Properties

 Label 49.4.a.e Level $49$ Weight $4$ Character orbit 49.a Self dual yes Analytic conductor $2.891$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$2.89109359028$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{65})$$ Defining polynomial: $$x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{2} + \beta_{2} q^{3} + ( 9 + \beta_{1} ) q^{4} -\beta_{3} q^{5} + ( -3 \beta_{2} + 2 \beta_{3} ) q^{6} + ( 17 + \beta_{1} ) q^{8} + ( 7 - 6 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{2} + \beta_{2} q^{3} + ( 9 + \beta_{1} ) q^{4} -\beta_{3} q^{5} + ( -3 \beta_{2} + 2 \beta_{3} ) q^{6} + ( 17 + \beta_{1} ) q^{8} + ( 7 - 6 \beta_{1} ) q^{9} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{10} + ( 22 - 6 \beta_{1} ) q^{11} + ( 5 \beta_{2} + 2 \beta_{3} ) q^{12} + ( 6 \beta_{2} + \beta_{3} ) q^{13} + ( -20 - 8 \beta_{1} ) q^{15} + ( -39 + 9 \beta_{1} ) q^{16} + ( -\beta_{2} + 9 \beta_{3} ) q^{17} + ( -89 + 7 \beta_{1} ) q^{18} + ( -11 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -2 \beta_{2} - 12 \beta_{3} ) q^{20} + ( -74 + 22 \beta_{1} ) q^{22} + ( 92 + 8 \beta_{1} ) q^{23} + ( 13 \beta_{2} + 2 \beta_{3} ) q^{24} + ( -21 + 22 \beta_{1} ) q^{25} + ( -16 \beta_{2} + 16 \beta_{3} ) q^{26} + ( 4 \beta_{2} - 12 \beta_{3} ) q^{27} + ( 58 - 14 \beta_{1} ) q^{29} + ( -148 - 20 \beta_{1} ) q^{30} + ( -38 \beta_{2} + 4 \beta_{3} ) q^{31} + ( -31 - 47 \beta_{1} ) q^{32} + ( 46 \beta_{2} - 12 \beta_{3} ) q^{33} + ( 21 \beta_{2} + 34 \beta_{3} ) q^{34} + ( -33 - 41 \beta_{1} ) q^{36} + ( 50 - 6 \beta_{1} ) q^{37} + ( 25 \beta_{2} - 38 \beta_{3} ) q^{38} + ( 224 - 28 \beta_{1} ) q^{39} + ( -2 \beta_{2} - 20 \beta_{3} ) q^{40} + ( -31 \beta_{2} + 3 \beta_{3} ) q^{41} + ( 170 + 70 \beta_{1} ) q^{43} + ( 102 - 26 \beta_{1} ) q^{44} + ( 12 \beta_{2} + 11 \beta_{3} ) q^{45} + ( 220 + 92 \beta_{1} ) q^{46} + ( -26 \beta_{2} + 32 \beta_{3} ) q^{47} + ( -75 \beta_{2} + 18 \beta_{3} ) q^{48} + ( 331 - 21 \beta_{1} ) q^{50} + ( 146 + 78 \beta_{1} ) q^{51} + ( 32 \beta_{2} + 24 \beta_{3} ) q^{52} + ( 22 + 36 \beta_{1} ) q^{53} + ( -36 \beta_{2} - 40 \beta_{3} ) q^{54} + ( 12 \beta_{2} - 4 \beta_{3} ) q^{55} + ( -454 + 34 \beta_{1} ) q^{57} + ( -166 + 58 \beta_{1} ) q^{58} + ( 49 \beta_{2} - 20 \beta_{3} ) q^{59} + ( -308 - 84 \beta_{1} ) q^{60} + ( 68 \beta_{2} - 27 \beta_{3} ) q^{61} + ( 122 \beta_{2} - 60 \beta_{3} ) q^{62} + ( -471 - 103 \beta_{1} ) q^{64} + ( -224 - 70 \beta_{1} ) q^{65} + ( -162 \beta_{2} + 44 \beta_{3} ) q^{66} + ( -524 - 76 \beta_{1} ) q^{67} + ( 13 \beta_{2} + 106 \beta_{3} ) q^{68} + ( 60 \beta_{2} + 16 \beta_{3} ) q^{69} + ( 548 - 28 \beta_{1} ) q^{71} + ( 23 - 89 \beta_{1} ) q^{72} + ( -37 \beta_{2} - 25 \beta_{3} ) q^{73} + ( -46 + 50 \beta_{1} ) q^{74} + ( -109 \beta_{2} + 44 \beta_{3} ) q^{75} + ( -63 \beta_{2} - 70 \beta_{3} ) q^{76} + ( -224 + 224 \beta_{1} ) q^{78} + ( -356 - 188 \beta_{1} ) q^{79} + ( -18 \beta_{2} + 12 \beta_{3} ) q^{80} + ( -293 + 42 \beta_{1} ) q^{81} + ( 99 \beta_{2} - 50 \beta_{3} ) q^{82} + ( \beta_{2} - 8 \beta_{3} ) q^{83} + ( -916 - 190 \beta_{1} ) q^{85} + ( 1290 + 170 \beta_{1} ) q^{86} + ( 114 \beta_{2} - 28 \beta_{3} ) q^{87} + ( 278 - 74 \beta_{1} ) q^{88} + ( -157 \beta_{2} - 75 \beta_{3} ) q^{89} + ( -14 \beta_{2} + 68 \beta_{3} ) q^{90} + ( 956 + 156 \beta_{1} ) q^{92} + ( -1212 + 260 \beta_{1} ) q^{93} + ( 142 \beta_{2} + 76 \beta_{3} ) q^{94} + ( 636 + 176 \beta_{1} ) q^{95} + ( 157 \beta_{2} - 94 \beta_{3} ) q^{96} + ( -189 \beta_{2} + 91 \beta_{3} ) q^{97} + ( 730 - 210 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 34q^{4} + 66q^{8} + 40q^{9} + O(q^{10})$$ $$4q + 2q^{2} + 34q^{4} + 66q^{8} + 40q^{9} + 100q^{11} - 64q^{15} - 174q^{16} - 370q^{18} - 340q^{22} + 352q^{23} - 128q^{25} + 260q^{29} - 552q^{30} - 30q^{32} - 50q^{36} + 212q^{37} + 952q^{39} + 540q^{43} + 460q^{44} + 696q^{46} + 1366q^{50} + 428q^{51} + 16q^{53} - 1884q^{57} - 780q^{58} - 1064q^{60} - 1678q^{64} - 756q^{65} - 1944q^{67} + 2248q^{71} + 270q^{72} - 284q^{74} - 1344q^{78} - 1048q^{79} - 1256q^{81} - 3284q^{85} + 4820q^{86} + 1260q^{88} + 3512q^{92} - 5368q^{93} + 2192q^{95} + 3340q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} + 3 \nu^{2} + 100 \nu - 79$$$$)/57$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 11 \nu^{2} + 12 \nu - 182$$$$)/19$$ $$\beta_{3}$$ $$=$$ $$($$$$11 \nu^{3} + 12 \nu^{2} - 265 \nu - 392$$$$)/57$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 7 \beta_{1} + 7$$$$)/7$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{3} + 10 \beta_{2} + 7 \beta_{1} + 133$$$$)/7$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{3} - 5 \beta_{2} + 23 \beta_{1} + 39$$

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
 −2.11692 −4.94534 3.11692 5.94534
−3.53113 −7.82220 4.46887 −2.07730 27.6212 0 12.4689 34.1868 7.33521
1.2 −3.53113 7.82220 4.46887 2.07730 −27.6212 0 12.4689 34.1868 −7.33521
1.3 4.53113 −3.57956 12.5311 13.4791 −16.2194 0 20.5311 −14.1868 61.0753
1.4 4.53113 3.57956 12.5311 −13.4791 16.2194 0 20.5311 −14.1868 −61.0753
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.4.a.e 4
3.b odd 2 1 441.4.a.u 4
4.b odd 2 1 784.4.a.bf 4
5.b even 2 1 1225.4.a.bb 4
7.b odd 2 1 inner 49.4.a.e 4
7.c even 3 2 49.4.c.e 8
7.d odd 6 2 49.4.c.e 8
21.c even 2 1 441.4.a.u 4
21.g even 6 2 441.4.e.y 8
21.h odd 6 2 441.4.e.y 8
28.d even 2 1 784.4.a.bf 4
35.c odd 2 1 1225.4.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 1.a even 1 1 trivial
49.4.a.e 4 7.b odd 2 1 inner
49.4.c.e 8 7.c even 3 2
49.4.c.e 8 7.d odd 6 2
441.4.a.u 4 3.b odd 2 1
441.4.a.u 4 21.c even 2 1
441.4.e.y 8 21.g even 6 2
441.4.e.y 8 21.h odd 6 2
784.4.a.bf 4 4.b odd 2 1
784.4.a.bf 4 28.d even 2 1
1225.4.a.bb 4 5.b even 2 1
1225.4.a.bb 4 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2}^{2} - T_{2} - 16$$ $$T_{3}^{4} - 74 T_{3}^{2} + 784$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -16 - T + T^{2} )^{2}$$
$3$ $$784 - 74 T^{2} + T^{4}$$
$5$ $$784 - 186 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 40 - 50 T + T^{2} )^{2}$$
$13$ $$2458624 - 3234 T^{2} + T^{4}$$
$17$ $$9746884 - 14564 T^{2} + T^{4}$$
$19$ $$52591504 - 14746 T^{2} + T^{4}$$
$23$ $$( 6704 - 176 T + T^{2} )^{2}$$
$29$ $$( 1040 - 130 T + T^{2} )^{2}$$
$31$ $$629407744 - 100104 T^{2} + T^{4}$$
$37$ $$( 2224 - 106 T + T^{2} )^{2}$$
$41$ $$307721764 - 66836 T^{2} + T^{4}$$
$43$ $$( -61400 - 270 T + T^{2} )^{2}$$
$47$ $$8332038400 - 187240 T^{2} + T^{4}$$
$53$ $$( -21044 - 8 T + T^{2} )^{2}$$
$59$ $$1600960144 - 189354 T^{2} + T^{4}$$
$61$ $$5022273424 - 360266 T^{2} + T^{4}$$
$67$ $$( 142336 + 972 T + T^{2} )^{2}$$
$71$ $$( 303104 - 1124 T + T^{2} )^{2}$$
$73$ $$12428236324 - 276756 T^{2} + T^{4}$$
$79$ $$( -505696 + 524 T + T^{2} )^{2}$$
$83$ $$6492304 - 11466 T^{2} + T^{4}$$
$89$ $$2844693770884 - 3623876 T^{2} + T^{4}$$
$97$ $$841222821124 - 3082884 T^{2} + T^{4}$$