Properties

Label 1600.3.b.s
Level $1600$
Weight $3$
Character orbit 1600.b
Analytic conductor $43.597$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + 4 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + ( -2 + 4 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{7} + ( 1 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( -2 + 4 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + ( -2 + 4 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{7} + ( 1 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{9} + ( -4 + 8 \zeta_{10} + 4 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{11} + ( -6 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{13} + ( -2 - 16 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{17} + ( -8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{19} + ( 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{21} + ( -6 + 12 \zeta_{10} - 10 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{23} + ( -12 + 24 \zeta_{10} - 4 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{27} + ( 6 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{29} + ( -20 + 40 \zeta_{10} - 28 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{31} + ( -32 - 24 \zeta_{10}^{2} + 24 \zeta_{10}^{3} ) q^{33} + ( -6 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{37} + ( 4 - 8 \zeta_{10} - 4 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{39} + ( -22 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{41} + ( 6 - 12 \zeta_{10} - 2 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{43} + ( -26 + 52 \zeta_{10} - 14 \zeta_{10}^{2} + 38 \zeta_{10}^{3} ) q^{47} + ( 9 + 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{49} + ( -28 + 56 \zeta_{10} - 60 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{51} + ( -54 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{53} + ( 16 + 32 \zeta_{10}^{2} - 32 \zeta_{10}^{3} ) q^{57} + ( 24 - 48 \zeta_{10} + 48 \zeta_{10}^{2} ) q^{59} + ( -58 - 52 \zeta_{10}^{2} + 52 \zeta_{10}^{3} ) q^{61} + ( 22 - 44 \zeta_{10} + 26 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{63} + ( -22 + 44 \zeta_{10} - 62 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{67} + ( -16 + 28 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{69} + ( -4 + 8 \zeta_{10} + 36 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{71} + ( -34 + 64 \zeta_{10}^{2} - 64 \zeta_{10}^{3} ) q^{73} + ( 80 + 40 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{77} + ( 40 - 80 \zeta_{10} + 72 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{79} + ( -55 + 28 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{81} + ( -26 + 52 \zeta_{10} - 2 \zeta_{10}^{2} + 50 \zeta_{10}^{3} ) q^{83} + ( 4 - 8 \zeta_{10} + 20 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{87} + ( -62 - 80 \zeta_{10}^{2} + 80 \zeta_{10}^{3} ) q^{89} + ( -12 + 24 \zeta_{10} + 4 \zeta_{10}^{2} + 28 \zeta_{10}^{3} ) q^{91} + ( -64 + 72 \zeta_{10}^{2} - 72 \zeta_{10}^{3} ) q^{93} + ( 78 + 24 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{97} + ( -20 + 40 \zeta_{10} + 4 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 16q^{13} + 24q^{17} - 40q^{21} + 8q^{29} - 80q^{33} + 16q^{37} - 112q^{41} - 4q^{49} - 176q^{53} - 128q^{61} - 120q^{69} - 264q^{73} + 240q^{77} - 276q^{81} - 88q^{89} - 400q^{93} + 264q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 3.80423i 0 0 0 8.50651i 0 −5.47214 0
1151.2 0 2.35114i 0 0 0 5.25731i 0 3.47214 0
1151.3 0 2.35114i 0 0 0 5.25731i 0 3.47214 0
1151.4 0 3.80423i 0 0 0 8.50651i 0 −5.47214 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
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 1600.3.b.s 4
4.b odd 2 1 inner 1600.3.b.s 4
5.b even 2 1 320.3.b.c 4
5.c odd 4 2 1600.3.h.n 8
8.b even 2 1 100.3.b.f 4
8.d odd 2 1 100.3.b.f 4
15.d odd 2 1 2880.3.e.e 4
20.d odd 2 1 320.3.b.c 4
20.e even 4 2 1600.3.h.n 8
24.f even 2 1 900.3.c.k 4
24.h odd 2 1 900.3.c.k 4
40.e odd 2 1 20.3.b.a 4
40.f even 2 1 20.3.b.a 4
40.i odd 4 2 100.3.d.b 8
40.k even 4 2 100.3.d.b 8
60.h even 2 1 2880.3.e.e 4
80.k odd 4 2 1280.3.g.e 8
80.q even 4 2 1280.3.g.e 8
120.i odd 2 1 180.3.c.a 4
120.m even 2 1 180.3.c.a 4
120.q odd 4 2 900.3.f.e 8
120.w even 4 2 900.3.f.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 40.e odd 2 1
20.3.b.a 4 40.f even 2 1
100.3.b.f 4 8.b even 2 1
100.3.b.f 4 8.d odd 2 1
100.3.d.b 8 40.i odd 4 2
100.3.d.b 8 40.k even 4 2
180.3.c.a 4 120.i odd 2 1
180.3.c.a 4 120.m even 2 1
320.3.b.c 4 5.b even 2 1
320.3.b.c 4 20.d odd 2 1
900.3.c.k 4 24.f even 2 1
900.3.c.k 4 24.h odd 2 1
900.3.f.e 8 120.q odd 4 2
900.3.f.e 8 120.w even 4 2
1280.3.g.e 8 80.k odd 4 2
1280.3.g.e 8 80.q even 4 2
1600.3.b.s 4 1.a even 1 1 trivial
1600.3.b.s 4 4.b odd 2 1 inner
1600.3.h.n 8 5.c odd 4 2
1600.3.h.n 8 20.e even 4 2
2880.3.e.e 4 15.d odd 2 1
2880.3.e.e 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} + 20 T_{3}^{2} + 80 \)
\( T_{7}^{4} + 100 T_{7}^{2} + 2000 \)
\( T_{13}^{2} + 8 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 80 + 20 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 2000 + 100 T^{2} + T^{4} \)
$11$ \( 1280 + 400 T^{2} + T^{4} \)
$13$ \( ( -4 + 8 T + T^{2} )^{2} \)
$17$ \( ( -284 - 12 T + T^{2} )^{2} \)
$19$ \( 20480 + 320 T^{2} + T^{4} \)
$23$ \( 80 + 260 T^{2} + T^{4} \)
$29$ \( ( -76 - 4 T + T^{2} )^{2} \)
$31$ \( 154880 + 2320 T^{2} + T^{4} \)
$37$ \( ( -484 - 8 T + T^{2} )^{2} \)
$41$ \( ( 604 + 56 T + T^{2} )^{2} \)
$43$ \( 2000 + 500 T^{2} + T^{4} \)
$47$ \( 3561680 + 4100 T^{2} + T^{4} \)
$53$ \( ( 1436 + 88 T + T^{2} )^{2} \)
$59$ \( 1658880 + 5760 T^{2} + T^{4} \)
$61$ \( ( -2356 + 64 T + T^{2} )^{2} \)
$67$ \( 19920080 + 10420 T^{2} + T^{4} \)
$71$ \( 10138880 + 8080 T^{2} + T^{4} \)
$73$ \( ( -764 + 132 T + T^{2} )^{2} \)
$79$ \( 2478080 + 13120 T^{2} + T^{4} \)
$83$ \( 2620880 + 6260 T^{2} + T^{4} \)
$89$ \( ( -7516 + 44 T + T^{2} )^{2} \)
$97$ \( ( 3636 - 132 T + T^{2} )^{2} \)
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