# Properties

 Label 24.3.b.a Level 24 Weight 3 Character orbit 24.b Analytic conductor 0.654 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 24.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.653952634465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.4752.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \beta_{1} - \beta_{2} ) q^{2}$$ $$+ \beta_{2} q^{3}$$ $$+ ( -2 - \beta_{2} - \beta_{3} ) q^{4}$$ $$+ 2 \beta_{3} q^{5}$$ $$+ ( -2 + \beta_{1} + \beta_{3} ) q^{6}$$ $$+ ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7}$$ $$+ ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{8}$$ $$+ 3 q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \beta_{1} - \beta_{2} ) q^{2}$$ $$+ \beta_{2} q^{3}$$ $$+ ( -2 - \beta_{2} - \beta_{3} ) q^{4}$$ $$+ 2 \beta_{3} q^{5}$$ $$+ ( -2 + \beta_{1} + \beta_{3} ) q^{6}$$ $$+ ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7}$$ $$+ ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{8}$$ $$+ 3 q^{9}$$ $$+ ( 4 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{10}$$ $$-8 q^{11}$$ $$+ ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{12}$$ $$+ ( -4 + 8 \beta_{1} - 4 \beta_{2} ) q^{13}$$ $$+ ( 8 + 2 \beta_{1} + 6 \beta_{2} ) q^{14}$$ $$+ ( -2 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{15}$$ $$+ ( -4 + 4 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{16}$$ $$+ ( -2 - 8 \beta_{2} ) q^{17}$$ $$+ ( 3 \beta_{1} - 3 \beta_{2} ) q^{18}$$ $$+ ( 8 - 4 \beta_{2} ) q^{19}$$ $$+ ( 20 - 4 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{20}$$ $$+ ( 4 - 8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{21}$$ $$+ ( -8 \beta_{1} + 8 \beta_{2} ) q^{22}$$ $$+ ( -4 + 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{23}$$ $$+ ( 12 - 6 \beta_{1} + 2 \beta_{2} ) q^{24}$$ $$+ ( -11 + 16 \beta_{2} ) q^{25}$$ $$+ ( -24 - 4 \beta_{2} - 4 \beta_{3} ) q^{26}$$ $$+ 3 \beta_{2} q^{27}$$ $$+ ( -20 + 16 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} ) q^{28}$$ $$+ ( 8 - 16 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} ) q^{29}$$ $$+ ( -16 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{30}$$ $$+ ( 6 - 12 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} ) q^{31}$$ $$+ ( -24 + 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{32}$$ $$-8 \beta_{2} q^{33}$$ $$+ ( 16 - 10 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{34}$$ $$+ ( 24 + 4 \beta_{2} ) q^{35}$$ $$+ ( -6 - 3 \beta_{2} - 3 \beta_{3} ) q^{36}$$ $$+ ( -4 + 8 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{37}$$ $$+ ( 8 + 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{38}$$ $$+ ( -4 + 8 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{39}$$ $$+ ( 24 + 12 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} ) q^{40}$$ $$+ ( 10 + 24 \beta_{2} ) q^{41}$$ $$+ ( 20 + 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{42}$$ $$+ ( 8 - 12 \beta_{2} ) q^{43}$$ $$+ ( 16 + 8 \beta_{2} + 8 \beta_{3} ) q^{44}$$ $$+ 6 \beta_{3} q^{45}$$ $$+ ( -16 - 4 \beta_{1} - 12 \beta_{2} ) q^{46}$$ $$+ ( -4 + 8 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} ) q^{47}$$ $$+ ( 20 + 8 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{48}$$ $$+ ( -11 - 32 \beta_{2} ) q^{49}$$ $$+ ( -32 + 5 \beta_{1} + 11 \beta_{2} + 16 \beta_{3} ) q^{50}$$ $$+ ( -24 - 2 \beta_{2} ) q^{51}$$ $$+ ( -24 \beta_{1} + 32 \beta_{2} - 8 \beta_{3} ) q^{52}$$ $$+ ( -8 + 16 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{53}$$ $$+ ( -6 + 3 \beta_{1} + 3 \beta_{3} ) q^{54}$$ $$-16 \beta_{3} q^{55}$$ $$+ ( -32 - 20 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{56}$$ $$+ ( -12 + 8 \beta_{2} ) q^{57}$$ $$+ ( 36 + 6 \beta_{1} + 20 \beta_{2} + 2 \beta_{3} ) q^{58}$$ $$+ ( -32 + 12 \beta_{2} ) q^{59}$$ $$+ ( -20 - 8 \beta_{1} + 22 \beta_{2} - 2 \beta_{3} ) q^{60}$$ $$+ ( 4 - 8 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} ) q^{61}$$ $$+ ( 56 - 10 \beta_{1} - 14 \beta_{2} + 16 \beta_{3} ) q^{62}$$ $$+ ( 6 - 12 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{63}$$ $$+ ( 8 - 32 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{64}$$ $$+ ( 24 - 40 \beta_{2} ) q^{65}$$ $$+ ( 16 - 8 \beta_{1} - 8 \beta_{3} ) q^{66}$$ $$+ ( -64 + 12 \beta_{2} ) q^{67}$$ $$+ ( 20 + 16 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{68}$$ $$+ ( -8 + 16 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{69}$$ $$+ ( -8 + 28 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} ) q^{70}$$ $$+ ( 20 - 40 \beta_{1} + 20 \beta_{2} - 4 \beta_{3} ) q^{71}$$ $$+ ( -6 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} ) q^{72}$$ $$+ ( 50 + 32 \beta_{2} ) q^{73}$$ $$+ ( -32 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{74}$$ $$+ ( 48 - 11 \beta_{2} ) q^{75}$$ $$+ ( -8 + 8 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} ) q^{76}$$ $$+ ( -16 + 32 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{77}$$ $$+ ( -8 - 8 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} ) q^{78}$$ $$+ ( -18 + 36 \beta_{1} - 18 \beta_{2} + 2 \beta_{3} ) q^{79}$$ $$+ ( -40 + 32 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} ) q^{80}$$ $$+ 9 q^{81}$$ $$+ ( -48 + 34 \beta_{1} - 10 \beta_{2} + 24 \beta_{3} ) q^{82}$$ $$+ ( 40 - 16 \beta_{2} ) q^{83}$$ $$+ ( -20 + 28 \beta_{1} - 26 \beta_{2} + 10 \beta_{3} ) q^{84}$$ $$+ ( 16 - 32 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{85}$$ $$+ ( 24 - 4 \beta_{1} - 8 \beta_{2} - 12 \beta_{3} ) q^{86}$$ $$+ ( 14 - 28 \beta_{1} + 14 \beta_{2} - 10 \beta_{3} ) q^{87}$$ $$+ ( 16 \beta_{1} - 32 \beta_{2} + 16 \beta_{3} ) q^{88}$$ $$+ ( -50 + 48 \beta_{2} ) q^{89}$$ $$+ ( 12 - 6 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} ) q^{90}$$ $$+ ( 72 + 56 \beta_{2} ) q^{91}$$ $$+ ( 40 - 32 \beta_{1} + 20 \beta_{2} - 12 \beta_{3} ) q^{92}$$ $$+ ( -4 + 8 \beta_{1} - 4 \beta_{2} - 22 \beta_{3} ) q^{93}$$ $$+ ( -48 + 12 \beta_{1} + 20 \beta_{2} - 16 \beta_{3} ) q^{94}$$ $$+ ( 8 - 16 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{95}$$ $$+ ( -16 + 20 \beta_{1} - 32 \beta_{2} - 4 \beta_{3} ) q^{96}$$ $$+ ( 14 - 48 \beta_{2} ) q^{97}$$ $$+ ( 64 - 43 \beta_{1} + 11 \beta_{2} - 32 \beta_{3} ) q^{98}$$ $$-24 q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 6q^{6}$$ $$\mathstrut -\mathstrut 4q^{8}$$ $$\mathstrut +\mathstrut 12q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 6q^{6}$$ $$\mathstrut -\mathstrut 4q^{8}$$ $$\mathstrut +\mathstrut 12q^{9}$$ $$\mathstrut +\mathstrut 12q^{10}$$ $$\mathstrut -\mathstrut 32q^{11}$$ $$\mathstrut -\mathstrut 12q^{12}$$ $$\mathstrut +\mathstrut 36q^{14}$$ $$\mathstrut -\mathstrut 8q^{16}$$ $$\mathstrut -\mathstrut 8q^{17}$$ $$\mathstrut +\mathstrut 6q^{18}$$ $$\mathstrut +\mathstrut 32q^{19}$$ $$\mathstrut +\mathstrut 72q^{20}$$ $$\mathstrut -\mathstrut 16q^{22}$$ $$\mathstrut +\mathstrut 36q^{24}$$ $$\mathstrut -\mathstrut 44q^{25}$$ $$\mathstrut -\mathstrut 96q^{26}$$ $$\mathstrut -\mathstrut 48q^{28}$$ $$\mathstrut -\mathstrut 60q^{30}$$ $$\mathstrut -\mathstrut 88q^{32}$$ $$\mathstrut +\mathstrut 44q^{34}$$ $$\mathstrut +\mathstrut 96q^{35}$$ $$\mathstrut -\mathstrut 24q^{36}$$ $$\mathstrut +\mathstrut 40q^{38}$$ $$\mathstrut +\mathstrut 120q^{40}$$ $$\mathstrut +\mathstrut 40q^{41}$$ $$\mathstrut +\mathstrut 84q^{42}$$ $$\mathstrut +\mathstrut 32q^{43}$$ $$\mathstrut +\mathstrut 64q^{44}$$ $$\mathstrut -\mathstrut 72q^{46}$$ $$\mathstrut +\mathstrut 96q^{48}$$ $$\mathstrut -\mathstrut 44q^{49}$$ $$\mathstrut -\mathstrut 118q^{50}$$ $$\mathstrut -\mathstrut 96q^{51}$$ $$\mathstrut -\mathstrut 48q^{52}$$ $$\mathstrut -\mathstrut 18q^{54}$$ $$\mathstrut -\mathstrut 168q^{56}$$ $$\mathstrut -\mathstrut 48q^{57}$$ $$\mathstrut +\mathstrut 156q^{58}$$ $$\mathstrut -\mathstrut 128q^{59}$$ $$\mathstrut -\mathstrut 96q^{60}$$ $$\mathstrut +\mathstrut 204q^{62}$$ $$\mathstrut -\mathstrut 32q^{64}$$ $$\mathstrut +\mathstrut 96q^{65}$$ $$\mathstrut +\mathstrut 48q^{66}$$ $$\mathstrut -\mathstrut 256q^{67}$$ $$\mathstrut +\mathstrut 112q^{68}$$ $$\mathstrut +\mathstrut 24q^{70}$$ $$\mathstrut -\mathstrut 12q^{72}$$ $$\mathstrut +\mathstrut 200q^{73}$$ $$\mathstrut -\mathstrut 120q^{74}$$ $$\mathstrut +\mathstrut 192q^{75}$$ $$\mathstrut -\mathstrut 16q^{76}$$ $$\mathstrut -\mathstrut 48q^{78}$$ $$\mathstrut -\mathstrut 96q^{80}$$ $$\mathstrut +\mathstrut 36q^{81}$$ $$\mathstrut -\mathstrut 124q^{82}$$ $$\mathstrut +\mathstrut 160q^{83}$$ $$\mathstrut -\mathstrut 24q^{84}$$ $$\mathstrut +\mathstrut 88q^{86}$$ $$\mathstrut +\mathstrut 32q^{88}$$ $$\mathstrut -\mathstrut 200q^{89}$$ $$\mathstrut +\mathstrut 36q^{90}$$ $$\mathstrut +\mathstrut 288q^{91}$$ $$\mathstrut +\mathstrut 96q^{92}$$ $$\mathstrut -\mathstrut 168q^{94}$$ $$\mathstrut -\mathstrut 24q^{96}$$ $$\mathstrut +\mathstrut 56q^{97}$$ $$\mathstrut +\mathstrut 170q^{98}$$ $$\mathstrut -\mathstrut 96q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$3$$ $$x^{2}\mathstrut -\mathstrut$$ $$6$$ $$x\mathstrut +\mathstrut$$ $$6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 3 \nu^{2} + 4 \nu + 2$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 6$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} + 8 \nu - 6$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$\beta_{3}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$8$$$$)/2$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.866025 − 0.719687i 0.866025 + 0.719687i −0.866025 + 1.99551i −0.866025 − 1.99551i
−0.366025 1.96622i 1.73205 −3.73205 + 1.43937i 2.87875i −0.633975 3.40559i 10.7436i 4.19615 + 6.81119i 3.00000 −5.66025 + 1.05369i
19.2 −0.366025 + 1.96622i 1.73205 −3.73205 1.43937i 2.87875i −0.633975 + 3.40559i 10.7436i 4.19615 6.81119i 3.00000 −5.66025 1.05369i
19.3 1.36603 1.46081i −1.73205 −0.267949 3.99102i 7.98203i −2.36603 + 2.53020i 2.13878i −6.19615 5.06040i 3.00000 11.6603 + 10.9037i
19.4 1.36603 + 1.46081i −1.73205 −0.267949 + 3.99102i 7.98203i −2.36603 2.53020i 2.13878i −6.19615 + 5.06040i 3.00000 11.6603 10.9037i
 $$n$$: e.g. 2-40 or 990-1000 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
8.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(24, [\chi])$$.