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Properties

Label 2471.1.d.b
Level $2471$
Weight $1$
Character orbit 2471.d
Self dual yes
Analytic conductor $1.233$
Analytic rank $0$
Dimension $15$
Projective image $D_{31}$
CM discriminant -2471
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2471,1,Mod(2470,2471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2471, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2471.2470");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2471 = 7 \cdot 353 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2471.d (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(1.23318964622\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\Q(\zeta_{62})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 14 x^{13} + 13 x^{12} + 78 x^{11} - 66 x^{10} - 220 x^{9} + 165 x^{8} + 330 x^{7} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{31}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{31} - \cdots)\)

$q$-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 basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{14} + \beta_{13} - \beta_{12} + \cdots - 1) q^{2}+ \cdots + ( - \beta_{7} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} + \beta_{13} - \beta_{12} + \cdots - 1) q^{2}+ \cdots + ( - \beta_{13} - \beta_{11} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{2} + q^{3} + 14 q^{4} + q^{5} + 2 q^{6} - 15 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{2} + q^{3} + 14 q^{4} + q^{5} + 2 q^{6} - 15 q^{7} - 2 q^{8} + 14 q^{9} + 2 q^{10} - q^{11} + 3 q^{12} + q^{13} + q^{14} - 2 q^{15} + 13 q^{16} - 3 q^{18} + 3 q^{20} - q^{21} - 2 q^{22} - q^{23} + 4 q^{24} + 14 q^{25} + 2 q^{26} + 2 q^{27} - 14 q^{28} - q^{29} - 4 q^{30} + q^{31} - 3 q^{32} + 2 q^{33} - q^{35} + 11 q^{36} - 2 q^{39} + 4 q^{40} - 2 q^{42} - q^{43} - 3 q^{44} + 3 q^{45} - 2 q^{46} + 5 q^{48} + 15 q^{49} - 3 q^{50} + 3 q^{52} + 4 q^{54} + 2 q^{55} + 2 q^{56} - 2 q^{58} + q^{59} - 6 q^{60} + 2 q^{62} - 14 q^{63} + 12 q^{64} - 2 q^{65} + 4 q^{66} + 2 q^{69} - 2 q^{70} - 6 q^{72} + 3 q^{75} + q^{77} - 4 q^{78} + 5 q^{80} + 13 q^{81} - 3 q^{84} - 2 q^{86} + 2 q^{87} - 4 q^{88} + q^{89} - 25 q^{90} - q^{91} - 3 q^{92} - 2 q^{93} + 6 q^{96} - q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{62} + \zeta_{62}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( \nu^{9} - 9\nu^{7} + 27\nu^{5} - 30\nu^{3} + 9\nu \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( \nu^{10} - 10\nu^{8} + 35\nu^{6} - 50\nu^{4} + 25\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \nu^{11} - 11\nu^{9} + 44\nu^{7} - 77\nu^{5} + 55\nu^{3} - 11\nu \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( \nu^{12} - 12\nu^{10} + 54\nu^{8} - 112\nu^{6} + 105\nu^{4} - 36\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( \nu^{13} - 13\nu^{11} + 65\nu^{9} - 156\nu^{7} + 182\nu^{5} - 91\nu^{3} + 13\nu \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( \nu^{14} - 14\nu^{12} + 77\nu^{10} - 210\nu^{8} + 294\nu^{6} - 196\nu^{4} + 49\nu^{2} - 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{9} + 9\beta_{7} + 36\beta_{5} + 84\beta_{3} + 126\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( \beta_{10} + 10\beta_{8} + 45\beta_{6} + 120\beta_{4} + 210\beta_{2} + 252 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( \beta_{11} + 11\beta_{9} + 55\beta_{7} + 165\beta_{5} + 330\beta_{3} + 462\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( \beta_{12} + 12\beta_{10} + 66\beta_{8} + 220\beta_{6} + 495\beta_{4} + 792\beta_{2} + 924 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( \beta_{13} + 13\beta_{11} + 78\beta_{9} + 286\beta_{7} + 715\beta_{5} + 1287\beta_{3} + 1716\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( \beta_{14} + 14\beta_{12} + 91\beta_{10} + 364\beta_{8} + 1001\beta_{6} + 2002\beta_{4} + 3003\beta_{2} + 3432 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2471\mathbb{Z}\right)^\times\).

\(n\) \(1060\) \(1415\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2470.1
−1.95906
−1.64153
−1.05793
−0.302856
0.501305
1.22421
1.74869
1.98974
1.90828
1.51752
0.880788
0.101298
−0.694611
−1.37793
−1.83792
−1.98974 1.51752 2.95906 0.501305 −3.01946 −1.00000 −3.89802 1.30286 −0.997466
2470.2 −1.90828 −1.05793 2.64153 −1.37793 2.01882 −1.00000 −3.13249 0.119212 2.62948
2470.3 −1.74869 −1.83792 2.05793 1.90828 3.21395 −1.00000 −1.84999 2.37793 −3.33699
2470.4 −1.51752 0.501305 1.30286 −1.95906 −0.760739 −1.00000 −0.459588 −0.748693 2.97291
2470.5 −1.22421 1.98974 0.498695 1.51752 −2.43586 −1.00000 0.613704 2.95906 −1.85776
2470.6 −0.880788 0.101298 −0.224212 −0.694611 −0.0892224 −1.00000 1.07827 −0.989739 0.611805
2470.7 −0.501305 −1.95906 −0.748693 −0.302856 0.982087 −1.00000 0.876629 2.83792 0.151823
2470.8 −0.101298 −0.694611 −0.989739 1.22421 0.0703629 −1.00000 0.201557 −0.517516 −0.124011
2470.9 0.302856 1.74869 −0.908279 −1.83792 0.529601 −1.00000 −0.577933 2.05793 −0.556623
2470.10 0.694611 1.22421 −0.517516 1.98974 0.850350 −1.00000 −1.05408 0.498695 1.38209
2470.11 1.05793 −1.37793 0.119212 −1.64153 −1.45775 −1.00000 −0.931811 0.898702 −1.73662
2470.12 1.37793 −1.64153 0.898702 0.880788 −2.26192 −1.00000 −0.139582 1.69461 1.21367
2470.13 1.64153 0.880788 1.69461 0.101298 1.44584 −1.00000 1.14022 −0.224212 0.166284
2470.14 1.83792 1.90828 2.37793 −1.05793 3.50725 −1.00000 2.53253 2.64153 −1.94438
2470.15 1.95906 −0.302856 2.83792 1.74869 −0.593312 −1.00000 3.60059 −0.908279 3.42579
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2470.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
2471.d odd 2 1 CM by \(\Q(\sqrt{-2471}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2471.1.d.b yes 15
7.b odd 2 1 2471.1.d.a ✓ 15
353.b even 2 1 2471.1.d.a ✓ 15
2471.d odd 2 1 CM 2471.1.d.b yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2471.1.d.a ✓ 15 7.b odd 2 1
2471.1.d.a ✓ 15 353.b even 2 1
2471.1.d.b yes 15 1.a even 1 1 trivial
2471.1.d.b yes 15 2471.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - T_{3}^{14} - 14 T_{3}^{13} + 13 T_{3}^{12} + 78 T_{3}^{11} - 66 T_{3}^{10} - 220 T_{3}^{9} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(2471, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + T^{14} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{15} - T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{15} - T^{14} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + T^{14} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{15} - T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{15} \) Copy content Toggle raw display
$19$ \( T^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + T^{14} + \cdots - 1 \) Copy content Toggle raw display
$29$ \( T^{15} + T^{14} + \cdots - 1 \) Copy content Toggle raw display
$31$ \( T^{15} - T^{14} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{15} \) Copy content Toggle raw display
$41$ \( T^{15} \) Copy content Toggle raw display
$43$ \( T^{15} + T^{14} + \cdots - 1 \) Copy content Toggle raw display
$47$ \( T^{15} \) Copy content Toggle raw display
$53$ \( T^{15} \) Copy content Toggle raw display
$59$ \( T^{15} - T^{14} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{15} \) Copy content Toggle raw display
$67$ \( T^{15} \) Copy content Toggle raw display
$71$ \( T^{15} \) Copy content Toggle raw display
$73$ \( T^{15} \) Copy content Toggle raw display
$79$ \( T^{15} \) Copy content Toggle raw display
$83$ \( T^{15} \) Copy content Toggle raw display
$89$ \( T^{15} - T^{14} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{15} \) Copy content Toggle raw display
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