# Properties

 Label 605.2.m.f Level $605$ Weight $2$ Character orbit 605.m Analytic conductor $4.831$ Analytic rank $0$ Dimension $80$ CM no Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.m (of order $$20$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 4q^{3} - 4q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 4q^{3} - 4q^{5} - 64q^{12} - 16q^{15} - 12q^{16} - 16q^{20} + 48q^{23} - 16q^{25} - 56q^{26} + 20q^{27} + 16q^{31} + 20q^{36} + 72q^{37} + 32q^{38} + 32q^{42} - 112q^{45} - 16q^{47} + 104q^{48} + 52q^{53} - 128q^{56} - 12q^{58} - 112q^{60} + 112q^{67} - 104q^{70} - 24q^{71} - 64q^{75} + 416q^{78} - 44q^{80} - 100q^{81} + 124q^{82} - 128q^{86} + 16q^{92} + 132q^{93} - 32q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
112.1 −1.13547 2.22849i −0.410525 + 2.59195i −2.50129 + 3.44274i −1.07122 1.96277i 6.24227 2.02824i 0.364179 0.0576804i 5.57165 + 0.882463i −3.69651 1.20107i −3.15768 + 4.61588i
112.2 −0.868868 1.70525i −0.0409975 + 0.258848i −0.977373 + 1.34524i −2.23450 + 0.0835971i 0.477022 0.154994i −4.53756 + 0.718679i −0.637391 0.100953i 2.78785 + 0.905827i 2.08404 + 3.73775i
112.3 −0.842891 1.65427i 0.241486 1.52468i −0.850561 + 1.17070i 0.954605 + 2.02206i −2.72577 + 0.885657i 2.03868 0.322896i −1.01396 0.160596i 0.586836 + 0.190675i 2.54040 3.28355i
112.4 −0.574811 1.12813i 0.355543 2.24481i 0.233302 0.321113i 1.94305 1.10660i −2.73681 + 0.889243i 2.53096 0.400865i −2.99744 0.474748i −2.05960 0.669206i −2.36528 1.55592i
112.5 −0.0806934 0.158370i −0.366739 + 2.31550i 1.15700 1.59248i −0.233963 + 2.22379i 0.396298 0.128765i 4.40946 0.698390i −0.696670 0.110342i −2.37386 0.771312i 0.371061 0.142393i
112.6 0.0806934 + 0.158370i −0.366739 + 2.31550i 1.15700 1.59248i −0.233963 + 2.22379i −0.396298 + 0.128765i −4.40946 + 0.698390i 0.696670 + 0.110342i −2.37386 0.771312i −0.371061 + 0.142393i
112.7 0.574811 + 1.12813i 0.355543 2.24481i 0.233302 0.321113i 1.94305 1.10660i 2.73681 0.889243i −2.53096 + 0.400865i 2.99744 + 0.474748i −2.05960 0.669206i 2.36528 + 1.55592i
112.8 0.842891 + 1.65427i 0.241486 1.52468i −0.850561 + 1.17070i 0.954605 + 2.02206i 2.72577 0.885657i −2.03868 + 0.322896i 1.01396 + 0.160596i 0.586836 + 0.190675i −2.54040 + 3.28355i
112.9 0.868868 + 1.70525i −0.0409975 + 0.258848i −0.977373 + 1.34524i −2.23450 + 0.0835971i −0.477022 + 0.154994i 4.53756 0.718679i 0.637391 + 0.100953i 2.78785 + 0.905827i −2.08404 3.73775i
112.10 1.13547 + 2.22849i −0.410525 + 2.59195i −2.50129 + 3.44274i −1.07122 1.96277i −6.24227 + 2.02824i −0.364179 + 0.0576804i −5.57165 0.882463i −3.69651 1.20107i 3.15768 4.61588i
118.1 −2.47030 0.391257i −1.19139 + 2.33823i 4.04718 + 1.31501i 1.53568 + 1.62532i 3.85794 5.30999i −0.328531 + 0.167395i −5.02626 2.56101i −2.28457 3.14444i −3.15768 4.61588i
118.2 −1.89028 0.299391i −0.118979 + 0.233510i 1.58142 + 0.513836i −0.770005 + 2.09931i 0.294816 0.405779i 4.09339 2.08569i 0.574999 + 0.292976i 1.72298 + 2.37149i 2.08404 3.73775i
118.3 −1.83377 0.290440i 0.700818 1.37543i 1.37624 + 0.447166i −1.62810 1.53273i −1.68462 + 2.31868i −1.83912 + 0.937080i 0.914708 + 0.466067i 0.362685 + 0.499192i 2.54040 + 3.28355i
118.4 −1.25054 0.198066i 1.03183 2.02507i −0.377491 0.122654i 1.65288 1.50599i −1.69144 + 2.32807i −2.28322 + 1.16336i 2.70403 + 1.37777i −1.27290 1.75200i −2.36528 + 1.55592i
118.5 −0.175554 0.0278050i −1.06432 + 2.08884i −1.87207 0.608271i −2.18725 0.464678i 0.244925 0.337111i −3.97783 + 2.02681i 0.628475 + 0.320224i −1.46712 2.01932i 0.371061 + 0.142393i
118.6 0.175554 + 0.0278050i −1.06432 + 2.08884i −1.87207 0.608271i −2.18725 0.464678i −0.244925 + 0.337111i 3.97783 2.02681i −0.628475 0.320224i −1.46712 2.01932i −0.371061 0.142393i
118.7 1.25054 + 0.198066i 1.03183 2.02507i −0.377491 0.122654i 1.65288 1.50599i 1.69144 2.32807i 2.28322 1.16336i −2.70403 1.37777i −1.27290 1.75200i 2.36528 1.55592i
118.8 1.83377 + 0.290440i 0.700818 1.37543i 1.37624 + 0.447166i −1.62810 1.53273i 1.68462 2.31868i 1.83912 0.937080i −0.914708 0.466067i 0.362685 + 0.499192i −2.54040 3.28355i
118.9 1.89028 + 0.299391i −0.118979 + 0.233510i 1.58142 + 0.513836i −0.770005 + 2.09931i −0.294816 + 0.405779i −4.09339 + 2.08569i −0.574999 0.292976i 1.72298 + 2.37149i −2.08404 + 3.73775i
118.10 2.47030 + 0.391257i −1.19139 + 2.33823i 4.04718 + 1.31501i 1.53568 + 1.62532i −3.85794 + 5.30999i 0.328531 0.167395i 5.02626 + 2.56101i −2.28457 3.14444i 3.15768 + 4.61588i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 602.10 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.c odd 4 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.e even 4 1 inner
55.k odd 20 3 inner
55.l even 20 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.m.f 80
5.c odd 4 1 inner 605.2.m.f 80
11.b odd 2 1 inner 605.2.m.f 80
11.c even 5 1 605.2.e.a 20
11.c even 5 3 inner 605.2.m.f 80
11.d odd 10 1 605.2.e.a 20
11.d odd 10 3 inner 605.2.m.f 80
55.e even 4 1 inner 605.2.m.f 80
55.k odd 20 1 605.2.e.a 20
55.k odd 20 3 inner 605.2.m.f 80
55.l even 20 1 605.2.e.a 20
55.l even 20 3 inner 605.2.m.f 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.e.a 20 11.c even 5 1
605.2.e.a 20 11.d odd 10 1
605.2.e.a 20 55.k odd 20 1
605.2.e.a 20 55.l even 20 1
605.2.m.f 80 1.a even 1 1 trivial
605.2.m.f 80 5.c odd 4 1 inner
605.2.m.f 80 11.b odd 2 1 inner
605.2.m.f 80 11.c even 5 3 inner
605.2.m.f 80 11.d odd 10 3 inner
605.2.m.f 80 55.e even 4 1 inner
605.2.m.f 80 55.k odd 20 3 inner
605.2.m.f 80 55.l even 20 3 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$52\!\cdots\!98$$$$T_{2}^{44} +$$$$49\!\cdots\!26$$$$T_{2}^{40} -$$$$40\!\cdots\!23$$$$T_{2}^{36} +$$$$25\!\cdots\!61$$$$T_{2}^{32} -$$$$65\!\cdots\!64$$$$T_{2}^{28} +$$$$16\!\cdots\!40$$$$T_{2}^{24} -$$$$37\!\cdots\!96$$$$T_{2}^{20} +$$$$66\!\cdots\!12$$$$T_{2}^{16} -$$$$65\!\cdots\!00$$$$T_{2}^{12} + 65826435072 T_{2}^{8} - 65699840 T_{2}^{4} + 65536$$">$$T_{2}^{80} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$.