# Properties

 Label 825.2.bx.h Level $825$ Weight $2$ Character orbit 825.bx Analytic conductor $6.588$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.bx (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - x^{14} + 15 x^{12} - 59 x^{10} + 104 x^{8} - 59 x^{6} + 15 x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{8} + \beta_{13} - 2 \beta_{15} ) q^{2} -\beta_{14} q^{3} + ( -1 - 2 \beta_{2} + 4 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{4} + ( -\beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{6} + ( \beta_{1} + \beta_{4} + 3 \beta_{13} + \beta_{14} - \beta_{15} ) q^{7} + ( 3 \beta_{1} + \beta_{4} - 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{11} - 4 \beta_{14} ) q^{8} -\beta_{12} q^{9} +O(q^{10})$$ $$q + ( -\beta_{8} + \beta_{13} - 2 \beta_{15} ) q^{2} -\beta_{14} q^{3} + ( -1 - 2 \beta_{2} + 4 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{4} + ( -\beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{6} + ( \beta_{1} + \beta_{4} + 3 \beta_{13} + \beta_{14} - \beta_{15} ) q^{7} + ( 3 \beta_{1} + \beta_{4} - 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{11} - 4 \beta_{14} ) q^{8} -\beta_{12} q^{9} + ( -3 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{12} ) q^{11} + ( -2 \beta_{4} + \beta_{8} - 4 \beta_{9} - 3 \beta_{11} - 2 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{12} + ( 2 \beta_{1} - \beta_{8} + \beta_{9} + \beta_{13} + \beta_{14} ) q^{13} + ( 1 - 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{7} - 2 \beta_{10} + \beta_{12} ) q^{14} + ( -5 \beta_{3} - 5 \beta_{5} - 5 \beta_{6} - 10 \beta_{7} - 5 \beta_{10} + 5 \beta_{12} ) q^{16} + 5 \beta_{9} q^{17} + ( -\beta_{1} - \beta_{4} - 2 \beta_{11} ) q^{18} + ( -1 + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{10} - \beta_{12} ) q^{19} + ( -3 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{10} + \beta_{12} ) q^{21} + ( 5 \beta_{1} + 2 \beta_{4} - 3 \beta_{8} + 4 \beta_{9} + 6 \beta_{11} + 3 \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{22} + ( \beta_{4} - \beta_{8} - \beta_{9} + \beta_{13} + \beta_{15} ) q^{23} + ( 3 + 3 \beta_{2} - 3 \beta_{3} + \beta_{6} + 3 \beta_{10} - 4 \beta_{12} ) q^{24} + ( -1 - 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{26} + \beta_{9} q^{27} + ( 5 \beta_{1} + 5 \beta_{4} - 5 \beta_{8} + 5 \beta_{9} + 9 \beta_{11} + 5 \beta_{13} - 9 \beta_{15} ) q^{28} + ( 4 \beta_{2} - \beta_{3} + 3 \beta_{5} + 4 \beta_{6} ) q^{29} + ( 2 - 2 \beta_{3} + 3 \beta_{6} ) q^{31} + ( -4 \beta_{4} + 8 \beta_{8} - 16 \beta_{9} - 7 \beta_{11} - 4 \beta_{13} - 7 \beta_{14} + 16 \beta_{15} ) q^{32} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{9} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{33} + ( -10 - 5 \beta_{6} - 5 \beta_{10} ) q^{34} + ( -3 + \beta_{2} + 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{7} + 2 \beta_{10} - 3 \beta_{12} ) q^{36} + ( -\beta_{1} - \beta_{4} + 5 \beta_{14} - 5 \beta_{15} ) q^{37} + ( -10 \beta_{1} - 7 \beta_{4} + 7 \beta_{8} - 5 \beta_{9} - 9 \beta_{11} - 10 \beta_{13} + 9 \beta_{15} ) q^{38} + ( -\beta_{3} - \beta_{5} - \beta_{6} + \beta_{10} + \beta_{12} ) q^{39} + ( -6 + 3 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} - \beta_{10} - 6 \beta_{12} ) q^{41} + ( 5 \beta_{1} - 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{13} - 4 \beta_{14} + 3 \beta_{15} ) q^{42} + ( \beta_{4} + 3 \beta_{9} - 3 \beta_{11} + \beta_{13} - 3 \beta_{14} - 3 \beta_{15} ) q^{43} + ( 4 - 5 \beta_{2} - 3 \beta_{3} - 7 \beta_{5} - 6 \beta_{6} - 14 \beta_{7} - 7 \beta_{10} + 10 \beta_{12} ) q^{44} + ( 1 + \beta_{2} - \beta_{3} + 3 \beta_{6} + \beta_{10} + \beta_{12} ) q^{46} + ( \beta_{1} - 4 \beta_{4} - \beta_{8} - 3 \beta_{9} - 3 \beta_{11} - 6 \beta_{14} ) q^{47} + ( -5 \beta_{1} - 5 \beta_{4} - 5 \beta_{11} + 5 \beta_{14} - 5 \beta_{15} ) q^{48} + ( -4 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{10} + 4 \beta_{12} ) q^{49} -5 \beta_{3} q^{51} + ( \beta_{4} - 2 \beta_{14} ) q^{52} + ( -3 \beta_{1} - 3 \beta_{8} + 5 \beta_{9} + 3 \beta_{13} + 5 \beta_{14} - 4 \beta_{15} ) q^{53} + ( -2 - \beta_{6} - \beta_{10} ) q^{54} + ( 6 - 4 \beta_{2} - 4 \beta_{5} - 8 \beta_{6} - 15 \beta_{7} - 8 \beta_{10} + 15 \beta_{12} ) q^{56} + ( -5 \beta_{1} + 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{57} + ( -4 \beta_{1} + 3 \beta_{4} + 4 \beta_{8} - \beta_{9} - \beta_{11} + 6 \beta_{14} ) q^{58} + ( -6 + \beta_{2} + \beta_{3} + \beta_{6} + 6 \beta_{7} ) q^{59} + ( 2 \beta_{3} - 6 \beta_{5} - 6 \beta_{6} - \beta_{7} - 3 \beta_{10} - 2 \beta_{12} ) q^{61} + ( \beta_{1} + \beta_{4} + 5 \beta_{13} + 7 \beta_{14} - 7 \beta_{15} ) q^{62} + ( \beta_{1} - 2 \beta_{4} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{63} + ( 16 + 2 \beta_{2} - 16 \beta_{3} + 11 \beta_{6} + 2 \beta_{10} - 8 \beta_{12} ) q^{64} + ( 6 - 4 \beta_{3} - 3 \beta_{5} - \beta_{6} - 4 \beta_{7} + 2 \beta_{10} - \beta_{12} ) q^{66} + ( 5 \beta_{4} + 5 \beta_{8} + 4 \beta_{9} + 5 \beta_{11} + 5 \beta_{13} + 5 \beta_{14} - 4 \beta_{15} ) q^{67} + ( -5 \beta_{1} + 10 \beta_{8} - 5 \beta_{9} - 10 \beta_{13} - 5 \beta_{14} + 20 \beta_{15} ) q^{68} + ( \beta_{3} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{69} + ( 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} + 3 \beta_{10} - 4 \beta_{12} ) q^{71} + ( -2 \beta_{1} + \beta_{4} - \beta_{8} + 7 \beta_{9} + 3 \beta_{11} - 2 \beta_{13} - 3 \beta_{15} ) q^{72} + ( 5 \beta_{1} + 5 \beta_{4} - 2 \beta_{11} + 8 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{73} + ( -1 - 4 \beta_{2} + 12 \beta_{3} + 5 \beta_{5} + 12 \beta_{7} + 5 \beta_{10} - \beta_{12} ) q^{74} + ( -9 + 12 \beta_{2} + 12 \beta_{5} + 11 \beta_{6} + 19 \beta_{7} + 11 \beta_{10} - 19 \beta_{12} ) q^{76} + ( -3 \beta_{1} + \beta_{4} + \beta_{8} + 6 \beta_{9} + 11 \beta_{11} - 2 \beta_{13} + 7 \beta_{14} - 9 \beta_{15} ) q^{77} + ( -2 \beta_{4} + \beta_{11} - 2 \beta_{13} + \beta_{14} ) q^{78} + ( -7 - 2 \beta_{2} + 7 \beta_{3} - 2 \beta_{6} - 2 \beta_{10} - 2 \beta_{12} ) q^{79} -\beta_{3} q^{81} + ( -7 \beta_{1} - 3 \beta_{4} + 3 \beta_{8} - 3 \beta_{9} - 14 \beta_{11} - 7 \beta_{13} + 14 \beta_{15} ) q^{82} + ( 3 \beta_{1} + 5 \beta_{4} - 5 \beta_{8} - 2 \beta_{9} + 8 \beta_{11} + 3 \beta_{13} - 8 \beta_{15} ) q^{83} + ( 9 - 5 \beta_{3} - 5 \beta_{5} - 9 \beta_{7} ) q^{84} + ( -12 - 4 \beta_{2} + 12 \beta_{3} - 8 \beta_{6} - 4 \beta_{10} - 5 \beta_{12} ) q^{86} + ( 4 \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} + 4 \beta_{13} + \beta_{14} - \beta_{15} ) q^{87} + ( 7 \beta_{1} - 2 \beta_{4} + 8 \beta_{8} - 23 \beta_{9} - 8 \beta_{11} - 4 \beta_{13} - 18 \beta_{14} + 18 \beta_{15} ) q^{88} + ( -2 + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{10} + 3 \beta_{12} ) q^{89} + ( 6 + 7 \beta_{2} - 2 \beta_{3} + 5 \beta_{5} - 2 \beta_{7} + 5 \beta_{10} + 6 \beta_{12} ) q^{91} + ( \beta_{11} + 2 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} ) q^{92} + ( 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + 3 \beta_{13} - 2 \beta_{15} ) q^{93} + ( 10 \beta_{3} - 8 \beta_{5} - 8 \beta_{6} - 3 \beta_{7} + \beta_{10} - 10 \beta_{12} ) q^{94} + ( -7 - 4 \beta_{2} + 16 \beta_{3} + 4 \beta_{5} + 16 \beta_{7} + 4 \beta_{10} - 7 \beta_{12} ) q^{96} + ( \beta_{1} + 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{97} + ( -3 \beta_{4} + 3 \beta_{8} + \beta_{9} + 7 \beta_{11} - 3 \beta_{13} + 7 \beta_{14} - \beta_{15} ) q^{98} + ( \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{7} + 3 \beta_{10} - \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{4} + 4q^{9} + O(q^{10})$$ $$16q + 4q^{4} + 4q^{9} - 6q^{11} + 20q^{14} - 40q^{16} - 12q^{19} - 8q^{21} + 40q^{24} - 16q^{26} + 6q^{31} - 100q^{34} - 4q^{36} - 12q^{39} - 50q^{41} - 14q^{44} - 12q^{46} - 42q^{49} - 20q^{51} - 20q^{54} + 40q^{56} - 70q^{59} + 42q^{61} + 154q^{64} + 50q^{66} + 10q^{69} + 50q^{71} + 58q^{74} - 28q^{76} - 60q^{79} - 4q^{81} + 68q^{84} - 68q^{86} - 64q^{89} + 74q^{91} + 78q^{94} + 20q^{96} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{14} + 15 x^{12} - 59 x^{10} + 104 x^{8} - 59 x^{6} + 15 x^{4} - x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$21 \nu^{14} + 2 \nu^{12} + 289 \nu^{10} - 908 \nu^{8} + 772 \nu^{6} + 1045 \nu^{4} - 622 \nu^{2} - 63$$$$)/384$$ $$\beta_{3}$$ $$=$$ $$($$$$21 \nu^{14} - 22 \nu^{12} + 329 \nu^{10} - 1260 \nu^{8} + 2436 \nu^{6} - 1995 \nu^{4} + 1498 \nu^{2} - 119$$$$)/384$$ $$\beta_{4}$$ $$=$$ $$($$$$-21 \nu^{15} + 22 \nu^{13} - 329 \nu^{11} + 1260 \nu^{9} - 2436 \nu^{7} + 1995 \nu^{5} - 1498 \nu^{3} + 119 \nu$$$$)/384$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{14} - 11 \nu^{12} + 76 \nu^{10} - 384 \nu^{8} + 804 \nu^{6} - 671 \nu^{4} + 137 \nu^{2} - 40$$$$)/96$$ $$\beta_{6}$$ $$=$$ $$($$$$-21 \nu^{14} + 10 \nu^{12} - 309 \nu^{10} + 1084 \nu^{8} - 1604 \nu^{6} + 475 \nu^{4} - 246 \nu^{2} + 91$$$$)/192$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{14} - 17 \nu^{12} + 138 \nu^{10} - 648 \nu^{8} + 1332 \nu^{6} - 1107 \nu^{4} + 155 \nu^{2} + 30$$$$)/96$$ $$\beta_{8}$$ $$=$$ $$($$$$9 \nu^{15} - 17 \nu^{13} + 138 \nu^{11} - 648 \nu^{9} + 1332 \nu^{7} - 1107 \nu^{5} + 155 \nu^{3} + 30 \nu$$$$)/96$$ $$\beta_{9}$$ $$=$$ $$($$$$-9 \nu^{15} + 23 \nu^{13} - 148 \nu^{11} + 736 \nu^{9} - 1748 \nu^{7} + 1867 \nu^{5} - 685 \nu^{3} - 16 \nu$$$$)/96$$ $$\beta_{10}$$ $$=$$ $$($$$$57 \nu^{14} - 30 \nu^{12} + 845 \nu^{10} - 2972 \nu^{8} + 4564 \nu^{6} - 1511 \nu^{4} + 466 \nu^{2} + 77$$$$)/384$$ $$\beta_{11}$$ $$=$$ $$($$$$-63 \nu^{15} + 42 \nu^{13} - 947 \nu^{11} + 3428 \nu^{9} - 5644 \nu^{7} + 2945 \nu^{5} - 1990 \nu^{3} + 685 \nu$$$$)/384$$ $$\beta_{12}$$ $$=$$ $$($$$$-17 \nu^{14} + 5 \nu^{12} - 246 \nu^{10} + 824 \nu^{8} - 1108 \nu^{6} - 101 \nu^{4} + 201 \nu^{2} - 58$$$$)/96$$ $$\beta_{13}$$ $$=$$ $$($$$$-17 \nu^{15} + 5 \nu^{13} - 246 \nu^{11} + 824 \nu^{9} - 1108 \nu^{7} - 101 \nu^{5} + 201 \nu^{3} - 58 \nu$$$$)/96$$ $$\beta_{14}$$ $$=$$ $$($$$$77 \nu^{15} - 134 \nu^{13} + 1185 \nu^{11} - 5388 \nu^{9} + 10980 \nu^{7} - 9107 \nu^{5} + 2666 \nu^{3} - 543 \nu$$$$)/384$$ $$\beta_{15}$$ $$=$$ $$($$$$40 \nu^{15} - 35 \nu^{13} + 589 \nu^{11} - 2284 \nu^{9} + 3776 \nu^{7} - 1556 \nu^{5} - 71 \nu^{3} + 97 \nu$$$$)/96$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{8} - 3 \beta_{4} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{12} + 5 \beta_{10} - 3 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$5 \beta_{15} - \beta_{14} + 6 \beta_{13} - \beta_{11} - 5 \beta_{9} - 12 \beta_{8} + 6 \beta_{4}$$ $$\nu^{6}$$ $$=$$ $$-6 \beta_{12} - 16 \beta_{10} - 23 \beta_{6} - 6 \beta_{3} - 16 \beta_{2} + 6$$ $$\nu^{7}$$ $$=$$ $$-16 \beta_{15} + 16 \beta_{14} - 22 \beta_{13} - 7 \beta_{11} + 29 \beta_{4} + 29 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-22 \beta_{12} - 67 \beta_{10} + 51 \beta_{7} - 67 \beta_{5} + 51 \beta_{3} + 36 \beta_{2} - 22$$ $$\nu^{9}$$ $$=$$ $$-67 \beta_{15} - 89 \beta_{13} + 67 \beta_{11} + 103 \beta_{9} + 221 \beta_{8} - 221 \beta_{4} - 89 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$132 \beta_{12} + 456 \beta_{10} - 132 \beta_{7} + 456 \beta_{6} + 168 \beta_{5} + 168 \beta_{2} - 89$$ $$\nu^{11}$$ $$=$$ $$456 \beta_{15} - 288 \beta_{14} + 588 \beta_{13} - 288 \beta_{9} - 588 \beta_{8} - 377 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-588 \beta_{7} - 1253 \beta_{6} + 756 \beta_{5} - 965 \beta_{3} - 1253 \beta_{2} + 588$$ $$\nu^{13}$$ $$=$$ $$756 \beta_{14} - 1253 \beta_{11} - 1253 \beta_{9} - 2597 \beta_{8} + 4227 \beta_{4} + 2597 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-2597 \beta_{12} - 8833 \beta_{10} + 4227 \beta_{7} - 5480 \beta_{6} - 5480 \beta_{5} + 2597 \beta_{3}$$ $$\nu^{15}$$ $$=$$ $$-8833 \beta_{15} + 3353 \beta_{14} - 11430 \beta_{13} + 3353 \beta_{11} + 8833 \beta_{9} + 18540 \beta_{8} - 11430 \beta_{4}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.701538 + 0.227943i 1.28932 − 0.418926i −1.28932 + 0.418926i 0.701538 − 0.227943i 0.280526 + 0.386111i 1.23158 + 1.69513i −1.23158 − 1.69513i −0.280526 − 0.386111i 0.280526 − 0.386111i 1.23158 − 1.69513i −1.23158 + 1.69513i −0.280526 + 0.386111i −0.701538 − 0.227943i 1.28932 + 0.418926i −1.28932 − 0.418926i 0.701538 + 0.227943i
−2.33569 + 0.758911i −0.587785 + 0.809017i 3.26145 2.36959i 0 0.758911 2.33569i −1.93196 2.65911i −2.93237 + 4.03606i −0.309017 0.951057i 0
49.2 −1.10527 + 0.359123i −0.587785 + 0.809017i −0.525387 + 0.381716i 0 0.359123 1.10527i 2.51974 + 3.46813i 1.80980 2.49097i −0.309017 0.951057i 0
49.3 1.10527 0.359123i 0.587785 0.809017i −0.525387 + 0.381716i 0 0.359123 1.10527i −2.51974 3.46813i −1.80980 + 2.49097i −0.309017 0.951057i 0
49.4 2.33569 0.758911i 0.587785 0.809017i 3.26145 2.36959i 0 0.758911 2.33569i 1.93196 + 2.65911i 2.93237 4.03606i −0.309017 0.951057i 0
124.1 −1.62947 2.24278i 0.951057 0.309017i −1.75683 + 5.40697i 0 −2.24278 1.62947i −2.16612 0.703814i 9.71623 3.15700i 0.809017 0.587785i 0
124.2 −0.817172 1.12474i −0.951057 + 0.309017i 0.0207616 0.0638975i 0 1.12474 + 0.817172i −1.21506 0.394797i −2.73326 + 0.888090i 0.809017 0.587785i 0
124.3 0.817172 + 1.12474i 0.951057 0.309017i 0.0207616 0.0638975i 0 1.12474 + 0.817172i 1.21506 + 0.394797i 2.73326 0.888090i 0.809017 0.587785i 0
124.4 1.62947 + 2.24278i −0.951057 + 0.309017i −1.75683 + 5.40697i 0 −2.24278 1.62947i 2.16612 + 0.703814i −9.71623 + 3.15700i 0.809017 0.587785i 0
499.1 −1.62947 + 2.24278i 0.951057 + 0.309017i −1.75683 5.40697i 0 −2.24278 + 1.62947i −2.16612 + 0.703814i 9.71623 + 3.15700i 0.809017 + 0.587785i 0
499.2 −0.817172 + 1.12474i −0.951057 0.309017i 0.0207616 + 0.0638975i 0 1.12474 0.817172i −1.21506 + 0.394797i −2.73326 0.888090i 0.809017 + 0.587785i 0
499.3 0.817172 1.12474i 0.951057 + 0.309017i 0.0207616 + 0.0638975i 0 1.12474 0.817172i 1.21506 0.394797i 2.73326 + 0.888090i 0.809017 + 0.587785i 0
499.4 1.62947 2.24278i −0.951057 0.309017i −1.75683 5.40697i 0 −2.24278 + 1.62947i 2.16612 0.703814i −9.71623 3.15700i 0.809017 + 0.587785i 0
724.1 −2.33569 0.758911i −0.587785 0.809017i 3.26145 + 2.36959i 0 0.758911 + 2.33569i −1.93196 + 2.65911i −2.93237 4.03606i −0.309017 + 0.951057i 0
724.2 −1.10527 0.359123i −0.587785 0.809017i −0.525387 0.381716i 0 0.359123 + 1.10527i 2.51974 3.46813i 1.80980 + 2.49097i −0.309017 + 0.951057i 0
724.3 1.10527 + 0.359123i 0.587785 + 0.809017i −0.525387 0.381716i 0 0.359123 + 1.10527i −2.51974 + 3.46813i −1.80980 2.49097i −0.309017 + 0.951057i 0
724.4 2.33569 + 0.758911i 0.587785 + 0.809017i 3.26145 + 2.36959i 0 0.758911 + 2.33569i 1.93196 2.65911i 2.93237 + 4.03606i −0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 724.4 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
11.c even 5 1 inner
55.j even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bx.h 16
5.b even 2 1 inner 825.2.bx.h 16
5.c odd 4 1 165.2.m.a 8
5.c odd 4 1 825.2.n.k 8
11.c even 5 1 inner 825.2.bx.h 16
15.e even 4 1 495.2.n.d 8
55.j even 10 1 inner 825.2.bx.h 16
55.k odd 20 1 165.2.m.a 8
55.k odd 20 1 825.2.n.k 8
55.k odd 20 1 1815.2.a.x 4
55.k odd 20 1 9075.2.a.cl 4
55.l even 20 1 1815.2.a.o 4
55.l even 20 1 9075.2.a.dj 4
165.u odd 20 1 5445.2.a.bv 4
165.v even 20 1 495.2.n.d 8
165.v even 20 1 5445.2.a.be 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 5.c odd 4 1
165.2.m.a 8 55.k odd 20 1
495.2.n.d 8 15.e even 4 1
495.2.n.d 8 165.v even 20 1
825.2.n.k 8 5.c odd 4 1
825.2.n.k 8 55.k odd 20 1
825.2.bx.h 16 1.a even 1 1 trivial
825.2.bx.h 16 5.b even 2 1 inner
825.2.bx.h 16 11.c even 5 1 inner
825.2.bx.h 16 55.j even 10 1 inner
1815.2.a.o 4 55.l even 20 1
1815.2.a.x 4 55.k odd 20 1
5445.2.a.be 4 165.v even 20 1
5445.2.a.bv 4 165.u odd 20 1
9075.2.a.cl 4 55.k odd 20 1
9075.2.a.dj 4 55.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{16} - \cdots$$ $$T_{13}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$14641 - 15609 T^{2} + 9087 T^{4} - 3647 T^{6} + 2730 T^{8} - 473 T^{10} + 57 T^{12} - 6 T^{14} + T^{16}$$
$3$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$2825761 - 3425878 T^{2} + 1742873 T^{4} - 297921 T^{6} + 25730 T^{8} - 1431 T^{10} + 383 T^{12} + 7 T^{14} + T^{16}$$
$11$ $$( 14641 + 3993 T + 968 T^{2} + 11 T^{3} - 85 T^{4} + T^{5} + 8 T^{6} + 3 T^{7} + T^{8} )^{2}$$
$13$ $$1 - 13 T^{2} + 75 T^{4} - 103 T^{6} + 2874 T^{8} + 683 T^{10} + 665 T^{12} - 42 T^{14} + T^{16}$$
$17$ $$( 390625 - 15625 T^{2} + 625 T^{4} - 25 T^{6} + T^{8} )^{2}$$
$19$ $$( 961 + 3813 T + 38671 T^{2} + 2379 T^{3} + 874 T^{4} - 123 T^{5} + 9 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$23$ $$( 1 + 23 T^{2} + 49 T^{4} + 27 T^{6} + T^{8} )^{2}$$
$29$ $$( 290521 - 18865 T + 34153 T^{2} + 5285 T^{3} + 844 T^{4} - 95 T^{5} + 17 T^{6} + T^{8} )^{2}$$
$31$ $$( 19321 + 4309 T + 2829 T^{2} + 253 T^{3} + 174 T^{4} + T^{5} + 11 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$37$ $$34507149121 + 1216920311 T^{2} + 694858955 T^{4} - 36085311 T^{6} + 2538464 T^{8} - 179451 T^{10} + 9275 T^{12} - 149 T^{14} + T^{16}$$
$41$ $$( 4289041 + 2091710 T + 708673 T^{2} + 150785 T^{3} + 23634 T^{4} + 2855 T^{5} + 327 T^{6} + 25 T^{7} + T^{8} )^{2}$$
$43$ $$( 3463321 + 346393 T^{2} + 12438 T^{4} + 188 T^{6} + T^{8} )^{2}$$
$47$ $$14641 + 51546 T^{2} + 3695817 T^{4} - 30497597 T^{6} + 95768130 T^{8} - 1868723 T^{10} + 15567 T^{12} - 21 T^{14} + T^{16}$$
$53$ $$2609649624481 - 797425294507 T^{2} + 101426643960 T^{4} - 4121797877 T^{6} + 66708839 T^{8} + 164587 T^{10} + 20160 T^{12} + 77 T^{14} + T^{16}$$
$59$ $$( 5285401 + 3253085 T + 1296507 T^{2} + 331435 T^{3} + 58754 T^{4} + 7285 T^{5} + 633 T^{6} + 35 T^{7} + T^{8} )^{2}$$
$61$ $$( 3575881 + 1760521 T + 459192 T^{2} + 41503 T^{3} + 8805 T^{4} - 1043 T^{5} + 242 T^{6} - 21 T^{7} + T^{8} )^{2}$$
$67$ $$( 11417641 + 2206736 T^{2} + 57706 T^{4} + 441 T^{6} + T^{8} )^{2}$$
$71$ $$( 5527201 - 2879975 T + 1218443 T^{2} - 269975 T^{3} + 37544 T^{4} - 3625 T^{5} + 347 T^{6} - 25 T^{7} + T^{8} )^{2}$$
$73$ $$7160815355803201 - 237551366137619 T^{2} + 4270977523600 T^{4} - 56542625141 T^{6} + 786710319 T^{8} - 7463101 T^{10} + 39360 T^{12} - 59 T^{14} + T^{16}$$
$79$ $$( 1437601 + 1402830 T + 669978 T^{2} + 183930 T^{3} + 37084 T^{4} + 4980 T^{5} + 487 T^{6} + 30 T^{7} + T^{8} )^{2}$$
$83$ $$22306271114771281 + 162048877954887 T^{2} + 16618769163095 T^{4} - 171228298263 T^{6} + 1787423584 T^{8} - 19706547 T^{10} + 158495 T^{12} - 597 T^{14} + T^{16}$$
$89$ $$( -271 - 132 T + 45 T^{2} + 16 T^{3} + T^{4} )^{4}$$
$97$ $$390625 - 12734375 T^{2} + 159687500 T^{4} - 58978125 T^{6} + 8136875 T^{8} + 165375 T^{10} + 8000 T^{12} + 65 T^{14} + T^{16}$$