Properties

Label 600.2.k.f
Level $600$
Weight $2$
Character orbit 600.k
Analytic conductor $4.791$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(12\)
Coefficient field: 12.0.180227832610816.1
Defining polynomial: \( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} - \beta_{5} q^{6} + (\beta_{10} + \beta_1) q^{7} + (\beta_{10} + \beta_{7}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} - \beta_{5} q^{6} + (\beta_{10} + \beta_1) q^{7} + (\beta_{10} + \beta_{7}) q^{8} - q^{9} + ( - \beta_{8} + \beta_{5} + \beta_{2}) q^{11} + \beta_{9} q^{12} + ( - \beta_{9} - \beta_{6} - 2 \beta_{3} + \beta_1) q^{13} + ( - \beta_{11} + \beta_{4} + \beta_{2} + 2) q^{14} + (\beta_{8} + \beta_{4}) q^{16} + (\beta_{9} - \beta_{6} - \beta_1) q^{17} - \beta_1 q^{18} + ( - \beta_{11} - \beta_{8} + 2 \beta_{4}) q^{19} + ( - \beta_{5} + \beta_{4}) q^{21} + (\beta_{10} - \beta_{9} + 2 \beta_{7} - \beta_{6} - 2 \beta_{3}) q^{22} + (2 \beta_{9} - \beta_{7} - \beta_{6}) q^{23} + ( - \beta_{8} + \beta_{4}) q^{24} + ( - \beta_{11} + 2 \beta_{5} - \beta_{4} + \beta_{2} - 2) q^{26} - \beta_{3} q^{27} + (2 \beta_{6} + 2 \beta_1) q^{28} + (\beta_{11} - \beta_{8} - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{29} + ( - \beta_{11} - \beta_{8} - 2 \beta_{5} - 2) q^{31} + ( - \beta_{10} - \beta_{7} + 2 \beta_{6} + 4 \beta_{3}) q^{32} + (\beta_{9} - \beta_{7} + \beta_1) q^{33} + ( - \beta_{11} - 2 \beta_{8} + \beta_{4} - \beta_{2} - 2) q^{34} - \beta_{2} q^{36} + ( - \beta_{9} - \beta_{6} + 2 \beta_{3} + \beta_1) q^{37} + ( - 2 \beta_{10} + 2 \beta_{6}) q^{38} + ( - \beta_{11} - \beta_{5} + \beta_{2} + 2) q^{39} + (2 \beta_{11} + 2 \beta_{8} + 4 \beta_{5} - 2) q^{41} + ( - \beta_{10} + \beta_{9} + \beta_{6} + 2 \beta_{3}) q^{42} + ( - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_1) q^{43} + (2 \beta_{8} + 2 \beta_{5} - 4) q^{44} + ( - 2 \beta_{11} - 4 \beta_{8} + 2 \beta_{4}) q^{46} + ( - 2 \beta_{10} - \beta_{7} + \beta_{6}) q^{47} + ( - \beta_{10} + \beta_{7}) q^{48} + (2 \beta_{8} + 2 \beta_{5} + 2 \beta_{2} + 1) q^{49} + ( - \beta_{11} + \beta_{5} - \beta_{2}) q^{51} + (2 \beta_{10} - 2 \beta_{9} - 4 \beta_{3} - 2 \beta_1) q^{52} + ( - \beta_{9} - \beta_{7} + 4 \beta_{3} - \beta_1) q^{53} + \beta_{5} q^{54} + (2 \beta_{11} + 2 \beta_{8} + 2 \beta_{2} + 4) q^{56} + ( - 2 \beta_{10} - \beta_{7} + \beta_{6}) q^{57} + (\beta_{10} + \beta_{9} + 4 \beta_{7} - \beta_{6} + 2 \beta_{3}) q^{58} + ( - \beta_{8} + 3 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{59} + (2 \beta_{11} + 2 \beta_{8} - 4 \beta_{5}) q^{61} + (2 \beta_{9} - 4 \beta_{3} - 2 \beta_1) q^{62} + ( - \beta_{10} - \beta_1) q^{63} + (2 \beta_{11} + \beta_{8} - 4 \beta_{5} - \beta_{4} + 4) q^{64} + ( - \beta_{11} - 2 \beta_{8} + \beta_{4} + \beta_{2} + 2) q^{66} + ( - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_1) q^{67} + ( - 2 \beta_{10} - 4 \beta_{3} - 2 \beta_1) q^{68} + ( - \beta_{11} + \beta_{8} - 2 \beta_{2}) q^{69} + ( - 2 \beta_{11} - 2 \beta_{8} - 4 \beta_{5} + 4) q^{71} + ( - \beta_{10} - \beta_{7}) q^{72} + (2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 4 \beta_{6} - 4 \beta_1) q^{73} + ( - \beta_{11} - 2 \beta_{5} - \beta_{4} + \beta_{2} - 2) q^{74} + (4 \beta_{11} + 2 \beta_{8} - 2 \beta_{4}) q^{76} - 4 \beta_{3} q^{77} + (\beta_{10} + \beta_{9} + \beta_{6} - 2 \beta_{3} + 2 \beta_1) q^{78} + ( - \beta_{11} - \beta_{8} - 2 \beta_{5} - 2) q^{79} + q^{81} + ( - 4 \beta_{9} + 8 \beta_{3} - 2 \beta_1) q^{82} + ( - 2 \beta_{9} - 2 \beta_{6} - 4 \beta_{3} + 2 \beta_1) q^{83} + (2 \beta_{11} - 2 \beta_{5}) q^{84} + ( - 2 \beta_{11} - 2 \beta_{4} - 2 \beta_{2} + 4) q^{86} + ( - \beta_{10} + 2 \beta_{9} - \beta_{7} - \beta_{6} - \beta_1) q^{87} + ( - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{3} - 4 \beta_1) q^{88} + (2 \beta_{11} - 2 \beta_{8} - 4 \beta_{2} + 2) q^{89} + ( - 4 \beta_{11} - 2 \beta_{8} + 6 \beta_{5} - 2 \beta_{2}) q^{91} + ( - 2 \beta_{10} + 2 \beta_{7} - 8 \beta_{3}) q^{92} + ( - \beta_{7} + \beta_{6} - 2 \beta_{3} - 2 \beta_1) q^{93} + (2 \beta_{11} - 2 \beta_{4}) q^{94} + (2 \beta_{11} + \beta_{8} - \beta_{4} - 4) q^{96} + (4 \beta_{10} - 2 \beta_{9} + 4 \beta_{7} - 2 \beta_{6} - 2 \beta_1) q^{97} + (2 \beta_{10} - 2 \beta_{9} + 2 \beta_{6} + 4 \beta_{3} + \beta_1) q^{98} + (\beta_{8} - \beta_{5} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4} - 2 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{4} - 2 q^{6} - 12 q^{9} + 20 q^{14} + 2 q^{16} + 2 q^{24} - 28 q^{26} - 32 q^{31} - 24 q^{34} + 2 q^{36} + 16 q^{39} - 8 q^{41} - 44 q^{44} - 4 q^{46} + 12 q^{49} + 2 q^{54} + 52 q^{56} + 46 q^{64} + 20 q^{66} + 32 q^{71} - 36 q^{74} + 12 q^{76} - 32 q^{79} + 12 q^{81} + 4 q^{84} + 40 q^{86} + 40 q^{89} + 4 q^{94} - 42 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 3\nu^{8} - 2\nu^{6} + 12\nu^{4} - 8\nu^{2} + 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - \nu^{8} - 6\nu^{6} - 4\nu^{4} + 8\nu^{2} + 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + \nu^{9} - 2\nu^{7} + 12\nu^{5} + 8\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 3\nu^{9} - 2\nu^{7} - 4\nu^{5} + 8\nu^{3} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 3\nu^{8} + 2\nu^{6} + 20\nu^{4} + 8\nu^{2} - 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + \nu^{9} + 6\nu^{7} + 4\nu^{5} - 8\nu^{3} - 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} - 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{10} + \nu^{8} + 2\nu^{6} - 12\nu^{4} + 8\nu^{2} + 32 ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} - \beta_{7} + 2\beta_{6} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{11} + \beta_{8} - 4\beta_{5} - \beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{10} + 4\beta_{9} + \beta_{7} - 2\beta_{6} + 4\beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2\beta_{11} - 5\beta_{8} - 4\beta_{5} + 5\beta_{4} + 4\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -\beta_{10} + 4\beta_{9} + 7\beta_{7} + 2\beta_{6} - 4\beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10\beta_{11} + 5\beta_{8} + 4\beta_{5} + 3\beta_{4} - 4\beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -7\beta_{10} - 4\beta_{9} + \beta_{7} - 2\beta_{6} + 36\beta_{3} - 12\beta_1 \) 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
−1.37729 0.321037i
−1.37729 + 0.321037i
−0.806504 1.16170i
−0.806504 + 1.16170i
−0.450129 1.34067i
−0.450129 + 1.34067i
0.450129 1.34067i
0.450129 + 1.34067i
0.806504 1.16170i
0.806504 + 1.16170i
1.37729 0.321037i
1.37729 + 0.321037i
−1.37729 0.321037i 1.00000i 1.79387 + 0.884323i 0 −0.321037 + 1.37729i −4.05705 −2.18678 1.79387i −1.00000 0
301.2 −1.37729 + 0.321037i 1.00000i 1.79387 0.884323i 0 −0.321037 1.37729i −4.05705 −2.18678 + 1.79387i −1.00000 0
301.3 −0.806504 1.16170i 1.00000i −0.699104 + 1.87383i 0 1.16170 0.806504i −0.746175 2.74067 0.699104i −1.00000 0
301.4 −0.806504 + 1.16170i 1.00000i −0.699104 1.87383i 0 1.16170 + 0.806504i −0.746175 2.74067 + 0.699104i −1.00000 0
301.5 −0.450129 1.34067i 1.00000i −1.59477 + 1.20695i 0 −1.34067 + 0.450129i 2.64265 2.33596 + 1.59477i −1.00000 0
301.6 −0.450129 + 1.34067i 1.00000i −1.59477 1.20695i 0 −1.34067 0.450129i 2.64265 2.33596 1.59477i −1.00000 0
301.7 0.450129 1.34067i 1.00000i −1.59477 1.20695i 0 −1.34067 0.450129i −2.64265 −2.33596 + 1.59477i −1.00000 0
301.8 0.450129 + 1.34067i 1.00000i −1.59477 + 1.20695i 0 −1.34067 + 0.450129i −2.64265 −2.33596 1.59477i −1.00000 0
301.9 0.806504 1.16170i 1.00000i −0.699104 1.87383i 0 1.16170 + 0.806504i 0.746175 −2.74067 0.699104i −1.00000 0
301.10 0.806504 + 1.16170i 1.00000i −0.699104 + 1.87383i 0 1.16170 0.806504i 0.746175 −2.74067 + 0.699104i −1.00000 0
301.11 1.37729 0.321037i 1.00000i 1.79387 0.884323i 0 −0.321037 1.37729i 4.05705 2.18678 1.79387i −1.00000 0
301.12 1.37729 + 0.321037i 1.00000i 1.79387 + 0.884323i 0 −0.321037 + 1.37729i 4.05705 2.18678 + 1.79387i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
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.k.f 12
3.b odd 2 1 1800.2.k.u 12
4.b odd 2 1 2400.2.k.f 12
5.b even 2 1 inner 600.2.k.f 12
5.c odd 4 1 120.2.d.a 6
5.c odd 4 1 120.2.d.b yes 6
8.b even 2 1 inner 600.2.k.f 12
8.d odd 2 1 2400.2.k.f 12
12.b even 2 1 7200.2.k.u 12
15.d odd 2 1 1800.2.k.u 12
15.e even 4 1 360.2.d.e 6
15.e even 4 1 360.2.d.f 6
20.d odd 2 1 2400.2.k.f 12
20.e even 4 1 480.2.d.a 6
20.e even 4 1 480.2.d.b 6
24.f even 2 1 7200.2.k.u 12
24.h odd 2 1 1800.2.k.u 12
40.e odd 2 1 2400.2.k.f 12
40.f even 2 1 inner 600.2.k.f 12
40.i odd 4 1 120.2.d.a 6
40.i odd 4 1 120.2.d.b yes 6
40.k even 4 1 480.2.d.a 6
40.k even 4 1 480.2.d.b 6
60.h even 2 1 7200.2.k.u 12
60.l odd 4 1 1440.2.d.e 6
60.l odd 4 1 1440.2.d.f 6
80.i odd 4 2 3840.2.f.l 12
80.j even 4 2 3840.2.f.m 12
80.s even 4 2 3840.2.f.m 12
80.t odd 4 2 3840.2.f.l 12
120.i odd 2 1 1800.2.k.u 12
120.m even 2 1 7200.2.k.u 12
120.q odd 4 1 1440.2.d.e 6
120.q odd 4 1 1440.2.d.f 6
120.w even 4 1 360.2.d.e 6
120.w even 4 1 360.2.d.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 5.c odd 4 1
120.2.d.a 6 40.i odd 4 1
120.2.d.b yes 6 5.c odd 4 1
120.2.d.b yes 6 40.i odd 4 1
360.2.d.e 6 15.e even 4 1
360.2.d.e 6 120.w even 4 1
360.2.d.f 6 15.e even 4 1
360.2.d.f 6 120.w even 4 1
480.2.d.a 6 20.e even 4 1
480.2.d.a 6 40.k even 4 1
480.2.d.b 6 20.e even 4 1
480.2.d.b 6 40.k even 4 1
600.2.k.f 12 1.a even 1 1 trivial
600.2.k.f 12 5.b even 2 1 inner
600.2.k.f 12 8.b even 2 1 inner
600.2.k.f 12 40.f even 2 1 inner
1440.2.d.e 6 60.l odd 4 1
1440.2.d.e 6 120.q odd 4 1
1440.2.d.f 6 60.l odd 4 1
1440.2.d.f 6 120.q odd 4 1
1800.2.k.u 12 3.b odd 2 1
1800.2.k.u 12 15.d odd 2 1
1800.2.k.u 12 24.h odd 2 1
1800.2.k.u 12 120.i odd 2 1
2400.2.k.f 12 4.b odd 2 1
2400.2.k.f 12 8.d odd 2 1
2400.2.k.f 12 20.d odd 2 1
2400.2.k.f 12 40.e odd 2 1
3840.2.f.l 12 80.i odd 4 2
3840.2.f.l 12 80.t odd 4 2
3840.2.f.m 12 80.j even 4 2
3840.2.f.m 12 80.s even 4 2
7200.2.k.u 12 12.b even 2 1
7200.2.k.u 12 24.f even 2 1
7200.2.k.u 12 60.h even 2 1
7200.2.k.u 12 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{10} - 8 T^{6} + 16 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 24 T^{4} + 128 T^{2} - 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 32 T^{4} + 96 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 48 T^{4} + 704 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 36 T^{4} + 368 T^{2} - 1024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 60 T^{4} + 512 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 92 T^{4} + 2304 T^{2} + \cdots - 16384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 8 T^{2} - 4 T - 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 64 T^{4} + 128 T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} - 100 T + 56)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 60 T^{4} + 512 T^{2} - 1024)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 80 T^{4} + 1216 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 8 T^{2} - 80 T + 128)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 384 T^{4} + 34560 T^{2} + \cdots - 16384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 8 T^{2} - 4 T - 64)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 10 T^{2} - 164 T + 1384)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} - 336 T^{4} + 28416 T^{2} + \cdots - 262144)^{2} \) Copy content Toggle raw display
show more
show less