Properties

Label 441.6.a.bb
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 59 x^{4} + 122 x^{3} + 941 x^{2} - 1856 x - 2338\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 25 - \beta_{1} + \beta_{4} ) q^{4} + ( 16 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( 28 - 9 \beta_{1} + 25 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 25 - \beta_{1} + \beta_{4} ) q^{4} + ( 16 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( 28 - 9 \beta_{1} + 25 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{8} + ( -138 + 19 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} - 3 \beta_{5} ) q^{10} + ( -107 - 6 \beta_{1} - 12 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} ) q^{11} + ( -215 - 12 \beta_{1} + 34 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} ) q^{13} + ( 801 - 27 \beta_{1} + 126 \beta_{2} + 18 \beta_{3} + 33 \beta_{4} + 12 \beta_{5} ) q^{16} + ( 521 - 31 \beta_{1} + 32 \beta_{2} - 7 \beta_{3} - 8 \beta_{4} - 17 \beta_{5} ) q^{17} + ( -231 + 55 \beta_{1} + 174 \beta_{2} - 12 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} ) q^{19} + ( -41 + 100 \beta_{1} - 335 \beta_{2} - 5 \beta_{3} - 7 \beta_{4} + 14 \beta_{5} ) q^{20} + ( -664 - 238 \beta_{1} + 30 \beta_{2} + 18 \beta_{3} - 32 \beta_{4} - 36 \beta_{5} ) q^{22} + ( 759 - 16 \beta_{2} - 45 \beta_{3} + 24 \beta_{4} - 51 \beta_{5} ) q^{23} + ( 883 + 290 \beta_{1} + 228 \beta_{2} - 36 \beta_{3} + 16 \beta_{4} - 18 \beta_{5} ) q^{25} + ( 2206 + 45 \beta_{1} - 443 \beta_{2} - 9 \beta_{3} + 7 \beta_{4} + 25 \beta_{5} ) q^{26} + ( 837 + 201 \beta_{1} - 86 \beta_{2} + 54 \beta_{3} + 24 \beta_{4} + 63 \beta_{5} ) q^{29} + ( -673 - 215 \beta_{1} - 94 \beta_{2} - 88 \beta_{3} + 56 \beta_{4} - 63 \beta_{5} ) q^{31} + ( 6490 - 687 \beta_{1} + 781 \beta_{2} + 3 \beta_{3} + 128 \beta_{4} - 26 \beta_{5} ) q^{32} + ( 2908 + 501 \beta_{1} + 655 \beta_{2} + 25 \beta_{3} + 139 \beta_{4} + 99 \beta_{5} ) q^{34} + ( 3758 - 300 \beta_{1} - 24 \beta_{2} - 54 \beta_{3} + 96 \beta_{4} + 42 \beta_{5} ) q^{37} + ( 8946 - 292 \beta_{1} + 318 \beta_{2} - 22 \beta_{3} + 62 \beta_{4} - 148 \beta_{5} ) q^{38} + ( -17155 - 446 \beta_{1} - 1195 \beta_{2} - 37 \beta_{3} - 277 \beta_{4} - 30 \beta_{5} ) q^{40} + ( 4961 + 23 \beta_{1} + 656 \beta_{2} + 65 \beta_{3} + 136 \beta_{4} + 151 \beta_{5} ) q^{41} + ( -1218 - 208 \beta_{1} - 120 \beta_{2} - 90 \beta_{3} - 32 \beta_{4} + 150 \beta_{5} ) q^{43} + ( 10718 + 990 \beta_{1} - 80 \beta_{2} + 132 \beta_{3} + 322 \beta_{4} + 236 \beta_{5} ) q^{44} + ( 876 + 1498 \beta_{1} + 2394 \beta_{2} + 54 \beta_{3} + 140 \beta_{4} + 36 \beta_{5} ) q^{46} + ( 8645 - 381 \beta_{1} + 206 \beta_{2} - 72 \beta_{3} - 184 \beta_{4} + 47 \beta_{5} ) q^{47} + ( 7180 + 312 \beta_{1} + 909 \beta_{2} - 276 \beta_{3} - 100 \beta_{4} - 320 \beta_{5} ) q^{50} + ( -19507 + 458 \beta_{1} + 881 \beta_{2} - \beta_{3} - 445 \beta_{4} - 246 \beta_{5} ) q^{52} + ( -13832 - 1614 \beta_{1} - 996 \beta_{2} - 132 \beta_{3} - 208 \beta_{4} + 58 \beta_{5} ) q^{53} + ( -2977 - 147 \beta_{1} - 2686 \beta_{2} + 536 \beta_{3} + 248 \beta_{4} + 417 \beta_{5} ) q^{55} + ( -11682 - 2452 \beta_{1} - 222 \beta_{2} - 162 \beta_{3} - 434 \beta_{4} - 408 \beta_{5} ) q^{58} + ( 2115 + 941 \beta_{1} + 1794 \beta_{2} - 100 \beta_{3} - 280 \beta_{4} - 115 \beta_{5} ) q^{59} + ( -8803 + 74 \beta_{1} - 1186 \beta_{2} - 328 \beta_{3} + 200 \beta_{4} - 225 \beta_{5} ) q^{61} + ( 2894 + 2492 \beta_{1} + 2918 \beta_{2} + 302 \beta_{3} + 178 \beta_{4} + 148 \beta_{5} ) q^{62} + ( 36125 - 125 \beta_{1} + 8580 \beta_{2} + 396 \beta_{3} + 737 \beta_{4} ) q^{64} + ( 467 + 3 \beta_{1} + 2942 \beta_{2} + 132 \beta_{3} - 392 \beta_{4} - 205 \beta_{5} ) q^{65} + ( 1794 + 148 \beta_{1} + 2496 \beta_{2} + 414 \beta_{3} - 304 \beta_{4} + 354 \beta_{5} ) q^{67} + ( 7407 - 3712 \beta_{1} + 2901 \beta_{2} - 73 \beta_{3} - 19 \beta_{4} - 646 \beta_{5} ) q^{68} + ( 19359 + 2364 \beta_{1} + 3200 \beta_{2} + 243 \beta_{3} + 120 \beta_{4} + 549 \beta_{5} ) q^{71} + ( -12854 - 787 \beta_{1} - 6584 \beta_{2} - 266 \beta_{3} - 32 \beta_{4} - 138 \beta_{5} ) q^{73} + ( 7008 - 324 \beta_{1} + 7124 \beta_{2} + 396 \beta_{3} - 192 \beta_{4} - 168 \beta_{5} ) q^{74} + ( 35078 + 528 \beta_{1} + 8234 \beta_{2} + 926 \beta_{3} + 830 \beta_{4} + 432 \beta_{5} ) q^{76} + ( 19592 + 3992 \beta_{1} + 3240 \beta_{2} + 468 \beta_{3} - 272 \beta_{4} + 312 \beta_{5} ) q^{79} + ( -55959 + 2424 \beta_{1} - 12525 \beta_{2} + 45 \beta_{3} - 777 \beta_{4} + 882 \beta_{5} ) q^{80} + ( 34012 - 5841 \beta_{1} + 6397 \beta_{2} + 163 \beta_{3} + 37 \beta_{4} - 819 \beta_{5} ) q^{82} + ( 17178 - 622 \beta_{1} + 828 \beta_{2} + 764 \beta_{3} - 16 \beta_{4} - 142 \beta_{5} ) q^{83} + ( -3110 + 514 \beta_{1} + 1428 \beta_{2} - 1134 \beta_{3} - 976 \beta_{4} - 888 \beta_{5} ) q^{85} + ( -1616 - 192 \beta_{1} - 3172 \beta_{2} - 96 \beta_{3} - 1444 \beta_{4} - 236 \beta_{5} ) q^{86} + ( -11784 - 3034 \beta_{1} + 12162 \beta_{2} - 1026 \beta_{3} - 716 \beta_{4} - 1380 \beta_{5} ) q^{88} + ( -7483 - 6481 \beta_{1} - 4588 \beta_{2} - 169 \beta_{3} + 1000 \beta_{4} + 355 \beta_{5} ) q^{89} + ( 75050 - 6930 \beta_{1} + 180 \beta_{2} + 240 \beta_{3} + 286 \beta_{4} - 340 \beta_{5} ) q^{92} + ( 21314 + 2584 \beta_{1} + 4154 \beta_{2} - 106 \beta_{3} - 286 \beta_{4} + 720 \beta_{5} ) q^{94} + ( 20578 + 9324 \beta_{1} + 1856 \beta_{2} - 366 \beta_{3} - 1264 \beta_{4} - 338 \beta_{5} ) q^{95} + ( -9896 + 6723 \beta_{1} + 6220 \beta_{2} + 22 \beta_{3} - 32 \beta_{4} + 432 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} + O(q^{10}) \) \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 59 x^{4} + 122 x^{3} + 941 x^{2} - 1856 x - 2338\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 28 \nu^{5} - 35 \nu^{4} - 546 \nu^{3} + 742 \nu^{2} - 7063 \nu + 11802 \)\()/1941\)
\(\beta_{2}\)\(=\)\((\)\( -87 \nu^{5} - 53 \nu^{4} + 3961 \nu^{3} + 2547 \nu^{2} - 42269 \nu - 27289 \)\()/1941\)
\(\beta_{3}\)\(=\)\((\)\( 158 \nu^{5} - 521 \nu^{4} - 6316 \nu^{3} + 21656 \nu^{2} + 33579 \nu - 143678 \)\()/1941\)
\(\beta_{4}\)\(=\)\((\)\( 108 \nu^{5} + 512 \nu^{4} - 5988 \nu^{3} - 17195 \nu^{2} + 81453 \nu + 71402 \)\()/647\)
\(\beta_{5}\)\(=\)\((\)\( -774 \nu^{5} - 1297 \nu^{4} + 39032 \nu^{3} + 55188 \nu^{2} - 447553 \nu - 433643 \)\()/1941\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{3} - 12 \beta_{2} - 4 \beta_{1} + 5\)\()/28\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 567\)\()/28\)
\(\nu^{3}\)\(=\)\((\)\(29 \beta_{5} + 4 \beta_{4} - 28 \beta_{3} - 302 \beta_{2} - 25 \beta_{1} - 129\)\()/28\)
\(\nu^{4}\)\(=\)\((\)\(99 \beta_{5} + 176 \beta_{4} + 155 \beta_{3} + 80 \beta_{2} + 74 \beta_{1} + 14843\)\()/28\)
\(\nu^{5}\)\(=\)\((\)\(862 \beta_{5} + 192 \beta_{4} - 737 \beta_{3} - 8710 \beta_{2} + 643 \beta_{1} - 9528\)\()/28\)

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
5.59680
3.75353
−5.10089
4.27213
−0.910122
−5.61145
−10.0089 0 68.1779 70.3512 0 0 −362.101 0 −704.138
1.2 −8.20863 0 35.3816 29.2259 0 0 −27.7583 0 −239.905
1.3 −3.38033 0 −20.5734 −54.5253 0 0 177.715 0 184.313
1.4 3.09163 0 −22.4418 13.7926 0 0 −168.314 0 42.6416
1.5 5.31815 0 −3.71724 103.471 0 0 −189.950 0 550.272
1.6 11.1881 0 93.1729 −62.3150 0 0 684.407 0 −697.185
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bb 6
3.b odd 2 1 147.6.a.n 6
7.b odd 2 1 441.6.a.ba 6
21.c even 2 1 147.6.a.o yes 6
21.g even 6 2 147.6.e.p 12
21.h odd 6 2 147.6.e.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.n 6 3.b odd 2 1
147.6.a.o yes 6 21.c even 2 1
147.6.e.p 12 21.g even 6 2
147.6.e.q 12 21.h odd 6 2
441.6.a.ba 6 7.b odd 2 1
441.6.a.bb 6 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{6} + 2 T_{2}^{5} - 169 T_{2}^{4} - 336 T_{2}^{3} + 6472 T_{2}^{2} + 4256 T_{2} - 51088 \)
\( T_{5}^{6} - 100 T_{5}^{5} - 6778 T_{5}^{4} + 651312 T_{5}^{3} + 9669292 T_{5}^{2} - 959211664 T_{5} + 9969962312 \)
\( T_{13}^{6} + 1352 T_{13}^{5} - 126598 T_{13}^{4} - 808285728 T_{13}^{3} - 312398022260 T_{13}^{2} - \)\(55\!\cdots\!04\)\( T_{13} + \)\(88\!\cdots\!04\)\( \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -51088 + 4256 T + 6472 T^{2} - 336 T^{3} - 169 T^{4} + 2 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( 9969962312 - 959211664 T + 9669292 T^{2} + 651312 T^{3} - 6778 T^{4} - 100 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( -55273527989696 + 15123294433216 T - 11867400976 T^{2} - 359951456 T^{3} - 500836 T^{4} + 604 T^{5} + T^{6} \)
$13$ \( 8818638530643704 - 5552830868704 T - 312398022260 T^{2} - 808285728 T^{3} - 126598 T^{4} + 1352 T^{5} + T^{6} \)
$17$ \( -77634357357142712 + 1817914266048944 T - 5595796869460 T^{2} + 4922692976 T^{3} + 960902 T^{4} - 3028 T^{5} + T^{6} \)
$19$ \( -3735293167739744768 + 16882924839579648 T + 16923734261952 T^{2} - 11346877440 T^{3} - 7968600 T^{4} + 1728 T^{5} + T^{6} \)
$23$ \( -10876910613896947648 - 251128931562936896 T + 49469900108272 T^{2} + 90302677920 T^{3} - 19751812 T^{4} - 4484 T^{5} + T^{6} \)
$29$ \( -\)\(45\!\cdots\!84\)\( - 413077597670801920 T + 394389617709760 T^{2} + 130079620992 T^{3} - 41146408 T^{4} - 5320 T^{5} + T^{6} \)
$31$ \( -\)\(32\!\cdots\!04\)\( - 763919493148671488 T + 1627711614738112 T^{2} - 140587032960 T^{3} - 88488040 T^{4} + 3976 T^{5} + T^{6} \)
$37$ \( -\)\(79\!\cdots\!96\)\( + 64747232745500190720 T - 17742287505662208 T^{2} + 1603803782912 T^{3} + 87265776 T^{4} - 22680 T^{5} + T^{6} \)
$41$ \( \)\(18\!\cdots\!56\)\( - 82163383248259387984 T - 15548374571673172 T^{2} + 3631066472048 T^{3} + 55056710 T^{4} - 28756 T^{5} + T^{6} \)
$43$ \( -\)\(28\!\cdots\!68\)\( + 59538852421323669504 T + 22779206566284288 T^{2} - 2231851972608 T^{3} - 387379104 T^{4} + 6768 T^{5} + T^{6} \)
$47$ \( \)\(13\!\cdots\!28\)\( + 16481230197039654912 T - 11579177448956736 T^{2} - 2130988958208 T^{3} + 735770472 T^{4} - 51552 T^{5} + T^{6} \)
$53$ \( \)\(93\!\cdots\!88\)\( - \)\(84\!\cdots\!28\)\( T - 942681098319975184 T^{2} - 27823564813472 T^{3} + 1498706108 T^{4} + 80884 T^{5} + T^{6} \)
$59$ \( -\)\(34\!\cdots\!52\)\( - \)\(45\!\cdots\!32\)\( T + 178489629727189696 T^{2} + 2643841282944 T^{3} - 1106900008 T^{4} - 8872 T^{5} + T^{6} \)
$61$ \( -\)\(18\!\cdots\!36\)\( + \)\(38\!\cdots\!40\)\( T + 42902272625339660 T^{2} - 29293770273344 T^{3} - 222122566 T^{4} + 50896 T^{5} + T^{6} \)
$67$ \( -\)\(24\!\cdots\!32\)\( - \)\(48\!\cdots\!56\)\( T + 1905614748297366528 T^{2} + 15790719633408 T^{3} - 3000969888 T^{4} - 6480 T^{5} + T^{6} \)
$71$ \( -\)\(29\!\cdots\!48\)\( + \)\(40\!\cdots\!88\)\( T - 19614262036831368464 T^{2} + 365269119773856 T^{3} + 858142556 T^{4} - 110852 T^{5} + T^{6} \)
$73$ \( \)\(32\!\cdots\!84\)\( + \)\(31\!\cdots\!12\)\( T + 3738292215205062988 T^{2} - 296839059037344 T^{3} - 5386018870 T^{4} + 64232 T^{5} + T^{6} \)
$79$ \( -\)\(10\!\cdots\!68\)\( + \)\(25\!\cdots\!16\)\( T - 18727181437041497088 T^{2} + 507917557775360 T^{3} - 1585863072 T^{4} - 111696 T^{5} + T^{6} \)
$83$ \( -\)\(19\!\cdots\!68\)\( - \)\(43\!\cdots\!52\)\( T + 2636280972634775296 T^{2} + 259543622746368 T^{3} - 2157797008 T^{4} - 101128 T^{5} + T^{6} \)
$89$ \( \)\(48\!\cdots\!72\)\( - \)\(41\!\cdots\!96\)\( T + 91546162688810429612 T^{2} + 151709255824976 T^{3} - 18350566330 T^{4} + 35012 T^{5} + T^{6} \)
$97$ \( -\)\(13\!\cdots\!88\)\( + \)\(30\!\cdots\!96\)\( T + 8495591547037465420 T^{2} - 119558814297504 T^{3} - 13346679478 T^{4} + 70952 T^{5} + T^{6} \)
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