# 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: $$x^{2} + 26$$ 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.

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

## 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

$p$ $F_p(T)$
$2$ $$T^{2} + 8424$$
$3$ $$T^{2} + 1350 T + 531441$$
$5$ $$T^{2} + 445629600$$
$7$ $$(T - 40250)^{2}$$
$11$ $$T^{2} + 1348029540000$$
$13$ $$(T - 1284050)^{2}$$
$17$ $$T^{2} + \cdots + 220359487855104$$
$19$ $$(T - 53343578)^{2}$$
$23$ $$T^{2} + 11\!\cdots\!44$$
$29$ $$T^{2} + 14\!\cdots\!00$$
$31$ $$(T - 66526202)^{2}$$
$37$ $$(T - 2228726450)^{2}$$
$41$ $$T^{2} + 67\!\cdots\!00$$
$43$ $$(T - 8977216250)^{2}$$
$47$ $$T^{2} + 11\!\cdots\!04$$
$53$ $$T^{2} + 16\!\cdots\!04$$
$59$ $$T^{2} + 21\!\cdots\!00$$
$61$ $$(T + 40679935918)^{2}$$
$67$ $$(T - 121176846650)^{2}$$
$71$ $$T^{2} + 20\!\cdots\!00$$
$73$ $$(T + 60956187550)^{2}$$
$79$ $$(T + 252324997702)^{2}$$
$83$ $$T^{2} + 16\!\cdots\!84$$
$89$ $$T^{2} + 12\!\cdots\!00$$
$97$ $$(T - 653817778850)^{2}$$