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Properties

Label 675.2.y.a
Level $675$
Weight $2$
Character orbit 675.y
Analytic conductor $5.390$
Analytic rank $0$
Dimension $224$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,2,Mod(19,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.y (of order \(30\), degree \(8\), not 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{30})\)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 5 q^{2} - 29 q^{4} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 5 q^{2} - 29 q^{4} + 20 q^{8} - 12 q^{10} - 5 q^{11} - 5 q^{13} + 23 q^{14} + 15 q^{16} + 20 q^{17} - 12 q^{19} + 17 q^{20} - 5 q^{22} + 5 q^{23} - 16 q^{25} - 72 q^{26} - 60 q^{28} + 15 q^{29} - 9 q^{31} - 7 q^{34} + 46 q^{35} - 20 q^{37} + 75 q^{38} - q^{40} - 13 q^{41} - 20 q^{44} - 4 q^{46} - 20 q^{47} + 56 q^{49} + 29 q^{50} - 15 q^{52} + 20 q^{53} - 44 q^{55} - 22 q^{56} - 5 q^{58} + 30 q^{59} - 3 q^{61} - 40 q^{62} - 12 q^{64} - 45 q^{65} + 10 q^{67} - 12 q^{70} + 106 q^{71} - 20 q^{73} - 82 q^{74} + 8 q^{76} + 115 q^{77} - 15 q^{79} + 22 q^{80} - 65 q^{83} - 21 q^{85} + 15 q^{86} - 5 q^{88} - 26 q^{89} - 54 q^{91} - 95 q^{92} + 41 q^{94} + 17 q^{95} - 5 q^{97} + 70 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.578019 + 2.71937i 0 −5.23375 − 2.33022i −2.07883 − 0.823700i 0 −1.90860 − 1.10193i 6.09369 − 8.38725i 0 3.44154 − 5.17697i
19.2 −0.539491 + 2.53811i 0 −4.32384 − 1.92510i 1.17379 + 1.90321i 0 1.26157 + 0.728365i 4.16839 − 5.73730i 0 −5.46381 + 1.95243i
19.3 −0.485498 + 2.28409i 0 −3.15426 − 1.40437i 1.74600 − 1.39696i 0 −1.06114 − 0.612649i 1.99399 − 2.74450i 0 2.34309 + 4.66624i
19.4 −0.454043 + 2.13611i 0 −2.52970 − 1.12630i −2.10220 − 0.762073i 0 2.11102 + 1.21880i 0.987242 − 1.35882i 0 2.58236 − 4.14451i
19.5 −0.401134 + 1.88719i 0 −1.57347 − 0.700555i 2.17003 + 0.539408i 0 2.05426 + 1.18603i −0.314833 + 0.433330i 0 −1.88844 + 3.87888i
19.6 −0.391880 + 1.84365i 0 −1.41839 − 0.631508i −0.424651 + 2.19537i 0 −0.724139 − 0.418082i −0.495642 + 0.682193i 0 −3.88109 − 1.64323i
19.7 −0.363246 + 1.70894i 0 −0.961428 − 0.428055i −2.04050 + 0.914526i 0 −3.25090 − 1.87691i −0.973104 + 1.33936i 0 −0.821664 − 3.81928i
19.8 −0.264379 + 1.24380i 0 0.349937 + 0.155802i −1.55641 − 1.60548i 0 1.19020 + 0.687160i −1.78115 + 2.45154i 0 2.40839 − 1.51142i
19.9 −0.250369 + 1.17789i 0 0.502342 + 0.223657i 1.04324 − 1.97779i 0 2.32094 + 1.33999i −1.80485 + 2.48416i 0 2.06843 + 1.72401i
19.10 −0.188119 + 0.885029i 0 1.07920 + 0.480492i 0.419997 − 2.19627i 0 −3.56731 − 2.05959i −1.69193 + 2.32874i 0 1.86475 + 0.784869i
19.11 −0.0965635 + 0.454296i 0 1.63003 + 0.725736i 1.56019 + 1.60181i 0 −3.50662 − 2.02455i −1.03309 + 1.42192i 0 −0.878353 + 0.554113i
19.12 −0.0963190 + 0.453145i 0 1.63103 + 0.726180i 2.22769 − 0.193424i 0 1.95191 + 1.12694i −1.03077 + 1.41873i 0 −0.126919 + 1.02810i
19.13 −0.0838034 + 0.394264i 0 1.67867 + 0.747392i 0.252792 + 2.22173i 0 4.03987 + 2.33242i −0.909187 + 1.25139i 0 −0.897134 − 0.0865220i
19.14 −0.0524738 + 0.246870i 0 1.76890 + 0.787565i −2.18454 − 0.477257i 0 −0.263893 − 0.152358i −0.583943 + 0.803728i 0 0.232451 − 0.514254i
19.15 0.0134680 − 0.0633621i 0 1.82326 + 0.811767i −1.33347 + 1.79495i 0 0.0617846 + 0.0356713i 0.152142 − 0.209405i 0 0.0957728 + 0.108666i
19.16 0.119971 − 0.564417i 0 1.52292 + 0.678046i 2.21766 − 0.286312i 0 −3.44219 − 1.98735i 1.24374 − 1.71186i 0 0.104455 − 1.28604i
19.17 0.182883 − 0.860395i 0 1.12026 + 0.498771i 0.712729 − 2.11944i 0 1.09544 + 0.632452i 1.66807 − 2.29590i 0 −1.69321 − 1.00084i
19.18 0.192241 − 0.904421i 0 1.04607 + 0.465740i 1.07309 + 1.96175i 0 0.805624 + 0.465127i 1.70929 − 2.35263i 0 1.98054 − 0.593396i
19.19 0.233920 − 1.10051i 0 0.670698 + 0.298614i −2.15310 + 0.603467i 0 −1.09191 − 0.630415i 1.80814 − 2.48869i 0 0.160466 + 2.51066i
19.20 0.255527 − 1.20216i 0 0.447195 + 0.199104i −0.173727 − 2.22931i 0 −1.60284 − 0.925401i 1.79842 − 2.47532i 0 −2.72438 − 0.360801i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.e even 10 1 inner
225.u even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.y.a 224
3.b odd 2 1 225.2.u.a ✓ 224
9.c even 3 1 inner 675.2.y.a 224
9.d odd 6 1 225.2.u.a ✓ 224
25.e even 10 1 inner 675.2.y.a 224
75.h odd 10 1 225.2.u.a ✓ 224
225.u even 30 1 inner 675.2.y.a 224
225.v odd 30 1 225.2.u.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.u.a ✓ 224 3.b odd 2 1
225.2.u.a ✓ 224 9.d odd 6 1
225.2.u.a ✓ 224 75.h odd 10 1
225.2.u.a ✓ 224 225.v odd 30 1
675.2.y.a 224 1.a even 1 1 trivial
675.2.y.a 224 9.c even 3 1 inner
675.2.y.a 224 25.e even 10 1 inner
675.2.y.a 224 225.u even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(675, [\chi])\).