# Properties

 Label 12.4.b.a Level $12$ Weight $4$ Character orbit 12.b Analytic conductor $0.708$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,4,Mod(11,12)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(12, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("12.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.708022920069$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + x^{2} + 4$$ x^4 + x^2 + 4 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + \beta_{2} - 2) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{5} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 6) q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{8} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 3) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b3 - b1) * q^3 + (b3 + b2 - 2) * q^4 + (b3 - b2 - 2*b1) * q^5 + (b3 - 3*b2 + b1 - 6) * q^6 + (b3 + b2) * q^7 + (-4*b3 + 4*b2 - 4*b1) * q^8 + (-3*b3 + 3*b2 + 6*b1 - 3) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + \beta_{2} - 2) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{5} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 6) q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{8} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 3) q^{9}+ \cdots + (105 \beta_{3} + 135 \beta_{2} - 30 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b3 - b1) * q^3 + (b3 + b2 - 2) * q^4 + (b3 - b2 - 2*b1) * q^5 + (b3 - 3*b2 + b1 - 6) * q^6 + (b3 + b2) * q^7 + (-4*b3 + 4*b2 - 4*b1) * q^8 + (-3*b3 + 3*b2 + 6*b1 - 3) * q^9 + (-2*b3 - 2*b2 + 20) * q^10 + (5*b3 - 5*b2 + 10*b1) * q^11 + (5*b3 - 3*b2 - 4*b1 + 30) * q^12 - 10 * q^13 + (-4*b3 + 4*b2 - 2*b1) * q^14 + (-b3 + 9*b2 - 10*b1) * q^15 + (-4*b3 - 4*b2 - 56) * q^16 + (4*b3 - 4*b2 - 8*b1) * q^17 + (6*b3 + 6*b2 - 3*b1 - 60) * q^18 + (-9*b3 - 9*b2) * q^19 + (8*b3 - 8*b2 + 24*b1) * q^20 + (3*b3 - 3*b2 - 6*b1 + 30) * q^21 + (10*b3 + 10*b2 + 60) * q^22 + (-14*b3 + 14*b2 - 28*b1) * q^23 + (-8*b3 + 28*b1 + 72) * q^24 + 45 * q^25 - 10*b1 * q^26 + (6*b3 - 27*b2 + 33*b1) * q^27 + (-2*b3 - 2*b2 - 60) * q^28 + (-17*b3 + 17*b2 + 34*b1) * q^29 + (-26*b3 + 6*b2 - 8*b1 - 60) * q^30 + (29*b3 + 29*b2) * q^31 + (16*b3 - 16*b2 - 48*b1) * q^32 + (15*b3 - 15*b2 - 30*b1 - 120) * q^33 + (-8*b3 - 8*b2 + 80) * q^34 + (10*b3 - 10*b2 + 20*b1) * q^35 + (-27*b3 + 21*b2 - 72*b1 + 6) * q^36 - 130 * q^37 + (36*b3 - 36*b2 + 18*b1) * q^38 + (10*b3 + 10*b1) * q^39 + (24*b3 + 24*b2 + 80) * q^40 + (-14*b3 + 14*b2 + 28*b1) * q^41 + (-6*b3 - 6*b2 + 30*b1 + 60) * q^42 + (-29*b3 - 29*b2) * q^43 + (-40*b3 + 40*b2 + 40*b1) * q^44 + (-3*b3 + 3*b2 + 6*b1 + 240) * q^45 + (-28*b3 - 28*b2 - 168) * q^46 + (-28*b3 + 28*b2 - 56*b1) * q^47 + (44*b3 + 12*b2 + 80*b1 - 120) * q^48 + 283 * q^49 + 45*b1 * q^50 + (-4*b3 + 36*b2 - 40*b1) * q^51 + (-10*b3 - 10*b2 + 20) * q^52 + (61*b3 - 61*b2 - 122*b1) * q^53 + (75*b3 - 9*b2 + 21*b1 + 198) * q^54 + (-40*b3 - 40*b2) * q^55 + (8*b3 - 8*b2 - 56*b1) * q^56 + (-27*b3 + 27*b2 + 54*b1 - 270) * q^57 + (34*b3 + 34*b2 - 340) * q^58 + (25*b3 - 25*b2 + 50*b1) * q^59 + (32*b3 - 48*b2 - 40*b1 - 240) * q^60 - 442 * q^61 + (-116*b3 + 116*b2 - 58*b1) * q^62 + (-33*b3 + 27*b2 - 60*b1) * q^63 + (-48*b3 - 48*b2 + 352) * q^64 + (-10*b3 + 10*b2 + 20*b1) * q^65 + (-30*b3 - 30*b2 - 120*b1 + 300) * q^66 + (95*b3 + 95*b2) * q^67 + (32*b3 - 32*b2 + 96*b1) * q^68 + (-42*b3 + 42*b2 + 84*b1 + 336) * q^69 + (20*b3 + 20*b2 + 120) * q^70 + (150*b3 - 150*b2 + 300*b1) * q^71 + (-60*b3 - 84*b2 + 12*b1 - 240) * q^72 + 410 * q^73 - 130*b1 * q^74 + (-45*b3 - 45*b1) * q^75 + (18*b3 + 18*b2 + 540) * q^76 + (-30*b3 + 30*b2 + 60*b1) * q^77 + (-10*b3 + 30*b2 - 10*b1 + 60) * q^78 + (-11*b3 - 11*b2) * q^79 + (-96*b3 + 96*b2 + 32*b1) * q^80 + (18*b3 - 18*b2 - 36*b1 - 711) * q^81 + (28*b3 + 28*b2 - 280) * q^82 + (-181*b3 + 181*b2 - 362*b1) * q^83 + (54*b3 + 6*b2 + 72*b1 - 60) * q^84 - 320 * q^85 + (116*b3 - 116*b2 + 58*b1) * q^86 + (17*b3 - 153*b2 + 170*b1) * q^87 + (40*b3 + 40*b2 - 720) * q^88 + (94*b3 - 94*b2 - 188*b1) * q^89 + (6*b3 + 6*b2 + 240*b1 - 60) * q^90 + (-10*b3 - 10*b2) * q^91 + (112*b3 - 112*b2 - 112*b1) * q^92 + (87*b3 - 87*b2 - 174*b1 + 870) * q^93 + (-56*b3 - 56*b2 - 336) * q^94 + (-90*b3 + 90*b2 - 180*b1) * q^95 + (-32*b3 + 192*b2 - 176*b1 + 96) * q^96 + 770 * q^97 + 283*b1 * q^98 + (105*b3 + 135*b2 - 30*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} - 24 q^{6} - 12 q^{9}+O(q^{10})$$ 4 * q - 8 * q^4 - 24 * q^6 - 12 * q^9 $$4 q - 8 q^{4} - 24 q^{6} - 12 q^{9} + 80 q^{10} + 120 q^{12} - 40 q^{13} - 224 q^{16} - 240 q^{18} + 120 q^{21} + 240 q^{22} + 288 q^{24} + 180 q^{25} - 240 q^{28} - 240 q^{30} - 480 q^{33} + 320 q^{34} + 24 q^{36} - 520 q^{37} + 320 q^{40} + 240 q^{42} + 960 q^{45} - 672 q^{46} - 480 q^{48} + 1132 q^{49} + 80 q^{52} + 792 q^{54} - 1080 q^{57} - 1360 q^{58} - 960 q^{60} - 1768 q^{61} + 1408 q^{64} + 1200 q^{66} + 1344 q^{69} + 480 q^{70} - 960 q^{72} + 1640 q^{73} + 2160 q^{76} + 240 q^{78} - 2844 q^{81} - 1120 q^{82} - 240 q^{84} - 1280 q^{85} - 2880 q^{88} - 240 q^{90} + 3480 q^{93} - 1344 q^{94} + 384 q^{96} + 3080 q^{97}+O(q^{100})$$ 4 * q - 8 * q^4 - 24 * q^6 - 12 * q^9 + 80 * q^10 + 120 * q^12 - 40 * q^13 - 224 * q^16 - 240 * q^18 + 120 * q^21 + 240 * q^22 + 288 * q^24 + 180 * q^25 - 240 * q^28 - 240 * q^30 - 480 * q^33 + 320 * q^34 + 24 * q^36 - 520 * q^37 + 320 * q^40 + 240 * q^42 + 960 * q^45 - 672 * q^46 - 480 * q^48 + 1132 * q^49 + 80 * q^52 + 792 * q^54 - 1080 * q^57 - 1360 * q^58 - 960 * q^60 - 1768 * q^61 + 1408 * q^64 + 1200 * q^66 + 1344 * q^69 + 480 * q^70 - 960 * q^72 + 1640 * q^73 + 2160 * q^76 + 240 * q^78 - 2844 * q^81 - 1120 * q^82 - 240 * q^84 - 1280 * q^85 - 2880 * q^88 - 240 * q^90 + 3480 * q^93 - 1344 * q^94 + 384 * q^96 + 3080 * q^97

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

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 2\nu^{2} + \nu + 1$$ v^3 + 2*v^2 + v + 1 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2\nu^{2} - \nu + 1$$ -v^3 + 2*v^2 - v + 1
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2 ) / 4$$ (b3 + b2 - 2) / 4 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (-b3 + b2 - b1) / 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

$p$ $F_p(T)$
$2$ $$T^{4} + 4T^{2} + 64$$
$3$ $$T^{4} + 6T^{2} + 729$$
$5$ $$(T^{2} + 80)^{2}$$
$7$ $$(T^{2} + 60)^{2}$$
$11$ $$(T^{2} - 1200)^{2}$$
$13$ $$(T + 10)^{4}$$
$17$ $$(T^{2} + 1280)^{2}$$
$19$ $$(T^{2} + 4860)^{2}$$
$23$ $$(T^{2} - 9408)^{2}$$
$29$ $$(T^{2} + 23120)^{2}$$
$31$ $$(T^{2} + 50460)^{2}$$
$37$ $$(T + 130)^{4}$$
$41$ $$(T^{2} + 15680)^{2}$$
$43$ $$(T^{2} + 50460)^{2}$$
$47$ $$(T^{2} - 37632)^{2}$$
$53$ $$(T^{2} + 297680)^{2}$$
$59$ $$(T^{2} - 30000)^{2}$$
$61$ $$(T + 442)^{4}$$
$67$ $$(T^{2} + 541500)^{2}$$
$71$ $$(T^{2} - 1080000)^{2}$$
$73$ $$(T - 410)^{4}$$
$79$ $$(T^{2} + 7260)^{2}$$
$83$ $$(T^{2} - 1572528)^{2}$$
$89$ $$(T^{2} + 706880)^{2}$$
$97$ $$(T - 770)^{4}$$