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Properties

Label 230.3.k.b
Level $230$
Weight $3$
Character orbit 230.k
Analytic conductor $6.267$
Analytic rank $0$
Dimension $240$
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] = [230,3,Mod(3,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([33, 32]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.k (of order \(44\), degree \(20\), 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: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8} + 16 q^{10} - 24 q^{11} - 28 q^{12} - 8 q^{13} + 8 q^{15} + 96 q^{16} + 44 q^{17} + 200 q^{18} - 24 q^{20} + 24 q^{21} + 24 q^{22} - 40 q^{23} + 240 q^{25} + 16 q^{26} - 76 q^{27} - 100 q^{28} - 216 q^{30} + 4 q^{31} - 96 q^{32} - 206 q^{33} + 136 q^{35} - 48 q^{36} + 556 q^{37} - 140 q^{38} + 16 q^{40} + 44 q^{41} - 24 q^{42} + 48 q^{43} + 12 q^{45} - 404 q^{46} - 24 q^{47} + 56 q^{48} - 138 q^{50} + 48 q^{51} - 16 q^{52} + 32 q^{53} + 64 q^{55} + 200 q^{56} - 920 q^{57} + 28 q^{58} + 152 q^{60} + 1800 q^{61} - 4 q^{62} - 406 q^{63} + 392 q^{65} + 104 q^{66} - 304 q^{67} - 88 q^{68} + 108 q^{70} - 1512 q^{71} + 48 q^{72} - 44 q^{73} - 252 q^{75} + 16 q^{76} - 492 q^{77} + 160 q^{78} + 16 q^{80} - 1344 q^{81} - 308 q^{82} - 516 q^{85} - 272 q^{86} + 814 q^{87} + 40 q^{88} + 670 q^{90} + 144 q^{91} + 8 q^{92} + 160 q^{93} - 670 q^{95} + 64 q^{96} - 242 q^{97} - 776 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 1.13214 + 0.847507i −5.13373 + 1.91478i 0.563465 + 1.91899i 4.33360 − 2.49398i −7.43487 − 2.18307i 2.35485 − 10.8251i −0.988434 + 2.65009i 15.8870 − 13.7662i 7.01989 + 0.849235i
3.2 1.13214 + 0.847507i −4.87737 + 1.81916i 0.563465 + 1.91899i −2.81617 − 4.13149i −7.06360 − 2.07406i −2.62119 + 12.0494i −0.988434 + 2.65009i 13.6776 − 11.8517i 0.313179 − 7.06413i
3.3 1.13214 + 0.847507i −3.78586 + 1.41205i 0.563465 + 1.91899i −0.916589 + 4.91527i −5.48284 − 1.60991i 0.413787 − 1.90215i −0.988434 + 2.65009i 5.53710 − 4.79793i −5.20343 + 4.78794i
3.4 1.13214 + 0.847507i −2.59640 + 0.968406i 0.563465 + 1.91899i 4.72347 + 1.63977i −3.76021 − 1.10410i −1.02169 + 4.69661i −0.988434 + 2.65009i −0.998275 + 0.865010i 3.95789 + 5.85962i
3.5 1.13214 + 0.847507i −1.46393 + 0.546016i 0.563465 + 1.91899i −4.98248 − 0.418219i −2.12011 − 0.622522i 2.78627 − 12.8083i −0.988434 + 2.65009i −4.95680 + 4.29509i −5.28640 − 4.69616i
3.6 1.13214 + 0.847507i −1.33310 + 0.497219i 0.563465 + 1.91899i −1.71782 − 4.69565i −1.93064 − 0.566888i −0.622339 + 2.86085i −0.988434 + 2.65009i −5.27183 + 4.56806i 2.03479 − 6.77197i
3.7 1.13214 + 0.847507i 0.583215 − 0.217528i 0.563465 + 1.91899i 0.599174 + 4.96397i 0.844635 + 0.248007i 0.414739 − 1.90653i −0.988434 + 2.65009i −6.50892 + 5.64002i −3.52865 + 6.12769i
3.8 1.13214 + 0.847507i 1.00233 − 0.373849i 0.563465 + 1.91899i −4.85970 + 1.17614i 1.45161 + 0.426232i −1.84838 + 8.49688i −0.988434 + 2.65009i −5.93685 + 5.14431i −6.49863 − 2.78707i
3.9 1.13214 + 0.847507i 3.11913 − 1.16338i 0.563465 + 1.91899i 4.81990 − 1.32988i 4.51725 + 1.32639i −2.04259 + 9.38963i −0.988434 + 2.65009i 1.57380 − 1.36371i 6.58386 + 2.57929i
3.10 1.13214 + 0.847507i 3.21185 − 1.19796i 0.563465 + 1.91899i 4.41997 + 2.33748i 4.65153 + 1.36581i 2.26142 − 10.3956i −0.988434 + 2.65009i 2.07914 − 1.80158i 3.02298 + 6.39231i
3.11 1.13214 + 0.847507i 3.86243 − 1.44061i 0.563465 + 1.91899i −0.740945 − 4.94480i 5.59372 + 1.64247i 1.21592 − 5.58950i −0.988434 + 2.65009i 6.04125 − 5.23477i 3.35190 − 6.22614i
3.12 1.13214 + 0.847507i 5.55917 − 2.07346i 0.563465 + 1.91899i −2.76117 + 4.16845i 8.05101 + 2.36399i −0.783489 + 3.60164i −0.988434 + 2.65009i 19.8034 − 17.1597i −6.65881 + 2.37914i
13.1 1.41061 + 0.100889i −0.933043 − 4.28913i 1.97964 + 0.284630i 4.55004 − 2.07296i −0.883435 − 6.14443i 1.77102 − 0.967050i 2.76379 + 0.601225i −9.33939 + 4.26516i 6.62747 − 2.46510i
13.2 1.41061 + 0.100889i −0.875157 − 4.02303i 1.97964 + 0.284630i −4.99252 + 0.273452i −0.828626 − 5.76322i −11.1890 + 6.10964i 2.76379 + 0.601225i −7.23219 + 3.30283i −7.07008 − 0.117955i
13.3 1.41061 + 0.100889i −0.800118 − 3.67808i 1.97964 + 0.284630i −2.59942 − 4.27118i −0.757577 − 5.26906i 6.22281 − 3.39791i 2.76379 + 0.601225i −4.70140 + 2.14706i −3.23586 − 6.28723i
13.4 1.41061 + 0.100889i −0.526675 − 2.42109i 1.97964 + 0.284630i 1.87475 + 4.63522i −0.498673 − 3.46835i 1.72156 − 0.940042i 2.76379 + 0.601225i 2.60241 − 1.18848i 2.17690 + 6.72764i
13.5 1.41061 + 0.100889i −0.110736 − 0.509045i 1.97964 + 0.284630i −4.91212 + 0.933335i −0.104848 − 0.729237i 5.72192 − 3.12441i 2.76379 + 0.601225i 7.93982 − 3.62600i −7.02324 + 0.820995i
13.6 1.41061 + 0.100889i 0.0578351 + 0.265863i 1.97964 + 0.284630i −0.330642 − 4.98906i 0.0547601 + 0.380865i −4.77713 + 2.60851i 2.76379 + 0.601225i 8.11935 − 3.70798i 0.0369324 − 7.07097i
13.7 1.41061 + 0.100889i 0.161368 + 0.741796i 1.97964 + 0.284630i 4.91560 − 0.914803i 0.152788 + 1.06267i 2.14811 − 1.17296i 2.76379 + 0.601225i 7.66247 − 3.49933i 7.02629 − 0.794501i
13.8 1.41061 + 0.100889i 0.302791 + 1.39191i 1.97964 + 0.284630i −2.63202 + 4.25117i 0.286693 + 1.99399i −5.20021 + 2.83953i 2.76379 + 0.601225i 6.34096 − 2.89582i −4.14165 + 5.73121i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.c even 11 1 inner
115.k odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.k.b ✓ 240
5.c odd 4 1 inner 230.3.k.b ✓ 240
23.c even 11 1 inner 230.3.k.b ✓ 240
115.k odd 44 1 inner 230.3.k.b ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.k.b ✓ 240 1.a even 1 1 trivial
230.3.k.b ✓ 240 5.c odd 4 1 inner
230.3.k.b ✓ 240 23.c even 11 1 inner
230.3.k.b ✓ 240 115.k odd 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{240} - 8 T_{3}^{239} + 32 T_{3}^{238} - 12 T_{3}^{237} - 1214 T_{3}^{236} + 6180 T_{3}^{235} + \cdots + 14\!\cdots\!01 \) acting on \(S_{3}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display