Properties

Label 2.26.a.b
Level 2
Weight 26
Character orbit 2.a
Self dual Yes
Analytic conductor 7.920
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 26 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(7.91993559904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4800\sqrt{106705}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 4096 q^{2} \) \( + ( 189924 - \beta ) q^{3} \) \( + 16777216 q^{4} \) \( + ( 370976550 + 324 \beta ) q^{5} \) \( + ( 777928704 - 4096 \beta ) q^{6} \) \( + ( -188268472 + 19278 \beta ) q^{7} \) \( + 68719476736 q^{8} \) \( + ( 1647265716333 - 379848 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \(+4096 q^{2}\) \(+(189924 - \beta) q^{3}\) \(+16777216 q^{4}\) \(+(370976550 + 324 \beta) q^{5}\) \(+(777928704 - 4096 \beta) q^{6}\) \(+(-188268472 + 19278 \beta) q^{7}\) \(+68719476736 q^{8}\) \(+(1647265716333 - 379848 \beta) q^{9}\) \(+(1519519948800 + 1327104 \beta) q^{10}\) \(+(4161517305132 + 1004157 \beta) q^{11}\) \(+(3186395971584 - 16777216 \beta) q^{12}\) \(+(-53233526576146 + 27104868 \beta) q^{13}\) \(+(-771147661312 + 78962688 \beta) q^{14}\) \(+(-726091206517800 - 309441174 \beta) q^{15}\) \(+281474976710656 q^{16}\) \(+(663939460056978 + 1473758712 \beta) q^{17}\) \(+(6747200374099968 - 1555857408 \beta) q^{18}\) \(+(-238539621474700 - 3309301413 \beta) q^{19}\) \(+(6223953710284800 + 5435817984 \beta) q^{20}\) \(+(-47430395830876128 + 3849623344 \beta) q^{21}\) \(+(17045574881820672 + 4113027072 \beta) q^{22}\) \(+(-57652512890736936 - 29903612886 \beta) q^{23}\) \(+(13051477899608064 - 68719476736 \beta) q^{24}\) \(+(97682109176149375 + 240392804400 \beta) q^{25}\) \(+(-218044524855894016 + 111021539328 \beta) q^{26}\) \(+(1085784778602576360 - 872119358442 \beta) q^{27}\) \(+(-3158620820733952 + 323431170048 \beta) q^{28}\) \(+(862206322603217790 + 1421989013844 \beta) q^{29}\) \(+(-2974069581896908800 - 1267471048704 \beta) q^{30}\) \(+(-4344041144175563488 + 48442188024 \beta) q^{31}\) \(+1152921504606846976 q^{32}\) \(+(-1678331102002510032 - 3970803791064 \beta) q^{33}\) \(+(2719496028393381888 + 6036515684352 \beta) q^{34}\) \(+(15286019889774068400 + 7090686945972 \beta) q^{35}\) \(+(27636532732313468928 - 6372791943168 \beta) q^{36}\) \(+(-17454182024875085242 - 17633536751628 \beta) q^{37}\) \(+(-977058289560371200 - 13554898587648 \beta) q^{38}\) \(+(-76747186917665552904 + 58381391526178 \beta) q^{39}\) \(+(25493314397326540800 + 22265110462464 \beta) q^{40}\) \(+(41507162177976734442 - 74734027646544 \beta) q^{41}\) \(+(-\)\(19\!\cdots\!88\)\( + 15768057217024 \beta) q^{42}\) \(+(22269804291735950924 - 102876456293163 \beta) q^{43}\) \(+(69818674715937472512 + 16846958886912 \beta) q^{44}\) \(+(\)\(30\!\cdots\!50\)\( + 392799391527492 \beta) q^{45}\) \(+(-\)\(23\!\cdots\!56\)\( - 122485198381056 \beta) q^{46}\) \(+(\)\(93\!\cdots\!88\)\( - 298064907420012 \beta) q^{47}\) \(+(53458853476794630144 - 281474976710656 \beta) q^{48}\) \(+(-\)\(42\!\cdots\!23\)\( - 7258879206432 \beta) q^{49}\) \(+(\)\(40\!\cdots\!00\)\( + 984648926822400 \beta) q^{50}\) \(+(-\)\(34\!\cdots\!28\)\( - 384037310439090 \beta) q^{51}\) \(+(-\)\(89\!\cdots\!36\)\( + 454744225087488 \beta) q^{52}\) \(+(-\)\(16\!\cdots\!66\)\( + 241040164382388 \beta) q^{53}\) \(+(\)\(44\!\cdots\!60\)\( - 3572200892178432 \beta) q^{54}\) \(+(\)\(23\!\cdots\!00\)\( + 1720850306381118 \beta) q^{55}\) \(+(-12937710881726267392 + 1324774072516608 \beta) q^{56}\) \(+(\)\(80\!\cdots\!00\)\( - 389976140087912 \beta) q^{57}\) \(+(\)\(35\!\cdots\!40\)\( + 5824467000705024 \beta) q^{58}\) \(+(-\)\(87\!\cdots\!20\)\( - 2895442853816367 \beta) q^{59}\) \(+(-\)\(12\!\cdots\!00\)\( - 5191561415491584 \beta) q^{60}\) \(+(\)\(16\!\cdots\!22\)\( - 10985591131211772 \beta) q^{61}\) \(+(-\)\(17\!\cdots\!48\)\( + 198419202146304 \beta) q^{62}\) \(+(-\)\(18\!\cdots\!76\)\( + 31827501882019830 \beta) q^{63}\) \(+\)\(47\!\cdots\!96\)\( q^{64}\) \(+(\)\(18\!\cdots\!00\)\( - 7192392191825904 \beta) q^{65}\) \(+(-\)\(68\!\cdots\!72\)\( - 16264412328198144 \beta) q^{66}\) \(+(\)\(16\!\cdots\!08\)\( - 44770819592512881 \beta) q^{67}\) \(+(\)\(11\!\cdots\!48\)\( + 24725568243105792 \beta) q^{68}\) \(+(\)\(62\!\cdots\!36\)\( + 51973099116976272 \beta) q^{69}\) \(+(\)\(62\!\cdots\!00\)\( + 29043453730701312 \beta) q^{70}\) \(+(-\)\(13\!\cdots\!28\)\( - 56033526802716162 \beta) q^{71}\) \(+(\)\(11\!\cdots\!88\)\( - 26102955799216128 \beta) q^{72}\) \(+(\)\(15\!\cdots\!74\)\( + 77685040590196248 \beta) q^{73}\) \(+(-\)\(71\!\cdots\!32\)\( - 72226966534668288 \beta) q^{74}\) \(+(-\)\(57\!\cdots\!00\)\( - 52025746193283775 \beta) q^{75}\) \(+(-\)\(40\!\cdots\!00\)\( - 55520864615006208 \beta) q^{76}\) \(+(\)\(46\!\cdots\!96\)\( + 80036679504296592 \beta) q^{77}\) \(+(-\)\(31\!\cdots\!84\)\( + 239130179691225088 \beta) q^{78}\) \(+(\)\(46\!\cdots\!60\)\( + 110134014723669132 \beta) q^{79}\) \(+(\)\(10\!\cdots\!00\)\( + 91197892454252544 \beta) q^{80}\) \(+(\)\(95\!\cdots\!21\)\( - 929580291915610104 \beta) q^{81}\) \(+(\)\(17\!\cdots\!32\)\( - 306110577240244224 \beta) q^{82}\) \(+(-\)\(22\!\cdots\!76\)\( + 907481271050993835 \beta) q^{83}\) \(+(-\)\(79\!\cdots\!48\)\( + 64585962360930304 \beta) q^{84}\) \(+(\)\(14\!\cdots\!00\)\( + 761846307568664472 \beta) q^{85}\) \(+(\)\(91\!\cdots\!04\)\( - 421381964976795648 \beta) q^{86}\) \(+(-\)\(33\!\cdots\!40\)\( - 592136481137909934 \beta) q^{87}\) \(+(\)\(28\!\cdots\!52\)\( + 69005143600791552 \beta) q^{88}\) \(+(-\)\(11\!\cdots\!10\)\( - 1306087630473617640 \beta) q^{89}\) \(+(\)\(12\!\cdots\!00\)\( + 1608906307696607232 \beta) q^{90}\) \(+(\)\(12\!\cdots\!12\)\( - 1031338917417064284 \beta) q^{91}\) \(+(-\)\(96\!\cdots\!76\)\( - 501699372568805376 \beta) q^{92}\) \(+(-\)\(94\!\cdots\!12\)\( + 4353241478293833664 \beta) q^{93}\) \(+(\)\(38\!\cdots\!48\)\( - 1220873860792369152 \beta) q^{94}\) \(+(-\)\(27\!\cdots\!00\)\( - 1304960058462667950 \beta) q^{95}\) \(+(\)\(21\!\cdots\!24\)\( - 1152921504606846976 \beta) q^{96}\) \(+(\)\(68\!\cdots\!18\)\( - 2065167228425089320 \beta) q^{97}\) \(+(-\)\(17\!\cdots\!08\)\( - 29732369229545472 \beta) q^{98}\) \(+(\)\(59\!\cdots\!56\)\( + 73369374596016345 \beta) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 8192q^{2} \) \(\mathstrut +\mathstrut 379848q^{3} \) \(\mathstrut +\mathstrut 33554432q^{4} \) \(\mathstrut +\mathstrut 741953100q^{5} \) \(\mathstrut +\mathstrut 1555857408q^{6} \) \(\mathstrut -\mathstrut 376536944q^{7} \) \(\mathstrut +\mathstrut 137438953472q^{8} \) \(\mathstrut +\mathstrut 3294531432666q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 8192q^{2} \) \(\mathstrut +\mathstrut 379848q^{3} \) \(\mathstrut +\mathstrut 33554432q^{4} \) \(\mathstrut +\mathstrut 741953100q^{5} \) \(\mathstrut +\mathstrut 1555857408q^{6} \) \(\mathstrut -\mathstrut 376536944q^{7} \) \(\mathstrut +\mathstrut 137438953472q^{8} \) \(\mathstrut +\mathstrut 3294531432666q^{9} \) \(\mathstrut +\mathstrut 3039039897600q^{10} \) \(\mathstrut +\mathstrut 8323034610264q^{11} \) \(\mathstrut +\mathstrut 6372791943168q^{12} \) \(\mathstrut -\mathstrut 106467053152292q^{13} \) \(\mathstrut -\mathstrut 1542295322624q^{14} \) \(\mathstrut -\mathstrut 1452182413035600q^{15} \) \(\mathstrut +\mathstrut 562949953421312q^{16} \) \(\mathstrut +\mathstrut 1327878920113956q^{17} \) \(\mathstrut +\mathstrut 13494400748199936q^{18} \) \(\mathstrut -\mathstrut 477079242949400q^{19} \) \(\mathstrut +\mathstrut 12447907420569600q^{20} \) \(\mathstrut -\mathstrut 94860791661752256q^{21} \) \(\mathstrut +\mathstrut 34091149763641344q^{22} \) \(\mathstrut -\mathstrut 115305025781473872q^{23} \) \(\mathstrut +\mathstrut 26102955799216128q^{24} \) \(\mathstrut +\mathstrut 195364218352298750q^{25} \) \(\mathstrut -\mathstrut 436089049711788032q^{26} \) \(\mathstrut +\mathstrut 2171569557205152720q^{27} \) \(\mathstrut -\mathstrut 6317241641467904q^{28} \) \(\mathstrut +\mathstrut 1724412645206435580q^{29} \) \(\mathstrut -\mathstrut 5948139163793817600q^{30} \) \(\mathstrut -\mathstrut 8688082288351126976q^{31} \) \(\mathstrut +\mathstrut 2305843009213693952q^{32} \) \(\mathstrut -\mathstrut 3356662204005020064q^{33} \) \(\mathstrut +\mathstrut 5438992056786763776q^{34} \) \(\mathstrut +\mathstrut 30572039779548136800q^{35} \) \(\mathstrut +\mathstrut 55273065464626937856q^{36} \) \(\mathstrut -\mathstrut 34908364049750170484q^{37} \) \(\mathstrut -\mathstrut 1954116579120742400q^{38} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!08\)\(q^{39} \) \(\mathstrut +\mathstrut 50986628794653081600q^{40} \) \(\mathstrut +\mathstrut 83014324355953468884q^{41} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!76\)\(q^{42} \) \(\mathstrut +\mathstrut 44539608583471901848q^{43} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!24\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!12\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!76\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!88\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(85\!\cdots\!46\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(80\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!56\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!72\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!32\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!20\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut 25875421763452534784q^{56} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!80\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!40\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!44\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!96\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!52\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!92\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!44\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!16\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!96\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!72\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!56\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!76\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!48\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!64\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!92\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!68\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!20\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!42\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!64\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!52\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!96\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!08\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!80\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!04\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!20\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!24\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!52\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!24\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!96\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!48\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!36\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!16\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!12\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
163.829
−162.829
4096.00 −1.37803e6 1.67772e7 8.78994e8 −5.64442e9 3.00388e10 6.87195e10 1.05168e12 3.60036e12
1.2 4096.00 1.75788e6 1.67772e7 −1.37041e8 7.20027e9 −3.04153e10 6.87195e10 2.24285e12 −5.61320e11
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut 379848 T_{3} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!24\)\( \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(2))\).