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: \( x^{2} - x - 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (2 \beta + 6) q^{4} + ( - 10 \beta - 19) q^{5} + (24 \beta - 48) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (2 \beta + 6) q^{4} + ( - 10 \beta - 19) q^{5} + (24 \beta - 48) q^{8} + (29 \beta + 389) q^{10} + ( - 23 \beta - 212) q^{11} + ( - 28 \beta + 462) q^{13} + ( - 40 \beta - 1032) q^{16} + (132 \beta - 1173) q^{17} + ( - 277 \beta + 180) q^{19} + ( - 98 \beta - 854) q^{20} + (235 \beta + 1063) q^{22} + (69 \beta + 6) q^{23} + (380 \beta + 936) q^{25} + ( - 434 \beta + 574) q^{26} + (700 \beta + 3526) q^{29} + (715 \beta - 1774) q^{31} + (304 \beta + 4048) q^{32} + (1041 \beta - 3711) q^{34} + ( - 790 \beta + 5545) q^{37} + (97 \beta + 10069) q^{38} + (24 \beta - 7968) q^{40} + (868 \beta + 1750) q^{41} + (1344 \beta - 6340) q^{43} + ( - 562 \beta - 2974) q^{44} + ( - 75 \beta - 2559) q^{46} + (1635 \beta - 11478) q^{47} + ( - 1316 \beta - 14996) q^{50} + (756 \beta + 700) q^{52} + (1818 \beta - 1521) q^{53} + (2557 \beta + 12538) q^{55} + ( - 4226 \beta - 29426) q^{58} + (531 \beta - 32904) q^{59} + (4154 \beta + 21243) q^{61} + (1059 \beta - 24681) q^{62} + ( - 3072 \beta + 17728) q^{64} + ( - 4088 \beta + 1582) q^{65} + (919 \beta + 21156) q^{67} + ( - 1554 \beta + 2730) q^{68} + (2184 \beta + 1104) q^{71} + (7372 \beta + 25253) q^{73} + ( - 4755 \beta + 23685) q^{74} + ( - 1302 \beta - 19418) q^{76} + ( - 5193 \beta + 4502) q^{79} + (11080 \beta + 34408) q^{80} + ( - 2618 \beta - 33866) q^{82} + ( - 4536 \beta - 52164) q^{83} + (9222 \beta - 26553) q^{85} + (4996 \beta - 43388) q^{86} + ( - 3984 \beta - 10248) q^{88} + (9356 \beta - 13333) q^{89} + (426 \beta + 5142) q^{92} + (9843 \beta - 49017) q^{94} + (3463 \beta + 99070) q^{95} + (196 \beta - 104566) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 12 q^{4} - 38 q^{5} - 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 12 q^{4} - 38 q^{5} - 96 q^{8} + 778 q^{10} - 424 q^{11} + 924 q^{13} - 2064 q^{16} - 2346 q^{17} + 360 q^{19} - 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} + 1148 q^{26} + 7052 q^{29} - 3548 q^{31} + 8096 q^{32} - 7422 q^{34} + 11090 q^{37} + 20138 q^{38} - 15936 q^{40} + 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} - 22956 q^{47} - 29992 q^{50} + 1400 q^{52} - 3042 q^{53} + 25076 q^{55} - 58852 q^{58} - 65808 q^{59} + 42486 q^{61} - 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} + 5460 q^{68} + 2208 q^{71} + 50506 q^{73} + 47370 q^{74} - 38836 q^{76} + 9004 q^{79} + 68816 q^{80} - 67732 q^{82} - 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} - 26666 q^{89} + 10284 q^{92} - 98034 q^{94} + 198140 q^{95} - 209132 q^{97}+O(q^{100}) \) 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
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} + 2T_{2} - 36 \) Copy content Toggle raw display
\( T_{5}^{2} + 38T_{5} - 3339 \) Copy content Toggle raw display
\( T_{13}^{2} - 924T_{13} + 184436 \) Copy content Toggle raw display

Hecke characteristic polynomials

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