# Properties

 Label 76.5.g.a Level $76$ Weight $5$ Character orbit 76.g Analytic conductor $7.856$ Analytic rank $0$ Dimension $76$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 76.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.85611719437$$ Analytic rank: $$0$$ Dimension: $$76$$ Relative dimension: $$38$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$76q - q^{2} + 13q^{4} - 2q^{5} - 15q^{6} - 94q^{8} + 972q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$76q - q^{2} + 13q^{4} - 2q^{5} - 15q^{6} - 94q^{8} + 972q^{9} + 180q^{10} - 166q^{12} - 178q^{13} - 30q^{14} + 237q^{16} - 50q^{17} + 356q^{18} + 2212q^{20} + 304q^{21} - 1035q^{22} + 1697q^{24} - 3752q^{25} + 3952q^{26} + 990q^{28} - 2q^{29} + 408q^{30} - 181q^{32} - 3000q^{33} + 2712q^{34} + 1078q^{36} - 2088q^{37} - 3198q^{38} - 6562q^{40} - 2138q^{41} - 2698q^{42} - 8631q^{44} + 2168q^{45} - 9380q^{46} + 10159q^{48} - 19380q^{49} + 9354q^{50} + 4706q^{52} - 818q^{53} + 6431q^{54} + 1476q^{56} - 6q^{57} - 13080q^{58} + 5626q^{60} + 5470q^{61} + 8892q^{62} - 30890q^{64} + 16316q^{65} - 14001q^{66} - 3284q^{68} + 38836q^{69} + 8532q^{70} - 2766q^{72} + 7222q^{73} - 10328q^{74} - 29613q^{76} + 16320q^{77} + 19028q^{78} + 12778q^{80} - 25574q^{81} + 30563q^{82} + 65132q^{84} - 21006q^{85} - 8430q^{86} + 44242q^{88} + 12910q^{89} - 48688q^{90} - 14052q^{92} - 4304q^{93} - 39664q^{94} + 15670q^{96} + 1174q^{97} + 2783q^{98} + O(q^{100})$$

## 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 −3.99928 + 0.0760910i −4.58614 + 2.64781i 15.9884 0.608618i −5.59416 9.68937i 18.1398 10.9383i 39.7837i −63.8958 + 3.65061i −26.4782 + 45.8616i 23.1099 + 38.3248i
7.2 −3.98849 0.303244i 0.697952 0.402963i 15.8161 + 2.41897i 22.3482 + 38.7083i −2.90597 + 1.39556i 79.0280i −62.3487 14.4442i −40.1752 + 69.5856i −77.3976 161.164i
7.3 −3.88155 + 0.966218i 8.18792 4.72730i 14.1328 7.50085i −10.0033 17.3262i −27.2142 + 26.2605i 65.6661i −47.6099 + 42.7703i 4.19464 7.26533i 55.5691 + 57.5871i
7.4 −3.77438 + 1.32442i 13.7242 7.92369i 12.4918 9.99773i 14.8198 + 25.6686i −41.3061 + 48.0836i 78.9843i −33.9076 + 54.2796i 85.0696 147.345i −89.9315 77.2553i
7.5 −3.71892 1.47296i 4.05898 2.34345i 11.6608 + 10.9556i −1.00987 1.74915i −18.5469 + 2.73642i 54.3250i −27.2284 57.9190i −29.5164 + 51.1240i 1.17921 + 7.99246i
7.6 −3.65546 + 1.62407i −12.2858 + 7.09323i 10.7248 11.8735i −5.51885 9.55894i 33.3905 45.8821i 41.7305i −19.9207 + 60.8208i 60.1279 104.145i 35.6983 + 25.9793i
7.7 −3.54548 1.85191i −14.2858 + 8.24790i 9.14084 + 13.1318i 15.0446 + 26.0580i 65.9243 2.78673i 53.2660i −8.08964 63.4867i 95.5557 165.507i −5.08315 120.250i
7.8 −3.25774 2.32102i 12.2124 7.05085i 5.22570 + 15.1226i −5.66259 9.80789i −56.1501 5.37552i 30.6646i 18.0759 61.3943i 58.9291 102.068i −4.31712 + 45.0945i
7.9 −3.19572 2.40570i −7.19063 + 4.15151i 4.42524 + 15.3759i −24.3962 42.2554i 32.9665 + 4.03141i 41.7489i 22.8479 59.7827i −6.02993 + 10.4441i −23.6904 + 193.726i
7.10 −2.88768 + 2.76791i −6.22437 + 3.59364i 0.677370 15.9857i 17.9743 + 31.1323i 8.02710 27.6057i 37.3202i 42.2908 + 48.0363i −14.6715 + 25.4118i −138.075 40.1490i
7.11 −2.87075 + 2.78547i 2.38822 1.37884i 0.482361 15.9927i −16.6192 28.7853i −3.01526 + 10.6106i 28.7543i 43.1625 + 47.2547i −36.6976 + 63.5621i 127.890 + 36.3431i
7.12 −2.44419 + 3.16638i 7.49168 4.32533i −4.05190 15.4784i 5.46741 + 9.46983i −4.61545 + 34.2934i 37.0844i 58.9142 + 25.0023i −3.08311 + 5.34010i −43.3484 5.83414i
7.13 −2.38146 3.21382i −5.15508 + 2.97629i −4.65733 + 15.3072i 8.40388 + 14.5559i 21.8419 + 9.47962i 36.4093i 60.2858 21.4855i −22.7834 + 39.4621i 26.7668 61.6729i
7.14 −1.59252 3.66931i 5.15508 2.97629i −10.9277 + 11.6869i 8.40388 + 14.5559i −19.1305 14.1758i 36.4093i 60.2858 + 21.4855i −22.7834 + 39.4621i 40.0269 54.0172i
7.15 −1.17048 + 3.82491i −8.74558 + 5.04926i −13.2599 8.95400i 1.11866 + 1.93758i −9.07644 39.3612i 77.4402i 49.7688 40.2376i 10.4901 18.1694i −8.72047 + 2.01089i
7.16 −0.707709 + 3.93690i −10.7755 + 6.22123i −14.9983 5.57236i −13.6139 23.5800i −16.8664 46.8248i 71.9337i 32.5522 55.1031i 36.9073 63.9254i 102.467 36.9088i
7.17 −0.485536 3.97042i 7.19063 4.15151i −15.5285 + 3.85556i −24.3962 42.2554i −19.9746 26.5341i 41.7489i 22.8479 + 59.7827i −6.02993 + 10.4441i −155.927 + 117.380i
7.18 −0.381196 3.98179i −12.2124 + 7.05085i −15.7094 + 3.03569i −5.66259 9.80789i 32.7304 + 45.9397i 30.6646i 18.0759 + 61.3943i 58.9291 102.068i −36.8945 + 26.2860i
7.19 −0.359633 + 3.98380i 4.35470 2.51418i −15.7413 2.86542i 9.73967 + 16.8696i 8.44992 + 18.2524i 18.1632i 17.0764 61.6798i −27.8578 + 48.2510i −70.7079 + 32.7340i
7.20 −0.244844 + 3.99250i 14.5224 8.38453i −15.8801 1.95508i −17.7086 30.6723i 29.9195 + 60.0337i 23.2025i 11.6938 62.9226i 100.101 173.380i 126.795 63.1918i
See all 76 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.38 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 76.5.g.a 76
4.b odd 2 1 inner 76.5.g.a 76
19.c even 3 1 inner 76.5.g.a 76
76.g odd 6 1 inner 76.5.g.a 76

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(76, [\chi])$$.