Properties

Label 2023.1.l.b
Level $2023$
Weight $1$
Character orbit 2023.l
Analytic conductor $1.010$
Analytic rank $0$
Dimension $16$
Projective image $D_{5}$
CM discriminant -119
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2023,1,Mod(468,2023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2023, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2023.468");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2023 = 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2023.l (of order \(8\), 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: \(1.00960852056\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: 16.0.109951162777600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 47x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 119)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.14161.1
Artin image: $D_5\times C_{16}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{80} - \cdots)\)

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} - \beta_1 q^{3} - \beta_{8} q^{4} + ( - \beta_{11} - \beta_{10}) q^{5} + (\beta_{15} + \beta_{14}) q^{6} - \beta_{7} q^{7} + \beta_{5} q^{8} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} - \beta_1 q^{3} - \beta_{8} q^{4} + ( - \beta_{11} - \beta_{10}) q^{5} + (\beta_{15} + \beta_{14}) q^{6} - \beta_{7} q^{7} + \beta_{5} q^{8} + \beta_{4} q^{9} - \beta_{7} q^{10} + (\beta_{11} - \beta_{10}) q^{12} + \beta_1 q^{14} + \beta_{13} q^{15} + (\beta_{2} - 1) q^{18} - \beta_{3} q^{20} + \beta_{8} q^{21} - \beta_{6} q^{24} + ( - \beta_{5} + \beta_{4}) q^{25} - \beta_{7} q^{27} - \beta_{14} q^{28} + \beta_{8} q^{30} + ( - \beta_{3} - \beta_1) q^{31} - \beta_{13} q^{32} + ( - \beta_{2} - 1) q^{35} + (\beta_{13} + \beta_{12}) q^{36} + (\beta_{15} - \beta_{14}) q^{40} + \beta_{6} q^{41} + ( - \beta_{5} - \beta_{4}) q^{42} - \beta_{4} q^{43} - \beta_{15} q^{45} - \beta_{13} q^{49} - q^{50} + ( - \beta_{13} + \beta_{12}) q^{53} + \beta_1 q^{54} - \beta_{11} q^{56} - \beta_{4} q^{60} - \beta_{6} q^{61} + \beta_{15} q^{62} + \beta_{10} q^{63} - \beta_{8} q^{64} + (\beta_{2} + 1) q^{67} - \beta_{13} q^{70} + \beta_{8} q^{72} + \beta_{10} q^{73} - \beta_{7} q^{75} + (\beta_{3} - \beta_1) q^{82} + ( - \beta_{2} + 1) q^{84} + ( - \beta_{2} + 1) q^{86} + \beta_{10} q^{90} + \beta_{5} q^{93} + \beta_{14} q^{96} + (\beta_{11} + \beta_{10}) q^{97} - \beta_{8} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{18} - 8 q^{35} - 16 q^{50} + 8 q^{67} + 24 q^{84} + 24 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 47x^{8} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 13 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 34\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} - 34\nu^{2} ) / 21 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} + 55\nu^{2} ) / 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 55\nu^{3} ) / 21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{11} - 89\nu^{3} ) / 21 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{12} - 89\nu^{4} ) / 21 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{12} - 48\nu^{4} ) / 7 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{13} - 48\nu^{5} ) / 7 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -5\nu^{13} - 233\nu^{5} ) / 21 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5\nu^{14} + 233\nu^{6} ) / 21 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -8\nu^{14} - 377\nu^{6} ) / 21 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -8\nu^{15} - 377\nu^{7} ) / 21 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\nu^{15} + 610\nu^{7} ) / 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{11} - 5\beta_{10} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{13} - 8\beta_{12} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{15} - 13\beta_{14} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 21\beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 21\beta_{3} - 34\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -34\beta_{5} - 55\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -55\beta_{7} - 89\beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 89\beta_{9} - 144\beta_{8} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -144\beta_{11} + 233\beta_{10} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 233\beta_{13} + 377\beta_{12} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 377\beta_{15} + 610\beta_{14} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(1737\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
468.1
0.236511 0.570989i
−0.236511 + 0.570989i
0.619195 1.49487i
−0.619195 + 1.49487i
1.49487 + 0.619195i
−1.49487 0.619195i
0.570989 + 0.236511i
−0.570989 0.236511i
1.49487 0.619195i
−1.49487 + 0.619195i
0.570989 0.236511i
−0.570989 + 0.236511i
0.236511 + 0.570989i
−0.236511 0.570989i
0.619195 + 1.49487i
−0.619195 1.49487i
−0.437016 + 0.437016i −0.236511 + 0.570989i 0.618034i 1.49487 + 0.619195i −0.146172 0.352891i −0.923880 + 0.382683i −0.707107 0.707107i 0.437016 + 0.437016i −0.923880 + 0.382683i
468.2 −0.437016 + 0.437016i 0.236511 0.570989i 0.618034i −1.49487 0.619195i 0.146172 + 0.352891i 0.923880 0.382683i −0.707107 0.707107i 0.437016 + 0.437016i 0.923880 0.382683i
468.3 1.14412 1.14412i −0.619195 + 1.49487i 1.61803i 0.570989 + 0.236511i 1.00188 + 2.41875i 0.923880 0.382683i −0.707107 0.707107i −1.14412 1.14412i 0.923880 0.382683i
468.4 1.14412 1.14412i 0.619195 1.49487i 1.61803i −0.570989 0.236511i −1.00188 2.41875i −0.923880 + 0.382683i −0.707107 0.707107i −1.14412 1.14412i −0.923880 + 0.382683i
1266.1 −1.14412 + 1.14412i −1.49487 0.619195i 1.61803i −0.236511 + 0.570989i 2.41875 1.00188i −0.382683 0.923880i 0.707107 + 0.707107i 1.14412 + 1.14412i −0.382683 0.923880i
1266.2 −1.14412 + 1.14412i 1.49487 + 0.619195i 1.61803i 0.236511 0.570989i −2.41875 + 1.00188i 0.382683 + 0.923880i 0.707107 + 0.707107i 1.14412 + 1.14412i 0.382683 + 0.923880i
1266.3 0.437016 0.437016i −0.570989 0.236511i 0.618034i −0.619195 + 1.49487i −0.352891 + 0.146172i 0.382683 + 0.923880i 0.707107 + 0.707107i −0.437016 0.437016i 0.382683 + 0.923880i
1266.4 0.437016 0.437016i 0.570989 + 0.236511i 0.618034i 0.619195 1.49487i 0.352891 0.146172i −0.382683 0.923880i 0.707107 + 0.707107i −0.437016 0.437016i −0.382683 0.923880i
1868.1 −1.14412 1.14412i −1.49487 + 0.619195i 1.61803i −0.236511 0.570989i 2.41875 + 1.00188i −0.382683 + 0.923880i 0.707107 0.707107i 1.14412 1.14412i −0.382683 + 0.923880i
1868.2 −1.14412 1.14412i 1.49487 0.619195i 1.61803i 0.236511 + 0.570989i −2.41875 1.00188i 0.382683 0.923880i 0.707107 0.707107i 1.14412 1.14412i 0.382683 0.923880i
1868.3 0.437016 + 0.437016i −0.570989 + 0.236511i 0.618034i −0.619195 1.49487i −0.352891 0.146172i 0.382683 0.923880i 0.707107 0.707107i −0.437016 + 0.437016i 0.382683 0.923880i
1868.4 0.437016 + 0.437016i 0.570989 0.236511i 0.618034i 0.619195 + 1.49487i 0.352891 + 0.146172i −0.382683 + 0.923880i 0.707107 0.707107i −0.437016 + 0.437016i −0.382683 + 0.923880i
1889.1 −0.437016 0.437016i −0.236511 0.570989i 0.618034i 1.49487 0.619195i −0.146172 + 0.352891i −0.923880 0.382683i −0.707107 + 0.707107i 0.437016 0.437016i −0.923880 0.382683i
1889.2 −0.437016 0.437016i 0.236511 + 0.570989i 0.618034i −1.49487 + 0.619195i 0.146172 0.352891i 0.923880 + 0.382683i −0.707107 + 0.707107i 0.437016 0.437016i 0.923880 + 0.382683i
1889.3 1.14412 + 1.14412i −0.619195 1.49487i 1.61803i 0.570989 0.236511i 1.00188 2.41875i 0.923880 + 0.382683i −0.707107 + 0.707107i −1.14412 + 1.14412i 0.923880 + 0.382683i
1889.4 1.14412 + 1.14412i 0.619195 + 1.49487i 1.61803i −0.570989 + 0.236511i −1.00188 + 2.41875i −0.923880 0.382683i −0.707107 + 0.707107i −1.14412 + 1.14412i −0.923880 0.382683i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 468.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
7.b odd 2 1 inner
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner
119.f odd 4 2 inner
119.l odd 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2023.1.l.b 16
7.b odd 2 1 inner 2023.1.l.b 16
17.b even 2 1 inner 2023.1.l.b 16
17.c even 4 2 inner 2023.1.l.b 16
17.d even 8 4 inner 2023.1.l.b 16
17.e odd 16 1 119.1.d.a 2
17.e odd 16 1 119.1.d.b yes 2
17.e odd 16 2 2023.1.c.e 4
17.e odd 16 4 2023.1.f.b 8
51.i even 16 1 1071.1.h.a 2
51.i even 16 1 1071.1.h.b 2
68.i even 16 1 1904.1.n.a 2
68.i even 16 1 1904.1.n.b 2
85.o even 16 1 2975.1.b.a 4
85.o even 16 1 2975.1.b.b 4
85.p odd 16 1 2975.1.h.c 2
85.p odd 16 1 2975.1.h.d 2
85.r even 16 1 2975.1.b.a 4
85.r even 16 1 2975.1.b.b 4
119.d odd 2 1 CM 2023.1.l.b 16
119.f odd 4 2 inner 2023.1.l.b 16
119.l odd 8 4 inner 2023.1.l.b 16
119.p even 16 1 119.1.d.a 2
119.p even 16 1 119.1.d.b yes 2
119.p even 16 2 2023.1.c.e 4
119.p even 16 4 2023.1.f.b 8
119.s even 48 2 833.1.h.a 4
119.s even 48 2 833.1.h.b 4
119.t odd 48 2 833.1.h.a 4
119.t odd 48 2 833.1.h.b 4
357.be odd 16 1 1071.1.h.a 2
357.be odd 16 1 1071.1.h.b 2
476.bf odd 16 1 1904.1.n.a 2
476.bf odd 16 1 1904.1.n.b 2
595.bx odd 16 1 2975.1.b.a 4
595.bx odd 16 1 2975.1.b.b 4
595.by even 16 1 2975.1.h.c 2
595.by even 16 1 2975.1.h.d 2
595.cd odd 16 1 2975.1.b.a 4
595.cd odd 16 1 2975.1.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.1.d.a 2 17.e odd 16 1
119.1.d.a 2 119.p even 16 1
119.1.d.b yes 2 17.e odd 16 1
119.1.d.b yes 2 119.p even 16 1
833.1.h.a 4 119.s even 48 2
833.1.h.a 4 119.t odd 48 2
833.1.h.b 4 119.s even 48 2
833.1.h.b 4 119.t odd 48 2
1071.1.h.a 2 51.i even 16 1
1071.1.h.a 2 357.be odd 16 1
1071.1.h.b 2 51.i even 16 1
1071.1.h.b 2 357.be odd 16 1
1904.1.n.a 2 68.i even 16 1
1904.1.n.a 2 476.bf odd 16 1
1904.1.n.b 2 68.i even 16 1
1904.1.n.b 2 476.bf odd 16 1
2023.1.c.e 4 17.e odd 16 2
2023.1.c.e 4 119.p even 16 2
2023.1.f.b 8 17.e odd 16 4
2023.1.f.b 8 119.p even 16 4
2023.1.l.b 16 1.a even 1 1 trivial
2023.1.l.b 16 7.b odd 2 1 inner
2023.1.l.b 16 17.b even 2 1 inner
2023.1.l.b 16 17.c even 4 2 inner
2023.1.l.b 16 17.d even 8 4 inner
2023.1.l.b 16 119.d odd 2 1 CM
2023.1.l.b 16 119.f odd 4 2 inner
2023.1.l.b 16 119.l odd 8 4 inner
2975.1.b.a 4 85.o even 16 1
2975.1.b.a 4 85.r even 16 1
2975.1.b.a 4 595.bx odd 16 1
2975.1.b.a 4 595.cd odd 16 1
2975.1.b.b 4 85.o even 16 1
2975.1.b.b 4 85.r even 16 1
2975.1.b.b 4 595.bx odd 16 1
2975.1.b.b 4 595.cd odd 16 1
2975.1.h.c 2 85.p odd 16 1
2975.1.h.c 2 595.by even 16 1
2975.1.h.d 2 85.p odd 16 1
2975.1.h.d 2 595.by even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 7T_{2}^{4} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2023, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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