# Properties

 Label 600.2.m.d Level 600 Weight 2 Character orbit 600.m Analytic conductor 4.791 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 600.m (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{11}$$ Twist minimal: no (minimal twist has level 120) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{6} q^{3} + \beta_{2} q^{4} -\beta_{14} q^{6} + ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{10} ) q^{7} + \beta_{3} q^{8} + ( -\beta_{2} + \beta_{8} + \beta_{14} - \beta_{15} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{6} q^{3} + \beta_{2} q^{4} -\beta_{14} q^{6} + ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{10} ) q^{7} + \beta_{3} q^{8} + ( -\beta_{2} + \beta_{8} + \beta_{14} - \beta_{15} ) q^{9} + ( -\beta_{2} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{11} + ( -\beta_{1} - \beta_{3} - \beta_{7} + \beta_{10} + \beta_{13} ) q^{12} + ( \beta_{1} - \beta_{5} + \beta_{7} + \beta_{13} ) q^{13} + ( 1 + 2 \beta_{8} - \beta_{9} - \beta_{11} + 2 \beta_{12} + \beta_{14} - \beta_{15} ) q^{14} + ( -1 - \beta_{2} + \beta_{8} + 2 \beta_{12} + \beta_{14} ) q^{16} + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{10} ) q^{17} + ( \beta_{1} - \beta_{5} + 2 \beta_{7} - \beta_{10} ) q^{18} + ( \beta_{2} - 2 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - 2 \beta_{14} ) q^{19} + ( -1 - \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{21} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{22} + ( \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{10} ) q^{23} + ( 1 + \beta_{2} - 3 \beta_{8} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{24} + ( 2 + 2 \beta_{2} - 2 \beta_{8} ) q^{26} + ( 2 \beta_{1} - \beta_{4} - \beta_{7} - 2 \beta_{13} ) q^{27} + ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{10} + \beta_{13} ) q^{28} + ( -3 \beta_{2} + \beta_{8} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{29} + ( -\beta_{2} + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{15} ) q^{31} + ( \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} ) q^{32} + ( 4 \beta_{1} + \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{13} ) q^{33} + ( 2 \beta_{8} - \beta_{9} + 2 \beta_{12} - \beta_{15} ) q^{34} + ( 2 + \beta_{2} - \beta_{9} - 2 \beta_{11} - \beta_{15} ) q^{36} + ( -\beta_{1} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{13} ) q^{37} + ( -\beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{10} + \beta_{13} ) q^{38} + ( -2 + 2 \beta_{2} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{39} + ( -\beta_{2} + \beta_{8} - \beta_{12} - \beta_{15} ) q^{41} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{13} ) q^{42} + ( -\beta_{6} - \beta_{7} ) q^{43} + ( 4 - 2 \beta_{8} - 2 \beta_{12} + 2 \beta_{15} ) q^{44} + ( -1 - \beta_{11} + 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{46} + ( -5 \beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{10} + 2 \beta_{13} ) q^{47} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{10} - \beta_{13} ) q^{48} + ( 3 - \beta_{2} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{49} + ( 2 - \beta_{2} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{51} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{52} + ( -\beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -5 + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{15} ) q^{54} + ( 4 - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{15} ) q^{56} + ( \beta_{1} - \beta_{4} + \beta_{5} + 3 \beta_{7} - 2 \beta_{10} + \beta_{13} ) q^{57} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} - 3 \beta_{13} ) q^{58} + ( \beta_{2} - \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{59} + ( -\beta_{2} - \beta_{8} - 3 \beta_{12} - \beta_{15} ) q^{61} + ( 2 \beta_{1} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{13} ) q^{62} + ( -\beta_{1} - \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{10} ) q^{63} + ( -3 - \beta_{2} - \beta_{8} - 2 \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{64} + ( -4 + 2 \beta_{2} - \beta_{9} - 2 \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{66} + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{10} + 2 \beta_{13} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{13} ) q^{68} + ( -3 + \beta_{2} - 2 \beta_{8} + 3 \beta_{11} - 2 \beta_{12} - \beta_{14} - \beta_{15} ) q^{69} + ( -4 - 2 \beta_{2} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{71} + ( \beta_{3} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{13} ) q^{72} + ( -6 \beta_{1} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{13} ) q^{73} + ( -4 - 2 \beta_{2} - 2 \beta_{8} - 2 \beta_{11} + 2 \beta_{14} ) q^{74} + ( -2 + 2 \beta_{2} - 4 \beta_{8} + 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{76} + ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 2 \beta_{10} - 2 \beta_{13} ) q^{77} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{13} ) q^{78} + ( \beta_{2} - 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{79} + ( 1 - 3 \beta_{2} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{12} + 3 \beta_{15} ) q^{81} + ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{13} ) q^{82} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{10} ) q^{83} + ( -8 + 2 \beta_{2} - 4 \beta_{8} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{84} + ( -1 + \beta_{11} + \beta_{14} ) q^{86} + ( 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{13} ) q^{87} + ( 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{13} ) q^{88} + ( -2 \beta_{2} + 2 \beta_{8} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{89} + ( 4 + 2 \beta_{2} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{91} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} + 3 \beta_{10} - \beta_{13} ) q^{92} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{13} ) q^{93} + ( 5 - 4 \beta_{2} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{14} + 4 \beta_{15} ) q^{94} + ( -5 + \beta_{2} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{14} ) q^{96} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} ) q^{97} + ( \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - \beta_{10} - \beta_{13} ) q^{98} + ( -2 + \beta_{2} + \beta_{8} + \beta_{9} + 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 2q^{4} + 2q^{6} + O(q^{10})$$ $$16q - 2q^{4} + 2q^{6} + 12q^{14} - 14q^{16} + 8q^{19} - 8q^{21} + 22q^{24} + 32q^{26} + 26q^{36} - 32q^{39} + 60q^{44} - 16q^{46} + 32q^{49} + 40q^{51} - 82q^{54} + 60q^{56} - 50q^{64} - 68q^{66} - 40q^{69} - 48q^{71} - 64q^{74} - 24q^{76} + 16q^{81} - 116q^{84} - 16q^{86} + 48q^{91} + 80q^{94} - 86q^{96} - 32q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + x^{14} + 4 x^{12} + 12 x^{10} + 16 x^{8} + 48 x^{6} + 64 x^{4} + 64 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} + 3 \nu^{13} + 10 \nu^{11} + 8 \nu^{9} + 8 \nu^{7} + 64 \nu^{5} + 64 \nu$$$$)/192$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{9} + \nu^{7} - 2 \nu^{5} - 4 \nu^{3} + 8 \nu$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} - 3 \nu^{13} + 2 \nu^{11} - 8 \nu^{9} + 4 \nu^{7} + 8 \nu^{5} - 48 \nu^{3} + 32 \nu$$$$)/192$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} - 3 \nu^{13} - 10 \nu^{11} - 20 \nu^{9} - 44 \nu^{7} - 40 \nu^{5} - 144 \nu^{3} - 160 \nu$$$$)/192$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{10} + \nu^{8} - 2 \nu^{6} - 4 \nu^{4} + 8 \nu^{2} - 16$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{12} + 4 \nu^{10} + 32 \nu^{8} + 32 \nu^{6} + 112 \nu^{4} + 192 \nu^{2} + 256$$$$)/192$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{15} + 3 \nu^{13} - 4 \nu^{11} + 4 \nu^{9} + 28 \nu^{7} + 56 \nu^{5} + 144 \nu^{3} + 224 \nu$$$$)/192$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{14} + 3 \nu^{12} + 4 \nu^{10} + 14 \nu^{8} - 4 \nu^{6} + 40 \nu^{4} + 48 \nu^{2} - 32$$$$)/96$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{14} + 3 \nu^{12} + 8 \nu^{10} + 4 \nu^{8} + 40 \nu^{6} + 128 \nu^{4} + 96 \nu^{2} + 320$$$$)/192$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{11} - 2 \nu^{9} - 8 \nu^{7} - 40 \nu^{5} + 32 \nu$$$$)/96$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{12} - 2 \nu^{10} - 10 \nu^{8} - 28 \nu^{6} - 8 \nu^{4} - 48 \nu^{2} - 128$$$$)/96$$ $$\beta_{15}$$ $$=$$ $$($$$$\nu^{14} + \nu^{12} + 8 \nu^{8} + 16 \nu^{6} + 16 \nu^{4} + 64 \nu^{2}$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$ $$\nu^{4}$$ $$=$$ $$\beta_{14} + 2 \beta_{12} + \beta_{8} - \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$-\beta_{13} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{15} - \beta_{14} - 2 \beta_{11} - \beta_{8} - \beta_{2} - 3$$ $$\nu^{7}$$ $$=$$ $$-\beta_{13} + \beta_{10} - 3 \beta_{7} + \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 4 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-2 \beta_{15} - 3 \beta_{14} - 4 \beta_{12} + 2 \beta_{11} + 4 \beta_{9} + \beta_{8} - 3 \beta_{2} - 1$$ $$\nu^{9}$$ $$=$$ $$\beta_{13} - \beta_{10} - 5 \beta_{7} - \beta_{6} - 11 \beta_{5} - 5 \beta_{4} - 6 \beta_{3} + 4 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-6 \beta_{15} - 5 \beta_{14} - 12 \beta_{12} + 6 \beta_{11} + 4 \beta_{9} - 17 \beta_{8} + 11 \beta_{2} - 7$$ $$\nu^{11}$$ $$=$$ $$7 \beta_{13} - 7 \beta_{10} - 3 \beta_{7} + 9 \beta_{6} + 3 \beta_{5} + 13 \beta_{4} + 6 \beta_{3} - 4 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$6 \beta_{15} - 3 \beta_{14} + 12 \beta_{12} + 10 \beta_{11} - 20 \beta_{9} + 9 \beta_{8} - 3 \beta_{2} + 15$$ $$\nu^{13}$$ $$=$$ $$17 \beta_{13} + 15 \beta_{10} + 11 \beta_{7} - 33 \beta_{6} + 21 \beta_{5} + 27 \beta_{4} - 6 \beta_{3} - 28 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$42 \beta_{15} + 27 \beta_{14} - 12 \beta_{12} + 6 \beta_{11} - 12 \beta_{9} - 17 \beta_{8} - 5 \beta_{2} + 57$$ $$\nu^{15}$$ $$=$$ $$-57 \beta_{13} - 39 \beta_{10} - 3 \beta_{7} - 55 \beta_{6} + 35 \beta_{5} - 19 \beta_{4} + 22 \beta_{3} + 60 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 −1.29041 − 0.578647i −1.29041 + 0.578647i −1.15595 − 0.814732i −1.15595 + 0.814732i −0.842022 − 1.13622i −0.842022 + 1.13622i −0.199044 − 1.40014i −0.199044 + 1.40014i 0.199044 − 1.40014i 0.199044 + 1.40014i 0.842022 − 1.13622i 0.842022 + 1.13622i 1.15595 − 0.814732i 1.15595 + 0.814732i 1.29041 − 0.578647i 1.29041 + 0.578647i
−1.29041 0.578647i 1.56044 0.751690i 1.33034 + 1.49339i 0 −2.44857 + 0.0670494i −4.28591 −0.852541 2.69688i 1.86993 2.34593i 0
299.2 −1.29041 + 0.578647i 1.56044 + 0.751690i 1.33034 1.49339i 0 −2.44857 0.0670494i −4.28591 −0.852541 + 2.69688i 1.86993 + 2.34593i 0
299.3 −1.15595 0.814732i −0.887900 + 1.48716i 0.672424 + 1.88357i 0 2.23800 0.995672i −0.797253 0.757320 2.72515i −1.42327 2.64089i 0
299.4 −1.15595 + 0.814732i −0.887900 1.48716i 0.672424 1.88357i 0 2.23800 + 0.995672i −0.797253 0.757320 + 2.72515i −1.42327 + 2.64089i 0
299.5 −0.842022 1.13622i 0.218455 + 1.71822i −0.581998 + 1.91345i 0 1.76833 1.69499i 3.64426 2.66415 0.949886i −2.90455 + 0.750707i 0
299.6 −0.842022 + 1.13622i 0.218455 1.71822i −0.581998 1.91345i 0 1.76833 + 1.69499i 3.64426 2.66415 + 0.949886i −2.90455 0.750707i 0
299.7 −0.199044 1.40014i 1.65195 0.520627i −1.92076 + 0.557378i 0 −1.05776 2.20933i 1.92736 1.16272 + 2.57839i 2.45790 1.72010i 0
299.8 −0.199044 + 1.40014i 1.65195 + 0.520627i −1.92076 0.557378i 0 −1.05776 + 2.20933i 1.92736 1.16272 2.57839i 2.45790 + 1.72010i 0
299.9 0.199044 1.40014i −1.65195 0.520627i −1.92076 0.557378i 0 −1.05776 + 2.20933i −1.92736 −1.16272 + 2.57839i 2.45790 + 1.72010i 0
299.10 0.199044 + 1.40014i −1.65195 + 0.520627i −1.92076 + 0.557378i 0 −1.05776 2.20933i −1.92736 −1.16272 2.57839i 2.45790 1.72010i 0
299.11 0.842022 1.13622i −0.218455 + 1.71822i −0.581998 1.91345i 0 1.76833 + 1.69499i −3.64426 −2.66415 0.949886i −2.90455 0.750707i 0
299.12 0.842022 + 1.13622i −0.218455 1.71822i −0.581998 + 1.91345i 0 1.76833 1.69499i −3.64426 −2.66415 + 0.949886i −2.90455 + 0.750707i 0
299.13 1.15595 0.814732i 0.887900 + 1.48716i 0.672424 1.88357i 0 2.23800 + 0.995672i 0.797253 −0.757320 2.72515i −1.42327 + 2.64089i 0
299.14 1.15595 + 0.814732i 0.887900 1.48716i 0.672424 + 1.88357i 0 2.23800 0.995672i 0.797253 −0.757320 + 2.72515i −1.42327 2.64089i 0
299.15 1.29041 0.578647i −1.56044 0.751690i 1.33034 1.49339i 0 −2.44857 0.0670494i 4.28591 0.852541 2.69688i 1.86993 + 2.34593i 0
299.16 1.29041 + 0.578647i −1.56044 + 0.751690i 1.33034 + 1.49339i 0 −2.44857 + 0.0670494i 4.28591 0.852541 + 2.69688i 1.86993 2.34593i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 299.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
24.f even 2 1 inner
120.m even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.m.d 16
3.b odd 2 1 600.2.m.c 16
4.b odd 2 1 2400.2.m.c 16
5.b even 2 1 inner 600.2.m.d 16
5.c odd 4 1 120.2.b.b yes 8
5.c odd 4 1 600.2.b.e 8
8.b even 2 1 2400.2.m.d 16
8.d odd 2 1 600.2.m.c 16
12.b even 2 1 2400.2.m.d 16
15.d odd 2 1 600.2.m.c 16
15.e even 4 1 120.2.b.a 8
15.e even 4 1 600.2.b.f 8
20.d odd 2 1 2400.2.m.c 16
20.e even 4 1 480.2.b.a 8
20.e even 4 1 2400.2.b.e 8
24.f even 2 1 inner 600.2.m.d 16
24.h odd 2 1 2400.2.m.c 16
40.e odd 2 1 600.2.m.c 16
40.f even 2 1 2400.2.m.d 16
40.i odd 4 1 480.2.b.b 8
40.i odd 4 1 2400.2.b.f 8
40.k even 4 1 120.2.b.a 8
40.k even 4 1 600.2.b.f 8
60.h even 2 1 2400.2.m.d 16
60.l odd 4 1 480.2.b.b 8
60.l odd 4 1 2400.2.b.f 8
120.i odd 2 1 2400.2.m.c 16
120.m even 2 1 inner 600.2.m.d 16
120.q odd 4 1 120.2.b.b yes 8
120.q odd 4 1 600.2.b.e 8
120.w even 4 1 480.2.b.a 8
120.w even 4 1 2400.2.b.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.b.a 8 15.e even 4 1
120.2.b.a 8 40.k even 4 1
120.2.b.b yes 8 5.c odd 4 1
120.2.b.b yes 8 120.q odd 4 1
480.2.b.a 8 20.e even 4 1
480.2.b.a 8 120.w even 4 1
480.2.b.b 8 40.i odd 4 1
480.2.b.b 8 60.l odd 4 1
600.2.b.e 8 5.c odd 4 1
600.2.b.e 8 120.q odd 4 1
600.2.b.f 8 15.e even 4 1
600.2.b.f 8 40.k even 4 1
600.2.m.c 16 3.b odd 2 1
600.2.m.c 16 8.d odd 2 1
600.2.m.c 16 15.d odd 2 1
600.2.m.c 16 40.e odd 2 1
600.2.m.d 16 1.a even 1 1 trivial
600.2.m.d 16 5.b even 2 1 inner
600.2.m.d 16 24.f even 2 1 inner
600.2.m.d 16 120.m even 2 1 inner
2400.2.b.e 8 20.e even 4 1
2400.2.b.e 8 120.w even 4 1
2400.2.b.f 8 40.i odd 4 1
2400.2.b.f 8 60.l odd 4 1
2400.2.m.c 16 4.b odd 2 1
2400.2.m.c 16 20.d odd 2 1
2400.2.m.c 16 24.h odd 2 1
2400.2.m.c 16 120.i odd 2 1
2400.2.m.d 16 8.b even 2 1
2400.2.m.d 16 12.b even 2 1
2400.2.m.d 16 40.f even 2 1
2400.2.m.d 16 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}^{8} - 36 T_{7}^{6} + 384 T_{7}^{4} - 1136 T_{7}^{2} + 576$$ $$T_{11}^{8} + 48 T_{11}^{6} + 672 T_{11}^{4} + 2560 T_{11}^{2} + 256$$ $$T_{29}^{4} - 64 T_{29}^{2} + 112 T_{29} - 48$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} + 4 T^{4} + 12 T^{6} + 16 T^{8} + 48 T^{10} + 64 T^{12} + 64 T^{14} + 256 T^{16}$$
$3$ $$1 - 4 T^{4} + 16 T^{6} + 70 T^{8} + 144 T^{10} - 324 T^{12} + 6561 T^{16}$$
$5$ 1
$7$ $$( 1 + 20 T^{2} + 244 T^{4} + 2364 T^{6} + 18678 T^{8} + 115836 T^{10} + 585844 T^{12} + 2352980 T^{14} + 5764801 T^{16} )^{2}$$
$11$ $$( 1 - 40 T^{2} + 892 T^{4} - 14424 T^{6} + 178918 T^{8} - 1745304 T^{10} + 13059772 T^{12} - 70862440 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 + 52 T^{2} + 1556 T^{4} + 31660 T^{6} + 477814 T^{8} + 5350540 T^{10} + 44440916 T^{12} + 250994068 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 84 T^{2} + 3604 T^{4} + 101164 T^{6} + 2018902 T^{8} + 29236396 T^{10} + 301009684 T^{12} + 2027555796 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 2 T + 40 T^{2} - 42 T^{3} + 766 T^{4} - 798 T^{5} + 14440 T^{6} - 13718 T^{7} + 130321 T^{8} )^{4}$$
$23$ $$( 1 - 92 T^{2} + 4420 T^{4} - 144852 T^{6} + 3702742 T^{8} - 76626708 T^{10} + 1236897220 T^{12} - 13619301788 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 + 52 T^{2} + 112 T^{3} + 1286 T^{4} + 3248 T^{5} + 43732 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 108 T^{2} + 5412 T^{4} - 170900 T^{6} + 4790966 T^{8} - 164234900 T^{10} + 4998095652 T^{12} - 95850397548 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + 68 T^{2} + 4788 T^{4} + 250460 T^{6} + 9311478 T^{8} + 342879740 T^{10} + 8973482868 T^{12} + 174469395812 T^{14} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 264 T^{2} + 32668 T^{4} - 2456248 T^{6} + 122337670 T^{8} - 4128952888 T^{10} + 92311960348 T^{12} - 1254027519624 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 320 T^{2} + 45756 T^{4} - 3816624 T^{6} + 203071110 T^{8} - 7056937776 T^{10} + 156430658556 T^{12} - 2022836175680 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$( 1 - 140 T^{2} + 13348 T^{4} - 854628 T^{6} + 45549910 T^{8} - 1887873252 T^{10} + 65133981988 T^{12} - 1509090146060 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 312 T^{2} + 45948 T^{4} - 4212040 T^{6} + 265479398 T^{8} - 11831620360 T^{10} + 362551820988 T^{12} - 6915280672248 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 - 312 T^{2} + 41500 T^{4} - 3304648 T^{6} + 204947494 T^{8} - 11503479688 T^{10} + 502870481500 T^{12} - 13160326495992 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 280 T^{2} + 39452 T^{4} - 3783016 T^{6} + 267380710 T^{8} - 14076602536 T^{10} + 546246119132 T^{12} - 14425704821080 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 176 T^{2} + 15964 T^{4} - 993216 T^{6} + 63719302 T^{8} - 4458546624 T^{10} + 321692495644 T^{12} - 15920675261744 T^{14} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 + 12 T + 220 T^{2} + 1564 T^{3} + 18854 T^{4} + 111044 T^{5} + 1109020 T^{6} + 4294932 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 344 T^{2} + 60316 T^{4} - 7095528 T^{6} + 604376710 T^{8} - 37812068712 T^{10} + 1712868304156 T^{12} - 52058973843416 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 524 T^{2} + 125604 T^{4} - 18135668 T^{6} + 1736460342 T^{8} - 113184703988 T^{10} + 4892285973924 T^{12} - 127377826693004 T^{14} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 + 512 T^{2} + 125148 T^{4} + 18790160 T^{6} + 1886502918 T^{8} + 129445412240 T^{10} + 5939313956508 T^{12} + 167393471164928 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 328 T^{2} + 61788 T^{4} - 8336760 T^{6} + 843121542 T^{8} - 66035475960 T^{10} + 3876717586908 T^{12} - 163009863435208 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 408 T^{2} + 72412 T^{4} - 7810088 T^{6} + 720805318 T^{8} - 73485117992 T^{10} + 6410582295772 T^{12} - 339852578011032 T^{14} + 7837433594376961 T^{16} )^{2}$$