# 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 $$2$$ 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{13}^{\mathrm{new}}(3, [\chi])$$.