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

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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 \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.180227832610816.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - \nu^{9} + 2 \nu^{7} + 4 \nu^{5} + 8 \nu^{3} \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{10} + 3 \nu^{8} - 2 \nu^{6} + 12 \nu^{4} - 8 \nu^{2} + 32 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{10} - \nu^{8} - 6 \nu^{6} - 4 \nu^{4} + 8 \nu^{2} + 32 \)\()/32\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + \nu^{9} - 2 \nu^{7} + 12 \nu^{5} + 8 \nu^{3} \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{11} + 3 \nu^{9} - 2 \nu^{7} - 4 \nu^{5} + 8 \nu^{3} + 32 \nu \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{10} - 3 \nu^{8} + 2 \nu^{6} + 20 \nu^{4} + 8 \nu^{2} - 32 \)\()/32\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{11} + \nu^{9} + 6 \nu^{7} + 4 \nu^{5} - 8 \nu^{3} - 32 \nu \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{11} - 3 \nu^{9} + 2 \nu^{7} + 4 \nu^{5} + 24 \nu^{3} - 32 \nu \)\()/32\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{10} + \nu^{8} + 2 \nu^{6} - 12 \nu^{4} + 8 \nu^{2} + 32 \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\)
\(\nu^{4}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{4}\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\)
\(\nu^{6}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\)
\(\nu^{7}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{8}\)\(=\)\(-\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\)
\(\nu^{9}\)\(=\)\(-\)\(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{10}\)\(=\)\(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(12\)
\(\nu^{11}\)\(=\)\(-\)\(7\) \(\beta_{10}\mathstrut -\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(36\) \(\beta_{3}\mathstrut -\mathstrut \) \(12\) \(\beta_{1}\)

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes
8.b Even 1 yes
40.f Even 1 yes

Hecke kernels

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