# Properties

 Label 336.2.s.c Level 336 Weight 2 Character orbit 336.s Analytic conductor 2.683 Analytic rank 0 Dimension 40 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 336.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 4q^{3} - 2q^{6} - 40q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 4q^{3} - 2q^{6} - 40q^{7} - 8q^{10} - 2q^{12} + 24q^{13} + 36q^{16} + 12q^{18} + 16q^{19} - 4q^{21} - 8q^{22} + 6q^{24} - 32q^{27} - 32q^{30} + 24q^{33} + 12q^{34} - 4q^{36} - 8q^{37} - 64q^{39} - 60q^{40} + 2q^{42} + 24q^{43} - 28q^{45} + 20q^{46} - 26q^{48} + 40q^{49} - 32q^{51} + 84q^{52} - 14q^{54} + 16q^{55} + 12q^{58} - 24q^{60} - 48q^{61} - 12q^{64} - 36q^{66} + 40q^{67} + 4q^{69} + 8q^{70} + 8q^{72} + 40q^{75} - 44q^{76} + 24q^{78} + 56q^{81} - 84q^{82} + 2q^{84} - 48q^{85} + 32q^{87} + 52q^{88} - 76q^{90} - 24q^{91} + 56q^{93} - 62q^{96} + 16q^{97} + 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
155.1 −1.41409 + 0.0184003i 0.827739 1.52146i 1.99932 0.0520394i −1.23935 + 1.23935i −1.14250 + 2.16672i −1.00000 −2.82627 + 0.110377i −1.62970 2.51875i 1.72975 1.77536i
155.2 −1.38145 0.302633i 1.42461 + 0.985133i 1.81683 + 0.836146i −0.766553 + 0.766553i −1.66990 1.79205i −1.00000 −2.25682 1.70493i 1.05903 + 2.80686i 1.29094 0.826974i
155.3 −1.37293 + 0.339195i −1.63649 + 0.567369i 1.76989 0.931385i 0.132854 0.132854i 2.05434 1.33405i −1.00000 −2.11403 + 1.87907i 2.35619 1.85698i −0.137336 + 0.227462i
155.4 −1.25684 0.648355i −0.629429 + 1.61364i 1.15927 + 1.62975i 3.08691 3.08691i 1.83730 1.61998i −1.00000 −0.400356 2.79995i −2.20764 2.03134i −5.88116 + 1.87833i
155.5 −1.14682 + 0.827528i −0.0404130 + 1.73158i 0.630395 1.89805i −0.0573500 + 0.0573500i −1.38658 2.01925i −1.00000 0.847742 + 2.69839i −2.99673 0.139957i 0.0183114 0.113229i
155.6 −0.806993 + 1.16136i −0.0806676 1.73017i −0.697526 1.87442i −1.66193 + 1.66193i 2.07445 + 1.30255i −1.00000 2.73978 + 0.702564i −2.98699 + 0.279137i −0.588940 3.27128i
155.7 −0.724332 1.21464i 1.72790 0.119805i −0.950686 + 1.75960i 0.793669 0.793669i −1.39709 2.01200i −1.00000 2.82589 0.119796i 2.97129 0.414021i −1.53890 0.389140i
155.8 −0.244056 + 1.39300i −1.68331 + 0.408017i −1.88087 0.679939i −2.26080 + 2.26080i −0.157543 2.44442i −1.00000 1.40619 2.45410i 2.66704 1.37363i −2.59752 3.70105i
155.9 −0.230305 1.39533i −1.57773 0.714681i −1.89392 + 0.642706i −1.31983 + 1.31983i −0.633860 + 2.36606i −1.00000 1.33297 + 2.49463i 1.97846 + 2.25515i 2.14556 + 1.53764i
155.10 −0.153777 + 1.40583i 1.18388 + 1.26429i −1.95271 0.432367i 2.84277 2.84277i −1.95942 + 1.46992i −1.00000 0.908115 2.67868i −0.196841 + 2.99354i 3.55929 + 4.43360i
155.11 0.153777 1.40583i 1.26429 + 1.18388i −1.95271 0.432367i −2.84277 + 2.84277i 1.85875 1.59532i −1.00000 −0.908115 + 2.67868i 0.196841 + 2.99354i 3.55929 + 4.43360i
155.12 0.230305 + 1.39533i −0.714681 1.57773i −1.89392 + 0.642706i 1.31983 1.31983i 2.03687 1.36058i −1.00000 −1.33297 2.49463i −1.97846 + 2.25515i 2.14556 + 1.53764i
155.13 0.244056 1.39300i 0.408017 1.68331i −1.88087 0.679939i 2.26080 2.26080i −2.24526 0.979187i −1.00000 −1.40619 + 2.45410i −2.66704 1.37363i −2.59752 3.70105i
155.14 0.724332 + 1.21464i −0.119805 + 1.72790i −0.950686 + 1.75960i −0.793669 + 0.793669i −2.18555 + 1.10606i −1.00000 −2.82589 + 0.119796i −2.97129 0.414021i −1.53890 0.389140i
155.15 0.806993 1.16136i −1.73017 0.0806676i −0.697526 1.87442i 1.66193 1.66193i −1.48992 + 1.94426i −1.00000 −2.73978 0.702564i 2.98699 + 0.279137i −0.588940 3.27128i
155.16 1.14682 0.827528i 1.73158 0.0404130i 0.630395 1.89805i 0.0573500 0.0573500i 1.95237 1.47928i −1.00000 −0.847742 2.69839i 2.99673 0.139957i 0.0183114 0.113229i
155.17 1.25684 + 0.648355i 1.61364 0.629429i 1.15927 + 1.62975i −3.08691 + 3.08691i 2.43617 + 0.255120i −1.00000 0.400356 + 2.79995i 2.20764 2.03134i −5.88116 + 1.87833i
155.18 1.37293 0.339195i 0.567369 1.63649i 1.76989 0.931385i −0.132854 + 0.132854i 0.223871 2.43924i −1.00000 2.11403 1.87907i −2.35619 1.85698i −0.137336 + 0.227462i
155.19 1.38145 + 0.302633i 0.985133 + 1.42461i 1.81683 + 0.836146i 0.766553 0.766553i 0.929782 + 2.26617i −1.00000 2.25682 + 1.70493i −1.05903 + 2.80686i 1.29094 0.826974i
155.20 1.41409 0.0184003i −1.52146 + 0.827739i 1.99932 0.0520394i 1.23935 1.23935i −2.13626 + 1.19850i −1.00000 2.82627 0.110377i 1.62970 2.51875i 1.72975 1.77536i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.s.c 40
3.b odd 2 1 inner 336.2.s.c 40
4.b odd 2 1 1344.2.s.c 40
12.b even 2 1 1344.2.s.c 40
16.e even 4 1 1344.2.s.c 40
16.f odd 4 1 inner 336.2.s.c 40
48.i odd 4 1 1344.2.s.c 40
48.k even 4 1 inner 336.2.s.c 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.s.c 40 1.a even 1 1 trivial
336.2.s.c 40 3.b odd 2 1 inner
336.2.s.c 40 16.f odd 4 1 inner
336.2.s.c 40 48.k even 4 1 inner
1344.2.s.c 40 4.b odd 2 1
1344.2.s.c 40 12.b even 2 1
1344.2.s.c 40 16.e even 4 1
1344.2.s.c 40 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database