Properties

 Label 28.3.g.a Level 28 Weight 3 Character orbit 28.g Analytic conductor 0.763 Analytic rank 0 Dimension 12 CM No Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 28.g (of order $$6$$ and degree $$2$$)

Newform invariants

 Self dual: No Analytic conductor: $$0.762944740209$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{6} q^{2}$$ $$+ \beta_{4} q^{3}$$ $$+ ( -\beta_{1} + \beta_{5} + \beta_{9} ) q^{4}$$ $$+ ( -1 - \beta_{5} + \beta_{7} - \beta_{9} ) q^{5}$$ $$+ ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{6}$$ $$+ ( -1 + \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{7}$$ $$+ ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{8}$$ $$+ ( 1 + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{6} q^{2}$$ $$+ \beta_{4} q^{3}$$ $$+ ( -\beta_{1} + \beta_{5} + \beta_{9} ) q^{4}$$ $$+ ( -1 - \beta_{5} + \beta_{7} - \beta_{9} ) q^{5}$$ $$+ ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{6}$$ $$+ ( -1 + \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{7}$$ $$+ ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{8}$$ $$+ ( 1 + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{9}$$ $$+ ( \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{10}$$ $$+ ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{11}$$ $$+ ( -6 - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{12}$$ $$+ ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} + \beta_{8} - \beta_{11} ) q^{13}$$ $$+ ( 3 - \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{14}$$ $$+ ( 2 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} + \beta_{8} + 3 \beta_{10} + 2 \beta_{11} ) q^{15}$$ $$+ ( 2 + 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{10} - 2 \beta_{11} ) q^{16}$$ $$+ ( -\beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{17}$$ $$+ ( 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - 11 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{18}$$ $$+ ( 1 + \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{19}$$ $$+ ( 13 - \beta_{1} - 2 \beta_{3} - 4 \beta_{10} + 2 \beta_{11} ) q^{20}$$ $$+ ( -7 - 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{21}$$ $$+ ( 3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} + 2 \beta_{8} - 2 \beta_{11} ) q^{22}$$ $$+ ( 3 - 3 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{23}$$ $$+ ( 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 9 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{24}$$ $$+ ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{25}$$ $$+ ( 9 + 2 \beta_{4} + 9 \beta_{5} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{26}$$ $$+ ( -5 - \beta_{1} - 10 \beta_{2} + 5 \beta_{3} - 10 \beta_{6} + \beta_{8} - \beta_{10} - 5 \beta_{11} ) q^{27}$$ $$+ ( -4 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 12 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 4 \beta_{10} + 6 \beta_{11} ) q^{28}$$ $$+ ( 7 + 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 6 \beta_{6} + 3 \beta_{8} + 3 \beta_{11} ) q^{29}$$ $$+ ( -15 - 8 \beta_{4} - 15 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + \beta_{9} - 8 \beta_{10} - 6 \beta_{11} ) q^{30}$$ $$+ ( 3 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} - 8 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{31}$$ $$+ ( 4 \beta_{1} - 4 \beta_{3} - 8 \beta_{4} + 16 \beta_{5} - 4 \beta_{9} ) q^{32}$$ $$+ ( -3 - 3 \beta_{5} - 4 \beta_{6} + 2 \beta_{11} ) q^{33}$$ $$+ ( -29 - 6 \beta_{1} - 5 \beta_{2} - \beta_{3} - 5 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{34}$$ $$+ ( -4 - \beta_{1} - 10 \beta_{2} + 5 \beta_{3} + 11 \beta_{4} + \beta_{5} - 8 \beta_{6} + 3 \beta_{7} + \beta_{8} + 3 \beta_{9} + 5 \beta_{10} - 4 \beta_{11} ) q^{35}$$ $$+ ( -10 + 2 \beta_{1} + 12 \beta_{2} + 12 \beta_{6} - 4 \beta_{8} + 8 \beta_{10} ) q^{36}$$ $$+ ( 10 + 10 \beta_{5} - 10 \beta_{6} + 4 \beta_{7} - 4 \beta_{9} + 5 \beta_{11} ) q^{37}$$ $$+ ( -\beta_{1} - 5 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 6 \beta_{7} + 6 \beta_{8} + \beta_{9} ) q^{38}$$ $$+ ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 6 \beta_{4} - \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{39}$$ $$+ ( -17 + 6 \beta_{4} - 17 \beta_{5} + 14 \beta_{6} - 2 \beta_{7} + 4 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} ) q^{40}$$ $$+ ( -1 - 5 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} + 10 \beta_{6} - 5 \beta_{8} - 5 \beta_{11} ) q^{41}$$ $$+ ( 23 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 31 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} - 6 \beta_{10} + 3 \beta_{11} ) q^{42}$$ $$+ ( -2 - 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} - 16 \beta_{10} - 2 \beta_{11} ) q^{43}$$ $$+ ( 10 - 2 \beta_{4} + 10 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - 5 \beta_{11} ) q^{44}$$ $$+ ( \beta_{1} - 10 \beta_{2} - 5 \beta_{3} - 25 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{45}$$ $$+ ( -9 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 24 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} + 9 \beta_{9} ) q^{46}$$ $$+ ( -3 \beta_{4} - 7 \beta_{7} - 7 \beta_{9} - 3 \beta_{10} ) q^{47}$$ $$+ ( 46 + 4 \beta_{1} - 6 \beta_{3} + 4 \beta_{10} + 6 \beta_{11} ) q^{48}$$ $$+ ( 14 + 9 \beta_{1} + 2 \beta_{2} + \beta_{3} + 10 \beta_{6} + 6 \beta_{7} + 9 \beta_{8} - 6 \beta_{9} - 5 \beta_{11} ) q^{49}$$ $$+ ( 18 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{6} - 4 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{50}$$ $$+ ( 7 + 17 \beta_{4} + 7 \beta_{5} + 14 \beta_{6} + 7 \beta_{7} + 7 \beta_{9} + 17 \beta_{10} + 7 \beta_{11} ) q^{51}$$ $$+ ( 2 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 14 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{52}$$ $$+ ( -6 \beta_{1} + 2 \beta_{2} + \beta_{3} + 16 \beta_{5} - 6 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} ) q^{53}$$ $$+ ( 27 + 27 \beta_{5} - \beta_{6} - 2 \beta_{7} - 9 \beta_{9} - 2 \beta_{11} ) q^{54}$$ $$+ ( 3 + 2 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 6 \beta_{6} - 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{11} ) q^{55}$$ $$+ ( -21 - 4 \beta_{1} + 14 \beta_{2} - \beta_{3} - 4 \beta_{4} - 46 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 10 \beta_{10} - \beta_{11} ) q^{56}$$ $$+ ( -23 - 6 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} - 6 \beta_{8} + 2 \beta_{11} ) q^{57}$$ $$+ ( -33 + 6 \beta_{4} - 33 \beta_{5} + 4 \beta_{6} + 6 \beta_{7} - 6 \beta_{9} + 6 \beta_{10} - 3 \beta_{11} ) q^{58}$$ $$+ ( -8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} + 8 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} ) q^{59}$$ $$+ ( -3 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 18 \beta_{4} + 40 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{60}$$ $$+ ( 20 + 20 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} + 8 \beta_{9} - \beta_{11} ) q^{61}$$ $$+ ( -39 + 13 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 3 \beta_{6} - 6 \beta_{8} + 6 \beta_{11} ) q^{62}$$ $$+ ( -1 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 18 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + 4 \beta_{8} - \beta_{9} - 6 \beta_{10} - \beta_{11} ) q^{63}$$ $$+ ( -28 - 8 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} - 24 \beta_{6} - 8 \beta_{8} - 8 \beta_{10} - 4 \beta_{11} ) q^{64}$$ $$+ ( -19 - 19 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} + 7 \beta_{9} + \beta_{11} ) q^{65}$$ $$+ ( 4 \beta_{1} + 5 \beta_{2} - 20 \beta_{5} - 4 \beta_{9} ) q^{66}$$ $$+ ( -2 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} + 9 \beta_{4} + 8 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{67}$$ $$+ ( -23 - 8 \beta_{4} - 23 \beta_{5} - 36 \beta_{6} - 12 \beta_{7} - 3 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} ) q^{68}$$ $$+ ( -19 + 9 \beta_{1} + 9 \beta_{8} ) q^{69}$$ $$+ ( 27 + 9 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 30 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} - 13 \beta_{9} + 10 \beta_{10} + 3 \beta_{11} ) q^{70}$$ $$+ ( 4 + 6 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 8 \beta_{6} - 6 \beta_{8} + 24 \beta_{10} + 4 \beta_{11} ) q^{71}$$ $$+ ( 22 - 12 \beta_{4} + 22 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} + 4 \beta_{9} - 12 \beta_{10} + 2 \beta_{11} ) q^{72}$$ $$+ ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 37 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{73}$$ $$+ ( 10 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 46 \beta_{5} - 8 \beta_{7} - 8 \beta_{8} - 10 \beta_{9} ) q^{74}$$ $$+ ( -2 - 8 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} - 6 \beta_{9} - 8 \beta_{10} - 2 \beta_{11} ) q^{75}$$ $$+ ( 46 - \beta_{1} + 2 \beta_{2} + 9 \beta_{3} + 2 \beta_{6} + 2 \beta_{8} + 6 \beta_{10} - 9 \beta_{11} ) q^{76}$$ $$+ ( -35 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 21 \beta_{5} - 10 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 5 \beta_{11} ) q^{77}$$ $$+ ( 3 - 4 \beta_{1} - 3 \beta_{3} - 6 \beta_{8} - 6 \beta_{10} + 3 \beta_{11} ) q^{78}$$ $$+ ( -7 - 27 \beta_{4} - 7 \beta_{5} - 14 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 27 \beta_{10} - 7 \beta_{11} ) q^{79}$$ $$+ ( -8 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} + 8 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} ) q^{80}$$ $$+ ( \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 24 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{81}$$ $$+ ( 55 - 10 \beta_{4} + 55 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} + 10 \beta_{9} - 10 \beta_{10} + 5 \beta_{11} ) q^{82}$$ $$+ ( 2 + 6 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} - 6 \beta_{8} + 8 \beta_{10} + 2 \beta_{11} ) q^{83}$$ $$+ ( -27 + 2 \beta_{1} - 36 \beta_{2} + 4 \beta_{3} + 20 \beta_{4} - 37 \beta_{5} - 12 \beta_{6} + 8 \beta_{7} + 12 \beta_{8} + \beta_{9} + 4 \beta_{10} - 6 \beta_{11} ) q^{84}$$ $$+ ( 18 - 10 \beta_{1} + 14 \beta_{2} + 7 \beta_{3} + 14 \beta_{6} - 10 \beta_{8} - 7 \beta_{11} ) q^{85}$$ $$+ ( 12 + 32 \beta_{4} + 12 \beta_{5} + 16 \beta_{6} + 12 \beta_{9} + 32 \beta_{10} + 16 \beta_{11} ) q^{86}$$ $$+ ( -3 \beta_{1} - 30 \beta_{2} + 15 \beta_{3} + 34 \beta_{4} + 15 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{87}$$ $$+ ( -4 \beta_{1} - 14 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 37 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} ) q^{88}$$ $$+ ( -1 - \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{9} - 2 \beta_{11} ) q^{89}$$ $$+ ( -51 - 10 \beta_{1} + 20 \beta_{2} + \beta_{3} + 20 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{90}$$ $$+ ( 3 - \beta_{1} + 18 \beta_{2} - 9 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} + \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{91}$$ $$+ ( -24 + 3 \beta_{1} + 18 \beta_{2} - \beta_{3} + 18 \beta_{6} + 18 \beta_{8} - 14 \beta_{10} + \beta_{11} ) q^{92}$$ $$+ ( 24 + 24 \beta_{5} + 22 \beta_{6} + 10 \beta_{7} - 10 \beta_{9} - 11 \beta_{11} ) q^{93}$$ $$+ ( -3 \beta_{1} + 11 \beta_{2} - 18 \beta_{3} - 8 \beta_{4} - 21 \beta_{5} - 14 \beta_{7} - 14 \beta_{8} + 3 \beta_{9} ) q^{94}$$ $$+ ( 14 \beta_{2} - 7 \beta_{3} + 13 \beta_{4} - 7 \beta_{5} ) q^{95}$$ $$+ ( -44 - 44 \beta_{5} + 40 \beta_{6} + 8 \beta_{7} - 4 \beta_{9} ) q^{96}$$ $$+ ( 55 - 3 \beta_{1} - 30 \beta_{2} - 15 \beta_{3} - 30 \beta_{6} - 3 \beta_{8} + 15 \beta_{11} ) q^{97}$$ $$+ ( 1 - 8 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 47 \beta_{5} + 15 \beta_{6} + 6 \beta_{7} - 12 \beta_{8} + 10 \beta_{9} + 18 \beta_{10} - 9 \beta_{11} ) q^{98}$$ $$+ ( 3 + 5 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 6 \beta_{6} - 5 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut -\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 12q^{6}$$ $$\mathstrut -\mathstrut 8q^{8}$$ $$\mathstrut +\mathstrut 4q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$12q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut -\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 12q^{6}$$ $$\mathstrut -\mathstrut 8q^{8}$$ $$\mathstrut +\mathstrut 4q^{9}$$ $$\mathstrut -\mathstrut 2q^{10}$$ $$\mathstrut -\mathstrut 24q^{12}$$ $$\mathstrut -\mathstrut 24q^{13}$$ $$\mathstrut +\mathstrut 2q^{14}$$ $$\mathstrut +\mathstrut 16q^{16}$$ $$\mathstrut -\mathstrut 2q^{17}$$ $$\mathstrut +\mathstrut 56q^{18}$$ $$\mathstrut +\mathstrut 152q^{20}$$ $$\mathstrut -\mathstrut 78q^{21}$$ $$\mathstrut +\mathstrut 44q^{22}$$ $$\mathstrut -\mathstrut 44q^{24}$$ $$\mathstrut +\mathstrut 56q^{26}$$ $$\mathstrut +\mathstrut 8q^{28}$$ $$\mathstrut +\mathstrut 72q^{29}$$ $$\mathstrut -\mathstrut 74q^{30}$$ $$\mathstrut -\mathstrut 112q^{32}$$ $$\mathstrut -\mathstrut 14q^{33}$$ $$\mathstrut -\mathstrut 316q^{34}$$ $$\mathstrut -\mathstrut 160q^{36}$$ $$\mathstrut +\mathstrut 86q^{37}$$ $$\mathstrut -\mathstrut 2q^{38}$$ $$\mathstrut -\mathstrut 148q^{40}$$ $$\mathstrut +\mathstrut 8q^{41}$$ $$\mathstrut +\mathstrut 68q^{42}$$ $$\mathstrut +\mathstrut 64q^{44}$$ $$\mathstrut +\mathstrut 156q^{45}$$ $$\mathstrut +\mathstrut 162q^{46}$$ $$\mathstrut +\mathstrut 512q^{48}$$ $$\mathstrut +\mathstrut 108q^{49}$$ $$\mathstrut +\mathstrut 208q^{50}$$ $$\mathstrut -\mathstrut 64q^{52}$$ $$\mathstrut -\mathstrut 74q^{53}$$ $$\mathstrut +\mathstrut 182q^{54}$$ $$\mathstrut +\mathstrut 16q^{56}$$ $$\mathstrut -\mathstrut 220q^{57}$$ $$\mathstrut -\mathstrut 176q^{58}$$ $$\mathstrut -\mathstrut 232q^{60}$$ $$\mathstrut +\mathstrut 86q^{61}$$ $$\mathstrut -\mathstrut 532q^{62}$$ $$\mathstrut -\mathstrut 160q^{64}$$ $$\mathstrut -\mathstrut 140q^{65}$$ $$\mathstrut +\mathstrut 102q^{66}$$ $$\mathstrut -\mathstrut 68q^{68}$$ $$\mathstrut -\mathstrut 300q^{69}$$ $$\mathstrut +\mathstrut 90q^{70}$$ $$\mathstrut +\mathstrut 152q^{72}$$ $$\mathstrut -\mathstrut 234q^{73}$$ $$\mathstrut +\mathstrut 290q^{74}$$ $$\mathstrut +\mathstrut 576q^{76}$$ $$\mathstrut -\mathstrut 262q^{77}$$ $$\mathstrut +\mathstrut 64q^{78}$$ $$\mathstrut +\mathstrut 146q^{81}$$ $$\mathstrut +\mathstrut 272q^{82}$$ $$\mathstrut -\mathstrut 28q^{84}$$ $$\mathstrut +\mathstrut 268q^{85}$$ $$\mathstrut -\mathstrut 16q^{86}$$ $$\mathstrut -\mathstrut 188q^{88}$$ $$\mathstrut +\mathstrut 6q^{89}$$ $$\mathstrut -\mathstrut 640q^{90}$$ $$\mathstrut -\mathstrut 448q^{92}$$ $$\mathstrut +\mathstrut 162q^{93}$$ $$\mathstrut +\mathstrut 102q^{94}$$ $$\mathstrut -\mathstrut 320q^{96}$$ $$\mathstrut +\mathstrut 744q^{97}$$ $$\mathstrut -\mathstrut 190q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12}\mathstrut -\mathstrut$$ $$3$$ $$x^{11}\mathstrut -\mathstrut$$ $$4$$ $$x^{10}\mathstrut +\mathstrut$$ $$3$$ $$x^{9}\mathstrut +\mathstrut$$ $$86$$ $$x^{8}\mathstrut -\mathstrut$$ $$163$$ $$x^{7}\mathstrut +\mathstrut$$ $$155$$ $$x^{6}\mathstrut -\mathstrut$$ $$166$$ $$x^{5}\mathstrut +\mathstrut$$ $$164$$ $$x^{4}\mathstrut -\mathstrut$$ $$116$$ $$x^{3}\mathstrut +\mathstrut$$ $$60$$ $$x^{2}\mathstrut -\mathstrut$$ $$20$$ $$x\mathstrut +\mathstrut$$ $$4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$548052$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$1422116$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$6191612$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$21007396$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$23867696$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$160225554$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$25497908$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$20918128$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$126880702$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$5442024$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1537456$$ $$\nu\mathstrut -\mathstrut$$ $$9243685$$$$)/5128417$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$2016555$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$4608465$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$11867940$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$1459025$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$175954938$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$201471321$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$122846749$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$215038026$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$177078348$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$48635510$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$46482276$$ $$\nu\mathstrut +\mathstrut$$ $$13204764$$$$)/10256834$$ $$\beta_{3}$$ $$=$$ $$($$$$6080681$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$174798$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$64448111$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$87898803$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$490503169$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$562380793$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$787680086$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$461315491$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$855472904$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$799718192$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$519820470$$ $$\nu\mathstrut +\mathstrut$$ $$199145960$$$$)/20513668$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$8092731$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$13062376$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$58450763$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$38422753$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$688812905$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$358968697$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$67350324$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$436258613$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$177419776$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$102428648$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$66312934$$ $$\nu\mathstrut -\mathstrut$$ $$38518016$$$$)/20513668$$ $$\beta_{5}$$ $$=$$ $$($$$$8864783$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$20455402$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$48834261$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$8785933$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$751559539$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$927115697$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$797312818$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$981317799$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$836256216$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$526622432$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$236999870$$ $$\nu\mathstrut -\mathstrut$$ $$70375800$$$$)/20513668$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$2781796$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$6284192$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$15510516$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$3410783$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$234558069$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$282488251$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$243549767$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$278083421$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$254949459$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$160381312$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$43781960$$ $$\nu\mathstrut +\mathstrut$$ $$15160294$$$$)/5128417$$ $$\beta_{7}$$ $$=$$ $$($$$$11588911$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$2778978$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$108332469$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$167711837$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$897253275$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$805776575$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$546730846$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$195583759$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$691987512$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$476417312$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$253397978$$ $$\nu\mathstrut +\mathstrut$$ $$51100076$$$$)/20513668$$ $$\beta_{8}$$ $$=$$ $$($$$$7577433$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$23003531$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$36322258$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$36406939$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$694698148$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$1232772201$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$650456281$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$847212044$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$921404444$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$415827090$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$151851740$$ $$\nu\mathstrut -\mathstrut$$ $$10748502$$$$)/10256834$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$18854063$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$54731322$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$82312181$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$56986907$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$1629791971$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$2949221097$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$2465476938$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$2338940247$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$2643323856$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1684290272$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$641927134$$ $$\nu\mathstrut +\mathstrut$$ $$162375652$$$$)/20513668$$ $$\beta_{10}$$ $$=$$ $$($$$$13906733$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$32333536$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$77826224$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$8906454$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$1190477049$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$1470041616$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$1124752022$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1425602332$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$1248314483$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$684256636$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$248435884$$ $$\nu\mathstrut -\mathstrut$$ $$51147512$$$$)/10256834$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$6254493$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$16328866$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$30974575$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$6021165$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$537499493$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$811952879$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$686835394$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$792739733$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$729655908$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$462671520$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$200722666$$ $$\nu\mathstrut +\mathstrut$$ $$44275732$$$$)/2930524$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$3$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$2$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$9$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$18$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$25$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$\beta_{11}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$18$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$54$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$75$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$75$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$52$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$293$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$122$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$81$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$52$$ $$\beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$71$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$462$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$122$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$482$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$71$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$482$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$70$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$379$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$651$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$622$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$532$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$462$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$462$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$2987$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$622$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2987$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$90$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$622$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$622$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$391$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$4734$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1283$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3830$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$90$$ $$\beta_{1}$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-$$$$4513$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$802$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$4734$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$7942$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4513$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$7942$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$4824$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$26985$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-$$$$15935$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$42942$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$4022$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$802$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$26154$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$23505$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$42942$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$23505$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$38920$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$42942$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$42942$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$216715$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$35802$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$24239$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$96854$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$38920$$ $$\beta_{1}$$$$)/4$$

Character Values

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

 $$n$$ $$15$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1 - \beta_{5}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.407369 + 0.812545i −2.29733 + 1.90372i 2.79733 − 1.03769i 0.121721 + 0.507075i 0.907369 + 0.0534805i 0.378279 + 0.358951i −0.407369 − 0.812545i −2.29733 − 1.90372i 2.79733 + 1.03769i 0.121721 − 0.507075i 0.907369 − 0.0534805i 0.378279 − 0.358951i
−1.98615 + 0.234945i 1.86796 + 1.07847i 3.88960 0.933271i 3.25304 + 5.63443i −3.96343 1.70313i −2.39669 6.57692i −7.50608 + 2.76746i −2.17382 3.76517i −7.78481 10.4265i
11.2 −1.51615 1.30434i −3.95004 2.28056i 0.597396 + 3.95514i −2.62655 4.54932i 3.01422 + 8.60985i 5.86799 3.81663i 4.25310 6.77577i 5.90188 + 10.2224i −1.95163 + 10.3234i
11.3 −0.371518 1.96519i 3.95004 + 2.28056i −3.72395 + 1.46021i −2.62655 4.54932i 3.01422 8.60985i −5.86799 + 3.81663i 4.25310 + 6.77577i 5.90188 + 10.2224i −7.96447 + 6.85183i
11.4 −0.104798 + 1.99725i 1.63031 + 0.941260i −3.97803 0.418616i −1.12649 1.95113i −2.05079 + 3.15750i 6.84270 + 1.47562i 1.25297 7.90127i −2.72806 4.72514i 4.01495 2.04540i
11.5 1.19654 1.60259i −1.86796 1.07847i −1.13656 3.83513i 3.25304 + 5.63443i −3.96343 + 1.70313i 2.39669 + 6.57692i −7.50608 2.76746i −2.17382 3.76517i 12.9221 + 1.52857i
11.6 1.78207 + 0.907869i −1.63031 0.941260i 2.35155 + 3.23577i −1.12649 1.95113i −2.05079 3.15750i −6.84270 1.47562i 1.25297 + 7.90127i −2.72806 4.72514i −0.236107 4.49975i
23.1 −1.98615 0.234945i 1.86796 1.07847i 3.88960 + 0.933271i 3.25304 5.63443i −3.96343 + 1.70313i −2.39669 + 6.57692i −7.50608 2.76746i −2.17382 + 3.76517i −7.78481 + 10.4265i
23.2 −1.51615 + 1.30434i −3.95004 + 2.28056i 0.597396 3.95514i −2.62655 + 4.54932i 3.01422 8.60985i 5.86799 + 3.81663i 4.25310 + 6.77577i 5.90188 10.2224i −1.95163 10.3234i
23.3 −0.371518 + 1.96519i 3.95004 2.28056i −3.72395 1.46021i −2.62655 + 4.54932i 3.01422 + 8.60985i −5.86799 3.81663i 4.25310 6.77577i 5.90188 10.2224i −7.96447 6.85183i
23.4 −0.104798 1.99725i 1.63031 0.941260i −3.97803 + 0.418616i −1.12649 + 1.95113i −2.05079 3.15750i 6.84270 1.47562i 1.25297 + 7.90127i −2.72806 + 4.72514i 4.01495 + 2.04540i
23.5 1.19654 + 1.60259i −1.86796 + 1.07847i −1.13656 + 3.83513i 3.25304 5.63443i −3.96343 1.70313i 2.39669 6.57692i −7.50608 + 2.76746i −2.17382 + 3.76517i 12.9221 1.52857i
23.6 1.78207 0.907869i −1.63031 + 0.941260i 2.35155 3.23577i −1.12649 + 1.95113i −2.05079 + 3.15750i −6.84270 + 1.47562i 1.25297 7.90127i −2.72806 + 4.72514i −0.236107 + 4.49975i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.6 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
7.c Even 1 yes
28.g Odd 1 yes

Hecke kernels

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