Properties

 Label 3234.2.a.x Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.a (trivial)

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: yes Analytic conductor: $$25.8236200137$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + b * q^5 + q^6 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{8} + q^{9} - \beta q^{10} + q^{11} - q^{12} - 2 q^{13} - \beta q^{15} + q^{16} - \beta q^{17} - q^{18} + ( - \beta - 2) q^{19} + \beta q^{20} - q^{22} + 2 \beta q^{23} + q^{24} + 7 q^{25} + 2 q^{26} - q^{27} - 6 q^{29} + \beta q^{30} + ( - \beta - 2) q^{31} - q^{32} - q^{33} + \beta q^{34} + q^{36} + ( - 2 \beta + 2) q^{37} + (\beta + 2) q^{38} + 2 q^{39} - \beta q^{40} - \beta q^{41} + ( - 2 \beta - 4) q^{43} + q^{44} + \beta q^{45} - 2 \beta q^{46} + ( - \beta - 6) q^{47} - q^{48} - 7 q^{50} + \beta q^{51} - 2 q^{52} + ( - 2 \beta + 6) q^{53} + q^{54} + \beta q^{55} + (\beta + 2) q^{57} + 6 q^{58} - 2 \beta q^{59} - \beta q^{60} - 2 q^{61} + (\beta + 2) q^{62} + q^{64} - 2 \beta q^{65} + q^{66} + (2 \beta + 8) q^{67} - \beta q^{68} - 2 \beta q^{69} - 12 q^{71} - q^{72} + ( - \beta + 4) q^{73} + (2 \beta - 2) q^{74} - 7 q^{75} + ( - \beta - 2) q^{76} - 2 q^{78} + ( - 2 \beta - 4) q^{79} + \beta q^{80} + q^{81} + \beta q^{82} + ( - 3 \beta + 6) q^{83} - 12 q^{85} + (2 \beta + 4) q^{86} + 6 q^{87} - q^{88} + (2 \beta - 6) q^{89} - \beta q^{90} + 2 \beta q^{92} + (\beta + 2) q^{93} + (\beta + 6) q^{94} + ( - 2 \beta - 12) q^{95} + q^{96} - 2 q^{97} + q^{99} +O(q^{100})$$ q - q^2 - q^3 + q^4 + b * q^5 + q^6 - q^8 + q^9 - b * q^10 + q^11 - q^12 - 2 * q^13 - b * q^15 + q^16 - b * q^17 - q^18 + (-b - 2) * q^19 + b * q^20 - q^22 + 2*b * q^23 + q^24 + 7 * q^25 + 2 * q^26 - q^27 - 6 * q^29 + b * q^30 + (-b - 2) * q^31 - q^32 - q^33 + b * q^34 + q^36 + (-2*b + 2) * q^37 + (b + 2) * q^38 + 2 * q^39 - b * q^40 - b * q^41 + (-2*b - 4) * q^43 + q^44 + b * q^45 - 2*b * q^46 + (-b - 6) * q^47 - q^48 - 7 * q^50 + b * q^51 - 2 * q^52 + (-2*b + 6) * q^53 + q^54 + b * q^55 + (b + 2) * q^57 + 6 * q^58 - 2*b * q^59 - b * q^60 - 2 * q^61 + (b + 2) * q^62 + q^64 - 2*b * q^65 + q^66 + (2*b + 8) * q^67 - b * q^68 - 2*b * q^69 - 12 * q^71 - q^72 + (-b + 4) * q^73 + (2*b - 2) * q^74 - 7 * q^75 + (-b - 2) * q^76 - 2 * q^78 + (-2*b - 4) * q^79 + b * q^80 + q^81 + b * q^82 + (-3*b + 6) * q^83 - 12 * q^85 + (2*b + 4) * q^86 + 6 * q^87 - q^88 + (2*b - 6) * q^89 - b * q^90 + 2*b * q^92 + (b + 2) * q^93 + (b + 6) * q^94 + (-2*b - 12) * q^95 + q^96 - 2 * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{18} - 4 q^{19} - 2 q^{22} + 2 q^{24} + 14 q^{25} + 4 q^{26} - 2 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{32} - 2 q^{33} + 2 q^{36} + 4 q^{37} + 4 q^{38} + 4 q^{39} - 8 q^{43} + 2 q^{44} - 12 q^{47} - 2 q^{48} - 14 q^{50} - 4 q^{52} + 12 q^{53} + 2 q^{54} + 4 q^{57} + 12 q^{58} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 2 q^{66} + 16 q^{67} - 24 q^{71} - 2 q^{72} + 8 q^{73} - 4 q^{74} - 14 q^{75} - 4 q^{76} - 4 q^{78} - 8 q^{79} + 2 q^{81} + 12 q^{83} - 24 q^{85} + 8 q^{86} + 12 q^{87} - 2 q^{88} - 12 q^{89} + 4 q^{93} + 12 q^{94} - 24 q^{95} + 2 q^{96} - 4 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^18 - 4 * q^19 - 2 * q^22 + 2 * q^24 + 14 * q^25 + 4 * q^26 - 2 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^32 - 2 * q^33 + 2 * q^36 + 4 * q^37 + 4 * q^38 + 4 * q^39 - 8 * q^43 + 2 * q^44 - 12 * q^47 - 2 * q^48 - 14 * q^50 - 4 * q^52 + 12 * q^53 + 2 * q^54 + 4 * q^57 + 12 * q^58 - 4 * q^61 + 4 * q^62 + 2 * q^64 + 2 * q^66 + 16 * q^67 - 24 * q^71 - 2 * q^72 + 8 * q^73 - 4 * q^74 - 14 * q^75 - 4 * q^76 - 4 * q^78 - 8 * q^79 + 2 * q^81 + 12 * q^83 - 24 * q^85 + 8 * q^86 + 12 * q^87 - 2 * q^88 - 12 * q^89 + 4 * q^93 + 12 * q^94 - 24 * q^95 + 2 * q^96 - 4 * q^97 + 2 * q^99

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.00000 −1.00000 1.00000 −3.46410 1.00000 0 −1.00000 1.00000 3.46410
1.2 −1.00000 −1.00000 1.00000 3.46410 1.00000 0 −1.00000 1.00000 −3.46410
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$+1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3234.2.a.x 2
3.b odd 2 1 9702.2.a.dd 2
7.b odd 2 1 462.2.a.h 2
21.c even 2 1 1386.2.a.p 2
28.d even 2 1 3696.2.a.bc 2
77.b even 2 1 5082.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.h 2 7.b odd 2 1
1386.2.a.p 2 21.c even 2 1
3234.2.a.x 2 1.a even 1 1 trivial
3696.2.a.bc 2 28.d even 2 1
5082.2.a.bu 2 77.b even 2 1
9702.2.a.dd 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{13} + 2$$ T13 + 2 $$T_{17}^{2} - 12$$ T17^2 - 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 12$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 12$$
$19$ $$T^{2} + 4T - 8$$
$23$ $$T^{2} - 48$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 4T - 8$$
$37$ $$T^{2} - 4T - 44$$
$41$ $$T^{2} - 12$$
$43$ $$T^{2} + 8T - 32$$
$47$ $$T^{2} + 12T + 24$$
$53$ $$T^{2} - 12T - 12$$
$59$ $$T^{2} - 48$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} - 16T + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 8T + 4$$
$79$ $$T^{2} + 8T - 32$$
$83$ $$T^{2} - 12T - 72$$
$89$ $$T^{2} + 12T - 12$$
$97$ $$(T + 2)^{2}$$