# Properties

 Label 430.2.k.d Level 430 Weight 2 Character orbit 430.k Analytic conductor 3.434 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.k (of order $$7$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{7})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{2} + 2q^{3} - 4q^{4} + 4q^{5} + 12q^{6} - 2q^{7} + 4q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{2} + 2q^{3} - 4q^{4} + 4q^{5} + 12q^{6} - 2q^{7} + 4q^{8} - 4q^{9} - 4q^{10} - 9q^{11} + 2q^{12} - 2q^{13} - 5q^{14} - 2q^{15} - 4q^{16} + 13q^{17} - 10q^{18} - 3q^{19} + 4q^{20} - 14q^{21} - 5q^{22} + 11q^{23} - 2q^{24} - 4q^{25} + 2q^{26} - 13q^{27} + 5q^{28} + 20q^{29} + 2q^{30} - 5q^{31} + 4q^{32} + 15q^{33} + q^{34} + 9q^{35} + 24q^{36} - 38q^{37} - 18q^{38} + 5q^{39} - 4q^{40} - 2q^{41} - 14q^{42} - 2q^{43} - 2q^{44} - 10q^{45} - 4q^{46} + 10q^{47} + 2q^{48} + 50q^{49} - 24q^{50} - 42q^{51} - 2q^{52} + 22q^{53} - 29q^{54} - 5q^{55} + 9q^{56} - 67q^{57} + 22q^{58} - 44q^{59} - 2q^{60} - 26q^{61} - 2q^{62} + 37q^{63} - 4q^{64} + 2q^{65} - 8q^{66} + 37q^{67} - 15q^{68} + 88q^{69} - 9q^{70} + 19q^{71} - 10q^{72} + 22q^{73} - 4q^{74} + 2q^{75} + 18q^{76} + 28q^{77} + 23q^{78} + 30q^{79} - 24q^{80} - 26q^{81} + 37q^{82} - 11q^{83} - 14q^{84} - 6q^{85} + 16q^{86} - 26q^{87} - 5q^{88} + 30q^{89} - 4q^{90} + 36q^{91} - 10q^{92} - 98q^{93} + 4q^{94} + 3q^{95} - 2q^{96} - 39q^{97} + 41q^{98} - 31q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −0.623490 0.781831i −2.05881 + 2.58167i −0.222521 + 0.974928i 0.900969 0.433884i 3.30207 3.39078 0.900969 0.433884i −1.75874 7.70553i −0.900969 0.433884i
11.2 −0.623490 0.781831i −1.03097 + 1.29280i −0.222521 + 0.974928i 0.900969 0.433884i 1.65355 −3.40711 0.900969 0.433884i 0.0591384 + 0.259102i −0.900969 0.433884i
11.3 −0.623490 0.781831i 0.0295669 0.0370758i −0.222521 + 0.974928i 0.900969 0.433884i −0.0474217 1.52167 0.900969 0.433884i 0.667062 + 2.92259i −0.900969 0.433884i
11.4 −0.623490 0.781831i 1.81324 2.27373i −0.222521 + 0.974928i 0.900969 0.433884i −2.90820 1.09852 0.900969 0.433884i −1.21444 5.32082i −0.900969 0.433884i
21.1 0.900969 + 0.433884i −1.69175 + 0.814703i 0.623490 + 0.781831i 0.222521 + 0.974928i −1.87770 4.12012 0.222521 + 0.974928i 0.327799 0.411047i −0.222521 + 0.974928i
21.2 0.900969 + 0.433884i −0.400237 + 0.192744i 0.623490 + 0.781831i 0.222521 + 0.974928i −0.444230 −4.95965 0.222521 + 0.974928i −1.74743 + 2.19121i −0.222521 + 0.974928i
21.3 0.900969 + 0.433884i 1.64572 0.792535i 0.623490 + 0.781831i 0.222521 + 0.974928i 1.82661 2.16285 0.222521 + 0.974928i 0.209800 0.263080i −0.222521 + 0.974928i
21.4 0.900969 + 0.433884i 2.24821 1.08268i 0.623490 + 0.781831i 0.222521 + 0.974928i 2.49532 −1.43324 0.222521 + 0.974928i 2.01177 2.52268i −0.222521 + 0.974928i
41.1 0.900969 0.433884i −1.69175 0.814703i 0.623490 0.781831i 0.222521 0.974928i −1.87770 4.12012 0.222521 0.974928i 0.327799 + 0.411047i −0.222521 0.974928i
41.2 0.900969 0.433884i −0.400237 0.192744i 0.623490 0.781831i 0.222521 0.974928i −0.444230 −4.95965 0.222521 0.974928i −1.74743 2.19121i −0.222521 0.974928i
41.3 0.900969 0.433884i 1.64572 + 0.792535i 0.623490 0.781831i 0.222521 0.974928i 1.82661 2.16285 0.222521 0.974928i 0.209800 + 0.263080i −0.222521 0.974928i
41.4 0.900969 0.433884i 2.24821 + 1.08268i 0.623490 0.781831i 0.222521 0.974928i 2.49532 −1.43324 0.222521 0.974928i 2.01177 + 2.52268i −0.222521 0.974928i
121.1 0.222521 0.974928i −0.273153 1.19676i −0.900969 0.433884i −0.623490 + 0.781831i −1.22754 −4.39678 −0.623490 + 0.781831i 1.34528 0.647852i 0.623490 + 0.781831i
121.2 0.222521 0.974928i −0.167393 0.733397i −0.900969 0.433884i −0.623490 + 0.781831i −0.752258 2.76599 −0.623490 + 0.781831i 2.19306 1.05612i 0.623490 + 0.781831i
121.3 0.222521 0.974928i 0.187922 + 0.823340i −0.900969 0.433884i −0.623490 + 0.781831i 0.844514 0.703699 −0.623490 + 0.781831i 2.06033 0.992204i 0.623490 + 0.781831i
121.4 0.222521 0.974928i 0.697666 + 3.05668i −0.900969 0.433884i −0.623490 + 0.781831i 3.13528 −2.56686 −0.623490 + 0.781831i −6.15362 + 2.96343i 0.623490 + 0.781831i
231.1 0.222521 + 0.974928i −0.273153 + 1.19676i −0.900969 + 0.433884i −0.623490 0.781831i −1.22754 −4.39678 −0.623490 0.781831i 1.34528 + 0.647852i 0.623490 0.781831i
231.2 0.222521 + 0.974928i −0.167393 + 0.733397i −0.900969 + 0.433884i −0.623490 0.781831i −0.752258 2.76599 −0.623490 0.781831i 2.19306 + 1.05612i 0.623490 0.781831i
231.3 0.222521 + 0.974928i 0.187922 0.823340i −0.900969 + 0.433884i −0.623490 0.781831i 0.844514 0.703699 −0.623490 0.781831i 2.06033 + 0.992204i 0.623490 0.781831i
231.4 0.222521 + 0.974928i 0.697666 3.05668i −0.900969 + 0.433884i −0.623490 0.781831i 3.13528 −2.56686 −0.623490 0.781831i −6.15362 2.96343i 0.623490 0.781831i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.k.d 24
43.e even 7 1 inner 430.2.k.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.d 24 1.a even 1 1 trivial
430.2.k.d 24 43.e even 7 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database