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Properties

Label 650.2.l.d
Level $650$
Weight $2$
Character orbit 650.l
Analytic conductor $5.190$
Analytic rank $0$
Dimension $36$
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(131,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.131");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.l (of order \(5\), degree \(4\), 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: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 9 q^{2} + 3 q^{3} - 9 q^{4} - 3 q^{5} + 3 q^{6} + 8 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 9 q^{2} + 3 q^{3} - 9 q^{4} - 3 q^{5} + 3 q^{6} + 8 q^{7} - 9 q^{8} - 18 q^{9} - 3 q^{10} - 4 q^{11} - 2 q^{12} + 9 q^{13} - 2 q^{14} - q^{15} - 9 q^{16} + q^{17} + 42 q^{18} - 22 q^{19} + 2 q^{20} + 7 q^{21} - 4 q^{22} + 10 q^{23} - 2 q^{24} - 17 q^{25} - 36 q^{26} - 12 q^{27} - 2 q^{28} + 12 q^{29} - 11 q^{30} - 13 q^{31} + 36 q^{32} + 18 q^{33} + 6 q^{34} + 27 q^{35} - 18 q^{36} + 15 q^{37} - 12 q^{38} + 2 q^{39} + 7 q^{40} + 6 q^{41} + 7 q^{42} - 30 q^{43} - 4 q^{44} - 58 q^{45} - 15 q^{46} + q^{47} - 2 q^{48} + 84 q^{49} + 13 q^{50} + 90 q^{51} + 9 q^{52} - 26 q^{53} + 13 q^{54} - 4 q^{55} - 2 q^{56} + 8 q^{57} + 12 q^{58} - 24 q^{59} + 14 q^{60} - 40 q^{61} - 3 q^{62} - 11 q^{63} - 9 q^{64} + 3 q^{65} - 22 q^{66} - 34 q^{67} - 14 q^{68} - 35 q^{69} - 18 q^{70} - 33 q^{71} - 3 q^{72} - q^{73} - 10 q^{74} + 16 q^{75} + 68 q^{76} - 21 q^{77} - 3 q^{78} + 32 q^{79} - 3 q^{80} + 20 q^{81} - 4 q^{82} - q^{83} - 23 q^{84} + 9 q^{85} + 10 q^{86} - 4 q^{88} + 37 q^{89} - 38 q^{90} + 2 q^{91} - 15 q^{92} + 60 q^{93} + q^{94} - 18 q^{95} + 3 q^{96} + 53 q^{97} + 29 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 0.309017 + 0.951057i −2.33879 + 1.69923i −0.809017 + 0.587785i −2.23074 + 0.154289i −2.33879 − 1.69923i 0.639592 −0.809017 − 0.587785i 1.65551 − 5.09514i −0.836073 − 2.07388i
131.2 0.309017 + 0.951057i −1.70722 + 1.24037i −0.809017 + 0.587785i 0.194897 + 2.22756i −1.70722 − 1.24037i 3.98562 −0.809017 − 0.587785i 0.449043 − 1.38201i −2.05831 + 0.873711i
131.3 0.309017 + 0.951057i −0.791723 + 0.575220i −0.809017 + 0.587785i −0.721553 − 2.11645i −0.791723 − 0.575220i 0.0472327 −0.809017 − 0.587785i −0.631104 + 1.94234i 1.78989 − 1.34026i
131.4 0.309017 + 0.951057i −0.653582 + 0.474855i −0.809017 + 0.587785i 1.24008 + 1.86070i −0.653582 − 0.474855i −3.62896 −0.809017 − 0.587785i −0.725369 + 2.23246i −1.38643 + 1.75437i
131.5 0.309017 + 0.951057i 0.0104546 − 0.00759568i −0.809017 + 0.587785i 1.51566 − 1.64401i 0.0104546 + 0.00759568i −1.25693 −0.809017 − 0.587785i −0.926999 + 2.85301i 2.03191 + 0.933453i
131.6 0.309017 + 0.951057i 0.970918 − 0.705413i −0.809017 + 0.587785i 2.10986 − 0.740590i 0.970918 + 0.705413i 4.65774 −0.809017 − 0.587785i −0.481977 + 1.48337i 1.35633 + 1.77774i
131.7 0.309017 + 0.951057i 1.44461 − 1.04957i −0.809017 + 0.587785i −1.58262 − 1.57966i 1.44461 + 1.04957i −2.59523 −0.809017 − 0.587785i 0.0582474 − 0.179267i 1.01329 − 1.99330i
131.8 0.309017 + 0.951057i 1.79156 − 1.30164i −0.809017 + 0.587785i −0.738946 + 2.11044i 1.79156 + 1.30164i −4.76570 −0.809017 − 0.587785i 0.588356 − 1.81077i −2.23549 − 0.0506178i
131.9 0.309017 + 0.951057i 2.58280 − 1.87651i −0.809017 + 0.587785i −2.21370 + 0.315508i 2.58280 + 1.87651i 4.91664 −0.809017 − 0.587785i 2.22250 − 6.84015i −0.984136 − 2.00785i
261.1 −0.809017 + 0.587785i −1.04672 + 3.22147i 0.309017 − 0.951057i 1.26558 − 1.84345i −1.04672 − 3.22147i −0.291606 0.309017 + 0.951057i −6.85522 − 4.98061i 0.0596820 + 2.23527i
261.2 −0.809017 + 0.587785i −0.589498 + 1.81429i 0.309017 − 0.951057i 1.50295 + 1.65564i −0.589498 − 1.81429i 1.04470 0.309017 + 0.951057i −0.517080 − 0.375681i −2.18907 − 0.456024i
261.3 −0.809017 + 0.587785i −0.551082 + 1.69606i 0.309017 − 0.951057i −1.22528 − 1.87048i −0.551082 − 1.69606i 0.192704 0.309017 + 0.951057i −0.145862 − 0.105975i 2.09071 + 0.793052i
261.4 −0.809017 + 0.587785i −0.269649 + 0.829895i 0.309017 − 0.951057i −2.12207 + 0.704867i −0.269649 − 0.829895i −4.17320 0.309017 + 0.951057i 1.81104 + 1.31579i 1.30248 − 1.81757i
261.5 −0.809017 + 0.587785i 0.0607683 − 0.187026i 0.309017 − 0.951057i 1.33939 − 1.79054i 0.0607683 + 0.187026i 1.65834 0.309017 + 0.951057i 2.39577 + 1.74063i −0.0311411 + 2.23585i
261.6 −0.809017 + 0.587785i 0.158687 − 0.488388i 0.309017 − 0.951057i 0.0912662 + 2.23420i 0.158687 + 0.488388i −2.77061 0.309017 + 0.951057i 2.21371 + 1.60835i −1.38707 − 1.75386i
261.7 −0.809017 + 0.587785i 0.573837 − 1.76609i 0.309017 − 0.951057i −0.740917 + 2.10975i 0.573837 + 1.76609i 5.19895 0.309017 + 0.951057i −0.362729 − 0.263538i −0.640665 − 2.14232i
261.8 −0.809017 + 0.587785i 0.884434 − 2.72201i 0.309017 − 0.951057i −1.38472 − 1.75572i 0.884434 + 2.72201i −0.995534 0.309017 + 0.951057i −4.20005 − 3.05152i 2.15225 + 0.606484i
261.9 −0.809017 + 0.587785i 0.970206 − 2.98599i 0.309017 − 0.951057i 2.20084 − 0.395326i 0.970206 + 2.98599i 2.13627 0.309017 + 0.951057i −5.54777 − 4.03069i −1.54815 + 1.61345i
391.1 −0.809017 − 0.587785i −1.04672 − 3.22147i 0.309017 + 0.951057i 1.26558 + 1.84345i −1.04672 + 3.22147i −0.291606 0.309017 − 0.951057i −6.85522 + 4.98061i 0.0596820 − 2.23527i
391.2 −0.809017 − 0.587785i −0.589498 − 1.81429i 0.309017 + 0.951057i 1.50295 − 1.65564i −0.589498 + 1.81429i 1.04470 0.309017 − 0.951057i −0.517080 + 0.375681i −2.18907 + 0.456024i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.l.d ✓ 36
25.d even 5 1 inner 650.2.l.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.l.d ✓ 36 1.a even 1 1 trivial
650.2.l.d ✓ 36 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 3 T_{3}^{35} + 27 T_{3}^{34} - 64 T_{3}^{33} + 376 T_{3}^{32} - 721 T_{3}^{31} + 3739 T_{3}^{30} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display