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 $2$

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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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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