# Properties

 Label 430.2.q.d Level 430 Weight 2 Character orbit 430.q Analytic conductor 3.434 Analytic rank 0 Dimension 60 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.q (of order $$21$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$5$$ over $$\Q(\zeta_{21})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$60q + 10q^{2} - q^{3} - 10q^{4} - 5q^{5} - 6q^{6} + q^{7} + 10q^{8} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q + 10q^{2} - q^{3} - 10q^{4} - 5q^{5} - 6q^{6} + q^{7} + 10q^{8} + 8q^{9} + 5q^{10} + 3q^{11} - q^{12} + 13q^{13} - 15q^{14} + q^{15} - 10q^{16} + 38q^{17} + 6q^{18} + 17q^{19} - 5q^{20} + 4q^{21} + 11q^{22} - 8q^{23} + q^{24} + 5q^{25} + 29q^{26} - 16q^{27} - 6q^{28} - 41q^{29} - q^{30} + 15q^{31} + 10q^{32} + 63q^{33} - 10q^{34} + 2q^{35} - 48q^{36} + 7q^{37} + 25q^{38} - 64q^{39} + 5q^{40} - 22q^{41} - 60q^{42} + 6q^{43} - 4q^{44} + 30q^{45} - 20q^{46} - 13q^{47} - q^{48} - 93q^{49} + 30q^{50} + 18q^{51} - 15q^{52} + 71q^{53} - 5q^{54} - 16q^{55} - q^{56} - 94q^{57} - 22q^{58} + 20q^{59} + q^{60} + 91q^{61} + 69q^{62} - 50q^{63} - 10q^{64} + 5q^{65} + 14q^{66} - 26q^{67} - 18q^{68} - 94q^{69} - 2q^{70} + 20q^{71} - 36q^{72} - 56q^{73} - 7q^{74} + 2q^{75} + 38q^{76} + 41q^{77} - 13q^{78} - 46q^{79} + 30q^{80} + 39q^{81} + 8q^{82} - 115q^{83} + 4q^{84} + 6q^{85} + 29q^{86} + 20q^{87} + 11q^{88} + 30q^{89} - 16q^{90} - 53q^{91} - 8q^{92} + 16q^{93} + 41q^{94} - 17q^{95} + q^{96} + 113q^{97} + 2q^{98} + 139q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 0.900969 0.433884i −2.40821 + 1.64189i 0.623490 0.781831i 0.733052 + 0.680173i −1.45733 + 2.52418i 0.522354 + 0.904743i 0.222521 0.974928i 2.00766 5.11543i 0.955573 + 0.294755i
31.2 0.900969 0.433884i −1.77519 + 1.21030i 0.623490 0.781831i 0.733052 + 0.680173i −1.07426 + 1.86067i −2.45123 4.24566i 0.222521 0.974928i 0.590433 1.50440i 0.955573 + 0.294755i
31.3 0.900969 0.433884i −0.206973 + 0.141112i 0.623490 0.781831i 0.733052 + 0.680173i −0.125250 + 0.216939i 0.851274 + 1.47445i 0.222521 0.974928i −1.07310 + 2.73421i 0.955573 + 0.294755i
31.4 0.900969 0.433884i 1.74260 1.18809i 0.623490 0.781831i 0.733052 + 0.680173i 1.05454 1.82652i −2.08827 3.61700i 0.222521 0.974928i 0.529092 1.34811i 0.955573 + 0.294755i
31.5 0.900969 0.433884i 1.82153 1.24190i 0.623490 0.781831i 0.733052 + 0.680173i 1.10230 1.90924i 1.71615 + 2.97246i 0.222521 0.974928i 0.679638 1.73169i 0.955573 + 0.294755i
81.1 0.222521 0.974928i −3.07767 0.949334i −0.900969 0.433884i −0.365341 0.930874i −1.61038 + 2.78926i 2.52339 + 4.37064i −0.623490 + 0.781831i 6.09208 + 4.15351i −0.988831 + 0.149042i
81.2 0.222521 0.974928i −2.45037 0.755840i −0.900969 0.433884i −0.365341 0.930874i −1.28215 + 2.22075i −2.38661 4.13373i −0.623490 + 0.781831i 2.95432 + 2.01422i −0.988831 + 0.149042i
81.3 0.222521 0.974928i −0.366458 0.113037i −0.900969 0.433884i −0.365341 0.930874i −0.191748 + 0.332117i −0.0855632 0.148200i −0.623490 + 0.781831i −2.35720 1.60711i −0.988831 + 0.149042i
81.4 0.222521 0.974928i 2.08443 + 0.642961i −0.900969 0.433884i −0.365341 0.930874i 1.09067 1.88910i 1.63836 + 2.83771i −0.623490 + 0.781831i 1.45273 + 0.990455i −0.988831 + 0.149042i
81.5 0.222521 0.974928i 2.85449 + 0.880495i −0.900969 0.433884i −0.365341 0.930874i 1.49360 2.58700i −1.74417 3.02100i −0.623490 + 0.781831i 4.89415 + 3.33678i −0.988831 + 0.149042i
101.1 −0.623490 + 0.781831i −1.04669 + 2.66692i −0.222521 0.974928i −0.0747301 0.997204i −1.43248 2.48113i 1.27853 2.21448i 0.900969 + 0.433884i −3.81773 3.54234i 0.826239 + 0.563320i
101.2 −0.623490 + 0.781831i −0.617972 + 1.57457i −0.222521 0.974928i −0.0747301 0.997204i −0.845746 1.46488i 0.932562 1.61524i 0.900969 + 0.433884i 0.101786 + 0.0944434i 0.826239 + 0.563320i
101.3 −0.623490 + 0.781831i −0.0969271 + 0.246966i −0.222521 0.974928i −0.0747301 0.997204i −0.132653 0.229762i −2.60514 + 4.51223i 0.900969 + 0.433884i 2.14756 + 1.99264i 0.826239 + 0.563320i
101.4 −0.623490 + 0.781831i 0.272539 0.694419i −0.222521 0.974928i −0.0747301 0.997204i 0.372993 + 0.646043i 0.186100 0.322335i 0.900969 + 0.433884i 1.79122 + 1.66201i 0.826239 + 0.563320i
101.5 −0.623490 + 0.781831i 1.12371 2.86316i −0.222521 0.974928i −0.0747301 0.997204i 1.53789 + 2.66370i 0.0651243 0.112799i 0.900969 + 0.433884i −4.73579 4.39418i 0.826239 + 0.563320i
111.1 0.900969 + 0.433884i −2.40821 1.64189i 0.623490 + 0.781831i 0.733052 0.680173i −1.45733 2.52418i 0.522354 0.904743i 0.222521 + 0.974928i 2.00766 + 5.11543i 0.955573 0.294755i
111.2 0.900969 + 0.433884i −1.77519 1.21030i 0.623490 + 0.781831i 0.733052 0.680173i −1.07426 1.86067i −2.45123 + 4.24566i 0.222521 + 0.974928i 0.590433 + 1.50440i 0.955573 0.294755i
111.3 0.900969 + 0.433884i −0.206973 0.141112i 0.623490 + 0.781831i 0.733052 0.680173i −0.125250 0.216939i 0.851274 1.47445i 0.222521 + 0.974928i −1.07310 2.73421i 0.955573 0.294755i
111.4 0.900969 + 0.433884i 1.74260 + 1.18809i 0.623490 + 0.781831i 0.733052 0.680173i 1.05454 + 1.82652i −2.08827 + 3.61700i 0.222521 + 0.974928i 0.529092 + 1.34811i 0.955573 0.294755i
111.5 0.900969 + 0.433884i 1.82153 + 1.24190i 0.623490 + 0.781831i 0.733052 0.680173i 1.10230 + 1.90924i 1.71615 2.97246i 0.222521 + 0.974928i 0.679638 + 1.73169i 0.955573 0.294755i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 411.5 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.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.q.d 60
43.g even 21 1 inner 430.2.q.d 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.q.d 60 1.a even 1 1 trivial
430.2.q.d 60 43.g even 21 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{60} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database