Properties

Label 441.2.p.b
Level $441$
Weight $2$
Character orbit 441.p
Analytic conductor $3.521$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{4} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{4} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{8} + ( \beta_{5} + \beta_{7} ) q^{11} + ( -3 \beta_{4} + 2 \beta_{6} ) q^{16} + ( 5 - \beta_{3} ) q^{22} + ( -3 \beta_{1} - \beta_{2} ) q^{23} + ( 5 - 5 \beta_{4} ) q^{25} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{5} + \beta_{7} ) q^{29} -3 \beta_{5} q^{32} -4 \beta_{6} q^{37} + 2 \beta_{3} q^{43} + ( \beta_{1} + \beta_{2} ) q^{44} + ( -11 - 5 \beta_{3} + 11 \beta_{4} - 5 \beta_{6} ) q^{46} + ( 5 \beta_{1} - 5 \beta_{5} ) q^{50} + ( -5 \beta_{5} + \beta_{7} ) q^{53} + ( 13 \beta_{4} - \beta_{6} ) q^{58} + ( -6 + \beta_{3} ) q^{64} + ( 4 - 4 \beta_{4} ) q^{67} + ( -\beta_{1} - 3 \beta_{2} + \beta_{5} - 3 \beta_{7} ) q^{71} + ( 8 \beta_{5} - 4 \beta_{7} ) q^{74} -8 \beta_{4} q^{79} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{86} + ( -7 + 5 \beta_{3} + 7 \beta_{4} + 5 \beta_{6} ) q^{88} + ( -15 \beta_{1} - 3 \beta_{2} + 15 \beta_{5} - 3 \beta_{7} ) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + O(q^{10}) \) \( 8q + 8q^{4} - 12q^{16} + 40q^{22} + 20q^{25} - 44q^{46} + 52q^{58} - 48q^{64} + 16q^{67} - 32q^{79} - 28q^{88} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 258 \nu \)\()/55\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 148 \)\()/55\)
\(\beta_{4}\)\(=\)\((\)\( -8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576 \)\()/495\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{7} + 55 \nu^{5} - 440 \nu^{3} + 576 \nu \)\()/495\)
\(\beta_{6}\)\(=\)\((\)\( -23 \nu^{6} + 220 \nu^{4} - 1265 \nu^{2} + 1656 \)\()/495\)
\(\beta_{7}\)\(=\)\((\)\( -13 \nu^{7} + 110 \nu^{5} - 715 \nu^{3} + 936 \nu \)\()/165\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 4 \beta_{4} + \beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 6 \beta_{5} + \beta_{2} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(8 \beta_{6} - 23 \beta_{4}\)
\(\nu^{5}\)\(=\)\(8 \beta_{7} - 39 \beta_{5}\)
\(\nu^{6}\)\(=\)\(-55 \beta_{3} - 148\)
\(\nu^{7}\)\(=\)\(-55 \beta_{2} - 258 \beta_{1}\)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\beta_{4}\) \(-1\)

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
−2.23256 + 1.28897i 0 2.32288 4.02334i 0 0 0 6.82058i 0 0
80.2 −1.00781 + 0.581861i 0 −0.322876 + 0.559237i 0 0 0 3.07892i 0 0
80.3 1.00781 0.581861i 0 −0.322876 + 0.559237i 0 0 0 3.07892i 0 0
80.4 2.23256 1.28897i 0 2.32288 4.02334i 0 0 0 6.82058i 0 0
215.1 −2.23256 1.28897i 0 2.32288 + 4.02334i 0 0 0 6.82058i 0 0
215.2 −1.00781 0.581861i 0 −0.322876 0.559237i 0 0 0 3.07892i 0 0
215.3 1.00781 + 0.581861i 0 −0.322876 0.559237i 0 0 0 3.07892i 0 0
215.4 2.23256 + 1.28897i 0 2.32288 + 4.02334i 0 0 0 6.82058i 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 441.2.p.b 8
3.b odd 2 1 inner 441.2.p.b 8
7.b odd 2 1 CM 441.2.p.b 8
7.c even 3 1 63.2.c.a 4
7.c even 3 1 inner 441.2.p.b 8
7.d odd 6 1 63.2.c.a 4
7.d odd 6 1 inner 441.2.p.b 8
21.c even 2 1 inner 441.2.p.b 8
21.g even 6 1 63.2.c.a 4
21.g even 6 1 inner 441.2.p.b 8
21.h odd 6 1 63.2.c.a 4
21.h odd 6 1 inner 441.2.p.b 8
28.f even 6 1 1008.2.k.a 4
28.g odd 6 1 1008.2.k.a 4
35.i odd 6 1 1575.2.b.a 4
35.j even 6 1 1575.2.b.a 4
35.k even 12 2 1575.2.g.d 8
35.l odd 12 2 1575.2.g.d 8
56.j odd 6 1 4032.2.k.c 4
56.k odd 6 1 4032.2.k.b 4
56.m even 6 1 4032.2.k.b 4
56.p even 6 1 4032.2.k.c 4
63.g even 3 1 567.2.o.f 8
63.h even 3 1 567.2.o.f 8
63.i even 6 1 567.2.o.f 8
63.j odd 6 1 567.2.o.f 8
63.k odd 6 1 567.2.o.f 8
63.n odd 6 1 567.2.o.f 8
63.s even 6 1 567.2.o.f 8
63.t odd 6 1 567.2.o.f 8
84.j odd 6 1 1008.2.k.a 4
84.n even 6 1 1008.2.k.a 4
105.o odd 6 1 1575.2.b.a 4
105.p even 6 1 1575.2.b.a 4
105.w odd 12 2 1575.2.g.d 8
105.x even 12 2 1575.2.g.d 8
168.s odd 6 1 4032.2.k.c 4
168.v even 6 1 4032.2.k.b 4
168.ba even 6 1 4032.2.k.c 4
168.be odd 6 1 4032.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a 4 7.c even 3 1
63.2.c.a 4 7.d odd 6 1
63.2.c.a 4 21.g even 6 1
63.2.c.a 4 21.h odd 6 1
441.2.p.b 8 1.a even 1 1 trivial
441.2.p.b 8 3.b odd 2 1 inner
441.2.p.b 8 7.b odd 2 1 CM
441.2.p.b 8 7.c even 3 1 inner
441.2.p.b 8 7.d odd 6 1 inner
441.2.p.b 8 21.c even 2 1 inner
441.2.p.b 8 21.g even 6 1 inner
441.2.p.b 8 21.h odd 6 1 inner
567.2.o.f 8 63.g even 3 1
567.2.o.f 8 63.h even 3 1
567.2.o.f 8 63.i even 6 1
567.2.o.f 8 63.j odd 6 1
567.2.o.f 8 63.k odd 6 1
567.2.o.f 8 63.n odd 6 1
567.2.o.f 8 63.s even 6 1
567.2.o.f 8 63.t odd 6 1
1008.2.k.a 4 28.f even 6 1
1008.2.k.a 4 28.g odd 6 1
1008.2.k.a 4 84.j odd 6 1
1008.2.k.a 4 84.n even 6 1
1575.2.b.a 4 35.i odd 6 1
1575.2.b.a 4 35.j even 6 1
1575.2.b.a 4 105.o odd 6 1
1575.2.b.a 4 105.p even 6 1
1575.2.g.d 8 35.k even 12 2
1575.2.g.d 8 35.l odd 12 2
1575.2.g.d 8 105.w odd 12 2
1575.2.g.d 8 105.x even 12 2
4032.2.k.b 4 56.k odd 6 1
4032.2.k.b 4 56.m even 6 1
4032.2.k.b 4 168.v even 6 1
4032.2.k.b 4 168.be odd 6 1
4032.2.k.c 4 56.j odd 6 1
4032.2.k.c 4 56.p even 6 1
4032.2.k.c 4 168.s odd 6 1
4032.2.k.c 4 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 8 T_{2}^{6} + 55 T_{2}^{4} - 72 T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 81 - 72 T^{2} + 55 T^{4} - 8 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( 1296 - 1584 T^{2} + 1900 T^{4} - 44 T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( 104976 - 29808 T^{2} + 8140 T^{4} - 92 T^{6} + T^{8} \)
$29$ \( ( 2916 + 116 T^{2} + T^{4} )^{2} \)
$31$ \( T^{8} \)
$37$ \( ( 12544 + 112 T^{2} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( ( -28 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( 1296 - 7632 T^{2} + 44908 T^{4} - 212 T^{6} + T^{8} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 16 - 4 T + T^{2} )^{4} \)
$71$ \( ( 12996 + 284 T^{2} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( ( 64 + 8 T + T^{2} )^{4} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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