Properties

Label 63.10.a.e
Level $63$
Weight $10$
Character orbit 63.a
Self dual yes
Analytic conductor $32.447$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.4472576783\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 426x + 2016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 7) q^{2} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + ( - 43 \beta_{2} + 13 \beta_1 - 518) q^{5} + 2401 q^{7} + (470 \beta_{2} - 147 \beta_1 - 4685) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 7) q^{2} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + ( - 43 \beta_{2} + 13 \beta_1 - 518) q^{5} + 2401 q^{7} + (470 \beta_{2} - 147 \beta_1 - 4685) q^{8} + (370 \beta_{2} - 470 \beta_1 - 32620) q^{10} + ( - 650 \beta_{2} - 658 \beta_1 + 1148) q^{11} + (3017 \beta_{2} - 175 \beta_1 - 6594) q^{13} + (2401 \beta_{2} - 16807) q^{14} + ( - 10614 \beta_{2} + 1617 \beta_1 + 160987) q^{16} + ( - 3030 \beta_{2} - 1574 \beta_1 - 338898) q^{17} + ( - 15371 \beta_{2} + 2437 \beta_1 + 74284) q^{19} + ( - 41524 \beta_{2} + 2044 \beta_1 + 640696) q^{20} + ( - 40972 \beta_{2} + 4004 \beta_1 - 949016) q^{22} + (24200 \beta_{2} + 3808 \beta_1 - 628544) q^{23} + ( - 15338 \beta_{2} - 4802 \beta_1 + 1024407) q^{25} + ( - 20986 \beta_{2} + 23394 \beta_1 + 2928352) q^{26} + ( - 19208 \beta_{2} + 16807 \beta_1 + 1246119) q^{28} + ( - 54866 \beta_{2} - 18914 \beta_1 - 1360606) q^{29} + (55698 \beta_{2} - 70302 \beta_1 + 956480) q^{31} + (36066 \beta_{2} - 20055 \beta_1 - 8407317) q^{32} + ( - 438178 \beta_{2} - 748 \beta_1 - 1327214) q^{34} + ( - 103243 \beta_{2} + 31213 \beta_1 - 1243718) q^{35} + (209418 \beta_{2} + 60522 \beta_1 + 465206) q^{37} + (248060 \beta_{2} - 139278 \beta_1 - 14493290) q^{38} + (625640 \beta_{2} - 76600 \beta_1 - 27619760) q^{40} + (163478 \beta_{2} + 131894 \beta_1 + 4806886) q^{41} + (121982 \beta_{2} + 65366 \beta_1 - 20543724) q^{43} + ( - 314984 \beta_{2} - 1960 \beta_1 - 32337328) q^{44} + ( - 405224 \beta_{2} + 119896 \beta_1 + 29915888) q^{46} + (534778 \beta_{2} - 83238 \beta_1 + 3456320) q^{47} + 5764801 q^{49} + (727615 \beta_{2} - 44940 \beta_1 - 24441685) q^{50} + (2925244 \beta_{2} - 361424 \beta_1 - 26969348) q^{52} + ( - 1553376 \beta_{2} + 450352 \beta_1 - 22500870) q^{53} + (1659356 \beta_{2} + 988364 \beta_1 - 35274344) q^{55} + (1128470 \beta_{2} - 352947 \beta_1 - 11248685) q^{56} + ( - 2535150 \beta_{2} - 138180 \beta_1 - 53054610) q^{58} + (2231195 \beta_{2} + 49659 \beta_1 + 14196700) q^{59} + (589107 \beta_{2} - 844773 \beta_1 + 63915614) q^{61} + ( - 3668848 \beta_{2} + 1303812 \beta_1 + 15661156) q^{62} + ( - 4312590 \beta_{2} - 314727 \beta_1 + 2617387) q^{64} + (958090 \beta_{2} - 1035790 \beta_1 - 121427740) q^{65} + (3939816 \beta_{2} + 47712 \beta_1 - 85058596) q^{67} + (613704 \beta_{2} - 2251634 \beta_1 - 247828602) q^{68} + (888370 \beta_{2} - 1128470 \beta_1 - 78320620) q^{70} + ( - 3499356 \beta_{2} + 526260 \beta_1 - 98838168) q^{71} + ( - 9544844 \beta_{2} + 118516 \beta_1 + 114737770) q^{73} + (4189718 \beta_{2} + 679140 \beta_1 + 230232154) q^{74} + ( - 15924468 \beta_{2} + 2299290 \beta_1 + 242946662) q^{76} + ( - 1560650 \beta_{2} - 1579858 \beta_1 + 2756348) q^{77} + ( - 7475532 \beta_{2} + 1679412 \beta_1 - 320137552) q^{79} + ( - 11964112 \beta_{2} + 4328752 \beta_1 + 444444448) q^{80} + (13216518 \beta_{2} - 570276 \beta_1 + 187558434) q^{82} + (4479559 \beta_{2} + 1977367 \beta_1 + 366839060) q^{83} + (18558254 \beta_{2} - 1518034 \beta_1 + 146059804) q^{85} + ( - 16416916 \beta_{2} + 4116 \beta_1 + 293660752) q^{86} + ( - 11172080 \beta_{2} - 4229456 \beta_1 + 402041600) q^{88} + (10612104 \beta_{2} - 1815976 \beta_1 - 168938826) q^{89} + (7243817 \beta_{2} - 420175 \beta_1 - 15832194) q^{91} + (25723952 \beta_{2} - 6344912 \beta_1 - 230374496) q^{92} + ( - 2488928 \beta_{2} + 4825540 \beta_1 + 462668276) q^{94} + ( - 10749416 \beta_{2} + 4098416 \beta_1 + 734357024) q^{95} + (33276782 \beta_{2} - 864850 \beta_1 - 215832750) q^{97} + (5764801 \beta_{2} - 40353607) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8} - 97860 q^{10} + 3444 q^{11} - 19782 q^{13} - 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22} - 1885632 q^{23} + 3073221 q^{25} + 8785056 q^{26} + 3738357 q^{28} - 4081818 q^{29} + 2869440 q^{31} - 25221951 q^{32} - 3981642 q^{34} - 3731154 q^{35} + 1395618 q^{37} - 43479870 q^{38} - 82859280 q^{40} + 14420658 q^{41} - 61631172 q^{43} - 97011984 q^{44} + 89747664 q^{46} + 10368960 q^{47} + 17294403 q^{49} - 73325055 q^{50} - 80908044 q^{52} - 67502610 q^{53} - 105823032 q^{55} - 33746055 q^{56} - 159163830 q^{58} + 42590100 q^{59} + 191746842 q^{61} + 46983468 q^{62} + 7852161 q^{64} - 364283220 q^{65} - 255175788 q^{67} - 743485806 q^{68} - 234961860 q^{70} - 296514504 q^{71} + 344213310 q^{73} + 690696462 q^{74} + 728839986 q^{76} + 8269044 q^{77} - 960412656 q^{79} + 1333333344 q^{80} + 562675302 q^{82} + 1100517180 q^{83} + 438179412 q^{85} + 880982256 q^{86} + 1206124800 q^{88} - 506816478 q^{89} - 47496582 q^{91} - 691123488 q^{92} + 1388004828 q^{94} + 2203071072 q^{95} - 647498250 q^{97} - 121060821 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 426x + 2016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 25\nu + 276 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 11\nu - 288 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 25\beta_{2} - 11\beta _1 + 1706 ) / 6 \) 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
4.96128
−22.2358
18.2745
−41.8019 0 1235.40 1791.89 0 2401.00 −30239.6 0 −74904.4
1.2 −13.3607 0 −333.491 −1922.19 0 2401.00 11296.4 0 25681.8
1.3 34.1627 0 655.088 −1423.70 0 2401.00 4888.28 0 −48637.4
\(n\): 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 63.10.a.e 3
3.b odd 2 1 7.10.a.b 3
12.b even 2 1 112.10.a.h 3
15.d odd 2 1 175.10.a.d 3
15.e even 4 2 175.10.b.d 6
21.c even 2 1 49.10.a.c 3
21.g even 6 2 49.10.c.e 6
21.h odd 6 2 49.10.c.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.b 3 3.b odd 2 1
49.10.a.c 3 21.c even 2 1
49.10.c.d 6 21.h odd 6 2
49.10.c.e 6 21.g even 6 2
63.10.a.e 3 1.a even 1 1 trivial
112.10.a.h 3 12.b even 2 1
175.10.a.d 3 15.d odd 2 1
175.10.b.d 6 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 21T_{2}^{2} - 1326T_{2} - 19080 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 21 T^{2} - 1326 T - 19080 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 1554 T^{2} + \cdots - 4903718400 \) Copy content Toggle raw display
$7$ \( (T - 2401)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots - 108859759460352 \) Copy content Toggle raw display
$13$ \( T^{3} + 19782 T^{2} + \cdots - 41548412541440 \) Copy content Toggle raw display
$17$ \( T^{3} + 1016694 T^{2} + \cdots + 21\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{3} - 222852 T^{2} + \cdots - 43\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{3} + 1885632 T^{2} + \cdots - 97\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{3} + 4081818 T^{2} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} - 2869440 T^{2} + \cdots - 74\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{3} - 1395618 T^{2} + \cdots - 34\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{3} - 14420658 T^{2} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{3} + 61631172 T^{2} + \cdots + 68\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{3} - 10368960 T^{2} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{3} + 67502610 T^{2} + \cdots - 23\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{3} - 42590100 T^{2} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} - 191746842 T^{2} + \cdots + 51\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{3} + 255175788 T^{2} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{3} + 296514504 T^{2} + \cdots - 16\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{3} - 344213310 T^{2} + \cdots + 19\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{3} + 960412656 T^{2} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} - 1100517180 T^{2} + \cdots - 18\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{3} + 506816478 T^{2} + \cdots - 19\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{3} + 647498250 T^{2} + \cdots - 49\!\cdots\!16 \) Copy content Toggle raw display
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