# Properties

 Label 1512.2.r.d 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: 8.0.508277025.1 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 21 x^{4} + 3 x^{3} - 22 x^{2} + 3 x + 19$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{5} + \beta_{5} q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{5} + \beta_{5} q^{7} + ( -\beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{13} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{17} + ( \beta_{1} - \beta_{3} - \beta_{6} ) q^{19} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} ) q^{23} + ( \beta_{2} - 2 \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{25} + ( 2 \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{29} + ( 3 - \beta_{2} + 3 \beta_{5} - \beta_{6} ) q^{31} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{35} + ( 7 - \beta_{6} ) q^{37} + ( -2 - \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{41} + ( -\beta_{2} + 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{7} ) q^{43} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{47} + ( -1 - \beta_{5} ) q^{49} + ( 1 + 3 \beta_{4} - 3 \beta_{6} ) q^{53} + ( 1 - \beta_{1} + \beta_{3} + 4 \beta_{4} - 3 \beta_{6} ) q^{55} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{59} + ( 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{61} + ( 3 \beta_{2} + 2 \beta_{3} - \beta_{5} - 6 \beta_{7} ) q^{65} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{67} + ( 4 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} ) q^{71} + ( 4 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} ) q^{73} + ( 2 + \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{77} + ( 5 \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{79} + ( -3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{83} + ( -3 + 5 \beta_{1} + 5 \beta_{2} + \beta_{4} - 3 \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{85} + ( 4 + 3 \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{89} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{91} + ( 6 - 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 6 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{95} + ( \beta_{3} + 2 \beta_{5} - 6 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{5} - 4q^{7} + O(q^{10})$$ $$8q - 4q^{5} - 4q^{7} + 6q^{11} - 3q^{13} + 16q^{17} - 4q^{19} + 5q^{23} - 14q^{25} - q^{29} + 11q^{31} + 8q^{35} + 54q^{37} - 2q^{41} - 11q^{43} - 7q^{47} - 4q^{49} + 8q^{53} + 12q^{55} - 9q^{59} - 7q^{61} + 9q^{65} - 12q^{67} + 24q^{71} + 26q^{73} + 6q^{77} - 22q^{79} + 6q^{83} - 11q^{85} + 28q^{89} + 6q^{91} + 23q^{95} - q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 21 x^{4} + 3 x^{3} - 22 x^{2} + 3 x + 19$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-43 \nu^{7} - 317 \nu^{6} - 270 \nu^{5} + 36 \nu^{4} + 2226 \nu^{3} + 5037 \nu^{2} - 3419 \nu - 2806$$$$)/2799$$ $$\beta_{2}$$ $$=$$ $$($$$$193 \nu^{7} + 620 \nu^{6} - 1023 \nu^{5} + 1878 \nu^{4} - 10143 \nu^{3} + 12564 \nu^{2} + 2696 \nu - 10991$$$$)/5598$$ $$\beta_{3}$$ $$=$$ $$($$$$130 \nu^{7} - 235 \nu^{6} - 312 \nu^{5} - 1389 \nu^{4} - 828 \nu^{3} + 6795 \nu^{2} + 2048 \nu - 3125$$$$)/2799$$ $$\beta_{4}$$ $$=$$ $$($$$$145 \nu^{7} - 298 \nu^{6} + 585 \nu^{5} - 1944 \nu^{4} + 1086 \nu^{3} + 1371 \nu^{2} - 2596 \nu + 3799$$$$)/2799$$ $$\beta_{5}$$ $$=$$ $$($$$$-145 \nu^{7} + 298 \nu^{6} - 585 \nu^{5} + 1944 \nu^{4} - 2019 \nu^{3} - 438 \nu^{2} + 730 \nu - 67$$$$)/1866$$ $$\beta_{6}$$ $$=$$ $$($$$$242 \nu^{7} - 581 \nu^{6} + 912 \nu^{5} - 3045 \nu^{4} + 3138 \nu^{3} + 1812 \nu^{2} - 6752 \nu + 929$$$$)/2799$$ $$\beta_{7}$$ $$=$$ $$($$$$-254 \nu^{7} + 818 \nu^{6} - 1443 \nu^{5} + 4422 \nu^{4} - 6162 \nu^{3} + 1221 \nu^{2} + 1697 \nu - 2363$$$$)/2799$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{1} - 2$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{7} - \beta_{6} - 12 \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 6$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-17 \beta_{7} + 6 \beta_{6} + 16 \beta_{5} - 19 \beta_{4} - 5 \beta_{3} + 12 \beta_{2} + \beta_{1} + 29$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{7} - 22 \beta_{6} + 5 \beta_{5} + 31 \beta_{4} - 17 \beta_{3} + 13 \beta_{2} + 10 \beta_{1} - 29$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$37 \beta_{7} - 13 \beta_{6} - 54 \beta_{5} + 19 \beta_{4} - 3 \beta_{3} - 14 \beta_{2} + 5 \beta_{1} - 18$$ $$\nu^{7}$$ $$=$$ $$($$$$-47 \beta_{7} + 107 \beta_{6} + 7 \beta_{5} - 238 \beta_{4} + 55 \beta_{2} - 30 \beta_{1} + 326$$$$)/3$$

## Character values

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

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{5}$$ $$1$$ $$1$$

## 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
 0.947217 − 0.807294i −0.577806 + 2.22188i 1.86526 − 0.199842i −0.734668 − 0.348716i 0.947217 + 0.807294i −0.577806 − 2.22188i 1.86526 + 0.199842i −0.734668 + 0.348716i
0 0 0 −1.87447 3.24667i 0 −0.500000 + 0.866025i 0 0 0
505.2 0 0 0 −1.81197 3.13842i 0 −0.500000 + 0.866025i 0 0 0
505.3 0 0 0 0.468293 + 0.811107i 0 −0.500000 + 0.866025i 0 0 0
505.4 0 0 0 1.21814 + 2.10988i 0 −0.500000 + 0.866025i 0 0 0
1009.1 0 0 0 −1.87447 + 3.24667i 0 −0.500000 0.866025i 0 0 0
1009.2 0 0 0 −1.81197 + 3.13842i 0 −0.500000 0.866025i 0 0 0
1009.3 0 0 0 0.468293 0.811107i 0 −0.500000 0.866025i 0 0 0
1009.4 0 0 0 1.21814 2.10988i 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.d 8
3.b odd 2 1 504.2.r.d 8
4.b odd 2 1 3024.2.r.l 8
9.c even 3 1 inner 1512.2.r.d 8
9.c even 3 1 4536.2.a.ba 4
9.d odd 6 1 504.2.r.d 8
9.d odd 6 1 4536.2.a.x 4
12.b even 2 1 1008.2.r.m 8
36.f odd 6 1 3024.2.r.l 8
36.f odd 6 1 9072.2.a.cl 4
36.h even 6 1 1008.2.r.m 8
36.h even 6 1 9072.2.a.ce 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.d 8 3.b odd 2 1
504.2.r.d 8 9.d odd 6 1
1008.2.r.m 8 12.b even 2 1
1008.2.r.m 8 36.h even 6 1
1512.2.r.d 8 1.a even 1 1 trivial
1512.2.r.d 8 9.c even 3 1 inner
3024.2.r.l 8 4.b odd 2 1
3024.2.r.l 8 36.f odd 6 1
4536.2.a.x 4 9.d odd 6 1
4536.2.a.ba 4 9.c even 3 1
9072.2.a.ce 4 36.h even 6 1
9072.2.a.cl 4 36.f odd 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
$5$ $$1 + 4 T + 5 T^{2} - 18 T^{3} - 94 T^{4} - 232 T^{5} - 110 T^{6} + 1011 T^{7} + 3826 T^{8} + 5055 T^{9} - 2750 T^{10} - 29000 T^{11} - 58750 T^{12} - 56250 T^{13} + 78125 T^{14} + 312500 T^{15} + 390625 T^{16}$$
$7$ $$( 1 + T + T^{2} )^{4}$$
$11$ $$1 - 6 T + T^{2} + 24 T^{3} + 49 T^{4} + 306 T^{5} - 2153 T^{6} + 1236 T^{7} + 7063 T^{8} + 13596 T^{9} - 260513 T^{10} + 407286 T^{11} + 717409 T^{12} + 3865224 T^{13} + 1771561 T^{14} - 116923026 T^{15} + 214358881 T^{16}$$
$13$ $$1 + 3 T - 16 T^{2} + 39 T^{3} + 337 T^{4} - 720 T^{5} + 2222 T^{6} + 11778 T^{7} - 44096 T^{8} + 153114 T^{9} + 375518 T^{10} - 1581840 T^{11} + 9625057 T^{12} + 14480427 T^{13} - 77228944 T^{14} + 188245551 T^{15} + 815730721 T^{16}$$
$17$ $$( 1 - 8 T + 35 T^{2} - 167 T^{3} + 886 T^{4} - 2839 T^{5} + 10115 T^{6} - 39304 T^{7} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 2 T + 55 T^{2} + 41 T^{3} + 1309 T^{4} + 779 T^{5} + 19855 T^{6} + 13718 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 - 5 T - 28 T^{2} + 177 T^{3} - 25 T^{4} + 494 T^{5} - 12137 T^{6} - 44331 T^{7} + 708646 T^{8} - 1019613 T^{9} - 6420473 T^{10} + 6010498 T^{11} - 6996025 T^{12} + 1139232711 T^{13} - 4145004892 T^{14} - 17024127235 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + T - 49 T^{2} - 294 T^{3} + 1136 T^{4} + 11204 T^{5} + 49096 T^{6} - 227337 T^{7} - 2198075 T^{8} - 6592773 T^{9} + 41289736 T^{10} + 273254356 T^{11} + 803471216 T^{12} - 6030277806 T^{13} - 29146342729 T^{14} + 17249876309 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 11 T - 39 T^{2} + 356 T^{3} + 5954 T^{4} - 25902 T^{5} - 215174 T^{6} + 3835 T^{7} + 11112081 T^{8} + 118885 T^{9} - 206782214 T^{10} - 771646482 T^{11} + 5498644034 T^{12} + 10191977756 T^{13} - 34612643559 T^{14} - 302638755221 T^{15} + 852891037441 T^{16}$$
$37$ $$( 1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 151848 T^{5} + 564028 T^{6} - 1367631 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 + 2 T - 37 T^{2} - 432 T^{3} - 841 T^{4} + 13042 T^{5} + 100489 T^{6} - 72294 T^{7} - 3776099 T^{8} - 2964054 T^{9} + 168922009 T^{10} + 898867682 T^{11} - 2376465001 T^{12} - 50049878832 T^{13} - 175753856917 T^{14} + 389508547762 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 11 T - 27 T^{2} - 596 T^{3} + 140 T^{4} + 2688 T^{5} - 152864 T^{6} + 330797 T^{7} + 13009029 T^{8} + 14224271 T^{9} - 282645536 T^{10} + 213714816 T^{11} + 478632140 T^{12} - 87617032028 T^{13} - 170676802323 T^{14} + 2990004722177 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 + 7 T - 61 T^{2} + 174 T^{3} + 5108 T^{4} - 21898 T^{5} - 62918 T^{6} + 808359 T^{7} - 1538879 T^{8} + 37992873 T^{9} - 138985862 T^{10} - 2273516054 T^{11} + 24925410548 T^{12} + 39906031218 T^{13} - 657532135069 T^{14} + 3546361843241 T^{15} + 23811286661761 T^{16}$$
$53$ $$( 1 - 4 T + 83 T^{2} + 197 T^{3} + 2722 T^{4} + 10441 T^{5} + 233147 T^{6} - 595508 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$1 + 9 T - 119 T^{2} - 1116 T^{3} + 10336 T^{4} + 79866 T^{5} - 564434 T^{6} - 2176353 T^{7} + 30131725 T^{8} - 128404827 T^{9} - 1964794754 T^{10} + 16402799214 T^{11} + 125245043296 T^{12} - 797855517684 T^{13} - 5019483503279 T^{14} + 22397863363371 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 + 7 T - 144 T^{2} - 1315 T^{3} + 11783 T^{4} + 112692 T^{5} - 470051 T^{6} - 3262925 T^{7} + 21742764 T^{8} - 199038425 T^{9} - 1749059771 T^{10} + 25578942852 T^{11} + 163145544503 T^{12} - 1110644135815 T^{13} - 7418933907984 T^{14} + 21999199852147 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 12 T - 139 T^{2} - 1338 T^{3} + 22729 T^{4} + 133506 T^{5} - 1954153 T^{6} - 2368722 T^{7} + 170018875 T^{8} - 158704374 T^{9} - 8772192817 T^{10} + 40153665078 T^{11} + 458014829209 T^{12} - 1806467393166 T^{13} - 12573715121491 T^{14} + 72728539263876 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 - 12 T + 167 T^{2} - 591 T^{3} + 7587 T^{4} - 41961 T^{5} + 841847 T^{6} - 4294932 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 109792 T^{5} + 1236328 T^{6} - 5057221 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$1 + 22 T + 237 T^{2} + 1616 T^{3} + 2882 T^{4} - 87342 T^{5} - 574370 T^{6} + 1050079 T^{7} + 29800944 T^{8} + 82956241 T^{9} - 3584643170 T^{10} - 43063012338 T^{11} + 112254133442 T^{12} + 4972523140784 T^{13} + 57611726958477 T^{14} + 422485997695498 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 - 6 T - 113 T^{2} + 1704 T^{3} + 3151 T^{4} - 166296 T^{5} + 1416277 T^{6} + 7918434 T^{7} - 172589093 T^{8} + 657230022 T^{9} + 9756732253 T^{10} - 95085890952 T^{11} + 149541169471 T^{12} + 6712125255672 T^{13} - 36944262190697 T^{14} - 162816305937762 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 14 T + 323 T^{2} - 2963 T^{3} + 41152 T^{4} - 263707 T^{5} + 2558483 T^{6} - 9869566 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 + T - 123 T^{2} - 856 T^{3} - 4138 T^{4} + 70242 T^{5} + 167026 T^{6} - 386291 T^{7} + 107246157 T^{8} - 37470227 T^{9} + 1571547634 T^{10} + 64107976866 T^{11} - 366334164778 T^{12} - 7350763259992 T^{13} - 102455556606267 T^{14} + 80798284478113 T^{15} + 7837433594376961 T^{16}$$