# Properties

 Label 169.4.i.a Level $169$ Weight $4$ Character orbit 169.i Analytic conductor $9.971$ Analytic rank $0$ Dimension $1080$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.i (of order $$39$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$1080$$ Relative dimension: $$45$$ over $$\Q(\zeta_{39})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10})$$ 1080 * q - 27 * q^2 - 19 * q^3 + 155 * q^4 - 30 * q^5 - 56 * q^6 + 9 * q^7 + 4 * q^8 + 364 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9} + q^{10} - 41 q^{11} - 314 q^{12} - 710 q^{13} - 54 q^{14} - 474 q^{15} + 675 q^{16} + 29 q^{17} - 1830 q^{18} - 124 q^{19} + 285 q^{20} - 232 q^{21} - 747 q^{22} - 1628 q^{23} - 1802 q^{24} - 2540 q^{25} - 263 q^{26} - 220 q^{27} + 214 q^{28} + 267 q^{29} + 5914 q^{30} - 1002 q^{31} - 320 q^{32} + 335 q^{33} - 1469 q^{34} + 278 q^{35} + 1677 q^{36} - 443 q^{37} + 265 q^{38} + 3197 q^{39} - 40 q^{40} + 347 q^{41} + 8306 q^{42} + 133 q^{43} - 770 q^{44} - 2902 q^{45} + 308 q^{46} - 4944 q^{47} + 4239 q^{48} + 508 q^{49} + 1080 q^{50} - 1450 q^{51} + 1961 q^{52} - 362 q^{53} - 2900 q^{54} - 1200 q^{55} - 188 q^{56} + 3151 q^{57} + 479 q^{58} + 7245 q^{59} - 2666 q^{60} + 423 q^{61} + 3407 q^{62} + 3674 q^{63} - 1752 q^{64} + 2076 q^{65} - 17211 q^{66} - 4475 q^{67} + 5648 q^{68} - 374 q^{69} - 13844 q^{70} + 6215 q^{71} - 9124 q^{72} - 1626 q^{73} + 4947 q^{74} + 3007 q^{75} - 3150 q^{76} + 840 q^{77} - 12274 q^{78} - 674 q^{79} - 2166 q^{80} + 3335 q^{81} - 9053 q^{82} + 10418 q^{83} - 13059 q^{84} + 12822 q^{85} + 6238 q^{86} + 7403 q^{87} + 13114 q^{88} - 9758 q^{89} + 4992 q^{90} - 2119 q^{91} + 38 q^{92} - 23142 q^{93} + 23236 q^{94} + 319 q^{95} + 1383 q^{96} + 3463 q^{97} + 1487 q^{98} - 5944 q^{99}+O(q^{100})$$ 1080 * q - 27 * q^2 - 19 * q^3 + 155 * q^4 - 30 * q^5 - 56 * q^6 + 9 * q^7 + 4 * q^8 + 364 * q^9 + q^10 - 41 * q^11 - 314 * q^12 - 710 * q^13 - 54 * q^14 - 474 * q^15 + 675 * q^16 + 29 * q^17 - 1830 * q^18 - 124 * q^19 + 285 * q^20 - 232 * q^21 - 747 * q^22 - 1628 * q^23 - 1802 * q^24 - 2540 * q^25 - 263 * q^26 - 220 * q^27 + 214 * q^28 + 267 * q^29 + 5914 * q^30 - 1002 * q^31 - 320 * q^32 + 335 * q^33 - 1469 * q^34 + 278 * q^35 + 1677 * q^36 - 443 * q^37 + 265 * q^38 + 3197 * q^39 - 40 * q^40 + 347 * q^41 + 8306 * q^42 + 133 * q^43 - 770 * q^44 - 2902 * q^45 + 308 * q^46 - 4944 * q^47 + 4239 * q^48 + 508 * q^49 + 1080 * q^50 - 1450 * q^51 + 1961 * q^52 - 362 * q^53 - 2900 * q^54 - 1200 * q^55 - 188 * q^56 + 3151 * q^57 + 479 * q^58 + 7245 * q^59 - 2666 * q^60 + 423 * q^61 + 3407 * q^62 + 3674 * q^63 - 1752 * q^64 + 2076 * q^65 - 17211 * q^66 - 4475 * q^67 + 5648 * q^68 - 374 * q^69 - 13844 * q^70 + 6215 * q^71 - 9124 * q^72 - 1626 * q^73 + 4947 * q^74 + 3007 * q^75 - 3150 * q^76 + 840 * q^77 - 12274 * q^78 - 674 * q^79 - 2166 * q^80 + 3335 * q^81 - 9053 * q^82 + 10418 * q^83 - 13059 * q^84 + 12822 * q^85 + 6238 * q^86 + 7403 * q^87 + 13114 * q^88 - 9758 * q^89 + 4992 * q^90 - 2119 * q^91 + 38 * q^92 - 23142 * q^93 + 23236 * q^94 + 319 * q^95 + 1383 * q^96 + 3463 * q^97 + 1487 * q^98 - 5944 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −4.36857 + 3.28276i 0.666615 3.26530i 6.08208 20.9978i 11.6758 + 6.12792i 7.80705 + 16.4530i −30.8182 5.00845i 26.8589 + 70.8210i 14.6217 + 6.22971i −71.1229 + 11.5586i
3.2 −4.25141 + 3.19473i −1.68365 + 8.24708i 5.64247 19.4801i −11.4022 5.98435i −19.1893 40.4405i −33.3973 5.42759i 23.1588 + 61.0648i −40.3401 17.1873i 67.5939 10.9851i
3.3 −4.12652 + 3.10088i 0.990669 4.85262i 5.18696 17.9075i −10.2304 5.36931i 10.9593 + 23.0963i 3.16043 + 0.513619i 19.4817 + 51.3691i 2.27298 + 0.968427i 58.8654 9.56655i
3.4 −4.11168 + 3.08973i −0.342888 + 1.67958i 5.13377 17.7238i −10.9675 5.75617i −3.77959 7.96533i 28.5869 + 4.64582i 19.0630 + 50.2651i 22.1360 + 9.43128i 62.8797 10.2189i
3.5 −3.78216 + 2.84211i −1.59681 + 7.82171i 4.00142 13.8145i 15.7312 + 8.25639i −16.1907 34.1213i 15.0706 + 2.44921i 10.7072 + 28.2326i −33.7899 14.3965i −82.9636 + 13.4829i
3.6 −3.76951 + 2.83260i 1.50425 7.36828i 3.95981 13.6708i 13.5333 + 7.10282i 15.2011 + 32.0357i 27.2448 + 4.42771i 10.4213 + 27.4788i −27.1894 11.5843i −71.1333 + 11.5603i
3.7 −3.59815 + 2.70384i −1.15347 + 5.65006i 3.41023 11.7735i 1.58751 + 0.833188i −11.1265 23.4486i 7.26213 + 1.18021i 6.79493 + 17.9167i −5.75321 2.45122i −7.96489 + 1.29442i
3.8 −3.30946 + 2.48690i −0.286302 + 1.40240i 2.54213 8.77646i 3.60691 + 1.89305i −2.54012 5.35320i −2.89951 0.471216i 1.66938 + 4.40180i 22.9547 + 9.78008i −16.6447 + 2.70503i
3.9 −3.14814 + 2.36567i 1.68249 8.24137i 2.08864 7.21081i −7.14936 3.75227i 14.1997 + 29.9252i −10.7212 1.74236i −0.688137 1.81447i −40.2500 17.1489i 31.3839 5.10037i
3.10 −3.11755 + 2.34268i 0.204779 1.00307i 2.00519 6.92271i 4.77311 + 2.50512i 1.71148 + 3.60686i −12.1144 1.96879i −1.09622 2.89048i 23.8752 + 10.1723i −20.7491 + 3.37206i
3.11 −2.50470 + 1.88216i 0.456815 2.23763i 0.505246 1.74431i −17.9736 9.43330i 3.06739 + 6.46438i −16.3193 2.65214i −6.87038 18.1157i 20.0411 + 8.53873i 62.7734 10.2017i
3.12 −2.34304 + 1.76068i −1.17092 + 5.73554i 0.164101 0.566542i −14.8440 7.79075i −7.35495 15.5002i −0.299423 0.0486610i −7.70132 20.3067i −6.68597 2.84863i 48.4972 7.88156i
3.13 −2.15092 + 1.61631i 1.23779 6.06311i −0.211743 + 0.731021i −2.40618 1.26286i 7.13748 + 15.0419i 23.6338 + 3.84087i −8.35871 22.0401i −10.3897 4.42663i 7.21667 1.17282i
3.14 −2.01266 + 1.51241i −1.46355 + 7.16893i −0.462348 + 1.59621i 12.1937 + 6.39972i −7.89677 16.6421i −29.4267 4.78231i −8.62553 22.7437i −24.4121 10.4010i −34.2206 + 5.56140i
3.15 −1.96807 + 1.47891i −1.90327 + 9.32284i −0.539605 + 1.86293i −2.56285 1.34509i −10.0419 21.1628i 15.5737 + 2.53097i −8.67688 22.8791i −58.4535 24.9047i 7.03313 1.14300i
3.16 −1.80785 + 1.35851i 1.80580 8.84539i −0.802965 + 2.77216i 13.2860 + 6.97301i 8.75196 + 18.4444i −29.1390 4.73555i −8.72958 23.0180i −50.1405 21.3629i −33.4920 + 5.44298i
3.17 −1.80419 + 1.35576i 0.148573 0.727759i −0.808720 + 2.79203i 17.9465 + 9.41905i 0.718615 + 1.51445i 10.7406 + 1.74552i −8.72846 23.0150i 24.3319 + 10.3668i −45.1489 + 7.33742i
3.18 −1.36129 + 1.02295i −0.443383 + 2.17184i −1.41904 + 4.89909i −2.67571 1.40432i −1.61809 3.41006i 33.0738 + 5.37501i −7.91035 20.8579i 20.3192 + 8.65719i 5.07897 0.825413i
3.19 −0.917761 + 0.689653i −1.14631 + 5.61498i −1.85907 + 6.41827i 0.345192 + 0.181171i −2.82035 5.94377i −16.0749 2.61243i −5.97689 15.7598i −5.37458 2.28989i −0.441748 + 0.0717911i
3.20 −0.869369 + 0.653288i 1.46258 7.16421i −1.89672 + 6.54825i 0.457687 + 0.240213i 3.40877 + 7.18382i 7.99891 + 1.29995i −5.71391 15.0664i −24.3473 10.3734i −0.554827 + 0.0901682i
See next 80 embeddings (of 1080 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 165.45 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.i even 39 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.i.a 1080
169.i even 39 1 inner 169.4.i.a 1080

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.4.i.a 1080 1.a even 1 1 trivial
169.4.i.a 1080 169.i even 39 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(169, [\chi])$$.