Properties

Label 4008.2.a.i
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 16 x^{7} + 45 x^{6} + 67 x^{5} - 166 x^{4} - 83 x^{3} + 152 x^{2} + 51 x - 10\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -2 + \beta_{5} + \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -2 + \beta_{5} + \beta_{7} ) q^{7} + q^{9} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{17} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{19} + ( -2 + \beta_{5} + \beta_{7} ) q^{21} + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{25} + q^{27} + ( -5 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{29} + ( -3 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{31} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{35} + ( 1 - \beta_{2} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{37} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{39} + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{41} + ( -4 - 2 \beta_{3} + \beta_{7} ) q^{43} + ( -1 + \beta_{1} ) q^{45} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{49} + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{51} + ( -3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{53} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{55} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{57} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{8} ) q^{59} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{61} + ( -2 + \beta_{5} + \beta_{7} ) q^{63} + ( 2 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{65} + ( 1 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{67} + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{69} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{8} ) q^{71} + ( 2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{73} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{75} + ( -3 + \beta_{1} - \beta_{2} - \beta_{7} + 2 \beta_{8} ) q^{77} + ( -4 + 2 \beta_{2} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{79} + q^{81} + ( -2 - 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{83} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{85} + ( -5 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{87} + ( -6 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} ) q^{89} + ( -3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{91} + ( -3 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{93} + ( -5 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} ) q^{95} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{97} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 9q^{3} - 6q^{5} - 11q^{7} + 9q^{9} + O(q^{10}) \) \( 9q + 9q^{3} - 6q^{5} - 11q^{7} + 9q^{9} + q^{11} - 4q^{13} - 6q^{15} - 9q^{17} - 8q^{19} - 11q^{21} - 7q^{23} - q^{25} + 9q^{27} - 9q^{29} - 25q^{31} + q^{33} + 5q^{35} - 6q^{37} - 4q^{39} - 4q^{41} - 24q^{43} - 6q^{45} - 16q^{47} + 4q^{49} - 9q^{51} - 26q^{53} - 29q^{55} - 8q^{57} - 4q^{59} - 20q^{61} - 11q^{63} - 8q^{65} - 25q^{67} - 7q^{69} - 15q^{71} - 10q^{73} - q^{75} - 20q^{77} - 34q^{79} + 9q^{81} - 4q^{83} - 13q^{85} - 9q^{87} + 13q^{89} - 21q^{91} - 25q^{93} - 7q^{95} - 4q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 16 x^{7} + 45 x^{6} + 67 x^{5} - 166 x^{4} - 83 x^{3} + 152 x^{2} + 51 x - 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -253 \nu^{8} + 10903 \nu^{7} - 17171 \nu^{6} - 175482 \nu^{5} + 247955 \nu^{4} + 716568 \nu^{3} - 563660 \nu^{2} - 635601 \nu + 68945 \)\()/102665\)
\(\beta_{3}\)\(=\)\((\)\( -428 \nu^{8} + 9923 \nu^{7} - 2266 \nu^{6} - 178372 \nu^{5} + 67645 \nu^{4} + 878658 \nu^{3} - 38080 \nu^{2} - 988001 \nu - 301330 \)\()/102665\)
\(\beta_{4}\)\(=\)\((\)\( -3031 \nu^{8} + 7666 \nu^{7} + 48718 \nu^{6} - 99334 \nu^{5} - 199070 \nu^{4} + 207921 \nu^{3} + 208150 \nu^{2} + 200063 \nu - 130065 \)\()/102665\)
\(\beta_{5}\)\(=\)\((\)\( 3811 \nu^{8} - 23831 \nu^{7} - 35953 \nu^{6} + 387944 \nu^{5} - 79645 \nu^{4} - 1646101 \nu^{3} + 520415 \nu^{2} + 1605297 \nu + 255125 \)\()/102665\)
\(\beta_{6}\)\(=\)\((\)\( 5684 \nu^{8} - 5129 \nu^{7} - 118627 \nu^{6} + 65121 \nu^{5} + 751965 \nu^{4} - 184819 \nu^{3} - 1469865 \nu^{2} + 296533 \nu + 445920 \)\()/102665\)
\(\beta_{7}\)\(=\)\((\)\( -6589 \nu^{8} + 20594 \nu^{7} + 101842 \nu^{6} - 311796 \nu^{5} - 367380 \nu^{4} + 1137454 \nu^{3} + 148730 \nu^{2} - 769633 \nu + 59190 \)\()/102665\)
\(\beta_{8}\)\(=\)\((\)\( 16128 \nu^{8} - 24668 \nu^{7} - 293609 \nu^{6} + 290982 \nu^{5} + 1500975 \nu^{4} - 446023 \nu^{3} - 2000020 \nu^{2} - 396149 \nu + 286160 \)\()/102665\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 8 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(-3 \beta_{8} - 12 \beta_{7} + 6 \beta_{6} - 12 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} - 17 \beta_{2} - 2 \beta_{1} + 54\)
\(\nu^{5}\)\(=\)\(-30 \beta_{8} - 45 \beta_{7} + 33 \beta_{6} - 27 \beta_{5} - 36 \beta_{4} + 34 \beta_{3} - 32 \beta_{2} + 73 \beta_{1} + 109\)
\(\nu^{6}\)\(=\)\(-56 \beta_{8} - 145 \beta_{7} + 106 \beta_{6} - 132 \beta_{5} + 54 \beta_{4} + 105 \beta_{3} - 225 \beta_{2} - 33 \beta_{1} + 635\)
\(\nu^{7}\)\(=\)\(-391 \beta_{8} - 576 \beta_{7} + 466 \beta_{6} - 324 \beta_{5} - 395 \beta_{4} + 506 \beta_{3} - 459 \beta_{2} + 691 \beta_{1} + 1496\)
\(\nu^{8}\)\(=\)\(-846 \beta_{8} - 1799 \beta_{7} + 1544 \beta_{6} - 1474 \beta_{5} + 418 \beta_{4} + 1662 \beta_{3} - 2818 \beta_{2} - 429 \beta_{1} + 7652\)

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
−3.17025
−1.99382
−0.985991
−0.482381
0.141931
1.41388
1.58856
2.94484
3.54323
0 1.00000 0 −4.17025 0 −1.78655 0 1.00000 0
1.2 0 1.00000 0 −2.99382 0 −4.65862 0 1.00000 0
1.3 0 1.00000 0 −1.98599 0 −0.0909950 0 1.00000 0
1.4 0 1.00000 0 −1.48238 0 −1.03908 0 1.00000 0
1.5 0 1.00000 0 −0.858069 0 2.33225 0 1.00000 0
1.6 0 1.00000 0 0.413878 0 2.72537 0 1.00000 0
1.7 0 1.00000 0 0.588564 0 −1.23518 0 1.00000 0
1.8 0 1.00000 0 1.94484 0 −3.19705 0 1.00000 0
1.9 0 1.00000 0 2.54323 0 −4.05015 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

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

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.i 9
4.b odd 2 1 8016.2.a.ba 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.i 9 1.a even 1 1 trivial
8016.2.a.ba 9 4.b odd 2 1

Hecke kernels

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

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)