Properties

Label 441.4.e.i
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
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.

\(f(q)\) \(=\) \( q + ( - 8 \zeta_{6} + 8) q^{4}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 8 \zeta_{6} + 8) q^{4} - 70 q^{13} - 64 \zeta_{6} q^{16} - 56 \zeta_{6} q^{19} + ( - 125 \zeta_{6} + 125) q^{25} + (308 \zeta_{6} - 308) q^{31} - 110 \zeta_{6} q^{37} - 520 q^{43} + (560 \zeta_{6} - 560) q^{52} - 182 \zeta_{6} q^{61} - 512 q^{64} + ( - 880 \zeta_{6} + 880) q^{67} + (1190 \zeta_{6} - 1190) q^{73} - 448 q^{76} - 884 \zeta_{6} q^{79} - 1330 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{4} - 140 q^{13} - 64 q^{16} - 56 q^{19} + 125 q^{25} - 308 q^{31} - 110 q^{37} - 1040 q^{43} - 560 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} - 1190 q^{73} - 896 q^{76} - 884 q^{79} - 2660 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 0 4.00000 + 6.92820i 0 0 0 0 0 0
361.1 0 0 4.00000 6.92820i 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 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.e.i 2
3.b odd 2 1 CM 441.4.e.i 2
7.b odd 2 1 441.4.e.j 2
7.c even 3 1 9.4.a.a 1
7.c even 3 1 inner 441.4.e.i 2
7.d odd 6 1 441.4.a.f 1
7.d odd 6 1 441.4.e.j 2
21.c even 2 1 441.4.e.j 2
21.g even 6 1 441.4.a.f 1
21.g even 6 1 441.4.e.j 2
21.h odd 6 1 9.4.a.a 1
21.h odd 6 1 inner 441.4.e.i 2
28.g odd 6 1 144.4.a.d 1
35.j even 6 1 225.4.a.d 1
35.l odd 12 2 225.4.b.g 2
56.k odd 6 1 576.4.a.l 1
56.p even 6 1 576.4.a.m 1
63.g even 3 1 81.4.c.b 2
63.h even 3 1 81.4.c.b 2
63.j odd 6 1 81.4.c.b 2
63.n odd 6 1 81.4.c.b 2
77.h odd 6 1 1089.4.a.g 1
84.n even 6 1 144.4.a.d 1
91.r even 6 1 1521.4.a.g 1
105.o odd 6 1 225.4.a.d 1
105.x even 12 2 225.4.b.g 2
168.s odd 6 1 576.4.a.m 1
168.v even 6 1 576.4.a.l 1
231.l even 6 1 1089.4.a.g 1
273.w odd 6 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 7.c even 3 1
9.4.a.a 1 21.h odd 6 1
81.4.c.b 2 63.g even 3 1
81.4.c.b 2 63.h even 3 1
81.4.c.b 2 63.j odd 6 1
81.4.c.b 2 63.n odd 6 1
144.4.a.d 1 28.g odd 6 1
144.4.a.d 1 84.n even 6 1
225.4.a.d 1 35.j even 6 1
225.4.a.d 1 105.o odd 6 1
225.4.b.g 2 35.l odd 12 2
225.4.b.g 2 105.x even 12 2
441.4.a.f 1 7.d odd 6 1
441.4.a.f 1 21.g even 6 1
441.4.e.i 2 1.a even 1 1 trivial
441.4.e.i 2 3.b odd 2 1 CM
441.4.e.i 2 7.c even 3 1 inner
441.4.e.i 2 21.h odd 6 1 inner
441.4.e.j 2 7.b odd 2 1
441.4.e.j 2 7.d odd 6 1
441.4.e.j 2 21.c even 2 1
441.4.e.j 2 21.g even 6 1
576.4.a.l 1 56.k odd 6 1
576.4.a.l 1 168.v even 6 1
576.4.a.m 1 56.p even 6 1
576.4.a.m 1 168.s odd 6 1
1089.4.a.g 1 77.h odd 6 1
1089.4.a.g 1 231.l even 6 1
1521.4.a.g 1 91.r even 6 1
1521.4.a.g 1 273.w odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13} + 70 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 56T + 3136 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 308T + 94864 \) Copy content Toggle raw display
$37$ \( T^{2} + 110T + 12100 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 520)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 182T + 33124 \) Copy content Toggle raw display
$67$ \( T^{2} - 880T + 774400 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1190 T + 1416100 \) Copy content Toggle raw display
$79$ \( T^{2} + 884T + 781456 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 1330)^{2} \) Copy content Toggle raw display
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