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Properties

Label 560.2.bb.d
Level $560$
Weight $2$
Character orbit 560.bb
Analytic conductor $4.472$
Analytic rank $0$
Dimension $70$
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] = [560,2,Mod(29,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bb (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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(35\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 70 q + 2 q^{2} + 2 q^{3} + 2 q^{5} + 70 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q + 2 q^{2} + 2 q^{3} + 2 q^{5} + 70 q^{7} + 8 q^{8} - 18 q^{10} - 2 q^{11} - 4 q^{12} + 6 q^{13} + 2 q^{14} - 6 q^{15} + 4 q^{16} - 18 q^{18} + 14 q^{19} + 12 q^{20} + 2 q^{21} - 12 q^{22} + 20 q^{24} + 6 q^{25} - 36 q^{26} + 8 q^{27} + 2 q^{29} + 8 q^{30} + 16 q^{31} - 8 q^{32} + 4 q^{34} + 2 q^{35} - 40 q^{36} + 10 q^{37} - 12 q^{38} - 24 q^{40} + 2 q^{43} - 24 q^{44} - 24 q^{45} - 16 q^{46} - 44 q^{48} + 70 q^{49} - 10 q^{50} + 8 q^{51} + 28 q^{52} - 30 q^{53} - 32 q^{54} + 6 q^{55} + 8 q^{56} - 76 q^{57} + 56 q^{58} + 2 q^{59} - 8 q^{60} + 30 q^{61} + 48 q^{62} + 12 q^{64} - 10 q^{65} + 80 q^{66} + 6 q^{67} - 36 q^{68} - 16 q^{69} - 18 q^{70} + 4 q^{72} - 36 q^{73} - 32 q^{74} - 2 q^{75} + 44 q^{76} - 2 q^{77} - 84 q^{78} - 40 q^{79} + 12 q^{80} - 82 q^{81} + 24 q^{82} + 10 q^{83} - 4 q^{84} + 32 q^{85} + 32 q^{86} - 4 q^{87} + 32 q^{88} + 18 q^{90} + 6 q^{91} - 92 q^{92} - 56 q^{93} - 20 q^{94} + 6 q^{95} + 16 q^{96} + 2 q^{98} - 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.41250 + 0.0695215i 0.861811 − 0.861811i 1.99033 − 0.196399i −1.76836 − 1.36854i −1.15740 + 1.27722i 1.00000 −2.79770 + 0.415785i 1.51457i 2.59295 + 1.81014i
29.2 −1.40611 − 0.151214i −1.45780 + 1.45780i 1.95427 + 0.425246i −2.23574 + 0.0381166i 2.27026 − 1.82938i 1.00000 −2.68361 − 0.893453i − 1.25035i 3.14946 + 0.284479i
29.3 −1.34972 + 0.422212i 1.56336 − 1.56336i 1.64347 − 1.13973i 2.04190 + 0.911403i −1.45003 + 2.77016i 1.00000 −1.73702 + 2.23221i − 1.88819i −3.14079 − 0.368023i
29.4 −1.33379 − 0.470109i 0.760413 − 0.760413i 1.55800 + 1.25405i 0.975010 − 2.01230i −1.37171 + 0.656755i 1.00000 −1.48850 − 2.40507i 1.84355i −2.24646 + 2.22563i
29.5 −1.31401 + 0.522843i −1.20613 + 1.20613i 1.45327 − 1.37405i 2.07792 − 0.825973i 0.954258 − 2.21549i 1.00000 −1.19121 + 2.56535i 0.0904875i −2.29857 + 2.17177i
29.6 −1.20051 − 0.747507i −1.99217 + 1.99217i 0.882467 + 1.79478i 0.520478 − 2.17465i 3.88079 − 0.902467i 1.00000 0.282200 − 2.81431i − 4.93747i −2.25041 + 2.22164i
29.7 −1.17664 − 0.784556i −0.0419138 + 0.0419138i 0.768944 + 1.84627i 0.673232 + 2.13231i 0.0822009 − 0.0164335i 1.00000 0.543737 − 2.77567i 2.99649i 0.880770 − 3.03714i
29.8 −1.08190 + 0.910767i 1.99977 − 1.99977i 0.341008 − 1.97071i −2.16539 + 0.557749i −0.342224 + 3.98487i 1.00000 1.42592 + 2.44269i − 4.99816i 1.83475 − 2.57559i
29.9 −0.868292 − 1.11627i 2.38527 − 2.38527i −0.492138 + 1.93850i −0.763308 − 2.10175i −4.73373 − 0.591506i 1.00000 2.59122 − 1.13383i − 8.37902i −1.68336 + 2.67700i
29.10 −0.866818 + 1.11742i 0.969878 − 0.969878i −0.497254 − 1.93720i 1.42081 + 1.72664i 0.243053 + 1.92447i 1.00000 2.59569 + 1.12356i 1.11867i −3.16097 + 0.0909610i
29.11 −0.744791 + 1.20220i −2.31279 + 2.31279i −0.890574 − 1.79078i −1.10225 + 1.94552i −1.05789 − 4.50298i 1.00000 2.81616 + 0.263104i − 7.69797i −1.51795 − 2.77413i
29.12 −0.735407 − 1.20796i −1.40023 + 1.40023i −0.918355 + 1.77669i 2.12747 + 0.688383i 2.72116 + 0.661689i 1.00000 2.82154 − 0.197250i − 0.921283i −0.733014 − 3.07615i
29.13 −0.639532 − 1.26135i −0.819103 + 0.819103i −1.18200 + 1.61335i −2.21530 − 0.304027i 1.55702 + 0.509331i 1.00000 2.79091 + 0.459123i 1.65814i 1.03327 + 2.98870i
29.14 −0.332249 + 1.37463i −1.50237 + 1.50237i −1.77922 − 0.913441i 2.16429 + 0.562013i −1.56605 − 2.56437i 1.00000 1.84679 − 2.14228i − 1.51426i −1.49164 + 2.78837i
29.15 −0.289368 − 1.38429i 1.70369 − 1.70369i −1.83253 + 0.801140i 0.0668319 + 2.23507i −2.85141 − 1.86542i 1.00000 1.63929 + 2.30494i − 2.80515i 3.07465 − 0.739272i
29.16 −0.239753 + 1.39374i 0.826272 − 0.826272i −1.88504 − 0.668309i −1.16974 + 1.90570i 0.953509 + 1.34971i 1.00000 1.38339 − 2.46703i 1.63455i −2.37561 − 2.08722i
29.17 −0.171434 − 1.40378i 0.104638 − 0.104638i −1.94122 + 0.481312i 2.23517 + 0.0634380i −0.164828 − 0.128951i 1.00000 1.00845 + 2.64254i 2.97810i −0.294130 − 3.14857i
29.18 −0.168286 + 1.40417i −0.0271728 + 0.0271728i −1.94336 − 0.472603i −1.76632 − 1.37117i −0.0335823 − 0.0427279i 1.00000 0.990653 − 2.64927i 2.99852i 2.22260 − 2.24946i
29.19 0.125099 + 1.40867i 1.73140 − 1.73140i −1.96870 + 0.352445i 1.71263 − 1.43768i 2.65556 + 2.22237i 1.00000 −0.742760 − 2.72916i − 2.99546i 2.23946 + 2.23267i
29.20 0.435695 − 1.34543i −2.28818 + 2.28818i −1.62034 − 1.17239i −1.81745 − 1.30264i 2.08163 + 4.07553i 1.00000 −2.28334 + 1.66924i − 7.47157i −2.54446 + 1.87769i
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 560.2.bb.d yes 70
5.b even 2 1 560.2.bb.c ✓ 70
16.e even 4 1 560.2.bb.c ✓ 70
80.q even 4 1 inner 560.2.bb.d yes 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bb.c ✓ 70 5.b even 2 1
560.2.bb.c ✓ 70 16.e even 4 1
560.2.bb.d yes 70 1.a even 1 1 trivial
560.2.bb.d yes 70 80.q even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{70} - 2 T_{3}^{69} + 2 T_{3}^{68} + 500 T_{3}^{66} - 1016 T_{3}^{65} + 1032 T_{3}^{64} + \cdots + 819200 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display