# Properties

 Label 690.3.k.a Level $690$ Weight $3$ Character orbit 690.k Analytic conductor $18.801$ Analytic rank $0$ Dimension $40$ 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: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 40q^{2} - 8q^{5} - 8q^{7} - 80q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 40q^{2} - 8q^{5} - 8q^{7} - 80q^{8} - 16q^{10} + 32q^{11} + 16q^{13} + 24q^{15} - 160q^{16} - 48q^{17} + 120q^{18} - 16q^{20} - 96q^{21} + 32q^{22} + 32q^{26} + 16q^{28} + 24q^{30} + 152q^{31} - 160q^{32} - 24q^{33} + 48q^{35} + 240q^{36} + 216q^{37} + 16q^{38} - 168q^{41} - 96q^{42} - 48q^{43} + 24q^{45} - 232q^{47} - 40q^{50} + 32q^{52} + 8q^{53} - 272q^{55} + 32q^{56} - 136q^{58} - 64q^{61} + 152q^{62} - 24q^{63} + 416q^{65} - 48q^{66} - 32q^{67} + 96q^{68} + 88q^{70} - 104q^{71} + 240q^{72} + 480q^{73} - 216q^{75} + 32q^{76} + 280q^{77} - 192q^{78} + 32q^{80} - 360q^{81} - 168q^{82} - 576q^{83} - 208q^{85} - 96q^{86} + 24q^{87} - 64q^{88} + 144q^{91} + 96q^{93} + 168q^{95} + 24q^{97} + 176q^{98} + O(q^{100})$$

## 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 −4.85719 1.18646i −2.44949 8.99906 + 8.99906i −2.00000 + 2.00000i 3.00000i −3.67074 6.04365i
277.2 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 2.69807 + 4.20956i −2.44949 −5.61863 5.61863i −2.00000 + 2.00000i 3.00000i −1.51149 + 6.90763i
277.3 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 3.78471 3.26741i −2.44949 3.72268 + 3.72268i −2.00000 + 2.00000i 3.00000i 7.05212 + 0.517301i
277.4 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −1.90173 + 4.62422i −2.44949 2.37572 + 2.37572i −2.00000 + 2.00000i 3.00000i −6.52595 + 2.72250i
277.5 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 0.0159424 4.99997i −2.44949 −0.165312 0.165312i −2.00000 + 2.00000i 3.00000i 5.01592 4.98403i
277.6 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 2.62353 + 4.25642i −2.44949 0.185981 + 0.185981i −2.00000 + 2.00000i 3.00000i −1.63289 + 6.87995i
277.7 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −4.90121 0.988992i −2.44949 2.14995 + 2.14995i −2.00000 + 2.00000i 3.00000i −3.91222 5.89021i
277.8 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −1.94861 4.60466i −2.44949 −2.18869 2.18869i −2.00000 + 2.00000i 3.00000i 2.65606 6.55327i
277.9 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 4.95981 + 0.632643i −2.44949 3.11187 + 3.11187i −2.00000 + 2.00000i 3.00000i 4.32717 + 5.59246i
277.10 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −4.92281 + 0.875159i −2.44949 −4.77467 4.77467i −2.00000 + 2.00000i 3.00000i −5.79797 4.04766i
277.11 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i −0.335340 + 4.98874i 2.44949 −8.75117 8.75117i −2.00000 + 2.00000i 3.00000i −5.32408 + 4.65340i
277.12 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i −4.34252 2.47842i 2.44949 −5.93628 5.93628i −2.00000 + 2.00000i 3.00000i −1.86410 6.82093i
277.13 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 3.72431 3.33609i 2.44949 6.04624 + 6.04624i −2.00000 + 2.00000i 3.00000i 7.06040 + 0.388220i
277.14 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 4.99686 0.177266i 2.44949 −4.63984 4.63984i −2.00000 + 2.00000i 3.00000i 5.17412 + 4.81959i
277.15 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 2.64969 4.24018i 2.44949 −5.13413 5.13413i −2.00000 + 2.00000i 3.00000i 6.88987 1.59048i
277.16 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i −3.81857 + 3.22778i 2.44949 3.99090 + 3.99090i −2.00000 + 2.00000i 3.00000i −7.04634 0.590789i
277.17 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 3.38409 + 3.68075i 2.44949 −0.957772 0.957772i −2.00000 + 2.00000i 3.00000i −0.296667 + 7.06484i
277.18 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i −2.41665 4.37719i 2.44949 0.381912 + 0.381912i −2.00000 + 2.00000i 3.00000i 1.96054 6.79384i
277.19 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i −4.45244 + 2.27502i 2.44949 −3.57605 3.57605i −2.00000 + 2.00000i 3.00000i −6.72747 2.17742i
277.20 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i 1.06006 + 4.88634i 2.44949 6.77823 + 6.77823i −2.00000 + 2.00000i 3.00000i −3.82628 + 5.94639i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 553.20 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.a 40
5.c odd 4 1 inner 690.3.k.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.3.k.a 40 1.a even 1 1 trivial
690.3.k.a 40 5.c odd 4 1 inner