Properties

Label 441.6.a.v
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 18) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{5} + (2 \beta_{3} - \beta_{2} + 27 \beta_1 - 38) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 18) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{5} + (2 \beta_{3} - \beta_{2} + 27 \beta_1 - 38) q^{8} + (\beta_{2} + 23 \beta_1 - 76) q^{10} + ( - 3 \beta_{3} - 9 \beta_{2} + 8 \beta_1 - 107) q^{11} + ( - 9 \beta_{3} + \beta_{2} - 76 \beta_1 - 96) q^{13} + ( - 6 \beta_{3} + \beta_{2} - 85 \beta_1 + 840) q^{16} + (4 \beta_{3} - 8 \beta_{2} - 108 \beta_1 + 92) q^{17} + (15 \beta_{3} + \beta_{2} - 220 \beta_1 - 72) q^{19} + ( - 30 \beta_{3} - 9 \beta_{2} + 29 \beta_1 + 1168) q^{20} + ( - 12 \beta_{3} - \beta_{2} - 419 \beta_1 + 448) q^{22} + ( - 20 \beta_{3} - 8 \beta_{2} + 300 \beta_1 - 1804) q^{23} + ( - 45 \beta_{3} - 95 \beta_{2} + 80 \beta_1 + 636) q^{25} + (20 \beta_{3} - 103 \beta_{2} + 116 \beta_1 - 3865) q^{26} + ( - 5 \beta_{3} + 103 \beta_{2} + 254 \beta_1 - 147) q^{29} + ( - 24 \beta_{3} - 94 \beta_{2} + 130 \beta_1 + 1523) q^{31} + ( - 50 \beta_{3} - 71 \beta_{2} + 131 \beta_1 - 3948) q^{32} + ( - 24 \beta_{3} - 96 \beta_{2} - 312 \beta_1 - 5256) q^{34} + ( - 9 \beta_{3} - 95 \beta_{2} + 1028 \beta_1 + 3508) q^{37} + ( - 28 \beta_{3} - 175 \beta_{2} - 316 \beta_1 - 10321) q^{38} + (42 \beta_{3} - 93 \beta_{2} + 633 \beta_1 + 1932) q^{40} + (142 \beta_{3} - 62 \beta_{2} - 328 \beta_1 + 1128) q^{41} + ( - 33 \beta_{3} + 93 \beta_{2} + 816 \beta_1 + 7142) q^{43} + (118 \beta_{3} - 167 \beta_{2} + 379 \beta_1 - 17864) q^{44} + (24 \beta_{3} + 240 \beta_{2} - 1752 \beta_1 + 16008) q^{46} + ( - 28 \beta_{3} + 56 \beta_{2} + 324 \beta_1 - 3818) q^{47} + ( - 100 \beta_{3} - 55 \beta_{2} - 2404 \beta_1 + 2399) q^{50} + (42 \beta_{3} + 144 \beta_{2} - 6036 \beta_1 + 13450) q^{52} + (239 \beta_{3} - 13 \beta_{2} - 1338 \beta_1 - 3095) q^{53} + (315 \beta_{3} + 335 \beta_{2} - 506 \beta_1 - 17987) q^{55} + (216 \beta_{3} + 239 \beta_{2} + 4171 \beta_1 + 12154) q^{58} + ( - 71 \beta_{3} + 163 \beta_{2} - 888 \beta_1 + 9011) q^{59} + ( - 240 \beta_{3} + 52 \beta_{2} - 796 \beta_1 + 1566) q^{61} + ( - 140 \beta_{3} + 58 \beta_{2} - 1875 \beta_1 + 4505) q^{62} + (150 \beta_{3} - 51 \beta_{2} - 3189 \beta_1 - 17600) q^{64} + ( - 246 \beta_{3} + 330 \beta_{2} - 1660 \beta_1 - 16136) q^{65} + (465 \beta_{3} - 193 \beta_{2} + 2764 \beta_1 - 2286) q^{67} + ( - 272 \beta_{3} - 128 \beta_{2} - 5280 \beta_1 - 13312) q^{68} + (180 \beta_{3} + 24 \beta_{2} - 3660 \beta_1 - 21390) q^{71} + (93 \beta_{3} + 143 \beta_{2} - 3056 \beta_1 - 14074) q^{73} + ( - 172 \beta_{3} + 1001 \beta_{2} - 216 \beta_1 + 46915) q^{74} + ( - 774 \beta_{3} - 432 \beta_{2} - 9924 \beta_1 - 3062) q^{76} + ( - 786 \beta_{3} - 96 \beta_{2} + 2286 \beta_1 - 11635) q^{79} + (690 \beta_{3} + 1047 \beta_{2} - 3607 \beta_1 - 6920) q^{80} + ( - 408 \beta_{3} + 98 \beta_{2} - 4112 \beta_1 - 13322) q^{82} + ( - 129 \beta_{3} + 1005 \beta_{2} - 6432 \beta_1 - 50001) q^{83} + (372 \beta_{3} - 192 \beta_{2} - 2988 \beta_1 + 9732) q^{85} + (252 \beta_{3} + 717 \beta_{2} + 11582 \beta_1 + 31705) q^{86} + ( - 186 \beta_{3} + 765 \beta_{2} - 13545 \beta_1 + 25668) q^{88} + (622 \beta_{3} + 1582 \beta_{2} + 9508 \beta_1 - 20966) q^{89} + (1072 \beta_{3} - 1424 \beta_{2} + 15792 \beta_1 - 44224) q^{92} + (168 \beta_{3} + 240 \beta_{2} - 990 \beta_1 + 18798) q^{94} + ( - 294 \beta_{3} - 1542 \beta_{2} - 3820 \beta_1 + 55528) q^{95} + (669 \beta_{3} + 863 \beta_{2} - 464 \beta_1 + 47705) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} - 283 q^{10} - 402 q^{11} - 462 q^{13} + 3273 q^{16} + 276 q^{17} - 510 q^{19} + 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} - 15138 q^{26} - 540 q^{29} + 6410 q^{31} - 15519 q^{32} - 21144 q^{34} + 15250 q^{37} - 41250 q^{38} + 8547 q^{40} + 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} - 15060 q^{47} + 7302 q^{50} + 47476 q^{52} - 13692 q^{53} - 73124 q^{55} + 52309 q^{58} + 34830 q^{59} + 5364 q^{61} + 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} - 58272 q^{68} - 89268 q^{71} - 59638 q^{73} + 185442 q^{74} - 21308 q^{76} - 44062 q^{79} - 33381 q^{80} - 57596 q^{82} - 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} - 77520 q^{89} - 158256 q^{92} + 73722 q^{94} + 221376 q^{95} + 188630 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 89\nu + 52 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + 91\beta _1 + 46 \) 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
−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.v 4
3.b odd 2 1 147.6.a.l 4
7.b odd 2 1 441.6.a.w 4
7.d odd 6 2 63.6.e.e 8
21.c even 2 1 147.6.a.m 4
21.g even 6 2 21.6.e.c 8
21.h odd 6 2 147.6.e.o 8
84.j odd 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.g even 6 2
63.6.e.e 8 7.d odd 6 2
147.6.a.l 4 3.b odd 2 1
147.6.a.m 4 21.c even 2 1
147.6.e.o 8 21.h odd 6 2
336.6.q.j 8 84.j odd 6 2
441.6.a.v 4 1.a even 1 1 trivial
441.6.a.w 4 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} - 94 T^{2} - 186 T + 204 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 7657 T^{2} + \cdots - 3188244 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 402 T^{3} + \cdots - 1682132124 \) Copy content Toggle raw display
$13$ \( T^{4} + 462 T^{3} + \cdots + 149501563456 \) Copy content Toggle raw display
$17$ \( T^{4} - 276 T^{3} + \cdots - 50104147968 \) Copy content Toggle raw display
$19$ \( T^{4} + 510 T^{3} + \cdots + 7391138416576 \) Copy content Toggle raw display
$23$ \( T^{4} + 6900 T^{3} + \cdots + 3007939608576 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 408027025117872 \) Copy content Toggle raw display
$31$ \( T^{4} - 6410 T^{3} + \cdots + 86716089209547 \) Copy content Toggle raw display
$37$ \( T^{4} - 15250 T^{3} + \cdots - 50\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{4} - 4308 T^{3} + \cdots - 18\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 991662745581932 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 270685655359056 \) Copy content Toggle raw display
$53$ \( T^{4} + 13692 T^{3} + \cdots - 85\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{4} - 34830 T^{3} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} - 5364 T^{3} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + 5994 T^{3} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{4} + 89268 T^{3} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{4} + 59638 T^{3} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + 44062 T^{3} + \cdots + 16\!\cdots\!59 \) Copy content Toggle raw display
$83$ \( T^{4} + 208446 T^{3} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{4} + 77520 T^{3} + \cdots - 47\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{4} - 188630 T^{3} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
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