# Properties

 Label 504.2.bk.a Level 504 Weight 2 Character orbit 504.bk Analytic conductor 4.024 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.bk (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.144054149089536.2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{3} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{4} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{7} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{3} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{4} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{7} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} + ( -\beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{10} + ( -\beta_{1} - \beta_{3} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{11} + ( -1 - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{13} + ( -\beta_{3} + 3 \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{14} + ( 2 + 2 \beta_{2} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{16} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{17} + ( -1 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{19} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{20} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} ) q^{22} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{8} + 2 \beta_{9} ) q^{25} + ( -3 - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{26} + ( 4 - 3 \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{28} + ( -1 - 2 \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{29} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{31} + ( 2 \beta_{3} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{32} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + 4 \beta_{7} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{34} + ( -2 - \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{8} + \beta_{9} + \beta_{11} ) q^{35} + ( -2 - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 4 \beta_{10} ) q^{37} + ( 4 - \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{38} + ( 1 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{40} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{41} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{44} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 5 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{46} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{47} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{49} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{50} + ( -4 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{9} - 2 \beta_{11} ) q^{52} + ( -2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + 5 \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{53} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{55} + ( 5 + \beta_{1} + \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + \beta_{5} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{56} + ( 3 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} ) q^{58} + ( -4 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{59} + ( -4 - 2 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{61} + ( 2 + \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 6 \beta_{7} + \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{62} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{64} + ( 1 - \beta_{1} - \beta_{4} + 3 \beta_{7} + \beta_{9} - 2 \beta_{11} ) q^{65} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - 5 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{67} + ( 4 + \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{68} + ( 7 - 3 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{70} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{71} + ( 2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{73} + ( 4 - \beta_{1} + 2 \beta_{2} + 6 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{10} ) q^{74} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 8 \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{76} + ( -1 - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{77} + ( 1 + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{10} ) q^{79} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{80} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{82} + ( -4 + 4 \beta_{1} + 2 \beta_{3} - 6 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{11} ) q^{83} + ( -\beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{85} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} - 5 \beta_{10} + 3 \beta_{11} ) q^{88} + ( -1 + \beta_{7} ) q^{89} + ( -7 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{91} + ( -9 + \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{92} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{94} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} + 5 \beta_{10} - 2 \beta_{11} ) q^{95} + ( -1 + 2 \beta_{1} + \beta_{3} + 5 \beta_{7} - \beta_{8} - 2 \beta_{9} - 7 \beta_{11} ) q^{97} + ( 1 + \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - \beta_{5} - 3 \beta_{6} + 9 \beta_{7} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 12q^{8} + O(q^{10})$$ $$12q + 12q^{8} + 6q^{10} + 6q^{11} - 6q^{14} + 6q^{17} - 6q^{19} + 24q^{22} - 6q^{26} + 6q^{28} - 18q^{35} + 24q^{38} + 42q^{40} - 6q^{44} - 18q^{46} - 12q^{49} + 48q^{50} - 24q^{52} + 18q^{58} - 42q^{59} - 72q^{64} + 12q^{65} + 30q^{67} + 36q^{68} + 30q^{70} + 18q^{73} - 12q^{74} - 36q^{80} + 54q^{82} + 6q^{88} - 18q^{89} - 72q^{91} - 60q^{92} - 12q^{94} + 6q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + x^{9} + 48 x^{8} - 189 x^{7} + 431 x^{6} - 654 x^{5} + 624 x^{4} - 340 x^{3} + 96 x^{2} - 12 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3057 \nu^{11} + 45996 \nu^{10} - 86008 \nu^{9} - 97831 \nu^{8} - 142271 \nu^{7} + 2511547 \nu^{6} - 6798430 \nu^{5} + 12502836 \nu^{4} - 15993902 \nu^{3} + 9818739 \nu^{2} - 3761958 \nu + 2179472$$$$)/1375087$$ $$\beta_{2}$$ $$=$$ $$($$$$-1900 \nu^{11} - 5021 \nu^{10} + 17481 \nu^{9} + 21324 \nu^{8} - 69565 \nu^{7} - 118672 \nu^{6} + 558099 \nu^{5} - 1557319 \nu^{4} + 1993822 \nu^{3} - 1625148 \nu^{2} + 523802 \nu - 210768$$$$)/289492$$ $$\beta_{3}$$ $$=$$ $$($$$$-89065 \nu^{11} + 12805 \nu^{10} + 165954 \nu^{9} + 1037153 \nu^{8} - 3089444 \nu^{7} + 5335597 \nu^{6} - 18686279 \nu^{5} + 29365260 \nu^{4} - 48405922 \nu^{3} + 63031010 \nu^{2} - 34487760 \nu + 1209724$$$$)/5500348$$ $$\beta_{4}$$ $$=$$ $$($$$$17437 \nu^{11} + 7066 \nu^{10} - 100969 \nu^{9} - 163381 \nu^{8} + 761375 \nu^{7} - 420189 \nu^{6} + 58962 \nu^{5} + 2030017 \nu^{4} - 3514016 \nu^{3} - 733302 \nu^{2} + 4532706 \nu - 1691468$$$$)/785764$$ $$\beta_{5}$$ $$=$$ $$($$$$-246051 \nu^{11} + 871028 \nu^{10} + 6163 \nu^{9} - 885875 \nu^{8} - 12628835 \nu^{7} + 52077825 \nu^{6} - 112773492 \nu^{5} + 167857153 \nu^{4} - 140407242 \nu^{3} + 50546510 \nu^{2} + 1803174 \nu + 3141260$$$$)/5500348$$ $$\beta_{6}$$ $$=$$ $$($$$$-146932 \nu^{11} + 319768 \nu^{10} + 291913 \nu^{9} + 229501 \nu^{8} - 7300724 \nu^{7} + 21292555 \nu^{6} - 44577244 \nu^{5} + 64482219 \nu^{4} - 57471147 \nu^{3} + 45859296 \nu^{2} - 31852978 \nu + 7497160$$$$)/2750174$$ $$\beta_{7}$$ $$=$$ $$($$$$-2236 \nu^{11} + 4785 \nu^{10} + 6307 \nu^{9} - 1316 \nu^{8} - 113135 \nu^{7} + 323728 \nu^{6} - 580379 \nu^{5} + 648191 \nu^{4} - 212358 \nu^{3} - 208368 \nu^{2} + 142154 \nu - 27316$$$$)/41356$$ $$\beta_{8}$$ $$=$$ $$($$$$228352 \nu^{11} - 272019 \nu^{10} - 940304 \nu^{9} - 525407 \nu^{8} + 10909881 \nu^{7} - 23124265 \nu^{6} + 35178775 \nu^{5} - 20414474 \nu^{4} - 27347851 \nu^{3} + 43381212 \nu^{2} - 16682018 \nu + 2200946$$$$)/2750174$$ $$\beta_{9}$$ $$=$$ $$($$$$475721 \nu^{11} - 1135999 \nu^{10} - 834872 \nu^{9} + 354939 \nu^{8} + 23253862 \nu^{7} - 75835685 \nu^{6} + 151374901 \nu^{5} - 194515274 \nu^{4} + 127509630 \nu^{3} - 11011742 \nu^{2} - 15651176 \nu + 5254820$$$$)/5500348$$ $$\beta_{10}$$ $$=$$ $$($$$$-26339 \nu^{11} + 54578 \nu^{10} + 59599 \nu^{9} + 15779 \nu^{8} - 1269817 \nu^{7} + 3777819 \nu^{6} - 7455618 \nu^{5} + 9211989 \nu^{4} - 5666228 \nu^{3} + 1295210 \nu^{2} + 102322 \nu - 105412$$$$)/289492$$ $$\beta_{11}$$ $$=$$ $$($$$$-191973 \nu^{11} + 499534 \nu^{10} + 212817 \nu^{9} - 108091 \nu^{8} - 9329027 \nu^{7} + 32474753 \nu^{6} - 69226294 \nu^{5} + 97308215 \nu^{4} - 80268948 \nu^{3} + 34473298 \nu^{2} - 8500670 \nu + 1072580$$$$)/785764$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{10} - 2 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 5 \beta_{7} - \beta_{6} - 2 \beta_{5} + 3 \beta_{3} - 5 \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 22 \beta_{7} + 8 \beta_{6} + \beta_{5} + 8 \beta_{4} - 8 \beta_{3} + 7 \beta_{2} - 7 \beta_{1} + 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{11} - 24 \beta_{10} - 16 \beta_{8} + 23 \beta_{7} - 5 \beta_{6} + 13 \beta_{5} + 25 \beta_{3} - 28 \beta_{2} + 24 \beta_{1} - 33$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-23 \beta_{11} + 93 \beta_{10} - 63 \beta_{9} + 73 \beta_{8} - 17 \beta_{7} + 9 \beta_{6} - 40 \beta_{5} - 18 \beta_{4} - 25 \beta_{3} + 11 \beta_{2} - 31 \beta_{1} + 78$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$43 \beta_{11} - 247 \beta_{10} + 69 \beta_{9} - 293 \beta_{8} - 342 \beta_{7} + 134 \beta_{6} + 193 \beta_{5} + 170 \beta_{4} - 50 \beta_{3} + 145 \beta_{2} - 29 \beta_{1} - 153$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-212 \beta_{11} + 474 \beta_{10} - 164 \beta_{9} + 666 \beta_{8} + 1329 \beta_{7} - 483 \beta_{6} - 287 \beta_{5} - 560 \beta_{4} + 487 \beta_{3} - 736 \beta_{2} + 490 \beta_{1} - 181$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$291 \beta_{11} + 501 \beta_{10} - 291 \beta_{9} - 467 \beta_{8} - 4069 \beta_{7} + 1347 \beta_{6} + 230 \beta_{5} + 1006 \beta_{4} - 2045 \beta_{3} + 2595 \beta_{2} - 2011 \beta_{1} + 1638$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$47 \beta_{11} - 4907 \beta_{10} + 2599 \beta_{9} - 2699 \beta_{8} + 6710 \beta_{7} - 2216 \beta_{6} + 2537 \beta_{5} - 896 \beta_{4} + 4722 \beta_{3} - 4489 \beta_{2} + 4897 \beta_{1} - 7147$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-3036 \beta_{11} + 22620 \beta_{10} - 8568 \beta_{9} + 20124 \beta_{8} + 4207 \beta_{7} - 2529 \beta_{6} - 12291 \beta_{5} - 7794 \beta_{4} - 6201 \beta_{3} + 2162 \beta_{2} - 7052 \beta_{1} + 18491$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$15225 \beta_{11} - 56695 \beta_{10} + 24185 \beta_{9} - 63533 \beta_{8} - 73831 \beta_{7} + 26995 \beta_{6} + 37488 \beta_{5} + 37188 \beta_{4} - 16347 \beta_{3} + 37499 \beta_{2} - 13697 \beta_{1} - 24794$$$$)/2$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$1 + \beta_{7}$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.186445 − 1.54034i 2.00233 + 0.854000i −2.37165 + 1.78079i 1.09935 − 0.468876i 0.609850 + 0.457915i −0.0263223 − 0.217464i 2.00233 − 0.854000i 0.186445 + 1.54034i 1.09935 + 0.468876i −2.37165 − 1.78079i −0.0263223 + 0.217464i 0.609850 − 0.457915i
−1.13090 0.849154i 0 0.557875 + 1.92062i −1.03926 + 1.80005i 0 1.25203 2.33076i 1.00000 2.64575i 0 2.70382 1.15319i
19.2 −1.13090 + 0.849154i 0 0.557875 1.92062i 1.59713 2.76632i 0 −0.694153 2.55307i 1.00000 + 2.64575i 0 0.542829 + 4.48465i
19.3 −0.169938 1.40397i 0 −1.94224 + 0.477176i −0.345107 + 0.597743i 0 −2.63639 + 0.222310i 1.00000 + 2.64575i 0 0.897858 + 0.382939i
19.4 −0.169938 + 1.40397i 0 −1.94224 0.477176i −1.59713 + 2.76632i 0 0.694153 + 2.55307i 1.00000 2.64575i 0 −3.61240 2.71243i
19.5 1.30084 0.554812i 0 1.38437 1.44344i 0.345107 0.597743i 0 2.63639 0.222310i 1.00000 2.64575i 0 0.117294 0.969037i
19.6 1.30084 + 0.554812i 0 1.38437 + 1.44344i 1.03926 1.80005i 0 −1.25203 + 2.33076i 1.00000 + 2.64575i 0 2.35060 1.76498i
451.1 −1.13090 0.849154i 0 0.557875 + 1.92062i 1.59713 + 2.76632i 0 −0.694153 + 2.55307i 1.00000 2.64575i 0 0.542829 4.48465i
451.2 −1.13090 + 0.849154i 0 0.557875 1.92062i −1.03926 1.80005i 0 1.25203 + 2.33076i 1.00000 + 2.64575i 0 2.70382 + 1.15319i
451.3 −0.169938 1.40397i 0 −1.94224 + 0.477176i −1.59713 2.76632i 0 0.694153 2.55307i 1.00000 + 2.64575i 0 −3.61240 + 2.71243i
451.4 −0.169938 + 1.40397i 0 −1.94224 0.477176i −0.345107 0.597743i 0 −2.63639 0.222310i 1.00000 2.64575i 0 0.897858 0.382939i
451.5 1.30084 0.554812i 0 1.38437 1.44344i 1.03926 + 1.80005i 0 −1.25203 2.33076i 1.00000 2.64575i 0 2.35060 + 1.76498i
451.6 1.30084 + 0.554812i 0 1.38437 + 1.44344i 0.345107 + 0.597743i 0 2.63639 + 0.222310i 1.00000 + 2.64575i 0 0.117294 + 0.969037i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.6 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.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.bk.a 12
3.b odd 2 1 56.2.m.a 12
4.b odd 2 1 2016.2.bs.a 12
7.d odd 6 1 inner 504.2.bk.a 12
8.b even 2 1 2016.2.bs.a 12
8.d odd 2 1 inner 504.2.bk.a 12
12.b even 2 1 224.2.q.a 12
21.c even 2 1 392.2.m.g 12
21.g even 6 1 56.2.m.a 12
21.g even 6 1 392.2.e.e 12
21.h odd 6 1 392.2.e.e 12
21.h odd 6 1 392.2.m.g 12
24.f even 2 1 56.2.m.a 12
24.h odd 2 1 224.2.q.a 12
28.f even 6 1 2016.2.bs.a 12
56.j odd 6 1 2016.2.bs.a 12
56.m even 6 1 inner 504.2.bk.a 12
84.h odd 2 1 1568.2.q.g 12
84.j odd 6 1 224.2.q.a 12
84.j odd 6 1 1568.2.e.e 12
84.n even 6 1 1568.2.e.e 12
84.n even 6 1 1568.2.q.g 12
168.e odd 2 1 392.2.m.g 12
168.i even 2 1 1568.2.q.g 12
168.s odd 6 1 1568.2.e.e 12
168.s odd 6 1 1568.2.q.g 12
168.v even 6 1 392.2.e.e 12
168.v even 6 1 392.2.m.g 12
168.ba even 6 1 224.2.q.a 12
168.ba even 6 1 1568.2.e.e 12
168.be odd 6 1 56.2.m.a 12
168.be odd 6 1 392.2.e.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.m.a 12 3.b odd 2 1
56.2.m.a 12 21.g even 6 1
56.2.m.a 12 24.f even 2 1
56.2.m.a 12 168.be odd 6 1
224.2.q.a 12 12.b even 2 1
224.2.q.a 12 24.h odd 2 1
224.2.q.a 12 84.j odd 6 1
224.2.q.a 12 168.ba even 6 1
392.2.e.e 12 21.g even 6 1
392.2.e.e 12 21.h odd 6 1
392.2.e.e 12 168.v even 6 1
392.2.e.e 12 168.be odd 6 1
392.2.m.g 12 21.c even 2 1
392.2.m.g 12 21.h odd 6 1
392.2.m.g 12 168.e odd 2 1
392.2.m.g 12 168.v even 6 1
504.2.bk.a 12 1.a even 1 1 trivial
504.2.bk.a 12 7.d odd 6 1 inner
504.2.bk.a 12 8.d odd 2 1 inner
504.2.bk.a 12 56.m even 6 1 inner
1568.2.e.e 12 84.j odd 6 1
1568.2.e.e 12 84.n even 6 1
1568.2.e.e 12 168.s odd 6 1
1568.2.e.e 12 168.ba even 6 1
1568.2.q.g 12 84.h odd 2 1
1568.2.q.g 12 84.n even 6 1
1568.2.q.g 12 168.i even 2 1
1568.2.q.g 12 168.s odd 6 1
2016.2.bs.a 12 4.b odd 2 1
2016.2.bs.a 12 8.b even 2 1
2016.2.bs.a 12 28.f even 6 1
2016.2.bs.a 12 56.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} + 15 T_{5}^{10} + 174 T_{5}^{8} + 723 T_{5}^{6} + 2286 T_{5}^{4} + 1071 T_{5}^{2} + 441$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{3} + 8 T^{6} )^{2}$$
$3$ 
$5$ $$1 - 15 T^{2} + 99 T^{4} - 412 T^{6} + 1641 T^{8} - 7989 T^{10} + 39846 T^{12} - 199725 T^{14} + 1025625 T^{16} - 6437500 T^{18} + 38671875 T^{20} - 146484375 T^{22} + 244140625 T^{24}$$
$7$ $$1 + 6 T^{2} - 33 T^{4} - 700 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12}$$
$11$ $$( 1 - 3 T - 21 T^{2} + 28 T^{3} + 393 T^{4} - 153 T^{5} - 4846 T^{6} - 1683 T^{7} + 47553 T^{8} + 37268 T^{9} - 307461 T^{10} - 483153 T^{11} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 + 42 T^{2} + 903 T^{4} + 13340 T^{6} + 152607 T^{8} + 1199562 T^{10} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 - 3 T + 39 T^{2} - 108 T^{3} + 729 T^{4} - 753 T^{5} + 12014 T^{6} - 12801 T^{7} + 210681 T^{8} - 530604 T^{9} + 3257319 T^{10} - 4259571 T^{11} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 + 3 T + 39 T^{2} + 108 T^{3} + 705 T^{4} + 2265 T^{5} + 12706 T^{6} + 43035 T^{7} + 254505 T^{8} + 740772 T^{9} + 5082519 T^{10} + 7428297 T^{11} + 47045881 T^{12} )^{2}$$
$23$ $$1 + 69 T^{2} + 1863 T^{4} + 44136 T^{6} + 1489365 T^{8} + 33541107 T^{10} + 591428630 T^{12} + 17743245603 T^{14} + 416785390965 T^{16} + 6533711996904 T^{18} + 145893365578503 T^{20} + 2858429273741781 T^{22} + 21914624432020321 T^{24}$$
$29$ $$( 1 - 126 T^{2} + 7623 T^{4} - 278868 T^{6} + 6410943 T^{8} - 89117406 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$1 - 87 T^{2} + 3867 T^{4} - 78332 T^{6} - 455595 T^{8} + 87513459 T^{10} - 3532899090 T^{12} + 84100434099 T^{14} - 420751549995 T^{16} - 69519938340092 T^{18} + 3298129641784347 T^{20} - 71307660967329687 T^{22} + 787662783788549761 T^{24}$$
$37$ $$1 + 105 T^{2} + 3339 T^{4} + 152764 T^{6} + 14203497 T^{8} + 501483003 T^{10} + 10611705558 T^{12} + 686530231107 T^{14} + 26619640141017 T^{16} + 391950629144476 T^{18} + 11728168896642219 T^{20} + 504901359103874145 T^{22} + 6582952005840035281 T^{24}$$
$41$ $$( 1 - 102 T^{2} + 6783 T^{4} - 298852 T^{6} + 11402223 T^{8} - 288227622 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 + 43 T^{2} )^{12}$$
$47$ $$1 - 159 T^{2} + 13131 T^{4} - 642364 T^{6} + 18269013 T^{8} - 74583429 T^{10} - 12209351154 T^{12} - 164754794661 T^{14} + 89146955624853 T^{16} - 6924179875597756 T^{18} + 312666005155583691 T^{20} - 8363262025496977791 T^{22} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 + 105 T^{2} - 693 T^{4} - 10308 T^{6} + 39259017 T^{8} + 1185419067 T^{10} - 43539019882 T^{12} + 3329842159203 T^{14} + 309772527717177 T^{16} - 228470234517732 T^{18} - 43145965455073173 T^{20} + 18363184388378870145 T^{22} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$( 1 + 21 T + 309 T^{2} + 3402 T^{3} + 29601 T^{4} + 225789 T^{5} + 1760606 T^{6} + 13321551 T^{7} + 103041081 T^{8} + 698699358 T^{9} + 3744264549 T^{10} + 15013410279 T^{11} + 42180533641 T^{12} )^{2}$$
$61$ $$1 - 159 T^{2} + 7419 T^{4} - 352004 T^{6} + 54786249 T^{8} - 3286587357 T^{10} + 113418284406 T^{12} - 12229391555397 T^{14} + 758561692640409 T^{16} - 18135377856569444 T^{18} + 1422276555126827739 T^{20} -$$$$11\!\cdots\!59$$$$T^{22} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$( 1 - 15 T - 3 T^{2} + 142 T^{3} + 9993 T^{4} - 15123 T^{5} - 719466 T^{6} - 1013241 T^{7} + 44858577 T^{8} + 42708346 T^{9} - 60453363 T^{10} - 20251876605 T^{11} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 - 16 T + 71 T^{2} )^{6}( 1 + 16 T + 71 T^{2} )^{6}$$
$73$ $$( 1 - 9 T + 183 T^{2} - 1404 T^{3} + 16701 T^{4} - 147195 T^{5} + 1376278 T^{6} - 10745235 T^{7} + 88999629 T^{8} - 546179868 T^{9} + 5196878103 T^{10} - 18657644337 T^{11} + 151334226289 T^{12} )^{2}$$
$79$ $$1 + 261 T^{2} + 30519 T^{4} + 2829272 T^{6} + 267713397 T^{8} + 18988265187 T^{10} + 1205673992502 T^{12} + 118505763032067 T^{14} + 10427458497935157 T^{16} + 687760531456810712 T^{18} + 46300643769538335159 T^{20} +$$$$24\!\cdots\!61$$$$T^{22} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$( 1 - 270 T^{2} + 28455 T^{4} - 2146852 T^{6} + 196026495 T^{8} - 12813746670 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 + 3 T + 92 T^{2} + 267 T^{3} + 7921 T^{4} )^{6}$$
$97$ $$( 1 - 222 T^{2} + 20703 T^{4} - 1529300 T^{6} + 194794527 T^{8} - 19653500382 T^{10} + 832972004929 T^{12} )^{2}$$