# Properties

 Label 28.3.c.a Level 28 Weight 3 Character orbit 28.c Analytic conductor 0.763 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 28.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.762944740209$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.1539727.2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{3} q^{3} + ( -\beta_{1} - \beta_{4} ) q^{4} + ( -1 + \beta_{2} + \beta_{5} ) q^{5} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{6} -\beta_{1} q^{7} + ( -2 - \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{8} + ( -1 - 2 \beta_{2} + 2 \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{3} q^{3} + ( -\beta_{1} - \beta_{4} ) q^{4} + ( -1 + \beta_{2} + \beta_{5} ) q^{5} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{6} -\beta_{1} q^{7} + ( -2 - \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{8} + ( -1 - 2 \beta_{2} + 2 \beta_{4} ) q^{9} + ( -5 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{10} + ( 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{11} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{12} + ( 1 + 3 \beta_{2} - 4 \beta_{4} - \beta_{5} ) q^{13} + ( 1 + \beta_{3} + \beta_{5} ) q^{14} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{15} + ( 2 - 3 \beta_{1} - 2 \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{16} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{17} + ( 6 + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{18} + ( -8 \beta_{2} + \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{19} + ( -6 + 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{20} + ( 1 - 3 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{21} + ( 10 - 2 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{22} + ( -4 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{23} + ( 20 + 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{24} + ( -3 - 6 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{25} + ( -7 - 2 \beta_{1} - \beta_{2} + 5 \beta_{3} + 8 \beta_{4} - 5 \beta_{5} ) q^{26} + ( 8 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -6 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{28} + ( -8 + 6 \beta_{2} + 6 \beta_{5} ) q^{29} + ( -12 - 4 \beta_{1} + 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{30} + ( 8 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{31} + ( -18 + 7 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{32} + ( 8 + 16 \beta_{2} - 8 \beta_{4} + 8 \beta_{5} ) q^{33} + ( 10 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{34} + ( 2 \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{35} + ( -20 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{36} + ( 8 - 10 \beta_{2} + 4 \beta_{4} - 6 \beta_{5} ) q^{37} + ( -33 - 10 \beta_{1} - \beta_{2} - \beta_{3} - 8 \beta_{4} + \beta_{5} ) q^{38} + ( -12 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{39} + ( 8 + 8 \beta_{1} + 8 \beta_{2} + 8 \beta_{4} - 8 \beta_{5} ) q^{40} + ( -4 + 2 \beta_{2} + 4 \beta_{4} + 6 \beta_{5} ) q^{41} + ( 11 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{42} + ( 12 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{43} + ( 28 + 12 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{44} + ( 3 - 3 \beta_{2} + 4 \beta_{4} + \beta_{5} ) q^{45} + ( 20 + 4 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{46} + ( -8 \beta_{1} - 16 \beta_{2} - 6 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} ) q^{47} + ( -12 - 10 \beta_{1} - 24 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{48} -7 q^{49} + ( 26 - 8 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{50} + ( 8 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{51} + ( 46 - 8 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} + 10 \beta_{5} ) q^{52} + ( 18 - 8 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} ) q^{53} + ( -6 + 4 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 10 \beta_{5} ) q^{54} + ( -16 \beta_{1} - 8 \beta_{2} + 12 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{55} + ( -10 + 3 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{56} + ( 26 + 2 \beta_{2} - 10 \beta_{4} - 8 \beta_{5} ) q^{57} + ( -30 + 12 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 6 \beta_{5} ) q^{58} + ( 8 \beta_{1} - 11 \beta_{3} ) q^{59} + ( -32 - 8 \beta_{1} + 8 \beta_{2} + 8 \beta_{5} ) q^{60} + ( -29 + 5 \beta_{2} - 8 \beta_{4} - 3 \beta_{5} ) q^{61} + ( 26 - 4 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} ) q^{62} + ( 3 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{63} + ( -38 - 3 \beta_{1} + 16 \beta_{2} - 10 \beta_{3} - \beta_{4} - 4 \beta_{5} ) q^{64} + ( -26 + 10 \beta_{2} - 10 \beta_{4} ) q^{65} + ( -64 + 16 \beta_{1} - 8 \beta_{2} + 16 \beta_{4} ) q^{66} + ( -4 \beta_{1} - 4 \beta_{2} + 24 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{67} + ( 12 - 6 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{68} + ( -4 - 12 \beta_{2} + 12 \beta_{4} ) q^{69} + ( 15 + 6 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{70} + ( 8 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{71} + ( 22 - 13 \beta_{1} + 16 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} ) q^{72} + ( -18 - 12 \beta_{2} + 8 \beta_{4} - 4 \beta_{5} ) q^{73} + ( 42 - 12 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} - 2 \beta_{5} ) q^{74} + ( 24 \beta_{1} + 16 \beta_{2} + 13 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} ) q^{75} + ( -18 - 6 \beta_{1} + 34 \beta_{2} + 18 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{76} + ( 18 + 2 \beta_{2} - 6 \beta_{4} - 4 \beta_{5} ) q^{77} + ( 40 + 28 \beta_{1} + 12 \beta_{2} + 24 \beta_{3} + 4 \beta_{4} ) q^{78} + ( 8 \beta_{1} - 12 \beta_{3} ) q^{79} + ( 56 + 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} ) q^{80} + ( -37 + 2 \beta_{2} + 6 \beta_{4} + 8 \beta_{5} ) q^{81} + ( -18 + 12 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} - 8 \beta_{4} + 10 \beta_{5} ) q^{82} + ( -24 \beta_{1} - 8 \beta_{2} - 23 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{83} + ( -14 - 2 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{84} + ( -42 + 10 \beta_{2} - 4 \beta_{4} + 6 \beta_{5} ) q^{85} + ( -26 + 2 \beta_{2} - 10 \beta_{3} - 4 \beta_{4} - 14 \beta_{5} ) q^{86} + ( -24 \beta_{1} - 24 \beta_{2} + 2 \beta_{3} - 12 \beta_{4} + 12 \beta_{5} ) q^{87} + ( -8 - 20 \beta_{1} - 32 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 16 \beta_{5} ) q^{88} + ( 66 - 24 \beta_{2} + 20 \beta_{4} - 4 \beta_{5} ) q^{89} + ( 7 + 2 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - 8 \beta_{4} + 5 \beta_{5} ) q^{90} + ( -6 \beta_{1} - 12 \beta_{2} - 11 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{91} + ( -16 - 24 \beta_{2} - 16 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} ) q^{92} + ( 44 + 12 \beta_{2} - 4 \beta_{4} + 8 \beta_{5} ) q^{93} + ( -50 - 4 \beta_{1} + 6 \beta_{2} + 14 \beta_{3} - 16 \beta_{4} + 2 \beta_{5} ) q^{94} + ( 36 \beta_{1} + 4 \beta_{2} - 16 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{95} + ( 20 - 14 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} - 26 \beta_{4} + 8 \beta_{5} ) q^{96} + ( 40 + 6 \beta_{2} - 12 \beta_{4} - 6 \beta_{5} ) q^{97} + 7 \beta_{2} q^{98} + ( -28 \beta_{1} + 4 \beta_{2} - 22 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{2} + q^{4} - 4q^{5} + 6q^{6} - 13q^{8} - 10q^{9} + O(q^{10})$$ $$6q - q^{2} + q^{4} - 4q^{5} + 6q^{6} - 13q^{8} - 10q^{9} - 28q^{10} + 6q^{12} + 12q^{13} + 7q^{14} + 17q^{16} - 4q^{17} + 43q^{18} - 32q^{20} + 52q^{22} + 122q^{24} - 30q^{25} - 56q^{26} - 35q^{28} - 36q^{29} - 64q^{30} - 101q^{32} + 80q^{33} + 58q^{34} - 131q^{36} + 28q^{37} - 190q^{38} + 40q^{40} - 20q^{41} + 70q^{42} + 164q^{44} + 12q^{45} + 120q^{46} - 98q^{48} - 42q^{49} + 161q^{50} + 292q^{52} + 92q^{53} - 44q^{54} - 49q^{56} + 160q^{57} - 166q^{58} - 176q^{60} - 164q^{61} + 148q^{62} - 215q^{64} - 136q^{65} - 408q^{66} + 62q^{68} - 48q^{69} + 84q^{70} + 151q^{72} - 132q^{73} + 250q^{74} - 78q^{76} + 112q^{77} + 248q^{78} + 312q^{80} - 218q^{81} - 86q^{82} - 98q^{84} - 232q^{85} - 164q^{86} - 100q^{88} + 348q^{89} + 52q^{90} - 104q^{92} + 288q^{93} - 276q^{94} + 170q^{96} + 252q^{97} + 7q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{3} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{2} - 2 \nu + 2$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{3} + \nu^{2} + 4 \nu - 4$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} + 3 \nu^{2} + 4 \nu - 8$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} + 2 \nu^{3} - \nu^{2} - 2 \nu + 6$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-\nu^{5} + \nu^{4} + \nu^{3} - 2 \nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{3} + \beta_{2} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_{1} + 3$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} + 4 \beta_{4} - \beta_{3} + 3 \beta_{2} - 5$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 6 \beta_{1} + 9$$$$)/4$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/28\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0.841985 − 1.13625i 0.841985 + 1.13625i 1.35935 + 0.390070i 1.35935 − 0.390070i −1.20134 − 0.746179i −1.20134 + 0.746179i
−1.92411 0.545716i 4.54500i 3.40439 + 2.10003i 1.36794 2.48028 8.74507i 2.64575i −5.40439 5.89853i −11.6570 −2.63206 0.746506i
15.2 −1.92411 + 0.545716i 4.54500i 3.40439 2.10003i 1.36794 2.48028 + 8.74507i 2.64575i −5.40439 + 5.89853i −11.6570 −2.63206 + 0.746506i
15.3 −0.163664 1.99329i 1.56028i −3.94643 + 0.652459i 3.43742 −3.11009 + 0.255361i 2.64575i 1.94643 + 7.75960i 6.56553 −0.562581 6.85178i
15.4 −0.163664 + 1.99329i 1.56028i −3.94643 0.652459i 3.43742 −3.11009 0.255361i 2.64575i 1.94643 7.75960i 6.56553 −0.562581 + 6.85178i
15.5 1.58777 1.21613i 2.98472i 1.04204 3.86188i −6.80536 3.62981 + 4.73905i 2.64575i −3.04204 7.39905i 0.0914622 −10.8054 + 8.27622i
15.6 1.58777 + 1.21613i 2.98472i 1.04204 + 3.86188i −6.80536 3.62981 4.73905i 2.64575i −3.04204 + 7.39905i 0.0914622 −10.8054 8.27622i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 15.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(28, [\chi])$$.