# Properties

 Label 800.6.c.f Level 800 Weight 6 Character orbit 800.c Analytic conductor 128.307 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$128.307055850$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{70})$$ Defining polynomial: $$x^{4} + 1225$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} - \beta_{2} ) q^{3} + ( -26 \beta_{1} - \beta_{2} ) q^{7} + ( -53 + 4 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} - \beta_{2} ) q^{3} + ( -26 \beta_{1} - \beta_{2} ) q^{7} + ( -53 + 4 \beta_{3} ) q^{9} + ( -160 - 5 \beta_{3} ) q^{11} + ( 25 \beta_{1} + 40 \beta_{2} ) q^{13} + ( 145 \beta_{1} - 40 \beta_{2} ) q^{17} + ( 360 + 20 \beta_{3} ) q^{19} + ( -72 - 24 \beta_{3} ) q^{21} + ( 422 \beta_{1} - 97 \beta_{2} ) q^{23} + ( -740 \beta_{1} - 158 \beta_{2} ) q^{27} + ( -54 - 40 \beta_{3} ) q^{29} + ( -4920 + 65 \beta_{3} ) q^{31} + ( 1080 \beta_{1} + 120 \beta_{2} ) q^{33} + ( 1635 \beta_{1} + 560 \beta_{2} ) q^{37} + ( 11000 - 55 \beta_{3} ) q^{39} + ( -5310 + 404 \beta_{3} ) q^{41} + ( 6418 \beta_{1} + 579 \beta_{2} ) q^{43} + ( -7074 \beta_{1} + 383 \beta_{2} ) q^{47} + ( 13823 - 52 \beta_{3} ) q^{49} + ( -12360 + 225 \beta_{3} ) q^{51} + ( -7835 \beta_{1} + 1080 \beta_{2} ) q^{53} + ( -4880 \beta_{1} - 200 \beta_{2} ) q^{57} + ( 15400 + 10 \beta_{3} ) q^{59} + ( 12270 + 592 \beta_{3} ) q^{61} + ( 258 \beta_{1} - 363 \beta_{2} ) q^{63} + ( -8646 \beta_{1} + 495 \beta_{2} ) q^{67} + ( -30536 + 616 \beta_{3} ) q^{69} + ( 6200 - 1495 \beta_{3} ) q^{71} + ( 1795 \beta_{1} + 3720 \beta_{2} ) q^{73} + ( 5560 \beta_{1} + 680 \beta_{2} ) q^{77} + ( 35920 + 110 \beta_{3} ) q^{79} + ( -51199 + 548 \beta_{3} ) q^{81} + ( -7982 \beta_{1} - 2817 \beta_{2} ) q^{83} + ( 11092 \beta_{1} - 266 \beta_{2} ) q^{87} + ( 20374 - 1640 \beta_{3} ) q^{89} + ( 13800 + 1065 \beta_{3} ) q^{91} + ( -28040 \beta_{1} + 5440 \beta_{2} ) q^{93} + ( -47535 \beta_{1} + 4840 \beta_{2} ) q^{97} + ( -13920 - 375 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 212q^{9} + O(q^{10})$$ $$4q - 212q^{9} - 640q^{11} + 1440q^{19} - 288q^{21} - 216q^{29} - 19680q^{31} + 44000q^{39} - 21240q^{41} + 55292q^{49} - 49440q^{51} + 61600q^{59} + 49080q^{61} - 122144q^{69} + 24800q^{71} + 143680q^{79} - 204796q^{81} + 81496q^{89} + 55200q^{91} - 55680q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 1225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/35$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{3} + 70 \nu$$$$)/35$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{3} + 140 \nu$$$$)/35$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$35 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-35 \beta_{3} + 70 \beta_{2}$$$$)/8$$

## 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
 −4.18330 + 4.18330i 4.18330 + 4.18330i 4.18330 − 4.18330i −4.18330 − 4.18330i
0 20.7332i 0 0 0 35.2668i 0 −186.866 0
449.2 0 12.7332i 0 0 0 68.7332i 0 80.8656 0
449.3 0 12.7332i 0 0 0 68.7332i 0 80.8656 0
449.4 0 20.7332i 0 0 0 35.2668i 0 −186.866 0
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.c.f 4
4.b odd 2 1 800.6.c.g 4
5.b even 2 1 inner 800.6.c.f 4
5.c odd 4 1 160.6.a.a 2
5.c odd 4 1 800.6.a.l 2
20.d odd 2 1 800.6.c.g 4
20.e even 4 1 160.6.a.e yes 2
20.e even 4 1 800.6.a.g 2
40.i odd 4 1 320.6.a.v 2
40.k even 4 1 320.6.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.a 2 5.c odd 4 1
160.6.a.e yes 2 20.e even 4 1
320.6.a.r 2 40.k even 4 1
320.6.a.v 2 40.i odd 4 1
800.6.a.g 2 20.e even 4 1
800.6.a.l 2 5.c odd 4 1
800.6.c.f 4 1.a even 1 1 trivial
800.6.c.f 4 5.b even 2 1 inner
800.6.c.g 4 4.b odd 2 1
800.6.c.g 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{4} + 592 T_{3}^{2} + 69696$$ $$T_{11}^{2} + 320 T_{11} - 2400$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 380 T^{2} + 136278 T^{4} - 22438620 T^{6} + 3486784401 T^{8}$$
$5$ 1
$7$ $$1 - 61260 T^{2} + 1500118918 T^{4} - 17304433753740 T^{6} + 79792266297612001 T^{8}$$
$11$ $$( 1 + 320 T + 319702 T^{2} + 51536320 T^{3} + 25937424601 T^{4} )^{2}$$
$13$ $$1 - 584172 T^{2} + 356551215094 T^{4} - 80533070900414028 T^{6} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$1 - 4615228 T^{2} + 9206362973894 T^{4} - 9304271497181437372 T^{6} +$$$$40\!\cdots\!01$$$$T^{8}$$
$19$ $$( 1 - 720 T + 4633798 T^{2} - 1782791280 T^{3} + 6131066257801 T^{4} )^{2}$$
$23$ $$1 - 19051660 T^{2} + 166087805861318 T^{4} -$$$$78\!\cdots\!40$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$( 1 + 108 T + 39233214 T^{2} + 2215204092 T^{3} + 420707233300201 T^{4} )^{2}$$
$31$ $$( 1 + 9840 T + 76732702 T^{2} + 281710845840 T^{3} + 819628286980801 T^{4} )^{2}$$
$37$ $$1 - 80374028 T^{2} + 7476476186271894 T^{4} -$$$$38\!\cdots\!72$$$$T^{6} +$$$$23\!\cdots\!01$$$$T^{8}$$
$41$ $$( 1 + 10620 T + 77106582 T^{2} + 1230392854620 T^{3} + 13422659310152401 T^{4} )^{2}$$
$43$ $$1 - 70773020 T^{2} - 17388341663539722 T^{4} -$$$$15\!\cdots\!80$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 434902380 T^{2} + 119597691857197478 T^{4} -$$$$22\!\cdots\!20$$$$T^{6} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$1 - 528500172 T^{2} + 98825823286833494 T^{4} -$$$$92\!\cdots\!28$$$$T^{6} +$$$$30\!\cdots\!01$$$$T^{8}$$
$59$ $$( 1 - 30800 T + 1666896598 T^{2} - 22019668409200 T^{3} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 - 24540 T + 1447225822 T^{2} - 20726393226540 T^{3} + 713342911662882601 T^{4} )^{2}$$
$67$ $$1 - 4665259900 T^{2} + 9004780480727533398 T^{4} -$$$$85\!\cdots\!00$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 - 12400 T + 1143670702 T^{2} - 22372443952400 T^{3} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$1 - 517006172 T^{2} + 8462322739873838694 T^{4} -$$$$22\!\cdots\!28$$$$T^{6} +$$$$18\!\cdots\!01$$$$T^{8}$$
$79$ $$( 1 - 71840 T + 7430807198 T^{2} - 221055731704160 T^{3} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 10802590140 T^{2} + 57941034568344378518 T^{4} -$$$$16\!\cdots\!60$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 - 40748 T + 8570866774 T^{2} - 227539254427852 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$1 - 3154415228 T^{2} - 87162631860647574906 T^{4} -$$$$23\!\cdots\!72$$$$T^{6} +$$$$54\!\cdots\!01$$$$T^{8}$$