Properties

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

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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 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.

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

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.a 2
3.b odd 2 1 1800.2.d.h 2
4.b odd 2 1 2400.2.d.d 2
5.b even 2 1 600.2.d.d 2
5.c odd 4 1 120.2.k.a 2
5.c odd 4 1 600.2.k.a 2
8.b even 2 1 600.2.d.d 2
8.d odd 2 1 2400.2.d.a 2
12.b even 2 1 7200.2.d.f 2
15.d odd 2 1 1800.2.d.c 2
15.e even 4 1 360.2.k.b 2
15.e even 4 1 1800.2.k.g 2
20.d odd 2 1 2400.2.d.a 2
20.e even 4 1 480.2.k.a 2
20.e even 4 1 2400.2.k.b 2
24.f even 2 1 7200.2.d.e 2
24.h odd 2 1 1800.2.d.c 2
40.e odd 2 1 2400.2.d.d 2
40.f even 2 1 inner 600.2.d.a 2
40.i odd 4 1 120.2.k.a 2
40.i odd 4 1 600.2.k.a 2
40.k even 4 1 480.2.k.a 2
40.k even 4 1 2400.2.k.b 2
60.h even 2 1 7200.2.d.e 2
60.l odd 4 1 1440.2.k.a 2
60.l odd 4 1 7200.2.k.f 2
80.i odd 4 1 3840.2.a.w 1
80.j even 4 1 3840.2.a.r 1
80.s even 4 1 3840.2.a.m 1
80.t odd 4 1 3840.2.a.d 1
120.i odd 2 1 1800.2.d.h 2
120.m even 2 1 7200.2.d.f 2
120.q odd 4 1 1440.2.k.a 2
120.q odd 4 1 7200.2.k.f 2
120.w even 4 1 360.2.k.b 2
120.w even 4 1 1800.2.k.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.a 2 5.c odd 4 1
120.2.k.a 2 40.i odd 4 1
360.2.k.b 2 15.e even 4 1
360.2.k.b 2 120.w even 4 1
480.2.k.a 2 20.e even 4 1
480.2.k.a 2 40.k even 4 1
600.2.d.a 2 1.a even 1 1 trivial
600.2.d.a 2 40.f even 2 1 inner
600.2.d.d 2 5.b even 2 1
600.2.d.d 2 8.b even 2 1
600.2.k.a 2 5.c odd 4 1
600.2.k.a 2 40.i odd 4 1
1440.2.k.a 2 60.l odd 4 1
1440.2.k.a 2 120.q odd 4 1
1800.2.d.c 2 15.d odd 2 1
1800.2.d.c 2 24.h odd 2 1
1800.2.d.h 2 3.b odd 2 1
1800.2.d.h 2 120.i odd 2 1
1800.2.k.g 2 15.e even 4 1
1800.2.k.g 2 120.w even 4 1
2400.2.d.a 2 8.d odd 2 1
2400.2.d.a 2 20.d odd 2 1
2400.2.d.d 2 4.b odd 2 1
2400.2.d.d 2 40.e odd 2 1
2400.2.k.b 2 20.e even 4 1
2400.2.k.b 2 40.k even 4 1
3840.2.a.d 1 80.t odd 4 1
3840.2.a.m 1 80.s even 4 1
3840.2.a.r 1 80.j even 4 1
3840.2.a.w 1 80.i odd 4 1
7200.2.d.e 2 24.f even 2 1
7200.2.d.e 2 60.h even 2 1
7200.2.d.f 2 12.b even 2 1
7200.2.d.f 2 120.m even 2 1
7200.2.k.f 2 60.l odd 4 1
7200.2.k.f 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 \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11}^{2} + 16 \)
\( T_{13} \)
\( T_{37} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ 1
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( 1 - 6 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( 1 - 30 T^{2} + 529 T^{4} \)
$29$ \( 1 - 22 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 10 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 4 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 78 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 10 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 54 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 58 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 + 12 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 4 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 46 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 14 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 14 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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