# Properties

 Label 234.4.a.b Level $234$ Weight $4$ Character orbit 234.a Self dual yes Analytic conductor $13.806$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,4,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 234.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: $$13.8064469413$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 4 q^{4} - 6 q^{5} + 20 q^{7} - 8 q^{8}+O(q^{10})$$ q - 2 * q^2 + 4 * q^4 - 6 * q^5 + 20 * q^7 - 8 * q^8 $$q - 2 q^{2} + 4 q^{4} - 6 q^{5} + 20 q^{7} - 8 q^{8} + 12 q^{10} - 24 q^{11} + 13 q^{13} - 40 q^{14} + 16 q^{16} + 30 q^{17} - 16 q^{19} - 24 q^{20} + 48 q^{22} + 72 q^{23} - 89 q^{25} - 26 q^{26} + 80 q^{28} + 282 q^{29} + 164 q^{31} - 32 q^{32} - 60 q^{34} - 120 q^{35} + 110 q^{37} + 32 q^{38} + 48 q^{40} + 126 q^{41} + 164 q^{43} - 96 q^{44} - 144 q^{46} + 204 q^{47} + 57 q^{49} + 178 q^{50} + 52 q^{52} + 738 q^{53} + 144 q^{55} - 160 q^{56} - 564 q^{58} - 120 q^{59} + 614 q^{61} - 328 q^{62} + 64 q^{64} - 78 q^{65} + 848 q^{67} + 120 q^{68} + 240 q^{70} - 132 q^{71} + 218 q^{73} - 220 q^{74} - 64 q^{76} - 480 q^{77} - 1096 q^{79} - 96 q^{80} - 252 q^{82} - 552 q^{83} - 180 q^{85} - 328 q^{86} + 192 q^{88} - 210 q^{89} + 260 q^{91} + 288 q^{92} - 408 q^{94} + 96 q^{95} - 1726 q^{97} - 114 q^{98}+O(q^{100})$$ q - 2 * q^2 + 4 * q^4 - 6 * q^5 + 20 * q^7 - 8 * q^8 + 12 * q^10 - 24 * q^11 + 13 * q^13 - 40 * q^14 + 16 * q^16 + 30 * q^17 - 16 * q^19 - 24 * q^20 + 48 * q^22 + 72 * q^23 - 89 * q^25 - 26 * q^26 + 80 * q^28 + 282 * q^29 + 164 * q^31 - 32 * q^32 - 60 * q^34 - 120 * q^35 + 110 * q^37 + 32 * q^38 + 48 * q^40 + 126 * q^41 + 164 * q^43 - 96 * q^44 - 144 * q^46 + 204 * q^47 + 57 * q^49 + 178 * q^50 + 52 * q^52 + 738 * q^53 + 144 * q^55 - 160 * q^56 - 564 * q^58 - 120 * q^59 + 614 * q^61 - 328 * q^62 + 64 * q^64 - 78 * q^65 + 848 * q^67 + 120 * q^68 + 240 * q^70 - 132 * q^71 + 218 * q^73 - 220 * q^74 - 64 * q^76 - 480 * q^77 - 1096 * q^79 - 96 * q^80 - 252 * q^82 - 552 * q^83 - 180 * q^85 - 328 * q^86 + 192 * q^88 - 210 * q^89 + 260 * q^91 + 288 * q^92 - 408 * q^94 + 96 * q^95 - 1726 * q^97 - 114 * q^98

## 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
 0
−2.00000 0 4.00000 −6.00000 0 20.0000 −8.00000 0 12.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 234.4.a.b 1
3.b odd 2 1 78.4.a.e 1
4.b odd 2 1 1872.4.a.e 1
12.b even 2 1 624.4.a.i 1
15.d odd 2 1 1950.4.a.c 1
24.f even 2 1 2496.4.a.b 1
24.h odd 2 1 2496.4.a.k 1
39.d odd 2 1 1014.4.a.b 1
39.f even 4 2 1014.4.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.e 1 3.b odd 2 1
234.4.a.b 1 1.a even 1 1 trivial
624.4.a.i 1 12.b even 2 1
1014.4.a.b 1 39.d odd 2 1
1014.4.b.c 2 39.f even 4 2
1872.4.a.e 1 4.b odd 2 1
1950.4.a.c 1 15.d odd 2 1
2496.4.a.b 1 24.f even 2 1
2496.4.a.k 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(234))$$:

 $$T_{5} + 6$$ T5 + 6 $$T_{7} - 20$$ T7 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T + 6$$
$7$ $$T - 20$$
$11$ $$T + 24$$
$13$ $$T - 13$$
$17$ $$T - 30$$
$19$ $$T + 16$$
$23$ $$T - 72$$
$29$ $$T - 282$$
$31$ $$T - 164$$
$37$ $$T - 110$$
$41$ $$T - 126$$
$43$ $$T - 164$$
$47$ $$T - 204$$
$53$ $$T - 738$$
$59$ $$T + 120$$
$61$ $$T - 614$$
$67$ $$T - 848$$
$71$ $$T + 132$$
$73$ $$T - 218$$
$79$ $$T + 1096$$
$83$ $$T + 552$$
$89$ $$T + 210$$
$97$ $$T + 1726$$