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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 $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(196,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.196");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + q^{2} + q^{3} - 15 q^{4} - 13 q^{5} + 7 q^{6} - 10 q^{7} - 12 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + q^{2} + q^{3} - 15 q^{4} - 13 q^{5} + 7 q^{6} - 10 q^{7} - 12 q^{8} + 7 q^{9} - 2 q^{10} - 11 q^{11} - 20 q^{12} + 13 q^{13} - 15 q^{14} - 2 q^{15} - 19 q^{16} + 6 q^{17} - 13 q^{18} + 30 q^{19} - 15 q^{20} - q^{21} - 12 q^{22} - 6 q^{23} + 20 q^{24} - 13 q^{25} + 2 q^{26} - 2 q^{27} + 10 q^{28} - 14 q^{29} - 2 q^{30} - 30 q^{31} + 43 q^{32} + 6 q^{33} - 19 q^{34} + 20 q^{35} - 39 q^{36} + 32 q^{37} + 2 q^{38} + 2 q^{39} + 6 q^{40} - 17 q^{41} + 83 q^{42} - 6 q^{43} + 46 q^{44} - 5 q^{45} + 46 q^{46} + 21 q^{47} - 7 q^{48} - 29 q^{49} + q^{50} + 55 q^{51} + 15 q^{52} + 2 q^{53} - 6 q^{54} + 22 q^{55} - 37 q^{56} + 33 q^{57} - 14 q^{58} - 13 q^{59} + 25 q^{60} - 22 q^{61} - 114 q^{62} - 44 q^{63} + 84 q^{64} + 13 q^{65} - 68 q^{66} - 35 q^{67} + 4 q^{68} - 38 q^{69} - 15 q^{70} + 24 q^{71} - 24 q^{72} + 64 q^{73} + 28 q^{74} + q^{75} - 54 q^{76} - 4 q^{77} + 2 q^{78} - 12 q^{79} + 38 q^{80} + 15 q^{81} + 46 q^{82} + 3 q^{83} + 27 q^{84} - 3 q^{85} + 40 q^{86} - 12 q^{87} - 29 q^{88} + 12 q^{89} - 10 q^{90} - 20 q^{91} + 16 q^{92} - 9 q^{93} - 44 q^{94} - 15 q^{95} + 51 q^{96} - 33 q^{97} + 70 q^{98} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 196.13
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{26} - T_{2}^{25} + 21 T_{2}^{24} - 14 T_{2}^{23} + 267 T_{2}^{22} - 149 T_{2}^{21} + 2136 T_{2}^{20} + \cdots + 4 \) Copy content Toggle raw display
\( T_{7}^{26} + 10 T_{7}^{25} + 110 T_{7}^{24} + 596 T_{7}^{23} + 4027 T_{7}^{22} + 16491 T_{7}^{21} + \cdots + 36864 \) Copy content Toggle raw display