Properties

Label 1859.1.k.c
Level $1859$
Weight $1$
Character orbit 1859.k
Analytic conductor $0.928$
Analytic rank $0$
Dimension $8$
Projective image $D_{5}$
CM discriminant -143
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1859,1,Mod(1374,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1859.1374");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1859.k (of order \(6\), degree \(2\), 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: \(0.927761858485\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.20449.1
Artin image: $D_5\times C_{12}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{60} - \cdots)\)

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{2} + ( - \beta_{5} - \beta_{4} + 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{4} + ( - 2 \beta_{3} - \beta_1) q^{6} + \beta_1 q^{7} - \beta_{7} q^{8} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{2} + ( - \beta_{5} - \beta_{4} + 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{4} + ( - 2 \beta_{3} - \beta_1) q^{6} + \beta_1 q^{7} - \beta_{7} q^{8} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{9} + ( - \beta_{7} - \beta_{3}) q^{11} + (\beta_{2} + 2) q^{12} - q^{14} + (2 \beta_{7} + \beta_{6} - \beta_1) q^{18} + (\beta_{3} + \beta_1) q^{19} - \beta_{7} q^{21} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{22} - \beta_{4} q^{23} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{24} - q^{25} - q^{27} + (\beta_{7} + \beta_{3}) q^{28} - \beta_{3} q^{32} + ( - \beta_{3} - \beta_1) q^{33} + ( - 2 \beta_{5} - \beta_{4} + 2) q^{36} + ( - \beta_{2} - 2) q^{38} + \beta_{6} q^{41} + (\beta_{5} + \beta_{4} - 1) q^{42} + ( - \beta_{7} - \beta_{6} + \beta_1) q^{44} - \beta_{3} q^{46} + \beta_{4} q^{49} + (\beta_{7} + \beta_{6} + \beta_{3}) q^{50} + ( - \beta_{2} - 1) q^{53} + (\beta_{7} + \beta_{6} + \beta_{3}) q^{54} + ( - \beta_{4} - \beta_{2}) q^{56} + ( - 2 \beta_{7} - \beta_{6} + \beta_1) q^{57} + ( - \beta_{7} - \beta_{3}) q^{63} + (\beta_{2} + 1) q^{64} + (\beta_{2} + 2) q^{66} - \beta_{5} q^{69} + ( - \beta_{3} - \beta_1) q^{72} + ( - \beta_{7} - \beta_{6} + \beta_1) q^{73} + (\beta_{5} + \beta_{4} - 1) q^{75} + (2 \beta_{7} + \beta_{6} + 2 \beta_{3}) q^{76} - \beta_{2} q^{77} - \beta_{5} q^{82} + ( - \beta_{6} + \beta_1) q^{83} + (\beta_{3} + \beta_1) q^{84} + (\beta_{5} - 1) q^{88} + q^{92} + (\beta_{7} + \beta_{6} - \beta_1) q^{96} + \beta_{3} q^{98} + (\beta_{7} + \beta_{6} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 2 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 2 q^{4} - 2 q^{9} + 12 q^{12} - 8 q^{14} + 2 q^{22} - 2 q^{23} - 8 q^{25} - 8 q^{27} + 6 q^{36} - 12 q^{38} - 2 q^{42} + 2 q^{49} - 4 q^{53} + 2 q^{56} + 4 q^{64} + 12 q^{66} - 4 q^{69} - 2 q^{75} + 4 q^{77} - 4 q^{82} - 4 q^{88} + 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 13\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 16\nu^{2} + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 24\nu^{3} + 9\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 3\beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{7} - 5\beta_{6} + 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{3} - 13\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times\).

\(n\) \(508\) \(1354\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1374.1
0.535233 0.309017i
1.40126 0.809017i
−1.40126 + 0.809017i
−0.535233 + 0.309017i
0.535233 + 0.309017i
1.40126 + 0.809017i
−1.40126 0.809017i
−0.535233 0.309017i
−1.40126 0.809017i 0.809017 1.40126i 0.809017 + 1.40126i 0 −2.26728 + 1.30902i 0.535233 0.309017i 1.00000i −0.809017 1.40126i 0
1374.2 −0.535233 0.309017i −0.309017 + 0.535233i −0.309017 0.535233i 0 0.330792 0.190983i 1.40126 0.809017i 1.00000i 0.309017 + 0.535233i 0
1374.3 0.535233 + 0.309017i −0.309017 + 0.535233i −0.309017 0.535233i 0 −0.330792 + 0.190983i −1.40126 + 0.809017i 1.00000i 0.309017 + 0.535233i 0
1374.4 1.40126 + 0.809017i 0.809017 1.40126i 0.809017 + 1.40126i 0 2.26728 1.30902i −0.535233 + 0.309017i 1.00000i −0.809017 1.40126i 0
1836.1 −1.40126 + 0.809017i 0.809017 + 1.40126i 0.809017 1.40126i 0 −2.26728 1.30902i 0.535233 + 0.309017i 1.00000i −0.809017 + 1.40126i 0
1836.2 −0.535233 + 0.309017i −0.309017 0.535233i −0.309017 + 0.535233i 0 0.330792 + 0.190983i 1.40126 + 0.809017i 1.00000i 0.309017 0.535233i 0
1836.3 0.535233 0.309017i −0.309017 0.535233i −0.309017 + 0.535233i 0 −0.330792 0.190983i −1.40126 0.809017i 1.00000i 0.309017 0.535233i 0
1836.4 1.40126 0.809017i 0.809017 + 1.40126i 0.809017 1.40126i 0 2.26728 + 1.30902i −0.535233 0.309017i 1.00000i −0.809017 + 1.40126i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1374.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
143.d odd 2 1 CM by \(\Q(\sqrt{-143}) \)
11.b odd 2 1 inner
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
143.i odd 6 1 inner
143.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.1.k.c 8
11.b odd 2 1 inner 1859.1.k.c 8
13.b even 2 1 inner 1859.1.k.c 8
13.c even 3 1 1859.1.c.c 4
13.c even 3 1 inner 1859.1.k.c 8
13.d odd 4 1 1859.1.i.a 4
13.d odd 4 1 1859.1.i.b 4
13.e even 6 1 1859.1.c.c 4
13.e even 6 1 inner 1859.1.k.c 8
13.f odd 12 1 143.1.d.a 2
13.f odd 12 1 143.1.d.b yes 2
13.f odd 12 1 1859.1.i.a 4
13.f odd 12 1 1859.1.i.b 4
39.k even 12 1 1287.1.g.a 2
39.k even 12 1 1287.1.g.b 2
52.l even 12 1 2288.1.m.a 2
52.l even 12 1 2288.1.m.b 2
65.o even 12 1 3575.1.c.c 4
65.o even 12 1 3575.1.c.d 4
65.s odd 12 1 3575.1.h.e 2
65.s odd 12 1 3575.1.h.f 2
65.t even 12 1 3575.1.c.c 4
65.t even 12 1 3575.1.c.d 4
143.d odd 2 1 CM 1859.1.k.c 8
143.g even 4 1 1859.1.i.a 4
143.g even 4 1 1859.1.i.b 4
143.i odd 6 1 1859.1.c.c 4
143.i odd 6 1 inner 1859.1.k.c 8
143.k odd 6 1 1859.1.c.c 4
143.k odd 6 1 inner 1859.1.k.c 8
143.o even 12 1 143.1.d.a 2
143.o even 12 1 143.1.d.b yes 2
143.o even 12 1 1859.1.i.a 4
143.o even 12 1 1859.1.i.b 4
143.w even 60 2 1573.1.l.a 4
143.w even 60 2 1573.1.l.b 4
143.w even 60 2 1573.1.l.c 4
143.w even 60 2 1573.1.l.d 4
143.x odd 60 2 1573.1.l.a 4
143.x odd 60 2 1573.1.l.b 4
143.x odd 60 2 1573.1.l.c 4
143.x odd 60 2 1573.1.l.d 4
429.bc odd 12 1 1287.1.g.a 2
429.bc odd 12 1 1287.1.g.b 2
572.be odd 12 1 2288.1.m.a 2
572.be odd 12 1 2288.1.m.b 2
715.bk odd 12 1 3575.1.c.c 4
715.bk odd 12 1 3575.1.c.d 4
715.bt even 12 1 3575.1.h.e 2
715.bt even 12 1 3575.1.h.f 2
715.bu odd 12 1 3575.1.c.c 4
715.bu odd 12 1 3575.1.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 13.f odd 12 1
143.1.d.a 2 143.o even 12 1
143.1.d.b yes 2 13.f odd 12 1
143.1.d.b yes 2 143.o even 12 1
1287.1.g.a 2 39.k even 12 1
1287.1.g.a 2 429.bc odd 12 1
1287.1.g.b 2 39.k even 12 1
1287.1.g.b 2 429.bc odd 12 1
1573.1.l.a 4 143.w even 60 2
1573.1.l.a 4 143.x odd 60 2
1573.1.l.b 4 143.w even 60 2
1573.1.l.b 4 143.x odd 60 2
1573.1.l.c 4 143.w even 60 2
1573.1.l.c 4 143.x odd 60 2
1573.1.l.d 4 143.w even 60 2
1573.1.l.d 4 143.x odd 60 2
1859.1.c.c 4 13.c even 3 1
1859.1.c.c 4 13.e even 6 1
1859.1.c.c 4 143.i odd 6 1
1859.1.c.c 4 143.k odd 6 1
1859.1.i.a 4 13.d odd 4 1
1859.1.i.a 4 13.f odd 12 1
1859.1.i.a 4 143.g even 4 1
1859.1.i.a 4 143.o even 12 1
1859.1.i.b 4 13.d odd 4 1
1859.1.i.b 4 13.f odd 12 1
1859.1.i.b 4 143.g even 4 1
1859.1.i.b 4 143.o even 12 1
1859.1.k.c 8 1.a even 1 1 trivial
1859.1.k.c 8 11.b odd 2 1 inner
1859.1.k.c 8 13.b even 2 1 inner
1859.1.k.c 8 13.c even 3 1 inner
1859.1.k.c 8 13.e even 6 1 inner
1859.1.k.c 8 143.d odd 2 1 CM
1859.1.k.c 8 143.i odd 6 1 inner
1859.1.k.c 8 143.k odd 6 1 inner
2288.1.m.a 2 52.l even 12 1
2288.1.m.a 2 572.be odd 12 1
2288.1.m.b 2 52.l even 12 1
2288.1.m.b 2 572.be odd 12 1
3575.1.c.c 4 65.o even 12 1
3575.1.c.c 4 65.t even 12 1
3575.1.c.c 4 715.bk odd 12 1
3575.1.c.c 4 715.bu odd 12 1
3575.1.c.d 4 65.o even 12 1
3575.1.c.d 4 65.t even 12 1
3575.1.c.d 4 715.bk odd 12 1
3575.1.c.d 4 715.bu odd 12 1
3575.1.h.e 2 65.s odd 12 1
3575.1.h.e 2 715.bt even 12 1
3575.1.h.f 2 65.s odd 12 1
3575.1.h.f 2 715.bt even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1859, [\chi])\):

\( T_{2}^{8} - 3T_{2}^{6} + 8T_{2}^{4} - 3T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

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