# Properties

 Label 234.10.a.c Level $234$ Weight $10$ Character orbit 234.a Self dual yes Analytic conductor $120.518$ Analytic rank $1$ 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] = [234,10,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, 10, names="a")

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ 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: $$120.518385662$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + 256 q^{4} + 1310 q^{5} - 5810 q^{7} + 4096 q^{8}+O(q^{10})$$ q + 16 * q^2 + 256 * q^4 + 1310 * q^5 - 5810 * q^7 + 4096 * q^8 $$q + 16 q^{2} + 256 q^{4} + 1310 q^{5} - 5810 q^{7} + 4096 q^{8} + 20960 q^{10} + 4498 q^{11} - 28561 q^{13} - 92960 q^{14} + 65536 q^{16} + 237498 q^{17} - 913014 q^{19} + 335360 q^{20} + 71968 q^{22} - 201544 q^{23} - 237025 q^{25} - 456976 q^{26} - 1487360 q^{28} - 1276834 q^{29} + 4163770 q^{31} + 1048576 q^{32} + 3799968 q^{34} - 7611100 q^{35} - 18442662 q^{37} - 14608224 q^{38} + 5365760 q^{40} + 22601670 q^{41} + 11726308 q^{43} + 1151488 q^{44} - 3224704 q^{46} - 59291534 q^{47} - 6597507 q^{49} - 3792400 q^{50} - 7311616 q^{52} - 108158694 q^{53} + 5892380 q^{55} - 23797760 q^{56} - 20429344 q^{58} + 14920154 q^{59} - 57003746 q^{61} + 66620320 q^{62} + 16777216 q^{64} - 37414910 q^{65} + 22074010 q^{67} + 60799488 q^{68} - 121777600 q^{70} - 44416250 q^{71} + 265794626 q^{73} - 295082592 q^{74} - 233731584 q^{76} - 26133380 q^{77} + 476755484 q^{79} + 85852160 q^{80} + 361626720 q^{82} + 505315830 q^{83} + 311122380 q^{85} + 187620928 q^{86} + 18423808 q^{88} - 890840634 q^{89} + 165939410 q^{91} - 51595264 q^{92} - 948664544 q^{94} - 1196048340 q^{95} - 802776958 q^{97} - 105560112 q^{98}+O(q^{100})$$ q + 16 * q^2 + 256 * q^4 + 1310 * q^5 - 5810 * q^7 + 4096 * q^8 + 20960 * q^10 + 4498 * q^11 - 28561 * q^13 - 92960 * q^14 + 65536 * q^16 + 237498 * q^17 - 913014 * q^19 + 335360 * q^20 + 71968 * q^22 - 201544 * q^23 - 237025 * q^25 - 456976 * q^26 - 1487360 * q^28 - 1276834 * q^29 + 4163770 * q^31 + 1048576 * q^32 + 3799968 * q^34 - 7611100 * q^35 - 18442662 * q^37 - 14608224 * q^38 + 5365760 * q^40 + 22601670 * q^41 + 11726308 * q^43 + 1151488 * q^44 - 3224704 * q^46 - 59291534 * q^47 - 6597507 * q^49 - 3792400 * q^50 - 7311616 * q^52 - 108158694 * q^53 + 5892380 * q^55 - 23797760 * q^56 - 20429344 * q^58 + 14920154 * q^59 - 57003746 * q^61 + 66620320 * q^62 + 16777216 * q^64 - 37414910 * q^65 + 22074010 * q^67 + 60799488 * q^68 - 121777600 * q^70 - 44416250 * q^71 + 265794626 * q^73 - 295082592 * q^74 - 233731584 * q^76 - 26133380 * q^77 + 476755484 * q^79 + 85852160 * q^80 + 361626720 * q^82 + 505315830 * q^83 + 311122380 * q^85 + 187620928 * q^86 + 18423808 * q^88 - 890840634 * q^89 + 165939410 * q^91 - 51595264 * q^92 - 948664544 * q^94 - 1196048340 * q^95 - 802776958 * q^97 - 105560112 * 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
16.0000 0 256.000 1310.00 0 −5810.00 4096.00 0 20960.0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 234.10.a.c 1
3.b odd 2 1 26.10.a.b 1
12.b even 2 1 208.10.a.a 1
39.d odd 2 1 338.10.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.10.a.b 1 3.b odd 2 1
208.10.a.a 1 12.b even 2 1
234.10.a.c 1 1.a even 1 1 trivial
338.10.a.d 1 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1310$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(234))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 16$$
$3$ $$T$$
$5$ $$T - 1310$$
$7$ $$T + 5810$$
$11$ $$T - 4498$$
$13$ $$T + 28561$$
$17$ $$T - 237498$$
$19$ $$T + 913014$$
$23$ $$T + 201544$$
$29$ $$T + 1276834$$
$31$ $$T - 4163770$$
$37$ $$T + 18442662$$
$41$ $$T - 22601670$$
$43$ $$T - 11726308$$
$47$ $$T + 59291534$$
$53$ $$T + 108158694$$
$59$ $$T - 14920154$$
$61$ $$T + 57003746$$
$67$ $$T - 22074010$$
$71$ $$T + 44416250$$
$73$ $$T - 265794626$$
$79$ $$T - 476755484$$
$83$ $$T - 505315830$$
$89$ $$T + 890840634$$
$97$ $$T + 802776958$$