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: \( x^{2} - 39 \) Copy content Toggle raw display
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.

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

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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 5616 \) Copy content Toggle raw display
\( T_{13}^{2} - 275184 \) Copy content Toggle raw display

Hecke characteristic polynomials

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