Properties

Label 441.4.p.a
Level $441$
Weight $4$
Character orbit 441.p
Analytic conductor $26.020$
Analytic rank $0$
Dimension $8$
CM no
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: 8.0.9948826238976.7
Defining polynomial: \(x^{8} + 36 x^{6} + 935 x^{4} + 12996 x^{2} + 130321\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 \beta_{4} q^{2} + ( \beta_{2} - \beta_{5} ) q^{5} + ( 16 \beta_{3} + 16 \beta_{4} ) q^{8} +O(q^{10})\) \( q -2 \beta_{4} q^{2} + ( \beta_{2} - \beta_{5} ) q^{5} + ( 16 \beta_{3} + 16 \beta_{4} ) q^{8} -2 \beta_{7} q^{10} -11 \beta_{3} q^{11} -\beta_{6} q^{13} + 64 \beta_{1} q^{16} + 3 \beta_{2} q^{17} + ( -3 \beta_{6} - 3 \beta_{7} ) q^{19} -44 q^{22} + 55 \beta_{4} q^{23} + ( -319 + 319 \beta_{1} ) q^{25} + ( -4 \beta_{2} + 4 \beta_{5} ) q^{26} + ( 89 \beta_{3} + 89 \beta_{4} ) q^{29} + 8 \beta_{7} q^{31} + 6 \beta_{6} q^{34} + 184 \beta_{1} q^{37} -12 \beta_{2} q^{38} + ( 16 \beta_{6} + 16 \beta_{7} ) q^{40} -5 \beta_{5} q^{41} -190 q^{43} + ( -220 + 220 \beta_{1} ) q^{46} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{47} + ( 638 \beta_{3} + 638 \beta_{4} ) q^{50} + 253 \beta_{3} q^{53} -11 \beta_{6} q^{55} + 356 \beta_{1} q^{58} -4 \beta_{2} q^{59} + ( -22 \beta_{6} - 22 \beta_{7} ) q^{61} + 32 \beta_{5} q^{62} -512 q^{64} -444 \beta_{4} q^{65} + ( -296 + 296 \beta_{1} ) q^{67} + ( 233 \beta_{3} + 233 \beta_{4} ) q^{71} -27 \beta_{7} q^{73} + 368 \beta_{3} q^{74} -836 \beta_{1} q^{79} + 64 \beta_{2} q^{80} + ( -10 \beta_{6} - 10 \beta_{7} ) q^{82} -58 \beta_{5} q^{83} -1332 q^{85} + 380 \beta_{4} q^{86} + ( 352 - 352 \beta_{1} ) q^{88} + ( 33 \beta_{2} - 33 \beta_{5} ) q^{89} -4 \beta_{7} q^{94} + 1332 \beta_{3} q^{95} + 19 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 256q^{16} - 352q^{22} - 1276q^{25} + 736q^{37} - 1520q^{43} - 880q^{46} + 1424q^{58} - 4096q^{64} - 1184q^{67} - 3344q^{79} - 10656q^{85} + 1408q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 36 x^{6} + 935 x^{4} + 12996 x^{2} + 130321\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 36 \nu^{6} + 935 \nu^{4} + 33660 \nu^{2} + 467856 \)\()/337535\)
\(\beta_{2}\)\(=\)\((\)\( -148 \nu^{6} + 33660 \nu^{4} + 536690 \nu^{2} + 10227852 \)\()/337535\)
\(\beta_{3}\)\(=\)\((\)\( 251 \nu^{7} + 15895 \nu^{5} + 234685 \nu^{3} + 3261996 \nu \)\()/6413165\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 2899 \nu \)\()/17765\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{6} + 72 \nu^{4} + 1148 \nu^{2} + 12996 \)\()/361\)
\(\beta_{6}\)\(=\)\((\)\( -148 \nu^{7} - 6050 \nu^{5} - 178090 \nu^{3} - 4107458 \nu \)\()/377245\)
\(\beta_{7}\)\(=\)\((\)\( -4682 \nu^{7} - 102850 \nu^{5} - 3027530 \nu^{3} + 13410428 \nu \)\()/6413165\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} + 6 \beta_{4}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} + \beta_{2} + 108 \beta_{1} - 108\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-34 \beta_{7} - 17 \beta_{6} - 330 \beta_{4} - 330 \beta_{3}\)\()/12\)
\(\nu^{4}\)\(=\)\(6 \beta_{5} + 6 \beta_{2} - 287 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(251 \beta_{7} + 502 \beta_{6} + 9714 \beta_{3}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(935 \beta_{5} - 1870 \beta_{2} + 23004\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(2899 \beta_{7} - 2899 \beta_{6} + 230574 \beta_{4}\)\()/12\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
−1.53821 4.07847i
2.76295 + 3.37136i
1.53821 + 4.07847i
−2.76295 3.37136i
−1.53821 + 4.07847i
2.76295 3.37136i
1.53821 4.07847i
−2.76295 + 3.37136i
−2.44949 + 1.41421i 0 0 −10.5357 18.2483i 0 0 22.6274i 0 51.6140 + 29.7993i
80.2 −2.44949 + 1.41421i 0 0 10.5357 + 18.2483i 0 0 22.6274i 0 −51.6140 29.7993i
80.3 2.44949 1.41421i 0 0 −10.5357 18.2483i 0 0 22.6274i 0 −51.6140 29.7993i
80.4 2.44949 1.41421i 0 0 10.5357 + 18.2483i 0 0 22.6274i 0 51.6140 + 29.7993i
215.1 −2.44949 1.41421i 0 0 −10.5357 + 18.2483i 0 0 22.6274i 0 51.6140 29.7993i
215.2 −2.44949 1.41421i 0 0 10.5357 18.2483i 0 0 22.6274i 0 −51.6140 + 29.7993i
215.3 2.44949 + 1.41421i 0 0 −10.5357 + 18.2483i 0 0 22.6274i 0 −51.6140 + 29.7993i
215.4 2.44949 + 1.41421i 0 0 10.5357 18.2483i 0 0 22.6274i 0 51.6140 29.7993i
\(n\): 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
3.b odd 2 1 inner
7.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.a 8
3.b odd 2 1 inner 441.4.p.a 8
7.b odd 2 1 inner 441.4.p.a 8
7.c even 3 1 63.4.c.b 4
7.c even 3 1 inner 441.4.p.a 8
7.d odd 6 1 63.4.c.b 4
7.d odd 6 1 inner 441.4.p.a 8
21.c even 2 1 inner 441.4.p.a 8
21.g even 6 1 63.4.c.b 4
21.g even 6 1 inner 441.4.p.a 8
21.h odd 6 1 63.4.c.b 4
21.h odd 6 1 inner 441.4.p.a 8
28.f even 6 1 1008.4.k.b 4
28.g odd 6 1 1008.4.k.b 4
84.j odd 6 1 1008.4.k.b 4
84.n even 6 1 1008.4.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.b 4 7.c even 3 1
63.4.c.b 4 7.d odd 6 1
63.4.c.b 4 21.g even 6 1
63.4.c.b 4 21.h odd 6 1
441.4.p.a 8 1.a even 1 1 trivial
441.4.p.a 8 3.b odd 2 1 inner
441.4.p.a 8 7.b odd 2 1 inner
441.4.p.a 8 7.c even 3 1 inner
441.4.p.a 8 7.d odd 6 1 inner
441.4.p.a 8 21.c even 2 1 inner
441.4.p.a 8 21.g even 6 1 inner
441.4.p.a 8 21.h odd 6 1 inner
1008.4.k.b 4 28.f even 6 1
1008.4.k.b 4 28.g odd 6 1
1008.4.k.b 4 84.j odd 6 1
1008.4.k.b 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 8 T_{2}^{2} + 64 \) acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 197136 + 444 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 58564 - 242 T^{2} + T^{4} )^{2} \)
$13$ \( ( 888 + T^{2} )^{4} \)
$17$ \( ( 15968016 + 3996 T^{2} + T^{4} )^{2} \)
$19$ \( ( 63872064 - 7992 T^{2} + T^{4} )^{2} \)
$23$ \( ( 36602500 - 6050 T^{2} + T^{4} )^{2} \)
$29$ \( ( 15842 + T^{2} )^{4} \)
$31$ \( ( 3229876224 - 56832 T^{2} + T^{4} )^{2} \)
$37$ \( ( 33856 - 184 T + T^{2} )^{4} \)
$41$ \( ( -11100 + T^{2} )^{4} \)
$43$ \( ( 190 + T )^{8} \)
$47$ \( ( 3154176 + 1776 T^{2} + T^{4} )^{2} \)
$53$ \( ( 16388608324 - 128018 T^{2} + T^{4} )^{2} \)
$59$ \( ( 50466816 + 7104 T^{2} + T^{4} )^{2} \)
$61$ \( ( 184721163264 - 429792 T^{2} + T^{4} )^{2} \)
$67$ \( ( 87616 + 296 T + T^{2} )^{4} \)
$71$ \( ( 108578 + T^{2} )^{4} \)
$73$ \( ( 419064611904 - 647352 T^{2} + T^{4} )^{2} \)
$79$ \( ( 698896 + 836 T + T^{2} )^{4} \)
$83$ \( ( -1493616 + T^{2} )^{4} \)
$89$ \( ( 233787722256 + 483516 T^{2} + T^{4} )^{2} \)
$97$ \( ( 320568 + T^{2} )^{4} \)
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