# Properties

 Label 1900.3.g.d Level $1900$ Weight $3$ Character orbit 1900.g Analytic conductor $51.771$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1900.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.7712502285$$ Analytic rank: $$0$$ Dimension: $$28$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 104q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 104q^{9} + 8q^{11} + 58q^{19} + 112q^{39} - 276q^{49} - 100q^{61} + 132q^{81} - 184q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1 0 −5.26531 0 0 0 12.1735i 0 18.7235 0
949.2 0 −5.26531 0 0 0 12.1735i 0 18.7235 0
949.3 0 −4.84257 0 0 0 6.85668i 0 14.4504 0
949.4 0 −4.84257 0 0 0 6.85668i 0 14.4504 0
949.5 0 −4.58743 0 0 0 1.52935i 0 12.0445 0
949.6 0 −4.58743 0 0 0 1.52935i 0 12.0445 0
949.7 0 −2.93221 0 0 0 5.62705i 0 −0.402160 0
949.8 0 −2.93221 0 0 0 5.62705i 0 −0.402160 0
949.9 0 −2.16913 0 0 0 8.18099i 0 −4.29489 0
949.10 0 −2.16913 0 0 0 8.18099i 0 −4.29489 0
949.11 0 −1.84332 0 0 0 10.5375i 0 −5.60216 0
949.12 0 −1.84332 0 0 0 10.5375i 0 −5.60216 0
949.13 0 −0.284180 0 0 0 2.19606i 0 −8.91924 0
949.14 0 −0.284180 0 0 0 2.19606i 0 −8.91924 0
949.15 0 0.284180 0 0 0 2.19606i 0 −8.91924 0
949.16 0 0.284180 0 0 0 2.19606i 0 −8.91924 0
949.17 0 1.84332 0 0 0 10.5375i 0 −5.60216 0
949.18 0 1.84332 0 0 0 10.5375i 0 −5.60216 0
949.19 0 2.16913 0 0 0 8.18099i 0 −4.29489 0
949.20 0 2.16913 0 0 0 8.18099i 0 −4.29489 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 949.28 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.b even 2 1 inner
19.b odd 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.3.g.d 28
5.b even 2 1 inner 1900.3.g.d 28
5.c odd 4 1 1900.3.e.g 14
5.c odd 4 1 1900.3.e.h yes 14
19.b odd 2 1 inner 1900.3.g.d 28
95.d odd 2 1 inner 1900.3.g.d 28
95.g even 4 1 1900.3.e.g 14
95.g even 4 1 1900.3.e.h yes 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.3.e.g 14 5.c odd 4 1
1900.3.e.g 14 95.g even 4 1
1900.3.e.h yes 14 5.c odd 4 1
1900.3.e.h yes 14 95.g even 4 1
1900.3.g.d 28 1.a even 1 1 trivial
1900.3.g.d 28 5.b even 2 1 inner
1900.3.g.d 28 19.b odd 2 1 inner
1900.3.g.d 28 95.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} - 89 T_{3}^{12} + 3026 T_{3}^{10} - 49092 T_{3}^{8} + 390297 T_{3}^{6} - 1440495 T_{3}^{4} + 1994425 T_{3}^{2} - 151875$$ acting on $$S_{3}^{\mathrm{new}}(1900, [\chi])$$.