# Properties

 Label 12.3.d.a Level $12$ Weight $3$ Character orbit 12.d Analytic conductor $0.327$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,3,Mod(7,12)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("12.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 12.d (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: $$0.326976317232$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + \beta q^{3} + (2 \beta - 2) q^{4} - 2 q^{5} + ( - \beta + 3) q^{6} - 4 \beta q^{7} + 8 q^{8} - 3 q^{9} +O(q^{10})$$ q + (-b - 1) * q^2 + b * q^3 + (2*b - 2) * q^4 - 2 * q^5 + (-b + 3) * q^6 - 4*b * q^7 + 8 * q^8 - 3 * q^9 $$q + ( - \beta - 1) q^{2} + \beta q^{3} + (2 \beta - 2) q^{4} - 2 q^{5} + ( - \beta + 3) q^{6} - 4 \beta q^{7} + 8 q^{8} - 3 q^{9} + (2 \beta + 2) q^{10} + 4 \beta q^{11} + ( - 2 \beta - 6) q^{12} + 2 q^{13} + (4 \beta - 12) q^{14} - 2 \beta q^{15} + ( - 8 \beta - 8) q^{16} + 10 q^{17} + (3 \beta + 3) q^{18} + 12 \beta q^{19} + ( - 4 \beta + 4) q^{20} + 12 q^{21} + ( - 4 \beta + 12) q^{22} - 16 \beta q^{23} + 8 \beta q^{24} - 21 q^{25} + ( - 2 \beta - 2) q^{26} - 3 \beta q^{27} + (8 \beta + 24) q^{28} - 26 q^{29} + (2 \beta - 6) q^{30} + 4 \beta q^{31} + (16 \beta - 16) q^{32} - 12 q^{33} + ( - 10 \beta - 10) q^{34} + 8 \beta q^{35} + ( - 6 \beta + 6) q^{36} + 26 q^{37} + ( - 12 \beta + 36) q^{38} + 2 \beta q^{39} - 16 q^{40} + 58 q^{41} + ( - 12 \beta - 12) q^{42} - 28 \beta q^{43} + ( - 8 \beta - 24) q^{44} + 6 q^{45} + (16 \beta - 48) q^{46} + 40 \beta q^{47} + ( - 8 \beta + 24) q^{48} + q^{49} + (21 \beta + 21) q^{50} + 10 \beta q^{51} + (4 \beta - 4) q^{52} - 74 q^{53} + (3 \beta - 9) q^{54} - 8 \beta q^{55} - 32 \beta q^{56} - 36 q^{57} + (26 \beta + 26) q^{58} - 52 \beta q^{59} + (4 \beta + 12) q^{60} + 26 q^{61} + ( - 4 \beta + 12) q^{62} + 12 \beta q^{63} + 64 q^{64} - 4 q^{65} + (12 \beta + 12) q^{66} + 4 \beta q^{67} + (20 \beta - 20) q^{68} + 48 q^{69} + ( - 8 \beta + 24) q^{70} - 24 q^{72} - 46 q^{73} + ( - 26 \beta - 26) q^{74} - 21 \beta q^{75} + ( - 24 \beta - 72) q^{76} + 48 q^{77} + ( - 2 \beta + 6) q^{78} + 68 \beta q^{79} + (16 \beta + 16) q^{80} + 9 q^{81} + ( - 58 \beta - 58) q^{82} + 28 \beta q^{83} + (24 \beta - 24) q^{84} - 20 q^{85} + (28 \beta - 84) q^{86} - 26 \beta q^{87} + 32 \beta q^{88} + 82 q^{89} + ( - 6 \beta - 6) q^{90} - 8 \beta q^{91} + (32 \beta + 96) q^{92} - 12 q^{93} + ( - 40 \beta + 120) q^{94} - 24 \beta q^{95} + ( - 16 \beta - 48) q^{96} + 2 q^{97} + ( - \beta - 1) q^{98} - 12 \beta q^{99} +O(q^{100})$$ q + (-b - 1) * q^2 + b * q^3 + (2*b - 2) * q^4 - 2 * q^5 + (-b + 3) * q^6 - 4*b * q^7 + 8 * q^8 - 3 * q^9 + (2*b + 2) * q^10 + 4*b * q^11 + (-2*b - 6) * q^12 + 2 * q^13 + (4*b - 12) * q^14 - 2*b * q^15 + (-8*b - 8) * q^16 + 10 * q^17 + (3*b + 3) * q^18 + 12*b * q^19 + (-4*b + 4) * q^20 + 12 * q^21 + (-4*b + 12) * q^22 - 16*b * q^23 + 8*b * q^24 - 21 * q^25 + (-2*b - 2) * q^26 - 3*b * q^27 + (8*b + 24) * q^28 - 26 * q^29 + (2*b - 6) * q^30 + 4*b * q^31 + (16*b - 16) * q^32 - 12 * q^33 + (-10*b - 10) * q^34 + 8*b * q^35 + (-6*b + 6) * q^36 + 26 * q^37 + (-12*b + 36) * q^38 + 2*b * q^39 - 16 * q^40 + 58 * q^41 + (-12*b - 12) * q^42 - 28*b * q^43 + (-8*b - 24) * q^44 + 6 * q^45 + (16*b - 48) * q^46 + 40*b * q^47 + (-8*b + 24) * q^48 + q^49 + (21*b + 21) * q^50 + 10*b * q^51 + (4*b - 4) * q^52 - 74 * q^53 + (3*b - 9) * q^54 - 8*b * q^55 - 32*b * q^56 - 36 * q^57 + (26*b + 26) * q^58 - 52*b * q^59 + (4*b + 12) * q^60 + 26 * q^61 + (-4*b + 12) * q^62 + 12*b * q^63 + 64 * q^64 - 4 * q^65 + (12*b + 12) * q^66 + 4*b * q^67 + (20*b - 20) * q^68 + 48 * q^69 + (-8*b + 24) * q^70 - 24 * q^72 - 46 * q^73 + (-26*b - 26) * q^74 - 21*b * q^75 + (-24*b - 72) * q^76 + 48 * q^77 + (-2*b + 6) * q^78 + 68*b * q^79 + (16*b + 16) * q^80 + 9 * q^81 + (-58*b - 58) * q^82 + 28*b * q^83 + (24*b - 24) * q^84 - 20 * q^85 + (28*b - 84) * q^86 - 26*b * q^87 + 32*b * q^88 + 82 * q^89 + (-6*b - 6) * q^90 - 8*b * q^91 + (32*b + 96) * q^92 - 12 * q^93 + (-40*b + 120) * q^94 - 24*b * q^95 + (-16*b - 48) * q^96 + 2 * q^97 + (-b - 1) * q^98 - 12*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 6 q^{6} + 16 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 - 4 * q^5 + 6 * q^6 + 16 * q^8 - 6 * q^9 $$2 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 6 q^{6} + 16 q^{8} - 6 q^{9} + 4 q^{10} - 12 q^{12} + 4 q^{13} - 24 q^{14} - 16 q^{16} + 20 q^{17} + 6 q^{18} + 8 q^{20} + 24 q^{21} + 24 q^{22} - 42 q^{25} - 4 q^{26} + 48 q^{28} - 52 q^{29} - 12 q^{30} - 32 q^{32} - 24 q^{33} - 20 q^{34} + 12 q^{36} + 52 q^{37} + 72 q^{38} - 32 q^{40} + 116 q^{41} - 24 q^{42} - 48 q^{44} + 12 q^{45} - 96 q^{46} + 48 q^{48} + 2 q^{49} + 42 q^{50} - 8 q^{52} - 148 q^{53} - 18 q^{54} - 72 q^{57} + 52 q^{58} + 24 q^{60} + 52 q^{61} + 24 q^{62} + 128 q^{64} - 8 q^{65} + 24 q^{66} - 40 q^{68} + 96 q^{69} + 48 q^{70} - 48 q^{72} - 92 q^{73} - 52 q^{74} - 144 q^{76} + 96 q^{77} + 12 q^{78} + 32 q^{80} + 18 q^{81} - 116 q^{82} - 48 q^{84} - 40 q^{85} - 168 q^{86} + 164 q^{89} - 12 q^{90} + 192 q^{92} - 24 q^{93} + 240 q^{94} - 96 q^{96} + 4 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 4 * q^5 + 6 * q^6 + 16 * q^8 - 6 * q^9 + 4 * q^10 - 12 * q^12 + 4 * q^13 - 24 * q^14 - 16 * q^16 + 20 * q^17 + 6 * q^18 + 8 * q^20 + 24 * q^21 + 24 * q^22 - 42 * q^25 - 4 * q^26 + 48 * q^28 - 52 * q^29 - 12 * q^30 - 32 * q^32 - 24 * q^33 - 20 * q^34 + 12 * q^36 + 52 * q^37 + 72 * q^38 - 32 * q^40 + 116 * q^41 - 24 * q^42 - 48 * q^44 + 12 * q^45 - 96 * q^46 + 48 * q^48 + 2 * q^49 + 42 * q^50 - 8 * q^52 - 148 * q^53 - 18 * q^54 - 72 * q^57 + 52 * q^58 + 24 * q^60 + 52 * q^61 + 24 * q^62 + 128 * q^64 - 8 * q^65 + 24 * q^66 - 40 * q^68 + 96 * q^69 + 48 * q^70 - 48 * q^72 - 92 * q^73 - 52 * q^74 - 144 * q^76 + 96 * q^77 + 12 * q^78 + 32 * q^80 + 18 * q^81 - 116 * q^82 - 48 * q^84 - 40 * q^85 - 168 * q^86 + 164 * q^89 - 12 * q^90 + 192 * q^92 - 24 * q^93 + 240 * q^94 - 96 * q^96 + 4 * q^97 - 2 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 1.73205i −2.00000 + 3.46410i −2.00000 3.00000 1.73205i 6.92820i 8.00000 −3.00000 2.00000 + 3.46410i
7.2 −1.00000 + 1.73205i 1.73205i −2.00000 3.46410i −2.00000 3.00000 + 1.73205i 6.92820i 8.00000 −3.00000 2.00000 3.46410i
 $$n$$: 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.3.d.a 2
3.b odd 2 1 36.3.d.c 2
4.b odd 2 1 inner 12.3.d.a 2
5.b even 2 1 300.3.c.b 2
5.c odd 4 2 300.3.f.a 4
7.b odd 2 1 588.3.g.b 2
8.b even 2 1 192.3.g.b 2
8.d odd 2 1 192.3.g.b 2
9.c even 3 1 324.3.f.d 2
9.c even 3 1 324.3.f.j 2
9.d odd 6 1 324.3.f.a 2
9.d odd 6 1 324.3.f.g 2
12.b even 2 1 36.3.d.c 2
15.d odd 2 1 900.3.c.e 2
15.e even 4 2 900.3.f.c 4
16.e even 4 2 768.3.b.c 4
16.f odd 4 2 768.3.b.c 4
20.d odd 2 1 300.3.c.b 2
20.e even 4 2 300.3.f.a 4
24.f even 2 1 576.3.g.e 2
24.h odd 2 1 576.3.g.e 2
28.d even 2 1 588.3.g.b 2
36.f odd 6 1 324.3.f.d 2
36.f odd 6 1 324.3.f.j 2
36.h even 6 1 324.3.f.a 2
36.h even 6 1 324.3.f.g 2
48.i odd 4 2 2304.3.b.l 4
48.k even 4 2 2304.3.b.l 4
60.h even 2 1 900.3.c.e 2
60.l odd 4 2 900.3.f.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 1.a even 1 1 trivial
12.3.d.a 2 4.b odd 2 1 inner
36.3.d.c 2 3.b odd 2 1
36.3.d.c 2 12.b even 2 1
192.3.g.b 2 8.b even 2 1
192.3.g.b 2 8.d odd 2 1
300.3.c.b 2 5.b even 2 1
300.3.c.b 2 20.d odd 2 1
300.3.f.a 4 5.c odd 4 2
300.3.f.a 4 20.e even 4 2
324.3.f.a 2 9.d odd 6 1
324.3.f.a 2 36.h even 6 1
324.3.f.d 2 9.c even 3 1
324.3.f.d 2 36.f odd 6 1
324.3.f.g 2 9.d odd 6 1
324.3.f.g 2 36.h even 6 1
324.3.f.j 2 9.c even 3 1
324.3.f.j 2 36.f odd 6 1
576.3.g.e 2 24.f even 2 1
576.3.g.e 2 24.h odd 2 1
588.3.g.b 2 7.b odd 2 1
588.3.g.b 2 28.d even 2 1
768.3.b.c 4 16.e even 4 2
768.3.b.c 4 16.f odd 4 2
900.3.c.e 2 15.d odd 2 1
900.3.c.e 2 60.h even 2 1
900.3.f.c 4 15.e even 4 2
900.3.f.c 4 60.l odd 4 2
2304.3.b.l 4 48.i odd 4 2
2304.3.b.l 4 48.k even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(12, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + 3$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2} + 48$$
$11$ $$T^{2} + 48$$
$13$ $$(T - 2)^{2}$$
$17$ $$(T - 10)^{2}$$
$19$ $$T^{2} + 432$$
$23$ $$T^{2} + 768$$
$29$ $$(T + 26)^{2}$$
$31$ $$T^{2} + 48$$
$37$ $$(T - 26)^{2}$$
$41$ $$(T - 58)^{2}$$
$43$ $$T^{2} + 2352$$
$47$ $$T^{2} + 4800$$
$53$ $$(T + 74)^{2}$$
$59$ $$T^{2} + 8112$$
$61$ $$(T - 26)^{2}$$
$67$ $$T^{2} + 48$$
$71$ $$T^{2}$$
$73$ $$(T + 46)^{2}$$
$79$ $$T^{2} + 13872$$
$83$ $$T^{2} + 2352$$
$89$ $$(T - 82)^{2}$$
$97$ $$(T - 2)^{2}$$
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