# Properties

 Label 504.2.p.g Level 504 Weight 2 Character orbit 504.p Analytic conductor 4.024 Analytic rank 0 Dimension 16 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.p (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + \beta_{3} q^{4} + \beta_{5} q^{5} + \beta_{8} q^{7} + ( 1 + \beta_{2} + \beta_{6} + \beta_{8} ) q^{8} +O(q^{10})$$ $$q + \beta_{6} q^{2} + \beta_{3} q^{4} + \beta_{5} q^{5} + \beta_{8} q^{7} + ( 1 + \beta_{2} + \beta_{6} + \beta_{8} ) q^{8} + ( \beta_{5} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{10} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{11} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{10} ) q^{13} + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} ) q^{14} + ( 1 + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{16} + \beta_{14} q^{17} + ( \beta_{1} + \beta_{10} ) q^{19} + ( -\beta_{2} - \beta_{4} + \beta_{10} - \beta_{12} - \beta_{13} ) q^{20} + ( 1 + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{22} + ( \beta_{3} - \beta_{6} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{23} + ( 1 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{25} + ( -\beta_{10} + \beta_{11} + \beta_{13} ) q^{26} + ( -1 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{28} + ( -\beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{29} + ( -\beta_{2} - \beta_{4} + \beta_{5} - \beta_{13} - \beta_{15} ) q^{31} + ( 1 + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{32} + ( -\beta_{1} - \beta_{5} - \beta_{10} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{34} + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{35} + ( -\beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{37} + ( \beta_{2} + \beta_{4} - \beta_{10} - \beta_{12} - \beta_{13} ) q^{38} + ( -\beta_{1} - 2 \beta_{5} - \beta_{11} + 2 \beta_{12} - \beta_{14} - \beta_{15} ) q^{40} + ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{14} ) q^{41} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{43} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{11} - \beta_{12} ) q^{44} + ( 3 - 3 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{46} + ( 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{10} ) q^{47} + ( -1 + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{11} + \beta_{13} - \beta_{15} ) q^{49} + ( 2 - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{11} + \beta_{12} ) q^{50} + ( \beta_{1} - 2 \beta_{5} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{52} + ( -\beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{53} + ( 3 \beta_{5} + \beta_{13} + \beta_{15} ) q^{55} + ( -3 + \beta_{2} + 2 \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{56} + ( 2 - 2 \beta_{4} - 2 \beta_{8} ) q^{58} + ( \beta_{1} + \beta_{5} + \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{15} ) q^{59} + ( -\beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{10} ) q^{61} + ( -3 \beta_{1} - 2 \beta_{5} + \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{62} + ( 1 + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} + 5 \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{64} + ( -2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{65} + ( -4 - \beta_{2} + \beta_{4} - 4 \beta_{6} - 2 \beta_{9} ) q^{67} + ( -3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{68} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{70} + ( -\beta_{3} + \beta_{6} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{71} -2 \beta_{14} q^{73} + ( 2 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{11} + \beta_{12} ) q^{74} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{76} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{77} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{11} - 2 \beta_{12} ) q^{79} + ( -\beta_{1} + 2 \beta_{5} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{80} + ( \beta_{1} - \beta_{5} + \beta_{10} - 3 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{82} + ( -\beta_{1} - 3 \beta_{5} - \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{83} + ( -3 \beta_{2} - 3 \beta_{4} - 2 \beta_{6} - \beta_{7} - 6 \beta_{8} + \beta_{9} + 2 \beta_{11} + 2 \beta_{12} ) q^{85} + ( 2 + \beta_{2} + \beta_{4} - 2 \beta_{7} + 2 \beta_{8} - \beta_{11} - \beta_{12} ) q^{86} + ( -6 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} ) q^{88} + ( -2 \beta_{1} - 2 \beta_{5} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{89} + ( -4 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} ) q^{91} + ( -2 - \beta_{2} + 2 \beta_{3} - \beta_{4} + 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{12} ) q^{92} + ( 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{11} + 2 \beta_{13} ) q^{94} + ( 2 \beta_{11} + 2 \beta_{12} ) q^{95} + ( 2 \beta_{5} - 4 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{97} + ( -4 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 2q^{2} + 2q^{4} + 10q^{8} + O(q^{10})$$ $$16q - 2q^{2} + 2q^{4} + 10q^{8} + 8q^{11} + 14q^{14} + 18q^{16} + 8q^{22} + 16q^{25} - 10q^{28} + 18q^{32} - 24q^{35} - 8q^{43} + 52q^{46} - 8q^{49} + 34q^{50} - 50q^{56} + 24q^{58} + 2q^{64} - 40q^{67} - 24q^{70} + 32q^{74} + 32q^{86} - 88q^{88} - 56q^{91} - 44q^{92} - 66q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + x^{14} - 4 x^{12} - 4 x^{10} + 16 x^{8} - 16 x^{6} - 64 x^{4} + 64 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - \nu^{5} - 2 \nu^{3}$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{13} + 4 \nu^{11} - 4 \nu^{9} - 8 \nu^{7} - 32 \nu^{6} + 64 \nu^{5} - 32 \nu^{4} + 32 \nu^{3} - 128 \nu$$$$)/128$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} + \nu^{12} - 4 \nu^{10} - 4 \nu^{8} + 16 \nu^{6} - 16 \nu^{4} - 64 \nu^{2} + 64$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{13} + 4 \nu^{11} - 4 \nu^{9} - 8 \nu^{7} + 32 \nu^{6} + 64 \nu^{5} + 32 \nu^{4} + 32 \nu^{3} - 128 \nu$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{9} + 2 \nu^{7} + 32 \nu$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{14} - \nu^{12} - 2 \nu^{10} + 8 \nu^{8} + 8 \nu^{6} - 32 \nu^{4} + 64$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{14} + \nu^{12} + 6 \nu^{10} - 4 \nu^{8} - 8 \nu^{6} + 32 \nu^{4} + 64 \nu^{2} - 128$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{15} + \nu^{13} - 4 \nu^{12} - 4 \nu^{11} + 4 \nu^{10} + 4 \nu^{9} - 8 \nu^{8} + 8 \nu^{7} - 16 \nu^{6} - 64 \nu^{5} - 32 \nu^{3} + 128 \nu^{2} + 128 \nu - 128$$$$)/128$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{14} - \nu^{12} - 6 \nu^{10} + 4 \nu^{8} + 8 \nu^{6} - 32 \nu^{4} + 64 \nu^{2} + 128$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{13} + 2 \nu^{11} + 32 \nu^{5} - 64 \nu^{3}$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{15} + \nu^{13} + 4 \nu^{11} + 8 \nu^{10} - 4 \nu^{9} - 8 \nu^{8} - 8 \nu^{7} - 16 \nu^{6} + 64 \nu^{4} + 32 \nu^{3} + 64 \nu^{2} - 128 \nu - 256$$$$)/128$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{13} - 4 \nu^{11} + 8 \nu^{10} + 4 \nu^{9} - 8 \nu^{8} + 8 \nu^{7} - 16 \nu^{6} + 64 \nu^{4} - 32 \nu^{3} + 64 \nu^{2} + 128 \nu - 256$$$$)/128$$ $$\beta_{13}$$ $$=$$ $$($$$$-3 \nu^{15} + \nu^{13} + 16 \nu^{11} - 8 \nu^{10} - 12 \nu^{9} + 8 \nu^{8} - 24 \nu^{7} + 16 \nu^{6} + 64 \nu^{5} - 64 \nu^{4} + 96 \nu^{3} - 64 \nu^{2} - 128 \nu + 256$$$$)/128$$ $$\beta_{14}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{13} + 4 \nu^{11} - 2 \nu^{9} - 4 \nu^{7} + 32 \nu^{5} - 128 \nu$$$$)/32$$ $$\beta_{15}$$ $$=$$ $$($$$$\nu^{15} - 7 \nu^{13} + 4 \nu^{11} + 8 \nu^{10} + 20 \nu^{9} - 8 \nu^{8} - 24 \nu^{7} - 16 \nu^{6} - 96 \nu^{5} + 64 \nu^{4} + 224 \nu^{3} + 64 \nu^{2} + 128 \nu - 256$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + \beta_{5}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9} + \beta_{7}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{14} + \beta_{13} + \beta_{11} - 3 \beta_{10} + 3 \beta_{5}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{12} + 2 \beta_{11} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{14} - \beta_{13} - 4 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 5 \beta_{5} + 4 \beta_{4} + 4 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{12} - 2 \beta_{11} - \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + 4 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$\beta_{14} + \beta_{13} - 4 \beta_{12} + 5 \beta_{11} - 3 \beta_{10} + 11 \beta_{5} + 4 \beta_{4} + 4 \beta_{2} + 16 \beta_{1}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$2 \beta_{12} + 2 \beta_{11} + \beta_{9} - 10 \beta_{8} - 3 \beta_{7} + 6 \beta_{6} - 4 \beta_{4} - 10 \beta_{3} - 6 \beta_{2} - 6$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$16 \beta_{15} + 15 \beta_{14} - 17 \beta_{13} - 28 \beta_{12} - 5 \beta_{11} + 19 \beta_{10} + 21 \beta_{5} - 4 \beta_{4} - 4 \beta_{2}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-2 \beta_{12} - 2 \beta_{11} - 17 \beta_{9} + 10 \beta_{8} + 19 \beta_{7} + 26 \beta_{6} + 4 \beta_{4} + 10 \beta_{3} + 6 \beta_{2} + 38$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$16 \beta_{15} - 15 \beta_{14} + 17 \beta_{13} - 36 \beta_{12} + 37 \beta_{11} + 45 \beta_{10} + 11 \beta_{5} + 4 \beta_{4} + 4 \beta_{2} + 32 \beta_{1}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$2 \beta_{12} + 2 \beta_{11} + 17 \beta_{9} - 42 \beta_{8} + 45 \beta_{7} + 6 \beta_{6} - 36 \beta_{4} + 22 \beta_{3} - 6 \beta_{2} - 6$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-16 \beta_{15} + 47 \beta_{14} + 15 \beta_{13} - 28 \beta_{12} + 59 \beta_{11} - 13 \beta_{10} + 85 \beta_{5} - 68 \beta_{4} - 68 \beta_{2} - 32 \beta_{1}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$62 \beta_{12} + 62 \beta_{11} + 15 \beta_{9} - 22 \beta_{8} - 13 \beta_{7} + 58 \beta_{6} - 28 \beta_{4} + 42 \beta_{3} + 6 \beta_{2} + 70$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$16 \beta_{15} - 79 \beta_{14} - 47 \beta_{13} - 228 \beta_{12} + 165 \beta_{11} + 109 \beta_{10} + 75 \beta_{5} + 68 \beta_{4} + 68 \beta_{2} + 32 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 0.474920 + 1.33209i −0.474920 − 1.33209i 0.474920 − 1.33209i −0.474920 + 1.33209i 1.20933 + 0.733159i −1.20933 − 0.733159i 1.20933 − 0.733159i −1.20933 + 0.733159i −1.40199 + 0.185533i 1.40199 − 0.185533i −1.40199 − 0.185533i 1.40199 + 0.185533i −0.310478 + 1.37971i 0.310478 − 1.37971i −0.310478 − 1.37971i 0.310478 + 1.37971i
−1.33209 0.474920i 0 1.54890 + 1.26527i −1.58069 0 −2.37995 1.15578i −1.46237 2.42105i 0 2.10562 + 0.750703i
307.2 −1.33209 0.474920i 0 1.54890 + 1.26527i 1.58069 0 2.37995 1.15578i −1.46237 2.42105i 0 −2.10562 0.750703i
307.3 −1.33209 + 0.474920i 0 1.54890 1.26527i −1.58069 0 −2.37995 + 1.15578i −1.46237 + 2.42105i 0 2.10562 0.750703i
307.4 −1.33209 + 0.474920i 0 1.54890 1.26527i 1.58069 0 2.37995 + 1.15578i −1.46237 + 2.42105i 0 −2.10562 + 0.750703i
307.5 −0.733159 1.20933i 0 −0.924955 + 1.77326i −1.12786 0 2.11337 + 1.59175i 2.82260 0.181508i 0 0.826905 + 1.36396i
307.6 −0.733159 1.20933i 0 −0.924955 + 1.77326i 1.12786 0 −2.11337 + 1.59175i 2.82260 0.181508i 0 −0.826905 1.36396i
307.7 −0.733159 + 1.20933i 0 −0.924955 1.77326i −1.12786 0 2.11337 1.59175i 2.82260 + 0.181508i 0 0.826905 1.36396i
307.8 −0.733159 + 1.20933i 0 −0.924955 1.77326i 1.12786 0 −2.11337 1.59175i 2.82260 + 0.181508i 0 −0.826905 + 1.36396i
307.9 0.185533 1.40199i 0 −1.93115 0.520231i −3.84444 0 1.62140 + 2.09071i −1.08765 + 2.61094i 0 −0.713272 + 5.38987i
307.10 0.185533 1.40199i 0 −1.93115 0.520231i 3.84444 0 −1.62140 + 2.09071i −1.08765 + 2.61094i 0 0.713272 5.38987i
307.11 0.185533 + 1.40199i 0 −1.93115 + 0.520231i −3.84444 0 1.62140 2.09071i −1.08765 2.61094i 0 −0.713272 5.38987i
307.12 0.185533 + 1.40199i 0 −1.93115 + 0.520231i 3.84444 0 −1.62140 2.09071i −1.08765 2.61094i 0 0.713272 + 5.38987i
307.13 1.37971 0.310478i 0 1.80721 0.856739i −2.33443 0 0.490487 2.59989i 2.22743 1.74315i 0 −3.22084 + 0.724789i
307.14 1.37971 0.310478i 0 1.80721 0.856739i 2.33443 0 −0.490487 2.59989i 2.22743 1.74315i 0 3.22084 0.724789i
307.15 1.37971 + 0.310478i 0 1.80721 + 0.856739i −2.33443 0 0.490487 + 2.59989i 2.22743 + 1.74315i 0 −3.22084 0.724789i
307.16 1.37971 + 0.310478i 0 1.80721 + 0.856739i 2.33443 0 −0.490487 + 2.59989i 2.22743 + 1.74315i 0 3.22084 + 0.724789i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.16 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.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.p.g 16
3.b odd 2 1 168.2.p.a 16
4.b odd 2 1 2016.2.p.g 16
7.b odd 2 1 inner 504.2.p.g 16
8.b even 2 1 2016.2.p.g 16
8.d odd 2 1 inner 504.2.p.g 16
12.b even 2 1 672.2.p.a 16
21.c even 2 1 168.2.p.a 16
24.f even 2 1 168.2.p.a 16
24.h odd 2 1 672.2.p.a 16
28.d even 2 1 2016.2.p.g 16
56.e even 2 1 inner 504.2.p.g 16
56.h odd 2 1 2016.2.p.g 16
84.h odd 2 1 672.2.p.a 16
168.e odd 2 1 168.2.p.a 16
168.i even 2 1 672.2.p.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.p.a 16 3.b odd 2 1
168.2.p.a 16 21.c even 2 1
168.2.p.a 16 24.f even 2 1
168.2.p.a 16 168.e odd 2 1
504.2.p.g 16 1.a even 1 1 trivial
504.2.p.g 16 7.b odd 2 1 inner
504.2.p.g 16 8.d odd 2 1 inner
504.2.p.g 16 56.e even 2 1 inner
672.2.p.a 16 12.b even 2 1
672.2.p.a 16 24.h odd 2 1
672.2.p.a 16 84.h odd 2 1
672.2.p.a 16 168.i even 2 1
2016.2.p.g 16 4.b odd 2 1
2016.2.p.g 16 8.b even 2 1
2016.2.p.g 16 28.d even 2 1
2016.2.p.g 16 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}^{8} - 24 T_{5}^{6} + 160 T_{5}^{4} - 368 T_{5}^{2} + 256$$ $$T_{11}^{4} - 2 T_{11}^{3} - 24 T_{11}^{2} + 60 T_{11} - 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T - 2 T^{3} - 4 T^{4} - 4 T^{5} + 8 T^{7} + 16 T^{8} )^{2}$$
$3$ 1
$5$ $$( 1 + 16 T^{2} + 140 T^{4} + 832 T^{6} + 4326 T^{8} + 20800 T^{10} + 87500 T^{12} + 250000 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$1 + 4 T^{2} + 68 T^{4} + 572 T^{6} + 3382 T^{8} + 28028 T^{10} + 163268 T^{12} + 470596 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 - 2 T + 20 T^{2} - 6 T^{3} + 182 T^{4} - 66 T^{5} + 2420 T^{6} - 2662 T^{7} + 14641 T^{8} )^{4}$$
$13$ $$( 1 + 36 T^{2} + 692 T^{4} + 12604 T^{6} + 195894 T^{8} + 2130076 T^{10} + 19764212 T^{12} + 173765124 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 - 72 T^{2} + 2652 T^{4} - 66024 T^{6} + 1253414 T^{8} - 19080936 T^{10} + 221497692 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 56 T^{2} + 1532 T^{4} - 36808 T^{6} + 804262 T^{8} - 13287688 T^{10} + 199651772 T^{12} - 2634569336 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 88 T^{2} + 3548 T^{4} - 96952 T^{6} + 2287782 T^{8} - 51287608 T^{10} + 992875868 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 164 T^{2} + 12980 T^{4} - 650812 T^{6} + 22500790 T^{8} - 547332892 T^{10} + 9180507380 T^{12} - 97551024644 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 + 68 T^{2} + 5252 T^{4} + 204988 T^{6} + 8264182 T^{8} + 196993468 T^{10} + 4850332292 T^{12} + 60350250308 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 232 T^{2} + 25148 T^{4} - 1670040 T^{6} + 74486502 T^{8} - 2286284760 T^{10} + 47131400828 T^{12} - 595248526888 T^{14} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 120 T^{2} + 10204 T^{4} - 627512 T^{6} + 28679654 T^{8} - 1054847672 T^{10} + 28834065244 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + 2 T + 120 T^{2} + 250 T^{3} + 6782 T^{4} + 10750 T^{5} + 221880 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 120 T^{2} + 10396 T^{4} + 591944 T^{6} + 31424966 T^{8} + 1307604296 T^{10} + 50729163676 T^{12} + 1293505839480 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 260 T^{2} + 33908 T^{4} - 2877468 T^{6} + 176791542 T^{8} - 8082807612 T^{10} + 267550429748 T^{12} - 5762733893540 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 - 264 T^{2} + 34076 T^{4} - 3023352 T^{6} + 203740198 T^{8} - 10524288312 T^{10} + 412911193436 T^{12} - 11135660881224 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 + 324 T^{2} + 46708 T^{4} + 4167580 T^{6} + 280236854 T^{8} + 15507565180 T^{10} + 646711541428 T^{12} + 16692601292964 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 10 T + 208 T^{2} + 1122 T^{3} + 16622 T^{4} + 75174 T^{5} + 933712 T^{6} + 3007630 T^{7} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 456 T^{2} + 94300 T^{4} - 11817416 T^{6} + 1002933542 T^{8} - 59571594056 T^{10} + 2396321518300 T^{12} - 58413729467976 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 328 T^{2} + 54492 T^{4} - 6043256 T^{6} + 500200454 T^{8} - 32204511224 T^{10} + 1547476948572 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 444 T^{2} + 95492 T^{4} - 12990660 T^{6} + 1221053302 T^{8} - 81074709060 T^{10} + 3719421134852 T^{12} - 107930830251324 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 - 120 T^{2} + 20284 T^{4} - 1531784 T^{6} + 164981030 T^{8} - 10552459976 T^{10} + 962644583164 T^{12} - 39232844804280 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 312 T^{2} + 51484 T^{4} - 5696504 T^{6} + 530357222 T^{8} - 45122008184 T^{10} + 3230221535644 T^{12} - 155058162779832 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 232 T^{2} + 17244 T^{4} + 409768 T^{6} - 157308730 T^{8} + 3855507112 T^{10} + 1526598921564 T^{12} - 193249505143528 T^{14} + 7837433594376961 T^{16} )^{2}$$