Properties

Label 441.2.e.d
Level 441
Weight 2
Character orbit 441.e
Analytic conductor 3.521
Analytic rank 0
Dimension 2
CM discriminant -7
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + 3 q^{8} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + 3 q^{8} + ( 4 - 4 \zeta_{6} ) q^{11} + \zeta_{6} q^{16} + 4 q^{22} + 8 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} -2 q^{29} + ( 5 - 5 \zeta_{6} ) q^{32} + 6 \zeta_{6} q^{37} -12 q^{43} -4 \zeta_{6} q^{44} + ( -8 + 8 \zeta_{6} ) q^{46} + 5 q^{50} + ( -10 + 10 \zeta_{6} ) q^{53} -2 \zeta_{6} q^{58} + 7 q^{64} + ( -4 + 4 \zeta_{6} ) q^{67} -16 q^{71} + ( -6 + 6 \zeta_{6} ) q^{74} -8 \zeta_{6} q^{79} -12 \zeta_{6} q^{86} + ( 12 - 12 \zeta_{6} ) q^{88} + 8 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{4} + 6q^{8} + O(q^{10}) \) \( 2q + q^{2} + q^{4} + 6q^{8} + 4q^{11} + q^{16} + 8q^{22} + 8q^{23} + 5q^{25} - 4q^{29} + 5q^{32} + 6q^{37} - 24q^{43} - 4q^{44} - 8q^{46} + 10q^{50} - 10q^{53} - 2q^{58} + 14q^{64} - 4q^{67} - 32q^{71} - 6q^{74} - 8q^{79} - 12q^{86} + 12q^{88} + 16q^{92} + O(q^{100}) \)

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 0 3.00000 0 0
361.1 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0 3.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d 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.e.d 2
3.b odd 2 1 49.2.c.a 2
7.b odd 2 1 CM 441.2.e.d 2
7.c even 3 1 441.2.a.c 1
7.c even 3 1 inner 441.2.e.d 2
7.d odd 6 1 441.2.a.c 1
7.d odd 6 1 inner 441.2.e.d 2
12.b even 2 1 784.2.i.f 2
21.c even 2 1 49.2.c.a 2
21.g even 6 1 49.2.a.a 1
21.g even 6 1 49.2.c.a 2
21.h odd 6 1 49.2.a.a 1
21.h odd 6 1 49.2.c.a 2
28.f even 6 1 7056.2.a.bg 1
28.g odd 6 1 7056.2.a.bg 1
84.h odd 2 1 784.2.i.f 2
84.j odd 6 1 784.2.a.f 1
84.j odd 6 1 784.2.i.f 2
84.n even 6 1 784.2.a.f 1
84.n even 6 1 784.2.i.f 2
105.o odd 6 1 1225.2.a.c 1
105.p even 6 1 1225.2.a.c 1
105.w odd 12 2 1225.2.b.c 2
105.x even 12 2 1225.2.b.c 2
168.s odd 6 1 3136.2.a.n 1
168.v even 6 1 3136.2.a.o 1
168.ba even 6 1 3136.2.a.n 1
168.be odd 6 1 3136.2.a.o 1
231.k odd 6 1 5929.2.a.c 1
231.l even 6 1 5929.2.a.c 1
273.w odd 6 1 8281.2.a.d 1
273.ba even 6 1 8281.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 21.g even 6 1
49.2.a.a 1 21.h odd 6 1
49.2.c.a 2 3.b odd 2 1
49.2.c.a 2 21.c even 2 1
49.2.c.a 2 21.g even 6 1
49.2.c.a 2 21.h odd 6 1
441.2.a.c 1 7.c even 3 1
441.2.a.c 1 7.d odd 6 1
441.2.e.d 2 1.a even 1 1 trivial
441.2.e.d 2 7.b odd 2 1 CM
441.2.e.d 2 7.c even 3 1 inner
441.2.e.d 2 7.d odd 6 1 inner
784.2.a.f 1 84.j odd 6 1
784.2.a.f 1 84.n even 6 1
784.2.i.f 2 12.b even 2 1
784.2.i.f 2 84.h odd 2 1
784.2.i.f 2 84.j odd 6 1
784.2.i.f 2 84.n even 6 1
1225.2.a.c 1 105.o odd 6 1
1225.2.a.c 1 105.p even 6 1
1225.2.b.c 2 105.w odd 12 2
1225.2.b.c 2 105.x even 12 2
3136.2.a.n 1 168.s odd 6 1
3136.2.a.n 1 168.ba even 6 1
3136.2.a.o 1 168.v even 6 1
3136.2.a.o 1 168.be odd 6 1
5929.2.a.c 1 231.k odd 6 1
5929.2.a.c 1 231.l even 6 1
7056.2.a.bg 1 28.f even 6 1
7056.2.a.bg 1 28.g odd 6 1
8281.2.a.d 1 273.w odd 6 1
8281.2.a.d 1 273.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \)
\( T_{5} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - T^{2} - 2 T^{3} + 4 T^{4} \)
$3$ 1
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 - 19 T^{2} + 361 T^{4} \)
$23$ \( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 12 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 10 T + 47 T^{2} + 530 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 61 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 16 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 73 T^{2} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 97 T^{2} )^{2} \)
show more
show less