# Properties

 Label 8820.2.d.a Level $8820$ Weight $2$ Character orbit 8820.d Analytic conductor $70.428$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8820.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$70.4280545828$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + 19894 x^{3} - 21609 x^{2} - 33614 x + 117649$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 1260) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} +O(q^{10})$$ $$q - q^{5} + ( \beta_{1} - \beta_{7} ) q^{11} -\beta_{8} q^{13} + \beta_{5} q^{17} + ( \beta_{1} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} + ( -\beta_{7} - \beta_{8} + \beta_{9} ) q^{23} + q^{25} -\beta_{11} q^{29} + ( \beta_{8} + \beta_{11} ) q^{31} + ( -\beta_{4} + \beta_{5} + \beta_{6} ) q^{37} + ( -\beta_{3} - \beta_{4} ) q^{41} + ( 4 + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{43} + ( -3 - \beta_{4} - \beta_{6} ) q^{47} + ( \beta_{1} - \beta_{7} - \beta_{11} ) q^{53} + ( -\beta_{1} + \beta_{7} ) q^{55} + ( 1 + \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{59} + ( \beta_{1} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} + \beta_{8} q^{65} + ( -1 - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{67} + ( -2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{71} + ( \beta_{1} + \beta_{9} ) q^{73} + ( -2 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{79} + ( -3 - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{83} -\beta_{5} q^{85} + ( -4 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{89} + ( -\beta_{1} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{95} -\beta_{9} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{5} + O(q^{10})$$ $$12 q - 12 q^{5} + 12 q^{25} + 4 q^{37} + 8 q^{41} + 36 q^{43} - 32 q^{47} - 4 q^{67} - 28 q^{79} - 40 q^{83} - 40 q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + 19894 x^{3} - 21609 x^{2} - 33614 x + 117649$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{10} + 9 \nu^{9} - 58 \nu^{8} + 78 \nu^{7} + 298 \nu^{6} - 1341 \nu^{5} + 2086 \nu^{4} + 3822 \nu^{3} - 19894 \nu^{2} + 4802 \nu + 33614$$$$)/16807$$ $$\beta_{2}$$ $$=$$ $$($$$$-10 \nu^{11} - 113 \nu^{10} + 748 \nu^{9} + 519 \nu^{8} - 9433 \nu^{7} + 25114 \nu^{6} - 2980 \nu^{5} - 188559 \nu^{4} + 533659 \nu^{3} - 387933 \nu^{2} - 2038449 \nu + 4386627$$$$)/84035$$ $$\beta_{3}$$ $$=$$ $$($$$$78 \nu^{11} - 16 \nu^{10} - 1423 \nu^{9} + 2088 \nu^{8} + 5319 \nu^{7} - 31616 \nu^{6} + 54744 \nu^{5} + 72072 \nu^{4} - 546644 \nu^{3} + 1011164 \nu^{2} + 1865577 \nu - 4437048$$$$)/420175$$ $$\beta_{4}$$ $$=$$ $$($$$$-103 \nu^{11} + 66 \nu^{10} + 1648 \nu^{9} - 3538 \nu^{8} - 3369 \nu^{7} + 39066 \nu^{6} - 88269 \nu^{5} - 19922 \nu^{4} + 642194 \nu^{3} - 1508514 \nu^{2} - 905177 \nu + 4857223$$$$)/420175$$ $$\beta_{5}$$ $$=$$ $$($$$$24 \nu^{11} - 111 \nu^{10} - 41 \nu^{9} + 1518 \nu^{8} - 5281 \nu^{7} + 3691 \nu^{6} + 27242 \nu^{5} - 105588 \nu^{4} + 157192 \nu^{3} + 189679 \nu^{2} - 823543 \nu + 1159683$$$$)/84035$$ $$\beta_{6}$$ $$=$$ $$($$$$326 \nu^{11} - 1177 \nu^{10} - 561 \nu^{9} + 18586 \nu^{8} - 60582 \nu^{7} + 37343 \nu^{6} + 312748 \nu^{5} - 1300516 \nu^{4} + 1613717 \nu^{3} + 1931433 \nu^{2} - 10098606 \nu + 13815354$$$$)/420175$$ $$\beta_{7}$$ $$=$$ $$($$$$2024 \nu^{11} - 9816 \nu^{10} + 6648 \nu^{9} + 104918 \nu^{8} - 431264 \nu^{7} + 512760 \nu^{6} + 1464432 \nu^{5} - 8068508 \nu^{4} + 13138664 \nu^{3} + 7322364 \nu^{2} - 63271152 \nu + 87816575$$$$)/1260525$$ $$\beta_{8}$$ $$=$$ $$($$$$-538 \nu^{11} + 2910 \nu^{10} - 2256 \nu^{9} - 29188 \nu^{8} + 124669 \nu^{7} - 156972 \nu^{6} - 396084 \nu^{5} + 2287180 \nu^{4} - 3863503 \nu^{3} - 1920114 \nu^{2} + 17928267 \nu - 24168466$$$$)/252105$$ $$\beta_{9}$$ $$=$$ $$($$$$-5459 \nu^{11} + 24666 \nu^{10} - 17418 \nu^{9} - 262778 \nu^{8} + 1126019 \nu^{7} - 1366950 \nu^{6} - 3581637 \nu^{5} + 20934158 \nu^{4} - 34869674 \nu^{3} - 15008994 \nu^{2} + 159827367 \nu - 253953770$$$$)/1260525$$ $$\beta_{10}$$ $$=$$ $$($$$$-5624 \nu^{11} + 26256 \nu^{10} - 19923 \nu^{9} - 275603 \nu^{8} + 1178894 \nu^{7} - 1475595 \nu^{6} - 3693282 \nu^{5} + 21672203 \nu^{4} - 36302189 \nu^{3} - 15266244 \nu^{2} + 165049542 \nu - 249415880$$$$)/1260525$$ $$\beta_{11}$$ $$=$$ $$($$$$-1594 \nu^{11} + 7221 \nu^{10} - 5088 \nu^{9} - 76858 \nu^{8} + 325759 \nu^{7} - 397035 \nu^{6} - 1031292 \nu^{5} + 6008398 \nu^{4} - 9970009 \nu^{3} - 4431609 \nu^{2} + 45330537 \nu - 69929125$$$$)/180075$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_{1} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{11} + 4 \beta_{10} + \beta_{9} + 2 \beta_{8} + 6 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{1} - 17$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} + 15 \beta_{5} - 12 \beta_{4} - 19 \beta_{3} - 2 \beta_{2} + 12 \beta_{1} + 18$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$32 \beta_{11} - 42 \beta_{10} - 23 \beta_{9} + 19 \beta_{8} + 24 \beta_{7} - 5 \beta_{6} + 15 \beta_{5} + 31 \beta_{4} + 16 \beta_{3} - 10 \beta_{2} - 22 \beta_{1} - 34$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$23 \beta_{11} - 98 \beta_{10} + 38 \beta_{9} + 11 \beta_{8} - 39 \beta_{7} - 8 \beta_{6} + 11 \beta_{5} - 78 \beta_{4} + 43 \beta_{3} + 40 \beta_{2} + 38 \beta_{1} + 271$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$40 \beta_{11} + 40 \beta_{10} - 5 \beta_{9} + 140 \beta_{8} + 540 \beta_{7} - 24 \beta_{6} - 152 \beta_{5} + 278 \beta_{4} + 441 \beta_{3} + 40 \beta_{2} + 73 \beta_{1} + 338$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-568 \beta_{11} + 398 \beta_{10} + 846 \beta_{9} + 58 \beta_{8} + 283 \beta_{7} + 162 \beta_{6} + 44 \beta_{5} - 60 \beta_{4} - 174 \beta_{3} + 62 \beta_{2} + 542 \beta_{1} + 3065$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$88 \beta_{11} + 376 \beta_{10} - 161 \beta_{9} + 1118 \beta_{8} + 2976 \beta_{7} - 414 \beta_{6} - 242 \beta_{5} + 2061 \beta_{4} + 254 \beta_{3} - 440 \beta_{2} - 2734 \beta_{1} + 231$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-2298 \beta_{11} - 897 \beta_{10} + 3955 \beta_{9} + 1307 \beta_{8} - 3513 \beta_{7} - 321 \beta_{6} + 3706 \beta_{5} - 4972 \beta_{4} - 3955 \beta_{3} + 1601 \beta_{2} - 1863 \beta_{1} - 7762$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$2194 \beta_{11} - 34 \beta_{10} - 4386 \beta_{9} + 11933 \beta_{8} + 20418 \beta_{7} - 7409 \beta_{6} + 673 \beta_{5} - 22 \beta_{4} + 3036 \beta_{3} + 30 \beta_{2} + 906 \beta_{1} - 53577$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times$$.

 $$n$$ $$1081$$ $$4411$$ $$7057$$ $$7841$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 1.63107 + 2.08318i 0.260926 + 2.63285i −2.61674 + 0.390758i 2.61827 + 0.380350i 1.75207 + 1.98249i −2.64559 + 0.0290059i −2.64559 − 0.0290059i 1.75207 − 1.98249i 2.61827 − 0.380350i −2.61674 − 0.390758i 0.260926 − 2.63285i 1.63107 − 2.08318i
0 0 0 −1.00000 0 0 0 0 0
881.2 0 0 0 −1.00000 0 0 0 0 0
881.3 0 0 0 −1.00000 0 0 0 0 0
881.4 0 0 0 −1.00000 0 0 0 0 0
881.5 0 0 0 −1.00000 0 0 0 0 0
881.6 0 0 0 −1.00000 0 0 0 0 0
881.7 0 0 0 −1.00000 0 0 0 0 0
881.8 0 0 0 −1.00000 0 0 0 0 0
881.9 0 0 0 −1.00000 0 0 0 0 0
881.10 0 0 0 −1.00000 0 0 0 0 0
881.11 0 0 0 −1.00000 0 0 0 0 0
881.12 0 0 0 −1.00000 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8820.2.d.a 12
3.b odd 2 1 8820.2.d.b 12
7.b odd 2 1 8820.2.d.b 12
7.c even 3 1 1260.2.cg.b yes 12
7.d odd 6 1 1260.2.cg.a 12
21.c even 2 1 inner 8820.2.d.a 12
21.g even 6 1 1260.2.cg.b yes 12
21.h odd 6 1 1260.2.cg.a 12
35.i odd 6 1 6300.2.ch.c 12
35.j even 6 1 6300.2.ch.b 12
35.k even 12 2 6300.2.dd.c 24
35.l odd 12 2 6300.2.dd.b 24
105.o odd 6 1 6300.2.ch.c 12
105.p even 6 1 6300.2.ch.b 12
105.w odd 12 2 6300.2.dd.b 24
105.x even 12 2 6300.2.dd.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.cg.a 12 7.d odd 6 1
1260.2.cg.a 12 21.h odd 6 1
1260.2.cg.b yes 12 7.c even 3 1
1260.2.cg.b yes 12 21.g even 6 1
6300.2.ch.b 12 35.j even 6 1
6300.2.ch.b 12 105.p even 6 1
6300.2.ch.c 12 35.i odd 6 1
6300.2.ch.c 12 105.o odd 6 1
6300.2.dd.b 24 35.l odd 12 2
6300.2.dd.b 24 105.w odd 12 2
6300.2.dd.c 24 35.k even 12 2
6300.2.dd.c 24 105.x even 12 2
8820.2.d.a 12 1.a even 1 1 trivial
8820.2.d.a 12 21.c even 2 1 inner
8820.2.d.b 12 3.b odd 2 1
8820.2.d.b 12 7.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 $$S_{2}^{\mathrm{new}}(8820, [\chi])$$:

 $$T_{11}^{12} + 68 T_{11}^{10} + 1572 T_{11}^{8} + 16988 T_{11}^{6} + 93880 T_{11}^{4} + 256320 T_{11}^{2} + 272484$$ $$T_{17}^{6} - 52 T_{17}^{4} - 108 T_{17}^{3} + 262 T_{17}^{2} + 372 T_{17} - 450$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$( 1 + T )^{12}$$
$7$ $$T^{12}$$
$11$ $$272484 + 256320 T^{2} + 93880 T^{4} + 16988 T^{6} + 1572 T^{8} + 68 T^{10} + T^{12}$$
$13$ $$189225 + 628242 T^{2} + 302563 T^{4} + 52392 T^{6} + 3575 T^{8} + 102 T^{10} + T^{12}$$
$17$ $$( -450 + 372 T + 262 T^{2} - 108 T^{3} - 52 T^{4} + T^{6} )^{2}$$
$19$ $$2064969 + 12114186 T^{2} + 3245431 T^{4} + 268308 T^{6} + 9743 T^{8} + 162 T^{10} + T^{12}$$
$23$ $$170302500 + 53539848 T^{2} + 6606868 T^{4} + 403372 T^{6} + 12564 T^{8} + 184 T^{10} + T^{12}$$
$29$ $$8100 + 148752 T^{2} + 552904 T^{4} + 121516 T^{6} + 7068 T^{8} + 148 T^{10} + T^{12}$$
$31$ $$9126441 + 12115410 T^{2} + 3920383 T^{4} + 370620 T^{6} + 13943 T^{8} + 210 T^{10} + T^{12}$$
$37$ $$( -1415 - 2966 T + 1917 T^{2} + 248 T^{3} - 105 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$41$ $$( -450 + 600 T + 688 T^{2} + 76 T^{3} - 46 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$43$ $$( 142465 - 27886 T - 7643 T^{2} + 1952 T^{3} - 25 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$47$ $$( 15264 - 4992 T - 7712 T^{2} - 1696 T^{3} - 40 T^{4} + 16 T^{5} + T^{6} )^{2}$$
$53$ $$82944 + 25279488 T^{2} + 4655104 T^{4} + 325056 T^{6} + 10736 T^{8} + 168 T^{10} + T^{12}$$
$59$ $$( -2610 - 5808 T + 3100 T^{2} + 228 T^{3} - 166 T^{4} + T^{6} )^{2}$$
$61$ $$1497690000 + 633520800 T^{2} + 72993136 T^{4} + 2859960 T^{6} + 47912 T^{8} + 360 T^{10} + T^{12}$$
$67$ $$( -66951 + 6318 T + 7959 T^{2} - 240 T^{3} - 179 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$71$ $$239073444 + 2127291696 T^{2} + 182441344 T^{4} + 5460252 T^{6} + 73976 T^{8} + 456 T^{10} + T^{12}$$
$73$ $$1172889 + 2472198 T^{2} + 658159 T^{4} + 71096 T^{6} + 3771 T^{8} + 98 T^{10} + T^{12}$$
$79$ $$( -166655 + 38522 T + 14007 T^{2} - 2216 T^{3} - 213 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$83$ $$( -13410 - 54444 T - 32006 T^{2} - 5144 T^{3} - 142 T^{4} + 20 T^{5} + T^{6} )^{2}$$
$89$ $$( -781650 + 191664 T + 16600 T^{2} - 4004 T^{3} - 196 T^{4} + 20 T^{5} + T^{6} )^{2}$$
$97$ $$1089936 + 1420128 T^{2} + 451456 T^{4} + 55880 T^{6} + 3276 T^{8} + 92 T^{10} + T^{12}$$