Properties

Label 600.2.d.e
Level 600
Weight 2
Character orbit 600.d
Analytic conductor 4.791
Analytic rank 0
Dimension 6
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 \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{3} q^{2} \) \(- q^{3}\) \( -\beta_{2} q^{4} \) \( -\beta_{3} q^{6} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{8} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \( + \beta_{3} q^{2} \) \(- q^{3}\) \( -\beta_{2} q^{4} \) \( -\beta_{3} q^{6} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{8} \) \(+ q^{9}\) \( + ( \beta_{4} - 2 \beta_{5} ) q^{11} \) \( + \beta_{2} q^{12} \) \( + ( \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{13} \) \( + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{14} \) \( + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{16} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{17} \) \( + \beta_{3} q^{18} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{19} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{21} \) \( + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{22} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} \) \( + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{24} \) \( + ( 4 - 2 \beta_{2} ) q^{26} \) \(- q^{27}\) \( + ( -3 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{28} \) \( -\beta_{4} q^{29} \) \( + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{31} \) \( + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{32} \) \( + ( -\beta_{4} + 2 \beta_{5} ) q^{33} \) \( + ( -2 + 2 \beta_{4} + 2 \beta_{5} ) q^{34} \) \( -\beta_{2} q^{36} \) \( + ( \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{37} \) \( + ( -4 - 2 \beta_{2} ) q^{38} \) \( + ( -\beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{39} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{42} \) \( + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{43} \) \( + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - \beta_{5} ) q^{44} \) \( + ( -4 - 2 \beta_{2} ) q^{46} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{47} \) \( + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{48} \) \( + ( -5 + 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} ) q^{49} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{51} \) \( + ( 2 + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{52} \) \( -2 q^{53} \) \( -\beta_{3} q^{54} \) \( + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{5} ) q^{56} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{57} \) \( + ( 1 + \beta_{1} - \beta_{5} ) q^{58} \) \( + ( -\beta_{4} - 2 \beta_{5} ) q^{59} \) \( + ( 2 \beta_{4} - 4 \beta_{5} ) q^{61} \) \( + ( 4 - 2 \beta_{2} - 2 \beta_{3} ) q^{62} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{63} \) \( + ( -4 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{64} \) \( + ( 3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{66} \) \( -4 q^{67} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{68} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{69} \) \( + ( 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{71} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{72} \) \( + 3 \beta_{4} q^{73} \) \( + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{74} \) \( + ( 2 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{76} \) \( + ( 8 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{77} \) \( + ( -4 + 2 \beta_{2} ) q^{78} \) \( + ( -6 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{79} \) \(+ q^{81}\) \( + ( 2 - 2 \beta_{1} - 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{82} \) \( + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} \) \( + ( 3 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{84} \) \( + ( -2 + 2 \beta_{1} + 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{86} \) \( + \beta_{4} q^{87} \) \( + ( 11 + \beta_{1} + 2 \beta_{2} - 3 \beta_{5} ) q^{88} \) \( + ( 6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} \) \( + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{91} \) \( + ( 2 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{92} \) \( + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{93} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{94} \) \( + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{96} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{97} \) \( + ( 8 - 4 \beta_{2} - 5 \beta_{3} ) q^{98} \) \( + ( \beta_{4} - 2 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut -\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 16q^{14} \) \(\mathstrut +\mathstrut 10q^{16} \) \(\mathstrut -\mathstrut 12q^{22} \) \(\mathstrut -\mathstrut 6q^{24} \) \(\mathstrut +\mathstrut 28q^{26} \) \(\mathstrut -\mathstrut 6q^{27} \) \(\mathstrut -\mathstrut 20q^{28} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 10q^{32} \) \(\mathstrut -\mathstrut 12q^{34} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut -\mathstrut 20q^{38} \) \(\mathstrut -\mathstrut 20q^{41} \) \(\mathstrut -\mathstrut 16q^{42} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 4q^{44} \) \(\mathstrut -\mathstrut 20q^{46} \) \(\mathstrut -\mathstrut 10q^{48} \) \(\mathstrut -\mathstrut 30q^{49} \) \(\mathstrut +\mathstrut 12q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut +\mathstrut 28q^{62} \) \(\mathstrut -\mathstrut 22q^{64} \) \(\mathstrut +\mathstrut 12q^{66} \) \(\mathstrut -\mathstrut 24q^{67} \) \(\mathstrut -\mathstrut 8q^{71} \) \(\mathstrut +\mathstrut 6q^{72} \) \(\mathstrut -\mathstrut 12q^{74} \) \(\mathstrut +\mathstrut 12q^{76} \) \(\mathstrut +\mathstrut 32q^{77} \) \(\mathstrut -\mathstrut 28q^{78} \) \(\mathstrut -\mathstrut 36q^{79} \) \(\mathstrut +\mathstrut 6q^{81} \) \(\mathstrut +\mathstrut 16q^{82} \) \(\mathstrut -\mathstrut 32q^{83} \) \(\mathstrut +\mathstrut 20q^{84} \) \(\mathstrut -\mathstrut 16q^{86} \) \(\mathstrut +\mathstrut 60q^{88} \) \(\mathstrut +\mathstrut 28q^{89} \) \(\mathstrut +\mathstrut 12q^{92} \) \(\mathstrut +\mathstrut 12q^{93} \) \(\mathstrut -\mathstrut 4q^{94} \) \(\mathstrut -\mathstrut 10q^{96} \) \(\mathstrut +\mathstrut 56q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut +\mathstrut \) \(3\) \(x^{4}\mathstrut -\mathstrut \) \(6\) \(x^{3}\mathstrut +\mathstrut \) \(6\) \(x^{2}\mathstrut -\mathstrut \) \(8\) \(x\mathstrut +\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + \nu^{3} - 4 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - \nu^{3} + 3 \nu^{2} + 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 3 \nu^{3} + 3 \nu^{2} - 2 \nu + 6 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8 \)\()/2\)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} + 2 \nu^{3} - 3 \nu^{2} + 3 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/2\)

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
−0.671462 1.24464i
−0.671462 + 1.24464i
1.40680 + 0.144584i
1.40680 0.144584i
0.264658 1.38923i
0.264658 + 1.38923i
−1.24464 0.671462i −1.00000 1.09828 + 1.67146i 0 1.24464 + 0.671462i 4.68585i −0.244644 2.81783i 1.00000 0
349.2 −1.24464 + 0.671462i −1.00000 1.09828 1.67146i 0 1.24464 0.671462i 4.68585i −0.244644 + 2.81783i 1.00000 0
349.3 −0.144584 1.40680i −1.00000 −1.95819 + 0.406803i 0 0.144584 + 1.40680i 3.62721i 0.855416 + 2.69597i 1.00000 0
349.4 −0.144584 + 1.40680i −1.00000 −1.95819 0.406803i 0 0.144584 1.40680i 3.62721i 0.855416 2.69597i 1.00000 0
349.5 1.38923 0.264658i −1.00000 1.85991 0.735342i 0 −1.38923 + 0.264658i 0.941367i 2.38923 1.51380i 1.00000 0
349.6 1.38923 + 0.264658i −1.00000 1.85991 + 0.735342i 0 −1.38923 0.264658i 0.941367i 2.38923 + 1.51380i 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
40.f Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\(T_{7}^{6} \) \(\mathstrut +\mathstrut 36 T_{7}^{4} \) \(\mathstrut +\mathstrut 320 T_{7}^{2} \) \(\mathstrut +\mathstrut 256 \)
\(T_{11}^{6} \) \(\mathstrut +\mathstrut 64 T_{11}^{4} \) \(\mathstrut +\mathstrut 1088 T_{11}^{2} \) \(\mathstrut +\mathstrut 4096 \)
\(T_{13}^{3} \) \(\mathstrut -\mathstrut 28 T_{13} \) \(\mathstrut -\mathstrut 16 \)
\(T_{37}^{3} \) \(\mathstrut +\mathstrut 4 T_{37}^{2} \) \(\mathstrut -\mathstrut 60 T_{37} \) \(\mathstrut -\mathstrut 256 \)