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Properties

Label 650.2.bj.b
Level $650$
Weight $2$
Character orbit 650.bj
Analytic conductor $5.190$
Analytic rank $0$
Dimension $288$
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] = [650,2,Mod(37,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([27, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.bj (of order \(60\), degree \(16\), 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(18\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 36 q^{4} + 2 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 36 q^{4} + 2 q^{5} + 8 q^{9} + 14 q^{15} + 36 q^{16} + 12 q^{17} - 328 q^{18} - 52 q^{19} + 32 q^{20} - 12 q^{21} + 14 q^{25} - 14 q^{26} + 42 q^{27} - 14 q^{29} - 6 q^{30} - 12 q^{31} - 24 q^{33} - 24 q^{34} + 80 q^{35} + 8 q^{36} + 30 q^{37} + 66 q^{39} - 34 q^{41} - 6 q^{42} + 30 q^{45} + 18 q^{47} - 184 q^{49} - 12 q^{52} - 32 q^{53} - 6 q^{54} + 26 q^{55} - 12 q^{58} + 32 q^{59} + 6 q^{61} - 12 q^{62} - 80 q^{63} + 72 q^{64} + 72 q^{65} - 44 q^{66} - 12 q^{67} + 6 q^{68} - 70 q^{69} - 8 q^{70} + 6 q^{71} + 36 q^{72} + 80 q^{73} - 24 q^{74} - 184 q^{75} + 18 q^{76} + 76 q^{77} - 34 q^{78} + 80 q^{79} - 10 q^{80} - 148 q^{81} - 94 q^{82} - 14 q^{83} + 24 q^{84} + 60 q^{85} + 16 q^{87} + 70 q^{89} - 32 q^{90} + 178 q^{91} + 34 q^{93} - 38 q^{94} - 34 q^{95} + 54 q^{97} - 56 q^{98} + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 0.994522 + 0.104528i −0.178150 − 3.39930i 0.978148 + 0.207912i 2.23383 + 0.100036i 0.178150 − 3.39930i 0.522349 − 0.904735i 0.951057 + 0.309017i −8.53992 + 0.897582i 2.21114 + 0.332986i
37.2 0.994522 + 0.104528i −0.139629 − 2.66429i 0.978148 + 0.207912i −2.05467 + 0.882231i 0.139629 − 2.66429i −1.38072 + 2.39147i 0.951057 + 0.309017i −4.09537 + 0.430441i −2.13563 + 0.662627i
37.3 0.994522 + 0.104528i −0.126657 − 2.41677i 0.978148 + 0.207912i 0.413829 + 2.19744i 0.126657 − 2.41677i 1.28459 − 2.22497i 0.951057 + 0.309017i −2.84115 + 0.298617i 0.181867 + 2.22866i
37.4 0.994522 + 0.104528i −0.114535 − 2.18546i 0.978148 + 0.207912i −1.40724 − 1.73772i 0.114535 − 2.18546i 1.58667 − 2.74819i 0.951057 + 0.309017i −1.77953 + 0.187036i −1.21789 − 1.87530i
37.5 0.994522 + 0.104528i −0.0909575 − 1.73557i 0.978148 + 0.207912i 2.07941 + 0.822221i 0.0909575 − 1.73557i −2.17865 + 3.77353i 0.951057 + 0.309017i −0.0203745 + 0.00214145i 1.98207 + 1.03507i
37.6 0.994522 + 0.104528i −0.0707893 − 1.35074i 0.978148 + 0.207912i 1.71187 − 1.43858i 0.0707893 − 1.35074i 1.09573 − 1.89786i 0.951057 + 0.309017i 1.16408 − 0.122349i 1.85286 − 1.25176i
37.7 0.994522 + 0.104528i −0.0619083 − 1.18128i 0.978148 + 0.207912i −1.93469 + 1.12116i 0.0619083 − 1.18128i −1.06085 + 1.83744i 0.951057 + 0.309017i 1.59197 − 0.167323i −2.04128 + 0.912784i
37.8 0.994522 + 0.104528i −0.0553227 − 1.05562i 0.978148 + 0.207912i 0.527290 − 2.17301i 0.0553227 − 1.05562i −0.792262 + 1.37224i 0.951057 + 0.309017i 1.87229 − 0.196786i 0.751542 − 2.10599i
37.9 0.994522 + 0.104528i 0.0160455 + 0.306166i 0.978148 + 0.207912i 0.666806 + 2.13433i −0.0160455 + 0.306166i −0.983345 + 1.70320i 0.951057 + 0.309017i 2.89009 − 0.303760i 0.440054 + 2.19234i
37.10 0.994522 + 0.104528i 0.0177030 + 0.337794i 0.978148 + 0.207912i 2.03283 − 0.931454i −0.0177030 + 0.337794i 0.149208 − 0.258435i 0.951057 + 0.309017i 2.86977 − 0.301625i 2.11906 − 0.713863i
37.11 0.994522 + 0.104528i 0.0352945 + 0.673458i 0.978148 + 0.207912i −1.66616 − 1.49128i −0.0352945 + 0.673458i 0.380784 − 0.659536i 0.951057 + 0.309017i 2.53127 − 0.266047i −1.50115 − 1.65727i
37.12 0.994522 + 0.104528i 0.0416548 + 0.794821i 0.978148 + 0.207912i 1.68606 + 1.46874i −0.0416548 + 0.794821i 1.91237 − 3.31233i 0.951057 + 0.309017i 2.35356 − 0.247369i 1.52330 + 1.63693i
37.13 0.994522 + 0.104528i 0.0799405 + 1.52536i 0.978148 + 0.207912i −1.73805 + 1.40683i −0.0799405 + 1.52536i 1.75971 − 3.04791i 0.951057 + 0.309017i 0.663244 − 0.0697098i −1.87559 + 1.21745i
37.14 0.994522 + 0.104528i 0.0939342 + 1.79237i 0.978148 + 0.207912i −0.751101 − 2.10615i −0.0939342 + 1.79237i −2.51563 + 4.35719i 0.951057 + 0.309017i −0.220208 + 0.0231448i −0.526834 − 2.17312i
37.15 0.994522 + 0.104528i 0.114962 + 2.19361i 0.978148 + 0.207912i −0.403883 − 2.19929i −0.114962 + 2.19361i 2.14415 − 3.71377i 0.951057 + 0.309017i −1.81513 + 0.190778i −0.171782 − 2.22946i
37.16 0.994522 + 0.104528i 0.133178 + 2.54119i 0.978148 + 0.207912i −1.92175 + 1.14320i −0.133178 + 2.54119i −1.72799 + 2.99296i 0.951057 + 0.309017i −3.45632 + 0.363274i −2.03071 + 0.936056i
37.17 0.994522 + 0.104528i 0.142284 + 2.71494i 0.978148 + 0.207912i 2.23490 − 0.0723985i −0.142284 + 2.71494i −1.03037 + 1.78466i 0.951057 + 0.309017i −4.36708 + 0.458998i 2.23022 + 0.161608i
37.18 0.994522 + 0.104528i 0.162953 + 3.10932i 0.978148 + 0.207912i 0.607126 + 2.15207i −0.162953 + 3.10932i 0.834242 − 1.44495i 0.951057 + 0.309017i −6.65778 + 0.699760i 0.378848 + 2.20374i
123.1 0.994522 − 0.104528i −0.178150 + 3.39930i 0.978148 − 0.207912i 2.23383 − 0.100036i 0.178150 + 3.39930i 0.522349 + 0.904735i 0.951057 − 0.309017i −8.53992 − 0.897582i 2.21114 − 0.332986i
123.2 0.994522 − 0.104528i −0.139629 + 2.66429i 0.978148 − 0.207912i −2.05467 − 0.882231i 0.139629 + 2.66429i −1.38072 − 2.39147i 0.951057 − 0.309017i −4.09537 − 0.430441i −2.13563 − 0.662627i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
325.bi even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.bj.b ✓ 288
13.f odd 12 1 650.2.bm.b yes 288
25.f odd 20 1 650.2.bm.b yes 288
325.bi even 60 1 inner 650.2.bj.b ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.bj.b ✓ 288 1.a even 1 1 trivial
650.2.bj.b ✓ 288 325.bi even 60 1 inner
650.2.bm.b yes 288 13.f odd 12 1
650.2.bm.b yes 288 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{288} - 4 T_{3}^{286} - 14 T_{3}^{285} + 288 T_{3}^{284} + 6 T_{3}^{283} - 1506 T_{3}^{282} + \cdots + 11\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display