# Properties

 Label 69.6.g.a Level $69$ Weight $6$ Character orbit 69.g Analytic conductor $11.066$ Analytic rank $0$ Dimension $380$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 69.g (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0664835671$$ Analytic rank: $$0$$ Dimension: $$380$$ Relative dimension: $$38$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$380q - 11q^{3} + 578q^{4} + 64q^{6} - 22q^{7} + 433q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$380q - 11q^{3} + 578q^{4} + 64q^{6} - 22q^{7} + 433q^{9} - 22q^{10} + 148q^{12} - 542q^{13} + 4620q^{15} - 11102q^{16} - 11526q^{18} - 22q^{19} + 12496q^{21} + 15878q^{24} - 17304q^{25} - 21014q^{27} - 22q^{28} - 33627q^{30} - 6566q^{31} + 41250q^{33} + 75988q^{34} + 12154q^{36} - 72138q^{37} + 45205q^{39} - 119372q^{40} - 11q^{42} + 59334q^{43} + 119802q^{46} + 5138q^{48} + 147060q^{49} - 11q^{51} - 246042q^{52} - 172143q^{54} - 95048q^{55} + 92422q^{57} + 277528q^{58} + 420629q^{60} + 125158q^{61} + 7194q^{63} - 100196q^{64} - 312378q^{66} - 94380q^{67} - 136956q^{69} - 395908q^{70} - 507803q^{72} + 83336q^{73} + 577756q^{75} + 415338q^{76} - 361512q^{78} + 342958q^{79} - 420607q^{81} + 247868q^{82} - 978505q^{84} - 634728q^{85} + 354721q^{87} + 22506q^{88} + 1636316q^{90} + 1162112q^{93} + 40940q^{94} - 102479q^{96} + 582824q^{97} - 674663q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −3.01088 + 10.2541i 2.12177 + 15.4434i −69.1616 44.4475i −4.96747 + 34.5495i −164.747 24.7413i −54.7844 + 25.0192i 405.553 351.413i −233.996 + 65.5346i −339.319 154.962i
5.2 −2.97006 + 10.1151i −14.6309 5.37922i −66.5741 42.7846i 8.40244 58.4402i 97.8662 132.017i −141.246 + 64.5047i 375.549 325.415i 185.128 + 157.406i 566.174 + 258.563i
5.3 −2.79794 + 9.52891i −3.60982 15.1647i −56.0515 36.0221i −6.21680 + 43.2388i 154.603 + 8.03237i 210.991 96.3564i 259.904 225.208i −216.938 + 109.484i −394.624 180.219i
5.4 −2.60390 + 8.86808i 14.0479 + 6.75686i −44.9425 28.8828i 14.8404 103.218i −96.4999 + 106.984i 66.5201 30.3787i 149.641 129.665i 151.690 + 189.840i 876.699 + 400.375i
5.5 −2.59429 + 8.83533i 15.4604 1.99434i −44.4125 28.5422i −6.70991 + 46.6684i −22.4879 + 141.771i −7.07279 + 3.23003i 144.705 125.387i 235.045 61.6665i −394.923 180.355i
5.6 −2.54608 + 8.67115i 8.15133 13.2874i −41.7861 26.8543i 0.168773 1.17384i 94.4634 + 104.512i −160.739 + 73.4069i 120.693 104.581i −110.112 216.620i 9.74886 + 4.45216i
5.7 −2.40375 + 8.18642i −12.7578 + 8.95766i −34.3194 22.0557i 4.22700 29.3994i −42.6647 125.972i 170.161 77.7098i 56.7143 49.1432i 82.5208 228.559i 230.515 + 105.273i
5.8 −1.97556 + 6.72815i −11.2628 + 10.7772i −14.4450 9.28323i −7.37109 + 51.2671i −50.2603 97.0690i −115.006 + 52.5213i −78.5868 + 68.0959i 10.7027 242.764i −330.370 150.875i
5.9 −1.91071 + 6.50727i −13.3402 8.06476i −11.7737 7.56648i −10.6354 + 73.9705i 77.9687 71.3986i −54.2822 + 24.7899i −92.2823 + 79.9631i 112.919 + 215.170i −461.025 210.543i
5.10 −1.75424 + 5.97440i −0.0984677 15.5881i −5.69594 3.66056i 11.1842 77.7880i 93.3025 + 26.7571i −8.66991 + 3.95942i −118.723 + 102.874i −242.981 + 3.06986i 445.117 + 203.278i
5.11 −1.54386 + 5.25792i 2.93920 + 15.3089i 1.65792 + 1.06548i 4.93387 34.3158i −85.0305 8.18070i −31.8114 + 14.5278i −140.687 + 121.906i −225.722 + 89.9917i 172.813 + 78.9209i
5.12 −1.50479 + 5.12484i 10.2377 + 11.7554i 2.92056 + 1.87693i −14.4825 + 100.728i −75.6499 + 34.7774i 170.076 77.6713i −143.185 + 124.071i −33.3771 + 240.697i −494.421 225.795i
5.13 −1.29764 + 4.41935i 13.8639 7.12680i 9.07333 + 5.83108i 3.05854 21.2726i 13.5055 + 70.5176i 139.466 63.6921i −148.933 + 129.051i 141.417 197.611i 90.0424 + 41.1210i
5.14 −1.13703 + 3.87238i −14.7589 5.01748i 13.2177 + 8.49447i 6.88235 47.8678i 36.2109 51.4470i −29.0361 + 13.2603i −145.526 + 126.099i 192.650 + 148.105i 177.537 + 81.0783i
5.15 −0.903285 + 3.07631i 14.5683 + 5.54656i 18.2724 + 11.7429i 0.440615 3.06454i −30.2222 + 39.8065i −221.259 + 101.046i −130.168 + 112.791i 181.471 + 161.608i 9.02947 + 4.12362i
5.16 −0.617954 + 2.10456i 7.38321 13.7291i 22.8728 + 14.6995i −12.1594 + 84.5703i 24.3312 + 24.0223i −68.9240 + 31.4765i −98.1155 + 85.0176i −133.976 202.730i −170.469 77.8506i
5.17 −0.327846 + 1.11654i −6.80137 14.0265i 25.7809 + 16.5684i −0.383188 + 2.66513i 17.8909 2.99549i 83.3649 38.0715i −55.0938 + 47.7391i −150.483 + 190.798i −2.85010 1.30160i
5.18 −0.235789 + 0.803025i −13.0330 + 8.55220i 26.3309 + 16.9218i 11.5441 80.2911i −3.79458 12.4824i −106.024 + 48.4193i −40.0374 + 34.6926i 96.7196 222.922i 61.7537 + 28.2020i
5.19 −0.190214 + 0.647810i −15.2666 + 3.15135i 26.5366 + 17.0541i −6.91235 + 48.0764i 0.862446 10.4893i 156.619 71.5256i −32.4235 + 28.0951i 223.138 96.2208i −29.8296 13.6227i
5.20 0.190214 0.647810i −0.946612 + 15.5597i 26.5366 + 17.0541i 6.91235 48.0764i 9.89966 + 3.57290i 156.619 71.5256i 32.4235 28.0951i −241.208 29.4580i −29.8296 13.6227i
See next 80 embeddings (of 380 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.38 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.6.g.a 380
3.b odd 2 1 inner 69.6.g.a 380
23.d odd 22 1 inner 69.6.g.a 380
69.g even 22 1 inner 69.6.g.a 380

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.g.a 380 1.a even 1 1 trivial
69.6.g.a 380 3.b odd 2 1 inner
69.6.g.a 380 23.d odd 22 1 inner
69.6.g.a 380 69.g even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(69, [\chi])$$.