Properties

Label 441.4.e.a
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
CM discriminant -7
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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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.

\(f(q)\) \(=\) \( q -5 \zeta_{6} q^{2} + ( -17 + 17 \zeta_{6} ) q^{4} + 45 q^{8} +O(q^{10})\) \( q -5 \zeta_{6} q^{2} + ( -17 + 17 \zeta_{6} ) q^{4} + 45 q^{8} + ( -68 + 68 \zeta_{6} ) q^{11} -89 \zeta_{6} q^{16} + 340 q^{22} -40 \zeta_{6} q^{23} + ( 125 - 125 \zeta_{6} ) q^{25} + 166 q^{29} + ( -85 + 85 \zeta_{6} ) q^{32} -450 \zeta_{6} q^{37} -180 q^{43} -1156 \zeta_{6} q^{44} + ( -200 + 200 \zeta_{6} ) q^{46} -625 q^{50} + ( 590 - 590 \zeta_{6} ) q^{53} -830 \zeta_{6} q^{58} -287 q^{64} + ( 740 - 740 \zeta_{6} ) q^{67} -688 q^{71} + ( -2250 + 2250 \zeta_{6} ) q^{74} + 1384 \zeta_{6} q^{79} + 900 \zeta_{6} q^{86} + ( -3060 + 3060 \zeta_{6} ) q^{88} + 680 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{2} - 17q^{4} + 90q^{8} + O(q^{10}) \) \( 2q - 5q^{2} - 17q^{4} + 90q^{8} - 68q^{11} - 89q^{16} + 680q^{22} - 40q^{23} + 125q^{25} + 332q^{29} - 85q^{32} - 450q^{37} - 360q^{43} - 1156q^{44} - 200q^{46} - 1250q^{50} + 590q^{53} - 830q^{58} - 574q^{64} + 740q^{67} - 1376q^{71} - 2250q^{74} + 1384q^{79} + 900q^{86} - 3060q^{88} + 1360q^{92} + O(q^{100}) \)

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
−2.50000 + 4.33013i 0 −8.50000 14.7224i 0 0 0 45.0000 0 0
361.1 −2.50000 4.33013i 0 −8.50000 + 14.7224i 0 0 0 45.0000 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 441.4.e.a 2
3.b odd 2 1 49.4.c.d 2
7.b odd 2 1 CM 441.4.e.a 2
7.c even 3 1 441.4.a.m 1
7.c even 3 1 inner 441.4.e.a 2
7.d odd 6 1 441.4.a.m 1
7.d odd 6 1 inner 441.4.e.a 2
21.c even 2 1 49.4.c.d 2
21.g even 6 1 49.4.a.a 1
21.g even 6 1 49.4.c.d 2
21.h odd 6 1 49.4.a.a 1
21.h odd 6 1 49.4.c.d 2
84.j odd 6 1 784.4.a.k 1
84.n even 6 1 784.4.a.k 1
105.o odd 6 1 1225.4.a.l 1
105.p even 6 1 1225.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 21.g even 6 1
49.4.a.a 1 21.h odd 6 1
49.4.c.d 2 3.b odd 2 1
49.4.c.d 2 21.c even 2 1
49.4.c.d 2 21.g even 6 1
49.4.c.d 2 21.h odd 6 1
441.4.a.m 1 7.c even 3 1
441.4.a.m 1 7.d odd 6 1
441.4.e.a 2 1.a even 1 1 trivial
441.4.e.a 2 7.b odd 2 1 CM
441.4.e.a 2 7.c even 3 1 inner
441.4.e.a 2 7.d odd 6 1 inner
784.4.a.k 1 84.j odd 6 1
784.4.a.k 1 84.n even 6 1
1225.4.a.l 1 105.o odd 6 1
1225.4.a.l 1 105.p even 6 1

Hecke kernels

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

\( T_{2}^{2} + 5 T_{2} + 25 \)
\( T_{5} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + 5 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 4624 + 68 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 1600 + 40 T + T^{2} \)
$29$ \( ( -166 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 202500 + 450 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 180 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( 348100 - 590 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( 547600 - 740 T + T^{2} \)
$71$ \( ( 688 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( 1915456 - 1384 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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