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Properties

Label 825.2.m.d
Level $825$
Weight $2$
Character orbit 825.m
Analytic conductor $6.588$
Analytic rank $0$
Dimension $116$
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(16,825)]
 
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, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.m (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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(29\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 116 q + 4 q^{2} + 29 q^{3} - 30 q^{4} - 3 q^{5} + q^{6} - 2 q^{7} + 11 q^{8} - 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q + 4 q^{2} + 29 q^{3} - 30 q^{4} - 3 q^{5} + q^{6} - 2 q^{7} + 11 q^{8} - 29 q^{9} + 2 q^{10} + 2 q^{11} + 30 q^{12} + 20 q^{13} - 2 q^{14} + 3 q^{15} - 26 q^{16} + 11 q^{17} - q^{18} - q^{19} + 17 q^{20} + 2 q^{21} - 14 q^{22} + 14 q^{23} - q^{24} - 13 q^{25} - 16 q^{26} + 29 q^{27} - 12 q^{28} + 5 q^{29} - 12 q^{30} + 62 q^{31} - 94 q^{32} + 13 q^{33} + 20 q^{34} + 19 q^{35} - 35 q^{36} - 10 q^{37} - 21 q^{38} - 20 q^{39} - 8 q^{40} + 4 q^{41} + 12 q^{42} - 46 q^{43} - 25 q^{44} + 2 q^{45} - 15 q^{46} - 15 q^{47} - 134 q^{48} - 5 q^{49} + 100 q^{50} - 11 q^{51} + 4 q^{52} - 6 q^{53} - 4 q^{54} - 21 q^{55} + 28 q^{56} + 21 q^{57} - 9 q^{58} - 29 q^{59} - 7 q^{60} - 10 q^{61} + 22 q^{62} + 8 q^{63} + 3 q^{64} + 10 q^{65} - 6 q^{66} - 35 q^{67} + 60 q^{68} - 14 q^{69} + 94 q^{70} + 50 q^{71} - 24 q^{72} - 60 q^{73} - 12 q^{74} - 17 q^{75} + 126 q^{76} + 76 q^{77} - 19 q^{78} + 11 q^{79} + q^{80} - 29 q^{81} - 21 q^{82} - 13 q^{83} - 13 q^{84} - 21 q^{85} + 6 q^{86} - 5 q^{87} - 61 q^{88} + 13 q^{89} - 8 q^{90} + 20 q^{91} + 97 q^{92} + 43 q^{93} + 39 q^{94} - 36 q^{95} - 26 q^{96} - 20 q^{97} - 34 q^{98} - 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.847576 + 2.60857i −0.309017 + 0.951057i −4.46822 − 3.24635i −1.98183 − 1.03555i −2.21898 − 1.61218i −0.0321751 + 0.0233766i 7.81752 − 5.67976i −0.809017 − 0.587785i 4.38105 − 4.29203i
16.2 −0.800260 + 2.46295i −0.309017 + 0.951057i −3.80765 − 2.76642i 0.723149 + 2.11591i −2.09511 − 1.52218i −2.42067 + 1.75872i 5.67044 − 4.11982i −0.809017 − 0.587785i −5.79007 + 0.0878035i
16.3 −0.776479 + 2.38976i −0.309017 + 0.951057i −3.48998 − 2.53562i −0.348283 + 2.20878i −2.03285 − 1.47695i 4.01919 − 2.92011i 4.70372 − 3.41745i −0.809017 − 0.587785i −5.00801 − 2.54738i
16.4 −0.757913 + 2.33262i −0.309017 + 0.951057i −3.24863 − 2.36027i 0.247603 − 2.22232i −1.98424 − 1.44164i −0.721549 + 0.524236i 3.99929 − 2.90566i −0.809017 − 0.587785i 4.99615 + 2.26189i
16.5 −0.727859 + 2.24012i −0.309017 + 0.951057i −2.87033 − 2.08542i 1.74945 − 1.39263i −1.90556 − 1.38447i 1.56703 − 1.13852i 2.94966 − 2.14305i −0.809017 − 0.587785i 1.84630 + 4.93262i
16.6 −0.588317 + 1.81065i −0.309017 + 0.951057i −1.31431 − 0.954903i −2.09234 + 0.788728i −1.54023 − 1.11904i −3.61705 + 2.62794i −0.578238 + 0.420115i −0.809017 − 0.587785i −0.197152 − 4.25253i
16.7 −0.554841 + 1.70762i −0.309017 + 0.951057i −0.990097 − 0.719347i −1.87831 − 1.21324i −1.45259 − 1.05537i −0.384319 + 0.279224i −1.12746 + 0.819150i −0.809017 − 0.587785i 3.11393 − 2.53429i
16.8 −0.483294 + 1.48743i −0.309017 + 0.951057i −0.360827 − 0.262156i 2.13956 − 0.649831i −1.26528 − 0.919279i 0.368134 − 0.267465i −1.96624 + 1.42855i −0.809017 − 0.587785i −0.0674617 + 3.49650i
16.9 −0.426854 + 1.31372i −0.309017 + 0.951057i 0.0743742 + 0.0540360i 1.03778 + 1.98066i −1.11752 − 0.811925i −0.334885 + 0.243308i −2.33777 + 1.69849i −0.809017 − 0.587785i −3.04502 + 0.517904i
16.10 −0.388389 + 1.19534i −0.309017 + 0.951057i 0.340048 + 0.247059i 1.09825 − 1.94778i −1.01681 − 0.738759i −3.43852 + 2.49823i −2.46102 + 1.78804i −0.809017 − 0.587785i 1.90171 + 2.06928i
16.11 −0.252495 + 0.777100i −0.309017 + 0.951057i 1.07790 + 0.783143i 0.151332 − 2.23094i −0.661041 − 0.480274i 3.23602 − 2.35111i −2.20283 + 1.60045i −0.809017 − 0.587785i 1.69545 + 0.680901i
16.12 −0.0929182 + 0.285973i −0.309017 + 0.951057i 1.54489 + 1.12243i −2.23508 + 0.0663282i −0.243263 − 0.176741i 2.68310 − 1.94939i −0.951057 + 0.690984i −0.809017 − 0.587785i 0.188712 − 0.645336i
16.13 −0.0908180 + 0.279509i −0.309017 + 0.951057i 1.54816 + 1.12480i −0.941014 + 2.02842i −0.237765 − 0.172746i −3.67252 + 2.66825i −0.930522 + 0.676064i −0.809017 − 0.587785i −0.481501 − 0.447239i
16.14 −0.0722725 + 0.222432i −0.309017 + 0.951057i 1.57378 + 1.14342i 2.17969 − 0.498934i −0.189212 − 0.137470i 1.81606 − 1.31944i −0.746498 + 0.542362i −0.809017 − 0.587785i −0.0465531 + 0.520893i
16.15 −0.0684947 + 0.210805i −0.309017 + 0.951057i 1.57829 + 1.14669i 1.73917 + 1.40545i −0.179321 − 0.130285i −0.403342 + 0.293045i −0.708476 + 0.514738i −0.809017 − 0.587785i −0.415399 + 0.270361i
16.16 −0.0210583 + 0.0648107i −0.309017 + 0.951057i 1.61428 + 1.17284i −1.19461 − 1.89022i −0.0551313 − 0.0400552i −1.33663 + 0.971117i −0.220269 + 0.160035i −0.809017 − 0.587785i 0.147663 − 0.0376185i
16.17 0.149586 − 0.460378i −0.309017 + 0.951057i 1.42846 + 1.03784i −0.470682 + 2.18597i 0.391621 + 0.284529i 1.20123 − 0.872745i 1.47472 − 1.07145i −0.809017 − 0.587785i 0.935965 + 0.543682i
16.18 0.300458 − 0.924714i −0.309017 + 0.951057i 0.853212 + 0.619895i 1.61330 − 1.54831i 0.786609 + 0.571505i −2.89935 + 2.10650i 2.40280 − 1.74574i −0.809017 − 0.587785i −0.947013 − 1.95704i
16.19 0.307072 − 0.945069i −0.309017 + 0.951057i 0.819171 + 0.595163i 0.405361 − 2.19902i 0.803924 + 0.584085i 1.55913 − 1.13277i 2.42186 − 1.75959i −0.809017 − 0.587785i −1.95375 − 1.05835i
16.20 0.362325 − 1.11512i −0.309017 + 0.951057i 0.505816 + 0.367497i −2.22061 + 0.262475i 0.948580 + 0.689183i −2.07711 + 1.50911i 2.49023 − 1.80926i −0.809017 − 0.587785i −0.511891 + 2.57135i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.g even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.m.d ✓ 116
11.c even 5 1 825.2.o.d yes 116
25.d even 5 1 825.2.o.d yes 116
275.g even 5 1 inner 825.2.m.d ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.m.d ✓ 116 1.a even 1 1 trivial
825.2.m.d ✓ 116 275.g even 5 1 inner
825.2.o.d yes 116 11.c even 5 1
825.2.o.d yes 116 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{116} - 4 T_{2}^{115} + 52 T_{2}^{114} - 189 T_{2}^{113} + 1457 T_{2}^{112} - 4858 T_{2}^{111} + \cdots + 132710400 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\). Copy content Toggle raw display