Properties

Label 4008.2.a.k
Level 4008
Weight 2
Character orbit 4008.a
Self dual yes
Analytic conductor 32.004
Analytic rank 0
Dimension 11
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + \beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + \beta_{3} q^{7} + q^{9} -\beta_{10} q^{11} + ( 1 - \beta_{4} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} ) q^{17} + ( -1 + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{19} -\beta_{3} q^{21} + ( -1 + \beta_{6} + \beta_{8} - \beta_{9} ) q^{23} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( 2 - \beta_{7} ) q^{29} + ( -\beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{31} + \beta_{10} q^{33} + ( 2 + 3 \beta_{3} - 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{35} + ( 1 - 2 \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{37} + ( -1 + \beta_{4} ) q^{39} + ( 2 + \beta_{3} - \beta_{5} + \beta_{9} ) q^{41} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{8} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -1 + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{47} + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{8} - \beta_{9} ) q^{49} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{51} + ( 4 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{53} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{55} + ( 1 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{57} + ( \beta_{3} - \beta_{4} - \beta_{6} + \beta_{10} ) q^{59} + ( 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{10} ) q^{61} + \beta_{3} q^{63} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{65} + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{67} + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} ) q^{69} + ( -2 - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{71} + ( 2 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{73} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{75} + ( 4 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{77} + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{79} + q^{81} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{7} + \beta_{8} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{85} + ( -2 + \beta_{7} ) q^{87} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{89} + ( -2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{91} + ( \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{93} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{10} ) q^{95} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{6} - \beta_{10} ) q^{97} -\beta_{10} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 11q^{3} + 10q^{5} - q^{7} + 11q^{9} + O(q^{10}) \) \( 11q - 11q^{3} + 10q^{5} - q^{7} + 11q^{9} - q^{11} + 10q^{13} - 10q^{15} + 17q^{17} + 2q^{19} + q^{21} - 3q^{23} + 21q^{25} - 11q^{27} + 17q^{29} - 15q^{31} + q^{33} + 11q^{35} + 4q^{37} - 10q^{39} + 16q^{41} + 10q^{43} + 10q^{45} - 16q^{47} + 22q^{49} - 17q^{51} + 42q^{53} - 5q^{55} - 2q^{57} - 2q^{59} + 12q^{61} - q^{63} + 10q^{65} - q^{67} + 3q^{69} - 9q^{71} + 24q^{73} - 21q^{75} + 22q^{77} - 30q^{79} + 11q^{81} + 16q^{83} + 25q^{85} - 17q^{87} + 37q^{89} + q^{91} + 15q^{93} + 5q^{95} + 4q^{97} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{3}\)\(=\)\((\)\(2623 \nu^{10} - 70944 \nu^{9} - 193391 \nu^{8} + 2378035 \nu^{7} + 2748114 \nu^{6} - 23458607 \nu^{5} - 15128561 \nu^{4} + 74577279 \nu^{3} + 34805559 \nu^{2} - 38238885 \nu - 7710406\)\()/10323192\)
\(\beta_{4}\)\(=\)\((\)\( -1379 \nu^{10} + 55664 \nu^{9} - 65533 \nu^{8} - 1226719 \nu^{7} + 1417390 \nu^{6} + 9091923 \nu^{5} - 7692731 \nu^{4} - 25269067 \nu^{3} + 11595133 \nu^{2} + 19499297 \nu - 2478234 \)\()/1876944\)
\(\beta_{5}\)\(=\)\((\)\(64985 \nu^{10} - 195192 \nu^{9} - 1989097 \nu^{8} + 5477813 \nu^{7} + 21384366 \nu^{6} - 52505737 \nu^{5} - 92700271 \nu^{4} + 195304833 \nu^{3} + 124569801 \nu^{2} - 220910235 \nu + 34175614\)\()/10323192\)
\(\beta_{6}\)\(=\)\((\)\(101765 \nu^{10} - 492056 \nu^{9} - 2608397 \nu^{8} + 12499993 \nu^{7} + 25094846 \nu^{6} - 110447061 \nu^{5} - 109201699 \nu^{4} + 387400045 \nu^{3} + 176633261 \nu^{2} - 427193567 \nu + 39160638\)\()/10323192\)
\(\beta_{7}\)\(=\)\((\)\(116141 \nu^{10} - 242652 \nu^{9} - 2909401 \nu^{8} + 5468837 \nu^{7} + 25357890 \nu^{6} - 43356601 \nu^{5} - 87108655 \nu^{4} + 142378425 \nu^{3} + 86939145 \nu^{2} - 163661007 \nu + 20257270\)\()/5161596\)
\(\beta_{8}\)\(=\)\((\)\(-522481 \nu^{10} + 1421984 \nu^{9} + 13457169 \nu^{8} - 32908293 \nu^{7} - 124595270 \nu^{6} + 263695489 \nu^{5} + 481600775 \nu^{4} - 850380841 \nu^{3} - 618583841 \nu^{2} + 928868747 \nu - 65756350\)\()/20646384\)
\(\beta_{9}\)\(=\)\((\)\(-596041 \nu^{10} + 2015712 \nu^{9} + 14695769 \nu^{8} - 46952653 \nu^{7} - 132016230 \nu^{6} + 379578137 \nu^{5} + 514603631 \nu^{4} - 1213924881 \nu^{3} - 743357145 \nu^{2} + 1176264339 \nu - 13787246\)\()/20646384\)
\(\beta_{10}\)\(=\)\((\)\(303385 \nu^{10} - 1032256 \nu^{9} - 7769641 \nu^{8} + 25611773 \nu^{7} + 71466478 \nu^{6} - 223268481 \nu^{5} - 278511527 \nu^{4} + 789881825 \nu^{3} + 385046857 \nu^{2} - 908813755 \nu + 47745918\)\()/10323192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + 8 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-2 \beta_{10} + \beta_{9} + 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + 11 \beta_{2} + 4 \beta_{1} + 48\)
\(\nu^{5}\)\(=\)\(-4 \beta_{10} + 14 \beta_{9} - 14 \beta_{8} + 3 \beta_{7} + 19 \beta_{6} - 14 \beta_{5} + 3 \beta_{3} + 17 \beta_{2} + 73 \beta_{1} + 50\)
\(\nu^{6}\)\(=\)\(-39 \beta_{10} + 17 \beta_{9} - 2 \beta_{8} + 35 \beta_{7} + 68 \beta_{6} + 20 \beta_{5} - 9 \beta_{4} - 16 \beta_{3} + 122 \beta_{2} + 74 \beta_{1} + 432\)
\(\nu^{7}\)\(=\)\(-89 \beta_{10} + 167 \beta_{9} - 151 \beta_{8} + 77 \beta_{7} + 294 \beta_{6} - 163 \beta_{5} + 4 \beta_{4} + 54 \beta_{3} + 241 \beta_{2} + 722 \beta_{1} + 656\)
\(\nu^{8}\)\(=\)\(-576 \beta_{10} + 244 \beta_{9} - 31 \beta_{8} + 516 \beta_{7} + 1101 \beta_{6} + 116 \beta_{5} - 55 \beta_{4} - 186 \beta_{3} + 1378 \beta_{2} + 1072 \beta_{1} + 4199\)
\(\nu^{9}\)\(=\)\(-1462 \beta_{10} + 1922 \beta_{9} - 1489 \beta_{8} + 1378 \beta_{7} + 4200 \beta_{6} - 1860 \beta_{5} + 147 \beta_{4} + 704 \beta_{3} + 3209 \beta_{2} + 7555 \beta_{1} + 8094\)
\(\nu^{10}\)\(=\)\(-7771 \beta_{10} + 3303 \beta_{9} - 341 \beta_{8} + 7202 \beta_{7} + 15782 \beta_{6} - 172 \beta_{5} - 44 \beta_{4} - 1868 \beta_{3} + 15861 \beta_{2} + 14338 \beta_{1} + 43210\)

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
3.54432
3.11942
2.80306
2.38119
1.09328
0.0427374
−1.60189
−2.17907
−2.34947
−2.83671
−3.01685
0 −1.00000 0 −2.54432 0 0.525270 0 1.00000 0
1.2 0 −1.00000 0 −2.11942 0 −0.802640 0 1.00000 0
1.3 0 −1.00000 0 −1.80306 0 −4.10861 0 1.00000 0
1.4 0 −1.00000 0 −1.38119 0 −0.260099 0 1.00000 0
1.5 0 −1.00000 0 −0.0932775 0 3.86231 0 1.00000 0
1.6 0 −1.00000 0 0.957263 0 −0.898491 0 1.00000 0
1.7 0 −1.00000 0 2.60189 0 −3.58131 0 1.00000 0
1.8 0 −1.00000 0 3.17907 0 0.651548 0 1.00000 0
1.9 0 −1.00000 0 3.34947 0 3.54581 0 1.00000 0
1.10 0 −1.00000 0 3.83671 0 4.48179 0 1.00000 0
1.11 0 −1.00000 0 4.01685 0 −4.41557 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.k 11
4.b odd 2 1 8016.2.a.be 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.k 11 1.a even 1 1 trivial
8016.2.a.be 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{11} - \cdots\)
\(T_{7}^{11} + \cdots\)