Properties

Label 1859.1.c.c
Level $1859$
Weight $1$
Character orbit 1859.c
Analytic conductor $0.928$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -143
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1859.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.927761858485\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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: $C_4\times D_5$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

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

\(n\) \(508\) \(1354\)
\(\chi(n)\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
846.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i −1.61803 −1.61803 0 2.61803i 0.618034i 1.00000i 1.61803 0
846.2 0.618034i 0.618034 0.618034 0 0.381966i 1.61803i 1.00000i −0.618034 0
846.3 0.618034i 0.618034 0.618034 0 0.381966i 1.61803i 1.00000i −0.618034 0
846.4 1.61803i −1.61803 −1.61803 0 2.61803i 0.618034i 1.00000i 1.61803 0
\(n\): e.g. 2-40 or 990-1000
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.1.c.c 4
11.b odd 2 1 inner 1859.1.c.c 4
13.b even 2 1 inner 1859.1.c.c 4
13.c even 3 2 1859.1.k.c 8
13.d odd 4 1 143.1.d.a 2
13.d odd 4 1 143.1.d.b yes 2
13.e even 6 2 1859.1.k.c 8
13.f odd 12 2 1859.1.i.a 4
13.f odd 12 2 1859.1.i.b 4
39.f even 4 1 1287.1.g.a 2
39.f even 4 1 1287.1.g.b 2
52.f even 4 1 2288.1.m.a 2
52.f even 4 1 2288.1.m.b 2
65.f even 4 1 3575.1.c.c 4
65.f even 4 1 3575.1.c.d 4
65.g odd 4 1 3575.1.h.e 2
65.g odd 4 1 3575.1.h.f 2
65.k even 4 1 3575.1.c.c 4
65.k even 4 1 3575.1.c.d 4
143.d odd 2 1 CM 1859.1.c.c 4
143.g even 4 1 143.1.d.a 2
143.g even 4 1 143.1.d.b yes 2
143.i odd 6 2 1859.1.k.c 8
143.k odd 6 2 1859.1.k.c 8
143.o even 12 2 1859.1.i.a 4
143.o even 12 2 1859.1.i.b 4
143.r odd 20 2 1573.1.l.a 4
143.r odd 20 2 1573.1.l.b 4
143.r odd 20 2 1573.1.l.c 4
143.r odd 20 2 1573.1.l.d 4
143.s even 20 2 1573.1.l.a 4
143.s even 20 2 1573.1.l.b 4
143.s even 20 2 1573.1.l.c 4
143.s even 20 2 1573.1.l.d 4
429.l odd 4 1 1287.1.g.a 2
429.l odd 4 1 1287.1.g.b 2
572.k odd 4 1 2288.1.m.a 2
572.k odd 4 1 2288.1.m.b 2
715.k odd 4 1 3575.1.c.c 4
715.k odd 4 1 3575.1.c.d 4
715.l even 4 1 3575.1.h.e 2
715.l even 4 1 3575.1.h.f 2
715.u odd 4 1 3575.1.c.c 4
715.u odd 4 1 3575.1.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 13.d odd 4 1
143.1.d.a 2 143.g even 4 1
143.1.d.b yes 2 13.d odd 4 1
143.1.d.b yes 2 143.g even 4 1
1287.1.g.a 2 39.f even 4 1
1287.1.g.a 2 429.l odd 4 1
1287.1.g.b 2 39.f even 4 1
1287.1.g.b 2 429.l odd 4 1
1573.1.l.a 4 143.r odd 20 2
1573.1.l.a 4 143.s even 20 2
1573.1.l.b 4 143.r odd 20 2
1573.1.l.b 4 143.s even 20 2
1573.1.l.c 4 143.r odd 20 2
1573.1.l.c 4 143.s even 20 2
1573.1.l.d 4 143.r odd 20 2
1573.1.l.d 4 143.s even 20 2
1859.1.c.c 4 1.a even 1 1 trivial
1859.1.c.c 4 11.b odd 2 1 inner
1859.1.c.c 4 13.b even 2 1 inner
1859.1.c.c 4 143.d odd 2 1 CM
1859.1.i.a 4 13.f odd 12 2
1859.1.i.a 4 143.o even 12 2
1859.1.i.b 4 13.f odd 12 2
1859.1.i.b 4 143.o even 12 2
1859.1.k.c 8 13.c even 3 2
1859.1.k.c 8 13.e even 6 2
1859.1.k.c 8 143.i odd 6 2
1859.1.k.c 8 143.k odd 6 2
2288.1.m.a 2 52.f even 4 1
2288.1.m.a 2 572.k odd 4 1
2288.1.m.b 2 52.f even 4 1
2288.1.m.b 2 572.k odd 4 1
3575.1.c.c 4 65.f even 4 1
3575.1.c.c 4 65.k even 4 1
3575.1.c.c 4 715.k odd 4 1
3575.1.c.c 4 715.u odd 4 1
3575.1.c.d 4 65.f even 4 1
3575.1.c.d 4 65.k even 4 1
3575.1.c.d 4 715.k odd 4 1
3575.1.c.d 4 715.u odd 4 1
3575.1.h.e 2 65.g odd 4 1
3575.1.h.e 2 715.l even 4 1
3575.1.h.f 2 65.g odd 4 1
3575.1.h.f 2 715.l even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

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