Properties

Label 441.6.a.s
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: \(\Q(\sqrt{249}) \)
Defining polynomial: \(x^{2} - x - 62\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{249})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 31 + 3 \beta ) q^{4} + ( -13 - 7 \beta ) q^{5} + ( 185 + 5 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 31 + 3 \beta ) q^{4} + ( -13 - 7 \beta ) q^{5} + ( 185 + 5 \beta ) q^{8} + ( -447 - 27 \beta ) q^{10} + ( 569 - \beta ) q^{11} + ( -458 - 9 \beta ) q^{13} + ( -497 + 99 \beta ) q^{16} + ( -236 + 148 \beta ) q^{17} + ( -1142 - 27 \beta ) q^{19} + ( -1705 - 277 \beta ) q^{20} + ( 507 + 567 \beta ) q^{22} + ( -644 - 308 \beta ) q^{23} + ( 82 + 231 \beta ) q^{25} + ( -1016 - 476 \beta ) q^{26} + ( 1131 - 45 \beta ) q^{29} + ( -1763 - 768 \beta ) q^{31} + ( -279 - 459 \beta ) q^{32} + ( 8940 + 60 \beta ) q^{34} + ( -9982 + 855 \beta ) q^{37} + ( -2816 - 1196 \beta ) q^{38} + ( -4575 - 1395 \beta ) q^{40} + ( -6852 + 846 \beta ) q^{41} + ( -364 - 2043 \beta ) q^{43} + ( 17453 + 1673 \beta ) q^{44} + ( -19740 - 1260 \beta ) q^{46} + ( -11278 - 604 \beta ) q^{47} + ( 14404 + 544 \beta ) q^{50} + ( -15872 - 1680 \beta ) q^{52} + ( 14951 + 1751 \beta ) q^{53} + ( -6963 - 3963 \beta ) q^{55} + ( -1659 + 1041 \beta ) q^{58} + ( 22507 - 3917 \beta ) q^{59} + ( -22298 + 2544 \beta ) q^{61} + ( -49379 - 3299 \beta ) q^{62} + ( -12833 - 4365 \beta ) q^{64} + ( 9860 + 3386 \beta ) q^{65} + ( 17612 - 4461 \beta ) q^{67} + ( 20212 + 4324 \beta ) q^{68} + ( -50346 - 1404 \beta ) q^{71} + ( 16912 - 5247 \beta ) q^{73} + ( 43028 - 8272 \beta ) q^{74} + ( -40424 - 4344 \beta ) q^{76} + ( -12649 + 6834 \beta ) q^{79} + ( -36505 + 1499 \beta ) q^{80} + ( 45600 - 5160 \beta ) q^{82} + ( 31539 - 1899 \beta ) q^{83} + ( -61164 - 1308 \beta ) q^{85} + ( -127030 - 4450 \beta ) q^{86} + ( 104955 + 2655 \beta ) q^{88} + ( 14726 - 130 \beta ) q^{89} + ( -77252 - 12404 \beta ) q^{92} + ( -48726 - 12486 \beta ) q^{94} + ( 26564 + 8534 \beta ) q^{95} + ( 4387 + 1017 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8} - 921 q^{10} + 1137 q^{11} - 925 q^{13} - 895 q^{16} - 324 q^{17} - 2311 q^{19} - 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} - 2508 q^{26} + 2217 q^{29} - 4294 q^{31} - 1017 q^{32} + 17940 q^{34} - 19109 q^{37} - 6828 q^{38} - 10545 q^{40} - 12858 q^{41} - 2771 q^{43} + 36579 q^{44} - 40740 q^{46} - 23160 q^{47} + 29352 q^{50} - 33424 q^{52} + 31653 q^{53} - 17889 q^{55} - 2277 q^{58} + 41097 q^{59} - 42052 q^{61} - 102057 q^{62} - 30031 q^{64} + 23106 q^{65} + 30763 q^{67} + 44748 q^{68} - 102096 q^{71} + 28577 q^{73} + 77784 q^{74} - 85192 q^{76} - 18464 q^{79} - 71511 q^{80} + 86040 q^{82} + 61179 q^{83} - 123636 q^{85} - 258510 q^{86} + 212565 q^{88} + 29322 q^{89} - 166908 q^{92} - 109938 q^{94} + 61662 q^{95} + 9791 q^{97} + O(q^{100}) \)

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
−7.38987
8.38987
−6.38987 0 8.83040 38.7291 0 0 148.051 0 −247.474
1.2 9.38987 0 56.1696 −71.7291 0 0 226.949 0 −673.526
\(n\): 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.s 2
3.b odd 2 1 147.6.a.k 2
7.b odd 2 1 441.6.a.t 2
7.d odd 6 2 63.6.e.c 4
21.c even 2 1 147.6.a.i 2
21.g even 6 2 21.6.e.b 4
21.h odd 6 2 147.6.e.l 4
84.j odd 6 2 336.6.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 21.g even 6 2
63.6.e.c 4 7.d odd 6 2
147.6.a.i 2 21.c even 2 1
147.6.a.k 2 3.b odd 2 1
147.6.e.l 4 21.h odd 6 2
336.6.q.e 4 84.j odd 6 2
441.6.a.s 2 1.a even 1 1 trivial
441.6.a.t 2 7.b odd 2 1

Hecke kernels

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

\( T_{2}^{2} - 3 T_{2} - 60 \)
\( T_{5}^{2} + 33 T_{5} - 2778 \)
\( T_{13}^{2} + 925 T_{13} + 208864 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -60 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -2778 + 33 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 323130 - 1137 T + T^{2} \)
$13$ \( 208864 + 925 T + T^{2} \)
$17$ \( -1337280 + 324 T + T^{2} \)
$19$ \( 1289800 + 2311 T + T^{2} \)
$23$ \( -5268480 + 1596 T + T^{2} \)
$29$ \( 1102716 - 2217 T + T^{2} \)
$31$ \( -32106935 + 4294 T + T^{2} \)
$37$ \( 45782164 + 19109 T + T^{2} \)
$41$ \( -3221280 + 12858 T + T^{2} \)
$43$ \( -257902490 + 2771 T + T^{2} \)
$47$ \( 111386604 + 23160 T + T^{2} \)
$53$ \( 59619540 - 31653 T + T^{2} \)
$59$ \( -532853988 - 41097 T + T^{2} \)
$61$ \( 39214660 + 42052 T + T^{2} \)
$67$ \( -1002216890 - 30763 T + T^{2} \)
$71$ \( 2483190108 + 102096 T + T^{2} \)
$73$ \( -1509644078 - 28577 T + T^{2} \)
$79$ \( -2822066537 + 18464 T + T^{2} \)
$83$ \( 711231498 - 61179 T + T^{2} \)
$89$ \( 213892896 - 29322 T + T^{2} \)
$97$ \( -40418570 - 9791 T + T^{2} \)
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