# Properties

 Label 605.2.e.c Level $605$ Weight $2$ Character orbit 605.e Analytic conductor $4.831$ Analytic rank $0$ Dimension $40$ 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: $$40$$ Relative dimension: $$20$$ over $$\Q(i)$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 8q^{3} - 4q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 8q^{3} - 4q^{5} - 32q^{12} + 32q^{15} - 24q^{16} + 32q^{20} - 24q^{23} - 52q^{25} - 8q^{26} + 32q^{27} + 16q^{31} - 40q^{36} + 36q^{37} + 32q^{38} + 32q^{42} + 64q^{45} + 32q^{47} - 16q^{48} - 68q^{53} + 80q^{56} + 132q^{58} - 64q^{60} - 88q^{67} - 8q^{70} - 16q^{75} - 248q^{78} - 164q^{80} - 40q^{81} + 100q^{82} - 80q^{86} - 96q^{91} - 56q^{92} + 24q^{93} - 68q^{97} + O(q^{100})$$

## 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 −1.91757 + 1.91757i 0.831561 0.831561i 5.35415i 1.62695 + 1.53396i 3.18915i −0.383282 + 0.383282i 6.43181 + 6.43181i 1.61701i −6.06127 + 0.178306i
362.2 −1.64949 + 1.64949i 1.43990 1.43990i 3.44167i 1.13534 1.92640i 4.75021i 3.25511 3.25511i 2.37802 + 2.37802i 1.14662i 1.30484 + 5.05032i
362.3 −1.63660 + 1.63660i −0.722598 + 0.722598i 3.35690i 0.360665 + 2.20679i 2.36520i 2.14665 2.14665i 2.22069 + 2.22069i 1.95570i −4.20189 3.02136i
362.4 −1.38689 + 1.38689i −2.07537 + 2.07537i 1.84694i −2.03181 0.933683i 5.75663i −1.48652 + 1.48652i −0.212274 0.212274i 5.61432i 4.11281 1.52298i
362.5 −1.29704 + 1.29704i 2.21032 2.21032i 1.36464i 0.668547 + 2.13379i 5.73375i 0.351575 0.351575i −0.824089 0.824089i 6.77099i −3.63475 1.90048i
362.6 −1.05784 + 1.05784i 1.53861 1.53861i 0.238041i −1.92587 1.13622i 3.25521i −1.66576 + 1.66576i −1.86387 1.86387i 1.73467i 3.23920 0.835322i
362.7 −0.848475 + 0.848475i −0.258670 + 0.258670i 0.560180i −0.152228 2.23088i 0.438950i 1.06376 1.06376i −2.17225 2.17225i 2.86618i 2.02201 + 1.76368i
362.8 −0.503119 + 0.503119i −1.98889 + 1.98889i 1.49374i −1.36457 + 1.77143i 2.00130i 2.43291 2.43291i −1.75777 1.75777i 4.91135i −0.204700 1.57778i
362.9 −0.436764 + 0.436764i 1.06258 1.06258i 1.61847i −1.07825 + 1.95892i 0.928194i −1.08812 + 1.08812i −1.58042 1.58042i 0.741845i −0.384643 1.32653i
362.10 −0.187165 + 0.187165i −0.0374447 + 0.0374447i 1.92994i 1.76123 1.37770i 0.0140167i −2.30359 + 2.30359i −0.735548 0.735548i 2.99720i −0.0717827 + 0.587500i
362.11 0.187165 0.187165i −0.0374447 + 0.0374447i 1.92994i 1.76123 1.37770i 0.0140167i 2.30359 2.30359i 0.735548 + 0.735548i 2.99720i 0.0717827 0.587500i
362.12 0.436764 0.436764i 1.06258 1.06258i 1.61847i −1.07825 + 1.95892i 0.928194i 1.08812 1.08812i 1.58042 + 1.58042i 0.741845i 0.384643 + 1.32653i
362.13 0.503119 0.503119i −1.98889 + 1.98889i 1.49374i −1.36457 + 1.77143i 2.00130i −2.43291 + 2.43291i 1.75777 + 1.75777i 4.91135i 0.204700 + 1.57778i
362.14 0.848475 0.848475i −0.258670 + 0.258670i 0.560180i −0.152228 2.23088i 0.438950i −1.06376 + 1.06376i 2.17225 + 2.17225i 2.86618i −2.02201 1.76368i
362.15 1.05784 1.05784i 1.53861 1.53861i 0.238041i −1.92587 1.13622i 3.25521i 1.66576 1.66576i 1.86387 + 1.86387i 1.73467i −3.23920 + 0.835322i
362.16 1.29704 1.29704i 2.21032 2.21032i 1.36464i 0.668547 + 2.13379i 5.73375i −0.351575 + 0.351575i 0.824089 + 0.824089i 6.77099i 3.63475 + 1.90048i
362.17 1.38689 1.38689i −2.07537 + 2.07537i 1.84694i −2.03181 0.933683i 5.75663i 1.48652 1.48652i 0.212274 + 0.212274i 5.61432i −4.11281 + 1.52298i
362.18 1.63660 1.63660i −0.722598 + 0.722598i 3.35690i 0.360665 + 2.20679i 2.36520i −2.14665 + 2.14665i −2.22069 2.22069i 1.95570i 4.20189 + 3.02136i
362.19 1.64949 1.64949i 1.43990 1.43990i 3.44167i 1.13534 1.92640i 4.75021i −3.25511 + 3.25511i −2.37802 2.37802i 1.14662i −1.30484 5.05032i
362.20 1.91757 1.91757i 0.831561 0.831561i 5.35415i 1.62695 + 1.53396i 3.18915i 0.383282 0.383282i −6.43181 6.43181i 1.61701i 6.06127 0.178306i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 483.20 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.c 40
5.c odd 4 1 inner 605.2.e.c 40
11.b odd 2 1 inner 605.2.e.c 40
11.c even 5 4 605.2.m.g 160
11.d odd 10 4 605.2.m.g 160
55.e even 4 1 inner 605.2.e.c 40
55.k odd 20 4 605.2.m.g 160
55.l even 20 4 605.2.m.g 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.e.c 40 1.a even 1 1 trivial
605.2.e.c 40 5.c odd 4 1 inner
605.2.e.c 40 11.b odd 2 1 inner
605.2.e.c 40 55.e even 4 1 inner
605.2.m.g 160 11.c even 5 4
605.2.m.g 160 11.d odd 10 4
605.2.m.g 160 55.k odd 20 4
605.2.m.g 160 55.l even 20 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$.