Properties

Label 4008.2.a.j
Level 4008
Weight 2
Character orbit 4008.a
Self dual yes
Analytic conductor 32.004
Analytic rank 1
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} + q^{9} + ( -\beta_{4} - \beta_{7} ) q^{11} + ( -1 + \beta_{1} + \beta_{8} ) q^{13} + ( 1 + \beta_{1} ) q^{15} + ( \beta_{3} - \beta_{8} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{19} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{21} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{23} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{25} - q^{27} + ( -1 - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{29} + ( 2 - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{31} + ( \beta_{4} + \beta_{7} ) q^{33} + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{35} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{7} ) q^{37} + ( 1 - \beta_{1} - \beta_{8} ) q^{39} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{41} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - \beta_{9} ) q^{43} + ( -1 - \beta_{1} ) q^{45} + ( 1 - \beta_{2} - \beta_{4} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} ) q^{49} + ( -\beta_{3} + \beta_{8} ) q^{51} + ( -2 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{53} + ( 3 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{57} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{59} + ( 1 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{61} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{63} + ( -1 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{65} + ( -2 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{67} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{69} + ( 3 + \beta_{1} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{71} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{75} + ( -4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{77} + ( 3 - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{79} + q^{81} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{85} + ( 1 + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{87} + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} ) q^{89} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{91} + ( -2 + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{93} + ( \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{95} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} + ( -\beta_{4} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 10q^{3} - 10q^{5} + q^{7} + 10q^{9} + O(q^{10}) \) \( 10q - 10q^{3} - 10q^{5} + q^{7} + 10q^{9} - q^{11} - 6q^{13} + 10q^{15} - 9q^{17} + 2q^{19} - q^{21} + 7q^{23} + 12q^{25} - 10q^{27} - 13q^{29} + 23q^{31} + q^{33} + q^{35} - 6q^{37} + 6q^{39} - 12q^{41} - 10q^{45} + 10q^{47} + 7q^{49} + 9q^{51} - 26q^{53} + 11q^{55} - 2q^{57} - 10q^{59} - 10q^{61} + q^{63} - 22q^{65} - 5q^{67} - 7q^{69} + 25q^{71} - 8q^{73} - 12q^{75} - 46q^{77} + 26q^{79} + 10q^{81} - 14q^{83} + 9q^{85} + 13q^{87} - 31q^{89} - 3q^{91} - 23q^{93} - 5q^{95} - 32q^{97} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 26 x^{8} - 3 x^{7} + 220 x^{6} + 42 x^{5} - 675 x^{4} - 67 x^{3} + 628 x^{2} - 48 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 31274 \nu^{9} + 188403 \nu^{8} - 799085 \nu^{7} - 4445172 \nu^{6} + 5651462 \nu^{5} + 31352276 \nu^{4} - 4209233 \nu^{3} - 62769314 \nu^{2} - 26831938 \nu + 17959625 \)\()/6386707\)
\(\beta_{3}\)\(=\)\((\)\( 45076 \nu^{9} - 26608 \nu^{8} - 1096194 \nu^{7} + 535650 \nu^{6} + 9006576 \nu^{5} - 3054363 \nu^{4} - 31921253 \nu^{3} + 4461694 \nu^{2} + 45128857 \nu + 3445799 \)\()/6386707\)
\(\beta_{4}\)\(=\)\((\)\(-140074 \nu^{9} - 197290 \nu^{8} + 4018521 \nu^{7} + 4959639 \nu^{6} - 37660725 \nu^{5} - 36535738 \nu^{4} + 125932814 \nu^{3} + 66704449 \nu^{2} - 128029606 \nu - 3074005\)\()/6386707\)
\(\beta_{5}\)\(=\)\((\)\(-252127 \nu^{9} - 49534 \nu^{8} + 6474592 \nu^{7} + 2278935 \nu^{6} - 53041455 \nu^{5} - 25138542 \nu^{4} + 150476284 \nu^{3} + 61255546 \nu^{2} - 122806757 \nu - 20051371\)\()/6386707\)
\(\beta_{6}\)\(=\)\((\)\( -400908 \nu^{9} + 156741 \nu^{8} + 10079448 \nu^{7} - 2045962 \nu^{6} - 81378952 \nu^{5} + 2450805 \nu^{4} + 233226318 \nu^{3} - 5489909 \nu^{2} - 195374082 \nu + 15210925 \)\()/6386707\)
\(\beta_{7}\)\(=\)\((\)\( -446372 \nu^{9} + 13553 \nu^{8} + 11479221 \nu^{7} + 1022842 \nu^{6} - 95483893 \nu^{5} - 17429855 \nu^{4} + 282874842 \nu^{3} + 36533942 \nu^{2} - 233347950 \nu + 3313559 \)\()/6386707\)
\(\beta_{8}\)\(=\)\((\)\( -581351 \nu^{9} + 4817 \nu^{8} + 14786968 \nu^{7} + 1678767 \nu^{6} - 120466276 \nu^{5} - 24247147 \nu^{4} + 344299563 \nu^{3} + 37019495 \nu^{2} - 291434843 \nu + 24062029 \)\()/6386707\)
\(\beta_{9}\)\(=\)\((\)\(676598 \nu^{9} + 130521 \nu^{8} - 17690130 \nu^{7} - 4911181 \nu^{6} + 151157770 \nu^{5} + 47856867 \nu^{4} - 472650162 \nu^{3} - 87928296 \nu^{2} + 458021977 \nu - 18894784\)\()/6386707\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{4} - 2 \beta_{3} - \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(12 \beta_{9} + \beta_{8} + 9 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 14 \beta_{4} - 24 \beta_{3} - \beta_{2} - 6 \beta_{1} + 33\)
\(\nu^{5}\)\(=\)\(4 \beta_{9} + 18 \beta_{8} - 11 \beta_{7} - 18 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} - 13 \beta_{3} + 17 \beta_{2} + 93 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(132 \beta_{9} + 6 \beta_{8} + 81 \beta_{7} + 59 \beta_{6} - 40 \beta_{5} + 167 \beta_{4} - 268 \beta_{3} - 20 \beta_{2} - 40 \beta_{1} + 341\)
\(\nu^{7}\)\(=\)\(81 \beta_{9} + 246 \beta_{8} - 103 \beta_{7} - 253 \beta_{6} + 212 \beta_{5} + 51 \beta_{4} - 139 \beta_{3} + 245 \beta_{2} + 1024 \beta_{1} + 45\)
\(\nu^{8}\)\(=\)\(1477 \beta_{9} - 11 \beta_{8} + 784 \beta_{7} + 876 \beta_{6} - 447 \beta_{5} + 1957 \beta_{4} - 2992 \beta_{3} - 265 \beta_{2} - 319 \beta_{1} + 3816\)
\(\nu^{9}\)\(=\)\(1188 \beta_{9} + 3084 \beta_{8} - 1004 \beta_{7} - 3245 \beta_{6} + 2675 \beta_{5} + 861 \beta_{4} - 1359 \beta_{3} + 3283 \beta_{2} + 11672 \beta_{1} + 459\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.44741
3.20371
1.76045
1.15141
0.108417
−0.0299246
−1.39234
−2.25873
−2.53098
−3.45943
0 −1.00000 0 −4.44741 0 2.42708 0 1.00000 0
1.2 0 −1.00000 0 −4.20371 0 −3.93441 0 1.00000 0
1.3 0 −1.00000 0 −2.76045 0 3.28542 0 1.00000 0
1.4 0 −1.00000 0 −2.15141 0 −2.22742 0 1.00000 0
1.5 0 −1.00000 0 −1.10842 0 −3.58606 0 1.00000 0
1.6 0 −1.00000 0 −0.970075 0 4.67245 0 1.00000 0
1.7 0 −1.00000 0 0.392343 0 0.0476950 0 1.00000 0
1.8 0 −1.00000 0 1.25873 0 −0.924269 0 1.00000 0
1.9 0 −1.00000 0 1.53098 0 1.95354 0 1.00000 0
1.10 0 −1.00000 0 2.45943 0 −0.714036 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4008.2.a.j 10
4.b odd 2 1 8016.2.a.bd 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.j 10 1.a even 1 1 trivial
8016.2.a.bd 10 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)