# Properties

 Label 605.2.e.b Level $605$ Weight $2$ Character orbit 605.e 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.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{3} + 8q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{3} + 8q^{5} + 12q^{12} - 36q^{15} - 8q^{16} - 64q^{20} - 24q^{23} + 16q^{25} - 16q^{27} - 8q^{31} + 24q^{36} + 32q^{37} - 40q^{38} + 60q^{42} - 28q^{45} - 28q^{47} + 56q^{48} + 116q^{53} - 80q^{56} - 80q^{58} + 104q^{60} - 8q^{67} - 80q^{70} + 24q^{71} - 76q^{75} + 60q^{78} + 8q^{80} + 8q^{81} - 20q^{82} - 40q^{86} + 80q^{91} + 52q^{92} + 32q^{93} + 92q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
362.1 −1.79273 + 1.79273i −1.41767 + 1.41767i 4.42777i 1.69208 1.46180i 5.08300i −1.27846 + 1.27846i 4.35233 + 4.35233i 1.01958i −0.412838 + 5.65406i
362.2 −1.72010 + 1.72010i 0.422864 0.422864i 3.91750i −0.759172 2.10325i 1.45474i −1.82555 + 1.82555i 3.29830 + 3.29830i 2.64237i 4.92366 + 2.31195i
362.3 −1.07485 + 1.07485i 0.563723 0.563723i 0.310591i 2.23083 + 0.152980i 1.21183i 0.135390 0.135390i −1.81586 1.81586i 2.36443i −2.56223 + 2.23337i
362.4 −1.03659 + 1.03659i 0.588647 0.588647i 0.149021i −2.18706 + 0.465567i 1.22037i 2.98069 2.98069i −1.91870 1.91870i 2.30699i 1.78448 2.74968i
362.5 −0.875624 + 0.875624i −1.79897 + 1.79897i 0.466567i 1.70992 + 1.44089i 3.15044i −2.45241 + 2.45241i −2.15978 2.15978i 3.47259i −2.75893 + 0.235572i
362.6 −0.738792 + 0.738792i 1.99135 1.99135i 0.908372i 0.742178 2.10931i 2.94239i −0.388787 + 0.388787i −2.14868 2.14868i 4.93096i 1.01002 + 2.10665i
362.7 −0.407176 + 0.407176i −0.544295 + 0.544295i 1.66842i 0.752803 + 2.10554i 0.443248i 0.843711 0.843711i −1.49369 1.49369i 2.40749i −1.16385 0.550801i
362.8 −0.345308 + 0.345308i −0.805651 + 0.805651i 1.76152i −2.18158 0.490620i 0.556396i −2.06222 + 2.06222i −1.29889 1.29889i 1.70185i 0.922733 0.583903i
362.9 0.345308 0.345308i −0.805651 + 0.805651i 1.76152i −2.18158 0.490620i 0.556396i 2.06222 2.06222i 1.29889 + 1.29889i 1.70185i −0.922733 + 0.583903i
362.10 0.407176 0.407176i −0.544295 + 0.544295i 1.66842i 0.752803 + 2.10554i 0.443248i −0.843711 + 0.843711i 1.49369 + 1.49369i 2.40749i 1.16385 + 0.550801i
362.11 0.738792 0.738792i 1.99135 1.99135i 0.908372i 0.742178 2.10931i 2.94239i 0.388787 0.388787i 2.14868 + 2.14868i 4.93096i −1.01002 2.10665i
362.12 0.875624 0.875624i −1.79897 + 1.79897i 0.466567i 1.70992 + 1.44089i 3.15044i 2.45241 2.45241i 2.15978 + 2.15978i 3.47259i 2.75893 0.235572i
362.13 1.03659 1.03659i 0.588647 0.588647i 0.149021i −2.18706 + 0.465567i 1.22037i −2.98069 + 2.98069i 1.91870 + 1.91870i 2.30699i −1.78448 + 2.74968i
362.14 1.07485 1.07485i 0.563723 0.563723i 0.310591i 2.23083 + 0.152980i 1.21183i −0.135390 + 0.135390i 1.81586 + 1.81586i 2.36443i 2.56223 2.23337i
362.15 1.72010 1.72010i 0.422864 0.422864i 3.91750i −0.759172 2.10325i 1.45474i 1.82555 1.82555i −3.29830 3.29830i 2.64237i −4.92366 2.31195i
362.16 1.79273 1.79273i −1.41767 + 1.41767i 4.42777i 1.69208 1.46180i 5.08300i 1.27846 1.27846i −4.35233 4.35233i 1.01958i 0.412838 5.65406i
483.1 −1.79273 1.79273i −1.41767 1.41767i 4.42777i 1.69208 + 1.46180i 5.08300i −1.27846 1.27846i 4.35233 4.35233i 1.01958i −0.412838 5.65406i
483.2 −1.72010 1.72010i 0.422864 + 0.422864i 3.91750i −0.759172 + 2.10325i 1.45474i −1.82555 1.82555i 3.29830 3.29830i 2.64237i 4.92366 2.31195i
483.3 −1.07485 1.07485i 0.563723 + 0.563723i 0.310591i 2.23083 0.152980i 1.21183i 0.135390 + 0.135390i −1.81586 + 1.81586i 2.36443i −2.56223 2.23337i
483.4 −1.03659 1.03659i 0.588647 + 0.588647i 0.149021i −2.18706 0.465567i 1.22037i 2.98069 + 2.98069i −1.91870 + 1.91870i 2.30699i 1.78448 + 2.74968i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 483.16 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
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.e.b 32
5.c odd 4 1 inner 605.2.e.b 32
11.b odd 2 1 inner 605.2.e.b 32
11.c even 5 1 55.2.l.a 32
11.c even 5 1 605.2.m.c 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.m.c 32
11.d odd 10 1 605.2.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 inner 605.2.e.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 55.2.l.a 32
55.k odd 20 1 275.2.bm.b 32
55.k odd 20 1 605.2.m.c 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.m.c 32
55.l even 20 1 605.2.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 1.a even 1 1 trivial
605.2.e.b 32 5.c odd 4 1 inner
605.2.e.b 32 11.b odd 2 1 inner
605.2.e.b 32 55.e even 4 1 inner
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 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 $$S_{2}^{\mathrm{new}}(605, [\chi])$$.