# Properties

 Label 2888.1.bm.a Level $2888$ Weight $1$ Character orbit 2888.bm Analytic conductor $1.441$ Analytic rank $0$ Dimension $36$ Projective image $D_{57}$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,1,Mod(11,2888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(114))

chi = DirichletCharacter(H, H._module([57, 57, 34]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2888.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2888 = 2^{3} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2888.bm (of order $$114$$, degree $$36$$, 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: $$1.44129975648$$ Analytic rank: $$0$$ Dimension: $$36$$ Coefficient field: $$\Q(\zeta_{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{36} - x^{35} + x^{33} - x^{32} + x^{30} - x^{29} + x^{27} - x^{26} + x^{24} - x^{23} + x^{21} + \cdots + 1$$ x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{57}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{57} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{114}^{25} q^{2} + (\zeta_{114}^{38} - \zeta_{114}^{15}) q^{3} + \zeta_{114}^{50} q^{4} + (\zeta_{114}^{40} + \zeta_{114}^{6}) q^{6} + \zeta_{114}^{18} q^{8} + ( - \zeta_{114}^{53} + \cdots - \zeta_{114}^{19}) q^{9} +O(q^{10})$$ q - z^25 * q^2 + (z^38 - z^15) * q^3 + z^50 * q^4 + (z^40 + z^6) * q^6 + z^18 * q^8 + (-z^53 + z^30 - z^19) * q^9 $$q - \zeta_{114}^{25} q^{2} + (\zeta_{114}^{38} - \zeta_{114}^{15}) q^{3} + \zeta_{114}^{50} q^{4} + (\zeta_{114}^{40} + \zeta_{114}^{6}) q^{6} + \zeta_{114}^{18} q^{8} + ( - \zeta_{114}^{53} + \cdots - \zeta_{114}^{19}) q^{9} + \cdots + (\zeta_{114}^{46} + \cdots - \zeta_{114}^{7}) q^{99} +O(q^{100})$$ q - z^25 * q^2 + (z^38 - z^15) * q^3 + z^50 * q^4 + (z^40 + z^6) * q^6 + z^18 * q^8 + (-z^53 + z^30 - z^19) * q^9 + (z^16 - z^11) * q^11 + (-z^31 + z^8) * q^12 - z^43 * q^16 + (z^40 - z^31) * q^17 + (-z^55 + z^44 - z^21) * q^18 + z^32 * q^19 + (-z^41 + z^36) * q^22 + (z^56 - z^33) * q^24 - z^29 * q^25 + (-z^45 - z^34 + z^11 + 1) * q^27 - z^11 * q^32 + (z^54 - z^49 - z^31 + z^26) * q^33 + (z^56 + z^8) * q^34 + (z^46 - z^23 + z^12) * q^36 + q^38 + (-z^39 - z^5) * q^41 + (z^52 + z^40) * q^43 + (-z^9 + z^4) * q^44 + (z^24 - z) * q^48 - z^33 * q^49 + z^54 * q^50 + (-z^55 + z^46 - z^21 + z^12) * q^51 + (z^36 - z^25 - z^13 + z^2) * q^54 + (-z^47 - z^13) * q^57 + (-z^35 - z^9) * q^59 + z^36 * q^64 + (z^56 - z^51 + z^22 - z^17) * q^66 + (z^52 - z^3) * q^67 + (-z^33 + z^24) * q^68 + (z^48 - z^37 + z^14) * q^72 + (z^42 - z^29) * q^73 + (z^44 + z^10) * q^75 - z^25 * q^76 + (z^49 + z^38 - z^26 + z^15 - z^3) * q^81 + (z^30 - z^7) * q^82 + (z^38 + z^28) * q^83 + (z^20 + z^8) * q^86 + (z^34 - z^29) * q^88 + (z^26 - z^17) * q^89 + (-z^49 + z^26) * q^96 + z * q^97 - z * q^98 + (z^46 - z^41 - z^35 + z^30 + z^12 - z^7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$36 q + q^{2} - 20 q^{3} + q^{4} - q^{6} - 2 q^{8} - 19 q^{9}+O(q^{10})$$ 36 * q + q^2 - 20 * q^3 + q^4 - q^6 - 2 * q^8 - 19 * q^9 $$36 q + q^{2} - 20 q^{3} + q^{4} - q^{6} - 2 q^{8} - 19 q^{9} + 2 q^{11} + 2 q^{12} + q^{16} + 2 q^{17} + q^{19} - q^{22} - q^{24} + q^{25} + 36 q^{27} + q^{32} + q^{33} + 2 q^{34} + 36 q^{38} - q^{41} + 2 q^{43} - q^{44} - q^{48} - 2 q^{49} - 2 q^{50} - 2 q^{51} + q^{54} + 2 q^{57} - q^{59} - 2 q^{64} + q^{66} - q^{67} - 4 q^{68} - q^{73} + 2 q^{75} + q^{76} - 20 q^{81} - q^{82} - 17 q^{83} + 2 q^{86} + 2 q^{88} + 2 q^{89} + 2 q^{96} - q^{97} + q^{98}+O(q^{100})$$ 36 * q + q^2 - 20 * q^3 + q^4 - q^6 - 2 * q^8 - 19 * q^9 + 2 * q^11 + 2 * q^12 + q^16 + 2 * q^17 + q^19 - q^22 - q^24 + q^25 + 36 * q^27 + q^32 + q^33 + 2 * q^34 + 36 * q^38 - q^41 + 2 * q^43 - q^44 - q^48 - 2 * q^49 - 2 * q^50 - 2 * q^51 + q^54 + 2 * q^57 - q^59 - 2 * q^64 + q^66 - q^67 - 4 * q^68 - q^73 + 2 * q^75 + q^76 - 20 * q^81 - q^82 - 17 * q^83 + 2 * q^86 + 2 * q^88 + 2 * q^89 + 2 * q^96 - q^97 + q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times$$.

 $$n$$ $$1445$$ $$2167$$ $$2529$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{114}^{50}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.754107 + 0.656752i −0.592235 + 0.805765i 0.851919 − 0.523673i 0.975796 + 0.218681i −0.926494 + 0.376309i −0.191711 − 0.981451i 0.975796 − 0.218681i −0.821778 + 0.569808i −0.998482 + 0.0550878i 0.851919 + 0.523673i 0.993931 + 0.110008i 0.137354 − 0.990522i −0.962268 − 0.272103i −0.962268 + 0.272103i 0.904357 + 0.426776i 0.635724 + 0.771917i −0.821778 − 0.569808i 0.451533 − 0.892254i 0.716783 + 0.697297i −0.998482 − 0.0550878i
−0.592235 0.805765i −0.254515 1.83543i −0.298515 + 0.954405i 0 −1.32819 + 1.29208i 0 0.945817 0.324699i −2.34174 + 0.662181i 0
83.1 0.137354 0.990522i −0.582579 + 1.86261i −0.962268 0.272103i 0 1.76494 + 0.832894i 0 −0.401695 + 0.915773i −2.30814 1.60043i 0
163.1 0.350638 0.936511i −0.901695 1.78180i −0.754107 0.656752i 0 −1.98484 + 0.219682i 0 −0.879474 + 0.475947i −1.76952 + 2.40751i 0
235.1 0.716783 0.697297i −1.48636 + 0.701431i 0.0275543 0.999620i 0 −0.576292 + 1.53921i 0 −0.677282 0.735724i 1.08154 1.31324i 0
315.1 0.975796 0.218681i −1.37947 1.34197i 0.904357 0.426776i 0 −1.63955 1.00783i 0 0.789141 0.614213i 0.0745025 + 2.70281i 0
387.1 0.993931 0.110008i −0.254515 0.103375i 0.975796 0.218681i 0 −0.264342 0.0747488i 0 0.945817 0.324699i −0.662691 0.644676i 0
467.1 0.716783 + 0.697297i −1.48636 0.701431i 0.0275543 + 0.999620i 0 −0.576292 1.53921i 0 −0.677282 + 0.735724i 1.08154 + 1.31324i 0
539.1 0.851919 + 0.523673i 0.445817 + 1.19072i 0.451533 + 0.892254i 0 −0.243750 + 1.24786i 0 −0.0825793 + 0.996584i −0.464966 + 0.404939i 0
619.1 −0.191711 + 0.981451i −1.17728 0.130301i −0.926494 0.376309i 0 0.353582 1.13046i 0 0.546948 0.837166i 0.393217 + 0.0881220i 0
691.1 0.350638 + 0.936511i −0.901695 + 1.78180i −0.754107 + 0.656752i 0 −1.98484 0.219682i 0 −0.879474 0.475947i −1.76952 2.40751i 0
771.1 −0.926494 + 0.376309i −0.582579 + 0.130559i 0.716783 0.697297i 0 0.490626 0.340192i 0 −0.401695 + 0.915773i −0.582004 + 0.274654i 0
843.1 −0.298515 + 0.954405i −1.37947 + 0.390078i −0.821778 0.569808i 0 0.0395009 1.43302i 0 0.789141 0.614213i 0.898868 0.552532i 0
923.1 −0.821778 0.569808i 0.0469482 0.0288589i 0.350638 + 0.936511i 0 −0.0550250 0.00303581i 0 0.245485 0.969400i −0.450162 + 0.889544i 0
995.1 −0.821778 + 0.569808i 0.0469482 + 0.0288589i 0.350638 0.936511i 0 −0.0550250 + 0.00303581i 0 0.245485 + 0.969400i −0.450162 0.889544i 0
1075.1 0.0275543 0.999620i 0.445817 0.541326i −0.998482 0.0550878i 0 −0.528836 0.460564i 0 −0.0825793 + 0.996584i 0.0974299 + 0.498787i 0
1147.1 −0.998482 0.0550878i 0.289141 + 1.48024i 0.993931 + 0.110008i 0 −0.207158 1.49392i 0 −0.986361 0.164595i −1.18101 + 0.479684i 0
1227.1 0.851919 0.523673i 0.445817 1.19072i 0.451533 0.892254i 0 −0.243750 1.24786i 0 −0.0825793 0.996584i −0.464966 0.404939i 0
1299.1 −0.754107 0.656752i −1.17728 + 1.60175i 0.137354 + 0.990522i 0 1.93975 0.434708i 0 0.546948 0.837166i −0.881094 2.81701i 0
1379.1 0.904357 + 0.426776i 0.0469482 1.70319i 0.635724 + 0.771917i 0 0.769340 1.52026i 0 0.245485 + 0.969400i −1.90018 0.104836i 0
1451.1 −0.191711 0.981451i −1.17728 + 0.130301i −0.926494 + 0.376309i 0 0.353582 + 1.13046i 0 0.546948 + 0.837166i 0.393217 0.0881220i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
361.i even 57 1 inner
2888.bm odd 114 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.bm.a 36
8.d odd 2 1 CM 2888.1.bm.a 36
361.i even 57 1 inner 2888.1.bm.a 36
2888.bm odd 114 1 inner 2888.1.bm.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2888.1.bm.a 36 1.a even 1 1 trivial
2888.1.bm.a 36 8.d odd 2 1 CM
2888.1.bm.a 36 361.i even 57 1 inner
2888.1.bm.a 36 2888.bm odd 114 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2888, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{36} - T^{35} + \cdots + 1$$
$3$ $$T^{36} + 20 T^{35} + \cdots + 1$$
$5$ $$T^{36}$$
$7$ $$T^{36}$$
$11$ $$T^{36} - 2 T^{35} + \cdots + 1$$
$13$ $$T^{36}$$
$17$ $$T^{36} - 2 T^{35} + \cdots + 1$$
$19$ $$T^{36} - T^{35} + \cdots + 1$$
$23$ $$T^{36}$$
$29$ $$T^{36}$$
$31$ $$T^{36}$$
$37$ $$T^{36}$$
$41$ $$T^{36} + T^{35} + \cdots + 1$$
$43$ $$T^{36} - 2 T^{35} + \cdots + 1$$
$47$ $$T^{36}$$
$53$ $$T^{36}$$
$59$ $$T^{36} + T^{35} + \cdots + 1$$
$61$ $$T^{36}$$
$67$ $$T^{36} + T^{35} + \cdots + 1$$
$71$ $$T^{36}$$
$73$ $$T^{36} + T^{35} + \cdots + 1$$
$79$ $$T^{36}$$
$83$ $$T^{36} + 17 T^{35} + \cdots + 1$$
$89$ $$T^{36} - 2 T^{35} + \cdots + 1$$
$97$ $$T^{36} + T^{35} + \cdots + 1$$