# Properties

 Label 731.2.z.a Level 731 Weight 2 Character orbit 731.z Analytic conductor 5.837 Analytic rank 0 Dimension 768 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.z (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$768$$ Relative dimension: $$64$$ over $$\Q(\zeta_{42})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$768q - 24q^{2} - 144q^{4} - 16q^{8} - 98q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$768q - 24q^{2} - 144q^{4} - 16q^{8} - 98q^{9} - 18q^{13} - 30q^{15} - 160q^{16} - 16q^{17} - 54q^{18} - 68q^{19} - 50q^{21} - 88q^{25} - 26q^{26} - 50q^{32} - 36q^{33} - 38q^{34} + 14q^{35} + 328q^{36} - 44q^{38} - 148q^{42} + 102q^{43} - 64q^{47} + 298q^{49} + 40q^{50} - 31q^{51} - 38q^{52} - 28q^{53} - 80q^{55} - 16q^{59} - 34q^{60} - 64q^{64} - 126q^{66} + 74q^{67} - 132q^{68} - 28q^{69} + 50q^{70} + 26q^{72} - 258q^{76} - 112q^{77} + 90q^{81} + 48q^{83} - 298q^{84} + 36q^{85} + 142q^{86} + 192q^{87} - 120q^{89} - 188q^{93} + 64q^{94} + 146q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1 −1.69621 + 2.12698i −0.339828 2.25461i −1.20187 5.26573i 0.0789863 + 0.115852i 5.37192 + 3.10148i −0.309020 + 0.178413i 8.33653 + 4.01466i −2.10106 + 0.648091i −0.380391 0.0285064i
67.2 −1.69621 + 2.12698i 0.339828 + 2.25461i −1.20187 5.26573i −0.0789863 0.115852i −5.37192 3.10148i 0.309020 0.178413i 8.33653 + 4.01466i −2.10106 + 0.648091i 0.380391 + 0.0285064i
67.3 −1.67449 + 2.09974i −0.149191 0.989818i −1.15996 5.08212i −2.44528 3.58657i 2.32818 + 1.34418i 0.298243 0.172191i 7.77406 + 3.74379i 1.90924 0.588921i 11.6255 + 0.871209i
67.4 −1.67449 + 2.09974i 0.149191 + 0.989818i −1.15996 5.08212i 2.44528 + 3.58657i −2.32818 1.34418i −0.298243 + 0.172191i 7.77406 + 3.74379i 1.90924 0.588921i −11.6255 0.871209i
67.5 −1.53756 + 1.92804i −0.193458 1.28351i −0.908197 3.97907i 0.441633 + 0.647756i 2.77211 + 1.60048i 4.06857 2.34899i 4.62452 + 2.22705i 1.25675 0.387655i −1.92793 0.144478i
67.6 −1.53756 + 1.92804i 0.193458 + 1.28351i −0.908197 3.97907i −0.441633 0.647756i −2.77211 1.60048i −4.06857 + 2.34899i 4.62452 + 2.22705i 1.25675 0.387655i 1.92793 + 0.144478i
67.7 −1.43695 + 1.80187i −0.348188 2.31008i −0.736893 3.22854i 0.523790 + 0.768259i 4.66280 + 2.69207i −2.81082 + 1.62283i 2.72340 + 1.31152i −2.34851 + 0.724419i −2.13696 0.160143i
67.8 −1.43695 + 1.80187i 0.348188 + 2.31008i −0.736893 3.22854i −0.523790 0.768259i −4.66280 2.69207i 2.81082 1.62283i 2.72340 + 1.31152i −2.34851 + 0.724419i 2.13696 + 0.160143i
67.9 −1.29990 + 1.63003i −0.470647 3.12254i −0.522200 2.28791i −1.19265 1.74929i 5.70163 + 3.29184i 0.150209 0.0867232i 0.651331 + 0.313664i −6.66204 + 2.05497i 4.40173 + 0.329864i
67.10 −1.29990 + 1.63003i 0.470647 + 3.12254i −0.522200 2.28791i 1.19265 + 1.74929i −5.70163 3.29184i −0.150209 + 0.0867232i 0.651331 + 0.313664i −6.66204 + 2.05497i −4.40173 0.329864i
67.11 −1.29543 + 1.62442i −0.00546009 0.0362253i −0.515560 2.25882i −2.06352 3.02662i 0.0659185 + 0.0380581i 0.0270939 0.0156427i 0.593233 + 0.285686i 2.86544 0.883870i 7.58967 + 0.568767i
67.12 −1.29543 + 1.62442i 0.00546009 + 0.0362253i −0.515560 2.25882i 2.06352 + 3.02662i −0.0659185 0.0380581i −0.0270939 + 0.0156427i 0.593233 + 0.285686i 2.86544 0.883870i −7.58967 0.568767i
67.13 −1.16355 + 1.45905i −0.00644316 0.0427476i −0.329925 1.44549i 0.740612 + 1.08628i 0.0698677 + 0.0403381i 2.40880 1.39072i −0.869832 0.418889i 2.86493 0.883715i −2.44667 0.183353i
67.14 −1.16355 + 1.45905i 0.00644316 + 0.0427476i −0.329925 1.44549i −0.740612 1.08628i −0.0698677 0.0403381i −2.40880 + 1.39072i −0.869832 0.418889i 2.86493 0.883715i 2.44667 + 0.183353i
67.15 −1.11327 + 1.39600i −0.311016 2.06346i −0.264394 1.15839i 1.52042 + 2.23004i 3.22682 + 1.86301i −2.35848 + 1.36167i −1.30600 0.628935i −1.29440 + 0.399269i −4.80577 0.360142i
67.16 −1.11327 + 1.39600i 0.311016 + 2.06346i −0.264394 1.15839i −1.52042 2.23004i −3.22682 1.86301i 2.35848 1.36167i −1.30600 0.628935i −1.29440 + 0.399269i 4.80577 + 0.360142i
67.17 −0.792393 + 0.993630i −0.413423 2.74288i 0.0856288 + 0.375164i 1.48606 + 2.17965i 3.05300 + 1.76265i 0.708079 0.408809i −2.73071 1.31504i −4.48577 + 1.38368i −3.34332 0.250547i
67.18 −0.792393 + 0.993630i 0.413423 + 2.74288i 0.0856288 + 0.375164i −1.48606 2.17965i −3.05300 1.76265i −0.708079 + 0.408809i −2.73071 1.31504i −4.48577 + 1.38368i 3.34332 + 0.250547i
67.19 −0.755649 + 0.947554i −0.215818 1.43186i 0.118189 + 0.517820i −1.58922 2.33095i 1.51984 + 0.877483i 1.01277 0.584720i −2.76386 1.33100i 0.863080 0.266225i 3.40959 + 0.255514i
67.20 −0.755649 + 0.947554i 0.215818 + 1.43186i 0.118189 + 0.517820i 1.58922 + 2.33095i −1.51984 0.877483i −1.01277 + 0.584720i −2.76386 1.33100i 0.863080 0.266225i −3.40959 0.255514i
See next 80 embeddings (of 768 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 713.64 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.g even 21 1 inner
731.z even 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.z.a 768
17.b even 2 1 inner 731.2.z.a 768
43.g even 21 1 inner 731.2.z.a 768
731.z even 42 1 inner 731.2.z.a 768

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.z.a 768 1.a even 1 1 trivial
731.2.z.a 768 17.b even 2 1 inner
731.2.z.a 768 43.g even 21 1 inner
731.2.z.a 768 731.z even 42 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