Properties

Label 4008.2.a.g
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{5} + \nu^{4} + 7 \nu^{3} - 2 \nu^{2} - 8 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 3 \nu^{2} + 8 \nu - 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + \nu^{5} + 7 \nu^{4} - 2 \nu^{3} - 9 \nu^{2} - \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - \nu^{5} - 7 \nu^{4} + 2 \nu^{3} + 9 \nu^{2} + 3 \nu - 1 \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + \nu^{5} + 8 \nu^{4} - 3 \nu^{3} - 15 \nu^{2} + 6 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - 2 \nu^{5} - 15 \nu^{4} + 6 \nu^{3} + 23 \nu^{2} - 2 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{2} + 7 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\((\)\(16 \beta_{6} + 18 \beta_{5} + 15 \beta_{4} + 29 \beta_{3} + 24 \beta_{2} + 22 \beta_{1} + 4\)\()/2\)
\(\nu^{6}\)\(=\)\(13 \beta_{6} + 21 \beta_{5} + 11 \beta_{4} + 15 \beta_{3} + 50 \beta_{2} + 49 \beta_{1} + 34\)

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
2.80982
1.47270
−0.674271
−2.05123
1.31154
−1.20126
0.332704
0 −1.00000 0 −3.20729 0 3.68779 0 1.00000 0
1.2 0 −1.00000 0 −3.01562 0 −1.84677 0 1.00000 0
1.3 0 −1.00000 0 −1.68509 0 −2.23045 0 1.00000 0
1.4 0 −1.00000 0 −0.597616 0 2.60994 0 1.00000 0
1.5 0 −1.00000 0 0.0621653 0 0.782308 0 1.00000 0
1.6 0 −1.00000 0 1.82640 0 2.26944 0 1.00000 0
1.7 0 −1.00000 0 3.61705 0 2.72774 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.g 7
4.b odd 2 1 8016.2.a.w 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.g 7 1.a even 1 1 trivial
8016.2.a.w 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\( T_{5}^{7} + 3 T_{5}^{6} - 15 T_{5}^{5} - 50 T_{5}^{4} + 23 T_{5}^{3} + 133 T_{5}^{2} + 56 T_{5} - 4 \)
\( T_{7}^{7} - 8 T_{7}^{6} + 11 T_{7}^{5} + 55 T_{7}^{4} - 147 T_{7}^{3} - 37 T_{7}^{2} + 336 T_{7} - 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{7} \)
$5$ \( 1 + 3 T + 20 T^{2} + 40 T^{3} + 173 T^{4} + 258 T^{5} + 1026 T^{6} + 1326 T^{7} + 5130 T^{8} + 6450 T^{9} + 21625 T^{10} + 25000 T^{11} + 62500 T^{12} + 46875 T^{13} + 78125 T^{14} \)
$7$ \( 1 - 8 T + 60 T^{2} - 281 T^{3} + 1267 T^{4} - 4377 T^{5} + 14644 T^{6} - 39420 T^{7} + 102508 T^{8} - 214473 T^{9} + 434581 T^{10} - 674681 T^{11} + 1008420 T^{12} - 941192 T^{13} + 823543 T^{14} \)
$11$ \( 1 - T + 46 T^{2} - 28 T^{3} + 1061 T^{4} - 414 T^{5} + 16252 T^{6} - 5010 T^{7} + 178772 T^{8} - 50094 T^{9} + 1412191 T^{10} - 409948 T^{11} + 7408346 T^{12} - 1771561 T^{13} + 19487171 T^{14} \)
$13$ \( 1 + 2 T + 48 T^{2} + 20 T^{3} + 854 T^{4} - 1667 T^{5} + 8225 T^{6} - 41310 T^{7} + 106925 T^{8} - 281723 T^{9} + 1876238 T^{10} + 571220 T^{11} + 17822064 T^{12} + 9653618 T^{13} + 62748517 T^{14} \)
$17$ \( 1 - 11 T + 125 T^{2} - 904 T^{3} + 6292 T^{4} - 34384 T^{5} + 176026 T^{6} - 750594 T^{7} + 2992442 T^{8} - 9936976 T^{9} + 30912596 T^{10} - 75502984 T^{11} + 177482125 T^{12} - 265513259 T^{13} + 410338673 T^{14} \)
$19$ \( 1 - 2 T + 68 T^{2} - 280 T^{3} + 2522 T^{4} - 11611 T^{5} + 70571 T^{6} - 268206 T^{7} + 1340849 T^{8} - 4191571 T^{9} + 17298398 T^{10} - 36489880 T^{11} + 168374732 T^{12} - 94091762 T^{13} + 893871739 T^{14} \)
$23$ \( 1 - 17 T + 264 T^{2} - 2580 T^{3} + 22869 T^{4} - 155172 T^{5} + 962930 T^{6} - 4821022 T^{7} + 22147390 T^{8} - 82085988 T^{9} + 278247123 T^{10} - 721989780 T^{11} + 1699194552 T^{12} - 2516610113 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 7 T + 146 T^{2} + 790 T^{3} + 9969 T^{4} + 44070 T^{5} + 422776 T^{6} + 1553690 T^{7} + 12260504 T^{8} + 37062870 T^{9} + 243133941 T^{10} + 558751990 T^{11} + 2994627754 T^{12} + 4163763247 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 - 10 T + 211 T^{2} - 1575 T^{3} + 18783 T^{4} - 110477 T^{5} + 946125 T^{6} - 4420900 T^{7} + 29329875 T^{8} - 106168397 T^{9} + 559564353 T^{10} - 1454545575 T^{11} + 6040750861 T^{12} - 8875036810 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 + 21 T + 360 T^{2} + 4288 T^{3} + 43987 T^{4} + 370784 T^{5} + 2772232 T^{6} + 17825150 T^{7} + 102572584 T^{8} + 507603296 T^{9} + 2228073511 T^{10} + 8036402368 T^{11} + 24963824520 T^{12} + 53880254589 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 - 8 T + 205 T^{2} - 1417 T^{3} + 20339 T^{4} - 118045 T^{5} + 1250235 T^{6} - 6008956 T^{7} + 51259635 T^{8} - 198433645 T^{9} + 1401784219 T^{10} - 4004103337 T^{11} + 23750521205 T^{12} - 38000833928 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 12 T + 145 T^{2} + 989 T^{3} + 7307 T^{4} + 31281 T^{5} + 187515 T^{6} + 541084 T^{7} + 8063145 T^{8} + 57838569 T^{9} + 580957649 T^{10} + 3381194189 T^{11} + 21316224235 T^{12} + 75856356588 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 - 25 T + 440 T^{2} - 5547 T^{3} + 60316 T^{4} - 549620 T^{5} + 4505677 T^{6} - 32294448 T^{7} + 211766819 T^{8} - 1214110580 T^{9} + 6262188068 T^{10} - 27067590507 T^{11} + 100911803080 T^{12} - 269480383225 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 + 7 T + 166 T^{2} + 1295 T^{3} + 19306 T^{4} + 115366 T^{5} + 1391231 T^{6} + 7778352 T^{7} + 73735243 T^{8} + 324063094 T^{9} + 2874219362 T^{10} + 10218172895 T^{11} + 69420451838 T^{12} + 155150527903 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 - 3 T + 362 T^{2} - 997 T^{3} + 58918 T^{4} - 141144 T^{5} + 5562009 T^{6} - 10948080 T^{7} + 328158531 T^{8} - 491322264 T^{9} + 12100519922 T^{10} - 12081008917 T^{11} + 258802596238 T^{12} - 126541600923 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 + 14 T + 304 T^{2} + 3390 T^{3} + 45370 T^{4} + 399265 T^{5} + 4077845 T^{6} + 29981926 T^{7} + 248748545 T^{8} + 1485665065 T^{9} + 10298127970 T^{10} + 46937400990 T^{11} + 256757275504 T^{12} + 721285241054 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 - 4 T + 382 T^{2} - 1324 T^{3} + 66982 T^{4} - 198497 T^{5} + 6988981 T^{6} - 17114094 T^{7} + 468261727 T^{8} - 891053033 T^{9} + 20145707266 T^{10} - 26680084204 T^{11} + 515747790874 T^{12} - 361833528676 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 27 T + 731 T^{2} - 12149 T^{3} + 188468 T^{4} - 2220242 T^{5} + 24216158 T^{6} - 212111532 T^{7} + 1719347218 T^{8} - 11192239922 T^{9} + 67454770348 T^{10} - 308726512469 T^{11} + 1318891655581 T^{12} - 3458707665867 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 + 12 T + 286 T^{2} + 1814 T^{3} + 32184 T^{4} + 167073 T^{5} + 3162525 T^{6} + 15750450 T^{7} + 230864325 T^{8} + 890332017 T^{9} + 12520123128 T^{10} + 51514409174 T^{11} + 592898475598 T^{12} + 1816010715468 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 - 8 T + 463 T^{2} - 2721 T^{3} + 93163 T^{4} - 415133 T^{5} + 11094341 T^{6} - 39502788 T^{7} + 876452939 T^{8} - 2590845053 T^{9} + 45932992357 T^{10} - 105983170401 T^{11} + 1424677112737 T^{12} - 1944699644168 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - 15 T + 252 T^{2} - 2335 T^{3} + 40204 T^{4} - 384018 T^{5} + 4378733 T^{6} - 30821664 T^{7} + 363434839 T^{8} - 2645500002 T^{9} + 22988124548 T^{10} - 110815179535 T^{11} + 992638242036 T^{12} - 4904105600535 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 - 14 T + 532 T^{2} - 6878 T^{3} + 130172 T^{4} - 1444175 T^{5} + 18617595 T^{6} - 167849250 T^{7} + 1656965955 T^{8} - 11439310175 T^{9} + 91767224668 T^{10} - 431541133598 T^{11} + 2970719626868 T^{12} - 6957738073454 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - 3 T + 475 T^{2} - 1207 T^{3} + 104508 T^{4} - 227430 T^{5} + 14447348 T^{6} - 26865048 T^{7} + 1401392756 T^{8} - 2139888870 T^{9} + 95381629884 T^{10} - 106854842167 T^{11} + 4078986622075 T^{12} - 2498916014787 T^{13} + 80798284478113 T^{14} \)
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