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Properties

Label 1848.1.er.b
Level $1848$
Weight $1$
Character orbit 1848.er
Analytic conductor $0.922$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1848,1,Mod(53,1848)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1848, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 15, 15, 20, 18]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1848.53");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1848.er (of order \(30\), degree \(8\), 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: \(0.922272143346\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{30}^{2} q^{2} + \zeta_{30} q^{3} + \zeta_{30}^{4} q^{4} + (\zeta_{30}^{9} + \zeta_{30}^{7}) q^{5} - \zeta_{30}^{3} q^{6} - \zeta_{30} q^{7} - \zeta_{30}^{6} q^{8} + \zeta_{30}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{30}^{2} q^{2} + \zeta_{30} q^{3} + \zeta_{30}^{4} q^{4} + (\zeta_{30}^{9} + \zeta_{30}^{7}) q^{5} - \zeta_{30}^{3} q^{6} - \zeta_{30} q^{7} - \zeta_{30}^{6} q^{8} + \zeta_{30}^{2} q^{9} + ( - \zeta_{30}^{11} - \zeta_{30}^{9}) q^{10} + \zeta_{30}^{5} q^{11} + \zeta_{30}^{5} q^{12} + \zeta_{30}^{3} q^{14} + (\zeta_{30}^{10} + \zeta_{30}^{8}) q^{15} + \zeta_{30}^{8} q^{16} - \zeta_{30}^{4} q^{18} + (\zeta_{30}^{13} + \zeta_{30}^{11}) q^{20} - \zeta_{30}^{2} q^{21} - \zeta_{30}^{7} q^{22} - \zeta_{30}^{7} q^{24} + (\zeta_{30}^{14} - \zeta_{30}^{3} - \zeta_{30}) q^{25} + \zeta_{30}^{3} q^{27} - \zeta_{30}^{5} q^{28} + (\zeta_{30}^{13} + \zeta_{30}^{5}) q^{29} + ( - \zeta_{30}^{12} - \zeta_{30}^{10}) q^{30} + ( - \zeta_{30}^{9} - \zeta_{30}^{5}) q^{31} - \zeta_{30}^{10} q^{32} + \zeta_{30}^{6} q^{33} + ( - \zeta_{30}^{10} - \zeta_{30}^{8}) q^{35} + \zeta_{30}^{6} q^{36} + ( - \zeta_{30}^{13} + 1) q^{40} + \zeta_{30}^{4} q^{42} + \zeta_{30}^{9} q^{44} + (\zeta_{30}^{11} + \zeta_{30}^{9}) q^{45} + \zeta_{30}^{9} q^{48} + \zeta_{30}^{2} q^{49} + (\zeta_{30}^{5} + \zeta_{30}^{3} + \zeta_{30}) q^{50} + ( - \zeta_{30}^{14} - 1) q^{53} - \zeta_{30}^{5} q^{54} + (\zeta_{30}^{14} + \zeta_{30}^{12}) q^{55} + \zeta_{30}^{7} q^{56} + ( - \zeta_{30}^{7} + 1) q^{58} + ( - \zeta_{30}^{12} + \zeta_{30}^{11}) q^{59} + (\zeta_{30}^{14} + \zeta_{30}^{12}) q^{60} + (\zeta_{30}^{11} + \zeta_{30}^{7}) q^{62} - \zeta_{30}^{3} q^{63} + \zeta_{30}^{12} q^{64} - \zeta_{30}^{8} q^{66} + (\zeta_{30}^{12} + \zeta_{30}^{10}) q^{70} - \zeta_{30}^{8} q^{72} + ( - \zeta_{30}^{13} + \zeta_{30}^{10}) q^{73} + ( - \zeta_{30}^{4} - \zeta_{30}^{2} - 1) q^{75} - \zeta_{30}^{6} q^{77} + ( - \zeta_{30}^{13} + \zeta_{30}^{6}) q^{79} + ( - \zeta_{30}^{2} - 1) q^{80} + \zeta_{30}^{4} q^{81} + ( - \zeta_{30}^{4} - \zeta_{30}^{2}) q^{83} - \zeta_{30}^{6} q^{84} + (\zeta_{30}^{14} + \zeta_{30}^{6}) q^{87} - \zeta_{30}^{11} q^{88} + ( - \zeta_{30}^{13} - \zeta_{30}^{11}) q^{90} + ( - \zeta_{30}^{10} - \zeta_{30}^{6}) q^{93} - \zeta_{30}^{11} q^{96} + (\zeta_{30}^{8} - \zeta_{30}) q^{97} - \zeta_{30}^{4} q^{98} + \zeta_{30}^{7} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} - q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + 2 q^{8} + q^{9} - q^{10} + 4 q^{11} + 4 q^{12} + 2 q^{14} - 3 q^{15} + q^{16} - q^{18} - 2 q^{20} - q^{21} + q^{22} + q^{24} + 2 q^{27} - 4 q^{28} + 3 q^{29} + 6 q^{30} - 6 q^{31} + 4 q^{32} - 2 q^{33} + 3 q^{35} - 2 q^{36} + 9 q^{40} + q^{42} + 2 q^{44} + q^{45} + 2 q^{48} + q^{49} + 5 q^{50} - 9 q^{53} - 4 q^{54} - q^{55} - q^{56} + 9 q^{58} + q^{59} - q^{60} - 2 q^{62} - 2 q^{63} - 2 q^{64} - q^{66} - 6 q^{70} - q^{72} - 3 q^{73} - 10 q^{75} + 2 q^{77} - q^{79} - 9 q^{80} + q^{81} - 2 q^{83} + 2 q^{84} - q^{87} + q^{88} + 2 q^{90} + 6 q^{93} + q^{96} + 2 q^{97} - q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times\).

\(n\) \(463\) \(617\) \(673\) \(925\) \(1585\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{30}^{9}\) \(-1\) \(-\zeta_{30}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.104528 − 0.994522i
0.913545 + 0.406737i
0.669131 + 0.743145i
−0.978148 + 0.207912i
0.913545 − 0.406737i
−0.104528 + 0.994522i
−0.978148 − 0.207912i
0.669131 − 0.743145i
0.978148 − 0.207912i 0.104528 + 0.994522i 0.913545 − 0.406737i 0.139886 − 0.155360i 0.309017 + 0.951057i −0.104528 − 0.994522i 0.809017 − 0.587785i −0.978148 + 0.207912i 0.104528 − 0.181049i
317.1 −0.669131 − 0.743145i −0.913545 − 0.406737i −0.104528 + 0.994522i 1.78716 + 0.379874i 0.309017 + 0.951057i 0.913545 + 0.406737i 0.809017 − 0.587785i 0.669131 + 0.743145i −0.913545 − 1.58231i
389.1 0.104528 − 0.994522i −0.669131 − 0.743145i −0.978148 − 0.207912i −1.22256 − 0.544320i −0.809017 + 0.587785i 0.669131 + 0.743145i −0.309017 + 0.951057i −0.104528 + 0.994522i −0.669131 + 1.15897i
653.1 −0.913545 + 0.406737i 0.978148 − 0.207912i 0.669131 − 0.743145i −0.204489 − 1.94558i −0.809017 + 0.587785i −0.978148 + 0.207912i −0.309017 + 0.951057i 0.913545 − 0.406737i 0.978148 + 1.69420i
1061.1 −0.669131 + 0.743145i −0.913545 + 0.406737i −0.104528 − 0.994522i 1.78716 − 0.379874i 0.309017 − 0.951057i 0.913545 − 0.406737i 0.809017 + 0.587785i 0.669131 − 0.743145i −0.913545 + 1.58231i
1325.1 0.978148 + 0.207912i 0.104528 − 0.994522i 0.913545 + 0.406737i 0.139886 + 0.155360i 0.309017 − 0.951057i −0.104528 + 0.994522i 0.809017 + 0.587785i −0.978148 − 0.207912i 0.104528 + 0.181049i
1565.1 −0.913545 − 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i −0.204489 + 1.94558i −0.809017 − 0.587785i −0.978148 − 0.207912i −0.309017 − 0.951057i 0.913545 + 0.406737i 0.978148 − 1.69420i
1829.1 0.104528 + 0.994522i −0.669131 + 0.743145i −0.978148 + 0.207912i −1.22256 + 0.544320i −0.809017 − 0.587785i 0.669131 − 0.743145i −0.309017 − 0.951057i −0.104528 − 0.994522i −0.669131 − 1.15897i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
77.m even 15 1 inner
1848.er odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1848.1.er.b yes 8
3.b odd 2 1 1848.1.er.c yes 8
7.c even 3 1 1848.1.er.a ✓ 8
8.b even 2 1 1848.1.er.c yes 8
11.c even 5 1 1848.1.er.a ✓ 8
21.h odd 6 1 1848.1.er.d yes 8
24.h odd 2 1 CM 1848.1.er.b yes 8
33.h odd 10 1 1848.1.er.d yes 8
56.p even 6 1 1848.1.er.d yes 8
77.m even 15 1 inner 1848.1.er.b yes 8
88.o even 10 1 1848.1.er.d yes 8
168.s odd 6 1 1848.1.er.a ✓ 8
231.z odd 30 1 1848.1.er.c yes 8
264.t odd 10 1 1848.1.er.a ✓ 8
616.ca even 30 1 1848.1.er.c yes 8
1848.er odd 30 1 inner 1848.1.er.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1848.1.er.a ✓ 8 7.c even 3 1
1848.1.er.a ✓ 8 11.c even 5 1
1848.1.er.a ✓ 8 168.s odd 6 1
1848.1.er.a ✓ 8 264.t odd 10 1
1848.1.er.b yes 8 1.a even 1 1 trivial
1848.1.er.b yes 8 24.h odd 2 1 CM
1848.1.er.b yes 8 77.m even 15 1 inner
1848.1.er.b yes 8 1848.er odd 30 1 inner
1848.1.er.c yes 8 3.b odd 2 1
1848.1.er.c yes 8 8.b even 2 1
1848.1.er.c yes 8 231.z odd 30 1
1848.1.er.c yes 8 616.ca even 30 1
1848.1.er.d yes 8 21.h odd 6 1
1848.1.er.d yes 8 33.h odd 10 1
1848.1.er.d yes 8 56.p even 6 1
1848.1.er.d yes 8 88.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - T_{5}^{7} - 4T_{5}^{5} - 6T_{5}^{4} + 11T_{5}^{3} + 20T_{5}^{2} - 6T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1848, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} - 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{8} + 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 9 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} - T^{7} - 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} + T^{7} + 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
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