# Properties

 Label 24.2.d.a Level 24 Weight 2 Character orbit 24.d Analytic conductor 0.192 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$0.191640964851$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 − 1.00000i 1.00000i
−1.00000 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 + 1.00000i −2.00000 2.00000 2.00000i −1.00000 2.00000 2.00000i
13.2 −1.00000 + 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 1.00000i −2.00000 2.00000 + 2.00000i −1.00000 2.00000 + 2.00000i
 $$n$$: 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.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(24, [\chi])$$.