# Properties

 Label 605.2.m.c 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 6q^{3} - 2q^{5} - 20q^{7} + 20q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 6q^{3} - 2q^{5} - 20q^{7} + 20q^{8} + 12q^{12} + 20q^{13} + 4q^{15} + 12q^{16} - 30q^{18} + 16q^{20} - 24q^{23} - 24q^{25} - 20q^{26} + 24q^{27} + 20q^{28} - 40q^{30} + 32q^{31} - 30q^{35} - 36q^{36} - 8q^{37} + 10q^{38} + 50q^{40} - 20q^{42} - 28q^{45} - 20q^{46} + 42q^{47} + 26q^{48} - 80q^{50} + 100q^{51} + 20q^{52} - 34q^{53} - 80q^{56} + 40q^{57} + 90q^{58} - 86q^{60} - 10q^{62} + 10q^{63} - 8q^{67} + 70q^{68} + 10q^{70} - 36q^{71} + 70q^{72} - 30q^{73} + 4q^{75} + 60q^{78} - 102q^{80} + 8q^{81} + 20q^{82} + 60q^{83} - 60q^{85} - 80q^{86} - 130q^{90} - 60q^{91} + 42q^{92} - 8q^{93} + 30q^{95} - 38q^{97} + O(q^{100})$$

## 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 −1.10437 2.16745i −0.0935509 + 0.590657i −2.30265 + 3.16932i 1.76571 1.37196i 1.38354 0.449539i −2.54993 + 0.403870i 4.60707 + 0.729688i 2.51305 + 0.816538i −4.92366 2.31195i
112.2 −0.562185 1.10335i 0.397989 2.51281i 0.274241 0.377461i −0.841972 + 2.07149i −2.99625 + 0.973540i −3.42554 + 0.542552i −3.01679 0.477813i −3.30263 1.07309i 2.75893 0.235572i
112.3 0.221702 + 0.435114i 0.178235 1.12533i 1.03540 1.42510i −0.207538 2.22642i 0.529164 0.171936i 2.88051 0.456227i 1.81429 + 0.287355i 1.61856 + 0.525902i 0.922733 0.583903i
112.4 0.690094 + 1.35439i −0.124714 + 0.787410i −0.182561 + 0.251273i 0.543872 + 2.16892i −1.15252 + 0.374477i −0.189114 + 0.0299527i 2.53639 + 0.401725i 2.24871 + 0.730650i −2.56223 + 2.23337i
118.1 −1.44791 0.229326i −0.377935 + 0.741739i 0.141727 + 0.0460500i 1.49572 + 1.66218i 0.717314 0.987298i −3.75588 + 1.91372i 2.41770 + 1.23188i 1.35601 + 1.86639i −1.78448 2.74968i
118.2 0.568744 + 0.0900803i 0.349459 0.685851i −1.58676 0.515569i −1.84663 + 1.26093i 0.260534 0.358595i 1.06314 0.541696i −1.88216 0.959009i 1.41508 + 1.94770i −1.16385 + 0.550801i
118.3 1.03195 + 0.163444i −1.27853 + 2.50925i −0.863913 0.280702i 0.639384 2.14271i −1.72949 + 2.38044i −0.489900 + 0.249616i −2.70750 1.37954i −2.89835 3.98923i 1.01002 2.10665i
118.4 2.50409 + 0.396609i 0.910200 1.78637i 4.21106 + 1.36825i −0.509700 2.17720i 2.98771 4.11224i −1.61096 + 0.820825i 5.48426 + 2.79437i −0.599293 0.824856i −0.412838 5.65406i
233.1 −2.16745 + 1.10437i −0.590657 0.0935509i 2.30265 3.16932i −2.23491 0.0720754i 1.38354 0.449539i 0.403870 + 2.54993i −0.729688 + 4.60707i −2.51305 0.816538i 4.92366 2.31195i
233.2 −1.10335 + 0.562185i 2.51281 + 0.397989i −0.274241 + 0.377461i 1.89876 + 1.18097i −2.99625 + 0.973540i 0.542552 + 3.42554i 0.477813 3.01679i 3.30263 + 1.07309i −2.75893 0.235572i
233.3 0.435114 0.221702i 1.12533 + 0.178235i −1.03540 + 1.42510i −1.14075 1.92320i 0.529164 0.171936i −0.456227 2.88051i −0.287355 + 1.81429i −1.61856 0.525902i −0.922733 0.583903i
233.4 1.35439 0.690094i −0.787410 0.124714i 0.182561 0.251273i 0.834856 + 2.07437i −1.15252 + 0.374477i 0.0299527 + 0.189114i −0.401725 + 2.53639i −2.24871 0.730650i 2.56223 + 2.23337i
282.1 −1.44791 + 0.229326i −0.377935 0.741739i 0.141727 0.0460500i 1.49572 1.66218i 0.717314 + 0.987298i −3.75588 1.91372i 2.41770 1.23188i 1.35601 1.86639i −1.78448 + 2.74968i
282.2 0.568744 0.0900803i 0.349459 + 0.685851i −1.58676 + 0.515569i −1.84663 1.26093i 0.260534 + 0.358595i 1.06314 + 0.541696i −1.88216 + 0.959009i 1.41508 1.94770i −1.16385 0.550801i
282.3 1.03195 0.163444i −1.27853 2.50925i −0.863913 + 0.280702i 0.639384 + 2.14271i −1.72949 2.38044i −0.489900 0.249616i −2.70750 + 1.37954i −2.89835 + 3.98923i 1.01002 + 2.10665i
282.4 2.50409 0.396609i 0.910200 + 1.78637i 4.21106 1.36825i −0.509700 + 2.17720i 2.98771 + 4.11224i −1.61096 0.820825i 5.48426 2.79437i −0.599293 + 0.824856i −0.412838 + 5.65406i
403.1 −0.396609 2.50409i −1.78637 + 0.910200i −4.21106 + 1.36825i −2.22815 0.188039i 2.98771 + 4.11224i −0.820825 + 1.61096i 2.79437 + 5.48426i 0.599293 0.824856i 0.412838 + 5.65406i
403.2 −0.163444 1.03195i 2.50925 1.27853i 0.863913 0.280702i −1.84025 1.27022i −1.72949 2.38044i −0.249616 + 0.489900i −1.37954 2.70750i 2.89835 3.98923i −1.01002 + 2.10665i
403.3 −0.0900803 0.568744i −0.685851 + 0.349459i 1.58676 0.515569i 0.628574 + 2.14590i 0.260534 + 0.358595i 0.541696 1.06314i −0.959009 1.88216i −1.41508 + 1.94770i 1.16385 0.550801i
403.4 0.229326 + 1.44791i 0.741739 0.377935i −0.141727 + 0.0460500i 2.04303 0.908872i 0.717314 + 0.987298i −1.91372 + 3.75588i 1.23188 + 2.41770i −1.35601 + 1.86639i 1.78448 + 2.74968i
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.c 32
5.c odd 4 1 inner 605.2.m.c 32
11.b odd 2 1 605.2.m.d 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.d 32
11.c even 5 1 605.2.m.e 32
11.d odd 10 1 55.2.l.a 32
11.d odd 10 1 605.2.e.b 32
11.d odd 10 1 inner 605.2.m.c 32
11.d odd 10 1 605.2.m.e 32
33.f even 10 1 495.2.bj.a 32
33.h odd 10 1 495.2.bj.a 32
44.g even 10 1 880.2.cm.a 32
44.h odd 10 1 880.2.cm.a 32
55.e even 4 1 605.2.m.d 32
55.h odd 10 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.d 32
55.k odd 20 1 605.2.m.e 32
55.l even 20 1 55.2.l.a 32
55.l even 20 1 275.2.bm.b 32
55.l even 20 1 605.2.e.b 32
55.l even 20 1 inner 605.2.m.c 32
55.l even 20 1 605.2.m.e 32
165.u odd 20 1 495.2.bj.a 32
165.v even 20 1 495.2.bj.a 32
220.v even 20 1 880.2.cm.a 32
220.w odd 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.c even 5 1
55.2.l.a 32 11.d odd 10 1
55.2.l.a 32 55.k odd 20 1
55.2.l.a 32 55.l even 20 1
275.2.bm.b 32 55.h odd 10 1
275.2.bm.b 32 55.j even 10 1
275.2.bm.b 32 55.k odd 20 1
275.2.bm.b 32 55.l even 20 1
495.2.bj.a 32 33.f even 10 1
495.2.bj.a 32 33.h odd 10 1
495.2.bj.a 32 165.u odd 20 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 1.a even 1 1 trivial
605.2.m.c 32 5.c odd 4 1 inner
605.2.m.c 32 11.d odd 10 1 inner
605.2.m.c 32 55.l even 20 1 inner
605.2.m.d 32 11.b odd 2 1
605.2.m.d 32 11.c even 5 1
605.2.m.d 32 55.e even 4 1
605.2.m.d 32 55.k odd 20 1
605.2.m.e 32 11.c even 5 1
605.2.m.e 32 11.d odd 10 1
605.2.m.e 32 55.k odd 20 1
605.2.m.e 32 55.l even 20 1
880.2.cm.a 32 44.g even 10 1
880.2.cm.a 32 44.h odd 10 1
880.2.cm.a 32 220.v even 20 1
880.2.cm.a 32 220.w odd 20 1

## Hecke kernels

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