Properties

Label 63.10.a.d
Level $63$
Weight $10$
Character orbit 63.a
Self dual yes
Analytic conductor $32.447$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{2} + (6 \beta - 310) q^{4} + (95 \beta + 1119) q^{5} - 2401 q^{7} + ( - 804 \beta - 1308) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 3) q^{2} + (6 \beta - 310) q^{4} + (95 \beta + 1119) q^{5} - 2401 q^{7} + ( - 804 \beta - 1308) q^{8} + (1404 \beta + 21692) q^{10} + (3326 \beta - 17658) q^{11} + (10899 \beta - 13265) q^{13} + ( - 2401 \beta - 7203) q^{14} + ( - 6792 \beta - 376) q^{16} + ( - 9426 \beta + 231960) q^{17} + ( - 1887 \beta - 462713) q^{19} + ( - 22736 \beta - 236880) q^{20} + ( - 7680 \beta + 588944) q^{22} + (38088 \beta - 389064) q^{23} + (212610 \beta + 1040861) q^{25} + (19432 \beta + 2063712) q^{26} + ( - 14406 \beta + 744310) q^{28} + (94682 \beta + 5001792) q^{29} + ( - 161430 \beta + 1233630) q^{31} + (390896 \beta - 642288) q^{32} + (203682 \beta - 1123338) q^{34} + ( - 228095 \beta - 2686719) q^{35} + ( - 248130 \beta + 15367776) q^{37} + ( - 468374 \beta - 1752330) q^{38} + ( - 1023936 \beta - 16204992) q^{40} + (860818 \beta + 9551724) q^{41} + (1048278 \beta + 2032550) q^{43} + ( - 1137008 \beta + 9325488) q^{44} + ( - 274800 \beta + 6183792) q^{46} + ( - 1033182 \beta + 41097510) q^{47} + 5764801 q^{49} + (1678691 \beta + 44156313) q^{50} + ( - 3458280 \beta + 16733192) q^{52} + ( - 4685568 \beta + 27594906) q^{53} + (2044284 \beta + 41222908) q^{55} + (1930404 \beta + 3140508) q^{56} + (5285838 \beta + 33279002) q^{58} + ( - 1563825 \beta + 3534609) q^{59} + ( - 3395319 \beta + 22158193) q^{61} + (749340 \beta - 27455100) q^{62} + (4007904 \beta + 73708576) q^{64} + (10935806 \beta + 184989630) q^{65} + ( - 7026216 \beta - 120960668) q^{67} + (4313820 \beta - 82822908) q^{68} + ( - 3371004 \beta - 52082492) q^{70} + ( - 15075900 \beta - 103246908) q^{71} + ( - 2840484 \beta - 249576594) q^{73} + (14623386 \beta - 1785762) q^{74} + ( - 2191308 \beta + 141255884) q^{76} + ( - 7985726 \beta + 42396858) q^{77} + (16873716 \beta + 234267548) q^{79} + ( - 7635968 \beta - 124952064) q^{80} + (12134178 \beta + 194793046) q^{82} + (21562275 \beta - 222011979) q^{83} + (11488506 \beta + 86737530) q^{85} + (5177384 \beta + 208415304) q^{86} + (9846624 \beta - 493005408) q^{88} + ( - 1406968 \beta - 318133698) q^{89} + ( - 26168499 \beta + 31849265) q^{91} + ( - 14141664 \beta + 164715744) q^{92} + (37997964 \beta - 76111596) q^{94} + ( - 46069288 \beta - 552373992) q^{95} + (5731530 \beta - 816358032) q^{97} + (5764801 \beta + 17294403) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} - 620 q^{4} + 2238 q^{5} - 4802 q^{7} - 2616 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} - 620 q^{4} + 2238 q^{5} - 4802 q^{7} - 2616 q^{8} + 43384 q^{10} - 35316 q^{11} - 26530 q^{13} - 14406 q^{14} - 752 q^{16} + 463920 q^{17} - 925426 q^{19} - 473760 q^{20} + 1177888 q^{22} - 778128 q^{23} + 2081722 q^{25} + 4127424 q^{26} + 1488620 q^{28} + 10003584 q^{29} + 2467260 q^{31} - 1284576 q^{32} - 2246676 q^{34} - 5373438 q^{35} + 30735552 q^{37} - 3504660 q^{38} - 32409984 q^{40} + 19103448 q^{41} + 4065100 q^{43} + 18650976 q^{44} + 12367584 q^{46} + 82195020 q^{47} + 11529602 q^{49} + 88312626 q^{50} + 33466384 q^{52} + 55189812 q^{53} + 82445816 q^{55} + 6281016 q^{56} + 66558004 q^{58} + 7069218 q^{59} + 44316386 q^{61} - 54910200 q^{62} + 147417152 q^{64} + 369979260 q^{65} - 241921336 q^{67} - 165645816 q^{68} - 104164984 q^{70} - 206493816 q^{71} - 499153188 q^{73} - 3571524 q^{74} + 282511768 q^{76} + 84793716 q^{77} + 468535096 q^{79} - 249904128 q^{80} + 389586092 q^{82} - 444023958 q^{83} + 173475060 q^{85} + 416830608 q^{86} - 986010816 q^{88} - 636267396 q^{89} + 63698530 q^{91} + 329431488 q^{92} - 152223192 q^{94} - 1104747984 q^{95} - 1632716064 q^{97} + 34588806 q^{98}+O(q^{100}) \) 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
−6.44622
7.44622
−10.8924 0 −393.355 −200.782 0 −2401.00 9861.52 0 2187.01
1.2 16.8924 0 −226.645 2438.78 0 −2401.00 −12477.5 0 41197.0
\(n\): 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.d 2
3.b odd 2 1 7.10.a.a 2
12.b even 2 1 112.10.a.e 2
15.d odd 2 1 175.10.a.b 2
15.e even 4 2 175.10.b.b 4
21.c even 2 1 49.10.a.b 2
21.g even 6 2 49.10.c.b 4
21.h odd 6 2 49.10.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.a 2 3.b odd 2 1
49.10.a.b 2 21.c even 2 1
49.10.c.b 4 21.g even 6 2
49.10.c.c 4 21.h odd 6 2
63.10.a.d 2 1.a even 1 1 trivial
112.10.a.e 2 12.b even 2 1
175.10.a.b 2 15.d odd 2 1
175.10.b.b 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 6T_{2} - 184 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T - 184 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2238 T - 489664 \) Copy content Toggle raw display
$7$ \( (T + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 35316 T - 1823214304 \) Copy content Toggle raw display
$13$ \( T^{2} + 26530 T - 22750162568 \) Copy content Toggle raw display
$17$ \( T^{2} - 463920 T + 36657492732 \) Copy content Toggle raw display
$19$ \( T^{2} + 925426 T + 213416091952 \) Copy content Toggle raw display
$23$ \( T^{2} + 778128 T - 128613482496 \) Copy content Toggle raw display
$29$ \( T^{2} - 10003584 T + 23287739754332 \) Copy content Toggle raw display
$31$ \( T^{2} - 2467260 T - 3507668488800 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 224285819284476 \) Copy content Toggle raw display
$41$ \( T^{2} - 19103448 T - 51779041048756 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 207953886197312 \) Copy content Toggle raw display
$47$ \( T^{2} - 82195020 T + 14\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{2} - 55189812 T - 34\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 459497424927744 \) Copy content Toggle raw display
$61$ \( T^{2} - 44316386 T - 17\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} + 241921336 T + 51\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{2} + 206493816 T - 33\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + 499153188 T + 60\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{2} - 468535096 T - 70118242258304 \) Copy content Toggle raw display
$83$ \( T^{2} + 444023958 T - 40\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + 636267396 T + 10\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{2} + 1632716064 T + 66\!\cdots\!24 \) Copy content Toggle raw display
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