# Properties

 Label 300.2.w.a Level $300$ Weight $2$ Character orbit 300.w Analytic conductor $2.396$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(67,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 0, 13]))

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

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

chi := DirichletCharacter("300.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.w (of order $$20$$, degree $$8$$, 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: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$30$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 12 q^{8}+O(q^{10})$$ 240 * q + 12 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 12 q^{8} + 8 q^{10} + 8 q^{12} + 4 q^{13} + 20 q^{17} - 20 q^{20} - 12 q^{22} + 20 q^{25} + 4 q^{28} - 8 q^{30} - 20 q^{32} - 8 q^{33} - 4 q^{37} - 76 q^{38} - 92 q^{40} - 20 q^{42} - 140 q^{44} - 4 q^{45} - 16 q^{48} - 164 q^{50} - 172 q^{52} - 4 q^{53} - 120 q^{58} + 20 q^{60} - 44 q^{62} - 60 q^{64} - 20 q^{65} + 16 q^{68} - 44 q^{70} + 12 q^{72} - 44 q^{73} - 48 q^{77} + 24 q^{78} - 4 q^{80} + 60 q^{81} + 24 q^{82} + 80 q^{84} - 64 q^{85} + 60 q^{88} - 260 q^{89} + 48 q^{90} + 144 q^{92} - 64 q^{93} + 40 q^{94} - 20 q^{96} - 180 q^{97} + 256 q^{98}+O(q^{100})$$ 240 * q + 12 * q^8 + 8 * q^10 + 8 * q^12 + 4 * q^13 + 20 * q^17 - 20 * q^20 - 12 * q^22 + 20 * q^25 + 4 * q^28 - 8 * q^30 - 20 * q^32 - 8 * q^33 - 4 * q^37 - 76 * q^38 - 92 * q^40 - 20 * q^42 - 140 * q^44 - 4 * q^45 - 16 * q^48 - 164 * q^50 - 172 * q^52 - 4 * q^53 - 120 * q^58 + 20 * q^60 - 44 * q^62 - 60 * q^64 - 20 * q^65 + 16 * q^68 - 44 * q^70 + 12 * q^72 - 44 * q^73 - 48 * q^77 + 24 * q^78 - 4 * q^80 + 60 * q^81 + 24 * q^82 + 80 * q^84 - 64 * q^85 + 60 * q^88 - 260 * q^89 + 48 * q^90 + 144 * q^92 - 64 * q^93 + 40 * q^94 - 20 * q^96 - 180 * q^97 + 256 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1 −1.38407 + 0.290430i −0.987688 0.156434i 1.83130 0.803952i 0.979302 + 2.01022i 1.41246 0.0703383i −0.0808033 + 0.0808033i −2.30116 + 1.64459i 0.951057 + 0.309017i −1.93925 2.49786i
67.2 −1.37859 0.315398i 0.987688 + 0.156434i 1.80105 + 0.869612i 2.22681 + 0.203301i −1.31228 0.527175i 2.91192 2.91192i −2.20864 1.76689i 0.951057 + 0.309017i −3.00574 0.982600i
67.3 −1.37682 0.323077i 0.987688 + 0.156434i 1.79124 + 0.889636i −2.11409 + 0.728446i −1.30932 0.534481i 0.284106 0.284106i −2.17879 1.80357i 0.951057 + 0.309017i 3.14605 0.319922i
67.4 −1.36970 0.352033i −0.987688 0.156434i 1.75215 + 0.964358i 1.21468 1.87738i 1.29776 + 0.561967i −0.466793 + 0.466793i −2.06042 1.93769i 0.951057 + 0.309017i −2.32465 + 2.14383i
67.5 −1.29457 + 0.569294i 0.987688 + 0.156434i 1.35181 1.47398i 2.22495 0.222669i −1.36769 + 0.359770i −3.70249 + 3.70249i −0.910882 + 2.67774i 0.951057 + 0.309017i −2.75359 + 1.55491i
67.6 −1.28400 + 0.592739i 0.987688 + 0.156434i 1.29732 1.52216i −1.17964 1.89959i −1.36092 + 0.384580i 1.39751 1.39751i −0.763519 + 2.72342i 0.951057 + 0.309017i 2.64062 + 1.73986i
67.7 −1.12003 0.863445i −0.987688 0.156434i 0.508925 + 1.93417i −1.70279 + 1.44932i 0.971166 + 1.02803i 1.87554 1.87554i 1.10004 2.60575i 0.951057 + 0.309017i 3.15857 0.153014i
67.8 −0.976661 + 1.02281i −0.987688 0.156434i −0.0922667 1.99787i −0.698282 2.12424i 1.12464 0.857431i −1.04258 + 1.04258i 2.13355 + 1.85687i 0.951057 + 0.309017i 2.85467 + 1.36046i
67.9 −0.723992 + 1.21484i 0.987688 + 0.156434i −0.951672 1.75907i −1.94762 + 1.09854i −0.905121 + 1.08663i −2.52433 + 2.52433i 2.82599 + 0.117421i 0.951057 + 0.309017i 0.0755097 3.16138i
67.10 −0.665012 1.24810i 0.987688 + 0.156434i −1.11552 + 1.66001i −0.495035 + 2.18058i −0.461578 1.33677i −0.602522 + 0.602522i 2.81369 + 0.288356i 0.951057 + 0.309017i 3.05079 0.832260i
67.11 −0.664619 1.24831i −0.987688 0.156434i −1.11656 + 1.65930i −1.95630 1.08300i 0.461157 + 1.33691i −2.24842 + 2.24842i 2.81342 + 0.291015i 0.951057 + 0.309017i −0.0517207 + 3.16185i
67.12 −0.557275 + 1.29979i −0.987688 0.156434i −1.37889 1.44868i −1.94762 + 1.09854i 0.753745 1.19661i 2.52433 2.52433i 2.65139 0.984953i 0.951057 + 0.309017i −0.342508 3.14367i
67.13 −0.344199 1.37169i 0.987688 + 0.156434i −1.76305 + 0.944266i 1.92573 1.13647i −0.125382 1.40864i 0.176718 0.176718i 1.90208 + 2.09335i 0.951057 + 0.309017i −2.22172 2.25032i
67.14 −0.253401 + 1.39133i 0.987688 + 0.156434i −1.87158 0.705127i −0.698282 2.12424i −0.467933 + 1.33456i 1.04258 1.04258i 1.45532 2.42529i 0.951057 + 0.309017i 3.13246 0.433253i
67.15 −0.194555 1.40077i −0.987688 0.156434i −1.92430 + 0.545051i 1.56105 + 1.60098i −0.0269690 + 1.41396i 1.26539 1.26539i 1.13787 + 2.58945i 0.951057 + 0.309017i 1.93889 2.49814i
67.16 0.170201 1.40393i −0.987688 0.156434i −1.94206 0.477901i 0.0997280 2.23384i −0.387729 + 1.36002i 2.03014 2.03014i −1.00148 + 2.64519i 0.951057 + 0.309017i −3.11920 0.520213i
67.17 0.275181 + 1.38718i −0.987688 0.156434i −1.84855 + 0.763453i −1.17964 1.89959i −0.0547899 1.41315i −1.39751 + 1.39751i −1.56773 2.35419i 0.951057 + 0.309017i 2.31047 2.15910i
67.18 0.300359 + 1.38195i −0.987688 0.156434i −1.81957 + 0.830163i 2.22495 0.222669i −0.0804769 1.41192i 3.70249 3.70249i −1.69377 2.26520i 0.951057 + 0.309017i 0.976003 + 3.00789i
67.19 0.578573 + 1.29045i 0.987688 + 0.156434i −1.33051 + 1.49324i 0.979302 + 2.01022i 0.369579 + 1.36507i 0.0808033 0.0808033i −2.69674 0.853002i 0.951057 + 0.309017i −2.02748 + 2.42679i
67.20 0.611319 1.27526i 0.987688 + 0.156434i −1.25258 1.55918i −0.726283 + 2.11483i 0.803287 1.16393i 3.39156 3.39156i −2.75409 + 0.644208i 0.951057 + 0.309017i 2.25297 + 2.21904i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.f odd 20 1 inner
100.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.w.a 240
3.b odd 2 1 900.2.bj.f 240
4.b odd 2 1 inner 300.2.w.a 240
12.b even 2 1 900.2.bj.f 240
25.f odd 20 1 inner 300.2.w.a 240
75.l even 20 1 900.2.bj.f 240
100.l even 20 1 inner 300.2.w.a 240
300.u odd 20 1 900.2.bj.f 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.w.a 240 1.a even 1 1 trivial
300.2.w.a 240 4.b odd 2 1 inner
300.2.w.a 240 25.f odd 20 1 inner
300.2.w.a 240 100.l even 20 1 inner
900.2.bj.f 240 3.b odd 2 1
900.2.bj.f 240 12.b even 2 1
900.2.bj.f 240 75.l even 20 1
900.2.bj.f 240 300.u odd 20 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(300, [\chi])$$.