 T_{2}^{4} - 2592 T_{2}^{2} + 6561 \) acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 6561 - 2592 T^{2} + 943 T^{4} - 32 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( 14818219109136 - 20494439856 T^{2} + 24495532 T^{4} - 5324 T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( 267118165009579536 - 25153313905008 T^{2} + 1851739468 T^{4} - 48668 T^{6} + T^{8} \)
$29$ \( ( 450373284 + 97556 T^{2} + T^{4} )^{2} \)
$31$ \( T^{8} \)
$37$ \( ( 12544 + 112 T^{2} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( ( -285628 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( 6424804038679120656 - 1509445868635728 T^{2} + 352095058348 T^{4} - 595508 T^{6} + T^{8} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 547600 + 740 T + T^{2} )^{4} \)
$71$ \( ( 58795580484 + 1431644 T^{2} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( ( 1915456 - 1384 T + T^{2} )^{4} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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