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Properties

Label 240.4.bl.a
Level $240$
Weight $4$
Character orbit 240.bl
Analytic conductor $14.160$
Analytic rank $0$
Dimension $144$
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] = [240,4,Mod(109,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.109");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.bl (of order \(4\), 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: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(72\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{10} + 208 q^{14} - 132 q^{16} - 24 q^{19} - 228 q^{24} + 40 q^{26} - 408 q^{30} + 1488 q^{31} + 636 q^{34} - 456 q^{35} + 108 q^{36} + 1456 q^{40} + 2296 q^{44} - 12 q^{46} + 7056 q^{49} - 2848 q^{50} + 744 q^{51} - 108 q^{54} - 336 q^{56} + 1376 q^{59} + 1224 q^{60} - 912 q^{61} + 6192 q^{64} - 976 q^{65} - 1368 q^{66} + 528 q^{69} + 1728 q^{70} - 6568 q^{74} - 1104 q^{75} - 2996 q^{76} + 3488 q^{79} - 7032 q^{80} - 11664 q^{81} - 4432 q^{86} - 648 q^{90} - 1696 q^{91} + 11012 q^{94} - 7728 q^{95} - 5160 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −2.82805 − 0.0459143i −2.12132 + 2.12132i 7.99578 + 0.259696i −9.00419 − 6.62756i 6.09661 − 5.90181i −23.4741 −22.6006 − 1.10156i − 9.00000i 25.1600 + 19.1565i
109.2 −2.82186 + 0.192626i −2.12132 + 2.12132i 7.92579 − 1.08713i −11.1631 + 0.620159i 5.57745 − 6.39469i 8.56742 −22.1561 + 4.59443i − 9.00000i 31.3813 − 3.90031i
109.3 −2.81874 − 0.233854i 2.12132 − 2.12132i 7.89063 + 1.31835i −11.1801 − 0.0661330i −6.47554 + 5.48338i 17.8644 −21.9333 − 5.56133i − 9.00000i 31.4985 + 2.80093i
109.4 −2.81781 − 0.244875i 2.12132 − 2.12132i 7.88007 + 1.38002i 1.14903 + 11.1211i −6.49693 + 5.45801i −28.0016 −21.8666 − 5.81827i − 9.00000i −0.514466 − 31.6186i
109.5 −2.81443 + 0.281051i 2.12132 − 2.12132i 7.84202 − 1.58200i 9.79718 − 5.38658i −5.37411 + 6.56650i −13.5093 −21.6262 + 6.65642i − 9.00000i −26.0596 + 17.9136i
109.6 −2.77176 + 0.563348i −2.12132 + 2.12132i 7.36528 − 3.12293i 4.26469 + 10.3350i 4.68474 − 7.07483i −10.0619 −18.6555 + 12.8052i − 9.00000i −17.6429 − 26.2436i
109.7 −2.71663 − 0.787361i 2.12132 − 2.12132i 6.76012 + 4.27793i 10.1466 + 4.69535i −7.43308 + 4.09259i 34.6148 −14.9965 − 16.9442i − 9.00000i −23.8676 − 20.7446i
109.8 −2.63649 + 1.02416i 2.12132 − 2.12132i 5.90218 − 5.40039i −0.980664 − 11.1372i −3.42027 + 7.76542i 0.741927 −10.0302 + 20.2829i − 9.00000i 13.9919 + 28.3589i
109.9 −2.63471 + 1.02874i −2.12132 + 2.12132i 5.88339 − 5.42086i 10.7792 − 2.96805i 3.40678 − 7.77135i 28.7747 −9.92440 + 20.3349i − 9.00000i −25.3467 + 18.9089i
109.10 −2.63282 − 1.03356i −2.12132 + 2.12132i 5.86350 + 5.44237i 4.67097 − 10.1579i 7.77757 − 3.39254i −1.17699 −9.81254 − 20.3891i − 9.00000i −22.7966 + 21.9161i
109.11 −2.57776 − 1.16412i −2.12132 + 2.12132i 5.28967 + 6.00162i −3.10771 + 10.7397i 7.93771 − 2.99879i 30.3163 −6.64889 − 21.6285i − 9.00000i 20.5132 − 24.0667i
109.12 −2.47040 − 1.37737i 2.12132 − 2.12132i 4.20571 + 6.80529i −6.41296 − 9.15827i −8.16234 + 2.31866i −25.7064 −1.01636 − 22.6046i − 9.00000i 3.22823 + 31.4576i
109.13 −2.37176 + 1.54103i 2.12132 − 2.12132i 3.25047 − 7.30989i −8.55780 + 7.19473i −1.76224 + 8.30027i 4.96305 3.55542 + 22.3463i − 9.00000i 9.20975 − 30.2520i
109.14 −2.34048 + 1.58813i −2.12132 + 2.12132i 2.95570 − 7.43396i 9.11227 − 6.47816i 1.59598 − 8.33384i −30.4448 4.88833 + 22.0931i − 9.00000i −11.0390 + 29.6334i
109.15 −2.24623 − 1.71885i 2.12132 − 2.12132i 2.09112 + 7.72187i 11.0698 − 1.56802i −8.41121 + 1.11875i 1.42347 8.57558 − 20.9394i − 9.00000i −27.5606 − 15.5052i
109.16 −2.00342 − 1.99658i −2.12132 + 2.12132i 0.0273714 + 7.99995i 9.80715 + 5.36840i 8.48527 − 0.0145159i −24.1724 15.9177 − 16.0819i − 9.00000i −8.92941 − 30.3359i
109.17 −2.00207 + 1.99793i −2.12132 + 2.12132i 0.0165447 − 7.99998i −9.28072 + 6.23443i 0.00877417 − 8.48528i 21.1327 15.9503 + 16.0496i − 9.00000i 6.12466 − 31.0240i
109.18 −1.98342 − 2.01644i −2.12132 + 2.12132i −0.132086 + 7.99891i −9.02412 + 6.60040i 8.48499 + 0.0700517i −28.2202 16.3913 − 15.5989i − 9.00000i 31.2080 + 5.10524i
109.19 −1.98265 − 2.01720i 2.12132 − 2.12132i −0.138176 + 7.99881i −2.09938 + 10.9815i −8.48496 − 0.0732818i −2.17616 16.4091 − 15.5801i − 9.00000i 26.3141 − 17.5376i
109.20 −1.88567 + 2.10814i −2.12132 + 2.12132i −0.888528 − 7.95050i −3.52123 − 10.6114i −0.471944 − 8.47215i 3.08691 18.4363 + 13.1189i − 9.00000i 29.0101 + 12.5862i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.bl.a ✓ 144
5.b even 2 1 inner 240.4.bl.a ✓ 144
16.e even 4 1 inner 240.4.bl.a ✓ 144
80.q even 4 1 inner 240.4.bl.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.bl.a ✓ 144 1.a even 1 1 trivial
240.4.bl.a ✓ 144 5.b even 2 1 inner
240.4.bl.a ✓ 144 16.e even 4 1 inner
240.4.bl.a ✓ 144 80.q even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(240, [\chi])\).