# 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

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.74198145183$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-26})$$ Defining polynomial: $$x^{2} + 26$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 1350q^{3} - 8656q^{4} + 50544q^{6} + 80500q^{7} + 759618q^{9} + O(q^{10})$$ $$2q - 1350q^{3} - 8656q^{4} + 50544q^{6} + 80500q^{7} + 759618q^{9} - 3875040q^{10} + 5842800q^{12} + 2568100q^{13} + 11625120q^{15} - 31546240q^{16} - 68234400q^{18} + 106687156q^{19} - 54337500q^{21} + 213127200q^{22} - 11726208q^{24} - 402977950q^{25} - 308038950q^{27} - 348404000q^{28} + 2615652000q^{30} + 133052404q^{31} - 639381600q^{33} - 2724928128q^{34} - 3287626704q^{36} + 4457452900q^{37} - 1733467500q^{39} + 899009280q^{40} + 2034396000q^{42} + 17954432500q^{43} - 15693912000q^{45} - 19727053632q^{46} + 21293712000q^{48} - 24442449402q^{49} + 8174784384q^{51} - 11114736800q^{52} + 65255286096q^{54} + 49019256000q^{55} - 72013830300q^{57} - 22071722400q^{58} - 50313519360q^{60} - 81359871836q^{61} + 30574624500q^{63} + 152542309376q^{64} - 143860860000q^{66} + 242353693300q^{67} + 59181160896q^{69} - 155970360000q^{70} + 15830380800q^{72} - 121912375100q^{73} + 272010116250q^{75} - 461742011168q^{76} + 64901023200q^{78} - 504649995404q^{79} + 12160432962q^{81} + 1507375396800q^{82} + 235172700000q^{84} - 626733469440q^{85} + 66215167200q^{87} - 49445510400q^{88} - 1471775067360q^{90} + 103366025000q^{91} - 89810372700q^{93} + 197685401472q^{94} - 845267125248q^{96} + 1307635557700q^{97} + 863165160000q^{99} + O(q^{100})$$

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 3.13.b.b 2
3.b odd 2 1 inner 3.13.b.b 2
4.b odd 2 1 48.13.e.b 2
5.b even 2 1 75.13.c.c 2
5.c odd 4 2 75.13.d.b 4
8.b even 2 1 192.13.e.d 2
8.d odd 2 1 192.13.e.c 2
9.c even 3 2 81.13.d.c 4
9.d odd 6 2 81.13.d.c 4
12.b even 2 1 48.13.e.b 2
15.d odd 2 1 75.13.c.c 2
15.e even 4 2 75.13.d.b 4
24.f even 2 1 192.13.e.c 2
24.h odd 2 1 192.13.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.13.b.b 2 1.a even 1 1 trivial
3.13.b.b 2 3.b odd 2 1 inner
48.13.e.b 2 4.b odd 2 1
48.13.e.b 2 12.b even 2 1
75.13.c.c 2 5.b even 2 1
75.13.c.c 2 15.d odd 2 1
75.13.d.b 4 5.c odd 4 2
75.13.d.b 4 15.e even 4 2
81.13.d.c 4 9.c even 3 2
81.13.d.c 4 9.d odd 6 2
192.13.e.c 2 8.d odd 2 1
192.13.e.c 2 24.f even 2 1
192.13.e.d 2 8.b even 2 1
192.13.e.d 2 24.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 232 T^{2} + 16777216 T^{4}$$
$3$ $$1 + 1350 T + 531441 T^{2}$$
$5$ $$1 - 42651650 T^{2} + 59604644775390625 T^{4}$$
$7$ $$( 1 - 40250 T + 13841287201 T^{2} )^{2}$$
$11$ $$1 - 4928827213442 T^{2} +$$$$98\!\cdots\!41$$$$T^{4}$$
$13$ $$( 1 - 1284050 T + 23298085122481 T^{2} )^{2}$$
$17$ $$1 - 944884986604418 T^{2} +$$$$33\!\cdots\!21$$$$T^{4}$$
$19$ $$( 1 - 53343578 T + 2213314919066161 T^{2} )^{2}$$
$23$ $$1 - 32280203131615298 T^{2} +$$$$48\!\cdots\!41$$$$T^{4}$$
$29$ $$1 - 693172036445878082 T^{2} +$$$$12\!\cdots\!81$$$$T^{4}$$
$31$ $$( 1 - 66526202 T + 787662783788549761 T^{2} )^{2}$$
$37$ $$( 1 - 2228726450 T + 6582952005840035281 T^{2} )^{2}$$
$41$ $$1 + 22304779456187067838 T^{2} +$$$$50\!\cdots\!61$$$$T^{4}$$
$43$ $$( 1 - 8977216250 T + 39959630797262576401 T^{2} )^{2}$$
$47$ $$1 -$$$$23\!\cdots\!78$$$$T^{2} +$$$$13\!\cdots\!81$$$$T^{4}$$
$53$ $$1 +$$$$71\!\cdots\!22$$$$T^{2} +$$$$24\!\cdots\!81$$$$T^{4}$$
$59$ $$1 -$$$$14\!\cdots\!62$$$$T^{2} +$$$$31\!\cdots\!61$$$$T^{4}$$
$61$ $$( 1 + 40679935918 T +$$$$26\!\cdots\!21$$$$T^{2} )^{2}$$
$67$ $$( 1 - 121176846650 T +$$$$81\!\cdots\!61$$$$T^{2} )^{2}$$
$71$ $$1 -$$$$30\!\cdots\!82$$$$T^{2} +$$$$26\!\cdots\!81$$$$T^{4}$$
$73$ $$( 1 + 60956187550 T +$$$$22\!\cdots\!21$$$$T^{2} )^{2}$$
$79$ $$( 1 + 252324997702 T +$$$$59\!\cdots\!41$$$$T^{2} )^{2}$$
$83$ $$1 -$$$$45\!\cdots\!38$$$$T^{2} +$$$$11\!\cdots\!21$$$$T^{4}$$
$89$ $$1 -$$$$48\!\cdots\!42$$$$T^{2} +$$$$61\!\cdots\!41$$$$T^{4}$$
$97$ $$( 1 - 653817778850 T +$$$$69\!\cdots\!41$$$$T^{2} )^{2}$$