# Properties

 Label 441.6.a.bb Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2 x^{5} - 59 x^{4} + 122 x^{3} + 941 x^{2} - 1856 x - 2338$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 7^{3}$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 25 - \beta_{1} + \beta_{4} ) q^{4} + ( 16 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( 28 - 9 \beta_{1} + 25 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 25 - \beta_{1} + \beta_{4} ) q^{4} + ( 16 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( 28 - 9 \beta_{1} + 25 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{8} + ( -138 + 19 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} - 3 \beta_{5} ) q^{10} + ( -107 - 6 \beta_{1} - 12 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} ) q^{11} + ( -215 - 12 \beta_{1} + 34 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} ) q^{13} + ( 801 - 27 \beta_{1} + 126 \beta_{2} + 18 \beta_{3} + 33 \beta_{4} + 12 \beta_{5} ) q^{16} + ( 521 - 31 \beta_{1} + 32 \beta_{2} - 7 \beta_{3} - 8 \beta_{4} - 17 \beta_{5} ) q^{17} + ( -231 + 55 \beta_{1} + 174 \beta_{2} - 12 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} ) q^{19} + ( -41 + 100 \beta_{1} - 335 \beta_{2} - 5 \beta_{3} - 7 \beta_{4} + 14 \beta_{5} ) q^{20} + ( -664 - 238 \beta_{1} + 30 \beta_{2} + 18 \beta_{3} - 32 \beta_{4} - 36 \beta_{5} ) q^{22} + ( 759 - 16 \beta_{2} - 45 \beta_{3} + 24 \beta_{4} - 51 \beta_{5} ) q^{23} + ( 883 + 290 \beta_{1} + 228 \beta_{2} - 36 \beta_{3} + 16 \beta_{4} - 18 \beta_{5} ) q^{25} + ( 2206 + 45 \beta_{1} - 443 \beta_{2} - 9 \beta_{3} + 7 \beta_{4} + 25 \beta_{5} ) q^{26} + ( 837 + 201 \beta_{1} - 86 \beta_{2} + 54 \beta_{3} + 24 \beta_{4} + 63 \beta_{5} ) q^{29} + ( -673 - 215 \beta_{1} - 94 \beta_{2} - 88 \beta_{3} + 56 \beta_{4} - 63 \beta_{5} ) q^{31} + ( 6490 - 687 \beta_{1} + 781 \beta_{2} + 3 \beta_{3} + 128 \beta_{4} - 26 \beta_{5} ) q^{32} + ( 2908 + 501 \beta_{1} + 655 \beta_{2} + 25 \beta_{3} + 139 \beta_{4} + 99 \beta_{5} ) q^{34} + ( 3758 - 300 \beta_{1} - 24 \beta_{2} - 54 \beta_{3} + 96 \beta_{4} + 42 \beta_{5} ) q^{37} + ( 8946 - 292 \beta_{1} + 318 \beta_{2} - 22 \beta_{3} + 62 \beta_{4} - 148 \beta_{5} ) q^{38} + ( -17155 - 446 \beta_{1} - 1195 \beta_{2} - 37 \beta_{3} - 277 \beta_{4} - 30 \beta_{5} ) q^{40} + ( 4961 + 23 \beta_{1} + 656 \beta_{2} + 65 \beta_{3} + 136 \beta_{4} + 151 \beta_{5} ) q^{41} + ( -1218 - 208 \beta_{1} - 120 \beta_{2} - 90 \beta_{3} - 32 \beta_{4} + 150 \beta_{5} ) q^{43} + ( 10718 + 990 \beta_{1} - 80 \beta_{2} + 132 \beta_{3} + 322 \beta_{4} + 236 \beta_{5} ) q^{44} + ( 876 + 1498 \beta_{1} + 2394 \beta_{2} + 54 \beta_{3} + 140 \beta_{4} + 36 \beta_{5} ) q^{46} + ( 8645 - 381 \beta_{1} + 206 \beta_{2} - 72 \beta_{3} - 184 \beta_{4} + 47 \beta_{5} ) q^{47} + ( 7180 + 312 \beta_{1} + 909 \beta_{2} - 276 \beta_{3} - 100 \beta_{4} - 320 \beta_{5} ) q^{50} + ( -19507 + 458 \beta_{1} + 881 \beta_{2} - \beta_{3} - 445 \beta_{4} - 246 \beta_{5} ) q^{52} + ( -13832 - 1614 \beta_{1} - 996 \beta_{2} - 132 \beta_{3} - 208 \beta_{4} + 58 \beta_{5} ) q^{53} + ( -2977 - 147 \beta_{1} - 2686 \beta_{2} + 536 \beta_{3} + 248 \beta_{4} + 417 \beta_{5} ) q^{55} + ( -11682 - 2452 \beta_{1} - 222 \beta_{2} - 162 \beta_{3} - 434 \beta_{4} - 408 \beta_{5} ) q^{58} + ( 2115 + 941 \beta_{1} + 1794 \beta_{2} - 100 \beta_{3} - 280 \beta_{4} - 115 \beta_{5} ) q^{59} + ( -8803 + 74 \beta_{1} - 1186 \beta_{2} - 328 \beta_{3} + 200 \beta_{4} - 225 \beta_{5} ) q^{61} + ( 2894 + 2492 \beta_{1} + 2918 \beta_{2} + 302 \beta_{3} + 178 \beta_{4} + 148 \beta_{5} ) q^{62} + ( 36125 - 125 \beta_{1} + 8580 \beta_{2} + 396 \beta_{3} + 737 \beta_{4} ) q^{64} + ( 467 + 3 \beta_{1} + 2942 \beta_{2} + 132 \beta_{3} - 392 \beta_{4} - 205 \beta_{5} ) q^{65} + ( 1794 + 148 \beta_{1} + 2496 \beta_{2} + 414 \beta_{3} - 304 \beta_{4} + 354 \beta_{5} ) q^{67} + ( 7407 - 3712 \beta_{1} + 2901 \beta_{2} - 73 \beta_{3} - 19 \beta_{4} - 646 \beta_{5} ) q^{68} + ( 19359 + 2364 \beta_{1} + 3200 \beta_{2} + 243 \beta_{3} + 120 \beta_{4} + 549 \beta_{5} ) q^{71} + ( -12854 - 787 \beta_{1} - 6584 \beta_{2} - 266 \beta_{3} - 32 \beta_{4} - 138 \beta_{5} ) q^{73} + ( 7008 - 324 \beta_{1} + 7124 \beta_{2} + 396 \beta_{3} - 192 \beta_{4} - 168 \beta_{5} ) q^{74} + ( 35078 + 528 \beta_{1} + 8234 \beta_{2} + 926 \beta_{3} + 830 \beta_{4} + 432 \beta_{5} ) q^{76} + ( 19592 + 3992 \beta_{1} + 3240 \beta_{2} + 468 \beta_{3} - 272 \beta_{4} + 312 \beta_{5} ) q^{79} + ( -55959 + 2424 \beta_{1} - 12525 \beta_{2} + 45 \beta_{3} - 777 \beta_{4} + 882 \beta_{5} ) q^{80} + ( 34012 - 5841 \beta_{1} + 6397 \beta_{2} + 163 \beta_{3} + 37 \beta_{4} - 819 \beta_{5} ) q^{82} + ( 17178 - 622 \beta_{1} + 828 \beta_{2} + 764 \beta_{3} - 16 \beta_{4} - 142 \beta_{5} ) q^{83} + ( -3110 + 514 \beta_{1} + 1428 \beta_{2} - 1134 \beta_{3} - 976 \beta_{4} - 888 \beta_{5} ) q^{85} + ( -1616 - 192 \beta_{1} - 3172 \beta_{2} - 96 \beta_{3} - 1444 \beta_{4} - 236 \beta_{5} ) q^{86} + ( -11784 - 3034 \beta_{1} + 12162 \beta_{2} - 1026 \beta_{3} - 716 \beta_{4} - 1380 \beta_{5} ) q^{88} + ( -7483 - 6481 \beta_{1} - 4588 \beta_{2} - 169 \beta_{3} + 1000 \beta_{4} + 355 \beta_{5} ) q^{89} + ( 75050 - 6930 \beta_{1} + 180 \beta_{2} + 240 \beta_{3} + 286 \beta_{4} - 340 \beta_{5} ) q^{92} + ( 21314 + 2584 \beta_{1} + 4154 \beta_{2} - 106 \beta_{3} - 286 \beta_{4} + 720 \beta_{5} ) q^{94} + ( 20578 + 9324 \beta_{1} + 1856 \beta_{2} - 366 \beta_{3} - 1264 \beta_{4} - 338 \beta_{5} ) q^{95} + ( -9896 + 6723 \beta_{1} + 6220 \beta_{2} + 22 \beta_{3} - 32 \beta_{4} + 432 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} + O(q^{10})$$ $$6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 59 x^{4} + 122 x^{3} + 941 x^{2} - 1856 x - 2338$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$28 \nu^{5} - 35 \nu^{4} - 546 \nu^{3} + 742 \nu^{2} - 7063 \nu + 11802$$$$)/1941$$ $$\beta_{2}$$ $$=$$ $$($$$$-87 \nu^{5} - 53 \nu^{4} + 3961 \nu^{3} + 2547 \nu^{2} - 42269 \nu - 27289$$$$)/1941$$ $$\beta_{3}$$ $$=$$ $$($$$$158 \nu^{5} - 521 \nu^{4} - 6316 \nu^{3} + 21656 \nu^{2} + 33579 \nu - 143678$$$$)/1941$$ $$\beta_{4}$$ $$=$$ $$($$$$108 \nu^{5} + 512 \nu^{4} - 5988 \nu^{3} - 17195 \nu^{2} + 81453 \nu + 71402$$$$)/647$$ $$\beta_{5}$$ $$=$$ $$($$$$-774 \nu^{5} - 1297 \nu^{4} + 39032 \nu^{3} + 55188 \nu^{2} - 447553 \nu - 433643$$$$)/1941$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{3} - 12 \beta_{2} - 4 \beta_{1} + 5$$$$)/28$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 567$$$$)/28$$ $$\nu^{3}$$ $$=$$ $$($$$$29 \beta_{5} + 4 \beta_{4} - 28 \beta_{3} - 302 \beta_{2} - 25 \beta_{1} - 129$$$$)/28$$ $$\nu^{4}$$ $$=$$ $$($$$$99 \beta_{5} + 176 \beta_{4} + 155 \beta_{3} + 80 \beta_{2} + 74 \beta_{1} + 14843$$$$)/28$$ $$\nu^{5}$$ $$=$$ $$($$$$862 \beta_{5} + 192 \beta_{4} - 737 \beta_{3} - 8710 \beta_{2} + 643 \beta_{1} - 9528$$$$)/28$$

## 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
 5.59680 3.75353 −5.10089 4.27213 −0.910122 −5.61145
−10.0089 0 68.1779 70.3512 0 0 −362.101 0 −704.138
1.2 −8.20863 0 35.3816 29.2259 0 0 −27.7583 0 −239.905
1.3 −3.38033 0 −20.5734 −54.5253 0 0 177.715 0 184.313
1.4 3.09163 0 −22.4418 13.7926 0 0 −168.314 0 42.6416
1.5 5.31815 0 −3.71724 103.471 0 0 −189.950 0 550.272
1.6 11.1881 0 93.1729 −62.3150 0 0 684.407 0 −697.185
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$

## 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 441.6.a.bb 6
3.b odd 2 1 147.6.a.n 6
7.b odd 2 1 441.6.a.ba 6
21.c even 2 1 147.6.a.o yes 6
21.g even 6 2 147.6.e.p 12
21.h odd 6 2 147.6.e.q 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.n 6 3.b odd 2 1
147.6.a.o yes 6 21.c even 2 1
147.6.e.p 12 21.g even 6 2
147.6.e.q 12 21.h odd 6 2
441.6.a.ba 6 7.b odd 2 1
441.6.a.bb 6 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{6} + 2 T_{2}^{5} - 169 T_{2}^{4} - 336 T_{2}^{3} + 6472 T_{2}^{2} + 4256 T_{2} - 51088$$ $$T_{5}^{6} - 100 T_{5}^{5} - 6778 T_{5}^{4} + 651312 T_{5}^{3} + 9669292 T_{5}^{2} - 959211664 T_{5} + 9969962312$$ $$T_{13}^{6} + 1352 T_{13}^{5} - 126598 T_{13}^{4} - 808285728 T_{13}^{3} - 312398022260 T_{13}^{2} -$$$$55\!\cdots\!04$$$$T_{13} +$$$$88\!\cdots\!04$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-51088 + 4256 T + 6472 T^{2} - 336 T^{3} - 169 T^{4} + 2 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$9969962312 - 959211664 T + 9669292 T^{2} + 651312 T^{3} - 6778 T^{4} - 100 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$-55273527989696 + 15123294433216 T - 11867400976 T^{2} - 359951456 T^{3} - 500836 T^{4} + 604 T^{5} + T^{6}$$
$13$ $$8818638530643704 - 5552830868704 T - 312398022260 T^{2} - 808285728 T^{3} - 126598 T^{4} + 1352 T^{5} + T^{6}$$
$17$ $$-77634357357142712 + 1817914266048944 T - 5595796869460 T^{2} + 4922692976 T^{3} + 960902 T^{4} - 3028 T^{5} + T^{6}$$
$19$ $$-3735293167739744768 + 16882924839579648 T + 16923734261952 T^{2} - 11346877440 T^{3} - 7968600 T^{4} + 1728 T^{5} + T^{6}$$
$23$ $$-10876910613896947648 - 251128931562936896 T + 49469900108272 T^{2} + 90302677920 T^{3} - 19751812 T^{4} - 4484 T^{5} + T^{6}$$
$29$ $$-$$$$45\!\cdots\!84$$$$- 413077597670801920 T + 394389617709760 T^{2} + 130079620992 T^{3} - 41146408 T^{4} - 5320 T^{5} + T^{6}$$
$31$ $$-$$$$32\!\cdots\!04$$$$- 763919493148671488 T + 1627711614738112 T^{2} - 140587032960 T^{3} - 88488040 T^{4} + 3976 T^{5} + T^{6}$$
$37$ $$-$$$$79\!\cdots\!96$$$$+ 64747232745500190720 T - 17742287505662208 T^{2} + 1603803782912 T^{3} + 87265776 T^{4} - 22680 T^{5} + T^{6}$$
$41$ $$18\!\cdots\!56$$$$- 82163383248259387984 T - 15548374571673172 T^{2} + 3631066472048 T^{3} + 55056710 T^{4} - 28756 T^{5} + T^{6}$$
$43$ $$-$$$$28\!\cdots\!68$$$$+ 59538852421323669504 T + 22779206566284288 T^{2} - 2231851972608 T^{3} - 387379104 T^{4} + 6768 T^{5} + T^{6}$$
$47$ $$13\!\cdots\!28$$$$+ 16481230197039654912 T - 11579177448956736 T^{2} - 2130988958208 T^{3} + 735770472 T^{4} - 51552 T^{5} + T^{6}$$
$53$ $$93\!\cdots\!88$$$$-$$$$84\!\cdots\!28$$$$T - 942681098319975184 T^{2} - 27823564813472 T^{3} + 1498706108 T^{4} + 80884 T^{5} + T^{6}$$
$59$ $$-$$$$34\!\cdots\!52$$$$-$$$$45\!\cdots\!32$$$$T + 178489629727189696 T^{2} + 2643841282944 T^{3} - 1106900008 T^{4} - 8872 T^{5} + T^{6}$$
$61$ $$-$$$$18\!\cdots\!36$$$$+$$$$38\!\cdots\!40$$$$T + 42902272625339660 T^{2} - 29293770273344 T^{3} - 222122566 T^{4} + 50896 T^{5} + T^{6}$$
$67$ $$-$$$$24\!\cdots\!32$$$$-$$$$48\!\cdots\!56$$$$T + 1905614748297366528 T^{2} + 15790719633408 T^{3} - 3000969888 T^{4} - 6480 T^{5} + T^{6}$$
$71$ $$-$$$$29\!\cdots\!48$$$$+$$$$40\!\cdots\!88$$$$T - 19614262036831368464 T^{2} + 365269119773856 T^{3} + 858142556 T^{4} - 110852 T^{5} + T^{6}$$
$73$ $$32\!\cdots\!84$$$$+$$$$31\!\cdots\!12$$$$T + 3738292215205062988 T^{2} - 296839059037344 T^{3} - 5386018870 T^{4} + 64232 T^{5} + T^{6}$$
$79$ $$-$$$$10\!\cdots\!68$$$$+$$$$25\!\cdots\!16$$$$T - 18727181437041497088 T^{2} + 507917557775360 T^{3} - 1585863072 T^{4} - 111696 T^{5} + T^{6}$$
$83$ $$-$$$$19\!\cdots\!68$$$$-$$$$43\!\cdots\!52$$$$T + 2636280972634775296 T^{2} + 259543622746368 T^{3} - 2157797008 T^{4} - 101128 T^{5} + T^{6}$$
$89$ $$48\!\cdots\!72$$$$-$$$$41\!\cdots\!96$$$$T + 91546162688810429612 T^{2} + 151709255824976 T^{3} - 18350566330 T^{4} + 35012 T^{5} + T^{6}$$
$97$ $$-$$$$13\!\cdots\!88$$$$+$$$$30\!\cdots\!96$$$$T + 8495591547037465420 T^{2} - 119558814297504 T^{3} - 13346679478 T^{4} + 70952 T^{5} + T^{6}$$