# Properties

 Label 585.2.i.g Level $585$ Weight $2$ Character orbit 585.i Analytic conductor $4.671$ Analytic rank $0$ Dimension $26$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$26$$ Relative dimension: $$13$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$26 q + q^{2} + q^{3} - 15 q^{4} - 13 q^{5} + 7 q^{6} - 10 q^{7} - 12 q^{8} + 7 q^{9}+O(q^{10})$$ 26 * q + q^2 + q^3 - 15 * q^4 - 13 * q^5 + 7 * q^6 - 10 * q^7 - 12 * q^8 + 7 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$26 q + q^{2} + q^{3} - 15 q^{4} - 13 q^{5} + 7 q^{6} - 10 q^{7} - 12 q^{8} + 7 q^{9} - 2 q^{10} - 11 q^{11} - 20 q^{12} + 13 q^{13} - 15 q^{14} - 2 q^{15} - 19 q^{16} + 6 q^{17} - 13 q^{18} + 30 q^{19} - 15 q^{20} - q^{21} - 12 q^{22} - 6 q^{23} + 20 q^{24} - 13 q^{25} + 2 q^{26} - 2 q^{27} + 10 q^{28} - 14 q^{29} - 2 q^{30} - 30 q^{31} + 43 q^{32} + 6 q^{33} - 19 q^{34} + 20 q^{35} - 39 q^{36} + 32 q^{37} + 2 q^{38} + 2 q^{39} + 6 q^{40} - 17 q^{41} + 83 q^{42} - 6 q^{43} + 46 q^{44} - 5 q^{45} + 46 q^{46} + 21 q^{47} - 7 q^{48} - 29 q^{49} + q^{50} + 55 q^{51} + 15 q^{52} + 2 q^{53} - 6 q^{54} + 22 q^{55} - 37 q^{56} + 33 q^{57} - 14 q^{58} - 13 q^{59} + 25 q^{60} - 22 q^{61} - 114 q^{62} - 44 q^{63} + 84 q^{64} + 13 q^{65} - 68 q^{66} - 35 q^{67} + 4 q^{68} - 38 q^{69} - 15 q^{70} + 24 q^{71} - 24 q^{72} + 64 q^{73} + 28 q^{74} + q^{75} - 54 q^{76} - 4 q^{77} + 2 q^{78} - 12 q^{79} + 38 q^{80} + 15 q^{81} + 46 q^{82} + 3 q^{83} + 27 q^{84} - 3 q^{85} + 40 q^{86} - 12 q^{87} - 29 q^{88} + 12 q^{89} - 10 q^{90} - 20 q^{91} + 16 q^{92} - 9 q^{93} - 44 q^{94} - 15 q^{95} + 51 q^{96} - 33 q^{97} + 70 q^{98} - 22 q^{99}+O(q^{100})$$ 26 * q + q^2 + q^3 - 15 * q^4 - 13 * q^5 + 7 * q^6 - 10 * q^7 - 12 * q^8 + 7 * q^9 - 2 * q^10 - 11 * q^11 - 20 * q^12 + 13 * q^13 - 15 * q^14 - 2 * q^15 - 19 * q^16 + 6 * q^17 - 13 * q^18 + 30 * q^19 - 15 * q^20 - q^21 - 12 * q^22 - 6 * q^23 + 20 * q^24 - 13 * q^25 + 2 * q^26 - 2 * q^27 + 10 * q^28 - 14 * q^29 - 2 * q^30 - 30 * q^31 + 43 * q^32 + 6 * q^33 - 19 * q^34 + 20 * q^35 - 39 * q^36 + 32 * q^37 + 2 * q^38 + 2 * q^39 + 6 * q^40 - 17 * q^41 + 83 * q^42 - 6 * q^43 + 46 * q^44 - 5 * q^45 + 46 * q^46 + 21 * q^47 - 7 * q^48 - 29 * q^49 + q^50 + 55 * q^51 + 15 * q^52 + 2 * q^53 - 6 * q^54 + 22 * q^55 - 37 * q^56 + 33 * q^57 - 14 * q^58 - 13 * q^59 + 25 * q^60 - 22 * q^61 - 114 * q^62 - 44 * q^63 + 84 * q^64 + 13 * q^65 - 68 * q^66 - 35 * q^67 + 4 * q^68 - 38 * q^69 - 15 * q^70 + 24 * q^71 - 24 * q^72 + 64 * q^73 + 28 * q^74 + q^75 - 54 * q^76 - 4 * q^77 + 2 * q^78 - 12 * q^79 + 38 * q^80 + 15 * q^81 + 46 * q^82 + 3 * q^83 + 27 * q^84 - 3 * q^85 + 40 * q^86 - 12 * q^87 - 29 * q^88 + 12 * q^89 - 10 * q^90 - 20 * q^91 + 16 * q^92 - 9 * q^93 - 44 * q^94 - 15 * q^95 + 51 * q^96 - 33 * q^97 + 70 * q^98 - 22 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
196.1 −1.23671 + 2.14204i −1.15563 1.29016i −2.05890 3.56612i −0.500000 0.866025i 4.19276 0.879859i −2.17152 + 3.76119i 5.23822 −0.329032 + 2.98190i 2.47342
196.2 −1.20515 + 2.08737i 1.64592 0.539392i −1.90475 3.29913i −0.500000 0.866025i −0.857662 + 4.08570i −0.497502 + 0.861700i 4.36143 2.41811 1.77559i 2.41029
196.3 −0.877561 + 1.51998i 0.0726124 1.73053i −0.540227 0.935700i −0.500000 0.866025i 2.56665 + 1.62901i 1.83503 3.17836i −1.61392 −2.98945 0.251316i 1.75512
196.4 −0.780853 + 1.35248i 0.0458444 + 1.73144i −0.219464 0.380123i −0.500000 0.866025i −2.37754 1.29000i −2.47307 + 4.28349i −2.43794 −2.99580 + 0.158754i 1.56171
196.5 −0.643841 + 1.11517i −1.73180 0.0293427i 0.170938 + 0.296073i −0.500000 0.866025i 1.14773 1.91235i 0.391685 0.678419i −3.01559 2.99828 + 0.101632i 1.28768
196.6 −0.0377835 + 0.0654430i −1.16946 + 1.27764i 0.997145 + 1.72711i −0.500000 0.866025i −0.0394262 0.124807i −0.0159135 + 0.0275629i −0.301837 −0.264725 2.98830i 0.0755671
196.7 0.0514441 0.0891038i 1.71492 + 0.243008i 0.994707 + 1.72288i −0.500000 0.866025i 0.109875 0.140305i −2.41664 + 4.18575i 0.410464 2.88189 + 0.833478i −0.102888
196.8 0.238506 0.413105i 1.59029 + 0.686267i 0.886230 + 1.53499i −0.500000 0.866025i 0.662795 0.493279i 0.335964 0.581906i 1.79951 2.05807 + 2.18273i −0.477012
196.9 0.669930 1.16035i 0.478101 1.66476i 0.102389 + 0.177343i −0.500000 0.866025i −1.61141 1.67004i −0.186958 + 0.323821i 2.95409 −2.54284 1.59184i −1.33986
196.10 0.768621 1.33129i −1.52975 0.812318i −0.181557 0.314466i −0.500000 0.866025i −2.25723 + 1.41218i −1.69912 + 2.94296i 2.51629 1.68028 + 2.48529i −1.53724
196.11 0.853509 1.47832i 0.733344 + 1.56914i −0.456955 0.791470i −0.500000 0.866025i 2.94561 + 0.255158i 1.35209 2.34188i 1.85397 −1.92441 + 2.30144i −1.70702
196.12 1.34334 2.32674i −1.55252 + 0.767915i −2.60914 4.51916i −0.500000 0.866025i −0.298825 + 4.64387i 1.78091 3.08463i −8.64648 1.82061 2.38440i −2.68668
196.13 1.35654 2.34960i 1.35813 1.07494i −2.68041 4.64261i −0.500000 0.866025i −0.683325 4.64925i −1.23495 + 2.13900i −9.11822 0.689008 2.91981i −2.71308
391.1 −1.23671 2.14204i −1.15563 + 1.29016i −2.05890 + 3.56612i −0.500000 + 0.866025i 4.19276 + 0.879859i −2.17152 3.76119i 5.23822 −0.329032 2.98190i 2.47342
391.2 −1.20515 2.08737i 1.64592 + 0.539392i −1.90475 + 3.29913i −0.500000 + 0.866025i −0.857662 4.08570i −0.497502 0.861700i 4.36143 2.41811 + 1.77559i 2.41029
391.3 −0.877561 1.51998i 0.0726124 + 1.73053i −0.540227 + 0.935700i −0.500000 + 0.866025i 2.56665 1.62901i 1.83503 + 3.17836i −1.61392 −2.98945 + 0.251316i 1.75512
391.4 −0.780853 1.35248i 0.0458444 1.73144i −0.219464 + 0.380123i −0.500000 + 0.866025i −2.37754 + 1.29000i −2.47307 4.28349i −2.43794 −2.99580 0.158754i 1.56171
391.5 −0.643841 1.11517i −1.73180 + 0.0293427i 0.170938 0.296073i −0.500000 + 0.866025i 1.14773 + 1.91235i 0.391685 + 0.678419i −3.01559 2.99828 0.101632i 1.28768
391.6 −0.0377835 0.0654430i −1.16946 1.27764i 0.997145 1.72711i −0.500000 + 0.866025i −0.0394262 + 0.124807i −0.0159135 0.0275629i −0.301837 −0.264725 + 2.98830i 0.0755671
391.7 0.0514441 + 0.0891038i 1.71492 0.243008i 0.994707 1.72288i −0.500000 + 0.866025i 0.109875 + 0.140305i −2.41664 4.18575i 0.410464 2.88189 0.833478i −0.102888
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.i.g 26
3.b odd 2 1 1755.2.i.g 26
9.c even 3 1 inner 585.2.i.g 26
9.c even 3 1 5265.2.a.bg 13
9.d odd 6 1 1755.2.i.g 26
9.d odd 6 1 5265.2.a.bh 13

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.g 26 1.a even 1 1 trivial
585.2.i.g 26 9.c even 3 1 inner
1755.2.i.g 26 3.b odd 2 1
1755.2.i.g 26 9.d odd 6 1
5265.2.a.bg 13 9.c even 3 1
5265.2.a.bh 13 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$:

 $$T_{2}^{26} - T_{2}^{25} + 21 T_{2}^{24} - 14 T_{2}^{23} + 267 T_{2}^{22} - 149 T_{2}^{21} + 2136 T_{2}^{20} - 954 T_{2}^{19} + 12399 T_{2}^{18} - 5148 T_{2}^{17} + 50685 T_{2}^{16} - 19869 T_{2}^{15} + 155359 T_{2}^{14} - 61123 T_{2}^{13} + \cdots + 4$$ T2^26 - T2^25 + 21*T2^24 - 14*T2^23 + 267*T2^22 - 149*T2^21 + 2136*T2^20 - 954*T2^19 + 12399*T2^18 - 5148*T2^17 + 50685*T2^16 - 19869*T2^15 + 155359*T2^14 - 61123*T2^13 + 341427*T2^12 - 131796*T2^11 + 552646*T2^10 - 215147*T2^9 + 584316*T2^8 - 214378*T2^7 + 398096*T2^6 - 145432*T2^5 + 73109*T2^4 - 2873*T2^3 + 559*T2^2 + 6*T2 + 4 $$T_{7}^{26} + 10 T_{7}^{25} + 110 T_{7}^{24} + 596 T_{7}^{23} + 4027 T_{7}^{22} + 16491 T_{7}^{21} + 92006 T_{7}^{20} + 299999 T_{7}^{19} + 1387381 T_{7}^{18} + 3533585 T_{7}^{17} + 14671332 T_{7}^{16} + \cdots + 36864$$ T7^26 + 10*T7^25 + 110*T7^24 + 596*T7^23 + 4027*T7^22 + 16491*T7^21 + 92006*T7^20 + 299999*T7^19 + 1387381*T7^18 + 3533585*T7^17 + 14671332*T7^16 + 29629349*T7^15 + 106529712*T7^14 + 151444445*T7^13 + 505428863*T7^12 + 516689838*T7^11 + 1435095536*T7^10 + 412615400*T7^9 + 1367404368*T7^8 + 163916336*T7^7 + 1012685408*T7^6 - 5516800*T7^5 + 262812160*T7^4 + 56736768*T7^3 + 38199552*T7^2 + 1207296*T7 + 36864