# Properties

 Label 49.4.c.e Level $49$ Weight $4$ Character orbit 49.c Analytic conductor $2.891$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.89109359028$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.5922408960000.19 Defining polynomial: $$x^{8} - 4 x^{7} - 54 x^{6} + 176 x^{5} + 1307 x^{4} - 2912 x^{3} - 15314 x^{2} + 16800 x + 86044$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} - \beta_{6} ) q^{2} + \beta_{3} q^{3} + ( -8 - 8 \beta_{1} + \beta_{6} ) q^{4} + ( \beta_{4} + \beta_{7} ) q^{5} + ( 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{6} + ( 17 - \beta_{2} ) q^{8} + ( 6 + 13 \beta_{1} - 6 \beta_{2} + 6 \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} - \beta_{6} ) q^{2} + \beta_{3} q^{3} + ( -8 - 8 \beta_{1} + \beta_{6} ) q^{4} + ( \beta_{4} + \beta_{7} ) q^{5} + ( 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{6} + ( 17 - \beta_{2} ) q^{8} + ( 6 + 13 \beta_{1} - 6 \beta_{2} + 6 \beta_{6} ) q^{9} + ( -2 \beta_{3} - 4 \beta_{4} ) q^{10} + ( -28 - 28 \beta_{1} - 6 \beta_{6} ) q^{11} + ( -2 \beta_{4} - 5 \beta_{5} - 2 \beta_{7} ) q^{12} + ( -6 \beta_{3} + 6 \beta_{5} + \beta_{7} ) q^{13} + ( -20 + 8 \beta_{2} ) q^{15} + ( -9 - 48 \beta_{1} + 9 \beta_{2} - 9 \beta_{6} ) q^{16} + ( -\beta_{3} + 9 \beta_{4} ) q^{17} + ( 96 + 96 \beta_{1} + 7 \beta_{6} ) q^{18} + ( 4 \beta_{4} + 11 \beta_{5} + 4 \beta_{7} ) q^{19} + ( 2 \beta_{3} - 2 \beta_{5} - 12 \beta_{7} ) q^{20} + ( -74 - 22 \beta_{2} ) q^{22} + ( -8 + 84 \beta_{1} + 8 \beta_{2} - 8 \beta_{6} ) q^{23} + ( 13 \beta_{3} + 2 \beta_{4} ) q^{24} + ( 43 + 43 \beta_{1} + 22 \beta_{6} ) q^{25} + ( -16 \beta_{4} + 16 \beta_{5} - 16 \beta_{7} ) q^{26} + ( -4 \beta_{3} + 4 \beta_{5} - 12 \beta_{7} ) q^{27} + ( 58 + 14 \beta_{2} ) q^{29} + ( 20 - 128 \beta_{1} - 20 \beta_{2} + 20 \beta_{6} ) q^{30} + ( -38 \beta_{3} + 4 \beta_{4} ) q^{31} + ( -16 - 16 \beta_{1} - 47 \beta_{6} ) q^{32} + ( 12 \beta_{4} - 46 \beta_{5} + 12 \beta_{7} ) q^{33} + ( -21 \beta_{3} + 21 \beta_{5} + 34 \beta_{7} ) q^{34} + ( -33 + 41 \beta_{2} ) q^{36} + ( 6 + 56 \beta_{1} - 6 \beta_{2} + 6 \beta_{6} ) q^{37} + ( 25 \beta_{3} - 38 \beta_{4} ) q^{38} + ( -252 - 252 \beta_{1} - 28 \beta_{6} ) q^{39} + ( 20 \beta_{4} + 2 \beta_{5} + 20 \beta_{7} ) q^{40} + ( 31 \beta_{3} - 31 \beta_{5} + 3 \beta_{7} ) q^{41} + ( 170 - 70 \beta_{2} ) q^{43} + ( 26 + 128 \beta_{1} - 26 \beta_{2} + 26 \beta_{6} ) q^{44} + ( 12 \beta_{3} + 11 \beta_{4} ) q^{45} + ( -128 - 128 \beta_{1} + 92 \beta_{6} ) q^{46} + ( -32 \beta_{4} + 26 \beta_{5} - 32 \beta_{7} ) q^{47} + ( 75 \beta_{3} - 75 \beta_{5} + 18 \beta_{7} ) q^{48} + ( 331 + 21 \beta_{2} ) q^{50} + ( -78 + 68 \beta_{1} + 78 \beta_{2} - 78 \beta_{6} ) q^{51} + ( 32 \beta_{3} + 24 \beta_{4} ) q^{52} + ( 14 + 14 \beta_{1} + 36 \beta_{6} ) q^{53} + ( 40 \beta_{4} + 36 \beta_{5} + 40 \beta_{7} ) q^{54} + ( -12 \beta_{3} + 12 \beta_{5} - 4 \beta_{7} ) q^{55} + ( -454 - 34 \beta_{2} ) q^{57} + ( -58 - 224 \beta_{1} + 58 \beta_{2} - 58 \beta_{6} ) q^{58} + ( 49 \beta_{3} - 20 \beta_{4} ) q^{59} + ( 224 + 224 \beta_{1} - 84 \beta_{6} ) q^{60} + ( 27 \beta_{4} - 68 \beta_{5} + 27 \beta_{7} ) q^{61} + ( -122 \beta_{3} + 122 \beta_{5} - 60 \beta_{7} ) q^{62} + ( -471 + 103 \beta_{2} ) q^{64} + ( 70 - 154 \beta_{1} - 70 \beta_{2} + 70 \beta_{6} ) q^{65} + ( -162 \beta_{3} + 44 \beta_{4} ) q^{66} + ( 448 + 448 \beta_{1} - 76 \beta_{6} ) q^{67} + ( -106 \beta_{4} - 13 \beta_{5} - 106 \beta_{7} ) q^{68} + ( -60 \beta_{3} + 60 \beta_{5} + 16 \beta_{7} ) q^{69} + ( 548 + 28 \beta_{2} ) q^{71} + ( 89 + 112 \beta_{1} - 89 \beta_{2} + 89 \beta_{6} ) q^{72} + ( -37 \beta_{3} - 25 \beta_{4} ) q^{73} + ( 96 + 96 \beta_{1} + 50 \beta_{6} ) q^{74} + ( -44 \beta_{4} + 109 \beta_{5} - 44 \beta_{7} ) q^{75} + ( 63 \beta_{3} - 63 \beta_{5} - 70 \beta_{7} ) q^{76} + ( -224 - 224 \beta_{2} ) q^{78} + ( 188 - 168 \beta_{1} - 188 \beta_{2} + 188 \beta_{6} ) q^{79} + ( -18 \beta_{3} + 12 \beta_{4} ) q^{80} + ( 335 + 335 \beta_{1} + 42 \beta_{6} ) q^{81} + ( 50 \beta_{4} - 99 \beta_{5} + 50 \beta_{7} ) q^{82} + ( -\beta_{3} + \beta_{5} - 8 \beta_{7} ) q^{83} + ( -916 + 190 \beta_{2} ) q^{85} + ( -170 + 1120 \beta_{1} + 170 \beta_{2} - 170 \beta_{6} ) q^{86} + ( 114 \beta_{3} - 28 \beta_{4} ) q^{87} + ( -352 - 352 \beta_{1} - 74 \beta_{6} ) q^{88} + ( 75 \beta_{4} + 157 \beta_{5} + 75 \beta_{7} ) q^{89} + ( 14 \beta_{3} - 14 \beta_{5} + 68 \beta_{7} ) q^{90} + ( 956 - 156 \beta_{2} ) q^{92} + ( -260 - 1472 \beta_{1} + 260 \beta_{2} - 260 \beta_{6} ) q^{93} + ( 142 \beta_{3} + 76 \beta_{4} ) q^{94} + ( -460 - 460 \beta_{1} + 176 \beta_{6} ) q^{95} + ( 94 \beta_{4} - 157 \beta_{5} + 94 \beta_{7} ) q^{96} + ( 189 \beta_{3} - 189 \beta_{5} + 91 \beta_{7} ) q^{97} + ( 730 + 210 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} - 34q^{4} + 132q^{8} - 40q^{9} + O(q^{10})$$ $$8q - 2q^{2} - 34q^{4} + 132q^{8} - 40q^{9} - 100q^{11} - 128q^{15} + 174q^{16} + 370q^{18} - 680q^{22} - 352q^{23} + 128q^{25} + 520q^{29} + 552q^{30} + 30q^{32} - 100q^{36} - 212q^{37} - 952q^{39} + 1080q^{43} - 460q^{44} - 696q^{46} + 2732q^{50} - 428q^{51} - 16q^{53} - 3768q^{57} + 780q^{58} + 1064q^{60} - 3356q^{64} + 756q^{65} + 1944q^{67} + 4496q^{71} - 270q^{72} + 284q^{74} - 2688q^{78} + 1048q^{79} + 1256q^{81} - 6568q^{85} - 4820q^{86} - 1260q^{88} + 7024q^{92} + 5368q^{93} - 2192q^{95} + 6680q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 54 x^{6} + 176 x^{5} + 1307 x^{4} - 2912 x^{3} - 15314 x^{2} + 16800 x + 86044$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{6} + 6 \nu^{5} + 149 \nu^{4} - 308 \nu^{3} - 3293 \nu^{2} + 3448 \nu + 22372$$$$)/8946$$ $$\beta_{2}$$ $$=$$ $$($$$$188 \nu^{7} - 658 \nu^{6} - 11168 \nu^{5} + 29565 \nu^{4} + 311680 \nu^{3} - 497414 \nu^{2} - 5795360 \nu + 4518059$$$$)/3072951$$ $$\beta_{3}$$ $$=$$ $$($$$$-94 \nu^{7} + 558 \nu^{6} + 4897 \nu^{5} - 31843 \nu^{4} - 120574 \nu^{3} + 1137914 \nu^{2} + 454250 \nu - 11478684$$$$)/1024317$$ $$\beta_{4}$$ $$=$$ $$($$$$1034 \nu^{7} - 2932 \nu^{6} - 63485 \nu^{5} + 111426 \nu^{4} + 1820038 \nu^{3} - 68156 \nu^{2} - 17694113 \nu - 24320296$$$$)/3072951$$ $$\beta_{5}$$ $$=$$ $$($$$$-685 \nu^{7} + 10069 \nu^{6} + 3667 \nu^{5} - 388145 \nu^{4} + 210780 \nu^{3} + 6478590 \nu^{2} - 3760032 \nu - 36898988$$$$)/2048634$$ $$\beta_{6}$$ $$=$$ $$($$$$2744 \nu^{7} - 8917 \nu^{6} - 155726 \nu^{5} + 356991 \nu^{4} + 3179236 \nu^{3} - 3891986 \nu^{2} - 26106296 \nu + 2554244$$$$)/6145902$$ $$\beta_{7}$$ $$=$$ $$($$$$5467 \nu^{7} + 2506 \nu^{6} - 235570 \nu^{5} - 369522 \nu^{4} + 3410879 \nu^{3} + 8909612 \nu^{2} - 8027680 \nu - 44176664$$$$)/6145902$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - \beta_{3} - 7 \beta_{2} + 7$$$$)/7$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{4} + 10 \beta_{3} - 7 \beta_{2} + 14 \beta_{1} + 119$$$$)/7$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 42 \beta_{6} - 2 \beta_{5} + 54 \beta_{4} - 31 \beta_{3} - 77 \beta_{2} + 189$$$$)/7$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{7} - 84 \beta_{6} + 40 \beta_{5} + 200 \beta_{4} + 236 \beta_{3} - 147 \beta_{2} + 1316 \beta_{1} + 2023$$$$)/7$$ $$\nu^{5}$$ $$=$$ $$($$$$356 \beta_{7} - 2240 \beta_{6} - 216 \beta_{5} + 1686 \beta_{4} - 315 \beta_{3} + 245 \beta_{2} + 2240 \beta_{1} + 3451$$$$)/7$$ $$\nu^{6}$$ $$=$$ $$($$$$1952 \beta_{7} - 6510 \beta_{6} + 2640 \beta_{5} + 6780 \beta_{4} + 3222 \beta_{3} + 1099 \beta_{2} + 50400 \beta_{1} + 26397$$$$)/7$$ $$\nu^{7}$$ $$=$$ $$($$$$22148 \beta_{7} - 73010 \beta_{6} - 6566 \beta_{5} + 44318 \beta_{4} + 2477 \beta_{3} + 49287 \beta_{2} + 139552 \beta_{1} + 28329$$$$)/7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 3.82402 + 1.22474i 5.23824 − 1.22474i −2.82402 − 1.22474i −4.23824 + 1.22474i 3.82402 − 1.22474i 5.23824 + 1.22474i −2.82402 + 1.22474i −4.23824 − 1.22474i
−2.26556 3.92407i −1.78978 + 3.09999i −6.26556 + 10.8523i 6.73953 + 11.6732i 16.2194 0 20.5311 7.09339 + 12.2861i 30.5377 52.8928i
18.2 −2.26556 3.92407i 1.78978 3.09999i −6.26556 + 10.8523i −6.73953 11.6732i −16.2194 0 20.5311 7.09339 + 12.2861i −30.5377 + 52.8928i
18.3 1.76556 + 3.05805i −3.91110 + 6.77422i −2.23444 + 3.87016i −1.03865 1.79899i −27.6212 0 12.4689 −17.0934 29.6066i 3.66760 6.35247i
18.4 1.76556 + 3.05805i 3.91110 6.77422i −2.23444 + 3.87016i 1.03865 + 1.79899i 27.6212 0 12.4689 −17.0934 29.6066i −3.66760 + 6.35247i
30.1 −2.26556 + 3.92407i −1.78978 3.09999i −6.26556 10.8523i 6.73953 11.6732i 16.2194 0 20.5311 7.09339 12.2861i 30.5377 + 52.8928i
30.2 −2.26556 + 3.92407i 1.78978 + 3.09999i −6.26556 10.8523i −6.73953 + 11.6732i −16.2194 0 20.5311 7.09339 12.2861i −30.5377 52.8928i
30.3 1.76556 3.05805i −3.91110 6.77422i −2.23444 3.87016i −1.03865 + 1.79899i −27.6212 0 12.4689 −17.0934 + 29.6066i 3.66760 + 6.35247i
30.4 1.76556 3.05805i 3.91110 + 6.77422i −2.23444 3.87016i 1.03865 1.79899i 27.6212 0 12.4689 −17.0934 + 29.6066i −3.66760 6.35247i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 30.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} + T_{2}^{3} + 17 T_{2}^{2} - 16 T_{2} + 256$$ $$T_{3}^{8} + 74 T_{3}^{6} + 4692 T_{3}^{4} + 58016 T_{3}^{2} + 614656$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 256 - 16 T + 17 T^{2} + T^{3} + T^{4} )^{2}$$
$3$ $$614656 + 58016 T^{2} + 4692 T^{4} + 74 T^{6} + T^{8}$$
$5$ $$614656 + 145824 T^{2} + 33812 T^{4} + 186 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 1600 + 2000 T + 2460 T^{2} + 50 T^{3} + T^{4} )^{2}$$
$13$ $$( 2458624 - 3234 T^{2} + T^{4} )^{2}$$
$17$ $$95001747709456 + 141953618576 T^{2} + 202363212 T^{4} + 14564 T^{6} + T^{8}$$
$19$ $$2765866292982016 + 775514317984 T^{2} + 164853012 T^{4} + 14746 T^{6} + T^{8}$$
$23$ $$( 44943616 + 1179904 T + 24272 T^{2} + 176 T^{3} + T^{4} )^{2}$$
$29$ $$( 1040 - 130 T + T^{2} )^{4}$$
$31$ $$396154108207169536 + 63006232805376 T^{2} + 9391403072 T^{4} + 100104 T^{6} + T^{8}$$
$37$ $$( 4946176 + 235744 T + 9012 T^{2} + 106 T^{3} + T^{4} )^{2}$$
$41$ $$( 307721764 - 66836 T^{2} + T^{4} )^{2}$$
$43$ $$( -61400 - 270 T + T^{2} )^{4}$$
$47$ $$69422863899074560000 + 1560090870016000 T^{2} + 26726779200 T^{4} + 187240 T^{6} + T^{8}$$
$53$ $$( 442849936 - 168352 T + 21108 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$59$ $$2563073382676500736 + 303148207106976 T^{2} + 34253977172 T^{4} + 189354 T^{6} + T^{8}$$
$61$ $$25223230345416683776 + 1809354357370784 T^{2} + 124769317332 T^{4} + 360266 T^{6} + T^{8}$$
$67$ $$( 20259536896 - 138350592 T + 802448 T^{2} - 972 T^{3} + T^{4} )^{2}$$
$71$ $$( 303104 - 1124 T + T^{2} )^{4}$$
$73$ $$15\!\cdots\!76$$$$+ 3439588972084944 T^{2} + 64165647212 T^{4} + 276756 T^{6} + T^{8}$$
$79$ $$( 255728444416 + 264984704 T + 780272 T^{2} - 524 T^{3} + T^{4} )^{2}$$
$83$ $$( 6492304 - 11466 T^{2} + T^{4} )^{2}$$
$89$ $$80\!\cdots\!56$$$$+ 10308817483656026384 T^{2} + 10287783492492 T^{4} + 3623876 T^{6} + T^{8}$$
$97$ $$( 841222821124 - 3082884 T^{2} + T^{4} )^{2}$$