Properties

Label 1652.1.g.b
Level 1652
Weight 1
Character orbit 1652.g
Self dual yes
Analytic conductor 0.824
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -1652
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1652 = 2^{2} \cdot 7 \cdot 59 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1652.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.824455400870\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.2729104.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( 1 - \beta ) q^{3} + q^{4} + ( -1 + \beta ) q^{6} - q^{7} - q^{8} + ( 1 - \beta ) q^{9} +O(q^{10})\) \( q - q^{2} + ( 1 - \beta ) q^{3} + q^{4} + ( -1 + \beta ) q^{6} - q^{7} - q^{8} + ( 1 - \beta ) q^{9} + ( 1 - \beta ) q^{11} + ( 1 - \beta ) q^{12} -\beta q^{13} + q^{14} + q^{16} + ( -1 + \beta ) q^{18} + \beta q^{19} + ( -1 + \beta ) q^{21} + ( -1 + \beta ) q^{22} + \beta q^{23} + ( -1 + \beta ) q^{24} + q^{25} + \beta q^{26} + q^{27} - q^{28} + ( -1 + \beta ) q^{29} - q^{32} + ( 2 - \beta ) q^{33} + ( 1 - \beta ) q^{36} -\beta q^{38} + q^{39} + ( 1 - \beta ) q^{42} + \beta q^{43} + ( 1 - \beta ) q^{44} -\beta q^{46} + ( 1 - \beta ) q^{48} + q^{49} - q^{50} -\beta q^{52} -\beta q^{53} - q^{54} + q^{56} - q^{57} + ( 1 - \beta ) q^{58} - q^{59} + 2 q^{61} + ( -1 + \beta ) q^{63} + q^{64} + ( -2 + \beta ) q^{66} + ( 1 - \beta ) q^{67} - q^{69} + ( -1 + \beta ) q^{72} + ( -1 + \beta ) q^{73} + ( 1 - \beta ) q^{75} + \beta q^{76} + ( -1 + \beta ) q^{77} - q^{78} + ( -1 + \beta ) q^{84} -\beta q^{86} + ( -2 + \beta ) q^{87} + ( -1 + \beta ) q^{88} + ( -1 + \beta ) q^{89} + \beta q^{91} + \beta q^{92} + ( -1 + \beta ) q^{96} + ( -1 + \beta ) q^{97} - q^{98} + ( 2 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + q^{3} + 2q^{4} - q^{6} - 2q^{7} - 2q^{8} + q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + q^{3} + 2q^{4} - q^{6} - 2q^{7} - 2q^{8} + q^{9} + q^{11} + q^{12} - q^{13} + 2q^{14} + 2q^{16} - q^{18} + q^{19} - q^{21} - q^{22} + q^{23} - q^{24} + 2q^{25} + q^{26} + 2q^{27} - 2q^{28} - q^{29} - 2q^{32} + 3q^{33} + q^{36} - q^{38} + 2q^{39} + q^{42} + q^{43} + q^{44} - q^{46} + q^{48} + 2q^{49} - 2q^{50} - q^{52} - q^{53} - 2q^{54} + 2q^{56} - 2q^{57} + q^{58} - 2q^{59} + 4q^{61} - q^{63} + 2q^{64} - 3q^{66} + q^{67} - 2q^{69} - q^{72} - q^{73} + q^{75} + q^{76} - q^{77} - 2q^{78} - q^{84} - q^{86} - 3q^{87} - q^{88} - q^{89} + q^{91} + q^{92} - q^{96} - q^{97} - 2q^{98} + 3q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1652\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(827\) \(1417\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1651.1
1.61803
−0.618034
−1.00000 −0.618034 1.00000 0 0.618034 −1.00000 −1.00000 −0.618034 0
1651.2 −1.00000 1.61803 1.00000 0 −1.61803 −1.00000 −1.00000 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
1652.g odd 2 1 CM by \(\Q(\sqrt{-413}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1652.1.g.b yes 2
4.b odd 2 1 1652.1.g.c yes 2
7.b odd 2 1 1652.1.g.a 2
28.d even 2 1 1652.1.g.d yes 2
59.b odd 2 1 1652.1.g.d yes 2
236.c even 2 1 1652.1.g.a 2
413.b even 2 1 1652.1.g.c yes 2
1652.g odd 2 1 CM 1652.1.g.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1652.1.g.a 2 7.b odd 2 1
1652.1.g.a 2 236.c even 2 1
1652.1.g.b yes 2 1.a even 1 1 trivial
1652.1.g.b yes 2 1652.g odd 2 1 CM
1652.1.g.c yes 2 4.b odd 2 1
1652.1.g.c yes 2 413.b even 2 1
1652.1.g.d yes 2 28.d even 2 1
1652.1.g.d yes 2 59.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1652, [\chi])\):

\( T_{3}^{2} - T_{3} - 1 \)
\( T_{11}^{2} - T_{11} - 1 \)

Hecke Characteristic Polynomials

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