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Properties

Label 237.2.l.a
Level $237$
Weight $2$
Character orbit 237.l
Analytic conductor $1.892$
Analytic rank $0$
Dimension $288$
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] = [237,2,Mod(14,237)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(237, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("237.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 237 = 3 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 237.l (of order \(26\), degree \(12\), 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: \(1.89245452790\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(24\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 13 q^{3} - 6 q^{4} - 13 q^{6} - 26 q^{7} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 13 q^{3} - 6 q^{4} - 13 q^{6} - 26 q^{7} - 15 q^{9} - 30 q^{10} - 13 q^{12} - 10 q^{13} + 13 q^{15} - 38 q^{16} - 56 q^{18} - 18 q^{19} - 9 q^{21} - 44 q^{22} + 56 q^{25} - 52 q^{27} - 26 q^{28} - 13 q^{30} - 68 q^{31} - 13 q^{33} - 26 q^{34} + 9 q^{36} + 26 q^{37} - 117 q^{39} - 110 q^{40} + 13 q^{42} - 26 q^{43} - 23 q^{45} + 164 q^{46} - 39 q^{48} + 38 q^{49} + 27 q^{51} - 94 q^{52} - 13 q^{54} - 92 q^{55} - 169 q^{57} - 26 q^{58} + 78 q^{60} - 26 q^{61} - 13 q^{63} - 2 q^{64} + 169 q^{66} + 162 q^{67} + 26 q^{69} + 52 q^{70} - 71 q^{72} - 52 q^{73} - 13 q^{75} + 256 q^{76} + 114 q^{79} - 15 q^{81} + 26 q^{82} - 7 q^{84} + 26 q^{85} + 36 q^{87} + 186 q^{88} - 111 q^{90} + 26 q^{91} - 13 q^{93} - 338 q^{94} - 143 q^{96} + 100 q^{97} + 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −0.646523 − 2.62305i 0.509171 + 1.65552i −4.69146 + 2.46227i −0.227484 + 0.157021i 4.01331 − 2.40591i −1.53535 + 1.73305i 5.90887 + 6.66973i −2.48149 + 1.68588i 0.558947 + 0.495183i
14.2 −0.594222 − 2.41085i 0.997192 − 1.41619i −3.68819 + 1.93571i −3.05135 + 2.10620i −4.00679 − 1.56255i −0.868195 + 0.979989i 3.56525 + 4.02434i −1.01122 − 2.82444i 6.89090 + 6.10481i
14.3 −0.530915 − 2.15401i −1.39084 + 1.03226i −2.58696 + 1.35774i 2.07727 − 1.43384i 2.96191 + 2.44784i 1.13879 − 1.28543i 1.35580 + 1.53038i 0.868886 − 2.87142i −4.19135 − 3.71321i
14.4 −0.500829 − 2.03194i −1.69679 − 0.347717i −2.10706 + 1.10587i −1.78939 + 1.23513i 0.143260 + 3.62193i −0.982563 + 1.10908i 0.526833 + 0.594671i 2.75819 + 1.18001i 3.40589 + 3.01736i
14.5 −0.500307 − 2.02983i 0.590745 − 1.62820i −2.09898 + 1.10163i 2.99046 − 2.06417i −3.60051 − 0.384511i −2.69978 + 3.04743i 0.513630 + 0.579768i −2.30204 − 1.92370i −5.68605 − 5.03740i
14.6 −0.462243 − 1.87539i 1.59490 + 0.675501i −1.53252 + 0.804327i −0.592725 + 0.409129i 0.529600 − 3.30330i 1.47111 − 1.66054i −0.344844 − 0.389248i 2.08740 + 2.15471i 1.04126 + 0.922476i
14.7 −0.416430 − 1.68952i −0.373686 − 1.69126i −0.910159 + 0.477688i 0.680333 − 0.469600i −2.70181 + 1.33564i 2.66708 − 3.01051i −1.12170 − 1.26614i −2.72072 + 1.26400i −1.07671 − 0.953883i
14.8 −0.275470 − 1.11762i 1.16264 + 1.28385i 0.597712 − 0.313703i 2.94428 − 2.03229i 1.11459 − 1.65305i −0.925835 + 1.04505i −2.04186 − 2.30478i −0.296548 + 2.98531i −3.08240 − 2.73077i
14.9 −0.257425 − 1.04441i −0.745751 + 1.56328i 0.746378 − 0.391729i −1.74624 + 1.20534i 1.82469 + 0.376445i −3.37962 + 3.81480i −2.02787 − 2.28899i −1.88771 − 2.33164i 1.70840 + 1.51351i
14.10 −0.193304 − 0.784265i 1.58055 − 0.708431i 1.19321 − 0.626244i −0.592230 + 0.408787i −0.861123 − 1.10262i −0.261157 + 0.294785i −1.79305 − 2.02393i 1.99625 − 2.23942i 0.435078 + 0.385445i
14.11 −0.0831041 − 0.337167i −1.63053 + 0.584259i 1.66414 − 0.873406i −1.68472 + 1.16288i 0.332497 + 0.501208i 2.06175 − 2.32724i −0.893329 − 1.00836i 2.31728 − 1.90531i 0.532091 + 0.471392i
14.12 −0.0824870 − 0.334663i −1.48042 − 0.899079i 1.66572 − 0.874236i 2.02953 − 1.40088i −0.178773 + 0.569606i −0.364991 + 0.411989i −0.887103 − 1.00133i 1.38331 + 2.66204i −0.636233 − 0.563654i
14.13 0.0824870 + 0.334663i 1.07097 + 1.36126i 1.66572 − 0.874236i −2.02953 + 1.40088i −0.367222 + 0.470700i −0.364991 + 0.411989i 0.887103 + 1.00133i −0.706050 + 2.91573i −0.636233 − 0.563654i
14.14 0.0831041 + 0.337167i −0.383460 + 1.68907i 1.66414 − 0.873406i 1.68472 − 1.16288i −0.601366 + 0.0110789i 2.06175 − 2.32724i 0.893329 + 1.00836i −2.70592 − 1.29538i 0.532091 + 0.471392i
14.15 0.193304 + 0.784265i 0.512752 − 1.65441i 1.19321 − 0.626244i 0.592230 − 0.408787i 1.39662 + 0.0823286i −0.261157 + 0.294785i 1.79305 + 2.02393i −2.47417 − 1.69661i 0.435078 + 0.385445i
14.16 0.257425 + 1.04441i −1.46199 + 0.928747i 0.746378 − 0.391729i 1.74624 − 1.20534i −1.34635 − 1.28785i −3.37962 + 3.81480i 2.02787 + 2.28899i 1.27486 − 2.71565i 1.70840 + 1.51351i
14.17 0.275470 + 1.11762i −1.41463 − 0.999409i 0.597712 − 0.313703i −2.94428 + 2.03229i 0.727276 − 1.85633i −0.925835 + 1.04505i 2.04186 + 2.30478i 1.00236 + 2.82759i −3.08240 − 2.73077i
14.18 0.416430 + 1.68952i 1.72397 + 0.167102i −0.910159 + 0.477688i −0.680333 + 0.469600i 0.435590 + 2.98227i 2.66708 − 3.01051i 1.12170 + 1.26614i 2.94415 + 0.576159i −1.07671 − 0.953883i
14.19 0.462243 + 1.87539i −0.862820 − 1.50185i −1.53252 + 0.804327i 0.592725 − 0.409129i 2.41772 − 2.31234i 1.47111 − 1.66054i 0.344844 + 0.389248i −1.51108 + 2.59165i 1.04126 + 0.922476i
14.20 0.500307 + 2.02983i 1.54512 − 0.782695i −2.09898 + 1.10163i −2.99046 + 2.06417i 2.36177 + 2.74473i −2.69978 + 3.04743i −0.513630 − 0.579768i 1.77478 − 2.41871i −5.68605 − 5.03740i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
79.f odd 26 1 inner
237.l even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 237.2.l.a ✓ 288
3.b odd 2 1 inner 237.2.l.a ✓ 288
79.f odd 26 1 inner 237.2.l.a ✓ 288
237.l even 26 1 inner 237.2.l.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
237.2.l.a ✓ 288 1.a even 1 1 trivial
237.2.l.a ✓ 288 3.b odd 2 1 inner
237.2.l.a ✓ 288 79.f odd 26 1 inner
237.2.l.a ✓ 288 237.l even 26 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(237, [\chi])\).