# Properties

 Label 76.3.g.c Level $76$ Weight $3$ Character orbit 76.g Analytic conductor $2.071$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 76.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07085000914$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 5q^{2} - 11q^{4} + 6q^{5} - 3q^{6} - 62q^{8} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 5q^{2} - 11q^{4} + 6q^{5} - 3q^{6} - 62q^{8} + 20q^{9} + 26q^{12} + 30q^{13} - 30q^{14} - 19q^{16} + 38q^{17} - 60q^{18} - 44q^{20} + 80q^{21} + 45q^{22} + 17q^{24} - 16q^{25} - 56q^{26} + 54q^{28} + 6q^{29} + 96q^{30} - 45q^{32} - 176q^{33} - 20q^{34} + 30q^{36} + 104q^{37} - 258q^{38} + 94q^{40} - 2q^{41} - 2q^{42} + 201q^{44} - 360q^{45} + 164q^{46} - 17q^{48} - 20q^{49} + 490q^{50} - 102q^{52} - 242q^{53} - 13q^{54} + 276q^{56} - 254q^{57} + 96q^{58} + 10q^{60} - 58q^{61} - 36q^{62} - 74q^{64} - 260q^{65} + 167q^{66} + 396q^{68} + 340q^{69} + 60q^{70} - 422q^{72} - 82q^{73} - 136q^{74} + 123q^{76} - 144q^{77} + 224q^{78} - 174q^{80} + 410q^{81} - 305q^{82} + 252q^{84} + 714q^{85} + 166q^{86} - 718q^{88} + 150q^{89} - 272q^{90} - 588q^{92} + 344q^{93} - 488q^{94} - 122q^{96} + 94q^{97} + 307q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.99453 + 0.147792i 3.58197 2.06805i 3.95632 0.589552i −3.72976 6.46014i −6.83872 + 4.65419i 3.06851i −7.80387 + 1.76059i 4.05369 7.02120i 8.39388 + 12.3337i
7.2 −1.93914 0.489616i −3.88623 + 2.24371i 3.52055 + 1.89887i −0.133773 0.231701i 8.63451 2.44812i 7.24937i −5.89713 5.40590i 5.56851 9.64495i 0.145960 + 0.514800i
7.3 −1.82726 + 0.813087i −0.851777 + 0.491774i 2.67778 2.97145i 1.71651 + 2.97309i 1.15657 1.59117i 2.43870i −2.47697 + 7.60688i −4.01632 + 6.95647i −5.55390 4.03694i
7.4 −1.69874 1.05559i 3.11547 1.79872i 1.77146 + 3.58635i 4.52938 + 7.84512i −7.19109 0.233096i 2.81904i 0.776454 7.96223i 1.97077 3.41347i 0.586964 18.1080i
7.5 −1.20067 1.59950i −1.67542 + 0.967303i −1.11680 + 3.84093i −1.80536 3.12697i 3.55882 + 1.51842i 13.2414i 7.48448 2.82535i −2.62865 + 4.55296i −2.83396 + 6.64212i
7.6 −0.784875 1.83956i 1.67542 0.967303i −2.76794 + 2.88764i −1.80536 3.12697i −3.09440 2.32282i 13.2414i 7.48448 + 2.82535i −2.62865 + 4.55296i −4.33527 + 5.77535i
7.7 −0.0647951 1.99895i −3.11547 + 1.79872i −3.99160 + 0.259045i 4.52938 + 7.84512i 3.79741 + 6.11112i 2.81904i 0.776454 + 7.96223i 1.97077 3.41347i 15.3885 9.56233i
7.8 0.00657150 + 1.99999i 0.443154 0.255855i −3.99991 + 0.0262858i 1.99023 + 3.44717i 0.514619 + 0.884621i 9.66827i −0.0788568 7.99961i −4.36908 + 7.56746i −6.88123 + 4.00309i
7.9 0.545551 1.92416i 3.88623 2.24371i −3.40475 2.09945i −0.133773 0.231701i −2.19712 8.70177i 7.24937i −5.89713 + 5.40590i 5.56851 9.64495i −0.518810 + 0.130995i
7.10 0.789173 + 1.83772i −3.65809 + 2.11200i −2.75441 + 2.90055i −1.06722 1.84849i −6.76812 5.05580i 1.82388i −7.50411 2.77279i 4.42107 7.65753i 2.55477 3.42003i
7.11 1.12526 1.65342i −3.58197 + 2.06805i −1.46759 3.72105i −3.72976 6.46014i −0.611284 + 8.24960i 3.06851i −7.80387 1.76059i 4.05369 7.02120i −14.8783 1.10246i
7.12 1.19692 + 1.60230i 3.65809 2.11200i −1.13475 + 3.83567i −1.06722 1.84849i 7.76251 + 3.33347i 1.82388i −7.50411 + 2.77279i 4.42107 7.65753i 1.68445 3.92252i
7.13 1.61779 1.17591i 0.851777 0.491774i 1.23446 3.80475i 1.71651 + 2.97309i 0.799709 1.79720i 2.43870i −2.47697 7.60688i −4.01632 + 6.95647i 6.27304 + 2.79135i
7.14 1.72876 + 1.00569i −0.443154 + 0.255855i 1.97719 + 3.47717i 1.99023 + 3.44717i −1.02341 0.00336270i 9.66827i −0.0788568 + 7.99961i −4.36908 + 7.56746i −0.0261575 + 7.96087i
11.1 −1.99453 0.147792i 3.58197 + 2.06805i 3.95632 + 0.589552i −3.72976 + 6.46014i −6.83872 4.65419i 3.06851i −7.80387 1.76059i 4.05369 + 7.02120i 8.39388 12.3337i
11.2 −1.93914 + 0.489616i −3.88623 2.24371i 3.52055 1.89887i −0.133773 + 0.231701i 8.63451 + 2.44812i 7.24937i −5.89713 + 5.40590i 5.56851 + 9.64495i 0.145960 0.514800i
11.3 −1.82726 0.813087i −0.851777 0.491774i 2.67778 + 2.97145i 1.71651 2.97309i 1.15657 + 1.59117i 2.43870i −2.47697 7.60688i −4.01632 6.95647i −5.55390 + 4.03694i
11.4 −1.69874 + 1.05559i 3.11547 + 1.79872i 1.77146 3.58635i 4.52938 7.84512i −7.19109 + 0.233096i 2.81904i 0.776454 + 7.96223i 1.97077 + 3.41347i 0.586964 + 18.1080i
11.5 −1.20067 + 1.59950i −1.67542 0.967303i −1.11680 3.84093i −1.80536 + 3.12697i 3.55882 1.51842i 13.2414i 7.48448 + 2.82535i −2.62865 4.55296i −2.83396 6.64212i
11.6 −0.784875 + 1.83956i 1.67542 + 0.967303i −2.76794 2.88764i −1.80536 + 3.12697i −3.09440 + 2.32282i 13.2414i 7.48448 2.82535i −2.62865 4.55296i −4.33527 5.77535i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.g.c 28
4.b odd 2 1 inner 76.3.g.c 28
19.c even 3 1 inner 76.3.g.c 28
76.g odd 6 1 inner 76.3.g.c 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.g.c 28 1.a even 1 1 trivial
76.3.g.c 28 4.b odd 2 1 inner
76.3.g.c 28 19.c even 3 1 inner
76.3.g.c 28 76.g odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{28} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(76, [\chi])$$.