# Properties

 Label 756.2.k.f Level 756 Weight 2 Character orbit 756.k Analytic conductor 6.037 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} ) q^{7} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{3} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{17} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{19} + ( -\beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{23} + ( -5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{25} + ( 2 + \beta_{2} + \beta_{5} ) q^{29} + ( -\beta_{2} - \beta_{3} + \beta_{5} ) q^{31} + ( 6 - \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{41} + ( -4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{43} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{47} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{49} + ( -3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{53} + ( 5 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} ) q^{55} + ( -2 + \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{59} + ( 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{61} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 9 \beta_{4} + 2 \beta_{5} ) q^{65} + ( 5 + \beta_{1} + 5 \beta_{4} + \beta_{5} ) q^{67} + ( -2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{71} + ( 2 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 7 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{77} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{79} + ( -5 - \beta_{1} + \beta_{3} ) q^{83} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{85} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 4 + 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{91} + ( 14 + 5 \beta_{2} + 5 \beta_{3} + 14 \beta_{4} - 5 \beta_{5} ) q^{95} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{5} + 2q^{7} + O(q^{10})$$ $$6q + q^{5} + 2q^{7} + 5q^{11} - 4q^{13} - 4q^{17} - 3q^{19} + 14q^{23} - 10q^{25} + 10q^{29} + 2q^{31} + 26q^{35} - 24q^{41} - 18q^{43} - 9q^{47} + 6q^{53} + 16q^{55} - 5q^{59} - 7q^{61} + 24q^{65} + 16q^{67} - 22q^{71} + q^{73} + 31q^{77} + 8q^{79} - 34q^{83} + 10q^{85} - 3q^{89} + 5q^{91} + 32q^{95} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 5 \nu^{3} - \nu^{2} + 3 \nu - 6$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 8 \nu^{3} + 8 \nu^{2} - 21 \nu + 12$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - 11 \nu^{3} + 20 \nu^{2} - 15 \nu + 9$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - 4 \nu^{4} + 14 \nu^{3} - 20 \nu^{2} + 30 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 5$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - 5 \beta_{1} - 5$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} + 11 \beta_{4} - 9 \beta_{3} - 7 \beta_{2} - 7 \beta_{1} + 16$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-17 \beta_{5} - 16 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} + 12 \beta_{1} + 31$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 − 1.41036i 0.5 + 2.05195i 0.5 + 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i 0.5 − 0.224437i
0 0 0 −1.21053 2.09671i 0 1.85185 + 1.88962i 0 0 0
109.2 0 0 0 −0.433463 0.750780i 0 −2.25729 1.38008i 0 0 0
109.3 0 0 0 2.14400 + 3.71351i 0 1.40545 2.24159i 0 0 0
541.1 0 0 0 −1.21053 + 2.09671i 0 1.85185 1.88962i 0 0 0
541.2 0 0 0 −0.433463 + 0.750780i 0 −2.25729 + 1.38008i 0 0 0
541.3 0 0 0 2.14400 3.71351i 0 1.40545 + 2.24159i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.k.f yes 6
3.b odd 2 1 756.2.k.e 6
7.c even 3 1 inner 756.2.k.f yes 6
7.c even 3 1 5292.2.a.u 3
7.d odd 6 1 5292.2.a.w 3
9.c even 3 1 2268.2.i.k 6
9.c even 3 1 2268.2.l.j 6
9.d odd 6 1 2268.2.i.j 6
9.d odd 6 1 2268.2.l.k 6
21.g even 6 1 5292.2.a.v 3
21.h odd 6 1 756.2.k.e 6
21.h odd 6 1 5292.2.a.x 3
63.g even 3 1 2268.2.i.k 6
63.h even 3 1 2268.2.l.j 6
63.j odd 6 1 2268.2.l.k 6
63.n odd 6 1 2268.2.i.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.e 6 3.b odd 2 1
756.2.k.e 6 21.h odd 6 1
756.2.k.f yes 6 1.a even 1 1 trivial
756.2.k.f yes 6 7.c even 3 1 inner
2268.2.i.j 6 9.d odd 6 1
2268.2.i.j 6 63.n odd 6 1
2268.2.i.k 6 9.c even 3 1
2268.2.i.k 6 63.g even 3 1
2268.2.l.j 6 9.c even 3 1
2268.2.l.j 6 63.h even 3 1
2268.2.l.k 6 9.d odd 6 1
2268.2.l.k 6 63.j odd 6 1
5292.2.a.u 3 7.c even 3 1
5292.2.a.v 3 21.g even 6 1
5292.2.a.w 3 7.d odd 6 1
5292.2.a.x 3 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$:

 $$T_{5}^{6} - T_{5}^{5} + 13 T_{5}^{4} + 30 T_{5}^{3} + 135 T_{5}^{2} + 108 T_{5} + 81$$ $$T_{13}^{3} + 2 T_{13}^{2} - 11 T_{13} - 21$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - T - 2 T^{2} + 35 T^{3} - 25 T^{4} - 52 T^{5} + 541 T^{6} - 260 T^{7} - 625 T^{8} + 4375 T^{9} - 1250 T^{10} - 3125 T^{11} + 15625 T^{12}$$
$7$ $$1 - 2 T + 2 T^{2} + 19 T^{3} + 14 T^{4} - 98 T^{5} + 343 T^{6}$$
$11$ $$1 - 5 T + 4 T^{2} - 11 T^{3} - 25 T^{4} + 718 T^{5} - 3005 T^{6} + 7898 T^{7} - 3025 T^{8} - 14641 T^{9} + 58564 T^{10} - 805255 T^{11} + 1771561 T^{12}$$
$13$ $$( 1 + 2 T + 28 T^{2} + 31 T^{3} + 364 T^{4} + 338 T^{5} + 2197 T^{6} )^{2}$$
$17$ $$1 + 4 T - 20 T^{2} - 146 T^{3} + 104 T^{4} + 1480 T^{5} + 4195 T^{6} + 25160 T^{7} + 30056 T^{8} - 717298 T^{9} - 1670420 T^{10} + 5679428 T^{11} + 24137569 T^{12}$$
$19$ $$1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12}$$
$23$ $$1 - 14 T + 88 T^{2} - 350 T^{3} + 1046 T^{4} - 602 T^{5} - 10061 T^{6} - 13846 T^{7} + 553334 T^{8} - 4258450 T^{9} + 24626008 T^{10} - 90108802 T^{11} + 148035889 T^{12}$$
$29$ $$( 1 - 5 T + 75 T^{2} - 227 T^{3} + 2175 T^{4} - 4205 T^{5} + 24389 T^{6} )^{2}$$
$31$ $$1 - 2 T - 78 T^{2} + 42 T^{3} + 3976 T^{4} - 200 T^{5} - 142097 T^{6} - 6200 T^{7} + 3820936 T^{8} + 1251222 T^{9} - 72034638 T^{10} - 57258302 T^{11} + 887503681 T^{12}$$
$37$ $$1 - 54 T^{2} - 274 T^{3} + 918 T^{4} + 7398 T^{5} + 12183 T^{6} + 273726 T^{7} + 1256742 T^{8} - 13878922 T^{9} - 101204694 T^{10} + 2565726409 T^{12}$$
$41$ $$( 1 + 12 T + 132 T^{2} + 903 T^{3} + 5412 T^{4} + 20172 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$( 1 + 9 T + 99 T^{2} + 493 T^{3} + 4257 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2}$$
$47$ $$1 + 9 T - 6 T^{2} - 423 T^{3} - 1947 T^{4} - 1260 T^{5} + 36583 T^{6} - 59220 T^{7} - 4300923 T^{8} - 43917129 T^{9} - 29278086 T^{10} + 2064105063 T^{11} + 10779215329 T^{12}$$
$53$ $$1 - 6 T - 24 T^{2} - 222 T^{3} + 6 T^{4} + 17166 T^{5} + 10591 T^{6} + 909798 T^{7} + 16854 T^{8} - 33050694 T^{9} - 189371544 T^{10} - 2509172958 T^{11} + 22164361129 T^{12}$$
$59$ $$1 + 5 T - 134 T^{2} - 223 T^{3} + 13355 T^{4} + 4526 T^{5} - 908765 T^{6} + 267034 T^{7} + 46488755 T^{8} - 45799517 T^{9} - 1623726374 T^{10} + 3574621495 T^{11} + 42180533641 T^{12}$$
$61$ $$1 + 7 T - 45 T^{2} + 84 T^{3} + 1093 T^{4} - 27251 T^{5} - 184994 T^{6} - 1662311 T^{7} + 4067053 T^{8} + 19066404 T^{9} - 623062845 T^{10} + 5912174107 T^{11} + 51520374361 T^{12}$$
$67$ $$1 - 16 T - 4 T^{2} + 234 T^{3} + 15028 T^{4} - 85604 T^{5} - 250049 T^{6} - 5735468 T^{7} + 67460692 T^{8} + 70378542 T^{9} - 80604484 T^{10} - 21602001712 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 + 11 T + 141 T^{2} + 1373 T^{3} + 10011 T^{4} + 55451 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 - T - 136 T^{2} + 477 T^{3} + 8461 T^{4} - 27386 T^{5} - 503183 T^{6} - 1999178 T^{7} + 45088669 T^{8} + 185561109 T^{9} - 3862160776 T^{10} - 2073071593 T^{11} + 151334226289 T^{12}$$
$79$ $$1 - 8 T - 18 T^{2} + 126 T^{3} - 2882 T^{4} + 21658 T^{5} + 208339 T^{6} + 1710982 T^{7} - 17986562 T^{8} + 62122914 T^{9} - 701101458 T^{10} - 24616451192 T^{11} + 243087455521 T^{12}$$
$83$ $$( 1 + 17 T + 333 T^{2} + 2921 T^{3} + 27639 T^{4} + 117113 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$1 + 3 T - 33 T^{2} + 1704 T^{3} + 393 T^{4} - 31803 T^{5} + 1804174 T^{6} - 2830467 T^{7} + 3112953 T^{8} + 1201267176 T^{9} - 2070493953 T^{10} + 16752178347 T^{11} + 496981290961 T^{12}$$
$97$ $$( 1 - 14 T + 251 T^{2} - 2660 T^{3} + 24347 T^{4} - 131726 T^{5} + 912673 T^{6} )^{2}$$