# Properties

 Label 12.4.b.a Level 12 Weight 4 Character orbit 12.b Analytic conductor 0.708 Analytic rank 0 Dimension 4 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 12.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.708022920069$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{-5})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -\beta_{1} - \beta_{3} ) q^{3}$$ $$+ ( -2 + \beta_{2} + \beta_{3} ) q^{4}$$ $$+ ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5}$$ $$+ ( -6 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{6}$$ $$+ ( \beta_{2} + \beta_{3} ) q^{7}$$ $$+ ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{8}$$ $$+ ( -3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -\beta_{1} - \beta_{3} ) q^{3}$$ $$+ ( -2 + \beta_{2} + \beta_{3} ) q^{4}$$ $$+ ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5}$$ $$+ ( -6 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{6}$$ $$+ ( \beta_{2} + \beta_{3} ) q^{7}$$ $$+ ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{8}$$ $$+ ( -3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{9}$$ $$+ ( 20 - 2 \beta_{2} - 2 \beta_{3} ) q^{10}$$ $$+ ( 10 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{11}$$ $$+ ( 30 - 4 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{12}$$ $$-10 q^{13}$$ $$+ ( -2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{14}$$ $$+ ( -10 \beta_{1} + 9 \beta_{2} - \beta_{3} ) q^{15}$$ $$+ ( -56 - 4 \beta_{2} - 4 \beta_{3} ) q^{16}$$ $$+ ( -8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{17}$$ $$+ ( -60 - 3 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{18}$$ $$+ ( -9 \beta_{2} - 9 \beta_{3} ) q^{19}$$ $$+ ( 24 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{20}$$ $$+ ( 30 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{21}$$ $$+ ( 60 + 10 \beta_{2} + 10 \beta_{3} ) q^{22}$$ $$+ ( -28 \beta_{1} + 14 \beta_{2} - 14 \beta_{3} ) q^{23}$$ $$+ ( 72 + 28 \beta_{1} - 8 \beta_{3} ) q^{24}$$ $$+ 45 q^{25}$$ $$-10 \beta_{1} q^{26}$$ $$+ ( 33 \beta_{1} - 27 \beta_{2} + 6 \beta_{3} ) q^{27}$$ $$+ ( -60 - 2 \beta_{2} - 2 \beta_{3} ) q^{28}$$ $$+ ( 34 \beta_{1} + 17 \beta_{2} - 17 \beta_{3} ) q^{29}$$ $$+ ( -60 - 8 \beta_{1} + 6 \beta_{2} - 26 \beta_{3} ) q^{30}$$ $$+ ( 29 \beta_{2} + 29 \beta_{3} ) q^{31}$$ $$+ ( -48 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{32}$$ $$+ ( -120 - 30 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{33}$$ $$+ ( 80 - 8 \beta_{2} - 8 \beta_{3} ) q^{34}$$ $$+ ( 20 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{35}$$ $$+ ( 6 - 72 \beta_{1} + 21 \beta_{2} - 27 \beta_{3} ) q^{36}$$ $$-130 q^{37}$$ $$+ ( 18 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{38}$$ $$+ ( 10 \beta_{1} + 10 \beta_{3} ) q^{39}$$ $$+ ( 80 + 24 \beta_{2} + 24 \beta_{3} ) q^{40}$$ $$+ ( 28 \beta_{1} + 14 \beta_{2} - 14 \beta_{3} ) q^{41}$$ $$+ ( 60 + 30 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{42}$$ $$+ ( -29 \beta_{2} - 29 \beta_{3} ) q^{43}$$ $$+ ( 40 \beta_{1} + 40 \beta_{2} - 40 \beta_{3} ) q^{44}$$ $$+ ( 240 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{45}$$ $$+ ( -168 - 28 \beta_{2} - 28 \beta_{3} ) q^{46}$$ $$+ ( -56 \beta_{1} + 28 \beta_{2} - 28 \beta_{3} ) q^{47}$$ $$+ ( -120 + 80 \beta_{1} + 12 \beta_{2} + 44 \beta_{3} ) q^{48}$$ $$+ 283 q^{49}$$ $$+ 45 \beta_{1} q^{50}$$ $$+ ( -40 \beta_{1} + 36 \beta_{2} - 4 \beta_{3} ) q^{51}$$ $$+ ( 20 - 10 \beta_{2} - 10 \beta_{3} ) q^{52}$$ $$+ ( -122 \beta_{1} - 61 \beta_{2} + 61 \beta_{3} ) q^{53}$$ $$+ ( 198 + 21 \beta_{1} - 9 \beta_{2} + 75 \beta_{3} ) q^{54}$$ $$+ ( -40 \beta_{2} - 40 \beta_{3} ) q^{55}$$ $$+ ( -56 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{56}$$ $$+ ( -270 + 54 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} ) q^{57}$$ $$+ ( -340 + 34 \beta_{2} + 34 \beta_{3} ) q^{58}$$ $$+ ( 50 \beta_{1} - 25 \beta_{2} + 25 \beta_{3} ) q^{59}$$ $$+ ( -240 - 40 \beta_{1} - 48 \beta_{2} + 32 \beta_{3} ) q^{60}$$ $$-442 q^{61}$$ $$+ ( -58 \beta_{1} + 116 \beta_{2} - 116 \beta_{3} ) q^{62}$$ $$+ ( -60 \beta_{1} + 27 \beta_{2} - 33 \beta_{3} ) q^{63}$$ $$+ ( 352 - 48 \beta_{2} - 48 \beta_{3} ) q^{64}$$ $$+ ( 20 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{65}$$ $$+ ( 300 - 120 \beta_{1} - 30 \beta_{2} - 30 \beta_{3} ) q^{66}$$ $$+ ( 95 \beta_{2} + 95 \beta_{3} ) q^{67}$$ $$+ ( 96 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{68}$$ $$+ ( 336 + 84 \beta_{1} + 42 \beta_{2} - 42 \beta_{3} ) q^{69}$$ $$+ ( 120 + 20 \beta_{2} + 20 \beta_{3} ) q^{70}$$ $$+ ( 300 \beta_{1} - 150 \beta_{2} + 150 \beta_{3} ) q^{71}$$ $$+ ( -240 + 12 \beta_{1} - 84 \beta_{2} - 60 \beta_{3} ) q^{72}$$ $$+ 410 q^{73}$$ $$-130 \beta_{1} q^{74}$$ $$+ ( -45 \beta_{1} - 45 \beta_{3} ) q^{75}$$ $$+ ( 540 + 18 \beta_{2} + 18 \beta_{3} ) q^{76}$$ $$+ ( 60 \beta_{1} + 30 \beta_{2} - 30 \beta_{3} ) q^{77}$$ $$+ ( 60 - 10 \beta_{1} + 30 \beta_{2} - 10 \beta_{3} ) q^{78}$$ $$+ ( -11 \beta_{2} - 11 \beta_{3} ) q^{79}$$ $$+ ( 32 \beta_{1} + 96 \beta_{2} - 96 \beta_{3} ) q^{80}$$ $$+ ( -711 - 36 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{81}$$ $$+ ( -280 + 28 \beta_{2} + 28 \beta_{3} ) q^{82}$$ $$+ ( -362 \beta_{1} + 181 \beta_{2} - 181 \beta_{3} ) q^{83}$$ $$+ ( -60 + 72 \beta_{1} + 6 \beta_{2} + 54 \beta_{3} ) q^{84}$$ $$-320 q^{85}$$ $$+ ( 58 \beta_{1} - 116 \beta_{2} + 116 \beta_{3} ) q^{86}$$ $$+ ( 170 \beta_{1} - 153 \beta_{2} + 17 \beta_{3} ) q^{87}$$ $$+ ( -720 + 40 \beta_{2} + 40 \beta_{3} ) q^{88}$$ $$+ ( -188 \beta_{1} - 94 \beta_{2} + 94 \beta_{3} ) q^{89}$$ $$+ ( -60 + 240 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{90}$$ $$+ ( -10 \beta_{2} - 10 \beta_{3} ) q^{91}$$ $$+ ( -112 \beta_{1} - 112 \beta_{2} + 112 \beta_{3} ) q^{92}$$ $$+ ( 870 - 174 \beta_{1} - 87 \beta_{2} + 87 \beta_{3} ) q^{93}$$ $$+ ( -336 - 56 \beta_{2} - 56 \beta_{3} ) q^{94}$$ $$+ ( -180 \beta_{1} + 90 \beta_{2} - 90 \beta_{3} ) q^{95}$$ $$+ ( 96 - 176 \beta_{1} + 192 \beta_{2} - 32 \beta_{3} ) q^{96}$$ $$+ 770 q^{97}$$ $$+ 283 \beta_{1} q^{98}$$ $$+ ( -30 \beta_{1} + 135 \beta_{2} + 105 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 24q^{6}$$ $$\mathstrut -\mathstrut 12q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 24q^{6}$$ $$\mathstrut -\mathstrut 12q^{9}$$ $$\mathstrut +\mathstrut 80q^{10}$$ $$\mathstrut +\mathstrut 120q^{12}$$ $$\mathstrut -\mathstrut 40q^{13}$$ $$\mathstrut -\mathstrut 224q^{16}$$ $$\mathstrut -\mathstrut 240q^{18}$$ $$\mathstrut +\mathstrut 120q^{21}$$ $$\mathstrut +\mathstrut 240q^{22}$$ $$\mathstrut +\mathstrut 288q^{24}$$ $$\mathstrut +\mathstrut 180q^{25}$$ $$\mathstrut -\mathstrut 240q^{28}$$ $$\mathstrut -\mathstrut 240q^{30}$$ $$\mathstrut -\mathstrut 480q^{33}$$ $$\mathstrut +\mathstrut 320q^{34}$$ $$\mathstrut +\mathstrut 24q^{36}$$ $$\mathstrut -\mathstrut 520q^{37}$$ $$\mathstrut +\mathstrut 320q^{40}$$ $$\mathstrut +\mathstrut 240q^{42}$$ $$\mathstrut +\mathstrut 960q^{45}$$ $$\mathstrut -\mathstrut 672q^{46}$$ $$\mathstrut -\mathstrut 480q^{48}$$ $$\mathstrut +\mathstrut 1132q^{49}$$ $$\mathstrut +\mathstrut 80q^{52}$$ $$\mathstrut +\mathstrut 792q^{54}$$ $$\mathstrut -\mathstrut 1080q^{57}$$ $$\mathstrut -\mathstrut 1360q^{58}$$ $$\mathstrut -\mathstrut 960q^{60}$$ $$\mathstrut -\mathstrut 1768q^{61}$$ $$\mathstrut +\mathstrut 1408q^{64}$$ $$\mathstrut +\mathstrut 1200q^{66}$$ $$\mathstrut +\mathstrut 1344q^{69}$$ $$\mathstrut +\mathstrut 480q^{70}$$ $$\mathstrut -\mathstrut 960q^{72}$$ $$\mathstrut +\mathstrut 1640q^{73}$$ $$\mathstrut +\mathstrut 2160q^{76}$$ $$\mathstrut +\mathstrut 240q^{78}$$ $$\mathstrut -\mathstrut 2844q^{81}$$ $$\mathstrut -\mathstrut 1120q^{82}$$ $$\mathstrut -\mathstrut 240q^{84}$$ $$\mathstrut -\mathstrut 1280q^{85}$$ $$\mathstrut -\mathstrut 2880q^{88}$$ $$\mathstrut -\mathstrut 240q^{90}$$ $$\mathstrut +\mathstrut 3480q^{93}$$ $$\mathstrut -\mathstrut 1344q^{94}$$ $$\mathstrut +\mathstrut 384q^{96}$$ $$\mathstrut +\mathstrut 3080q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

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

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.866025 − 1.11803i −0.866025 + 1.11803i 0.866025 − 1.11803i 0.866025 + 1.11803i
−1.73205 2.23607i 3.46410 3.87298i −2.00000 + 7.74597i 8.94427i −14.6603 1.03776i 7.74597i 20.7846 8.94427i −3.00000 26.8328i 20.0000 15.4919i
11.2 −1.73205 + 2.23607i 3.46410 + 3.87298i −2.00000 7.74597i 8.94427i −14.6603 + 1.03776i 7.74597i 20.7846 + 8.94427i −3.00000 + 26.8328i 20.0000 + 15.4919i
11.3 1.73205 2.23607i −3.46410 + 3.87298i −2.00000 7.74597i 8.94427i 2.66025 + 14.4542i 7.74597i −20.7846 8.94427i −3.00000 26.8328i 20.0000 + 15.4919i
11.4 1.73205 + 2.23607i −3.46410 3.87298i −2.00000 + 7.74597i 8.94427i 2.66025 14.4542i 7.74597i −20.7846 + 8.94427i −3.00000 + 26.8328i 20.0000 15.4919i
 $$n$$: 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
4.b Odd 1 yes
12.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{4}^{\mathrm{new}}(12, [\chi])$$.