# Properties

 Label 690.2.i.e Level $690$ Weight $2$ Character orbit 690.i Analytic conductor $5.510$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

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## 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{3} - 4q^{6} - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{3} - 4q^{6} - 8q^{7} - 8q^{10} - 4q^{12} - 4q^{15} - 32q^{16} + 8q^{18} - 32q^{21} - 8q^{22} + 4q^{27} - 8q^{28} + 20q^{30} - 24q^{31} + 20q^{36} - 32q^{37} - 16q^{40} + 8q^{42} + 144q^{43} + 36q^{45} - 32q^{46} - 4q^{48} + 12q^{51} - 64q^{55} + 52q^{57} + 16q^{58} + 4q^{60} - 24q^{61} - 116q^{63} + 12q^{66} - 16q^{67} - 80q^{70} - 8q^{72} + 40q^{73} + 44q^{75} + 24q^{76} - 36q^{78} - 108q^{81} - 32q^{82} - 80q^{85} + 68q^{87} - 8q^{88} + 16q^{90} + 120q^{91} + 12q^{93} + 4q^{96} - 8q^{97} + O(q^{100})$$

## 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 −1.42268 0.987925i 1.00000i 2.16582 + 0.556076i 1.70455 0.307415i 3.32202 + 3.32202i 0.707107 + 0.707107i 1.04801 + 2.81099i −1.92467 + 1.13826i
47.2 −0.707107 + 0.707107i −1.35428 + 1.07978i 1.00000i −1.85580 + 1.24740i 0.194102 1.72114i −2.75926 2.75926i 0.707107 + 0.707107i 0.668152 2.92465i 0.430209 2.19429i
47.3 −0.707107 + 0.707107i −0.639676 1.60960i 1.00000i −1.63199 + 1.52859i 1.59048 + 0.685841i 0.772441 + 0.772441i 0.707107 + 0.707107i −2.18163 + 2.05924i 0.0731134 2.23487i
47.4 −0.707107 + 0.707107i 0.451971 + 1.67204i 1.00000i 0.664484 + 2.13506i −1.50190 0.862720i 2.34139 + 2.34139i 0.707107 + 0.707107i −2.59144 + 1.51143i −1.97957 1.03985i
47.5 −0.707107 + 0.707107i 0.471935 1.66652i 1.00000i −1.55923 1.60274i 0.844697 + 1.51211i −3.44975 3.44975i 0.707107 + 0.707107i −2.55455 1.57298i 2.23586 + 0.0307665i
47.6 −0.707107 + 0.707107i 1.31856 + 1.12312i 1.00000i −1.11299 1.93940i −1.72653 + 0.138191i 1.56700 + 1.56700i 0.707107 + 0.707107i 0.477181 + 2.96181i 2.15836 + 0.584357i
47.7 −0.707107 + 0.707107i 1.40859 + 1.00790i 1.00000i −0.318132 + 2.21332i −1.70872 + 0.283328i −2.49652 2.49652i 0.707107 + 0.707107i 0.968258 + 2.83945i −1.34010 1.79001i
47.8 −0.707107 + 0.707107i 1.47268 0.911701i 1.00000i 2.23363 + 0.104337i −0.396675 + 1.68602i −1.29732 1.29732i 0.707107 + 0.707107i 1.33760 2.68530i −1.65319 + 1.50564i
47.9 0.707107 0.707107i −1.67204 0.451971i 1.00000i −0.664484 2.13506i −1.50190 + 0.862720i 2.34139 + 2.34139i −0.707107 0.707107i 2.59144 + 1.51143i −1.97957 1.03985i
47.10 0.707107 0.707107i −1.12312 1.31856i 1.00000i 1.11299 + 1.93940i −1.72653 0.138191i 1.56700 + 1.56700i −0.707107 0.707107i −0.477181 + 2.96181i 2.15836 + 0.584357i
47.11 0.707107 0.707107i −1.07978 + 1.35428i 1.00000i 1.85580 1.24740i 0.194102 + 1.72114i −2.75926 2.75926i −0.707107 0.707107i −0.668152 2.92465i 0.430209 2.19429i
47.12 0.707107 0.707107i −1.00790 1.40859i 1.00000i 0.318132 2.21332i −1.70872 0.283328i −2.49652 2.49652i −0.707107 0.707107i −0.968258 + 2.83945i −1.34010 1.79001i
47.13 0.707107 0.707107i 0.911701 1.47268i 1.00000i −2.23363 0.104337i −0.396675 1.68602i −1.29732 1.29732i −0.707107 0.707107i −1.33760 2.68530i −1.65319 + 1.50564i
47.14 0.707107 0.707107i 0.987925 + 1.42268i 1.00000i −2.16582 0.556076i 1.70455 + 0.307415i 3.32202 + 3.32202i −0.707107 0.707107i −1.04801 + 2.81099i −1.92467 + 1.13826i
47.15 0.707107 0.707107i 1.60960 + 0.639676i 1.00000i 1.63199 1.52859i 1.59048 0.685841i 0.772441 + 0.772441i −0.707107 0.707107i 2.18163 + 2.05924i 0.0731134 2.23487i
47.16 0.707107 0.707107i 1.66652 0.471935i 1.00000i 1.55923 + 1.60274i 0.844697 1.51211i −3.44975 3.44975i −0.707107 0.707107i 2.55455 1.57298i 2.23586 + 0.0307665i
323.1 −0.707107 0.707107i −1.42268 + 0.987925i 1.00000i 2.16582 0.556076i 1.70455 + 0.307415i 3.32202 3.32202i 0.707107 0.707107i 1.04801 2.81099i −1.92467 1.13826i
323.2 −0.707107 0.707107i −1.35428 1.07978i 1.00000i −1.85580 1.24740i 0.194102 + 1.72114i −2.75926 + 2.75926i 0.707107 0.707107i 0.668152 + 2.92465i 0.430209 + 2.19429i
323.3 −0.707107 0.707107i −0.639676 + 1.60960i 1.00000i −1.63199 1.52859i 1.59048 0.685841i 0.772441 0.772441i 0.707107 0.707107i −2.18163 2.05924i 0.0731134 + 2.23487i
323.4 −0.707107 0.707107i 0.451971 1.67204i 1.00000i 0.664484 2.13506i −1.50190 + 0.862720i 2.34139 2.34139i 0.707107 0.707107i −2.59144 1.51143i −1.97957 + 1.03985i
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.e 32
3.b odd 2 1 inner 690.2.i.e 32
5.c odd 4 1 inner 690.2.i.e 32
15.e even 4 1 inner 690.2.i.e 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.i.e 32 1.a even 1 1 trivial
690.2.i.e 32 3.b odd 2 1 inner
690.2.i.e 32 5.c odd 4 1 inner
690.2.i.e 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 $$S_{2}^{\mathrm{new}}(690, [\chi])$$.