# Properties

 Label 804.2.i.d Level 804 Weight 2 Character orbit 804.i Analytic conductor 6.420 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$804 = 2^{2} \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 804.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta_{6} q^{5} + ( -\beta_{1} + \beta_{6} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta_{6} q^{5} + ( -\beta_{1} + \beta_{6} ) q^{7} + q^{9} + 2 \beta_{2} q^{11} + ( \beta_{3} + \beta_{4} ) q^{13} -\beta_{6} q^{15} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( \beta_{1} - \beta_{6} ) q^{21} + \beta_{5} q^{23} + ( \beta_{3} + \beta_{6} ) q^{25} - q^{27} + ( \beta_{1} - \beta_{6} - \beta_{7} ) q^{29} + ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{6} - \beta_{7} ) q^{31} -2 \beta_{2} q^{33} + ( -\beta_{1} + 5 \beta_{2} - \beta_{4} + \beta_{6} ) q^{35} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{37} + ( -\beta_{3} - \beta_{4} ) q^{39} + ( 3 \beta_{2} - \beta_{4} - \beta_{7} ) q^{41} + ( -\beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{43} + \beta_{6} q^{45} + ( 3 \beta_{2} - \beta_{4} - \beta_{7} ) q^{47} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{49} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{51} + ( 5 - \beta_{3} ) q^{53} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{57} + ( 3 + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( 5 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{61} + ( -\beta_{1} + \beta_{6} ) q^{63} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{65} + ( -5 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{67} -\beta_{5} q^{69} + ( -4 \beta_{1} + 2 \beta_{2} + 4 \beta_{6} - \beta_{7} ) q^{71} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( -\beta_{3} - \beta_{6} ) q^{75} -2 \beta_{1} q^{77} + ( 4 \beta_{1} + 4 \beta_{2} + \beta_{4} - 4 \beta_{6} ) q^{79} + q^{81} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{83} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{85} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{87} + ( -2 - \beta_{5} + \beta_{6} - \beta_{7} ) q^{89} + ( -2 + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{91} + ( -2 \beta_{1} - 3 \beta_{2} + 2 \beta_{6} + \beta_{7} ) q^{93} + ( -3 - 5 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{95} + ( 5 + \beta_{1} - 5 \beta_{2} - \beta_{5} ) q^{97} + 2 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{3} + 2q^{5} + q^{7} + 8q^{9} + O(q^{10})$$ $$8q - 8q^{3} + 2q^{5} + q^{7} + 8q^{9} + 8q^{11} - 2q^{15} + 4q^{17} - 5q^{19} - q^{21} + q^{23} + 2q^{25} - 8q^{27} - 2q^{29} + 9q^{31} - 8q^{33} + 21q^{35} - 5q^{37} + 11q^{41} + 6q^{43} + 2q^{45} + 11q^{47} + 7q^{49} - 4q^{51} + 40q^{53} + 2q^{55} + 5q^{57} + 28q^{59} + 20q^{61} + q^{63} - 4q^{65} - 33q^{67} - q^{69} + 11q^{71} - 7q^{73} - 2q^{75} - 2q^{77} + 12q^{79} + 8q^{81} - 4q^{83} + 5q^{85} + 2q^{87} - 16q^{89} - 8q^{91} - 9q^{93} - 18q^{95} + 20q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 11 x^{6} + 4 x^{5} + 91 x^{4} - 6 x^{3} + 129 x^{2} + 36 x + 144$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$47 \nu^{7} - 199 \nu^{6} + 597 \nu^{5} - 1256 \nu^{4} + 2985 \nu^{3} - 14726 \nu^{2} + 3555 \nu - 1044$$$$)/18060$$ $$\beta_{3}$$ $$=$$ $$($$$$20 \nu^{7} + 37 \nu^{6} + 190 \nu^{5} + 170 \nu^{4} + 2756 \nu^{3} + 330 \nu^{2} + 360 \nu - 15411$$$$)/4515$$ $$\beta_{4}$$ $$=$$ $$($$$$83 \nu^{7} - 915 \nu^{6} + 1541 \nu^{5} - 7572 \nu^{4} + 481 \nu^{3} - 58078 \nu^{2} - 3021 \nu - 11988$$$$)/18060$$ $$\beta_{5}$$ $$=$$ $$($$$$67 \nu^{7} + 139 \nu^{6} - 417 \nu^{5} + 3128 \nu^{4} - 2085 \nu^{3} + 10286 \nu^{2} - 62907 \nu + 13344$$$$)/9030$$ $$\beta_{6}$$ $$=$$ $$($$$$-38 \nu^{7} + 20 \nu^{6} - 361 \nu^{5} - 323 \nu^{4} - 3611 \nu^{3} - 627 \nu^{2} - 684 \nu - 1692$$$$)/4515$$ $$\beta_{7}$$ $$=$$ $$($$$$577 \nu^{7} - 573 \nu^{6} + 6535 \nu^{5} + 2346 \nu^{4} + 52541 \nu^{3} + 340 \nu^{2} + 74499 \nu + 20844$$$$)/9030$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{4} - 5 \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} - 9 \beta_{6} - \beta_{5} - \beta_{3} - 3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{5} - 11 \beta_{4} - 11 \beta_{3} + 41 \beta_{2} - 16 \beta_{1} - 41$$ $$\nu^{5}$$ $$=$$ $$11 \beta_{7} + 91 \beta_{6} - 18 \beta_{4} + 56 \beta_{2} - 91 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$18 \beta_{7} + 212 \beta_{6} + 18 \beta_{5} + 113 \beta_{3} + 397$$ $$\nu^{7}$$ $$=$$ $$113 \beta_{5} + 248 \beta_{4} + 248 \beta_{3} - 798 \beta_{2} + 966 \beta_{1} + 798$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$$.

 $$n$$ $$269$$ $$337$$ $$403$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −1.30177 − 2.25473i −0.531167 − 0.920008i 0.641129 + 1.11047i 1.69181 + 2.93030i −1.30177 + 2.25473i −0.531167 + 0.920008i 0.641129 − 1.11047i 1.69181 − 2.93030i
0 −1.00000 0 −2.60354 0 −1.30177 + 2.25473i 0 1.00000 0
37.2 0 −1.00000 0 −1.06233 0 −0.531167 + 0.920008i 0 1.00000 0
37.3 0 −1.00000 0 1.28226 0 0.641129 1.11047i 0 1.00000 0
37.4 0 −1.00000 0 3.38361 0 1.69181 2.93030i 0 1.00000 0
565.1 0 −1.00000 0 −2.60354 0 −1.30177 2.25473i 0 1.00000 0
565.2 0 −1.00000 0 −1.06233 0 −0.531167 0.920008i 0 1.00000 0
565.3 0 −1.00000 0 1.28226 0 0.641129 + 1.11047i 0 1.00000 0
565.4 0 −1.00000 0 3.38361 0 1.69181 + 2.93030i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 565.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.i.d 8
3.b odd 2 1 2412.2.l.e 8
67.c even 3 1 inner 804.2.i.d 8
201.g odd 6 1 2412.2.l.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.i.d 8 1.a even 1 1 trivial
804.2.i.d 8 67.c even 3 1 inner
2412.2.l.e 8 3.b odd 2 1
2412.2.l.e 8 201.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{3} - 10 T_{5}^{2} + 3 T_{5} + 12$$ $$T_{7}^{8} - \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 + T )^{8}$$
$5$ $$( 1 - T + 10 T^{2} - 12 T^{3} + 62 T^{4} - 60 T^{5} + 250 T^{6} - 125 T^{7} + 625 T^{8} )^{2}$$
$7$ $$1 - T - 17 T^{2} + 18 T^{3} + 140 T^{4} - 118 T^{5} - 1026 T^{6} + 323 T^{7} + 7697 T^{8} + 2261 T^{9} - 50274 T^{10} - 40474 T^{11} + 336140 T^{12} + 302526 T^{13} - 2000033 T^{14} - 823543 T^{15} + 5764801 T^{16}$$
$11$ $$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{4}$$
$13$ $$1 - 23 T^{2} - 6 T^{3} + 94 T^{4} + 108 T^{5} - 2222 T^{6} - 489 T^{7} + 71146 T^{8} - 6357 T^{9} - 375518 T^{10} + 237276 T^{11} + 2684734 T^{12} - 2227758 T^{13} - 111016607 T^{14} + 815730721 T^{16}$$
$17$ $$1 - 4 T - 29 T^{2} + 126 T^{3} + 311 T^{4} - 670 T^{5} - 10575 T^{6} - 6188 T^{7} + 304921 T^{8} - 105196 T^{9} - 3056175 T^{10} - 3291710 T^{11} + 25975031 T^{12} + 178901982 T^{13} - 699989501 T^{14} - 1641354692 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 5 T - 25 T^{2} - 242 T^{3} - 48 T^{4} + 3794 T^{5} + 7346 T^{6} - 18627 T^{7} - 102551 T^{8} - 353913 T^{9} + 2651906 T^{10} + 26023046 T^{11} - 6255408 T^{12} - 599215958 T^{13} - 1176147025 T^{14} + 4469358695 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - T - 24 T^{2} + 31 T^{3} + 279 T^{4} - 708 T^{5} + 18640 T^{6} - 2096 T^{7} - 503040 T^{8} - 48208 T^{9} + 9860560 T^{10} - 8614236 T^{11} + 78075639 T^{12} + 199526633 T^{13} - 3552861336 T^{14} - 3404825447 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + 2 T - 43 T^{2} - 96 T^{3} - 115 T^{4} - 1316 T^{5} - 17729 T^{6} + 71592 T^{7} + 1922293 T^{8} + 2076168 T^{9} - 14910089 T^{10} - 32095924 T^{11} - 81337315 T^{12} - 1969070304 T^{13} - 25577402803 T^{14} + 34499752618 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 9 T + 21 T^{2} + 334 T^{3} - 3603 T^{4} + 19093 T^{5} - 11834 T^{6} - 590409 T^{7} + 5215882 T^{8} - 18302679 T^{9} - 11372474 T^{10} + 568799563 T^{11} - 3327446163 T^{12} + 9562136434 T^{13} + 18637577301 T^{14} - 247613526999 T^{15} + 852891037441 T^{16}$$
$37$ $$1 + 5 T - 23 T^{2} + 836 T^{3} + 4710 T^{4} - 16170 T^{5} + 319326 T^{6} + 1913289 T^{7} - 4942379 T^{8} + 70791693 T^{9} + 437157294 T^{10} - 819059010 T^{11} + 8827298310 T^{12} + 57971548052 T^{13} - 59011707407 T^{14} + 474659385665 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 11 T - 5 T^{2} - 66 T^{3} + 3342 T^{4} + 5808 T^{5} - 102972 T^{6} + 250283 T^{7} - 2878959 T^{8} + 10261603 T^{9} - 173095932 T^{10} + 400293168 T^{11} + 9443693262 T^{12} - 7646509266 T^{13} - 23750521205 T^{14} - 2142297012691 T^{15} + 7984925229121 T^{16}$$
$43$ $$( 1 - 3 T + 57 T^{2} + 133 T^{3} + 1551 T^{4} + 5719 T^{5} + 105393 T^{6} - 238521 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 11 T - 29 T^{2} + 66 T^{3} + 5076 T^{4} + 6666 T^{5} - 242622 T^{6} + 458909 T^{7} - 280527 T^{8} + 21568723 T^{9} - 535951998 T^{10} + 692084118 T^{11} + 24769260756 T^{12} + 15136770462 T^{13} - 312597244541 T^{14} - 5572854325093 T^{15} + 23811286661761 T^{16}$$
$53$ $$( 1 - 20 T + 333 T^{2} - 3393 T^{3} + 29770 T^{4} - 179829 T^{5} + 935397 T^{6} - 2977540 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 14 T + 201 T^{2} - 1563 T^{3} + 14548 T^{4} - 92217 T^{5} + 699681 T^{6} - 2875306 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$1 - 20 T + 241 T^{2} - 2946 T^{3} + 27920 T^{4} - 224900 T^{5} + 1845888 T^{6} - 13520291 T^{7} + 97623770 T^{8} - 824737751 T^{9} + 6868549248 T^{10} - 51048026900 T^{11} + 386575880720 T^{12} - 2488180702746 T^{13} + 12416410221001 T^{14} - 62854856720420 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 33 T + 545 T^{2} + 6105 T^{3} + 54459 T^{4} + 409035 T^{5} + 2446505 T^{6} + 9925179 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 11 T + 54 T^{2} + 1111 T^{3} - 19219 T^{4} + 170456 T^{5} - 99822 T^{6} - 11375958 T^{7} + 161395572 T^{8} - 807693018 T^{9} - 503202702 T^{10} + 61008077416 T^{11} - 488387097139 T^{12} + 2004498808961 T^{13} + 6917415331734 T^{14} - 100046321742301 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 + 7 T - 130 T^{2} - 2149 T^{3} + 4271 T^{4} + 206570 T^{5} + 1092495 T^{6} - 8070615 T^{7} - 120604568 T^{8} - 589154895 T^{9} + 5821905855 T^{10} + 80359241690 T^{11} + 121288887311 T^{12} - 4455030853357 T^{13} - 19673449417570 T^{14} + 77331789633679 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 12 T - 47 T^{2} - 486 T^{3} + 12854 T^{4} + 63174 T^{5} - 99654 T^{6} - 4520379 T^{7} - 25440400 T^{8} - 357109941 T^{9} - 621940614 T^{10} + 31147245786 T^{11} + 500664341174 T^{12} - 1495449409914 T^{13} - 11425110409487 T^{14} - 230446907833908 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 + 4 T - 293 T^{2} - 642 T^{3} + 54065 T^{4} + 66808 T^{5} - 6809565 T^{6} - 2000374 T^{7} + 658163011 T^{8} - 166031042 T^{9} - 46911093285 T^{10} + 38199945896 T^{11} + 2565834124865 T^{12} - 2528864092806 T^{13} - 95793529397117 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 8 T + 295 T^{2} + 1841 T^{3} + 36468 T^{4} + 163849 T^{5} + 2336695 T^{6} + 5639752 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 20 T - 53 T^{2} + 1854 T^{3} + 28472 T^{4} - 297470 T^{5} - 2904618 T^{6} + 7196245 T^{7} + 391603694 T^{8} + 698035765 T^{9} - 27329550762 T^{10} - 271492837310 T^{11} + 2520605688632 T^{12} + 15920928836478 T^{13} - 44147516261237 T^{14} - 1615965689562260 T^{15} + 7837433594376961 T^{16}$$