# Properties

 Label 3234.2.a.z 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + q^{11} + q^{12} - \beta q^{13} + q^{16} - 4 q^{17} - q^{18} + ( - \beta - 4) q^{19} - q^{22} + (4 \beta + 2) q^{23} - q^{24} - 5 q^{25} + \beta q^{26} + q^{27} - 4 \beta q^{29} + (5 \beta - 4) q^{31} - q^{32} + q^{33} + 4 q^{34} + q^{36} + (4 \beta - 2) q^{37} + (\beta + 4) q^{38} - \beta q^{39} + ( - 4 \beta - 4) q^{41} - 10 q^{43} + q^{44} + ( - 4 \beta - 2) q^{46} + \beta q^{47} + q^{48} + 5 q^{50} - 4 q^{51} - \beta q^{52} + ( - 4 \beta + 2) q^{53} - q^{54} + ( - \beta - 4) q^{57} + 4 \beta q^{58} + ( - 2 \beta - 4) q^{59} + \beta q^{61} + ( - 5 \beta + 4) q^{62} + q^{64} - q^{66} - 8 \beta q^{67} - 4 q^{68} + (4 \beta + 2) q^{69} + (4 \beta + 8) q^{71} - q^{72} - 4 q^{73} + ( - 4 \beta + 2) q^{74} - 5 q^{75} + ( - \beta - 4) q^{76} + \beta q^{78} + (4 \beta - 4) q^{79} + q^{81} + (4 \beta + 4) q^{82} + (5 \beta + 8) q^{83} + 10 q^{86} - 4 \beta q^{87} - q^{88} + (\beta - 4) q^{89} + (4 \beta + 2) q^{92} + (5 \beta - 4) q^{93} - \beta q^{94} - q^{96} - 9 \beta q^{97} + q^{99} +O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - q^8 + q^9 + q^11 + q^12 - b * q^13 + q^16 - 4 * q^17 - q^18 + (-b - 4) * q^19 - q^22 + (4*b + 2) * q^23 - q^24 - 5 * q^25 + b * q^26 + q^27 - 4*b * q^29 + (5*b - 4) * q^31 - q^32 + q^33 + 4 * q^34 + q^36 + (4*b - 2) * q^37 + (b + 4) * q^38 - b * q^39 + (-4*b - 4) * q^41 - 10 * q^43 + q^44 + (-4*b - 2) * q^46 + b * q^47 + q^48 + 5 * q^50 - 4 * q^51 - b * q^52 + (-4*b + 2) * q^53 - q^54 + (-b - 4) * q^57 + 4*b * q^58 + (-2*b - 4) * q^59 + b * q^61 + (-5*b + 4) * q^62 + q^64 - q^66 - 8*b * q^67 - 4 * q^68 + (4*b + 2) * q^69 + (4*b + 8) * q^71 - q^72 - 4 * q^73 + (-4*b + 2) * q^74 - 5 * q^75 + (-b - 4) * q^76 + b * q^78 + (4*b - 4) * q^79 + q^81 + (4*b + 4) * q^82 + (5*b + 8) * q^83 + 10 * q^86 - 4*b * q^87 - q^88 + (b - 4) * q^89 + (4*b + 2) * q^92 + (5*b - 4) * q^93 - b * q^94 - q^96 - 9*b * 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 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{22} + 4 q^{23} - 2 q^{24} - 10 q^{25} + 2 q^{27} - 8 q^{31} - 2 q^{32} + 2 q^{33} + 8 q^{34} + 2 q^{36} - 4 q^{37} + 8 q^{38} - 8 q^{41} - 20 q^{43} + 2 q^{44} - 4 q^{46} + 2 q^{48} + 10 q^{50} - 8 q^{51} + 4 q^{53} - 2 q^{54} - 8 q^{57} - 8 q^{59} + 8 q^{62} + 2 q^{64} - 2 q^{66} - 8 q^{68} + 4 q^{69} + 16 q^{71} - 2 q^{72} - 8 q^{73} + 4 q^{74} - 10 q^{75} - 8 q^{76} - 8 q^{79} + 2 q^{81} + 8 q^{82} + 16 q^{83} + 20 q^{86} - 2 q^{88} - 8 q^{89} + 4 q^{92} - 8 q^{93} - 2 q^{96} + 2 q^{99}+O(q^{100})$$ 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 + 2 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^22 + 4 * q^23 - 2 * q^24 - 10 * q^25 + 2 * q^27 - 8 * q^31 - 2 * q^32 + 2 * q^33 + 8 * q^34 + 2 * q^36 - 4 * q^37 + 8 * q^38 - 8 * q^41 - 20 * q^43 + 2 * q^44 - 4 * q^46 + 2 * q^48 + 10 * q^50 - 8 * q^51 + 4 * q^53 - 2 * q^54 - 8 * q^57 - 8 * q^59 + 8 * q^62 + 2 * q^64 - 2 * q^66 - 8 * q^68 + 4 * q^69 + 16 * q^71 - 2 * q^72 - 8 * q^73 + 4 * q^74 - 10 * q^75 - 8 * q^76 - 8 * q^79 + 2 * q^81 + 8 * q^82 + 16 * q^83 + 20 * q^86 - 2 * q^88 - 8 * q^89 + 4 * q^92 - 8 * q^93 - 2 * q^96 + 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
 1.41421 −1.41421
−1.00000 1.00000 1.00000 0 −1.00000 0 −1.00000 1.00000 0
1.2 −1.00000 1.00000 1.00000 0 −1.00000 0 −1.00000 1.00000 0
 $$n$$: 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.z yes 2
3.b odd 2 1 9702.2.a.dl 2
7.b odd 2 1 3234.2.a.y 2
21.c even 2 1 9702.2.a.de 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.y 2 7.b odd 2 1
3234.2.a.z yes 2 1.a even 1 1 trivial
9702.2.a.de 2 21.c even 2 1
9702.2.a.dl 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5}$$ T5 $$T_{13}^{2} - 2$$ T13^2 - 2 $$T_{17} + 4$$ T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2$$
$17$ $$(T + 4)^{2}$$
$19$ $$T^{2} + 8T + 14$$
$23$ $$T^{2} - 4T - 28$$
$29$ $$T^{2} - 32$$
$31$ $$T^{2} + 8T - 34$$
$37$ $$T^{2} + 4T - 28$$
$41$ $$T^{2} + 8T - 16$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 2$$
$53$ $$T^{2} - 4T - 28$$
$59$ $$T^{2} + 8T + 8$$
$61$ $$T^{2} - 2$$
$67$ $$T^{2} - 128$$
$71$ $$T^{2} - 16T + 32$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} + 8T - 16$$
$83$ $$T^{2} - 16T + 14$$
$89$ $$T^{2} + 8T + 14$$
$97$ $$T^{2} - 162$$