Properties

Label 252.9.c.a
Level $252$
Weight $9$
Character orbit 252.c
Analytic conductor $102.659$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + 5613923146579405376 x^{10} + 2733728904154246859079616 x^{8} + 757873148017661341349205888000 x^{6} + 113644318422397913452531577312640000 x^{4} + 8098650340007618970326973663348480000000 x^{2} + 199066230990417435753898292645889849600000000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{37}\cdot 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} -\beta_{2} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} -\beta_{2} q^{7} + ( 3 \beta_{1} + 5 \beta_{9} + \beta_{10} ) q^{11} + ( -5968 + 2 \beta_{2} - \beta_{4} ) q^{13} + ( 8 \beta_{1} + 13 \beta_{9} - \beta_{12} ) q^{17} + ( -17972 - 7 \beta_{2} + \beta_{4} + \beta_{6} ) q^{19} + ( -25 \beta_{1} + 249 \beta_{9} - 6 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{23} + ( -109657 + 75 \beta_{2} + \beta_{3} + \beta_{4} ) q^{25} + ( 37 \beta_{1} - 15 \beta_{9} + 5 \beta_{10} + 4 \beta_{12} + 4 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{29} + ( -222140 + 182 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{31} + ( 113 \beta_{1} - 277 \beta_{9} + 7 \beta_{10} - 2 \beta_{11} + 7 \beta_{13} ) q^{35} + ( -11440 + 175 \beta_{2} + 2 \beta_{3} - 13 \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{37} + ( 397 \beta_{1} - 2385 \beta_{9} + 52 \beta_{10} - 3 \beta_{11} - 29 \beta_{12} + 4 \beta_{13} + 11 \beta_{14} + 2 \beta_{15} ) q^{41} + ( 529524 - 322 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 25 \beta_{6} + \beta_{7} ) q^{43} + ( -51 \beta_{1} - 137 \beta_{9} - 47 \beta_{10} - 14 \beta_{11} - \beta_{12} + 5 \beta_{13} - 2 \beta_{14} - 27 \beta_{15} ) q^{47} + 823543 q^{49} + ( 106 \beta_{1} - 3501 \beta_{9} + 191 \beta_{10} + 9 \beta_{11} + 74 \beta_{12} - 8 \beta_{13} - 24 \beta_{14} + 27 \beta_{15} ) q^{53} + ( -1168324 + 2152 \beta_{2} + 4 \beta_{3} - 155 \beta_{4} + 4 \beta_{5} - 22 \beta_{6} - 15 \beta_{7} + 16 \beta_{8} ) q^{55} + ( -1417 \beta_{1} + 1093 \beta_{9} - 317 \beta_{10} + 24 \beta_{11} - 35 \beta_{12} - 79 \beta_{13} - 70 \beta_{14} + 15 \beta_{15} ) q^{59} + ( 2139118 + 493 \beta_{2} - 12 \beta_{3} + 177 \beta_{4} + 11 \beta_{5} - 74 \beta_{6} + 14 \beta_{7} - 12 \beta_{8} ) q^{61} + ( -4553 \beta_{1} + 8056 \beta_{9} - 628 \beta_{10} - 77 \beta_{11} - 8 \beta_{12} + 4 \beta_{13} + 157 \beta_{14} - 46 \beta_{15} ) q^{65} + ( 1417835 - 852 \beta_{2} - 20 \beta_{3} + 274 \beta_{4} - 9 \beta_{5} - 58 \beta_{6} + 56 \beta_{7} + 9 \beta_{8} ) q^{67} + ( -5797 \beta_{1} + 19069 \beta_{9} + 470 \beta_{10} + 19 \beta_{11} + 44 \beta_{12} - 140 \beta_{13} - 16 \beta_{14} + 93 \beta_{15} ) q^{71} + ( 133846 + 3698 \beta_{2} - 48 \beta_{3} + 560 \beta_{4} - 11 \beta_{5} + 35 \beta_{6} - 75 \beta_{7} - 30 \beta_{8} ) q^{73} + ( 3674 \beta_{1} - 283 \beta_{9} + 413 \beta_{10} - 12 \beta_{11} - 98 \beta_{12} - 28 \beta_{13} - 49 \beta_{14} - 49 \beta_{15} ) q^{77} + ( -5640793 + 7110 \beta_{2} - 76 \beta_{3} - 852 \beta_{4} + 47 \beta_{5} - 120 \beta_{6} - 8 \beta_{7} - 31 \beta_{8} ) q^{79} + ( -921 \beta_{1} + 35145 \beta_{9} + 83 \beta_{10} - 216 \beta_{11} - 117 \beta_{12} - 21 \beta_{13} + 186 \beta_{14} - 103 \beta_{15} ) q^{83} + ( -3513030 + 18881 \beta_{2} + 115 \beta_{3} - 575 \beta_{4} - 178 \beta_{5} + 126 \beta_{6} + 18 \beta_{7} + 72 \beta_{8} ) q^{85} + ( 12261 \beta_{1} - 50351 \beta_{9} + 224 \beta_{10} - 173 \beta_{11} + 185 \beta_{12} + 192 \beta_{13} - 275 \beta_{14} + 162 \beta_{15} ) q^{89} + ( -1603721 + 5994 \beta_{2} + 49 \beta_{3} + 245 \beta_{4} + 49 \beta_{5} + 49 \beta_{6} + 49 \beta_{8} ) q^{91} + ( -6992 \beta_{1} + 12620 \beta_{9} + 2004 \beta_{10} - 384 \beta_{11} - 8 \beta_{12} - 12 \beta_{13} - 168 \beta_{14} - 220 \beta_{15} ) q^{95} + ( 8376986 + 36710 \beta_{2} + 116 \beta_{3} - 1138 \beta_{4} + 163 \beta_{5} + 333 \beta_{6} + 107 \beta_{7} - 34 \beta_{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 95480q^{13} - 287560q^{19} - 1754520q^{25} - 3554264q^{31} - 182920q^{37} + 8472416q^{43} + 13176688q^{49} - 18692072q^{55} + 34224568q^{61} + 22683096q^{67} + 2137296q^{73} - 90245624q^{79} - 56204456q^{85} - 25661888q^{91} + 134041152q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + 5613923146579405376 x^{10} + 2733728904154246859079616 x^{8} + 757873148017661341349205888000 x^{6} + 113644318422397913452531577312640000 x^{4} + 8098650340007618970326973663348480000000 x^{2} + 199066230990417435753898292645889849600000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-71173663779390023467793462232458441 \nu^{14} - 242104188371258784610229706855597574948340 \nu^{12} - 314902528771961841476819840094458429066618967396 \nu^{10} - 193745781204085695875675850361395817074968610107984896 \nu^{8} - 55895231301428433747023175003241069137052455993022307550336 \nu^{6} - 6204011324979878958172841529724215367458896960156087463650042880 \nu^{4} - 52637061029614541929216935639161106503544739578405220709926055040000 \nu^{2} + 13188579339515187011894893589407560749858000069808994059959306035302400000\)\()/ \)\(70\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(28\!\cdots\!67\)\( \nu^{14} + \)\(69\!\cdots\!20\)\( \nu^{12} - \)\(18\!\cdots\!48\)\( \nu^{10} - \)\(34\!\cdots\!08\)\( \nu^{8} - \)\(24\!\cdots\!28\)\( \nu^{6} - \)\(76\!\cdots\!00\)\( \nu^{4} - \)\(91\!\cdots\!00\)\( \nu^{2} - \)\(54\!\cdots\!00\)\(\)\()/ \)\(64\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(45\!\cdots\!33\)\( \nu^{14} + \)\(16\!\cdots\!80\)\( \nu^{12} + \)\(23\!\cdots\!48\)\( \nu^{10} + \)\(16\!\cdots\!08\)\( \nu^{8} + \)\(62\!\cdots\!28\)\( \nu^{6} + \)\(11\!\cdots\!00\)\( \nu^{4} + \)\(10\!\cdots\!00\)\( \nu^{2} + \)\(28\!\cdots\!00\)\(\)\()/ \)\(64\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(73\!\cdots\!19\)\( \nu^{14} + \)\(25\!\cdots\!40\)\( \nu^{12} + \)\(34\!\cdots\!64\)\( \nu^{10} + \)\(23\!\cdots\!44\)\( \nu^{8} + \)\(76\!\cdots\!04\)\( \nu^{6} + \)\(12\!\cdots\!00\)\( \nu^{4} + \)\(85\!\cdots\!00\)\( \nu^{2} + \)\(21\!\cdots\!00\)\(\)\()/ \)\(57\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(43\!\cdots\!59\)\( \nu^{14} + \)\(15\!\cdots\!56\)\( \nu^{12} + \)\(22\!\cdots\!64\)\( \nu^{10} + \)\(15\!\cdots\!80\)\( \nu^{8} + \)\(57\!\cdots\!60\)\( \nu^{6} + \)\(10\!\cdots\!56\)\( \nu^{4} + \)\(84\!\cdots\!00\)\( \nu^{2} + \)\(16\!\cdots\!00\)\(\)\()/ \)\(25\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(24\!\cdots\!27\)\( \nu^{14} + \)\(86\!\cdots\!20\)\( \nu^{12} + \)\(12\!\cdots\!12\)\( \nu^{10} + \)\(88\!\cdots\!52\)\( \nu^{8} + \)\(34\!\cdots\!32\)\( \nu^{6} + \)\(70\!\cdots\!00\)\( \nu^{4} + \)\(71\!\cdots\!00\)\( \nu^{2} + \)\(24\!\cdots\!00\)\(\)\()/ \)\(57\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(11\!\cdots\!93\)\( \nu^{14} + \)\(44\!\cdots\!30\)\( \nu^{12} + \)\(66\!\cdots\!08\)\( \nu^{10} + \)\(49\!\cdots\!68\)\( \nu^{8} + \)\(19\!\cdots\!88\)\( \nu^{6} + \)\(37\!\cdots\!00\)\( \nu^{4} + \)\(32\!\cdots\!00\)\( \nu^{2} + \)\(90\!\cdots\!00\)\(\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(11\!\cdots\!07\)\( \nu^{15} + \)\(40\!\cdots\!40\)\( \nu^{13} + \)\(53\!\cdots\!92\)\( \nu^{11} + \)\(33\!\cdots\!52\)\( \nu^{9} + \)\(10\!\cdots\!32\)\( \nu^{7} + \)\(15\!\cdots\!20\)\( \nu^{5} + \)\(11\!\cdots\!00\)\( \nu^{3} + \)\(63\!\cdots\!00\)\( \nu\)\()/ \)\(20\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(10\!\cdots\!53\)\( \nu^{15} + \)\(39\!\cdots\!30\)\( \nu^{13} + \)\(57\!\cdots\!68\)\( \nu^{11} + \)\(42\!\cdots\!28\)\( \nu^{9} + \)\(17\!\cdots\!48\)\( \nu^{7} + \)\(41\!\cdots\!00\)\( \nu^{5} + \)\(46\!\cdots\!00\)\( \nu^{3} + \)\(18\!\cdots\!00\)\( \nu\)\()/ \)\(44\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(89\!\cdots\!81\)\( \nu^{15} + \)\(36\!\cdots\!60\)\( \nu^{13} + \)\(61\!\cdots\!36\)\( \nu^{11} + \)\(53\!\cdots\!56\)\( \nu^{9} + \)\(25\!\cdots\!96\)\( \nu^{7} + \)\(68\!\cdots\!00\)\( \nu^{5} + \)\(87\!\cdots\!00\)\( \nu^{3} + \)\(36\!\cdots\!00\)\( \nu\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(14\!\cdots\!09\)\( \nu^{15} + \)\(55\!\cdots\!00\)\( \nu^{13} + \)\(88\!\cdots\!04\)\( \nu^{11} + \)\(72\!\cdots\!44\)\( \nu^{9} + \)\(32\!\cdots\!04\)\( \nu^{7} + \)\(75\!\cdots\!60\)\( \nu^{5} + \)\(84\!\cdots\!00\)\( \nu^{3} + \)\(32\!\cdots\!00\)\( \nu\)\()/ \)\(99\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(14\!\cdots\!61\)\( \nu^{15} + \)\(48\!\cdots\!60\)\( \nu^{13} + \)\(64\!\cdots\!16\)\( \nu^{11} + \)\(40\!\cdots\!36\)\( \nu^{9} + \)\(12\!\cdots\!76\)\( \nu^{7} + \)\(16\!\cdots\!00\)\( \nu^{5} + \)\(69\!\cdots\!00\)\( \nu^{3} + \)\(43\!\cdots\!00\)\( \nu\)\()/ \)\(89\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(47\!\cdots\!07\)\( \nu^{15} - \)\(17\!\cdots\!20\)\( \nu^{13} - \)\(24\!\cdots\!92\)\( \nu^{11} - \)\(16\!\cdots\!32\)\( \nu^{9} - \)\(59\!\cdots\!12\)\( \nu^{7} - \)\(10\!\cdots\!00\)\( \nu^{5} - \)\(69\!\cdots\!00\)\( \nu^{3} - \)\(11\!\cdots\!00\)\( \nu\)\()/ \)\(89\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(84\!\cdots\!87\)\( \nu^{15} - \)\(31\!\cdots\!20\)\( \nu^{13} - \)\(45\!\cdots\!72\)\( \nu^{11} - \)\(33\!\cdots\!12\)\( \nu^{9} - \)\(12\!\cdots\!92\)\( \nu^{7} - \)\(25\!\cdots\!00\)\( \nu^{5} - \)\(23\!\cdots\!00\)\( \nu^{3} - \)\(58\!\cdots\!00\)\( \nu\)\()/ \)\(14\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 75 \beta_{2} - 500282\)
\(\nu^{3}\)\(=\)\(502 \beta_{15} - 31 \beta_{14} + 460 \beta_{13} + 1454 \beta_{12} - 1513 \beta_{11} + 2748 \beta_{10} + 3882 \beta_{9} - 735213 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-2788 \beta_{8} - 27244 \beta_{7} - 545604 \beta_{6} + 558072 \beta_{5} - 175942 \beta_{4} - 1126350 \beta_{3} - 51984002 \beta_{2} + 368645611096\)
\(\nu^{5}\)\(=\)\(-454235776 \beta_{15} - 131527410 \beta_{14} - 890878772 \beta_{13} - 1720652240 \beta_{12} + 2101192890 \beta_{11} - 4021873940 \beta_{10} + 74457983824 \beta_{9} + 625133110974 \beta_{1}\)
\(\nu^{6}\)\(=\)\(12957774648 \beta_{8} + 76508573832 \beta_{7} + 755360666664 \beta_{6} - 996846903360 \beta_{5} - 376664708772 \beta_{4} + 1133759094108 \beta_{3} + 20970015303524 \beta_{2} - 311561442089858368\)
\(\nu^{7}\)\(=\)\(419814568152048 \beta_{15} + 242553119836164 \beta_{14} + 1272036012192088 \beta_{13} + 1822295094507024 \beta_{12} - 2325503133142964 \beta_{11} + 4720449650485144 \beta_{10} - 142795125562167088 \beta_{9} - 565229652740887628 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-24300550853282320 \beta_{8} - 115730134049941296 \beta_{7} - 810552288052644880 \beta_{6} + 1353239350272672480 \beta_{5} + 638930265334840776 \beta_{4} - 1120566624945267928 \beta_{3} + 8436963296110730776 \beta_{2} + 279732038816012714757984\)
\(\nu^{9}\)\(=\)\(-419844729271930170176 \beta_{15} - 301449942340117738984 \beta_{14} - 1596532712934071597648 \beta_{13} - 1887306013763475128704 \beta_{12} + 2411812329057741419528 \beta_{11} - 5155874869627231021008 \beta_{10} + 202991333690869679066624 \beta_{9} + 527675197972752739783128 \beta_{1}\)
\(\nu^{10}\)\(=\)\(34151585857558388607200 \beta_{8} + 145095124722096531367712 \beta_{7} + 808222054135278332716320 \beta_{6} - 1660357773360629278883712 \beta_{5} - 730942869928178755835152 \beta_{4} + 1105617541548661372258416 \beta_{3} - 34438222040119161109853552 \beta_{2} - 259257570553545648325229474816\)
\(\nu^{11}\)\(=\)\(439271077232477982505908416 \beta_{15} + 329835753158374584500199312 \beta_{14} + 1872792439599262038315655264 \beta_{13} + 1939777505886012994684132672 \beta_{12} - 2445949442235071971798133328 \beta_{11} + 5500097080850594751175413088 \beta_{10} - 257247730206750326248644717248 \beta_{9} - 502129741360547241253966058928 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-41615046277868231748108352320 \beta_{8} - 168255734613429200062581876672 \beta_{7} - 787205822230314478986660755520 \beta_{6} + 1932491479844842889616247621248 \beta_{5} + 733125811486095164872699154208 \beta_{4} - 1093457578428140521406042364000 \beta_{3} + 57136563170499626210327172267872 \beta_{2} + 244940412902260940478140508255490688\)
\(\nu^{13}\)\(=\)\(-467883656638019677319311316199936 \beta_{15} - 342202748791706764352531662607520 \beta_{14} - 2111374569899557173867148565530688 \beta_{13} - 1985650637182304365876556888666880 \beta_{12} + 2462629740575091385637901078312736 \beta_{11} - 5829208644364858660615420414838336 \beta_{10} + 306447048514414161805980011220381440 \beta_{9} + 484039561325792091921298417854665056 \beta_{1}\)
\(\nu^{14}\)\(=\)\(46673225177018123929881679035681152 \beta_{8} + 187701160155906473825235472375304832 \beta_{7} + 762636284332136368450364957164459136 \beta_{6} - 2176944720626448807317959800484758528 \beta_{5} - 688659851211179286903517876257916992 \beta_{4} + 1085197475932710527211573168112416704 \beta_{3} - 77044489276884416196280379720444485568 \beta_{2} - 234495423149875137169825655953308061678080\)
\(\nu^{15}\)\(=\)\(500440623325978058417375784357990614272 \beta_{15} + 346298417468215031944643904188915405888 \beta_{14} + 2321059610219648473993029293230883671936 \beta_{13} + 2027321206761073699688044917147344492288 \beta_{12} - 2476217717507576804056997690851674354496 \beta_{11} + 6171047471759462917712088906448029553536 \beta_{10} - 351188771919751586634913856939116052201728 \beta_{9} - 471142407663789631927311528662499586260672 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1013.87i
941.588i
899.394i
675.810i
640.282i
491.437i
348.567i
221.696i
221.696i
348.567i
491.437i
640.282i
675.810i
899.394i
941.588i
1013.87i
0 0 0 1013.87i 0 −907.493 0 0 0
197.2 0 0 0 941.588i 0 907.493 0 0 0
197.3 0 0 0 899.394i 0 907.493 0 0 0
197.4 0 0 0 675.810i 0 907.493 0 0 0
197.5 0 0 0 640.282i 0 −907.493 0 0 0
197.6 0 0 0 491.437i 0 −907.493 0 0 0
197.7 0 0 0 348.567i 0 907.493 0 0 0
197.8 0 0 0 221.696i 0 −907.493 0 0 0
197.9 0 0 0 221.696i 0 −907.493 0 0 0
197.10 0 0 0 348.567i 0 907.493 0 0 0
197.11 0 0 0 491.437i 0 −907.493 0 0 0
197.12 0 0 0 640.282i 0 −907.493 0 0 0
197.13 0 0 0 675.810i 0 907.493 0 0 0
197.14 0 0 0 899.394i 0 907.493 0 0 0
197.15 0 0 0 941.588i 0 907.493 0 0 0
197.16 0 0 0 1013.87i 0 −907.493 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.c.a 16
3.b odd 2 1 inner 252.9.c.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.9.c.a 16 1.a even 1 1 trivial
252.9.c.a 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( \)\(19\!\cdots\!00\)\( + \)\(80\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{4} + \)\(75\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!16\)\( T^{8} + 5613923146579405376 T^{10} + 6534459751956 T^{12} + 4002260 T^{14} + T^{16} \)
$7$ \( ( -823543 + T^{2} )^{8} \)
$11$ \( \)\(12\!\cdots\!64\)\( + \)\(87\!\cdots\!16\)\( T^{2} + \)\(23\!\cdots\!20\)\( T^{4} + \)\(30\!\cdots\!24\)\( T^{6} + \)\(21\!\cdots\!28\)\( T^{8} + \)\(86\!\cdots\!68\)\( T^{10} + 1908139166863216476 T^{12} + 2178125420 T^{14} + T^{16} \)
$13$ \( ( \)\(25\!\cdots\!84\)\( - \)\(98\!\cdots\!36\)\( T - \)\(23\!\cdots\!56\)\( T^{2} + \)\(47\!\cdots\!52\)\( T^{3} + 1772033110952777776 T^{4} - 130023762630944 T^{5} - 3027093540 T^{6} + 47740 T^{7} + T^{8} )^{2} \)
$17$ \( \)\(19\!\cdots\!04\)\( + \)\(13\!\cdots\!40\)\( T^{2} + \)\(23\!\cdots\!96\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{6} + \)\(66\!\cdots\!56\)\( T^{8} + \)\(12\!\cdots\!16\)\( T^{10} + \)\(12\!\cdots\!24\)\( T^{12} + 56225030804 T^{14} + T^{16} \)
$19$ \( ( -\)\(33\!\cdots\!88\)\( + \)\(92\!\cdots\!48\)\( T + \)\(25\!\cdots\!96\)\( T^{2} + \)\(87\!\cdots\!92\)\( T^{3} + \)\(41\!\cdots\!24\)\( T^{4} - 7317713037741568 T^{5} - 48262861352 T^{6} + 143780 T^{7} + T^{8} )^{2} \)
$23$ \( \)\(37\!\cdots\!36\)\( + \)\(39\!\cdots\!68\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{4} + \)\(19\!\cdots\!96\)\( T^{6} + \)\(11\!\cdots\!44\)\( T^{8} + \)\(22\!\cdots\!96\)\( T^{10} + \)\(18\!\cdots\!76\)\( T^{12} + 705510724076 T^{14} + T^{16} \)
$29$ \( \)\(17\!\cdots\!36\)\( + \)\(89\!\cdots\!20\)\( T^{2} + \)\(39\!\cdots\!96\)\( T^{4} + \)\(58\!\cdots\!60\)\( T^{6} + \)\(39\!\cdots\!96\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{10} + \)\(27\!\cdots\!64\)\( T^{12} + 2612912974304 T^{14} + T^{16} \)
$31$ \( ( -\)\(23\!\cdots\!16\)\( + \)\(11\!\cdots\!04\)\( T + \)\(31\!\cdots\!36\)\( T^{2} + \)\(10\!\cdots\!12\)\( T^{3} - \)\(30\!\cdots\!08\)\( T^{4} - 2759908676721155456 T^{5} - 944893972488 T^{6} + 1777132 T^{7} + T^{8} )^{2} \)
$37$ \( ( \)\(14\!\cdots\!76\)\( + \)\(12\!\cdots\!92\)\( T - \)\(29\!\cdots\!84\)\( T^{2} - \)\(75\!\cdots\!52\)\( T^{3} + \)\(31\!\cdots\!20\)\( T^{4} + 114585627391285952 T^{5} - 10347323581004 T^{6} + 91460 T^{7} + T^{8} )^{2} \)
$41$ \( \)\(21\!\cdots\!76\)\( + \)\(29\!\cdots\!80\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{4} + \)\(15\!\cdots\!72\)\( T^{6} + \)\(83\!\cdots\!80\)\( T^{8} + \)\(17\!\cdots\!16\)\( T^{10} + \)\(16\!\cdots\!88\)\( T^{12} + 67902135964596 T^{14} + T^{16} \)
$43$ \( ( -\)\(16\!\cdots\!52\)\( - \)\(67\!\cdots\!92\)\( T + \)\(18\!\cdots\!64\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!72\)\( T^{4} + \)\(15\!\cdots\!08\)\( T^{5} - 34995094103120 T^{6} - 4236208 T^{7} + T^{8} )^{2} \)
$47$ \( \)\(10\!\cdots\!16\)\( + \)\(15\!\cdots\!04\)\( T^{2} + \)\(81\!\cdots\!04\)\( T^{4} + \)\(20\!\cdots\!76\)\( T^{6} + \)\(27\!\cdots\!24\)\( T^{8} + \)\(18\!\cdots\!60\)\( T^{10} + \)\(71\!\cdots\!96\)\( T^{12} + 134018826761808 T^{14} + T^{16} \)
$53$ \( \)\(21\!\cdots\!36\)\( + \)\(92\!\cdots\!24\)\( T^{2} + \)\(70\!\cdots\!44\)\( T^{4} + \)\(21\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!36\)\( T^{8} + \)\(20\!\cdots\!00\)\( T^{10} + \)\(47\!\cdots\!36\)\( T^{12} + 384163508021040 T^{14} + T^{16} \)
$59$ \( \)\(41\!\cdots\!56\)\( + \)\(31\!\cdots\!48\)\( T^{2} + \)\(14\!\cdots\!64\)\( T^{4} + \)\(27\!\cdots\!12\)\( T^{6} + \)\(28\!\cdots\!88\)\( T^{8} + \)\(17\!\cdots\!52\)\( T^{10} + \)\(62\!\cdots\!60\)\( T^{12} + 1218346300712144 T^{14} + T^{16} \)
$61$ \( ( \)\(30\!\cdots\!56\)\( - \)\(47\!\cdots\!92\)\( T - \)\(46\!\cdots\!60\)\( T^{2} + \)\(65\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!92\)\( T^{4} + \)\(34\!\cdots\!00\)\( T^{5} - 585031183295904 T^{6} - 17112284 T^{7} + T^{8} )^{2} \)
$67$ \( ( -\)\(37\!\cdots\!08\)\( + \)\(62\!\cdots\!80\)\( T - \)\(10\!\cdots\!04\)\( T^{2} - \)\(45\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!00\)\( T^{4} + \)\(15\!\cdots\!88\)\( T^{5} - 2115446744573908 T^{6} - 11341548 T^{7} + T^{8} )^{2} \)
$71$ \( \)\(31\!\cdots\!04\)\( + \)\(31\!\cdots\!08\)\( T^{2} + \)\(22\!\cdots\!56\)\( T^{4} + \)\(54\!\cdots\!24\)\( T^{6} + \)\(49\!\cdots\!16\)\( T^{8} + \)\(20\!\cdots\!68\)\( T^{10} + \)\(41\!\cdots\!28\)\( T^{12} + 3536901649565612 T^{14} + T^{16} \)
$73$ \( ( \)\(26\!\cdots\!08\)\( + \)\(56\!\cdots\!96\)\( T - \)\(52\!\cdots\!84\)\( T^{2} - \)\(80\!\cdots\!80\)\( T^{3} + \)\(92\!\cdots\!76\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{5} - 5373802513161172 T^{6} - 1068648 T^{7} + T^{8} )^{2} \)
$79$ \( ( -\)\(18\!\cdots\!36\)\( + \)\(43\!\cdots\!60\)\( T + \)\(23\!\cdots\!76\)\( T^{2} + \)\(21\!\cdots\!24\)\( T^{3} + \)\(40\!\cdots\!64\)\( T^{4} - \)\(18\!\cdots\!16\)\( T^{5} - 4860777234507252 T^{6} + 45122812 T^{7} + T^{8} )^{2} \)
$83$ \( \)\(10\!\cdots\!44\)\( + \)\(30\!\cdots\!52\)\( T^{2} + \)\(12\!\cdots\!68\)\( T^{4} + \)\(11\!\cdots\!60\)\( T^{6} + \)\(35\!\cdots\!52\)\( T^{8} + \)\(47\!\cdots\!04\)\( T^{10} + \)\(30\!\cdots\!88\)\( T^{12} + 8886612474957440 T^{14} + T^{16} \)
$89$ \( \)\(20\!\cdots\!24\)\( + \)\(14\!\cdots\!20\)\( T^{2} + \)\(15\!\cdots\!16\)\( T^{4} + \)\(59\!\cdots\!80\)\( T^{6} + \)\(85\!\cdots\!96\)\( T^{8} + \)\(51\!\cdots\!52\)\( T^{10} + \)\(14\!\cdots\!40\)\( T^{12} + 19937169420745428 T^{14} + T^{16} \)
$97$ \( ( -\)\(21\!\cdots\!04\)\( - \)\(31\!\cdots\!80\)\( T - \)\(95\!\cdots\!16\)\( T^{2} + \)\(29\!\cdots\!44\)\( T^{3} + \)\(35\!\cdots\!60\)\( T^{4} + \)\(94\!\cdots\!84\)\( T^{5} - 32999447726614276 T^{6} - 67020576 T^{7} + T^{8} )^{2} \)
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