# Properties

 Label 1512.2.s.n Level 1512 Weight 2 Character orbit 1512.s 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.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3 x^{7} + 4 x^{6} + 28 x^{5} + 14 x^{4} - 52 x^{3} + 306 x^{2} + 1052 x + 1051$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( -\beta_{4} - \beta_{6} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( -\beta_{4} - \beta_{6} ) q^{7} + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{11} + ( -3 + \beta_{2} + \beta_{7} ) q^{13} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{17} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{19} + ( -\beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{23} + ( -5 + \beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} + ( 1 + 2 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{31} + ( 3 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{35} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{37} + ( -2 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{43} + ( \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{47} + ( -1 + 2 \beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{49} + ( 1 + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{53} + ( -5 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{55} + ( 1 + \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( -\beta_{1} - \beta_{2} - 6 \beta_{4} + \beta_{5} ) q^{61} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} - 9 \beta_{4} - \beta_{5} - \beta_{6} ) q^{65} + ( -8 - \beta_{1} + \beta_{3} + 8 \beta_{4} + \beta_{7} ) q^{67} + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( 2 - \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{73} + ( -5 - \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{77} + ( -2 \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{79} + ( 8 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{83} + ( 1 + \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{85} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{89} + ( -1 - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{91} + ( -3 + 4 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{95} + ( -2 + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{5} - 4q^{7} + O(q^{10})$$ $$8q - 2q^{5} - 4q^{7} - q^{11} - 20q^{13} - 4q^{17} + q^{19} + 12q^{23} - 14q^{25} + 12q^{29} + 8q^{31} + 9q^{35} - 12q^{41} + 2q^{43} - 9q^{47} + 6q^{49} + 7q^{53} - 36q^{55} - 4q^{59} - 25q^{61} - 28q^{65} - 30q^{67} - 22q^{71} + 4q^{73} - 37q^{77} + 7q^{79} + 58q^{83} + 14q^{85} + 9q^{89} + 15q^{91} - 4q^{95} - 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 4 x^{6} + 28 x^{5} + 14 x^{4} - 52 x^{3} + 306 x^{2} + 1052 x + 1051$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 16 \nu^{6} + 212 \nu^{5} - 2728 \nu^{4} + 7695 \nu^{3} - 16738 \nu^{2} - 4364 \nu - 25565$$$$)/27783$$ $$\beta_{3}$$ $$=$$ $$($$$$-29 \nu^{7} + 86 \nu^{6} + 845 \nu^{5} - 4615 \nu^{4} + 3267 \nu^{3} + 11201 \nu^{2} - 15194 \nu - 88514$$$$)/27783$$ $$\beta_{4}$$ $$=$$ $$($$$$-31 \nu^{7} + 307 \nu^{6} - 1091 \nu^{5} + 1030 \nu^{4} + 1674 \nu^{3} - 2006 \nu^{2} - 14971 \nu + 14402$$$$)/27783$$ $$\beta_{5}$$ $$=$$ $$($$$$-83 \nu^{7} + 383 \nu^{6} - 2098 \nu^{5} + 3215 \nu^{4} - 5157 \nu^{3} - 12559 \nu^{2} - 23915 \nu + 21349$$$$)/27783$$ $$\beta_{6}$$ $$=$$ $$($$$$-214 \nu^{7} + 967 \nu^{6} - 1898 \nu^{5} - 2108 \nu^{4} + 3618 \nu^{3} + 5485 \nu^{2} - 47014 \nu - 32581$$$$)/27783$$ $$\beta_{7}$$ $$=$$ $$($$$$-58 \nu^{7} + 226 \nu^{6} - 443 \nu^{5} - 1238 \nu^{4} + 837 \nu^{3} + 2260 \nu^{2} - 20344 \nu - 33523$$$$)/3969$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} - 7 \beta_{5} + \beta_{4} - 9 \beta_{3} + \beta_{2} - 6$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{7} + 9 \beta_{6} - 7 \beta_{5} - 6 \beta_{4} - 12 \beta_{3} - \beta_{2} - 11 \beta_{1} - 37$$ $$\nu^{5}$$ $$=$$ $$-27 \beta_{7} + 46 \beta_{6} + 15 \beta_{5} - 33 \beta_{4} + 31 \beta_{3} + 3 \beta_{2} - 48 \beta_{1} - 67$$ $$\nu^{6}$$ $$=$$ $$-113 \beta_{7} + 112 \beta_{6} + 144 \beta_{5} + 63 \beta_{4} + 276 \beta_{3} - \beta_{2} - 81 \beta_{1} - 88$$ $$\nu^{7}$$ $$=$$ $$-192 \beta_{7} - 146 \beta_{6} + 417 \beta_{5} + 679 \beta_{4} + 887 \beta_{3} - 30 \beta_{2} - 26 \beta_{1} + 333$$

## Character values

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

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 −1.30724 + 1.29485i 2.45973 − 2.20662i 1.89574 + 2.48951i −1.54823 − 0.711712i −1.30724 − 1.29485i 2.45973 + 2.20662i 1.89574 − 2.48951i −1.54823 + 0.711712i
0 0 0 −1.70873 2.95960i 0 −2.27499 1.35071i 0 0 0
865.2 0 0 0 −1.26913 2.19820i 0 2.64086 + 0.160857i 0 0 0
865.3 0 0 0 −0.0144658 0.0250554i 0 −1.70811 + 2.02048i 0 0 0
865.4 0 0 0 1.99232 + 3.45080i 0 −0.657753 2.56269i 0 0 0
1297.1 0 0 0 −1.70873 + 2.95960i 0 −2.27499 + 1.35071i 0 0 0
1297.2 0 0 0 −1.26913 + 2.19820i 0 2.64086 0.160857i 0 0 0
1297.3 0 0 0 −0.0144658 + 0.0250554i 0 −1.70811 2.02048i 0 0 0
1297.4 0 0 0 1.99232 3.45080i 0 −0.657753 + 2.56269i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1297.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.s.n 8
3.b odd 2 1 1512.2.s.o yes 8
7.c even 3 1 inner 1512.2.s.n 8
21.h odd 6 1 1512.2.s.o yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.s.n 8 1.a even 1 1 trivial
1512.2.s.n 8 7.c even 3 1 inner
1512.2.s.o yes 8 3.b odd 2 1
1512.2.s.o yes 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$:

 $$T_{5}^{8} + \cdots$$ $$T_{11}^{8} + \cdots$$ $$T_{13}^{4} + 10 T_{13}^{3} + 21 T_{13}^{2} - T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 2 T - T^{2} + 20 T^{3} + 36 T^{4} - 46 T^{5} + 330 T^{6} + 755 T^{7} - 374 T^{8} + 3775 T^{9} + 8250 T^{10} - 5750 T^{11} + 22500 T^{12} + 62500 T^{13} - 15625 T^{14} + 156250 T^{15} + 390625 T^{16}$$
$7$ $$1 + 4 T + 5 T^{2} - 33 T^{3} - 136 T^{4} - 231 T^{5} + 245 T^{6} + 1372 T^{7} + 2401 T^{8}$$
$11$ $$1 + T - 15 T^{2} + 108 T^{3} + 240 T^{4} - 1366 T^{5} + 5502 T^{6} + 16607 T^{7} - 62387 T^{8} + 182677 T^{9} + 665742 T^{10} - 1818146 T^{11} + 3513840 T^{12} + 17393508 T^{13} - 26573415 T^{14} + 19487171 T^{15} + 214358881 T^{16}$$
$13$ $$( 1 + 10 T + 73 T^{2} + 389 T^{3} + 1556 T^{4} + 5057 T^{5} + 12337 T^{6} + 21970 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 + 4 T - 29 T^{2} - 246 T^{3} + 203 T^{4} + 5446 T^{5} + 13245 T^{6} - 47608 T^{7} - 341903 T^{8} - 809336 T^{9} + 3827805 T^{10} + 26756198 T^{11} + 16954763 T^{12} - 349284822 T^{13} - 699989501 T^{14} + 1641354692 T^{15} + 6975757441 T^{16}$$
$19$ $$1 - T + 3 T^{2} + 94 T^{3} - 40 T^{4} - 786 T^{5} - 206 T^{6} - 9541 T^{7} - 158247 T^{8} - 181279 T^{9} - 74366 T^{10} - 5391174 T^{11} - 5212840 T^{12} + 232753306 T^{13} + 141137643 T^{14} - 893871739 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 12 T + 61 T^{2} - 194 T^{3} + 271 T^{4} + 838 T^{5} - 11959 T^{6} + 133924 T^{7} - 856813 T^{8} + 3080252 T^{9} - 6326311 T^{10} + 10195946 T^{11} + 75836911 T^{12} - 1248650542 T^{13} + 9030189229 T^{14} - 40857905364 T^{15} + 78310985281 T^{16}$$
$29$ $$( 1 - 6 T + 59 T^{2} - 387 T^{3} + 2091 T^{4} - 11223 T^{5} + 49619 T^{6} - 146334 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$1 - 8 T + 9 T^{2} + 102 T^{3} - 1287 T^{4} + 10998 T^{5} - 5977 T^{6} - 318082 T^{7} + 2302767 T^{8} - 9860542 T^{9} - 5743897 T^{10} + 327641418 T^{11} - 1188571527 T^{12} + 2920173402 T^{13} + 7987533129 T^{14} - 220100912888 T^{15} + 852891037441 T^{16}$$
$37$ $$1 - 63 T^{2} - 114 T^{3} + 1781 T^{4} + 5700 T^{5} + 37899 T^{6} - 141018 T^{7} - 2640591 T^{8} - 5217666 T^{9} + 51883731 T^{10} + 288722100 T^{11} + 3337880741 T^{12} - 7905211098 T^{13} - 161640763767 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 + 6 T + 127 T^{2} + 453 T^{3} + 6650 T^{4} + 18573 T^{5} + 213487 T^{6} + 413526 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - T + 144 T^{2} - 50 T^{3} + 8638 T^{4} - 2150 T^{5} + 266256 T^{6} - 79507 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 + 9 T - 67 T^{2} - 660 T^{3} + 2564 T^{4} + 7452 T^{5} - 311076 T^{6} + 354939 T^{7} + 24590333 T^{8} + 16682133 T^{9} - 687166884 T^{10} + 773688996 T^{11} + 12511502084 T^{12} - 151367704620 T^{13} - 722207427043 T^{14} + 4559608084167 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 7 T - 158 T^{2} + 729 T^{3} + 18893 T^{4} - 54478 T^{5} - 1436103 T^{6} + 995821 T^{7} + 90249700 T^{8} + 52778513 T^{9} - 4034013327 T^{10} - 8110521206 T^{11} + 149074857533 T^{12} + 304864514397 T^{13} - 3501969058382 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 4 T - 21 T^{2} - 738 T^{3} - 6108 T^{4} - 3112 T^{5} + 138210 T^{6} + 1929881 T^{7} + 10657612 T^{8} + 113862979 T^{9} + 481109010 T^{10} - 639139448 T^{11} - 74012840988 T^{12} - 527614132662 T^{13} - 885791206461 T^{14} + 9954605939276 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 + 25 T + 196 T^{2} + 1097 T^{3} + 19941 T^{4} + 171360 T^{5} + 257114 T^{6} + 4239618 T^{7} + 85475992 T^{8} + 258616698 T^{9} + 956721194 T^{10} + 38895464160 T^{11} + 276099915381 T^{12} + 926522142197 T^{13} + 10097993374756 T^{14} + 78568570900525 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 30 T + 335 T^{2} + 2816 T^{3} + 39697 T^{4} + 415264 T^{5} + 2662719 T^{6} + 27218576 T^{7} + 300754815 T^{8} + 1823644592 T^{9} + 11952945591 T^{10} + 124896046432 T^{11} + 799939050337 T^{12} + 3801952301312 T^{13} + 30303558026615 T^{14} + 181821348159690 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 + 11 T + 194 T^{2} + 1324 T^{3} + 16844 T^{4} + 94004 T^{5} + 977954 T^{6} + 3937021 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$1 - 4 T - 177 T^{2} + 378 T^{3} + 16890 T^{4} - 8478 T^{5} - 1265668 T^{6} + 148369 T^{7} + 87542238 T^{8} + 10830937 T^{9} - 6744744772 T^{10} - 3298086126 T^{11} + 479646290490 T^{12} + 783621062154 T^{13} - 26786158053153 T^{14} - 44189594076388 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 7 T + 6 T^{2} + 609 T^{3} - 7809 T^{4} + 35112 T^{5} + 213797 T^{6} - 5183759 T^{7} + 23372886 T^{8} - 409516961 T^{9} + 1334307077 T^{10} + 17311585368 T^{11} - 304161182529 T^{12} + 1873927346991 T^{13} + 1458524733126 T^{14} - 134427362903113 T^{15} + 1517108809906561 T^{16}$$
$83$ $$( 1 - 29 T + 560 T^{2} - 7232 T^{3} + 75754 T^{4} - 600256 T^{5} + 3857840 T^{6} - 16581823 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 - 9 T - 128 T^{2} - 549 T^{3} + 20701 T^{4} + 107280 T^{5} - 329006 T^{6} - 7780266 T^{7} - 45486752 T^{8} - 692443674 T^{9} - 2606056526 T^{10} + 75629074320 T^{11} + 1298827130941 T^{12} - 3065648637501 T^{13} - 63613605243008 T^{14} - 398082014059761 T^{15} + 3936588805702081 T^{16}$$
$97$ $$( 1 + 4 T + 192 T^{2} - 412 T^{3} + 15886 T^{4} - 39964 T^{5} + 1806528 T^{6} + 3650692 T^{7} + 88529281 T^{8} )^{2}$$