# Properties

 Label 49.6.a.f Level 49 Weight 6 Character orbit 49.a Self dual yes Analytic conductor 7.859 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.85880717084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 5 - \beta ) q^{2} + ( 6 - 6 \beta ) q^{3} + ( 7 - 9 \beta ) q^{4} + ( 4 + 10 \beta ) q^{5} + ( 114 - 30 \beta ) q^{6} + ( 1 - 11 \beta ) q^{8} + ( 297 - 36 \beta ) q^{9} +O(q^{10})$$ $$q + ( 5 - \beta ) q^{2} + ( 6 - 6 \beta ) q^{3} + ( 7 - 9 \beta ) q^{4} + ( 4 + 10 \beta ) q^{5} + ( 114 - 30 \beta ) q^{6} + ( 1 - 11 \beta ) q^{8} + ( 297 - 36 \beta ) q^{9} + ( -120 + 36 \beta ) q^{10} + ( 136 + 124 \beta ) q^{11} + ( 798 - 42 \beta ) q^{12} + ( 112 + 126 \beta ) q^{13} + ( -816 - 24 \beta ) q^{15} + ( -65 + 243 \beta ) q^{16} + ( -862 - 76 \beta ) q^{17} + ( 1989 - 441 \beta ) q^{18} + ( 1642 - 18 \beta ) q^{19} + ( -1232 - 56 \beta ) q^{20} + ( -1056 + 360 \beta ) q^{22} + ( 1328 - 568 \beta ) q^{23} + ( 930 - 6 \beta ) q^{24} + ( -1709 + 180 \beta ) q^{25} + ( -1204 + 392 \beta ) q^{26} + ( 3348 - 324 \beta ) q^{27} + ( 3474 - 252 \beta ) q^{29} + ( -3744 + 720 \beta ) q^{30} + ( -260 + 540 \beta ) q^{31} + ( -3759 + 1389 \beta ) q^{32} + ( -9600 - 816 \beta ) q^{33} + ( -3246 + 558 \beta ) q^{34} + ( 6615 - 2601 \beta ) q^{36} + ( 3386 - 540 \beta ) q^{37} + ( 8462 - 1714 \beta ) q^{38} + ( -9912 - 672 \beta ) q^{39} + ( -1536 - 144 \beta ) q^{40} + ( 3570 - 1092 \beta ) q^{41} + ( -3904 + 4788 \beta ) q^{43} + ( -14672 - 1472 \beta ) q^{44} + ( -3852 + 2466 \beta ) q^{45} + ( 14592 - 3600 \beta ) q^{46} + ( -7724 + 3748 \beta ) q^{47} + ( -20802 + 390 \beta ) q^{48} + ( -11065 + 2429 \beta ) q^{50} + ( 1212 + 5172 \beta ) q^{51} + ( -15092 - 1260 \beta ) q^{52} + ( 4630 + 208 \beta ) q^{53} + ( 21276 - 4644 \beta ) q^{54} + ( 17904 + 3096 \beta ) q^{55} + ( 11364 - 9852 \beta ) q^{57} + ( 20898 - 4482 \beta ) q^{58} + ( 22994 - 2050 \beta ) q^{59} + ( -2688 + 7392 \beta ) q^{60} + ( 34780 - 4806 \beta ) q^{61} + ( -8860 + 2420 \beta ) q^{62} + ( -36161 + 1539 \beta ) q^{64} + ( 18088 + 2884 \beta ) q^{65} + ( -36576 + 6336 \beta ) q^{66} + ( 11420 + 1944 \beta ) q^{67} + ( 3542 + 7910 \beta ) q^{68} + ( 55680 - 7968 \beta ) q^{69} + ( 46608 + 4200 \beta ) q^{71} + ( 5841 - 2907 \beta ) q^{72} + ( -6098 - 5256 \beta ) q^{73} + ( 24490 - 5546 \beta ) q^{74} + ( -25374 + 10254 \beta ) q^{75} + ( 13762 - 14742 \beta ) q^{76} + ( -40152 + 7224 \beta ) q^{78} + ( 33080 - 14904 \beta ) q^{79} + ( 33760 + 2752 \beta ) q^{80} + ( -24867 - 11340 \beta ) q^{81} + ( 33138 - 7938 \beta ) q^{82} + ( -66654 + 15750 \beta ) q^{83} + ( -14088 - 9684 \beta ) q^{85} + ( -86552 + 23056 \beta ) q^{86} + ( 42012 - 20844 \beta ) q^{87} + ( -18960 - 2736 \beta ) q^{88} + ( -31034 - 22208 \beta ) q^{89} + ( -53784 + 13716 \beta ) q^{90} + ( 80864 - 10816 \beta ) q^{92} + ( -46920 + 1560 \beta ) q^{93} + ( -91092 + 22716 \beta ) q^{94} + ( 4048 + 16168 \beta ) q^{95} + ( -139230 + 22554 \beta ) q^{96} + ( -14798 + 8820 \beta ) q^{97} + ( -22104 + 27468 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 9q^{2} + 6q^{3} + 5q^{4} + 18q^{5} + 198q^{6} - 9q^{8} + 558q^{9} + O(q^{10})$$ $$2q + 9q^{2} + 6q^{3} + 5q^{4} + 18q^{5} + 198q^{6} - 9q^{8} + 558q^{9} - 204q^{10} + 396q^{11} + 1554q^{12} + 350q^{13} - 1656q^{15} + 113q^{16} - 1800q^{17} + 3537q^{18} + 3266q^{19} - 2520q^{20} - 1752q^{22} + 2088q^{23} + 1854q^{24} - 3238q^{25} - 2016q^{26} + 6372q^{27} + 6696q^{29} - 6768q^{30} + 20q^{31} - 6129q^{32} - 20016q^{33} - 5934q^{34} + 10629q^{36} + 6232q^{37} + 15210q^{38} - 20496q^{39} - 3216q^{40} + 6048q^{41} - 3020q^{43} - 30816q^{44} - 5238q^{45} + 25584q^{46} - 11700q^{47} - 41214q^{48} - 19701q^{50} + 7596q^{51} - 31444q^{52} + 9468q^{53} + 37908q^{54} + 38904q^{55} + 12876q^{57} + 37314q^{58} + 43938q^{59} + 2016q^{60} + 64754q^{61} - 15300q^{62} - 70783q^{64} + 39060q^{65} - 66816q^{66} + 24784q^{67} + 14994q^{68} + 103392q^{69} + 97416q^{71} + 8775q^{72} - 17452q^{73} + 43434q^{74} - 40494q^{75} + 12782q^{76} - 73080q^{78} + 51256q^{79} + 70272q^{80} - 61074q^{81} + 58338q^{82} - 117558q^{83} - 37860q^{85} - 150048q^{86} + 63180q^{87} - 40656q^{88} - 84276q^{89} - 93852q^{90} + 150912q^{92} - 92280q^{93} - 159468q^{94} + 24264q^{95} - 255906q^{96} - 20776q^{97} - 16740q^{99} + O(q^{100})$$

## 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
 4.27492 −3.27492
0.725083 −19.6495 −31.4743 46.7492 −14.2475 0 −46.0241 143.103 33.8970
1.2 8.27492 25.6495 36.4743 −28.7492 212.248 0 37.0241 414.897 −237.897
 $$n$$: e.g. 2-40 or 990-1000 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 49.6.a.f 2
3.b odd 2 1 441.6.a.l 2
4.b odd 2 1 784.6.a.v 2
7.b odd 2 1 7.6.a.b 2
7.c even 3 2 49.6.c.d 4
7.d odd 6 2 49.6.c.e 4
21.c even 2 1 63.6.a.f 2
28.d even 2 1 112.6.a.h 2
35.c odd 2 1 175.6.a.c 2
35.f even 4 2 175.6.b.c 4
56.e even 2 1 448.6.a.u 2
56.h odd 2 1 448.6.a.w 2
77.b even 2 1 847.6.a.c 2
84.h odd 2 1 1008.6.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 7.b odd 2 1
49.6.a.f 2 1.a even 1 1 trivial
49.6.c.d 4 7.c even 3 2
49.6.c.e 4 7.d odd 6 2
63.6.a.f 2 21.c even 2 1
112.6.a.h 2 28.d even 2 1
175.6.a.c 2 35.c odd 2 1
175.6.b.c 4 35.f even 4 2
441.6.a.l 2 3.b odd 2 1
448.6.a.u 2 56.e even 2 1
448.6.a.w 2 56.h odd 2 1
784.6.a.v 2 4.b odd 2 1
847.6.a.c 2 77.b even 2 1
1008.6.a.bq 2 84.h odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2}^{2} - 9 T_{2} + 6$$ $$T_{3}^{2} - 6 T_{3} - 504$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 9 T + 70 T^{2} - 288 T^{3} + 1024 T^{4}$$
$3$ $$1 - 6 T - 18 T^{2} - 1458 T^{3} + 59049 T^{4}$$
$5$ $$1 - 18 T + 4906 T^{2} - 56250 T^{3} + 9765625 T^{4}$$
$7$ 1
$11$ $$1 - 396 T + 142198 T^{2} - 63776196 T^{3} + 25937424601 T^{4}$$
$13$ $$1 - 350 T + 546978 T^{2} - 129952550 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 1800 T + 3567406 T^{2} + 2555742600 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 - 3266 T + 7614270 T^{2} - 8086939334 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 - 2088 T + 9365230 T^{2} - 13439084184 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 - 6696 T + 51326470 T^{2} - 137342653704 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 - 20 T + 53103102 T^{2} - 572583020 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 - 6232 T + 144242070 T^{2} - 432151540024 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 - 6048 T + 223864366 T^{2} - 700698303648 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 + 3020 T - 30383466 T^{2} + 443965497860 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 + 11700 T + 292735582 T^{2} + 2683336581900 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 - 9468 T + 858185230 T^{2} - 3959474927724 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 - 43938 T + 1852599934 T^{2} - 31412343849462 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 - 64754 T + 2408321418 T^{2} - 54690988874954 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 - 24784 T + 2799959190 T^{2} - 33461500651888 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 - 97416 T + 5729557966 T^{2} - 175760806457016 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 + 17452 T + 3828622374 T^{2} + 36179245441036 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 - 51256 T + 3645565854 T^{2} - 157717602787144 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 + 117558 T + 7798161502 T^{2} + 463065739909794 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 + 84276 T + 5915697430 T^{2} + 470602194123924 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 + 20776 T + 16174049358 T^{2} + 178410581179432 T^{3} + 73742412689492826049 T^{4}$$