Properties

Label 441.6.a.u
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{39}) \)
Defining polynomial: \(x^{2} - 39\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{39}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} -28 q^{4} -\beta q^{5} -120 q^{8} +O(q^{10})\) \( q + 2 q^{2} -28 q^{4} -\beta q^{5} -120 q^{8} -2 \beta q^{10} + 284 q^{11} + 7 \beta q^{13} + 656 q^{16} + 2 \beta q^{17} + 29 \beta q^{19} + 28 \beta q^{20} + 568 q^{22} -1496 q^{23} + 2491 q^{25} + 14 \beta q^{26} + 4366 q^{29} -86 \beta q^{31} + 5152 q^{32} + 4 \beta q^{34} -12630 q^{37} + 58 \beta q^{38} + 120 \beta q^{40} + 126 \beta q^{41} -1356 q^{43} -7952 q^{44} -2992 q^{46} -134 \beta q^{47} + 4982 q^{50} -196 \beta q^{52} -14150 q^{53} -284 \beta q^{55} + 8732 q^{58} + 499 \beta q^{59} -475 \beta q^{61} -172 \beta q^{62} -10688 q^{64} -39312 q^{65} -3644 q^{67} -56 \beta q^{68} -35632 q^{71} + 544 \beta q^{73} -25260 q^{74} -812 \beta q^{76} -54616 q^{79} -656 \beta q^{80} + 252 \beta q^{82} + 7 \beta q^{83} -11232 q^{85} -2712 q^{86} -34080 q^{88} + 272 \beta q^{89} + 41888 q^{92} -268 \beta q^{94} -162864 q^{95} + 2450 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 56 q^{4} - 240 q^{8} + O(q^{10}) \) \( 2 q + 4 q^{2} - 56 q^{4} - 240 q^{8} + 568 q^{11} + 1312 q^{16} + 1136 q^{22} - 2992 q^{23} + 4982 q^{25} + 8732 q^{29} + 10304 q^{32} - 25260 q^{37} - 2712 q^{43} - 15904 q^{44} - 5984 q^{46} + 9964 q^{50} - 28300 q^{53} + 17464 q^{58} - 21376 q^{64} - 78624 q^{65} - 7288 q^{67} - 71264 q^{71} - 50520 q^{74} - 109232 q^{79} - 22464 q^{85} - 5424 q^{86} - 68160 q^{88} + 83776 q^{92} - 325728 q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.24500
−6.24500
2.00000 0 −28.0000 −74.9400 0 0 −120.000 0 −149.880
1.2 2.00000 0 −28.0000 74.9400 0 0 −120.000 0 149.880
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.u 2
3.b odd 2 1 49.6.a.c 2
7.b odd 2 1 inner 441.6.a.u 2
12.b even 2 1 784.6.a.z 2
21.c even 2 1 49.6.a.c 2
21.g even 6 2 49.6.c.g 4
21.h odd 6 2 49.6.c.g 4
84.h odd 2 1 784.6.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.c 2 3.b odd 2 1
49.6.a.c 2 21.c even 2 1
49.6.c.g 4 21.g even 6 2
49.6.c.g 4 21.h odd 6 2
441.6.a.u 2 1.a even 1 1 trivial
441.6.a.u 2 7.b odd 2 1 inner
784.6.a.z 2 12.b even 2 1
784.6.a.z 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} - 2 \)
\( T_{5}^{2} - 5616 \)
\( T_{13}^{2} - 275184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -5616 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -284 + T )^{2} \)
$13$ \( -275184 + T^{2} \)
$17$ \( -22464 + T^{2} \)
$19$ \( -4723056 + T^{2} \)
$23$ \( ( 1496 + T )^{2} \)
$29$ \( ( -4366 + T )^{2} \)
$31$ \( -41535936 + T^{2} \)
$37$ \( ( 12630 + T )^{2} \)
$41$ \( -89159616 + T^{2} \)
$43$ \( ( 1356 + T )^{2} \)
$47$ \( -100840896 + T^{2} \)
$53$ \( ( 14150 + T )^{2} \)
$59$ \( -1398389616 + T^{2} \)
$61$ \( -1267110000 + T^{2} \)
$67$ \( ( 3644 + T )^{2} \)
$71$ \( ( 35632 + T )^{2} \)
$73$ \( -1661976576 + T^{2} \)
$79$ \( ( 54616 + T )^{2} \)
$83$ \( -275184 + T^{2} \)
$89$ \( -415494144 + T^{2} \)
$97$ \( -33710040000 + T^{2} \)
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