Properties

Label 441.6.a.w
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 97 x^{2} + 7 x + 294\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 18 - \beta_{1} + \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -38 + 27 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 18 - \beta_{1} + \beta_{2} ) q^{4} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -38 + 27 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{8} + ( 76 - 23 \beta_{1} - \beta_{2} ) q^{10} + ( -107 + 8 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{11} + ( 96 + 76 \beta_{1} - \beta_{2} + 9 \beta_{3} ) q^{13} + ( 840 - 85 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{16} + ( -92 + 108 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 72 + 220 \beta_{1} - \beta_{2} - 15 \beta_{3} ) q^{19} + ( -1168 - 29 \beta_{1} + 9 \beta_{2} + 30 \beta_{3} ) q^{20} + ( 448 - 419 \beta_{1} - \beta_{2} - 12 \beta_{3} ) q^{22} + ( -1804 + 300 \beta_{1} - 8 \beta_{2} - 20 \beta_{3} ) q^{23} + ( 636 + 80 \beta_{1} - 95 \beta_{2} - 45 \beta_{3} ) q^{25} + ( 3865 - 116 \beta_{1} + 103 \beta_{2} - 20 \beta_{3} ) q^{26} + ( -147 + 254 \beta_{1} + 103 \beta_{2} - 5 \beta_{3} ) q^{29} + ( -1523 - 130 \beta_{1} + 94 \beta_{2} + 24 \beta_{3} ) q^{31} + ( -3948 + 131 \beta_{1} - 71 \beta_{2} - 50 \beta_{3} ) q^{32} + ( 5256 + 312 \beta_{1} + 96 \beta_{2} + 24 \beta_{3} ) q^{34} + ( 3508 + 1028 \beta_{1} - 95 \beta_{2} - 9 \beta_{3} ) q^{37} + ( 10321 + 316 \beta_{1} + 175 \beta_{2} + 28 \beta_{3} ) q^{38} + ( -1932 - 633 \beta_{1} + 93 \beta_{2} - 42 \beta_{3} ) q^{40} + ( -1128 + 328 \beta_{1} + 62 \beta_{2} - 142 \beta_{3} ) q^{41} + ( 7142 + 816 \beta_{1} + 93 \beta_{2} - 33 \beta_{3} ) q^{43} + ( -17864 + 379 \beta_{1} - 167 \beta_{2} + 118 \beta_{3} ) q^{44} + ( 16008 - 1752 \beta_{1} + 240 \beta_{2} + 24 \beta_{3} ) q^{46} + ( 3818 - 324 \beta_{1} - 56 \beta_{2} + 28 \beta_{3} ) q^{47} + ( 2399 - 2404 \beta_{1} - 55 \beta_{2} - 100 \beta_{3} ) q^{50} + ( -13450 + 6036 \beta_{1} - 144 \beta_{2} - 42 \beta_{3} ) q^{52} + ( -3095 - 1338 \beta_{1} - 13 \beta_{2} + 239 \beta_{3} ) q^{53} + ( 17987 + 506 \beta_{1} - 335 \beta_{2} - 315 \beta_{3} ) q^{55} + ( 12154 + 4171 \beta_{1} + 239 \beta_{2} + 216 \beta_{3} ) q^{58} + ( -9011 + 888 \beta_{1} - 163 \beta_{2} + 71 \beta_{3} ) q^{59} + ( -1566 + 796 \beta_{1} - 52 \beta_{2} + 240 \beta_{3} ) q^{61} + ( -4505 + 1875 \beta_{1} - 58 \beta_{2} + 140 \beta_{3} ) q^{62} + ( -17600 - 3189 \beta_{1} - 51 \beta_{2} + 150 \beta_{3} ) q^{64} + ( -16136 - 1660 \beta_{1} + 330 \beta_{2} - 246 \beta_{3} ) q^{65} + ( -2286 + 2764 \beta_{1} - 193 \beta_{2} + 465 \beta_{3} ) q^{67} + ( 13312 + 5280 \beta_{1} + 128 \beta_{2} + 272 \beta_{3} ) q^{68} + ( -21390 - 3660 \beta_{1} + 24 \beta_{2} + 180 \beta_{3} ) q^{71} + ( 14074 + 3056 \beta_{1} - 143 \beta_{2} - 93 \beta_{3} ) q^{73} + ( 46915 - 216 \beta_{1} + 1001 \beta_{2} - 172 \beta_{3} ) q^{74} + ( 3062 + 9924 \beta_{1} + 432 \beta_{2} + 774 \beta_{3} ) q^{76} + ( -11635 + 2286 \beta_{1} - 96 \beta_{2} - 786 \beta_{3} ) q^{79} + ( 6920 + 3607 \beta_{1} - 1047 \beta_{2} - 690 \beta_{3} ) q^{80} + ( 13322 + 4112 \beta_{1} - 98 \beta_{2} + 408 \beta_{3} ) q^{82} + ( 50001 + 6432 \beta_{1} - 1005 \beta_{2} + 129 \beta_{3} ) q^{83} + ( 9732 - 2988 \beta_{1} - 192 \beta_{2} + 372 \beta_{3} ) q^{85} + ( 31705 + 11582 \beta_{1} + 717 \beta_{2} + 252 \beta_{3} ) q^{86} + ( 25668 - 13545 \beta_{1} + 765 \beta_{2} - 186 \beta_{3} ) q^{88} + ( 20966 - 9508 \beta_{1} - 1582 \beta_{2} - 622 \beta_{3} ) q^{89} + ( -44224 + 15792 \beta_{1} - 1424 \beta_{2} + 1072 \beta_{3} ) q^{92} + ( -18798 + 990 \beta_{1} - 240 \beta_{2} - 168 \beta_{3} ) q^{94} + ( 55528 - 3820 \beta_{1} - 1542 \beta_{2} - 294 \beta_{3} ) q^{95} + ( -47705 + 464 \beta_{1} - 863 \beta_{2} - 669 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + O(q^{10}) \) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 97 x^{2} + 7 x + 294\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 49 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} - 89 \nu + 52 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 49\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 2 \beta_{2} + 91 \beta_{1} + 46\)

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
−9.22385
−1.74818
1.79080
10.1812
−10.2239 0 72.5272 −23.7528 0 0 −414.344 0 242.845
1.2 −2.74818 0 −24.4475 −58.3673 0 0 155.128 0 160.404
1.3 0.790805 0 −31.3746 104.192 0 0 −50.1170 0 82.3953
1.4 9.18123 0 52.2950 −22.0716 0 0 186.333 0 −202.644
\(n\): 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.w 4
3.b odd 2 1 147.6.a.m 4
7.b odd 2 1 441.6.a.v 4
7.c even 3 2 63.6.e.e 8
21.c even 2 1 147.6.a.l 4
21.g even 6 2 147.6.e.o 8
21.h odd 6 2 21.6.e.c 8
84.n even 6 2 336.6.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 21.h odd 6 2
63.6.e.e 8 7.c even 3 2
147.6.a.l 4 21.c even 2 1
147.6.a.m 4 3.b odd 2 1
147.6.e.o 8 21.g even 6 2
336.6.q.j 8 84.n even 6 2
441.6.a.v 4 7.b odd 2 1
441.6.a.w 4 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{4} + 3 T_{2}^{3} - 94 T_{2}^{2} - 186 T_{2} + 204 \)
\( T_{5}^{4} - 7657 T_{5}^{2} - 302700 T_{5} - 3188244 \)
\( T_{13}^{4} - 462 T_{13}^{3} - 1148423 T_{13}^{2} + 515112852 T_{13} + 149501563456 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 204 - 186 T - 94 T^{2} + 3 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( -3188244 - 302700 T - 7657 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( -1682132124 - 100810572 T - 238363 T^{2} + 402 T^{3} + T^{4} \)
$13$ \( 149501563456 + 515112852 T - 1148423 T^{2} - 462 T^{3} + T^{4} \)
$17$ \( -50104147968 - 894878208 T - 1594752 T^{2} + 276 T^{3} + T^{4} \)
$19$ \( 7391138416576 + 797823780 T - 5896871 T^{2} - 510 T^{3} + T^{4} \)
$23$ \( 3007939608576 - 17122936320 T + 7121856 T^{2} + 6900 T^{3} + T^{4} \)
$29$ \( 408027025117872 - 62527747272 T - 52650397 T^{2} + 540 T^{3} + T^{4} \)
$31$ \( 86716089209547 - 58618529034 T - 17713868 T^{2} + 6410 T^{3} + T^{4} \)
$37$ \( -5042926288839456 + 1497067500180 T - 46856207 T^{2} - 15250 T^{3} + T^{4} \)
$41$ \( -1856858915261952 + 1101575496480 T - 192741244 T^{2} + 4308 T^{3} + T^{4} \)
$43$ \( -991662745581932 - 199921376588 T + 199961493 T^{2} - 29198 T^{3} + T^{4} \)
$47$ \( -270685655359056 + 44937987408 T + 50649744 T^{2} - 15060 T^{3} + T^{4} \)
$53$ \( -8505482723267472 - 8038879393320 T - 497907429 T^{2} + 13692 T^{3} + T^{4} \)
$59$ \( -2578852214901936 - 461620404360 T + 188256441 T^{2} + 34830 T^{3} + T^{4} \)
$61$ \( 17942190625624624 - 3379722031440 T - 532596176 T^{2} + 5364 T^{3} + T^{4} \)
$67$ \( 550087288501666684 + 4941755739000 T - 2845994891 T^{2} + 5994 T^{3} + T^{4} \)
$71$ \( 21932335650275568 - 16377596837712 T + 1521744768 T^{2} + 89268 T^{3} + T^{4} \)
$73$ \( -122130292613870700 + 20327373037020 T + 362940181 T^{2} - 59638 T^{3} + T^{4} \)
$79$ \( 165231063841623259 + 14857064631634 T - 3781804908 T^{2} + 44062 T^{3} + T^{4} \)
$83$ \( -41533908097096407132 + 738604511000820 T + 7607249829 T^{2} - 208446 T^{3} + T^{4} \)
$89$ \( -47322044296216531968 + 1866206095720704 T - 16806213508 T^{2} - 77520 T^{3} + T^{4} \)
$97$ \( -11638556269792123644 - 99054118022220 T + 9271508101 T^{2} + 188630 T^{3} + T^{4} \)
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