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Properties

Label 145.3.s.a
Level $145$
Weight $3$
Character orbit 145.s
Analytic conductor $3.951$
Analytic rank $0$
Dimension $336$
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] = [145,3,Mod(14,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([14, 13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.14");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 145.s (of order \(28\), degree \(12\), 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: \(3.95096383322\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{4} - 14 q^{5} - 28 q^{6} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 28 q^{4} - 14 q^{5} - 28 q^{6} - 28 q^{9} - 24 q^{10} - 36 q^{11} - 64 q^{14} + 36 q^{15} + 156 q^{16} + 16 q^{19} - 194 q^{20} + 56 q^{21} - 388 q^{24} - 154 q^{25} + 132 q^{26} + 108 q^{29} - 272 q^{30} - 192 q^{31} + 392 q^{34} + 182 q^{35} - 412 q^{36} + 292 q^{39} + 524 q^{40} - 260 q^{41} - 4 q^{44} - 294 q^{45} - 24 q^{46} + 756 q^{49} - 622 q^{50} - 28 q^{51} - 516 q^{54} - 124 q^{55} - 536 q^{56} + 40 q^{59} - 8 q^{60} - 268 q^{61} - 28 q^{64} + 862 q^{65} - 216 q^{66} + 100 q^{69} + 2034 q^{70} - 1288 q^{71} + 1068 q^{74} + 366 q^{75} - 2112 q^{76} + 132 q^{79} + 1162 q^{80} - 340 q^{81} - 1984 q^{84} + 94 q^{85} + 612 q^{89} - 1100 q^{90} + 812 q^{91} - 804 q^{94} - 1282 q^{95} + 3892 q^{96} - 1640 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −3.85038 + 0.433834i −1.01590 + 0.638332i 10.7375 − 2.45077i −0.998472 − 4.89929i 3.63467 − 2.89856i 3.65359 + 0.833908i −25.6512 + 8.97573i −3.28037 + 6.81176i 5.96998 + 18.4310i
14.2 −3.52806 + 0.397517i 3.48277 − 2.18837i 8.38948 − 1.91484i −4.17158 + 2.75643i −11.4175 + 9.10517i −9.81356 − 2.23988i −15.4328 + 5.40017i 3.43579 − 7.13449i 13.6219 − 11.3831i
14.3 −3.26339 + 0.367695i 1.09699 − 0.689283i 6.61477 − 1.50978i 4.32149 + 2.51490i −3.32645 + 2.65275i −0.136475 − 0.0311495i −8.63243 + 3.02062i −3.17668 + 6.59645i −15.0274 − 6.61810i
14.4 −2.93224 + 0.330384i 4.35726 − 2.73785i 4.58918 − 1.04745i 3.41728 − 3.64996i −11.8720 + 9.46761i 7.30181 + 1.66659i −1.96969 + 0.689223i 7.58497 − 15.7504i −8.81441 + 11.8316i
14.5 −2.91400 + 0.328329i −1.76404 + 1.10842i 4.48389 − 1.02342i −1.52392 + 4.76211i 4.77650 − 3.80913i 7.02402 + 1.60319i −1.65853 + 0.580344i −2.02171 + 4.19811i 2.87717 − 14.3771i
14.6 −2.66893 + 0.300716i −4.31263 + 2.70980i 3.13304 − 0.715095i 4.34812 − 2.46857i 10.6952 − 8.52915i 2.68458 + 0.612739i 1.99358 − 0.697583i 7.35076 − 15.2640i −10.8625 + 7.89599i
14.7 −2.47841 + 0.279249i −2.58159 + 1.62212i 2.16480 − 0.494103i −4.78429 − 1.45277i 5.94525 − 4.74118i −8.33939 − 1.90341i 4.18923 − 1.46587i 0.128373 − 0.266570i 12.2631 + 2.26454i
14.8 −1.86101 + 0.209686i 3.09225 − 1.94299i −0.480315 + 0.109629i −4.89917 + 0.999073i −5.34729 + 4.26432i 7.61772 + 1.73870i 7.94166 − 2.77891i 1.88184 − 3.90768i 8.90792 − 2.88657i
14.9 −1.85134 + 0.208596i 0.937114 − 0.588828i −0.515764 + 0.117720i 4.27846 − 2.58743i −1.61209 + 1.28560i −12.5954 − 2.87482i 7.96432 − 2.78684i −3.37349 + 7.00512i −7.38116 + 5.68268i
14.10 −1.34327 + 0.151350i 0.435808 − 0.273836i −2.11825 + 0.483477i −2.52119 − 4.31783i −0.543962 + 0.433795i 5.44079 + 1.24182i 7.87585 − 2.75588i −3.79001 + 7.87004i 4.04013 + 5.41841i
14.11 −1.15793 + 0.130467i 2.24873 − 1.41297i −2.57594 + 0.587941i 1.89450 + 4.62719i −2.41952 + 1.92950i 0.227968 + 0.0520323i 7.30549 − 2.55630i −0.844648 + 1.75393i −2.79739 − 5.11077i
14.12 −0.756022 + 0.0851832i −3.45294 + 2.16962i −3.33540 + 0.761283i 1.60138 + 4.73662i 2.42568 − 1.93442i −5.75129 − 1.31269i 5.32923 − 1.86478i 3.31056 − 6.87444i −1.61416 − 3.44458i
14.13 −0.398939 + 0.0449496i −1.27000 + 0.797991i −3.74258 + 0.854219i 2.48777 − 4.33717i 0.470781 − 0.375435i 5.26443 + 1.20157i 2.97040 − 1.03939i −2.92886 + 6.08183i −0.797513 + 1.84209i
14.14 −0.0495626 + 0.00558437i 4.70919 − 2.95898i −3.89729 + 0.889530i −2.19083 − 4.49447i −0.216876 + 0.172953i −10.1258 − 2.31116i 0.376502 − 0.131744i 9.51594 − 19.7600i 0.133682 + 0.210523i
14.15 0.0495626 − 0.00558437i −4.70919 + 2.95898i −3.89729 + 0.889530i −4.86929 − 1.13579i −0.216876 + 0.172953i 10.1258 + 2.31116i −0.376502 + 0.131744i 9.51594 − 19.7600i −0.247677 − 0.0291007i
14.16 0.398939 − 0.0449496i 1.27000 − 0.797991i −3.74258 + 0.854219i −3.67485 + 3.39050i 0.470781 − 0.375435i −5.26443 − 1.20157i −2.97040 + 1.03939i −2.92886 + 6.08183i −1.31364 + 1.51779i
14.17 0.756022 − 0.0851832i 3.45294 − 2.16962i −3.33540 + 0.761283i 4.97421 + 0.507232i 2.42568 − 1.93442i 5.75129 + 1.31269i −5.32923 + 1.86478i 3.31056 − 6.87444i 3.80382 − 0.0402405i
14.18 1.15793 − 0.130467i −2.24873 + 1.41297i −2.57594 + 0.587941i 4.93274 + 0.817359i −2.41952 + 1.92950i −0.227968 − 0.0520323i −7.30549 + 2.55630i −0.844648 + 1.75393i 5.81839 + 0.302882i
14.19 1.34327 − 0.151350i −0.435808 + 0.273836i −2.11825 + 0.483477i −4.77059 − 1.49717i −0.543962 + 0.433795i −5.44079 − 1.24182i −7.87585 + 2.75588i −3.79001 + 7.87004i −6.63477 − 1.28907i
14.20 1.85134 − 0.208596i −0.937114 + 0.588828i −0.515764 + 0.117720i −1.57051 + 4.74695i −1.61209 + 1.28560i 12.5954 + 2.87482i −7.96432 + 2.78684i −3.37349 + 7.00512i −1.91735 + 9.11581i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.f odd 28 1 inner
145.s odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.3.s.a ✓ 336
5.b even 2 1 inner 145.3.s.a ✓ 336
29.f odd 28 1 inner 145.3.s.a ✓ 336
145.s odd 28 1 inner 145.3.s.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.3.s.a ✓ 336 1.a even 1 1 trivial
145.3.s.a ✓ 336 5.b even 2 1 inner
145.3.s.a ✓ 336 29.f odd 28 1 inner
145.3.s.a ✓ 336 145.s odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(145, [\chi])\).