# Properties

 Label 430.2.g.b Level 430 Weight 2 Character orbit 430.g Analytic conductor 3.434 Analytic rank 0 Dimension 40 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{10} + 16q^{11} - 24q^{13} + 8q^{15} - 40q^{16} - 12q^{17} - 16q^{21} + 44q^{23} + 24q^{25} + 32q^{31} - 64q^{35} + 48q^{36} - 28q^{38} - 4q^{40} + 8q^{41} - 16q^{43} - 28q^{47} + 24q^{52} - 80q^{53} + 24q^{56} + 64q^{57} + 12q^{58} + 24q^{67} - 12q^{68} + 40q^{78} - 120q^{81} + 48q^{83} + 28q^{87} - 44q^{92} - 16q^{95} - 60q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1 −0.707107 + 0.707107i −2.27071 2.27071i 1.00000i −1.31680 + 1.80721i 3.21127 −0.136059 + 0.136059i 0.707107 + 0.707107i 7.31224i −0.346772 2.20902i
257.2 −0.707107 + 0.707107i −1.88501 1.88501i 1.00000i 1.60089 1.56114i 2.66581 3.34265 3.34265i 0.707107 + 0.707107i 4.10653i −0.0281087 + 2.23589i
257.3 −0.707107 + 0.707107i −1.12833 1.12833i 1.00000i −1.89019 1.19465i 1.59570 0.600193 0.600193i 0.707107 + 0.707107i 0.453751i 2.18131 0.491820i
257.4 −0.707107 + 0.707107i −0.515696 0.515696i 1.00000i 0.417939 + 2.19666i 0.729304 −1.42532 + 1.42532i 0.707107 + 0.707107i 2.46812i −1.84880 1.25775i
257.5 −0.707107 + 0.707107i −0.325226 0.325226i 1.00000i 2.06254 0.863667i 0.459939 −2.30721 + 2.30721i 0.707107 + 0.707107i 2.78846i −0.847732 + 2.06914i
257.6 −0.707107 + 0.707107i 0.146163 + 0.146163i 1.00000i 1.92089 + 1.14462i −0.206706 2.59530 2.59530i 0.707107 + 0.707107i 2.95727i −2.16765 + 0.548906i
257.7 −0.707107 + 0.707107i 0.427501 + 0.427501i 1.00000i −2.20958 + 0.343186i −0.604578 0.978156 0.978156i 0.707107 + 0.707107i 2.63449i 1.31974 1.80508i
257.8 −0.707107 + 0.707107i 1.53278 + 1.53278i 1.00000i −1.06852 + 1.96425i −2.16769 −1.64724 + 1.64724i 0.707107 + 0.707107i 1.69886i −0.633377 2.14449i
257.9 −0.707107 + 0.707107i 1.91446 + 1.91446i 1.00000i 2.21975 + 0.269648i −2.70746 −1.02061 + 1.02061i 0.707107 + 0.707107i 4.33033i −1.76027 + 1.37893i
257.10 −0.707107 + 0.707107i 2.10406 + 2.10406i 1.00000i −1.02982 1.98481i −2.97559 3.26278 3.26278i 0.707107 + 0.707107i 5.85412i 2.13167 + 0.675277i
257.11 0.707107 0.707107i −2.10406 2.10406i 1.00000i 1.02982 + 1.98481i −2.97559 −3.26278 + 3.26278i −0.707107 0.707107i 5.85412i 2.13167 + 0.675277i
257.12 0.707107 0.707107i −1.91446 1.91446i 1.00000i −2.21975 0.269648i −2.70746 1.02061 1.02061i −0.707107 0.707107i 4.33033i −1.76027 + 1.37893i
257.13 0.707107 0.707107i −1.53278 1.53278i 1.00000i 1.06852 1.96425i −2.16769 1.64724 1.64724i −0.707107 0.707107i 1.69886i −0.633377 2.14449i
257.14 0.707107 0.707107i −0.427501 0.427501i 1.00000i 2.20958 0.343186i −0.604578 −0.978156 + 0.978156i −0.707107 0.707107i 2.63449i 1.31974 1.80508i
257.15 0.707107 0.707107i −0.146163 0.146163i 1.00000i −1.92089 1.14462i −0.206706 −2.59530 + 2.59530i −0.707107 0.707107i 2.95727i −2.16765 + 0.548906i
257.16 0.707107 0.707107i 0.325226 + 0.325226i 1.00000i −2.06254 + 0.863667i 0.459939 2.30721 2.30721i −0.707107 0.707107i 2.78846i −0.847732 + 2.06914i
257.17 0.707107 0.707107i 0.515696 + 0.515696i 1.00000i −0.417939 2.19666i 0.729304 1.42532 1.42532i −0.707107 0.707107i 2.46812i −1.84880 1.25775i
257.18 0.707107 0.707107i 1.12833 + 1.12833i 1.00000i 1.89019 + 1.19465i 1.59570 −0.600193 + 0.600193i −0.707107 0.707107i 0.453751i 2.18131 0.491820i
257.19 0.707107 0.707107i 1.88501 + 1.88501i 1.00000i −1.60089 + 1.56114i 2.66581 −3.34265 + 3.34265i −0.707107 0.707107i 4.10653i −0.0281087 + 2.23589i
257.20 0.707107 0.707107i 2.27071 + 2.27071i 1.00000i 1.31680 1.80721i 3.21127 0.136059 0.136059i −0.707107 0.707107i 7.31224i −0.346772 2.20902i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 343.20 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
43.b odd 2 1 inner
215.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.g.b 40
5.c odd 4 1 inner 430.2.g.b 40
43.b odd 2 1 inner 430.2.g.b 40
215.g even 4 1 inner 430.2.g.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.g.b 40 1.a even 1 1 trivial
430.2.g.b 40 5.c odd 4 1 inner
430.2.g.b 40 43.b odd 2 1 inner
430.2.g.b 40 215.g even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database