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Properties

Label 675.2.r.a
Level $675$
Weight $2$
Character orbit 675.r
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(46,675)]
 
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, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.r (of order \(15\), 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 \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28} + 15 q^{29} + 3 q^{31} - 12 q^{32} + q^{34} - 14 q^{35} - 24 q^{37} - 55 q^{38} + q^{40} + 19 q^{41} - 8 q^{43} - 4 q^{44} - 20 q^{46} + 10 q^{47} - 72 q^{49} + 3 q^{50} - 25 q^{52} + 12 q^{53} - 20 q^{55} + 60 q^{56} - 23 q^{58} + 30 q^{59} - 3 q^{61} + 44 q^{62} - 44 q^{64} - 51 q^{65} - 12 q^{67} + 156 q^{68} - 16 q^{70} - 42 q^{71} - 12 q^{73} - 90 q^{74} - 8 q^{76} - 31 q^{77} - 15 q^{79} - 298 q^{80} + 8 q^{82} - 59 q^{83} - 11 q^{85} - 9 q^{86} - 23 q^{88} - 106 q^{89} + 30 q^{91} - 11 q^{92} + 25 q^{94} - 7 q^{95} - 21 q^{97} - 146 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.77417 − 1.97042i 0 −0.525803 + 5.00268i 1.61775 + 1.54366i 0 1.02644 − 1.77785i 6.50010 − 4.72260i 0 0.171493 − 5.92636i
46.2 −1.69452 − 1.88195i 0 −0.461298 + 4.38896i 2.12223 − 0.704374i 0 −1.03406 + 1.79105i 4.94396 − 3.59200i 0 −4.92175 − 2.80036i
46.3 −1.64688 − 1.82905i 0 −0.424138 + 4.03540i −0.681844 − 2.12957i 0 2.55879 − 4.43196i 4.09710 − 2.97672i 0 −2.77218 + 4.75428i
46.4 −1.55906 − 1.73151i 0 −0.358407 + 3.41002i −2.08227 + 0.814962i 0 −0.0399999 + 0.0692819i 2.69327 − 1.95678i 0 4.65750 + 2.33489i
46.5 −1.53828 − 1.70844i 0 −0.343384 + 3.26708i −0.322490 + 2.21269i 0 −1.62043 + 2.80667i 2.39008 − 1.73649i 0 4.27633 − 2.85279i
46.6 −1.15183 − 1.27924i 0 −0.100677 + 0.957875i 0.0644453 − 2.23514i 0 −2.11497 + 3.66323i −1.44394 + 1.04909i 0 −2.93350 + 2.49206i
46.7 −0.956880 − 1.06272i 0 −0.00470356 + 0.0447514i 2.07212 − 0.840433i 0 1.92019 − 3.32586i −2.26178 + 1.64328i 0 −2.87591 − 1.39789i
46.8 −0.955162 − 1.06081i 0 −0.00393676 + 0.0374558i −0.397449 + 2.20046i 0 1.20474 − 2.08666i −2.26620 + 1.64649i 0 2.71391 − 1.68018i
46.9 −0.797750 − 0.885991i 0 0.0604816 − 0.575444i −1.30610 − 1.81497i 0 0.157578 − 0.272934i −2.48714 + 1.80701i 0 −0.566103 + 2.60508i
46.10 −0.721003 − 0.800755i 0 0.0876938 − 0.834351i −2.23594 + 0.0235208i 0 0.316483 − 0.548165i −2.47480 + 1.79805i 0 1.63096 + 1.77348i
46.11 −0.513717 − 0.570540i 0 0.147446 − 1.40285i 1.29212 + 1.82494i 0 −1.54115 + 2.66935i −2.11835 + 1.53907i 0 0.377417 − 1.67471i
46.12 −0.510289 − 0.566733i 0 0.148265 − 1.41065i 0.965605 + 2.01683i 0 −0.476660 + 0.825599i −2.10906 + 1.53232i 0 0.650268 − 1.57641i
46.13 −0.349998 − 0.388712i 0 0.180458 − 1.71695i 0.130882 − 2.23223i 0 −1.68864 + 2.92481i −1.57689 + 1.14568i 0 −0.913505 + 0.730402i
46.14 0.110043 + 0.122216i 0 0.206230 − 1.96215i 2.06082 + 0.867759i 0 2.06663 − 3.57951i 0.528597 − 0.384048i 0 0.120726 + 0.347356i
46.15 0.136671 + 0.151788i 0 0.204696 − 1.94755i 1.79725 − 1.33037i 0 0.530047 − 0.918069i 0.654078 − 0.475215i 0 0.447567 + 0.0909796i
46.16 0.220390 + 0.244767i 0 0.197717 − 1.88116i −2.09765 + 0.774498i 0 −1.22023 + 2.11349i 1.03695 − 0.753387i 0 −0.651873 − 0.342746i
46.17 0.251657 + 0.279494i 0 0.194272 − 1.84837i 0.102616 − 2.23371i 0 1.17975 − 2.04339i 1.17403 − 0.852985i 0 0.650132 − 0.533449i
46.18 0.293899 + 0.326408i 0 0.188891 − 1.79718i −2.06853 + 0.849220i 0 −0.888066 + 1.53818i 1.35281 − 0.982874i 0 −0.885131 − 0.425600i
46.19 0.727367 + 0.807823i 0 0.0855417 − 0.813875i −0.934297 + 2.03152i 0 1.73969 − 3.01324i 2.47854 − 1.80077i 0 −2.32069 + 0.722917i
46.20 0.942959 + 1.04726i 0 0.00147092 − 0.0139948i 2.16654 − 0.553283i 0 −2.48652 + 4.30678i 2.29622 − 1.66830i 0 2.62239 + 1.74721i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.d even 5 1 inner
225.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.r.a 224
3.b odd 2 1 225.2.q.a ✓ 224
9.c even 3 1 inner 675.2.r.a 224
9.d odd 6 1 225.2.q.a ✓ 224
25.d even 5 1 inner 675.2.r.a 224
75.j odd 10 1 225.2.q.a ✓ 224
225.q even 15 1 inner 675.2.r.a 224
225.t odd 30 1 225.2.q.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.q.a ✓ 224 3.b odd 2 1
225.2.q.a ✓ 224 9.d odd 6 1
225.2.q.a ✓ 224 75.j odd 10 1
225.2.q.a ✓ 224 225.t odd 30 1
675.2.r.a 224 1.a even 1 1 trivial
675.2.r.a 224 9.c even 3 1 inner
675.2.r.a 224 25.d even 5 1 inner
675.2.r.a 224 225.q even 15 1 inner

Hecke kernels

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