# Properties

 Label 1216.4.a.bd Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - 2 x^{4} - 28 x^{3} - 8 x^{2} + 73 x - 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{3} + ( -1 + \beta_{2} - \beta_{3} ) q^{5} + ( 1 + 2 \beta_{2} - \beta_{4} ) q^{7} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{3} + ( -1 + \beta_{2} - \beta_{3} ) q^{5} + ( 1 + 2 \beta_{2} - \beta_{4} ) q^{7} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{9} + ( 3 + \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{11} + ( 14 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{13} + ( 15 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{15} + ( -13 - 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{17} -19 q^{19} + ( 45 + 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{21} + ( 8 + 13 \beta_{2} - \beta_{3} + \beta_{4} ) q^{23} + ( -18 + 3 \beta_{1} + 10 \beta_{2} - 3 \beta_{3} ) q^{25} + ( 16 + 3 \beta_{1} - 14 \beta_{2} - 9 \beta_{3} ) q^{27} + ( 19 - 11 \beta_{1} + 16 \beta_{2} + 6 \beta_{3} - 5 \beta_{4} ) q^{29} + ( -27 - 5 \beta_{1} + 25 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} ) q^{31} + ( -15 + 3 \beta_{1} + 9 \beta_{2} + \beta_{3} - 6 \beta_{4} ) q^{33} + ( -5 + \beta_{1} - 16 \beta_{2} - 3 \beta_{3} + 10 \beta_{4} ) q^{35} + ( 87 - 3 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} ) q^{37} + ( 66 - 6 \beta_{1} + 21 \beta_{2} - \beta_{3} - 13 \beta_{4} ) q^{39} + ( -171 - 13 \beta_{1} + 27 \beta_{2} + 11 \beta_{3} + 16 \beta_{4} ) q^{41} + ( -29 - 17 \beta_{1} - 15 \beta_{3} + 2 \beta_{4} ) q^{43} + ( 111 + 4 \beta_{1} + 33 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} ) q^{45} + ( 139 + 18 \beta_{1} + 5 \beta_{2} - 21 \beta_{3} - 14 \beta_{4} ) q^{47} + ( -86 + 6 \beta_{1} + 10 \beta_{2} + 20 \beta_{3} - 16 \beta_{4} ) q^{49} + ( 121 + 22 \beta_{1} - 23 \beta_{2} + 10 \beta_{3} ) q^{51} + ( 179 + 11 \beta_{1} + 16 \beta_{2} - 2 \beta_{3} + 7 \beta_{4} ) q^{53} + ( 3 - 12 \beta_{1} - 9 \beta_{2} - \beta_{3} + 14 \beta_{4} ) q^{55} + ( -19 - 19 \beta_{2} ) q^{57} + ( 30 - 21 \beta_{1} + 18 \beta_{2} + 7 \beta_{3} + 4 \beta_{4} ) q^{59} + ( 205 - 10 \beta_{1} + 57 \beta_{2} - 13 \beta_{3} + 4 \beta_{4} ) q^{61} + ( 27 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} + 8 \beta_{4} ) q^{63} + ( -328 - 6 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} + 22 \beta_{4} ) q^{65} + ( 151 - 22 \beta_{1} + 57 \beta_{2} - 2 \beta_{3} - 36 \beta_{4} ) q^{67} + ( 330 + 14 \beta_{1} + 27 \beta_{2} - 19 \beta_{3} + 5 \beta_{4} ) q^{69} + ( 176 + 6 \beta_{1} - 66 \beta_{2} + 26 \beta_{3} + 2 \beta_{4} ) q^{71} + ( -285 - 2 \beta_{1} + 34 \beta_{2} + 14 \beta_{3} + 24 \beta_{4} ) q^{73} + ( 214 + 13 \beta_{1} + 28 \beta_{2} - 31 \beta_{3} ) q^{75} + ( 279 - 10 \beta_{1} + 39 \beta_{2} + 29 \beta_{3} - 14 \beta_{4} ) q^{77} + ( 41 - 35 \beta_{1} - 71 \beta_{2} - 15 \beta_{3} + 10 \beta_{4} ) q^{79} + ( -379 - 26 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} + 12 \beta_{4} ) q^{81} + ( 444 + 46 \beta_{1} + 50 \beta_{3} + 14 \beta_{4} ) q^{83} + ( 381 + 8 \beta_{1} - 87 \beta_{2} - 3 \beta_{3} + 24 \beta_{4} ) q^{85} + ( 410 + 20 \beta_{1} - 97 \beta_{2} + 71 \beta_{3} - 5 \beta_{4} ) q^{87} + ( -87 + 73 \beta_{1} + 15 \beta_{2} - 49 \beta_{3} - 6 \beta_{4} ) q^{89} + ( 346 - 7 \beta_{1} + 116 \beta_{2} + 37 \beta_{3} - 60 \beta_{4} ) q^{91} + ( 574 + 34 \beta_{1} - 14 \beta_{2} - 34 \beta_{3} + 38 \beta_{4} ) q^{93} + ( 19 - 19 \beta_{2} + 19 \beta_{3} ) q^{95} + ( -298 + 28 \beta_{1} + 26 \beta_{2} - 36 \beta_{3} - 48 \beta_{4} ) q^{97} + ( 111 - 17 \beta_{1} + 29 \beta_{3} + 28 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 6q^{3} - 5q^{5} + 7q^{7} - 5q^{9} + O(q^{10})$$ $$5q + 6q^{3} - 5q^{5} + 7q^{7} - 5q^{9} + 13q^{11} + 72q^{13} + 72q^{15} - 59q^{17} - 95q^{19} + 224q^{21} + 52q^{23} - 86q^{25} + 54q^{27} + 128q^{29} - 110q^{31} - 68q^{33} - 45q^{35} + 436q^{37} + 356q^{39} - 804q^{41} - 143q^{43} + 579q^{45} + 661q^{47} - 406q^{49} + 570q^{51} + 898q^{53} + 17q^{55} - 114q^{57} + 196q^{59} + 1079q^{61} + 143q^{63} - 1632q^{65} + 832q^{67} + 1644q^{69} + 834q^{71} - 1375q^{73} + 1054q^{75} + 1473q^{77} + 154q^{79} - 1859q^{81} + 2224q^{83} + 1807q^{85} + 2004q^{87} - 542q^{89} + 1890q^{91} + 2788q^{93} + 95q^{95} - 1528q^{97} + 601q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 28 x^{3} - 8 x^{2} + 73 x - 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{4} - 7 \nu^{3} - 15 \nu^{2} + 177 \nu + 24$$$$)/22$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - 7 \nu^{3} - 15 \nu^{2} + 89 \nu + 68$$$$)/22$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{4} - 13 \nu^{3} - 119 \nu^{2} - 17 \nu + 153$$$$)/11$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{4} - \nu^{3} + 111 \nu^{2} + 151 \nu - 270$$$$)/11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - 5 \beta_{2} + \beta_{1} + 25$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{4} + 6 \beta_{3} - 61 \beta_{2} + 25 \beta_{1} + 176$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$29 \beta_{4} + 36 \beta_{3} - 200 \beta_{2} + 58 \beta_{1} + 766$$$$)/2$$

## 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
 6.33779 −2.31928 −3.54663 0.169415 1.35870
0 −5.31962 0 −9.41363 0 −16.2508 0 1.29831 0
1.2 0 −3.67449 0 7.12795 0 0.511313 0 −13.4981 0
1.3 0 2.55330 0 −7.40075 0 −10.4962 0 −20.4807 0
1.4 0 4.75519 0 −10.5762 0 30.4413 0 −4.38817 0
1.5 0 7.68562 0 15.2627 0 2.79440 0 32.0687 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.bd 5
4.b odd 2 1 1216.4.a.y 5
8.b even 2 1 608.4.a.e 5
8.d odd 2 1 608.4.a.h yes 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.e 5 8.b even 2 1
608.4.a.h yes 5 8.d odd 2 1
1216.4.a.y 5 4.b odd 2 1
1216.4.a.bd 5 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{5} - 6 T_{3}^{4} - 47 T_{3}^{3} + 228 T_{3}^{2} + 496 T_{3} - 1824$$ $$T_{5}^{5} + 5 T_{5}^{4} - 257 T_{5}^{3} - 1825 T_{5}^{2} + 10428 T_{5} + 80160$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$-1824 + 496 T + 228 T^{2} - 47 T^{3} - 6 T^{4} + T^{5}$$
$5$ $$80160 + 10428 T - 1825 T^{2} - 257 T^{3} + 5 T^{4} + T^{5}$$
$7$ $$-7419 + 16245 T - 3070 T^{2} - 630 T^{3} - 7 T^{4} + T^{5}$$
$11$ $$-122580 - 246756 T + 47205 T^{2} - 1945 T^{3} - 13 T^{4} + T^{5}$$
$13$ $$-527040 + 742032 T + 180028 T^{2} - 3293 T^{3} - 72 T^{4} + T^{5}$$
$17$ $$-297647925 + 27836849 T - 159558 T^{2} - 11378 T^{3} + 59 T^{4} + T^{5}$$
$19$ $$( 19 + T )^{5}$$
$23$ $$-304895232 + 15022080 T + 188592 T^{2} - 9981 T^{3} - 52 T^{4} + T^{5}$$
$29$ $$-106742590368 + 819078416 T + 12394020 T^{2} - 85253 T^{3} - 128 T^{4} + T^{5}$$
$31$ $$53136006400 + 921248096 T - 5213304 T^{2} - 67884 T^{3} + 110 T^{4} + T^{5}$$
$37$ $$-4385970560 - 233373376 T + 1574256 T^{2} + 45708 T^{3} - 436 T^{4} + T^{5}$$
$41$ $$-855201369600 - 24972060672 T - 111605664 T^{2} + 24544 T^{3} + 804 T^{4} + T^{5}$$
$43$ $$197931733952 + 4990172380 T - 29832187 T^{2} - 192905 T^{3} + 143 T^{4} + T^{5}$$
$47$ $$2970610340352 - 37596704184 T + 147459257 T^{2} - 75429 T^{3} - 661 T^{4} + T^{5}$$
$53$ $$173906788200 - 2703478428 T - 4906162 T^{2} + 218623 T^{3} - 898 T^{4} + T^{5}$$
$59$ $$-131722274520 + 1737447516 T + 25779390 T^{2} - 195967 T^{3} - 196 T^{4} + T^{5}$$
$61$ $$2650589198500 - 34224946276 T + 98105399 T^{2} + 198143 T^{3} - 1079 T^{4} + T^{5}$$
$67$ $$-823051244968 + 33202255612 T + 428884778 T^{2} - 687695 T^{3} - 832 T^{4} + T^{5}$$
$71$ $$679526470240 - 22927513776 T + 139258480 T^{2} - 88280 T^{3} - 834 T^{4} + T^{5}$$
$73$ $$2709187037875 - 35969795771 T - 110154050 T^{2} + 382954 T^{3} + 1375 T^{4} + T^{5}$$
$79$ $$17190104220000 + 190301187792 T - 62553712 T^{2} - 1043960 T^{3} - 154 T^{4} + T^{5}$$
$83$ $$297963542605824 - 1557570574848 T + 2188003968 T^{2} + 278012 T^{3} - 2224 T^{4} + T^{5}$$
$89$ $$414712220295168 + 1205910041760 T - 703344888 T^{2} - 2520764 T^{3} + 542 T^{4} + T^{5}$$
$97$ $$212723907840 - 6123481664 T - 219463376 T^{2} - 339868 T^{3} + 1528 T^{4} + T^{5}$$