# Properties

 Label 688.2.bg.c Level $688$ Weight $2$ Character orbit 688.bg Analytic conductor $5.494$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$688 = 2^{4} \cdot 43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 688.bg (of order $$21$$, degree $$12$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.49370765906$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$3$$ over $$\Q(\zeta_{21})$$ Twist minimal: no (minimal twist has level 43) Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 16q^{3} - 17q^{5} - 6q^{7} - q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 16q^{3} - 17q^{5} - 6q^{7} - q^{9} + 4q^{11} + 3q^{15} - 10q^{17} - 10q^{19} - 21q^{21} - 4q^{23} - 2q^{25} + 4q^{27} + 9q^{29} - 40q^{31} - 11q^{33} - 11q^{35} - 19q^{37} + q^{39} - 28q^{41} + 8q^{43} - 46q^{45} + 30q^{47} + 6q^{49} - 57q^{51} - 24q^{53} - 14q^{55} + 52q^{57} + q^{59} - 14q^{61} - 47q^{63} + 38q^{65} - 66q^{67} - 7q^{69} + 33q^{71} + 29q^{73} + 55q^{75} - 27q^{77} + 17q^{79} + 38q^{81} + 23q^{83} - 56q^{85} + 86q^{87} - 19q^{89} + 13q^{91} - 30q^{93} - q^{95} - 31q^{97} + 31q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −1.28860 + 0.397480i 0 −1.48781 + 3.79089i 0 −1.38920 + 2.40617i 0 −0.976224 + 0.665578i 0
17.2 0 0.711521 0.219475i 0 0.511972 1.30448i 0 2.37928 4.12103i 0 −2.02062 + 1.37764i 0
17.3 0 1.63701 0.504949i 0 0.140805 0.358765i 0 −1.74586 + 3.02391i 0 −0.0539035 + 0.0367508i 0
81.1 0 −1.28860 0.397480i 0 −1.48781 3.79089i 0 −1.38920 2.40617i 0 −0.976224 0.665578i 0
81.2 0 0.711521 + 0.219475i 0 0.511972 + 1.30448i 0 2.37928 + 4.12103i 0 −2.02062 1.37764i 0
81.3 0 1.63701 + 0.504949i 0 0.140805 + 0.358765i 0 −1.74586 3.02391i 0 −0.0539035 0.0367508i 0
225.1 0 0.0129300 + 0.0119973i 0 −3.39819 + 0.512194i 0 0.134521 + 0.232998i 0 −0.224167 2.99130i 0
225.2 0 0.240184 + 0.222858i 0 0.0260188 0.00392170i 0 −1.56464 2.71003i 0 −0.216168 2.88456i 0
225.3 0 1.40948 + 1.30780i 0 2.95419 0.445272i 0 0.339884 + 0.588696i 0 0.0520854 + 0.695032i 0
273.1 0 −0.922165 + 2.34964i 0 −0.0373507 0.498411i 0 1.65334 2.86367i 0 −2.47126 2.29299i 0
273.2 0 0.528255 1.34597i 0 0.0684907 + 0.913945i 0 0.971539 1.68276i 0 0.666571 + 0.618487i 0
273.3 0 0.863608 2.20044i 0 −0.284956 3.80248i 0 −1.30981 + 2.26866i 0 −1.89695 1.76011i 0
289.1 0 −1.09131 + 0.744044i 0 −1.30537 1.21121i 0 0.00749281 + 0.0129779i 0 −0.458662 + 1.16865i 0
289.2 0 1.90223 1.29692i 0 2.37710 + 2.20563i 0 1.38418 + 2.39748i 0 0.840455 2.14144i 0
289.3 0 2.41097 1.64377i 0 −2.00127 1.85691i 0 −1.01083 1.75082i 0 2.01476 5.13352i 0
353.1 0 −2.66973 0.402397i 0 −2.09843 1.43068i 0 1.09365 1.89425i 0 4.09883 + 1.26432i 0
353.2 0 1.14296 + 0.172273i 0 −1.45700 0.993363i 0 0.297283 0.514909i 0 −1.59004 0.490464i 0
353.3 0 2.93359 + 0.442167i 0 −0.492700 0.335917i 0 −2.18154 + 3.77854i 0 5.54371 + 1.71001i 0
369.1 0 −1.09131 0.744044i 0 −1.30537 + 1.21121i 0 0.00749281 0.0129779i 0 −0.458662 1.16865i 0
369.2 0 1.90223 + 1.29692i 0 2.37710 2.20563i 0 1.38418 2.39748i 0 0.840455 + 2.14144i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 625.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.bg.c 36
4.b odd 2 1 43.2.g.a 36
12.b even 2 1 387.2.y.c 36
43.g even 21 1 inner 688.2.bg.c 36
172.o odd 42 1 43.2.g.a 36
172.o odd 42 1 1849.2.a.n 18
172.p even 42 1 1849.2.a.o 18
516.bb even 42 1 387.2.y.c 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.g.a 36 4.b odd 2 1
43.2.g.a 36 172.o odd 42 1
387.2.y.c 36 12.b even 2 1
387.2.y.c 36 516.bb even 42 1
688.2.bg.c 36 1.a even 1 1 trivial
688.2.bg.c 36 43.g even 21 1 inner
1849.2.a.n 18 172.o odd 42 1
1849.2.a.o 18 172.p even 42 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{36} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(688, [\chi])$$.