# Properties

 Label 690.2.h.b Level $690$ Weight $2$ Character orbit 690.h Analytic conductor $5.510$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{2} - 2q^{3} + 24q^{4} - 2q^{6} + 24q^{8} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{2} - 2q^{3} + 24q^{4} - 2q^{6} + 24q^{8} + 6q^{9} - 2q^{12} + 24q^{16} + 6q^{18} + 4q^{23} - 2q^{24} + 12q^{25} - 2q^{27} - 28q^{31} + 24q^{32} - 8q^{35} + 6q^{36} + 4q^{46} - 16q^{47} - 2q^{48} - 4q^{49} + 12q^{50} - 2q^{54} + 4q^{55} - 28q^{62} + 24q^{64} - 8q^{69} - 8q^{70} + 6q^{72} - 6q^{75} - 8q^{77} + 14q^{81} - 44q^{85} - 28q^{87} + 4q^{92} + 4q^{93} - 16q^{94} - 4q^{95} - 2q^{96} - 4q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
689.1 1.00000 −1.68494 0.401202i 1.00000 −1.83064 + 1.28404i −1.68494 0.401202i −1.73603 1.00000 2.67807 + 1.35201i −1.83064 + 1.28404i
689.2 1.00000 −1.68494 0.401202i 1.00000 1.83064 1.28404i −1.68494 0.401202i 1.73603 1.00000 2.67807 + 1.35201i 1.83064 1.28404i
689.3 1.00000 −1.68494 + 0.401202i 1.00000 −1.83064 1.28404i −1.68494 + 0.401202i −1.73603 1.00000 2.67807 1.35201i −1.83064 1.28404i
689.4 1.00000 −1.68494 + 0.401202i 1.00000 1.83064 + 1.28404i −1.68494 + 0.401202i 1.73603 1.00000 2.67807 1.35201i 1.83064 + 1.28404i
689.5 1.00000 −1.44698 0.951967i 1.00000 −1.42959 1.71938i −1.44698 0.951967i 4.73682 1.00000 1.18752 + 2.75496i −1.42959 1.71938i
689.6 1.00000 −1.44698 0.951967i 1.00000 1.42959 + 1.71938i −1.44698 0.951967i −4.73682 1.00000 1.18752 + 2.75496i 1.42959 + 1.71938i
689.7 1.00000 −1.44698 + 0.951967i 1.00000 −1.42959 + 1.71938i −1.44698 + 0.951967i 4.73682 1.00000 1.18752 2.75496i −1.42959 + 1.71938i
689.8 1.00000 −1.44698 + 0.951967i 1.00000 1.42959 1.71938i −1.44698 + 0.951967i −4.73682 1.00000 1.18752 2.75496i 1.42959 1.71938i
689.9 1.00000 −0.571841 1.63493i 1.00000 −2.12720 + 0.689226i −0.571841 1.63493i −1.28874 1.00000 −2.34600 + 1.86984i −2.12720 + 0.689226i
689.10 1.00000 −0.571841 1.63493i 1.00000 2.12720 0.689226i −0.571841 1.63493i 1.28874 1.00000 −2.34600 + 1.86984i 2.12720 0.689226i
689.11 1.00000 −0.571841 + 1.63493i 1.00000 −2.12720 0.689226i −0.571841 + 1.63493i −1.28874 1.00000 −2.34600 1.86984i −2.12720 0.689226i
689.12 1.00000 −0.571841 + 1.63493i 1.00000 2.12720 + 0.689226i −0.571841 + 1.63493i 1.28874 1.00000 −2.34600 1.86984i 2.12720 + 0.689226i
689.13 1.00000 0.250553 1.71383i 1.00000 −0.545034 2.16863i 0.250553 1.71383i −1.86244 1.00000 −2.87445 0.858813i −0.545034 2.16863i
689.14 1.00000 0.250553 1.71383i 1.00000 0.545034 + 2.16863i 0.250553 1.71383i 1.86244 1.00000 −2.87445 0.858813i 0.545034 + 2.16863i
689.15 1.00000 0.250553 + 1.71383i 1.00000 −0.545034 + 2.16863i 0.250553 + 1.71383i −1.86244 1.00000 −2.87445 + 0.858813i −0.545034 + 2.16863i
689.16 1.00000 0.250553 + 1.71383i 1.00000 0.545034 2.16863i 0.250553 + 1.71383i 1.86244 1.00000 −2.87445 + 0.858813i 0.545034 2.16863i
689.17 1.00000 1.29400 1.15133i 1.00000 −2.22484 0.223804i 1.29400 1.15133i −0.666856 1.00000 0.348885 2.97964i −2.22484 0.223804i
689.18 1.00000 1.29400 1.15133i 1.00000 2.22484 + 0.223804i 1.29400 1.15133i 0.666856 1.00000 0.348885 2.97964i 2.22484 + 0.223804i
689.19 1.00000 1.29400 + 1.15133i 1.00000 −2.22484 + 0.223804i 1.29400 + 1.15133i −0.666856 1.00000 0.348885 + 2.97964i −2.22484 + 0.223804i
689.20 1.00000 1.29400 + 1.15133i 1.00000 2.22484 0.223804i 1.29400 + 1.15133i 0.666856 1.00000 0.348885 + 2.97964i 2.22484 0.223804i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 689.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
23.b odd 2 1 inner
345.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.h.b yes 24
3.b odd 2 1 690.2.h.a 24
5.b even 2 1 690.2.h.a 24
15.d odd 2 1 inner 690.2.h.b yes 24
23.b odd 2 1 inner 690.2.h.b yes 24
69.c even 2 1 690.2.h.a 24
115.c odd 2 1 690.2.h.a 24
345.h even 2 1 inner 690.2.h.b yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.h.a 24 3.b odd 2 1
690.2.h.a 24 5.b even 2 1
690.2.h.a 24 69.c even 2 1
690.2.h.a 24 115.c odd 2 1
690.2.h.b yes 24 1.a even 1 1 trivial
690.2.h.b yes 24 15.d odd 2 1 inner
690.2.h.b yes 24 23.b odd 2 1 inner
690.2.h.b yes 24 345.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{47}^{6} + 4 T_{47}^{5} - 132 T_{47}^{4} - 472 T_{47}^{3} + 2368 T_{47}^{2} + 12288 T_{47} + 13824$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.