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Properties

Label 240.4.s.b
Level $240$
Weight $4$
Character orbit 240.s
Analytic conductor $14.160$
Analytic rank $0$
Dimension $52$
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] = [240,4,Mod(61,240)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.61");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.s (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: \(52\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 20 q^{4} - 84 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 20 q^{4} - 84 q^{8} - 40 q^{10} + 40 q^{11} + 244 q^{14} - 780 q^{15} + 108 q^{16} - 408 q^{17} + 36 q^{18} + 316 q^{19} - 40 q^{20} - 488 q^{22} - 96 q^{24} + 308 q^{26} + 344 q^{28} - 400 q^{29} + 280 q^{32} + 528 q^{33} + 180 q^{34} + 144 q^{36} - 16 q^{37} - 1384 q^{38} + 300 q^{40} + 324 q^{42} - 808 q^{43} - 1704 q^{44} + 3028 q^{46} - 528 q^{48} - 4116 q^{49} - 100 q^{50} - 444 q^{51} + 1928 q^{52} - 752 q^{53} - 216 q^{54} + 128 q^{56} + 792 q^{58} + 688 q^{59} - 300 q^{60} - 260 q^{61} + 2924 q^{62} + 504 q^{63} - 1528 q^{64} - 336 q^{66} + 408 q^{67} + 4928 q^{68} - 420 q^{69} + 460 q^{70} + 612 q^{72} + 4308 q^{74} - 164 q^{76} + 2008 q^{77} - 852 q^{78} - 200 q^{79} - 1040 q^{80} - 4212 q^{81} - 4088 q^{82} - 1104 q^{83} - 1800 q^{84} + 740 q^{85} - 2464 q^{86} + 6952 q^{88} + 2648 q^{91} - 4072 q^{92} - 2352 q^{93} - 2036 q^{94} - 1920 q^{96} + 5432 q^{97} - 9264 q^{98} + 360 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −2.82621 − 0.112058i 2.12132 + 2.12132i 7.97489 + 0.633396i −3.53553 + 3.53553i −5.75758 − 6.23300i 33.8922i −22.4677 − 2.68376i 9.00000i 10.3883 − 9.59597i
61.2 −2.82131 − 0.200455i −2.12132 − 2.12132i 7.91964 + 1.13109i 3.53553 − 3.53553i 5.55968 + 6.41014i − 31.9806i −22.1171 − 4.77870i 9.00000i −10.6836 + 9.26614i
61.3 −2.72649 − 0.752512i 2.12132 + 2.12132i 6.86745 + 4.10343i −3.53553 + 3.53553i −4.18743 − 7.38007i − 9.24942i −15.6361 − 16.3558i 9.00000i 12.3001 − 6.97905i
61.4 −2.60532 − 1.10106i −2.12132 − 2.12132i 5.57535 + 5.73720i 3.53553 − 3.53553i 3.19102 + 7.86240i 16.5280i −8.20859 − 21.0860i 9.00000i −13.1040 + 5.31837i
61.5 −2.56250 + 1.19733i −2.12132 − 2.12132i 5.13281 − 6.13630i 3.53553 − 3.53553i 7.97580 + 2.89597i − 5.34769i −5.80568 + 21.8699i 9.00000i −4.82662 + 13.2930i
61.6 −2.47397 + 1.37093i 2.12132 + 2.12132i 4.24109 − 6.78330i −3.53553 + 3.53553i −8.15628 − 2.33990i − 14.0467i −1.19290 + 22.5960i 9.00000i 3.89984 − 13.5938i
61.7 −1.99476 − 2.00522i 2.12132 + 2.12132i −0.0418486 + 7.99989i −3.53553 + 3.53553i 0.0221937 − 8.48525i − 5.78064i 16.1251 − 15.8740i 9.00000i 14.1421 + 0.0369895i
61.8 −1.99210 + 2.00787i −2.12132 − 2.12132i −0.0630969 − 7.99975i 3.53553 − 3.53553i 8.48522 − 0.0334625i 13.9877i 16.1882 + 15.8096i 9.00000i 0.0557708 + 14.1420i
61.9 −1.67722 − 2.27749i −2.12132 − 2.12132i −2.37388 + 7.63968i 3.53553 − 3.53553i −1.27336 + 8.38919i − 9.91614i 21.3808 − 7.40691i 9.00000i −13.9820 − 2.12227i
61.10 −1.28108 + 2.52167i 2.12132 + 2.12132i −4.71766 − 6.46093i −3.53553 + 3.53553i −8.06686 + 2.63169i − 10.4604i 22.3361 − 3.61943i 9.00000i −4.38615 − 13.4448i
61.11 −0.698684 − 2.74077i −2.12132 − 2.12132i −7.02368 + 3.82987i 3.53553 − 3.53553i −4.33193 + 7.29619i 18.2986i 15.4041 + 16.5745i 9.00000i −12.1603 − 7.21988i
61.12 −0.417304 + 2.79747i −2.12132 − 2.12132i −7.65171 − 2.33479i 3.53553 − 3.53553i 6.81957 − 5.04910i − 15.0328i 9.72462 − 20.4311i 9.00000i 8.41517 + 11.3660i
61.13 0.0617444 − 2.82775i 2.12132 + 2.12132i −7.99238 − 0.349196i −3.53553 + 3.53553i 6.12955 − 5.86759i − 5.35310i −1.48092 + 22.5789i 9.00000i 9.77932 + 10.2159i
61.14 0.350930 + 2.80657i 2.12132 + 2.12132i −7.75370 + 1.96982i −3.53553 + 3.53553i −5.20920 + 6.69807i 24.5183i −8.24944 − 21.0700i 9.00000i −11.1635 − 8.68201i
61.15 0.913783 − 2.67675i −2.12132 − 2.12132i −6.33000 − 4.89194i 3.53553 − 3.53553i −7.61668 + 3.73982i − 22.3132i −18.8788 + 12.4737i 9.00000i −6.23303 − 12.6945i
61.16 0.969437 + 2.65710i 2.12132 + 2.12132i −6.12038 + 5.15179i −3.53553 + 3.53553i −3.58008 + 7.69305i − 32.5362i −19.6221 − 11.2682i 9.00000i −12.8218 − 5.96680i
61.17 0.970191 + 2.65683i −2.12132 − 2.12132i −6.11746 + 5.15526i 3.53553 − 3.53553i 3.57789 − 7.69407i 6.51896i −19.6317 − 11.2514i 9.00000i 12.8234 + 5.96316i
61.18 1.31352 − 2.50493i 2.12132 + 2.12132i −4.54931 − 6.58056i −3.53553 + 3.53553i 8.10016 − 2.52734i 5.83627i −22.4594 + 2.75195i 9.00000i 4.21224 + 13.5003i
61.19 1.64280 − 2.30244i −2.12132 − 2.12132i −2.60244 − 7.56487i 3.53553 − 3.53553i −8.36911 + 1.39931i 35.4606i −21.6929 − 6.43560i 9.00000i −2.33218 − 13.9485i
61.20 2.17362 − 1.80980i 2.12132 + 2.12132i 1.44926 − 7.86763i −3.53553 + 3.53553i 8.45011 + 0.771785i 28.2829i −11.0887 − 19.7241i 9.00000i −1.28631 + 14.0835i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e 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.s.b ✓ 52
4.b odd 2 1 960.4.s.b 52
16.e even 4 1 inner 240.4.s.b ✓ 52
16.f odd 4 1 960.4.s.b 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.s.b ✓ 52 1.a even 1 1 trivial
240.4.s.b ✓ 52 16.e even 4 1 inner
960.4.s.b 52 4.b odd 2 1
960.4.s.b 52 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{52} + 10976 T_{7}^{50} + 55717944 T_{7}^{48} + 173770991008 T_{7}^{46} + 373120056503920 T_{7}^{44} + \cdots + 18\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display