# Properties

 Label 4334.2.a.h Level $4334$ Weight $2$ Character orbit 4334.a Self dual yes Analytic conductor $34.607$ Analytic rank $0$ Dimension $27$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4334 = 2 \cdot 11 \cdot 197$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4334.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: $$34.6071642360$$ Analytic rank: $$0$$ Dimension: $$27$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$27 q - 27 q^{2} + 27 q^{4} + 9 q^{5} + q^{7} - 27 q^{8} + 43 q^{9}+O(q^{10})$$ 27 * q - 27 * q^2 + 27 * q^4 + 9 * q^5 + q^7 - 27 * q^8 + 43 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$27 q - 27 q^{2} + 27 q^{4} + 9 q^{5} + q^{7} - 27 q^{8} + 43 q^{9} - 9 q^{10} + 27 q^{11} + 4 q^{13} - q^{14} + 8 q^{15} + 27 q^{16} + 3 q^{17} - 43 q^{18} + 30 q^{19} + 9 q^{20} + 11 q^{21} - 27 q^{22} + 13 q^{23} + 50 q^{25} - 4 q^{26} - 3 q^{27} + q^{28} + 5 q^{29} - 8 q^{30} + 40 q^{31} - 27 q^{32} - 3 q^{34} - 16 q^{35} + 43 q^{36} + 21 q^{37} - 30 q^{38} + 5 q^{39} - 9 q^{40} + 13 q^{41} - 11 q^{42} + 10 q^{43} + 27 q^{44} + 48 q^{45} - 13 q^{46} + 78 q^{49} - 50 q^{50} + 8 q^{51} + 4 q^{52} + 8 q^{53} + 3 q^{54} + 9 q^{55} - q^{56} - 16 q^{57} - 5 q^{58} + 24 q^{59} + 8 q^{60} + 28 q^{61} - 40 q^{62} - 18 q^{63} + 27 q^{64} - q^{65} + 24 q^{67} + 3 q^{68} - 3 q^{69} + 16 q^{70} - 3 q^{71} - 43 q^{72} + 9 q^{73} - 21 q^{74} + 26 q^{75} + 30 q^{76} + q^{77} - 5 q^{78} + 12 q^{79} + 9 q^{80} + 99 q^{81} - 13 q^{82} - 11 q^{83} + 11 q^{84} + 15 q^{85} - 10 q^{86} - 34 q^{87} - 27 q^{88} + 69 q^{89} - 48 q^{90} + q^{91} + 13 q^{92} - 24 q^{93} - 31 q^{95} + 41 q^{97} - 78 q^{98} + 43 q^{99}+O(q^{100})$$ 27 * q - 27 * q^2 + 27 * q^4 + 9 * q^5 + q^7 - 27 * q^8 + 43 * q^9 - 9 * q^10 + 27 * q^11 + 4 * q^13 - q^14 + 8 * q^15 + 27 * q^16 + 3 * q^17 - 43 * q^18 + 30 * q^19 + 9 * q^20 + 11 * q^21 - 27 * q^22 + 13 * q^23 + 50 * q^25 - 4 * q^26 - 3 * q^27 + q^28 + 5 * q^29 - 8 * q^30 + 40 * q^31 - 27 * q^32 - 3 * q^34 - 16 * q^35 + 43 * q^36 + 21 * q^37 - 30 * q^38 + 5 * q^39 - 9 * q^40 + 13 * q^41 - 11 * q^42 + 10 * q^43 + 27 * q^44 + 48 * q^45 - 13 * q^46 + 78 * q^49 - 50 * q^50 + 8 * q^51 + 4 * q^52 + 8 * q^53 + 3 * q^54 + 9 * q^55 - q^56 - 16 * q^57 - 5 * q^58 + 24 * q^59 + 8 * q^60 + 28 * q^61 - 40 * q^62 - 18 * q^63 + 27 * q^64 - q^65 + 24 * q^67 + 3 * q^68 - 3 * q^69 + 16 * q^70 - 3 * q^71 - 43 * q^72 + 9 * q^73 - 21 * q^74 + 26 * q^75 + 30 * q^76 + q^77 - 5 * q^78 + 12 * q^79 + 9 * q^80 + 99 * q^81 - 13 * q^82 - 11 * q^83 + 11 * q^84 + 15 * q^85 - 10 * q^86 - 34 * q^87 - 27 * q^88 + 69 * q^89 - 48 * q^90 + q^91 + 13 * q^92 - 24 * q^93 - 31 * q^95 + 41 * q^97 - 78 * q^98 + 43 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −1.00000 −3.32608 1.00000 3.07836 3.32608 −4.99294 −1.00000 8.06282 −3.07836
1.2 −1.00000 −3.27909 1.00000 −0.172349 3.27909 4.49591 −1.00000 7.75244 0.172349
1.3 −1.00000 −3.12559 1.00000 4.00012 3.12559 −0.0131838 −1.00000 6.76933 −4.00012
1.4 −1.00000 −2.76463 1.00000 −0.249776 2.76463 −2.52527 −1.00000 4.64317 0.249776
1.5 −1.00000 −2.46207 1.00000 −2.57755 2.46207 −3.72083 −1.00000 3.06177 2.57755
1.6 −1.00000 −2.45740 1.00000 −4.36120 2.45740 2.66113 −1.00000 3.03881 4.36120
1.7 −1.00000 −2.09175 1.00000 −0.380034 2.09175 −1.49841 −1.00000 1.37543 0.380034
1.8 −1.00000 −1.37214 1.00000 0.332573 1.37214 4.01205 −1.00000 −1.11723 −0.332573
1.9 −1.00000 −1.36128 1.00000 1.41439 1.36128 1.06104 −1.00000 −1.14691 −1.41439
1.10 −1.00000 −1.15238 1.00000 −0.0196651 1.15238 3.79263 −1.00000 −1.67203 0.0196651
1.11 −1.00000 −0.982227 1.00000 −0.607693 0.982227 −0.935926 −1.00000 −2.03523 0.607693
1.12 −1.00000 −0.593027 1.00000 0.744487 0.593027 −4.68885 −1.00000 −2.64832 −0.744487
1.13 −1.00000 −0.127044 1.00000 −4.01526 0.127044 0.758073 −1.00000 −2.98386 4.01526
1.14 −1.00000 −0.0408121 1.00000 3.97341 0.0408121 4.79303 −1.00000 −2.99833 −3.97341
1.15 −1.00000 0.181771 1.00000 4.29102 −0.181771 −3.68727 −1.00000 −2.96696 −4.29102
1.16 −1.00000 0.309146 1.00000 2.79911 −0.309146 0.293952 −1.00000 −2.90443 −2.79911
1.17 −1.00000 1.05524 1.00000 −3.32834 −1.05524 −3.79792 −1.00000 −1.88648 3.32834
1.18 −1.00000 1.14059 1.00000 −0.375030 −1.14059 0.176944 −1.00000 −1.69906 0.375030
1.19 −1.00000 1.56203 1.00000 −0.643018 −1.56203 3.99306 −1.00000 −0.560072 0.643018
1.20 −1.00000 1.80604 1.00000 −3.59945 −1.80604 3.05375 −1.00000 0.261765 3.59945
See all 27 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.27 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$11$$ $$-1$$
$$197$$ $$+1$$

## 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 4334.2.a.h 27

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4334.2.a.h 27 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{27} - 62 T_{3}^{25} + T_{3}^{24} + 1679 T_{3}^{23} - 49 T_{3}^{22} - 26122 T_{3}^{21} + \cdots - 1472$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4334))$$.