Properties

 Label 800.4.c.o Level $800$ Weight $4$ Character orbit 800.c Analytic conductor $47.202$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$47.2015280046$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1135425807366400.12 Defining polynomial: $$x^{8} + 52x^{6} + 664x^{4} + 337x^{2} + 36$$ x^8 + 52*x^6 + 664*x^4 + 337*x^2 + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + (\beta_{7} + \beta_{3}) q^{7} + (\beta_{4} - 24) q^{9}+O(q^{10})$$ q - b3 * q^3 + (b7 + b3) * q^7 + (b4 - 24) * q^9 $$q - \beta_{3} q^{3} + (\beta_{7} + \beta_{3}) q^{7} + (\beta_{4} - 24) q^{9} + (3 \beta_{5} - 4 \beta_1) q^{11} + (\beta_{6} + 18 \beta_{2}) q^{13} + (\beta_{6} + 15 \beta_{2}) q^{17} + (2 \beta_{5} - 5 \beta_1) q^{19} + ( - \beta_{4} + 56) q^{21} + ( - 6 \beta_{7} - 16 \beta_{3}) q^{23} + ( - \beta_{7} + 48 \beta_{3}) q^{27} + ( - 3 \beta_{4} + 18) q^{29} + ( - 10 \beta_{5} - 16 \beta_1) q^{31} + ( - 4 \beta_{6} - 189 \beta_{2}) q^{33} + ( - 3 \beta_{6} - 232 \beta_{2}) q^{37} + (\beta_{5} - 69 \beta_1) q^{39} + (4 \beta_{4} + 181) q^{41} + (11 \beta_{7} + 13 \beta_{3}) q^{43} + ( - 11 \beta_{7} + 31 \beta_{3}) q^{47} + ( - 4 \beta_{4} + 27) q^{49} + (\beta_{5} - 66 \beta_1) q^{51} - 222 \beta_{2} q^{53} + ( - 5 \beta_{6} - 245 \beta_{2}) q^{57} + ( - 25 \beta_{5} + 53 \beta_1) q^{59} + (\beta_{4} + 444) q^{61} + (28 \beta_{7} - 80 \beta_{3}) q^{63} + (25 \beta_{7} + 46 \beta_{3}) q^{67} + (16 \beta_{4} - 846) q^{69} + (29 \beta_{5} - \beta_1) q^{71} + (17 \beta_{6} - 265 \beta_{2}) q^{73} + (19 \beta_{6} - 556 \beta_{2}) q^{77} + (9 \beta_{5} - 107 \beta_1) q^{79} + ( - 21 \beta_{4} + 1795) q^{81} + ( - 37 \beta_{7} + 114 \beta_{3}) q^{83} + (3 \beta_{7} - 171 \beta_{3}) q^{87} + (5 \beta_{4} - 225) q^{89} + ( - 34 \beta_{5} + 74 \beta_1) q^{91} + ( - 16 \beta_{6} - 866 \beta_{2}) q^{93} + (10 \beta_{6} + 126 \beta_{2}) q^{97} + (77 \beta_{5} + 285 \beta_1) q^{99}+O(q^{100})$$ q - b3 * q^3 + (b7 + b3) * q^7 + (b4 - 24) * q^9 + (3*b5 - 4*b1) * q^11 + (b6 + 18*b2) * q^13 + (b6 + 15*b2) * q^17 + (2*b5 - 5*b1) * q^19 + (-b4 + 56) * q^21 + (-6*b7 - 16*b3) * q^23 + (-b7 + 48*b3) * q^27 + (-3*b4 + 18) * q^29 + (-10*b5 - 16*b1) * q^31 + (-4*b6 - 189*b2) * q^33 + (-3*b6 - 232*b2) * q^37 + (b5 - 69*b1) * q^39 + (4*b4 + 181) * q^41 + (11*b7 + 13*b3) * q^43 + (-11*b7 + 31*b3) * q^47 + (-4*b4 + 27) * q^49 + (b5 - 66*b1) * q^51 - 222*b2 * q^53 + (-5*b6 - 245*b2) * q^57 + (-25*b5 + 53*b1) * q^59 + (b4 + 444) * q^61 + (28*b7 - 80*b3) * q^63 + (25*b7 + 46*b3) * q^67 + (16*b4 - 846) * q^69 + (29*b5 - b1) * q^71 + (17*b6 - 265*b2) * q^73 + (19*b6 - 556*b2) * q^77 + (9*b5 - 107*b1) * q^79 + (-21*b4 + 1795) * q^81 + (-37*b7 + 114*b3) * q^83 + (3*b7 - 171*b3) * q^87 + (5*b4 - 225) * q^89 + (-34*b5 + 74*b1) * q^91 + (-16*b6 - 866*b2) * q^93 + (10*b6 + 126*b2) * q^97 + (77*b5 + 285*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 192 q^{9}+O(q^{10})$$ 8 * q - 192 * q^9 $$8 q - 192 q^{9} + 448 q^{21} + 144 q^{29} + 1448 q^{41} + 216 q^{49} + 3552 q^{61} - 6768 q^{69} + 14360 q^{81} - 1800 q^{89}+O(q^{100})$$ 8 * q - 192 * q^9 + 448 * q^21 + 144 * q^29 + 1448 * q^41 + 216 * q^49 + 3552 * q^61 - 6768 * q^69 + 14360 * q^81 - 1800 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 52x^{6} + 664x^{4} + 337x^{2} + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( 4\nu^{6} + 158\nu^{4} + 1280\nu^{2} + 323 ) / 599$$ (4*v^6 + 158*v^4 + 1280*v^2 + 323) / 599 $$\beta_{2}$$ $$=$$ $$( 23\nu^{7} + 1208\nu^{5} + 15746\nu^{3} + 11591\nu ) / 3594$$ (23*v^7 + 1208*v^5 + 15746*v^3 + 11591*v) / 3594 $$\beta_{3}$$ $$=$$ $$( -23\nu^{7} - 1208\nu^{5} - 15746\nu^{3} - 4403\nu ) / 3594$$ (-23*v^7 - 1208*v^5 - 15746*v^3 - 4403*v) / 3594 $$\beta_{4}$$ $$=$$ $$( -8\nu^{6} - 316\nu^{4} - 164\nu^{2} + 30502 ) / 599$$ (-8*v^6 - 316*v^4 - 164*v^2 + 30502) / 599 $$\beta_{5}$$ $$=$$ $$( 196\nu^{6} + 10138\nu^{4} + 127412\nu^{2} + 32599 ) / 599$$ (196*v^6 + 10138*v^4 + 127412*v^2 + 32599) / 599 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} - 52\nu^{5} - 670\nu^{3} - 493\nu ) / 3$$ (-v^7 - 52*v^5 - 670*v^3 - 493*v) / 3 $$\beta_{7}$$ $$=$$ $$( -2467\nu^{7} - 127696\nu^{5} - 1607674\nu^{3} - 449143\nu ) / 3594$$ (-2467*v^7 - 127696*v^5 - 1607674*v^3 - 449143*v) / 3594
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{4} + 2\beta _1 - 52 ) / 4$$ (b4 + 2*b1 - 52) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - 3\beta_{6} - 105\beta_{3} - 154\beta_{2} ) / 8$$ (b7 - 3*b6 - 105*b3 - 154*b2) / 8 $$\nu^{4}$$ $$=$$ $$( \beta_{5} - 27\beta_{4} - 103\beta _1 + 1376 ) / 4$$ (b5 - 27*b4 - 103*b1 + 1376) / 4 $$\nu^{5}$$ $$=$$ $$( -14\beta_{7} + 65\beta_{6} + 1428\beta_{3} + 3312\beta_{2} ) / 4$$ (-14*b7 + 65*b6 + 1428*b3 + 3312*b2) / 4 $$\nu^{6}$$ $$=$$ $$( -79\beta_{5} + 1493\beta_{4} + 8055\beta _1 - 76070 ) / 8$$ (-79*b5 + 1493*b4 + 8055*b1 - 76070) / 8 $$\nu^{7}$$ $$=$$ $$( 393\beta_{7} - 2387\beta_{6} - 40067\beta_{3} - 121620\beta_{2} ) / 4$$ (393*b7 - 2387*b6 - 40067*b3 - 121620*b2) / 4

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 4.54854i 5.54854i 0.610728i − 0.389272i − 0.610728i 0.389272i − 4.54854i − 5.54854i
0 10.0971i 0 0 0 10.5923i 0 −74.9510 0
449.2 0 10.0971i 0 0 0 10.5923i 0 −74.9510 0
449.3 0 0.221457i 0 0 0 22.7992i 0 26.9510 0
449.4 0 0.221457i 0 0 0 22.7992i 0 26.9510 0
449.5 0 0.221457i 0 0 0 22.7992i 0 26.9510 0
449.6 0 0.221457i 0 0 0 22.7992i 0 26.9510 0
449.7 0 10.0971i 0 0 0 10.5923i 0 −74.9510 0
449.8 0 10.0971i 0 0 0 10.5923i 0 −74.9510 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.c.o 8
4.b odd 2 1 inner 800.4.c.o 8
5.b even 2 1 inner 800.4.c.o 8
5.c odd 4 1 800.4.a.ba 4
5.c odd 4 1 800.4.a.bb yes 4
20.d odd 2 1 inner 800.4.c.o 8
20.e even 4 1 800.4.a.ba 4
20.e even 4 1 800.4.a.bb yes 4
40.i odd 4 1 1600.4.a.cw 4
40.i odd 4 1 1600.4.a.cx 4
40.k even 4 1 1600.4.a.cw 4
40.k even 4 1 1600.4.a.cx 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.ba 4 5.c odd 4 1
800.4.a.ba 4 20.e even 4 1
800.4.a.bb yes 4 5.c odd 4 1
800.4.a.bb yes 4 20.e even 4 1
800.4.c.o 8 1.a even 1 1 trivial
800.4.c.o 8 4.b odd 2 1 inner
800.4.c.o 8 5.b even 2 1 inner
800.4.c.o 8 20.d odd 2 1 inner
1600.4.a.cw 4 40.i odd 4 1
1600.4.a.cw 4 40.k even 4 1
1600.4.a.cx 4 40.i odd 4 1
1600.4.a.cx 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{4} + 102T_{3}^{2} + 5$$ T3^4 + 102*T3^2 + 5 $$T_{11}^{4} - 5982T_{11}^{2} + 6762845$$ T11^4 - 5982*T11^2 + 6762845

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 102 T^{2} + 5)^{2}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 632 T^{2} + 58320)^{2}$$
$11$ $$(T^{4} - 5982 T^{2} + 6762845)^{2}$$
$13$ $$(T^{4} + 5840 T^{2} + 5161984)^{2}$$
$17$ $$(T^{4} + 5642 T^{2} + 5621641)^{2}$$
$19$ $$(T^{4} - 4390 T^{2} + 4753125)^{2}$$
$23$ $$(T^{4} + 46392 T^{2} + \cdots + 523059920)^{2}$$
$29$ $$(T^{2} - 36 T - 23040)^{4}$$
$31$ $$(T^{4} - 80312 T^{2} + \cdots + 1457948880)^{2}$$
$37$ $$(T^{4} + 154376 T^{2} + \cdots + 927811600)^{2}$$
$41$ $$(T^{2} - 362 T - 8775)^{4}$$
$43$ $$(T^{4} + 81808 T^{2} + \cdots + 1179648000)^{2}$$
$47$ $$(T^{4} + 152912 T^{2} + \cdots + 5516513280)^{2}$$
$53$ $$(T^{2} + 49284)^{4}$$
$59$ $$(T^{4} - 578768 T^{2} + \cdots + 83483873280)^{2}$$
$61$ $$(T^{2} - 888 T + 194540)^{4}$$
$67$ $$(T^{4} + 557582 T^{2} + \cdots + 75081483405)^{2}$$
$71$ $$(T^{4} - 428432 T^{2} + 7787520)^{2}$$
$73$ $$(T^{4} + 1640938 T^{2} + \cdots + 462425840361)^{2}$$
$79$ $$(T^{4} - 1189848 T^{2} + \cdots + 37300611920)^{2}$$
$83$ $$(T^{4} + 1939422 T^{2} + \cdots + 842120280125)^{2}$$
$89$ $$(T^{2} + 450 T - 14275)^{4}$$
$97$ $$(T^{4} + 550952 T^{2} + \cdots + 59401388176)^{2}$$