# Properties

 Label 1050.3.q.c Level $1050$ Weight $3$ Character orbit 1050.q Analytic conductor $28.610$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: 16.0.11007531417600000000.1 Defining polynomial: $$x^{16} - 7 x^{12} + 48 x^{8} - 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{13} q^{2} + ( \beta_{1} + 2 \beta_{8} ) q^{3} -2 \beta_{6} q^{4} + ( -\beta_{4} - 2 \beta_{12} ) q^{6} + ( -2 \beta_{2} + 2 \beta_{5} + 3 \beta_{10} - 3 \beta_{13} ) q^{7} -2 \beta_{5} q^{8} + ( -3 - 3 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{13} q^{2} + ( \beta_{1} + 2 \beta_{8} ) q^{3} -2 \beta_{6} q^{4} + ( -\beta_{4} - 2 \beta_{12} ) q^{6} + ( -2 \beta_{2} + 2 \beta_{5} + 3 \beta_{10} - 3 \beta_{13} ) q^{7} -2 \beta_{5} q^{8} + ( -3 - 3 \beta_{6} ) q^{9} + ( -\beta_{3} - 3 \beta_{4} + \beta_{6} + 2 \beta_{9} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{15} ) q^{11} + ( 4 \beta_{1} + 2 \beta_{8} ) q^{12} + ( 2 \beta_{1} + \beta_{2} - 4 \beta_{5} + \beta_{7} - 2 \beta_{8} + 8 \beta_{13} ) q^{13} + ( 4 - 2 \beta_{6} + 3 \beta_{9} + \beta_{15} ) q^{14} + ( -4 - 4 \beta_{6} ) q^{16} + ( -7 \beta_{1} - 3 \beta_{2} + \beta_{5} - 14 \beta_{8} + 3 \beta_{10} + \beta_{13} - 2 \beta_{14} ) q^{17} + ( -3 \beta_{5} + 3 \beta_{13} ) q^{18} + ( -9 + 2 \beta_{3} + 14 \beta_{4} + 9 \beta_{6} + 2 \beta_{11} + 7 \beta_{12} - \beta_{15} ) q^{19} + ( 4 \beta_{3} - 5 \beta_{4} - \beta_{11} - 4 \beta_{12} ) q^{21} + ( 6 \beta_{1} - 4 \beta_{2} + \beta_{5} + \beta_{7} + 6 \beta_{8} + 8 \beta_{10} - 2 \beta_{14} ) q^{22} + ( -6 \beta_{2} - 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{10} - \beta_{13} - 2 \beta_{14} ) q^{23} + ( 2 \beta_{4} - 2 \beta_{12} ) q^{24} + ( -8 + 2 \beta_{3} + 4 \beta_{4} + 8 \beta_{6} + 2 \beta_{11} + 2 \beta_{12} + \beta_{15} ) q^{26} + ( 3 \beta_{1} - 3 \beta_{8} ) q^{27} + ( 2 \beta_{2} - 2 \beta_{5} + 4 \beta_{10} - 4 \beta_{13} ) q^{28} + ( -9 + 2 \beta_{3} - 13 \beta_{4} - 2 \beta_{9} + \beta_{11} - 4 \beta_{15} ) q^{29} + ( -22 - 2 \beta_{3} - \beta_{4} - 11 \beta_{6} + 5 \beta_{9} + \beta_{12} + 5 \beta_{15} ) q^{31} + ( -4 \beta_{5} + 4 \beta_{13} ) q^{32} + ( -2 \beta_{1} - 6 \beta_{5} + 6 \beta_{7} - \beta_{8} + 3 \beta_{10} + 3 \beta_{13} - 6 \beta_{14} ) q^{33} + ( 2 + 7 \beta_{4} + 4 \beta_{6} + 3 \beta_{9} - 4 \beta_{11} + 14 \beta_{12} ) q^{34} -6 q^{36} + ( -10 \beta_{2} - \beta_{7} + 24 \beta_{8} + 5 \beta_{10} - 2 \beta_{13} - \beta_{14} ) q^{37} + ( -14 \beta_{1} - 2 \beta_{2} + 9 \beta_{5} - 28 \beta_{8} + 2 \beta_{10} + 9 \beta_{13} + 2 \beta_{14} ) q^{38} + ( \beta_{3} + 12 \beta_{4} + 6 \beta_{6} - \beta_{9} + 2 \beta_{11} + 12 \beta_{12} + \beta_{15} ) q^{39} + ( 3 + \beta_{4} + 6 \beta_{6} + 4 \beta_{9} + 19 \beta_{11} + 2 \beta_{12} ) q^{41} + ( 8 \beta_{1} + 5 \beta_{7} + 10 \beta_{8} - \beta_{14} ) q^{42} + ( -14 \beta_{1} + 9 \beta_{2} - 5 \beta_{5} - 4 \beta_{7} - 14 \beta_{8} - 18 \beta_{10} + 8 \beta_{14} ) q^{43} + ( 2 + 2 \beta_{3} + 2 \beta_{6} + 8 \beta_{9} - 2 \beta_{11} - 6 \beta_{12} + 4 \beta_{15} ) q^{44} + ( -4 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + 3 \beta_{9} - 8 \beta_{11} - 3 \beta_{12} - 3 \beta_{15} ) q^{46} + ( 4 \beta_{1} - 40 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 8 \beta_{10} + 20 \beta_{13} + 2 \beta_{14} ) q^{47} + ( 4 \beta_{1} - 4 \beta_{8} ) q^{48} + ( -24 - 9 \beta_{6} + 10 \beta_{9} - 6 \beta_{15} ) q^{49} + ( 21 + 3 \beta_{3} + 21 \beta_{6} + 4 \beta_{9} - 3 \beta_{11} + 3 \beta_{12} + 2 \beta_{15} ) q^{51} + ( -4 \beta_{1} + 2 \beta_{2} + 8 \beta_{5} - 8 \beta_{8} - 2 \beta_{10} + 8 \beta_{13} + 2 \beta_{14} ) q^{52} + ( 8 \beta_{1} - 4 \beta_{2} + 26 \beta_{5} + 16 \beta_{7} - 4 \beta_{10} - 26 \beta_{13} - 8 \beta_{14} ) q^{53} + ( 6 \beta_{4} + 3 \beta_{12} ) q^{54} + ( -4 - 12 \beta_{6} + 4 \beta_{9} + 6 \beta_{15} ) q^{56} + ( -27 \beta_{1} + 2 \beta_{2} + 21 \beta_{5} + \beta_{7} - 27 \beta_{8} - 4 \beta_{10} - 2 \beta_{14} ) q^{57} + ( -8 \beta_{2} + \beta_{7} + 26 \beta_{8} + 4 \beta_{10} + 9 \beta_{13} + \beta_{14} ) q^{58} + ( -22 + 21 \beta_{3} - 7 \beta_{4} - 11 \beta_{6} + 8 \beta_{9} + 7 \beta_{12} + 8 \beta_{15} ) q^{59} + ( 8 - 16 \beta_{3} - 4 \beta_{4} - 8 \beta_{6} - 16 \beta_{11} - 2 \beta_{12} - 16 \beta_{15} ) q^{61} + ( -2 \beta_{1} + 10 \beta_{2} - 11 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 22 \beta_{13} ) q^{62} + ( 9 \beta_{2} - 9 \beta_{5} - 3 \beta_{10} + 3 \beta_{13} ) q^{63} -8 q^{64} + ( -12 + 12 \beta_{3} - \beta_{4} - 6 \beta_{6} + 3 \beta_{9} + \beta_{12} + 3 \beta_{15} ) q^{66} + ( 30 \beta_{1} - 3 \beta_{2} - 9 \beta_{5} - 16 \beta_{7} - 3 \beta_{10} + 9 \beta_{13} + 8 \beta_{14} ) q^{67} + ( -28 \beta_{1} + 4 \beta_{5} + 4 \beta_{7} - 14 \beta_{8} + 6 \beta_{10} - 2 \beta_{13} - 4 \beta_{14} ) q^{68} + ( -3 - \beta_{4} - 6 \beta_{6} + 6 \beta_{9} - 9 \beta_{11} - 2 \beta_{12} ) q^{69} + ( 1 - 14 \beta_{3} + 29 \beta_{4} + 6 \beta_{9} - 7 \beta_{11} + 12 \beta_{15} ) q^{71} + 6 \beta_{13} q^{72} + ( 2 \beta_{1} - 25 \beta_{2} - 10 \beta_{5} + 4 \beta_{8} + 25 \beta_{10} - 10 \beta_{13} + 15 \beta_{14} ) q^{73} + ( -2 \beta_{3} - 24 \beta_{4} - 4 \beta_{6} + 5 \beta_{9} - 4 \beta_{11} - 24 \beta_{12} - 5 \beta_{15} ) q^{74} + ( 18 + 14 \beta_{4} + 36 \beta_{6} + 2 \beta_{9} + 4 \beta_{11} + 28 \beta_{12} ) q^{76} + ( 31 \beta_{1} - 21 \beta_{2} - 49 \beta_{5} + 8 \beta_{7} + 23 \beta_{8} + 14 \beta_{10} + 42 \beta_{13} - 10 \beta_{14} ) q^{77} + ( -24 \beta_{1} + 2 \beta_{2} + 6 \beta_{5} - \beta_{7} - 24 \beta_{8} - 4 \beta_{10} + 2 \beta_{14} ) q^{78} + ( -3 + 2 \beta_{3} - 3 \beta_{6} + 38 \beta_{9} - 2 \beta_{11} - 13 \beta_{12} + 19 \beta_{15} ) q^{79} + 9 \beta_{6} q^{81} + ( -4 \beta_{1} + 6 \beta_{5} - 19 \beta_{7} - 2 \beta_{8} + 8 \beta_{10} - 3 \beta_{13} + 19 \beta_{14} ) q^{82} + ( -3 \beta_{1} - 7 \beta_{2} + 15 \beta_{5} + 34 \beta_{7} + 3 \beta_{8} - 30 \beta_{13} ) q^{83} + ( 10 \beta_{3} - 2 \beta_{4} + 8 \beta_{11} - 10 \beta_{12} ) q^{84} + ( -10 - 8 \beta_{3} - 10 \beta_{6} - 18 \beta_{9} + 8 \beta_{11} + 14 \beta_{12} - 9 \beta_{15} ) q^{86} + ( -9 \beta_{1} + 3 \beta_{2} - 13 \beta_{5} - 18 \beta_{8} - 3 \beta_{10} - 13 \beta_{13} - 6 \beta_{14} ) q^{87} + ( 12 \beta_{1} + 8 \beta_{2} + 2 \beta_{5} + 4 \beta_{7} + 8 \beta_{10} - 2 \beta_{13} - 2 \beta_{14} ) q^{88} + ( -41 + 19 \beta_{3} - 22 \beta_{4} + 41 \beta_{6} + 19 \beta_{11} - 11 \beta_{12} + 4 \beta_{15} ) q^{89} + ( -35 + 8 \beta_{3} + 18 \beta_{4} + 7 \beta_{6} - 14 \beta_{9} + 12 \beta_{11} + 13 \beta_{12} + 7 \beta_{15} ) q^{91} + ( 6 \beta_{1} - 6 \beta_{2} - 2 \beta_{5} + 4 \beta_{7} + 6 \beta_{8} + 12 \beta_{10} - 8 \beta_{14} ) q^{92} + ( -4 \beta_{2} + 5 \beta_{7} - 33 \beta_{8} + 2 \beta_{10} - 3 \beta_{13} + 5 \beta_{14} ) q^{93} + ( -80 - 4 \beta_{3} + 2 \beta_{4} - 40 \beta_{6} - 8 \beta_{9} - 2 \beta_{12} - 8 \beta_{15} ) q^{94} + ( 8 \beta_{4} + 4 \beta_{12} ) q^{96} + ( -36 \beta_{1} + 8 \beta_{2} + 20 \beta_{5} - 18 \beta_{7} + 36 \beta_{8} - 40 \beta_{13} ) q^{97} + ( -12 \beta_{2} - 9 \beta_{5} + 32 \beta_{10} + 24 \beta_{13} ) q^{98} + ( 3 + 6 \beta_{3} + 9 \beta_{4} + 6 \beta_{9} + 3 \beta_{11} + 12 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 16 q^{4} - 24 q^{9} + O(q^{10})$$ $$16 q + 16 q^{4} - 24 q^{9} - 8 q^{11} + 80 q^{14} - 32 q^{16} - 216 q^{19} - 192 q^{26} - 144 q^{29} - 264 q^{31} - 96 q^{36} - 48 q^{39} + 16 q^{44} + 16 q^{46} - 312 q^{49} + 168 q^{51} + 32 q^{56} - 264 q^{59} + 192 q^{61} - 128 q^{64} - 144 q^{66} + 16 q^{71} + 32 q^{74} - 24 q^{79} - 72 q^{81} - 80 q^{86} - 984 q^{89} - 616 q^{91} - 960 q^{94} + 48 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7 x^{12} + 48 x^{8} - 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{14} + 377 \nu^{2}$$$$)/144$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{12} - 161$$$$)/72$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} + 281 \nu^{2}$$$$)/48$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} + 17 \nu^{13} - 120 \nu^{9} + 816 \nu^{5} - 305 \nu^{3} - 119 \nu$$$$)/72$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} - 17 \nu^{13} + 120 \nu^{9} - 816 \nu^{5} - 305 \nu^{3} + 119 \nu$$$$)/72$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{12} - 48 \nu^{8} + 336 \nu^{4} - 49$$$$)/48$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} + 19 \nu^{13} - 132 \nu^{9} + 912 \nu^{5} - 341 \nu^{3} - 133 \nu$$$$)/36$$ $$\beta_{8}$$ $$=$$ $$($$$$55 \nu^{14} - 384 \nu^{10} + 2640 \nu^{6} - 385 \nu^{2}$$$$)/144$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{15} + 19 \nu^{13} - 132 \nu^{9} + 912 \nu^{5} + 341 \nu^{3} - 133 \nu$$$$)/36$$ $$\beta_{10}$$ $$=$$ $$($$$$-49 \nu^{12} + 336 \nu^{8} - 2256 \nu^{4} + 7$$$$)/144$$ $$\beta_{11}$$ $$=$$ $$($$$$-7 \nu^{14} + 48 \nu^{10} - 328 \nu^{6} + \nu^{2}$$$$)/8$$ $$\beta_{12}$$ $$=$$ $$($$$$91 \nu^{15} + \nu^{13} - 624 \nu^{11} + 4272 \nu^{7} - 13 \nu^{3} + 233 \nu$$$$)/144$$ $$\beta_{13}$$ $$=$$ $$($$$$-91 \nu^{15} + \nu^{13} + 624 \nu^{11} - 4272 \nu^{7} + 13 \nu^{3} + 233 \nu$$$$)/144$$ $$\beta_{14}$$ $$=$$ $$($$$$199 \nu^{15} + 77 \nu^{13} - 1392 \nu^{11} - 528 \nu^{9} + 9552 \nu^{7} + 3648 \nu^{5} - 1393 \nu^{3} - 11 \nu$$$$)/144$$ $$\beta_{15}$$ $$=$$ $$($$$$-203 \nu^{15} + \nu^{13} + 1392 \nu^{11} - 9552 \nu^{7} + 29 \nu^{3} + 521 \nu$$$$)/144$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{7}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 3 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{9} - \beta_{7} + 2 \beta_{5} + 2 \beta_{4}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{10} + 7 \beta_{6} + 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{15} + 5 \beta_{14} - 11 \beta_{13} - 11 \beta_{12} + 5 \beta_{9} + 11 \beta_{5} - 11 \beta_{4}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{11} + 9 \beta_{8} + 9 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{15} + 13 \beta_{14} + 29 \beta_{13} - 29 \beta_{12} - 13 \beta_{7}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$21 \beta_{10} + 47 \beta_{6} - 21 \beta_{2}$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$17 \beta_{9} + 17 \beta_{7} + 38 \beta_{5} - 38 \beta_{4}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$55 \beta_{11} + 123 \beta_{8} + 55 \beta_{3}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-89 \beta_{15} + 89 \beta_{14} + 199 \beta_{13} - 199 \beta_{12} - 89 \beta_{9} - 199 \beta_{5} - 199 \beta_{4}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$-72 \beta_{2} - 161$$ $$\nu^{13}$$ $$=$$ $$($$$$-233 \beta_{15} - 233 \beta_{14} + 521 \beta_{13} + 521 \beta_{12} + 233 \beta_{7}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$377 \beta_{3} - 843 \beta_{1}$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-305 \beta_{9} + 305 \beta_{7} - 682 \beta_{5} - 682 \beta_{4}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −1.56290 − 0.418778i 0.596975 + 0.159959i −0.418778 + 1.56290i 0.159959 − 0.596975i 1.56290 + 0.418778i −0.596975 − 0.159959i 0.418778 − 1.56290i −0.159959 + 0.596975i −1.56290 + 0.418778i 0.596975 − 0.159959i −0.418778 − 1.56290i 0.159959 + 0.596975i 1.56290 − 0.418778i −0.596975 + 0.159959i 0.418778 + 1.56290i −0.159959 − 0.596975i
−1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −4.79227 + 5.10237i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −2.55620 6.51658i 2.82843i −1.50000 2.59808i 0
199.3 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −4.79227 + 5.10237i 2.82843i −1.50000 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −2.55620 6.51658i 2.82843i −1.50000 2.59808i 0
199.5 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 2.55620 + 6.51658i 2.82843i −1.50000 2.59808i 0
199.6 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 4.79227 5.10237i 2.82843i −1.50000 2.59808i 0
199.7 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 2.55620 + 6.51658i 2.82843i −1.50000 2.59808i 0
199.8 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 4.79227 5.10237i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −4.79227 5.10237i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −2.55620 + 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −4.79227 5.10237i 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −2.55620 + 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.5 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 2.55620 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.6 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 4.79227 + 5.10237i 2.82843i −1.50000 + 2.59808i 0
649.7 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 2.55620 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.8 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 4.79227 + 5.10237i 2.82843i −1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.q.c 16
5.b even 2 1 inner 1050.3.q.c 16
5.c odd 4 1 210.3.o.a 8
5.c odd 4 1 1050.3.p.b 8
7.d odd 6 1 inner 1050.3.q.c 16
15.e even 4 1 630.3.v.b 8
35.i odd 6 1 inner 1050.3.q.c 16
35.k even 12 1 210.3.o.a 8
35.k even 12 1 1050.3.p.b 8
35.k even 12 1 1470.3.f.a 8
35.l odd 12 1 1470.3.f.a 8
105.w odd 12 1 630.3.v.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a 8 5.c odd 4 1
210.3.o.a 8 35.k even 12 1
630.3.v.b 8 15.e even 4 1
630.3.v.b 8 105.w odd 12 1
1050.3.p.b 8 5.c odd 4 1
1050.3.p.b 8 35.k even 12 1
1050.3.q.c 16 1.a even 1 1 trivial
1050.3.q.c 16 5.b even 2 1 inner
1050.3.q.c 16 7.d odd 6 1 inner
1050.3.q.c 16 35.i odd 6 1 inner
1470.3.f.a 8 35.k even 12 1
1470.3.f.a 8 35.l odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T^{2} + T^{4} )^{4}$$
$3$ $$( 9 + 3 T^{2} + T^{4} )^{4}$$
$5$ $$T^{16}$$
$7$ $$( 5764801 + 187278 T^{2} + 5243 T^{4} + 78 T^{6} + T^{8} )^{2}$$
$11$ $$( 22505536 - 9715712 T + 5617504 T^{2} + 576448 T^{3} + 93448 T^{4} + 2896 T^{5} + 316 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$13$ $$( 33189121 - 2396652 T^{2} + 56678 T^{4} - 492 T^{6} + T^{8} )^{2}$$
$17$ $$249911075147776 + 110921063206912 T^{2} + 46798690502656 T^{4} + 1049657469952 T^{6} + 16985988544 T^{8} + 132468352 T^{10} + 752416 T^{12} + 952 T^{14} + T^{16}$$
$19$ $$( 3571138081 + 723561972 T + 463098 T^{2} - 9807480 T^{3} + 279971 T^{4} + 87480 T^{5} + 4698 T^{6} + 108 T^{7} + T^{8} )^{2}$$
$23$ $$1065552449536 - 1470527123456 T^{2} + 1775035869184 T^{4} - 348864472064 T^{6} + 59211949504 T^{8} - 259354496 T^{10} + 885664 T^{12} - 1064 T^{14} + T^{16}$$
$29$ $$( 20104 - 7872 T - 460 T^{2} + 36 T^{3} + T^{4} )^{4}$$
$31$ $$( 56085121 + 103977276 T + 76432266 T^{2} + 22575384 T^{3} + 3247283 T^{4} + 214632 T^{5} + 7434 T^{6} + 132 T^{7} + T^{8} )^{2}$$
$37$ $$471847657494245761 - 129050759692330484 T^{2} + 32888490519073834 T^{4} - 652884175474256 T^{6} + 11534826381139 T^{8} - 13486485584 T^{10} + 12145834 T^{12} - 3956 T^{14} + T^{16}$$
$41$ $$( 6360766467136 + 21661234752 T^{2} + 21449888 T^{4} + 7992 T^{6} + T^{8} )^{2}$$
$43$ $$( 769478085601 + 12825766164 T^{2} + 18153926 T^{4} + 7764 T^{6} + T^{8} )^{2}$$
$47$ $$31\!\cdots\!56$$$$+$$$$80\!\cdots\!32$$$$T^{2} +$$$$14\!\cdots\!00$$$$T^{4} + 1344386875751350272 T^{6} + 911406690226944 T^{8} + 328708666368 T^{10} + 85040800 T^{12} + 11088 T^{14} + T^{16}$$
$53$ $$2159867877514018816 - 10983051048964325376 T^{2} + 55793159420663824384 T^{4} - 286145340602056704 T^{6} + 1359601472385024 T^{8} - 532823924736 T^{10} + 166309504 T^{12} - 14304 T^{14} + T^{16}$$
$59$ $$( 19263180552256 + 2847011029248 T + 158253288928 T^{2} + 2659555200 T^{3} - 7342584 T^{4} - 541200 T^{5} + 1708 T^{6} + 132 T^{7} + T^{8} )^{2}$$
$61$ $$( 981652934656 + 227515711488 T + 11061556224 T^{2} - 1510060032 T^{3} + 34904768 T^{4} + 631296 T^{5} - 3504 T^{6} - 96 T^{7} + T^{8} )^{2}$$
$67$ $$72\!\cdots\!01$$$$-$$$$80\!\cdots\!32$$$$T^{2} + 66235798468756028970 T^{4} - 247301598317024592 T^{6} + 679557055843539 T^{8} - 333394064208 T^{10} + 127187370 T^{12} - 12468 T^{14} + T^{16}$$
$71$ $$( -2872184 - 278336 T - 6988 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$73$ $$27\!\cdots\!21$$$$+$$$$10\!\cdots\!32$$$$T^{2} +$$$$28\!\cdots\!90$$$$T^{4} + 24692431707306932912 T^{6} + 15914101584265459 T^{8} + 3050256636848 T^{10} + 430265290 T^{12} + 23948 T^{14} + T^{16}$$
$79$ $$( 11859027266503921 - 1749791718948 T + 2439817185346 T^{2} - 2253624528 T^{3} + 393143259 T^{4} - 236688 T^{5} + 22546 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$83$ $$( 8967746052897856 - 5541444773952 T^{2} + 905020064 T^{4} - 52728 T^{6} + T^{8} )^{2}$$
$89$ $$( 91315148362816 - 9140634983424 T + 67279672416 T^{2} + 23794988544 T^{3} + 785244488 T^{4} + 12238992 T^{5} + 105564 T^{6} + 492 T^{7} + T^{8} )^{2}$$
$97$ $$( 3470767019462656 - 2182950729728 T^{2} + 469171584 T^{4} - 39392 T^{6} + T^{8} )^{2}$$