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Properties

Label 585.2.dd.a
Level $585$
Weight $2$
Character orbit 585.dd
Analytic conductor $4.671$
Analytic rank $0$
Dimension $224$
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(41,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.dd (of order \(12\), degree \(4\), 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: \(224\)
Relative dimension: \(56\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 12 q^{6} - 8 q^{7} + 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 12 q^{6} - 8 q^{7} + 36 q^{8} - 12 q^{11} - 4 q^{15} + 112 q^{16} + 44 q^{18} - 8 q^{19} - 36 q^{21} - 8 q^{24} - 48 q^{26} + 24 q^{27} - 16 q^{28} - 48 q^{30} - 16 q^{31} - 72 q^{32} + 4 q^{33} - 4 q^{37} + 8 q^{39} - 24 q^{41} - 128 q^{42} - 16 q^{45} - 72 q^{47} - 56 q^{48} - 4 q^{52} + 8 q^{54} + 36 q^{57} + 36 q^{58} - 72 q^{59} + 32 q^{60} + 36 q^{62} - 8 q^{63} - 108 q^{66} + 28 q^{67} - 48 q^{71} - 204 q^{72} + 28 q^{73} + 16 q^{76} + 144 q^{77} - 60 q^{78} - 12 q^{79} - 48 q^{80} + 12 q^{81} - 36 q^{83} - 16 q^{84} - 12 q^{85} - 108 q^{86} - 88 q^{87} + 60 q^{89} + 16 q^{91} - 120 q^{92} + 108 q^{93} + 16 q^{96} - 4 q^{97} + 96 q^{98} + 116 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −2.63134 + 0.705065i −1.57219 − 0.726783i 4.69478 − 2.71053i −0.965926 + 0.258819i 4.64940 + 0.803915i −0.762413 − 0.762413i −6.58990 + 6.58990i 1.94357 + 2.28528i 2.35919 − 1.36208i
41.2 −2.61595 + 0.700942i 1.02288 + 1.39775i 4.61983 − 2.66726i 0.965926 − 0.258819i −3.65555 − 2.93948i −2.36954 − 2.36954i −6.38565 + 6.38565i −0.907438 + 2.85947i −2.34540 + 1.35412i
41.3 −2.51603 + 0.674168i 1.72636 + 0.140333i 4.14384 − 2.39245i −0.965926 + 0.258819i −4.43817 + 0.810772i 0.557406 + 0.557406i −5.12940 + 5.12940i 2.96061 + 0.484530i 2.25581 − 1.30239i
41.4 −2.37759 + 0.637074i 0.286340 − 1.70822i 3.51504 − 2.02941i −0.965926 + 0.258819i 0.407461 + 4.24387i 2.62190 + 2.62190i −3.58341 + 3.58341i −2.83602 − 0.978264i 2.13169 − 1.23073i
41.5 −2.33010 + 0.624348i −0.643338 − 1.60814i 3.30750 − 1.90959i 0.965926 − 0.258819i 2.50308 + 3.34546i −3.35526 − 3.35526i −3.10305 + 3.10305i −2.17223 + 2.06916i −2.08911 + 1.20615i
41.6 −2.32002 + 0.621648i 1.63389 − 0.574817i 3.26400 − 1.88447i 0.965926 − 0.258819i −3.43332 + 2.34929i 1.69373 + 1.69373i −3.00432 + 3.00432i 2.33917 − 1.87837i −2.08007 + 1.20093i
41.7 −2.19862 + 0.589118i −1.70645 + 0.296724i 2.75481 − 1.59049i 0.965926 − 0.258819i 3.57702 − 1.65768i 0.239385 + 0.239385i −1.90079 + 1.90079i 2.82391 − 1.01269i −1.97123 + 1.13809i
41.8 −2.14844 + 0.575674i −0.298319 + 1.70617i 2.55236 − 1.47360i −0.965926 + 0.258819i −0.341273 − 3.83734i 2.15447 + 2.15447i −1.48974 + 1.48974i −2.82201 − 1.01797i 1.92624 − 1.11212i
41.9 −2.01247 + 0.539241i 1.24023 + 1.20906i 2.02722 − 1.17042i 0.965926 − 0.258819i −3.14791 − 1.76441i 3.20298 + 3.20298i −0.502129 + 0.502129i 0.0763573 + 2.99903i −1.80434 + 1.04173i
41.10 −1.93722 + 0.519077i 0.257107 − 1.71286i 1.75134 − 1.01114i 0.965926 − 0.258819i 0.391035 + 3.45165i 1.38486 + 1.38486i −0.0315896 + 0.0315896i −2.86779 − 0.880776i −1.73687 + 1.00278i
41.11 −1.92423 + 0.515595i 1.72898 − 0.103149i 1.70476 − 0.984242i −0.965926 + 0.258819i −3.27376 + 1.08993i −2.69815 − 2.69815i 0.0443954 − 0.0443954i 2.97872 − 0.356683i 1.72521 − 0.996053i
41.12 −1.92097 + 0.514722i −1.58422 + 0.700174i 1.69314 − 0.977533i −0.965926 + 0.258819i 2.68285 − 2.16045i −1.48966 − 1.48966i 0.0631888 − 0.0631888i 2.01951 − 2.21846i 1.72229 − 0.994367i
41.13 −1.70353 + 0.456459i −1.23799 − 1.21135i 0.961603 − 0.555182i −0.965926 + 0.258819i 2.66188 + 1.49849i −0.0263174 − 0.0263174i 1.10944 − 1.10944i 0.0652383 + 2.99929i 1.52734 − 0.881811i
41.14 −1.69282 + 0.453591i −0.270104 + 1.71086i 0.927854 − 0.535697i 0.965926 − 0.258819i −0.318791 − 3.01870i −2.25816 − 2.25816i 1.15076 − 1.15076i −2.85409 − 0.924222i −1.51774 + 0.876270i
41.15 −1.35599 + 0.363336i −0.619884 + 1.61733i −0.0253567 + 0.0146397i 0.965926 − 0.258819i 0.252923 − 2.41830i 0.411149 + 0.411149i 2.01437 − 2.01437i −2.23149 − 2.00511i −1.21575 + 0.701912i
41.16 −1.22167 + 0.327347i 1.04859 − 1.37857i −0.346718 + 0.200178i −0.965926 + 0.258819i −0.829760 + 2.02742i −2.25656 − 2.25656i 2.14670 − 2.14670i −0.800932 − 2.89111i 1.09532 − 0.632385i
41.17 −1.18182 + 0.316668i 1.18056 − 1.26739i −0.435625 + 0.251508i 0.965926 − 0.258819i −0.993869 + 1.87168i 0.251873 + 0.251873i 2.16550 − 2.16550i −0.212559 − 2.99246i −1.05959 + 0.611757i
41.18 −1.16984 + 0.313458i −1.49310 + 0.877863i −0.461781 + 0.266609i −0.965926 + 0.258819i 1.47152 − 1.49498i 0.482341 + 0.482341i 2.16940 − 2.16940i 1.45871 − 2.62148i 1.04885 − 0.605554i
41.19 −1.13687 + 0.304624i 1.57773 + 0.714686i −0.532366 + 0.307362i −0.965926 + 0.258819i −2.01139 − 0.331893i 1.92694 + 1.92694i 2.17610 − 2.17610i 1.97845 + 2.25516i 1.01929 − 0.588489i
41.20 −0.972552 + 0.260594i −0.606566 − 1.62237i −0.854103 + 0.493117i 0.965926 − 0.258819i 1.01270 + 1.41977i −0.808400 − 0.808400i 2.12607 − 2.12607i −2.26415 + 1.96815i −0.871966 + 0.503430i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.dd.a yes 224
9.d odd 6 1 585.2.cm.a ✓ 224
13.f odd 12 1 585.2.cm.a ✓ 224
117.bc even 12 1 inner 585.2.dd.a yes 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.cm.a ✓ 224 9.d odd 6 1
585.2.cm.a ✓ 224 13.f odd 12 1
585.2.dd.a yes 224 1.a even 1 1 trivial
585.2.dd.a yes 224 117.bc even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).