# Properties

 Label 1216.4.a.g Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$71.7463225670$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ Defining polynomial: $$x^{2} - x - 18$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 - \beta ) q^{3} + ( 3 + 3 \beta ) q^{5} + ( -7 - 4 \beta ) q^{7} + ( 7 + 9 \beta ) q^{9} +O(q^{10})$$ $$q + ( -4 - \beta ) q^{3} + ( 3 + 3 \beta ) q^{5} + ( -7 - 4 \beta ) q^{7} + ( 7 + 9 \beta ) q^{9} + ( 9 - \beta ) q^{11} + ( -2 - 13 \beta ) q^{13} + ( -66 - 18 \beta ) q^{15} + ( -39 - 2 \beta ) q^{17} -19 q^{19} + ( 100 + 27 \beta ) q^{21} + ( 30 + 13 \beta ) q^{23} + ( 46 + 27 \beta ) q^{25} + ( -82 - 25 \beta ) q^{27} + ( -12 + 21 \beta ) q^{29} + ( 128 - 44 \beta ) q^{31} + ( -18 - 4 \beta ) q^{33} + ( -237 - 45 \beta ) q^{35} + ( -110 + 28 \beta ) q^{37} + ( 242 + 67 \beta ) q^{39} + ( -30 + 10 \beta ) q^{41} + ( -335 - 7 \beta ) q^{43} + ( 507 + 75 \beta ) q^{45} + ( -159 - 71 \beta ) q^{47} + ( -6 + 72 \beta ) q^{49} + ( 192 + 49 \beta ) q^{51} + ( 618 - 17 \beta ) q^{53} + ( -27 + 21 \beta ) q^{55} + ( 76 + 19 \beta ) q^{57} + ( 156 - 25 \beta ) q^{59} + ( -101 - 111 \beta ) q^{61} + ( -697 - 127 \beta ) q^{63} + ( -708 - 84 \beta ) q^{65} + ( -650 + 77 \beta ) q^{67} + ( -354 - 95 \beta ) q^{69} + ( 42 + 116 \beta ) q^{71} + ( 281 - 184 \beta ) q^{73} + ( -670 - 181 \beta ) q^{75} + ( 9 - 25 \beta ) q^{77} + ( 704 - 58 \beta ) q^{79} + ( 589 - 36 \beta ) q^{81} + ( 432 - 194 \beta ) q^{83} + ( -225 - 129 \beta ) q^{85} + ( -330 - 93 \beta ) q^{87} + ( -24 - 188 \beta ) q^{89} + ( 950 + 151 \beta ) q^{91} + ( 280 + 92 \beta ) q^{93} + ( -57 - 57 \beta ) q^{95} + ( 596 + 102 \beta ) q^{97} + ( -99 + 65 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{3} + 9q^{5} - 18q^{7} + 23q^{9} + O(q^{10})$$ $$2q - 9q^{3} + 9q^{5} - 18q^{7} + 23q^{9} + 17q^{11} - 17q^{13} - 150q^{15} - 80q^{17} - 38q^{19} + 227q^{21} + 73q^{23} + 119q^{25} - 189q^{27} - 3q^{29} + 212q^{31} - 40q^{33} - 519q^{35} - 192q^{37} + 551q^{39} - 50q^{41} - 677q^{43} + 1089q^{45} - 389q^{47} + 60q^{49} + 433q^{51} + 1219q^{53} - 33q^{55} + 171q^{57} + 287q^{59} - 313q^{61} - 1521q^{63} - 1500q^{65} - 1223q^{67} - 803q^{69} + 200q^{71} + 378q^{73} - 1521q^{75} - 7q^{77} + 1350q^{79} + 1142q^{81} + 670q^{83} - 579q^{85} - 753q^{87} - 236q^{89} + 2051q^{91} + 652q^{93} - 171q^{95} + 1294q^{97} - 133q^{99} + O(q^{100})$$

## 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
 4.77200 −3.77200
0 −8.77200 0 17.3160 0 −26.0880 0 49.9480 0
1.2 0 −0.227998 0 −8.31601 0 8.08801 0 −26.9480 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$19$$ $$1$$

## 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 1216.4.a.g 2
4.b odd 2 1 1216.4.a.p 2
8.b even 2 1 38.4.a.c 2
8.d odd 2 1 304.4.a.c 2
24.h odd 2 1 342.4.a.h 2
40.f even 2 1 950.4.a.e 2
40.i odd 4 2 950.4.b.i 4
56.h odd 2 1 1862.4.a.e 2
152.g odd 2 1 722.4.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.c 2 8.b even 2 1
304.4.a.c 2 8.d odd 2 1
342.4.a.h 2 24.h odd 2 1
722.4.a.f 2 152.g odd 2 1
950.4.a.e 2 40.f even 2 1
950.4.b.i 4 40.i odd 4 2
1216.4.a.g 2 1.a even 1 1 trivial
1216.4.a.p 2 4.b odd 2 1
1862.4.a.e 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{2} + 9 T_{3} + 2$$ $$T_{5}^{2} - 9 T_{5} - 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + 9 T + T^{2}$$
$5$ $$-144 - 9 T + T^{2}$$
$7$ $$-211 + 18 T + T^{2}$$
$11$ $$54 - 17 T + T^{2}$$
$13$ $$-3012 + 17 T + T^{2}$$
$17$ $$1527 + 80 T + T^{2}$$
$19$ $$( 19 + T )^{2}$$
$23$ $$-1752 - 73 T + T^{2}$$
$29$ $$-8046 + 3 T + T^{2}$$
$31$ $$-24096 - 212 T + T^{2}$$
$37$ $$-5092 + 192 T + T^{2}$$
$41$ $$-1200 + 50 T + T^{2}$$
$43$ $$113688 + 677 T + T^{2}$$
$47$ $$-54168 + 389 T + T^{2}$$
$53$ $$366216 - 1219 T + T^{2}$$
$59$ $$9186 - 287 T + T^{2}$$
$61$ $$-200366 + 313 T + T^{2}$$
$67$ $$265728 + 1223 T + T^{2}$$
$71$ $$-235572 - 200 T + T^{2}$$
$73$ $$-582151 - 378 T + T^{2}$$
$79$ $$394232 - 1350 T + T^{2}$$
$83$ $$-574632 - 670 T + T^{2}$$
$89$ $$-631104 + 236 T + T^{2}$$
$97$ $$228736 - 1294 T + T^{2}$$