Properties

Label 600.2.d.c
Level 600
Weight 2
Character orbit 600.d
Analytic conductor 4.791
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 600.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{2} - q^{3} + 2 i q^{4} + ( -1 - i ) q^{6} + 2 i q^{7} + ( -2 + 2 i ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 + i ) q^{2} - q^{3} + 2 i q^{4} + ( -1 - i ) q^{6} + 2 i q^{7} + ( -2 + 2 i ) q^{8} + q^{9} -2 i q^{12} -4 q^{13} + ( -2 + 2 i ) q^{14} -4 q^{16} + 2 i q^{17} + ( 1 + i ) q^{18} + 4 i q^{19} -2 i q^{21} + 4 i q^{23} + ( 2 - 2 i ) q^{24} + ( -4 - 4 i ) q^{26} - q^{27} -4 q^{28} -6 i q^{29} + 2 q^{31} + ( -4 - 4 i ) q^{32} + ( -2 + 2 i ) q^{34} + 2 i q^{36} -8 q^{37} + ( -4 + 4 i ) q^{38} + 4 q^{39} + 2 q^{41} + ( 2 - 2 i ) q^{42} -4 q^{43} + ( -4 + 4 i ) q^{46} + 12 i q^{47} + 4 q^{48} + 3 q^{49} -2 i q^{51} -8 i q^{52} + 6 q^{53} + ( -1 - i ) q^{54} + ( -4 - 4 i ) q^{56} -4 i q^{57} + ( 6 - 6 i ) q^{58} + 4 i q^{59} + ( 2 + 2 i ) q^{62} + 2 i q^{63} -8 i q^{64} + 12 q^{67} -4 q^{68} -4 i q^{69} + 12 q^{71} + ( -2 + 2 i ) q^{72} -6 i q^{73} + ( -8 - 8 i ) q^{74} -8 q^{76} + ( 4 + 4 i ) q^{78} -10 q^{79} + q^{81} + ( 2 + 2 i ) q^{82} + 16 q^{83} + 4 q^{84} + ( -4 - 4 i ) q^{86} + 6 i q^{87} + 10 q^{89} -8 i q^{91} -8 q^{92} -2 q^{93} + ( -12 + 12 i ) q^{94} + ( 4 + 4 i ) q^{96} + 2 i q^{97} + ( 3 + 3 i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} - 2q^{6} - 4q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} - 2q^{6} - 4q^{8} + 2q^{9} - 8q^{13} - 4q^{14} - 8q^{16} + 2q^{18} + 4q^{24} - 8q^{26} - 2q^{27} - 8q^{28} + 4q^{31} - 8q^{32} - 4q^{34} - 16q^{37} - 8q^{38} + 8q^{39} + 4q^{41} + 4q^{42} - 8q^{43} - 8q^{46} + 8q^{48} + 6q^{49} + 12q^{53} - 2q^{54} - 8q^{56} + 12q^{58} + 4q^{62} + 24q^{67} - 8q^{68} + 24q^{71} - 4q^{72} - 16q^{74} - 16q^{76} + 8q^{78} - 20q^{79} + 2q^{81} + 4q^{82} + 32q^{83} + 8q^{84} - 8q^{86} + 20q^{89} - 16q^{92} - 4q^{93} - 24q^{94} + 8q^{96} + 6q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
1.00000i
1.00000i
1.00000 1.00000i −1.00000 2.00000i 0 −1.00000 + 1.00000i 2.00000i −2.00000 2.00000i 1.00000 0
349.2 1.00000 + 1.00000i −1.00000 2.00000i 0 −1.00000 1.00000i 2.00000i −2.00000 + 2.00000i 1.00000 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
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.d.c 2
3.b odd 2 1 1800.2.d.b 2
4.b odd 2 1 2400.2.d.c 2
5.b even 2 1 600.2.d.b 2
5.c odd 4 1 24.2.d.a 2
5.c odd 4 1 600.2.k.b 2
8.b even 2 1 600.2.d.b 2
8.d odd 2 1 2400.2.d.b 2
12.b even 2 1 7200.2.d.d 2
15.d odd 2 1 1800.2.d.i 2
15.e even 4 1 72.2.d.b 2
15.e even 4 1 1800.2.k.a 2
20.d odd 2 1 2400.2.d.b 2
20.e even 4 1 96.2.d.a 2
20.e even 4 1 2400.2.k.a 2
24.f even 2 1 7200.2.d.g 2
24.h odd 2 1 1800.2.d.i 2
35.f even 4 1 1176.2.c.a 2
40.e odd 2 1 2400.2.d.c 2
40.f even 2 1 inner 600.2.d.c 2
40.i odd 4 1 24.2.d.a 2
40.i odd 4 1 600.2.k.b 2
40.k even 4 1 96.2.d.a 2
40.k even 4 1 2400.2.k.a 2
45.k odd 12 2 648.2.n.k 4
45.l even 12 2 648.2.n.c 4
60.h even 2 1 7200.2.d.g 2
60.l odd 4 1 288.2.d.b 2
60.l odd 4 1 7200.2.k.d 2
80.i odd 4 1 768.2.a.h 1
80.j even 4 1 768.2.a.e 1
80.s even 4 1 768.2.a.d 1
80.t odd 4 1 768.2.a.a 1
120.i odd 2 1 1800.2.d.b 2
120.m even 2 1 7200.2.d.d 2
120.q odd 4 1 288.2.d.b 2
120.q odd 4 1 7200.2.k.d 2
120.w even 4 1 72.2.d.b 2
120.w even 4 1 1800.2.k.a 2
140.j odd 4 1 4704.2.c.a 2
180.v odd 12 2 2592.2.r.g 4
180.x even 12 2 2592.2.r.f 4
240.z odd 4 1 2304.2.a.b 1
240.bb even 4 1 2304.2.a.e 1
240.bd odd 4 1 2304.2.a.l 1
240.bf even 4 1 2304.2.a.o 1
280.s even 4 1 1176.2.c.a 2
280.y odd 4 1 4704.2.c.a 2
360.bo even 12 2 2592.2.r.f 4
360.br even 12 2 648.2.n.c 4
360.bt odd 12 2 2592.2.r.g 4
360.bu odd 12 2 648.2.n.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 5.c odd 4 1
24.2.d.a 2 40.i odd 4 1
72.2.d.b 2 15.e even 4 1
72.2.d.b 2 120.w even 4 1
96.2.d.a 2 20.e even 4 1
96.2.d.a 2 40.k even 4 1
288.2.d.b 2 60.l odd 4 1
288.2.d.b 2 120.q odd 4 1
600.2.d.b 2 5.b even 2 1
600.2.d.b 2 8.b even 2 1
600.2.d.c 2 1.a even 1 1 trivial
600.2.d.c 2 40.f even 2 1 inner
600.2.k.b 2 5.c odd 4 1
600.2.k.b 2 40.i odd 4 1
648.2.n.c 4 45.l even 12 2
648.2.n.c 4 360.br even 12 2
648.2.n.k 4 45.k odd 12 2
648.2.n.k 4 360.bu odd 12 2
768.2.a.a 1 80.t odd 4 1
768.2.a.d 1 80.s even 4 1
768.2.a.e 1 80.j even 4 1
768.2.a.h 1 80.i odd 4 1
1176.2.c.a 2 35.f even 4 1
1176.2.c.a 2 280.s even 4 1
1800.2.d.b 2 3.b odd 2 1
1800.2.d.b 2 120.i odd 2 1
1800.2.d.i 2 15.d odd 2 1
1800.2.d.i 2 24.h odd 2 1
1800.2.k.a 2 15.e even 4 1
1800.2.k.a 2 120.w even 4 1
2304.2.a.b 1 240.z odd 4 1
2304.2.a.e 1 240.bb even 4 1
2304.2.a.l 1 240.bd odd 4 1
2304.2.a.o 1 240.bf even 4 1
2400.2.d.b 2 8.d odd 2 1
2400.2.d.b 2 20.d odd 2 1
2400.2.d.c 2 4.b odd 2 1
2400.2.d.c 2 40.e odd 2 1
2400.2.k.a 2 20.e even 4 1
2400.2.k.a 2 40.k even 4 1
2592.2.r.f 4 180.x even 12 2
2592.2.r.f 4 360.bo even 12 2
2592.2.r.g 4 180.v odd 12 2
2592.2.r.g 4 360.bt odd 12 2
4704.2.c.a 2 140.j odd 4 1
4704.2.c.a 2 280.y odd 4 1
7200.2.d.d 2 12.b even 2 1
7200.2.d.d 2 120.m even 2 1
7200.2.d.g 2 24.f even 2 1
7200.2.d.g 2 60.h even 2 1
7200.2.k.d 2 60.l odd 4 1
7200.2.k.d 2 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11} \)
\( T_{13} + 4 \)
\( T_{37} + 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ 1
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( 1 - 30 T^{2} + 529 T^{4} \)
$29$ \( 1 - 22 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 8 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 50 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 102 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( ( 1 - 12 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 + 10 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 16 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
show more
show less