Properties

Label 8008.2.a.y
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{10} q^{5} + q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{10} q^{5} + q^{7} + ( 2 + \beta_{2} ) q^{9} - q^{11} - q^{13} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{15} -\beta_{8} q^{17} + ( -1 - \beta_{6} ) q^{19} -\beta_{1} q^{21} + ( \beta_{9} + \beta_{12} ) q^{23} + ( 1 + \beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{25} + ( -3 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{27} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{11} - \beta_{13} ) q^{29} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} - \beta_{10} ) q^{31} + \beta_{1} q^{33} + \beta_{10} q^{35} + ( 1 + \beta_{2} - \beta_{7} - \beta_{13} ) q^{37} + \beta_{1} q^{39} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{41} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{43} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} + \beta_{12} ) q^{45} + ( -1 + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{47} + q^{49} + ( -4 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{51} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{12} ) q^{53} -\beta_{10} q^{55} + ( -3 + 4 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{13} ) q^{57} + ( -5 - \beta_{3} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{12} ) q^{59} + ( -2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{61} + ( 2 + \beta_{2} ) q^{63} -\beta_{10} q^{65} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{9} - \beta_{12} + \beta_{13} ) q^{67} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{69} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{71} + ( 3 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{73} + ( 3 - 8 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} ) q^{75} - q^{77} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{79} + ( 3 + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{81} + ( -5 + \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{8} + \beta_{12} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{85} + ( 2 - 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{87} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{89} - q^{91} + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{93} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{95} + ( -2 - \beta_{5} - \beta_{9} + \beta_{11} + \beta_{13} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 3q^{3} - 6q^{5} + 14q^{7} + 21q^{9} + O(q^{10}) \) \( 14q - 3q^{3} - 6q^{5} + 14q^{7} + 21q^{9} - 14q^{11} - 14q^{13} - 6q^{15} - 6q^{17} - 13q^{19} - 3q^{21} - 9q^{23} + 22q^{25} - 18q^{27} + 2q^{29} - 2q^{31} + 3q^{33} - 6q^{35} - q^{37} + 3q^{39} - 16q^{41} - 15q^{43} - 44q^{45} - 8q^{47} + 14q^{49} - 14q^{51} - 6q^{53} + 6q^{55} - 10q^{57} - 36q^{59} - 19q^{61} + 21q^{63} + 6q^{65} - 34q^{67} - q^{69} - 10q^{71} + 9q^{73} - 44q^{75} - 14q^{77} - q^{79} + 42q^{81} - 56q^{83} + 21q^{85} - 5q^{87} - 14q^{89} - 14q^{91} - 20q^{93} + q^{95} - 14q^{97} - 21q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + 2996 x^{6} - 7275 x^{5} - 2804 x^{4} + 6417 x^{3} + 538 x^{2} - 1032 x - 128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(-11074521 \nu^{13} + 106614016 \nu^{12} + 202633696 \nu^{11} - 2843591948 \nu^{10} - 1105297503 \nu^{9} + 27555851639 \nu^{8} + 3080027798 \nu^{7} - 120227610741 \nu^{6} - 11970806879 \nu^{5} + 239059287029 \nu^{4} + 81567084 \nu^{3} - 180363225628 \nu^{2} + 99775185884 \nu - 16683138390\)\()/ 15572760974 \)
\(\beta_{4}\)\(=\)\((\)\(-58724453 \nu^{13} + 415732687 \nu^{12} + 429387489 \nu^{11} - 9589273014 \nu^{10} + 12008354815 \nu^{9} + 76649830082 \nu^{8} - 161921785412 \nu^{7} - 265081043224 \nu^{6} + 685142582580 \nu^{5} + 431622615285 \nu^{4} - 1032089287458 \nu^{3} - 437413157891 \nu^{2} + 297904178158 \nu + 264264384876\)\()/ 31145521948 \)
\(\beta_{5}\)\(=\)\((\)\(-84573421 \nu^{13} + 56007919 \nu^{12} + 1990802275 \nu^{11} + 1134603604 \nu^{10} - 16590727105 \nu^{9} - 46010891430 \nu^{8} + 64820773162 \nu^{7} + 415744840748 \nu^{6} - 158509912178 \nu^{5} - 1451593421773 \nu^{4} + 342029081890 \nu^{3} + 1859229248461 \nu^{2} - 443394945668 \nu - 403244409496\)\()/ 31145521948 \)
\(\beta_{6}\)\(=\)\((\)\(-86197394 \nu^{13} - 519312690 \nu^{12} + 4042783649 \nu^{11} + 14135363599 \nu^{10} - 62980268748 \nu^{9} - 143598407974 \nu^{8} + 424769570879 \nu^{7} + 681805991838 \nu^{6} - 1291874241299 \nu^{5} - 1505481346651 \nu^{4} + 1580032475418 \nu^{3} + 1262115684952 \nu^{2} - 467654431587 \nu - 148105342534\)\()/ 15572760974 \)
\(\beta_{7}\)\(=\)\((\)\(-100221529 \nu^{13} + 20088978 \nu^{12} + 3433412919 \nu^{11} - 872022317 \nu^{10} - 43636307733 \nu^{9} + 14040705209 \nu^{8} + 252001600285 \nu^{7} - 104822338105 \nu^{6} - 640067051370 \nu^{5} + 362130604330 \nu^{4} + 492483408208 \nu^{3} - 465460641758 \nu^{2} + 207142479617 \nu + 51792914190\)\()/ 15572760974 \)
\(\beta_{8}\)\(=\)\((\)\(-204529955 \nu^{13} + 624487815 \nu^{12} + 5370665871 \nu^{11} - 15034911330 \nu^{10} - 53506698243 \nu^{9} + 127831458804 \nu^{8} + 259840610114 \nu^{7} - 471825438700 \nu^{6} - 618802526998 \nu^{5} + 721519658405 \nu^{4} + 562295462232 \nu^{3} - 319648593677 \nu^{2} - 32398485750 \nu - 12295329000\)\()/ 31145521948 \)
\(\beta_{9}\)\(=\)\((\)\(-217441493 \nu^{13} + 681733073 \nu^{12} + 6654223007 \nu^{11} - 21109853824 \nu^{10} - 75443571221 \nu^{9} + 245864340184 \nu^{8} + 385128276528 \nu^{7} - 1330673265748 \nu^{6} - 832031771460 \nu^{5} + 3310520475965 \nu^{4} + 436856651496 \nu^{3} - 3075140922419 \nu^{2} + 331065135556 \nu + 306825120252\)\()/ 31145521948 \)
\(\beta_{10}\)\(=\)\((\)\(208926672 \nu^{13} - 491561333 \nu^{12} - 5735162063 \nu^{11} + 12048495611 \nu^{10} + 58834255396 \nu^{9} - 105792814085 \nu^{8} - 282276492982 \nu^{7} + 415135249248 \nu^{6} + 630297354346 \nu^{5} - 721447644136 \nu^{4} - 522691012619 \nu^{3} + 434273622092 \nu^{2} + 10035841931 \nu - 9943373676\)\()/ 15572760974 \)
\(\beta_{11}\)\(=\)\((\)\(488425337 \nu^{13} - 1340879233 \nu^{12} - 12295355455 \nu^{11} + 31978291166 \nu^{10} + 110106082525 \nu^{9} - 271168142180 \nu^{8} - 418235692408 \nu^{7} + 1033264260798 \nu^{6} + 540080988466 \nu^{5} - 1837139952593 \nu^{4} + 267466107302 \nu^{3} + 1348441940209 \nu^{2} - 668143324930 \nu - 141640358616\)\()/ 31145521948 \)
\(\beta_{12}\)\(=\)\((\)\(-282478821 \nu^{13} + 525449102 \nu^{12} + 7980519403 \nu^{11} - 11978040794 \nu^{10} - 86063793263 \nu^{9} + 93824334855 \nu^{8} + 449176063003 \nu^{7} - 309731889602 \nu^{6} - 1152974583333 \nu^{5} + 424114073096 \nu^{4} + 1239699101363 \nu^{3} - 211254743701 \nu^{2} - 245962834998 \nu + 816058514\)\()/ 15572760974 \)
\(\beta_{13}\)\(=\)\((\)\(-461482718 \nu^{13} + 1119246863 \nu^{12} + 12534518457 \nu^{11} - 27398869301 \nu^{10} - 127270246922 \nu^{9} + 241925316093 \nu^{8} + 607199002972 \nu^{7} - 976096110556 \nu^{6} - 1367160452244 \nu^{5} + 1851226663240 \nu^{4} + 1215628784765 \nu^{3} - 1405372442586 \nu^{2} - 180279716217 \nu + 105424374118\)\()/ 15572760974 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 12 \beta_{2} + 39\)
\(\nu^{5}\)\(=\)\(14 \beta_{13} - 15 \beta_{12} + 16 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} + 13 \beta_{8} + 17 \beta_{7} - \beta_{6} + 14 \beta_{5} + 15 \beta_{4} + \beta_{3} + 90 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-15 \beta_{13} + 13 \beta_{12} + 18 \beta_{11} - 38 \beta_{10} - 3 \beta_{9} - \beta_{8} + 3 \beta_{7} - 2 \beta_{6} - 15 \beta_{5} + 29 \beta_{4} - 19 \beta_{3} + 131 \beta_{2} + 4 \beta_{1} + 353\)
\(\nu^{7}\)\(=\)\(167 \beta_{13} - 186 \beta_{12} + 205 \beta_{11} + 77 \beta_{10} - 43 \beta_{9} + 139 \beta_{8} + 215 \beta_{7} - 18 \beta_{6} + 156 \beta_{5} + 181 \beta_{4} + 16 \beta_{3} + 6 \beta_{2} + 932 \beta_{1} - 16\)
\(\nu^{8}\)\(=\)\(-160 \beta_{13} + 116 \beta_{12} + 257 \beta_{11} - 531 \beta_{10} - 80 \beta_{9} - 17 \beta_{8} + 76 \beta_{7} - 42 \beta_{6} - 187 \beta_{5} + 345 \beta_{4} - 261 \beta_{3} + 1403 \beta_{2} + 106 \beta_{1} + 3423\)
\(\nu^{9}\)\(=\)\(1906 \beta_{13} - 2198 \beta_{12} + 2435 \beta_{11} + 482 \beta_{10} - 667 \beta_{9} + 1449 \beta_{8} + 2498 \beta_{7} - 238 \beta_{6} + 1627 \beta_{5} + 2054 \beta_{4} + 185 \beta_{3} + 160 \beta_{2} + 9828 \beta_{1} - 200\)
\(\nu^{10}\)\(=\)\(-1452 \beta_{13} + 725 \beta_{12} + 3377 \beta_{11} - 6685 \beta_{10} - 1435 \beta_{9} - 153 \beta_{8} + 1309 \beta_{7} - 611 \beta_{6} - 2214 \beta_{5} + 3921 \beta_{4} - 3166 \beta_{3} + 14968 \beta_{2} + 1990 \beta_{1} + 34307\)
\(\nu^{11}\)\(=\)\(21440 \beta_{13} - 25547 \beta_{12} + 28043 \beta_{11} + 1700 \beta_{10} - 9180 \beta_{9} + 15259 \beta_{8} + 28129 \beta_{7} - 2822 \beta_{6} + 16505 \beta_{5} + 22781 \beta_{4} + 1878 \beta_{3} + 2896 \beta_{2} + 104779 \beta_{1} - 2231\)
\(\nu^{12}\)\(=\)\(-11353 \beta_{13} + 597 \beta_{12} + 42620 \beta_{11} - 80507 \beta_{10} - 21831 \beta_{9} - 402 \beta_{8} + 19277 \beta_{7} - 7711 \beta_{6} - 25713 \beta_{5} + 44080 \beta_{4} - 36193 \beta_{3} + 159733 \beta_{2} + 32051 \beta_{1} + 349673\)
\(\nu^{13}\)\(=\)\(240060 \beta_{13} - 294913 \beta_{12} + 318651 \beta_{11} - 16807 \beta_{10} - 119388 \beta_{9} + 163166 \beta_{8} + 312802 \beta_{7} - 31947 \beta_{6} + 164782 \beta_{5} + 250307 \beta_{4} + 17619 \beta_{3} + 44674 \beta_{2} + 1125192 \beta_{1} - 23048\)

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
3.38813
3.22236
2.52493
1.97367
1.53687
1.36889
0.528665
−0.129238
−0.374242
−1.44419
−1.85932
−2.05490
−2.46105
−3.22058
0 −3.38813 0 −4.16385 0 1.00000 0 8.47945 0
1.2 0 −3.22236 0 1.94503 0 1.00000 0 7.38359 0
1.3 0 −2.52493 0 3.43935 0 1.00000 0 3.37528 0
1.4 0 −1.97367 0 −1.63177 0 1.00000 0 0.895375 0
1.5 0 −1.53687 0 −0.216819 0 1.00000 0 −0.638026 0
1.6 0 −1.36889 0 −3.92349 0 1.00000 0 −1.12615 0
1.7 0 −0.528665 0 0.933973 0 1.00000 0 −2.72051 0
1.8 0 0.129238 0 −0.197815 0 1.00000 0 −2.98330 0
1.9 0 0.374242 0 3.66821 0 1.00000 0 −2.85994 0
1.10 0 1.44419 0 −3.20437 0 1.00000 0 −0.914318 0
1.11 0 1.85932 0 1.13919 0 1.00000 0 0.457073 0
1.12 0 2.05490 0 0.929320 0 1.00000 0 1.22261 0
1.13 0 2.46105 0 −1.12857 0 1.00000 0 3.05676 0
1.14 0 3.22058 0 −3.58838 0 1.00000 0 7.37212 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{14} + \cdots\)
\(T_{5}^{14} + \cdots\)
\(T_{17}^{14} + \cdots\)