# Properties

 Label 4334.2.a.e Level $4334$ Weight $2$ Character orbit 4334.a Self dual yes Analytic conductor $34.607$ Analytic rank $0$ Dimension $24$ 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: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{2} - 4 q^{3} + 24 q^{4} - 4 q^{5} + 4 q^{6} + 7 q^{7} - 24 q^{8} + 28 q^{9}+O(q^{10})$$ 24 * q - 24 * q^2 - 4 * q^3 + 24 * q^4 - 4 * q^5 + 4 * q^6 + 7 * q^7 - 24 * q^8 + 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{2} - 4 q^{3} + 24 q^{4} - 4 q^{5} + 4 q^{6} + 7 q^{7} - 24 q^{8} + 28 q^{9} + 4 q^{10} - 24 q^{11} - 4 q^{12} + 21 q^{13} - 7 q^{14} - 2 q^{15} + 24 q^{16} + 15 q^{17} - 28 q^{18} + 21 q^{19} - 4 q^{20} + 15 q^{21} + 24 q^{22} - 17 q^{23} + 4 q^{24} + 46 q^{25} - 21 q^{26} - 19 q^{27} + 7 q^{28} + 9 q^{29} + 2 q^{30} + 27 q^{31} - 24 q^{32} + 4 q^{33} - 15 q^{34} - 2 q^{35} + 28 q^{36} + 5 q^{37} - 21 q^{38} + 17 q^{39} + 4 q^{40} + 16 q^{41} - 15 q^{42} + 3 q^{43} - 24 q^{44} - 21 q^{45} + 17 q^{46} - 24 q^{47} - 4 q^{48} + 55 q^{49} - 46 q^{50} - 12 q^{51} + 21 q^{52} - 26 q^{53} + 19 q^{54} + 4 q^{55} - 7 q^{56} + 30 q^{57} - 9 q^{58} - 17 q^{59} - 2 q^{60} + 44 q^{61} - 27 q^{62} + 4 q^{63} + 24 q^{64} + 35 q^{65} - 4 q^{66} + 10 q^{67} + 15 q^{68} + 3 q^{69} + 2 q^{70} - 6 q^{71} - 28 q^{72} + 77 q^{73} - 5 q^{74} - 32 q^{75} + 21 q^{76} - 7 q^{77} - 17 q^{78} + 43 q^{79} - 4 q^{80} + 48 q^{81} - 16 q^{82} - 20 q^{83} + 15 q^{84} + 35 q^{85} - 3 q^{86} + 36 q^{87} + 24 q^{88} + 3 q^{89} + 21 q^{90} + 63 q^{91} - 17 q^{92} + 36 q^{93} + 24 q^{94} - 3 q^{95} + 4 q^{96} + 16 q^{97} - 55 q^{98} - 28 q^{99}+O(q^{100})$$ 24 * q - 24 * q^2 - 4 * q^3 + 24 * q^4 - 4 * q^5 + 4 * q^6 + 7 * q^7 - 24 * q^8 + 28 * q^9 + 4 * q^10 - 24 * q^11 - 4 * q^12 + 21 * q^13 - 7 * q^14 - 2 * q^15 + 24 * q^16 + 15 * q^17 - 28 * q^18 + 21 * q^19 - 4 * q^20 + 15 * q^21 + 24 * q^22 - 17 * q^23 + 4 * q^24 + 46 * q^25 - 21 * q^26 - 19 * q^27 + 7 * q^28 + 9 * q^29 + 2 * q^30 + 27 * q^31 - 24 * q^32 + 4 * q^33 - 15 * q^34 - 2 * q^35 + 28 * q^36 + 5 * q^37 - 21 * q^38 + 17 * q^39 + 4 * q^40 + 16 * q^41 - 15 * q^42 + 3 * q^43 - 24 * q^44 - 21 * q^45 + 17 * q^46 - 24 * q^47 - 4 * q^48 + 55 * q^49 - 46 * q^50 - 12 * q^51 + 21 * q^52 - 26 * q^53 + 19 * q^54 + 4 * q^55 - 7 * q^56 + 30 * q^57 - 9 * q^58 - 17 * q^59 - 2 * q^60 + 44 * q^61 - 27 * q^62 + 4 * q^63 + 24 * q^64 + 35 * q^65 - 4 * q^66 + 10 * q^67 + 15 * q^68 + 3 * q^69 + 2 * q^70 - 6 * q^71 - 28 * q^72 + 77 * q^73 - 5 * q^74 - 32 * q^75 + 21 * q^76 - 7 * q^77 - 17 * q^78 + 43 * q^79 - 4 * q^80 + 48 * q^81 - 16 * q^82 - 20 * q^83 + 15 * q^84 + 35 * q^85 - 3 * q^86 + 36 * q^87 + 24 * q^88 + 3 * q^89 + 21 * q^90 + 63 * q^91 - 17 * q^92 + 36 * q^93 + 24 * q^94 - 3 * q^95 + 4 * q^96 + 16 * q^97 - 55 * q^98 - 28 * 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 −3.29945 1.00000 −3.72842 3.29945 −3.54518 −1.00000 7.88640 3.72842
1.2 −1.00000 −3.08498 1.00000 1.01853 3.08498 −2.58850 −1.00000 6.51711 −1.01853
1.3 −1.00000 −3.05385 1.00000 3.30243 3.05385 4.28183 −1.00000 6.32597 −3.30243
1.4 −1.00000 −2.81176 1.00000 −2.87657 2.81176 −0.0915560 −1.00000 4.90602 2.87657
1.5 −1.00000 −2.34418 1.00000 −2.76933 2.34418 1.79797 −1.00000 2.49516 2.76933
1.6 −1.00000 −2.15745 1.00000 1.30451 2.15745 3.75589 −1.00000 1.65457 −1.30451
1.7 −1.00000 −2.03428 1.00000 3.12875 2.03428 −4.32049 −1.00000 1.13829 −3.12875
1.8 −1.00000 −1.47170 1.00000 3.58412 1.47170 −0.948099 −1.00000 −0.834090 −3.58412
1.9 −1.00000 −1.15892 1.00000 −2.65435 1.15892 4.05701 −1.00000 −1.65690 2.65435
1.10 −1.00000 −1.02498 1.00000 −1.91029 1.02498 0.691966 −1.00000 −1.94941 1.91029
1.11 −1.00000 −0.392540 1.00000 2.96246 0.392540 −1.89262 −1.00000 −2.84591 −2.96246
1.12 −1.00000 −0.257887 1.00000 0.105378 0.257887 −1.45976 −1.00000 −2.93349 −0.105378
1.13 −1.00000 0.0886475 1.00000 −3.04967 −0.0886475 −1.08307 −1.00000 −2.99214 3.04967
1.14 −1.00000 0.140715 1.00000 0.341585 −0.140715 −2.48538 −1.00000 −2.98020 −0.341585
1.15 −1.00000 0.626459 1.00000 −4.16763 −0.626459 4.47360 −1.00000 −2.60755 4.16763
1.16 −1.00000 0.745887 1.00000 2.52923 −0.745887 4.14909 −1.00000 −2.44365 −2.52923
1.17 −1.00000 1.08316 1.00000 −1.14511 −1.08316 2.88226 −1.00000 −1.82677 1.14511
1.18 −1.00000 1.21094 1.00000 −0.613687 −1.21094 −4.17411 −1.00000 −1.53363 0.613687
1.19 −1.00000 2.01603 1.00000 2.21808 −2.01603 1.91345 −1.00000 1.06437 −2.21808
1.20 −1.00000 2.05410 1.00000 4.08690 −2.05410 0.841598 −1.00000 1.21932 −4.08690
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.24 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.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4334.2.a.e 24 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} + 4 T_{3}^{23} - 42 T_{3}^{22} - 175 T_{3}^{21} + 735 T_{3}^{20} + 3231 T_{3}^{19} + \cdots + 292$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4334))$$.