Properties

Label 600.4.a.q
Level $600$
Weight $4$
Character orbit 600.a
Self dual yes
Analytic conductor $35.401$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.4011460034\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3 q^{3} + 16 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + 16 q^{7} + 9 q^{9} - 28 q^{11} + 26 q^{13} + 62 q^{17} - 68 q^{19} + 48 q^{21} + 208 q^{23} + 27 q^{27} - 58 q^{29} + 160 q^{31} - 84 q^{33} - 270 q^{37} + 78 q^{39} + 282 q^{41} - 76 q^{43} + 280 q^{47} - 87 q^{49} + 186 q^{51} + 210 q^{53} - 204 q^{57} + 196 q^{59} + 742 q^{61} + 144 q^{63} - 836 q^{67} + 624 q^{69} - 504 q^{71} + 1062 q^{73} - 448 q^{77} + 768 q^{79} + 81 q^{81} + 1052 q^{83} - 174 q^{87} - 726 q^{89} + 416 q^{91} + 480 q^{93} + 1406 q^{97} - 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
0
0 3.00000 0 0 0 16.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.4.a.q 1
3.b odd 2 1 1800.4.a.bb 1
4.b odd 2 1 1200.4.a.c 1
5.b even 2 1 120.4.a.c 1
5.c odd 4 2 600.4.f.c 2
15.d odd 2 1 360.4.a.b 1
15.e even 4 2 1800.4.f.r 2
20.d odd 2 1 240.4.a.l 1
20.e even 4 2 1200.4.f.o 2
40.e odd 2 1 960.4.a.h 1
40.f even 2 1 960.4.a.u 1
60.h even 2 1 720.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.c 1 5.b even 2 1
240.4.a.l 1 20.d odd 2 1
360.4.a.b 1 15.d odd 2 1
600.4.a.q 1 1.a even 1 1 trivial
600.4.f.c 2 5.c odd 4 2
720.4.a.l 1 60.h even 2 1
960.4.a.h 1 40.e odd 2 1
960.4.a.u 1 40.f even 2 1
1200.4.a.c 1 4.b odd 2 1
1200.4.f.o 2 20.e even 4 2
1800.4.a.bb 1 3.b odd 2 1
1800.4.f.r 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(600))\):

\( T_{7} - 16 \) Copy content Toggle raw display
\( T_{11} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 16 \) Copy content Toggle raw display
$11$ \( T + 28 \) Copy content Toggle raw display
$13$ \( T - 26 \) Copy content Toggle raw display
$17$ \( T - 62 \) Copy content Toggle raw display
$19$ \( T + 68 \) Copy content Toggle raw display
$23$ \( T - 208 \) Copy content Toggle raw display
$29$ \( T + 58 \) Copy content Toggle raw display
$31$ \( T - 160 \) Copy content Toggle raw display
$37$ \( T + 270 \) Copy content Toggle raw display
$41$ \( T - 282 \) Copy content Toggle raw display
$43$ \( T + 76 \) Copy content Toggle raw display
$47$ \( T - 280 \) Copy content Toggle raw display
$53$ \( T - 210 \) Copy content Toggle raw display
$59$ \( T - 196 \) Copy content Toggle raw display
$61$ \( T - 742 \) Copy content Toggle raw display
$67$ \( T + 836 \) Copy content Toggle raw display
$71$ \( T + 504 \) Copy content Toggle raw display
$73$ \( T - 1062 \) Copy content Toggle raw display
$79$ \( T - 768 \) Copy content Toggle raw display
$83$ \( T - 1052 \) Copy content Toggle raw display
$89$ \( T + 726 \) Copy content Toggle raw display
$97$ \( T - 1406 \) Copy content Toggle raw display
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