# Properties

 Label 605.2.m.e Level $605$ Weight $2$ Character orbit 605.m Analytic conductor $4.831$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# 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: $$32$$ Relative dimension: $$4$$ over $$\Q(\zeta_{20})$$ Twist minimal: no (minimal twist has level 55) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 10q^{2} - 4q^{3} - 2q^{5} + 20q^{6} + 10q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 10q^{2} - 4q^{3} - 2q^{5} + 20q^{6} + 10q^{8} + 12q^{12} + 10q^{13} + 14q^{15} - 8q^{16} + 10q^{18} + 16q^{20} - 24q^{23} + 16q^{25} + 20q^{26} - 16q^{27} - 50q^{28} - 30q^{30} - 28q^{31} + 10q^{35} + 24q^{36} - 8q^{37} + 10q^{38} + 50q^{40} - 40q^{41} - 10q^{42} - 28q^{45} - 60q^{46} - 28q^{47} - 54q^{48} + 50q^{50} - 20q^{51} + 50q^{52} - 24q^{53} - 80q^{56} - 30q^{57} - 50q^{58} + 34q^{60} + 60q^{61} - 100q^{62} + 30q^{63} - 8q^{67} + 30q^{68} + 30q^{70} + 24q^{71} - 80q^{72} - 50q^{73} + 34q^{75} + 60q^{78} + 98q^{80} - 12q^{81} - 10q^{82} - 90q^{83} - 30q^{85} + 100q^{86} + 20q^{90} + 20q^{91} - 68q^{92} - 8q^{93} + 40q^{95} - 8q^{97} + O(q^{100})$$

## 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 −0.665529 1.30617i −0.130227 + 0.822224i −0.0875924 + 0.120561i −1.11862 1.93615i 1.16064 0.377114i 4.16343 0.659422i −2.68004 0.424477i 2.19408 + 0.712899i −1.78448 + 2.74968i
112.2 0.261423 + 0.513072i 0.120415 0.760272i 0.980670 1.34978i −1.76986 + 1.36660i 0.421554 0.136971i −1.17850 + 0.186656i 2.08639 + 0.330452i 2.28966 + 0.743954i −1.16385 0.550801i
112.3 0.474334 + 0.930933i −0.440550 + 2.78152i 0.533928 0.734888i 2.23541 + 0.0540419i −2.79838 + 0.909249i 0.543058 0.0860119i 3.00128 + 0.475357i −4.68963 1.52375i 1.01002 + 2.10665i
112.4 1.15100 + 2.25897i 0.313634 1.98021i −2.60258 + 3.58214i 1.91314 + 1.15755i 4.83422 1.57073i 1.78576 0.282837i −6.07934 0.962874i −0.969677 0.315067i −0.412838 + 5.65406i
118.1 −1.50135 0.237790i −0.361933 + 0.710333i 0.295389 + 0.0959778i −1.89470 1.18749i 0.712297 0.980393i −0.170602 + 0.0869260i 2.28811 + 1.16585i 1.38978 + 1.91287i 2.56223 + 2.23337i
118.2 −0.482327 0.0763931i 0.517260 1.01518i −1.67531 0.544341i 2.05331 + 0.885381i −0.327041 + 0.450133i 2.59854 1.32402i 1.63669 + 0.833936i 1.00032 + 1.37683i −0.922733 0.583903i
118.3 1.22307 + 0.193716i 1.15501 2.26684i −0.443732 0.144177i −2.23029 + 0.160637i 1.85178 2.54876i −3.09022 + 1.57455i −2.72149 1.38667i −2.04114 2.80938i −2.75893 0.235572i
118.4 2.40264 + 0.380541i −0.271495 + 0.532840i 3.72576 + 1.21057i 1.85044 1.25533i −0.855073 + 1.17691i −2.30033 + 1.17208i 4.15610 + 2.11764i 1.55315 + 2.13772i 4.92366 2.31195i
233.1 −1.30617 + 0.665529i −0.822224 0.130227i 0.0875924 0.120561i −0.233059 2.22389i 1.16064 0.377114i −0.659422 4.16343i 0.424477 2.68004i −2.19408 0.712899i 1.78448 + 2.74968i
233.2 0.513072 0.261423i 0.760272 + 0.120415i −0.980670 + 1.34978i 2.23511 + 0.0653109i 0.421554 0.136971i 0.186656 + 1.17850i −0.330452 + 2.08639i −2.28966 0.743954i 1.16385 0.550801i
233.3 0.930933 0.474334i −2.78152 0.440550i −0.533928 + 0.734888i −1.77672 + 1.35766i −2.79838 + 0.909249i −0.0860119 0.543058i −0.475357 + 3.00128i 4.68963 + 1.52375i −1.01002 + 2.10665i
233.4 2.25897 1.15100i 1.98021 + 0.313634i 2.60258 3.58214i −0.867371 + 2.06099i 4.83422 1.57073i −0.282837 1.78576i 0.962874 6.07934i 0.969677 + 0.315067i 0.412838 + 5.65406i
282.1 −1.50135 + 0.237790i −0.361933 0.710333i 0.295389 0.0959778i −1.89470 + 1.18749i 0.712297 + 0.980393i −0.170602 0.0869260i 2.28811 1.16585i 1.38978 1.91287i 2.56223 2.23337i
282.2 −0.482327 + 0.0763931i 0.517260 + 1.01518i −1.67531 + 0.544341i 2.05331 0.885381i −0.327041 0.450133i 2.59854 + 1.32402i 1.63669 0.833936i 1.00032 1.37683i −0.922733 + 0.583903i
282.3 1.22307 0.193716i 1.15501 + 2.26684i −0.443732 + 0.144177i −2.23029 0.160637i 1.85178 + 2.54876i −3.09022 1.57455i −2.72149 + 1.38667i −2.04114 + 2.80938i −2.75893 + 0.235572i
282.4 2.40264 0.380541i −0.271495 0.532840i 3.72576 1.21057i 1.85044 + 1.25533i −0.855073 1.17691i −2.30033 1.17208i 4.15610 2.11764i 1.55315 2.13772i 4.92366 + 2.31195i
403.1 −0.380541 2.40264i 0.532840 0.271495i −3.72576 + 1.21057i −0.622076 2.14779i −0.855073 1.17691i −1.17208 + 2.30033i 2.11764 + 4.15610i −1.55315 + 2.13772i −4.92366 + 2.31195i
403.2 −0.193716 1.22307i −2.26684 + 1.15501i 0.443732 0.144177i −0.536423 + 2.17077i 1.85178 + 2.54876i −1.57455 + 3.09022i −1.38667 2.72149i 2.04114 2.80938i 2.75893 + 0.235572i
403.3 0.0763931 + 0.482327i −1.01518 + 0.517260i 1.67531 0.544341i 1.47656 1.67922i −0.327041 0.450133i 1.32402 2.59854i 0.833936 + 1.63669i −1.00032 + 1.37683i 0.922733 + 0.583903i
403.4 0.237790 + 1.50135i 0.710333 0.361933i −0.295389 + 0.0959778i −1.71486 + 1.43501i 0.712297 + 0.980393i −0.0869260 + 0.170602i 1.16585 + 2.28811i −1.38978 + 1.91287i −2.56223 2.23337i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 602.4 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.d odd 10 1 inner
55.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.m.e 32
5.c odd 4 1 inner 605.2.m.e 32
11.b odd 2 1 55.2.l.a 32
11.c even 5 1 55.2.l.a 32
11.c even 5 1 605.2.e.b 32
11.c even 5 1 605.2.m.c 32
11.c even 5 1 605.2.m.d 32
11.d odd 10 1 605.2.e.b 32
11.d odd 10 1 605.2.m.c 32
11.d odd 10 1 605.2.m.d 32
11.d odd 10 1 inner 605.2.m.e 32
33.d even 2 1 495.2.bj.a 32
33.h odd 10 1 495.2.bj.a 32
44.c even 2 1 880.2.cm.a 32
44.h odd 10 1 880.2.cm.a 32
55.d odd 2 1 275.2.bm.b 32
55.e even 4 1 55.2.l.a 32
55.e even 4 1 275.2.bm.b 32
55.j even 10 1 275.2.bm.b 32
55.k odd 20 1 55.2.l.a 32
55.k odd 20 1 275.2.bm.b 32
55.k odd 20 1 605.2.e.b 32
55.k odd 20 1 605.2.m.c 32
55.k odd 20 1 605.2.m.d 32
55.l even 20 1 605.2.e.b 32
55.l even 20 1 605.2.m.c 32
55.l even 20 1 605.2.m.d 32
55.l even 20 1 inner 605.2.m.e 32
165.l odd 4 1 495.2.bj.a 32
165.v even 20 1 495.2.bj.a 32
220.i odd 4 1 880.2.cm.a 32
220.v even 20 1 880.2.cm.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.l.a 32 11.b odd 2 1
55.2.l.a 32 11.c even 5 1
55.2.l.a 32 55.e even 4 1
55.2.l.a 32 55.k odd 20 1
275.2.bm.b 32 55.d odd 2 1
275.2.bm.b 32 55.e even 4 1
275.2.bm.b 32 55.j even 10 1
275.2.bm.b 32 55.k odd 20 1
495.2.bj.a 32 33.d even 2 1
495.2.bj.a 32 33.h odd 10 1
495.2.bj.a 32 165.l odd 4 1
495.2.bj.a 32 165.v even 20 1
605.2.e.b 32 11.c even 5 1
605.2.e.b 32 11.d odd 10 1
605.2.e.b 32 55.k odd 20 1
605.2.e.b 32 55.l even 20 1
605.2.m.c 32 11.c even 5 1
605.2.m.c 32 11.d odd 10 1
605.2.m.c 32 55.k odd 20 1
605.2.m.c 32 55.l even 20 1
605.2.m.d 32 11.c even 5 1
605.2.m.d 32 11.d odd 10 1
605.2.m.d 32 55.k odd 20 1
605.2.m.d 32 55.l even 20 1
605.2.m.e 32 1.a even 1 1 trivial
605.2.m.e 32 5.c odd 4 1 inner
605.2.m.e 32 11.d odd 10 1 inner
605.2.m.e 32 55.l even 20 1 inner
880.2.cm.a 32 44.c even 2 1
880.2.cm.a 32 44.h odd 10 1
880.2.cm.a 32 220.i odd 4 1
880.2.cm.a 32 220.v even 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$.