# Properties

 Label 7448.2.a.br Level $7448$ Weight $2$ Character orbit 7448.a Self dual yes Analytic conductor $59.473$ Analytic rank $0$ Dimension $8$ 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: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 15 x^{6} - x^{5} + 66 x^{4} + 4 x^{3} - 76 x^{2} + 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1064) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( 1 + \beta_{2} ) q^{9} + ( 1 + \beta_{3} + \beta_{4} - \beta_{7} ) q^{11} + \beta_{7} q^{13} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{15} + ( \beta_{2} - \beta_{3} + \beta_{5} ) q^{17} - q^{19} + ( 3 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{23} + ( 2 - \beta_{5} - \beta_{6} - \beta_{7} ) q^{25} + ( -\beta_{3} - \beta_{6} + \beta_{7} ) q^{27} + ( \beta_{4} + \beta_{6} ) q^{29} + ( \beta_{2} - 2 \beta_{3} + \beta_{7} ) q^{31} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{33} + ( 2 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{37} + ( -1 + \beta_{5} ) q^{39} + 2 q^{41} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{43} + ( -3 - 2 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{45} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} + 3 \beta_{7} ) q^{51} + ( -3 \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{53} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{55} + \beta_{1} q^{57} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{61} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{65} + ( -\beta_{1} - \beta_{2} + \beta_{7} ) q^{67} + ( 2 - 4 \beta_{1} + \beta_{3} - \beta_{6} ) q^{69} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{71} + ( -2 - 3 \beta_{1} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{73} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{75} + ( 2 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{7} ) q^{79} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{81} + ( 5 + 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{83} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{85} + ( -2 - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{87} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{89} + ( 1 + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{93} + \beta_{3} q^{95} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{97} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{5} + 6 q^{9} + O(q^{10})$$ $$8 q + q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} - 4 q^{17} - 8 q^{19} + 25 q^{23} + 15 q^{25} - 3 q^{27} + 6 q^{29} - 8 q^{33} + 13 q^{37} - 11 q^{39} + 16 q^{41} + 17 q^{43} - 17 q^{45} + 24 q^{47} + 5 q^{51} + 2 q^{53} - 5 q^{55} - 2 q^{59} + 13 q^{61} - 26 q^{65} + 2 q^{67} + 11 q^{69} + 10 q^{71} - 5 q^{73} + 20 q^{75} + 16 q^{79} - 12 q^{81} + 43 q^{83} + 24 q^{85} - 20 q^{87} - 8 q^{89} + 2 q^{93} - q^{95} + 12 q^{97} + 37 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 15 x^{6} - x^{5} + 66 x^{4} + 4 x^{3} - 76 x^{2} + 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{5} - 11 \nu^{4} + 17 \nu^{3} + 32 \nu^{2} - 24 \nu - 16$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - 13 \nu^{5} - 16 \nu^{4} + 43 \nu^{3} + 64 \nu^{2} - 12 \nu - 32$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 13 \nu^{5} + 5 \nu^{4} - 46 \nu^{3} - 34 \nu^{2} + 28 \nu + 18$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{6} - 28 \nu^{5} - 17 \nu^{4} + 109 \nu^{3} + 68 \nu^{2} - 84 \nu - 24$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - 15 \nu^{5} - 14 \nu^{4} + 61 \nu^{3} + 50 \nu^{2} - 42 \nu - 20$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} + \beta_{3} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} - \beta_{3} + 8 \beta_{2} + \beta_{1} + 25$$ $$\nu^{5}$$ $$=$$ $$-10 \beta_{7} + 10 \beta_{6} + \beta_{5} + 2 \beta_{4} + 8 \beta_{3} + \beta_{2} + 40 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{7} + 14 \beta_{6} + 13 \beta_{5} + 4 \beta_{4} - 8 \beta_{3} + 58 \beta_{2} + 13 \beta_{1} + 169$$ $$\nu^{7}$$ $$=$$ $$-84 \beta_{7} + 89 \beta_{6} + 16 \beta_{5} + 26 \beta_{4} + 53 \beta_{3} + 19 \beta_{2} + 277 \beta_{1} + 46$$

## 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
 2.85665 2.39016 0.912296 0.684189 −0.436576 −1.34151 −2.46993 −2.59528
0 −2.85665 0 −0.827910 0 0 0 5.16044 0
1.2 0 −2.39016 0 −3.25179 0 0 0 2.71288 0
1.3 0 −0.912296 0 1.66528 0 0 0 −2.16772 0
1.4 0 −0.684189 0 3.65096 0 0 0 −2.53189 0
1.5 0 0.436576 0 0.299640 0 0 0 −2.80940 0
1.6 0 1.34151 0 −2.90869 0 0 0 −1.20034 0
1.7 0 2.46993 0 4.03871 0 0 0 3.10055 0
1.8 0 2.59528 0 −1.66621 0 0 0 3.73546 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.br 8
7.b odd 2 1 7448.2.a.bq 8
7.d odd 6 2 1064.2.q.n 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.n 16 7.d odd 6 2
7448.2.a.bq 8 7.b odd 2 1
7448.2.a.br 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7448))$$:

 $$T_{3}^{8} - 15 T_{3}^{6} + T_{3}^{5} + 66 T_{3}^{4} - 4 T_{3}^{3} - 76 T_{3}^{2} - 8 T_{3} + 16$$ $$T_{5}^{8} - \cdots$$ $$T_{11}^{8} - \cdots$$ $$T_{13}^{8} - 41 T_{13}^{6} - T_{13}^{5} + 310 T_{13}^{4} - 200 T_{13}^{3} - 516 T_{13}^{2} + 488 T_{13} - 16$$ $$T_{17}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$16 - 8 T - 76 T^{2} - 4 T^{3} + 66 T^{4} + T^{5} - 15 T^{6} + T^{8}$$
$5$ $$96 - 192 T - 464 T^{2} + 52 T^{3} + 222 T^{4} + 9 T^{5} - 27 T^{6} - T^{7} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$-8 + 12 T + 276 T^{2} + 219 T^{3} - 575 T^{4} + 234 T^{5} - 6 T^{6} - 9 T^{7} + T^{8}$$
$13$ $$-16 + 488 T - 516 T^{2} - 200 T^{3} + 310 T^{4} - T^{5} - 41 T^{6} + T^{8}$$
$17$ $$-24632 - 49060 T - 13866 T^{2} + 6693 T^{3} + 1921 T^{4} - 291 T^{5} - 78 T^{6} + 4 T^{7} + T^{8}$$
$19$ $$( 1 + T )^{8}$$
$23$ $$8831 - 8540 T - 6571 T^{2} + 5239 T^{3} + 254 T^{4} - 842 T^{5} + 229 T^{6} - 25 T^{7} + T^{8}$$
$29$ $$6672 + 3024 T - 19204 T^{2} - 420 T^{3} + 3630 T^{4} + 329 T^{5} - 105 T^{6} - 6 T^{7} + T^{8}$$
$31$ $$9344 - 2944 T - 9856 T^{2} - 384 T^{3} + 1944 T^{4} + 116 T^{5} - 100 T^{6} + T^{8}$$
$37$ $$-536000 + 84416 T + 128620 T^{2} - 38244 T^{3} - 2172 T^{4} + 1617 T^{5} - 89 T^{6} - 13 T^{7} + T^{8}$$
$41$ $$( -2 + T )^{8}$$
$43$ $$-51712 - 8256 T + 33288 T^{2} - 2180 T^{3} - 4964 T^{4} + 1081 T^{5} + 11 T^{6} - 17 T^{7} + T^{8}$$
$47$ $$13177 - 94228 T + 99216 T^{2} + 49952 T^{3} - 28762 T^{4} + 3532 T^{5} + 16 T^{6} - 24 T^{7} + T^{8}$$
$53$ $$-1827792 + 2630088 T - 589620 T^{2} - 92016 T^{3} + 25732 T^{4} + 779 T^{5} - 295 T^{6} - 2 T^{7} + T^{8}$$
$59$ $$952576 + 293696 T - 601712 T^{2} - 18452 T^{3} + 26810 T^{4} - 155 T^{5} - 311 T^{6} + 2 T^{7} + T^{8}$$
$61$ $$124920 + 1003116 T + 102976 T^{2} - 116073 T^{3} + 4949 T^{4} + 2330 T^{5} - 158 T^{6} - 13 T^{7} + T^{8}$$
$67$ $$14848 - 3968 T - 9248 T^{2} + 684 T^{3} + 1740 T^{4} + 69 T^{5} - 83 T^{6} - 2 T^{7} + T^{8}$$
$71$ $$-712576 + 651456 T + 99968 T^{2} - 147712 T^{3} + 12656 T^{4} + 2476 T^{5} - 240 T^{6} - 10 T^{7} + T^{8}$$
$73$ $$-282879 - 713790 T - 412001 T^{2} + 69965 T^{3} + 26580 T^{4} - 1796 T^{5} - 357 T^{6} + 5 T^{7} + T^{8}$$
$79$ $$-2067840 + 2325312 T - 24000 T^{2} - 178176 T^{3} + 11056 T^{4} + 3404 T^{5} - 220 T^{6} - 16 T^{7} + T^{8}$$
$83$ $$95768 - 192580 T - 42608 T^{2} + 86413 T^{3} - 9059 T^{4} - 2854 T^{5} + 626 T^{6} - 43 T^{7} + T^{8}$$
$89$ $$-512640 + 744384 T - 305888 T^{2} + 4960 T^{3} + 17280 T^{4} - 1404 T^{5} - 264 T^{6} + 8 T^{7} + T^{8}$$
$97$ $$580864 + 820352 T + 135104 T^{2} - 117344 T^{3} + 4944 T^{4} + 2336 T^{5} - 168 T^{6} - 12 T^{7} + T^{8}$$