Properties

Label 8008.2.a.z
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 15
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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{11} q^{5} - q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{11} q^{5} - q^{7} + ( 2 + \beta_{2} ) q^{9} - q^{11} + q^{13} + \beta_{3} q^{15} + ( 1 + \beta_{9} ) q^{17} + ( -1 + \beta_{6} - \beta_{11} ) q^{19} + \beta_{1} q^{21} + ( 1 - \beta_{1} + \beta_{2} + \beta_{13} ) q^{23} + ( 2 + \beta_{4} - \beta_{10} - \beta_{11} ) q^{25} + ( 1 - 2 \beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{27} + ( 1 + \beta_{1} + \beta_{8} + \beta_{12} ) q^{29} + ( -1 + \beta_{5} - \beta_{11} - \beta_{12} ) q^{31} + \beta_{1} q^{33} + \beta_{11} q^{35} + ( \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} ) q^{37} -\beta_{1} q^{39} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{41} + ( -1 + \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{43} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{45} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{47} + q^{49} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{51} + ( 3 + \beta_{3} - \beta_{5} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{53} + \beta_{11} q^{55} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{10} - \beta_{13} + 2 \beta_{14} ) q^{57} + ( -1 + \beta_{3} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{59} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{12} ) q^{61} + ( -2 - \beta_{2} ) q^{63} -\beta_{11} q^{65} + ( -\beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{67} + ( 3 - 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{69} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{71} + ( -\beta_{3} - \beta_{4} + \beta_{5} - \beta_{11} + \beta_{13} ) q^{73} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{75} + q^{77} + ( \beta_{1} - \beta_{3} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{79} + ( 4 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} ) q^{81} + ( -3 - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{14} ) q^{83} + ( 2 - 2 \beta_{1} - \beta_{4} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{85} + ( -2 - 2 \beta_{2} - \beta_{5} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{87} + ( 1 - \beta_{1} + \beta_{2} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{89} - q^{91} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{9} - 3 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{93} + ( 5 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{95} + ( 1 + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - q^{3} + 4q^{5} - 15q^{7} + 26q^{9} + O(q^{10}) \) \( 15q - q^{3} + 4q^{5} - 15q^{7} + 26q^{9} - 15q^{11} + 15q^{13} - 6q^{15} + 8q^{17} - 17q^{19} + q^{21} + 7q^{23} + 33q^{25} - 4q^{27} + 14q^{29} - 4q^{31} + q^{33} - 4q^{35} + 3q^{37} - q^{39} - 13q^{43} + 20q^{45} + 6q^{47} + 15q^{49} + 8q^{51} + 38q^{53} - 4q^{55} + 24q^{57} - 18q^{59} + 23q^{61} - 26q^{63} + 4q^{65} - 8q^{67} + 43q^{69} - 12q^{71} + 11q^{73} + 12q^{75} + 15q^{77} - q^{79} + 51q^{81} - 16q^{83} + 13q^{85} - 25q^{87} + 28q^{89} - 15q^{91} - 14q^{93} + 49q^{95} + 30q^{97} - 26q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + 11690 x^{7} - 7119 x^{6} - 22246 x^{5} + 11137 x^{4} + 20034 x^{3} - 8392 x^{2} - 6016 x + 2560\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(1523579 \nu^{14} + 213661034 \nu^{13} - 179139600 \nu^{12} - 6986479725 \nu^{11} + 4541129757 \nu^{10} + 85561543273 \nu^{9} - 48599699344 \nu^{8} - 488772294898 \nu^{7} + 252291530370 \nu^{6} + 1334713440565 \nu^{5} - 626492952357 \nu^{4} - 1586347082289 \nu^{3} + 678120858481 \nu^{2} + 568931122604 \nu - 253585799040\)\()/ 10828273438 \)
\(\beta_{4}\)\(=\)\((\)\(204728997 \nu^{14} - 5797949397 \nu^{13} + 1200404289 \nu^{12} + 186669594656 \nu^{11} - 160969626719 \nu^{10} - 2248818957080 \nu^{9} + 2312689549068 \nu^{8} + 12635569411410 \nu^{7} - 13379380540814 \nu^{6} - 34174617664635 \nu^{5} + 34513885202850 \nu^{4} + 42027402294341 \nu^{3} - 37011471642118 \nu^{2} - 18367614659000 \nu + 11235942086336\)\()/ 173252375008 \)
\(\beta_{5}\)\(=\)\((\)\(157141431 \nu^{14} - 1230552919 \nu^{13} - 3155705101 \nu^{12} + 40387483536 \nu^{11} + 1702785579 \nu^{10} - 504234185256 \nu^{9} + 342178471052 \nu^{8} + 3010380395486 \nu^{7} - 2722381354490 \nu^{6} - 8922472657697 \nu^{5} + 8079082533374 \nu^{4} + 12254264192927 \nu^{3} - 9827207416642 \nu^{2} - 5792078145608 \nu + 3811771653184\)\()/ 86626187504 \)
\(\beta_{6}\)\(=\)\((\)\(413408383 \nu^{14} + 1544421129 \nu^{13} - 15746220005 \nu^{12} - 49653568744 \nu^{11} + 228419477571 \nu^{10} + 588397861744 \nu^{9} - 1564264168284 \nu^{8} - 3156065445642 \nu^{7} + 5007460939446 \nu^{6} + 7618648102207 \nu^{5} - 5948270274850 \nu^{4} - 6815219183521 \nu^{3} + 347842922614 \nu^{2} + 98512228088 \nu + 545872471392\)\()/ 173252375008 \)
\(\beta_{7}\)\(=\)\((\)\(-742937461 \nu^{14} - 1826318179 \nu^{13} + 30835167495 \nu^{12} + 59173290264 \nu^{11} - 502014930785 \nu^{10} - 714535204112 \nu^{9} + 4061739875940 \nu^{8} + 3991707328670 \nu^{7} - 17068026173042 \nu^{6} - 10549349624229 \nu^{5} + 35541387263846 \nu^{4} + 12283735275291 \nu^{3} - 31984400742130 \nu^{2} - 5198752858728 \nu + 8430631658656\)\()/ 173252375008 \)
\(\beta_{8}\)\(=\)\((\)\(1890437133 \nu^{14} - 1073055941 \nu^{13} - 66111397935 \nu^{12} + 31796527976 \nu^{11} + 904394485593 \nu^{10} - 346151330752 \nu^{9} - 6238137460212 \nu^{8} + 1668210629874 \nu^{7} + 23572513434290 \nu^{6} - 3218674662259 \nu^{5} - 49012994686182 \nu^{4} + 934503342013 \nu^{3} + 49331444492578 \nu^{2} + 1799763702344 \nu - 14280187531456\)\()/ 346504750016 \)
\(\beta_{9}\)\(=\)\((\)\(-1938443511 \nu^{14} + 1484217919 \nu^{13} + 68173131037 \nu^{12} - 44865881400 \nu^{11} - 931234396411 \nu^{10} + 498828247584 \nu^{9} + 6276390141532 \nu^{8} - 2491001622102 \nu^{7} - 21963128480886 \nu^{6} + 5412559613929 \nu^{5} + 37978280261778 \nu^{4} - 3729507797895 \nu^{3} - 26682788237798 \nu^{2} - 727396710904 \nu + 4190172143104\)\()/ 346504750016 \)
\(\beta_{10}\)\(=\)\((\)\(-1184655527 \nu^{14} + 40599471 \nu^{13} + 42697917709 \nu^{12} + 530650360 \nu^{11} - 604962621531 \nu^{10} - 33239466432 \nu^{9} + 4319125883612 \nu^{8} + 423236823962 \nu^{7} - 16606302188950 \nu^{6} - 2490295041815 \nu^{5} + 33684876907586 \nu^{4} + 7018914045929 \nu^{3} - 31804427646502 \nu^{2} - 7104276001064 \nu + 9057739992992\)\()/ 173252375008 \)
\(\beta_{11}\)\(=\)\((\)\(396227811 \nu^{14} - 390133495 \nu^{13} - 13013329249 \nu^{12} + 11962731552 \nu^{11} + 161054746947 \nu^{10} - 137156782884 \nu^{9} - 939947023304 \nu^{8} + 728812002254 \nu^{7} + 2676813930998 \nu^{6} - 1811579665029 \nu^{5} - 3475630121246 \nu^{4} + 1906817321679 \nu^{3} + 1592639636418 \nu^{2} - 612660355988 \nu - 107982020560\)\()/ 43313093752 \)
\(\beta_{12}\)\(=\)\((\)\(2388875051 \nu^{14} - 8261584387 \nu^{13} - 71865979993 \nu^{12} + 259474132888 \nu^{11} + 775871452191 \nu^{10} - 3032322946288 \nu^{9} - 3545121573980 \nu^{8} + 16349472328158 \nu^{7} + 5728168812302 \nu^{6} - 41488112322565 \nu^{5} + 2209043811558 \nu^{4} + 45742051602491 \nu^{3} - 12659651595602 \nu^{2} - 15947603806024 \nu + 7139114332064\)\()/ 173252375008 \)
\(\beta_{13}\)\(=\)\((\)\(-2665656839 \nu^{14} + 4559868439 \nu^{13} + 84068282789 \nu^{12} - 139774279616 \nu^{11} - 980978775387 \nu^{10} + 1587508125288 \nu^{9} + 5227482261084 \nu^{8} - 8262828912134 \nu^{7} - 12866431836518 \nu^{6} + 20039589961337 \nu^{5} + 13182923217098 \nu^{4} - 21042920256135 \nu^{3} - 3528585941854 \nu^{2} + 7004343553608 \nu - 486592259136\)\()/ 173252375008 \)
\(\beta_{14}\)\(=\)\((\)\(-3526798371 \nu^{14} + 4709151755 \nu^{13} + 117063051249 \nu^{12} - 143804690696 \nu^{11} - 1480208505991 \nu^{10} + 1622561315744 \nu^{9} + 9029091617148 \nu^{8} - 8308488511342 \nu^{7} - 28251013727598 \nu^{6} + 19095872506589 \nu^{5} + 45139985518970 \nu^{4} - 16157472133315 \nu^{3} - 33181232975262 \nu^{2} + 592595901752 \nu + 7092433199008\)\()/ 173252375008 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 8 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{6} - 2 \beta_{4} - \beta_{3} + 12 \beta_{2} - \beta_{1} + 40\)
\(\nu^{5}\)\(=\)\(-\beta_{14} - \beta_{13} - \beta_{12} - 15 \beta_{11} - 14 \beta_{9} + 12 \beta_{8} + 14 \beta_{7} + 13 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 74 \beta_{1} - 14\)
\(\nu^{6}\)\(=\)\(18 \beta_{14} - 3 \beta_{13} + 16 \beta_{12} + 31 \beta_{11} + 14 \beta_{10} + 3 \beta_{9} + \beta_{8} - 14 \beta_{6} - \beta_{5} - 33 \beta_{4} - 19 \beta_{3} + 129 \beta_{2} - 13 \beta_{1} + 374\)
\(\nu^{7}\)\(=\)\(-18 \beta_{14} - 21 \beta_{13} - 21 \beta_{12} - 184 \beta_{11} + 3 \beta_{10} - 167 \beta_{9} + 129 \beta_{8} + 161 \beta_{7} + 144 \beta_{6} + 18 \beta_{5} - 15 \beta_{4} - 17 \beta_{3} + 17 \beta_{2} + 733 \beta_{1} - 157\)
\(\nu^{8}\)\(=\)\(252 \beta_{14} - 71 \beta_{13} + 202 \beta_{12} + 375 \beta_{11} + 148 \beta_{10} + 55 \beta_{9} + 17 \beta_{8} + 2 \beta_{7} - 154 \beta_{6} - 21 \beta_{5} - 423 \beta_{4} - 266 \beta_{3} + 1362 \beta_{2} - 123 \beta_{1} + 3719\)
\(\nu^{9}\)\(=\)\(-231 \beta_{14} - 310 \beta_{13} - 314 \beta_{12} - 2079 \beta_{11} + 70 \beta_{10} - 1908 \beta_{9} + 1362 \beta_{8} + 1762 \beta_{7} + 1520 \beta_{6} + 228 \beta_{5} - 172 \beta_{4} - 222 \beta_{3} + 222 \beta_{2} + 7485 \beta_{1} - 1611\)
\(\nu^{10}\)\(=\)\(3231 \beta_{14} - 1178 \beta_{13} + 2365 \beta_{12} + 4177 \beta_{11} + 1396 \beta_{10} + 696 \beta_{9} + 222 \beta_{8} + 61 \beta_{7} - 1578 \beta_{6} - 304 \beta_{5} - 4992 \beta_{4} - 3330 \beta_{3} + 14343 \beta_{2} - 968 \beta_{1} + 37990\)
\(\nu^{11}\)\(=\)\(-2559 \beta_{14} - 4056 \beta_{13} - 4140 \beta_{12} - 22536 \beta_{11} + 1102 \beta_{10} - 21413 \beta_{9} + 14343 \beta_{8} + 18970 \beta_{7} + 15716 \beta_{6} + 2557 \beta_{5} - 1838 \beta_{4} - 2713 \beta_{3} + 2681 \beta_{2} + 77567 \beta_{1} - 15851\)
\(\nu^{12}\)\(=\)\(39686 \beta_{14} - 16983 \beta_{13} + 26769 \beta_{12} + 44927 \beta_{11} + 12188 \beta_{10} + 7551 \beta_{9} + 2681 \beta_{8} + 1224 \beta_{7} - 15767 \beta_{6} - 3816 \beta_{5} - 56875 \beta_{4} - 39609 \beta_{3} + 151136 \beta_{2} - 5951 \beta_{1} + 393237\)
\(\nu^{13}\)\(=\)\(-25891 \beta_{14} - 50465 \beta_{13} - 51380 \beta_{12} - 238942 \beta_{11} + 14779 \beta_{10} - 238140 \beta_{9} + 151136 \beta_{8} + 203010 \beta_{7} + 160838 \beta_{6} + 27223 \beta_{5} - 19447 \beta_{4} - 32565 \beta_{3} + 31400 \beta_{2} + 810362 \beta_{1} - 152360\)
\(\nu^{14}\)\(=\)\(475419 \beta_{14} - 227599 \beta_{13} + 297511 \beta_{12} + 475676 \beta_{11} + 98135 \beta_{10} + 74828 \beta_{9} + 31400 \beta_{8} + 20288 \beta_{7} - 156157 \beta_{6} - 44832 \beta_{5} - 637087 \beta_{4} - 459067 \beta_{3} + 1594630 \beta_{2} - 14591 \beta_{1} + 4100536\)

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
3.34164
3.21278
2.38300
2.04397
1.74947
1.40500
0.619952
0.455829
−0.804723
−1.32542
−1.58213
−1.69218
−2.42593
−3.11300
−3.26824
0 −3.34164 0 1.86211 0 −1.00000 0 8.16655 0
1.2 0 −3.21278 0 0.0641301 0 −1.00000 0 7.32193 0
1.3 0 −2.38300 0 −4.12953 0 −1.00000 0 2.67869 0
1.4 0 −2.04397 0 4.25466 0 −1.00000 0 1.17780 0
1.5 0 −1.74947 0 −0.876486 0 −1.00000 0 0.0606454 0
1.6 0 −1.40500 0 3.56716 0 −1.00000 0 −1.02598 0
1.7 0 −0.619952 0 −1.29231 0 −1.00000 0 −2.61566 0
1.8 0 −0.455829 0 0.836884 0 −1.00000 0 −2.79222 0
1.9 0 0.804723 0 −0.337817 0 −1.00000 0 −2.35242 0
1.10 0 1.32542 0 2.70237 0 −1.00000 0 −1.24326 0
1.11 0 1.58213 0 −3.58630 0 −1.00000 0 −0.496866 0
1.12 0 1.69218 0 −2.62665 0 −1.00000 0 −0.136527 0
1.13 0 2.42593 0 3.43967 0 −1.00000 0 2.88513 0
1.14 0 3.11300 0 −2.63092 0 −1.00000 0 6.69079 0
1.15 0 3.26824 0 2.75302 0 −1.00000 0 7.68141 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{15} + \cdots\)
\(T_{5}^{15} - \cdots\)
\(T_{17}^{15} - \cdots\)