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 $2$

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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: \( x^{2} + 26 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - 3 \beta - 675) q^{3} - 4328 q^{4} + 230 \beta q^{5} + ( - 675 \beta + 25272) q^{6} + 40250 q^{7} - 232 \beta q^{8} + (4050 \beta + 379809) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( 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}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1350 q^{3} - 8656 q^{4} + 50544 q^{6} + 80500 q^{7} + 759618 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1350 q^{3} - 8656 q^{4} + 50544 q^{6} + 80500 q^{7} + 759618 q^{9} - 3875040 q^{10} + 5842800 q^{12} + 2568100 q^{13} + 11625120 q^{15} - 31546240 q^{16} - 68234400 q^{18} + 106687156 q^{19} - 54337500 q^{21} + 213127200 q^{22} - 11726208 q^{24} - 402977950 q^{25} - 308038950 q^{27} - 348404000 q^{28} + 2615652000 q^{30} + 133052404 q^{31} - 639381600 q^{33} - 2724928128 q^{34} - 3287626704 q^{36} + 4457452900 q^{37} - 1733467500 q^{39} + 899009280 q^{40} + 2034396000 q^{42} + 17954432500 q^{43} - 15693912000 q^{45} - 19727053632 q^{46} + 21293712000 q^{48} - 24442449402 q^{49} + 8174784384 q^{51} - 11114736800 q^{52} + 65255286096 q^{54} + 49019256000 q^{55} - 72013830300 q^{57} - 22071722400 q^{58} - 50313519360 q^{60} - 81359871836 q^{61} + 30574624500 q^{63} + 152542309376 q^{64} - 143860860000 q^{66} + 242353693300 q^{67} + 59181160896 q^{69} - 155970360000 q^{70} + 15830380800 q^{72} - 121912375100 q^{73} + 272010116250 q^{75} - 461742011168 q^{76} + 64901023200 q^{78} - 504649995404 q^{79} + 12160432962 q^{81} + 1507375396800 q^{82} + 235172700000 q^{84} - 626733469440 q^{85} + 66215167200 q^{87} - 49445510400 q^{88} - 1471775067360 q^{90} + 103366025000 q^{91} - 89810372700 q^{93} + 197685401472 q^{94} - 845267125248 q^{96} + 1307635557700 q^{97} + 863165160000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 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 \(S_{13}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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