# Properties

 Label 675.2.r.a Level $675$ Weight $2$ Character orbit 675.r Analytic conductor $5.390$ Analytic rank $0$ Dimension $224$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(46,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([20, 18]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("675.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.r (of order $$15$$, degree $$8$$, 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: $$5.38990213644$$ Analytic rank: $$0$$ Dimension: $$224$$ Relative dimension: $$28$$ over $$\Q(\zeta_{15})$$ Twist minimal: no (minimal twist has level 225) Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8}+O(q^{10})$$ 224 * q + 3 * q^2 + 23 * q^4 + 8 * q^5 - 8 * q^7 + 20 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28} + 15 q^{29} + 3 q^{31} - 12 q^{32} + q^{34} - 14 q^{35} - 24 q^{37} - 55 q^{38} + q^{40} + 19 q^{41} - 8 q^{43} - 4 q^{44} - 20 q^{46} + 10 q^{47} - 72 q^{49} + 3 q^{50} - 25 q^{52} + 12 q^{53} - 20 q^{55} + 60 q^{56} - 23 q^{58} + 30 q^{59} - 3 q^{61} + 44 q^{62} - 44 q^{64} - 51 q^{65} - 12 q^{67} + 156 q^{68} - 16 q^{70} - 42 q^{71} - 12 q^{73} - 90 q^{74} - 8 q^{76} - 31 q^{77} - 15 q^{79} - 298 q^{80} + 8 q^{82} - 59 q^{83} - 11 q^{85} - 9 q^{86} - 23 q^{88} - 106 q^{89} + 30 q^{91} - 11 q^{92} + 25 q^{94} - 7 q^{95} - 21 q^{97} - 146 q^{98}+O(q^{100})$$ 224 * q + 3 * q^2 + 23 * q^4 + 8 * q^5 - 8 * q^7 + 20 * q^8 - 20 * q^10 + 11 * q^11 - 3 * q^13 - q^14 + 23 * q^16 + 24 * q^17 - 12 * q^19 - q^20 - 11 * q^22 - q^23 - 16 * q^25 + 136 * q^26 + 4 * q^28 + 15 * q^29 + 3 * q^31 - 12 * q^32 + q^34 - 14 * q^35 - 24 * q^37 - 55 * q^38 + q^40 + 19 * q^41 - 8 * q^43 - 4 * q^44 - 20 * q^46 + 10 * q^47 - 72 * q^49 + 3 * q^50 - 25 * q^52 + 12 * q^53 - 20 * q^55 + 60 * q^56 - 23 * q^58 + 30 * q^59 - 3 * q^61 + 44 * q^62 - 44 * q^64 - 51 * q^65 - 12 * q^67 + 156 * q^68 - 16 * q^70 - 42 * q^71 - 12 * q^73 - 90 * q^74 - 8 * q^76 - 31 * q^77 - 15 * q^79 - 298 * q^80 + 8 * q^82 - 59 * q^83 - 11 * q^85 - 9 * q^86 - 23 * q^88 - 106 * q^89 + 30 * q^91 - 11 * q^92 + 25 * q^94 - 7 * q^95 - 21 * q^97 - 146 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1 −1.77417 1.97042i 0 −0.525803 + 5.00268i 1.61775 + 1.54366i 0 1.02644 1.77785i 6.50010 4.72260i 0 0.171493 5.92636i
46.2 −1.69452 1.88195i 0 −0.461298 + 4.38896i 2.12223 0.704374i 0 −1.03406 + 1.79105i 4.94396 3.59200i 0 −4.92175 2.80036i
46.3 −1.64688 1.82905i 0 −0.424138 + 4.03540i −0.681844 2.12957i 0 2.55879 4.43196i 4.09710 2.97672i 0 −2.77218 + 4.75428i
46.4 −1.55906 1.73151i 0 −0.358407 + 3.41002i −2.08227 + 0.814962i 0 −0.0399999 + 0.0692819i 2.69327 1.95678i 0 4.65750 + 2.33489i
46.5 −1.53828 1.70844i 0 −0.343384 + 3.26708i −0.322490 + 2.21269i 0 −1.62043 + 2.80667i 2.39008 1.73649i 0 4.27633 2.85279i
46.6 −1.15183 1.27924i 0 −0.100677 + 0.957875i 0.0644453 2.23514i 0 −2.11497 + 3.66323i −1.44394 + 1.04909i 0 −2.93350 + 2.49206i
46.7 −0.956880 1.06272i 0 −0.00470356 + 0.0447514i 2.07212 0.840433i 0 1.92019 3.32586i −2.26178 + 1.64328i 0 −2.87591 1.39789i
46.8 −0.955162 1.06081i 0 −0.00393676 + 0.0374558i −0.397449 + 2.20046i 0 1.20474 2.08666i −2.26620 + 1.64649i 0 2.71391 1.68018i
46.9 −0.797750 0.885991i 0 0.0604816 0.575444i −1.30610 1.81497i 0 0.157578 0.272934i −2.48714 + 1.80701i 0 −0.566103 + 2.60508i
46.10 −0.721003 0.800755i 0 0.0876938 0.834351i −2.23594 + 0.0235208i 0 0.316483 0.548165i −2.47480 + 1.79805i 0 1.63096 + 1.77348i
46.11 −0.513717 0.570540i 0 0.147446 1.40285i 1.29212 + 1.82494i 0 −1.54115 + 2.66935i −2.11835 + 1.53907i 0 0.377417 1.67471i
46.12 −0.510289 0.566733i 0 0.148265 1.41065i 0.965605 + 2.01683i 0 −0.476660 + 0.825599i −2.10906 + 1.53232i 0 0.650268 1.57641i
46.13 −0.349998 0.388712i 0 0.180458 1.71695i 0.130882 2.23223i 0 −1.68864 + 2.92481i −1.57689 + 1.14568i 0 −0.913505 + 0.730402i
46.14 0.110043 + 0.122216i 0 0.206230 1.96215i 2.06082 + 0.867759i 0 2.06663 3.57951i 0.528597 0.384048i 0 0.120726 + 0.347356i
46.15 0.136671 + 0.151788i 0 0.204696 1.94755i 1.79725 1.33037i 0 0.530047 0.918069i 0.654078 0.475215i 0 0.447567 + 0.0909796i
46.16 0.220390 + 0.244767i 0 0.197717 1.88116i −2.09765 + 0.774498i 0 −1.22023 + 2.11349i 1.03695 0.753387i 0 −0.651873 0.342746i
46.17 0.251657 + 0.279494i 0 0.194272 1.84837i 0.102616 2.23371i 0 1.17975 2.04339i 1.17403 0.852985i 0 0.650132 0.533449i
46.18 0.293899 + 0.326408i 0 0.188891 1.79718i −2.06853 + 0.849220i 0 −0.888066 + 1.53818i 1.35281 0.982874i 0 −0.885131 0.425600i
46.19 0.727367 + 0.807823i 0 0.0855417 0.813875i −0.934297 + 2.03152i 0 1.73969 3.01324i 2.47854 1.80077i 0 −2.32069 + 0.722917i
46.20 0.942959 + 1.04726i 0 0.00147092 0.0139948i 2.16654 0.553283i 0 −2.48652 + 4.30678i 2.29622 1.66830i 0 2.62239 + 1.74721i
See next 80 embeddings (of 224 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.d even 5 1 inner
225.q even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.r.a 224
3.b odd 2 1 225.2.q.a 224
9.c even 3 1 inner 675.2.r.a 224
9.d odd 6 1 225.2.q.a 224
25.d even 5 1 inner 675.2.r.a 224
75.j odd 10 1 225.2.q.a 224
225.q even 15 1 inner 675.2.r.a 224
225.t odd 30 1 225.2.q.a 224

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.q.a 224 3.b odd 2 1
225.2.q.a 224 9.d odd 6 1
225.2.q.a 224 75.j odd 10 1
225.2.q.a 224 225.t odd 30 1
675.2.r.a 224 1.a even 1 1 trivial
675.2.r.a 224 9.c even 3 1 inner
675.2.r.a 224 25.d even 5 1 inner
675.2.r.a 224 225.q even 15 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(675, [\chi])$$.