# Properties

 Label 338.4.e.i Level $338$ Weight $4$ Character orbit 338.e Analytic conductor $19.943$ Analytic rank $0$ Dimension $24$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(23,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

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

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

chi := DirichletCharacter("338.23");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.e (of order $$6$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$19.9426455819$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 18 q^{3} + 48 q^{4} - 226 q^{9}+O(q^{10})$$ 24 * q - 18 * q^3 + 48 * q^4 - 226 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 18 q^{3} + 48 q^{4} - 226 q^{9} + 72 q^{10} - 144 q^{12} + 200 q^{14} - 192 q^{16} + 198 q^{17} - 148 q^{22} + 534 q^{23} - 1472 q^{25} + 2676 q^{27} + 238 q^{29} + 472 q^{30} - 1228 q^{35} + 904 q^{36} + 648 q^{38} + 576 q^{40} + 104 q^{42} + 856 q^{43} - 288 q^{48} + 1798 q^{49} - 1156 q^{51} + 356 q^{53} - 2252 q^{55} + 400 q^{56} - 3408 q^{61} - 2500 q^{62} - 1536 q^{64} + 6096 q^{66} - 792 q^{68} + 2336 q^{69} - 1096 q^{74} + 3596 q^{75} - 11160 q^{77} - 3500 q^{79} - 6676 q^{81} + 4560 q^{82} - 3204 q^{87} + 592 q^{88} + 9720 q^{90} + 4272 q^{92} + 3944 q^{94} - 8186 q^{95}+O(q^{100})$$ 24 * q - 18 * q^3 + 48 * q^4 - 226 * q^9 + 72 * q^10 - 144 * q^12 + 200 * q^14 - 192 * q^16 + 198 * q^17 - 148 * q^22 + 534 * q^23 - 1472 * q^25 + 2676 * q^27 + 238 * q^29 + 472 * q^30 - 1228 * q^35 + 904 * q^36 + 648 * q^38 + 576 * q^40 + 104 * q^42 + 856 * q^43 - 288 * q^48 + 1798 * q^49 - 1156 * q^51 + 356 * q^53 - 2252 * q^55 + 400 * q^56 - 3408 * q^61 - 2500 * q^62 - 1536 * q^64 + 6096 * q^66 - 792 * q^68 + 2336 * q^69 - 1096 * q^74 + 3596 * q^75 - 11160 * q^77 - 3500 * q^79 - 6676 * q^81 + 4560 * q^82 - 3204 * q^87 + 592 * q^88 + 9720 * q^90 + 4272 * q^92 + 3944 * q^94 - 8186 * q^95

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −1.73205 + 1.00000i −5.00428 8.66767i 2.00000 3.46410i 13.8136i 17.3353 + 10.0086i 0.227146 + 0.131143i 8.00000i −36.5857 + 63.3682i −13.8136 23.9258i
23.2 −1.73205 + 1.00000i 0.622927 + 1.07894i 2.00000 3.46410i 20.5002i −2.15788 1.24585i −18.4070 10.6273i 8.00000i 12.7239 22.0385i 20.5002 + 35.5074i
23.3 −1.73205 + 1.00000i 1.96372 + 3.40126i 2.00000 3.46410i 0.152827i −6.80251 3.92743i −29.4373 16.9956i 8.00000i 5.78763 10.0245i −0.152827 0.264703i
23.4 −1.73205 + 1.00000i −4.47729 7.75489i 2.00000 3.46410i 17.3267i 15.5098 + 8.95458i 14.5068 + 8.37550i 8.00000i −26.5923 + 46.0591i 17.3267 + 30.0108i
23.5 −1.73205 + 1.00000i 4.07401 + 7.05638i 2.00000 3.46410i 12.6646i −14.1128 8.14801i 24.9160 + 14.3853i 8.00000i −19.6950 + 34.1128i −12.6646 21.9357i
23.6 −1.73205 + 1.00000i −1.67908 2.90825i 2.00000 3.46410i 6.80407i 5.81649 + 3.35815i −13.4563 7.76902i 8.00000i 7.86140 13.6163i 6.80407 + 11.7850i
23.7 1.73205 1.00000i −4.47729 7.75489i 2.00000 3.46410i 17.3267i −15.5098 8.95458i −14.5068 8.37550i 8.00000i −26.5923 + 46.0591i 17.3267 + 30.0108i
23.8 1.73205 1.00000i 0.622927 + 1.07894i 2.00000 3.46410i 20.5002i 2.15788 + 1.24585i 18.4070 + 10.6273i 8.00000i 12.7239 22.0385i 20.5002 + 35.5074i
23.9 1.73205 1.00000i −1.67908 2.90825i 2.00000 3.46410i 6.80407i −5.81649 3.35815i 13.4563 + 7.76902i 8.00000i 7.86140 13.6163i 6.80407 + 11.7850i
23.10 1.73205 1.00000i −5.00428 8.66767i 2.00000 3.46410i 13.8136i −17.3353 10.0086i −0.227146 0.131143i 8.00000i −36.5857 + 63.3682i −13.8136 23.9258i
23.11 1.73205 1.00000i 1.96372 + 3.40126i 2.00000 3.46410i 0.152827i 6.80251 + 3.92743i 29.4373 + 16.9956i 8.00000i 5.78763 10.0245i −0.152827 0.264703i
23.12 1.73205 1.00000i 4.07401 + 7.05638i 2.00000 3.46410i 12.6646i 14.1128 + 8.14801i −24.9160 14.3853i 8.00000i −19.6950 + 34.1128i −12.6646 21.9357i
147.1 −1.73205 1.00000i −5.00428 + 8.66767i 2.00000 + 3.46410i 13.8136i 17.3353 10.0086i 0.227146 0.131143i 8.00000i −36.5857 63.3682i −13.8136 + 23.9258i
147.2 −1.73205 1.00000i 0.622927 1.07894i 2.00000 + 3.46410i 20.5002i −2.15788 + 1.24585i −18.4070 + 10.6273i 8.00000i 12.7239 + 22.0385i 20.5002 35.5074i
147.3 −1.73205 1.00000i 1.96372 3.40126i 2.00000 + 3.46410i 0.152827i −6.80251 + 3.92743i −29.4373 + 16.9956i 8.00000i 5.78763 + 10.0245i −0.152827 + 0.264703i
147.4 −1.73205 1.00000i −4.47729 + 7.75489i 2.00000 + 3.46410i 17.3267i 15.5098 8.95458i 14.5068 8.37550i 8.00000i −26.5923 46.0591i 17.3267 30.0108i
147.5 −1.73205 1.00000i 4.07401 7.05638i 2.00000 + 3.46410i 12.6646i −14.1128 + 8.14801i 24.9160 14.3853i 8.00000i −19.6950 34.1128i −12.6646 + 21.9357i
147.6 −1.73205 1.00000i −1.67908 + 2.90825i 2.00000 + 3.46410i 6.80407i 5.81649 3.35815i −13.4563 + 7.76902i 8.00000i 7.86140 + 13.6163i 6.80407 11.7850i
147.7 1.73205 + 1.00000i −4.47729 + 7.75489i 2.00000 + 3.46410i 17.3267i −15.5098 + 8.95458i −14.5068 + 8.37550i 8.00000i −26.5923 46.0591i 17.3267 30.0108i
147.8 1.73205 + 1.00000i 0.622927 1.07894i 2.00000 + 3.46410i 20.5002i 2.15788 1.24585i 18.4070 10.6273i 8.00000i 12.7239 + 22.0385i 20.5002 35.5074i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.e.i 24
13.b even 2 1 inner 338.4.e.i 24
13.c even 3 1 338.4.b.h 12
13.c even 3 1 inner 338.4.e.i 24
13.d odd 4 1 338.4.c.o 12
13.d odd 4 1 338.4.c.p 12
13.e even 6 1 338.4.b.h 12
13.e even 6 1 inner 338.4.e.i 24
13.f odd 12 1 338.4.a.n 6
13.f odd 12 1 338.4.a.o yes 6
13.f odd 12 1 338.4.c.o 12
13.f odd 12 1 338.4.c.p 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.4.a.n 6 13.f odd 12 1
338.4.a.o yes 6 13.f odd 12 1
338.4.b.h 12 13.c even 3 1
338.4.b.h 12 13.e even 6 1
338.4.c.o 12 13.d odd 4 1
338.4.c.o 12 13.f odd 12 1
338.4.c.p 12 13.d odd 4 1
338.4.c.p 12 13.f odd 12 1
338.4.e.i 24 1.a even 1 1 trivial
338.4.e.i 24 13.b even 2 1 inner
338.4.e.i 24 13.c even 3 1 inner
338.4.e.i 24 13.e even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(338, [\chi])$$:

 $$T_{3}^{12} + 9 T_{3}^{11} + 178 T_{3}^{10} + 589 T_{3}^{9} + 13675 T_{3}^{8} + 37320 T_{3}^{7} + \cdots + 143976001$$ T3^12 + 9*T3^11 + 178*T3^10 + 589*T3^9 + 13675*T3^8 + 37320*T3^7 + 662301*T3^6 - 237676*T3^5 + 10068423*T3^4 + 1070173*T3^3 + 92507896*T3^2 - 96555953*T3 + 143976001 $$T_{5}^{12} + 1118T_{5}^{10} + 459449T_{5}^{8} + 85344405T_{5}^{6} + 6935622164T_{5}^{4} + 178926938144T_{5}^{2} + 4175227456$$ T5^12 + 1118*T5^10 + 459449*T5^8 + 85344405*T5^6 + 6935622164*T5^4 + 178926938144*T5^2 + 4175227456