# Properties

 Label 504.2.cc.a Level 504 Weight 2 Character orbit 504.cc Analytic conductor 4.024 Analytic rank 0 Dimension 16 CM discriminant -56 Inner twists 8

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{11} q^{2} -\beta_{14} q^{3} + ( 2 - 2 \beta_{4} ) q^{4} + ( \beta_{8} - \beta_{9} + \beta_{10} + \beta_{15} ) q^{5} + \beta_{1} q^{6} -\beta_{12} q^{7} + 2 \beta_{2} q^{8} + ( \beta_{2} - \beta_{12} + \beta_{13} ) q^{9} +O(q^{10})$$ $$q + \beta_{11} q^{2} -\beta_{14} q^{3} + ( 2 - 2 \beta_{4} ) q^{4} + ( \beta_{8} - \beta_{9} + \beta_{10} + \beta_{15} ) q^{5} + \beta_{1} q^{6} -\beta_{12} q^{7} + 2 \beta_{2} q^{8} + ( \beta_{2} - \beta_{12} + \beta_{13} ) q^{9} + ( \beta_{6} + \beta_{9} - \beta_{10} - \beta_{15} ) q^{10} + ( -2 \beta_{6} - 2 \beta_{14} ) q^{12} + ( 2 \beta_{6} - \beta_{10} + 2 \beta_{14} + \beta_{15} ) q^{13} + \beta_{7} q^{14} + ( -1 - \beta_{5} + \beta_{7} + 2 \beta_{11} + \beta_{12} ) q^{15} -4 \beta_{4} q^{16} + ( -2 \beta_{4} + \beta_{5} ) q^{18} + ( -\beta_{1} - \beta_{3} - \beta_{6} - \beta_{10} - \beta_{14} - 2 \beta_{15} ) q^{19} + ( -2 \beta_{9} + 2 \beta_{10} ) q^{20} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{8} + \beta_{10} ) q^{21} + ( 3 + 3 \beta_{4} - \beta_{7} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{10} ) q^{24} + ( 6 \beta_{2} + 5 \beta_{4} - 3 \beta_{11} + \beta_{12} ) q^{25} + ( -3 \beta_{1} + \beta_{3} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{14} ) q^{26} + ( -\beta_{1} + 2 \beta_{3} - \beta_{8} ) q^{27} + ( -2 \beta_{12} + 2 \beta_{13} ) q^{28} + ( 4 - 4 \beta_{4} - \beta_{7} - \beta_{11} - 2 \beta_{12} ) q^{30} + ( 4 \beta_{2} - 4 \beta_{11} ) q^{32} + ( -\beta_{3} - 3 \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{15} ) q^{35} + ( 2 \beta_{2} - 2 \beta_{11} + 2 \beta_{13} ) q^{36} + ( 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{6} + \beta_{8} - 2 \beta_{10} + 3 \beta_{14} - \beta_{15} ) q^{38} + ( -5 - 4 \beta_{2} + 5 \beta_{4} - \beta_{7} + 4 \beta_{11} - \beta_{13} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{14} ) q^{40} + ( \beta_{3} + \beta_{6} + \beta_{8} + \beta_{10} + \beta_{14} + 3 \beta_{15} ) q^{42} + ( 4 \beta_{1} - 2 \beta_{3} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{14} ) q^{45} + ( -3 \beta_{2} + 6 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{46} -4 \beta_{6} q^{48} + ( -7 + 7 \beta_{4} ) q^{49} + ( -6 - 5 \beta_{2} - 6 \beta_{4} - \beta_{7} + 5 \beta_{11} ) q^{50} + ( -4 \beta_{1} + 2 \beta_{3} + 4 \beta_{6} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{15} ) q^{52} + ( 2 \beta_{3} - \beta_{6} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + 3 \beta_{15} ) q^{54} + 2 \beta_{5} q^{56} + ( 5 \beta_{2} + \beta_{4} - 2 \beta_{5} - 5 \beta_{11} - \beta_{13} ) q^{57} + ( -\beta_{3} - 3 \beta_{6} - \beta_{8} + \beta_{10} - 3 \beta_{14} + 2 \beta_{15} ) q^{59} + ( -2 + 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{12} - 2 \beta_{13} ) q^{60} + ( 2 \beta_{3} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 4 \beta_{14} ) q^{61} + ( 7 \beta_{4} + \beta_{5} ) q^{63} -8 q^{64} + ( -6 + 3 \beta_{4} - 2 \beta_{5} - \beta_{11} - 3 \beta_{12} + 6 \beta_{13} ) q^{65} + ( -\beta_{3} + 2 \beta_{6} - \beta_{8} - \beta_{10} - 4 \beta_{14} - 3 \beta_{15} ) q^{69} + ( -\beta_{3} - \beta_{8} - \beta_{9} - 2 \beta_{10} - 5 \beta_{14} ) q^{70} + ( -2 \beta_{2} - 3 \beta_{12} - 3 \beta_{13} ) q^{71} + ( -4 + 2 \beta_{5} - 2 \beta_{7} ) q^{72} + ( 5 \beta_{1} - \beta_{3} + 5 \beta_{6} + 2 \beta_{8} - 7 \beta_{10} ) q^{75} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} - 4 \beta_{15} ) q^{76} + ( 8 - 5 \beta_{2} - \beta_{5} + \beta_{7} + 2 \beta_{12} - 2 \beta_{13} ) q^{78} + ( -12 \beta_{2} + 6 \beta_{11} + \beta_{12} ) q^{79} + ( -4 \beta_{8} - 4 \beta_{15} ) q^{80} + ( 5 + 2 \beta_{5} - 2 \beta_{7} ) q^{81} + ( 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{6} + 2 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} + 4 \beta_{15} ) q^{83} + ( -4 \beta_{1} + 4 \beta_{3} - 2 \beta_{8} + 2 \beta_{10} ) q^{84} + ( -5 \beta_{1} + \beta_{3} - 5 \beta_{6} - 5 \beta_{8} + \beta_{10} - 5 \beta_{14} - 3 \beta_{15} ) q^{90} + ( \beta_{1} - 2 \beta_{3} + 3 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + \beta_{14} ) q^{91} + ( 12 - 6 \beta_{4} - 2 \beta_{5} ) q^{92} + ( -4 \beta_{2} + 2 \beta_{7} + 4 \beta_{11} + 6 \beta_{12} - 3 \beta_{13} ) q^{95} -4 \beta_{10} q^{96} -7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{4} + O(q^{10})$$ $$16q + 16q^{4} - 16q^{15} - 32q^{16} - 16q^{18} + 72q^{23} + 40q^{25} + 32q^{30} - 40q^{39} - 56q^{49} - 144q^{50} + 8q^{57} - 16q^{60} + 56q^{63} - 128q^{64} - 72q^{65} - 64q^{72} + 128q^{78} + 80q^{81} + 144q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10 x^{12} + 19 x^{8} + 810 x^{4} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$10 \nu^{13} + 19 \nu^{9} - 1349 \nu^{5} + 6561 \nu$$$$)/9234$$ $$\beta_{2}$$ $$=$$ $$($$$$10 \nu^{14} + 19 \nu^{10} - 1349 \nu^{6} + 6561 \nu^{2}$$$$)/27702$$ $$\beta_{3}$$ $$=$$ $$($$$$-11 \nu^{13} + 133 \nu^{9} - 209 \nu^{5} + 324 \nu$$$$)/9234$$ $$\beta_{4}$$ $$=$$ $$($$$$10 \nu^{12} + 19 \nu^{8} + 190 \nu^{4} + 8100$$$$)/1539$$ $$\beta_{5}$$ $$=$$ $$($$$$-31 \nu^{12} + 95 \nu^{8} - 589 \nu^{4} - 25110$$$$)/3078$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} - 10 \nu^{11} - 19 \nu^{7} - 810 \nu^{3}$$$$)/2187$$ $$\beta_{7}$$ $$=$$ $$($$$$50 \nu^{12} + 95 \nu^{8} - 589 \nu^{4} + 32805$$$$)/3078$$ $$\beta_{8}$$ $$=$$ $$($$$$-70 \nu^{13} - 133 \nu^{9} + 209 \nu^{5} - 36693 \nu$$$$)/9234$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{15} - 1133 \nu^{3} + 1026 \nu$$$$)/1026$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{13} + 791 \nu$$$$)/114$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{14} + 791 \nu^{2}$$$$)/342$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{14} - 449 \nu^{2}$$$$)/342$$ $$\beta_{13}$$ $$=$$ $$($$$$109 \nu^{14} + 361 \nu^{10} + 2071 \nu^{6} + 88290 \nu^{2}$$$$)/27702$$ $$\beta_{14}$$ $$=$$ $$($$$$-\nu^{15} - 620 \nu^{3}$$$$)/513$$ $$\beta_{15}$$ $$=$$ $$($$$$170 \nu^{15} + 323 \nu^{11} + 4769 \nu^{7} + 111537 \nu^{3} - 83106 \nu$$$$)/83106$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + \beta_{8} + \beta_{3}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{11}$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{14} + 2 \beta_{10} - 6 \beta_{9} + 2 \beta_{8} + 2 \beta_{3}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{7} + 5 \beta_{4} - 5$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{10} - 2 \beta_{8} + \beta_{3} - 21 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$\beta_{13} - \beta_{12} - 19 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$($$$$60 \beta_{15} + 51 \beta_{14} + 20 \beta_{10} + 20 \beta_{8} + 51 \beta_{6} + 20 \beta_{3}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$20 \beta_{5} + 31 \beta_{4}$$ $$\nu^{9}$$ $$=$$ $$($$$$-38 \beta_{10} - 71 \beta_{8} + 142 \beta_{3} - 33 \beta_{1}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$71 \beta_{13} - 109 \beta_{11} + 109 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$($$$$-114 \beta_{15} - 76 \beta_{10} + 114 \beta_{9} - 76 \beta_{8} - 753 \beta_{6} - 76 \beta_{3}$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$38 \beta_{7} - 38 \beta_{5} - 715$$ $$\nu^{13}$$ $$=$$ $$($$$$-449 \beta_{10} - 791 \beta_{8} - 791 \beta_{3}$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$-791 \beta_{12} - 449 \beta_{11}$$ $$\nu^{15}$$ $$=$$ $$($$$$-3399 \beta_{14} - 1240 \beta_{10} + 3720 \beta_{9} - 1240 \beta_{8} - 1240 \beta_{3}$$$$)/3$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
293.1
 −0.475594 + 1.66548i 1.24461 + 1.20455i −1.24461 − 1.20455i 0.475594 − 1.66548i −1.20455 + 1.24461i −1.66548 − 0.475594i 1.66548 + 0.475594i 1.20455 − 1.24461i −0.475594 − 1.66548i 1.24461 − 1.20455i −1.24461 + 1.20455i 0.475594 + 1.66548i −1.20455 − 1.24461i −1.66548 + 0.475594i 1.66548 − 0.475594i 1.20455 + 1.24461i
−1.22474 + 0.707107i −1.68014 0.420861i 1.00000 1.73205i 3.32070 + 1.91721i 2.35534 0.672592i 1.32288 + 2.29129i 2.82843i 2.64575 + 1.41421i −5.42268
293.2 −1.22474 + 0.707107i −0.420861 1.68014i 1.00000 1.73205i −3.87242 2.23574i 1.70349 + 1.76015i −1.32288 2.29129i 2.82843i −2.64575 + 1.41421i 6.32364
293.3 −1.22474 + 0.707107i 0.420861 + 1.68014i 1.00000 1.73205i 3.87242 + 2.23574i −1.70349 1.76015i −1.32288 2.29129i 2.82843i −2.64575 + 1.41421i −6.32364
293.4 −1.22474 + 0.707107i 1.68014 + 0.420861i 1.00000 1.73205i −3.32070 1.91721i −2.35534 + 0.672592i 1.32288 + 2.29129i 2.82843i 2.64575 + 1.41421i 5.42268
293.5 1.22474 0.707107i −1.68014 + 0.420861i 1.00000 1.73205i −0.0658376 0.0380114i −1.76015 + 1.70349i 1.32288 + 2.29129i 2.82843i 2.64575 1.41421i −0.107512
293.6 1.22474 0.707107i −0.420861 + 1.68014i 1.00000 1.73205i −1.99323 1.15079i 0.672592 + 2.35534i −1.32288 2.29129i 2.82843i −2.64575 1.41421i −3.25493
293.7 1.22474 0.707107i 0.420861 1.68014i 1.00000 1.73205i 1.99323 + 1.15079i −0.672592 2.35534i −1.32288 2.29129i 2.82843i −2.64575 1.41421i 3.25493
293.8 1.22474 0.707107i 1.68014 0.420861i 1.00000 1.73205i 0.0658376 + 0.0380114i 1.76015 1.70349i 1.32288 + 2.29129i 2.82843i 2.64575 1.41421i 0.107512
461.1 −1.22474 0.707107i −1.68014 + 0.420861i 1.00000 + 1.73205i 3.32070 1.91721i 2.35534 + 0.672592i 1.32288 2.29129i 2.82843i 2.64575 1.41421i −5.42268
461.2 −1.22474 0.707107i −0.420861 + 1.68014i 1.00000 + 1.73205i −3.87242 + 2.23574i 1.70349 1.76015i −1.32288 + 2.29129i 2.82843i −2.64575 1.41421i 6.32364
461.3 −1.22474 0.707107i 0.420861 1.68014i 1.00000 + 1.73205i 3.87242 2.23574i −1.70349 + 1.76015i −1.32288 + 2.29129i 2.82843i −2.64575 1.41421i −6.32364
461.4 −1.22474 0.707107i 1.68014 0.420861i 1.00000 + 1.73205i −3.32070 + 1.91721i −2.35534 0.672592i 1.32288 2.29129i 2.82843i 2.64575 1.41421i 5.42268
461.5 1.22474 + 0.707107i −1.68014 0.420861i 1.00000 + 1.73205i −0.0658376 + 0.0380114i −1.76015 1.70349i 1.32288 2.29129i 2.82843i 2.64575 + 1.41421i −0.107512
461.6 1.22474 + 0.707107i −0.420861 1.68014i 1.00000 + 1.73205i −1.99323 + 1.15079i 0.672592 2.35534i −1.32288 + 2.29129i 2.82843i −2.64575 + 1.41421i −3.25493
461.7 1.22474 + 0.707107i 0.420861 + 1.68014i 1.00000 + 1.73205i 1.99323 1.15079i −0.672592 + 2.35534i −1.32288 + 2.29129i 2.82843i −2.64575 + 1.41421i 3.25493
461.8 1.22474 + 0.707107i 1.68014 + 0.420861i 1.00000 + 1.73205i 0.0658376 0.0380114i 1.76015 + 1.70349i 1.32288 2.29129i 2.82843i 2.64575 + 1.41421i 0.107512
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 461.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
8.b even 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner
72.j odd 6 1 inner
504.cc 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.cc.a 16
7.b odd 2 1 inner 504.2.cc.a 16
8.b even 2 1 inner 504.2.cc.a 16
9.d odd 6 1 inner 504.2.cc.a 16
56.h odd 2 1 CM 504.2.cc.a 16
63.o even 6 1 inner 504.2.cc.a 16
72.j odd 6 1 inner 504.2.cc.a 16
504.cc even 6 1 inner 504.2.cc.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cc.a 16 1.a even 1 1 trivial
504.2.cc.a 16 7.b odd 2 1 inner
504.2.cc.a 16 8.b even 2 1 inner
504.2.cc.a 16 9.d odd 6 1 inner
504.2.cc.a 16 56.h odd 2 1 CM
504.2.cc.a 16 63.o even 6 1 inner
504.2.cc.a 16 72.j odd 6 1 inner
504.2.cc.a 16 504.cc even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{4}$$
$3$ $$( 1 - 10 T^{4} + 81 T^{8} )^{2}$$
$5$ $$( 1 + 22 T^{4} + 625 T^{8} )^{2}( 1 - 22 T^{4} - 141 T^{8} - 13750 T^{12} + 390625 T^{16} )$$
$7$ $$( 1 + 7 T^{2} + 49 T^{4} )^{4}$$
$11$ $$( 1 - 11 T^{2} + 121 T^{4} )^{8}$$
$13$ $$( 1 - 310 T^{4} + 67539 T^{8} - 8853910 T^{12} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{16}$$
$19$ $$( 1 + 650 T^{4} + 292179 T^{8} + 84708650 T^{12} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 6 T + 23 T^{2} )^{8}( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 29 T^{2} + 841 T^{4} )^{8}$$
$31$ $$( 1 + 31 T^{2} + 961 T^{4} )^{8}$$
$37$ $$( 1 - 37 T^{2} )^{16}$$
$41$ $$( 1 - 41 T^{2} + 1681 T^{4} )^{8}$$
$43$ $$( 1 + 43 T^{2} + 1849 T^{4} )^{8}$$
$47$ $$( 1 - 47 T^{2} + 2209 T^{4} )^{8}$$
$53$ $$( 1 + 53 T^{2} )^{16}$$
$59$ $$( 1 + 1130 T^{4} - 10840461 T^{8} + 13692617930 T^{12} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 7370 T^{4} + 13845841 T^{8} )^{2}( 1 + 7370 T^{4} + 40471059 T^{8} + 102043848170 T^{12} + 191707312997281 T^{16} )$$
$67$ $$( 1 + 67 T^{2} + 4489 T^{4} )^{8}$$
$71$ $$( 1 + 110 T^{2} + 7059 T^{4} + 554510 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 73 T^{2} )^{16}$$
$79$ $$( 1 + 130 T^{2} + 6241 T^{4} )^{4}( 1 - 130 T^{2} + 10659 T^{4} - 811330 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 13130 T^{4} + 124938579 T^{8} + 623127754730 T^{12} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{16}$$
$97$ $$( 1 + 97 T^{2} + 9409 T^{4} )^{8}$$
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