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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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1859.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

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
Defining polynomial: \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\)
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: $C_{12}\times D_5$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{60} - \cdots)\)

$q$-expansion

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 5 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{6} + 8 \nu^{4} - 16 \nu^{2} + 1 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 9 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{5} - 24 \nu^{3} + 9 \nu \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 2 \beta_{6} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{5} + 3 \beta_{4} + 3 \beta_{2}\)
\(\nu^{5}\)\(=\)\(3 \beta_{7} - 5 \beta_{6} + 3 \beta_{3}\)
\(\nu^{6}\)\(=\)\(8 \beta_{2} - 5\)
\(\nu^{7}\)\(=\)\(8 \beta_{3} - 13 \beta_{1}\)

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.

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 1836.4
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 \(S_{1}^{\mathrm{new}}(1859, [\chi])\):

\( T_{2}^{8} - 3 T_{2}^{6} + 8 T_{2}^{4} - 3 T_{2}^{2} + 1 \)
\( T_{5} \)

Hecke characteristic polynomials

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