Properties

Label 441.6.a.m
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: \(\Q(\sqrt{37}) \)
Defining polynomial: \(x^{2} - x - 9\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} + ( 6 + 2 \beta ) q^{4} + ( -19 - 10 \beta ) q^{5} + ( -48 + 24 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} + ( 6 + 2 \beta ) q^{4} + ( -19 - 10 \beta ) q^{5} + ( -48 + 24 \beta ) q^{8} + ( 389 + 29 \beta ) q^{10} + ( -212 - 23 \beta ) q^{11} + ( 462 - 28 \beta ) q^{13} + ( -1032 - 40 \beta ) q^{16} + ( -1173 + 132 \beta ) q^{17} + ( 180 - 277 \beta ) q^{19} + ( -854 - 98 \beta ) q^{20} + ( 1063 + 235 \beta ) q^{22} + ( 6 + 69 \beta ) q^{23} + ( 936 + 380 \beta ) q^{25} + ( 574 - 434 \beta ) q^{26} + ( 3526 + 700 \beta ) q^{29} + ( -1774 + 715 \beta ) q^{31} + ( 4048 + 304 \beta ) q^{32} + ( -3711 + 1041 \beta ) q^{34} + ( 5545 - 790 \beta ) q^{37} + ( 10069 + 97 \beta ) q^{38} + ( -7968 + 24 \beta ) q^{40} + ( 1750 + 868 \beta ) q^{41} + ( -6340 + 1344 \beta ) q^{43} + ( -2974 - 562 \beta ) q^{44} + ( -2559 - 75 \beta ) q^{46} + ( -11478 + 1635 \beta ) q^{47} + ( -14996 - 1316 \beta ) q^{50} + ( 700 + 756 \beta ) q^{52} + ( -1521 + 1818 \beta ) q^{53} + ( 12538 + 2557 \beta ) q^{55} + ( -29426 - 4226 \beta ) q^{58} + ( -32904 + 531 \beta ) q^{59} + ( 21243 + 4154 \beta ) q^{61} + ( -24681 + 1059 \beta ) q^{62} + ( 17728 - 3072 \beta ) q^{64} + ( 1582 - 4088 \beta ) q^{65} + ( 21156 + 919 \beta ) q^{67} + ( 2730 - 1554 \beta ) q^{68} + ( 1104 + 2184 \beta ) q^{71} + ( 25253 + 7372 \beta ) q^{73} + ( 23685 - 4755 \beta ) q^{74} + ( -19418 - 1302 \beta ) q^{76} + ( 4502 - 5193 \beta ) q^{79} + ( 34408 + 11080 \beta ) q^{80} + ( -33866 - 2618 \beta ) q^{82} + ( -52164 - 4536 \beta ) q^{83} + ( -26553 + 9222 \beta ) q^{85} + ( -43388 + 4996 \beta ) q^{86} + ( -10248 - 3984 \beta ) q^{88} + ( -13333 + 9356 \beta ) q^{89} + ( 5142 + 426 \beta ) q^{92} + ( -49017 + 9843 \beta ) q^{94} + ( 99070 + 3463 \beta ) q^{95} + ( -104566 + 196 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 12 q^{4} - 38 q^{5} - 96 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 12 q^{4} - 38 q^{5} - 96 q^{8} + 778 q^{10} - 424 q^{11} + 924 q^{13} - 2064 q^{16} - 2346 q^{17} + 360 q^{19} - 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} + 1148 q^{26} + 7052 q^{29} - 3548 q^{31} + 8096 q^{32} - 7422 q^{34} + 11090 q^{37} + 20138 q^{38} - 15936 q^{40} + 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} - 22956 q^{47} - 29992 q^{50} + 1400 q^{52} - 3042 q^{53} + 25076 q^{55} - 58852 q^{58} - 65808 q^{59} + 42486 q^{61} - 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} + 5460 q^{68} + 2208 q^{71} + 50506 q^{73} + 47370 q^{74} - 38836 q^{76} + 9004 q^{79} + 68816 q^{80} - 67732 q^{82} - 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} - 26666 q^{89} + 10284 q^{92} - 98034 q^{94} + 198140 q^{95} - 209132 q^{97} + O(q^{100}) \)

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
3.54138
−2.54138
−7.08276 0 18.1655 −79.8276 0 0 97.9863 0 565.400
1.2 5.08276 0 −6.16553 41.8276 0 0 −193.986 0 212.600
\(n\): 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.m 2
3.b odd 2 1 49.6.a.e 2
7.b odd 2 1 441.6.a.n 2
7.d odd 6 2 63.6.e.d 4
12.b even 2 1 784.6.a.t 2
21.c even 2 1 49.6.a.d 2
21.g even 6 2 7.6.c.a 4
21.h odd 6 2 49.6.c.f 4
84.h odd 2 1 784.6.a.ba 2
84.j odd 6 2 112.6.i.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.c.a 4 21.g even 6 2
49.6.a.d 2 21.c even 2 1
49.6.a.e 2 3.b odd 2 1
49.6.c.f 4 21.h odd 6 2
63.6.e.d 4 7.d odd 6 2
112.6.i.c 4 84.j odd 6 2
441.6.a.m 2 1.a even 1 1 trivial
441.6.a.n 2 7.b odd 2 1
784.6.a.t 2 12.b even 2 1
784.6.a.ba 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} + 2 T_{2} - 36 \)
\( T_{5}^{2} + 38 T_{5} - 3339 \)
\( T_{13}^{2} - 924 T_{13} + 184436 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -36 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -3339 + 38 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 25371 + 424 T + T^{2} \)
$13$ \( 184436 - 924 T + T^{2} \)
$17$ \( 731241 + 2346 T + T^{2} \)
$19$ \( -2806573 - 360 T + T^{2} \)
$23$ \( -176121 - 12 T + T^{2} \)
$29$ \( -5697324 - 7052 T + T^{2} \)
$31$ \( -15768249 + 3548 T + T^{2} \)
$37$ \( 7655325 - 11090 T + T^{2} \)
$41$ \( -24814188 - 3500 T + T^{2} \)
$43$ \( -26638832 + 12680 T + T^{2} \)
$47$ \( 32835159 + 22956 T + T^{2} \)
$53$ \( -119976147 + 3042 T + T^{2} \)
$59$ \( 1072240659 + 65808 T + T^{2} \)
$61$ \( -187196443 - 42486 T + T^{2} \)
$67$ \( 416327579 - 42312 T + T^{2} \)
$71$ \( -175265856 - 2208 T + T^{2} \)
$73$ \( -1373102199 - 50506 T + T^{2} \)
$79$ \( -977520209 - 9004 T + T^{2} \)
$83$ \( 1959796944 + 104328 T + T^{2} \)
$89$ \( -3061016343 + 26666 T + T^{2} \)
$97$ \( 10932626964 + 209132 T + T^{2} \)
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