# Properties

 Label 684.3.t.a Level $684$ Weight $3$ Character orbit 684.t Analytic conductor $18.638$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 2 q^{7} + 4 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80 q + 2 q^{7} + 4 q^{9} + 12 q^{11} - 12 q^{17} - 2 q^{19} - 48 q^{23} - 200 q^{25} - 216 q^{35} + 102 q^{39} + 28 q^{43} + 2 q^{45} - 174 q^{47} - 306 q^{49} + 213 q^{57} + 14 q^{61} + 62 q^{63} + 220 q^{73} - 60 q^{77} + 340 q^{81} + 150 q^{83} - 252 q^{87} - 252 q^{93} + 360 q^{95} + 542 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
265.1 0 −2.99885 0.0829166i 0 −2.98656 + 5.17287i 0 −4.37090 7.57062i 0 8.98625 + 0.497310i 0
265.2 0 −2.90955 0.731127i 0 −3.02121 + 5.23289i 0 6.41336 + 11.1083i 0 7.93091 + 4.25449i 0
265.3 0 −2.87088 0.870675i 0 1.28913 2.23284i 0 2.75015 + 4.76339i 0 7.48385 + 4.99920i 0
265.4 0 −2.83737 + 0.974328i 0 3.33339 5.77360i 0 4.83336 + 8.37163i 0 7.10137 5.52906i 0
265.5 0 −2.83582 + 0.978843i 0 −1.74243 + 3.01798i 0 −0.449830 0.779128i 0 7.08373 5.55164i 0
265.6 0 −2.79131 1.09935i 0 −2.57011 + 4.45156i 0 −2.98315 5.16697i 0 6.58286 + 6.13726i 0
265.7 0 −2.77003 1.15194i 0 3.04651 5.27672i 0 −4.19196 7.26070i 0 6.34609 + 6.38179i 0
265.8 0 −2.74539 + 1.20947i 0 3.78362 6.55342i 0 −5.25651 9.10454i 0 6.07437 6.64094i 0
265.9 0 −2.57066 + 1.54651i 0 1.45280 2.51632i 0 −0.663126 1.14857i 0 4.21662 7.95111i 0
265.10 0 −2.19396 2.04610i 0 2.17299 3.76373i 0 1.82107 + 3.15419i 0 0.626916 + 8.97814i 0
265.11 0 −1.74945 + 2.43709i 0 −1.72643 + 2.99026i 0 4.42424 + 7.66301i 0 −2.87882 8.52716i 0
265.12 0 −1.72468 + 2.45468i 0 −4.82077 + 8.34982i 0 −0.153959 0.266665i 0 −3.05094 8.46710i 0
265.13 0 −1.45850 2.62160i 0 0.370923 0.642458i 0 0.938087 + 1.62481i 0 −4.74554 + 7.64721i 0
265.14 0 −1.21404 2.74338i 0 −3.81342 + 6.60503i 0 1.86164 + 3.22446i 0 −6.05222 + 6.66113i 0
265.15 0 −1.08441 + 2.79715i 0 0.475788 0.824090i 0 −4.90114 8.48903i 0 −6.64810 6.06653i 0
265.16 0 −1.05443 2.80859i 0 −1.91223 + 3.31208i 0 −5.72810 9.92136i 0 −6.77637 + 5.92290i 0
265.17 0 −0.960366 2.84213i 0 4.33807 7.51375i 0 −2.48195 4.29886i 0 −7.15540 + 5.45897i 0
265.18 0 −0.913509 + 2.85753i 0 3.79806 6.57843i 0 3.58363 + 6.20703i 0 −7.33100 5.22077i 0
265.19 0 −0.775474 + 2.89804i 0 −0.775683 + 1.34352i 0 −0.749140 1.29755i 0 −7.79728 4.49471i 0
265.20 0 −0.0369083 + 2.99977i 0 −0.692448 + 1.19936i 0 5.80422 + 10.0532i 0 −8.99728 0.221433i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
19.b odd 2 1 inner
171.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.t.a 80
3.b odd 2 1 2052.3.t.a 80
9.c even 3 1 inner 684.3.t.a 80
9.d odd 6 1 2052.3.t.a 80
19.b odd 2 1 inner 684.3.t.a 80
57.d even 2 1 2052.3.t.a 80
171.l even 6 1 2052.3.t.a 80
171.o odd 6 1 inner 684.3.t.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.t.a 80 1.a even 1 1 trivial
684.3.t.a 80 9.c even 3 1 inner
684.3.t.a 80 19.b odd 2 1 inner
684.3.t.a 80 171.o odd 6 1 inner
2052.3.t.a 80 3.b odd 2 1
2052.3.t.a 80 9.d odd 6 1
2052.3.t.a 80 57.d even 2 1
2052.3.t.a 80 171.l even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(684, [\chi])$$.