Properties

Label 567.2.g.k
Level $567$
Weight $2$
Character orbit 567.g
Analytic conductor $4.528$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1767277521.3
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{2}) q^{2} + (\beta_{7} - \beta_{5}) q^{4} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{3} - \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{2}) q^{2} + (\beta_{7} - \beta_{5}) q^{4} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{3} - \beta_{2} + 1) q^{8} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_1 + 1) q^{10} + ( - 2 \beta_{3} - \beta_1 - 1) q^{11} + (\beta_{7} - \beta_{5} + \beta_1 + 1) q^{13} + ( - \beta_{7} + 3 \beta_{5} - \beta_{3} - 2 \beta_1 - 2) q^{14} + (\beta_{6} - \beta_{4} + \beta_{2}) q^{16} + (2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_1 + 1) q^{17} + (\beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{19} + (\beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{20} + ( - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 3) q^{22} + ( - \beta_{2} + \beta_1 + 3) q^{23} + (3 \beta_{2} + \beta_1 + 1) q^{25} + (3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{26} + ( - \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{28} + ( - 2 \beta_{7} - 3 \beta_{5}) q^{29} + (2 \beta_{6} + 5 \beta_{5} + \beta_{4} - \beta_{3}) q^{31} + (2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{32} + ( - \beta_{7} + 5 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{34} + (\beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 1) q^{35} + ( - 3 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{37} + ( - 2 \beta_{3} + 5) q^{38} + (\beta_{3} - \beta_{2} - \beta_1 - 4) q^{40} + ( - 2 \beta_{7} - 3 \beta_{6} - 3 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{43} + ( - 4 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{44} + ( - \beta_{7} + 5 \beta_{6} + 4 \beta_{5} + \beta_{4} + 5 \beta_{2} - \beta_1 - 4) q^{46} + ( - \beta_{6} + 5 \beta_{5} - \beta_{4} - \beta_{2} - 5) q^{47} + (3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 2) q^{49} + (3 \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + \beta_{4} + 3 \beta_{2} + 3 \beta_1 + 8) q^{50} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 6) q^{52} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{2} - \beta_1 - 2) q^{53} + (2 \beta_{3} - 5 \beta_{2} - \beta_1 - 5) q^{55} + (\beta_{7} + 4 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 + 2) q^{56} + (2 \beta_{3} + \beta_{2} - 2) q^{58} + (3 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{59} + ( - 4 \beta_{6} - 6 \beta_{5} + \beta_{4} - 4 \beta_{2} + 6) q^{61} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 5) q^{62} + ( - 3 \beta_{3} - 3 \beta_1 - 5) q^{64} + (\beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} + 3 \beta_{2} + \beta_1 + 1) q^{65} + ( - 3 \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{3}) q^{67} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 13) q^{68} + ( - 3 \beta_{7} - 5 \beta_{6} - \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + \cdots - 4) q^{70}+ \cdots + (\beta_{7} + 4 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 5 q^{4} - 4 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 5 q^{4} - 4 q^{5} - 2 q^{7} + 6 q^{8} + 7 q^{10} - 10 q^{11} + 5 q^{13} - 7 q^{14} + q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} + 7 q^{22} + 24 q^{23} + 16 q^{25} + q^{26} + 5 q^{28} - 10 q^{29} + 18 q^{31} - 10 q^{32} - 23 q^{35} + 40 q^{38} - 36 q^{40} - 5 q^{41} + 7 q^{43} + 13 q^{44} - 12 q^{46} - 21 q^{47} + 2 q^{49} + 38 q^{50} - 50 q^{52} - 12 q^{53} - 52 q^{55} + 33 q^{56} - 14 q^{58} + 6 q^{59} + 20 q^{61} - 36 q^{62} - 46 q^{64} + 8 q^{65} + 5 q^{67} - 102 q^{68} - 46 q^{70} + 18 q^{71} + 6 q^{73} - 5 q^{76} + 16 q^{77} + 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} + 9 q^{85} + 44 q^{86} + 36 q^{88} - 22 q^{89} + 13 q^{91} - 36 q^{92} + 15 q^{94} - 16 q^{95} + 9 q^{97} - 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -14\nu^{7} + 688\nu^{6} - 619\nu^{5} - 193\nu^{4} - 6480\nu^{3} + 17846\nu^{2} - 10595\nu + 23150 ) / 8102 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -139\nu^{7} + 465\nu^{6} - 648\nu^{5} + 688\nu^{4} - 5887\nu^{3} + 15724\nu^{2} - 9416\nu - 2218 ) / 8102 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1269\nu^{7} + 2176\nu^{6} - 262\nu^{5} + 11731\nu^{4} - 43953\nu^{3} + 36565\nu^{2} - 52069\nu + 14432 ) / 8102 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} - 241837 \nu + 88979 ) / 8102 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4805\nu^{7} - 8118\nu^{6} + 2087\nu^{5} - 47477\nu^{4} + 167278\nu^{3} - 139939\nu^{2} + 265634\nu - 98045 ) / 8102 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - 4\beta_{2} - 4\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - 6\beta_{5} + 2\beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} + 5\beta_{6} - 38\beta_{5} + 16\beta_{4} - 23\beta_{3} - 17\beta_{2} - 17\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -70\beta_{7} - 70\beta_{6} - 11\beta_{5} - 14\beta_{4} + 4\beta_{3} - 14\beta_{2} - 8\beta _1 - 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{7} + 37\beta_{6} - 60\beta_{5} + 35\beta_{4} - 16\beta_{3} + 50\beta_{2} + 46\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -44\beta_{7} - 38\beta_{6} - 22\beta_{5} - 7\beta_{4} - 301\beta_{3} - 283\beta_{2} - 97\beta _1 - 565 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-\beta_{5}\) \(-1 + \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.373419 0.0835272i
2.11692 0.978886i
0.0512865 + 1.21608i
−1.54162 1.88572i
0.373419 + 0.0835272i
2.11692 + 0.978886i
0.0512865 1.21608i
−1.54162 + 1.88572i
−1.18584 + 2.05393i 0 −1.81242 3.13920i −1.22875 0 2.61263 + 0.417345i 3.85358 0 1.45709 2.52376i
109.2 −0.186423 + 0.322894i 0 0.930493 + 1.61166i −1.42143 0 −1.03335 2.43561i −1.43955 0 0.264988 0.458972i
109.3 0.768262 1.33067i 0 −0.180452 0.312552i −3.15761 0 −0.00900690 + 2.64574i 2.51851 0 −2.42587 + 4.20173i
109.4 1.10400 1.91218i 0 −1.43762 2.49004i 3.80779 0 −2.57027 0.627473i −1.93254 0 4.20379 7.28117i
541.1 −1.18584 2.05393i 0 −1.81242 + 3.13920i −1.22875 0 2.61263 0.417345i 3.85358 0 1.45709 + 2.52376i
541.2 −0.186423 0.322894i 0 0.930493 1.61166i −1.42143 0 −1.03335 + 2.43561i −1.43955 0 0.264988 + 0.458972i
541.3 0.768262 + 1.33067i 0 −0.180452 + 0.312552i −3.15761 0 −0.00900690 2.64574i 2.51851 0 −2.42587 4.20173i
541.4 1.10400 + 1.91218i 0 −1.43762 + 2.49004i 3.80779 0 −2.57027 + 0.627473i −1.93254 0 4.20379 + 7.28117i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.g.k 8
3.b odd 2 1 567.2.g.j 8
7.c even 3 1 567.2.h.j 8
9.c even 3 1 567.2.e.d yes 8
9.c even 3 1 567.2.h.j 8
9.d odd 6 1 567.2.e.c 8
9.d odd 6 1 567.2.h.k 8
21.h odd 6 1 567.2.h.k 8
63.g even 3 1 inner 567.2.g.k 8
63.g even 3 1 3969.2.a.s 4
63.h even 3 1 567.2.e.d yes 8
63.j odd 6 1 567.2.e.c 8
63.k odd 6 1 3969.2.a.t 4
63.n odd 6 1 567.2.g.j 8
63.n odd 6 1 3969.2.a.x 4
63.s even 6 1 3969.2.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.e.c 8 9.d odd 6 1
567.2.e.c 8 63.j odd 6 1
567.2.e.d yes 8 9.c even 3 1
567.2.e.d yes 8 63.h even 3 1
567.2.g.j 8 3.b odd 2 1
567.2.g.j 8 63.n odd 6 1
567.2.g.k 8 1.a even 1 1 trivial
567.2.g.k 8 63.g even 3 1 inner
567.2.h.j 8 7.c even 3 1
567.2.h.j 8 9.c even 3 1
567.2.h.k 8 9.d odd 6 1
567.2.h.k 8 21.h odd 6 1
3969.2.a.s 4 63.g even 3 1
3969.2.a.t 4 63.k odd 6 1
3969.2.a.w 4 63.s even 6 1
3969.2.a.x 4 63.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} - 6T_{2}^{5} + 39T_{2}^{4} - 30T_{2}^{3} + 54T_{2}^{2} + 18T_{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{8} - 5T_{13}^{7} + 25T_{13}^{6} - 40T_{13}^{5} + 107T_{13}^{4} - 70T_{13}^{3} + 400T_{13}^{2} - 140T_{13} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 7 T^{6} - 6 T^{5} + 39 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} - 12 T^{2} - 33 T - 21)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + T^{6} - 14 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 5 T^{3} - 24 T^{2} - 174 T - 249)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + 25 T^{6} - 40 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( T^{8} - 6 T^{7} + 81 T^{6} + \cdots + 321489 \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + 64 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{4} - 12 T^{3} + 36 T^{2} - 33 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{7} + 100 T^{6} + \cdots + 3969 \) Copy content Toggle raw display
$31$ \( T^{8} - 18 T^{7} + 231 T^{6} + \cdots + 441 \) Copy content Toggle raw display
$37$ \( T^{8} + 78 T^{6} + 74 T^{5} + \cdots + 904401 \) Copy content Toggle raw display
$41$ \( T^{8} + 5 T^{7} + 97 T^{6} + \cdots + 194481 \) Copy content Toggle raw display
$43$ \( T^{8} - 7 T^{7} + 79 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$47$ \( T^{8} + 21 T^{7} + 288 T^{6} + \cdots + 194481 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + 144 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + 144 T^{6} + \cdots + 35721 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} + 373 T^{6} + \cdots + 1087849 \) Copy content Toggle raw display
$67$ \( T^{8} - 5 T^{7} + 97 T^{6} + \cdots + 124609 \) Copy content Toggle raw display
$71$ \( (T^{4} - 9 T^{3} - 27 T^{2} + 135 T + 243)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 6 T^{7} + 186 T^{6} + \cdots + 5239521 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} + 184 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$83$ \( T^{8} - 9 T^{7} + 207 T^{6} + \cdots + 26040609 \) Copy content Toggle raw display
$89$ \( T^{8} + 22 T^{7} + 370 T^{6} + \cdots + 441 \) Copy content Toggle raw display
$97$ \( T^{8} - 9 T^{7} + 339 T^{6} + \cdots + 8277129 \) Copy content Toggle raw display
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