# Properties

 Label 735.2.v.a Level 735 Weight 2 Character orbit 735.v Analytic conductor 5.869 Analytic rank 0 Dimension 32 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.v (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.86900454856$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 48q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 48q^{8} + 16q^{11} + 16q^{15} + 48q^{16} - 32q^{22} + 40q^{23} + 8q^{30} - 48q^{32} - 32q^{36} - 32q^{37} - 32q^{43} - 64q^{46} - 144q^{50} + 16q^{51} - 24q^{53} + 16q^{57} - 32q^{58} - 40q^{60} - 40q^{65} + 32q^{67} + 128q^{71} - 24q^{72} - 16q^{78} + 16q^{81} + 96q^{85} - 64q^{86} + 64q^{88} - 80q^{92} - 24q^{93} + 72q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
178.1 −2.03317 0.544785i −0.258819 0.965926i 2.10492 + 1.21528i −2.22675 0.203934i 2.10489i 0 −0.640825 0.640825i −0.866025 + 0.500000i 4.41625 + 1.62773i
178.2 −2.03317 0.544785i 0.258819 + 0.965926i 2.10492 + 1.21528i 2.22675 + 0.203934i 2.10489i 0 −0.640825 0.640825i −0.866025 + 0.500000i −4.41625 1.62773i
178.3 −0.737849 0.197706i −0.258819 0.965926i −1.22672 0.708245i 1.19764 1.88830i 0.763878i 0 1.84539 + 1.84539i −0.866025 + 0.500000i −1.25700 + 1.15650i
178.4 −0.737849 0.197706i 0.258819 + 0.965926i −1.22672 0.708245i −1.19764 + 1.88830i 0.763878i 0 1.84539 + 1.84539i −0.866025 + 0.500000i 1.25700 1.15650i
178.5 0.228203 + 0.0611467i −0.258819 0.965926i −1.68371 0.972092i 1.18965 + 1.89334i 0.236253i 0 −0.658899 0.658899i −0.866025 + 0.500000i 0.155711 + 0.504808i
178.6 0.228203 + 0.0611467i 0.258819 + 0.965926i −1.68371 0.972092i −1.18965 1.89334i 0.236253i 0 −0.658899 0.658899i −0.866025 + 0.500000i −0.155711 0.504808i
178.7 2.54281 + 0.681344i −0.258819 0.965926i 4.26961 + 2.46506i −0.678180 + 2.13074i 2.63251i 0 5.45433 + 5.45433i −0.866025 + 0.500000i −3.17625 + 4.95601i
178.8 2.54281 + 0.681344i 0.258819 + 0.965926i 4.26961 + 2.46506i 0.678180 2.13074i 2.63251i 0 5.45433 + 5.45433i −0.866025 + 0.500000i 3.17625 4.95601i
313.1 −0.681344 2.54281i −0.965926 0.258819i −4.26961 + 2.46506i −2.18437 + 0.478051i 2.63251i 0 5.45433 + 5.45433i 0.866025 + 0.500000i 2.70390 + 5.22872i
313.2 −0.681344 2.54281i 0.965926 + 0.258819i −4.26961 + 2.46506i 2.18437 0.478051i 2.63251i 0 5.45433 + 5.45433i 0.866025 + 0.500000i −2.70390 5.22872i
313.3 −0.0611467 0.228203i −0.965926 0.258819i 1.68371 0.972092i −1.04485 + 1.97694i 0.236253i 0 −0.658899 0.658899i 0.866025 + 0.500000i 0.515032 + 0.117555i
313.4 −0.0611467 0.228203i 0.965926 + 0.258819i 1.68371 0.972092i 1.04485 1.97694i 0.236253i 0 −0.658899 0.658899i 0.866025 + 0.500000i −0.515032 0.117555i
313.5 0.197706 + 0.737849i −0.965926 0.258819i 1.22672 0.708245i 2.23413 + 0.0930365i 0.763878i 0 1.84539 + 1.84539i 0.866025 + 0.500000i 0.373055 + 1.66685i
313.6 0.197706 + 0.737849i 0.965926 + 0.258819i 1.22672 0.708245i −2.23413 0.0930365i 0.763878i 0 1.84539 + 1.84539i 0.866025 + 0.500000i −0.373055 1.66685i
313.7 0.544785 + 2.03317i −0.965926 0.258819i −2.10492 + 1.21528i −0.936763 2.03039i 2.10489i 0 −0.640825 0.640825i 0.866025 + 0.500000i 3.61778 3.01072i
313.8 0.544785 + 2.03317i 0.965926 + 0.258819i −2.10492 + 1.21528i 0.936763 + 2.03039i 2.10489i 0 −0.640825 0.640825i 0.866025 + 0.500000i −3.61778 + 3.01072i
472.1 −0.681344 + 2.54281i −0.965926 + 0.258819i −4.26961 2.46506i −2.18437 0.478051i 2.63251i 0 5.45433 5.45433i 0.866025 0.500000i 2.70390 5.22872i
472.2 −0.681344 + 2.54281i 0.965926 0.258819i −4.26961 2.46506i 2.18437 + 0.478051i 2.63251i 0 5.45433 5.45433i 0.866025 0.500000i −2.70390 + 5.22872i
472.3 −0.0611467 + 0.228203i −0.965926 + 0.258819i 1.68371 + 0.972092i −1.04485 1.97694i 0.236253i 0 −0.658899 + 0.658899i 0.866025 0.500000i 0.515032 0.117555i
472.4 −0.0611467 + 0.228203i 0.965926 0.258819i 1.68371 + 0.972092i 1.04485 + 1.97694i 0.236253i 0 −0.658899 + 0.658899i 0.866025 0.500000i −0.515032 + 0.117555i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.8 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.v.a 32
5.c odd 4 1 inner 735.2.v.a 32
7.b odd 2 1 inner 735.2.v.a 32
7.c even 3 1 105.2.m.a 16
7.c even 3 1 inner 735.2.v.a 32
7.d odd 6 1 105.2.m.a 16
7.d odd 6 1 inner 735.2.v.a 32
21.g even 6 1 315.2.p.e 16
21.h odd 6 1 315.2.p.e 16
28.f even 6 1 1680.2.cz.d 16
28.g odd 6 1 1680.2.cz.d 16
35.f even 4 1 inner 735.2.v.a 32
35.i odd 6 1 525.2.m.b 16
35.j even 6 1 525.2.m.b 16
35.k even 12 1 105.2.m.a 16
35.k even 12 1 525.2.m.b 16
35.k even 12 1 inner 735.2.v.a 32
35.l odd 12 1 105.2.m.a 16
35.l odd 12 1 525.2.m.b 16
35.l odd 12 1 inner 735.2.v.a 32
105.w odd 12 1 315.2.p.e 16
105.x even 12 1 315.2.p.e 16
140.w even 12 1 1680.2.cz.d 16
140.x odd 12 1 1680.2.cz.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.m.a 16 7.c even 3 1
105.2.m.a 16 7.d odd 6 1
105.2.m.a 16 35.k even 12 1
105.2.m.a 16 35.l odd 12 1
315.2.p.e 16 21.g even 6 1
315.2.p.e 16 21.h odd 6 1
315.2.p.e 16 105.w odd 12 1
315.2.p.e 16 105.x even 12 1
525.2.m.b 16 35.i odd 6 1
525.2.m.b 16 35.j even 6 1
525.2.m.b 16 35.k even 12 1
525.2.m.b 16 35.l odd 12 1
735.2.v.a 32 1.a even 1 1 trivial
735.2.v.a 32 5.c odd 4 1 inner
735.2.v.a 32 7.b odd 2 1 inner
735.2.v.a 32 7.c even 3 1 inner
735.2.v.a 32 7.d odd 6 1 inner
735.2.v.a 32 35.f even 4 1 inner
735.2.v.a 32 35.k even 12 1 inner
735.2.v.a 32 35.l odd 12 1 inner
1680.2.cz.d 16 28.f even 6 1
1680.2.cz.d 16 28.g odd 6 1
1680.2.cz.d 16 140.w even 12 1
1680.2.cz.d 16 140.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(735, [\chi])$$:

 $$T_{2}^{16} - \cdots$$ $$T_{13}^{16} + 736 T_{13}^{12} + 8576 T_{13}^{8} + 18432 T_{13}^{4} + 4096$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database