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Properties

Label 237.2.n.b
Level $237$
Weight $2$
Character orbit 237.n
Analytic conductor $1.892$
Analytic rank $0$
Dimension $576$
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(29,237)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(237, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([39, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("237.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 237 = 3 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 237.n (of order \(78\), degree \(24\), 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: \(576\)
Relative dimension: \(24\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 576 q - 26 q^{3} - 72 q^{4} - 23 q^{6} - 40 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q - 26 q^{3} - 72 q^{4} - 23 q^{6} - 40 q^{7} - 24 q^{9} - 48 q^{10} - 26 q^{12} - 56 q^{13} - 52 q^{15} - 52 q^{16} - q^{18} - 60 q^{19} - 30 q^{21} - 46 q^{22} - 33 q^{24} - 158 q^{25} + 13 q^{27} + 44 q^{28} + 7 q^{30} - 34 q^{31} - 26 q^{33} - 70 q^{34} - 21 q^{36} - 116 q^{37} + 99 q^{39} + 26 q^{40} + 5 q^{42} - 22 q^{43} - 16 q^{45} - 230 q^{46} - 54 q^{48} - 68 q^{49} - 6 q^{51} - 8 q^{52} + 61 q^{54} + 50 q^{55} + 130 q^{57} - 52 q^{58} - 114 q^{60} - 52 q^{61} + 139 q^{63} - 16 q^{64} - 148 q^{66} - 6 q^{67} - 65 q^{69} + 452 q^{70} - 85 q^{72} + 88 q^{73} - 23 q^{75} + 20 q^{76} + 216 q^{79} - 24 q^{81} + 112 q^{82} - 149 q^{84} + 16 q^{85} - 129 q^{87} + 354 q^{88} - 42 q^{90} + 52 q^{91} - 221 q^{93} + 416 q^{94} + 104 q^{96} - 274 q^{97} - 49 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −2.72614 − 0.556546i −1.54171 + 0.789378i 5.28216 + 2.25052i 2.54140 + 2.44105i 4.64226 − 1.29392i 0.203441 + 2.52007i −8.56772 − 5.91387i 1.75376 − 2.43399i −5.56967 − 8.06907i
29.2 −2.45200 − 0.500579i 0.849450 − 1.50945i 3.92175 + 1.67090i −0.193709 − 0.186060i −2.83845 + 3.27595i −0.184511 − 2.28558i −4.66055 − 3.21694i −1.55687 − 2.56440i 0.381836 + 0.553185i
29.3 −2.03225 − 0.414887i 1.60803 + 0.643607i 2.11795 + 0.902373i 1.56747 + 1.50558i −3.00090 − 1.97512i −0.0456779 − 0.565822i −0.515801 − 0.356032i 2.17154 + 2.06988i −2.56085 − 3.71003i
29.4 −2.03176 − 0.414786i −0.905386 − 1.47658i 2.11604 + 0.901559i −0.908380 − 0.872511i 1.22706 + 3.37559i 0.355489 + 4.40352i −0.512133 − 0.353500i −1.36055 + 2.67374i 1.48370 + 2.14952i
29.5 −2.01617 − 0.411605i −1.61759 + 0.619200i 2.05558 + 0.875800i −2.37473 − 2.28096i 3.51620 − 0.582607i −0.165464 − 2.04964i −0.396914 − 0.273970i 2.23318 − 2.00322i 3.84901 + 5.57626i
29.6 −1.58659 − 0.323905i −1.47631 − 0.905815i 0.572393 + 0.243874i 1.81325 + 1.74165i 2.04890 + 1.91534i −0.271988 − 3.36917i 1.83618 + 1.26742i 1.35900 + 2.67453i −2.31276 − 3.35061i
29.7 −1.30939 − 0.267314i 1.38553 − 1.03938i −0.196908 − 0.0838945i −1.82201 − 1.75006i −2.09204 + 0.990582i −0.00950347 − 0.117722i 2.43507 + 1.68081i 0.839387 − 2.88018i 1.91791 + 2.77857i
29.8 −0.960157 − 0.196018i 1.49205 + 0.879654i −0.956480 − 0.407518i −1.49014 − 1.43130i −1.26017 − 1.13707i 0.232350 + 2.87817i 2.45148 + 1.69213i 1.45242 + 2.62497i 1.15021 + 1.66636i
29.9 −0.910505 − 0.185881i −0.277799 + 1.70963i −1.04549 − 0.445443i 2.36421 + 2.27085i 0.570724 − 1.50499i 0.0660252 + 0.817869i 2.39870 + 1.65570i −2.84566 − 0.949866i −1.73051 − 2.50708i
29.10 −0.773873 − 0.157987i 0.823638 − 1.52369i −1.26604 − 0.539409i 2.75728 + 2.64841i −0.878115 + 1.04902i 0.357602 + 4.42970i 2.19458 + 1.51481i −1.64324 − 2.50993i −1.71537 − 2.48515i
29.11 −0.440118 − 0.0898507i −1.45674 + 0.936970i −1.65433 − 0.704843i 0.102481 + 0.0984348i 0.725324 − 0.281488i −0.0670482 − 0.830541i 1.40413 + 0.969201i 1.24418 − 2.72984i −0.0362595 − 0.0525309i
29.12 −0.346609 − 0.0707608i 0.750623 + 1.56095i −1.72483 − 0.734881i −1.61150 − 1.54787i −0.149719 − 0.594154i −0.382070 − 4.73278i 1.12812 + 0.778682i −1.87313 + 2.34337i 0.449034 + 0.650538i
29.13 0.346609 + 0.0707608i 1.68385 − 0.405779i −1.72483 − 0.734881i 1.61150 + 1.54787i 0.612351 − 0.0214963i −0.382070 − 4.73278i −1.12812 − 0.778682i 2.67069 − 1.36654i 0.449034 + 0.650538i
29.14 0.440118 + 0.0898507i −0.195526 − 1.72098i −1.65433 − 0.704843i −0.102481 − 0.0984348i 0.0685765 − 0.775002i −0.0670482 − 0.830541i −1.40413 − 0.969201i −2.92354 + 0.672994i −0.0362595 − 0.0525309i
29.15 0.773873 + 0.157987i −0.659349 + 1.60164i −1.26604 − 0.539409i −2.75728 − 2.64841i −0.763292 + 1.13530i 0.357602 + 4.42970i −2.19458 − 1.51481i −2.13052 − 2.11208i −1.71537 − 2.48515i
29.16 0.910505 + 0.185881i 1.14859 − 1.29643i −1.04549 − 0.445443i −2.36421 − 2.27085i 1.28678 − 0.966905i 0.0660252 + 0.817869i −2.39870 − 1.65570i −0.361466 − 2.97814i −1.73051 − 2.50708i
29.17 0.960157 + 0.196018i 1.62502 + 0.599415i −0.956480 − 0.407518i 1.49014 + 1.43130i 1.44278 + 0.894066i 0.232350 + 2.87817i −2.45148 − 1.69213i 2.28140 + 1.94813i 1.15021 + 1.66636i
29.18 1.30939 + 0.267314i 0.0711646 + 1.73059i −0.196908 − 0.0838945i 1.82201 + 1.75006i −0.369429 + 2.28504i −0.00950347 − 0.117722i −2.43507 − 1.68081i −2.98987 + 0.246313i 1.91791 + 2.77857i
29.19 1.58659 + 0.323905i −1.63534 − 0.570681i 0.572393 + 0.243874i −1.81325 − 1.74165i −2.40976 − 1.43513i −0.271988 − 3.36917i −1.83618 − 1.26742i 2.34865 + 1.86651i −2.31276 − 3.35061i
29.20 2.01617 + 0.411605i −0.543401 − 1.64460i 2.05558 + 0.875800i 2.37473 + 2.28096i −0.418665 − 3.53947i −0.165464 − 2.04964i 0.396914 + 0.273970i −2.40943 + 1.78736i 3.84901 + 5.57626i
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
79.h odd 78 1 inner
237.n even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 237.2.n.b ✓ 576
3.b odd 2 1 inner 237.2.n.b ✓ 576
79.h odd 78 1 inner 237.2.n.b ✓ 576
237.n even 78 1 inner 237.2.n.b ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
237.2.n.b ✓ 576 1.a even 1 1 trivial
237.2.n.b ✓ 576 3.b odd 2 1 inner
237.2.n.b ✓ 576 79.h odd 78 1 inner
237.2.n.b ✓ 576 237.n even 78 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{576} + 60 T_{2}^{574} + 1735 T_{2}^{572} + 31456 T_{2}^{570} + 386779 T_{2}^{568} + \cdots + 82\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(237, [\chi])\). Copy content Toggle raw display