# Properties

 Label 200.6.a.e Level $200$ Weight $6$ Character orbit 200.a Self dual yes Analytic conductor $32.077$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0767639626$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ x^2 - x - 60 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{241}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 4) q^{3} + (2 \beta + 4) q^{7} + (8 \beta + 14) q^{9}+O(q^{10})$$ q + (-b - 4) * q^3 + (2*b + 4) * q^7 + (8*b + 14) * q^9 $$q + ( - \beta - 4) q^{3} + (2 \beta + 4) q^{7} + (8 \beta + 14) q^{9} + ( - 21 \beta - 100) q^{11} + (52 \beta + 296) q^{13} + (16 \beta - 139) q^{17} + (59 \beta - 420) q^{19} + ( - 12 \beta - 498) q^{21} + (98 \beta + 976) q^{23} + (197 \beta - 1012) q^{27} + (148 \beta - 2340) q^{29} + ( - 354 \beta - 2504) q^{31} + (184 \beta + 5461) q^{33} + ( - 484 \beta - 6250) q^{37} + ( - 504 \beta - 13716) q^{39} + (936 \beta - 2667) q^{41} + ( - 1076 \beta - 112) q^{43} - 13036 q^{47} + (16 \beta - 15827) q^{49} + (75 \beta - 3300) q^{51} + ( - 360 \beta + 23406) q^{53} + (184 \beta - 12539) q^{57} + ( - 416 \beta - 40888) q^{59} + (44 \beta - 23466) q^{61} + (60 \beta + 3912) q^{63} + ( - 323 \beta - 34404) q^{67} + ( - 1368 \beta - 27522) q^{69} + ( - 344 \beta + 3724) q^{71} + ( - 72 \beta + 54411) q^{73} + ( - 284 \beta - 10522) q^{77} + (3062 \beta - 54052) q^{79} + ( - 1720 \beta - 46831) q^{81} + (3841 \beta - 13612) q^{83} + (1748 \beta - 26308) q^{87} + (5024 \beta + 35495) q^{89} + (800 \beta + 26248) q^{91} + (3920 \beta + 95330) q^{93} + (3800 \beta + 48426) q^{97} + ( - 1094 \beta - 41888) q^{99}+O(q^{100})$$ q + (-b - 4) * q^3 + (2*b + 4) * q^7 + (8*b + 14) * q^9 + (-21*b - 100) * q^11 + (52*b + 296) * q^13 + (16*b - 139) * q^17 + (59*b - 420) * q^19 + (-12*b - 498) * q^21 + (98*b + 976) * q^23 + (197*b - 1012) * q^27 + (148*b - 2340) * q^29 + (-354*b - 2504) * q^31 + (184*b + 5461) * q^33 + (-484*b - 6250) * q^37 + (-504*b - 13716) * q^39 + (936*b - 2667) * q^41 + (-1076*b - 112) * q^43 - 13036 * q^47 + (16*b - 15827) * q^49 + (75*b - 3300) * q^51 + (-360*b + 23406) * q^53 + (184*b - 12539) * q^57 + (-416*b - 40888) * q^59 + (44*b - 23466) * q^61 + (60*b + 3912) * q^63 + (-323*b - 34404) * q^67 + (-1368*b - 27522) * q^69 + (-344*b + 3724) * q^71 + (-72*b + 54411) * q^73 + (-284*b - 10522) * q^77 + (3062*b - 54052) * q^79 + (-1720*b - 46831) * q^81 + (3841*b - 13612) * q^83 + (1748*b - 26308) * q^87 + (5024*b + 35495) * q^89 + (800*b + 26248) * q^91 + (3920*b + 95330) * q^93 + (3800*b + 48426) * q^97 + (-1094*b - 41888) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{3} + 8 q^{7} + 28 q^{9}+O(q^{10})$$ 2 * q - 8 * q^3 + 8 * q^7 + 28 * q^9 $$2 q - 8 q^{3} + 8 q^{7} + 28 q^{9} - 200 q^{11} + 592 q^{13} - 278 q^{17} - 840 q^{19} - 996 q^{21} + 1952 q^{23} - 2024 q^{27} - 4680 q^{29} - 5008 q^{31} + 10922 q^{33} - 12500 q^{37} - 27432 q^{39} - 5334 q^{41} - 224 q^{43} - 26072 q^{47} - 31654 q^{49} - 6600 q^{51} + 46812 q^{53} - 25078 q^{57} - 81776 q^{59} - 46932 q^{61} + 7824 q^{63} - 68808 q^{67} - 55044 q^{69} + 7448 q^{71} + 108822 q^{73} - 21044 q^{77} - 108104 q^{79} - 93662 q^{81} - 27224 q^{83} - 52616 q^{87} + 70990 q^{89} + 52496 q^{91} + 190660 q^{93} + 96852 q^{97} - 83776 q^{99}+O(q^{100})$$ 2 * q - 8 * q^3 + 8 * q^7 + 28 * q^9 - 200 * q^11 + 592 * q^13 - 278 * q^17 - 840 * q^19 - 996 * q^21 + 1952 * q^23 - 2024 * q^27 - 4680 * q^29 - 5008 * q^31 + 10922 * q^33 - 12500 * q^37 - 27432 * q^39 - 5334 * q^41 - 224 * q^43 - 26072 * q^47 - 31654 * q^49 - 6600 * q^51 + 46812 * q^53 - 25078 * q^57 - 81776 * q^59 - 46932 * q^61 + 7824 * q^63 - 68808 * q^67 - 55044 * q^69 + 7448 * q^71 + 108822 * q^73 - 21044 * q^77 - 108104 * q^79 - 93662 * q^81 - 27224 * q^83 - 52616 * q^87 + 70990 * q^89 + 52496 * q^91 + 190660 * q^93 + 96852 * q^97 - 83776 * q^99

## 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
 8.26209 −7.26209
0 −19.5242 0 0 0 35.0483 0 138.193 0
1.2 0 11.5242 0 0 0 −27.0483 0 −110.193 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
$$2$$ $$-1$$
$$5$$ $$-1$$

## 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 200.6.a.e 2
4.b odd 2 1 400.6.a.u 2
5.b even 2 1 200.6.a.f yes 2
5.c odd 4 2 200.6.c.f 4
20.d odd 2 1 400.6.a.r 2
20.e even 4 2 400.6.c.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.a.e 2 1.a even 1 1 trivial
200.6.a.f yes 2 5.b even 2 1
200.6.c.f 4 5.c odd 4 2
400.6.a.r 2 20.d odd 2 1
400.6.a.u 2 4.b odd 2 1
400.6.c.o 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 8T_{3} - 225$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(200))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 8T - 225$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 8T - 948$$
$11$ $$T^{2} + 200T - 96281$$
$13$ $$T^{2} - 592T - 564048$$
$17$ $$T^{2} + 278T - 42375$$
$19$ $$T^{2} + 840T - 662521$$
$23$ $$T^{2} - 1952 T - 1361988$$
$29$ $$T^{2} + 4680 T + 196736$$
$31$ $$T^{2} + 5008 T - 23931140$$
$37$ $$T^{2} + 12500 T - 17393196$$
$41$ $$T^{2} + 5334 T - 204026247$$
$43$ $$T^{2} + 224 T - 279011472$$
$47$ $$(T + 13036)^{2}$$
$53$ $$T^{2} - 46812 T + 516607236$$
$59$ $$T^{2} + 81776 T + 1630122048$$
$61$ $$T^{2} + 46932 T + 550186580$$
$67$ $$T^{2} + 68808 T + 1158491927$$
$71$ $$T^{2} - 7448 T - 14650800$$
$73$ $$T^{2} - 108822 T + 2959307577$$
$79$ $$T^{2} + 108104 T + 662040300$$
$83$ $$T^{2} + 27224 T - 3370254177$$
$89$ $$T^{2} - 70990 T - 4823083791$$
$97$ $$T^{2} - 96852 T - 1134962524$$