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Properties

Label 825.2.by.a
Level $825$
Weight $2$
Character orbit 825.by
Analytic conductor $6.588$
Analytic rank $0$
Dimension $240$
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] = [825,2,Mod(4,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.by (of order \(10\), 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(60\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 60 q^{4} - 4 q^{5} + 4 q^{6} - 10 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 60 q^{4} - 4 q^{5} + 4 q^{6} - 10 q^{7} + 60 q^{9} + 8 q^{10} + 4 q^{11} - 20 q^{12} - 64 q^{16} + 10 q^{17} - 14 q^{19} + 46 q^{20} + 8 q^{21} + 10 q^{22} + 40 q^{23} - 12 q^{24} - 36 q^{25} + 40 q^{26} - 16 q^{30} - 12 q^{31} + 20 q^{33} - 60 q^{36} - 16 q^{39} - 42 q^{40} + 24 q^{41} - 44 q^{44} + 4 q^{45} + 12 q^{46} - 30 q^{47} + 58 q^{49} + 62 q^{50} + 16 q^{51} + 20 q^{52} - 30 q^{53} - 4 q^{54} + 40 q^{57} - 70 q^{58} + 6 q^{59} + 48 q^{60} + 38 q^{61} - 30 q^{62} + 104 q^{64} + 60 q^{65} - 12 q^{66} - 40 q^{67} + 250 q^{68} - 8 q^{69} - 14 q^{70} + 64 q^{71} + 22 q^{74} - 168 q^{76} - 10 q^{77} - 48 q^{79} + 252 q^{80} - 60 q^{81} - 40 q^{82} + 20 q^{83} - 24 q^{84} - 30 q^{85} + 88 q^{86} - 40 q^{87} - 100 q^{88} + 12 q^{90} + 40 q^{91} - 150 q^{92} - 14 q^{94} - 42 q^{95} + 28 q^{96} - 70 q^{97} - 140 q^{98} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.62702 − 2.23940i −0.587785 − 0.809017i −1.74969 + 5.38501i −2.23606 + 0.00369733i −0.855376 + 2.63258i −3.66791 + 1.19178i 9.64084 − 3.13250i −0.309017 + 0.951057i 3.64641 + 5.00144i
4.2 −1.61147 − 2.21800i 0.587785 + 0.809017i −1.70464 + 5.24636i −1.32539 − 1.80093i 0.847200 − 2.60741i 3.97271 − 1.29081i 9.16856 − 2.97905i −0.309017 + 0.951057i −1.85864 + 5.84185i
4.3 −1.58214 − 2.17763i 0.587785 + 0.809017i −1.62086 + 4.98850i −0.405324 + 2.19903i 0.831779 − 2.55995i −0.542994 + 0.176430i 8.30761 − 2.69931i −0.309017 + 0.951057i 5.42993 − 2.59652i
4.4 −1.52978 − 2.10556i −0.587785 − 0.809017i −1.47513 + 4.53997i 1.11498 − 1.93825i −0.804252 + 2.47523i 0.220156 − 0.0715332i 6.86532 − 2.23068i −0.309017 + 0.951057i −5.78678 + 0.617448i
4.5 −1.52010 − 2.09224i −0.587785 − 0.809017i −1.44873 + 4.45872i 1.35053 + 1.78216i −0.799165 + 2.45958i −0.283269 + 0.0920396i 6.61178 − 2.14830i −0.309017 + 0.951057i 1.67576 − 5.53468i
4.6 −1.30303 − 1.79346i 0.587785 + 0.809017i −0.900593 + 2.77174i 2.23555 + 0.0482506i 0.685041 − 2.10834i 3.97500 − 1.29155i 1.92783 − 0.626389i −0.309017 + 0.951057i −2.82644 − 4.07224i
4.7 −1.29571 − 1.78339i −0.587785 − 0.809017i −0.883577 + 2.71937i 0.399664 − 2.20006i −0.681193 + 2.09650i 3.07728 − 0.999868i 1.80155 − 0.585359i −0.309017 + 0.951057i −4.44140 + 2.13788i
4.8 −1.28637 − 1.77054i 0.587785 + 0.809017i −0.862018 + 2.65302i −1.77424 + 1.36091i 0.676285 − 2.08139i 1.00018 − 0.324978i 1.64336 − 0.533960i −0.309017 + 0.951057i 4.69187 + 1.39072i
4.9 −1.25828 − 1.73187i 0.587785 + 0.809017i −0.798079 + 2.45623i −2.12526 − 0.695176i 0.661516 − 2.03594i −4.70550 + 1.52891i 1.18621 − 0.385423i −0.309017 + 0.951057i 1.47021 + 4.55540i
4.10 −1.21171 − 1.66777i −0.587785 − 0.809017i −0.695197 + 2.13959i −0.0511483 + 2.23548i −0.637033 + 1.96058i −1.60997 + 0.523111i 0.489564 − 0.159069i −0.309017 + 0.951057i 3.79025 − 2.62345i
4.11 −1.20850 − 1.66336i −0.587785 − 0.809017i −0.688253 + 2.11822i −0.466635 + 2.18684i −0.635346 + 1.95539i 4.36107 − 1.41700i 0.444331 − 0.144372i −0.309017 + 0.951057i 4.20142 − 1.86661i
4.12 −1.20723 − 1.66162i −0.587785 − 0.809017i −0.685517 + 2.10980i 2.11994 − 0.711234i −0.634681 + 1.95335i −4.36588 + 1.41856i 0.426571 − 0.138601i −0.309017 + 0.951057i −3.74106 − 2.66390i
4.13 −1.12310 − 1.54581i 0.587785 + 0.809017i −0.510153 + 1.57009i −1.07799 − 1.95907i 0.590448 − 1.81721i −1.82631 + 0.593403i −0.634409 + 0.206132i −0.309017 + 0.951057i −1.81767 + 3.86659i
4.14 −1.08192 − 1.48913i 0.587785 + 0.809017i −0.428937 + 1.32013i 1.32008 − 1.80483i 0.568798 − 1.75058i −0.161477 + 0.0524671i −1.07124 + 0.348066i −0.309017 + 0.951057i −4.11584 − 0.0130915i
4.15 −0.882893 − 1.21520i 0.587785 + 0.809017i −0.0791716 + 0.243665i 1.43185 + 1.71750i 0.464164 − 1.42855i −4.03713 + 1.31174i −2.49110 + 0.809407i −0.309017 + 0.951057i 0.822929 − 3.25635i
4.16 −0.873307 − 1.20200i −0.587785 − 0.809017i −0.0641141 + 0.197323i −0.621823 − 2.14787i −0.459125 + 1.41304i −1.08303 + 0.351897i −2.53291 + 0.822991i −0.309017 + 0.951057i −2.03870 + 2.62318i
4.17 −0.787431 − 1.08381i 0.587785 + 0.809017i 0.0634471 − 0.195270i 1.86350 + 1.23586i 0.413977 − 1.27409i 0.114201 − 0.0371061i −2.80977 + 0.912951i −0.309017 + 0.951057i −0.127946 − 2.99283i
4.18 −0.734437 − 1.01087i −0.587785 − 0.809017i 0.135582 − 0.417277i −2.09294 + 0.787161i −0.386117 + 1.18834i 1.93521 − 0.628789i −2.89808 + 0.941642i −0.309017 + 0.951057i 2.33284 + 1.53756i
4.19 −0.730422 − 1.00534i −0.587785 − 0.809017i 0.140842 − 0.433468i 2.02211 + 0.954494i −0.384006 + 1.18185i 2.77555 − 0.901831i −2.90235 + 0.943032i −0.309017 + 0.951057i −0.517406 − 2.73009i
4.20 −0.686976 − 0.945541i −0.587785 − 0.809017i 0.195922 − 0.602986i −2.21471 − 0.308288i −0.361164 + 1.11155i −2.43045 + 0.789701i −2.92784 + 0.951313i −0.309017 + 0.951057i 1.22996 + 2.30589i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.ba even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.by.a ✓ 240
11.c even 5 1 825.2.cb.a yes 240
25.e even 10 1 825.2.cb.a yes 240
275.ba even 10 1 inner 825.2.by.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.by.a ✓ 240 1.a even 1 1 trivial
825.2.by.a ✓ 240 275.ba even 10 1 inner
825.2.cb.a yes 240 11.c even 5 1
825.2.cb.a yes 240 25.e even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(825, [\chi])\).