Properties

Label 4008.2.a.m
Level 4008
Weight 2
Character orbit 4008.a
Self dual yes
Analytic conductor 32.004
Analytic rank 0
Dimension 13
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} -\beta_{9} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} -\beta_{9} q^{7} + q^{9} + ( 1 + \beta_{5} ) q^{11} + ( 1 + \beta_{3} ) q^{13} + \beta_{1} q^{15} + ( 1 - \beta_{2} ) q^{17} + ( 1 - \beta_{9} + \beta_{10} ) q^{19} -\beta_{9} q^{21} + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{23} + ( 3 - \beta_{5} + \beta_{6} ) q^{25} + q^{27} -\beta_{7} q^{29} + ( -1 + \beta_{4} + \beta_{10} ) q^{31} + ( 1 + \beta_{5} ) q^{33} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{35} + ( 1 - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{37} + ( 1 + \beta_{3} ) q^{39} + ( \beta_{1} - \beta_{4} + \beta_{10} - \beta_{12} ) q^{41} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{43} + \beta_{1} q^{45} + ( -1 - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} ) q^{47} + ( \beta_{1} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{49} + ( 1 - \beta_{2} ) q^{51} + ( -1 + \beta_{1} + \beta_{9} + \beta_{10} ) q^{53} + ( -2 \beta_{4} - \beta_{7} - 2 \beta_{9} ) q^{55} + ( 1 - \beta_{9} + \beta_{10} ) q^{57} + ( 2 - \beta_{1} + \beta_{4} - \beta_{8} + \beta_{9} ) q^{59} + ( 3 + \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{61} -\beta_{9} q^{63} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{65} + ( 2 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{67} + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{69} + ( 1 - \beta_{1} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{71} + ( 4 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{12} ) q^{73} + ( 3 - \beta_{5} + \beta_{6} ) q^{75} + ( 2 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{77} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{79} + q^{81} + ( 4 - 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{83} + ( 3 + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{85} -\beta_{7} q^{87} + ( 3 + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{89} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{91} + ( -1 + \beta_{4} + \beta_{10} ) q^{93} + ( -2 + 2 \beta_{1} - \beta_{4} + \beta_{7} + 2 \beta_{10} ) q^{95} + ( 3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{97} + ( 1 + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{3} + 2q^{5} + q^{7} + 13q^{9} + O(q^{10}) \) \( 13q + 13q^{3} + 2q^{5} + q^{7} + 13q^{9} + 11q^{11} + 12q^{13} + 2q^{15} + 15q^{17} + 14q^{19} + q^{21} + 9q^{23} + 37q^{25} + 13q^{27} - 3q^{29} - 17q^{31} + 11q^{33} + 15q^{35} + 16q^{37} + 12q^{39} + 12q^{41} + 20q^{43} + 2q^{45} - 6q^{47} + 26q^{49} + 15q^{51} - 12q^{53} + 7q^{55} + 14q^{57} + 14q^{59} + 24q^{61} + q^{63} + 8q^{65} + 3q^{67} + 9q^{69} + 17q^{71} + 34q^{73} + 37q^{75} + 30q^{77} + 10q^{79} + 13q^{81} + 44q^{83} + 25q^{85} - 3q^{87} + 25q^{89} + 29q^{91} - 17q^{93} - 15q^{95} + 38q^{97} + 11q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + 26911 x^{5} - 72240 x^{4} - 14532 x^{3} + 112850 x^{2} - 72184 x + 12144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-6207348777 \nu^{12} - 12640530894 \nu^{11} + 576861363065 \nu^{10} + 295838273673 \nu^{9} - 16627326024925 \nu^{8} + 5016057310831 \nu^{7} + 201417980697138 \nu^{6} - 196205928405368 \nu^{5} - 1037037612348531 \nu^{4} + 1724063123712248 \nu^{3} + 1650853947740980 \nu^{2} - 4373968597429142 \nu + 1211669325949820\)\()/ 111451400959900 \)
\(\beta_{3}\)\(=\)\((\)\(126516435801 \nu^{12} - 351643713690 \nu^{11} - 4532098501229 \nu^{10} + 14015645726951 \nu^{9} + 48247617829913 \nu^{8} - 193085396549923 \nu^{7} - 90263735212158 \nu^{6} + 1012871433621172 \nu^{5} - 714691137288825 \nu^{4} - 993417294727940 \nu^{3} + 598916455112988 \nu^{2} - 3707736754746094 \nu + 3605634089719788\)\()/ 557257004799500 \)
\(\beta_{4}\)\(=\)\((\)\(8155241826 \nu^{12} - 24439858116 \nu^{11} - 363301449706 \nu^{10} + 1102737562371 \nu^{9} + 5918945839452 \nu^{8} - 18714943276923 \nu^{7} - 42939671149010 \nu^{6} + 149114772131121 \nu^{5} + 129168526434186 \nu^{4} - 552823809736903 \nu^{3} - 61985566741568 \nu^{2} + 719430531278666 \nu - 261781210799648\)\()/ 27862850239975 \)
\(\beta_{5}\)\(=\)\((\)\(42253210649 \nu^{12} - 42904580362 \nu^{11} - 2192557864275 \nu^{10} + 1857640717639 \nu^{9} + 43420560752365 \nu^{8} - 30214877144177 \nu^{7} - 406570644479846 \nu^{6} + 236986143576626 \nu^{5} + 1772258179746447 \nu^{4} - 967840084997036 \nu^{3} - 2780474603245950 \nu^{2} + 1829362384230014 \nu - 9603799562310\)\()/ 55725700479950 \)
\(\beta_{6}\)\(=\)\((\)\(42253210649 \nu^{12} - 42904580362 \nu^{11} - 2192557864275 \nu^{10} + 1857640717639 \nu^{9} + 43420560752365 \nu^{8} - 30214877144177 \nu^{7} - 406570644479846 \nu^{6} + 236986143576626 \nu^{5} + 1772258179746447 \nu^{4} - 967840084997036 \nu^{3} - 2724748902766000 \nu^{2} + 1829362384230014 \nu - 455409403401910\)\()/ 55725700479950 \)
\(\beta_{7}\)\(=\)\((\)\(138385219703 \nu^{12} - 105435570025 \nu^{11} - 6791649579897 \nu^{10} + 5771355044103 \nu^{9} + 127066606088584 \nu^{8} - 121899916781369 \nu^{7} - 1131795821929059 \nu^{6} + 1217996115089036 \nu^{5} + 4728856566083805 \nu^{4} - 5762733210647085 \nu^{3} - 6707972583182866 \nu^{2} + 10417867358098218 \nu - 2901404480931816\)\()/ 139314251199875 \)
\(\beta_{8}\)\(=\)\((\)\(111023509273 \nu^{12} - 148746957474 \nu^{11} - 5533191476745 \nu^{10} + 7117312024823 \nu^{9} + 104000386651945 \nu^{8} - 129559753690019 \nu^{7} - 912294162144422 \nu^{6} + 1109495790415652 \nu^{5} + 3686070397711199 \nu^{4} - 4408101894281592 \nu^{3} - 5320985228785100 \nu^{2} + 6347787481933558 \nu - 629228580752360\)\()/ 111451400959900 \)
\(\beta_{9}\)\(=\)\((\)\(-323942240303 \nu^{12} + 655210507370 \nu^{11} + 16238231918487 \nu^{10} - 30174775561853 \nu^{9} - 309203328646339 \nu^{8} + 524566352898419 \nu^{7} + 2761108370208174 \nu^{6} - 4297098906328266 \nu^{5} - 11231871461142475 \nu^{4} + 16707098673252820 \nu^{3} + 14675109344372636 \nu^{2} - 25352491816897518 \nu + 6802024064231936\)\()/ 278628502399750 \)
\(\beta_{10}\)\(=\)\((\)\(208422301596 \nu^{12} + 101272023955 \nu^{11} - 9810853937919 \nu^{10} - 3015289235129 \nu^{9} + 173349897433318 \nu^{8} + 10485608743617 \nu^{7} - 1435122721494978 \nu^{6} + 348943217873482 \nu^{5} + 5522030332256130 \nu^{4} - 3276596851987855 \nu^{3} - 7596810398586632 \nu^{2} + 7530986189108326 \nu - 1313465392184157\)\()/ 139314251199875 \)
\(\beta_{11}\)\(=\)\((\)\(889692540773 \nu^{12} - 1710596413190 \nu^{11} - 43690158347557 \nu^{10} + 79678278406623 \nu^{9} + 807810238699029 \nu^{8} - 1397243777337779 \nu^{7} - 6930554296860074 \nu^{6} + 11473935236033836 \nu^{5} + 26745397139049795 \nu^{4} - 44471075457654780 \nu^{3} - 31856351450553896 \nu^{2} + 67210663867287138 \nu - 20950363454455596\)\()/ 557257004799500 \)
\(\beta_{12}\)\(=\)\((\)\(224267911141 \nu^{12} + 63390661414 \nu^{11} - 11028950583521 \nu^{10} - 1496522255749 \nu^{9} + 205661421513157 \nu^{8} - 13254732421323 \nu^{7} - 1812887391077730 \nu^{6} + 558179494985656 \nu^{5} + 7475418034130291 \nu^{4} - 4406202874244528 \nu^{3} - 10980877487070868 \nu^{2} + 10312125439960266 \nu - 2219186123392628\)\()/ 111451400959900 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + 8\)
\(\nu^{3}\)\(=\)\(-\beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 11 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{12} - \beta_{11} + 5 \beta_{10} - \beta_{9} + 2 \beta_{8} + 19 \beta_{6} - 17 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 91\)
\(\nu^{5}\)\(=\)\(-30 \beta_{12} - 18 \beta_{11} + 29 \beta_{10} + 7 \beta_{9} + 20 \beta_{8} + 3 \beta_{7} + 46 \beta_{6} - 16 \beta_{5} + 43 \beta_{4} - 21 \beta_{3} - 7 \beta_{2} + 143 \beta_{1} - 7\)
\(\nu^{6}\)\(=\)\(-137 \beta_{12} - 28 \beta_{11} + 144 \beta_{10} - 42 \beta_{9} + 41 \beta_{8} - 12 \beta_{7} + 335 \beta_{6} - 280 \beta_{5} - 100 \beta_{4} + 40 \beta_{3} - 102 \beta_{2} + 49 \beta_{1} + 1184\)
\(\nu^{7}\)\(=\)\(-665 \beta_{12} - 281 \beta_{11} + 632 \beta_{10} + 213 \beta_{9} + 350 \beta_{8} + 97 \beta_{7} + 905 \beta_{6} - 247 \beta_{5} + 753 \beta_{4} - 381 \beta_{3} - 199 \beta_{2} + 2061 \beta_{1} + 6\)
\(\nu^{8}\)\(=\)\(-2931 \beta_{12} - 676 \beta_{11} + 3142 \beta_{10} - 1097 \beta_{9} + 716 \beta_{8} - 320 \beta_{7} + 5857 \beta_{6} - 4654 \beta_{5} - 1956 \beta_{4} + 668 \beta_{3} - 2055 \beta_{2} + 1017 \beta_{1} + 16711\)
\(\nu^{9}\)\(=\)\(-13266 \beta_{12} - 4390 \beta_{11} + 12482 \beta_{10} + 4644 \beta_{9} + 6123 \beta_{8} + 2314 \beta_{7} + 16957 \beta_{6} - 4186 \beta_{5} + 12395 \beta_{4} - 6687 \beta_{3} - 4266 \beta_{2} + 31753 \beta_{1} + 1836\)
\(\nu^{10}\)\(=\)\(-57319 \beta_{12} - 14661 \beta_{11} + 61925 \beta_{10} - 23592 \beta_{9} + 12118 \beta_{8} - 6102 \beta_{7} + 102280 \beta_{6} - 78127 \beta_{5} - 34966 \beta_{4} + 10790 \beta_{3} - 38346 \beta_{2} + 20530 \beta_{1} + 249558\)
\(\nu^{11}\)\(=\)\(-251329 \beta_{12} - 70391 \beta_{11} + 235453 \beta_{10} + 89347 \beta_{9} + 107880 \beta_{8} + 48175 \beta_{7} + 310864 \beta_{6} - 76539 \beta_{5} + 199760 \beta_{4} - 116067 \beta_{3} - 83231 \beta_{2} + 510061 \beta_{1} + 60252\)
\(\nu^{12}\)\(=\)\(-1073038 \beta_{12} - 294937 \beta_{11} + 1162932 \beta_{10} - 459487 \beta_{9} + 205019 \beta_{8} - 103543 \beta_{7} + 1786032 \beta_{6} - 1320651 \beta_{5} - 597828 \beta_{4} + 173012 \beta_{3} - 691732 \beta_{2} + 410169 \beta_{1} + 3880740\)

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
−4.02595
−3.60065
−3.05600
−2.71099
−2.04519
0.284022
0.649069
1.25667
1.68837
2.79706
3.18209
3.38131
4.20017
0 1.00000 0 −4.02595 0 0.910115 0 1.00000 0
1.2 0 1.00000 0 −3.60065 0 −4.85776 0 1.00000 0
1.3 0 1.00000 0 −3.05600 0 2.59989 0 1.00000 0
1.4 0 1.00000 0 −2.71099 0 −0.775439 0 1.00000 0
1.5 0 1.00000 0 −2.04519 0 1.81275 0 1.00000 0
1.6 0 1.00000 0 0.284022 0 −3.90646 0 1.00000 0
1.7 0 1.00000 0 0.649069 0 4.19646 0 1.00000 0
1.8 0 1.00000 0 1.25667 0 −4.55521 0 1.00000 0
1.9 0 1.00000 0 1.68837 0 0.722089 0 1.00000 0
1.10 0 1.00000 0 2.79706 0 4.56518 0 1.00000 0
1.11 0 1.00000 0 3.18209 0 −1.39408 0 1.00000 0
1.12 0 1.00000 0 3.38131 0 2.18014 0 1.00000 0
1.13 0 1.00000 0 4.20017 0 −0.497670 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.m 13
4.b odd 2 1 8016.2.a.bg 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.m 13 1.a even 1 1 trivial
8016.2.a.bg 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{13} - \cdots\)
\(T_{7}^{13} - \cdots\)