# Properties

 Label 825.2.bx.f 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.

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

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 $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$ $$-1$$

## 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
 −0.280526 + 0.386111i −1.23158 + 1.69513i 1.23158 − 1.69513i 0.280526 − 0.386111i 1.28932 − 0.418926i −0.701538 + 0.227943i 0.701538 − 0.227943i −1.28932 + 0.418926i 1.28932 + 0.418926i −0.701538 − 0.227943i 0.701538 + 0.227943i −1.28932 − 0.418926i −0.280526 − 0.386111i −1.23158 − 1.69513i 1.23158 + 1.69513i 0.280526 + 0.386111i
−1.40496 + 0.456498i 0.587785 0.809017i 0.147481 0.107152i 0 −0.456498 + 1.40496i −1.34895 1.85666i 1.57833 2.17239i −0.309017 0.951057i 0
49.2 −1.04169 + 0.338464i −0.587785 + 0.809017i −0.647481 + 0.470423i 0 0.338464 1.04169i 0.414410 + 0.570387i 1.80285 2.48141i −0.309017 0.951057i 0
49.3 1.04169 0.338464i 0.587785 0.809017i −0.647481 + 0.470423i 0 0.338464 1.04169i −0.414410 0.570387i −1.80285 + 2.48141i −0.309017 0.951057i 0
49.4 1.40496 0.456498i −0.587785 + 0.809017i 0.147481 0.107152i 0 −0.456498 + 1.40496i 1.34895 + 1.85666i −1.57833 + 2.17239i −0.309017 0.951057i 0
124.1 −1.38463 1.90578i −0.951057 + 0.309017i −1.09676 + 3.37549i 0 1.90578 + 1.38463i −0.184055 0.0598032i 3.47080 1.12773i 0.809017 0.587785i 0
124.2 −0.154211 0.212253i −0.951057 + 0.309017i 0.596764 1.83665i 0 0.212253 + 0.154211i 3.03722 + 0.986854i −0.980901 + 0.318714i 0.809017 0.587785i 0
124.3 0.154211 + 0.212253i 0.951057 0.309017i 0.596764 1.83665i 0 0.212253 + 0.154211i −3.03722 0.986854i 0.980901 0.318714i 0.809017 0.587785i 0
124.4 1.38463 + 1.90578i 0.951057 0.309017i −1.09676 + 3.37549i 0 1.90578 + 1.38463i 0.184055 + 0.0598032i −3.47080 + 1.12773i 0.809017 0.587785i 0
499.1 −1.38463 + 1.90578i −0.951057 0.309017i −1.09676 3.37549i 0 1.90578 1.38463i −0.184055 + 0.0598032i 3.47080 + 1.12773i 0.809017 + 0.587785i 0
499.2 −0.154211 + 0.212253i −0.951057 0.309017i 0.596764 + 1.83665i 0 0.212253 0.154211i 3.03722 0.986854i −0.980901 0.318714i 0.809017 + 0.587785i 0
499.3 0.154211 0.212253i 0.951057 + 0.309017i 0.596764 + 1.83665i 0 0.212253 0.154211i −3.03722 + 0.986854i 0.980901 + 0.318714i 0.809017 + 0.587785i 0
499.4 1.38463 1.90578i 0.951057 + 0.309017i −1.09676 3.37549i 0 1.90578 1.38463i 0.184055 0.0598032i −3.47080 1.12773i 0.809017 + 0.587785i 0
724.1 −1.40496 0.456498i 0.587785 + 0.809017i 0.147481 + 0.107152i 0 −0.456498 1.40496i −1.34895 + 1.85666i 1.57833 + 2.17239i −0.309017 + 0.951057i 0
724.2 −1.04169 0.338464i −0.587785 0.809017i −0.647481 0.470423i 0 0.338464 + 1.04169i 0.414410 0.570387i 1.80285 + 2.48141i −0.309017 + 0.951057i 0
724.3 1.04169 + 0.338464i 0.587785 + 0.809017i −0.647481 0.470423i 0 0.338464 + 1.04169i −0.414410 + 0.570387i −1.80285 2.48141i −0.309017 + 0.951057i 0
724.4 1.40496 + 0.456498i −0.587785 0.809017i 0.147481 + 0.107152i 0 −0.456498 1.40496i 1.34895 1.85666i −1.57833 2.17239i −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.f 16
5.b even 2 1 inner 825.2.bx.f 16
5.c odd 4 1 165.2.m.d 8
5.c odd 4 1 825.2.n.g 8
11.c even 5 1 inner 825.2.bx.f 16
15.e even 4 1 495.2.n.a 8
55.j even 10 1 inner 825.2.bx.f 16
55.k odd 20 1 165.2.m.d 8
55.k odd 20 1 825.2.n.g 8
55.k odd 20 1 1815.2.a.p 4
55.k odd 20 1 9075.2.a.di 4
55.l even 20 1 1815.2.a.w 4
55.l even 20 1 9075.2.a.cm 4
165.u odd 20 1 5445.2.a.bf 4
165.v even 20 1 495.2.n.a 8
165.v even 20 1 5445.2.a.bt 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.d 8 5.c odd 4 1
165.2.m.d 8 55.k odd 20 1
495.2.n.a 8 15.e even 4 1
495.2.n.a 8 165.v even 20 1
825.2.n.g 8 5.c odd 4 1
825.2.n.g 8 55.k odd 20 1
825.2.bx.f 16 1.a even 1 1 trivial
825.2.bx.f 16 5.b even 2 1 inner
825.2.bx.f 16 11.c even 5 1 inner
825.2.bx.f 16 55.j even 10 1 inner
1815.2.a.p 4 55.k odd 20 1
1815.2.a.w 4 55.l even 20 1
5445.2.a.bf 4 165.u odd 20 1
5445.2.a.bt 4 165.v even 20 1
9075.2.a.cm 4 55.l even 20 1
9075.2.a.di 4 55.k odd 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

$p$ $F_p(T)$
$2$ $$1 + 7 T^{2} + 195 T^{4} - 403 T^{6} + 354 T^{8} - 137 T^{10} + 25 T^{12} - 2 T^{14} + T^{16}$$
$3$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$1 - 42 T^{2} + 665 T^{4} + 683 T^{6} + 2874 T^{8} - 103 T^{10} + 75 T^{12} - 13 T^{14} + T^{16}$$
$11$ $$( 14641 - 3993 T - 2662 T^{2} + 209 T^{3} + 335 T^{4} + 19 T^{5} - 22 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$13$ $$14641 + 10527 T^{2} + 24275 T^{4} + 2397 T^{6} + 6574 T^{8} - 3477 T^{10} + 725 T^{12} - 42 T^{14} + T^{16}$$
$17$ $$1 + 74 T^{2} + 16087 T^{4} - 88228 T^{6} + 182805 T^{8} + 9628 T^{10} + 3327 T^{12} - 94 T^{14} + T^{16}$$
$19$ $$( 1 - 7 T + 195 T^{2} + 403 T^{3} + 354 T^{4} + 137 T^{5} + 25 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$23$ $$( 201601 + 66031 T^{2} + 4697 T^{4} + 119 T^{6} + T^{8} )^{2}$$
$29$ $$( 249001 - 237025 T + 84767 T^{2} + 10335 T^{3} + 11144 T^{4} + 1125 T^{5} + 113 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$31$ $$( 1560001 - 1287719 T + 498345 T^{2} - 97231 T^{3} + 17274 T^{4} - 2071 T^{5} + 235 T^{6} - 19 T^{7} + T^{8} )^{2}$$
$37$ $$157218521219761 - 8848208637313 T^{2} + 239380989003 T^{4} - 3516201371 T^{6} + 73477880 T^{8} - 961331 T^{10} + 9843 T^{12} - 73 T^{14} + T^{16}$$
$41$ $$( 1681 - 12956 T + 40225 T^{2} - 26019 T^{3} + 6434 T^{4} + 441 T^{5} + 205 T^{6} + 9 T^{7} + T^{8} )^{2}$$
$43$ $$( 75625 + 201125 T^{2} + 12650 T^{4} + 220 T^{6} + T^{8} )^{2}$$
$47$ $$923521 + 2177626 T^{2} + 38740485 T^{4} - 83073841 T^{6} + 66941054 T^{8} - 333571 T^{10} + 10695 T^{12} - 149 T^{14} + T^{16}$$
$53$ $$168823196161 + 130409109709 T^{2} + 286174748472 T^{4} - 11451840553 T^{6} + 213068855 T^{8} - 2046217 T^{10} + 28332 T^{12} - 239 T^{14} + T^{16}$$
$59$ $$( 346921 + 325717 T + 228195 T^{2} + 86697 T^{3} + 18894 T^{4} + 2053 T^{5} + 165 T^{6} + 13 T^{7} + T^{8} )^{2}$$
$61$ $$( 1324801 + 1545793 T + 639460 T^{2} - 110177 T^{3} + 22389 T^{4} - 2123 T^{5} + 210 T^{6} - 13 T^{7} + T^{8} )^{2}$$
$67$ $$( 383161 + 136344 T^{2} + 11462 T^{4} + 201 T^{6} + T^{8} )^{2}$$
$71$ $$( 78961 - 75589 T + 176745 T^{2} - 4771 T^{3} - 3276 T^{4} + 189 T^{5} + 225 T^{6} + 11 T^{7} + T^{8} )^{2}$$
$73$ $$751274631121 - 108853913707 T^{2} + 6983327708 T^{4} - 157957429 T^{6} + 12540455 T^{8} - 253709 T^{10} + 3848 T^{12} - 47 T^{14} + T^{16}$$
$79$ $$( 28561 - 57122 T + 47658 T^{2} - 8554 T^{3} + 9780 T^{4} - 2044 T^{5} + 283 T^{6} - 22 T^{7} + T^{8} )^{2}$$
$83$ $$2825761 + 27519651 T^{2} + 745882295 T^{4} - 280290751 T^{6} + 40797384 T^{8} - 399151 T^{10} + 7335 T^{12} - 109 T^{14} + T^{16}$$
$89$ $$( 209 + 310 T - 89 T^{2} - 10 T^{3} + T^{4} )^{4}$$
$97$ $$1804403844961 - 294418986299 T^{2} + 24006035600 T^{4} - 1219729221 T^{6} + 86003999 T^{8} - 3231501 T^{10} + 56120 T^{12} + 101 T^{14} + T^{16}$$