# Properties

 Label 1470.2.d.h Level $1470$ Weight $2$ Character orbit 1470.d Analytic conductor $11.738$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{2} + 24q^{4} + 24q^{8} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{2} + 24q^{4} + 24q^{8} + 8q^{9} - 16q^{15} + 24q^{16} + 8q^{18} + 16q^{23} + 8q^{25} - 16q^{30} + 24q^{32} + 8q^{36} + 16q^{39} + 16q^{46} + 8q^{50} + 16q^{51} - 16q^{53} - 16q^{57} - 16q^{60} + 24q^{64} + 48q^{65} + 8q^{72} + 16q^{78} - 48q^{79} - 24q^{81} + 16q^{85} + 16q^{92} - 64q^{93} + 112q^{95} - 64q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1469.1 1.00000 −1.71521 0.240923i 1.00000 −0.263182 + 2.22053i −1.71521 0.240923i 0 1.00000 2.88391 + 0.826470i −0.263182 + 2.22053i
1469.2 1.00000 −1.71521 + 0.240923i 1.00000 −0.263182 2.22053i −1.71521 + 0.240923i 0 1.00000 2.88391 0.826470i −0.263182 2.22053i
1469.3 1.00000 −1.67905 0.425213i 1.00000 2.18793 + 0.461477i −1.67905 0.425213i 0 1.00000 2.63839 + 1.42790i 2.18793 + 0.461477i
1469.4 1.00000 −1.67905 + 0.425213i 1.00000 2.18793 0.461477i −1.67905 + 0.425213i 0 1.00000 2.63839 1.42790i 2.18793 0.461477i
1469.5 1.00000 −1.29908 1.14560i 1.00000 1.07275 1.96194i −1.29908 1.14560i 0 1.00000 0.375199 + 2.97645i 1.07275 1.96194i
1469.6 1.00000 −1.29908 + 1.14560i 1.00000 1.07275 + 1.96194i −1.29908 + 1.14560i 0 1.00000 0.375199 2.97645i 1.07275 + 1.96194i
1469.7 1.00000 −1.17052 1.27667i 1.00000 −1.07909 1.95846i −1.17052 1.27667i 0 1.00000 −0.259782 + 2.98873i −1.07909 1.95846i
1469.8 1.00000 −1.17052 + 1.27667i 1.00000 −1.07909 + 1.95846i −1.17052 + 1.27667i 0 1.00000 −0.259782 2.98873i −1.07909 + 1.95846i
1469.9 1.00000 −1.03805 1.38652i 1.00000 −1.98784 + 1.02395i −1.03805 1.38652i 0 1.00000 −0.844897 + 2.87857i −1.98784 + 1.02395i
1469.10 1.00000 −1.03805 + 1.38652i 1.00000 −1.98784 1.02395i −1.03805 + 1.38652i 0 1.00000 −0.844897 2.87857i −1.98784 1.02395i
1469.11 1.00000 −0.321854 1.70188i 1.00000 2.20838 + 0.350775i −0.321854 1.70188i 0 1.00000 −2.79282 + 1.09552i 2.20838 + 0.350775i
1469.12 1.00000 −0.321854 + 1.70188i 1.00000 2.20838 0.350775i −0.321854 + 1.70188i 0 1.00000 −2.79282 1.09552i 2.20838 0.350775i
1469.13 1.00000 0.321854 1.70188i 1.00000 −2.20838 + 0.350775i 0.321854 1.70188i 0 1.00000 −2.79282 1.09552i −2.20838 + 0.350775i
1469.14 1.00000 0.321854 + 1.70188i 1.00000 −2.20838 0.350775i 0.321854 + 1.70188i 0 1.00000 −2.79282 + 1.09552i −2.20838 0.350775i
1469.15 1.00000 1.03805 1.38652i 1.00000 1.98784 + 1.02395i 1.03805 1.38652i 0 1.00000 −0.844897 2.87857i 1.98784 + 1.02395i
1469.16 1.00000 1.03805 + 1.38652i 1.00000 1.98784 1.02395i 1.03805 + 1.38652i 0 1.00000 −0.844897 + 2.87857i 1.98784 1.02395i
1469.17 1.00000 1.17052 1.27667i 1.00000 1.07909 1.95846i 1.17052 1.27667i 0 1.00000 −0.259782 2.98873i 1.07909 1.95846i
1469.18 1.00000 1.17052 + 1.27667i 1.00000 1.07909 + 1.95846i 1.17052 + 1.27667i 0 1.00000 −0.259782 + 2.98873i 1.07909 + 1.95846i
1469.19 1.00000 1.29908 1.14560i 1.00000 −1.07275 1.96194i 1.29908 1.14560i 0 1.00000 0.375199 2.97645i −1.07275 1.96194i
1469.20 1.00000 1.29908 + 1.14560i 1.00000 −1.07275 + 1.96194i 1.29908 + 1.14560i 0 1.00000 0.375199 + 2.97645i −1.07275 + 1.96194i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1469.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.d.h yes 24
3.b odd 2 1 1470.2.d.g 24
5.b even 2 1 1470.2.d.g 24
7.b odd 2 1 inner 1470.2.d.h yes 24
15.d odd 2 1 inner 1470.2.d.h yes 24
21.c even 2 1 1470.2.d.g 24
35.c odd 2 1 1470.2.d.g 24
105.g even 2 1 inner 1470.2.d.h yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.d.g 24 3.b odd 2 1
1470.2.d.g 24 5.b even 2 1
1470.2.d.g 24 21.c even 2 1
1470.2.d.g 24 35.c odd 2 1
1470.2.d.h yes 24 1.a even 1 1 trivial
1470.2.d.h yes 24 7.b odd 2 1 inner
1470.2.d.h yes 24 15.d odd 2 1 inner
1470.2.d.h yes 24 105.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{12} + 92 T_{11}^{10} + 3066 T_{11}^{8} + 44976 T_{11}^{6} + 266528 T_{11}^{4} + 330880 T_{11}^{2} + 4384$$ $$T_{13}^{12} - 108 T_{13}^{10} + 4204 T_{13}^{8} - 71072 T_{13}^{6} + 486448 T_{13}^{4} - 900544 T_{13}^{2} + 429632$$ $$T_{23}^{6} - 4 T_{23}^{5} - 62 T_{23}^{4} + 96 T_{23}^{3} + 1192 T_{23}^{2} + 1024 T_{23} - 1264$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database