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: \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{4} - \beta_1 + 25) q^{4} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 16) q^{5} + ( - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 25 \beta_{2} - 9 \beta_1 + 28) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{4} - \beta_1 + 25) q^{4} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 16) q^{5} + ( - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 25 \beta_{2} - 9 \beta_1 + 28) q^{8} + ( - 3 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} + 15 \beta_{2} + 19 \beta_1 - 138) q^{10} + (7 \beta_{5} + 8 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} - 6 \beta_1 - 107) q^{11} + (3 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} + 34 \beta_{2} - 12 \beta_1 - 215) q^{13} + (12 \beta_{5} + 33 \beta_{4} + 18 \beta_{3} + 126 \beta_{2} - 27 \beta_1 + 801) q^{16} + ( - 17 \beta_{5} - 8 \beta_{4} - 7 \beta_{3} + 32 \beta_{2} - 31 \beta_1 + 521) q^{17} + (3 \beta_{5} + 24 \beta_{4} - 12 \beta_{3} + 174 \beta_{2} + 55 \beta_1 - 231) q^{19} + (14 \beta_{5} - 7 \beta_{4} - 5 \beta_{3} - 335 \beta_{2} + 100 \beta_1 - 41) q^{20} + ( - 36 \beta_{5} - 32 \beta_{4} + 18 \beta_{3} + 30 \beta_{2} - 238 \beta_1 - 664) q^{22} + ( - 51 \beta_{5} + 24 \beta_{4} - 45 \beta_{3} - 16 \beta_{2} + 759) q^{23} + ( - 18 \beta_{5} + 16 \beta_{4} - 36 \beta_{3} + 228 \beta_{2} + \cdots + 883) q^{25}+ \cdots + (432 \beta_{5} - 32 \beta_{4} + 22 \beta_{3} + 6220 \beta_{2} + \cdots - 9896) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 28\nu^{5} - 35\nu^{4} - 546\nu^{3} + 742\nu^{2} - 7063\nu + 11802 ) / 1941 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -87\nu^{5} - 53\nu^{4} + 3961\nu^{3} + 2547\nu^{2} - 42269\nu - 27289 ) / 1941 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 158\nu^{5} - 521\nu^{4} - 6316\nu^{3} + 21656\nu^{2} + 33579\nu - 143678 ) / 1941 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 108\nu^{5} + 512\nu^{4} - 5988\nu^{3} - 17195\nu^{2} + 81453\nu + 71402 ) / 647 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -774\nu^{5} - 1297\nu^{4} + 39032\nu^{3} + 55188\nu^{2} - 447553\nu - 433643 ) / 1941 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} - 12\beta_{2} - 4\beta _1 + 5 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + 4\beta_{4} + 5\beta_{3} - 4\beta_{2} - 4\beta _1 + 567 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29\beta_{5} + 4\beta_{4} - 28\beta_{3} - 302\beta_{2} - 25\beta _1 - 129 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 99\beta_{5} + 176\beta_{4} + 155\beta_{3} + 80\beta_{2} + 74\beta _1 + 14843 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 862\beta_{5} + 192\beta_{4} - 737\beta_{3} - 8710\beta_{2} + 643\beta _1 - 9528 ) / 28 \) 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
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{6} + 2T_{2}^{5} - 169T_{2}^{4} - 336T_{2}^{3} + 6472T_{2}^{2} + 4256T_{2} - 51088 \) Copy content Toggle raw display
\( T_{5}^{6} - 100T_{5}^{5} - 6778T_{5}^{4} + 651312T_{5}^{3} + 9669292T_{5}^{2} - 959211664T_{5} + 9969962312 \) Copy content Toggle raw display
\( T_{13}^{6} + 1352 T_{13}^{5} - 126598 T_{13}^{4} - 808285728 T_{13}^{3} - 312398022260 T_{13}^{2} - 5552830868704 T_{13} + 88\!\cdots\!04 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} - 169 T^{4} + \cdots - 51088 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 100 T^{5} + \cdots + 9969962312 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 604 T^{5} + \cdots - 55273527989696 \) Copy content Toggle raw display
$13$ \( T^{6} + 1352 T^{5} + \cdots + 88\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{6} - 3028 T^{5} + \cdots - 77\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{6} + 1728 T^{5} + \cdots - 37\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{6} - 4484 T^{5} + \cdots - 10\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{6} - 5320 T^{5} + \cdots - 45\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{6} + 3976 T^{5} + \cdots - 32\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{6} - 22680 T^{5} + \cdots - 79\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{6} - 28756 T^{5} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{6} + 6768 T^{5} + \cdots - 28\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{6} - 51552 T^{5} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{6} + 80884 T^{5} + \cdots + 93\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{6} - 8872 T^{5} + \cdots - 34\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{6} + 50896 T^{5} + \cdots - 18\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{6} - 6480 T^{5} + \cdots - 24\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{6} - 110852 T^{5} + \cdots - 29\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{6} + 64232 T^{5} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{6} - 111696 T^{5} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{6} - 101128 T^{5} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{6} + 35012 T^{5} + \cdots + 48\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{6} + 70952 T^{5} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
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