Properties

Label 3.13.b.b
Level 3
Weight 13
Character orbit 3.b
Analytic conductor 2.742
Analytic rank 0
Dimension 2
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 13 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.74198145183\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-26}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 18\sqrt{-26}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + ( -675 - 3 \beta ) q^{3} \) \( -4328 q^{4} \) \( + 230 \beta q^{5} \) \( + ( 25272 - 675 \beta ) q^{6} \) \( + 40250 q^{7} \) \( -232 \beta q^{8} \) \( + ( 379809 + 4050 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + ( -675 - 3 \beta ) q^{3} \) \( -4328 q^{4} \) \( + 230 \beta q^{5} \) \( + ( 25272 - 675 \beta ) q^{6} \) \( + 40250 q^{7} \) \( -232 \beta q^{8} \) \( + ( 379809 + 4050 \beta ) q^{9} \) \( -1937520 q^{10} \) \( -12650 \beta q^{11} \) \( + ( 2921400 + 12984 \beta ) q^{12} \) \( + 1284050 q^{13} \) \( + 40250 \beta q^{14} \) \( + ( 5812560 - 155250 \beta ) q^{15} \) \( -15773120 q^{16} \) \( + 161736 \beta q^{17} \) \( + ( -34117200 + 379809 \beta ) q^{18} \) \( + 53343578 q^{19} \) \( -995440 \beta q^{20} \) \( + ( -27168750 - 120750 \beta ) q^{21} \) \( + 106563600 q^{22} \) \( + 1170884 \beta q^{23} \) \( + ( -5863104 + 156600 \beta ) q^{24} \) \( -201488975 q^{25} \) \( + 1284050 \beta q^{26} \) \( + ( -154019475 - 3873177 \beta ) q^{27} \) \( -174202000 q^{28} \) \( + 1310050 \beta q^{29} \) \( + ( 1307826000 + 5812560 \beta ) q^{30} \) \( + 66526202 q^{31} \) \( -16723392 \beta q^{32} \) \( + ( -319690800 + 8538750 \beta ) q^{33} \) \( -1362464064 q^{34} \) \( + 9257500 \beta q^{35} \) \( + ( -1643813352 - 17528400 \beta ) q^{36} \) \( + 2228726450 q^{37} \) \( + 53343578 \beta q^{38} \) \( + ( -866733750 - 3852150 \beta ) q^{39} \) \( + 449504640 q^{40} \) \( -89469100 \beta q^{41} \) \( + ( 1017198000 - 27168750 \beta ) q^{42} \) \( + 8977216250 q^{43} \) \( + 54749200 \beta q^{44} \) \( + ( -7846956000 + 87356070 \beta ) q^{45} \) \( -9863526816 q^{46} \) \( -11733464 \beta q^{47} \) \( + ( 10646856000 + 47319360 \beta ) q^{48} \) \( -12221224701 q^{49} \) \( -201488975 \beta q^{50} \) \( + ( 4087392192 - 109171800 \beta ) q^{51} \) \( -5557368400 q^{52} \) \( + 448279614 \beta q^{53} \) \( + ( 32627643048 - 154019475 \beta ) q^{54} \) \( + 24509628000 q^{55} \) \( -9338000 \beta q^{56} \) \( + ( -36006915150 - 160030734 \beta ) q^{57} \) \( -11035861200 q^{58} \) \( -502355650 \beta q^{59} \) \( + ( -25156759680 + 671922000 \beta ) q^{60} \) \( -40679935918 q^{61} \) \( + 66526202 \beta q^{62} \) \( + ( 15287312250 + 163012500 \beta ) q^{63} \) \( + 76271154688 q^{64} \) \( + 295331500 \beta q^{65} \) \( + ( -71930430000 - 319690800 \beta ) q^{66} \) \( + 121176846650 q^{67} \) \( -699993408 \beta q^{68} \) \( + ( 29590580448 - 790346700 \beta ) q^{69} \) \( -77985180000 q^{70} \) \( + 488726700 \beta q^{71} \) \( + ( 7915190400 - 88115688 \beta ) q^{72} \) \( -60956187550 q^{73} \) \( + 2228726450 \beta q^{74} \) \( + ( 136005058125 + 604466925 \beta ) q^{75} \) \( -230871005584 q^{76} \) \( -509162500 \beta q^{77} \) \( + ( 32450511600 - 866733750 \beta ) q^{78} \) \( -252324997702 q^{79} \) \( -3627817600 \beta q^{80} \) \( + ( 6080216481 + 3076452900 \beta ) q^{81} \) \( + 753687698400 q^{82} \) \( -4475910446 \beta q^{83} \) \( + ( 117586350000 + 522606000 \beta ) q^{84} \) \( -313366734720 q^{85} \) \( + 8977216250 \beta q^{86} \) \( + ( 33107583600 - 884283750 \beta ) q^{87} \) \( -24722755200 q^{88} \) \( + 1225929900 \beta q^{89} \) \( + ( -735887533680 - 7846956000 \beta ) q^{90} \) \( + 51683012500 q^{91} \) \( -5067585952 \beta q^{92} \) \( + ( -44905186350 - 199578606 \beta ) q^{93} \) \( + 98842700736 q^{94} \) \( + 12269022940 \beta q^{95} \) \( + ( -422633562624 + 11288289600 \beta ) q^{96} \) \( + 653817778850 q^{97} \) \( -12221224701 \beta q^{98} \) \( + ( 431582580000 - 4804583850 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 1350q^{3} \) \(\mathstrut -\mathstrut 8656q^{4} \) \(\mathstrut +\mathstrut 50544q^{6} \) \(\mathstrut +\mathstrut 80500q^{7} \) \(\mathstrut +\mathstrut 759618q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 1350q^{3} \) \(\mathstrut -\mathstrut 8656q^{4} \) \(\mathstrut +\mathstrut 50544q^{6} \) \(\mathstrut +\mathstrut 80500q^{7} \) \(\mathstrut +\mathstrut 759618q^{9} \) \(\mathstrut -\mathstrut 3875040q^{10} \) \(\mathstrut +\mathstrut 5842800q^{12} \) \(\mathstrut +\mathstrut 2568100q^{13} \) \(\mathstrut +\mathstrut 11625120q^{15} \) \(\mathstrut -\mathstrut 31546240q^{16} \) \(\mathstrut -\mathstrut 68234400q^{18} \) \(\mathstrut +\mathstrut 106687156q^{19} \) \(\mathstrut -\mathstrut 54337500q^{21} \) \(\mathstrut +\mathstrut 213127200q^{22} \) \(\mathstrut -\mathstrut 11726208q^{24} \) \(\mathstrut -\mathstrut 402977950q^{25} \) \(\mathstrut -\mathstrut 308038950q^{27} \) \(\mathstrut -\mathstrut 348404000q^{28} \) \(\mathstrut +\mathstrut 2615652000q^{30} \) \(\mathstrut +\mathstrut 133052404q^{31} \) \(\mathstrut -\mathstrut 639381600q^{33} \) \(\mathstrut -\mathstrut 2724928128q^{34} \) \(\mathstrut -\mathstrut 3287626704q^{36} \) \(\mathstrut +\mathstrut 4457452900q^{37} \) \(\mathstrut -\mathstrut 1733467500q^{39} \) \(\mathstrut +\mathstrut 899009280q^{40} \) \(\mathstrut +\mathstrut 2034396000q^{42} \) \(\mathstrut +\mathstrut 17954432500q^{43} \) \(\mathstrut -\mathstrut 15693912000q^{45} \) \(\mathstrut -\mathstrut 19727053632q^{46} \) \(\mathstrut +\mathstrut 21293712000q^{48} \) \(\mathstrut -\mathstrut 24442449402q^{49} \) \(\mathstrut +\mathstrut 8174784384q^{51} \) \(\mathstrut -\mathstrut 11114736800q^{52} \) \(\mathstrut +\mathstrut 65255286096q^{54} \) \(\mathstrut +\mathstrut 49019256000q^{55} \) \(\mathstrut -\mathstrut 72013830300q^{57} \) \(\mathstrut -\mathstrut 22071722400q^{58} \) \(\mathstrut -\mathstrut 50313519360q^{60} \) \(\mathstrut -\mathstrut 81359871836q^{61} \) \(\mathstrut +\mathstrut 30574624500q^{63} \) \(\mathstrut +\mathstrut 152542309376q^{64} \) \(\mathstrut -\mathstrut 143860860000q^{66} \) \(\mathstrut +\mathstrut 242353693300q^{67} \) \(\mathstrut +\mathstrut 59181160896q^{69} \) \(\mathstrut -\mathstrut 155970360000q^{70} \) \(\mathstrut +\mathstrut 15830380800q^{72} \) \(\mathstrut -\mathstrut 121912375100q^{73} \) \(\mathstrut +\mathstrut 272010116250q^{75} \) \(\mathstrut -\mathstrut 461742011168q^{76} \) \(\mathstrut +\mathstrut 64901023200q^{78} \) \(\mathstrut -\mathstrut 504649995404q^{79} \) \(\mathstrut +\mathstrut 12160432962q^{81} \) \(\mathstrut +\mathstrut 1507375396800q^{82} \) \(\mathstrut +\mathstrut 235172700000q^{84} \) \(\mathstrut -\mathstrut 626733469440q^{85} \) \(\mathstrut +\mathstrut 66215167200q^{87} \) \(\mathstrut -\mathstrut 49445510400q^{88} \) \(\mathstrut -\mathstrut 1471775067360q^{90} \) \(\mathstrut +\mathstrut 103366025000q^{91} \) \(\mathstrut -\mathstrut 89810372700q^{93} \) \(\mathstrut +\mathstrut 197685401472q^{94} \) \(\mathstrut -\mathstrut 845267125248q^{96} \) \(\mathstrut +\mathstrut 1307635557700q^{97} \) \(\mathstrut +\mathstrut 863165160000q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
5.09902i
5.09902i
91.7824i −675.000 + 275.347i −4328.00 21109.9i 25272.0 + 61953.1i 40250.0 21293.5i 379809. 371719.i −1.93752e6
2.2 91.7824i −675.000 275.347i −4328.00 21109.9i 25272.0 61953.1i 40250.0 21293.5i 379809. + 371719.i −1.93752e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 8424 \) acting on \(S_{13}^{\mathrm{new}}(3, [\chi])\).