# Properties

 Label 7448.2.a.bm Level $7448$ Weight $2$ Character orbit 7448.a Self dual yes Analytic conductor $59.473$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7448 = 2^{3} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7448.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$59.4725794254$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.98211824.1 Defining polynomial: $$x^{6} - 3 x^{5} - 6 x^{4} + 15 x^{3} + 8 x^{2} - 9 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{3} + \beta_{3} q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{3} + \beta_{3} q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{11} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{13} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{15} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + q^{19} + ( -2 \beta_{3} - \beta_{4} ) q^{23} + ( 2 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{25} + ( -1 + \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{27} + ( -3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 - \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{31} + ( -4 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{33} + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{37} + ( 6 - 5 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{39} + ( 4 + 2 \beta_{1} - 3 \beta_{5} ) q^{41} + ( \beta_{2} - \beta_{3} + \beta_{5} ) q^{43} + ( 1 - 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{45} + ( -5 - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{47} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{51} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{53} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{55} + ( -1 + \beta_{1} ) q^{57} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{59} + ( -3 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{61} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 2 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{67} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{69} + ( 3 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{71} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{73} + ( -5 - 2 \beta_{2} + \beta_{4} ) q^{75} + ( -6 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{79} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{81} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{83} + ( 10 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{85} + ( -5 - 3 \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{87} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{89} + ( 2 - 6 \beta_{1} - 4 \beta_{2} - \beta_{4} ) q^{93} + \beta_{3} q^{95} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{97} + ( 8 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} - q^{5} + 3 q^{9} + O(q^{10})$$ $$6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} - 6 x^{4} + 15 x^{3} + 8 x^{2} - 9 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 3 \nu^{4} - 5 \nu^{3} + 13 \nu^{2} + 3 \nu - 5$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 9 \nu^{3} + 10 \nu^{2} + 19 \nu - 4$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} + 7 \nu^{3} - 17 \nu^{2} - 13 \nu + 11$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 3 \nu^{4} + 6 \nu^{3} - 14 \nu^{2} - 9 \nu + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 3 \beta_{2} + 7 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$11 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + 13 \beta_{2} + 15 \beta_{1} + 20$$ $$\nu^{5}$$ $$=$$ $$30 \beta_{5} - 13 \beta_{4} + 6 \beta_{3} + 43 \beta_{2} + 64 \beta_{1} + 41$$

## 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
 −2.04436 −1.04143 0.129411 0.511631 2.15897 3.28578
0 −3.04436 0 2.60221 0 0 0 6.26815 0
1.2 0 −2.04143 0 −3.17676 0 0 0 1.16742 0
1.3 0 −0.870589 0 −0.696873 0 0 0 −2.24208 0
1.4 0 −0.488369 0 3.51566 0 0 0 −2.76150 0
1.5 0 1.15897 0 −1.74190 0 0 0 −1.65679 0
1.6 0 2.28578 0 −1.50233 0 0 0 2.22478 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

## 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 7448.2.a.bm 6
7.b odd 2 1 7448.2.a.bn yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7448.2.a.bm 6 1.a even 1 1 trivial
7448.2.a.bn yes 6 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7448))$$:

 $$T_{3}^{6} + 3 T_{3}^{5} - 6 T_{3}^{4} - 19 T_{3}^{3} + 2 T_{3}^{2} + 19 T_{3} + 7$$ $$T_{5}^{6} + T_{5}^{5} - 17 T_{5}^{4} - 24 T_{5}^{3} + 59 T_{5}^{2} + 123 T_{5} + 53$$ $$T_{11}^{6} - 3 T_{11}^{5} - 28 T_{11}^{4} + 121 T_{11}^{3} + 4 T_{11}^{2} - 539 T_{11} + 575$$ $$T_{13}^{6} + 6 T_{13}^{5} - 40 T_{13}^{4} - 194 T_{13}^{3} + 639 T_{13}^{2} + 1544 T_{13} - 3920$$ $$T_{17}^{6} - 2 T_{17}^{5} - 55 T_{17}^{4} + 138 T_{17}^{3} + 91 T_{17}^{2} - 322 T_{17} + 148$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$7 + 19 T + 2 T^{2} - 19 T^{3} - 6 T^{4} + 3 T^{5} + T^{6}$$
$5$ $$53 + 123 T + 59 T^{2} - 24 T^{3} - 17 T^{4} + T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$575 - 539 T + 4 T^{2} + 121 T^{3} - 28 T^{4} - 3 T^{5} + T^{6}$$
$13$ $$-3920 + 1544 T + 639 T^{2} - 194 T^{3} - 40 T^{4} + 6 T^{5} + T^{6}$$
$17$ $$148 - 322 T + 91 T^{2} + 138 T^{3} - 55 T^{4} - 2 T^{5} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$-6860 - 1178 T + 1369 T^{2} + 140 T^{3} - 78 T^{4} - 2 T^{5} + T^{6}$$
$29$ $$9521 - 8729 T + 2112 T^{2} + 147 T^{3} - 90 T^{4} + T^{5} + T^{6}$$
$31$ $$3148 + 6410 T - 773 T^{2} - 596 T^{3} - 3 T^{4} + 14 T^{5} + T^{6}$$
$37$ $$-7 + 3 T + 93 T^{2} - 66 T^{3} - 27 T^{4} + 7 T^{5} + T^{6}$$
$41$ $$-43133 - 52561 T - 1954 T^{2} + 1921 T^{3} - 6 T^{4} - 21 T^{5} + T^{6}$$
$43$ $$-1472 - 224 T + 544 T^{2} + 52 T^{3} - 51 T^{4} - T^{5} + T^{6}$$
$47$ $$-4855 - 10983 T - 3633 T^{2} + 20 T^{3} + 159 T^{4} + 23 T^{5} + T^{6}$$
$53$ $$4549 + 3249 T - 1382 T^{2} - 1021 T^{3} - 36 T^{4} + 15 T^{5} + T^{6}$$
$59$ $$-79 + 871 T + 1469 T^{2} + 856 T^{3} + 221 T^{4} + 25 T^{5} + T^{6}$$
$61$ $$-5867 + 2083 T + 2247 T^{2} - 1036 T^{3} + 11 T^{4} + 21 T^{5} + T^{6}$$
$67$ $$4396 - 20032 T + 11549 T^{2} + 496 T^{3} - 253 T^{4} - 2 T^{5} + T^{6}$$
$71$ $$1813 - 2249 T - 459 T^{2} + 398 T^{3} + T^{4} - 13 T^{5} + T^{6}$$
$73$ $$-71932 + 53374 T + 15735 T^{2} - 326 T^{3} - 245 T^{4} - 2 T^{5} + T^{6}$$
$79$ $$-35696 + 163568 T + 14268 T^{2} - 3908 T^{3} - 245 T^{4} + 17 T^{5} + T^{6}$$
$83$ $$-32732 - 1600 T + 4927 T^{2} - 180 T^{3} - 133 T^{4} + 4 T^{5} + T^{6}$$
$89$ $$-360640 - 90656 T + 35672 T^{2} + 840 T^{3} - 379 T^{4} - T^{5} + T^{6}$$
$97$ $$74851 - 33099 T - 547 T^{2} + 1312 T^{3} - 69 T^{4} - 13 T^{5} + T^{6}$$