Properties

Label 625.2.e.j
Level $625$
Weight $2$
Character orbit 625.e
Analytic conductor $4.991$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,2,Mod(124,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.124");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.e (of order \(10\), degree \(4\), not 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: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 6 q^{4} + 14 q^{6} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 6 q^{4} + 14 q^{6} + 24 q^{9} - 6 q^{11} + 2 q^{14} + 2 q^{16} - 20 q^{19} + 14 q^{21} - 20 q^{24} + 44 q^{26} - 16 q^{31} + 12 q^{34} + 2 q^{36} - 2 q^{39} - 16 q^{41} + 62 q^{44} + 84 q^{46} + 16 q^{49} - 56 q^{51} - 100 q^{54} + 70 q^{56} + 30 q^{59} + 34 q^{61} - 74 q^{64} + 88 q^{66} + 18 q^{69} - 26 q^{71} + 72 q^{74} - 40 q^{76} + 110 q^{79} - 38 q^{81} - 118 q^{84} + 14 q^{86} - 56 q^{91} - 8 q^{94} - 86 q^{96} + 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1 −2.53458 0.823534i 0.446178 0.614111i 4.12784 + 2.99905i 0 −1.63661 + 1.18907i 2.04213i −4.85960 6.68866i 0.748993 + 2.30516i 0
124.2 −1.91511 0.622258i −1.77844 + 2.44781i 1.66242 + 1.20782i 0 4.92909 3.58119i 0.369971i −0.0649413 0.0893841i −1.90189 5.85342i 0
124.3 −1.00562 0.326747i −0.404046 + 0.556121i −0.713519 0.518402i 0 0.588029 0.427228i 1.01199i 1.79116 + 2.46533i 0.781033 + 2.40377i 0
124.4 −0.310759 0.100972i 1.00827 1.38777i −1.53166 1.11281i 0 −0.453455 + 0.329455i 3.42409i 0.747732 + 1.02917i 0.0177620 + 0.0546659i 0
124.5 0.310759 + 0.100972i −1.00827 + 1.38777i −1.53166 1.11281i 0 −0.453455 + 0.329455i 3.42409i −0.747732 1.02917i 0.0177620 + 0.0546659i 0
124.6 1.00562 + 0.326747i 0.404046 0.556121i −0.713519 0.518402i 0 0.588029 0.427228i 1.01199i −1.79116 2.46533i 0.781033 + 2.40377i 0
124.7 1.91511 + 0.622258i 1.77844 2.44781i 1.66242 + 1.20782i 0 4.92909 3.58119i 0.369971i 0.0649413 + 0.0893841i −1.90189 5.85342i 0
124.8 2.53458 + 0.823534i −0.446178 + 0.614111i 4.12784 + 2.99905i 0 −1.63661 + 1.18907i 2.04213i 4.85960 + 6.68866i 0.748993 + 2.30516i 0
249.1 −1.45439 + 2.00179i −2.01315 + 0.654112i −1.27390 3.92066i 0 1.61850 4.98124i 0.973070i 4.99460 + 1.62284i 1.19786 0.870294i 0
249.2 −1.36725 + 1.88186i 2.18963 0.711454i −1.05398 3.24381i 0 −1.65491 + 5.09330i 3.59425i 3.12093 + 1.01405i 1.86126 1.35229i 0
249.3 −0.989484 + 1.36191i 0.675574 0.219507i −0.257680 0.793058i 0 −0.369521 + 1.13727i 4.59110i −1.86699 0.606623i −2.01883 + 1.46677i 0
249.4 −0.294474 + 0.405309i −2.94186 + 0.955869i 0.540474 + 1.66341i 0 0.478880 1.47384i 0.0237879i −1.78629 0.580400i 5.31382 3.86071i 0
249.5 0.294474 0.405309i 2.94186 0.955869i 0.540474 + 1.66341i 0 0.478880 1.47384i 0.0237879i 1.78629 + 0.580400i 5.31382 3.86071i 0
249.6 0.989484 1.36191i −0.675574 + 0.219507i −0.257680 0.793058i 0 −0.369521 + 1.13727i 4.59110i 1.86699 + 0.606623i −2.01883 + 1.46677i 0
249.7 1.36725 1.88186i −2.18963 + 0.711454i −1.05398 3.24381i 0 −1.65491 + 5.09330i 3.59425i −3.12093 1.01405i 1.86126 1.35229i 0
249.8 1.45439 2.00179i 2.01315 0.654112i −1.27390 3.92066i 0 1.61850 4.98124i 0.973070i −4.99460 1.62284i 1.19786 0.870294i 0
374.1 −1.45439 2.00179i −2.01315 0.654112i −1.27390 + 3.92066i 0 1.61850 + 4.98124i 0.973070i 4.99460 1.62284i 1.19786 + 0.870294i 0
374.2 −1.36725 1.88186i 2.18963 + 0.711454i −1.05398 + 3.24381i 0 −1.65491 5.09330i 3.59425i 3.12093 1.01405i 1.86126 + 1.35229i 0
374.3 −0.989484 1.36191i 0.675574 + 0.219507i −0.257680 + 0.793058i 0 −0.369521 1.13727i 4.59110i −1.86699 + 0.606623i −2.01883 1.46677i 0
374.4 −0.294474 0.405309i −2.94186 0.955869i 0.540474 1.66341i 0 0.478880 + 1.47384i 0.0237879i −1.78629 + 0.580400i 5.31382 + 3.86071i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.j 32
5.b even 2 1 inner 625.2.e.j 32
5.c odd 4 1 625.2.d.m 16
5.c odd 4 1 625.2.d.q 16
25.d even 5 1 625.2.b.d 16
25.d even 5 1 inner 625.2.e.j 32
25.d even 5 2 625.2.e.k 32
25.e even 10 1 625.2.b.d 16
25.e even 10 1 inner 625.2.e.j 32
25.e even 10 2 625.2.e.k 32
25.f odd 20 1 625.2.a.e 8
25.f odd 20 1 625.2.a.g yes 8
25.f odd 20 1 625.2.d.m 16
25.f odd 20 2 625.2.d.n 16
25.f odd 20 2 625.2.d.p 16
25.f odd 20 1 625.2.d.q 16
75.l even 20 1 5625.2.a.s 8
75.l even 20 1 5625.2.a.be 8
100.l even 20 1 10000.2.a.be 8
100.l even 20 1 10000.2.a.bn 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e 8 25.f odd 20 1
625.2.a.g yes 8 25.f odd 20 1
625.2.b.d 16 25.d even 5 1
625.2.b.d 16 25.e even 10 1
625.2.d.m 16 5.c odd 4 1
625.2.d.m 16 25.f odd 20 1
625.2.d.n 16 25.f odd 20 2
625.2.d.p 16 25.f odd 20 2
625.2.d.q 16 5.c odd 4 1
625.2.d.q 16 25.f odd 20 1
625.2.e.j 32 1.a even 1 1 trivial
625.2.e.j 32 5.b even 2 1 inner
625.2.e.j 32 25.d even 5 1 inner
625.2.e.j 32 25.e even 10 1 inner
625.2.e.k 32 25.d even 5 2
625.2.e.k 32 25.e even 10 2
5625.2.a.s 8 75.l even 20 1
5625.2.a.be 8 75.l even 20 1
10000.2.a.be 8 100.l even 20 1
10000.2.a.bn 8 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{32} - 11 T_{2}^{30} + 100 T_{2}^{28} - 790 T_{2}^{26} + 6030 T_{2}^{24} - 27593 T_{2}^{22} + \cdots + 6561 \) Copy content Toggle raw display
\( T_{3}^{32} - 24 T_{3}^{30} + 320 T_{3}^{28} - 3270 T_{3}^{26} + 31080 T_{3}^{24} - 222072 T_{3}^{22} + \cdots + 707281 \) Copy content Toggle raw display