# Properties

 Label 690.3.k.b Level $690$ Weight $3$ Character orbit 690.k Analytic conductor $18.801$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 690.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.8011382409$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 48q^{2} - 8q^{5} - 8q^{7} + 96q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 48q^{2} - 8q^{5} - 8q^{7} + 96q^{8} + 8q^{10} - 32q^{11} - 24q^{13} + 24q^{15} - 192q^{16} + 72q^{17} - 144q^{18} + 32q^{22} + 24q^{25} + 48q^{26} + 16q^{28} - 24q^{30} + 24q^{31} + 192q^{32} - 24q^{33} + 288q^{36} - 128q^{37} - 16q^{38} - 16q^{40} - 40q^{41} + 48q^{43} - 136q^{47} - 80q^{50} - 48q^{52} + 144q^{53} - 144q^{55} - 32q^{56} + 96q^{57} + 8q^{58} + 128q^{61} - 24q^{62} - 24q^{63} + 184q^{65} + 48q^{66} - 144q^{68} + 40q^{70} - 40q^{71} - 288q^{72} + 40q^{73} - 72q^{75} + 32q^{76} - 104q^{77} + 96q^{78} + 32q^{80} - 432q^{81} + 40q^{82} - 88q^{85} - 96q^{86} + 120q^{87} - 64q^{88} + 24q^{90} + 144q^{91} - 96q^{93} + 312q^{95} + 480q^{97} + 584q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 4.77937 1.46889i 2.44949 8.57539 + 8.57539i 2.00000 2.00000i 3.00000i −6.24826 3.31048i
277.2 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −4.35673 + 2.45335i 2.44949 −3.40648 3.40648i 2.00000 2.00000i 3.00000i 6.81008 + 1.90338i
277.3 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −2.46815 + 4.34836i 2.44949 −1.62328 1.62328i 2.00000 2.00000i 3.00000i 6.81651 1.88022i
277.4 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 3.38334 + 3.68144i 2.44949 −2.06971 2.06971i 2.00000 2.00000i 3.00000i 0.298094 7.06478i
277.5 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 0.0428394 4.99982i 2.44949 0.599576 + 0.599576i 2.00000 2.00000i 3.00000i −5.04266 + 4.95698i
277.6 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −3.65385 3.41312i 2.44949 0.283627 + 0.283627i 2.00000 2.00000i 3.00000i 0.240725 + 7.06697i
277.7 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 4.29808 2.55470i 2.44949 4.10164 + 4.10164i 2.00000 2.00000i 3.00000i −6.85278 1.74338i
277.8 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −2.55939 + 4.29529i 2.44949 5.47277 + 5.47277i 2.00000 2.00000i 3.00000i 6.85468 1.73590i
277.9 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 4.00430 + 2.99426i 2.44949 −4.65996 4.65996i 2.00000 2.00000i 3.00000i −1.01004 6.99856i
277.10 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −4.46596 2.24837i 2.44949 −8.35401 8.35401i 2.00000 2.00000i 3.00000i 2.21759 + 6.71434i
277.11 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i −4.94131 0.763816i 2.44949 8.62845 + 8.62845i 2.00000 2.00000i 3.00000i 4.17750 + 5.70513i
277.12 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 1.48797 4.77346i 2.44949 −9.54800 9.54800i 2.00000 2.00000i 3.00000i −6.26143 + 3.28550i
277.13 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −4.51369 + 2.15095i −2.44949 −8.14300 8.14300i 2.00000 2.00000i 3.00000i 6.66464 + 2.36274i
277.14 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −1.86672 + 4.63846i −2.44949 2.71586 + 2.71586i 2.00000 2.00000i 3.00000i 6.50519 2.77174i
277.15 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −4.84935 + 1.21811i −2.44949 −0.117427 0.117427i 2.00000 2.00000i 3.00000i 6.06746 + 3.63124i
277.16 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 3.93589 3.08362i −2.44949 −1.42240 1.42240i 2.00000 2.00000i 3.00000i −7.01952 0.852269i
277.17 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 2.08074 4.54648i −2.44949 −1.45838 1.45838i 2.00000 2.00000i 3.00000i −6.62723 + 2.46574i
277.18 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −2.00232 4.58156i −2.44949 −5.25476 5.25476i 2.00000 2.00000i 3.00000i −2.57924 + 6.58388i
277.19 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 0.815565 4.93304i −2.44949 5.39766 + 5.39766i 2.00000 2.00000i 3.00000i −5.74860 + 4.11747i
277.20 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −4.82966 1.29396i −2.44949 5.84802 + 5.84802i 2.00000 2.00000i 3.00000i 3.53570 + 6.12363i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 553.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.3.k.b 48
5.c odd 4 1 inner 690.3.k.b 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.3.k.b 48 1.a even 1 1 trivial
690.3.k.b 48 5.c odd 4 1 inner