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Properties

Label 230.3.k.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 24 q^{2} - 4 q^{5} - 74 q^{7} + 48 q^{8} - 16 q^{10} + 8 q^{11} + 44 q^{12} + 24 q^{13} + 24 q^{15} + 96 q^{16} + 12 q^{17} + 88 q^{18} - 24 q^{20} + 24 q^{21} + 8 q^{22} - 44 q^{23} - 128 q^{25} + 48 q^{26} - 60 q^{27} - 116 q^{28} + 120 q^{30} - 12 q^{31} + 96 q^{32} - 334 q^{33} - 224 q^{35} - 176 q^{36} + 188 q^{37} + 76 q^{38} - 16 q^{40} - 116 q^{41} + 24 q^{42} + 120 q^{43} + 204 q^{45} + 396 q^{46} - 144 q^{47} - 88 q^{48} + 170 q^{50} - 176 q^{51} + 48 q^{52} + 192 q^{53} - 312 q^{55} + 296 q^{56} + 88 q^{57} - 28 q^{58} - 72 q^{60} - 552 q^{61} - 12 q^{62} - 122 q^{63} - 392 q^{65} - 8 q^{66} - 72 q^{67} - 24 q^{68} + 100 q^{70} + 424 q^{71} - 176 q^{72} + 452 q^{73} + 604 q^{75} - 112 q^{76} + 356 q^{77} + 32 q^{78} + 16 q^{80} - 704 q^{81} + 148 q^{82} - 360 q^{83} + 428 q^{85} - 376 q^{86} - 462 q^{87} - 104 q^{88} - 510 q^{90} + 432 q^{91} - 192 q^{93} - 166 q^{95} - 1042 q^{97} - 88 q^{98}+O(q^{100}) \) 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 −5.47365 + 2.04157i 0.563465 + 1.91899i 1.48407 + 4.77468i 7.92716 + 2.32763i −2.01723 + 9.27304i 0.988434 − 2.65009i 18.9911 − 16.4559i 2.36640 − 6.66334i
3.2 −1.13214 − 0.847507i −4.48436 + 1.67258i 0.563465 + 1.91899i −1.64232 − 4.72258i 6.49443 + 1.90694i 1.22526 − 5.63243i 0.988434 − 2.65009i 10.5102 − 9.10714i −2.14308 + 6.73849i
3.3 −1.13214 − 0.847507i −2.51818 + 0.939231i 0.563465 + 1.91899i 2.35609 − 4.41008i 3.64692 + 1.07083i −1.24199 + 5.70935i 0.988434 − 2.65009i −1.34269 + 1.16345i −6.40499 + 2.99601i
3.4 −1.13214 − 0.847507i −2.44177 + 0.910732i 0.563465 + 1.91899i −4.63287 + 1.88057i 3.53626 + 1.03834i 0.738198 − 3.39344i 0.988434 − 2.65009i −1.66895 + 1.44616i 6.83883 + 1.79732i
3.5 −1.13214 − 0.847507i −1.89358 + 0.706270i 0.563465 + 1.91899i 3.30772 + 3.74953i 2.74236 + 0.805231i 1.94574 − 8.94444i 0.988434 − 2.65009i −3.71490 + 3.21898i −0.567036 − 7.04830i
3.6 −1.13214 − 0.847507i −1.16785 + 0.435585i 0.563465 + 1.91899i −0.970697 + 4.90487i 1.69133 + 0.496618i −1.47212 + 6.76723i 0.988434 − 2.65009i −5.62761 + 4.87635i 5.25587 − 4.73031i
3.7 −1.13214 − 0.847507i −0.0190211 + 0.00709452i 0.563465 + 1.91899i 4.84193 − 1.24727i 0.0275472 + 0.00808858i −1.98672 + 9.13280i 0.988434 − 2.65009i −6.80143 + 5.89348i −6.53880 − 2.69149i
3.8 −1.13214 − 0.847507i 1.09130 − 0.407033i 0.563465 + 1.91899i −4.93104 − 0.827554i −1.58046 − 0.464066i 0.600939 − 2.76247i 0.988434 − 2.65009i −5.77649 + 5.00536i 4.88125 + 5.11599i
3.9 −1.13214 − 0.847507i 1.90726 − 0.711370i 0.563465 + 1.91899i 1.62652 − 4.72805i −2.76216 − 0.811045i 2.04047 − 9.37989i 0.988434 − 2.65009i −3.67017 + 3.18022i −5.84849 + 3.97431i
3.10 −1.13214 − 0.847507i 3.55876 − 1.32735i 0.563465 + 1.91899i −2.88474 + 4.08390i −5.15393 − 1.51333i −1.27831 + 5.87629i 0.988434 − 2.65009i 4.10115 − 3.55367i 6.72706 − 2.17869i
3.11 −1.13214 − 0.847507i 4.17989 − 1.55902i 0.563465 + 1.91899i 4.74689 + 1.57069i −6.05348 − 1.77746i 0.338789 − 1.55739i 0.988434 − 2.65009i 8.23916 − 7.13928i −4.04295 − 5.80125i
3.12 −1.13214 − 0.847507i 4.65466 − 1.73610i 0.563465 + 1.91899i −3.20030 − 3.84162i −6.74107 − 1.97936i −1.76266 + 8.10282i 0.988434 − 2.65009i 11.8501 − 10.2682i 0.367373 + 7.06152i
13.1 −1.41061 − 0.100889i −1.09457 − 5.03168i 1.97964 + 0.284630i 3.91427 + 3.11103i 1.03638 + 7.20817i −4.48066 + 2.44662i −2.76379 − 0.601225i −15.9330 + 7.27636i −5.20764 − 4.78336i
13.2 −1.41061 − 0.100889i −1.02391 − 4.70683i 1.97964 + 0.284630i −0.635666 − 4.95943i 0.969469 + 6.74281i −8.25197 + 4.50592i −2.76379 − 0.601225i −12.9192 + 5.90000i 0.396327 + 7.05995i
13.3 −1.41061 − 0.100889i −0.745890 − 3.42880i 1.97964 + 0.284630i 1.45154 − 4.78467i 0.706232 + 4.91195i 10.4743 − 5.71940i −2.76379 − 0.601225i −3.01364 + 1.37628i −2.53027 + 6.60286i
13.4 −1.41061 − 0.100889i −0.730822 − 3.35954i 1.97964 + 0.284630i −1.24754 + 4.84186i 0.691966 + 4.81273i 0.518845 − 0.283311i −2.76379 − 0.601225i −2.56569 + 1.17171i 2.24828 − 6.70412i
13.5 −1.41061 − 0.100889i −0.160640 − 0.738452i 1.97964 + 0.284630i −4.82720 − 1.30312i 0.152100 + 1.05788i 3.79539 − 2.07244i −2.76379 − 0.601225i 7.66718 − 3.50148i 6.67783 + 2.32520i
13.6 −1.41061 − 0.100889i −0.0934166 − 0.429429i 1.97964 + 0.284630i 4.81942 − 1.33161i 0.0884498 + 0.615182i −0.371231 + 0.202708i −2.76379 − 0.601225i 8.01101 − 3.65850i −6.93267 + 1.39216i
13.7 −1.41061 − 0.100889i 0.203005 + 0.933200i 1.97964 + 0.284630i −4.34220 − 2.47897i −0.192212 − 1.33686i −8.01935 + 4.37890i −2.76379 − 0.601225i 7.35704 − 3.35985i 5.87505 + 3.93494i
13.8 −1.41061 − 0.100889i 0.313117 + 1.43938i 1.97964 + 0.284630i 0.477701 + 4.97713i −0.296470 − 2.06199i 7.52566 − 4.10932i −2.76379 − 0.601225i 6.21292 − 2.83735i −0.171714 − 7.06898i
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.a ✓ 240
5.c odd 4 1 inner 230.3.k.a ✓ 240
23.c even 11 1 inner 230.3.k.a ✓ 240
115.k odd 44 1 inner 230.3.k.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.k.a ✓ 240 1.a even 1 1 trivial
230.3.k.a ✓ 240 5.c odd 4 1 inner
230.3.k.a ✓ 240 23.c even 11 1 inner
230.3.k.a ✓ 240 115.k odd 44 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{240} + 20 T_{3}^{237} - 1390 T_{3}^{236} - 4356 T_{3}^{235} + 200 T_{3}^{234} + \cdots + 11\!\cdots\!81 \) acting on \(S_{3}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display