# Properties

 Label 504.2.r.c Level 504 Weight 2 Character orbit 504.r Analytic conductor 4.024 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.r (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{18} + \zeta_{18}^{4} ) q^{3} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{5} -\zeta_{18}^{3} q^{7} + ( 3 \zeta_{18}^{2} - 3 \zeta_{18}^{5} ) q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{18} + \zeta_{18}^{4} ) q^{3} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{5} -\zeta_{18}^{3} q^{7} + ( 3 \zeta_{18}^{2} - 3 \zeta_{18}^{5} ) q^{9} + ( -\zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( 1 - 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{13} + ( 2 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{15} + ( 2 - \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{17} + ( -3 - 3 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{19} + ( \zeta_{18} + \zeta_{18}^{4} ) q^{21} + ( -2 - 5 \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{23} + ( 3 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{25} + ( -6 + 3 \zeta_{18}^{3} ) q^{27} + ( 3 \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{29} + ( 3 - 2 \zeta_{18} - \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{31} + ( -3 - 3 \zeta_{18}^{5} ) q^{33} + ( -1 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{4} ) q^{35} + ( -5 - 3 \zeta_{18} - 3 \zeta_{18}^{2} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{37} + ( 4 - \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{39} + ( -2 + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{41} + ( -\zeta_{18} - 2 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{43} + ( -3 \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{45} + ( -2 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{47} + ( -1 + \zeta_{18}^{3} ) q^{49} + ( -4 \zeta_{18} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{51} + ( -2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{53} + ( -\zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{55} + ( 4 + 6 \zeta_{18} + 4 \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{57} + ( -1 - 7 \zeta_{18} + 7 \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{59} + ( 3 \zeta_{18} + \zeta_{18}^{2} + 4 \zeta_{18}^{3} + \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{61} -3 \zeta_{18}^{2} q^{63} + ( 5 \zeta_{18} - 2 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{65} + ( 4 + 5 \zeta_{18} - 6 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{67} + ( -1 + 2 \zeta_{18} + 8 \zeta_{18}^{2} - 7 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{69} + ( 7 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 7 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{71} + ( 1 - 5 \zeta_{18} - 5 \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{73} + ( 4 - 2 \zeta_{18} - 5 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{75} + ( 2 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{77} + ( -2 \zeta_{18} - 5 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{79} + ( 9 \zeta_{18} - 9 \zeta_{18}^{4} ) q^{81} + ( 4 \zeta_{18} + 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{83} + ( 5 + 2 \zeta_{18} - 6 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{85} + ( 2 - \zeta_{18} - 7 \zeta_{18}^{2} - 7 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{87} + ( -6 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{89} + ( -1 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} ) q^{91} + ( 4 - 3 \zeta_{18} + \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 6 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{93} + ( 2 + 2 \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{95} + ( 9 \zeta_{18} - 11 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - 11 \zeta_{18}^{4} + 9 \zeta_{18}^{5} ) q^{97} + ( -3 + 6 \zeta_{18} + 6 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{5} - 3q^{7} + O(q^{10})$$ $$6q + 3q^{5} - 3q^{7} + 3q^{13} + 9q^{15} + 12q^{17} - 18q^{19} - 6q^{23} + 6q^{25} - 27q^{27} + 3q^{29} + 9q^{31} - 18q^{33} - 6q^{35} - 30q^{37} + 18q^{39} - 6q^{41} + 15q^{43} + 9q^{45} - 9q^{47} - 3q^{49} + 9q^{51} - 12q^{53} + 27q^{57} - 3q^{59} + 12q^{61} - 15q^{65} + 12q^{67} - 27q^{69} + 42q^{71} + 6q^{73} + 9q^{75} + 15q^{79} + 6q^{83} + 15q^{85} - 9q^{87} - 36q^{89} - 6q^{91} + 9q^{93} + 6q^{95} - 15q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 + \zeta_{18}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 0.939693 − 0.342020i −0.766044 − 0.642788i −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 + 0.642788i −0.173648 − 0.984808i
0 −1.70574 0.300767i 0 −0.266044 0.460802i 0 −0.500000 + 0.866025i 0 2.81908 + 1.02606i 0
169.2 0 0.592396 + 1.62760i 0 0.326352 + 0.565258i 0 −0.500000 + 0.866025i 0 −2.29813 + 1.92836i 0
169.3 0 1.11334 1.32683i 0 1.43969 + 2.49362i 0 −0.500000 + 0.866025i 0 −0.520945 2.95442i 0
337.1 0 −1.70574 + 0.300767i 0 −0.266044 + 0.460802i 0 −0.500000 0.866025i 0 2.81908 1.02606i 0
337.2 0 0.592396 1.62760i 0 0.326352 0.565258i 0 −0.500000 0.866025i 0 −2.29813 1.92836i 0
337.3 0 1.11334 + 1.32683i 0 1.43969 2.49362i 0 −0.500000 0.866025i 0 −0.520945 + 2.95442i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.3 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 504.2.r.c 6
3.b odd 2 1 1512.2.r.c 6
4.b odd 2 1 1008.2.r.i 6
9.c even 3 1 inner 504.2.r.c 6
9.c even 3 1 4536.2.a.s 3
9.d odd 6 1 1512.2.r.c 6
9.d odd 6 1 4536.2.a.v 3
12.b even 2 1 3024.2.r.h 6
36.f odd 6 1 1008.2.r.i 6
36.f odd 6 1 9072.2.a.br 3
36.h even 6 1 3024.2.r.h 6
36.h even 6 1 9072.2.a.cc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.c 6 1.a even 1 1 trivial
504.2.r.c 6 9.c even 3 1 inner
1008.2.r.i 6 4.b odd 2 1
1008.2.r.i 6 36.f odd 6 1
1512.2.r.c 6 3.b odd 2 1
1512.2.r.c 6 9.d odd 6 1
3024.2.r.h 6 12.b even 2 1
3024.2.r.h 6 36.h even 6 1
4536.2.a.s 3 9.c even 3 1
4536.2.a.v 3 9.d odd 6 1
9072.2.a.br 3 36.f odd 6 1
9072.2.a.cc 3 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 3 T_{5}^{5} + 9 T_{5}^{4} - 2 T_{5}^{3} + 3 T_{5}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 9 T^{3} + 27 T^{6}$$
$5$ $$1 - 3 T - 6 T^{2} + 13 T^{3} + 63 T^{4} - 60 T^{5} - 259 T^{6} - 300 T^{7} + 1575 T^{8} + 1625 T^{9} - 3750 T^{10} - 9375 T^{11} + 15625 T^{12}$$
$7$ $$( 1 + T + T^{2} )^{3}$$
$11$ $$1 - 24 T^{2} - 18 T^{3} + 312 T^{4} + 216 T^{5} - 3593 T^{6} + 2376 T^{7} + 37752 T^{8} - 23958 T^{9} - 351384 T^{10} + 1771561 T^{12}$$
$13$ $$1 - 3 T - 21 T^{2} + 60 T^{3} + 285 T^{4} - 417 T^{5} - 3202 T^{6} - 5421 T^{7} + 48165 T^{8} + 131820 T^{9} - 599781 T^{10} - 1113879 T^{11} + 4826809 T^{12}$$
$17$ $$( 1 - 6 T + 54 T^{2} - 203 T^{3} + 918 T^{4} - 1734 T^{5} + 4913 T^{6} )^{2}$$
$19$ $$( 1 + 9 T + 63 T^{2} + 323 T^{3} + 1197 T^{4} + 3249 T^{5} + 6859 T^{6} )^{2}$$
$23$ $$1 + 6 T + 12 T^{2} + 130 T^{3} - 18 T^{4} - 3282 T^{5} - 7909 T^{6} - 75486 T^{7} - 9522 T^{8} + 1581710 T^{9} + 3358092 T^{10} + 38618058 T^{11} + 148035889 T^{12}$$
$29$ $$1 - 3 T - 42 T^{2} + 157 T^{3} + 657 T^{4} - 1554 T^{5} - 10195 T^{6} - 45066 T^{7} + 552537 T^{8} + 3829073 T^{9} - 29705802 T^{10} - 61533447 T^{11} + 594823321 T^{12}$$
$31$ $$1 - 9 T - 18 T^{2} + 79 T^{3} + 2367 T^{4} - 1422 T^{5} - 86865 T^{6} - 44082 T^{7} + 2274687 T^{8} + 2353489 T^{9} - 16623378 T^{10} - 257662359 T^{11} + 887503681 T^{12}$$
$37$ $$( 1 + 15 T + 165 T^{2} + 1167 T^{3} + 6105 T^{4} + 20535 T^{5} + 50653 T^{6} )^{2}$$
$41$ $$1 + 6 T - 72 T^{2} - 298 T^{3} + 4398 T^{4} + 8034 T^{5} - 177169 T^{6} + 329394 T^{7} + 7393038 T^{8} - 20538458 T^{9} - 203454792 T^{10} + 695137206 T^{11} + 4750104241 T^{12}$$
$43$ $$1 - 15 T + 42 T^{2} - 199 T^{3} + 6525 T^{4} - 24624 T^{5} - 76509 T^{6} - 1058832 T^{7} + 12064725 T^{8} - 15821893 T^{9} + 143589642 T^{10} - 2205126645 T^{11} + 6321363049 T^{12}$$
$47$ $$1 + 9 T - 66 T^{2} - 371 T^{3} + 7077 T^{4} + 20028 T^{5} - 269497 T^{6} + 941316 T^{7} + 15633093 T^{8} - 38518333 T^{9} - 322058946 T^{10} + 2064105063 T^{11} + 10779215329 T^{12}$$
$53$ $$( 1 + 6 T + 162 T^{2} + 617 T^{3} + 8586 T^{4} + 16854 T^{5} + 148877 T^{6} )^{2}$$
$59$ $$1 + 3 T - 24 T^{2} + 369 T^{3} - 453 T^{4} - 5694 T^{5} + 333403 T^{6} - 335946 T^{7} - 1576893 T^{8} + 75784851 T^{9} - 290816664 T^{10} + 2144772897 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 12 T - 48 T^{2} + 478 T^{3} + 8460 T^{4} - 30636 T^{5} - 395685 T^{6} - 1868796 T^{7} + 31479660 T^{8} + 108496918 T^{9} - 664600368 T^{10} - 10135155612 T^{11} + 51520374361 T^{12}$$
$67$ $$1 - 12 T - 12 T^{2} + 690 T^{3} - 1488 T^{4} - 2856 T^{5} - 34441 T^{6} - 191352 T^{7} - 6679632 T^{8} + 207526470 T^{9} - 241813452 T^{10} - 16201501284 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 - 21 T + 249 T^{2} - 2115 T^{3} + 17679 T^{4} - 105861 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$( 1 - 3 T + 105 T^{2} - 475 T^{3} + 7665 T^{4} - 15987 T^{5} + 389017 T^{6} )^{2}$$
$79$ $$1 - 15 T + 30 T^{2} - 59 T^{3} + 1125 T^{4} + 89100 T^{5} - 1368633 T^{6} + 7038900 T^{7} + 7021125 T^{8} - 29089301 T^{9} + 1168502430 T^{10} - 46155845985 T^{11} + 243087455521 T^{12}$$
$83$ $$1 - 6 T - 42 T^{2} + 2610 T^{3} - 9624 T^{4} - 83688 T^{5} + 2759119 T^{6} - 6946104 T^{7} - 66299736 T^{8} + 1492364070 T^{9} - 1993249482 T^{10} - 23634243858 T^{11} + 326940373369 T^{12}$$
$89$ $$( 1 + 18 T + 318 T^{2} + 3241 T^{3} + 28302 T^{4} + 142578 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$1 + 15 T + 168 T^{2} + 1477 T^{3} + 2385 T^{4} - 42462 T^{5} - 510111 T^{6} - 4118814 T^{7} + 22440465 T^{8} + 1348018021 T^{9} + 14872919208 T^{10} + 128810103855 T^{11} + 832972004929 T^{12}$$