Properties

Label 570.2.q.b
Level $570$
Weight $2$
Character orbit 570.q
Analytic conductor $4.551$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3}) q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} + ( - \beta_{8} + 1) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_{6} + \beta_1) q^{5} + ( - \beta_{8} + 1) q^{6} + (\beta_{9} - 3 \beta_{4}) q^{7} - \beta_{4} q^{8} + ( - \beta_{8} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3}) q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} + ( - \beta_{8} + 1) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_{6} + \beta_1) q^{5} + ( - \beta_{8} + 1) q^{6} + (\beta_{9} - 3 \beta_{4}) q^{7} - \beta_{4} q^{8} + ( - \beta_{8} + 1) q^{9} + ( - \beta_{10} + \beta_{5}) q^{10} + ( - 2 \beta_{2} - 1) q^{11} - \beta_{4} q^{12} + (\beta_{6} + \beta_{5} + \beta_{3}) q^{13} + (\beta_{11} - 3 \beta_{8} - \beta_1) q^{14} + ( - \beta_{10} + \beta_{5}) q^{15} - \beta_{8} q^{16} + ( - \beta_{9} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5}) q^{17} - \beta_{4} q^{18} + ( - \beta_{11} + 2 \beta_{10} - \beta_{8} + \beta_{2} + 2 \beta_1 + 1) q^{19} + ( - \beta_{7} + \beta_1) q^{20} + (\beta_{11} - 3 \beta_{8} - \beta_1) q^{21} + (2 \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3}) q^{22} + ( - 5 \beta_{6} - 3 \beta_{5} - 4 \beta_{3}) q^{23} - \beta_{8} q^{24} + (2 \beta_{11} + 2 \beta_{10} + \beta_{8} - 2 \beta_{6} - 2 \beta_{3} - 1) q^{25} + ( - \beta_{2} + \beta_1 - 1) q^{26} - \beta_{4} q^{27} + ( - \beta_{5} + 3 \beta_{3}) q^{28} + ( - 3 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} - 2) q^{29} + ( - \beta_{7} + \beta_1) q^{30} + 2 \beta_1 q^{31} + \beta_{3} q^{32} + (2 \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3}) q^{33} + ( - \beta_{11} + 2 \beta_{10}) q^{34} + ( - 4 \beta_{10} + 2 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + \cdots - 4 \beta_{2}) q^{35}+ \cdots + (2 \beta_{10} + \beta_{8} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} + 6 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} + 6 q^{6} + 6 q^{9} - 2 q^{10} - 4 q^{11} - 18 q^{14} - 2 q^{15} - 6 q^{16} + 6 q^{19} - 18 q^{21} - 6 q^{24} - 2 q^{25} - 8 q^{26} - 16 q^{29} + 4 q^{34} + 2 q^{35} - 6 q^{36} - 8 q^{39} + 2 q^{40} + 10 q^{41} - 2 q^{44} + 28 q^{46} - 56 q^{49} + 16 q^{50} + 4 q^{51} - 6 q^{54} - 8 q^{55} - 36 q^{56} + 8 q^{59} + 2 q^{60} - 28 q^{61} - 12 q^{64} - 16 q^{65} - 2 q^{66} + 28 q^{69} + 16 q^{70} + 44 q^{71} + 14 q^{74} + 16 q^{75} - 12 q^{76} - 36 q^{79} - 6 q^{81} - 36 q^{84} - 32 q^{85} + 24 q^{86} + 6 q^{89} + 2 q^{90} + 64 q^{94} - 12 q^{95} - 12 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + 144 \nu^{3} - 48 \nu^{2} + 48 \nu + 748 ) / 460 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} - 454 \nu^{4} + 444 \nu^{3} - 148 \nu^{2} + 148 \nu + 612 ) / 460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + 110 \nu^{4} - 50 \nu^{3} - 1370 \nu^{2} + 12 \nu - 12 ) / 460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + 524 \nu^{4} + 1148 \nu^{3} + 154 \nu^{2} - 154 \nu - 142 ) / 230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} - 110 \nu^{4} + 50 \nu^{3} + 1140 \nu^{2} - 12 \nu + 12 ) / 230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + 278 \nu^{4} - 118 \nu^{3} - 1982 \nu^{2} + 32 \nu - 32 ) / 460 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + 1614 \nu^{4} + 3424 \nu^{3} + 484 \nu^{2} - 484 \nu - 420 ) / 460 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} - 1774 \nu^{4} - 3856 \nu^{3} - 524 \nu^{2} + 524 \nu + 476 ) / 460 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + 2510 \nu^{4} + 2790 \nu^{3} - 1294 \nu^{2} + 1060 \nu - 1060 ) / 460 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} - 1424 \nu^{4} - 1636 \nu^{3} + 732 \nu^{2} - 612 \nu + 612 ) / 230 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{7} + 2\beta_{4} + \beta_{2} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} + \beta_{10} + 7\beta_{8} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} + 3\beta_{10} + 9\beta_{8} - 3\beta_{6} - 8\beta_{5} - 9\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{9} + 6\beta_{7} + 28\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + 33 \beta_{5} + 39 \beta_{4} + 39 \beta_{3} + 11 \beta_{2} - 33 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94\beta_{11} + 28\beta_{10} + 116\beta_{8} - 116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 138\beta_{9} + 44\beta_{7} + 166\beta_{4} - 44\beta_{2} + 138\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 398\beta_{9} + 122\beta_{7} + 122\beta_{6} + 398\beta_{5} + 486\beta_{4} + 486\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 580\beta_{11} + 182\beta_{10} + 702\beta_{8} + 182\beta_{6} + 580\beta_{5} + 702\beta_{3} - 702 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-1 + \beta_{8}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.531325 1.98293i
−0.147520 0.550552i
0.312819 + 1.16746i
−1.16746 + 0.312819i
1.98293 0.531325i
0.550552 0.147520i
−0.531325 + 1.98293i
−0.147520 + 0.550552i
0.312819 1.16746i
−1.16746 0.312819i
1.98293 + 0.531325i
0.550552 + 0.147520i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −0.837733 + 2.07321i 0.500000 + 0.866025i 0.785680i 1.00000i 0.500000 + 0.866025i 1.76210 1.37659i
49.2 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −0.445186 2.19130i 0.500000 + 0.866025i 4.67513i 1.00000i 0.500000 + 0.866025i −0.710109 + 2.12032i
49.3 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 2.14894 + 0.618092i 0.500000 + 0.866025i 3.53919i 1.00000i 0.500000 + 0.866025i −1.55199 1.60976i
49.4 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.60976 1.55199i 0.500000 + 0.866025i 3.53919i 1.00000i 0.500000 + 0.866025i −0.618092 2.14894i
49.5 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.37659 + 1.76210i 0.500000 + 0.866025i 0.785680i 1.00000i 0.500000 + 0.866025i −2.07321 + 0.837733i
49.6 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 2.12032 0.710109i 0.500000 + 0.866025i 4.67513i 1.00000i 0.500000 + 0.866025i 2.19130 + 0.445186i
349.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −0.837733 2.07321i 0.500000 0.866025i 0.785680i 1.00000i 0.500000 0.866025i 1.76210 + 1.37659i
349.2 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −0.445186 + 2.19130i 0.500000 0.866025i 4.67513i 1.00000i 0.500000 0.866025i −0.710109 2.12032i
349.3 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 2.14894 0.618092i 0.500000 0.866025i 3.53919i 1.00000i 0.500000 0.866025i −1.55199 + 1.60976i
349.4 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.60976 + 1.55199i 0.500000 0.866025i 3.53919i 1.00000i 0.500000 0.866025i −0.618092 + 2.14894i
349.5 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.37659 1.76210i 0.500000 0.866025i 0.785680i 1.00000i 0.500000 0.866025i −2.07321 0.837733i
349.6 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 2.12032 + 0.710109i 0.500000 0.866025i 4.67513i 1.00000i 0.500000 0.866025i 2.19130 0.445186i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.q.b 12
3.b odd 2 1 1710.2.t.b 12
5.b even 2 1 inner 570.2.q.b 12
15.d odd 2 1 1710.2.t.b 12
19.c even 3 1 inner 570.2.q.b 12
57.h odd 6 1 1710.2.t.b 12
95.i even 6 1 inner 570.2.q.b 12
285.n odd 6 1 1710.2.t.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.q.b 12 1.a even 1 1 trivial
570.2.q.b 12 5.b even 2 1 inner
570.2.q.b 12 19.c even 3 1 inner
570.2.q.b 12 95.i even 6 1 inner
1710.2.t.b 12 3.b odd 2 1
1710.2.t.b 12 15.d odd 2 1
1710.2.t.b 12 57.h odd 6 1
1710.2.t.b 12 285.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 35T_{7}^{4} + 295T_{7}^{2} + 169 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + T^{10} - 32 T^{9} + 6 T^{8} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 35 T^{4} + 295 T^{2} + 169)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 13 T - 5)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} - 12 T^{10} + 112 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{12} - 44 T^{10} + 1336 T^{8} + \cdots + 6250000 \) Copy content Toggle raw display
$19$ \( (T^{6} - 3 T^{5} + 24 T^{4} - 25 T^{3} + \cdots + 6859)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 195 T^{10} + \cdots + 58120048561 \) Copy content Toggle raw display
$29$ \( (T^{6} + 8 T^{5} + 80 T^{4} + 132 T^{3} + \cdots + 16900)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 16 T + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 27 T^{4} + 107 T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 5 T^{5} + 96 T^{4} + 633 T^{3} + \cdots + 19321)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 80 T^{10} + 4992 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$47$ \( T^{12} - 264 T^{10} + \cdots + 147763360000 \) Copy content Toggle raw display
$53$ \( T^{12} - 207 T^{10} + \cdots + 332150625 \) Copy content Toggle raw display
$59$ \( (T^{6} - 4 T^{5} + 104 T^{4} + \cdots + 204304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 14 T^{5} + 214 T^{4} + \cdots + 77284)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 208 T^{10} + \cdots + 102627966736 \) Copy content Toggle raw display
$71$ \( (T^{6} - 22 T^{5} + 346 T^{4} + \cdots + 42436)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 180 T^{10} + \cdots + 623201296 \) Copy content Toggle raw display
$79$ \( (T^{6} + 18 T^{5} + 232 T^{4} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 3 T^{5} + 286 T^{4} + \cdots + 4133089)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 200 T^{10} + \cdots + 3906250000 \) Copy content Toggle raw display
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