# Properties

 Label 1512.2.j.a Level 1512 Weight 2 Character orbit 1512.j Analytic conductor 12.073 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1512.j (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.56070144.2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{3} - \beta_{5} ) q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + \beta_{3} q^{7} + ( -2 + 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta_{3} - \beta_{5} ) q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + \beta_{3} q^{7} + ( -2 + 2 \beta_{3} ) q^{8} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{11} + ( 1 + \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( 1 + \beta_{1} ) q^{14} + ( 4 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{16} + 2 \beta_{3} q^{17} + ( 2 - \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{20} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{22} + ( 2 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{23} + ( 4 + 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{25} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{26} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{28} + ( -3 + \beta_{1} - \beta_{5} ) q^{29} + ( -1 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} -4 \beta_{5} q^{32} + ( 2 + 2 \beta_{1} ) q^{34} + ( -\beta_{2} - \beta_{3} ) q^{35} + ( 2 + 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{37} + ( -3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} ) q^{38} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{40} + ( -2 - 2 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{41} + ( 1 + 3 \beta_{1} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{43} + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{44} + ( -1 - \beta_{1} - 4 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{46} + ( -1 + 3 \beta_{1} - 2 \beta_{4} - 3 \beta_{5} ) q^{47} - q^{49} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} ) q^{50} + ( -2 \beta_{2} - 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{52} + ( 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{55} + ( -2 - 2 \beta_{3} ) q^{56} + ( 4 - \beta_{1} + \beta_{3} + 3 \beta_{5} ) q^{58} + ( 1 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{5} ) q^{59} + ( -2 - 2 \beta_{1} + 6 \beta_{3} - 2 \beta_{5} ) q^{61} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{62} -8 \beta_{3} q^{64} + ( -7 - 7 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 7 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{65} + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{68} + ( -1 - \beta_{1} - \beta_{4} - \beta_{6} ) q^{70} + ( 4 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{71} + ( 1 + 3 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{74} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{76} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{77} + ( -1 - \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{79} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{80} + ( -5 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{82} + ( -1 - \beta_{1} - 6 \beta_{3} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 2 - 3 \beta_{1} + \beta_{2} - 7 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{86} + ( 2 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{88} + ( \beta_{2} + 9 \beta_{3} + \beta_{6} + \beta_{7} ) q^{89} + ( -1 - \beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{91} + ( -4 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{92} + ( 4 - 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{94} + ( 14 + 5 \beta_{1} + \beta_{4} - 5 \beta_{5} + \beta_{6} - \beta_{7} ) q^{95} + ( -4 + 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} + ( 1 + \beta_{3} + \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} - 8q^{5} - 16q^{8} + O(q^{10})$$ $$8q - 4q^{2} - 8q^{5} - 16q^{8} + 4q^{10} + 4q^{14} + 16q^{16} + 16q^{19} - 12q^{22} + 16q^{23} + 32q^{25} + 16q^{26} + 8q^{28} - 24q^{29} + 16q^{32} + 8q^{34} - 20q^{38} + 16q^{40} + 8q^{43} + 4q^{46} - 8q^{47} - 8q^{49} + 8q^{50} + 8q^{52} - 16q^{56} + 24q^{58} - 12q^{62} + 8q^{67} + 16q^{68} - 4q^{70} + 32q^{71} + 8q^{73} + 28q^{74} + 24q^{76} - 16q^{80} - 36q^{82} + 32q^{86} - 8q^{91} - 24q^{92} + 40q^{94} + 112q^{95} - 32q^{97} + 4q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{7} - \nu^{6} - 25 \nu^{5} - 46 \nu^{4} - 5 \nu^{3} - 132 \nu^{2} + 28 \nu - 92$$$$)/37$$ $$\beta_{2}$$ $$=$$ $$($$$$-6 \nu^{7} + 21 \nu^{6} - 67 \nu^{5} + 115 \nu^{4} - 117 \nu^{3} + 71 \nu^{2} + 115 \nu - 66$$$$)/37$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{7} + 21 \nu^{6} - 67 \nu^{5} + 115 \nu^{4} - 117 \nu^{3} + 71 \nu^{2} + 41 \nu - 29$$$$)/37$$ $$\beta_{4}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} - 11 \nu^{4} + 17 \nu^{3} - 23 \nu^{2} + 15 \nu - 4$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 36 \nu^{6} - 136 \nu^{5} + 361 \nu^{4} - 634 \nu^{3} + 793 \nu^{2} - 601 \nu + 241$$$$)/37$$ $$\beta_{6}$$ $$=$$ $$($$$$-42 \nu^{7} + 147 \nu^{6} - 543 \nu^{5} + 953 \nu^{4} - 1485 \nu^{3} + 1163 \nu^{2} - 749 \nu + 56$$$$)/37$$ $$\beta_{7}$$ $$=$$ $$($$$$-42 \nu^{7} + 147 \nu^{6} - 543 \nu^{5} + 1027 \nu^{4} - 1633 \nu^{3} + 1681 \nu^{2} - 1193 \nu + 500$$$$)/37$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} + 6 \beta_{5} + 3 \beta_{4} + 14 \beta_{3} - 5 \beta_{2} - 10$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{6} - \beta_{5} - 4 \beta_{4} + 15 \beta_{3} - 6 \beta_{2} + 7 \beta_{1} + 12$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{7} - \beta_{6} - 40 \beta_{5} - 25 \beta_{4} - 42 \beta_{3} + 11 \beta_{2} + 10 \beta_{1} + 52$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{7} + 12 \beta_{6} - 21 \beta_{5} + 7 \beta_{4} - 101 \beta_{3} + 32 \beta_{2} - 39 \beta_{1} - 40$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-23 \beta_{7} + 19 \beta_{6} + 84 \beta_{5} + 70 \beta_{4} + \beta_{3} + 3 \beta_{2} - 70 \beta_{1} - 153$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1
 0.5 − 0.564882i 0.5 + 1.56488i 0.5 + 0.564882i 0.5 − 1.56488i 0.5 + 1.19293i 0.5 − 2.19293i 0.5 − 1.19293i 0.5 + 2.19293i
−1.36603 0.366025i 0 1.73205 + 1.00000i −3.12976 0 1.00000i −2.00000 2.00000i 0 4.27534 + 1.14557i
323.2 −1.36603 0.366025i 0 1.73205 + 1.00000i 1.12976 0 1.00000i −2.00000 2.00000i 0 −1.54329 0.413523i
323.3 −1.36603 + 0.366025i 0 1.73205 1.00000i −3.12976 0 1.00000i −2.00000 + 2.00000i 0 4.27534 1.14557i
323.4 −1.36603 + 0.366025i 0 1.73205 1.00000i 1.12976 0 1.00000i −2.00000 + 2.00000i 0 −1.54329 + 0.413523i
323.5 0.366025 1.36603i 0 −1.73205 1.00000i −4.38587 0 1.00000i −2.00000 + 2.00000i 0 −1.60534 + 5.99121i
323.6 0.366025 1.36603i 0 −1.73205 1.00000i 2.38587 0 1.00000i −2.00000 + 2.00000i 0 0.873288 3.25916i
323.7 0.366025 + 1.36603i 0 −1.73205 + 1.00000i −4.38587 0 1.00000i −2.00000 2.00000i 0 −1.60534 5.99121i
323.8 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 2.38587 0 1.00000i −2.00000 2.00000i 0 0.873288 + 3.25916i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.j.a 8
3.b odd 2 1 1512.2.j.b yes 8
4.b odd 2 1 6048.2.j.a 8
8.b even 2 1 6048.2.j.b 8
8.d odd 2 1 1512.2.j.b yes 8
12.b even 2 1 6048.2.j.b 8
24.f even 2 1 inner 1512.2.j.a 8
24.h odd 2 1 6048.2.j.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.j.a 8 1.a even 1 1 trivial
1512.2.j.a 8 24.f even 2 1 inner
1512.2.j.b yes 8 3.b odd 2 1
1512.2.j.b yes 8 8.d odd 2 1
6048.2.j.a 8 4.b odd 2 1
6048.2.j.a 8 24.h odd 2 1
6048.2.j.b 8 8.b even 2 1
6048.2.j.b 8 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 4 T_{5}^{3} - 10 T_{5}^{2} - 28 T_{5} + 37$$ acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$
$3$ 1
$5$ $$( 1 + 4 T + 10 T^{2} + 32 T^{3} + 87 T^{4} + 160 T^{5} + 250 T^{6} + 500 T^{7} + 625 T^{8} )^{2}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$1 - 44 T^{2} + 994 T^{4} - 15968 T^{6} + 198763 T^{8} - 1932128 T^{10} + 14553154 T^{12} - 77948684 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 8 T^{2} + 228 T^{4} - 4504 T^{6} + 20006 T^{8} - 761176 T^{10} + 6511908 T^{12} - 38614472 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4}$$
$19$ $$( 1 - 8 T + 30 T^{2} - 112 T^{3} + 611 T^{4} - 2128 T^{5} + 10830 T^{6} - 54872 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 8 T + 46 T^{2} - 88 T^{3} + 231 T^{4} - 2024 T^{5} + 24334 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 6 T + 64 T^{2} + 174 T^{3} + 841 T^{4} )^{4}$$
$31$ $$1 - 156 T^{2} + 12106 T^{4} - 610416 T^{6} + 22035219 T^{8} - 586609776 T^{10} + 11180145226 T^{12} - 138450574236 T^{14} + 852891037441 T^{16}$$
$37$ $$1 - 164 T^{2} + 14154 T^{4} - 829168 T^{6} + 35617331 T^{8} - 1135130992 T^{10} + 26526874794 T^{12} - 420779131076 T^{14} + 3512479453921 T^{16}$$
$41$ $$1 - 116 T^{2} + 11362 T^{4} - 660608 T^{6} + 32960203 T^{8} - 1110482048 T^{10} + 32106296482 T^{12} - 551012091956 T^{14} + 7984925229121 T^{16}$$
$43$ $$( 1 - 4 T + 84 T^{2} - 44 T^{3} + 3002 T^{4} - 1892 T^{5} + 155316 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 4 T + 76 T^{2} + 44 T^{3} + 2154 T^{4} + 2068 T^{5} + 167884 T^{6} + 415292 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 124 T^{2} - 192 T^{3} + 7734 T^{4} - 10176 T^{5} + 348316 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$1 - 232 T^{2} + 28836 T^{4} - 2582456 T^{6} + 175948646 T^{8} - 8989529336 T^{10} + 349416221796 T^{12} - 9785883804712 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 - 92 T^{2} + 6486 T^{4} - 342332 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 4 T + 228 T^{2} - 620 T^{3} + 21530 T^{4} - 41540 T^{5} + 1023492 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 16 T + 310 T^{2} - 3320 T^{3} + 33807 T^{4} - 235720 T^{5} + 1562710 T^{6} - 5726576 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 4 T + 156 T^{2} - 1460 T^{3} + 11402 T^{4} - 106580 T^{5} + 831324 T^{6} - 1556068 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$1 - 248 T^{2} + 37188 T^{4} - 3949096 T^{6} + 344639558 T^{8} - 24646308136 T^{10} + 1448475612228 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 - 232 T^{2} + 34596 T^{4} - 3949112 T^{6} + 355979174 T^{8} - 27205432568 T^{10} + 1641868073316 T^{12} - 75850166621608 T^{14} + 2252292232139041 T^{16}$$
$89$ $$1 - 292 T^{2} + 47890 T^{4} - 6052768 T^{6} + 614802619 T^{8} - 47943975328 T^{10} + 3004725921490 T^{12} - 145118536960612 T^{14} + 3936588805702081 T^{16}$$
$97$ $$( 1 + 16 T + 300 T^{2} + 4208 T^{3} + 41318 T^{4} + 408176 T^{5} + 2822700 T^{6} + 14602768 T^{7} + 88529281 T^{8} )^{2}$$