Properties

Label 63.6.e.c
Level $63$
Weight $6$
Character orbit 63.e
Analytic conductor $10.104$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-83})\)
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 2 \beta_1 - 2) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 34 \beta_1) q^{4} + ( - 7 \beta_{3} - 20 \beta_1 - 20) q^{5} + ( - 14 \beta_{3} + 7 \beta_{2} - 7 \beta_1 - 91) q^{7} + (5 \beta_{2} + 190) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 2 \beta_1 - 2) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 34 \beta_1) q^{4} + ( - 7 \beta_{3} - 20 \beta_1 - 20) q^{5} + ( - 14 \beta_{3} + 7 \beta_{2} - 7 \beta_1 - 91) q^{7} + (5 \beta_{2} + 190) q^{8} + (27 \beta_{3} - 27 \beta_{2} + 474 \beta_1) q^{10} + ( - \beta_{3} + \beta_{2} + 568 \beta_1) q^{11} + (9 \beta_{2} + 467) q^{13} + (98 \beta_{3} - 21 \beta_{2} + 616 \beta_1 - 266) q^{14} + ( - 99 \beta_{3} + 398 \beta_1 + 398) q^{16} + ( - 148 \beta_{3} + 148 \beta_{2} + 88 \beta_1) q^{17} + ( - 27 \beta_{3} - 1169 \beta_1 - 1169) q^{19} + (277 \beta_{2} + 1982) q^{20} + (567 \beta_{2} + 1074) q^{22} + (308 \beta_{3} + 952 \beta_1 + 952) q^{23} + (231 \beta_{3} - 231 \beta_{2} + 313 \beta_1) q^{25} + ( - 476 \beta_{3} - 1492 \beta_1 - 1492) q^{26} + ( - 35 \beta_{3} + 490 \beta_{2} - 1554 \beta_1 + 2842) q^{28} + ( - 45 \beta_{2} + 1086) q^{29} + (768 \beta_{3} - 768 \beta_{2} + 2531 \beta_1) q^{31} + ( - 459 \beta_{3} + 459 \beta_{2} - 738 \beta_1) q^{32} + ( - 60 \beta_{2} - 9000) q^{34} + (728 \beta_{3} - 231 \beta_{2} + 4858 \beta_1 - 1358) q^{35} + ( - 855 \beta_{3} + 9127 \beta_1 + 9127) q^{37} + (1196 \beta_{3} - 1196 \beta_{2} + 4012 \beta_1) q^{38} + ( - 1395 \beta_{3} - 5970 \beta_1 - 5970) q^{40} + ( - 846 \beta_{2} + 6006) q^{41} + ( - 2043 \beta_{2} - 2407) q^{43} + ( - 1673 \beta_{3} - 19126 \beta_1 - 19126) q^{44} + ( - 1260 \beta_{3} + 1260 \beta_{2} - 21000 \beta_1) q^{46} + ( - 604 \beta_{3} - 11882 \beta_1 - 11882) q^{47} + (2450 \beta_{3} - 1225 \beta_{2} + 1225 \beta_1 - 882) q^{49} + (544 \beta_{2} + 14948) q^{50} + (1680 \beta_{3} - 1680 \beta_{2} + 17552 \beta_1) q^{52} + (1751 \beta_{3} - 1751 \beta_{2} + 16702 \beta_1) q^{53} + (3963 \beta_{2} + 10926) q^{55} + ( - 2625 \beta_{3} + 875 \beta_{2} - 5670 \beta_1 - 19460) q^{56} + ( - 1041 \beta_{3} + 618 \beta_1 + 618) q^{58} + (3917 \beta_{3} - 3917 \beta_{2} - 18590 \beta_1) q^{59} + (2544 \beta_{3} - 19754 \beta_1 - 19754) q^{61} + (3299 \beta_{2} + 52678) q^{62} + ( - 4365 \beta_{2} - 17198) q^{64} + ( - 3386 \beta_{3} - 13246 \beta_1 - 13246) q^{65} + ( - 4461 \beta_{3} + 4461 \beta_{2} + 13151 \beta_1) q^{67} + (4324 \beta_{3} + 24536 \beta_1 + 24536) q^{68} + (861 \beta_{3} + 5586 \beta_{2} - 28098 \beta_1 + 26754) q^{70} + ( - 1404 \beta_{2} - 51750) q^{71} + (5247 \beta_{3} - 5247 \beta_{2} - 11665 \beta_1) q^{73} + ( - 8272 \beta_{3} + 8272 \beta_{2} + 34756 \beta_1) q^{74} + (4344 \beta_{2} + 44768) q^{76} + (4067 \beta_{3} + 3892 \beta_{2} - 48146 \beta_1 + 3108) q^{77} + ( - 6834 \beta_{3} + 5815 \beta_1 + 5815) q^{79} + ( - 1499 \beta_{3} + 1499 \beta_{2} + 35006 \beta_1) q^{80} + ( - 5160 \beta_{3} + 40440 \beta_1 + 40440) q^{82} + (1899 \beta_{2} - 29640) q^{83} + ( - 1308 \beta_{2} - 62472) q^{85} + (4450 \beta_{3} + 131480 \beta_1 + 131480) q^{86} + (2655 \beta_{3} - 2655 \beta_{2} + 107610 \beta_1) q^{88} + ( - 130 \beta_{3} + 14596 \beta_1 + 14596) q^{89} + ( - 6475 \beta_{3} + 2450 \beta_{2} - 11081 \beta_1 - 46403) q^{91} + ( - 12404 \beta_{2} - 89656) q^{92} + (12486 \beta_{3} - 12486 \beta_{2} + 61212 \beta_1) q^{94} + (8534 \beta_{3} - 8534 \beta_{2} + 35098 \beta_1) q^{95} + ( - 1017 \beta_{2} - 5404) q^{97} + ( - 343 \beta_{3} + 3675 \beta_{2} - 74186 \beta_1 + 80164) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 65 q^{4} - 33 q^{5} - 350 q^{7} + 750 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 65 q^{4} - 33 q^{5} - 350 q^{7} + 750 q^{8} - 921 q^{10} - 1137 q^{11} + 1850 q^{13} - 2352 q^{14} + 895 q^{16} - 324 q^{17} - 2311 q^{19} + 7374 q^{20} + 3162 q^{22} + 1596 q^{23} - 395 q^{25} - 2508 q^{26} + 13531 q^{28} + 4434 q^{29} - 4294 q^{31} + 1017 q^{32} - 35880 q^{34} - 15414 q^{35} + 19109 q^{37} - 6828 q^{38} - 10545 q^{40} + 25716 q^{41} - 5542 q^{43} - 36579 q^{44} + 40740 q^{46} - 23160 q^{47} - 5978 q^{49} + 58704 q^{50} - 33424 q^{52} - 31653 q^{53} + 35778 q^{55} - 65625 q^{56} + 2277 q^{58} + 41097 q^{59} - 42052 q^{61} + 204114 q^{62} - 60062 q^{64} - 23106 q^{65} - 30763 q^{67} + 44748 q^{68} + 151179 q^{70} - 204192 q^{71} + 28577 q^{73} - 77784 q^{74} + 170384 q^{76} + 96873 q^{77} + 18464 q^{79} - 71511 q^{80} + 86040 q^{82} - 122358 q^{83} - 247272 q^{85} + 258510 q^{86} - 212565 q^{88} + 29322 q^{89} - 161875 q^{91} - 333816 q^{92} - 109938 q^{94} - 61662 q^{95} - 19582 q^{97} + 462021 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 20x^{2} - 21x + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 41\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 20\nu - 41 ) / 20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 - \beta_{1}\) \(1\)

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
4.19493 1.84460i
−3.69493 + 2.71062i
4.19493 + 1.84460i
−3.69493 2.71062i
−4.69493 + 8.13186i 0 −28.0848 48.6443i −35.8645 + 62.1192i 0 −87.5000 + 95.6596i 226.949 0 −336.763 583.291i
37.2 3.19493 5.53379i 0 −4.41520 7.64735i 19.3645 33.5404i 0 −87.5000 95.6596i 148.051 0 −123.737 214.318i
46.1 −4.69493 8.13186i 0 −28.0848 + 48.6443i −35.8645 62.1192i 0 −87.5000 95.6596i 226.949 0 −336.763 + 583.291i
46.2 3.19493 + 5.53379i 0 −4.41520 + 7.64735i 19.3645 + 33.5404i 0 −87.5000 + 95.6596i 148.051 0 −123.737 + 214.318i
\(n\): 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.c 4
3.b odd 2 1 21.6.e.b 4
7.c even 3 1 inner 63.6.e.c 4
7.c even 3 1 441.6.a.t 2
7.d odd 6 1 441.6.a.s 2
12.b even 2 1 336.6.q.e 4
21.c even 2 1 147.6.e.l 4
21.g even 6 1 147.6.a.k 2
21.g even 6 1 147.6.e.l 4
21.h odd 6 1 21.6.e.b 4
21.h odd 6 1 147.6.a.i 2
84.n even 6 1 336.6.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 3.b odd 2 1
21.6.e.b 4 21.h odd 6 1
63.6.e.c 4 1.a even 1 1 trivial
63.6.e.c 4 7.c even 3 1 inner
147.6.a.i 2 21.h odd 6 1
147.6.a.k 2 21.g even 6 1
147.6.e.l 4 21.c even 2 1
147.6.e.l 4 21.g even 6 1
336.6.q.e 4 12.b even 2 1
336.6.q.e 4 84.n even 6 1
441.6.a.s 2 7.d odd 6 1
441.6.a.t 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{3} + 69T_{2}^{2} - 180T_{2} + 3600 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + 69 T^{2} + \cdots + 3600 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 33 T^{3} + 3867 T^{2} + \cdots + 7717284 \) Copy content Toggle raw display
$7$ \( (T^{2} + 175 T + 16807)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 1137 T^{3} + \cdots + 104412996900 \) Copy content Toggle raw display
$13$ \( (T^{2} - 925 T + 208864)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 324 T^{3} + \cdots + 1788317798400 \) Copy content Toggle raw display
$19$ \( T^{4} + 2311 T^{3} + \cdots + 1663584040000 \) Copy content Toggle raw display
$23$ \( T^{4} - 1596 T^{3} + \cdots + 27756881510400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2217 T + 1102716)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 4294 T^{3} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{4} - 19109 T^{3} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12858 T - 3221280)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2771 T - 257902490)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 23160 T^{3} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + 31653 T^{3} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} - 41097 T^{3} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{4} + 42052 T^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + 30763 T^{3} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + 102096 T + 2483190108)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 28577 T^{3} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{4} - 18464 T^{3} + \cdots + 79\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{2} + 61179 T + 711231498)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 29322 T^{3} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{2} + 9791 T - 40418570)^{2} \) Copy content Toggle raw display
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