# Properties

 Label 605.2.g.q Level $605$ Weight $2$ Character orbit 605.g Analytic conductor $4.831$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.g (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{3} - 6q^{4} - 6q^{5} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{3} - 6q^{4} - 6q^{5} - 12q^{9} + 72q^{12} + 18q^{14} - 6q^{15} - 18q^{16} - 6q^{20} + 48q^{23} - 6q^{25} + 36q^{26} - 30q^{27} + 96q^{34} + 12q^{36} + 30q^{42} + 48q^{45} - 42q^{47} + 6q^{48} + 24q^{49} - 24q^{53} - 120q^{56} + 24q^{58} - 18q^{60} - 30q^{64} + 120q^{67} + 24q^{69} + 18q^{70} - 6q^{75} - 288q^{78} - 18q^{80} - 30q^{81} - 42q^{82} + 6q^{86} - 120q^{89} - 36q^{91} - 36q^{92} + 60q^{93} - 24q^{97} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1 −2.12272 1.54225i 0.414045 1.27430i 1.50939 + 4.64542i −0.809017 + 0.587785i −2.84419 + 2.06642i −0.810808 2.49541i 2.33876 7.19797i 0.974647 + 0.708122i 2.62383
81.2 −1.11052 0.806841i 1.00795 3.10216i −0.0357685 0.110084i −0.809017 + 0.587785i −3.62230 + 2.63176i −0.424181 1.30550i −0.897462 + 2.76210i −6.18037 4.49030i 1.37268
81.3 −0.389057 0.282666i −0.494946 + 1.52329i −0.546569 1.68217i −0.809017 + 0.587785i 0.623145 0.452741i −0.148607 0.457364i −0.560059 + 1.72368i 0.351618 + 0.255466i 0.480901
81.4 0.389057 + 0.282666i −0.494946 + 1.52329i −0.546569 1.68217i −0.809017 + 0.587785i −0.623145 + 0.452741i 0.148607 + 0.457364i 0.560059 1.72368i 0.351618 + 0.255466i −0.480901
81.5 1.11052 + 0.806841i 1.00795 3.10216i −0.0357685 0.110084i −0.809017 + 0.587785i 3.62230 2.63176i 0.424181 + 1.30550i 0.897462 2.76210i −6.18037 4.49030i −1.37268
81.6 2.12272 + 1.54225i 0.414045 1.27430i 1.50939 + 4.64542i −0.809017 + 0.587785i 2.84419 2.06642i 0.810808 + 2.49541i −2.33876 + 7.19797i 0.974647 + 0.708122i −2.62383
251.1 −0.810808 + 2.49541i −1.08398 + 0.787560i −3.95163 2.87103i 0.309017 + 0.951057i −1.08638 3.34354i −2.12272 1.54225i 6.12296 4.44859i −0.372282 + 1.14577i −2.62383
251.2 −0.424181 + 1.30550i −2.63885 + 1.91724i 0.0936432 + 0.0680358i 0.309017 + 0.951057i −1.38360 4.25827i −1.11052 0.806841i −2.34959 + 1.70707i 2.36069 7.26546i −1.37268
251.3 −0.148607 + 0.457364i 1.29579 0.941443i 1.43094 + 1.03964i 0.309017 + 0.951057i 0.238020 + 0.732550i −0.389057 0.282666i −1.46625 + 1.06529i −0.134306 + 0.413352i −0.480901
251.4 0.148607 0.457364i 1.29579 0.941443i 1.43094 + 1.03964i 0.309017 + 0.951057i −0.238020 0.732550i 0.389057 + 0.282666i 1.46625 1.06529i −0.134306 + 0.413352i 0.480901
251.5 0.424181 1.30550i −2.63885 + 1.91724i 0.0936432 + 0.0680358i 0.309017 + 0.951057i 1.38360 + 4.25827i 1.11052 + 0.806841i 2.34959 1.70707i 2.36069 7.26546i 1.37268
251.6 0.810808 2.49541i −1.08398 + 0.787560i −3.95163 2.87103i 0.309017 + 0.951057i 1.08638 + 3.34354i 2.12272 + 1.54225i −6.12296 + 4.44859i −0.372282 + 1.14577i 2.62383
366.1 −2.12272 + 1.54225i 0.414045 + 1.27430i 1.50939 4.64542i −0.809017 0.587785i −2.84419 2.06642i −0.810808 + 2.49541i 2.33876 + 7.19797i 0.974647 0.708122i 2.62383
366.2 −1.11052 + 0.806841i 1.00795 + 3.10216i −0.0357685 + 0.110084i −0.809017 0.587785i −3.62230 2.63176i −0.424181 + 1.30550i −0.897462 2.76210i −6.18037 + 4.49030i 1.37268
366.3 −0.389057 + 0.282666i −0.494946 1.52329i −0.546569 + 1.68217i −0.809017 0.587785i 0.623145 + 0.452741i −0.148607 + 0.457364i −0.560059 1.72368i 0.351618 0.255466i 0.480901
366.4 0.389057 0.282666i −0.494946 1.52329i −0.546569 + 1.68217i −0.809017 0.587785i −0.623145 0.452741i 0.148607 0.457364i 0.560059 + 1.72368i 0.351618 0.255466i −0.480901
366.5 1.11052 0.806841i 1.00795 + 3.10216i −0.0357685 + 0.110084i −0.809017 0.587785i 3.62230 + 2.63176i 0.424181 1.30550i 0.897462 + 2.76210i −6.18037 + 4.49030i −1.37268
366.6 2.12272 1.54225i 0.414045 + 1.27430i 1.50939 4.64542i −0.809017 0.587785i 2.84419 + 2.06642i 0.810808 2.49541i −2.33876 7.19797i 0.974647 0.708122i −2.62383
511.1 −0.810808 2.49541i −1.08398 0.787560i −3.95163 + 2.87103i 0.309017 0.951057i −1.08638 + 3.34354i −2.12272 + 1.54225i 6.12296 + 4.44859i −0.372282 1.14577i −2.62383
511.2 −0.424181 1.30550i −2.63885 1.91724i 0.0936432 0.0680358i 0.309017 0.951057i −1.38360 + 4.25827i −1.11052 + 0.806841i −2.34959 1.70707i 2.36069 + 7.26546i −1.37268
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 511.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.q 24
11.b odd 2 1 inner 605.2.g.q 24
11.c even 5 1 605.2.a.m 6
11.c even 5 3 inner 605.2.g.q 24
11.d odd 10 1 605.2.a.m 6
11.d odd 10 3 inner 605.2.g.q 24
33.f even 10 1 5445.2.a.bx 6
33.h odd 10 1 5445.2.a.bx 6
44.g even 10 1 9680.2.a.cw 6
44.h odd 10 1 9680.2.a.cw 6
55.h odd 10 1 3025.2.a.bg 6
55.j even 10 1 3025.2.a.bg 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.m 6 11.c even 5 1
605.2.a.m 6 11.d odd 10 1
605.2.g.q 24 1.a even 1 1 trivial
605.2.g.q 24 11.b odd 2 1 inner
605.2.g.q 24 11.c even 5 3 inner
605.2.g.q 24 11.d odd 10 3 inner
3025.2.a.bg 6 55.h odd 10 1
3025.2.a.bg 6 55.j even 10 1
5445.2.a.bx 6 33.f even 10 1
5445.2.a.bx 6 33.h odd 10 1
9680.2.a.cw 6 44.g even 10 1
9680.2.a.cw 6 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{24} + \cdots$$ $$T_{3}^{12} + \cdots$$