Properties

Label 600.2.k.a
Level 600
Weight 2
Character orbit 600.k
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.k (of order \(2\), degree \(1\), 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 120)
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.

\(f(q)\) \(=\) \( q + ( -1 + i ) q^{2} -i q^{3} -2 i q^{4} + ( 1 + i ) q^{6} -2 q^{7} + ( 2 + 2 i ) q^{8} - q^{9} +O(q^{10})\) \( q + ( -1 + i ) q^{2} -i q^{3} -2 i q^{4} + ( 1 + i ) q^{6} -2 q^{7} + ( 2 + 2 i ) q^{8} - q^{9} + 4 i q^{11} -2 q^{12} + ( 2 - 2 i ) q^{14} -4 q^{16} + 6 q^{17} + ( 1 - i ) q^{18} + 4 i q^{19} + 2 i q^{21} + ( -4 - 4 i ) q^{22} + 4 q^{23} + ( 2 - 2 i ) q^{24} + i q^{27} + 4 i q^{28} -6 i q^{29} + 10 q^{31} + ( 4 - 4 i ) q^{32} + 4 q^{33} + ( -6 + 6 i ) q^{34} + 2 i q^{36} -4 i q^{37} + ( -4 - 4 i ) q^{38} + 10 q^{41} + ( -2 - 2 i ) q^{42} + 4 i q^{43} + 8 q^{44} + ( -4 + 4 i ) q^{46} + 4 q^{47} + 4 i q^{48} -3 q^{49} -6 i q^{51} + 10 i q^{53} + ( -1 - i ) q^{54} + ( -4 - 4 i ) q^{56} + 4 q^{57} + ( 6 + 6 i ) q^{58} + 8 i q^{59} -8 i q^{61} + ( -10 + 10 i ) q^{62} + 2 q^{63} + 8 i q^{64} + ( -4 + 4 i ) q^{66} + 12 i q^{67} -12 i q^{68} -4 i q^{69} -4 q^{71} + ( -2 - 2 i ) q^{72} -10 q^{73} + ( 4 + 4 i ) q^{74} + 8 q^{76} -8 i q^{77} -14 q^{79} + q^{81} + ( -10 + 10 i ) q^{82} + 4 q^{84} + ( -4 - 4 i ) q^{86} -6 q^{87} + ( -8 + 8 i ) q^{88} + 14 q^{89} -8 i q^{92} -10 i q^{93} + ( -4 + 4 i ) q^{94} + ( -4 - 4 i ) q^{96} + 10 q^{97} + ( 3 - 3 i ) q^{98} -4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{6} - 4q^{7} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{6} - 4q^{7} + 4q^{8} - 2q^{9} - 4q^{12} + 4q^{14} - 8q^{16} + 12q^{17} + 2q^{18} - 8q^{22} + 8q^{23} + 4q^{24} + 20q^{31} + 8q^{32} + 8q^{33} - 12q^{34} - 8q^{38} + 20q^{41} - 4q^{42} + 16q^{44} - 8q^{46} + 8q^{47} - 6q^{49} - 2q^{54} - 8q^{56} + 8q^{57} + 12q^{58} - 20q^{62} + 4q^{63} - 8q^{66} - 8q^{71} - 4q^{72} - 20q^{73} + 8q^{74} + 16q^{76} - 28q^{79} + 2q^{81} - 20q^{82} + 8q^{84} - 8q^{86} - 12q^{87} - 16q^{88} + 28q^{89} - 8q^{94} - 8q^{96} + 20q^{97} + 6q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000i 2.00000i 0 1.00000 1.00000i −2.00000 2.00000 2.00000i −1.00000 0
301.2 −1.00000 + 1.00000i 1.00000i 2.00000i 0 1.00000 + 1.00000i −2.00000 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
8.b 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.k.a 2
3.b odd 2 1 1800.2.k.g 2
4.b odd 2 1 2400.2.k.b 2
5.b even 2 1 120.2.k.a 2
5.c odd 4 1 600.2.d.a 2
5.c odd 4 1 600.2.d.d 2
8.b even 2 1 inner 600.2.k.a 2
8.d odd 2 1 2400.2.k.b 2
12.b even 2 1 7200.2.k.f 2
15.d odd 2 1 360.2.k.b 2
15.e even 4 1 1800.2.d.c 2
15.e even 4 1 1800.2.d.h 2
20.d odd 2 1 480.2.k.a 2
20.e even 4 1 2400.2.d.a 2
20.e even 4 1 2400.2.d.d 2
24.f even 2 1 7200.2.k.f 2
24.h odd 2 1 1800.2.k.g 2
40.e odd 2 1 480.2.k.a 2
40.f even 2 1 120.2.k.a 2
40.i odd 4 1 600.2.d.a 2
40.i odd 4 1 600.2.d.d 2
40.k even 4 1 2400.2.d.a 2
40.k even 4 1 2400.2.d.d 2
60.h even 2 1 1440.2.k.a 2
60.l odd 4 1 7200.2.d.e 2
60.l odd 4 1 7200.2.d.f 2
80.k odd 4 1 3840.2.a.m 1
80.k odd 4 1 3840.2.a.r 1
80.q even 4 1 3840.2.a.d 1
80.q even 4 1 3840.2.a.w 1
120.i odd 2 1 360.2.k.b 2
120.m even 2 1 1440.2.k.a 2
120.q odd 4 1 7200.2.d.e 2
120.q odd 4 1 7200.2.d.f 2
120.w even 4 1 1800.2.d.c 2
120.w even 4 1 1800.2.d.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.a 2 5.b even 2 1
120.2.k.a 2 40.f even 2 1
360.2.k.b 2 15.d odd 2 1
360.2.k.b 2 120.i odd 2 1
480.2.k.a 2 20.d odd 2 1
480.2.k.a 2 40.e odd 2 1
600.2.d.a 2 5.c odd 4 1
600.2.d.a 2 40.i odd 4 1
600.2.d.d 2 5.c odd 4 1
600.2.d.d 2 40.i odd 4 1
600.2.k.a 2 1.a even 1 1 trivial
600.2.k.a 2 8.b even 2 1 inner
1440.2.k.a 2 60.h even 2 1
1440.2.k.a 2 120.m even 2 1
1800.2.d.c 2 15.e even 4 1
1800.2.d.c 2 120.w even 4 1
1800.2.d.h 2 15.e even 4 1
1800.2.d.h 2 120.w even 4 1
1800.2.k.g 2 3.b odd 2 1
1800.2.k.g 2 24.h odd 2 1
2400.2.d.a 2 20.e even 4 1
2400.2.d.a 2 40.k even 4 1
2400.2.d.d 2 20.e even 4 1
2400.2.d.d 2 40.k even 4 1
2400.2.k.b 2 4.b odd 2 1
2400.2.k.b 2 8.d odd 2 1
3840.2.a.d 1 80.q even 4 1
3840.2.a.m 1 80.k odd 4 1
3840.2.a.r 1 80.k odd 4 1
3840.2.a.w 1 80.q even 4 1
7200.2.d.e 2 60.l odd 4 1
7200.2.d.e 2 120.q odd 4 1
7200.2.d.f 2 60.l odd 4 1
7200.2.d.f 2 120.q odd 4 1
7200.2.k.f 2 12.b even 2 1
7200.2.k.f 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 2 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( ( 1 + 2 T + 7 T^{2} )^{2} \)
$11$ \( 1 - 6 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 22 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 10 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 58 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 10 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 4 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 6 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 54 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 58 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 4 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 10 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 14 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{2} \)
$89$ \( ( 1 - 14 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
show more
show less