Properties

Label 63.6.e.a
Level $63$
Weight $6$
Character orbit 63.e
Analytic conductor $10.104$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + 28 \zeta_{6} q^{4} + ( - 11 \zeta_{6} + 11) q^{5} + (7 \zeta_{6} + 126) q^{7} - 120 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + 28 \zeta_{6} q^{4} + ( - 11 \zeta_{6} + 11) q^{5} + (7 \zeta_{6} + 126) q^{7} - 120 q^{8} + 22 \zeta_{6} q^{10} + 269 \zeta_{6} q^{11} - 308 q^{13} + (252 \zeta_{6} - 266) q^{14} + (656 \zeta_{6} - 656) q^{16} + 1896 \zeta_{6} q^{17} + ( - 164 \zeta_{6} + 164) q^{19} + 308 q^{20} - 538 q^{22} + (3264 \zeta_{6} - 3264) q^{23} + 3004 \zeta_{6} q^{25} + ( - 616 \zeta_{6} + 616) q^{26} + (3724 \zeta_{6} - 196) q^{28} - 2417 q^{29} - 2841 \zeta_{6} q^{31} - 5152 \zeta_{6} q^{32} - 3792 q^{34} + ( - 1386 \zeta_{6} + 1463) q^{35} + ( - 11328 \zeta_{6} + 11328) q^{37} + 328 \zeta_{6} q^{38} + (1320 \zeta_{6} - 1320) q^{40} + 16856 q^{41} - 7894 q^{43} + (7532 \zeta_{6} - 7532) q^{44} - 6528 \zeta_{6} q^{46} + ( - 21102 \zeta_{6} + 21102) q^{47} + (1813 \zeta_{6} + 15827) q^{49} - 6008 q^{50} - 8624 \zeta_{6} q^{52} - 29691 \zeta_{6} q^{53} + 2959 q^{55} + ( - 840 \zeta_{6} - 15120) q^{56} + ( - 4834 \zeta_{6} + 4834) q^{58} - 8163 \zeta_{6} q^{59} + (15166 \zeta_{6} - 15166) q^{61} + 5682 q^{62} - 10688 q^{64} + (3388 \zeta_{6} - 3388) q^{65} + 32078 \zeta_{6} q^{67} + (53088 \zeta_{6} - 53088) q^{68} + (2926 \zeta_{6} - 154) q^{70} + 38274 q^{71} - 34866 \zeta_{6} q^{73} + 22656 \zeta_{6} q^{74} + 4592 q^{76} + (35777 \zeta_{6} - 1883) q^{77} + (13529 \zeta_{6} - 13529) q^{79} + 7216 \zeta_{6} q^{80} + (33712 \zeta_{6} - 33712) q^{82} + 68103 q^{83} + 20856 q^{85} + ( - 15788 \zeta_{6} + 15788) q^{86} - 32280 \zeta_{6} q^{88} + (114922 \zeta_{6} - 114922) q^{89} + ( - 2156 \zeta_{6} - 38808) q^{91} - 91392 q^{92} + 42204 \zeta_{6} q^{94} - 1804 \zeta_{6} q^{95} + 154959 q^{97} + (31654 \zeta_{6} - 35280) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 28 q^{4} + 11 q^{5} + 259 q^{7} - 240 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 28 q^{4} + 11 q^{5} + 259 q^{7} - 240 q^{8} + 22 q^{10} + 269 q^{11} - 616 q^{13} - 280 q^{14} - 656 q^{16} + 1896 q^{17} + 164 q^{19} + 616 q^{20} - 1076 q^{22} - 3264 q^{23} + 3004 q^{25} + 616 q^{26} + 3332 q^{28} - 4834 q^{29} - 2841 q^{31} - 5152 q^{32} - 7584 q^{34} + 1540 q^{35} + 11328 q^{37} + 328 q^{38} - 1320 q^{40} + 33712 q^{41} - 15788 q^{43} - 7532 q^{44} - 6528 q^{46} + 21102 q^{47} + 33467 q^{49} - 12016 q^{50} - 8624 q^{52} - 29691 q^{53} + 5918 q^{55} - 31080 q^{56} + 4834 q^{58} - 8163 q^{59} - 15166 q^{61} + 11364 q^{62} - 21376 q^{64} - 3388 q^{65} + 32078 q^{67} - 53088 q^{68} + 2618 q^{70} + 76548 q^{71} - 34866 q^{73} + 22656 q^{74} + 9184 q^{76} + 32011 q^{77} - 13529 q^{79} + 7216 q^{80} - 33712 q^{82} + 136206 q^{83} + 41712 q^{85} + 15788 q^{86} - 32280 q^{88} - 114922 q^{89} - 79772 q^{91} - 182784 q^{92} + 42204 q^{94} - 1804 q^{95} + 309918 q^{97} - 38906 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 + 1.73205i 0 14.0000 + 24.2487i 5.50000 9.52628i 0 129.500 + 6.06218i −120.000 0 11.0000 + 19.0526i
46.1 −1.00000 1.73205i 0 14.0000 24.2487i 5.50000 + 9.52628i 0 129.500 6.06218i −120.000 0 11.0000 19.0526i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.e.a 2
3.b odd 2 1 21.6.e.a 2
7.c even 3 1 inner 63.6.e.a 2
7.c even 3 1 441.6.a.g 1
7.d odd 6 1 441.6.a.h 1
12.b even 2 1 336.6.q.b 2
21.c even 2 1 147.6.e.g 2
21.g even 6 1 147.6.a.d 1
21.g even 6 1 147.6.e.g 2
21.h odd 6 1 21.6.e.a 2
21.h odd 6 1 147.6.a.c 1
84.n even 6 1 336.6.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 3.b odd 2 1
21.6.e.a 2 21.h odd 6 1
63.6.e.a 2 1.a even 1 1 trivial
63.6.e.a 2 7.c even 3 1 inner
147.6.a.c 1 21.h odd 6 1
147.6.a.d 1 21.g even 6 1
147.6.e.g 2 21.c even 2 1
147.6.e.g 2 21.g even 6 1
336.6.q.b 2 12.b even 2 1
336.6.q.b 2 84.n even 6 1
441.6.a.g 1 7.c even 3 1
441.6.a.h 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} + 4 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$7$ \( T^{2} - 259T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} - 269T + 72361 \) Copy content Toggle raw display
$13$ \( (T + 308)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1896 T + 3594816 \) Copy content Toggle raw display
$19$ \( T^{2} - 164T + 26896 \) Copy content Toggle raw display
$23$ \( T^{2} + 3264 T + 10653696 \) Copy content Toggle raw display
$29$ \( (T + 2417)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2841 T + 8071281 \) Copy content Toggle raw display
$37$ \( T^{2} - 11328 T + 128323584 \) Copy content Toggle raw display
$41$ \( (T - 16856)^{2} \) Copy content Toggle raw display
$43$ \( (T + 7894)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 21102 T + 445294404 \) Copy content Toggle raw display
$53$ \( T^{2} + 29691 T + 881555481 \) Copy content Toggle raw display
$59$ \( T^{2} + 8163 T + 66634569 \) Copy content Toggle raw display
$61$ \( T^{2} + 15166 T + 230007556 \) Copy content Toggle raw display
$67$ \( T^{2} - 32078 T + 1028998084 \) Copy content Toggle raw display
$71$ \( (T - 38274)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 34866 T + 1215637956 \) Copy content Toggle raw display
$79$ \( T^{2} + 13529 T + 183033841 \) Copy content Toggle raw display
$83$ \( (T - 68103)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 114922 T + 13207066084 \) Copy content Toggle raw display
$97$ \( (T - 154959)^{2} \) Copy content Toggle raw display
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