# Properties

 Label 18.12.a.e Level 18 Weight 12 Character orbit 18.a Self dual yes Analytic conductor 13.830 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$18 = 2 \cdot 3^{2}$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 18.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.8301772501$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 32q^{2} + 1024q^{4} + 11730q^{5} - 50008q^{7} + 32768q^{8} + O(q^{10})$$ $$q + 32q^{2} + 1024q^{4} + 11730q^{5} - 50008q^{7} + 32768q^{8} + 375360q^{10} + 531420q^{11} + 1332566q^{13} - 1600256q^{14} + 1048576q^{16} + 5109678q^{17} + 2901404q^{19} + 12011520q^{20} + 17005440q^{22} - 30597000q^{23} + 88764775q^{25} + 42642112q^{26} - 51208192q^{28} + 77006634q^{29} - 239418352q^{31} + 33554432q^{32} + 163509696q^{34} - 586593840q^{35} - 785041666q^{37} + 92844928q^{38} + 384368640q^{40} - 411252954q^{41} + 351233348q^{43} + 544174080q^{44} - 979104000q^{46} - 95821680q^{47} + 523473321q^{49} + 2840472800q^{50} + 1364547584q^{52} + 1465857378q^{53} + 6233556600q^{55} - 1638662144q^{56} + 2464212288q^{58} - 5621152020q^{59} - 10473587770q^{61} - 7661387264q^{62} + 1073741824q^{64} + 15630999180q^{65} + 4515307532q^{67} + 5232310272q^{68} - 18771002880q^{70} + 8509579560q^{71} + 2012496986q^{73} - 25121333312q^{74} + 2971037696q^{76} - 26575251360q^{77} - 22238409568q^{79} + 12299796480q^{80} - 13160094528q^{82} - 6328647516q^{83} + 59936522940q^{85} + 11239467136q^{86} + 17413570560q^{88} + 50123706678q^{89} - 66638960528q^{91} - 31331328000q^{92} - 3066293760q^{94} + 34033468920q^{95} + 94805961314q^{97} + 16751146272q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
32.0000 0 1024.00 11730.0 0 −50008.0 32768.0 0 375360.
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.12.a.e 1
3.b odd 2 1 6.12.a.b 1
4.b odd 2 1 144.12.a.o 1
9.c even 3 2 162.12.c.a 2
9.d odd 6 2 162.12.c.j 2
12.b even 2 1 48.12.a.a 1
15.d odd 2 1 150.12.a.f 1
15.e even 4 2 150.12.c.b 2
24.f even 2 1 192.12.a.t 1
24.h odd 2 1 192.12.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.b 1 3.b odd 2 1
18.12.a.e 1 1.a even 1 1 trivial
48.12.a.a 1 12.b even 2 1
144.12.a.o 1 4.b odd 2 1
150.12.a.f 1 15.d odd 2 1
150.12.c.b 2 15.e even 4 2
162.12.c.a 2 9.c even 3 2
162.12.c.j 2 9.d odd 6 2
192.12.a.j 1 24.h odd 2 1
192.12.a.t 1 24.f even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 11730$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(18))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 32 T$$
$3$ 
$5$ $$1 - 11730 T + 48828125 T^{2}$$
$7$ $$1 + 50008 T + 1977326743 T^{2}$$
$11$ $$1 - 531420 T + 285311670611 T^{2}$$
$13$ $$1 - 1332566 T + 1792160394037 T^{2}$$
$17$ $$1 - 5109678 T + 34271896307633 T^{2}$$
$19$ $$1 - 2901404 T + 116490258898219 T^{2}$$
$23$ $$1 + 30597000 T + 952809757913927 T^{2}$$
$29$ $$1 - 77006634 T + 12200509765705829 T^{2}$$
$31$ $$1 + 239418352 T + 25408476896404831 T^{2}$$
$37$ $$1 + 785041666 T + 177917621779460413 T^{2}$$
$41$ $$1 + 411252954 T + 550329031716248441 T^{2}$$
$43$ $$1 - 351233348 T + 929293739471222707 T^{2}$$
$47$ $$1 + 95821680 T + 2472159215084012303 T^{2}$$
$53$ $$1 - 1465857378 T + 9269035929372191597 T^{2}$$
$59$ $$1 + 5621152020 T + 30155888444737842659 T^{2}$$
$61$ $$1 + 10473587770 T + 43513917611435838661 T^{2}$$
$67$ $$1 - 4515307532 T +$$$$12\!\cdots\!83$$$$T^{2}$$
$71$ $$1 - 8509579560 T +$$$$23\!\cdots\!71$$$$T^{2}$$
$73$ $$1 - 2012496986 T +$$$$31\!\cdots\!77$$$$T^{2}$$
$79$ $$1 + 22238409568 T +$$$$74\!\cdots\!79$$$$T^{2}$$
$83$ $$1 + 6328647516 T +$$$$12\!\cdots\!67$$$$T^{2}$$
$89$ $$1 - 50123706678 T +$$$$27\!\cdots\!89$$$$T^{2}$$
$97$ $$1 - 94805961314 T +$$$$71\!\cdots\!53$$$$T^{2}$$