# Properties

 Label 731.2.p.a Level 731 Weight 2 Character orbit 731.p Analytic conductor 5.837 Analytic rank 0 Dimension 384 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.p (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$384$$ Relative dimension: $$64$$ over $$\Q(\zeta_{14})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$384q - 12q^{2} - 72q^{4} - 8q^{8} + 62q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$384q - 12q^{2} - 72q^{4} - 8q^{8} + 62q^{9} - 18q^{13} - 12q^{15} - 20q^{16} - 14q^{17} + 30q^{18} + 8q^{19} + 20q^{21} + 46q^{25} + 2q^{26} - 30q^{30} + 50q^{32} - 90q^{33} + 8q^{34} + 52q^{35} - 328q^{36} - 46q^{38} + 184q^{42} + 60q^{43} - 2q^{47} - 340q^{49} - 172q^{50} - 68q^{51} + 38q^{52} - 8q^{53} - 28q^{55} + 10q^{59} + 58q^{60} - 44q^{64} + 90q^{66} - 104q^{67} + 60q^{68} - 2q^{69} + 40q^{70} - 164q^{72} - 72q^{76} + 94q^{77} + 120q^{81} - 66q^{83} + 112q^{84} - 96q^{85} - 136q^{86} + 72q^{87} - 6q^{89} - 220q^{93} - 142q^{94} + 124q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −0.608972 2.66808i −0.149961 0.0342275i −4.94587 + 2.38180i −1.75767 + 1.40169i 0.420950i 1.31934i 5.95414 + 7.46625i −2.68159 1.29139i 4.81020 + 3.83601i
16.2 −0.608972 2.66808i 0.149961 + 0.0342275i −4.94587 + 2.38180i 1.75767 1.40169i 0.420950i 1.31934i 5.95414 + 7.46625i −2.68159 1.29139i −4.81020 3.83601i
16.3 −0.594326 2.60391i −2.90090 0.662112i −4.62520 + 2.22738i −1.09262 + 0.871338i 7.94720i 2.92823i 5.21824 + 6.54347i 5.27392 + 2.53979i 2.91826 + 2.32724i
16.4 −0.594326 2.60391i 2.90090 + 0.662112i −4.62520 + 2.22738i 1.09262 0.871338i 7.94720i 2.92823i 5.21824 + 6.54347i 5.27392 + 2.53979i −2.91826 2.32724i
16.5 −0.534495 2.34178i −2.00623 0.457909i −3.39629 + 1.63557i 1.96143 1.56419i 4.94289i 1.32734i 2.65019 + 3.32324i 1.11237 + 0.535690i −4.71134 3.75717i
16.6 −0.534495 2.34178i 2.00623 + 0.457909i −3.39629 + 1.63557i −1.96143 + 1.56419i 4.94289i 1.32734i 2.65019 + 3.32324i 1.11237 + 0.535690i 4.71134 + 3.75717i
16.7 −0.521303 2.28398i −1.00960 0.230435i −3.14286 + 1.51352i −2.97746 + 2.37445i 2.42604i 3.68726i 2.17391 + 2.72600i −1.73671 0.836354i 6.97534 + 5.56265i
16.8 −0.521303 2.28398i 1.00960 + 0.230435i −3.14286 + 1.51352i 2.97746 2.37445i 2.42604i 3.68726i 2.17391 + 2.72600i −1.73671 0.836354i −6.97534 5.56265i
16.9 −0.501288 2.19629i −1.02764 0.234553i −2.77045 + 1.33418i 1.62476 1.29571i 2.37458i 4.11644i 1.50987 + 1.89332i −1.70187 0.819578i −3.66022 2.91893i
16.10 −0.501288 2.19629i 1.02764 + 0.234553i −2.77045 + 1.33418i −1.62476 + 1.29571i 2.37458i 4.11644i 1.50987 + 1.89332i −1.70187 0.819578i 3.66022 + 2.91893i
16.11 −0.473651 2.07520i −3.20126 0.730666i −2.28018 + 1.09808i −0.606394 + 0.483583i 6.98933i 4.01427i 0.704462 + 0.883367i 7.01126 + 3.37644i 1.29075 + 1.02934i
16.12 −0.473651 2.07520i 3.20126 + 0.730666i −2.28018 + 1.09808i 0.606394 0.483583i 6.98933i 4.01427i 0.704462 + 0.883367i 7.01126 + 3.37644i −1.29075 1.02934i
16.13 −0.456593 2.00046i −2.15206 0.491193i −1.99144 + 0.959029i 0.261751 0.208739i 4.52939i 2.19293i 0.269092 + 0.337431i 1.68717 + 0.812500i −0.537089 0.428314i
16.14 −0.456593 2.00046i 2.15206 + 0.491193i −1.99144 + 0.959029i −0.261751 + 0.208739i 4.52939i 2.19293i 0.269092 + 0.337431i 1.68717 + 0.812500i 0.537089 + 0.428314i
16.15 −0.359164 1.57360i −1.14363 0.261026i −0.545279 + 0.262592i −0.961694 + 0.766926i 1.89337i 0.0287002i −1.40365 1.76012i −1.46315 0.704615i 1.55224 + 1.23787i
16.16 −0.359164 1.57360i 1.14363 + 0.261026i −0.545279 + 0.262592i 0.961694 0.766926i 1.89337i 0.0287002i −1.40365 1.76012i −1.46315 0.704615i −1.55224 1.23787i
16.17 −0.343890 1.50668i −2.15737 0.492406i −0.349888 + 0.168497i 2.57313 2.05201i 3.41980i 1.34370i −1.55292 1.94730i 1.70888 + 0.822954i −3.97659 3.17123i
16.18 −0.343890 1.50668i 2.15737 + 0.492406i −0.349888 + 0.168497i −2.57313 + 2.05201i 3.41980i 1.34370i −1.55292 1.94730i 1.70888 + 0.822954i 3.97659 + 3.17123i
16.19 −0.332663 1.45749i −2.23027 0.509045i −0.211677 + 0.101938i −3.41321 + 2.72194i 3.41994i 3.16066i −1.64521 2.06303i 2.01209 + 0.968970i 5.10265 + 4.06923i
16.20 −0.332663 1.45749i 2.23027 + 0.509045i −0.211677 + 0.101938i 3.41321 2.72194i 3.41994i 3.16066i −1.64521 2.06303i 2.01209 + 0.968970i −5.10265 4.06923i
See next 80 embeddings (of 384 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 594.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
43.e even 7 1 inner
731.p even 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.p.a 384
17.b even 2 1 inner 731.2.p.a 384
43.e even 7 1 inner 731.2.p.a 384
731.p even 14 1 inner 731.2.p.a 384

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.p.a 384 1.a even 1 1 trivial
731.2.p.a 384 17.b even 2 1 inner
731.2.p.a 384 43.e even 7 1 inner
731.2.p.a 384 731.p even 14 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(731, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database