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Properties

Label 425.2.q.a
Level $425$
Weight $2$
Character orbit 425.q
Analytic conductor $3.394$
Analytic rank $0$
Dimension $176$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(84,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.84");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.q (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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 10 q^{2} + 38 q^{4} - 10 q^{8} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 10 q^{2} + 38 q^{4} - 10 q^{8} - 50 q^{9} - 10 q^{13} + 6 q^{15} - 58 q^{16} - 8 q^{19} - 44 q^{21} - 20 q^{25} - 64 q^{26} + 78 q^{30} - 20 q^{33} - 23 q^{34} - 46 q^{35} + 52 q^{36} - 10 q^{38} - 20 q^{42} + 40 q^{47} + 164 q^{49} + 84 q^{50} + 20 q^{51} - 30 q^{52} - 40 q^{53} - 18 q^{55} - 18 q^{59} - 48 q^{60} + 66 q^{64} + 80 q^{66} - 50 q^{67} + 68 q^{69} - 152 q^{70} + 60 q^{72} - 72 q^{76} - 40 q^{77} - 38 q^{81} + 50 q^{83} - 52 q^{84} + 47 q^{85} - 94 q^{86} - 120 q^{87} + 60 q^{89} + 50 q^{94} - 250 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
84.1 −1.61286 + 2.21990i −0.861202 − 2.65051i −1.70864 − 5.25866i 0.297302 − 2.21622i 7.27287 + 2.36310i 1.90625 9.21020 + 2.99258i −3.85647 + 2.80189i 4.44028 + 4.23442i
84.2 −1.61286 + 2.21990i 0.861202 + 2.65051i −1.70864 − 5.25866i −0.297302 + 2.21622i −7.27287 − 2.36310i −1.90625 9.21020 + 2.99258i −3.85647 + 2.80189i −4.44028 − 4.23442i
84.3 −1.47509 + 2.03029i −0.532962 − 1.64029i −1.32814 − 4.08760i 0.886990 + 2.05262i 4.11642 + 1.33751i −3.49211 5.48463 + 1.78207i 0.0205526 − 0.0149323i −5.47580 − 1.22696i
84.4 −1.47509 + 2.03029i 0.532962 + 1.64029i −1.32814 − 4.08760i −0.886990 − 2.05262i −4.11642 − 1.33751i 3.49211 5.48463 + 1.78207i 0.0205526 − 0.0149323i 5.47580 + 1.22696i
84.5 −1.39460 + 1.91951i −0.152608 − 0.469678i −1.12155 − 3.45179i 1.97789 − 1.04304i 1.11438 + 0.362083i −1.83697 3.67682 + 1.19467i 2.22974 − 1.62000i −0.756252 + 5.25121i
84.6 −1.39460 + 1.91951i 0.152608 + 0.469678i −1.12155 − 3.45179i −1.97789 + 1.04304i −1.11438 − 0.362083i 1.83697 3.67682 + 1.19467i 2.22974 − 1.62000i 0.756252 − 5.25121i
84.7 −1.27397 + 1.75347i −0.347412 − 1.06922i −0.833625 − 2.56564i −2.16832 − 0.546269i 2.31745 + 0.752985i −2.57428 1.43812 + 0.467275i 1.40451 − 1.02043i 3.72024 − 3.10615i
84.8 −1.27397 + 1.75347i 0.347412 + 1.06922i −0.833625 − 2.56564i 2.16832 + 0.546269i −2.31745 − 0.752985i 2.57428 1.43812 + 0.467275i 1.40451 − 1.02043i −3.72024 + 3.10615i
84.9 −1.14277 + 1.57289i −0.782761 − 2.40909i −0.550027 − 1.69281i 1.07788 + 1.95913i 4.68376 + 1.52185i 3.64131 −0.406932 − 0.132220i −2.76396 + 2.00813i −4.31327 − 0.543447i
84.10 −1.14277 + 1.57289i 0.782761 + 2.40909i −0.550027 − 1.69281i −1.07788 − 1.95913i −4.68376 − 1.52185i −3.64131 −0.406932 − 0.132220i −2.76396 + 2.00813i 4.31327 + 0.543447i
84.11 −0.857096 + 1.17969i −0.707067 − 2.17613i −0.0390246 − 0.120105i −1.58143 − 1.58085i 3.17318 + 1.03103i 2.12040 −2.59849 − 0.844299i −1.80854 + 1.31398i 3.22035 − 0.510657i
84.12 −0.857096 + 1.17969i 0.707067 + 2.17613i −0.0390246 − 0.120105i 1.58143 + 1.58085i −3.17318 − 1.03103i −2.12040 −2.59849 − 0.844299i −1.80854 + 1.31398i −3.22035 + 0.510657i
84.13 −0.753629 + 1.03728i −0.0324420 − 0.0998462i 0.110039 + 0.338664i 0.863935 − 2.06243i 0.128018 + 0.0415955i −3.13290 −2.87301 − 0.933498i 2.41813 − 1.75688i 1.48823 + 2.45045i
84.14 −0.753629 + 1.03728i 0.0324420 + 0.0998462i 0.110039 + 0.338664i −0.863935 + 2.06243i −0.128018 − 0.0415955i 3.13290 −2.87301 − 0.933498i 2.41813 − 1.75688i −1.48823 − 2.45045i
84.15 −0.677444 + 0.932422i −1.03346 − 3.18065i 0.207554 + 0.638786i 2.01329 − 0.972965i 3.66582 + 1.19110i −4.10770 −2.92848 − 0.951521i −6.62147 + 4.81078i −0.456678 + 2.53636i
84.16 −0.677444 + 0.932422i 1.03346 + 3.18065i 0.207554 + 0.638786i −2.01329 + 0.972965i −3.66582 − 1.19110i 4.10770 −2.92848 − 0.951521i −6.62147 + 4.81078i 0.456678 − 2.53636i
84.17 −0.568609 + 0.782624i −0.724084 − 2.22850i 0.328851 + 1.01210i −1.46866 + 1.68613i 2.15580 + 0.700461i −0.961359 −2.81914 − 0.915994i −2.01486 + 1.46388i −0.484512 − 2.10816i
84.18 −0.568609 + 0.782624i 0.724084 + 2.22850i 0.328851 + 1.01210i 1.46866 − 1.68613i −2.15580 − 0.700461i 0.961359 −2.81914 − 0.915994i −2.01486 + 1.46388i 0.484512 + 2.10816i
84.19 −0.453067 + 0.623593i −0.240491 − 0.740156i 0.434435 + 1.33705i 2.23529 − 0.0588141i 0.570514 + 0.185371i 2.18212 −2.49676 − 0.811247i 1.93706 − 1.40735i −0.976061 + 1.42056i
84.20 −0.453067 + 0.623593i 0.240491 + 0.740156i 0.434435 + 1.33705i −2.23529 + 0.0588141i −0.570514 − 0.185371i −2.18212 −2.49676 − 0.811247i 1.93706 − 1.40735i 0.976061 − 1.42056i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 84.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
25.e even 10 1 inner
425.q even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.q.a ✓ 176
17.b even 2 1 inner 425.2.q.a ✓ 176
25.e even 10 1 inner 425.2.q.a ✓ 176
425.q even 10 1 inner 425.2.q.a ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.q.a ✓ 176 1.a even 1 1 trivial
425.2.q.a ✓ 176 17.b even 2 1 inner
425.2.q.a ✓ 176 25.e even 10 1 inner
425.2.q.a ✓ 176 425.q even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(425, [\chi])\).