# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$0.653952634465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-7})$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -\beta_{2} - \beta_{3} ) q^{3}$$ $$+ ( -3 + \beta_{3} ) q^{4}$$ $$+ 4 \beta_{2} q^{5}$$ $$+ ( 1 - \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{6}$$ $$+ 4 q^{7}$$ $$+ ( -2 \beta_{1} + 4 \beta_{2} ) q^{8}$$ $$+ ( -5 - 4 \beta_{1} - 2 \beta_{2} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -\beta_{2} - \beta_{3} ) q^{3}$$ $$+ ( -3 + \beta_{3} ) q^{4}$$ $$+ 4 \beta_{2} q^{5}$$ $$+ ( 1 - \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{6}$$ $$+ 4 q^{7}$$ $$+ ( -2 \beta_{1} + 4 \beta_{2} ) q^{8}$$ $$+ ( -5 - 4 \beta_{1} - 2 \beta_{2} ) q^{9}$$ $$+ ( -4 - 4 \beta_{3} ) q^{10}$$ $$-6 \beta_{2} q^{11}$$ $$+ ( 7 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{12}$$ $$+ 4 \beta_{3} q^{13}$$ $$+ 4 \beta_{1} q^{14}$$ $$+ ( -8 + 8 \beta_{1} + 4 \beta_{2} ) q^{15}$$ $$+ ( 2 - 6 \beta_{3} ) q^{16}$$ $$+ ( 8 \beta_{1} + 4 \beta_{2} ) q^{17}$$ $$+ ( 14 - 5 \beta_{1} - 2 \beta_{3} ) q^{18}$$ $$+ 2 \beta_{3} q^{19}$$ $$+ ( -8 \beta_{1} - 16 \beta_{2} ) q^{20}$$ $$+ ( -4 \beta_{2} - 4 \beta_{3} ) q^{21}$$ $$+ ( 6 + 6 \beta_{3} ) q^{22}$$ $$+ ( -16 \beta_{1} - 8 \beta_{2} ) q^{23}$$ $$+ ( -10 + 10 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{24}$$ $$+ 7 q^{25}$$ $$+ ( 4 \beta_{1} + 16 \beta_{2} ) q^{26}$$ $$+ ( 19 \beta_{2} + \beta_{3} ) q^{27}$$ $$+ ( -12 + 4 \beta_{3} ) q^{28}$$ $$-12 \beta_{2} q^{29}$$ $$+ ( -28 - 8 \beta_{1} + 4 \beta_{3} ) q^{30}$$ $$-4 q^{31}$$ $$+ ( -4 \beta_{1} - 24 \beta_{2} ) q^{32}$$ $$+ ( 12 - 12 \beta_{1} - 6 \beta_{2} ) q^{33}$$ $$+ ( -28 + 4 \beta_{3} ) q^{34}$$ $$+ 16 \beta_{2} q^{35}$$ $$+ ( 15 + 12 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{36}$$ $$-20 \beta_{3} q^{37}$$ $$+ ( 2 \beta_{1} + 8 \beta_{2} ) q^{38}$$ $$+ ( 28 + 8 \beta_{1} + 4 \beta_{2} ) q^{39}$$ $$+ ( 40 + 8 \beta_{3} ) q^{40}$$ $$+ ( 16 \beta_{1} + 8 \beta_{2} ) q^{41}$$ $$+ ( 4 - 4 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} ) q^{42}$$ $$-2 \beta_{3} q^{43}$$ $$+ ( 12 \beta_{1} + 24 \beta_{2} ) q^{44}$$ $$+ ( -20 \beta_{2} + 16 \beta_{3} ) q^{45}$$ $$+ ( 56 - 8 \beta_{3} ) q^{46}$$ $$+ ( -42 - 12 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{48}$$ $$-33 q^{49}$$ $$+ 7 \beta_{1} q^{50}$$ $$+ ( -28 \beta_{2} + 8 \beta_{3} ) q^{51}$$ $$+ ( -28 - 12 \beta_{3} ) q^{52}$$ $$+ 36 \beta_{2} q^{53}$$ $$+ ( -19 + \beta_{1} + 4 \beta_{2} - 19 \beta_{3} ) q^{54}$$ $$-48 q^{55}$$ $$+ ( -8 \beta_{1} + 16 \beta_{2} ) q^{56}$$ $$+ ( 14 + 4 \beta_{1} + 2 \beta_{2} ) q^{57}$$ $$+ ( 12 + 12 \beta_{3} ) q^{58}$$ $$-34 \beta_{2} q^{59}$$ $$+ ( 24 - 24 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{60}$$ $$+ 36 \beta_{3} q^{61}$$ $$-4 \beta_{1} q^{62}$$ $$+ ( -20 - 16 \beta_{1} - 8 \beta_{2} ) q^{63}$$ $$+ ( 36 + 20 \beta_{3} ) q^{64}$$ $$+ ( -32 \beta_{1} - 16 \beta_{2} ) q^{65}$$ $$+ ( 42 + 12 \beta_{1} - 6 \beta_{3} ) q^{66}$$ $$-18 \beta_{3} q^{67}$$ $$+ ( -24 \beta_{1} + 16 \beta_{2} ) q^{68}$$ $$+ ( 56 \beta_{2} - 16 \beta_{3} ) q^{69}$$ $$+ ( -16 - 16 \beta_{3} ) q^{70}$$ $$+ ( 48 \beta_{1} + 24 \beta_{2} ) q^{71}$$ $$+ ( -28 + 10 \beta_{1} - 20 \beta_{2} + 20 \beta_{3} ) q^{72}$$ $$-6 q^{73}$$ $$+ ( -20 \beta_{1} - 80 \beta_{2} ) q^{74}$$ $$+ ( -7 \beta_{2} - 7 \beta_{3} ) q^{75}$$ $$+ ( -14 - 6 \beta_{3} ) q^{76}$$ $$-24 \beta_{2} q^{77}$$ $$+ ( -28 + 28 \beta_{1} + 4 \beta_{3} ) q^{78}$$ $$+ 124 q^{79}$$ $$+ ( 48 \beta_{1} + 32 \beta_{2} ) q^{80}$$ $$+ ( -31 + 40 \beta_{1} + 20 \beta_{2} ) q^{81}$$ $$+ ( -56 + 8 \beta_{3} ) q^{82}$$ $$+ 2 \beta_{2} q^{83}$$ $$+ ( 28 + 8 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{84}$$ $$-32 \beta_{3} q^{85}$$ $$+ ( -2 \beta_{1} - 8 \beta_{2} ) q^{86}$$ $$+ ( 24 - 24 \beta_{1} - 12 \beta_{2} ) q^{87}$$ $$+ ( -60 - 12 \beta_{3} ) q^{88}$$ $$+ ( -56 \beta_{1} - 28 \beta_{2} ) q^{89}$$ $$+ ( 20 + 16 \beta_{1} + 64 \beta_{2} + 20 \beta_{3} ) q^{90}$$ $$+ 16 \beta_{3} q^{91}$$ $$+ ( 48 \beta_{1} - 32 \beta_{2} ) q^{92}$$ $$+ ( 4 \beta_{2} + 4 \beta_{3} ) q^{93}$$ $$+ ( -16 \beta_{1} - 8 \beta_{2} ) q^{95}$$ $$+ ( 44 - 44 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{96}$$ $$+ 118 q^{97}$$ $$-33 \beta_{1} q^{98}$$ $$+ ( 30 \beta_{2} - 24 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 12q^{4}$$ $$\mathstrut +\mathstrut 4q^{6}$$ $$\mathstrut +\mathstrut 16q^{7}$$ $$\mathstrut -\mathstrut 20q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 12q^{4}$$ $$\mathstrut +\mathstrut 4q^{6}$$ $$\mathstrut +\mathstrut 16q^{7}$$ $$\mathstrut -\mathstrut 20q^{9}$$ $$\mathstrut -\mathstrut 16q^{10}$$ $$\mathstrut +\mathstrut 28q^{12}$$ $$\mathstrut -\mathstrut 32q^{15}$$ $$\mathstrut +\mathstrut 8q^{16}$$ $$\mathstrut +\mathstrut 56q^{18}$$ $$\mathstrut +\mathstrut 24q^{22}$$ $$\mathstrut -\mathstrut 40q^{24}$$ $$\mathstrut +\mathstrut 28q^{25}$$ $$\mathstrut -\mathstrut 48q^{28}$$ $$\mathstrut -\mathstrut 112q^{30}$$ $$\mathstrut -\mathstrut 16q^{31}$$ $$\mathstrut +\mathstrut 48q^{33}$$ $$\mathstrut -\mathstrut 112q^{34}$$ $$\mathstrut +\mathstrut 60q^{36}$$ $$\mathstrut +\mathstrut 112q^{39}$$ $$\mathstrut +\mathstrut 160q^{40}$$ $$\mathstrut +\mathstrut 16q^{42}$$ $$\mathstrut +\mathstrut 224q^{46}$$ $$\mathstrut -\mathstrut 168q^{48}$$ $$\mathstrut -\mathstrut 132q^{49}$$ $$\mathstrut -\mathstrut 112q^{52}$$ $$\mathstrut -\mathstrut 76q^{54}$$ $$\mathstrut -\mathstrut 192q^{55}$$ $$\mathstrut +\mathstrut 56q^{57}$$ $$\mathstrut +\mathstrut 48q^{58}$$ $$\mathstrut +\mathstrut 96q^{60}$$ $$\mathstrut -\mathstrut 80q^{63}$$ $$\mathstrut +\mathstrut 144q^{64}$$ $$\mathstrut +\mathstrut 168q^{66}$$ $$\mathstrut -\mathstrut 64q^{70}$$ $$\mathstrut -\mathstrut 112q^{72}$$ $$\mathstrut -\mathstrut 24q^{73}$$ $$\mathstrut -\mathstrut 56q^{76}$$ $$\mathstrut -\mathstrut 112q^{78}$$ $$\mathstrut +\mathstrut 496q^{79}$$ $$\mathstrut -\mathstrut 124q^{81}$$ $$\mathstrut -\mathstrut 224q^{82}$$ $$\mathstrut +\mathstrut 112q^{84}$$ $$\mathstrut +\mathstrut 96q^{87}$$ $$\mathstrut -\mathstrut 240q^{88}$$ $$\mathstrut +\mathstrut 80q^{90}$$ $$\mathstrut +\mathstrut 176q^{96}$$ $$\mathstrut +\mathstrut 472q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$6$$ $$x^{2}\mathstrut +\mathstrut$$ $$16$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −0.707107 − 1.87083i −0.707107 + 1.87083i 0.707107 − 1.87083i 0.707107 + 1.87083i
−0.707107 1.87083i −1.41421 2.64575i −3.00000 + 2.64575i 5.65685 −3.94975 + 4.51658i 4.00000 7.07107 + 3.74166i −5.00000 + 7.48331i −4.00000 10.5830i
5.2 −0.707107 + 1.87083i −1.41421 + 2.64575i −3.00000 2.64575i 5.65685 −3.94975 4.51658i 4.00000 7.07107 3.74166i −5.00000 7.48331i −4.00000 + 10.5830i
5.3 0.707107 1.87083i 1.41421 + 2.64575i −3.00000 2.64575i −5.65685 5.94975 0.774923i 4.00000 −7.07107 + 3.74166i −5.00000 + 7.48331i −4.00000 + 10.5830i
5.4 0.707107 + 1.87083i 1.41421 2.64575i −3.00000 + 2.64575i −5.65685 5.94975 + 0.774923i 4.00000 −7.07107 3.74166i −5.00000 7.48331i −4.00000 10.5830i
 $$n$$: 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
3.b Odd 1 yes
8.b Even 1 yes
24.h Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}^{2}$$ $$\mathstrut -\mathstrut 32$$ acting on $$S_{3}^{\mathrm{new}}(24, [\chi])$$.