# 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 $$2$$, degree $$1$$, minimal)

## 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: $$2^{4}$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} - 24q^{6} - 12q^{9} + O(q^{10})$$ $$4q - 8q^{4} - 24q^{6} - 12q^{9} + 80q^{10} + 120q^{12} - 40q^{13} - 224q^{16} - 240q^{18} + 120q^{21} + 240q^{22} + 288q^{24} + 180q^{25} - 240q^{28} - 240q^{30} - 480q^{33} + 320q^{34} + 24q^{36} - 520q^{37} + 320q^{40} + 240q^{42} + 960q^{45} - 672q^{46} - 480q^{48} + 1132q^{49} + 80q^{52} + 792q^{54} - 1080q^{57} - 1360q^{58} - 960q^{60} - 1768q^{61} + 1408q^{64} + 1200q^{66} + 1344q^{69} + 480q^{70} - 960q^{72} + 1640q^{73} + 2160q^{76} + 240q^{78} - 2844q^{81} - 1120q^{82} - 240q^{84} - 1280q^{85} - 2880q^{88} - 240q^{90} + 3480q^{93} - 1344q^{94} + 384q^{96} + 3080q^{97} + O(q^{100})$$

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

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

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.4.b.a 4
3.b odd 2 1 inner 12.4.b.a 4
4.b odd 2 1 inner 12.4.b.a 4
8.b even 2 1 192.4.c.b 4
8.d odd 2 1 192.4.c.b 4
12.b even 2 1 inner 12.4.b.a 4
16.e even 4 2 768.4.f.c 8
16.f odd 4 2 768.4.f.c 8
24.f even 2 1 192.4.c.b 4
24.h odd 2 1 192.4.c.b 4
48.i odd 4 2 768.4.f.c 8
48.k even 4 2 768.4.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.b.a 4 1.a even 1 1 trivial
12.4.b.a 4 3.b odd 2 1 inner
12.4.b.a 4 4.b odd 2 1 inner
12.4.b.a 4 12.b even 2 1 inner
192.4.c.b 4 8.b even 2 1
192.4.c.b 4 8.d odd 2 1
192.4.c.b 4 24.f even 2 1
192.4.c.b 4 24.h odd 2 1
768.4.f.c 8 16.e even 4 2
768.4.f.c 8 16.f odd 4 2
768.4.f.c 8 48.i odd 4 2
768.4.f.c 8 48.k even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(12, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + 64 T^{4}$$
$3$ $$1 + 6 T^{2} + 729 T^{4}$$
$5$ $$( 1 - 170 T^{2} + 15625 T^{4} )^{2}$$
$7$ $$( 1 - 626 T^{2} + 117649 T^{4} )^{2}$$
$11$ $$( 1 + 1462 T^{2} + 1771561 T^{4} )^{2}$$
$13$ $$( 1 + 10 T + 2197 T^{2} )^{4}$$
$17$ $$( 1 - 8546 T^{2} + 24137569 T^{4} )^{2}$$
$19$ $$( 1 - 8858 T^{2} + 47045881 T^{4} )^{2}$$
$23$ $$( 1 + 14926 T^{2} + 148035889 T^{4} )^{2}$$
$29$ $$( 1 - 25658 T^{2} + 594823321 T^{4} )^{2}$$
$31$ $$( 1 - 9122 T^{2} + 887503681 T^{4} )^{2}$$
$37$ $$( 1 + 130 T + 50653 T^{2} )^{4}$$
$41$ $$( 1 - 122162 T^{2} + 4750104241 T^{4} )^{2}$$
$43$ $$( 1 - 108554 T^{2} + 6321363049 T^{4} )^{2}$$
$47$ $$( 1 + 170014 T^{2} + 10779215329 T^{4} )^{2}$$
$53$ $$( 1 - 74 T^{2} + 22164361129 T^{4} )^{2}$$
$59$ $$( 1 + 380758 T^{2} + 42180533641 T^{4} )^{2}$$
$61$ $$( 1 + 442 T + 226981 T^{2} )^{4}$$
$67$ $$( 1 - 60026 T^{2} + 90458382169 T^{4} )^{2}$$
$71$ $$( 1 - 364178 T^{2} + 128100283921 T^{4} )^{2}$$
$73$ $$( 1 - 410 T + 389017 T^{2} )^{4}$$
$79$ $$( 1 - 978818 T^{2} + 243087455521 T^{4} )^{2}$$
$83$ $$( 1 - 428954 T^{2} + 326940373369 T^{4} )^{2}$$
$89$ $$( 1 - 703058 T^{2} + 496981290961 T^{4} )^{2}$$
$97$ $$( 1 - 770 T + 912673 T^{2} )^{4}$$