Properties

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

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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: \(\Q(\sqrt{193}) \)
Defining polynomial: \(x^{2} - x - 48\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{193})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 17 + 3 \beta ) q^{4} + 36 q^{5} + ( 129 - 9 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 17 + 3 \beta ) q^{4} + 36 q^{5} + ( 129 - 9 \beta ) q^{8} + ( 36 + 36 \beta ) q^{10} + ( -236 - 8 \beta ) q^{11} + ( -612 - 72 \beta ) q^{13} + ( -847 + 15 \beta ) q^{16} + ( 576 - 216 \beta ) q^{17} + ( -36 + 288 \beta ) q^{19} + ( 612 + 108 \beta ) q^{20} + ( -620 - 252 \beta ) q^{22} + ( 224 + 56 \beta ) q^{23} -1829 q^{25} + ( -4068 - 756 \beta ) q^{26} + ( -2866 - 640 \beta ) q^{29} + ( -5256 + 576 \beta ) q^{31} + ( -4255 - 529 \beta ) q^{32} + ( -9792 + 144 \beta ) q^{34} + ( 6090 - 1056 \beta ) q^{37} + ( 13788 + 540 \beta ) q^{38} + ( 4644 - 324 \beta ) q^{40} + ( 10368 + 216 \beta ) q^{41} + ( -1932 - 2400 \beta ) q^{43} + ( -5164 - 868 \beta ) q^{44} + ( 2912 + 336 \beta ) q^{46} + ( 2232 + 3456 \beta ) q^{47} + ( -1829 - 1829 \beta ) q^{50} + ( -20772 - 3276 \beta ) q^{52} + ( -1702 + 1184 \beta ) q^{53} + ( -8496 - 288 \beta ) q^{55} + ( -33586 - 4146 \beta ) q^{58} + ( 14436 + 864 \beta ) q^{59} + ( 10980 - 4680 \beta ) q^{61} + ( 22392 - 4104 \beta ) q^{62} + ( -2543 - 5793 \beta ) q^{64} + ( -22032 - 2592 \beta ) q^{65} + ( -13580 + 6480 \beta ) q^{67} + ( -21312 - 2592 \beta ) q^{68} + ( 47416 - 2552 \beta ) q^{71} + ( -29232 + 1872 \beta ) q^{73} + ( -44598 + 3978 \beta ) q^{74} + ( 40860 + 5652 \beta ) q^{76} + ( -24808 - 6480 \beta ) q^{79} + ( -30492 + 540 \beta ) q^{80} + ( 20736 + 10800 \beta ) q^{82} + ( -40428 + 9504 \beta ) q^{83} + ( 20736 - 7776 \beta ) q^{85} + ( -117132 - 6732 \beta ) q^{86} + ( -26988 + 1164 \beta ) q^{88} + ( 63432 - 3672 \beta ) q^{89} + ( 11872 + 1792 \beta ) q^{92} + ( 168120 + 9144 \beta ) q^{94} + ( -1296 + 10368 \beta ) q^{95} + ( -8136 - 19584 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8} + 108 q^{10} - 480 q^{11} - 1296 q^{13} - 1679 q^{16} + 936 q^{17} + 216 q^{19} + 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} - 8892 q^{26} - 6372 q^{29} - 9936 q^{31} - 9039 q^{32} - 19440 q^{34} + 11124 q^{37} + 28116 q^{38} + 8964 q^{40} + 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} + 7920 q^{47} - 5487 q^{50} - 44820 q^{52} - 2220 q^{53} - 17280 q^{55} - 71318 q^{58} + 29736 q^{59} + 17280 q^{61} + 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} - 45216 q^{68} + 92280 q^{71} - 56592 q^{73} - 85218 q^{74} + 87372 q^{76} - 56096 q^{79} - 60444 q^{80} + 52272 q^{82} - 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} + 123192 q^{89} + 25536 q^{92} + 345384 q^{94} + 7776 q^{95} - 35856 q^{97} + O(q^{100}) \)

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
−6.44622
7.44622
−5.44622 0 −2.33867 36.0000 0 0 187.016 0 −196.064
1.2 8.44622 0 39.3387 36.0000 0 0 61.9840 0 304.064
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.r 2
3.b odd 2 1 147.6.a.j yes 2
7.b odd 2 1 441.6.a.q 2
21.c even 2 1 147.6.a.h 2
21.g even 6 2 147.6.e.n 4
21.h odd 6 2 147.6.e.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.h 2 21.c even 2 1
147.6.a.j yes 2 3.b odd 2 1
147.6.e.m 4 21.h odd 6 2
147.6.e.n 4 21.g even 6 2
441.6.a.q 2 7.b odd 2 1
441.6.a.r 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{2} - 3 T_{2} - 46 \)
\( T_{5} - 36 \)
\( T_{13}^{2} + 1296 T_{13} + 169776 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -46 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -36 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( 54512 + 480 T + T^{2} \)
$13$ \( 169776 + 1296 T + T^{2} \)
$17$ \( -2032128 - 936 T + T^{2} \)
$19$ \( -3990384 - 216 T + T^{2} \)
$23$ \( -87808 - 504 T + T^{2} \)
$29$ \( -9612604 + 6372 T + T^{2} \)
$31$ \( 8672832 + 9936 T + T^{2} \)
$37$ \( -22869468 - 11124 T + T^{2} \)
$41$ \( 107495424 - 20952 T + T^{2} \)
$43$ \( -268110576 + 6264 T + T^{2} \)
$47$ \( -560613312 - 7920 T + T^{2} \)
$53$ \( -66407452 + 2220 T + T^{2} \)
$59$ \( 185038992 - 29736 T + T^{2} \)
$61$ \( -982141200 - 17280 T + T^{2} \)
$67$ \( -1919121200 + 20680 T + T^{2} \)
$71$ \( 1814661632 - 92280 T + T^{2} \)
$73$ \( 631577088 + 56592 T + T^{2} \)
$79$ \( -1239346496 + 56096 T + T^{2} \)
$83$ \( -3085453296 + 71352 T + T^{2} \)
$89$ \( 3143484288 - 123192 T + T^{2} \)
$97$ \( -18184056768 + 35856 T + T^{2} \)
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