# Properties

 Label 1764.2.b.m Level $1764$ Weight $2$ Character orbit 1764.b Analytic conductor $14.086$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.15911316233388032.1 Defining polynomial: $$x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 35 x^{8} - 56 x^{7} + 84 x^{6} - 112 x^{5} + 140 x^{4} - 160 x^{3} + 160 x^{2} - 128 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 588) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 35 x^{8} - 56 x^{7} + 84 x^{6} - 112 x^{5} + 140 x^{4} - 160 x^{3} + 160 x^{2} - 128 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{11} + 2 \nu^{9} + 3 \nu^{7} + 4 \nu^{6} - 4 \nu^{5} + 4 \nu^{4} - 4 \nu^{3} + 40 \nu^{2} + 32$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} - 3 \nu^{10} + 8 \nu^{9} - 16 \nu^{8} + 23 \nu^{7} - 37 \nu^{6} + 58 \nu^{5} - 70 \nu^{4} + 88 \nu^{3} - 80 \nu^{2} + 80 \nu - 64$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 31 \nu^{7} - 51 \nu^{6} + 72 \nu^{5} - 102 \nu^{4} + 116 \nu^{3} - 136 \nu^{2} + 128 \nu - 96$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} - 4 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 35 \nu^{7} - 56 \nu^{6} + 84 \nu^{5} - 112 \nu^{4} + 140 \nu^{3} - 160 \nu^{2} + 160 \nu - 128$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 34 \nu^{7} - 55 \nu^{6} + 78 \nu^{5} - 90 \nu^{4} + 132 \nu^{3} - 112 \nu^{2} + 112 \nu - 64$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{11} + 5 \nu^{10} - 12 \nu^{9} + 24 \nu^{8} - 34 \nu^{7} + 63 \nu^{6} - 84 \nu^{5} + 110 \nu^{4} - 128 \nu^{3} + 136 \nu^{2} - 128 \nu + 96$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{11} - 3 \nu^{10} + 7 \nu^{9} - 12 \nu^{8} + 19 \nu^{7} - 33 \nu^{6} + 47 \nu^{5} - 54 \nu^{4} + 70 \nu^{3} - 72 \nu^{2} + 80 \nu - 48$$$$)/16$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{11} + 3 \nu^{10} - 8 \nu^{9} + 14 \nu^{8} - 23 \nu^{7} + 37 \nu^{6} - 50 \nu^{5} + 64 \nu^{4} - 80 \nu^{3} + 84 \nu^{2} - 80 \nu + 48$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$3 \nu^{11} - 8 \nu^{10} + 18 \nu^{9} - 28 \nu^{8} + 49 \nu^{7} - 76 \nu^{6} + 104 \nu^{5} - 136 \nu^{4} + 164 \nu^{3} - 160 \nu^{2} + 144 \nu - 64$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-4 \nu^{11} + 9 \nu^{10} - 20 \nu^{9} + 36 \nu^{8} - 56 \nu^{7} + 83 \nu^{6} - 120 \nu^{5} + 146 \nu^{4} - 192 \nu^{3} + 192 \nu^{2} - 160 \nu + 96$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$3 \nu^{11} - 11 \nu^{10} + 24 \nu^{9} - 44 \nu^{8} + 77 \nu^{7} - 117 \nu^{6} + 166 \nu^{5} - 210 \nu^{4} + 248 \nu^{3} - 280 \nu^{2} + 240 \nu - 128$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{9} + \beta_{7} + \beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{10} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{10} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{11} + \beta_{9} + 6 \beta_{8} + 3 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-3 \beta_{10} + 4 \beta_{8} + 4 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + \beta_{3} + 5 \beta_{2} + \beta_{1} + 3$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$6 \beta_{11} - 5 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - 5 \beta_{2} + 5 \beta_{1} - 9$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$5 \beta_{10} + 14 \beta_{9} + 6 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 9 \beta_{4} - 11 \beta_{3} - 3 \beta_{2} - 3 \beta_{1} - 7$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-10 \beta_{11} + 15 \beta_{9} - 14 \beta_{8} + 5 \beta_{7} + 12 \beta_{6} - 14 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 11 \beta_{2} - 7 \beta_{1} - 1$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-\beta_{10} - 4 \beta_{9} - 16 \beta_{8} - 12 \beta_{7} - 16 \beta_{6} - 13 \beta_{5} + 21 \beta_{4} - 37 \beta_{3} + 7 \beta_{2} + 3 \beta_{1} + 13$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$2 \beta_{11} - 24 \beta_{10} + \beta_{9} + 42 \beta_{8} - \beta_{7} - 46 \beta_{6} - 10 \beta_{5} + 36 \beta_{4} - 39 \beta_{3} - 39 \beta_{2} + 15 \beta_{1} + 5$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 1.34902 − 0.424442i 1.34902 + 0.424442i 1.22594 − 0.705031i 1.22594 + 0.705031i 0.639847 − 1.26119i 0.639847 + 1.26119i 0.250649 − 1.39182i 0.250649 + 1.39182i −0.476589 − 1.33149i −0.476589 + 1.33149i −0.988865 − 1.01101i −0.988865 + 1.01101i
−1.34902 0.424442i 0 1.63970 + 1.14516i 0.127929i 0 0 −1.72593 2.24080i 0 −0.0542984 + 0.172579i
1567.2 −1.34902 + 0.424442i 0 1.63970 1.14516i 0.127929i 0 0 −1.72593 + 2.24080i 0 −0.0542984 0.172579i
1567.3 −1.22594 0.705031i 0 1.00586 + 1.72865i 3.64758i 0 0 −0.0143727 2.82839i 0 −2.57166 + 4.47172i
1567.4 −1.22594 + 0.705031i 0 1.00586 1.72865i 3.64758i 0 0 −0.0143727 + 2.82839i 0 −2.57166 4.47172i
1567.5 −0.639847 1.26119i 0 −1.18119 + 1.61393i 3.10455i 0 0 2.79126 + 0.457034i 0 −3.91542 + 1.98644i
1567.6 −0.639847 + 1.26119i 0 −1.18119 1.61393i 3.10455i 0 0 2.79126 0.457034i 0 −3.91542 1.98644i
1567.7 −0.250649 1.39182i 0 −1.87435 + 0.697718i 3.39209i 0 0 1.44090 + 2.43388i 0 4.72119 0.850222i
1567.8 −0.250649 + 1.39182i 0 −1.87435 0.697718i 3.39209i 0 0 1.44090 2.43388i 0 4.72119 + 0.850222i
1567.9 0.476589 1.33149i 0 −1.54572 1.26915i 0.509876i 0 0 −2.42653 + 1.45325i 0 0.678894 + 0.243001i
1567.10 0.476589 + 1.33149i 0 −1.54572 + 1.26915i 0.509876i 0 0 −2.42653 1.45325i 0 0.678894 0.243001i
1567.11 0.988865 1.01101i 0 −0.0442929 1.99951i 1.12886i 0 0 −2.06533 1.93246i 0 1.14129 + 1.11629i
1567.12 0.988865 + 1.01101i 0 −0.0442929 + 1.99951i 1.12886i 0 0 −2.06533 + 1.93246i 0 1.14129 1.11629i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.m 12
3.b odd 2 1 588.2.b.d yes 12
4.b odd 2 1 1764.2.b.l 12
7.b odd 2 1 1764.2.b.l 12
12.b even 2 1 588.2.b.c 12
21.c even 2 1 588.2.b.c 12
21.g even 6 2 588.2.o.f 24
21.h odd 6 2 588.2.o.e 24
28.d even 2 1 inner 1764.2.b.m 12
84.h odd 2 1 588.2.b.d yes 12
84.j odd 6 2 588.2.o.e 24
84.n even 6 2 588.2.o.f 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.b.c 12 12.b even 2 1
588.2.b.c 12 21.c even 2 1
588.2.b.d yes 12 3.b odd 2 1
588.2.b.d yes 12 84.h odd 2 1
588.2.o.e 24 21.h odd 6 2
588.2.o.e 24 84.j odd 6 2
588.2.o.f 24 21.g even 6 2
588.2.o.f 24 84.n even 6 2
1764.2.b.l 12 4.b odd 2 1
1764.2.b.l 12 7.b odd 2 1
1764.2.b.m 12 1.a even 1 1 trivial
1764.2.b.m 12 28.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{12} + 36 T_{5}^{10} + 446 T_{5}^{8} + 2096 T_{5}^{6} + 2428 T_{5}^{4} + 528 T_{5}^{2} + 8$$ $$T_{11}^{12} + 104 T_{11}^{10} + 4040 T_{11}^{8} + 72672 T_{11}^{6} + 603920 T_{11}^{4} + 1962496 T_{11}^{2} + 1968128$$ $$T_{19}^{6} - 72 T_{19}^{4} - 32 T_{19}^{3} + 1288 T_{19}^{2} + 1280 T_{19} - 256$$ $$T_{29}^{6} + 16 T_{29}^{5} + 26 T_{29}^{4} - 592 T_{29}^{3} - 2268 T_{29}^{2} + 1440 T_{29} + 9544$$ $$T_{53}^{6} - 16 T_{53}^{5} + 24 T_{53}^{4} + 288 T_{53}^{3} + 144 T_{53}^{2} - 384 T_{53} - 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 128 T + 160 T^{2} + 160 T^{3} + 140 T^{4} + 112 T^{5} + 84 T^{6} + 56 T^{7} + 35 T^{8} + 20 T^{9} + 10 T^{10} + 4 T^{11} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$8 + 528 T^{2} + 2428 T^{4} + 2096 T^{6} + 446 T^{8} + 36 T^{10} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$1968128 + 1962496 T^{2} + 603920 T^{4} + 72672 T^{6} + 4040 T^{8} + 104 T^{10} + T^{12}$$
$13$ $$40328 + 231280 T^{2} + 261356 T^{4} + 45488 T^{6} + 3094 T^{8} + 92 T^{10} + T^{12}$$
$17$ $$8405000 + 5426000 T^{2} + 1170556 T^{4} + 111344 T^{6} + 5182 T^{8} + 116 T^{10} + T^{12}$$
$19$ $$( -256 + 1280 T + 1288 T^{2} - 32 T^{3} - 72 T^{4} + T^{6} )^{2}$$
$23$ $$518162432 + 148759040 T^{2} + 15486736 T^{4} + 745504 T^{6} + 18120 T^{8} + 216 T^{10} + T^{12}$$
$29$ $$( 9544 + 1440 T - 2268 T^{2} - 592 T^{3} + 26 T^{4} + 16 T^{5} + T^{6} )^{2}$$
$31$ $$( -6272 - 4032 T + 776 T^{2} + 416 T^{3} - 56 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$37$ $$( 10424 - 5568 T - 4148 T^{2} + 1280 T^{3} - 22 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$41$ $$84872 + 3872656 T^{2} + 2756604 T^{4} + 359408 T^{6} + 14878 T^{8} + 228 T^{10} + T^{12}$$
$43$ $$6422528 + 10633216 T^{2} + 3342592 T^{4} + 385792 T^{6} + 17504 T^{8} + 240 T^{10} + T^{12}$$
$47$ $$( -62336 - 9280 T + 9144 T^{2} + 160 T^{3} - 192 T^{4} + T^{6} )^{2}$$
$53$ $$( -256 - 384 T + 144 T^{2} + 288 T^{3} + 24 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$59$ $$( 256 - 768 T + 184 T^{2} + 736 T^{3} - 144 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$61$ $$7688 + 206064 T^{2} + 681324 T^{4} + 245552 T^{6} + 11574 T^{8} + 188 T^{10} + T^{12}$$
$67$ $$8388608 + 156237824 T^{2} + 57819136 T^{4} + 3080192 T^{6} + 59648 T^{8} + 448 T^{10} + T^{12}$$
$71$ $$4917248 + 9593344 T^{2} + 5741840 T^{4} + 1007328 T^{6} + 29000 T^{8} + 296 T^{10} + T^{12}$$
$73$ $$19046792 + 36105776 T^{2} + 13724652 T^{4} + 1446512 T^{6} + 43158 T^{8} + 396 T^{10} + T^{12}$$
$79$ $$1038221312 + 811810816 T^{2} + 104620288 T^{4} + 4009728 T^{6} + 63584 T^{8} + 432 T^{10} + T^{12}$$
$83$ $$( 82432 - 71424 T + 8928 T^{2} + 1472 T^{3} - 208 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$89$ $$146068232 + 737059664 T^{2} + 78033532 T^{4} + 2963696 T^{6} + 49790 T^{8} + 372 T^{10} + T^{12}$$
$97$ $$1057448072 + 582577072 T^{2} + 92381420 T^{4} + 4243440 T^{6} + 75094 T^{8} + 492 T^{10} + T^{12}$$