Properties

Label 375.2.i.a
Level $375$
Weight $2$
Character orbit 375.i
Analytic conductor $2.994$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [375,2,Mod(49,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.i (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: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} - \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \cdots - 1) q^{6} + \cdots - \zeta_{20}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} - \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \cdots - 1) q^{6} + \cdots + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} - 2 q^{6} + 2 q^{9} - 12 q^{11} + 20 q^{14} + 2 q^{16} + 8 q^{19} + 20 q^{21} + 24 q^{24} + 36 q^{26} + 16 q^{29} - 14 q^{34} + 2 q^{36} + 4 q^{39} + 8 q^{44} - 20 q^{46} - 104 q^{49} + 4 q^{51} + 2 q^{54} - 60 q^{56} - 8 q^{59} + 4 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} + 16 q^{71} - 20 q^{74} + 8 q^{76} - 2 q^{81} - 20 q^{84} + 28 q^{86} + 18 q^{89} - 80 q^{91} + 4 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(\zeta_{20}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.587785 + 0.809017i −0.951057 + 0.309017i 0.309017 + 0.951057i 0 0.309017 0.951057i 4.47214i −2.85317 0.927051i 0.809017 0.587785i 0
49.2 0.587785 0.809017i 0.951057 0.309017i 0.309017 + 0.951057i 0 0.309017 0.951057i 4.47214i 2.85317 + 0.927051i 0.809017 0.587785i 0
199.1 −0.587785 0.809017i −0.951057 0.309017i 0.309017 0.951057i 0 0.309017 + 0.951057i 4.47214i −2.85317 + 0.927051i 0.809017 + 0.587785i 0
199.2 0.587785 + 0.809017i 0.951057 + 0.309017i 0.309017 0.951057i 0 0.309017 + 0.951057i 4.47214i 2.85317 0.927051i 0.809017 + 0.587785i 0
274.1 −0.951057 0.309017i 0.587785 0.809017i −0.809017 0.587785i 0 −0.809017 + 0.587785i 4.47214i 1.76336 + 2.42705i −0.309017 0.951057i 0
274.2 0.951057 + 0.309017i −0.587785 + 0.809017i −0.809017 0.587785i 0 −0.809017 + 0.587785i 4.47214i −1.76336 2.42705i −0.309017 0.951057i 0
349.1 −0.951057 + 0.309017i 0.587785 + 0.809017i −0.809017 + 0.587785i 0 −0.809017 0.587785i 4.47214i 1.76336 2.42705i −0.309017 + 0.951057i 0
349.2 0.951057 0.309017i −0.587785 0.809017i −0.809017 + 0.587785i 0 −0.809017 0.587785i 4.47214i −1.76336 + 2.42705i −0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.2
Significant digits:
Format:

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 375.2.i.a 8
5.b even 2 1 inner 375.2.i.a 8
5.c odd 4 1 75.2.g.a 4
5.c odd 4 1 375.2.g.a 4
15.e even 4 1 225.2.h.a 4
25.d even 5 1 inner 375.2.i.a 8
25.d even 5 1 1875.2.b.b 4
25.e even 10 1 inner 375.2.i.a 8
25.e even 10 1 1875.2.b.b 4
25.f odd 20 1 75.2.g.a 4
25.f odd 20 1 375.2.g.a 4
25.f odd 20 1 1875.2.a.a 2
25.f odd 20 1 1875.2.a.d 2
75.l even 20 1 225.2.h.a 4
75.l even 20 1 5625.2.a.a 2
75.l even 20 1 5625.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 5.c odd 4 1
75.2.g.a 4 25.f odd 20 1
225.2.h.a 4 15.e even 4 1
225.2.h.a 4 75.l even 20 1
375.2.g.a 4 5.c odd 4 1
375.2.g.a 4 25.f odd 20 1
375.2.i.a 8 1.a even 1 1 trivial
375.2.i.a 8 5.b even 2 1 inner
375.2.i.a 8 25.d even 5 1 inner
375.2.i.a 8 25.e even 10 1 inner
1875.2.a.a 2 25.f odd 20 1
1875.2.a.d 2 25.f odd 20 1
1875.2.b.b 4 25.d even 5 1
1875.2.b.b 4 25.e even 10 1
5625.2.a.a 2 75.l even 20 1
5625.2.a.h 2 75.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + 16 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 44 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + 16 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 20 T^{6} + \cdots + 160000 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots + 841)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 40 T^{2} + \cdots + 400)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 25 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$41$ \( (T^{4} + 10 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 92 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 44 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{8} + 5 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 44 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} + 24 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 100 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 76 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$89$ \( (T^{4} - 9 T^{3} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 124 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
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