Properties

Label 441.2.p.c
Level $441$
Weight $2$
Character orbit 441.p
Analytic conductor $3.521$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{48}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{48}^{2} + \zeta_{48}^{4} - \zeta_{48}^{12} - \zeta_{48}^{14} ) q^{2} + ( 1 + 2 \zeta_{48}^{6} - \zeta_{48}^{8} - 2 \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{4} + ( 2 \zeta_{48} - \zeta_{48}^{5} - 2 \zeta_{48}^{7} - 2 \zeta_{48}^{9} - \zeta_{48}^{11} + 2 \zeta_{48}^{15} ) q^{5} + ( \zeta_{48}^{2} - \zeta_{48}^{6} - \zeta_{48}^{10} - 3 \zeta_{48}^{12} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{48}^{2} + \zeta_{48}^{4} - \zeta_{48}^{12} - \zeta_{48}^{14} ) q^{2} + ( 1 + 2 \zeta_{48}^{6} - \zeta_{48}^{8} - 2 \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{4} + ( 2 \zeta_{48} - \zeta_{48}^{5} - 2 \zeta_{48}^{7} - 2 \zeta_{48}^{9} - \zeta_{48}^{11} + 2 \zeta_{48}^{15} ) q^{5} + ( \zeta_{48}^{2} - \zeta_{48}^{6} - \zeta_{48}^{10} - 3 \zeta_{48}^{12} ) q^{8} + ( -4 \zeta_{48} - \zeta_{48}^{3} - 4 \zeta_{48}^{7} + \zeta_{48}^{11} - \zeta_{48}^{13} ) q^{10} + ( 2 \zeta_{48}^{4} - 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} + 2 \zeta_{48}^{14} ) q^{11} + ( 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} + \zeta_{48}^{9} + 2 \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{13} -3 \zeta_{48}^{8} q^{16} + ( 3 \zeta_{48} - 2 \zeta_{48}^{3} - 3 \zeta_{48}^{7} + 2 \zeta_{48}^{11} + 2 \zeta_{48}^{13} ) q^{17} + ( 2 \zeta_{48} + 4 \zeta_{48}^{5} + 2 \zeta_{48}^{7} - 2 \zeta_{48}^{9} - 4 \zeta_{48}^{11} - 2 \zeta_{48}^{15} ) q^{19} + ( -7 \zeta_{48}^{3} - 7 \zeta_{48}^{5} + 7 \zeta_{48}^{13} ) q^{20} -2 q^{22} + ( 2 \zeta_{48}^{4} - 2 \zeta_{48}^{12} ) q^{23} + ( -5 - \zeta_{48}^{6} + 5 \zeta_{48}^{8} + \zeta_{48}^{10} + \zeta_{48}^{14} ) q^{25} + ( -\zeta_{48} + 4 \zeta_{48}^{5} + \zeta_{48}^{7} + \zeta_{48}^{9} + 4 \zeta_{48}^{11} - \zeta_{48}^{15} ) q^{26} + ( 2 \zeta_{48}^{2} - 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} - 2 \zeta_{48}^{12} ) q^{29} + ( 2 \zeta_{48} + 4 \zeta_{48}^{3} + 2 \zeta_{48}^{7} - 4 \zeta_{48}^{11} + 4 \zeta_{48}^{13} ) q^{31} + ( 3 \zeta_{48}^{4} - \zeta_{48}^{6} - \zeta_{48}^{10} + \zeta_{48}^{14} ) q^{32} + ( 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} + \zeta_{48}^{9} + 2 \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{34} + ( -\zeta_{48}^{2} + 4 \zeta_{48}^{8} - \zeta_{48}^{14} ) q^{37} + ( 2 \zeta_{48} + 8 \zeta_{48}^{3} - 2 \zeta_{48}^{7} - 8 \zeta_{48}^{11} - 8 \zeta_{48}^{13} ) q^{38} + ( -6 \zeta_{48} - 5 \zeta_{48}^{5} - 6 \zeta_{48}^{7} + 6 \zeta_{48}^{9} + 5 \zeta_{48}^{11} + 6 \zeta_{48}^{15} ) q^{40} + ( 4 \zeta_{48}^{3} + 4 \zeta_{48}^{5} + \zeta_{48}^{9} - 4 \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{41} + ( -4 - 6 \zeta_{48}^{2} - 6 \zeta_{48}^{6} + 6 \zeta_{48}^{10} ) q^{43} + ( 2 \zeta_{48}^{2} - 6 \zeta_{48}^{4} + 6 \zeta_{48}^{12} - 2 \zeta_{48}^{14} ) q^{44} + ( 2 + 2 \zeta_{48}^{6} - 2 \zeta_{48}^{8} - 2 \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{46} + ( 4 \zeta_{48} - 2 \zeta_{48}^{5} - 4 \zeta_{48}^{7} - 4 \zeta_{48}^{9} - 2 \zeta_{48}^{11} + 4 \zeta_{48}^{15} ) q^{47} + ( -6 \zeta_{48}^{2} + 6 \zeta_{48}^{6} + 6 \zeta_{48}^{10} + 7 \zeta_{48}^{12} ) q^{50} + ( 7 \zeta_{48} + 7 \zeta_{48}^{7} ) q^{52} + ( 5 \zeta_{48}^{6} + 5 \zeta_{48}^{10} - 5 \zeta_{48}^{14} ) q^{53} + ( -6 \zeta_{48}^{3} + 6 \zeta_{48}^{5} + 4 \zeta_{48}^{9} - 6 \zeta_{48}^{13} + 4 \zeta_{48}^{15} ) q^{55} + ( -4 \zeta_{48}^{2} - 6 \zeta_{48}^{8} - 4 \zeta_{48}^{14} ) q^{58} + ( -4 \zeta_{48} - 2 \zeta_{48}^{3} + 4 \zeta_{48}^{7} + 2 \zeta_{48}^{11} + 2 \zeta_{48}^{13} ) q^{59} + ( -4 \zeta_{48} - \zeta_{48}^{5} - 4 \zeta_{48}^{7} + 4 \zeta_{48}^{9} + \zeta_{48}^{11} + 4 \zeta_{48}^{15} ) q^{61} + ( 8 \zeta_{48}^{3} + 8 \zeta_{48}^{5} + 2 \zeta_{48}^{9} - 8 \zeta_{48}^{13} - 2 \zeta_{48}^{15} ) q^{62} + ( 7 + 2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} ) q^{64} + ( \zeta_{48}^{2} + 10 \zeta_{48}^{4} - 10 \zeta_{48}^{12} - \zeta_{48}^{14} ) q^{65} + ( -6 \zeta_{48}^{6} + 6 \zeta_{48}^{10} + 6 \zeta_{48}^{14} ) q^{67} + ( -7 \zeta_{48} + 7 \zeta_{48}^{7} + 7 \zeta_{48}^{9} - 7 \zeta_{48}^{15} ) q^{68} + ( -2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{10} + 2 \zeta_{48}^{12} ) q^{71} + ( 2 \zeta_{48} - 3 \zeta_{48}^{3} + 2 \zeta_{48}^{7} + 3 \zeta_{48}^{11} - 3 \zeta_{48}^{13} ) q^{73} + ( 2 \zeta_{48}^{4} + 3 \zeta_{48}^{6} + 3 \zeta_{48}^{10} - 3 \zeta_{48}^{14} ) q^{74} + ( -14 \zeta_{48}^{9} - 14 \zeta_{48}^{15} ) q^{76} + ( 4 \zeta_{48}^{2} + 4 \zeta_{48}^{8} + 4 \zeta_{48}^{14} ) q^{79} + ( -6 \zeta_{48} - 3 \zeta_{48}^{3} + 6 \zeta_{48}^{7} + 3 \zeta_{48}^{11} + 3 \zeta_{48}^{13} ) q^{80} + ( 9 \zeta_{48} + 4 \zeta_{48}^{5} + 9 \zeta_{48}^{7} - 9 \zeta_{48}^{9} - 4 \zeta_{48}^{11} - 9 \zeta_{48}^{15} ) q^{82} + ( -4 \zeta_{48}^{3} - 4 \zeta_{48}^{5} - 8 \zeta_{48}^{9} + 4 \zeta_{48}^{13} + 8 \zeta_{48}^{15} ) q^{83} + ( -8 + 9 \zeta_{48}^{2} + 9 \zeta_{48}^{6} - 9 \zeta_{48}^{10} ) q^{85} + ( -10 \zeta_{48}^{2} - 16 \zeta_{48}^{4} + 16 \zeta_{48}^{12} + 10 \zeta_{48}^{14} ) q^{86} + ( 2 - 4 \zeta_{48}^{6} - 2 \zeta_{48}^{8} + 4 \zeta_{48}^{10} + 4 \zeta_{48}^{14} ) q^{88} + ( -4 \zeta_{48} - 5 \zeta_{48}^{5} + 4 \zeta_{48}^{7} + 4 \zeta_{48}^{9} - 5 \zeta_{48}^{11} - 4 \zeta_{48}^{15} ) q^{89} + ( 4 \zeta_{48}^{2} - 4 \zeta_{48}^{6} - 4 \zeta_{48}^{10} - 2 \zeta_{48}^{12} ) q^{92} + ( -8 \zeta_{48} - 2 \zeta_{48}^{3} - 8 \zeta_{48}^{7} + 2 \zeta_{48}^{11} - 2 \zeta_{48}^{13} ) q^{94} + ( -20 \zeta_{48}^{4} - 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} + 2 \zeta_{48}^{14} ) q^{95} + ( -3 \zeta_{48}^{3} + 3 \zeta_{48}^{5} + 2 \zeta_{48}^{9} - 3 \zeta_{48}^{13} + 2 \zeta_{48}^{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{4} + O(q^{10}) \) \( 16q + 8q^{4} - 24q^{16} - 32q^{22} - 40q^{25} + 32q^{37} - 64q^{43} + 16q^{46} - 48q^{58} + 112q^{64} + 32q^{79} - 128q^{85} + 16q^{88} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\zeta_{48}^{2}\) \(-1\)

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
−0.608761 0.793353i
0.608761 + 0.793353i
0.130526 0.991445i
−0.130526 + 0.991445i
0.793353 0.608761i
−0.793353 + 0.608761i
0.991445 + 0.130526i
−0.991445 0.130526i
−0.608761 + 0.793353i
0.608761 0.793353i
0.130526 + 0.991445i
−0.130526 0.991445i
0.793353 + 0.608761i
−0.793353 0.608761i
0.991445 0.130526i
−0.991445 + 0.130526i
−2.09077 + 1.20711i 0 1.91421 3.31552i −1.68925 2.92586i 0 0 4.41421i 0 7.06365 + 4.07820i
80.2 −2.09077 + 1.20711i 0 1.91421 3.31552i 1.68925 + 2.92586i 0 0 4.41421i 0 −7.06365 4.07820i
80.3 −0.358719 + 0.207107i 0 −0.914214 + 1.58346i −1.46508 2.53759i 0 0 1.58579i 0 1.05110 + 0.606854i
80.4 −0.358719 + 0.207107i 0 −0.914214 + 1.58346i 1.46508 + 2.53759i 0 0 1.58579i 0 −1.05110 0.606854i
80.5 0.358719 0.207107i 0 −0.914214 + 1.58346i −1.46508 2.53759i 0 0 1.58579i 0 −1.05110 0.606854i
80.6 0.358719 0.207107i 0 −0.914214 + 1.58346i 1.46508 + 2.53759i 0 0 1.58579i 0 1.05110 + 0.606854i
80.7 2.09077 1.20711i 0 1.91421 3.31552i −1.68925 2.92586i 0 0 4.41421i 0 −7.06365 4.07820i
80.8 2.09077 1.20711i 0 1.91421 3.31552i 1.68925 + 2.92586i 0 0 4.41421i 0 7.06365 + 4.07820i
215.1 −2.09077 1.20711i 0 1.91421 + 3.31552i −1.68925 + 2.92586i 0 0 4.41421i 0 7.06365 4.07820i
215.2 −2.09077 1.20711i 0 1.91421 + 3.31552i 1.68925 2.92586i 0 0 4.41421i 0 −7.06365 + 4.07820i
215.3 −0.358719 0.207107i 0 −0.914214 1.58346i −1.46508 + 2.53759i 0 0 1.58579i 0 1.05110 0.606854i
215.4 −0.358719 0.207107i 0 −0.914214 1.58346i 1.46508 2.53759i 0 0 1.58579i 0 −1.05110 + 0.606854i
215.5 0.358719 + 0.207107i 0 −0.914214 1.58346i −1.46508 + 2.53759i 0 0 1.58579i 0 −1.05110 + 0.606854i
215.6 0.358719 + 0.207107i 0 −0.914214 1.58346i 1.46508 2.53759i 0 0 1.58579i 0 1.05110 0.606854i
215.7 2.09077 + 1.20711i 0 1.91421 + 3.31552i −1.68925 + 2.92586i 0 0 4.41421i 0 −7.06365 + 4.07820i
215.8 2.09077 + 1.20711i 0 1.91421 + 3.31552i 1.68925 2.92586i 0 0 4.41421i 0 7.06365 4.07820i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.8
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 441.2.p.c 16
3.b odd 2 1 inner 441.2.p.c 16
7.b odd 2 1 inner 441.2.p.c 16
7.c even 3 1 441.2.c.b 8
7.c even 3 1 inner 441.2.p.c 16
7.d odd 6 1 441.2.c.b 8
7.d odd 6 1 inner 441.2.p.c 16
21.c even 2 1 inner 441.2.p.c 16
21.g even 6 1 441.2.c.b 8
21.g even 6 1 inner 441.2.p.c 16
21.h odd 6 1 441.2.c.b 8
21.h odd 6 1 inner 441.2.p.c 16
28.f even 6 1 7056.2.k.g 8
28.g odd 6 1 7056.2.k.g 8
84.j odd 6 1 7056.2.k.g 8
84.n even 6 1 7056.2.k.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.c.b 8 7.c even 3 1
441.2.c.b 8 7.d odd 6 1
441.2.c.b 8 21.g even 6 1
441.2.c.b 8 21.h odd 6 1
441.2.p.c 16 1.a even 1 1 trivial
441.2.p.c 16 3.b odd 2 1 inner
441.2.p.c 16 7.b odd 2 1 inner
441.2.p.c 16 7.c even 3 1 inner
441.2.p.c 16 7.d odd 6 1 inner
441.2.p.c 16 21.c even 2 1 inner
441.2.p.c 16 21.g even 6 1 inner
441.2.p.c 16 21.h odd 6 1 inner
7056.2.k.g 8 28.f even 6 1
7056.2.k.g 8 28.g odd 6 1
7056.2.k.g 8 84.j odd 6 1
7056.2.k.g 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6 T_{2}^{6} + 35 T_{2}^{4} - 6 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 6 T^{2} + 35 T^{4} - 6 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 9604 + 1960 T^{2} + 302 T^{4} + 20 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 256 - 384 T^{2} + 560 T^{4} - 24 T^{6} + T^{8} )^{2} \)
$13$ \( ( 98 + 20 T^{2} + T^{4} )^{4} \)
$17$ \( ( 9604 + 5096 T^{2} + 2606 T^{4} + 52 T^{6} + T^{8} )^{2} \)
$19$ \( ( 2458624 - 125440 T^{2} + 4832 T^{4} - 80 T^{6} + T^{8} )^{2} \)
$23$ \( ( 16 - 4 T^{2} + T^{4} )^{4} \)
$29$ \( ( 16 + 24 T^{2} + T^{4} )^{4} \)
$31$ \( ( 2458624 - 125440 T^{2} + 4832 T^{4} - 80 T^{6} + T^{8} )^{2} \)
$37$ \( ( 196 - 112 T + 50 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$41$ \( ( 98 - 68 T^{2} + T^{4} )^{4} \)
$43$ \( ( -56 + 8 T + T^{2} )^{8} \)
$47$ \( ( 2458624 + 125440 T^{2} + 4832 T^{4} + 80 T^{6} + T^{8} )^{2} \)
$53$ \( ( 2500 - 50 T^{2} + T^{4} )^{4} \)
$59$ \( ( 2458624 + 125440 T^{2} + 4832 T^{4} + 80 T^{6} + T^{8} )^{2} \)
$61$ \( ( 9604 - 6664 T^{2} + 4526 T^{4} - 68 T^{6} + T^{8} )^{2} \)
$67$ \( ( 5184 + 72 T^{2} + T^{4} )^{4} \)
$71$ \( ( 16 + 24 T^{2} + T^{4} )^{4} \)
$73$ \( ( 9604 - 5096 T^{2} + 2606 T^{4} - 52 T^{6} + T^{8} )^{2} \)
$79$ \( ( 256 + 128 T + 80 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$83$ \( ( 25088 - 320 T^{2} + T^{4} )^{4} \)
$89$ \( ( 23059204 + 787528 T^{2} + 22094 T^{4} + 164 T^{6} + T^{8} )^{2} \)
$97$ \( ( 98 + 52 T^{2} + T^{4} )^{4} \)
show more
show less