# Properties

 Label 18.12.a.a Level 18 Weight 12 Character orbit 18.a Self dual yes Analytic conductor 13.830 Analytic rank 1 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 32q^{2} + 1024q^{4} - 3630q^{5} + 32936q^{7} - 32768q^{8} + O(q^{10})$$ $$q - 32q^{2} + 1024q^{4} - 3630q^{5} + 32936q^{7} - 32768q^{8} + 116160q^{10} + 758748q^{11} - 2482858q^{13} - 1053952q^{14} + 1048576q^{16} - 8290386q^{17} - 10867300q^{19} - 3717120q^{20} - 24279936q^{22} - 20539272q^{23} - 35651225q^{25} + 79451456q^{26} + 33726464q^{28} - 28814550q^{29} + 150501392q^{31} - 33554432q^{32} + 265292352q^{34} - 119557680q^{35} - 319891714q^{37} + 347753600q^{38} + 118947840q^{40} + 368008998q^{41} + 620469572q^{43} + 776957952q^{44} + 657256704q^{46} - 2763110256q^{47} - 892546647q^{49} + 1140839200q^{50} - 2542446592q^{52} + 268284258q^{53} - 2754255240q^{55} - 1079246848q^{56} + 922065600q^{58} - 1672894740q^{59} - 7787197498q^{61} - 4816044544q^{62} + 1073741824q^{64} + 9012774540q^{65} + 18706694156q^{67} - 8489355264q^{68} + 3825845760q^{70} + 8346990888q^{71} + 19641746522q^{73} + 10236534848q^{74} - 11128115200q^{76} + 24990124128q^{77} - 5873807200q^{79} - 3806330880q^{80} - 11776287936q^{82} - 8492558172q^{83} + 30094101180q^{85} - 19855026304q^{86} - 24862654464q^{88} - 75527864010q^{89} - 81775411088q^{91} - 21032214528q^{92} + 88419528192q^{94} + 39448299000q^{95} - 82356782494q^{97} + 28561492704q^{98} + O(q^{100})$$

## 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 −3630.00 0 32936.0 −32768.0 0 116160.
 $$n$$: 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.a 1
3.b odd 2 1 6.12.a.c 1
4.b odd 2 1 144.12.a.e 1
9.c even 3 2 162.12.c.i 2
9.d odd 6 2 162.12.c.b 2
12.b even 2 1 48.12.a.d 1
15.d odd 2 1 150.12.a.a 1
15.e even 4 2 150.12.c.e 2
24.f even 2 1 192.12.a.m 1
24.h odd 2 1 192.12.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.c 1 3.b odd 2 1
18.12.a.a 1 1.a even 1 1 trivial
48.12.a.d 1 12.b even 2 1
144.12.a.e 1 4.b odd 2 1
150.12.a.a 1 15.d odd 2 1
150.12.c.e 2 15.e even 4 2
162.12.c.b 2 9.d odd 6 2
162.12.c.i 2 9.c even 3 2
192.12.a.c 1 24.h odd 2 1
192.12.a.m 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 $$T_{5} + 3630$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(18))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 32 T$$
$3$ 1
$5$ $$1 + 3630 T + 48828125 T^{2}$$
$7$ $$1 - 32936 T + 1977326743 T^{2}$$
$11$ $$1 - 758748 T + 285311670611 T^{2}$$
$13$ $$1 + 2482858 T + 1792160394037 T^{2}$$
$17$ $$1 + 8290386 T + 34271896307633 T^{2}$$
$19$ $$1 + 10867300 T + 116490258898219 T^{2}$$
$23$ $$1 + 20539272 T + 952809757913927 T^{2}$$
$29$ $$1 + 28814550 T + 12200509765705829 T^{2}$$
$31$ $$1 - 150501392 T + 25408476896404831 T^{2}$$
$37$ $$1 + 319891714 T + 177917621779460413 T^{2}$$
$41$ $$1 - 368008998 T + 550329031716248441 T^{2}$$
$43$ $$1 - 620469572 T + 929293739471222707 T^{2}$$
$47$ $$1 + 2763110256 T + 2472159215084012303 T^{2}$$
$53$ $$1 - 268284258 T + 9269035929372191597 T^{2}$$
$59$ $$1 + 1672894740 T + 30155888444737842659 T^{2}$$
$61$ $$1 + 7787197498 T + 43513917611435838661 T^{2}$$
$67$ $$1 - 18706694156 T +$$$$12\!\cdots\!83$$$$T^{2}$$
$71$ $$1 - 8346990888 T +$$$$23\!\cdots\!71$$$$T^{2}$$
$73$ $$1 - 19641746522 T +$$$$31\!\cdots\!77$$$$T^{2}$$
$79$ $$1 + 5873807200 T +$$$$74\!\cdots\!79$$$$T^{2}$$
$83$ $$1 + 8492558172 T +$$$$12\!\cdots\!67$$$$T^{2}$$
$89$ $$1 + 75527864010 T +$$$$27\!\cdots\!89$$$$T^{2}$$
$97$ $$1 + 82356782494 T +$$$$71\!\cdots\!53$$$$T^{2}$$