# Properties

 Label 448.6.a.bc Level $448$ Weight $6$ Character orbit 448.a Self dual yes Analytic conductor $71.852$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 448.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$71.8519512762$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 198x^{2} + 43x + 5999$$ x^4 - x^3 - 198*x^2 + 43*x + 5999 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 224) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 4) q^{3} + (\beta_{2} + \beta_1 + 7) q^{5} + 49 q^{7} + (\beta_{3} + 3 \beta_{2} + 9 \beta_1 + 170) q^{9}+O(q^{10})$$ q + (-b1 - 4) * q^3 + (b2 + b1 + 7) * q^5 + 49 * q^7 + (b3 + 3*b2 + 9*b1 + 170) * q^9 $$q + ( - \beta_1 - 4) q^{3} + (\beta_{2} + \beta_1 + 7) q^{5} + 49 q^{7} + (\beta_{3} + 3 \beta_{2} + 9 \beta_1 + 170) q^{9} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 123) q^{11} + (2 \beta_{3} - 3 \beta_{2} - 19 \beta_1 - 161) q^{13} + ( - 3 \beta_{3} - 23 \beta_{2} - 29 \beta_1 - 533) q^{15} + ( - 2 \beta_{3} - 14 \beta_{2} + 44 \beta_1 - 448) q^{17} + (4 \beta_{3} - 2 \beta_{2} - 7 \beta_1 - 158) q^{19} + ( - 49 \beta_1 - 196) q^{21} + (\beta_{3} - 25 \beta_{2} - 45 \beta_1 + 57) q^{23} + (5 \beta_{3} + 15 \beta_{2} + 185 \beta_1 + 736) q^{25} + ( - 4 \beta_{3} - 96 \beta_{2} - 110 \beta_1 - 3544) q^{27} + ( - 24 \beta_{3} - 2 \beta_{2} + 22 \beta_1 - 976) q^{29} + (6 \beta_{3} + 46 \beta_{2} + 88 \beta_1 + 3146) q^{31} + ( - 26 \beta_{3} - 8 \beta_{2} + 270 \beta_1 - 532) q^{33} + (49 \beta_{2} + 49 \beta_1 + 343) q^{35} + (26 \beta_{3} - 48 \beta_{2} + 164 \beta_1 - 274) q^{37} + (47 \beta_{3} + 99 \beta_{2} + 133 \beta_1 + 8633) q^{39} + (30 \beta_{3} + 44 \beta_{2} + 140 \beta_1 + 5530) q^{41} + (40 \beta_{3} - 69 \beta_{2} - 44 \beta_1 - 8189) q^{43} + (42 \beta_{3} + 331 \beta_{2} + 1087 \beta_1 + 14245) q^{45} + ( - 96 \beta_{3} - 148 \beta_{2} - 22 \beta_1 - 692) q^{47} + 2401 q^{49} + ( - 38 \beta_{3} + 166 \beta_{2} + 640 \beta_1 - 14286) q^{51} + ( - 54 \beta_{3} - 414 \beta_{2} + 438 \beta_1 - 5992) q^{53} + ( - 10 \beta_{3} - 142 \beta_{2} - 386 \beta_1 + 6278) q^{55} + (55 \beta_{3} + 25 \beta_{2} - 121 \beta_1 + 3871) q^{57} + ( - 72 \beta_{3} + 232 \beta_{2} - 47 \beta_1 - 27052) q^{59} + ( - 40 \beta_{3} - 347 \beta_{2} + 897 \beta_1 - 1533) q^{61} + (49 \beta_{3} + 147 \beta_{2} + 441 \beta_1 + 8330) q^{63} + ( - 45 \beta_{3} - 443 \beta_{2} - 629 \beta_1 - 23465) q^{65} + (116 \beta_{3} + 383 \beta_{2} - 278 \beta_1 - 27177) q^{67} + (106 \beta_{3} + 626 \beta_{2} + 506 \beta_1 + 20398) q^{69} + (98 \beta_{3} - 98 \beta_{2} - 462 \beta_1 + 13930) q^{71} + ( - 118 \beta_{3} - 212 \beta_{2} - 906 \beta_1 - 2702) q^{73} + ( - 160 \beta_{3} - 900 \beta_{2} - 2351 \beta_1 - 77704) q^{75} + ( - 98 \beta_{3} + 49 \beta_{2} + 98 \beta_1 - 6027) q^{77} + ( - 14 \beta_{3} + 1036 \beta_{2} - 1258 \beta_1 + 20168) q^{79} + (15 \beta_{3} + 1557 \beta_{2} + 3887 \beta_1 + 26660) q^{81} + (116 \beta_{3} + 726 \beta_{2} + 2705 \beta_1 - 30190) q^{83} + (86 \beta_{3} + 496 \beta_{2} - 1956 \beta_1 - 24392) q^{85} + ( - 282 \beta_{3} + 190 \beta_{2} + 2988 \beta_1 - 6078) q^{87} + ( - 48 \beta_{3} + 14 \beta_{2} - 820 \beta_1 - 7504) q^{89} + (98 \beta_{3} - 147 \beta_{2} - 931 \beta_1 - 7889) q^{91} + ( - 114 \beta_{3} - 1238 \beta_{2} - 4890 \beta_1 - 52122) q^{93} + (11 \beta_{3} - 177 \beta_{2} + 293 \beta_1 - 16899) q^{95} + (468 \beta_{3} - 916 \beta_{2} - 418 \beta_1 + 10710) q^{97} + ( - 54 \beta_{3} - 659 \beta_{2} + 1094 \beta_1 - 75895) q^{99}+O(q^{100})$$ q + (-b1 - 4) * q^3 + (b2 + b1 + 7) * q^5 + 49 * q^7 + (b3 + 3*b2 + 9*b1 + 170) * q^9 + (-2*b3 + b2 + 2*b1 - 123) * q^11 + (2*b3 - 3*b2 - 19*b1 - 161) * q^13 + (-3*b3 - 23*b2 - 29*b1 - 533) * q^15 + (-2*b3 - 14*b2 + 44*b1 - 448) * q^17 + (4*b3 - 2*b2 - 7*b1 - 158) * q^19 + (-49*b1 - 196) * q^21 + (b3 - 25*b2 - 45*b1 + 57) * q^23 + (5*b3 + 15*b2 + 185*b1 + 736) * q^25 + (-4*b3 - 96*b2 - 110*b1 - 3544) * q^27 + (-24*b3 - 2*b2 + 22*b1 - 976) * q^29 + (6*b3 + 46*b2 + 88*b1 + 3146) * q^31 + (-26*b3 - 8*b2 + 270*b1 - 532) * q^33 + (49*b2 + 49*b1 + 343) * q^35 + (26*b3 - 48*b2 + 164*b1 - 274) * q^37 + (47*b3 + 99*b2 + 133*b1 + 8633) * q^39 + (30*b3 + 44*b2 + 140*b1 + 5530) * q^41 + (40*b3 - 69*b2 - 44*b1 - 8189) * q^43 + (42*b3 + 331*b2 + 1087*b1 + 14245) * q^45 + (-96*b3 - 148*b2 - 22*b1 - 692) * q^47 + 2401 * q^49 + (-38*b3 + 166*b2 + 640*b1 - 14286) * q^51 + (-54*b3 - 414*b2 + 438*b1 - 5992) * q^53 + (-10*b3 - 142*b2 - 386*b1 + 6278) * q^55 + (55*b3 + 25*b2 - 121*b1 + 3871) * q^57 + (-72*b3 + 232*b2 - 47*b1 - 27052) * q^59 + (-40*b3 - 347*b2 + 897*b1 - 1533) * q^61 + (49*b3 + 147*b2 + 441*b1 + 8330) * q^63 + (-45*b3 - 443*b2 - 629*b1 - 23465) * q^65 + (116*b3 + 383*b2 - 278*b1 - 27177) * q^67 + (106*b3 + 626*b2 + 506*b1 + 20398) * q^69 + (98*b3 - 98*b2 - 462*b1 + 13930) * q^71 + (-118*b3 - 212*b2 - 906*b1 - 2702) * q^73 + (-160*b3 - 900*b2 - 2351*b1 - 77704) * q^75 + (-98*b3 + 49*b2 + 98*b1 - 6027) * q^77 + (-14*b3 + 1036*b2 - 1258*b1 + 20168) * q^79 + (15*b3 + 1557*b2 + 3887*b1 + 26660) * q^81 + (116*b3 + 726*b2 + 2705*b1 - 30190) * q^83 + (86*b3 + 496*b2 - 1956*b1 - 24392) * q^85 + (-282*b3 + 190*b2 + 2988*b1 - 6078) * q^87 + (-48*b3 + 14*b2 - 820*b1 - 7504) * q^89 + (98*b3 - 147*b2 - 931*b1 - 7889) * q^91 + (-114*b3 - 1238*b2 - 4890*b1 - 52122) * q^93 + (11*b3 - 177*b2 + 293*b1 - 16899) * q^95 + (468*b3 - 916*b2 - 418*b1 + 10710) * q^97 + (-54*b3 - 659*b2 + 1094*b1 - 75895) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{3} + 30 q^{5} + 196 q^{7} + 696 q^{9}+O(q^{10})$$ 4 * q - 18 * q^3 + 30 * q^5 + 196 * q^7 + 696 * q^9 $$4 q - 18 q^{3} + 30 q^{5} + 196 q^{7} + 696 q^{9} - 484 q^{11} - 686 q^{13} - 2184 q^{15} - 1700 q^{17} - 654 q^{19} - 882 q^{21} + 136 q^{23} + 3304 q^{25} - 14388 q^{27} - 3812 q^{29} + 12748 q^{31} - 1536 q^{33} + 1470 q^{35} - 820 q^{37} + 34704 q^{39} + 22340 q^{41} - 32924 q^{43} + 59070 q^{45} - 2620 q^{47} + 9604 q^{49} - 55788 q^{51} - 22984 q^{53} + 24360 q^{55} + 15132 q^{57} - 108158 q^{59} - 4258 q^{61} + 34104 q^{63} - 95028 q^{65} - 109496 q^{67} + 82392 q^{69} + 54600 q^{71} - 12384 q^{73} - 315198 q^{75} - 23716 q^{77} + 78184 q^{79} + 114384 q^{81} - 115582 q^{83} - 101652 q^{85} - 17772 q^{87} - 31560 q^{89} - 33614 q^{91} - 218040 q^{93} - 67032 q^{95} + 41068 q^{97} - 301284 q^{99}+O(q^{100})$$ 4 * q - 18 * q^3 + 30 * q^5 + 196 * q^7 + 696 * q^9 - 484 * q^11 - 686 * q^13 - 2184 * q^15 - 1700 * q^17 - 654 * q^19 - 882 * q^21 + 136 * q^23 + 3304 * q^25 - 14388 * q^27 - 3812 * q^29 + 12748 * q^31 - 1536 * q^33 + 1470 * q^35 - 820 * q^37 + 34704 * q^39 + 22340 * q^41 - 32924 * q^43 + 59070 * q^45 - 2620 * q^47 + 9604 * q^49 - 55788 * q^51 - 22984 * q^53 + 24360 * q^55 + 15132 * q^57 - 108158 * q^59 - 4258 * q^61 + 34104 * q^63 - 95028 * q^65 - 109496 * q^67 + 82392 * q^69 + 54600 * q^71 - 12384 * q^73 - 315198 * q^75 - 23716 * q^77 + 78184 * q^79 + 114384 * q^81 - 115582 * q^83 - 101652 * q^85 - 17772 * q^87 - 31560 * q^89 - 33614 * q^91 - 218040 * q^93 - 67032 * q^95 + 41068 * q^97 - 301284 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 198x^{2} + 43x + 5999$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 8\nu^{2} - 272\nu - 959 ) / 21$$ (2*v^3 + 8*v^2 - 272*v - 959) / 21 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 20\nu^{2} + 258\nu - 1820 ) / 7$$ (-2*v^3 + 20*v^2 + 258*v - 1820) / 7
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 3\beta_{2} + \beta _1 + 397 ) / 4$$ (b3 + 3*b2 + b1 + 397) / 4 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 15\beta_{2} + 134\beta _1 + 165 ) / 2$$ (-2*b3 + 15*b2 + 134*b1 + 165) / 2

## 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
 13.1634 6.13256 −6.08658 −12.2094
0 −30.3268 0 100.401 0 49.0000 0 676.717 0
1.2 0 −16.2651 0 −69.5406 0 49.0000 0 21.5544 0
1.3 0 8.17317 0 20.6340 0 49.0000 0 −176.199 0
1.4 0 20.4188 0 −21.4943 0 49.0000 0 173.928 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$

## 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 448.6.a.bc 4
4.b odd 2 1 448.6.a.bd 4
8.b even 2 1 224.6.a.h yes 4
8.d odd 2 1 224.6.a.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.g 4 8.d odd 2 1
224.6.a.h yes 4 8.b even 2 1
448.6.a.bc 4 1.a even 1 1 trivial
448.6.a.bd 4 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(448))$$:

 $$T_{3}^{4} + 18T_{3}^{3} - 672T_{3}^{2} - 6328T_{3} + 82320$$ T3^4 + 18*T3^3 - 672*T3^2 - 6328*T3 + 82320 $$T_{5}^{4} - 30T_{5}^{3} - 7452T_{5}^{2} + 7680T_{5} + 3096576$$ T5^4 - 30*T5^3 - 7452*T5^2 + 7680*T5 + 3096576

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 18 T^{3} - 672 T^{2} + \cdots + 82320$$
$5$ $$T^{4} - 30 T^{3} - 7452 T^{2} + \cdots + 3096576$$
$7$ $$(T - 49)^{4}$$
$11$ $$T^{4} + 484 T^{3} + \cdots - 9325648640$$
$13$ $$T^{4} + 686 T^{3} + \cdots - 40778482496$$
$17$ $$T^{4} + 1700 T^{3} + \cdots - 670305393104$$
$19$ $$T^{4} + 654 T^{3} + \cdots - 9671002992$$
$23$ $$T^{4} - 136 T^{3} + \cdots + 2509965337600$$
$29$ $$T^{4} + 3812 T^{3} + \cdots - 69185366122448$$
$31$ $$T^{4} - 12748 T^{3} + \cdots + 5102153647360$$
$37$ $$T^{4} + \cdots - 386774083587152$$
$41$ $$T^{4} - 22340 T^{3} + \cdots - 20\!\cdots\!12$$
$43$ $$T^{4} + 32924 T^{3} + \cdots - 14\!\cdots\!00$$
$47$ $$T^{4} + 2620 T^{3} + \cdots + 11\!\cdots\!44$$
$53$ $$T^{4} + 22984 T^{3} + \cdots - 12\!\cdots\!88$$
$59$ $$T^{4} + 108158 T^{3} + \cdots - 80\!\cdots\!16$$
$61$ $$T^{4} + 4258 T^{3} + \cdots - 43\!\cdots\!20$$
$67$ $$T^{4} + 109496 T^{3} + \cdots - 14\!\cdots\!68$$
$71$ $$T^{4} - 54600 T^{3} + \cdots - 22\!\cdots\!40$$
$73$ $$T^{4} + 12384 T^{3} + \cdots + 60\!\cdots\!40$$
$79$ $$T^{4} - 78184 T^{3} + \cdots - 14\!\cdots\!20$$
$83$ $$T^{4} + 115582 T^{3} + \cdots - 34\!\cdots\!04$$
$89$ $$T^{4} + 31560 T^{3} + \cdots + 12\!\cdots\!40$$
$97$ $$T^{4} - 41068 T^{3} + \cdots + 56\!\cdots\!76$$