Properties

 Label 65.5.g.a Level $65$ Weight $5$ Character orbit 65.g Analytic conductor $6.719$ Analytic rank $0$ Dimension $52$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,5,Mod(34,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("65.34");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 65.g (of order $$4$$, degree $$2$$, 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: $$6.71904760045$$ Analytic rank: $$0$$ Dimension: $$52$$ Relative dimension: $$26$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$52 q + 58 q^{5} + 60 q^{6} - 1196 q^{9}+O(q^{10})$$ 52 * q + 58 * q^5 + 60 * q^6 - 1196 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$52 q + 58 q^{5} + 60 q^{6} - 1196 q^{9} + 8 q^{11} + 424 q^{14} - 508 q^{15} - 3128 q^{16} + 268 q^{19} + 2116 q^{20} - 668 q^{21} + 2032 q^{24} + 1100 q^{26} - 3704 q^{29} - 1660 q^{31} + 1252 q^{34} + 6692 q^{35} + 7408 q^{39} + 3644 q^{40} + 5960 q^{41} + 1444 q^{44} - 8674 q^{45} + 17620 q^{46} - 3928 q^{50} - 30588 q^{54} + 14784 q^{55} - 5548 q^{59} - 13976 q^{60} - 20008 q^{61} - 15614 q^{65} + 24752 q^{66} + 26908 q^{70} - 16360 q^{71} - 7304 q^{74} - 5760 q^{76} - 9536 q^{79} - 62348 q^{80} - 36516 q^{81} + 66564 q^{84} - 11604 q^{85} - 5264 q^{86} - 940 q^{89} + 37644 q^{91} + 55448 q^{94} - 61644 q^{96} + 16140 q^{99}+O(q^{100})$$ 52 * q + 58 * q^5 + 60 * q^6 - 1196 * q^9 + 8 * q^11 + 424 * q^14 - 508 * q^15 - 3128 * q^16 + 268 * q^19 + 2116 * q^20 - 668 * q^21 + 2032 * q^24 + 1100 * q^26 - 3704 * q^29 - 1660 * q^31 + 1252 * q^34 + 6692 * q^35 + 7408 * q^39 + 3644 * q^40 + 5960 * q^41 + 1444 * q^44 - 8674 * q^45 + 17620 * q^46 - 3928 * q^50 - 30588 * q^54 + 14784 * q^55 - 5548 * q^59 - 13976 * q^60 - 20008 * q^61 - 15614 * q^65 + 24752 * q^66 + 26908 * q^70 - 16360 * q^71 - 7304 * q^74 - 5760 * q^76 - 9536 * q^79 - 62348 * q^80 - 36516 * q^81 + 66564 * q^84 - 11604 * q^85 - 5264 * q^86 - 940 * q^89 + 37644 * q^91 + 55448 * q^94 - 61644 * q^96 + 16140 * q^99

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1 −5.15266 + 5.15266i 15.6320i 37.0999i 22.4978 + 10.9017i 80.5463 + 80.5463i −43.9050 43.9050i 108.720 + 108.720i −163.359 −172.097 + 59.7510i
34.2 −4.99838 + 4.99838i 5.28046i 33.9676i 24.9466 1.63388i −26.3937 26.3937i 65.3572 + 65.3572i 89.8089 + 89.8089i 53.1168 −116.526 + 132.859i
34.3 −4.94726 + 4.94726i 3.40815i 32.9507i −22.9591 9.89351i 16.8610 + 16.8610i 9.32308 + 9.32308i 83.8595 + 83.8595i 69.3845 162.530 64.6387i
34.4 −4.82935 + 4.82935i 11.4208i 30.6453i −1.40781 + 24.9603i −55.1549 55.1549i −48.2799 48.2799i 70.7273 + 70.7273i −49.4340 −113.743 127.341i
34.5 −3.80385 + 3.80385i 13.5404i 12.9386i −6.62279 24.1068i −51.5057 51.5057i −14.0955 14.0955i −11.6451 11.6451i −102.342 116.891 + 66.5067i
34.6 −3.49855 + 3.49855i 7.86015i 8.47972i −17.4903 + 17.8631i 27.4991 + 27.4991i −0.666849 0.666849i −26.3101 26.3101i 19.2180 −1.30419 123.686i
34.7 −2.95432 + 2.95432i 14.1332i 1.45602i −0.166336 24.9994i 41.7539 + 41.7539i 27.9563 + 27.9563i −42.9676 42.9676i −118.747 74.3478 + 73.3650i
34.8 −2.80141 + 2.80141i 1.58015i 0.304183i 20.5382 14.2543i 4.42666 + 4.42666i −31.3088 31.3088i −45.6747 45.6747i 78.5031 −17.6038 + 97.4679i
34.9 −2.29180 + 2.29180i 1.09931i 5.49535i 14.5107 + 20.3578i −2.51939 2.51939i 7.15078 + 7.15078i −49.2629 49.2629i 79.7915 −79.9114 13.4005i
34.10 −1.96266 + 1.96266i 11.7087i 8.29597i −20.0793 + 14.8937i −22.9802 22.9802i 49.2380 + 49.2380i −47.6846 47.6846i −56.0939 10.1776 68.6399i
34.11 −0.479269 + 0.479269i 2.62394i 15.5406i −22.0613 11.7601i −1.25757 1.25757i −51.9642 51.9642i −15.1164 15.1164i 74.1150 16.2095 4.93705i
34.12 −0.212553 + 0.212553i 13.8459i 15.9096i 19.7160 + 15.3714i 2.94299 + 2.94299i 38.8793 + 38.8793i −6.78250 6.78250i −110.709 −7.45794 + 0.923458i
34.13 −0.0630863 + 0.0630863i 12.3873i 15.9920i −17.7387 + 17.6164i 0.781468 + 0.781468i −27.4701 27.4701i −2.01826 2.01826i −72.4449 0.00771715 2.23042i
34.14 0.0630863 0.0630863i 12.3873i 15.9920i 17.6164 17.7387i 0.781468 + 0.781468i 27.4701 + 27.4701i 2.01826 + 2.01826i −72.4449 −0.00771715 2.23042i
34.15 0.212553 0.212553i 13.8459i 15.9096i 15.3714 + 19.7160i 2.94299 + 2.94299i −38.8793 38.8793i 6.78250 + 6.78250i −110.709 7.45794 + 0.923458i
34.16 0.479269 0.479269i 2.62394i 15.5406i −11.7601 22.0613i −1.25757 1.25757i 51.9642 + 51.9642i 15.1164 + 15.1164i 74.1150 −16.2095 4.93705i
34.17 1.96266 1.96266i 11.7087i 8.29597i 14.8937 20.0793i −22.9802 22.9802i −49.2380 49.2380i 47.6846 + 47.6846i −56.0939 −10.1776 68.6399i
34.18 2.29180 2.29180i 1.09931i 5.49535i 20.3578 + 14.5107i −2.51939 2.51939i −7.15078 7.15078i 49.2629 + 49.2629i 79.7915 79.9114 13.4005i
34.19 2.80141 2.80141i 1.58015i 0.304183i −14.2543 + 20.5382i 4.42666 + 4.42666i 31.3088 + 31.3088i 45.6747 + 45.6747i 78.5031 17.6038 + 97.4679i
34.20 2.95432 2.95432i 14.1332i 1.45602i −24.9994 0.166336i 41.7539 + 41.7539i −27.9563 27.9563i 42.9676 + 42.9676i −118.747 −74.3478 + 73.3650i
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 44.26 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.d odd 4 1 inner
65.g odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.5.g.a 52
5.b even 2 1 inner 65.5.g.a 52
13.d odd 4 1 inner 65.5.g.a 52
65.g odd 4 1 inner 65.5.g.a 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.5.g.a 52 1.a even 1 1 trivial
65.5.g.a 52 5.b even 2 1 inner
65.5.g.a 52 13.d odd 4 1 inner
65.5.g.a 52 65.g odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(65, [\chi])$$.