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Properties

Label 31.2.d.a
Level $31$
Weight $2$
Character orbit 31.d
Analytic conductor $0.248$
Analytic rank $0$
Dimension $4$
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] = [31,2,Mod(2,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 31.d (of order \(5\), 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: \(0.247536246266\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{2} - 1) q^{2} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{3}+ \cdots + 2 \zeta_{10}^{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{2} - 1) q^{2} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{3}+ \cdots + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + q^{3} + 3 q^{4} - 6 q^{5} - 2 q^{6} - 3 q^{7} - 5 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} + q^{3} + 3 q^{4} - 6 q^{5} - 2 q^{6} - 3 q^{7} - 5 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} + 6 q^{14} + q^{15} + 9 q^{16} - 3 q^{17} + 6 q^{18} - 5 q^{19} - 7 q^{20} + 3 q^{21} - 6 q^{22} + 11 q^{23} + 5 q^{24} - 6 q^{25} - 12 q^{26} - 5 q^{27} - 6 q^{28} + 5 q^{29} - 2 q^{30} - 11 q^{31} - 18 q^{32} - 8 q^{33} + 11 q^{34} + 12 q^{35} + 4 q^{36} - 8 q^{37} + 10 q^{38} + 9 q^{39} + 10 q^{40} + 8 q^{41} - 6 q^{42} + q^{43} + 6 q^{44} - 8 q^{45} - 7 q^{46} + 7 q^{47} + 6 q^{48} - 2 q^{49} + 12 q^{50} + 3 q^{51} - 3 q^{52} + 21 q^{53} + 10 q^{54} - 2 q^{55} - 20 q^{57} - 15 q^{58} + 5 q^{59} - 3 q^{60} + 8 q^{61} - 8 q^{62} - 24 q^{63} - 7 q^{64} - 9 q^{65} + 6 q^{66} - 8 q^{67} - 6 q^{68} - 11 q^{69} - 9 q^{70} - 7 q^{71} - 10 q^{72} + 21 q^{73} + 11 q^{74} - 9 q^{75} + 15 q^{76} + 24 q^{77} - 3 q^{78} - 6 q^{80} - q^{81} + 4 q^{82} - 14 q^{83} - 9 q^{84} + 2 q^{85} - 7 q^{86} + 30 q^{87} + 20 q^{88} + 5 q^{89} - 4 q^{90} + 18 q^{91} + 22 q^{92} - 4 q^{93} - 14 q^{94} - 5 q^{95} - 2 q^{96} - 3 q^{97} + 4 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.809017 + 0.587785i
−0.309017 − 0.951057i
−0.309017 + 0.951057i
0.809017 − 0.587785i
−1.30902 − 0.951057i 0.809017 − 0.587785i 0.190983 + 0.587785i −0.381966 −1.61803 0.927051 + 2.85317i −0.690983 + 2.12663i −0.618034 + 1.90211i 0.500000 + 0.363271i
4.1 −0.190983 − 0.587785i −0.309017 + 0.951057i 1.30902 − 0.951057i −2.61803 0.618034 −2.42705 + 1.76336i −1.80902 − 1.31433i 1.61803 + 1.17557i 0.500000 + 1.53884i
8.1 −0.190983 + 0.587785i −0.309017 − 0.951057i 1.30902 + 0.951057i −2.61803 0.618034 −2.42705 − 1.76336i −1.80902 + 1.31433i 1.61803 − 1.17557i 0.500000 − 1.53884i
16.1 −1.30902 + 0.951057i 0.809017 + 0.587785i 0.190983 − 0.587785i −0.381966 −1.61803 0.927051 − 2.85317i −0.690983 − 2.12663i −0.618034 − 1.90211i 0.500000 − 0.363271i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.2.d.a ✓ 4
3.b odd 2 1 279.2.i.a 4
4.b odd 2 1 496.2.n.b 4
5.b even 2 1 775.2.k.c 4
5.c odd 4 2 775.2.bf.a 8
31.b odd 2 1 961.2.d.b 4
31.c even 3 2 961.2.g.b 8
31.d even 5 1 inner 31.2.d.a ✓ 4
31.d even 5 1 961.2.a.d 2
31.d even 5 2 961.2.d.f 4
31.e odd 6 2 961.2.g.c 8
31.f odd 10 1 961.2.a.e 2
31.f odd 10 1 961.2.d.b 4
31.f odd 10 2 961.2.d.e 4
31.g even 15 2 961.2.c.f 4
31.g even 15 2 961.2.g.b 8
31.g even 15 4 961.2.g.f 8
31.h odd 30 2 961.2.c.d 4
31.h odd 30 2 961.2.g.c 8
31.h odd 30 4 961.2.g.g 8
93.k even 10 1 8649.2.a.f 2
93.l odd 10 1 279.2.i.a 4
93.l odd 10 1 8649.2.a.g 2
124.l odd 10 1 496.2.n.b 4
155.n even 10 1 775.2.k.c 4
155.s odd 20 2 775.2.bf.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.d.a ✓ 4 1.a even 1 1 trivial
31.2.d.a ✓ 4 31.d even 5 1 inner
279.2.i.a 4 3.b odd 2 1
279.2.i.a 4 93.l odd 10 1
496.2.n.b 4 4.b odd 2 1
496.2.n.b 4 124.l odd 10 1
775.2.k.c 4 5.b even 2 1
775.2.k.c 4 155.n even 10 1
775.2.bf.a 8 5.c odd 4 2
775.2.bf.a 8 155.s odd 20 2
961.2.a.d 2 31.d even 5 1
961.2.a.e 2 31.f odd 10 1
961.2.c.d 4 31.h odd 30 2
961.2.c.f 4 31.g even 15 2
961.2.d.b 4 31.b odd 2 1
961.2.d.b 4 31.f odd 10 1
961.2.d.e 4 31.f odd 10 2
961.2.d.f 4 31.d even 5 2
961.2.g.b 8 31.c even 3 2
961.2.g.b 8 31.g even 15 2
961.2.g.c 8 31.e odd 6 2
961.2.g.c 8 31.h odd 30 2
961.2.g.f 8 31.g even 15 4
961.2.g.g 8 31.h odd 30 4
8649.2.a.f 2 93.k even 10 1
8649.2.a.g 2 93.l odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(31, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( T^{4} - 11 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( T^{4} - 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$31$ \( T^{4} + 11 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{4} - T^{3} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( T^{4} - 7 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$53$ \( T^{4} - 21 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 7 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{4} - 21 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 14 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( T^{4} - 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$97$ \( T^{4} + 3 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
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