# Properties

 Label 76.6.d.a Level $76$ Weight $6$ Character orbit 76.d Analytic conductor $12.189$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 76.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.1891703058$$ Analytic rank: $$0$$ Dimension: $$48$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 22q^{4} - 4q^{5} - 246q^{6} + 3516q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 22q^{4} - 4q^{5} - 246q^{6} + 3516q^{9} - 1222q^{16} + 1000q^{17} - 300q^{20} + 17158q^{24} + 24996q^{25} + 12102q^{26} + 6382q^{28} - 16528q^{30} + 27524q^{36} + 3598q^{38} + 35966q^{42} - 65408q^{44} - 13476q^{45} - 143096q^{49} - 148446q^{54} - 53980q^{57} - 117018q^{58} + 133740q^{61} + 28068q^{62} - 78130q^{64} + 128204q^{66} - 48338q^{68} - 104080q^{73} + 48932q^{74} - 251688q^{76} + 184844q^{77} + 80080q^{80} + 375128q^{81} + 175628q^{82} - 384748q^{85} + 139810q^{92} + 183272q^{93} + 601258q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
75.1 −5.61709 0.669578i 15.7189 31.1033 + 7.52215i −23.2669 −88.2945 10.5250i 110.037i −169.673 63.0787i 4.08426 130.692 + 15.5790i
75.2 −5.61709 + 0.669578i 15.7189 31.1033 7.52215i −23.2669 −88.2945 + 10.5250i 110.037i −169.673 + 63.0787i 4.08426 130.692 15.5790i
75.3 −5.48835 1.37043i −24.7554 28.2439 + 15.0427i −13.4021 135.866 + 33.9254i 215.359i −134.397 121.266i 369.829 73.5554 + 18.3666i
75.4 −5.48835 + 1.37043i −24.7554 28.2439 15.0427i −13.4021 135.866 33.9254i 215.359i −134.397 + 121.266i 369.829 73.5554 18.3666i
75.5 −5.30781 1.95631i −7.37371 24.3457 + 20.7674i 74.1521 39.1383 + 14.4252i 80.5452i −88.5950 157.857i −188.628 −393.585 145.064i
75.6 −5.30781 + 1.95631i −7.37371 24.3457 20.7674i 74.1521 39.1383 14.4252i 80.5452i −88.5950 + 157.857i −188.628 −393.585 + 145.064i
75.7 −5.06482 2.51944i −9.05324 19.3048 + 25.5211i −89.6042 45.8530 + 22.8091i 100.840i −33.4765 177.897i −161.039 453.829 + 225.753i
75.8 −5.06482 + 2.51944i −9.05324 19.3048 25.5211i −89.6042 45.8530 22.8091i 100.840i −33.4765 + 177.897i −161.039 453.829 225.753i
75.9 −4.77452 3.03380i 25.5646 13.5921 + 28.9699i 94.0869 −122.059 77.5580i 166.433i 22.9933 179.553i 410.550 −449.220 285.441i
75.10 −4.77452 + 3.03380i 25.5646 13.5921 28.9699i 94.0869 −122.059 + 77.5580i 166.433i 22.9933 + 179.553i 410.550 −449.220 + 285.441i
75.11 −3.92565 4.07299i 5.43850 −1.17856 + 31.9783i 4.82210 −21.3497 22.1510i 68.2624i 134.874 120.735i −213.423 −18.9299 19.6404i
75.12 −3.92565 + 4.07299i 5.43850 −1.17856 31.9783i 4.82210 −21.3497 + 22.1510i 68.2624i 134.874 + 120.735i −213.423 −18.9299 + 19.6404i
75.13 −3.73141 4.25166i 29.4761 −4.15316 + 31.7293i −71.3638 −109.988 125.322i 160.370i 150.399 100.737i 625.843 266.288 + 303.414i
75.14 −3.73141 + 4.25166i 29.4761 −4.15316 31.7293i −71.3638 −109.988 + 125.322i 160.370i 150.399 + 100.737i 625.843 266.288 303.414i
75.15 −3.09334 4.73617i −0.774530 −12.8625 + 29.3011i −32.1016 2.39588 + 3.66830i 237.436i 178.563 29.7192i −242.400 99.3011 + 152.038i
75.16 −3.09334 + 4.73617i −0.774530 −12.8625 29.3011i −32.1016 2.39588 3.66830i 237.436i 178.563 + 29.7192i −242.400 99.3011 152.038i
75.17 −2.99929 4.79628i −24.6139 −14.0085 + 28.7708i 37.3653 73.8243 + 118.055i 35.5589i 180.009 19.1033i 362.846 −112.069 179.214i
75.18 −2.99929 + 4.79628i −24.6139 −14.0085 28.7708i 37.3653 73.8243 118.055i 35.5589i 180.009 + 19.1033i 362.846 −112.069 + 179.214i
75.19 −1.28503 5.50897i 16.4423 −28.6974 + 14.1583i 16.6706 −21.1288 90.5803i 54.8970i 114.875 + 139.899i 27.3504 −21.4221 91.8375i
75.20 −1.28503 + 5.50897i 16.4423 −28.6974 14.1583i 16.6706 −21.1288 + 90.5803i 54.8970i 114.875 139.899i 27.3504 −21.4221 + 91.8375i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 75.48 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.b odd 2 1 inner
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.6.d.a 48
4.b odd 2 1 inner 76.6.d.a 48
19.b odd 2 1 inner 76.6.d.a 48
76.d even 2 1 inner 76.6.d.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.d.a 48 1.a even 1 1 trivial
76.6.d.a 48 4.b odd 2 1 inner
76.6.d.a 48 19.b odd 2 1 inner
76.6.d.a 48 76.d even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(76, [\chi])$$.