Properties

Label 4008.2.a.l
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + 1411 x^{4} - 11022 x^{3} - 2450 x^{2} + 6960 x + 3008\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} + ( 1 - \beta_{9} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} + ( 1 - \beta_{9} ) q^{7} + q^{9} + ( \beta_{1} - \beta_{10} ) q^{11} + ( 1 - \beta_{5} ) q^{13} + \beta_{1} q^{15} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( 1 - \beta_{11} ) q^{19} + ( 1 - \beta_{9} ) q^{21} + ( \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{23} + ( 2 + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{25} + q^{27} + ( \beta_{1} + \beta_{7} - \beta_{10} ) q^{29} + ( 3 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{31} + ( \beta_{1} - \beta_{10} ) q^{33} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{35} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{11} ) q^{37} + ( 1 - \beta_{5} ) q^{39} + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{41} + ( 2 + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{43} + \beta_{1} q^{45} + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{47} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{49} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{51} + ( 3 + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{10} ) q^{53} + ( 4 + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{55} + ( 1 - \beta_{11} ) q^{57} + ( -1 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{8} + \beta_{9} ) q^{59} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{61} + ( 1 - \beta_{9} ) q^{63} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{65} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{67} + ( \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{69} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{11} ) q^{71} + ( 2 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} + ( 2 + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{75} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{77} + ( 5 + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{79} + q^{81} + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{83} + ( 2 - 3 \beta_{1} + 2 \beta_{5} - 2 \beta_{7} + \beta_{10} ) q^{85} + ( \beta_{1} + \beta_{7} - \beta_{10} ) q^{87} + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} ) q^{89} + ( 1 - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{91} + ( 3 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{93} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{95} + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{97} + ( \beta_{1} - \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{3} + 4q^{5} + 11q^{7} + 12q^{9} + O(q^{10}) \) \( 12q + 12q^{3} + 4q^{5} + 11q^{7} + 12q^{9} + q^{11} + 8q^{13} + 4q^{15} + 3q^{17} + 12q^{19} + 11q^{21} + 7q^{23} + 18q^{25} + 12q^{27} + 5q^{29} + 33q^{31} + q^{33} + 15q^{35} + 8q^{37} + 8q^{39} - 6q^{41} + 16q^{43} + 4q^{45} + 18q^{47} + 25q^{49} + 3q^{51} + 20q^{53} + 39q^{55} + 12q^{57} + 4q^{59} + 10q^{61} + 11q^{63} + 9q^{67} + 7q^{69} + 11q^{71} + 22q^{73} + 18q^{75} + 24q^{77} + 56q^{79} + 12q^{81} + 26q^{83} + 15q^{85} + 5q^{87} - 15q^{89} + 11q^{91} + 33q^{93} + 3q^{95} + 8q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + 1411 x^{4} - 11022 x^{3} - 2450 x^{2} + 6960 x + 3008\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-2612041 \nu^{11} + 172266458 \nu^{10} - 431149861 \nu^{9} - 5604811029 \nu^{8} + 15598783254 \nu^{7} + 61532588360 \nu^{6} - 170251220041 \nu^{5} - 269226343929 \nu^{4} + 657292058507 \nu^{3} + 528815192804 \nu^{2} - 748850317708 \nu - 506734859797\)\()/ 6887737669 \)
\(\beta_{3}\)\(=\)\((\)\(48270257 \nu^{11} - 393971708 \nu^{10} - 803884765 \nu^{9} + 12731441603 \nu^{8} - 7245605191 \nu^{7} - 136791943933 \nu^{6} + 180010099322 \nu^{5} + 557383751450 \nu^{4} - 796092987441 \nu^{3} - 886580239346 \nu^{2} + 843664925302 \nu + 623931970190\)\()/ 13775475338 \)
\(\beta_{4}\)\(=\)\((\)\(433554181 \nu^{11} - 1027661208 \nu^{10} - 15151502491 \nu^{9} + 32370295155 \nu^{8} + 187572944809 \nu^{7} - 331978409981 \nu^{6} - 1009649126248 \nu^{5} + 1181123521726 \nu^{4} + 2516794688379 \nu^{3} - 933458223074 \nu^{2} - 2492427824318 \nu - 810005934996\)\()/ 27550950676 \)
\(\beta_{5}\)\(=\)\((\)\(-924523625 \nu^{11} + 3335040372 \nu^{10} + 28727151823 \nu^{9} - 105506939067 \nu^{8} - 286439821885 \nu^{7} + 1096386897181 \nu^{6} + 971661553200 \nu^{5} - 4121786159918 \nu^{4} - 965164697859 \nu^{3} + 4869923492678 \nu^{2} + 619274447522 \nu - 1194286516536\)\()/ 55101901352 \)
\(\beta_{6}\)\(=\)\((\)\(497566513 \nu^{11} - 583717328 \nu^{10} - 19421435387 \nu^{9} + 18043299379 \nu^{8} + 279234229797 \nu^{7} - 177472297861 \nu^{6} - 1817022049764 \nu^{5} + 523553323290 \nu^{4} + 5297394080111 \nu^{3} + 403005890338 \nu^{2} - 5364621239478 \nu - 2291097916892\)\()/ 27550950676 \)
\(\beta_{7}\)\(=\)\((\)\(-608147925 \nu^{11} + 2540267068 \nu^{10} + 17866911943 \nu^{9} - 80927081243 \nu^{8} - 156734313797 \nu^{7} + 851051501325 \nu^{6} + 322943033664 \nu^{5} - 3288682206998 \nu^{4} + 476137596409 \nu^{3} + 4250845840470 \nu^{2} - 751582083438 \nu - 1570311572784\)\()/ 27550950676 \)
\(\beta_{8}\)\(=\)\((\)\(1276718751 \nu^{11} - 4447291652 \nu^{10} - 40378997177 \nu^{9} + 141144188077 \nu^{8} + 420342586963 \nu^{7} - 1472464513643 \nu^{6} - 1640879715920 \nu^{5} + 5556472146210 \nu^{4} + 2723364140149 \nu^{3} - 6468711211946 \nu^{2} - 2707550657158 \nu + 1003995641592\)\()/ 55101901352 \)
\(\beta_{9}\)\(=\)\((\)\(-672160257 \nu^{11} + 2096323188 \nu^{10} + 22136844839 \nu^{9} - 66600085467 \nu^{8} - 248395598785 \nu^{7} + 696545389205 \nu^{6} + 1130315957180 \nu^{5} - 2631112008562 \nu^{4} - 2304461795323 \nu^{3} + 2886830776382 \nu^{2} + 2120611331722 \nu + 103637063844\)\()/ 27550950676 \)
\(\beta_{10}\)\(=\)\((\)\(825574067 \nu^{11} - 1232246264 \nu^{10} - 31068710789 \nu^{9} + 37574547953 \nu^{8} + 427577508547 \nu^{7} - 358636986823 \nu^{6} - 2660357207468 \nu^{5} + 975184226114 \nu^{4} + 7596696525145 \nu^{3} + 904825188458 \nu^{2} - 7836145638226 \nu - 3936944027888\)\()/ 27550950676 \)
\(\beta_{11}\)\(=\)\((\)\(-1494430261 \nu^{11} + 4167534940 \nu^{10} + 50046579515 \nu^{9} - 131337919583 \nu^{8} - 578035323361 \nu^{7} + 1352380306305 \nu^{6} + 2776874943512 \nu^{5} - 4915467010946 \nu^{4} - 6107654784727 \nu^{3} + 4699818876386 \nu^{2} + 5874998351882 \nu + 1214478641340\)\()/ 27550950676 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + 7\)
\(\nu^{3}\)\(=\)\(-\beta_{10} + \beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 12 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{11} - 13 \beta_{9} + 2 \beta_{8} + 15 \beta_{7} - 15 \beta_{6} + 19 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 78\)
\(\nu^{5}\)\(=\)\(2 \beta_{11} - 15 \beta_{10} + \beta_{9} - 3 \beta_{8} + 11 \beta_{7} - \beta_{6} - 31 \beta_{5} + 23 \beta_{4} + 12 \beta_{3} + 30 \beta_{2} + 144 \beta_{1} + 31\)
\(\nu^{6}\)\(=\)\(26 \beta_{11} + 6 \beta_{10} - 157 \beta_{9} + 34 \beta_{8} + 188 \beta_{7} - 202 \beta_{6} + 4 \beta_{5} + 292 \beta_{4} - 38 \beta_{3} - 41 \beta_{2} - 46 \beta_{1} + 976\)
\(\nu^{7}\)\(=\)\(49 \beta_{11} - 199 \beta_{10} + 29 \beta_{9} - 57 \beta_{8} + 96 \beta_{7} - 32 \beta_{6} - 401 \beta_{5} + 406 \beta_{4} + 108 \beta_{3} + 394 \beta_{2} + 1764 \beta_{1} + 484\)
\(\nu^{8}\)\(=\)\(465 \beta_{11} + 136 \beta_{10} - 1897 \beta_{9} + 446 \beta_{8} + 2264 \beta_{7} - 2645 \beta_{6} + 150 \beta_{5} + 4221 \beta_{4} - 533 \beta_{3} - 623 \beta_{2} - 730 \beta_{1} + 12688\)
\(\nu^{9}\)\(=\)\(810 \beta_{11} - 2629 \beta_{10} + 581 \beta_{9} - 879 \beta_{8} + 663 \beta_{7} - 715 \beta_{6} - 4895 \beta_{5} + 6479 \beta_{4} + 759 \beta_{3} + 5122 \beta_{2} + 22012 \beta_{1} + 7620\)
\(\nu^{10}\)\(=\)\(7141 \beta_{11} + 2177 \beta_{10} - 22913 \beta_{9} + 5288 \beta_{8} + 26926 \beta_{7} - 34360 \beta_{6} + 3630 \beta_{5} + 59518 \beta_{4} - 6696 \beta_{3} - 8532 \beta_{2} - 9985 \beta_{1} + 167606\)
\(\nu^{11}\)\(=\)\(11158 \beta_{11} - 35307 \beta_{10} + 10456 \beta_{9} - 13224 \beta_{8} + 1478 \beta_{7} - 13396 \beta_{6} - 57692 \beta_{5} + 98497 \beta_{4} + 2215 \beta_{3} + 67138 \beta_{2} + 278574 \beta_{1} + 119224\)

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
−3.64003
−3.49351
−2.09997
−1.14709
−0.716151
−0.619387
1.53322
1.56937
2.31305
2.93317
3.59055
3.77676
0 1.00000 0 −3.64003 0 −1.45249 0 1.00000 0
1.2 0 1.00000 0 −3.49351 0 4.58171 0 1.00000 0
1.3 0 1.00000 0 −2.09997 0 2.97416 0 1.00000 0
1.4 0 1.00000 0 −1.14709 0 −3.32442 0 1.00000 0
1.5 0 1.00000 0 −0.716151 0 −3.33002 0 1.00000 0
1.6 0 1.00000 0 −0.619387 0 0.954126 0 1.00000 0
1.7 0 1.00000 0 1.53322 0 −0.915565 0 1.00000 0
1.8 0 1.00000 0 1.56937 0 3.21458 0 1.00000 0
1.9 0 1.00000 0 2.31305 0 4.58648 0 1.00000 0
1.10 0 1.00000 0 2.93317 0 1.35675 0 1.00000 0
1.11 0 1.00000 0 3.59055 0 4.10828 0 1.00000 0
1.12 0 1.00000 0 3.77676 0 −1.75359 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.l 12
4.b odd 2 1 8016.2.a.bf 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.l 12 1.a even 1 1 trivial
8016.2.a.bf 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)