Properties

Label 441.2.e.b
Level 441
Weight 2
Character orbit 441.e
Analytic conductor 3.521
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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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 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} -3 q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} -3 q^{8} + ( 2 - 2 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} + 2 q^{13} + \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 2 q^{20} -4 q^{22} + ( 1 - \zeta_{6} ) q^{25} -2 \zeta_{6} q^{26} + 2 q^{29} + ( -5 + 5 \zeta_{6} ) q^{32} -6 q^{34} -6 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} -6 \zeta_{6} q^{40} + 2 q^{41} -4 q^{43} -4 \zeta_{6} q^{44} - q^{50} + ( 2 - 2 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} + 8 q^{55} -2 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} + 7 q^{64} + 4 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + ( -6 + 6 \zeta_{6} ) q^{73} + ( -6 + 6 \zeta_{6} ) q^{74} + 4 q^{76} + 16 \zeta_{6} q^{79} + ( -2 + 2 \zeta_{6} ) q^{80} -2 \zeta_{6} q^{82} -12 q^{83} + 12 q^{85} + 4 \zeta_{6} q^{86} + ( -12 + 12 \zeta_{6} ) q^{88} + 14 \zeta_{6} q^{89} + ( -8 + 8 \zeta_{6} ) q^{95} -18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{4} + 2q^{5} - 6q^{8} + O(q^{10}) \) \( 2q - q^{2} + q^{4} + 2q^{5} - 6q^{8} + 2q^{10} + 4q^{11} + 4q^{13} + q^{16} + 6q^{17} + 4q^{19} + 4q^{20} - 8q^{22} + q^{25} - 2q^{26} + 4q^{29} - 5q^{32} - 12q^{34} - 6q^{37} + 4q^{38} - 6q^{40} + 4q^{41} - 8q^{43} - 4q^{44} - 2q^{50} + 2q^{52} + 6q^{53} + 16q^{55} - 2q^{58} - 12q^{59} - 2q^{61} + 14q^{64} + 4q^{65} - 4q^{67} - 6q^{68} - 6q^{73} - 6q^{74} + 8q^{76} + 16q^{79} - 2q^{80} - 2q^{82} - 24q^{83} + 24q^{85} + 4q^{86} - 12q^{88} + 14q^{89} - 8q^{95} - 36q^{97} + O(q^{100}) \)

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 1.00000 1.73205i 0 0 −3.00000 0 1.00000 + 1.73205i
361.1 −0.500000 0.866025i 0 0.500000 0.866025i 1.00000 + 1.73205i 0 0 −3.00000 0 1.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.b 2
3.b odd 2 1 147.2.e.c 2
7.b odd 2 1 441.2.e.a 2
7.c even 3 1 441.2.a.f 1
7.c even 3 1 inner 441.2.e.b 2
7.d odd 6 1 63.2.a.a 1
7.d odd 6 1 441.2.e.a 2
12.b even 2 1 2352.2.q.e 2
21.c even 2 1 147.2.e.b 2
21.g even 6 1 21.2.a.a 1
21.g even 6 1 147.2.e.b 2
21.h odd 6 1 147.2.a.a 1
21.h odd 6 1 147.2.e.c 2
28.f even 6 1 1008.2.a.l 1
28.g odd 6 1 7056.2.a.p 1
35.i odd 6 1 1575.2.a.c 1
35.k even 12 2 1575.2.d.a 2
56.j odd 6 1 4032.2.a.h 1
56.m even 6 1 4032.2.a.k 1
63.i even 6 1 567.2.f.g 2
63.k odd 6 1 567.2.f.b 2
63.s even 6 1 567.2.f.g 2
63.t odd 6 1 567.2.f.b 2
77.i even 6 1 7623.2.a.g 1
84.h odd 2 1 2352.2.q.x 2
84.j odd 6 1 336.2.a.a 1
84.j odd 6 1 2352.2.q.x 2
84.n even 6 1 2352.2.a.v 1
84.n even 6 1 2352.2.q.e 2
105.o odd 6 1 3675.2.a.n 1
105.p even 6 1 525.2.a.d 1
105.w odd 12 2 525.2.d.a 2
168.s odd 6 1 9408.2.a.bv 1
168.v even 6 1 9408.2.a.m 1
168.ba even 6 1 1344.2.a.g 1
168.be odd 6 1 1344.2.a.s 1
231.k odd 6 1 2541.2.a.j 1
273.ba even 6 1 3549.2.a.c 1
336.bo even 12 2 5376.2.c.r 2
336.br odd 12 2 5376.2.c.l 2
357.s even 6 1 6069.2.a.b 1
399.s odd 6 1 7581.2.a.d 1
420.be odd 6 1 8400.2.a.bn 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 21.g even 6 1
63.2.a.a 1 7.d odd 6 1
147.2.a.a 1 21.h odd 6 1
147.2.e.b 2 21.c even 2 1
147.2.e.b 2 21.g even 6 1
147.2.e.c 2 3.b odd 2 1
147.2.e.c 2 21.h odd 6 1
336.2.a.a 1 84.j odd 6 1
441.2.a.f 1 7.c even 3 1
441.2.e.a 2 7.b odd 2 1
441.2.e.a 2 7.d odd 6 1
441.2.e.b 2 1.a even 1 1 trivial
441.2.e.b 2 7.c even 3 1 inner
525.2.a.d 1 105.p even 6 1
525.2.d.a 2 105.w odd 12 2
567.2.f.b 2 63.k odd 6 1
567.2.f.b 2 63.t odd 6 1
567.2.f.g 2 63.i even 6 1
567.2.f.g 2 63.s even 6 1
1008.2.a.l 1 28.f even 6 1
1344.2.a.g 1 168.ba even 6 1
1344.2.a.s 1 168.be odd 6 1
1575.2.a.c 1 35.i odd 6 1
1575.2.d.a 2 35.k even 12 2
2352.2.a.v 1 84.n even 6 1
2352.2.q.e 2 12.b even 2 1
2352.2.q.e 2 84.n even 6 1
2352.2.q.x 2 84.h odd 2 1
2352.2.q.x 2 84.j odd 6 1
2541.2.a.j 1 231.k odd 6 1
3549.2.a.c 1 273.ba even 6 1
3675.2.a.n 1 105.o odd 6 1
4032.2.a.h 1 56.j odd 6 1
4032.2.a.k 1 56.m even 6 1
5376.2.c.l 2 336.br odd 12 2
5376.2.c.r 2 336.bo even 12 2
6069.2.a.b 1 357.s even 6 1
7056.2.a.p 1 28.g odd 6 1
7581.2.a.d 1 399.s odd 6 1
7623.2.a.g 1 77.i even 6 1
8400.2.a.bn 1 420.be odd 6 1
9408.2.a.m 1 168.v even 6 1
9408.2.a.bv 1 168.s odd 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ 1
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 14 T + 107 T^{2} - 1246 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 18 T + 97 T^{2} )^{2} \)
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