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Properties

Label 560.2.bb.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 70 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - 70 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - 70 q^{7} - 8 q^{8} + 6 q^{10} - 2 q^{11} + 4 q^{12} - 6 q^{13} + 2 q^{14} - 6 q^{15} + 4 q^{16} + 18 q^{18} + 14 q^{19} + 20 q^{20} + 2 q^{21} + 12 q^{22} + 20 q^{24} - 6 q^{25} - 36 q^{26} - 8 q^{27} + 2 q^{29} + 28 q^{30} + 16 q^{31} + 8 q^{32} + 4 q^{34} + 4 q^{35} - 40 q^{36} - 10 q^{37} + 12 q^{38} + 44 q^{40} - 2 q^{43} - 24 q^{44} + 22 q^{45} - 16 q^{46} + 44 q^{48} + 70 q^{49} - 74 q^{50} + 8 q^{51} - 28 q^{52} + 30 q^{53} - 32 q^{54} - 6 q^{55} + 8 q^{56} + 76 q^{57} - 56 q^{58} + 2 q^{59} - 64 q^{60} + 30 q^{61} - 48 q^{62} + 12 q^{64} - 10 q^{65} + 80 q^{66} - 6 q^{67} + 36 q^{68} - 16 q^{69} - 6 q^{70} - 4 q^{72} + 36 q^{73} - 32 q^{74} - 98 q^{75} + 44 q^{76} + 2 q^{77} + 84 q^{78} - 40 q^{79} - 24 q^{80} - 82 q^{81} - 24 q^{82} - 10 q^{83} - 4 q^{84} - 32 q^{85} + 32 q^{86} + 4 q^{87} - 32 q^{88} + 158 q^{90} + 6 q^{91} + 92 q^{92} + 56 q^{93} - 20 q^{94} + 6 q^{95} + 16 q^{96} - 2 q^{98} - 34 q^{99}+O(q^{100}) \) 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 −1.41420 − 0.00610828i 1.35224 − 1.35224i 1.99993 + 0.0172767i −0.516149 + 2.17568i −1.92059 + 1.90407i −1.00000 −2.82819 − 0.0366488i − 0.657087i 0.743228 − 3.07370i
29.2 −1.40637 − 0.148759i −0.279102 + 0.279102i 1.95574 + 0.418421i 2.19030 − 0.450071i 0.434038 − 0.351001i −1.00000 −2.68825 − 0.879388i 2.84420i −3.14733 + 0.307138i
29.3 −1.36233 + 0.379542i 2.24950 − 2.24950i 1.71190 − 1.03412i 0.718756 − 2.11740i −2.21079 + 3.91835i −1.00000 −1.93968 + 2.05856i − 7.12050i −0.175541 + 3.15740i
29.4 −1.32540 + 0.493263i −2.08897 + 2.08897i 1.51338 − 1.30754i −2.21442 − 0.310380i 1.73832 − 3.79914i −1.00000 −1.36088 + 2.47952i − 5.72761i 3.08810 − 0.680914i
29.5 −1.31948 − 0.508904i 1.51552 − 1.51552i 1.48203 + 1.34297i −2.14663 − 0.626067i −2.77094 + 1.22844i −1.00000 −1.27206 − 2.52623i − 1.59360i 2.51382 + 1.91851i
29.6 −1.30469 − 0.545695i −1.97585 + 1.97585i 1.40443 + 1.42393i 1.90686 + 1.16786i 3.65608 − 1.49966i −1.00000 −1.05532 − 2.62418i − 4.80794i −1.85056 − 2.56426i
29.7 −1.25189 − 0.657851i −1.16502 + 1.16502i 1.13446 + 1.64712i −1.61836 + 1.54302i 2.22489 − 0.692068i −1.00000 −0.336668 − 2.80832i 0.285460i 3.04109 − 0.867061i
29.8 −1.16286 + 0.804830i 0.441211 − 0.441211i 0.704499 − 1.87181i −2.22133 − 0.256291i −0.157968 + 0.868166i −1.00000 0.687255 + 2.74366i 2.61067i 2.78937 − 1.48976i
29.9 −1.13277 + 0.846660i −1.23809 + 1.23809i 0.566334 − 1.91814i 1.09503 + 1.94959i 0.354228 − 2.45070i −1.00000 0.982488 + 2.65230i − 0.0657107i −2.89106 − 1.28132i
29.10 −0.954551 − 1.04347i −0.512166 + 0.512166i −0.177666 + 1.99209i −0.780394 − 2.09547i 1.02332 + 0.0455422i −1.00000 2.24828 − 1.71616i 2.47537i −1.44164 + 2.81455i
29.11 −0.795266 − 1.16942i 0.629715 − 0.629715i −0.735102 + 1.86001i 2.03423 − 0.928390i −1.23720 − 0.235612i −1.00000 2.75974 − 0.619555i 2.20692i −2.70344 − 1.64056i
29.12 −0.734791 + 1.20834i 0.510771 − 0.510771i −0.920165 − 1.77575i 2.08147 − 0.816996i 0.241875 + 0.992494i −1.00000 2.82184 + 0.192935i 2.47823i −0.542236 + 3.11544i
29.13 −0.543124 + 1.30576i −0.0768379 + 0.0768379i −1.41003 − 1.41838i −1.44155 + 1.70937i −0.0585996 − 0.142065i −1.00000 2.61789 − 1.07081i 2.98819i −1.44908 − 2.81072i
29.14 −0.536830 + 1.30836i −1.88282 + 1.88282i −1.42363 − 1.40474i 0.665150 − 2.13485i −1.45266 − 3.47417i −1.00000 2.60215 − 1.10852i − 4.09005i 2.43608 + 2.01631i
29.15 −0.525764 − 1.31285i 1.37854 − 1.37854i −1.44714 + 1.38050i −1.88536 + 1.20226i −2.53459 − 1.08502i −1.00000 2.57324 + 1.17407i − 0.800723i 2.56964 + 1.84308i
29.16 −0.435695 + 1.34543i 2.28818 − 2.28818i −1.62034 − 1.17239i 1.30264 + 1.81745i 2.08163 + 4.07553i −1.00000 2.28334 − 1.66924i − 7.47157i −3.01280 + 0.960759i
29.17 −0.125099 − 1.40867i −1.73140 + 1.73140i −1.96870 + 0.352445i 1.43768 − 1.71263i 2.65556 + 2.22237i −1.00000 0.742760 + 2.72916i − 2.99546i −2.59238 − 1.81096i
29.18 0.168286 − 1.40417i 0.0271728 − 0.0271728i −1.94336 − 0.472603i 1.37117 + 1.76632i −0.0335823 − 0.0427279i −1.00000 −0.990653 + 2.64927i 2.99852i 2.71096 − 1.62810i
29.19 0.171434 + 1.40378i −0.104638 + 0.104638i −1.94122 + 0.481312i −0.0634380 − 2.23517i −0.164828 − 0.128951i −1.00000 −1.00845 − 2.64254i 2.97810i 3.12682 − 0.472237i
29.20 0.239753 − 1.39374i −0.826272 + 0.826272i −1.88504 − 0.668309i −1.90570 + 1.16974i 0.953509 + 1.34971i −1.00000 −1.38339 + 2.46703i 1.63455i 1.17342 + 2.93651i
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.35
Significant digits:
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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.c ✓ 70
5.b even 2 1 560.2.bb.d yes 70
16.e even 4 1 560.2.bb.d yes 70
80.q even 4 1 inner 560.2.bb.c ✓ 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bb.c ✓ 70 1.a even 1 1 trivial
560.2.bb.c ✓ 70 80.q even 4 1 inner
560.2.bb.d yes 70 5.b even 2 1
560.2.bb.d yes 70 16.e even 4 1

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