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Properties

Label 825.2.o.d
Level $825$
Weight $2$
Character orbit 825.o
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(421,825)]
 
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, 6, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.421");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.o (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.

\(\operatorname{Tr}(f)(q) = \) \( 116 q - q^{2} + 29 q^{3} - 35 q^{4} - 3 q^{5} + 6 q^{6} - 2 q^{7} + q^{8} - 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q - q^{2} + 29 q^{3} - 35 q^{4} - 3 q^{5} + 6 q^{6} - 2 q^{7} + q^{8} - 29 q^{9} + 2 q^{10} - 13 q^{11} + 30 q^{12} - 12 q^{14} + 3 q^{15} - 41 q^{16} - 14 q^{17} - q^{18} - 21 q^{19} + 7 q^{20} + 2 q^{21} + 16 q^{22} + 14 q^{23} - q^{24} + 17 q^{25} - 16 q^{26} + 29 q^{27} - 7 q^{28} - 20 q^{29} + 8 q^{30} + 12 q^{31} - 94 q^{32} - 7 q^{33} + 20 q^{34} - 46 q^{35} + 130 q^{36} + 40 q^{37} + 29 q^{38} - 18 q^{40} - 16 q^{41} - 8 q^{42} - 46 q^{43} + 15 q^{44} + 2 q^{45} + 60 q^{46} + 5 q^{47} + 26 q^{48} - 5 q^{49} - 45 q^{50} - 11 q^{51} - 96 q^{52} + 44 q^{53} - 4 q^{54} - 6 q^{55} + 28 q^{56} + 21 q^{57} - 19 q^{58} + 26 q^{59} - 7 q^{60} + 5 q^{61} - 128 q^{62} - 2 q^{63} - 17 q^{64} + 10 q^{65} + 14 q^{66} - 35 q^{67} + 60 q^{68} + 6 q^{69} - 26 q^{70} - 10 q^{71} + 11 q^{72} - 40 q^{73} - 12 q^{74} - 27 q^{75} + 126 q^{76} + q^{77} - 19 q^{78} + 126 q^{79} - 114 q^{80} - 29 q^{81} - 21 q^{82} - 78 q^{83} + 12 q^{84} - 6 q^{85} - 54 q^{86} - 5 q^{87} - 41 q^{88} + 13 q^{89} + 2 q^{90} - 25 q^{91} - 28 q^{92} + 43 q^{93} - 66 q^{94} + 49 q^{95} - 21 q^{96} - 80 q^{97} - 34 q^{98} - 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1 −0.839125 − 2.58256i −0.309017 + 0.951057i −4.34746 + 3.15861i −0.753856 + 2.10516i 2.71547 −0.651553 + 2.00527i 7.41165 + 5.38488i −0.809017 − 0.587785i 6.06928 + 0.180385i
421.2 −0.833458 − 2.56512i −0.309017 + 0.951057i −4.26716 + 3.10027i −1.51475 − 1.64485i 2.69713 0.539414 − 1.66015i 7.14503 + 5.19117i −0.809017 − 0.587785i −2.95676 + 5.25643i
421.3 −0.803963 − 2.47434i −0.309017 + 0.951057i −3.85799 + 2.80299i 2.21915 − 0.274568i 2.60168 0.638454 − 1.96496i 5.82765 + 4.23404i −0.809017 − 0.587785i −2.46349 − 5.27019i
421.4 −0.689957 − 2.12347i −0.309017 + 0.951057i −2.41505 + 1.75464i 1.66835 − 1.48883i 2.23275 −0.785339 + 2.41703i 1.77953 + 1.29291i −0.809017 − 0.587785i −4.31258 − 2.51545i
421.5 −0.631377 − 1.94318i −0.309017 + 0.951057i −1.75928 + 1.27819i 1.15189 + 1.91655i 2.04318 0.760032 − 2.33914i 0.288586 + 0.209670i −0.809017 − 0.587785i 2.99693 − 3.44839i
421.6 −0.627062 − 1.92990i −0.309017 + 0.951057i −1.71327 + 1.24476i −0.386380 − 2.20243i 2.02922 −1.22448 + 3.76856i 0.193258 + 0.140410i −0.809017 − 0.587785i −4.00819 + 2.12674i
421.7 −0.624954 − 1.92341i −0.309017 + 0.951057i −1.69091 + 1.22852i −2.08682 + 0.803233i 2.02239 −0.908261 + 2.79534i 0.147379 + 0.107077i −0.809017 − 0.587785i 2.84911 + 3.51183i
421.8 −0.544750 − 1.67657i −0.309017 + 0.951057i −0.896097 + 0.651053i −2.17936 + 0.500372i 1.76285 0.998056 − 3.07170i −1.27267 − 0.924646i −0.809017 − 0.587785i 2.02612 + 3.38128i
421.9 −0.511863 − 1.57535i −0.309017 + 0.951057i −0.601696 + 0.437158i −0.445511 − 2.19124i 1.65642 1.42210 − 4.37677i −1.68349 − 1.22312i −0.809017 − 0.587785i −3.22393 + 1.82345i
421.10 −0.398217 − 1.22559i −0.309017 + 0.951057i 0.274547 − 0.199470i 2.21054 − 0.336905i 1.28866 −0.0680796 + 0.209528i −2.43889 − 1.77196i −0.809017 − 0.587785i −1.29318 − 2.57505i
421.11 −0.369397 − 1.13689i −0.309017 + 0.951057i 0.461976 − 0.335645i −0.0892724 + 2.23429i 1.19539 −0.246448 + 0.758490i −2.48643 − 1.80650i −0.809017 − 0.587785i 2.57311 − 0.723845i
421.12 −0.174625 − 0.537442i −0.309017 + 0.951057i 1.35968 − 0.987868i −1.21538 − 1.87693i 0.565100 −1.09958 + 3.38416i −1.68271 − 1.22256i −0.809017 − 0.587785i −0.796503 + 0.980954i
421.13 −0.133504 − 0.410882i −0.309017 + 0.951057i 1.46703 − 1.06586i 1.79895 + 1.32807i 0.432027 −1.07730 + 3.31558i −1.33283 − 0.968360i −0.809017 − 0.587785i 0.305515 − 0.916459i
421.14 −0.116010 − 0.357043i −0.309017 + 0.951057i 1.50401 − 1.09273i 2.15586 + 0.593531i 0.375417 0.966374 − 2.97419i −1.17207 − 0.851559i −0.809017 − 0.587785i −0.0381856 − 0.838589i
421.15 −0.107304 − 0.330248i −0.309017 + 0.951057i 1.52048 − 1.10470i −0.606863 − 2.15214i 0.347244 −0.241278 + 0.742576i −1.08983 − 0.791809i −0.809017 − 0.587785i −0.645623 + 0.431349i
421.16 −0.0437163 − 0.134545i −0.309017 + 0.951057i 1.60184 − 1.16381i −0.922956 + 2.03670i 0.141469 1.03256 − 3.17790i −0.455513 − 0.330949i −0.809017 − 0.587785i 0.314376 + 0.0351421i
421.17 0.0173609 + 0.0534314i −0.309017 + 0.951057i 1.61548 − 1.17372i −2.18802 + 0.461070i −0.0561811 0.656952 − 2.02189i 0.181662 + 0.131985i −0.809017 − 0.587785i −0.0626216 − 0.108904i
421.18 0.230605 + 0.709730i −0.309017 + 0.951057i 1.16750 − 0.848236i 1.16947 − 1.90587i −0.746254 0.546149 − 1.68088i 2.07871 + 1.51027i −0.809017 − 0.587785i 1.62234 + 0.390501i
421.19 0.294500 + 0.906377i −0.309017 + 0.951057i 0.883245 − 0.641715i −2.05705 − 0.876676i −0.953021 −0.406116 + 1.24990i 2.38377 + 1.73191i −0.809017 − 0.587785i 0.188800 − 2.12264i
421.20 0.330263 + 1.01644i −0.309017 + 0.951057i 0.693949 − 0.504184i 0.408183 + 2.19850i −1.06875 −0.878536 + 2.70386i 2.47094 + 1.79524i −0.809017 − 0.587785i −2.09984 + 1.14098i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.29
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{116} + T_{2}^{115} + 47 T_{2}^{114} + 41 T_{2}^{113} + 1212 T_{2}^{112} + 1017 T_{2}^{111} + \cdots + 132710400 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\). Copy content Toggle raw display