# Properties

 Label 690.2.i.f Level $690$ Weight $2$ Character orbit 690.i Analytic conductor $5.510$ Analytic rank $0$ Dimension $32$ 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.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{3} + 12q^{6} + 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{3} + 12q^{6} + 8q^{7} - 4q^{12} - 12q^{15} - 32q^{16} + 8q^{18} - 40q^{22} + 32q^{25} + 4q^{27} + 8q^{28} - 20q^{30} + 8q^{31} + 8q^{33} + 20q^{36} - 16q^{37} + 8q^{40} - 8q^{42} - 80q^{43} - 4q^{45} + 32q^{46} - 4q^{48} + 36q^{51} + 12q^{57} - 16q^{58} - 4q^{60} + 8q^{61} + 44q^{63} + 52q^{66} + 64q^{67} + 64q^{70} - 8q^{72} - 56q^{73} - 68q^{75} - 8q^{76} + 60q^{78} - 44q^{81} - 48q^{85} - 60q^{87} - 40q^{88} - 64q^{90} + 40q^{91} + 92q^{93} - 12q^{96} - 40q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −0.707107 + 0.707107i −1.68613 0.396187i 1.00000i 2.10894 0.743206i 1.47242 0.912127i −2.17056 2.17056i 0.707107 + 0.707107i 2.68607 + 1.33605i −0.965722 + 2.01677i
47.2 −0.707107 + 0.707107i −1.51580 + 0.838057i 1.00000i −2.23509 0.0661068i 0.479239 1.66443i 2.39575 + 2.39575i 0.707107 + 0.707107i 1.59532 2.54066i 1.62719 1.53370i
47.3 −0.707107 + 0.707107i −1.16628 1.28054i 1.00000i −1.31868 1.80585i 1.73017 + 0.0807905i 1.43599 + 1.43599i 0.707107 + 0.707107i −0.279562 + 2.98695i 2.20937 + 0.344476i
47.4 −0.707107 + 0.707107i −0.144577 1.72601i 1.00000i 1.25275 + 1.85219i 1.32270 + 1.11824i −2.31531 2.31531i 0.707107 + 0.707107i −2.95819 + 0.499082i −2.19553 0.423868i
47.5 −0.707107 + 0.707107i −0.0456124 1.73145i 1.00000i 1.54063 1.62063i 1.25657 + 1.19207i 0.528026 + 0.528026i 0.707107 + 0.707107i −2.99584 + 0.157951i 0.0565714 + 2.23535i
47.6 −0.707107 + 0.707107i 0.906765 + 1.47573i 1.00000i −1.96911 1.05954i −1.68468 0.402318i −0.621187 0.621187i 0.707107 + 0.707107i −1.35555 + 2.67628i 2.14157 0.643162i
47.7 −0.707107 + 0.707107i 1.12660 1.31559i 1.00000i −0.192443 + 2.22777i 0.133641 + 1.72689i 1.96010 + 1.96010i 0.707107 + 0.707107i −0.461565 2.96428i −1.43919 1.71135i
47.8 −0.707107 + 0.707107i 1.40373 + 1.01467i 1.00000i 2.22721 0.198845i −1.71006 + 0.275105i 0.787190 + 0.787190i 0.707107 + 0.707107i 0.940895 + 2.84863i −1.43427 + 1.71548i
47.9 0.707107 0.707107i −1.47573 0.906765i 1.00000i 1.96911 + 1.05954i −1.68468 + 0.402318i −0.621187 0.621187i −0.707107 0.707107i 1.35555 + 2.67628i 2.14157 0.643162i
47.10 0.707107 0.707107i −1.01467 1.40373i 1.00000i −2.22721 + 0.198845i −1.71006 0.275105i 0.787190 + 0.787190i −0.707107 0.707107i −0.940895 + 2.84863i −1.43427 + 1.71548i
47.11 0.707107 0.707107i −0.838057 + 1.51580i 1.00000i 2.23509 + 0.0661068i 0.479239 + 1.66443i 2.39575 + 2.39575i −0.707107 0.707107i −1.59532 2.54066i 1.62719 1.53370i
47.12 0.707107 0.707107i 0.396187 + 1.68613i 1.00000i −2.10894 + 0.743206i 1.47242 + 0.912127i −2.17056 2.17056i −0.707107 0.707107i −2.68607 + 1.33605i −0.965722 + 2.01677i
47.13 0.707107 0.707107i 1.28054 + 1.16628i 1.00000i 1.31868 + 1.80585i 1.73017 0.0807905i 1.43599 + 1.43599i −0.707107 0.707107i 0.279562 + 2.98695i 2.20937 + 0.344476i
47.14 0.707107 0.707107i 1.31559 1.12660i 1.00000i 0.192443 2.22777i 0.133641 1.72689i 1.96010 + 1.96010i −0.707107 0.707107i 0.461565 2.96428i −1.43919 1.71135i
47.15 0.707107 0.707107i 1.72601 + 0.144577i 1.00000i −1.25275 1.85219i 1.32270 1.11824i −2.31531 2.31531i −0.707107 0.707107i 2.95819 + 0.499082i −2.19553 0.423868i
47.16 0.707107 0.707107i 1.73145 + 0.0456124i 1.00000i −1.54063 + 1.62063i 1.25657 1.19207i 0.528026 + 0.528026i −0.707107 0.707107i 2.99584 + 0.157951i 0.0565714 + 2.23535i
323.1 −0.707107 0.707107i −1.68613 + 0.396187i 1.00000i 2.10894 + 0.743206i 1.47242 + 0.912127i −2.17056 + 2.17056i 0.707107 0.707107i 2.68607 1.33605i −0.965722 2.01677i
323.2 −0.707107 0.707107i −1.51580 0.838057i 1.00000i −2.23509 + 0.0661068i 0.479239 + 1.66443i 2.39575 2.39575i 0.707107 0.707107i 1.59532 + 2.54066i 1.62719 + 1.53370i
323.3 −0.707107 0.707107i −1.16628 + 1.28054i 1.00000i −1.31868 + 1.80585i 1.73017 0.0807905i 1.43599 1.43599i 0.707107 0.707107i −0.279562 2.98695i 2.20937 0.344476i
323.4 −0.707107 0.707107i −0.144577 + 1.72601i 1.00000i 1.25275 1.85219i 1.32270 1.11824i −2.31531 + 2.31531i 0.707107 0.707107i −2.95819 0.499082i −2.19553 + 0.423868i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.i.f 32
3.b odd 2 1 inner 690.2.i.f 32
5.c odd 4 1 inner 690.2.i.f 32
15.e even 4 1 inner 690.2.i.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.i.f 32 1.a even 1 1 trivial
690.2.i.f 32 3.b odd 2 1 inner
690.2.i.f 32 5.c odd 4 1 inner
690.2.i.f 32 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.