# Properties

 Label 49.6.c.g Level $49$ Weight $6$ Character orbit 49.c Analytic conductor $7.859$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.85880717084$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-13})$$ Defining polynomial: $$x^{4} - 13x^{2} + 169$$ x^4 - 13*x^2 + 169 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_1 + 2) q^{2} + \beta_{2} q^{3} + 28 \beta_1 q^{4} + (3 \beta_{3} - 3 \beta_{2}) q^{5} + 2 \beta_{3} q^{6} + 120 q^{8} + (381 \beta_1 - 381) q^{9}+O(q^{10})$$ q + (-2*b1 + 2) * q^2 + b2 * q^3 + 28*b1 * q^4 + (3*b3 - 3*b2) * q^5 + 2*b3 * q^6 + 120 * q^8 + (381*b1 - 381) * q^9 $$q + ( - 2 \beta_1 + 2) q^{2} + \beta_{2} q^{3} + 28 \beta_1 q^{4} + (3 \beta_{3} - 3 \beta_{2}) q^{5} + 2 \beta_{3} q^{6} + 120 q^{8} + (381 \beta_1 - 381) q^{9} - 6 \beta_{2} q^{10} + 284 \beta_1 q^{11} + ( - 28 \beta_{3} + 28 \beta_{2}) q^{12} - 21 \beta_{3} q^{13} + 1872 q^{15} + (656 \beta_1 - 656) q^{16} - 6 \beta_{2} q^{17} + 762 \beta_1 q^{18} + (87 \beta_{3} - 87 \beta_{2}) q^{19} + 84 \beta_{3} q^{20} + 568 q^{22} + (1496 \beta_1 - 1496) q^{23} + 120 \beta_{2} q^{24} - 2491 \beta_1 q^{25} + ( - 42 \beta_{3} + 42 \beta_{2}) q^{26} - 138 \beta_{3} q^{27} - 4366 q^{29} + ( - 3744 \beta_1 + 3744) q^{30} - 258 \beta_{2} q^{31} + 5152 \beta_1 q^{32} + ( - 284 \beta_{3} + 284 \beta_{2}) q^{33} - 12 \beta_{3} q^{34} - 10668 q^{36} + ( - 12630 \beta_1 + 12630) q^{37} - 174 \beta_{2} q^{38} - 13104 \beta_1 q^{39} + (360 \beta_{3} - 360 \beta_{2}) q^{40} + 378 \beta_{3} q^{41} - 1356 q^{43} + (7952 \beta_1 - 7952) q^{44} + 1143 \beta_{2} q^{45} + 2992 \beta_1 q^{46} + (402 \beta_{3} - 402 \beta_{2}) q^{47} - 656 \beta_{3} q^{48} - 4982 q^{50} + ( - 3744 \beta_1 + 3744) q^{51} - 588 \beta_{2} q^{52} - 14150 \beta_1 q^{53} + ( - 276 \beta_{3} + 276 \beta_{2}) q^{54} + 852 \beta_{3} q^{55} + 54288 q^{57} + (8732 \beta_1 - 8732) q^{58} - 1497 \beta_{2} q^{59} + 52416 \beta_1 q^{60} + ( - 1425 \beta_{3} + 1425 \beta_{2}) q^{61} - 516 \beta_{3} q^{62} - 10688 q^{64} + (39312 \beta_1 - 39312) q^{65} + 568 \beta_{2} q^{66} + 3644 \beta_1 q^{67} + (168 \beta_{3} - 168 \beta_{2}) q^{68} - 1496 \beta_{3} q^{69} + 35632 q^{71} + (45720 \beta_1 - 45720) q^{72} + 1632 \beta_{2} q^{73} - 25260 \beta_1 q^{74} + (2491 \beta_{3} - 2491 \beta_{2}) q^{75} + 2436 \beta_{3} q^{76} - 26208 q^{78} + ( - 54616 \beta_1 + 54616) q^{79} + 1968 \beta_{2} q^{80} + 6471 \beta_1 q^{81} + (756 \beta_{3} - 756 \beta_{2}) q^{82} + 21 \beta_{3} q^{83} - 11232 q^{85} + (2712 \beta_1 - 2712) q^{86} - 4366 \beta_{2} q^{87} + 34080 \beta_1 q^{88} + ( - 816 \beta_{3} + 816 \beta_{2}) q^{89} + 2286 \beta_{3} q^{90} - 41888 q^{92} + ( - 160992 \beta_1 + 160992) q^{93} - 804 \beta_{2} q^{94} - 162864 \beta_1 q^{95} + ( - 5152 \beta_{3} + 5152 \beta_{2}) q^{96} - 7350 \beta_{3} q^{97} - 108204 q^{99}+O(q^{100})$$ q + (-2*b1 + 2) * q^2 + b2 * q^3 + 28*b1 * q^4 + (3*b3 - 3*b2) * q^5 + 2*b3 * q^6 + 120 * q^8 + (381*b1 - 381) * q^9 - 6*b2 * q^10 + 284*b1 * q^11 + (-28*b3 + 28*b2) * q^12 - 21*b3 * q^13 + 1872 * q^15 + (656*b1 - 656) * q^16 - 6*b2 * q^17 + 762*b1 * q^18 + (87*b3 - 87*b2) * q^19 + 84*b3 * q^20 + 568 * q^22 + (1496*b1 - 1496) * q^23 + 120*b2 * q^24 - 2491*b1 * q^25 + (-42*b3 + 42*b2) * q^26 - 138*b3 * q^27 - 4366 * q^29 + (-3744*b1 + 3744) * q^30 - 258*b2 * q^31 + 5152*b1 * q^32 + (-284*b3 + 284*b2) * q^33 - 12*b3 * q^34 - 10668 * q^36 + (-12630*b1 + 12630) * q^37 - 174*b2 * q^38 - 13104*b1 * q^39 + (360*b3 - 360*b2) * q^40 + 378*b3 * q^41 - 1356 * q^43 + (7952*b1 - 7952) * q^44 + 1143*b2 * q^45 + 2992*b1 * q^46 + (402*b3 - 402*b2) * q^47 - 656*b3 * q^48 - 4982 * q^50 + (-3744*b1 + 3744) * q^51 - 588*b2 * q^52 - 14150*b1 * q^53 + (-276*b3 + 276*b2) * q^54 + 852*b3 * q^55 + 54288 * q^57 + (8732*b1 - 8732) * q^58 - 1497*b2 * q^59 + 52416*b1 * q^60 + (-1425*b3 + 1425*b2) * q^61 - 516*b3 * q^62 - 10688 * q^64 + (39312*b1 - 39312) * q^65 + 568*b2 * q^66 + 3644*b1 * q^67 + (168*b3 - 168*b2) * q^68 - 1496*b3 * q^69 + 35632 * q^71 + (45720*b1 - 45720) * q^72 + 1632*b2 * q^73 - 25260*b1 * q^74 + (2491*b3 - 2491*b2) * q^75 + 2436*b3 * q^76 - 26208 * q^78 + (-54616*b1 + 54616) * q^79 + 1968*b2 * q^80 + 6471*b1 * q^81 + (756*b3 - 756*b2) * q^82 + 21*b3 * q^83 - 11232 * q^85 + (2712*b1 - 2712) * q^86 - 4366*b2 * q^87 + 34080*b1 * q^88 + (-816*b3 + 816*b2) * q^89 + 2286*b3 * q^90 - 41888 * q^92 + (-160992*b1 + 160992) * q^93 - 804*b2 * q^94 - 162864*b1 * q^95 + (-5152*b3 + 5152*b2) * q^96 - 7350*b3 * q^97 - 108204 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 56 q^{4} + 480 q^{8} - 762 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 + 56 * q^4 + 480 * q^8 - 762 * q^9 $$4 q + 4 q^{2} + 56 q^{4} + 480 q^{8} - 762 q^{9} + 568 q^{11} + 7488 q^{15} - 1312 q^{16} + 1524 q^{18} + 2272 q^{22} - 2992 q^{23} - 4982 q^{25} - 17464 q^{29} + 7488 q^{30} + 10304 q^{32} - 42672 q^{36} + 25260 q^{37} - 26208 q^{39} - 5424 q^{43} - 15904 q^{44} + 5984 q^{46} - 19928 q^{50} + 7488 q^{51} - 28300 q^{53} + 217152 q^{57} - 17464 q^{58} + 104832 q^{60} - 42752 q^{64} - 78624 q^{65} + 7288 q^{67} + 142528 q^{71} - 91440 q^{72} - 50520 q^{74} - 104832 q^{78} + 109232 q^{79} + 12942 q^{81} - 44928 q^{85} - 5424 q^{86} + 68160 q^{88} - 167552 q^{92} + 321984 q^{93} - 325728 q^{95} - 432816 q^{99}+O(q^{100})$$ 4 * q + 4 * q^2 + 56 * q^4 + 480 * q^8 - 762 * q^9 + 568 * q^11 + 7488 * q^15 - 1312 * q^16 + 1524 * q^18 + 2272 * q^22 - 2992 * q^23 - 4982 * q^25 - 17464 * q^29 + 7488 * q^30 + 10304 * q^32 - 42672 * q^36 + 25260 * q^37 - 26208 * q^39 - 5424 * q^43 - 15904 * q^44 + 5984 * q^46 - 19928 * q^50 + 7488 * q^51 - 28300 * q^53 + 217152 * q^57 - 17464 * q^58 + 104832 * q^60 - 42752 * q^64 - 78624 * q^65 + 7288 * q^67 + 142528 * q^71 - 91440 * q^72 - 50520 * q^74 - 104832 * q^78 + 109232 * q^79 + 12942 * q^81 - 44928 * q^85 - 5424 * q^86 + 68160 * q^88 - 167552 * q^92 + 321984 * q^93 - 325728 * q^95 - 432816 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 13x^{2} + 169$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 13$$ (v^2) / 13 $$\beta_{2}$$ $$=$$ $$( 4\nu^{3} + 52\nu ) / 13$$ (4*v^3 + 52*v) / 13 $$\beta_{3}$$ $$=$$ $$( -4\nu^{3} + 104\nu ) / 13$$ (-4*v^3 + 104*v) / 13
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 12$$ (b3 + b2) / 12 $$\nu^{2}$$ $$=$$ $$13\beta_1$$ 13*b1 $$\nu^{3}$$ $$=$$ $$( -13\beta_{3} + 26\beta_{2} ) / 12$$ (-13*b3 + 26*b2) / 12

## Character values

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

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

## 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
 −3.12250 + 1.80278i 3.12250 − 1.80278i −3.12250 − 1.80278i 3.12250 + 1.80278i
1.00000 + 1.73205i −12.4900 + 21.6333i 14.0000 24.2487i −37.4700 64.8999i −49.9600 0 120.000 −190.500 329.956i 74.9400 129.800i
18.2 1.00000 + 1.73205i 12.4900 21.6333i 14.0000 24.2487i 37.4700 + 64.8999i 49.9600 0 120.000 −190.500 329.956i −74.9400 + 129.800i
30.1 1.00000 1.73205i −12.4900 21.6333i 14.0000 + 24.2487i −37.4700 + 64.8999i −49.9600 0 120.000 −190.500 + 329.956i 74.9400 + 129.800i
30.2 1.00000 1.73205i 12.4900 + 21.6333i 14.0000 + 24.2487i 37.4700 64.8999i 49.9600 0 120.000 −190.500 + 329.956i −74.9400 129.800i
 $$n$$: e.g. 2-40 or 990-1000 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 49.6.c.g 4
7.b odd 2 1 inner 49.6.c.g 4
7.c even 3 1 49.6.a.c 2
7.c even 3 1 inner 49.6.c.g 4
7.d odd 6 1 49.6.a.c 2
7.d odd 6 1 inner 49.6.c.g 4
21.g even 6 1 441.6.a.u 2
21.h odd 6 1 441.6.a.u 2
28.f even 6 1 784.6.a.z 2
28.g odd 6 1 784.6.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.c 2 7.c even 3 1
49.6.a.c 2 7.d odd 6 1
49.6.c.g 4 1.a even 1 1 trivial
49.6.c.g 4 7.b odd 2 1 inner
49.6.c.g 4 7.c even 3 1 inner
49.6.c.g 4 7.d odd 6 1 inner
441.6.a.u 2 21.g even 6 1
441.6.a.u 2 21.h odd 6 1
784.6.a.z 2 28.f even 6 1
784.6.a.z 2 28.g odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{3}^{4} + 624T_{3}^{2} + 389376$$ T3^4 + 624*T3^2 + 389376

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{2}$$
$3$ $$T^{4} + 624 T^{2} + 389376$$
$5$ $$T^{4} + 5616 T^{2} + \cdots + 31539456$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 284 T + 80656)^{2}$$
$13$ $$(T^{2} - 275184)^{2}$$
$17$ $$T^{4} + 22464 T^{2} + \cdots + 504631296$$
$19$ $$T^{4} + 4723056 T^{2} + \cdots + 22307257979136$$
$23$ $$(T^{2} + 1496 T + 2238016)^{2}$$
$29$ $$(T + 4366)^{4}$$
$31$ $$T^{4} + 41535936 T^{2} + \cdots + 17\!\cdots\!96$$
$37$ $$(T^{2} - 12630 T + 159516900)^{2}$$
$41$ $$(T^{2} - 89159616)^{2}$$
$43$ $$(T + 1356)^{4}$$
$47$ $$T^{4} + 100840896 T^{2} + \cdots + 10\!\cdots\!16$$
$53$ $$(T^{2} + 14150 T + 200222500)^{2}$$
$59$ $$T^{4} + 1398389616 T^{2} + \cdots + 19\!\cdots\!56$$
$61$ $$T^{4} + 1267110000 T^{2} + \cdots + 16\!\cdots\!00$$
$67$ $$(T^{2} - 3644 T + 13278736)^{2}$$
$71$ $$(T - 35632)^{4}$$
$73$ $$T^{4} + 1661976576 T^{2} + \cdots + 27\!\cdots\!76$$
$79$ $$(T^{2} - 54616 T + 2982907456)^{2}$$
$83$ $$(T^{2} - 275184)^{2}$$
$89$ $$T^{4} + 415494144 T^{2} + \cdots + 17\!\cdots\!36$$
$97$ $$(T^{2} - 33710040000)^{2}$$