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Properties

Label 8004.2.a.j
Level 8004
Weight 2
Character orbit 8004.a
Self dual Yes
Analytic conductor 63.912
Analytic rank 0
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} -\beta_{7} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} -\beta_{7} q^{7} + q^{9} -\beta_{6} q^{11} + \beta_{4} q^{13} + \beta_{1} q^{15} + \beta_{9} q^{17} + ( 1 + \beta_{8} ) q^{19} -\beta_{7} q^{21} + q^{23} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{25} + q^{27} - q^{29} + ( 1 - \beta_{7} + \beta_{10} ) q^{31} -\beta_{6} q^{33} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{14} ) q^{35} + ( \beta_{1} + \beta_{14} ) q^{37} + \beta_{4} q^{39} + ( -\beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{41} + ( 1 + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{14} ) q^{43} + \beta_{1} q^{45} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{15} ) q^{47} + ( 2 - \beta_{1} - \beta_{2} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{49} + \beta_{9} q^{51} + ( 1 - \beta_{8} - \beta_{11} - \beta_{12} ) q^{53} + ( 2 + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{12} - \beta_{14} ) q^{55} + ( 1 + \beta_{8} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{12} - 2 \beta_{14} ) q^{59} + ( 2 + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{13} ) q^{61} -\beta_{7} q^{63} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{7} - \beta_{12} + 2 \beta_{14} ) q^{65} + ( 3 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{15} ) q^{67} + q^{69} + ( \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{71} + ( 2 - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{73} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{75} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{77} + ( 1 - \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{79} + q^{81} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{83} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{85} - q^{87} + ( \beta_{7} - \beta_{11} + \beta_{12} ) q^{89} + ( 1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{91} + ( 1 - \beta_{7} + \beta_{10} ) q^{93} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{95} + ( 2 + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{97} -\beta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{3} + 3q^{5} + 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q + 16q^{3} + 3q^{5} + 4q^{7} + 16q^{9} + 5q^{11} + 6q^{13} + 3q^{15} + 3q^{17} + 11q^{19} + 4q^{21} + 16q^{23} + 27q^{25} + 16q^{27} - 16q^{29} + 14q^{31} + 5q^{33} + 11q^{35} + 4q^{37} + 6q^{39} + 11q^{41} + 23q^{43} + 3q^{45} - 2q^{47} + 34q^{49} + 3q^{51} + 19q^{53} + 31q^{55} + 11q^{57} + 32q^{59} + 19q^{61} + 4q^{63} + 6q^{65} + 33q^{67} + 16q^{69} - 5q^{71} + 23q^{73} + 27q^{75} + 42q^{77} + 24q^{79} + 16q^{81} + 7q^{83} - 16q^{87} - 2q^{89} + 25q^{91} + 14q^{93} + 7q^{95} + 33q^{97} + 5q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + 46229 x^{8} - 47547 x^{7} - 122604 x^{6} + 65278 x^{5} + 151028 x^{4} - 17988 x^{3} - 67608 x^{2} - 8424 x + 3888\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(128089526929863711 \nu^{15} - 1780989647760093687 \nu^{14} - 2343590043301536281 \nu^{13} + 82428009307722432792 \nu^{12} - 39868293740417462800 \nu^{11} - 1434850612960902643364 \nu^{10} + 1061787358987892121509 \nu^{9} + 12059212146105833185194 \nu^{8} - 6849714181952244815581 \nu^{7} - 49998222735113343315999 \nu^{6} + 12943701821822810175386 \nu^{5} + 87862981358437561842546 \nu^{4} - 985493942779777495944 \nu^{3} - 44010301598073058173068 \nu^{2} - 5734576008998611033216 \nu + 1436096932911927060552\)\()/ \)\(60\!\cdots\!08\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-958087900246403592 \nu^{15} + 2000567955267874333 \nu^{14} + 49357290400200766149 \nu^{13} - 88974714684163101079 \nu^{12} - 963378769466844341132 \nu^{11} + 1415048553601951450082 \nu^{10} + 8893034704374805484880 \nu^{9} - 9960991537155374353069 \nu^{8} - 38488851452268663931116 \nu^{7} + 31123533131595448591055 \nu^{6} + 63124294156284455139465 \nu^{5} - 39843524562808100614170 \nu^{4} - 17036408551707978644276 \nu^{3} + 2778605399574854232452 \nu^{2} + 4140098642180225184060 \nu + 10187796734063718568848\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{4}\)\(=\)\((\)\(4486141389040236929 \nu^{15} - 16191524997021342828 \nu^{14} - 211861740633534736190 \nu^{13} + 721401882595358991197 \nu^{12} + 3820388029937201339968 \nu^{11} - 11804652191031635223666 \nu^{10} - 34218304176590055106445 \nu^{9} + 91025874133171328629977 \nu^{8} + 160820231928040018855981 \nu^{7} - 348051881480837120172498 \nu^{6} - 374464283660322591966429 \nu^{5} + 631086737220708958112600 \nu^{4} + 373450758905607661475524 \nu^{3} - 436002491424203250589680 \nu^{2} - 119911680097050427614324 \nu + 62507843402239991683728\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-2170353582856352493 \nu^{15} + 6394679216719266718 \nu^{14} + 105192616124931196248 \nu^{13} - 273011890888993264339 \nu^{12} - 1963933514692243560836 \nu^{11} + 4159658356999503153722 \nu^{10} + 18290876230917575384385 \nu^{9} - 28530527789909789588287 \nu^{8} - 88888441699606672234641 \nu^{7} + 90582649286250348855860 \nu^{6} + 210029308309103838088587 \nu^{5} - 128266188808560877480848 \nu^{4} - 209073282062352857065304 \nu^{3} + 66867424884872498482256 \nu^{2} + 63961903110921104602836 \nu - 6638130556491638529144\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{6}\)\(=\)\((\)\(1181647930064972053 \nu^{15} - 4488671540266881279 \nu^{14} - 53334067423612189198 \nu^{13} + 193579150520849398669 \nu^{12} + 901332302367398689427 \nu^{11} - 3015546321065278419096 \nu^{10} - 7391050976096937823363 \nu^{9} + 21665705786600716578792 \nu^{8} + 30526909733759416311410 \nu^{7} - 74930972648825634056871 \nu^{6} - 56119572043326164111031 \nu^{5} + 117837296847755457978037 \nu^{4} + 33444989655224619931886 \nu^{3} - 61721460041205402182268 \nu^{2} - 7863407220333139897548 \nu + 3287618491616138537532\)\()/ \)\(90\!\cdots\!12\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-8599410738229651769 \nu^{15} + 31111659866916402345 \nu^{14} + 397851089569802580887 \nu^{13} - 1357434095246951691440 \nu^{12} - 6950409487206069267520 \nu^{11} + 21480555688367312916876 \nu^{10} + 59301366750646584632885 \nu^{9} - 157147318825198786079022 \nu^{8} - 257042371188445739715205 \nu^{7} + 552656578883574074980929 \nu^{6} + 511728863085525731674458 \nu^{5} - 875606750046801999312014 \nu^{4} - 366980592422283677211496 \nu^{3} + 458996610472572057501996 \nu^{2} + 63732795840155393435880 \nu - 41744720482603951993704\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{8}\)\(=\)\((\)\(10166089516631326123 \nu^{15} - 39636043673251564446 \nu^{14} - 463149040304201162734 \nu^{13} + 1741577168182876800871 \nu^{12} + 7937358719018169248978 \nu^{11} - 27920257611472853652366 \nu^{10} - 66526615038749754213403 \nu^{9} + 209053689477697038444789 \nu^{8} + 286804021630695488156393 \nu^{7} - 763778398499676259472184 \nu^{6} - 590218313888419955236599 \nu^{5} + 1275883347371748259536286 \nu^{4} + 501014971463436711617372 \nu^{3} - 734875152379202721919584 \nu^{2} - 188592356674960101516348 \nu + 75381322683826038993408\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{9}\)\(=\)\((\)\(3833924860594975845 \nu^{15} - 15405971411407983416 \nu^{14} - 174748567588173763710 \nu^{13} + 683105279786040092525 \nu^{12} + 3000960197217652805056 \nu^{11} - 11102524477032892698670 \nu^{10} - 25318495554936950579361 \nu^{9} + 84691399626259266906305 \nu^{8} + 111080659953374901792933 \nu^{7} - 315727483540994364481618 \nu^{6} - 237361474884041971348545 \nu^{5} + 530394989727383036599272 \nu^{4} + 206406504455500178785888 \nu^{3} - 290455416050807336719528 \nu^{2} - 58482275477158954969404 \nu + 24368330568281801524128\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{10}\)\(=\)\((\)\(6055690973521494323 \nu^{15} - 21065591888740470630 \nu^{14} - 282138336758875451579 \nu^{13} + 904938036516128835734 \nu^{12} + 5019613746164753586637 \nu^{11} - 13994381788245243532422 \nu^{10} - 44614067074995671927885 \nu^{9} + 99213383165647751861865 \nu^{8} + 210015545205224104585822 \nu^{7} - 335507138819282232469092 \nu^{6} - 494990400061097230473426 \nu^{5} + 513309222385173256168109 \nu^{4} + 516577122221256491326894 \nu^{3} - 270657506289607141947036 \nu^{2} - 186121404779846364836280 \nu + 18853135064703346138200\)\()/ \)\(27\!\cdots\!36\)\( \)
\(\beta_{11}\)\(=\)\((\)\(2103742732645796968 \nu^{15} - 7587807286478686338 \nu^{14} - 98318269336507859293 \nu^{13} + 333589527564189827935 \nu^{12} + 1746467731155074504273 \nu^{11} - 5342141610410401144176 \nu^{10} - 15313513871554454343490 \nu^{9} + 39771223605111128968242 \nu^{8} + 69581428527495728527523 \nu^{7} - 143065840395346199349174 \nu^{6} - 151733582643722727860283 \nu^{5} + 230632231887950394137473 \nu^{4} + 131479917376481033077250 \nu^{3} - 118630530405786786024672 \nu^{2} - 36722337309588432352164 \nu + 7696294671096247442904\)\()/ \)\(90\!\cdots\!12\)\( \)
\(\beta_{12}\)\(=\)\((\)\(15643727938311237652 \nu^{15} - 56547423258025579917 \nu^{14} - 736811796288429578563 \nu^{13} + 2511604996502047287625 \nu^{12} + 13220075965007561551802 \nu^{11} - 40821134148960918199182 \nu^{10} - 117349102552007926729696 \nu^{9} + 310045122176464699783689 \nu^{8} + 543132566223574961590886 \nu^{7} - 1144928931914839786999539 \nu^{6} - 1234507192266499635469335 \nu^{5} + 1908106630392058166642008 \nu^{4} + 1205285884230039076834004 \nu^{3} - 1029539354898820273092228 \nu^{2} - 431698814922534389321508 \nu + 60226142682101943655944\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{13}\)\(=\)\((\)\(16902750278691257515 \nu^{15} - 66100600868765845944 \nu^{14} - 765501985844103172942 \nu^{13} + 2878724443289246873407 \nu^{12} + 13050836157168499238396 \nu^{11} - 45539064879024801708138 \nu^{10} - 109453059483503010262423 \nu^{9} + 334561429000218091255911 \nu^{8} + 478818628546944868872983 \nu^{7} - 1190123853407848150127478 \nu^{6} - 1026586069984405062827043 \nu^{5} + 1916731753201095741369916 \nu^{4} + 932234121437022270337808 \nu^{3} - 1043599212580558264999584 \nu^{2} - 292239039888885554789364 \nu + 83307899572874489830104\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{14}\)\(=\)\((\)\(18310254666719483314 \nu^{15} - 64997796606289655715 \nu^{14} - 854944160891350069327 \nu^{13} + 2842406557507315492831 \nu^{12} + 15152969014719198018848 \nu^{11} - 45130633610269192266030 \nu^{10} - 132199513307530952648122 \nu^{9} + 331680872425345273285917 \nu^{8} + 594566658421572741407942 \nu^{7} - 1173217721169586183526145 \nu^{6} - 1274906221868961120611337 \nu^{5} + 1870018247056491778135546 \nu^{4} + 1105580567571758766871040 \nu^{3} - 982373741782699986430740 \nu^{2} - 330829154134196740304148 \nu + 73093928798098161025416\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-28101973564733554969 \nu^{15} + 108682923995574038268 \nu^{14} + 1292569355600345279962 \nu^{13} - 4807355654391981949669 \nu^{12} - 22457115817186195403960 \nu^{11} + 77765457791517503993622 \nu^{10} + 191723502462768280879765 \nu^{9} - 588180818937536644168137 \nu^{8} - 848072989308638681641817 \nu^{7} + 2164766731936135949379246 \nu^{6} + 1814874225627403113533529 \nu^{5} - 3591447951250514794518160 \nu^{4} - 1594204885835619991562096 \nu^{3} + 1969768145562148310520360 \nu^{2} + 517658891675791265035020 \nu - 193169117447690193359208\)\()/ \)\(54\!\cdots\!72\)\( \)
Display \(\nu^j\) in terms of \(\beta_i\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{14} + \beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{13} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + 13 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(17 \beta_{15} - 16 \beta_{14} - 5 \beta_{13} + 16 \beta_{11} + \beta_{10} - \beta_{9} + 21 \beta_{8} - 22 \beta_{7} - 18 \beta_{5} + 16 \beta_{4} - 14 \beta_{3} + 18 \beta_{2} + \beta_{1} + 92\)
\(\nu^{5}\)\(=\)\(6 \beta_{14} + 27 \beta_{13} + 3 \beta_{12} + 42 \beta_{11} - 27 \beta_{10} - 43 \beta_{9} + 18 \beta_{8} + 45 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} - 6 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + 198 \beta_{1} + 48\)
\(\nu^{6}\)\(=\)\(292 \beta_{15} - 263 \beta_{14} - 140 \beta_{13} + 20 \beta_{12} + 250 \beta_{11} + 27 \beta_{10} - 22 \beta_{9} + 385 \beta_{8} - 435 \beta_{7} + 18 \beta_{6} - 310 \beta_{5} + 250 \beta_{4} - 196 \beta_{3} + 312 \beta_{2} + 2 \beta_{1} + 1422\)
\(\nu^{7}\)\(=\)\(-18 \beta_{15} + 218 \beta_{14} + 566 \beta_{13} + 97 \beta_{12} + 742 \beta_{11} - 578 \beta_{10} - 802 \beta_{9} + 301 \beta_{8} + 923 \beta_{7} - 114 \beta_{6} + 160 \beta_{5} - 222 \beta_{4} + 266 \beta_{3} - 204 \beta_{2} + 3255 \beta_{1} + 626\)
\(\nu^{8}\)\(=\)\(5100 \beta_{15} - 4491 \beta_{14} - 3030 \beta_{13} + 664 \beta_{12} + 3993 \beta_{11} + 555 \beta_{10} - 404 \beta_{9} + 6878 \beta_{8} - 8259 \beta_{7} + 636 \beta_{6} - 5445 \beta_{5} + 4089 \beta_{4} - 2905 \beta_{3} + 5467 \beta_{2} - 413 \beta_{1} + 23542\)
\(\nu^{9}\)\(=\)\(-819 \beta_{15} + 5553 \beta_{14} + 10895 \beta_{13} + 2315 \beta_{12} + 12537 \beta_{11} - 11449 \beta_{10} - 14473 \beta_{9} + 4797 \beta_{8} + 18229 \beta_{7} - 2238 \beta_{6} + 3752 \beta_{5} - 5663 \beta_{4} + 6505 \beta_{3} - 5024 \beta_{2} + 55717 \beta_{1} + 6965\)
\(\nu^{10}\)\(=\)\(90066 \beta_{15} - 78119 \beta_{14} - 60472 \beta_{13} + 15735 \beta_{12} + 65104 \beta_{11} + 10695 \beta_{10} - 6786 \beta_{9} + 122233 \beta_{8} - 153768 \beta_{7} + 16003 \beta_{6} - 97105 \beta_{5} + 69364 \beta_{4} - 45408 \beta_{3} + 96976 \beta_{2} - 15564 \beta_{1} + 403524\)
\(\nu^{11}\)\(=\)\(-24915 \beta_{15} + 124569 \beta_{14} + 202728 \beta_{13} + 48849 \beta_{12} + 209058 \beta_{11} - 218564 \beta_{10} - 259251 \beta_{9} + 72776 \beta_{8} + 352935 \beta_{7} - 38377 \beta_{6} + 80018 \beta_{5} - 125013 \beta_{4} + 141337 \beta_{3} - 109660 \beta_{2} + 975756 \beta_{1} + 55139\)
\(\nu^{12}\)\(=\)\(1602574 \beta_{15} - 1369817 \beta_{14} - 1167210 \beta_{13} + 327646 \beta_{12} + 1078384 \beta_{11} + 203896 \beta_{10} - 105992 \beta_{9} + 2172396 \beta_{8} - 2833616 \beta_{7} + 353182 \beta_{6} - 1745053 \beta_{5} + 1206681 \beta_{4} - 739735 \beta_{3} + 1735350 \beta_{2} - 414872 \beta_{1} + 7050086\)
\(\nu^{13}\)\(=\)\(-638200 \beta_{15} + 2634590 \beta_{14} + 3719888 \beta_{13} + 964071 \beta_{12} + 3477248 \beta_{11} - 4089626 \beta_{10} - 4643295 \beta_{9} + 1043482 \beta_{8} + 6758749 \beta_{7} - 620343 \beta_{6} + 1646430 \beta_{5} - 2570854 \beta_{4} + 2894128 \beta_{3} - 2258798 \beta_{2} + 17322068 \beta_{1} - 143779\)
\(\nu^{14}\)\(=\)\(28665480 \beta_{15} - 24111775 \beta_{14} - 22171021 \beta_{13} + 6402010 \beta_{12} + 18076637 \beta_{11} + 3905179 \beta_{10} - 1539030 \beta_{9} + 38666159 \beta_{8} - 51935720 \beta_{7} + 7302225 \beta_{6} - 31465887 \beta_{5} + 21356178 \beta_{4} - 12425342 \beta_{3} + 31239529 \beta_{2} - 9671295 \beta_{1} + 124565501\)
\(\nu^{15}\)\(=\)\(-14893147 \beta_{15} + 53944933 \beta_{14} + 67887756 \beta_{13} + 18262187 \beta_{12} + 57915120 \beta_{11} - 75643357 \beta_{10} - 83307736 \beta_{9} + 13834904 \beta_{8} + 128598654 \beta_{7} - 9769310 \beta_{6} + 33307089 \beta_{5} - 50852857 \beta_{4} + 57298007 \beta_{3} - 45079355 \beta_{2} + 310116137 \beta_{1} - 21232931\)
Display \(\beta_i\) in terms of \(\nu^j\)

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
−4.29726
−3.12066
−2.45101
−2.34509
−1.86734
−0.900032
−0.812118
−0.434018
0.187793
0.897331
1.63007
1.72426
2.96949
3.50323
4.08902
4.22634
0 1.00000 0 −4.29726 0 3.54512 0 1.00000 0
1.2 0 1.00000 0 −3.12066 0 −0.760812 0 1.00000 0
1.3 0 1.00000 0 −2.45101 0 3.23607 0 1.00000 0
1.4 0 1.00000 0 −2.34509 0 −1.25807 0 1.00000 0
1.5 0 1.00000 0 −1.86734 0 −4.84365 0 1.00000 0
1.6 0 1.00000 0 −0.900032 0 0.226627 0 1.00000 0
1.7 0 1.00000 0 −0.812118 0 −4.94593 0 1.00000 0
1.8 0 1.00000 0 −0.434018 0 −2.25969 0 1.00000 0
1.9 0 1.00000 0 0.187793 0 3.11113 0 1.00000 0
1.10 0 1.00000 0 0.897331 0 3.81467 0 1.00000 0
1.11 0 1.00000 0 1.63007 0 −0.577524 0 1.00000 0
1.12 0 1.00000 0 1.72426 0 4.78449 0 1.00000 0
1.13 0 1.00000 0 2.96949 0 −3.61534 0 1.00000 0
1.14 0 1.00000 0 3.50323 0 0.271751 0 1.00000 0
1.15 0 1.00000 0 4.08902 0 2.57555 0 1.00000 0
1.16 0 1.00000 0 4.22634 0 0.695618 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{16} - \cdots\)
\(T_{7}^{16} - \cdots\)