Properties

Label 8034.2.a.bc
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} -\beta_{12} q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} -\beta_{12} q^{7} + q^{8} + q^{9} -\beta_{1} q^{10} -\beta_{13} q^{11} - q^{12} - q^{13} -\beta_{12} q^{14} + \beta_{1} q^{15} + q^{16} + ( \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{17} + q^{18} + ( \beta_{6} + \beta_{10} ) q^{19} -\beta_{1} q^{20} + \beta_{12} q^{21} -\beta_{13} q^{22} + ( \beta_{4} - \beta_{6} + \beta_{8} - \beta_{13} + \beta_{14} ) q^{23} - q^{24} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{25} - q^{26} - q^{27} -\beta_{12} q^{28} + ( 2 - \beta_{4} + \beta_{11} - \beta_{14} ) q^{29} + \beta_{1} q^{30} + ( \beta_{6} - \beta_{10} ) q^{31} + q^{32} + \beta_{13} q^{33} + ( \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{34} + ( -1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{35} + q^{36} + ( 2 - \beta_{8} + \beta_{13} ) q^{37} + ( \beta_{6} + \beta_{10} ) q^{38} + q^{39} -\beta_{1} q^{40} + ( -\beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{41} + \beta_{12} q^{42} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{9} - 2 \beta_{12} - \beta_{14} ) q^{43} -\beta_{13} q^{44} -\beta_{1} q^{45} + ( \beta_{4} - \beta_{6} + \beta_{8} - \beta_{13} + \beta_{14} ) q^{46} + ( -\beta_{4} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{47} - q^{48} + ( 2 - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{49} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{50} + ( -\beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{51} - q^{52} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{6} - \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{53} - q^{54} + ( -1 - \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{55} -\beta_{12} q^{56} + ( -\beta_{6} - \beta_{10} ) q^{57} + ( 2 - \beta_{4} + \beta_{11} - \beta_{14} ) q^{58} + ( 2 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{8} ) q^{59} + \beta_{1} q^{60} + ( 2 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{14} ) q^{61} + ( \beta_{6} - \beta_{10} ) q^{62} -\beta_{12} q^{63} + q^{64} + \beta_{1} q^{65} + \beta_{13} q^{66} + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{67} + ( \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{68} + ( -\beta_{4} + \beta_{6} - \beta_{8} + \beta_{13} - \beta_{14} ) q^{69} + ( -1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{70} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{71} + q^{72} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{9} - \beta_{11} + \beta_{13} ) q^{73} + ( 2 - \beta_{8} + \beta_{13} ) q^{74} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{75} + ( \beta_{6} + \beta_{10} ) q^{76} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{77} + q^{78} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{79} -\beta_{1} q^{80} + q^{81} + ( -\beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{82} + ( 1 + \beta_{1} - \beta_{5} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{83} + \beta_{12} q^{84} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{85} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{9} - 2 \beta_{12} - \beta_{14} ) q^{86} + ( -2 + \beta_{4} - \beta_{11} + \beta_{14} ) q^{87} -\beta_{13} q^{88} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{12} ) q^{89} -\beta_{1} q^{90} + \beta_{12} q^{91} + ( \beta_{4} - \beta_{6} + \beta_{8} - \beta_{13} + \beta_{14} ) q^{92} + ( -\beta_{6} + \beta_{10} ) q^{93} + ( -\beta_{4} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{94} + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} ) q^{95} - q^{96} + ( 2 + \beta_{1} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{14} ) q^{97} + ( 2 - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{98} -\beta_{13} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(2537687267978126305 \nu^{14} - 4451260489557971801 \nu^{13} - 118607585529944247160 \nu^{12} + 201894771958174626540 \nu^{11} + 2065774167255360616712 \nu^{10} - 3389041191171673838947 \nu^{9} - 16731997871850458694196 \nu^{8} + 26184651556522014754456 \nu^{7} + 64612006487857426157383 \nu^{6} - 94142583432419588002804 \nu^{5} - 109525350370822082766017 \nu^{4} + 145425424067694665372939 \nu^{3} + 53532513774968079841586 \nu^{2} - 75481126501892671808752 \nu + 10315145682746966682944\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-3699895265701566353 \nu^{14} + 9148574385842311585 \nu^{13} + 173370522950640285632 \nu^{12} - 401147448883605874380 \nu^{11} - 3040455429814694571304 \nu^{10} + 6347593527480726095411 \nu^{9} + 25017731332666579088492 \nu^{8} - 44104795536205132068824 \nu^{7} - 100220807092977766199447 \nu^{6} + 130523696906252529491036 \nu^{5} + 189558836334691265479921 \nu^{4} - 141476191243349227852339 \nu^{3} - 143327565524240360562922 \nu^{2} + 51082916732053674000704 \nu + 31232388687353964064640\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{4}\)\(=\)\((\)\(3773756680062282389 \nu^{14} - 6621368153969086837 \nu^{13} - 173187246246377101328 \nu^{12} + 299029290976494333660 \nu^{11} + 2937458974583126918344 \nu^{10} - 4977240979565848871999 \nu^{9} - 22880252731934338749548 \nu^{8} + 37839894450753416967704 \nu^{7} + 83503603148532481698995 \nu^{6} - 132396296199577483382588 \nu^{5} - 131442940707426684242005 \nu^{4} + 201052630181593672596943 \nu^{3} + 56590561866592878591298 \nu^{2} - 105730885302223976198336 \nu + 15999879528543583420672\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-480421702545662123 \nu^{14} + 43110065350197505 \nu^{13} + 23213090564296575506 \nu^{12} - 740490576502024596 \nu^{11} - 423613518372852698680 \nu^{10} - 14892506323265156095 \nu^{9} + 3671032048907900349578 \nu^{8} + 411319014825229416136 \nu^{7} - 15679149848322058779965 \nu^{6} - 3172708855946150385058 \nu^{5} + 30948944380910040502771 \nu^{4} + 8259757592004157683545 \nu^{3} - 22217767129638255474988 \nu^{2} - 4580081011050082891228 \nu + 2332369560757484941472\)\()/ \)\(55\!\cdots\!92\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-3904290949793122853 \nu^{14} + 14888092941463260925 \nu^{13} + 169062781332764512088 \nu^{12} - 668437526636551236252 \nu^{11} - 2592166138155870798184 \nu^{10} + 10967030382881649513455 \nu^{9} + 16479152638452021523460 \nu^{8} - 80762745305159870682104 \nu^{7} - 35045824171083536765987 \nu^{6} + 263743776775326664831940 \nu^{5} - 17153063243581124000891 \nu^{4} - 329708970008761859686711 \nu^{3} + 86096965550968147850822 \nu^{2} + 85313542688545755485744 \nu - 17231425109937506795776\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{7}\)\(=\)\((\)\(1437348996203833439 \nu^{14} - 3946760310112449751 \nu^{13} - 62994684520973032856 \nu^{12} + 176671191969983900820 \nu^{11} + 984697502339295110968 \nu^{10} - 2894763636328689608669 \nu^{9} - 6497133196312926368636 \nu^{8} + 21396335118373983609512 \nu^{7} + 15463140566778105369785 \nu^{6} - 71147018611462724250524 \nu^{5} + 668266301521186841729 \nu^{4} + 96396192676307847030949 \nu^{3} - 26343781779899371656962 \nu^{2} - 36576815788443985896752 \nu + 4137311857191453969280\)\()/ \)\(14\!\cdots\!12\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-216776146664850899 \nu^{14} + 833519197187931691 \nu^{13} + 9225298037107150072 \nu^{12} - 37446683496667123780 \nu^{11} - 136909874293211051608 \nu^{10} + 615042765945550203385 \nu^{9} + 807482146801725193676 \nu^{8} - 4539938442156400956040 \nu^{7} - 1259609430605682009317 \nu^{6} + 14947524072875232440620 \nu^{5} - 2596195015386327990477 \nu^{4} - 19543885760794156101025 \nu^{3} + 5767233907927992512410 \nu^{2} + 6876707545146179250864 \nu - 1099608980575046137664\)\()/ \)\(16\!\cdots\!68\)\( \)
\(\beta_{9}\)\(=\)\((\)\(2262060993304819351 \nu^{14} - 3898546964586732887 \nu^{13} - 102677561674421271760 \nu^{12} + 172180740620193204084 \nu^{11} + 1709275636723449924632 \nu^{10} - 2762375590140558319621 \nu^{9} - 12869509104005418846724 \nu^{8} + 19631778341601514731400 \nu^{7} + 44047896643808091309313 \nu^{6} - 59268935086500033928756 \nu^{5} - 62006168022988305824471 \nu^{4} + 60676860335814627002837 \nu^{3} + 22877990863326377997014 \nu^{2} - 5496566174053900711552 \nu + 5241129959594824984640\)\()/ \)\(14\!\cdots\!12\)\( \)
\(\beta_{10}\)\(=\)\((\)\(7389528242931052763 \nu^{14} - 17619842819561824147 \nu^{13} - 335494621667721213464 \nu^{12} + 781347790842433004388 \nu^{11} + 5596799573243534456152 \nu^{10} - 12587537843796393862097 \nu^{9} - 42434297297760917373068 \nu^{8} + 89945009312798312263112 \nu^{7} + 148333866313014220649117 \nu^{6} - 276567011587519356139052 \nu^{5} - 225796791826086771929275 \nu^{4} + 305614434437315646882457 \nu^{3} + 122418852737580306892054 \nu^{2} - 64975688162446145854064 \nu - 20760519604905515508992\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{11}\)\(=\)\((\)\(8766750062989708633 \nu^{14} - 22529199141324906641 \nu^{13} - 383811818206987097848 \nu^{12} + 995941970920453001676 \nu^{11} + 6001036075358757513992 \nu^{10} - 15968681453075819058955 \nu^{9} - 39886372998134487110740 \nu^{8} + 113304212701430619470680 \nu^{7} + 99684232238791936686799 \nu^{6} - 345811571939604474832852 \nu^{5} - 22639092092040808747577 \nu^{4} + 392231796537565988390195 \nu^{3} - 156334774693565882680990 \nu^{2} - 114807959618689508407408 \nu + 54269842840724729422208\)\()/ \)\(44\!\cdots\!36\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-4417538992890062809 \nu^{14} + 10268326608661339349 \nu^{13} + 197311809783773428132 \nu^{12} - 456611053302482919612 \nu^{11} - 3196413318423942416024 \nu^{10} + 7398646108103984300491 \nu^{9} + 22849031189715444748168 \nu^{8} - 53545307775003849319096 \nu^{7} - 69555529379053943147695 \nu^{6} + 169932974581872220205488 \nu^{5} + 70235966360674007667737 \nu^{4} - 207964216939141801609367 \nu^{3} + 8344259772351224143546 \nu^{2} + 71793846729075368713576 \nu - 11807950134708017239328\)\()/ \)\(22\!\cdots\!68\)\( \)
\(\beta_{13}\)\(=\)\((\)\(7599288743075757563 \nu^{14} - 14675949069150921499 \nu^{13} - 343394734237158555296 \nu^{12} + 648617517135946752852 \nu^{11} + 5675871771319530293080 \nu^{10} - 10416374257422677241905 \nu^{9} - 42209961752351543724788 \nu^{8} + 74291750278280117421560 \nu^{7} + 141106197202116037886717 \nu^{6} - 228640259637922398414884 \nu^{5} - 190386748714222964816683 \nu^{4} + 258839300934634508381089 \nu^{3} + 69730647157650718059982 \nu^{2} - 62443733500550485578896 \nu - 2219445897514458889376\)\()/ \)\(22\!\cdots\!68\)\( \)
\(\beta_{14}\)\(=\)\((\)\(5118564908078654033 \nu^{14} - 13012426933683220879 \nu^{13} - 226696578225165112178 \nu^{12} + 578099458562396288700 \nu^{11} + 3621424480522273550704 \nu^{10} - 9348618128120933493683 \nu^{9} - 25231217957008947225314 \nu^{8} + 67395099414387524537648 \nu^{7} + 72665931767149837846055 \nu^{6} - 212442650490802176795278 \nu^{5} - 62438278530719538088105 \nu^{4} + 256098528986214242270833 \nu^{3} - 19685540454780823356992 \nu^{2} - 79244364779543304621452 \nu + 9700285797997532462848\)\()/ \)\(11\!\cdots\!84\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{14} + 2 \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(-4 \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{7} - \beta_{4} + 12 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-34 \beta_{14} + 32 \beta_{13} + 5 \beta_{12} + 19 \beta_{11} + 15 \beta_{10} - 10 \beta_{9} - 19 \beta_{8} + 18 \beta_{7} + 19 \beta_{6} + 18 \beta_{5} - 16 \beta_{4} - 3 \beta_{3} + 19 \beta_{2} - 16 \beta_{1} + 85\)
\(\nu^{5}\)\(=\)\(-\beta_{14} + 8 \beta_{13} - 75 \beta_{12} - 20 \beta_{11} - 27 \beta_{10} - 22 \beta_{9} - 45 \beta_{7} - 8 \beta_{6} + \beta_{5} - 23 \beta_{4} - \beta_{3} + 4 \beta_{2} + 168 \beta_{1} - 15\)
\(\nu^{6}\)\(=\)\(-556 \beta_{14} + 502 \beta_{13} + 115 \beta_{12} + 317 \beta_{11} + 229 \beta_{10} - 96 \beta_{9} - 333 \beta_{8} + 313 \beta_{7} + 337 \beta_{6} + 302 \beta_{5} - 252 \beta_{4} - 79 \beta_{3} + 310 \beta_{2} - 247 \beta_{1} + 1198\)
\(\nu^{7}\)\(=\)\(-31 \beta_{14} + 239 \beta_{13} - 1254 \beta_{12} - 346 \beta_{11} - 562 \beta_{10} - 432 \beta_{9} - 10 \beta_{8} - 849 \beta_{7} - 229 \beta_{6} + 18 \beta_{5} - 414 \beta_{4} - 35 \beta_{3} + 99 \beta_{2} + 2518 \beta_{1} - 180\)
\(\nu^{8}\)\(=\)\(-9097 \beta_{14} + 7967 \beta_{13} + 2088 \beta_{12} + 5193 \beta_{11} + 3576 \beta_{10} - 888 \beta_{9} - 5611 \beta_{8} + 5351 \beta_{7} + 5729 \beta_{6} + 5021 \beta_{5} - 3952 \beta_{4} - 1609 \beta_{3} + 4887 \beta_{2} - 3873 \beta_{1} + 17955\)
\(\nu^{9}\)\(=\)\(-619 \beta_{14} + 5095 \beta_{13} - 20805 \beta_{12} - 5891 \beta_{11} - 10625 \beta_{10} - 8055 \beta_{9} - 308 \beta_{8} - 15316 \beta_{7} - 5067 \beta_{6} + 61 \beta_{5} - 6877 \beta_{4} - 750 \beta_{3} + 1621 \beta_{2} + 39150 \beta_{1} - 2196\)
\(\nu^{10}\)\(=\)\(-149357 \beta_{14} + 127959 \beta_{13} + 35595 \beta_{12} + 84969 \beta_{11} + 56709 \beta_{10} - 7087 \beta_{9} - 93089 \beta_{8} + 90567 \beta_{7} + 95731 \beta_{6} + 83612 \beta_{5} - 62019 \beta_{4} - 30076 \beta_{3} + 76569 \beta_{2} - 62051 \beta_{1} + 277897\)
\(\nu^{11}\)\(=\)\(-10176 \beta_{14} + 95830 \beta_{13} - 346912 \beta_{12} - 100736 \beta_{11} - 192217 \beta_{10} - 145094 \beta_{9} - 6660 \beta_{8} - 270979 \beta_{7} - 101706 \beta_{6} - 5067 \beta_{5} - 110424 \beta_{4} - 13520 \beta_{3} + 21534 \beta_{2} + 622037 \beta_{1} - 31788\)
\(\nu^{12}\)\(=\)\(-2460611 \beta_{14} + 2074863 \beta_{13} + 595649 \beta_{12} + 1393860 \beta_{11} + 908995 \beta_{10} - 28519 \beta_{9} - 1536904 \beta_{8} + 1524931 \beta_{7} + 1589921 \beta_{6} + 1395762 \beta_{5} - 977347 \beta_{4} - 540506 \beta_{3} + 1203101 \beta_{2} - 1013593 \beta_{1} + 4386294\)
\(\nu^{13}\)\(=\)\(-143953 \beta_{14} + 1697102 \beta_{13} - 5809478 \beta_{12} - 1731649 \beta_{11} - 3399748 \beta_{10} - 2552103 \beta_{9} - 122438 \beta_{8} - 4738578 \beta_{7} - 1938971 \beta_{6} - 196689 \beta_{5} - 1745788 \beta_{4} - 226830 \beta_{3} + 231759 \beta_{2} + 10022933 \beta_{1} - 564758\)
\(\nu^{14}\)\(=\)\(-40658804 \beta_{14} + 33887265 \beta_{13} + 9929787 \beta_{12} + 22941355 \beta_{11} + 14689662 \beta_{10} + 623231 \beta_{9} - 25363596 \beta_{8} + 25613853 \beta_{7} + 26363690 \beta_{6} + 23335400 \beta_{5} - 15489614 \beta_{4} - 9506066 \beta_{3} + 19015737 \beta_{2} - 16809082 \beta_{1} + 70152806\)

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
4.06882
3.77963
2.75355
2.43337
2.05101
1.17201
0.676512
0.113262
−0.656329
−1.34829
−1.40985
−1.97610
−3.01123
−3.53778
−4.10857
1.00000 −1.00000 1.00000 −4.06882 −1.00000 4.31915 1.00000 1.00000 −4.06882
1.2 1.00000 −1.00000 1.00000 −3.77963 −1.00000 2.13919 1.00000 1.00000 −3.77963
1.3 1.00000 −1.00000 1.00000 −2.75355 −1.00000 0.824838 1.00000 1.00000 −2.75355
1.4 1.00000 −1.00000 1.00000 −2.43337 −1.00000 −2.53842 1.00000 1.00000 −2.43337
1.5 1.00000 −1.00000 1.00000 −2.05101 −1.00000 −1.63593 1.00000 1.00000 −2.05101
1.6 1.00000 −1.00000 1.00000 −1.17201 −1.00000 −4.40989 1.00000 1.00000 −1.17201
1.7 1.00000 −1.00000 1.00000 −0.676512 −1.00000 −2.72085 1.00000 1.00000 −0.676512
1.8 1.00000 −1.00000 1.00000 −0.113262 −1.00000 1.75034 1.00000 1.00000 −0.113262
1.9 1.00000 −1.00000 1.00000 0.656329 −1.00000 2.89208 1.00000 1.00000 0.656329
1.10 1.00000 −1.00000 1.00000 1.34829 −1.00000 −1.25683 1.00000 1.00000 1.34829
1.11 1.00000 −1.00000 1.00000 1.40985 −1.00000 4.86981 1.00000 1.00000 1.40985
1.12 1.00000 −1.00000 1.00000 1.97610 −1.00000 4.65524 1.00000 1.00000 1.97610
1.13 1.00000 −1.00000 1.00000 3.01123 −1.00000 −0.871929 1.00000 1.00000 3.01123
1.14 1.00000 −1.00000 1.00000 3.53778 −1.00000 −4.09818 1.00000 1.00000 3.53778
1.15 1.00000 −1.00000 1.00000 4.10857 −1.00000 1.08136 1.00000 1.00000 4.10857
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(1\)
\(103\) \(1\)

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 8034.2.a.bc 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.bc 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{15} + \cdots\)
\(T_{7}^{15} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{15} \)
$3$ \( ( 1 + T )^{15} \)
$5$ \( -2048 - 17600 T + 11840 T^{2} + 65434 T^{3} - 21747 T^{4} - 71361 T^{5} + 16428 T^{6} + 33191 T^{7} - 5112 T^{8} - 7580 T^{9} + 707 T^{10} + 872 T^{11} - 44 T^{12} - 48 T^{13} + T^{14} + T^{15} \)
$7$ \( 211634 - 81965 T - 603653 T^{2} + 198660 T^{3} + 590741 T^{4} - 164378 T^{5} - 255993 T^{6} + 63048 T^{7} + 53846 T^{8} - 12093 T^{9} - 5625 T^{10} + 1187 T^{11} + 275 T^{12} - 56 T^{13} - 5 T^{14} + T^{15} \)
$11$ \( 12059584 - 26416752 T - 10896792 T^{2} + 20631220 T^{3} + 4172728 T^{4} - 6334617 T^{5} - 871537 T^{6} + 989273 T^{7} + 104586 T^{8} - 85010 T^{9} - 6959 T^{10} + 4049 T^{11} + 233 T^{12} - 100 T^{13} - 3 T^{14} + T^{15} \)
$13$ \( ( 1 + T )^{15} \)
$17$ \( -19422208 + 6641856 T + 49568000 T^{2} + 2454736 T^{3} - 36241488 T^{4} - 10412508 T^{5} + 6405224 T^{6} + 2262971 T^{7} - 467737 T^{8} - 188053 T^{9} + 16523 T^{10} + 7434 T^{11} - 285 T^{12} - 140 T^{13} + 2 T^{14} + T^{15} \)
$19$ \( -186761216 - 835273216 T - 435859520 T^{2} + 322671264 T^{3} + 208704000 T^{4} - 44973064 T^{5} - 31898808 T^{6} + 4016080 T^{7} + 2283443 T^{8} - 252310 T^{9} - 81280 T^{10} + 9271 T^{11} + 1332 T^{12} - 160 T^{13} - 8 T^{14} + T^{15} \)
$23$ \( -98264192 - 25657684 T + 507218156 T^{2} + 669085305 T^{3} + 148143115 T^{4} - 138242288 T^{5} - 40556213 T^{6} + 14657341 T^{7} + 2649737 T^{8} - 763725 T^{9} - 69372 T^{10} + 19293 T^{11} + 772 T^{12} - 227 T^{13} - 3 T^{14} + T^{15} \)
$29$ \( -1382105088 + 3171667200 T + 852181376 T^{2} - 2793311962 T^{3} - 623918523 T^{4} + 542028763 T^{5} + 59541620 T^{6} - 49846291 T^{7} + 598863 T^{8} + 1882716 T^{9} - 149313 T^{10} - 27039 T^{11} + 3678 T^{12} + 57 T^{13} - 26 T^{14} + T^{15} \)
$31$ \( 274726912 - 1793216512 T + 2848839680 T^{2} + 321641536 T^{3} - 1622679040 T^{4} - 398085192 T^{5} + 143483680 T^{6} + 43772014 T^{7} - 4457097 T^{8} - 1762934 T^{9} + 56422 T^{10} + 33069 T^{11} - 248 T^{12} - 294 T^{13} + T^{15} \)
$37$ \( 800768 - 24411264 T - 60870064 T^{2} - 12991088 T^{3} + 53743844 T^{4} + 27920318 T^{5} - 9963249 T^{6} - 6481973 T^{7} + 882362 T^{8} + 517128 T^{9} - 65149 T^{10} - 14194 T^{11} + 2217 T^{12} + 82 T^{13} - 25 T^{14} + T^{15} \)
$41$ \( -557822512 + 2117967748 T - 1667961000 T^{2} - 1761234527 T^{3} + 2629810936 T^{4} - 730947253 T^{5} - 161337307 T^{6} + 76521862 T^{7} + 2785481 T^{8} - 2717604 T^{9} - 8028 T^{10} + 44370 T^{11} - 163 T^{12} - 341 T^{13} + T^{14} + T^{15} \)
$43$ \( 777363968 + 3013408192 T + 4082915152 T^{2} + 2022396648 T^{3} - 164892500 T^{4} - 414998320 T^{5} - 50285743 T^{6} + 30139991 T^{7} + 4649569 T^{8} - 1130435 T^{9} - 149182 T^{10} + 23094 T^{11} + 2040 T^{12} - 241 T^{13} - 10 T^{14} + T^{15} \)
$47$ \( 251671604384 - 179973101476 T - 227647683061 T^{2} + 120061365158 T^{3} + 13347130444 T^{4} - 10762718210 T^{5} - 155256926 T^{6} + 408347005 T^{7} - 4600742 T^{8} - 8062187 T^{9} + 131445 T^{10} + 86319 T^{11} - 1119 T^{12} - 470 T^{13} + 3 T^{14} + T^{15} \)
$53$ \( -4553643008 - 1928338560 T + 6586494464 T^{2} + 5761846288 T^{3} + 667666000 T^{4} - 653654724 T^{5} - 146053630 T^{6} + 32552869 T^{7} + 8424949 T^{8} - 977557 T^{9} - 221098 T^{10} + 18948 T^{11} + 2741 T^{12} - 212 T^{13} - 13 T^{14} + T^{15} \)
$59$ \( 38526802432 - 36910800688 T - 9515961416 T^{2} + 15327670470 T^{3} - 1827735943 T^{4} - 1798493462 T^{5} + 569353163 T^{6} + 18108123 T^{7} - 33559566 T^{8} + 5277827 T^{9} - 23680 T^{10} - 71469 T^{11} + 6265 T^{12} + 43 T^{13} - 28 T^{14} + T^{15} \)
$61$ \( 158418815744 - 24496646816 T - 111450023808 T^{2} + 5435849840 T^{3} + 22184404668 T^{4} - 278244174 T^{5} - 1705861163 T^{6} + 1276149 T^{7} + 58953863 T^{8} - 407661 T^{9} - 990316 T^{10} + 20798 T^{11} + 7752 T^{12} - 271 T^{13} - 22 T^{14} + T^{15} \)
$67$ \( -756505865536 + 223907115568 T + 285838306892 T^{2} - 78667916521 T^{3} - 34612525792 T^{4} + 10119958710 T^{5} + 1431237998 T^{6} - 577384949 T^{7} + 387044 T^{8} + 13270678 T^{9} - 1010234 T^{10} - 72487 T^{11} + 10727 T^{12} - 106 T^{13} - 29 T^{14} + T^{15} \)
$71$ \( 290069168128 + 263299407872 T - 190714375552 T^{2} - 36005173056 T^{3} + 26558326082 T^{4} + 1341824049 T^{5} - 1538916877 T^{6} + 2182184 T^{7} + 45162651 T^{8} - 1200115 T^{9} - 707824 T^{10} + 28756 T^{11} + 5646 T^{12} - 279 T^{13} - 18 T^{14} + T^{15} \)
$73$ \( -2146208123936 - 1269354477672 T + 900628113572 T^{2} + 818830120359 T^{3} + 128965617924 T^{4} - 31040332813 T^{5} - 7699476724 T^{6} + 518071572 T^{7} + 168860455 T^{8} - 5959296 T^{9} - 1868678 T^{10} + 58654 T^{11} + 10442 T^{12} - 381 T^{13} - 23 T^{14} + T^{15} \)
$79$ \( 4164592205824 + 620204482560 T - 1111888407552 T^{2} - 126660170624 T^{3} + 108926072336 T^{4} + 9374914456 T^{5} - 5166508228 T^{6} - 288365088 T^{7} + 131587989 T^{8} + 3170417 T^{9} - 1787311 T^{10} + 7681 T^{11} + 11458 T^{12} - 275 T^{13} - 27 T^{14} + T^{15} \)
$83$ \( -96663104 - 1278306032 T - 5769787408 T^{2} - 10483274530 T^{3} - 7835490789 T^{4} - 1590526002 T^{5} + 499633473 T^{6} + 195247938 T^{7} - 2498400 T^{8} - 5812524 T^{9} - 155368 T^{10} + 74486 T^{11} + 2056 T^{12} - 442 T^{13} - 7 T^{14} + T^{15} \)
$89$ \( 3996849127424 - 4884918315008 T + 1514398846720 T^{2} + 386497753312 T^{3} - 305035852448 T^{4} + 40491036096 T^{5} + 7603515372 T^{6} - 2185093306 T^{7} + 37910079 T^{8} + 30075603 T^{9} - 2140063 T^{10} - 124724 T^{11} + 16291 T^{12} - 103 T^{13} - 35 T^{14} + T^{15} \)
$97$ \( 15882900389888 + 4053247485184 T - 3036495735936 T^{2} - 491363201888 T^{3} + 242631108432 T^{4} + 18269234200 T^{5} - 9528325688 T^{6} - 170103672 T^{7} + 189942955 T^{8} - 2881663 T^{9} - 1942876 T^{10} + 66089 T^{11} + 9742 T^{12} - 442 T^{13} - 19 T^{14} + T^{15} \)
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