Properties

Label 1600.4.a.cg
Level $1600$
Weight $4$
Character orbit 1600.a
Self dual yes
Analytic conductor $94.403$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(94.4030560092\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta q^{3} -7 \beta q^{7} -7 q^{9} +O(q^{10})\) \( q -\beta q^{3} -7 \beta q^{7} -7 q^{9} -2 \beta q^{11} -62 q^{13} + 46 q^{17} + 24 \beta q^{19} + 140 q^{21} + 43 \beta q^{23} + 34 \beta q^{27} + 90 q^{29} + 34 \beta q^{31} + 40 q^{33} -214 q^{37} + 62 \beta q^{39} -10 q^{41} + 15 \beta q^{43} + 89 \beta q^{47} + 637 q^{49} -46 \beta q^{51} -678 q^{53} -480 q^{57} -92 \beta q^{59} -250 q^{61} + 49 \beta q^{63} -11 \beta q^{67} -860 q^{69} + 82 \beta q^{71} -522 q^{73} + 280 q^{77} -196 \beta q^{79} -491 q^{81} -85 \beta q^{83} -90 \beta q^{87} + 970 q^{89} + 434 \beta q^{91} -680 q^{93} + 934 q^{97} + 14 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{9} + O(q^{10}) \) \( 2q - 14q^{9} - 124q^{13} + 92q^{17} + 280q^{21} + 180q^{29} + 80q^{33} - 428q^{37} - 20q^{41} + 1274q^{49} - 1356q^{53} - 960q^{57} - 500q^{61} - 1720q^{69} - 1044q^{73} + 560q^{77} - 982q^{81} + 1940q^{89} - 1360q^{93} + 1868q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
0 −4.47214 0 0 0 −31.3050 0 −7.00000 0
1.2 0 4.47214 0 0 0 31.3050 0 −7.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.4.a.cg 2
4.b odd 2 1 inner 1600.4.a.cg 2
5.b even 2 1 320.4.a.q 2
8.b even 2 1 800.4.a.o 2
8.d odd 2 1 800.4.a.o 2
20.d odd 2 1 320.4.a.q 2
40.e odd 2 1 160.4.a.d 2
40.f even 2 1 160.4.a.d 2
40.i odd 4 2 800.4.c.l 4
40.k even 4 2 800.4.c.l 4
80.k odd 4 2 1280.4.d.v 4
80.q even 4 2 1280.4.d.v 4
120.i odd 2 1 1440.4.a.bb 2
120.m even 2 1 1440.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 40.e odd 2 1
160.4.a.d 2 40.f even 2 1
320.4.a.q 2 5.b even 2 1
320.4.a.q 2 20.d odd 2 1
800.4.a.o 2 8.b even 2 1
800.4.a.o 2 8.d odd 2 1
800.4.c.l 4 40.i odd 4 2
800.4.c.l 4 40.k even 4 2
1280.4.d.v 4 80.k odd 4 2
1280.4.d.v 4 80.q even 4 2
1440.4.a.bb 2 120.i odd 2 1
1440.4.a.bb 2 120.m even 2 1
1600.4.a.cg 2 1.a even 1 1 trivial
1600.4.a.cg 2 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\):

\( T_{3}^{2} - 20 \)
\( T_{7}^{2} - 980 \)
\( T_{11}^{2} - 80 \)
\( T_{13} + 62 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -20 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -980 + T^{2} \)
$11$ \( -80 + T^{2} \)
$13$ \( ( 62 + T )^{2} \)
$17$ \( ( -46 + T )^{2} \)
$19$ \( -11520 + T^{2} \)
$23$ \( -36980 + T^{2} \)
$29$ \( ( -90 + T )^{2} \)
$31$ \( -23120 + T^{2} \)
$37$ \( ( 214 + T )^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( -4500 + T^{2} \)
$47$ \( -158420 + T^{2} \)
$53$ \( ( 678 + T )^{2} \)
$59$ \( -169280 + T^{2} \)
$61$ \( ( 250 + T )^{2} \)
$67$ \( -2420 + T^{2} \)
$71$ \( -134480 + T^{2} \)
$73$ \( ( 522 + T )^{2} \)
$79$ \( -768320 + T^{2} \)
$83$ \( -144500 + T^{2} \)
$89$ \( ( -970 + T )^{2} \)
$97$ \( ( -934 + T )^{2} \)
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