Properties

Label 605.2.g.a
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(81,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(10))
 
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("605.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), 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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{4} - q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + q^{4} - q^{5} + 3 q^{8} + 3 q^{9} + 4 q^{10} - 2 q^{13} + q^{16} - 6 q^{17} + 3 q^{18} + 4 q^{19} + q^{20} + 16 q^{23} - q^{25} - 2 q^{26} - 6 q^{29} + 8 q^{31} + 20 q^{32} + 24 q^{34} - 3 q^{36} + 2 q^{37} + 4 q^{38} + 3 q^{40} - 2 q^{41} + 16 q^{43} - 12 q^{45} - 4 q^{46} + 12 q^{47} + 7 q^{49} - q^{50} + 2 q^{52} + 2 q^{53} - 6 q^{58} - 4 q^{59} + 10 q^{61} + 8 q^{62} - 7 q^{64} + 8 q^{65} - 64 q^{67} + 6 q^{68} - 8 q^{71} - 9 q^{72} - 14 q^{73} + 2 q^{74} + 16 q^{76} - 8 q^{79} + q^{80} - 9 q^{81} - 2 q^{82} + 4 q^{83} - 6 q^{85} - 4 q^{86} + 40 q^{89} + 3 q^{90} + 4 q^{92} + 12 q^{94} + 4 q^{95} - 10 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.809017 0.587785i
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.809017 0.587785i 0 −0.309017 0.951057i −0.809017 + 0.587785i 0 0 −0.927051 + 2.85317i 2.42705 + 1.76336i 1.00000
251.1 0.309017 0.951057i 0 0.809017 + 0.587785i 0.309017 + 0.951057i 0 0 2.42705 1.76336i −0.927051 + 2.85317i 1.00000
366.1 −0.809017 + 0.587785i 0 −0.309017 + 0.951057i −0.809017 0.587785i 0 0 −0.927051 2.85317i 2.42705 1.76336i 1.00000
511.1 0.309017 + 0.951057i 0 0.809017 0.587785i 0.309017 0.951057i 0 0 2.42705 + 1.76336i −0.927051 2.85317i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.a 4
11.b odd 2 1 605.2.g.c 4
11.c even 5 1 55.2.a.a 1
11.c even 5 3 inner 605.2.g.a 4
11.d odd 10 1 605.2.a.b 1
11.d odd 10 3 605.2.g.c 4
33.f even 10 1 5445.2.a.i 1
33.h odd 10 1 495.2.a.a 1
44.g even 10 1 9680.2.a.r 1
44.h odd 10 1 880.2.a.h 1
55.h odd 10 1 3025.2.a.f 1
55.j even 10 1 275.2.a.a 1
55.k odd 20 2 275.2.b.b 2
77.j odd 10 1 2695.2.a.c 1
88.l odd 10 1 3520.2.a.n 1
88.o even 10 1 3520.2.a.p 1
132.o even 10 1 7920.2.a.i 1
143.n even 10 1 9295.2.a.b 1
165.o odd 10 1 2475.2.a.i 1
165.v even 20 2 2475.2.c.f 2
220.n odd 10 1 4400.2.a.p 1
220.v even 20 2 4400.2.b.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 11.c even 5 1
275.2.a.a 1 55.j even 10 1
275.2.b.b 2 55.k odd 20 2
495.2.a.a 1 33.h odd 10 1
605.2.a.b 1 11.d odd 10 1
605.2.g.a 4 1.a even 1 1 trivial
605.2.g.a 4 11.c even 5 3 inner
605.2.g.c 4 11.b odd 2 1
605.2.g.c 4 11.d odd 10 3
880.2.a.h 1 44.h odd 10 1
2475.2.a.i 1 165.o odd 10 1
2475.2.c.f 2 165.v even 20 2
2695.2.a.c 1 77.j odd 10 1
3025.2.a.f 1 55.h odd 10 1
3520.2.a.n 1 88.l odd 10 1
3520.2.a.p 1 88.o even 10 1
4400.2.a.p 1 220.n odd 10 1
4400.2.b.n 2 220.v even 20 2
5445.2.a.i 1 33.f even 10 1
7920.2.a.i 1 132.o even 10 1
9295.2.a.b 1 143.n even 10 1
9680.2.a.r 1 44.g even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + \cdots + 10000 \) Copy content Toggle raw display
$67$ \( (T + 16)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} + \cdots + 10000 \) Copy content Toggle raw display
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