# Properties

 Label 126.3.r.a Level $126$ Weight $3$ Character orbit 126.r Analytic conductor $3.433$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 126.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43325133094$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 64 q^{4} + 8 q^{6} + 2 q^{7} - 20 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32 q - 64 q^{4} + 8 q^{6} + 2 q^{7} - 20 q^{9} - 36 q^{11} + 10 q^{13} + 36 q^{14} + 10 q^{15} + 128 q^{16} - 54 q^{17} + 28 q^{19} + 28 q^{21} - 126 q^{23} - 16 q^{24} + 80 q^{25} - 72 q^{26} - 126 q^{27} - 4 q^{28} + 36 q^{29} + 76 q^{30} + 16 q^{31} - 40 q^{33} - 90 q^{35} + 40 q^{36} + 22 q^{37} + 46 q^{39} + 72 q^{41} + 120 q^{42} + 16 q^{43} + 72 q^{44} + 464 q^{45} - 12 q^{46} + 2 q^{49} - 288 q^{50} - 286 q^{51} - 20 q^{52} - 72 q^{53} - 160 q^{54} - 24 q^{55} - 72 q^{56} - 282 q^{57} - 24 q^{58} - 20 q^{60} + 124 q^{61} + 66 q^{63} - 256 q^{64} - 16 q^{66} - 140 q^{67} + 108 q^{68} + 218 q^{69} + 72 q^{70} + 196 q^{73} + 216 q^{74} + 658 q^{75} - 56 q^{76} + 486 q^{77} + 32 q^{78} + 76 q^{79} - 380 q^{81} - 56 q^{84} + 60 q^{85} - 144 q^{86} - 740 q^{87} - 486 q^{89} + 296 q^{90} - 122 q^{91} + 252 q^{92} + 238 q^{93} - 336 q^{94} + 32 q^{96} - 38 q^{97} + 288 q^{98} + 394 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 1.41421i −2.92705 + 0.657548i −2.00000 2.90307 + 1.67609i 0.929913 + 4.13948i 5.64984 + 4.13271i 2.82843i 8.13526 3.84935i 2.37035 4.10556i
11.2 1.41421i −2.82827 1.00044i −2.00000 0.857116 + 0.494856i −1.41483 + 3.99978i −6.76024 1.81637i 2.82843i 6.99824 + 5.65903i 0.699833 1.21215i
11.3 1.41421i −0.991189 2.83153i −2.00000 −6.36825 3.67671i −4.00438 + 1.40175i 2.82364 + 6.40524i 2.82843i −7.03509 + 5.61316i −5.19965 + 9.00606i
11.4 1.41421i −0.677712 + 2.92245i −2.00000 −5.54397 3.20081i 4.13297 + 0.958429i 4.41544 5.43175i 2.82843i −8.08141 3.96116i −4.52663 + 7.84035i
11.5 1.41421i 0.146807 + 2.99641i −2.00000 2.36201 + 1.36371i 4.23756 0.207617i −2.54299 + 6.52175i 2.82843i −8.95690 + 0.879787i 1.92857 3.34039i
11.6 1.41421i 2.09605 2.14630i −2.00000 −2.07383 1.19732i −3.03532 2.96426i −5.45012 4.39274i 2.82843i −0.213187 8.99747i −1.69327 + 2.93283i
11.7 1.41421i 2.36405 + 1.84696i −2.00000 1.22482 + 0.707152i 2.61199 3.34328i 3.10205 6.27513i 2.82843i 2.17750 + 8.73261i 1.00006 1.73216i
11.8 1.41421i 2.81732 1.03088i −2.00000 6.63902 + 3.83304i −1.45789 3.98429i 2.93662 + 6.35423i 2.82843i 6.87456 5.80865i 5.42073 9.38899i
11.9 1.41421i −2.92348 + 0.673248i −2.00000 −1.51694 0.875808i −0.952117 4.13443i −1.24611 6.88819i 2.82843i 8.09347 3.93646i 1.23858 2.14528i
11.10 1.41421i −2.24880 1.98568i −2.00000 1.84316 + 1.06415i 2.80817 3.18028i 6.16730 + 3.31125i 2.82843i 1.11417 + 8.93077i −1.50493 + 2.60662i
11.11 1.41421i −1.59142 + 2.54310i −2.00000 5.46142 + 3.15315i −3.59649 2.25061i −3.23416 + 6.20807i 2.82843i −3.93476 8.09430i −4.45923 + 7.72361i
11.12 1.41421i −0.664523 2.92548i −2.00000 −2.12968 1.22957i 4.13725 0.939777i −6.56056 + 2.44111i 2.82843i −8.11682 + 3.88809i 1.73888 3.01183i
11.13 1.41421i 1.16644 + 2.76395i −2.00000 −7.41683 4.28211i −3.90882 + 1.64959i −6.97339 0.609798i 2.82843i −6.27884 + 6.44795i 6.05582 10.4890i
11.14 1.41421i 1.98625 2.24829i −2.00000 8.39861 + 4.84894i 3.17956 + 2.80898i −3.70991 5.93604i 2.82843i −1.10962 8.93133i −6.85744 + 11.8774i
11.15 1.41421i 2.02240 2.21583i −2.00000 −7.20455 4.15955i 3.13365 + 2.86011i 5.54044 4.27827i 2.82843i −0.819791 8.96259i 5.88249 10.1888i
11.16 1.41421i 2.25313 + 1.98076i −2.00000 2.56482 + 1.48080i −2.80121 + 3.18641i 6.84216 1.47812i 2.82843i 1.15321 + 8.92581i −2.09417 + 3.62721i
23.1 1.41421i −2.92348 0.673248i −2.00000 −1.51694 + 0.875808i −0.952117 + 4.13443i −1.24611 + 6.88819i 2.82843i 8.09347 + 3.93646i 1.23858 + 2.14528i
23.2 1.41421i −2.24880 + 1.98568i −2.00000 1.84316 1.06415i 2.80817 + 3.18028i 6.16730 3.31125i 2.82843i 1.11417 8.93077i −1.50493 2.60662i
23.3 1.41421i −1.59142 2.54310i −2.00000 5.46142 3.15315i −3.59649 + 2.25061i −3.23416 6.20807i 2.82843i −3.93476 + 8.09430i −4.45923 7.72361i
23.4 1.41421i −0.664523 + 2.92548i −2.00000 −2.12968 + 1.22957i 4.13725 + 0.939777i −6.56056 2.44111i 2.82843i −8.11682 3.88809i 1.73888 + 3.01183i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.r.a yes 32
3.b odd 2 1 378.3.r.a 32
7.c even 3 1 126.3.i.a 32
9.c even 3 1 378.3.i.a 32
9.d odd 6 1 126.3.i.a 32
21.h odd 6 1 378.3.i.a 32
63.h even 3 1 378.3.r.a 32
63.j odd 6 1 inner 126.3.r.a yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.i.a 32 7.c even 3 1
126.3.i.a 32 9.d odd 6 1
126.3.r.a yes 32 1.a even 1 1 trivial
126.3.r.a yes 32 63.j odd 6 1 inner
378.3.i.a 32 9.c even 3 1
378.3.i.a 32 21.h odd 6 1
378.3.r.a 32 3.b odd 2 1
378.3.r.a 32 63.h even 3 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(126, [\chi])$$.