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Properties

Label 240.4.bf.a
Level $240$
Weight $4$
Character orbit 240.bf
Analytic conductor $14.160$
Analytic rank $0$
Dimension $280$
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(53,240)]
 
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, 1, 2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.53");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.bf (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: \(280\)
Relative dimension: \(140\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 4 q^{3} - 12 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 4 q^{3} - 12 q^{4} - 4 q^{6} + 28 q^{10} - 56 q^{12} - 8 q^{13} - 4 q^{15} - 140 q^{16} + 252 q^{18} - 24 q^{19} - 4 q^{21} - 188 q^{22} + 108 q^{24} - 4 q^{27} - 692 q^{28} + 524 q^{30} - 16 q^{31} - 4 q^{33} + 924 q^{34} - 260 q^{36} - 8 q^{37} - 216 q^{39} + 276 q^{40} + 640 q^{42} + 248 q^{45} - 92 q^{46} - 256 q^{48} - 4 q^{51} - 1136 q^{52} + 1104 q^{54} - 108 q^{57} + 2340 q^{58} - 1076 q^{60} + 904 q^{61} - 1376 q^{63} + 1596 q^{64} + 1644 q^{66} + 108 q^{69} + 1692 q^{70} - 1608 q^{72} + 1596 q^{75} - 2612 q^{76} - 2172 q^{78} - 8 q^{81} + 1776 q^{82} + 2880 q^{84} + 496 q^{85} + 108 q^{87} - 6132 q^{88} - 2024 q^{90} - 8 q^{91} - 112 q^{93} - 420 q^{94} + 1392 q^{96} - 8 q^{97} - 2656 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −2.82831 + 0.0259598i 2.54029 − 4.53287i 7.99865 − 0.146844i −6.83181 + 8.85022i −7.06705 + 12.8863i 3.16052 + 3.16052i −22.6188 + 0.622964i −14.0938 − 23.0296i 19.0927 − 25.2085i
53.2 −2.82750 − 0.0725253i −1.14388 + 5.06868i 7.98948 + 0.410130i 3.65025 + 10.5677i 3.60192 − 14.2487i 2.44889 + 2.44889i −22.5605 − 1.73908i −24.3831 − 11.5959i −9.55466 − 30.1448i
53.3 −2.82712 + 0.0858352i −3.85172 − 3.48773i 7.98526 − 0.485334i 5.73933 + 9.59479i 11.1887 + 9.52962i 17.2283 + 17.2283i −22.5337 + 2.05752i 2.67153 + 26.8675i −17.0494 − 26.6330i
53.4 −2.82507 + 0.137779i −1.70327 + 4.90906i 7.96203 − 0.778473i 5.66433 − 9.63926i 4.13548 − 14.1031i 22.5252 + 22.5252i −22.3860 + 3.29625i −21.1978 − 16.7229i −14.6740 + 28.0120i
53.5 −2.82057 + 0.210727i 5.14156 − 0.751210i 7.91119 − 1.18874i −9.65846 − 5.63154i −14.3438 + 3.20230i 0.694297 + 0.694297i −22.0635 + 5.02001i 25.8714 − 7.72479i 28.4290 + 13.8488i
53.6 −2.81222 + 0.302361i −0.538080 − 5.16822i 7.81716 − 1.70061i 10.0322 − 4.93502i 3.07587 + 14.3715i −11.0779 − 11.0779i −21.4694 + 7.14609i −26.4209 + 5.56183i −26.7207 + 16.9117i
53.7 −2.78670 + 0.484050i 4.65091 + 2.31712i 7.53139 − 2.69780i 9.70304 + 5.55436i −14.0823 − 4.20585i −18.9352 − 18.9352i −19.6819 + 11.1635i 16.2619 + 21.5534i −29.7281 − 10.7816i
53.8 −2.77916 − 0.525619i −4.90214 − 1.72308i 7.44745 + 2.92156i −3.91292 − 10.4733i 12.7181 + 7.36538i 5.79675 + 5.79675i −19.1620 − 12.0340i 21.0620 + 16.8936i 5.36970 + 31.1635i
53.9 −2.77604 + 0.541854i 4.48784 + 2.61902i 7.41279 − 3.00842i 9.05296 − 6.56079i −13.8776 − 4.83876i 14.4951 + 14.4951i −18.9481 + 12.3681i 13.2814 + 23.5075i −21.5764 + 23.1184i
53.10 −2.77480 − 0.548170i −5.01163 + 1.37244i 7.39902 + 3.04213i 10.1126 − 4.76821i 14.6586 − 1.06102i −15.4438 − 15.4438i −18.8632 − 12.4972i 23.2328 − 13.7563i −30.6742 + 7.68740i
53.11 −2.76599 − 0.591036i 2.93634 + 4.28695i 7.30135 + 3.26960i −7.05182 − 8.67594i −5.58812 − 13.5931i −8.10086 − 8.10086i −18.2630 − 13.3590i −9.75586 + 25.1758i 14.3774 + 28.1654i
53.12 −2.74641 + 0.676170i −3.00368 − 4.24004i 7.08559 − 3.71409i −11.1110 − 1.24350i 11.1163 + 9.61390i −20.9282 − 20.9282i −16.9486 + 14.9915i −8.95584 + 25.4714i 31.3562 − 4.09776i
53.13 −2.73427 − 0.723711i −5.19584 − 0.0565838i 6.95249 + 3.95764i −4.32882 + 10.3083i 14.1659 + 3.91500i −18.3161 − 18.3161i −16.1458 − 15.8529i 26.9936 + 0.588002i 19.2964 − 25.0529i
53.14 −2.66161 + 0.956992i −4.80696 + 1.97309i 6.16833 − 5.09428i −10.5291 + 3.76016i 10.9060 − 9.85183i 10.9554 + 10.9554i −11.5425 + 19.4620i 19.2138 − 18.9692i 24.4258 − 20.0843i
53.15 −2.65777 − 0.967597i 4.57722 − 2.45949i 6.12751 + 5.14331i 10.8078 + 2.86210i −14.5450 + 2.10787i 9.18541 + 9.18541i −11.3089 − 19.5987i 14.9018 − 22.5152i −25.9553 − 18.0644i
53.16 −2.64952 − 0.989980i 0.964944 − 5.10577i 6.03988 + 5.24594i 1.28828 − 11.1059i −7.61124 + 12.5725i 6.69843 + 6.69843i −10.8094 − 19.8786i −25.1378 − 9.85356i −14.4079 + 28.1498i
53.17 −2.64632 + 0.998497i 2.03890 + 4.77942i 6.00601 − 5.28468i −7.41505 + 8.36762i −10.1678 − 10.6120i −5.54040 − 5.54040i −10.6171 + 19.9819i −18.6858 + 19.4895i 11.2675 − 29.5473i
53.18 −2.59817 + 1.11782i −2.52874 + 4.53932i 5.50097 − 5.80856i −1.84960 − 11.0263i 1.49597 − 14.6206i −20.6204 − 20.6204i −7.79954 + 21.2407i −14.2109 − 22.9576i 17.1310 + 26.5806i
53.19 −2.52727 − 1.27000i −3.29912 + 4.01445i 4.77422 + 6.41925i −10.8091 − 2.85729i 13.4361 − 5.95575i 5.55121 + 5.55121i −3.91333 − 22.2865i −5.23165 − 26.4883i 23.6887 + 20.9486i
53.20 −2.52593 − 1.27267i 3.64937 + 3.69894i 4.76063 + 6.42934i −8.20591 + 7.59362i −4.51052 − 13.9877i 23.9789 + 23.9789i −3.84262 − 22.2988i −0.364252 + 26.9975i 30.3917 − 8.73756i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
80.t odd 4 1 inner
240.bf 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.bf.a yes 280
3.b odd 2 1 inner 240.4.bf.a yes 280
5.c odd 4 1 240.4.bb.a ✓ 280
15.e even 4 1 240.4.bb.a ✓ 280
16.e even 4 1 240.4.bb.a ✓ 280
48.i odd 4 1 240.4.bb.a ✓ 280
80.t odd 4 1 inner 240.4.bf.a yes 280
240.bf even 4 1 inner 240.4.bf.a yes 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.bb.a ✓ 280 5.c odd 4 1
240.4.bb.a ✓ 280 15.e even 4 1
240.4.bb.a ✓ 280 16.e even 4 1
240.4.bb.a ✓ 280 48.i odd 4 1
240.4.bf.a yes 280 1.a even 1 1 trivial
240.4.bf.a yes 280 3.b odd 2 1 inner
240.4.bf.a yes 280 80.t odd 4 1 inner
240.4.bf.a yes 280 240.bf even 4 1 inner

Hecke kernels

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