# Properties

 Label 6348.2.a.t Level $6348$ Weight $2$ Character orbit 6348.a Self dual yes Analytic conductor $50.689$ Analytic rank $0$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6348 = 2^{2} \cdot 3 \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6348.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.6890352031$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 4 x^{9} - 14 x^{8} + 65 x^{7} + 57 x^{6} - 354 x^{5} - 46 x^{4} + 714 x^{3} - 74 x^{2} - 323 x - 23$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 276) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} -\beta_{4} q^{5} + ( 2 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} -\beta_{4} q^{5} + ( 2 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{7} + q^{9} + ( 1 + \beta_{5} - \beta_{6} ) q^{11} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{13} -\beta_{4} q^{15} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} ) q^{17} + ( 4 - 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{19} + ( 2 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{21} + ( 3 + \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{25} + q^{27} + ( \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{29} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{31} + ( 1 + \beta_{5} - \beta_{6} ) q^{33} + ( -4 - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - 4 \beta_{8} ) q^{35} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{37} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{39} + ( 5 - \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} ) q^{41} + ( 5 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} + 3 \beta_{8} + \beta_{9} ) q^{43} -\beta_{4} q^{45} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{47} + ( 3 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{8} ) q^{49} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} ) q^{51} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{7} - \beta_{9} ) q^{53} + ( \beta_{1} + 4 \beta_{3} - \beta_{5} + 2 \beta_{8} + \beta_{9} ) q^{55} + ( 4 - 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{57} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} - 5 \beta_{7} - \beta_{8} - \beta_{9} ) q^{59} + ( 1 + \beta_{1} - 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 5 \beta_{7} - 4 \beta_{8} + \beta_{9} ) q^{61} + ( 2 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{63} + ( 1 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{65} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{67} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 5 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{71} + ( -4 - 2 \beta_{1} + 3 \beta_{3} + \beta_{5} - 2 \beta_{8} + \beta_{9} ) q^{73} + ( 3 + \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{75} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{77} + ( 5 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{79} + q^{81} + ( 2 + \beta_{1} - 2 \beta_{3} + 3 \beta_{6} + \beta_{7} + \beta_{9} ) q^{83} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{85} + ( \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{87} + ( 6 - 2 \beta_{1} - \beta_{2} + 7 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{89} + ( -\beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - 6 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{91} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{93} + ( -5 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{95} + ( 4 - 2 \beta_{2} - 3 \beta_{3} - \beta_{5} + \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{97} + ( 1 + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 10q^{3} + 2q^{5} + 11q^{7} + 10q^{9} + O(q^{10})$$ $$10q + 10q^{3} + 2q^{5} + 11q^{7} + 10q^{9} + 11q^{11} + 2q^{15} + 13q^{17} + 18q^{19} + 11q^{21} + 10q^{25} + 10q^{27} + 5q^{29} + 15q^{31} + 11q^{33} - 13q^{35} + 5q^{37} + 24q^{41} + 40q^{43} + 2q^{45} - 9q^{47} + 5q^{49} + 13q^{51} + 6q^{53} - 14q^{55} + 18q^{57} + 28q^{59} + 39q^{61} + 11q^{63} + 14q^{65} + 32q^{67} - 33q^{71} - 50q^{73} + 10q^{75} + 19q^{77} + 33q^{79} + 10q^{81} + 29q^{83} - 21q^{85} + 5q^{87} + 17q^{89} + 36q^{91} + 15q^{93} - 42q^{95} + 46q^{97} + 11q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4 x^{9} - 14 x^{8} + 65 x^{7} + 57 x^{6} - 354 x^{5} - 46 x^{4} + 714 x^{3} - 74 x^{2} - 323 x - 23$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-9 \nu^{9} - 4 \nu^{8} + 167 \nu^{7} + 216 \nu^{6} - 1140 \nu^{5} - 2057 \nu^{4} + 3380 \nu^{3} + 5481 \nu^{2} - 3540 \nu - 2070$$$$)/529$$ $$\beta_{3}$$ $$=$$ $$($$$$16 \nu^{9} - 44 \nu^{8} - 279 \nu^{7} + 559 \nu^{6} + 1743 \nu^{5} - 1766 \nu^{4} - 4266 \nu^{3} + 8 \nu^{2} + 2000 \nu + 1035$$$$)/529$$ $$\beta_{4}$$ $$=$$ $$($$$$20 \nu^{9} - 55 \nu^{8} - 481 \nu^{7} + 831 \nu^{6} + 3898 \nu^{5} - 3530 \nu^{4} - 11416 \nu^{3} + 3184 \nu^{2} + 6203 \nu - 161$$$$)/529$$ $$\beta_{5}$$ $$=$$ $$($$$$50 \nu^{9} + 12 \nu^{8} - 1030 \nu^{7} - 165 \nu^{6} + 7123 \nu^{5} + 927 \nu^{4} - 18006 \nu^{3} - 2896 \nu^{2} + 9815 \nu + 2507$$$$)/529$$ $$\beta_{6}$$ $$=$$ $$($$$$-76 \nu^{9} + 117 \nu^{8} + 1331 \nu^{7} - 1672 \nu^{6} - 7917 \nu^{5} + 7250 \nu^{4} + 17423 \nu^{3} - 9146 \nu^{2} - 7706 \nu + 506$$$$)/529$$ $$\beta_{7}$$ $$=$$ $$($$$$-95 \nu^{9} + 152 \nu^{8} + 1589 \nu^{7} - 2044 \nu^{6} - 9051 \nu^{5} + 7993 \nu^{4} + 19427 \nu^{3} - 8086 \nu^{2} - 8885 \nu - 161$$$$)/529$$ $$\beta_{8}$$ $$=$$ $$($$$$-99 \nu^{9} + 163 \nu^{8} + 1791 \nu^{7} - 2316 \nu^{6} - 11206 \nu^{5} + 9757 \nu^{4} + 26577 \nu^{3} - 10733 \nu^{2} - 13617 \nu - 1081$$$$)/529$$ $$\beta_{9}$$ $$=$$ $$($$$$-233 \nu^{9} + 405 \nu^{8} + 4119 \nu^{7} - 5563 \nu^{6} - 25289 \nu^{5} + 22023 \nu^{4} + 59953 \nu^{3} - 20943 \nu^{2} - 33058 \nu - 2277$$$$)/529$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{4} + 6 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} + 9 \beta_{8} - 6 \beta_{7} + \beta_{5} + 6 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 25$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{9} - 12 \beta_{8} - 22 \beta_{7} + 17 \beta_{6} + 10 \beta_{4} - 5 \beta_{3} + \beta_{2} + 40 \beta_{1} + 8$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{9} + 69 \beta_{8} - 38 \beta_{7} + 5 \beta_{6} + 12 \beta_{5} + 36 \beta_{4} - 17 \beta_{3} + 27 \beta_{2} + 61 \beta_{1} + 166$$ $$\nu^{7}$$ $$=$$ $$69 \beta_{9} - 107 \beta_{8} - 196 \beta_{7} + 180 \beta_{6} + 2 \beta_{5} + 80 \beta_{4} - 61 \beta_{3} + 20 \beta_{2} + 277 \beta_{1} + 59$$ $$\nu^{8}$$ $$=$$ $$-107 \beta_{9} + 504 \beta_{8} - 270 \beta_{7} + 95 \beta_{6} + 116 \beta_{5} + 225 \beta_{4} - 84 \beta_{3} + 269 \beta_{2} + 463 \beta_{1} + 1132$$ $$\nu^{9}$$ $$=$$ $$504 \beta_{9} - 857 \beta_{8} - 1631 \beta_{7} + 1640 \beta_{6} + 45 \beta_{5} + 595 \beta_{4} - 564 \beta_{3} + 257 \beta_{2} + 1972 \beta_{1} + 430$$

## 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
 2.75582 −1.92165 −2.66501 2.46393 −2.12473 −0.0733115 −0.633097 2.84010 1.00266 2.35529
0 1.00000 0 −3.63668 0 4.36939 0 1.00000 0
1.2 0 1.00000 0 −2.68761 0 0.168413 0 1.00000 0
1.3 0 1.00000 0 −2.49042 0 4.04500 0 1.00000 0
1.4 0 1.00000 0 −1.04710 0 −1.46307 0 1.00000 0
1.5 0 1.00000 0 0.395239 0 −3.09253 0 1.00000 0
1.6 0 1.00000 0 1.12335 0 −1.05967 0 1.00000 0
1.7 0 1.00000 0 1.52600 0 3.74770 0 1.00000 0
1.8 0 1.00000 0 1.80838 0 3.41002 0 1.00000 0
1.9 0 1.00000 0 2.92409 0 1.00076 0 1.00000 0
1.10 0 1.00000 0 4.08477 0 −0.126013 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$23$$ $$-1$$

## 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 6348.2.a.t 10
23.b odd 2 1 6348.2.a.s 10
23.d odd 22 2 276.2.i.a 20
69.g even 22 2 828.2.q.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.i.a 20 23.d odd 22 2
828.2.q.c 20 69.g even 22 2
6348.2.a.s 10 23.b odd 2 1
6348.2.a.t 10 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6348))$$:

 $$T_{5}^{10} - \cdots$$ $$T_{7}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$( -1 + T )^{10}$$
$5$ $$373 - 1198 T + 172 T^{2} + 1528 T^{3} - 709 T^{4} - 457 T^{5} + 245 T^{6} + 52 T^{7} - 28 T^{8} - 2 T^{9} + T^{10}$$
$7$ $$23 + 44 T - 1119 T^{2} + 33 T^{3} + 1493 T^{4} - 154 T^{5} - 455 T^{6} + 121 T^{7} + 23 T^{8} - 11 T^{9} + T^{10}$$
$11$ $$-6049 + 14509 T - 9268 T^{2} - 2607 T^{3} + 4487 T^{4} - 693 T^{5} - 595 T^{6} + 187 T^{7} + 16 T^{8} - 11 T^{9} + T^{10}$$
$13$ $$-27323 - 41008 T + 48284 T^{2} + 26202 T^{3} - 16721 T^{4} - 3762 T^{5} + 2155 T^{6} + 88 T^{7} - 83 T^{8} + T^{10}$$
$17$ $$13463 - 9303 T - 26168 T^{2} + 15337 T^{3} + 9802 T^{4} - 4772 T^{5} - 862 T^{6} + 466 T^{7} + 2 T^{8} - 13 T^{9} + T^{10}$$
$19$ $$22639 + 169475 T + 72147 T^{2} - 238111 T^{3} + 95263 T^{4} - 576 T^{5} - 5947 T^{6} + 887 T^{7} + 47 T^{8} - 18 T^{9} + T^{10}$$
$23$ $$T^{10}$$
$29$ $$307 - 2113 T - 4703 T^{2} + 9719 T^{3} + 11014 T^{4} - 10265 T^{5} + 1337 T^{6} + 442 T^{7} - 81 T^{8} - 5 T^{9} + T^{10}$$
$31$ $$-36961 + 189646 T - 192853 T^{2} - 7652 T^{3} + 54119 T^{4} - 9399 T^{5} - 3285 T^{6} + 922 T^{7} - 2 T^{8} - 15 T^{9} + T^{10}$$
$37$ $$-204113471 - 9425538 T + 28228293 T^{2} + 1467538 T^{3} - 1321999 T^{4} - 62471 T^{5} + 27561 T^{6} + 970 T^{7} - 268 T^{8} - 5 T^{9} + T^{10}$$
$41$ $$19742141 + 8060464 T - 6663533 T^{2} - 595492 T^{3} + 704262 T^{4} - 55164 T^{5} - 17045 T^{6} + 2478 T^{7} + 59 T^{8} - 24 T^{9} + T^{10}$$
$43$ $$24034781 - 28235851 T + 11074643 T^{2} - 626382 T^{3} - 734449 T^{4} + 204721 T^{5} - 12544 T^{6} - 2861 T^{7} + 578 T^{8} - 40 T^{9} + T^{10}$$
$47$ $$-337853 - 112793 T + 691965 T^{2} + 815546 T^{3} + 361609 T^{4} + 63239 T^{5} - 1114 T^{6} - 1665 T^{7} - 125 T^{8} + 9 T^{9} + T^{10}$$
$53$ $$-109070807 - 984055 T + 18784451 T^{2} + 913129 T^{3} - 998975 T^{4} - 51242 T^{5} + 23277 T^{6} + 965 T^{7} - 249 T^{8} - 6 T^{9} + T^{10}$$
$59$ $$1346311 + 5482445 T - 1440588 T^{2} - 2337594 T^{3} + 797679 T^{4} + 97297 T^{5} - 56919 T^{6} + 5674 T^{7} + 29 T^{8} - 28 T^{9} + T^{10}$$
$61$ $$-4112263 + 71930 T + 3198156 T^{2} - 752307 T^{3} - 561866 T^{4} + 269007 T^{5} - 36679 T^{6} - 200 T^{7} + 469 T^{8} - 39 T^{9} + T^{10}$$
$67$ $$4947251 - 5691787 T + 172938 T^{2} + 2458760 T^{3} - 1475796 T^{4} + 391754 T^{5} - 50231 T^{6} + 1799 T^{7} + 274 T^{8} - 32 T^{9} + T^{10}$$
$71$ $$3825889 + 5768169 T - 69217287 T^{2} - 745030 T^{3} + 4616625 T^{4} + 384373 T^{5} - 58272 T^{6} - 7227 T^{7} + 75 T^{8} + 33 T^{9} + T^{10}$$
$73$ $$-204421579 - 18135253 T + 31249180 T^{2} + 5547664 T^{3} - 1250539 T^{4} - 389785 T^{5} - 11463 T^{6} + 6466 T^{7} + 921 T^{8} + 50 T^{9} + T^{10}$$
$79$ $$-26171441 + 19513384 T - 232902 T^{2} - 3152787 T^{3} + 776167 T^{4} + 32890 T^{5} - 29736 T^{6} + 2156 T^{7} + 234 T^{8} - 33 T^{9} + T^{10}$$
$83$ $$-35374439 + 61909400 T - 13531448 T^{2} - 3524083 T^{3} + 1251755 T^{4} - 2836 T^{5} - 31220 T^{6} + 2588 T^{7} + 164 T^{8} - 29 T^{9} + T^{10}$$
$89$ $$82902247 - 302319404 T - 2128860 T^{2} + 22835561 T^{3} - 629380 T^{4} - 570249 T^{5} + 25921 T^{6} + 5384 T^{7} - 291 T^{8} - 17 T^{9} + T^{10}$$
$97$ $$-242728531 + 85281175 T + 33114594 T^{2} - 17684558 T^{3} + 1255711 T^{4} + 429995 T^{5} - 70925 T^{6} + 74 T^{7} + 643 T^{8} - 46 T^{9} + T^{10}$$