# Properties

 Label 289.4.b.f Level $289$ Weight $4$ Character orbit 289.b Analytic conductor $17.052$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(288,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("289.288");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 289.b (of order $$2$$, degree $$1$$, 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: $$17.0515519917$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10})$$ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100})$$ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
288.1 −4.44354 0.537336i 11.7450 17.2592i 2.38767i 6.40763i −16.6411 26.7113 76.6918i
288.2 −4.44354 0.537336i 11.7450 17.2592i 2.38767i 6.40763i −16.6411 26.7113 76.6918i
288.3 −4.42326 9.44971i 11.5652 8.63042i 41.7985i 12.6513i −15.7698 −62.2970 38.1746i
288.4 −4.42326 9.44971i 11.5652 8.63042i 41.7985i 12.6513i −15.7698 −62.2970 38.1746i
288.5 −4.28857 2.85039i 10.3918 6.36629i 12.2241i 29.4308i −10.2575 18.8753 27.3023i
288.6 −4.28857 2.85039i 10.3918 6.36629i 12.2241i 29.4308i −10.2575 18.8753 27.3023i
288.7 −3.25752 6.51757i 2.61146 19.1073i 21.2311i 22.0995i 17.5533 −15.4787 62.2426i
288.8 −3.25752 6.51757i 2.61146 19.1073i 21.2311i 22.0995i 17.5533 −15.4787 62.2426i
288.9 −1.26162 6.09538i −6.40830 13.2627i 7.69007i 27.2593i 18.1779 −10.1536 16.7325i
288.10 −1.26162 6.09538i −6.40830 13.2627i 7.69007i 27.2593i 18.1779 −10.1536 16.7325i
288.11 −0.447590 9.51519i −7.79966 11.7425i 4.25890i 2.27797i 7.07177 −63.5388 5.25584i
288.12 −0.447590 9.51519i −7.79966 11.7425i 4.25890i 2.27797i 7.07177 −63.5388 5.25584i
288.13 0.820352 0.130026i −7.32702 8.70710i 0.106667i 11.4024i −12.5736 26.9831 7.14288i
288.14 0.820352 0.130026i −7.32702 8.70710i 0.106667i 11.4024i −12.5736 26.9831 7.14288i
288.15 1.07125 5.02012i −6.85243 4.32398i 5.37778i 4.44673i −15.9106 1.79838 4.63204i
288.16 1.07125 5.02012i −6.85243 4.32398i 5.37778i 4.44673i −15.9106 1.79838 4.63204i
288.17 2.16473 9.14133i −3.31395 16.2298i 19.7885i 12.2246i −24.4916 −56.5639 35.1331i
288.18 2.16473 9.14133i −3.31395 16.2298i 19.7885i 12.2246i −24.4916 −56.5639 35.1331i
288.19 3.73276 3.65511i 5.93351 14.5154i 13.6437i 12.9694i −7.71372 13.6402 54.1823i
288.20 3.73276 3.65511i 5.93351 14.5154i 13.6437i 12.9694i −7.71372 13.6402 54.1823i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 288.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.4.b.f 24
17.b even 2 1 inner 289.4.b.f 24
17.c even 4 1 289.4.a.h 12
17.c even 4 1 289.4.a.i yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.4.a.h 12 17.c even 4 1
289.4.a.i yes 12 17.c even 4 1
289.4.b.f 24 1.a even 1 1 trivial
289.4.b.f 24 17.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 72 T_{2}^{10} - 17 T_{2}^{9} + 1872 T_{2}^{8} + 627 T_{2}^{7} - 20922 T_{2}^{6} - 5163 T_{2}^{5} + 93255 T_{2}^{4} - 4607 T_{2}^{3} - 117822 T_{2}^{2} + 21960 T_{2} + 29352$$ acting on $$S_{4}^{\mathrm{new}}(289, [\chi])$$.