Properties

Label 8036.2.a.q
Level 8036
Weight 2
Character orbit 8036.a
Self dual Yes
Analytic conductor 64.168
Analytic rank 0
Dimension 15
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8036.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( 2 + \beta_{2} ) q^{9} + ( 1 - \beta_{3} + \beta_{5} - \beta_{12} ) q^{11} + \beta_{14} q^{13} + ( \beta_{2} + \beta_{5} + \beta_{8} - \beta_{12} + \beta_{14} ) q^{15} -\beta_{9} q^{17} + ( 1 - \beta_{2} + \beta_{3} + \beta_{10} - \beta_{11} ) q^{19} + ( \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{23} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{25} + ( 1 - \beta_{1} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{27} + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} ) q^{29} + ( 2 - \beta_{1} + \beta_{3} + \beta_{12} ) q^{31} + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{33} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{10} - \beta_{12} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{39} - q^{41} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{43} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{9} - \beta_{10} - \beta_{13} ) q^{45} + ( -2 - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{12} + \beta_{13} ) q^{47} + ( 3 + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{12} + \beta_{14} ) q^{51} + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{53} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{55} + ( 2 + \beta_{1} - 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{57} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{10} + \beta_{13} ) q^{59} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 3 \beta_{7} - \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{65} + ( 1 - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{67} + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{69} + ( -\beta_{1} - \beta_{2} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{71} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{73} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{6} - \beta_{8} + \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{14} ) q^{75} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - 2 \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{79} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{81} + ( 1 + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{83} + ( 3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{13} ) q^{85} + ( -2 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{13} ) q^{87} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} ) q^{89} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{93} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{5} - 3 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{95} + ( -\beta_{1} - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{97} + ( \beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - q^{3} + 3q^{5} + 30q^{9} + O(q^{10}) \) \( 15q - q^{3} + 3q^{5} + 30q^{9} + 9q^{11} + 7q^{13} + 2q^{15} + 3q^{17} + 7q^{19} - q^{23} + 32q^{25} + 11q^{27} + 18q^{29} + 30q^{31} - 16q^{33} + 23q^{37} + 5q^{39} - 15q^{41} + 12q^{43} - 13q^{45} - 16q^{47} + 29q^{51} + 33q^{53} + 37q^{55} + 16q^{57} - 10q^{59} + q^{61} + 16q^{65} + 20q^{67} + 21q^{69} + 5q^{71} - 3q^{73} - 51q^{75} + 25q^{79} + 43q^{81} + 18q^{83} + 36q^{85} - 53q^{87} - 11q^{89} + 65q^{93} - 30q^{95} + 16q^{97} - 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + 13971 x^{7} - 20311 x^{6} - 22309 x^{5} + 38415 x^{4} + 8429 x^{3} - 22584 x^{2} - 399 x + 3381\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(21568415171 \nu^{14} + 33366538546 \nu^{13} - 780431594905 \nu^{12} - 1076896328346 \nu^{11} + 11058300703621 \nu^{10} + 12550722332099 \nu^{9} - 78350006830010 \nu^{8} - 63368265036347 \nu^{7} + 290964255861732 \nu^{6} + 125086901622927 \nu^{5} - 518478043052718 \nu^{4} - 34154729402677 \nu^{3} + 315331689321588 \nu^{2} - 33588397215636 \nu - 53909708702913\)\()/ 8954190428232 \)
\(\beta_{4}\)\(=\)\((\)\(207367915591 \nu^{14} + 75537732154 \nu^{13} - 6864511642829 \nu^{12} - 2264494596922 \nu^{11} + 84630564241705 \nu^{10} + 19997951254183 \nu^{9} - 471109920542042 \nu^{8} - 26134945978879 \nu^{7} + 1061905945918156 \nu^{6} - 337765061235149 \nu^{5} - 14256388065710 \nu^{4} + 1086025882031975 \nu^{3} - 2504020310672772 \nu^{2} - 382307010614964 \nu + 964719687046731\)\()/ 53725142569392 \)
\(\beta_{5}\)\(=\)\((\)\(375675670889 \nu^{14} + 197307815606 \nu^{13} - 13458627125155 \nu^{12} - 5416632772358 \nu^{11} + 187920186390263 \nu^{10} + 41971019614937 \nu^{9} - 1297715569983766 \nu^{8} + 1604986959871 \nu^{7} + 4597551204850484 \nu^{6} - 1062369256372771 \nu^{5} - 7504847710757362 \nu^{4} + 2957473201007065 \nu^{3} + 3676894843258596 \nu^{2} - 925571855654028 \nu - 394220916659787\)\()/ 53725142569392 \)
\(\beta_{6}\)\(=\)\((\)\(563847518531 \nu^{14} + 271104011042 \nu^{13} - 20576643380593 \nu^{12} - 7746814774274 \nu^{11} + 294326513071613 \nu^{10} + 65601128365283 \nu^{9} - 2097712145040178 \nu^{8} - 64560226435499 \nu^{7} + 7768694051421020 \nu^{6} - 1280840055975121 \nu^{5} - 13752243630567190 \nu^{4} + 4090409019404467 \nu^{3} + 8604345326302092 \nu^{2} - 1583632566957636 \nu - 1413258738046953\)\()/ 53725142569392 \)
\(\beta_{7}\)\(=\)\((\)\(82318075711 \nu^{14} + 110890419958 \nu^{13} - 2903659812299 \nu^{12} - 3508288802422 \nu^{11} + 39739641971875 \nu^{10} + 39661374923485 \nu^{9} - 268865569213064 \nu^{8} - 191585391007741 \nu^{7} + 943989351011626 \nu^{6} + 359635444491391 \nu^{5} - 1587383470520204 \nu^{4} - 134303162335843 \nu^{3} + 916971510040800 \nu^{2} + 51742333507248 \nu - 123529112322861\)\()/ 6715642821174 \)
\(\beta_{8}\)\(=\)\((\)\(57933555161 \nu^{14} + 19902931966 \nu^{13} - 2038445185159 \nu^{12} - 636604082118 \nu^{11} + 27895029041707 \nu^{10} + 6543610028837 \nu^{9} - 188732383570406 \nu^{8} - 21783985515293 \nu^{7} + 661794329691888 \nu^{6} - 12151856350959 \nu^{5} - 1127435686751094 \nu^{4} + 128904705950189 \nu^{3} + 743133409360908 \nu^{2} - 36292597607964 \nu - 132293240921499\)\()/ 4477095214116 \)
\(\beta_{9}\)\(=\)\((\)\(263299916363 \nu^{14} + 284951510738 \nu^{13} - 9037992446569 \nu^{12} - 9319581543506 \nu^{11} + 119443657067957 \nu^{10} + 110071063789547 \nu^{9} - 772667003836402 \nu^{8} - 576510242603315 \nu^{7} + 2574917570991548 \nu^{6} + 1337367111898247 \nu^{5} - 4158684124869190 \nu^{4} - 1188084234706517 \nu^{3} + 2549675888785068 \nu^{2} + 395328021911436 \nu - 369589303008177\)\()/ 17908380856464 \)
\(\beta_{10}\)\(=\)\((\)\(-803771605457 \nu^{14} + 55554951370 \nu^{13} + 29873873602171 \nu^{12} - 2773231651450 \nu^{11} - 437414324713775 \nu^{10} + 64143993774511 \nu^{9} + 3202662100683478 \nu^{8} - 708242318071255 \nu^{7} - 12199681840975508 \nu^{6} + 3575923666457563 \nu^{5} + 22280485010412706 \nu^{4} - 7066331684287153 \nu^{3} - 14659545336879396 \nu^{2} + 2212465125784428 \nu + 2285567947233123\)\()/ 53725142569392 \)
\(\beta_{11}\)\(=\)\((\)\(-527518007087 \nu^{14} - 247737754298 \nu^{13} + 19237736393941 \nu^{12} + 7202431283834 \nu^{11} - 274960302342017 \nu^{10} - 63910333187519 \nu^{9} + 1958752009116682 \nu^{8} + 105751315240871 \nu^{7} - 7259458623838364 \nu^{6} + 925907072139781 \nu^{5} + 12894254808423982 \nu^{4} - 3195696128916367 \nu^{3} - 8114892589426428 \nu^{2} + 1033679062742580 \nu + 1300628388320541\)\()/ 17908380856464 \)
\(\beta_{12}\)\(=\)\((\)\(-1991383769819 \nu^{14} - 970794976610 \nu^{13} + 72904083529129 \nu^{12} + 29524283624978 \nu^{11} - 1045949435856725 \nu^{10} - 291365982179867 \nu^{9} + 7472488513327378 \nu^{8} + 870143948269859 \nu^{7} - 27720511397395436 \nu^{6} + 1482781049982937 \nu^{5} + 49218218809824022 \nu^{4} - 8617048669258075 \nu^{3} - 31166858944757532 \nu^{2} + 2619386748959364 \nu + 5020217462879745\)\()/ 53725142569392 \)
\(\beta_{13}\)\(=\)\((\)\(154414062065 \nu^{14} + 69132074966 \nu^{13} - 5550942035752 \nu^{12} - 2103794805215 \nu^{11} + 77954228306660 \nu^{10} + 20374588096121 \nu^{9} - 543496021568218 \nu^{8} - 55206873249419 \nu^{7} + 1963368044552309 \nu^{6} - 139122061304389 \nu^{5} - 3386733041584858 \nu^{4} + 660872437651219 \nu^{3} + 2062137225183060 \nu^{2} - 196898900419467 \nu - 312802565393409\)\()/ 3357821410587 \)
\(\beta_{14}\)\(=\)\((\)\(-1366326190169 \nu^{14} - 650669688638 \nu^{13} + 49657832878051 \nu^{12} + 19718267961758 \nu^{11} - 706794387141431 \nu^{10} - 191670870598865 \nu^{9} + 5008459392835390 \nu^{8} + 533869176269825 \nu^{7} - 18440308332790964 \nu^{6} + 1233561484274971 \nu^{5} + 32538031186207330 \nu^{4} - 6160094617298017 \nu^{3} - 20547001848439524 \nu^{2} + 1854323154805428 \nu + 3416523056642787\)\()/ 26862571284696 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 7 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + 11 \beta_{2} + \beta_{1} + 38\)
\(\nu^{5}\)\(=\)\(-3 \beta_{14} + 2 \beta_{12} + \beta_{10} + 14 \beta_{9} - 16 \beta_{8} - 14 \beta_{7} + 12 \beta_{6} - \beta_{5} - 4 \beta_{3} + 59 \beta_{1} - 5\)
\(\nu^{6}\)\(=\)\(14 \beta_{14} + 14 \beta_{13} + \beta_{12} + 2 \beta_{11} - 17 \beta_{10} + 17 \beta_{9} - 32 \beta_{8} + \beta_{7} + 23 \beta_{6} - 19 \beta_{5} + \beta_{4} - 21 \beta_{3} + 116 \beta_{2} + 27 \beta_{1} + 333\)
\(\nu^{7}\)\(=\)\(-62 \beta_{14} - 10 \beta_{13} + 44 \beta_{12} - 7 \beta_{11} + 12 \beta_{10} + 170 \beta_{9} - 202 \beta_{8} - 166 \beta_{7} + 132 \beta_{6} - 29 \beta_{5} + 10 \beta_{4} - 67 \beta_{3} + 7 \beta_{2} + 558 \beta_{1} + 38\)
\(\nu^{8}\)\(=\)\(147 \beta_{14} + 152 \beta_{13} + 30 \beta_{12} + 60 \beta_{11} - 243 \beta_{10} + 234 \beta_{9} - 419 \beta_{8} + 27 \beta_{7} + 370 \beta_{6} - 263 \beta_{5} + 28 \beta_{4} - 320 \beta_{3} + 1240 \beta_{2} + 451 \beta_{1} + 3159\)
\(\nu^{9}\)\(=\)\(-927 \beta_{14} - 256 \beta_{13} + 685 \beta_{12} - 156 \beta_{11} + 52 \beta_{10} + 2001 \beta_{9} - 2390 \beta_{8} - 1858 \beta_{7} + 1452 \beta_{6} - 508 \beta_{5} + 250 \beta_{4} - 903 \beta_{3} + 228 \beta_{2} + 5654 \beta_{1} + 1342\)
\(\nu^{10}\)\(=\)\(1327 \beta_{14} + 1492 \beta_{13} + 627 \beta_{12} + 1096 \beta_{11} - 3269 \beta_{10} + 3039 \beta_{9} - 5210 \beta_{8} + 459 \beta_{7} + 5154 \beta_{6} - 3302 \beta_{5} + 521 \beta_{4} - 4318 \beta_{3} + 13447 \beta_{2} + 6366 \beta_{1} + 31539\)
\(\nu^{11}\)\(=\)\(-12261 \beta_{14} - 4464 \beta_{13} + 9349 \beta_{12} - 2401 \beta_{11} - 866 \beta_{10} + 23339 \beta_{9} - 27666 \beta_{8} - 20283 \beta_{7} + 16290 \beta_{6} - 7403 \beta_{5} + 4286 \beta_{4} - 11529 \beta_{3} + 4656 \beta_{2} + 59734 \beta_{1} + 23366\)
\(\nu^{12}\)\(=\)\(10196 \beta_{14} + 13647 \beta_{13} + 10931 \beta_{12} + 16261 \beta_{11} - 42318 \beta_{10} + 38527 \beta_{9} - 63562 \beta_{8} + 6361 \beta_{7} + 66741 \beta_{6} - 40030 \beta_{5} + 8268 \beta_{4} - 54924 \beta_{3} + 147413 \beta_{2} + 83308 \beta_{1} + 326176\)
\(\nu^{13}\)\(=\)\(-152923 \beta_{14} - 66240 \beta_{13} + 119702 \beta_{12} - 31483 \beta_{11} - 28360 \beta_{10} + 271267 \beta_{9} - 318032 \beta_{8} - 218879 \beta_{7} + 186813 \beta_{6} - 99003 \beta_{5} + 62906 \beta_{4} - 144488 \beta_{3} + 77660 \beta_{2} + 647792 \beta_{1} + 337884\)
\(\nu^{14}\)\(=\)\(56795 \beta_{14} + 115353 \beta_{13} + 170227 \beta_{12} + 216472 \beta_{11} - 532792 \beta_{10} + 481792 \beta_{9} - 768045 \beta_{8} + 78391 \beta_{7} + 829524 \beta_{6} - 478852 \beta_{5} + 120857 \beta_{4} - 676566 \beta_{3} + 1629219 \beta_{2} + 1048772 \beta_{1} + 3460210\)

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
3.42372
3.10047
2.29181
1.93225
1.72996
1.40418
1.27444
0.515082
−0.449590
−0.741382
−2.11769
−2.47735
−2.73742
−2.90634
−3.24213
0 −3.42372 0 2.35740 0 0 0 8.72183 0
1.2 0 −3.10047 0 −3.31369 0 0 0 6.61290 0
1.3 0 −2.29181 0 −3.92303 0 0 0 2.25239 0
1.4 0 −1.93225 0 −0.475465 0 0 0 0.733587 0
1.5 0 −1.72996 0 −0.138351 0 0 0 −0.00725198 0
1.6 0 −1.40418 0 3.61567 0 0 0 −1.02828 0
1.7 0 −1.27444 0 4.37429 0 0 0 −1.37580 0
1.8 0 −0.515082 0 2.19964 0 0 0 −2.73469 0
1.9 0 0.449590 0 −0.787803 0 0 0 −2.79787 0
1.10 0 0.741382 0 −2.48663 0 0 0 −2.45035 0
1.11 0 2.11769 0 3.93656 0 0 0 1.48461 0
1.12 0 2.47735 0 −3.49844 0 0 0 3.13725 0
1.13 0 2.73742 0 −0.863054 0 0 0 4.49344 0
1.14 0 2.90634 0 0.506103 0 0 0 5.44682 0
1.15 0 3.24213 0 1.49679 0 0 0 7.51143 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
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\(T_{3}^{15} + \cdots\)
\(T_{5}^{15} - \cdots\)
\(T_{11}^{15} - \cdots\)