# Properties

 Label 3.9.b.a Level 3 Weight 9 Character orbit 3.b Analytic conductor 1.222 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.22213583018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-14})$$ Defining polynomial: $$x^{2} + 14$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{-14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 45 - 3 \beta ) q^{3} -248 q^{4} -10 \beta q^{5} + ( 1512 + 45 \beta ) q^{6} -1750 q^{7} + 8 \beta q^{8} + ( -2511 - 270 \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( 45 - 3 \beta ) q^{3} -248 q^{4} -10 \beta q^{5} + ( 1512 + 45 \beta ) q^{6} -1750 q^{7} + 8 \beta q^{8} + ( -2511 - 270 \beta ) q^{9} + 5040 q^{10} + 310 \beta q^{11} + ( -11160 + 744 \beta ) q^{12} + 25730 q^{13} -1750 \beta q^{14} + ( -15120 - 450 \beta ) q^{15} -67520 q^{16} + 3336 \beta q^{17} + ( 136080 - 2511 \beta ) q^{18} + 18938 q^{19} + 2480 \beta q^{20} + ( -78750 + 5250 \beta ) q^{21} -156240 q^{22} -20956 \beta q^{23} + ( 12096 + 360 \beta ) q^{24} + 340225 q^{25} + 25730 \beta q^{26} + ( -521235 - 4617 \beta ) q^{27} + 434000 q^{28} + 20530 \beta q^{29} + ( 226800 - 15120 \beta ) q^{30} -351478 q^{31} -65472 \beta q^{32} + ( 468720 + 13950 \beta ) q^{33} -1681344 q^{34} + 17500 \beta q^{35} + ( 622728 + 66960 \beta ) q^{36} + 1335170 q^{37} + 18938 \beta q^{38} + ( 1157850 - 77190 \beta ) q^{39} + 40320 q^{40} + 83540 \beta q^{41} + ( -2646000 - 78750 \beta ) q^{42} -3526150 q^{43} -76880 \beta q^{44} + ( -1360800 + 25110 \beta ) q^{45} + 10561824 q^{46} -181784 \beta q^{47} + ( -3038400 + 202560 \beta ) q^{48} -2702301 q^{49} + 340225 \beta q^{50} + ( 5044032 + 150120 \beta ) q^{51} -6381040 q^{52} -294066 \beta q^{53} + ( 2326968 - 521235 \beta ) q^{54} + 1562400 q^{55} -14000 \beta q^{56} + ( 852210 - 56814 \beta ) q^{57} -10347120 q^{58} + 610910 \beta q^{59} + ( 3749760 + 111600 \beta ) q^{60} + 753602 q^{61} -351478 \beta q^{62} + ( 4394250 + 472500 \beta ) q^{63} + 15712768 q^{64} -257300 \beta q^{65} + ( -7030800 + 468720 \beta ) q^{66} + 2268890 q^{67} -827328 \beta q^{68} + ( -31685472 - 943020 \beta ) q^{69} -8820000 q^{70} + 758220 \beta q^{71} + ( 1088640 - 20088 \beta ) q^{72} + 27672770 q^{73} + 1335170 \beta q^{74} + ( 15310125 - 1020675 \beta ) q^{75} -4696624 q^{76} -542500 \beta q^{77} + ( 38903760 + 1157850 \beta ) q^{78} -22980982 q^{79} + 675200 \beta q^{80} + ( -30436479 + 1355940 \beta ) q^{81} -42104160 q^{82} -2066606 \beta q^{83} + ( 19530000 - 1302000 \beta ) q^{84} + 16813440 q^{85} -3526150 \beta q^{86} + ( 31041360 + 923850 \beta ) q^{87} -1249920 q^{88} + 3234540 \beta q^{89} + ( -12655440 - 1360800 \beta ) q^{90} -45027500 q^{91} + 5197088 \beta q^{92} + ( -15816510 + 1054434 \beta ) q^{93} + 91619136 q^{94} -189380 \beta q^{95} + ( -98993664 - 2946240 \beta ) q^{96} + 147271010 q^{97} -2702301 \beta q^{98} + ( 42184800 - 778410 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 90q^{3} - 496q^{4} + 3024q^{6} - 3500q^{7} - 5022q^{9} + O(q^{10})$$ $$2q + 90q^{3} - 496q^{4} + 3024q^{6} - 3500q^{7} - 5022q^{9} + 10080q^{10} - 22320q^{12} + 51460q^{13} - 30240q^{15} - 135040q^{16} + 272160q^{18} + 37876q^{19} - 157500q^{21} - 312480q^{22} + 24192q^{24} + 680450q^{25} - 1042470q^{27} + 868000q^{28} + 453600q^{30} - 702956q^{31} + 937440q^{33} - 3362688q^{34} + 1245456q^{36} + 2670340q^{37} + 2315700q^{39} + 80640q^{40} - 5292000q^{42} - 7052300q^{43} - 2721600q^{45} + 21123648q^{46} - 6076800q^{48} - 5404602q^{49} + 10088064q^{51} - 12762080q^{52} + 4653936q^{54} + 3124800q^{55} + 1704420q^{57} - 20694240q^{58} + 7499520q^{60} + 1507204q^{61} + 8788500q^{63} + 31425536q^{64} - 14061600q^{66} + 4537780q^{67} - 63370944q^{69} - 17640000q^{70} + 2177280q^{72} + 55345540q^{73} + 30620250q^{75} - 9393248q^{76} + 77807520q^{78} - 45961964q^{79} - 60872958q^{81} - 84208320q^{82} + 39060000q^{84} + 33626880q^{85} + 62082720q^{87} - 2499840q^{88} - 25310880q^{90} - 90055000q^{91} - 31633020q^{93} + 183238272q^{94} - 197987328q^{96} + 294542020q^{97} + 84369600q^{99} + O(q^{100})$$

## 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
 − 3.74166i 3.74166i
22.4499i 45.0000 + 67.3498i −248.000 224.499i 1512.00 1010.25i −1750.00 179.600i −2511.00 + 6061.48i 5040.00
2.2 22.4499i 45.0000 67.3498i −248.000 224.499i 1512.00 + 1010.25i −1750.00 179.600i −2511.00 6061.48i 5040.00
 $$n$$: 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 3.9.b.a 2
3.b odd 2 1 inner 3.9.b.a 2
4.b odd 2 1 48.9.e.b 2
5.b even 2 1 75.9.c.c 2
5.c odd 4 2 75.9.d.b 4
8.b even 2 1 192.9.e.e 2
8.d odd 2 1 192.9.e.f 2
9.c even 3 2 81.9.d.d 4
9.d odd 6 2 81.9.d.d 4
12.b even 2 1 48.9.e.b 2
15.d odd 2 1 75.9.c.c 2
15.e even 4 2 75.9.d.b 4
24.f even 2 1 192.9.e.f 2
24.h odd 2 1 192.9.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 1.a even 1 1 trivial
3.9.b.a 2 3.b odd 2 1 inner
48.9.e.b 2 4.b odd 2 1
48.9.e.b 2 12.b even 2 1
75.9.c.c 2 5.b even 2 1
75.9.c.c 2 15.d odd 2 1
75.9.d.b 4 5.c odd 4 2
75.9.d.b 4 15.e even 4 2
81.9.d.d 4 9.c even 3 2
81.9.d.d 4 9.d odd 6 2
192.9.e.e 2 8.b even 2 1
192.9.e.e 2 24.h odd 2 1
192.9.e.f 2 8.d odd 2 1
192.9.e.f 2 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 8 T^{2} + 65536 T^{4}$$
$3$ $$1 - 90 T + 6561 T^{2}$$
$5$ $$1 - 730850 T^{2} + 152587890625 T^{4}$$
$7$ $$( 1 + 1750 T + 5764801 T^{2} )^{2}$$
$11$ $$1 - 380283362 T^{2} + 45949729863572161 T^{4}$$
$13$ $$( 1 - 25730 T + 815730721 T^{2} )^{2}$$
$17$ $$1 - 8342551298 T^{2} + 48661191875666868481 T^{4}$$
$19$ $$( 1 - 18938 T + 16983563041 T^{2} )^{2}$$
$23$ $$1 + 64711613182 T^{2} +$$$$61\!\cdots\!61$$$$T^{4}$$
$29$ $$1 - 788066452322 T^{2} +$$$$25\!\cdots\!21$$$$T^{4}$$
$31$ $$( 1 + 351478 T + 852891037441 T^{2} )^{2}$$
$37$ $$( 1 - 1335170 T + 3512479453921 T^{2} )^{2}$$
$41$ $$1 - 12452468931842 T^{2} +$$$$63\!\cdots\!41$$$$T^{4}$$
$43$ $$( 1 + 3526150 T + 11688200277601 T^{2} )^{2}$$
$47$ $$1 - 30967680304898 T^{2} +$$$$56\!\cdots\!21$$$$T^{4}$$
$53$ $$1 - 80936075395298 T^{2} +$$$$38\!\cdots\!21$$$$T^{4}$$
$59$ $$1 - 105562517046242 T^{2} +$$$$21\!\cdots\!41$$$$T^{4}$$
$61$ $$( 1 - 753602 T + 191707312997281 T^{2} )^{2}$$
$67$ $$( 1 - 2268890 T + 406067677556641 T^{2} )^{2}$$
$71$ $$1 - 1001758688017922 T^{2} +$$$$41\!\cdots\!21$$$$T^{4}$$
$73$ $$( 1 - 27672770 T + 806460091894081 T^{2} )^{2}$$
$79$ $$( 1 + 22980982 T + 1517108809906561 T^{2} )^{2}$$
$83$ $$1 - 2352070843223138 T^{2} +$$$$50\!\cdots\!81$$$$T^{4}$$
$89$ $$1 - 2600204109557762 T^{2} +$$$$15\!\cdots\!61$$$$T^{4}$$
$97$ $$( 1 - 147271010 T + 7837433594376961 T^{2} )^{2}$$