# Properties

 Label 300.2.n.a Level $300$ Weight $2$ Character orbit 300.n Analytic conductor $2.396$ Analytic rank $0$ Dimension $224$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(11,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 8]))

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

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

chi := DirichletCharacter("300.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.n (of order $$10$$, degree $$4$$, 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: $$224$$ Relative dimension: $$56$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10})$$ 224 * q - 6 * q^4 + q^6 - 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$224 q - 6 q^{4} + q^{6} - 6 q^{9} - 8 q^{10} - 9 q^{12} - 12 q^{13} - 18 q^{16} - 26 q^{18} + 12 q^{21} - 6 q^{22} - 16 q^{24} - 12 q^{25} + 2 q^{28} - 13 q^{30} + 6 q^{33} - 30 q^{34} + 35 q^{36} + 12 q^{37} - 24 q^{40} - 13 q^{42} - 6 q^{45} - 18 q^{46} - 34 q^{48} - 168 q^{49} - 28 q^{52} - 38 q^{54} - 44 q^{57} - 34 q^{58} - 76 q^{60} + 4 q^{61} + 18 q^{64} - 46 q^{66} - 18 q^{69} + 72 q^{70} - 29 q^{72} - 20 q^{73} + 16 q^{76} + 5 q^{78} - 30 q^{81} - 20 q^{82} - 18 q^{84} - 76 q^{85} + 6 q^{88} + 2 q^{90} - 52 q^{93} + 96 q^{94} - 50 q^{96} - 72 q^{97}+O(q^{100})$$ 224 * q - 6 * q^4 + q^6 - 6 * q^9 - 8 * q^10 - 9 * q^12 - 12 * q^13 - 18 * q^16 - 26 * q^18 + 12 * q^21 - 6 * q^22 - 16 * q^24 - 12 * q^25 + 2 * q^28 - 13 * q^30 + 6 * q^33 - 30 * q^34 + 35 * q^36 + 12 * q^37 - 24 * q^40 - 13 * q^42 - 6 * q^45 - 18 * q^46 - 34 * q^48 - 168 * q^49 - 28 * q^52 - 38 * q^54 - 44 * q^57 - 34 * q^58 - 76 * q^60 + 4 * q^61 + 18 * q^64 - 46 * q^66 - 18 * q^69 + 72 * q^70 - 29 * q^72 - 20 * q^73 + 16 * q^76 + 5 * q^78 - 30 * q^81 - 20 * q^82 - 18 * q^84 - 76 * q^85 + 6 * q^88 + 2 * q^90 - 52 * q^93 + 96 * q^94 - 50 * q^96 - 72 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.40873 + 0.124406i 0.673811 1.59561i 1.96905 0.350510i −2.20342 0.380687i −0.750715 + 2.33161i 0.0822219i −2.73025 + 0.738736i −2.09196 2.15028i 3.15139 + 0.262166i
11.2 −1.40679 + 0.144680i −1.60959 + 0.639710i 1.95814 0.407070i 2.17904 + 0.501802i 2.17180 1.13282i 2.67645i −2.69580 + 0.855967i 2.18154 2.05934i −3.13805 0.390668i
11.3 −1.39445 + 0.235585i −1.04145 1.38397i 1.88900 0.657024i 1.89530 1.18653i 1.77830 + 1.68454i 4.72716i −2.47934 + 1.36121i −0.830762 + 2.88268i −2.36338 + 2.10106i
11.4 −1.39367 0.240196i −0.354401 + 1.69541i 1.88461 + 0.669506i −0.834949 + 2.07433i 0.901146 2.27770i 4.41884i −2.46571 1.38574i −2.74880 1.20171i 1.66189 2.69038i
11.5 −1.38528 0.284622i 1.64016 + 0.556662i 1.83798 + 0.788561i 1.45398 + 1.69881i −2.11364 1.23796i 2.78321i −2.32167 1.61550i 2.38025 + 1.82603i −1.53064 2.76716i
11.6 −1.31876 0.510760i −1.38129 1.04500i 1.47825 + 1.34714i −1.62369 + 1.53741i 1.28785 + 2.08361i 1.57078i −1.26139 2.53158i 0.815942 + 2.88691i 2.92650 1.19816i
11.7 −1.29913 0.558792i 1.06350 1.36711i 1.37550 + 1.45189i 1.51671 1.64305i −2.14555 + 1.18178i 1.88972i −0.975659 2.65482i −0.737954 2.90782i −2.88853 + 1.28702i
11.8 −1.25692 + 0.648193i 1.55000 0.772971i 1.15969 1.62945i 0.411823 + 2.19782i −1.44720 + 1.97626i 3.34784i −0.401439 + 2.79979i 1.80503 2.39622i −1.94224 2.49554i
11.9 −1.24841 + 0.664442i 1.66858 + 0.464581i 1.11703 1.65899i 0.345709 2.20918i −2.39175 + 0.528689i 0.309856i −0.292213 + 2.81329i 2.56833 + 1.55038i 1.03629 + 2.98766i
11.10 −1.24461 + 0.671529i −1.27306 + 1.17444i 1.09810 1.67158i −2.10329 0.759058i 0.795787 2.31662i 0.738888i −0.244189 + 2.81787i 0.241361 2.99028i 3.12750 0.467689i
11.11 −1.21555 0.722800i −1.65104 + 0.523516i 0.955121 + 1.75720i −0.573135 2.16137i 2.38532 + 0.557011i 1.39878i 0.109105 2.82632i 2.45186 1.72869i −0.865563 + 3.04151i
11.12 −1.03327 0.965583i 0.539210 + 1.64598i 0.135297 + 1.99542i −1.86451 1.23434i 1.03218 2.22140i 3.63805i 1.78694 2.19245i −2.41850 + 1.77506i 0.734688 + 3.07575i
11.13 −1.02327 + 0.976179i 0.339606 + 1.69843i 0.0941493 1.99778i 2.10329 + 0.759058i −2.00548 1.40643i 0.738888i 1.85385 + 2.13617i −2.76934 + 1.15359i −2.89320 + 1.27647i
11.14 −1.01770 + 0.981981i −1.62298 0.604914i 0.0714277 1.99872i −0.345709 + 2.20918i 2.24573 0.978119i 0.309856i 1.89002 + 2.10424i 2.26816 + 1.96353i −1.81755 2.58776i
11.15 −1.00488 + 0.995099i −0.799639 1.53642i 0.0195579 1.99990i −0.411823 2.19782i 2.33243 + 0.748191i 3.34784i 1.97045 + 2.02912i −1.72115 + 2.45716i 2.60088 + 1.79873i
11.16 −0.923787 1.07080i 1.20611 + 1.24310i −0.293236 + 1.97839i 2.16410 0.562738i 0.216925 2.43987i 3.72853i 2.38935 1.51361i −0.0905939 + 2.99863i −2.60175 1.79747i
11.17 −0.884246 1.10368i 0.846655 1.51102i −0.436219 + 1.95185i 0.448288 + 2.19067i −2.41633 + 0.401675i 2.41810i 2.53994 1.24447i −1.56635 2.55862i 2.02140 2.43186i
11.18 −0.776415 1.18202i −0.846655 + 1.51102i −0.794359 + 1.83548i 0.448288 + 2.19067i 2.44341 0.172411i 2.41810i 2.78634 0.486145i −1.56635 2.55862i 2.24137 2.23076i
11.19 −0.732928 1.20947i −1.20611 1.24310i −0.925633 + 1.77291i 2.16410 0.562738i −0.619499 + 2.36986i 3.72853i 2.82270 0.179888i −0.0905939 + 2.99863i −2.26674 2.20497i
11.20 −0.654964 + 1.25340i 1.65603 0.507508i −1.14204 1.64187i −1.89530 + 1.18653i −0.448527 + 2.40807i 4.72716i 2.80592 0.356077i 2.48487 1.68090i −0.245847 3.15271i
See next 80 embeddings (of 224 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.56 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner
100.j odd 10 1 inner
300.n even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.n.a 224
3.b odd 2 1 inner 300.2.n.a 224
4.b odd 2 1 inner 300.2.n.a 224
12.b even 2 1 inner 300.2.n.a 224
25.d even 5 1 inner 300.2.n.a 224
75.j odd 10 1 inner 300.2.n.a 224
100.j odd 10 1 inner 300.2.n.a 224
300.n even 10 1 inner 300.2.n.a 224

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.n.a 224 1.a even 1 1 trivial
300.2.n.a 224 3.b odd 2 1 inner
300.2.n.a 224 4.b odd 2 1 inner
300.2.n.a 224 12.b even 2 1 inner
300.2.n.a 224 25.d even 5 1 inner
300.2.n.a 224 75.j odd 10 1 inner
300.2.n.a 224 100.j odd 10 1 inner
300.2.n.a 224 300.n even 10 1 inner

## Hecke kernels

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