↑

Properties

Label 726.2.m.a
Level $726$
Weight $2$
Character orbit 726.m
Analytic conductor $5.797$
Analytic rank $0$
Dimension $200$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [726,2,Mod(25,726)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(726, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([0, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("726.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.m (of order \(55\), degree \(40\), 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: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(200\)
Relative dimension: \(5\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 200 q - 5 q^{2} + 50 q^{3} + 5 q^{4} + 5 q^{6} + q^{7} - 5 q^{8} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 200 q - 5 q^{2} + 50 q^{3} + 5 q^{4} + 5 q^{6} + q^{7} - 5 q^{8} - 50 q^{9} + 11 q^{11} + 20 q^{12} + 21 q^{13} + 10 q^{14} + 5 q^{16} + 12 q^{17} - 5 q^{18} + 4 q^{21} + 11 q^{22} - 22 q^{23} + 5 q^{24} + 3 q^{25} + q^{26} + 50 q^{27} + q^{28} - 55 q^{30} - 36 q^{31} + 20 q^{32} - 11 q^{33} - 7 q^{34} - 154 q^{35} + 5 q^{36} + 19 q^{37} - 22 q^{38} + 34 q^{39} - 23 q^{41} - 10 q^{42} + 44 q^{43} - 11 q^{44} + 44 q^{46} - 20 q^{47} - 5 q^{48} + 24 q^{49} - 3 q^{50} + 43 q^{51} - q^{52} + 176 q^{53} - 20 q^{54} + 33 q^{55} - 7 q^{56} - 166 q^{59} - 38 q^{61} + 14 q^{62} + q^{63} + 5 q^{64} - 6 q^{65} + 44 q^{66} + 74 q^{67} + q^{68} - 33 q^{69} + 55 q^{70} - 24 q^{71} - 5 q^{72} - 35 q^{73} + 14 q^{74} - 3 q^{75} - 22 q^{76} + 187 q^{77} + 4 q^{78} + 32 q^{79} - 50 q^{81} + 12 q^{82} - 47 q^{83} - q^{84} + 5 q^{85} + 11 q^{88} + 62 q^{89} + 165 q^{91} + 66 q^{92} - 74 q^{93} + 64 q^{94} - 57 q^{95} + 5 q^{96} - 33 q^{97} + 52 q^{98} + 11 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0.466667 + 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i −0.558900 + 2.75828i 0.696938 − 0.717132i 0.0141030 + 0.0536535i −0.993482 − 0.113991i −0.809017 + 0.587785i −2.70034 + 0.792890i
25.2 0.466667 + 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i −0.223091 + 1.10100i 0.696938 − 0.717132i −0.660395 − 2.51240i −0.993482 − 0.113991i −0.809017 + 0.587785i −1.07787 + 0.316490i
25.3 0.466667 + 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i 0.0927210 − 0.457596i 0.696938 − 0.717132i −0.985878 − 3.75067i −0.993482 − 0.113991i −0.809017 + 0.587785i 0.447983 − 0.131540i
25.4 0.466667 + 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i 0.386070 − 1.90533i 0.696938 − 0.717132i 0.653883 + 2.48763i −0.993482 − 0.113991i −0.809017 + 0.587785i 1.86530 − 0.547702i
25.5 0.466667 + 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i 0.868915 − 4.28826i 0.696938 − 0.717132i −0.551782 − 2.09920i −0.993482 − 0.113991i −0.809017 + 0.587785i 4.19817 − 1.23270i
31.1 −0.774142 + 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i −3.28591 + 1.61557i −0.254218 + 0.967147i 0.371183 + 4.32161i 0.466667 + 0.884433i 0.309017 − 0.951057i 1.52108 − 3.33070i
31.2 −0.774142 + 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i −2.15772 + 1.06088i −0.254218 + 0.967147i −0.0196196 − 0.228426i 0.466667 + 0.884433i 0.309017 − 0.951057i 0.998832 − 2.18714i
31.3 −0.774142 + 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i −1.00552 + 0.494380i −0.254218 + 0.967147i −0.0292996 − 0.341129i 0.466667 + 0.884433i 0.309017 − 0.951057i 0.465465 − 1.01923i
31.4 −0.774142 + 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i 1.30134 − 0.639828i −0.254218 + 0.967147i −0.171242 − 1.99374i 0.466667 + 0.884433i 0.309017 − 0.951057i −0.602406 + 1.31908i
31.5 −0.774142 + 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i 2.26676 − 1.11449i −0.254218 + 0.967147i 0.367322 + 4.27666i 0.466667 + 0.884433i 0.309017 − 0.951057i −1.04931 + 2.29766i
37.1 0.362808 + 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i −1.33037 − 1.36892i −0.998369 + 0.0570888i 1.36125 − 2.25738i −0.897398 − 0.441221i −0.809017 − 0.587785i 0.792976 − 1.73637i
37.2 0.362808 + 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i −0.990650 − 1.01935i −0.998369 + 0.0570888i 0.916339 − 1.51958i −0.897398 − 0.441221i −0.809017 − 0.587785i 0.590484 − 1.29298i
37.3 0.362808 + 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i 0.164177 + 0.168934i −0.998369 + 0.0570888i −2.34603 + 3.89045i −0.897398 − 0.441221i −0.809017 − 0.587785i −0.0978591 + 0.214281i
37.4 0.362808 + 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i 1.76554 + 1.81670i −0.998369 + 0.0570888i −1.42021 + 2.35515i −0.897398 − 0.441221i −0.809017 − 0.587785i −1.05237 + 2.30436i
37.5 0.362808 + 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i 2.34137 + 2.40921i −0.998369 + 0.0570888i 2.17791 − 3.61165i −0.897398 − 0.441221i −0.809017 − 0.587785i −1.39559 + 3.05591i
49.1 0.921124 + 0.389270i 0.809017 + 0.587785i 0.696938 + 0.717132i −0.559259 − 2.12764i 0.516397 + 0.856349i 0.223019 − 0.0795729i 0.362808 + 0.931864i 0.309017 + 0.951057i 0.313081 − 2.17753i
49.2 0.921124 + 0.389270i 0.809017 + 0.587785i 0.696938 + 0.717132i 0.119237 + 0.453625i 0.516397 + 0.856349i −3.67610 + 1.31163i 0.362808 + 0.931864i 0.309017 + 0.951057i −0.0667505 + 0.464260i
49.3 0.921124 + 0.389270i 0.809017 + 0.587785i 0.696938 + 0.717132i 0.297189 + 1.13062i 0.516397 + 0.856349i 1.89112 − 0.674751i 0.362808 + 0.931864i 0.309017 + 0.951057i −0.166370 + 1.15713i
49.4 0.921124 + 0.389270i 0.809017 + 0.587785i 0.696938 + 0.717132i 0.707254 + 2.69067i 0.516397 + 0.856349i 4.19536 − 1.49690i 0.362808 + 0.931864i 0.309017 + 0.951057i −0.395930 + 2.75376i
49.5 0.921124 + 0.389270i 0.809017 + 0.587785i 0.696938 + 0.717132i 0.906907 + 3.45023i 0.516397 + 0.856349i −4.21317 + 1.50325i 0.362808 + 0.931864i 0.309017 + 0.951057i −0.507699 + 3.53112i
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.m.a ✓ 200
121.g even 55 1 inner 726.2.m.a ✓ 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.2.m.a ✓ 200 1.a even 1 1 trivial
726.2.m.a ✓ 200 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{200} - 14 T_{5}^{198} + 273 T_{5}^{197} - 99 T_{5}^{196} - 4194 T_{5}^{195} + \cdots + 21\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\). Copy content Toggle raw display