# Properties

 Label 700.2.k.b Level $700$ Weight $2$ Character orbit 700.k Analytic conductor $5.590$ Analytic rank $0$ Dimension $36$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 140) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 8q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 8q^{6} + 16q^{12} + 4q^{13} - 8q^{16} + 20q^{17} - 28q^{18} - 4q^{22} - 32q^{26} - 20q^{37} + 20q^{42} + 16q^{46} + 24q^{48} - 16q^{52} + 44q^{53} - 24q^{56} + 16q^{57} + 4q^{58} - 64q^{61} - 40q^{62} + 32q^{66} - 80q^{68} - 80q^{72} - 52q^{73} + 8q^{76} + 76q^{78} - 36q^{81} - 56q^{82} + 56q^{86} + 40q^{88} + 56q^{92} - 32q^{93} + 120q^{96} - 20q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −1.35755 0.396294i −0.945787 + 0.945787i 1.68590 + 1.07598i 0 1.65877 0.909146i 0.707107 + 0.707107i −1.86230 2.12881i 1.21097i 0
43.2 −1.34141 + 0.447909i 0.396892 0.396892i 1.59875 1.20166i 0 −0.354623 + 0.710167i 0.707107 + 0.707107i −1.60635 + 2.32801i 2.68495i 0
43.3 −1.29055 + 0.578354i 1.27396 1.27396i 1.33101 1.49278i 0 −0.907305 + 2.38090i −0.707107 0.707107i −0.854378 + 2.69630i 0.245954i 0
43.4 −1.26992 0.622342i 2.09607 2.09607i 1.22538 + 1.58065i 0 −3.96631 + 1.35737i −0.707107 0.707107i −0.572433 2.76990i 5.78704i 0
43.5 −0.903055 1.08834i 1.00798 1.00798i −0.368985 + 1.96567i 0 −2.00729 0.186768i 0.707107 + 0.707107i 2.47254 1.37352i 0.967954i 0
43.6 −0.802007 1.16481i −1.75731 + 1.75731i −0.713568 + 1.86837i 0 3.45630 + 0.637557i −0.707107 0.707107i 2.74859 0.667278i 3.17626i 0
43.7 −0.578354 + 1.29055i −1.27396 + 1.27396i −1.33101 1.49278i 0 −0.907305 2.38090i 0.707107 + 0.707107i 2.69630 0.854378i 0.245954i 0
43.8 −0.447909 + 1.34141i −0.396892 + 0.396892i −1.59875 1.20166i 0 −0.354623 0.710167i −0.707107 0.707107i 2.32801 1.60635i 2.68495i 0
43.9 −0.297828 1.38250i −0.137886 + 0.137886i −1.82260 + 0.823494i 0 0.231693 + 0.149560i −0.707107 0.707107i 1.68130 + 2.27447i 2.96198i 0
43.10 0.361308 1.36728i −1.26588 + 1.26588i −1.73891 0.988019i 0 1.27344 + 2.18818i 0.707107 + 0.707107i −1.97918 + 2.02060i 0.204893i 0
43.11 0.396294 + 1.35755i 0.945787 0.945787i −1.68590 + 1.07598i 0 1.65877 + 0.909146i −0.707107 0.707107i −2.12881 1.86230i 1.21097i 0
43.12 0.622342 + 1.26992i −2.09607 + 2.09607i −1.22538 + 1.58065i 0 −3.96631 1.35737i 0.707107 + 0.707107i −2.76990 0.572433i 5.78704i 0
43.13 0.649412 1.25629i 2.28163 2.28163i −1.15653 1.63170i 0 −1.38467 4.34811i 0.707107 + 0.707107i −2.80095 + 0.393286i 7.41170i 0
43.14 1.08834 + 0.903055i −1.00798 + 1.00798i 0.368985 + 1.96567i 0 −2.00729 + 0.186768i −0.707107 0.707107i −1.37352 + 2.47254i 0.967954i 0
43.15 1.16481 + 0.802007i 1.75731 1.75731i 0.713568 + 1.86837i 0 3.45630 0.637557i 0.707107 + 0.707107i −0.667278 + 2.74859i 3.17626i 0
43.16 1.25629 0.649412i −2.28163 + 2.28163i 1.15653 1.63170i 0 −1.38467 + 4.34811i −0.707107 0.707107i 0.393286 2.80095i 7.41170i 0
43.17 1.36728 0.361308i 1.26588 1.26588i 1.73891 0.988019i 0 1.27344 2.18818i −0.707107 0.707107i 2.02060 1.97918i 0.204893i 0
43.18 1.38250 + 0.297828i 0.137886 0.137886i 1.82260 + 0.823494i 0 0.231693 0.149560i 0.707107 + 0.707107i 2.27447 + 1.68130i 2.96198i 0
407.1 −1.35755 + 0.396294i −0.945787 0.945787i 1.68590 1.07598i 0 1.65877 + 0.909146i 0.707107 0.707107i −1.86230 + 2.12881i 1.21097i 0
407.2 −1.34141 0.447909i 0.396892 + 0.396892i 1.59875 + 1.20166i 0 −0.354623 0.710167i 0.707107 0.707107i −1.60635 2.32801i 2.68495i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 407.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.k.b 36
4.b odd 2 1 inner 700.2.k.b 36
5.b even 2 1 140.2.k.a 36
5.c odd 4 1 140.2.k.a 36
5.c odd 4 1 inner 700.2.k.b 36
20.d odd 2 1 140.2.k.a 36
20.e even 4 1 140.2.k.a 36
20.e even 4 1 inner 700.2.k.b 36
35.c odd 2 1 980.2.k.l 36
35.f even 4 1 980.2.k.l 36
35.i odd 6 2 980.2.x.l 72
35.j even 6 2 980.2.x.k 72
35.k even 12 2 980.2.x.l 72
35.l odd 12 2 980.2.x.k 72
140.c even 2 1 980.2.k.l 36
140.j odd 4 1 980.2.k.l 36
140.p odd 6 2 980.2.x.k 72
140.s even 6 2 980.2.x.l 72
140.w even 12 2 980.2.x.k 72
140.x odd 12 2 980.2.x.l 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.k.a 36 5.b even 2 1
140.2.k.a 36 5.c odd 4 1
140.2.k.a 36 20.d odd 2 1
140.2.k.a 36 20.e even 4 1
700.2.k.b 36 1.a even 1 1 trivial
700.2.k.b 36 4.b odd 2 1 inner
700.2.k.b 36 5.c odd 4 1 inner
700.2.k.b 36 20.e even 4 1 inner
980.2.k.l 36 35.c odd 2 1
980.2.k.l 36 35.f even 4 1
980.2.k.l 36 140.c even 2 1
980.2.k.l 36 140.j odd 4 1
980.2.x.k 72 35.j even 6 2
980.2.x.k 72 35.l odd 12 2
980.2.x.k 72 140.p odd 6 2
980.2.x.k 72 140.w even 12 2
980.2.x.l 72 35.i odd 6 2
980.2.x.l 72 35.k even 12 2
980.2.x.l 72 140.s even 6 2
980.2.x.l 72 140.x odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{36} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$.