# Properties

 Label 441.7.d.c Level $441$ Weight $7$ Character orbit 441.d Analytic conductor $101.454$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 441.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$101.453850876$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{3} ) q^{2} + ( 42 - 4 \beta_{3} - \beta_{4} ) q^{4} + ( 25 \beta_{1} - 3 \beta_{2} + \beta_{5} ) q^{5} + ( -404 + 50 \beta_{3} + 5 \beta_{4} - 4 \beta_{7} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{3} ) q^{2} + ( 42 - 4 \beta_{3} - \beta_{4} ) q^{4} + ( 25 \beta_{1} - 3 \beta_{2} + \beta_{5} ) q^{5} + ( -404 + 50 \beta_{3} + 5 \beta_{4} - 4 \beta_{7} ) q^{8} + ( 290 \beta_{1} - 32 \beta_{2} + 9 \beta_{5} + 5 \beta_{6} ) q^{10} + ( -55 + 103 \beta_{3} - 11 \beta_{4} + \beta_{7} ) q^{11} + ( 264 \beta_{1} + 3 \beta_{2} + 19 \beta_{5} + 20 \beta_{6} ) q^{13} + ( 3020 - 646 \beta_{3} - 35 \beta_{4} + 20 \beta_{7} ) q^{16} + ( -442 \beta_{1} - 82 \beta_{2} + 6 \beta_{5} + 24 \beta_{6} ) q^{17} + ( -1550 \beta_{1} + 85 \beta_{2} + 37 \beta_{5} - 44 \beta_{6} ) q^{19} + ( 1440 \beta_{1} - 366 \beta_{2} + 27 \beta_{5} + 15 \beta_{6} ) q^{20} + ( 10798 + 308 \beta_{3} - 81 \beta_{4} - 44 \beta_{7} ) q^{22} + ( 1088 + 280 \beta_{3} + 16 \beta_{4} + 64 \beta_{7} ) q^{23} + ( 4030 - 1107 \beta_{3} + 103 \beta_{4} + 43 \beta_{7} ) q^{25} + ( -699 \beta_{1} - 1457 \beta_{2} + 131 \beta_{5} - 105 \beta_{6} ) q^{26} + ( 605 - 1247 \beta_{3} - 193 \beta_{4} - 145 \beta_{7} ) q^{29} + ( 7331 \beta_{1} + 530 \beta_{2} - 18 \beta_{5} - 44 \beta_{6} ) q^{31} + ( -45324 + 4098 \beta_{3} + 581 \beta_{4} + 116 \beta_{7} ) q^{32} + ( 8908 \beta_{1} - 464 \beta_{2} + 142 \beta_{5} - 74 \beta_{6} ) q^{34} + ( -16510 - 1255 \beta_{3} + 147 \beta_{4} - 113 \beta_{7} ) q^{37} + ( -7111 \beta_{1} + 2639 \beta_{2} + 93 \beta_{5} + 297 \beta_{6} ) q^{38} + ( 18340 \beta_{1} + 614 \beta_{2} - 33 \beta_{5} - 5 \beta_{6} ) q^{40} + ( 7782 \beta_{1} + 1052 \beta_{2} + 652 \beta_{5} + 432 \beta_{6} ) q^{41} + ( 6182 + 1901 \beta_{3} + 399 \beta_{4} - 221 \beta_{7} ) q^{43} + ( 24840 + 7614 \beta_{3} - 7 \beta_{4} - 388 \beta_{7} ) q^{44} + ( 28024 + 56 \beta_{3} + 408 \beta_{4} + 64 \beta_{7} ) q^{46} + ( -39992 \beta_{1} - 82 \beta_{2} - 1042 \beta_{5} + 96 \beta_{6} ) q^{47} + ( -119905 + 1687 \beta_{3} + 1477 \beta_{4} + 412 \beta_{7} ) q^{50} + ( 137418 \beta_{1} + 7402 \beta_{2} + 922 \beta_{5} + 1174 \beta_{6} ) q^{52} + ( -137125 - 6857 \beta_{3} + 353 \beta_{4} - 511 \beta_{7} ) q^{53} + ( 43985 \beta_{1} - 7433 \beta_{2} + 1251 \beta_{5} + 920 \beta_{6} ) q^{55} + ( -131828 + 14186 \beta_{3} - 155 \beta_{4} - 772 \beta_{7} ) q^{58} + ( -15871 \beta_{1} + 571 \beta_{2} + 2323 \beta_{5} + 756 \beta_{6} ) q^{59} + ( 38980 \beta_{1} - 15208 \beta_{2} - 1304 \beta_{5} + 720 \beta_{6} ) q^{61} + ( -62717 \beta_{1} - 6697 \beta_{2} - 682 \beta_{5} - 258 \beta_{6} ) q^{62} + ( 285124 - 49742 \beta_{3} - 1163 \beta_{4} + 1044 \beta_{7} ) q^{64} + ( -109080 - 24294 \beta_{3} + 2086 \beta_{4} + 46 \beta_{7} ) q^{65} + ( 291068 + 6357 \beta_{3} + 1103 \beta_{4} - 1045 \beta_{7} ) q^{67} + ( 68544 \beta_{1} - 1284 \beta_{2} + 858 \beta_{5} - 270 \beta_{6} ) q^{68} + ( -186172 - 22382 \beta_{3} - 2650 \beta_{4} + 758 \beta_{7} ) q^{71} + ( 35228 \beta_{1} - 14317 \beta_{2} - 373 \beta_{5} - 1924 \beta_{6} ) q^{73} + ( -113705 - 22921 \beta_{3} - 135 \beta_{4} + 588 \beta_{7} ) q^{74} + ( -172566 \beta_{1} - 20802 \beta_{2} - 4152 \beta_{5} - 1716 \beta_{6} ) q^{76} + ( -152983 - 7998 \beta_{3} + 2018 \beta_{4} + 2750 \beta_{7} ) q^{79} + ( -174940 \beta_{1} + 3990 \beta_{2} - 2545 \beta_{5} - 1605 \beta_{6} ) q^{80} + ( -120834 \beta_{1} - 40378 \beta_{2} + 3292 \beta_{5} - 2772 \beta_{6} ) q^{82} + ( 95781 \beta_{1} + 15973 \beta_{2} + 197 \beta_{5} + 1092 \beta_{6} ) q^{83} + ( -39210 - 21786 \beta_{3} + 3154 \beta_{4} - 326 \beta_{7} ) q^{85} + ( 197143 - 26113 \beta_{3} - 4731 \beta_{4} + 1596 \beta_{7} ) q^{86} + ( 85844 - 21950 \beta_{3} - 6691 \beta_{4} + 2788 \beta_{7} ) q^{88} + ( 81874 \beta_{1} + 3154 \beta_{2} - 918 \beta_{5} - 192 \beta_{6} ) q^{89} + ( -89712 - 13776 \beta_{3} - 784 \beta_{4} - 2464 \beta_{7} ) q^{92} + ( 48026 \beta_{1} + 52686 \beta_{2} - 6074 \beta_{5} - 2674 \beta_{6} ) q^{94} + ( -143310 + 27108 \beta_{3} - 5972 \beta_{4} - 1292 \beta_{7} ) q^{95} + ( -114285 \beta_{1} - 33555 \beta_{2} - 4027 \beta_{5} + 4 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8} + O(q^{10})$$ $$8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8} - 628 q^{11} + 25442 q^{16} + 86106 q^{22} + 7856 q^{23} + 34076 q^{25} + 8300 q^{29} - 372414 q^{32} - 129412 q^{37} + 45740 q^{43} + 185058 q^{44} + 223008 q^{46} - 967216 q^{50} - 1081948 q^{53} - 1079598 q^{58} + 2378626 q^{64} - 828408 q^{65} + 2317804 q^{67} - 1442344 q^{71} - 865880 q^{74} - 1222904 q^{79} - 275112 q^{85} + 1632448 q^{86} + 732882 q^{88} - 678720 q^{92} - 1183584 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$405518177 \nu^{7} + 12595814413 \nu^{6} + 65484675874 \nu^{5} + 2673339327091 \nu^{4} + 20860832470090 \nu^{3} + 529666993003657 \nu^{2} + 423382833624317 \nu + 6787296640242135$$$$)/ 5770013111021565$$ $$\beta_{2}$$ $$=$$ $$($$$$8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + 336428371378 \nu^{3} + 15479370553 \nu^{2} + 15909261436572 \nu - 5583985297290$$$$)/ 7542500798721$$ $$\beta_{3}$$ $$=$$ $$($$$$8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + 336428371378 \nu^{3} + 15479370553 \nu^{2} + 824259839130 \nu - 5583985297290$$$$)/ 7542500798721$$ $$\beta_{4}$$ $$=$$ $$($$$$-7295536 \nu^{7} - 50933099 \nu^{6} - 1392479248 \nu^{5} - 2649247696 \nu^{4} - 323422537422 \nu^{3} - 13289665936 \nu^{2} - 707660422560 \nu + 208034068231995$$$$)/ 2514166932907$$ $$\beta_{5}$$ $$=$$ $$($$$$65881610383 \nu^{7} - 648926656693 \nu^{6} + 19199653600241 \nu^{5} - 98352920361781 \nu^{4} + 3069708627551885 \nu^{3} - 15721266306367057 \nu^{2} + 162140643786549358 \nu - 277910537962726260$$$$)/ 7693350814695420$$ $$\beta_{6}$$ $$=$$ $$($$$$-135000035243 \nu^{7} - 904004937787 \nu^{6} - 30723933002041 \nu^{5} - 280397778215839 \nu^{4} - 6373179964075405 \nu^{3} - 43067900538846163 \nu^{2} - 253385183003352218 \nu - 478310744905115040$$$$)/ 7693350814695420$$ $$\beta_{7}$$ $$=$$ $$($$$$1392321691 \nu^{7} - 2568340439 \nu^{6} + 265748679913 \nu^{5} + 505597536901 \nu^{4} + 47373358739629 \nu^{3} + 2536275627841 \nu^{2} + 135053950277610 \nu - 1324669278373776$$$$)/ 30170003194884$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 105 \beta_{1} - 105$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{7} + 2 \beta_{4} + 169 \beta_{3} - 216$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{7} - 217 \beta_{6} - 205 \beta_{5} + 213 \beta_{4} + 722 \beta_{3} - 722 \beta_{2} - 17781 \beta_{1} - 17781$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$848 \beta_{7} - 1614 \beta_{6} + 930 \beta_{5} - 766 \beta_{4} - 32053 \beta_{3} - 32053 \beta_{2} - 77112 \beta_{1} + 77112$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$2216 \beta_{7} - 41381 \beta_{4} - 197354 \beta_{3} + 3375249$$ $$\nu^{7}$$ $$=$$ $$($$$$163308 \beta_{7} + 385038 \beta_{6} - 104886 \beta_{5} - 221730 \beta_{4} - 6279409 \beta_{3} + 6279409 \beta_{2} + 20983752 \beta_{1} + 20983752$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 7.29767 + 12.6399i 7.29767 − 12.6399i 2.26350 − 3.92050i 2.26350 + 3.92050i −2.75320 − 4.76869i −2.75320 + 4.76869i −6.30797 + 10.9257i −6.30797 − 10.9257i
−15.5953 0 179.215 25.8331i 0 0 −1796.81 0 402.876i
244.2 −15.5953 0 179.215 25.8331i 0 0 −1796.81 0 402.876i
244.3 −5.52700 0 −33.4522 66.9661i 0 0 538.619 0 370.122i
244.4 −5.52700 0 −33.4522 66.9661i 0 0 538.619 0 370.122i
244.5 4.50641 0 −43.6923 62.2665i 0 0 −485.305 0 280.598i
244.6 4.50641 0 −43.6923 62.2665i 0 0 −485.305 0 280.598i
244.7 11.6159 0 70.9299 190.875i 0 0 80.4975 0 2217.19i
244.8 11.6159 0 70.9299 190.875i 0 0 80.4975 0 2217.19i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 244.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 441.7.d.c 8
3.b odd 2 1 147.7.d.b 8
7.b odd 2 1 inner 441.7.d.c 8
7.c even 3 1 63.7.m.d 8
7.d odd 6 1 63.7.m.d 8
21.c even 2 1 147.7.d.b 8
21.g even 6 1 21.7.f.a 8
21.g even 6 1 147.7.f.d 8
21.h odd 6 1 21.7.f.a 8
21.h odd 6 1 147.7.f.d 8
84.j odd 6 1 336.7.bh.d 8
84.n even 6 1 336.7.bh.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a 8 21.g even 6 1
21.7.f.a 8 21.h odd 6 1
63.7.m.d 8 7.c even 3 1
63.7.m.d 8 7.d odd 6 1
147.7.d.b 8 3.b odd 2 1
147.7.d.b 8 21.c even 2 1
147.7.f.d 8 21.g even 6 1
147.7.f.d 8 21.h odd 6 1
336.7.bh.d 8 84.j odd 6 1
336.7.bh.d 8 84.n even 6 1
441.7.d.c 8 1.a even 1 1 trivial
441.7.d.c 8 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 5 T_{2}^{3} - 202 T_{2}^{2} - 284 T_{2} + 4512$$ acting on $$S_{7}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4512 - 284 T - 202 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$422734160250000 + 848355795000 T^{2} + 351918225 T^{4} + 45462 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( -43002975612 - 2285742356 T - 3746347 T^{2} + 314 T^{3} + T^{4} )^{2}$$
$13$ $$21\!\cdots\!56$$$$+$$$$14\!\cdots\!08$$$$T^{2} + 331924469476329 T^{4} + 30553062 T^{6} + T^{8}$$
$17$ $$14\!\cdots\!04$$$$+ 45391770927437501952 T^{2} + 313740876622608 T^{4} + 46058328 T^{6} + T^{8}$$
$19$ $$40\!\cdots\!56$$$$+$$$$46\!\cdots\!68$$$$T^{2} + 16571381197978233 T^{4} + 221900526 T^{6} + T^{8}$$
$23$ $$( 3970225373184 - 88355144192 T - 145891648 T^{2} - 3928 T^{3} + T^{4} )^{2}$$
$29$ $$( -131918946476762880 + 26271861557728 T - 1346898667 T^{2} - 4150 T^{3} + T^{4} )^{2}$$
$31$ $$16\!\cdots\!01$$$$+$$$$23\!\cdots\!04$$$$T^{2} + 354646576456751070 T^{4} + 1135098420 T^{6} + T^{8}$$
$37$ $$( -483334842751696444 - 34229202109340 T + 453289125 T^{2} + 64706 T^{3} + T^{4} )^{2}$$
$41$ $$36\!\cdots\!44$$$$+$$$$43\!\cdots\!00$$$$T^{2} +$$$$15\!\cdots\!32$$$$T^{4} + 21593029296 T^{6} + T^{8}$$
$43$ $$( -293597400770976796 + 85125133057276 T - 4841210811 T^{2} - 22870 T^{3} + T^{4} )^{2}$$
$47$ $$99\!\cdots\!96$$$$+$$$$50\!\cdots\!80$$$$T^{2} +$$$$84\!\cdots\!48$$$$T^{4} + 50915681112 T^{6} + T^{8}$$
$53$ $$( -85899317769490075392 + 3583685222151280 T + 88674528125 T^{2} + 540974 T^{3} + T^{4} )^{2}$$
$59$ $$20\!\cdots\!00$$$$+$$$$67\!\cdots\!12$$$$T^{2} +$$$$61\!\cdots\!89$$$$T^{4} + 158965820454 T^{6} + T^{8}$$
$61$ $$68\!\cdots\!00$$$$+$$$$32\!\cdots\!00$$$$T^{2} +$$$$55\!\cdots\!00$$$$T^{4} + 398165036352 T^{6} + T^{8}$$
$67$ $$($$$$17\!\cdots\!52$$$$- 58128338394558464 T + 435008587497 T^{2} - 1158902 T^{3} + T^{4} )^{2}$$
$71$ $$( -$$$$19\!\cdots\!12$$$$- 115286062429846880 T - 36558200716 T^{2} + 721172 T^{3} + T^{4} )^{2}$$
$73$ $$77\!\cdots\!24$$$$+$$$$42\!\cdots\!48$$$$T^{2} +$$$$76\!\cdots\!01$$$$T^{4} + 514015226550 T^{6} + T^{8}$$
$79$ $$( -$$$$99\!\cdots\!51$$$$- 96707301792932732 T - 147338397318 T^{2} + 611452 T^{3} + T^{4} )^{2}$$
$83$ $$52\!\cdots\!24$$$$+$$$$31\!\cdots\!16$$$$T^{2} +$$$$85\!\cdots\!29$$$$T^{4} + 524747194014 T^{6} + T^{8}$$
$89$ $$87\!\cdots\!56$$$$+$$$$52\!\cdots\!48$$$$T^{2} +$$$$43\!\cdots\!28$$$$T^{4} + 116188680600 T^{6} + T^{8}$$
$97$ $$90\!\cdots\!76$$$$+$$$$26\!\cdots\!08$$$$T^{2} +$$$$12\!\cdots\!73$$$$T^{4} + 2056709392566 T^{6} + T^{8}$$