# Properties

 Label 336.2.u.a Level 336 Weight 2 Character orbit 336.u Analytic conductor 2.683 Analytic rank 0 Dimension 64 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.u (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ over $$\Q(i)$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 4q^{4} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 4q^{4} + 24q^{8} + 8q^{11} - 16q^{14} + 4q^{16} - 4q^{18} - 28q^{22} - 16q^{23} + 32q^{28} + 16q^{29} + 24q^{35} + 16q^{37} + 20q^{42} - 8q^{43} - 36q^{44} - 40q^{46} - 52q^{50} + 16q^{53} - 28q^{56} - 92q^{58} + 24q^{60} - 52q^{64} + 56q^{67} - 40q^{70} - 128q^{71} + 4q^{72} - 60q^{74} - 64q^{81} - 24q^{84} + 92q^{86} - 84q^{88} + 8q^{91} + 136q^{92} - 64q^{98} - 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1 −1.38346 + 0.293324i −0.707107 + 0.707107i 1.82792 0.811604i 2.49549 2.49549i 0.770842 1.18567i 0.134305 + 2.64234i −2.29079 + 1.65899i 1.00000i −2.72043 + 4.18440i
139.2 −1.38346 + 0.293324i 0.707107 0.707107i 1.82792 0.811604i −2.49549 + 2.49549i −0.770842 + 1.18567i 0.134305 2.64234i −2.29079 + 1.65899i 1.00000i 2.72043 4.18440i
139.3 −1.27023 0.621708i −0.707107 + 0.707107i 1.22696 + 1.57942i 1.90169 1.90169i 1.33780 0.458573i 2.22978 1.42411i −0.576579 2.76904i 1.00000i −3.59788 + 1.23329i
139.4 −1.27023 0.621708i 0.707107 0.707107i 1.22696 + 1.57942i −1.90169 + 1.90169i −1.33780 + 0.458573i 2.22978 + 1.42411i −0.576579 2.76904i 1.00000i 3.59788 1.23329i
139.5 −1.20526 + 0.739825i −0.707107 + 0.707107i 0.905318 1.78337i −1.32544 + 1.32544i 0.329115 1.37538i 2.64192 + 0.142388i 0.228233 + 2.81920i 1.00000i 0.616912 2.57810i
139.6 −1.20526 + 0.739825i 0.707107 0.707107i 0.905318 1.78337i 1.32544 1.32544i −0.329115 + 1.37538i 2.64192 0.142388i 0.228233 + 2.81920i 1.00000i −0.616912 + 2.57810i
139.7 −1.08962 0.901509i −0.707107 + 0.707107i 0.374563 + 1.96461i 0.167468 0.167468i 1.40794 0.133018i −2.61783 0.383395i 1.36298 2.47836i 1.00000i −0.333451 + 0.0315033i
139.8 −1.08962 0.901509i 0.707107 0.707107i 0.374563 + 1.96461i −0.167468 + 0.167468i −1.40794 + 0.133018i −2.61783 + 0.383395i 1.36298 2.47836i 1.00000i 0.333451 0.0315033i
139.9 −0.777418 1.18136i −0.707107 + 0.707107i −0.791242 + 1.83683i −3.13913 + 3.13913i 1.38507 + 0.285633i 1.14645 2.38446i 2.78509 0.493238i 1.00000i 6.14886 + 1.26804i
139.10 −0.777418 1.18136i 0.707107 0.707107i −0.791242 + 1.83683i 3.13913 3.13913i −1.38507 0.285633i 1.14645 + 2.38446i 2.78509 0.493238i 1.00000i −6.14886 1.26804i
139.11 −0.618822 + 1.27164i −0.707107 + 0.707107i −1.23412 1.57383i 1.80101 1.80101i −0.461610 1.33676i −1.67026 + 2.05189i 2.76504 0.595430i 1.00000i 1.17573 + 3.40473i
139.12 −0.618822 + 1.27164i 0.707107 0.707107i −1.23412 1.57383i −1.80101 + 1.80101i 0.461610 + 1.33676i −1.67026 2.05189i 2.76504 0.595430i 1.00000i −1.17573 3.40473i
139.13 −0.453489 + 1.33953i −0.707107 + 0.707107i −1.58870 1.21493i 0.308965 0.308965i −0.626528 1.26786i 1.19726 2.35936i 2.34789 1.57715i 1.00000i 0.273757 + 0.553981i
139.14 −0.453489 + 1.33953i 0.707107 0.707107i −1.58870 1.21493i −0.308965 + 0.308965i 0.626528 + 1.26786i 1.19726 + 2.35936i 2.34789 1.57715i 1.00000i −0.273757 0.553981i
139.15 −0.135637 1.40769i −0.707107 + 0.707107i −1.96321 + 0.381870i 2.46638 2.46638i 1.09130 + 0.899481i −2.50032 0.865115i 0.803838 + 2.71180i 1.00000i −3.80643 3.13737i
139.16 −0.135637 1.40769i 0.707107 0.707107i −1.96321 + 0.381870i −2.46638 + 2.46638i −1.09130 0.899481i −2.50032 + 0.865115i 0.803838 + 2.71180i 1.00000i 3.80643 + 3.13737i
139.17 0.0533274 + 1.41321i −0.707107 + 0.707107i −1.99431 + 0.150725i −1.82248 + 1.82248i −1.03700 0.961581i −1.96782 + 1.76852i −0.319358 2.81034i 1.00000i −2.67273 2.47836i
139.18 0.0533274 + 1.41321i 0.707107 0.707107i −1.99431 + 0.150725i 1.82248 1.82248i 1.03700 + 0.961581i −1.96782 1.76852i −0.319358 2.81034i 1.00000i 2.67273 + 2.47836i
139.19 0.0748365 1.41223i −0.707107 + 0.707107i −1.98880 0.211373i −0.353074 + 0.353074i 0.945681 + 1.05152i 2.14763 + 1.54521i −0.447343 + 2.79283i 1.00000i 0.472200 + 0.525045i
139.20 0.0748365 1.41223i 0.707107 0.707107i −1.98880 0.211373i 0.353074 0.353074i −0.945681 1.05152i 2.14763 1.54521i −0.447343 + 2.79283i 1.00000i −0.472200 0.525045i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
16.f odd 4 1 inner
112.j 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.u.a 64
4.b odd 2 1 1344.2.u.a 64
7.b odd 2 1 inner 336.2.u.a 64
16.e even 4 1 1344.2.u.a 64
16.f odd 4 1 inner 336.2.u.a 64
28.d even 2 1 1344.2.u.a 64
112.j even 4 1 inner 336.2.u.a 64
112.l odd 4 1 1344.2.u.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.u.a 64 1.a even 1 1 trivial
336.2.u.a 64 7.b odd 2 1 inner
336.2.u.a 64 16.f odd 4 1 inner
336.2.u.a 64 112.j even 4 1 inner
1344.2.u.a 64 4.b odd 2 1
1344.2.u.a 64 16.e even 4 1
1344.2.u.a 64 28.d even 2 1
1344.2.u.a 64 112.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database