# Properties

 Label 234.10.a.a 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)

 $$f(q)$$ $$=$$ $$q - 16 q^{2} + 256 q^{4} + 1979 q^{5} - 10115 q^{7} - 4096 q^{8}+O(q^{10})$$ q - 16 * q^2 + 256 * q^4 + 1979 * q^5 - 10115 * q^7 - 4096 * q^8 $$q - 16 q^{2} + 256 q^{4} + 1979 q^{5} - 10115 q^{7} - 4096 q^{8} - 31664 q^{10} - 18850 q^{11} + 28561 q^{13} + 161840 q^{14} + 65536 q^{16} + 142403 q^{17} + 83302 q^{19} + 506624 q^{20} + 301600 q^{22} + 536544 q^{23} + 1963316 q^{25} - 456976 q^{26} - 2589440 q^{28} + 2600442 q^{29} - 2214004 q^{31} - 1048576 q^{32} - 2278448 q^{34} - 20017585 q^{35} + 18099241 q^{37} - 1332832 q^{38} - 8105984 q^{40} - 26812240 q^{41} - 42253475 q^{43} - 4825600 q^{44} - 8584704 q^{46} - 35914993 q^{47} + 61959618 q^{49} - 31413056 q^{50} + 7311616 q^{52} + 66514064 q^{53} - 37304150 q^{55} + 41431040 q^{56} - 41607072 q^{58} + 108164002 q^{59} - 207449912 q^{61} + 35424064 q^{62} + 16777216 q^{64} + 56522219 q^{65} + 193015514 q^{67} + 36455168 q^{68} + 320281360 q^{70} + 201833497 q^{71} - 121628110 q^{73} - 289587856 q^{74} + 21325312 q^{76} + 190667750 q^{77} + 112871912 q^{79} + 129695744 q^{80} + 428995840 q^{82} - 308254212 q^{83} + 281815537 q^{85} + 676055600 q^{86} + 77209600 q^{88} + 6374870 q^{89} - 288894515 q^{91} + 137355264 q^{92} + 574639888 q^{94} + 164854658 q^{95} + 871266886 q^{97} - 991353888 q^{98}+O(q^{100})$$ q - 16 * q^2 + 256 * q^4 + 1979 * q^5 - 10115 * q^7 - 4096 * q^8 - 31664 * q^10 - 18850 * q^11 + 28561 * q^13 + 161840 * q^14 + 65536 * q^16 + 142403 * q^17 + 83302 * q^19 + 506624 * q^20 + 301600 * q^22 + 536544 * q^23 + 1963316 * q^25 - 456976 * q^26 - 2589440 * q^28 + 2600442 * q^29 - 2214004 * q^31 - 1048576 * q^32 - 2278448 * q^34 - 20017585 * q^35 + 18099241 * q^37 - 1332832 * q^38 - 8105984 * q^40 - 26812240 * q^41 - 42253475 * q^43 - 4825600 * q^44 - 8584704 * q^46 - 35914993 * q^47 + 61959618 * q^49 - 31413056 * q^50 + 7311616 * q^52 + 66514064 * q^53 - 37304150 * q^55 + 41431040 * q^56 - 41607072 * q^58 + 108164002 * q^59 - 207449912 * q^61 + 35424064 * q^62 + 16777216 * q^64 + 56522219 * q^65 + 193015514 * q^67 + 36455168 * q^68 + 320281360 * q^70 + 201833497 * q^71 - 121628110 * q^73 - 289587856 * q^74 + 21325312 * q^76 + 190667750 * q^77 + 112871912 * q^79 + 129695744 * q^80 + 428995840 * q^82 - 308254212 * q^83 + 281815537 * q^85 + 676055600 * q^86 + 77209600 * q^88 + 6374870 * q^89 - 288894515 * q^91 + 137355264 * q^92 + 574639888 * q^94 + 164854658 * q^95 + 871266886 * q^97 - 991353888 * 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 1979.00 0 −10115.0 −4096.00 0 −31664.0
 $$n$$: 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.a 1
3.b odd 2 1 26.10.a.c 1
12.b even 2 1 208.10.a.b 1
39.d odd 2 1 338.10.a.b 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1979$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(234))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 16$$
$3$ $$T$$
$5$ $$T - 1979$$
$7$ $$T + 10115$$
$11$ $$T + 18850$$
$13$ $$T - 28561$$
$17$ $$T - 142403$$
$19$ $$T - 83302$$
$23$ $$T - 536544$$
$29$ $$T - 2600442$$
$31$ $$T + 2214004$$
$37$ $$T - 18099241$$
$41$ $$T + 26812240$$
$43$ $$T + 42253475$$
$47$ $$T + 35914993$$
$53$ $$T - 66514064$$
$59$ $$T - 108164002$$
$61$ $$T + 207449912$$
$67$ $$T - 193015514$$
$71$ $$T - 201833497$$
$73$ $$T + 121628110$$
$79$ $$T - 112871912$$
$83$ $$T + 308254212$$
$89$ $$T - 6374870$$
$97$ $$T - 871266886$$