# Properties

 Label 9800.2.a.db Level $9800$ Weight $2$ Character orbit 9800.a Self dual yes Analytic conductor $78.253$ Analytic rank $1$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9800 = 2^{3} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$78.2533939809$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 22 x^{8} - 16 x^{7} + 146 x^{6} + 200 x^{5} - 206 x^{4} - 440 x^{3} - 124 x^{2} + 72 x + 28$$ Coefficient ring: $$\Z[a_1, \ldots, a_{41}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 1960) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} + ( 1 - \beta_{4} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{3} + ( 1 - \beta_{4} - \beta_{8} ) q^{9} + ( -1 + \beta_{1} ) q^{11} + ( -\beta_{3} - \beta_{5} + \beta_{6} ) q^{13} + ( \beta_{3} - \beta_{7} + \beta_{9} ) q^{17} + ( \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{19} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{23} + ( -2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{27} + ( 1 - \beta_{1} + 2 \beta_{8} ) q^{29} + ( \beta_{3} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{31} + ( 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{33} + ( -4 - \beta_{2} + \beta_{8} ) q^{37} + ( -2 - \beta_{2} + \beta_{4} - \beta_{8} ) q^{39} + ( \beta_{3} - 2 \beta_{9} ) q^{41} + ( -2 - \beta_{2} + 2 \beta_{4} + 2 \beta_{8} ) q^{43} + ( \beta_{5} - \beta_{6} - \beta_{9} ) q^{47} + ( -3 - \beta_{1} + 2 \beta_{4} + 2 \beta_{8} ) q^{51} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{8} ) q^{57} + ( -\beta_{3} + \beta_{5} - 4 \beta_{7} ) q^{59} + ( 2 \beta_{3} + \beta_{5} - 2 \beta_{9} ) q^{61} + ( -4 - 2 \beta_{1} + \beta_{2} + \beta_{8} ) q^{67} + ( -3 \beta_{3} - \beta_{6} - 5 \beta_{7} + 3 \beta_{9} ) q^{69} + ( -1 + \beta_{1} - \beta_{4} + \beta_{8} ) q^{71} + ( \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{8} ) q^{79} + ( 4 - \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{8} ) q^{81} + ( -2 \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{9} ) q^{83} + ( 8 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{87} + ( -2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{89} + ( -4 + 2 \beta_{1} + \beta_{2} + 3 \beta_{8} ) q^{93} + ( -3 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} + \beta_{9} ) q^{97} + ( -6 + \beta_{2} + 2 \beta_{4} + 6 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 14q^{9} + O(q^{10})$$ $$10q + 14q^{9} - 12q^{11} - 16q^{23} + 12q^{29} - 36q^{37} - 20q^{39} - 24q^{43} - 36q^{51} - 8q^{53} - 16q^{57} - 40q^{67} - 8q^{71} - 4q^{79} + 50q^{81} - 48q^{93} - 72q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 22 x^{8} - 16 x^{7} + 146 x^{6} + 200 x^{5} - 206 x^{4} - 440 x^{3} - 124 x^{2} + 72 x + 28$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$134 \nu^{9} - 1139 \nu^{8} - 724 \nu^{7} + 18925 \nu^{6} - 5572 \nu^{5} - 98852 \nu^{4} + 19160 \nu^{3} + 154038 \nu^{2} + 1496 \nu - 15000$$$$)/5966$$ $$\beta_{2}$$ $$=$$ $$($$$$-96 \nu^{9} - 2167 \nu^{8} + 7108 \nu^{7} + 39557 \nu^{6} - 71340 \nu^{5} - 236118 \nu^{4} + 138540 \nu^{3} + 403878 \nu^{2} + 6408 \nu - 73312$$$$)/2983$$ $$\beta_{3}$$ $$=$$ $$($$$$-424 \nu^{9} + 621 \nu^{8} + 8524 \nu^{7} - 6010 \nu^{6} - 55564 \nu^{5} - 296 \nu^{4} + 107758 \nu^{3} + 31282 \nu^{2} - 37324 \nu - 5608$$$$)/5966$$ $$\beta_{4}$$ $$=$$ $$($$$$1151 \nu^{9} - 2326 \nu^{8} - 20466 \nu^{7} + 24293 \nu^{6} + 115180 \nu^{5} - 26944 \nu^{4} - 163198 \nu^{3} - 52936 \nu^{2} - 38796 \nu - 4982$$$$)/5966$$ $$\beta_{5}$$ $$=$$ $$($$$$-1431 \nu^{9} - 1260 \nu^{8} + 36226 \nu^{7} + 40122 \nu^{6} - 272544 \nu^{5} - 355976 \nu^{4} + 522528 \nu^{3} + 722312 \nu^{2} - 55868 \nu - 135264$$$$)/5966$$ $$\beta_{6}$$ $$=$$ $$($$$$-1450 \nu^{9} + 393 \nu^{8} + 33034 \nu^{7} + 10881 \nu^{6} - 234088 \nu^{5} - 185508 \nu^{4} + 443678 \nu^{3} + 407558 \nu^{2} - 71752 \nu - 49346$$$$)/5966$$ $$\beta_{7}$$ $$=$$ $$($$$$2516 \nu^{9} - 3488 \nu^{8} - 49568 \nu^{7} + 26095 \nu^{6} + 314068 \nu^{5} + 94736 \nu^{4} - 547972 \nu^{3} - 392804 \nu^{2} + 61748 \nu + 61982$$$$)/5966$$ $$\beta_{8}$$ $$=$$ $$($$$$2773 \nu^{9} - 4181 \nu^{8} - 53806 \nu^{7} + 34142 \nu^{6} + 338376 \nu^{5} + 76736 \nu^{4} - 603030 \nu^{3} - 398834 \nu^{2} + 113948 \nu + 84236$$$$)/5966$$ $$\beta_{9}$$ $$=$$ $$($$$$2773 \nu^{9} - 4181 \nu^{8} - 53806 \nu^{7} + 34142 \nu^{6} + 338376 \nu^{5} + 76736 \nu^{4} - 603030 \nu^{3} - 398834 \nu^{2} + 102016 \nu + 84236$$$$)/5966$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{9} + \beta_{8}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{4} + \beta_{2} - 3 \beta_{1} + 17$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$-5 \beta_{9} + 4 \beta_{8} + \beta_{7} - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_{1} + 5$$ $$\nu^{4}$$ $$=$$ $$($$$$-28 \beta_{9} + 10 \beta_{8} + 24 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 6 \beta_{4} + 7 \beta_{2} - 16 \beta_{1} + 74$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-111 \beta_{9} + 81 \beta_{8} + 32 \beta_{7} + 2 \beta_{6} - 27 \beta_{5} - 3 \beta_{4} + 45 \beta_{3} + 29 \beta_{2} - 37 \beta_{1} + 155$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-176 \beta_{9} + 75 \beta_{8} + 130 \beta_{7} + 28 \beta_{6} - 34 \beta_{5} - 29 \beta_{4} + 2 \beta_{3} + 45 \beta_{2} - 91 \beta_{1} + 391$$ $$\nu^{7}$$ $$=$$ $$-648 \beta_{9} + 444 \beta_{8} + 223 \beta_{7} + 31 \beta_{6} - 164 \beta_{5} - 22 \beta_{4} + 213 \beta_{3} + 179 \beta_{2} - 262 \beta_{1} + 1024$$ $$\nu^{8}$$ $$=$$ $$($$$$-4304 \beta_{9} + 2055 \beta_{8} + 2784 \beta_{7} + 672 \beta_{6} - 944 \beta_{5} - 523 \beta_{4} + 80 \beta_{3} + 1127 \beta_{2} - 2181 \beta_{1} + 8839$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$-7702 \beta_{9} + 5048 \beta_{8} + 2902 \beta_{7} + 586 \beta_{6} - 1990 \beta_{5} - 234 \beta_{4} + 1858 \beta_{3} + 2156 \beta_{2} - 3454 \beta_{1} + 12906$$

## 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
 −1.84294 0.422497 −0.609577 3.50994 1.61569 −0.432569 −1.92302 −1.30040 3.39389 −2.83351
0 −3.30674 0 0 0 0 0 7.93455 0
1.2 0 −2.69859 0 0 0 0 0 4.28239 0
1.3 0 −1.35056 0 0 0 0 0 −1.17599 0
1.4 0 −1.12212 0 0 0 0 0 −1.74084 0
1.5 0 −0.836591 0 0 0 0 0 −2.30012 0
1.6 0 0.836591 0 0 0 0 0 −2.30012 0
1.7 0 1.12212 0 0 0 0 0 −1.74084 0
1.8 0 1.35056 0 0 0 0 0 −1.17599 0
1.9 0 2.69859 0 0 0 0 0 4.28239 0
1.10 0 3.30674 0 0 0 0 0 7.93455 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9800.2.a.db 10
5.b even 2 1 9800.2.a.dc 10
5.c odd 4 2 1960.2.g.g 20
7.b odd 2 1 inner 9800.2.a.db 10
35.c odd 2 1 9800.2.a.dc 10
35.f even 4 2 1960.2.g.g 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.g.g 20 5.c odd 4 2
1960.2.g.g 20 35.f even 4 2
9800.2.a.db 10 1.a even 1 1 trivial
9800.2.a.db 10 7.b odd 2 1 inner
9800.2.a.dc 10 5.b even 2 1
9800.2.a.dc 10 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9800))$$:

 $$T_{3}^{10} - 22 T_{3}^{8} + 153 T_{3}^{6} - 384 T_{3}^{4} + 384 T_{3}^{2} - 128$$ $$T_{11}^{5} + 6 T_{11}^{4} - 15 T_{11}^{3} - 76 T_{11}^{2} + 112 T_{11} - 32$$ $$T_{13}^{10} - 92 T_{13}^{8} + 2873 T_{13}^{6} - 34134 T_{13}^{4} + 110844 T_{13}^{2} - 95048$$ $$T_{19}^{10} - 112 T_{19}^{8} + 3776 T_{19}^{6} - 39040 T_{19}^{4} + 153600 T_{19}^{2} - 204800$$ $$T_{23}^{5} + 8 T_{23}^{4} - 70 T_{23}^{3} - 528 T_{23}^{2} + 1056 T_{23} + 7232$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$-128 + 384 T^{2} - 384 T^{4} + 153 T^{6} - 22 T^{8} + T^{10}$$
$5$ $$T^{10}$$
$7$ $$T^{10}$$
$11$ $$( -32 + 112 T - 76 T^{2} - 15 T^{3} + 6 T^{4} + T^{5} )^{2}$$
$13$ $$-95048 + 110844 T^{2} - 34134 T^{4} + 2873 T^{6} - 92 T^{8} + T^{10}$$
$17$ $$-245000 + 131100 T^{2} - 24758 T^{4} + 2057 T^{6} - 76 T^{8} + T^{10}$$
$19$ $$-204800 + 153600 T^{2} - 39040 T^{4} + 3776 T^{6} - 112 T^{8} + T^{10}$$
$23$ $$( 7232 + 1056 T - 528 T^{2} - 70 T^{3} + 8 T^{4} + T^{5} )^{2}$$
$29$ $$( -2848 + 1072 T + 348 T^{2} - 71 T^{3} - 6 T^{4} + T^{5} )^{2}$$
$31$ $$-6422528 + 1765376 T^{2} - 178560 T^{4} + 8196 T^{6} - 164 T^{8} + T^{10}$$
$37$ $$( -976 - 968 T - 132 T^{2} + 70 T^{3} + 18 T^{4} + T^{5} )^{2}$$
$41$ $$-3135008 + 1201744 T^{2} - 161872 T^{4} + 8872 T^{6} - 170 T^{8} + T^{10}$$
$43$ $$( 2560 - 2560 T - 1440 T^{2} - 96 T^{3} + 12 T^{4} + T^{5} )^{2}$$
$47$ $$-131072 + 143360 T^{2} - 48640 T^{4} + 5785 T^{6} - 150 T^{8} + T^{10}$$
$53$ $$( -3904 + 3264 T - 88 T^{2} - 118 T^{3} + 4 T^{4} + T^{5} )^{2}$$
$59$ $$-685388288 + 79078912 T^{2} - 2903552 T^{4} + 47364 T^{6} - 356 T^{8} + T^{10}$$
$61$ $$-8000000 + 8080000 T^{2} - 640200 T^{4} + 18700 T^{6} - 230 T^{8} + T^{10}$$
$67$ $$( -11648 - 7456 T - 1192 T^{2} + 42 T^{3} + 20 T^{4} + T^{5} )^{2}$$
$71$ $$( -640 - 1600 T - 800 T^{2} - 108 T^{3} + 4 T^{4} + T^{5} )^{2}$$
$73$ $$-8388608 + 2146304 T^{2} - 194632 T^{4} + 8012 T^{6} - 150 T^{8} + T^{10}$$
$79$ $$( -2144 + 3792 T - 1000 T^{2} - 223 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$83$ $$-52428800 + 12288000 T^{2} - 970880 T^{4} + 32256 T^{6} - 416 T^{8} + T^{10}$$
$89$ $$-51200 + 108800 T^{2} - 68680 T^{4} + 11916 T^{6} - 214 T^{8} + T^{10}$$
$97$ $$-41332232 + 19890844 T^{2} - 2251766 T^{4} + 56553 T^{6} - 428 T^{8} + T^{10}$$