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Properties

Label 925.2.p.e
Level $925$
Weight $2$
Character orbit 925.p
Analytic conductor $7.386$
Analytic rank $0$
Dimension $78$
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] = [925,2,Mod(201,925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(925, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("925.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.p (of order \(9\), degree \(6\), 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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(78\)
Relative dimension: \(13\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 78 q - 3 q^{2} + 3 q^{4} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 78 q - 3 q^{2} + 3 q^{4} + 6 q^{7} - 6 q^{8} - 15 q^{12} + 27 q^{13} + 6 q^{14} + 21 q^{16} - 15 q^{18} - 12 q^{19} + 15 q^{21} + 30 q^{22} - 18 q^{23} + 15 q^{24} + 36 q^{27} - 3 q^{28} - 3 q^{29} - 36 q^{31} + 36 q^{32} + 30 q^{33} - 21 q^{34} + 108 q^{36} + 54 q^{37} - 60 q^{38} + 9 q^{39} - 3 q^{41} + 6 q^{42} + 12 q^{43} - 15 q^{44} - 36 q^{46} + 63 q^{47} + 54 q^{48} - 30 q^{49} - 30 q^{51} - 105 q^{52} + 24 q^{53} + 27 q^{54} + 120 q^{56} - 27 q^{57} - 33 q^{58} + 15 q^{59} - 6 q^{61} + 24 q^{62} + 48 q^{63} - 54 q^{64} - 6 q^{66} + 48 q^{67} - 246 q^{68} + 45 q^{69} + 72 q^{71} + 132 q^{72} - 36 q^{73} + 96 q^{74} + 21 q^{76} + 3 q^{77} - 186 q^{78} + 48 q^{79} + 48 q^{81} - 66 q^{82} - 18 q^{83} + 27 q^{84} + 108 q^{86} + 72 q^{87} - 42 q^{88} - 21 q^{89} - 27 q^{91} + 75 q^{92} + 48 q^{93} - 78 q^{94} - 126 q^{96} + 12 q^{97} + 30 q^{98} + 141 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1 −2.49124 − 0.906737i −1.29415 + 0.471033i 3.85202 + 3.23223i 0 3.65114 −0.700893 + 3.97496i −4.01440 − 6.95314i −0.845177 + 0.709188i 0
201.2 −2.16856 − 0.789290i 0.692999 − 0.252231i 2.54757 + 2.13767i 0 −1.70189 0.729160 − 4.13527i −1.52959 − 2.64932i −1.88151 + 1.57877i 0
201.3 −2.16600 − 0.788359i 3.14581 − 1.14498i 2.53795 + 2.12959i 0 −7.71647 −0.535095 + 3.03467i −1.51330 − 2.62111i 6.28699 − 5.27541i 0
201.4 −1.45372 − 0.529110i −2.83129 + 1.03050i 0.301247 + 0.252777i 0 4.66114 0.0382977 − 0.217197i 1.24283 + 2.15265i 4.65612 − 3.90695i 0
201.5 −0.984986 − 0.358506i −1.06803 + 0.388732i −0.690417 − 0.579329i 0 1.19136 0.166656 − 0.945152i 1.52056 + 2.63369i −1.30855 + 1.09801i 0
201.6 −0.375513 − 0.136676i 2.52020 − 0.917280i −1.40976 − 1.18293i 0 −1.07174 0.655367 − 3.71677i 0.767318 + 1.32903i 3.21190 − 2.69510i 0
201.7 −0.245476 − 0.0893459i 1.54899 − 0.563785i −1.47981 − 1.24171i 0 −0.430611 −0.563378 + 3.19508i 0.513547 + 0.889489i −0.216629 + 0.181773i 0
201.8 0.283650 + 0.103240i −1.07237 + 0.390309i −1.46229 − 1.22701i 0 −0.344472 0.133707 − 0.758288i −0.589956 − 1.02183i −1.30051 + 1.09125i 0
201.9 1.40182 + 0.510220i 1.31249 − 0.477707i 0.172680 + 0.144895i 0 2.08360 −0.700650 + 3.97358i −1.32365 − 2.29262i −0.803711 + 0.674394i 0
201.10 1.49659 + 0.544715i 0.936054 − 0.340696i 0.410987 + 0.344859i 0 1.58647 0.562278 − 3.18884i −1.16541 − 2.01855i −1.53801 + 1.29054i 0
201.11 1.72753 + 0.628768i −1.98287 + 0.721706i 1.05691 + 0.886852i 0 −3.87925 −0.504129 + 2.85906i −0.570181 − 0.987582i 1.11279 − 0.933739i 0
201.12 2.23433 + 0.813229i 2.54674 − 0.926939i 2.79879 + 2.34846i 0 6.44407 0.133648 − 0.757955i 1.96585 + 3.40496i 3.32856 − 2.79299i 0
201.13 2.41522 + 0.879069i −1.63550 + 0.595272i 3.52844 + 2.96072i 0 −4.47337 0.543143 − 3.08032i 3.34908 + 5.80077i 0.0223662 − 0.0187675i 0
451.1 −2.49124 + 0.906737i −1.29415 − 0.471033i 3.85202 − 3.23223i 0 3.65114 −0.700893 − 3.97496i −4.01440 + 6.95314i −0.845177 − 0.709188i 0
451.2 −2.16856 + 0.789290i 0.692999 + 0.252231i 2.54757 − 2.13767i 0 −1.70189 0.729160 + 4.13527i −1.52959 + 2.64932i −1.88151 − 1.57877i 0
451.3 −2.16600 + 0.788359i 3.14581 + 1.14498i 2.53795 − 2.12959i 0 −7.71647 −0.535095 − 3.03467i −1.51330 + 2.62111i 6.28699 + 5.27541i 0
451.4 −1.45372 + 0.529110i −2.83129 − 1.03050i 0.301247 − 0.252777i 0 4.66114 0.0382977 + 0.217197i 1.24283 − 2.15265i 4.65612 + 3.90695i 0
451.5 −0.984986 + 0.358506i −1.06803 − 0.388732i −0.690417 + 0.579329i 0 1.19136 0.166656 + 0.945152i 1.52056 − 2.63369i −1.30855 − 1.09801i 0
451.6 −0.375513 + 0.136676i 2.52020 + 0.917280i −1.40976 + 1.18293i 0 −1.07174 0.655367 + 3.71677i 0.767318 − 1.32903i 3.21190 + 2.69510i 0
451.7 −0.245476 + 0.0893459i 1.54899 + 0.563785i −1.47981 + 1.24171i 0 −0.430611 −0.563378 − 3.19508i 0.513547 − 0.889489i −0.216629 − 0.181773i 0
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.p.e ✓ 78
5.b even 2 1 925.2.p.f yes 78
5.c odd 4 2 925.2.bc.e 156
37.f even 9 1 inner 925.2.p.e ✓ 78
185.x even 18 1 925.2.p.f yes 78
185.bd odd 36 2 925.2.bc.e 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.p.e ✓ 78 1.a even 1 1 trivial
925.2.p.e ✓ 78 37.f even 9 1 inner
925.2.p.f yes 78 5.b even 2 1
925.2.p.f yes 78 185.x even 18 1
925.2.bc.e 156 5.c odd 4 2
925.2.bc.e 156 185.bd odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{78} + 3 T_{2}^{77} + 3 T_{2}^{76} + 2 T_{2}^{75} - 6 T_{2}^{74} - 75 T_{2}^{73} + 423 T_{2}^{72} + \cdots + 59049 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display