# Properties

 Label 1512.2.r.e Level 1512 Weight 2 Character orbit 1512.r 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.r (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.2091141441.1 Defining polynomial: $$x^{8} - x^{7} + x^{6} + 3 x^{5} - 15 x^{4} + 9 x^{3} + 9 x^{2} - 27 x + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 504) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} + \beta_{3} ) q^{5} -\beta_{3} q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{2} + \beta_{3} ) q^{5} -\beta_{3} q^{7} + ( \beta_{1} + 2 \beta_{3} ) q^{11} + ( 1 + \beta_{3} - \beta_{7} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} ) q^{19} + ( \beta_{4} - \beta_{7} ) q^{23} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{25} + ( -3 \beta_{3} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{29} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} ) q^{35} + ( 1 - \beta_{1} - \beta_{2} ) q^{37} + ( -2 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{41} + ( \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{43} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{47} + ( -1 - \beta_{3} ) q^{49} + ( -4 \beta_{1} - 4 \beta_{2} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{53} + ( -8 + 3 \beta_{1} + 3 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{55} + ( -3 - 3 \beta_{3} - 3 \beta_{4} - \beta_{7} ) q^{59} + ( -6 \beta_{3} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{61} + ( 3 \beta_{1} + \beta_{3} ) q^{65} + ( 4 - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - \beta_{7} ) q^{67} + ( 1 + \beta_{4} + \beta_{6} ) q^{71} + ( -5 - \beta_{1} - \beta_{2} + 3 \beta_{4} + 3 \beta_{6} ) q^{73} + ( 2 - \beta_{2} + 2 \beta_{3} ) q^{77} + ( \beta_{1} - 5 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} - \beta_{6} ) q^{83} + ( 3 - 2 \beta_{2} + 3 \beta_{3} - \beta_{7} ) q^{85} + ( -3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{89} + ( 1 + \beta_{5} ) q^{91} + ( -10 + 3 \beta_{2} - 10 \beta_{3} + 2 \beta_{4} + 3 \beta_{7} ) q^{95} + ( 2 \beta_{1} - 3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 3q^{5} + 4q^{7} + O(q^{10})$$ $$8q + 3q^{5} + 4q^{7} - 7q^{11} + 3q^{13} - 6q^{17} - 8q^{19} - 2q^{23} - 5q^{25} + 9q^{29} + 3q^{31} + 6q^{35} + 6q^{37} - 9q^{41} + 8q^{43} - 3q^{47} - 4q^{49} - 12q^{53} - 56q^{55} - 10q^{59} + 20q^{61} - q^{65} + 11q^{67} + 6q^{71} - 48q^{73} + 7q^{77} + 21q^{79} - 8q^{83} + 9q^{85} - 12q^{89} + 6q^{91} - 36q^{95} + 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + x^{6} + 3 x^{5} - 15 x^{4} + 9 x^{3} + 9 x^{2} - 27 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{6} + 2 \nu^{5} - 6 \nu^{4} + 6 \nu^{3} + 9 \nu^{2} - 9 \nu$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - \nu^{5} + \nu^{4} + 3 \nu^{3} - 6 \nu^{2} + 9 \nu + 9$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} - 2 \nu^{5} - 3 \nu^{4} + 3 \nu^{3} - 18 \nu^{2} + 9 \nu + 54$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} - 2 \nu^{6} - 7 \nu^{5} + 15 \nu^{4} - 12 \nu^{3} - 9 \nu^{2} + 72 \nu - 54$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{7} - 5 \nu^{6} - 4 \nu^{5} + 12 \nu^{4} - 21 \nu^{3} + 36 \nu^{2} + 72 \nu - 81$$$$)/27$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} - \nu^{5} + 5 \nu^{4} - 12 \nu^{3} + 6 \nu^{2} + 36 \nu - 36$$$$)/9$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 8 \nu^{6} - 8 \nu^{5} - 15 \nu^{4} + 12 \nu^{3} - 72 \nu^{2} + 9 \nu + 216$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{4} + 2 \beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 4 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{7} - \beta_{6} - 3 \beta_{5} + 4 \beta_{4} + \beta_{2} - 4 \beta_{1} + 18$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-9 \beta_{7} + 10 \beta_{6} - 12 \beta_{5} - 7 \beta_{4} + 18 \beta_{3} + 8 \beta_{2} + \beta_{1} + 18$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{7} + 2 \beta_{6} + 9 \beta_{5} - 20 \beta_{4} - 9 \beta_{3} + 16 \beta_{2} - 22 \beta_{1} - 27$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-24 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} + 108 \beta_{3} - 19 \beta_{2} - 5 \beta_{1} - 18$$$$)/3$$

## Character values

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

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
505.1
 1.65525 + 0.510048i 0.335492 + 1.69925i 0.199732 − 1.72050i −1.69047 + 0.377226i 1.65525 − 0.510048i 0.335492 − 1.69925i 0.199732 + 1.72050i −1.69047 − 0.377226i
0 0 0 −1.15525 2.00095i 0 0.500000 0.866025i 0 0 0
505.2 0 0 0 0.164508 + 0.284936i 0 0.500000 0.866025i 0 0 0
505.3 0 0 0 0.300268 + 0.520080i 0 0.500000 0.866025i 0 0 0
505.4 0 0 0 2.19047 + 3.79401i 0 0.500000 0.866025i 0 0 0
1009.1 0 0 0 −1.15525 + 2.00095i 0 0.500000 + 0.866025i 0 0 0
1009.2 0 0 0 0.164508 0.284936i 0 0.500000 + 0.866025i 0 0 0
1009.3 0 0 0 0.300268 0.520080i 0 0.500000 + 0.866025i 0 0 0
1009.4 0 0 0 2.19047 3.79401i 0 0.500000 + 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1009.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.r.e 8
3.b odd 2 1 504.2.r.e 8
4.b odd 2 1 3024.2.r.m 8
9.c even 3 1 inner 1512.2.r.e 8
9.c even 3 1 4536.2.a.y 4
9.d odd 6 1 504.2.r.e 8
9.d odd 6 1 4536.2.a.z 4
12.b even 2 1 1008.2.r.l 8
36.f odd 6 1 3024.2.r.m 8
36.f odd 6 1 9072.2.a.cg 4
36.h even 6 1 1008.2.r.l 8
36.h even 6 1 9072.2.a.cj 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.e 8 3.b odd 2 1
504.2.r.e 8 9.d odd 6 1
1008.2.r.l 8 12.b even 2 1
1008.2.r.l 8 36.h even 6 1
1512.2.r.e 8 1.a even 1 1 trivial
1512.2.r.e 8 9.c even 3 1 inner
3024.2.r.m 8 4.b odd 2 1
3024.2.r.m 8 36.f odd 6 1
4536.2.a.y 4 9.c even 3 1
4536.2.a.z 4 9.d odd 6 1
9072.2.a.cg 4 36.f odd 6 1
9072.2.a.cj 4 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 3 T - 3 T^{2} + 36 T^{3} - 32 T^{4} - 204 T^{5} + 540 T^{6} + 597 T^{7} - 3831 T^{8} + 2985 T^{9} + 13500 T^{10} - 25500 T^{11} - 20000 T^{12} + 112500 T^{13} - 46875 T^{14} - 234375 T^{15} + 390625 T^{16}$$
$7$ $$( 1 - T + T^{2} )^{4}$$
$11$ $$1 + 7 T - 2 T^{2} - 81 T^{3} + 203 T^{4} + 1468 T^{5} - 675 T^{6} - 3769 T^{7} + 21730 T^{8} - 41459 T^{9} - 81675 T^{10} + 1953908 T^{11} + 2972123 T^{12} - 13045131 T^{13} - 3543122 T^{14} + 136410197 T^{15} + 214358881 T^{16}$$
$13$ $$1 - 3 T - 32 T^{2} + 57 T^{3} + 673 T^{4} - 432 T^{5} - 11810 T^{6} + 2142 T^{7} + 166240 T^{8} + 27846 T^{9} - 1995890 T^{10} - 949104 T^{11} + 19221553 T^{12} + 21163701 T^{13} - 154457888 T^{14} - 188245551 T^{15} + 815730721 T^{16}$$
$17$ $$( 1 + 3 T + 41 T^{2} + 189 T^{3} + 807 T^{4} + 3213 T^{5} + 11849 T^{6} + 14739 T^{7} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 4 T + 49 T^{2} + 193 T^{3} + 1297 T^{4} + 3667 T^{5} + 17689 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 + 2 T - 59 T^{2} - 192 T^{3} + 1775 T^{4} + 6170 T^{5} - 33567 T^{6} - 71510 T^{7} + 655807 T^{8} - 1644730 T^{9} - 17756943 T^{10} + 75070390 T^{11} + 496717775 T^{12} - 1235777856 T^{13} - 8734117451 T^{14} + 6809650894 T^{15} + 78310985281 T^{16}$$
$29$ $$1 - 9 T + 13 T^{2} - 72 T^{3} + 904 T^{4} - 2970 T^{5} + 29086 T^{6} - 17613 T^{7} - 1095047 T^{8} - 510777 T^{9} + 24461326 T^{10} - 72435330 T^{11} + 639382024 T^{12} - 1476802728 T^{13} + 7732703173 T^{14} - 155248886781 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 3 T - 41 T^{2} + 354 T^{3} - 128 T^{4} - 9180 T^{5} + 42568 T^{6} + 75933 T^{7} - 1254023 T^{8} + 2353923 T^{9} + 40907848 T^{10} - 273481380 T^{11} - 118210688 T^{12} + 10134719454 T^{13} - 36387650921 T^{14} - 82537842333 T^{15} + 852891037441 T^{16}$$
$37$ $$( 1 - 3 T + 140 T^{2} - 324 T^{3} + 7620 T^{4} - 11988 T^{5} + 191660 T^{6} - 151959 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 + 9 T - 36 T^{2} - 381 T^{3} + 895 T^{4} - 852 T^{5} - 99027 T^{6} + 57981 T^{7} + 4458864 T^{8} + 2377221 T^{9} - 166464387 T^{10} - 58720692 T^{11} + 2529056095 T^{12} - 44141212581 T^{13} - 171003752676 T^{14} + 1752788464929 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 - 8 T - 73 T^{2} + 914 T^{3} + 2642 T^{4} - 51508 T^{5} + 63012 T^{6} + 1050915 T^{7} - 5977292 T^{8} + 45189345 T^{9} + 116509188 T^{10} - 4095246556 T^{11} + 9032472242 T^{12} + 134365716902 T^{13} - 461459502577 T^{14} - 2174548888856 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 + 3 T - 141 T^{2} - 486 T^{3} + 11422 T^{4} + 34152 T^{5} - 635364 T^{6} - 765849 T^{7} + 30578781 T^{8} - 35994903 T^{9} - 1403519076 T^{10} + 3545763096 T^{11} + 55735716382 T^{12} - 111461673402 T^{13} - 1519869361389 T^{14} + 1519869361389 T^{15} + 23811286661761 T^{16}$$
$53$ $$( 1 + 6 T + 59 T^{2} + 621 T^{3} + 4704 T^{4} + 32913 T^{5} + 165731 T^{6} + 893262 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$1 + 10 T - 71 T^{2} - 1536 T^{3} - 346 T^{4} + 106150 T^{5} + 583014 T^{6} - 3064183 T^{7} - 49833644 T^{8} - 180786797 T^{9} + 2029471734 T^{10} + 21800980850 T^{11} - 4192606906 T^{12} - 1098123723264 T^{13} - 2994817888511 T^{14} + 24886514848190 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 20 T + 161 T^{2} - 562 T^{3} - 931 T^{4} + 2180 T^{5} - 139869 T^{6} + 5694174 T^{7} - 64208639 T^{8} + 347344614 T^{9} - 520452549 T^{10} + 494818580 T^{11} - 12890477971 T^{12} - 474663121162 T^{13} + 8294780272121 T^{14} - 62854856720420 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 - 11 T - 82 T^{2} + 689 T^{3} + 8039 T^{4} - 18292 T^{5} - 590799 T^{6} + 1428153 T^{7} + 17009218 T^{8} + 95686251 T^{9} - 2652096711 T^{10} - 5501556796 T^{11} + 161994861719 T^{12} + 930236198723 T^{13} - 7417587337858 T^{14} - 66667827658553 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 - 3 T + 276 T^{2} - 630 T^{3} + 29126 T^{4} - 44730 T^{5} + 1391316 T^{6} - 1073733 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 24 T + 425 T^{2} + 4923 T^{3} + 48495 T^{4} + 359379 T^{5} + 2264825 T^{6} + 9336408 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$1 - 21 T + 133 T^{2} + 252 T^{3} - 9896 T^{4} + 136080 T^{5} - 973784 T^{6} - 2121567 T^{7} + 81625693 T^{8} - 167603793 T^{9} - 6077385944 T^{10} + 67092747120 T^{11} - 385450001576 T^{12} + 775418212548 T^{13} + 32330631584293 T^{14} - 403282088709339 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 + 8 T - 259 T^{2} - 1146 T^{3} + 49577 T^{4} + 119878 T^{5} - 6175577 T^{6} - 3335418 T^{7} + 604903483 T^{8} - 276839694 T^{9} - 42543549953 T^{10} + 68544681986 T^{11} + 2352841180217 T^{12} - 4514140576878 T^{13} - 84677556702571 T^{14} + 217088407917016 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 6 T + 263 T^{2} + 1377 T^{3} + 32646 T^{4} + 122553 T^{5} + 2083223 T^{6} + 4229814 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 16 T - 175 T^{2} + 2086 T^{3} + 49490 T^{4} - 309968 T^{5} - 6453258 T^{6} + 6735477 T^{7} + 823825414 T^{8} + 653341269 T^{9} - 60718704522 T^{10} - 282899424464 T^{11} + 4381314116690 T^{12} + 17913191776102 T^{13} - 145770100862575 T^{14} - 1292772551649808 T^{15} + 7837433594376961 T^{16}$$