# Properties

 Label 6012.2.a.d Level 6012 Weight 2 Character orbit 6012.a Self dual yes Analytic conductor 48.006 Analytic rank 0 Dimension 5 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6012 = 2^{2} \cdot 3^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6012.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.0060616952$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.826865.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 668) 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.

 $$f(q)$$ $$=$$ $$q -2 q^{5} + ( 2 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q -2 q^{5} + ( 2 - \beta_{3} ) q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{11} -2 \beta_{1} q^{13} + ( 2 - 2 \beta_{1} + 2 \beta_{4} ) q^{17} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{19} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{23} - q^{25} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{29} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{31} + ( -4 + 2 \beta_{3} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{41} + 4 \beta_{1} q^{43} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{47} + ( 4 - \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{49} + ( -2 + 4 \beta_{1} - 2 \beta_{4} ) q^{53} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{55} + ( -2 + 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{59} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{61} + 4 \beta_{1} q^{65} + ( 6 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{67} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{71} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{73} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{77} + ( -6 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{79} + ( 2 - 6 \beta_{1} ) q^{83} + ( -4 + 4 \beta_{1} - 4 \beta_{4} ) q^{85} + ( 5 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{89} + ( -6 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{91} + ( -2 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{95} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 10q^{5} + 9q^{7} + O(q^{10})$$ $$5q - 10q^{5} + 9q^{7} - 5q^{11} - 4q^{13} + 2q^{17} + 5q^{19} - 6q^{23} - 5q^{25} + 5q^{29} + 9q^{31} - 18q^{35} + 8q^{37} + 4q^{41} + 8q^{43} - 13q^{47} + 14q^{49} + 2q^{53} + 10q^{55} - 4q^{59} + 11q^{61} + 8q^{65} + 28q^{67} - 2q^{71} + 8q^{73} + 12q^{77} - 10q^{79} - 2q^{83} - 4q^{85} + 17q^{89} - 12q^{91} - 10q^{95} - 27q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2 \beta_{3} + 8 \beta_{2} + 9 \beta_{1} + 8$$

## 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
 −0.873948 0.147687 2.75474 −1.69135 1.66287
0 0 0 −2.00000 0 −1.42676 0 0 0
1.2 0 0 0 −2.00000 0 −0.516539 0 0 0
1.3 0 0 0 −2.00000 0 1.53681 0 0 0
1.4 0 0 0 −2.00000 0 4.48567 0 0 0
1.5 0 0 0 −2.00000 0 4.92082 0 0 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

## 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 6012.2.a.d 5
3.b odd 2 1 668.2.a.b 5
12.b even 2 1 2672.2.a.j 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.a.b 5 3.b odd 2 1
2672.2.a.j 5 12.b even 2 1
6012.2.a.d 5 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$167$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6012))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 2 T + 5 T^{2} )^{5}$$
$7$ $$1 - 9 T + 51 T^{2} - 223 T^{3} + 787 T^{4} - 2265 T^{5} + 5509 T^{6} - 10927 T^{7} + 17493 T^{8} - 21609 T^{9} + 16807 T^{10}$$
$11$ $$1 + 5 T + 43 T^{2} + 191 T^{3} + 849 T^{4} + 3017 T^{5} + 9339 T^{6} + 23111 T^{7} + 57233 T^{8} + 73205 T^{9} + 161051 T^{10}$$
$13$ $$1 + 4 T + 45 T^{2} + 160 T^{3} + 1006 T^{4} + 2840 T^{5} + 13078 T^{6} + 27040 T^{7} + 98865 T^{8} + 114244 T^{9} + 371293 T^{10}$$
$17$ $$1 - 2 T + 25 T^{2} - 96 T^{3} + 662 T^{4} - 1308 T^{5} + 11254 T^{6} - 27744 T^{7} + 122825 T^{8} - 167042 T^{9} + 1419857 T^{10}$$
$19$ $$1 - 5 T + 53 T^{2} - 279 T^{3} + 1821 T^{4} - 6547 T^{5} + 34599 T^{6} - 100719 T^{7} + 363527 T^{8} - 651605 T^{9} + 2476099 T^{10}$$
$23$ $$1 + 6 T + 63 T^{2} + 192 T^{3} + 1414 T^{4} + 2452 T^{5} + 32522 T^{6} + 101568 T^{7} + 766521 T^{8} + 1679046 T^{9} + 6436343 T^{10}$$
$29$ $$1 - 5 T + 117 T^{2} - 541 T^{3} + 6005 T^{4} - 22981 T^{5} + 174145 T^{6} - 454981 T^{7} + 2853513 T^{8} - 3536405 T^{9} + 20511149 T^{10}$$
$31$ $$1 - 9 T + 119 T^{2} - 909 T^{3} + 6257 T^{4} - 39425 T^{5} + 193967 T^{6} - 873549 T^{7} + 3545129 T^{8} - 8311689 T^{9} + 28629151 T^{10}$$
$37$ $$1 - 8 T + 141 T^{2} - 1104 T^{3} + 8998 T^{4} - 60080 T^{5} + 332926 T^{6} - 1511376 T^{7} + 7142073 T^{8} - 14993288 T^{9} + 69343957 T^{10}$$
$41$ $$1 - 4 T + 109 T^{2} - 424 T^{3} + 6474 T^{4} - 20392 T^{5} + 265434 T^{6} - 712744 T^{7} + 7512389 T^{8} - 11303044 T^{9} + 115856201 T^{10}$$
$43$ $$1 - 8 T + 135 T^{2} - 992 T^{3} + 9706 T^{4} - 56752 T^{5} + 417358 T^{6} - 1834208 T^{7} + 10733445 T^{8} - 27350408 T^{9} + 147008443 T^{10}$$
$47$ $$1 + 13 T + 261 T^{2} + 2283 T^{3} + 25333 T^{4} + 157087 T^{5} + 1190651 T^{6} + 5043147 T^{7} + 27097803 T^{8} + 63435853 T^{9} + 229345007 T^{10}$$
$53$ $$1 - 2 T + 141 T^{2} - 216 T^{3} + 11910 T^{4} - 17068 T^{5} + 631230 T^{6} - 606744 T^{7} + 20991657 T^{8} - 15780962 T^{9} + 418195493 T^{10}$$
$59$ $$1 + 4 T + 59 T^{2} + 120 T^{3} + 4414 T^{4} + 36008 T^{5} + 260426 T^{6} + 417720 T^{7} + 12117361 T^{8} + 48469444 T^{9} + 714924299 T^{10}$$
$61$ $$1 - 11 T + 229 T^{2} - 2491 T^{3} + 24489 T^{4} - 221163 T^{5} + 1493829 T^{6} - 9269011 T^{7} + 51978649 T^{8} - 152304251 T^{9} + 844596301 T^{10}$$
$67$ $$1 - 28 T + 491 T^{2} - 5616 T^{3} + 53302 T^{4} - 436232 T^{5} + 3571234 T^{6} - 25210224 T^{7} + 147674633 T^{8} - 564231388 T^{9} + 1350125107 T^{10}$$
$71$ $$1 + 2 T + 175 T^{2} + 320 T^{3} + 17542 T^{4} + 36060 T^{5} + 1245482 T^{6} + 1613120 T^{7} + 62634425 T^{8} + 50823362 T^{9} + 1804229351 T^{10}$$
$73$ $$1 - 8 T + 241 T^{2} - 616 T^{3} + 19206 T^{4} + 4320 T^{5} + 1402038 T^{6} - 3282664 T^{7} + 93753097 T^{8} - 227185928 T^{9} + 2073071593 T^{10}$$
$79$ $$1 + 10 T + 139 T^{2} + 1552 T^{3} + 15962 T^{4} + 195500 T^{5} + 1260998 T^{6} + 9686032 T^{7} + 68532421 T^{8} + 389500810 T^{9} + 3077056399 T^{10}$$
$83$ $$1 + 2 T + 179 T^{2} + 656 T^{3} + 20622 T^{4} + 69980 T^{5} + 1711626 T^{6} + 4519184 T^{7} + 102349873 T^{8} + 94916642 T^{9} + 3939040643 T^{10}$$
$89$ $$1 - 17 T + 485 T^{2} - 5689 T^{3} + 89097 T^{4} - 743149 T^{5} + 7929633 T^{6} - 45062569 T^{7} + 341909965 T^{8} - 1066618097 T^{9} + 5584059449 T^{10}$$
$97$ $$1 + 27 T + 521 T^{2} + 8059 T^{3} + 104425 T^{4} + 1093259 T^{5} + 10129225 T^{6} + 75827131 T^{7} + 475502633 T^{8} + 2390290587 T^{9} + 8587340257 T^{10}$$