# Properties

 Label 605.2.m.d 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 −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
112.2 −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
112.3 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
112.4 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
118.1 −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
118.2 −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
118.3 −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
118.4 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
233.1 −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
233.2 −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
233.3 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
233.4 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
282.1 −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
282.2 −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
282.3 −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
282.4 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
403.1 −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
403.2 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
403.3 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
403.4 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
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.d 32
5.c odd 4 1 inner 605.2.m.d 32
11.b odd 2 1 605.2.m.c 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.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.d 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.c 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.c 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.d 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 11.b odd 2 1
605.2.m.c 32 11.c even 5 1
605.2.m.c 32 55.e even 4 1
605.2.m.c 32 55.k odd 20 1
605.2.m.d 32 1.a even 1 1 trivial
605.2.m.d 32 5.c odd 4 1 inner
605.2.m.d 32 11.d odd 10 1 inner
605.2.m.d 32 55.l even 20 1 inner
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 $$S_{2}^{\mathrm{new}}(605, [\chi])$$.