# Properties

 Label 147.7.f.a Level $147$ Weight $7$ Character orbit 147.f Analytic conductor $33.818$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 147.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.8179502921$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( -9 - 9 \beta_{2} ) q^{3} + ( -43 + 43 \beta_{2} - \beta_{5} + \beta_{6} ) q^{4} + ( 6 - 3 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{5} + ( -9 + 18 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} ) q^{6} + ( -61 + 2 \beta_{1} + 14 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 4 \beta_{7} ) q^{8} + 243 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( -9 - 9 \beta_{2} ) q^{3} + ( -43 + 43 \beta_{2} - \beta_{5} + \beta_{6} ) q^{4} + ( 6 - 3 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{5} + ( -9 + 18 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} ) q^{6} + ( -61 + 2 \beta_{1} + 14 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 4 \beta_{7} ) q^{8} + 243 \beta_{2} q^{9} + ( -9 - 48 \beta_{1} - 9 \beta_{2} - 34 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} - 14 \beta_{7} ) q^{10} + ( -265 - \beta_{1} + 265 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} - \beta_{7} ) q^{11} + ( 774 - 387 \beta_{2} + 18 \beta_{5} - 9 \beta_{6} ) q^{12} + ( -864 + 11 \beta_{1} + 1728 \beta_{2} - 14 \beta_{3} + 17 \beta_{4} - 5 \beta_{5} + 10 \beta_{6} ) q^{13} + ( -81 - 9 \beta_{1} - 9 \beta_{4} - 27 \beta_{5} - 18 \beta_{7} ) q^{15} + ( 184 \beta_{1} + 933 \beta_{2} + 10 \beta_{4} + 5 \beta_{6} - 10 \beta_{7} ) q^{16} + ( -626 + 124 \beta_{1} - 626 \beta_{2} + 70 \beta_{3} - 24 \beta_{5} - 24 \beta_{6} + 54 \beta_{7} ) q^{17} + ( 243 - 243 \beta_{1} - 243 \beta_{2} + 243 \beta_{3} ) q^{18} + ( 3908 - 360 \beta_{1} - 1954 \beta_{2} + 720 \beta_{3} + 27 \beta_{4} - 2 \beta_{5} + \beta_{6} + 27 \beta_{7} ) q^{19} + ( -3945 + 114 \beta_{1} + 7890 \beta_{2} - 24 \beta_{3} - 66 \beta_{4} - 45 \beta_{5} + 90 \beta_{6} ) q^{20} + ( -169 + 26 \beta_{1} + 570 \beta_{3} + 26 \beta_{4} - 11 \beta_{5} + 52 \beta_{7} ) q^{22} + ( 1302 \beta_{1} - 4144 \beta_{2} + 4 \beta_{4} + 54 \beta_{6} - 4 \beta_{7} ) q^{23} + ( 549 - 180 \beta_{1} + 549 \beta_{2} - 126 \beta_{3} + 27 \beta_{5} + 27 \beta_{6} - 54 \beta_{7} ) q^{24} + ( 5198 - 1399 \beta_{1} - 5198 \beta_{2} + 1398 \beta_{3} + 2 \beta_{4} + 81 \beta_{5} - 81 \beta_{6} + \beta_{7} ) q^{25} + ( 2732 - 1093 \beta_{1} - 1366 \beta_{2} + 2186 \beta_{3} - 106 \beta_{4} - 206 \beta_{5} + 103 \beta_{6} - 106 \beta_{7} ) q^{26} + ( 2187 - 4374 \beta_{2} ) q^{27} + ( 3949 - 61 \beta_{1} + 466 \beta_{3} - 61 \beta_{4} + 183 \beta_{5} - 122 \beta_{7} ) q^{29} + ( 1170 \beta_{1} + 243 \beta_{2} - 126 \beta_{4} + 81 \beta_{6} + 126 \beta_{7} ) q^{30} + ( -3401 - 78 \beta_{1} - 3401 \beta_{2} - 122 \beta_{3} + 282 \beta_{5} + 282 \beta_{6} + 44 \beta_{7} ) q^{31} + ( 17073 - 1978 \beta_{1} - 17073 \beta_{2} + 1920 \beta_{3} + 116 \beta_{4} - 183 \beta_{5} + 183 \beta_{6} + 58 \beta_{7} ) q^{32} + ( 4770 + 18 \beta_{1} - 2385 \beta_{2} - 36 \beta_{3} + 27 \beta_{4} + 162 \beta_{5} - 81 \beta_{6} + 27 \beta_{7} ) q^{33} + ( 10238 + 3216 \beta_{1} - 20476 \beta_{2} - 1752 \beta_{3} + 288 \beta_{4} + 466 \beta_{5} - 932 \beta_{6} ) q^{34} + ( -10449 - 243 \beta_{5} ) q^{36} + ( 2756 \beta_{1} - 11970 \beta_{2} + 71 \beta_{4} - 475 \beta_{6} - 71 \beta_{7} ) q^{37} + ( -41396 - 1661 \beta_{1} - 41396 \beta_{2} - 1451 \beta_{3} - 525 \beta_{5} - 525 \beta_{6} - 210 \beta_{7} ) q^{38} + ( 23328 - 225 \beta_{1} - 23328 \beta_{2} + 378 \beta_{3} - 306 \beta_{4} + 135 \beta_{5} - 135 \beta_{6} - 153 \beta_{7} ) q^{39} + ( 14298 - 3056 \beta_{1} - 7149 \beta_{2} + 6112 \beta_{3} - 98 \beta_{4} + 186 \beta_{5} - 93 \beta_{6} - 98 \beta_{7} ) q^{40} + ( -6362 - 424 \beta_{1} + 12724 \beta_{2} + 340 \beta_{3} - 256 \beta_{4} - 436 \beta_{5} + 872 \beta_{6} ) q^{41} + ( -57678 + 257 \beta_{1} + 6060 \beta_{3} + 257 \beta_{4} + 1051 \beta_{5} + 514 \beta_{7} ) q^{43} + ( 1064 \beta_{1} - 47319 \beta_{2} + 122 \beta_{4} - 495 \beta_{6} - 122 \beta_{7} ) q^{44} + ( 729 + 243 \beta_{1} + 729 \beta_{2} + 243 \beta_{5} + 243 \beta_{6} + 243 \beta_{7} ) q^{45} + ( 132004 + 3440 \beta_{1} - 132004 \beta_{2} - 3516 \beta_{3} + 152 \beta_{4} + 1076 \beta_{5} - 1076 \beta_{6} + 76 \beta_{7} ) q^{46} + ( -13376 + 1702 \beta_{1} + 6688 \beta_{2} - 3404 \beta_{3} + 770 \beta_{4} + 112 \beta_{5} - 56 \beta_{6} + 770 \beta_{7} ) q^{47} + ( 8397 - 3402 \beta_{1} - 16794 \beta_{2} + 1836 \beta_{3} - 270 \beta_{4} + 45 \beta_{5} - 90 \beta_{6} ) q^{48} + ( -150284 - 170 \beta_{1} - 6623 \beta_{3} - 170 \beta_{4} - 1173 \beta_{5} - 340 \beta_{7} ) q^{50} + ( -2862 \beta_{1} + 16902 \beta_{2} + 486 \beta_{4} + 648 \beta_{6} - 486 \beta_{7} ) q^{51} + ( -61172 + 4182 \beta_{1} - 61172 \beta_{2} + 3804 \beta_{3} - 446 \beta_{5} - 446 \beta_{6} + 378 \beta_{7} ) q^{52} + ( -3055 - 1947 \beta_{1} + 3055 \beta_{2} + 2096 \beta_{3} - 298 \beta_{4} - 2463 \beta_{5} + 2463 \beta_{6} - 149 \beta_{7} ) q^{53} + ( -4374 + 2187 \beta_{1} + 2187 \beta_{2} - 4374 \beta_{3} ) q^{54} + ( -44475 + 3319 \beta_{1} + 88950 \beta_{2} - 1474 \beta_{3} - 371 \beta_{4} - 345 \beta_{5} + 690 \beta_{6} ) q^{55} + ( -52758 - 243 \beta_{1} - 9720 \beta_{3} - 243 \beta_{4} + 27 \beta_{5} - 486 \beta_{7} ) q^{57} + ( -10376 \beta_{1} - 37607 \beta_{2} - 122 \beta_{4} + 1181 \beta_{6} + 122 \beta_{7} ) q^{58} + ( -151147 + 757 \beta_{1} - 151147 \beta_{2} + 1460 \beta_{3} + 377 \beta_{5} + 377 \beta_{6} - 703 \beta_{7} ) q^{59} + ( 106515 - 1242 \beta_{1} - 106515 \beta_{2} + 648 \beta_{3} + 1188 \beta_{4} + 1215 \beta_{5} - 1215 \beta_{6} + 594 \beta_{7} ) q^{60} + ( 67056 + 2144 \beta_{1} - 33528 \beta_{2} - 4288 \beta_{3} - 964 \beta_{4} - 104 \beta_{5} + 52 \beta_{6} - 964 \beta_{7} ) q^{61} + ( -25589 - 14874 \beta_{1} + 51178 \beta_{2} + 6415 \beta_{3} + 2044 \beta_{4} - 704 \beta_{5} + 1408 \beta_{6} ) q^{62} + ( -176205 - 738 \beta_{1} + 5004 \beta_{3} - 738 \beta_{4} - 3193 \beta_{5} - 1476 \beta_{7} ) q^{64} + ( -17582 \beta_{1} - 93648 \beta_{2} - 1586 \beta_{4} + 1584 \beta_{6} + 1586 \beta_{7} ) q^{65} + ( 1521 - 5832 \beta_{1} + 1521 \beta_{2} - 5130 \beta_{3} + 99 \beta_{5} + 99 \beta_{6} - 702 \beta_{7} ) q^{66} + ( 101116 + 18853 \beta_{1} - 101116 \beta_{2} - 19020 \beta_{3} + 334 \beta_{4} - 1851 \beta_{5} + 1851 \beta_{6} + 167 \beta_{7} ) q^{67} + ( 281604 + 22668 \beta_{1} - 140802 \beta_{2} - 45336 \beta_{3} - 1644 \beta_{4} - 228 \beta_{5} + 114 \beta_{6} - 1644 \beta_{7} ) q^{68} + ( -37296 - 23472 \beta_{1} + 74592 \beta_{2} + 11790 \beta_{3} - 108 \beta_{4} + 486 \beta_{5} - 972 \beta_{6} ) q^{69} + ( 28372 + 1706 \beta_{1} - 7094 \beta_{3} + 1706 \beta_{4} + 1332 \beta_{5} + 3412 \beta_{7} ) q^{71} + ( 4374 \beta_{1} - 14823 \beta_{2} - 486 \beta_{4} - 729 \beta_{6} + 486 \beta_{7} ) q^{72} + ( -77984 + 10023 \beta_{1} - 77984 \beta_{2} + 10218 \beta_{3} + 887 \beta_{5} + 887 \beta_{6} - 195 \beta_{7} ) q^{73} + ( 304420 + 32159 \beta_{1} - 304420 \beta_{2} - 30641 \beta_{3} - 3036 \beta_{4} + 3045 \beta_{5} - 3045 \beta_{6} - 1518 \beta_{7} ) q^{74} + ( -93564 + 12582 \beta_{1} + 46782 \beta_{2} - 25164 \beta_{3} - 27 \beta_{4} - 1458 \beta_{5} + 729 \beta_{6} - 27 \beta_{7} ) q^{75} + ( -58806 + 71982 \beta_{1} + 117612 \beta_{2} - 34440 \beta_{3} - 3102 \beta_{4} - 1200 \beta_{5} + 2400 \beta_{6} ) q^{76} + ( -36882 + 954 \beta_{1} - 29511 \beta_{3} + 954 \beta_{4} + 2781 \beta_{5} + 1908 \beta_{7} ) q^{78} + ( -51686 \beta_{1} - 127061 \beta_{2} + 484 \beta_{4} + 822 \beta_{6} - 484 \beta_{7} ) q^{79} + ( -70377 - 1754 \beta_{1} - 70377 \beta_{2} - 6204 \beta_{3} + 691 \beta_{5} + 691 \beta_{6} + 4450 \beta_{7} ) q^{80} + ( -59049 + 59049 \beta_{2} ) q^{81} + ( -70684 - 20938 \beta_{1} + 35342 \beta_{2} + 41876 \beta_{3} + 4664 \beta_{4} - 224 \beta_{5} + 112 \beta_{6} + 4664 \beta_{7} ) q^{82} + ( 73913 - 49367 \beta_{1} - 147826 \beta_{2} + 25496 \beta_{3} - 1625 \beta_{4} + 4843 \beta_{5} - 9686 \beta_{6} ) q^{83} + ( 225666 - 1618 \beta_{1} + 77016 \beta_{3} - 1618 \beta_{4} + 1962 \beta_{5} - 3236 \beta_{7} ) q^{85} + ( 21721 \beta_{1} - 578108 \beta_{2} + 4158 \beta_{4} - 7533 \beta_{6} - 4158 \beta_{7} ) q^{86} + ( -35541 - 2547 \beta_{1} - 35541 \beta_{2} - 4194 \beta_{3} - 1647 \beta_{5} - 1647 \beta_{6} + 1647 \beta_{7} ) q^{87} + ( 83477 + 28310 \beta_{1} - 83477 \beta_{2} - 28008 \beta_{3} - 604 \beta_{4} - 107 \beta_{5} + 107 \beta_{6} - 302 \beta_{7} ) q^{88} + ( 296972 + 3980 \beta_{1} - 148486 \beta_{2} - 7960 \beta_{3} - 78 \beta_{4} + 252 \beta_{5} - 126 \beta_{6} - 78 \beta_{7} ) q^{89} + ( 2187 - 19926 \beta_{1} - 4374 \beta_{2} + 8262 \beta_{3} + 3402 \beta_{4} + 729 \beta_{5} - 1458 \beta_{6} ) q^{90} + ( 8028 - 3016 \beta_{1} - 88428 \beta_{3} - 3016 \beta_{4} + 8832 \beta_{5} - 6032 \beta_{7} ) q^{92} + ( 2502 \beta_{1} + 91827 \beta_{2} + 396 \beta_{4} - 7614 \beta_{6} - 396 \beta_{7} ) q^{93} + ( 153920 - 390 \beta_{1} + 153920 \beta_{2} + 6106 \beta_{3} - 2750 \beta_{5} - 2750 \beta_{6} - 6496 \beta_{7} ) q^{94} + ( 265014 - 78386 \beta_{1} - 265014 \beta_{2} + 72450 \beta_{3} + 11872 \beta_{4} + 9798 \beta_{5} - 9798 \beta_{6} + 5936 \beta_{7} ) q^{95} + ( -307314 + 17280 \beta_{1} + 153657 \beta_{2} - 34560 \beta_{3} - 1566 \beta_{4} + 3294 \beta_{5} - 1647 \beta_{6} - 1566 \beta_{7} ) q^{96} + ( -135633 + 217 \beta_{1} + 271266 \beta_{2} - 3574 \beta_{3} + 6931 \beta_{4} - 1627 \beta_{5} + 3254 \beta_{6} ) q^{97} + ( -64395 - 243 \beta_{1} + 486 \beta_{3} - 243 \beta_{4} - 2187 \beta_{5} - 486 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 5q^{2} - 108q^{3} - 173q^{4} + 42q^{5} - 454q^{8} + 972q^{9} + O(q^{10})$$ $$8q - 5q^{2} - 108q^{3} - 173q^{4} + 42q^{5} - 454q^{8} + 972q^{9} - 261q^{10} - 1070q^{11} + 4671q^{12} - 756q^{15} + 3911q^{16} - 7212q^{17} + 1215q^{18} + 24606q^{19} - 78q^{22} - 15224q^{23} + 6129q^{24} + 22274q^{25} + 19044q^{26} + 32524q^{29} + 2349q^{30} - 40200q^{31} + 70203q^{32} + 28890q^{33} - 84078q^{36} - 45670q^{37} - 503310q^{38} + 93366q^{39} + 94941q^{40} - 445660q^{43} - 188829q^{44} + 10206q^{45} + 525804q^{46} - 82884q^{47} - 1218884q^{50} + 64908q^{51} - 722856q^{52} - 13034q^{53} - 32805q^{54} - 442908q^{57} - 159501q^{58} - 1810362q^{59} + 429705q^{60} + 392856q^{61} - 1410446q^{64} - 389004q^{65} + 1053q^{66} + 384094q^{67} + 1616346q^{68} + 225688q^{71} - 55161q^{72} - 903078q^{73} + 1185530q^{74} - 601398q^{75} - 342792q^{78} - 559592q^{79} - 847713q^{80} - 236196q^{81} - 347634q^{82} + 1953576q^{85} - 2302402q^{86} - 439074q^{87} + 304887q^{88} + 1770036q^{89} - 113064q^{92} + 361800q^{93} + 1837620q^{94} + 1160112q^{95} - 1895481q^{96} - 520020q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-1585013359 \nu^{7} + 18232571539 \nu^{6} - 303603349712 \nu^{5} + 4245938445433 \nu^{4} - 65749585575908 \nu^{3} + 780366646056751 \nu^{2} - 2109023702500351 \nu + 22182278913510768$$$$)/ 23802631772961636$$ $$\beta_{3}$$ $$=$$ $$($$$$-8019055 \nu^{7} - 15616321 \nu^{6} - 1444380025 \nu^{5} + 2053828205 \nu^{4} - 305021724904 \nu^{3} - 177516832405 \nu^{2} - 1592248206780 \nu - 29614389599556$$$$)/ 11465622241311$$ $$\beta_{4}$$ $$=$$ $$($$$$27322708193 \nu^{7} + 1000103862583 \nu^{6} + 22563530233273 \nu^{5} + 219959885092027 \nu^{4} + 2723907868150765 \nu^{3} + 19609334703803473 \nu^{2} + 313020438506182988 \nu + 699314583065439780$$$$)/ 11901315886480818$$ $$\beta_{5}$$ $$=$$ $$($$$$-39673486 \nu^{7} + 224426993 \nu^{6} - 7145928130 \nu^{5} + 10161113066 \nu^{4} - 1531993215028 \nu^{3} - 878247070906 \nu^{2} - 7877491417656 \nu - 963541759742958$$$$)/ 11465622241311$$ $$\beta_{6}$$ $$=$$ $$($$$$59472187739 \nu^{7} - 700951590437 \nu^{6} + 11672037067696 \nu^{5} - 218751249599021 \nu^{4} + 2527744179224764 \nu^{3} - 30001211870054633 \nu^{2} + 130709559191228591 \nu - 852798172253616144$$$$)/ 11901315886480818$$ $$\beta_{7}$$ $$=$$ $$($$$$226304020214 \nu^{7} + 436984015237 \nu^{6} + 31940434175641 \nu^{5} - 171439510285889 \nu^{4} + 4754608612331077 \nu^{3} - 4492583429983241 \nu^{2} - 114813811933007845 \nu + 14206534053079428$$$$)/ 11901315886480818$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + 2 \beta_{3} + 106 \beta_{2} - 2 \beta_{1} - 106$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{7} + 6 \beta_{5} - 2 \beta_{4} - 145 \beta_{3} - 2 \beta_{1} + 252$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{7} - 205 \beta_{6} + 2 \beta_{4} - 15886 \beta_{2} + 772 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$424 \beta_{7} + 1560 \beta_{6} - 1560 \beta_{5} + 848 \beta_{4} + 24589 \beta_{3} + 90180 \beta_{2} - 25013 \beta_{1} - 90180$$ $$\nu^{6}$$ $$=$$ $$-304 \beta_{7} + 38461 \beta_{5} - 152 \beta_{4} - 199046 \beta_{3} - 152 \beta_{1} + 2727346$$ $$\nu^{7}$$ $$=$$ $$75858 \beta_{7} - 355626 \beta_{6} - 75858 \beta_{4} - 22658292 \beta_{2} + 4625113 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 5.73828 − 9.93899i 4.15432 − 7.19549i −2.30325 + 3.98935i −7.08935 + 12.2791i 5.73828 + 9.93899i 4.15432 + 7.19549i −2.30325 − 3.98935i −7.08935 − 12.2791i
−6.23828 + 10.8050i −13.5000 + 7.79423i −45.8323 79.3839i 175.367 + 101.248i 194.490i 0 345.159 121.500 210.444i −2187.98 + 1263.23i
19.2 −4.65432 + 8.06151i −13.5000 + 7.79423i −11.3253 19.6160i −151.343 87.3778i 145.107i 0 −384.906 121.500 210.444i 1408.79 813.368i
19.3 1.80325 3.12332i −13.5000 + 7.79423i 25.4966 + 44.1614i −71.9311 41.5295i 56.2198i 0 414.723 121.500 210.444i −259.420 + 149.776i
19.4 6.58935 11.4131i −13.5000 + 7.79423i −54.8390 94.9839i 68.9069 + 39.7834i 205.435i 0 −601.976 121.500 210.444i 908.103 524.293i
31.1 −6.23828 10.8050i −13.5000 7.79423i −45.8323 + 79.3839i 175.367 101.248i 194.490i 0 345.159 121.500 + 210.444i −2187.98 1263.23i
31.2 −4.65432 8.06151i −13.5000 7.79423i −11.3253 + 19.6160i −151.343 + 87.3778i 145.107i 0 −384.906 121.500 + 210.444i 1408.79 + 813.368i
31.3 1.80325 + 3.12332i −13.5000 7.79423i 25.4966 44.1614i −71.9311 + 41.5295i 56.2198i 0 414.723 121.500 + 210.444i −259.420 149.776i
31.4 6.58935 + 11.4131i −13.5000 7.79423i −54.8390 + 94.9839i 68.9069 39.7834i 205.435i 0 −601.976 121.500 + 210.444i 908.103 + 524.293i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b 8 7.b odd 2 1
21.7.f.b 8 7.c even 3 1
63.7.m.c 8 21.c even 2 1
63.7.m.c 8 21.h odd 6 1
147.7.d.a 8 7.c even 3 1
147.7.d.a 8 7.d odd 6 1
147.7.f.a 8 1.a even 1 1 trivial
147.7.f.a 8 7.d odd 6 1 inner
336.7.bh.b 8 28.d even 2 1
336.7.bh.b 8 28.g odd 6 1
441.7.d.d 8 21.g even 6 1
441.7.d.d 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{8} + \cdots$$ $$98\!\cdots\!00$$$$T_{5}^{2} +$$$$24\!\cdots\!00$$$$T_{5} +$$$$54\!\cdots\!00$$">$$T_{5}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$30470400 - 5045280 T + 1950436 T^{2} + 129428 T^{3} + 39854 T^{4} + 818 T^{5} + 227 T^{6} + 5 T^{7} + T^{8}$$
$3$ $$( 243 + 27 T + T^{2} )^{4}$$
$5$ $$54693259182810000 + 24205120650000 T - 9840546578700 T^{2} - 4356625500 T^{3} + 1536505749 T^{4} + 1767906 T^{5} - 41505 T^{6} - 42 T^{7} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$34\!\cdots\!00$$$$- 18488408967989050800 T + 113509429921169980 T^{2} - 43785002211340 T^{3} + 345504620141 T^{4} + 343064750 T^{5} + 1409111 T^{6} + 1070 T^{7} + T^{8}$$
$13$ $$34\!\cdots\!00$$$$+ 53257929335845404768 T^{2} + 105787465152777 T^{4} + 22315686 T^{6} + T^{8}$$
$17$ $$22\!\cdots\!00$$$$-$$$$46\!\cdots\!60$$$$T -$$$$83\!\cdots\!72$$$$T^{2} + 23263548118111082880 T^{3} + 3514480347514032 T^{4} - 547654326480 T^{5} - 58598892 T^{6} + 7212 T^{7} + T^{8}$$
$19$ $$12\!\cdots\!96$$$$-$$$$11\!\cdots\!92$$$$T +$$$$36\!\cdots\!68$$$$T^{2} - 27399467456974423980 T^{3} - 6359170662393867 T^{4} + 670975723710 T^{5} + 174549627 T^{6} - 24606 T^{7} + T^{8}$$
$23$ $$63\!\cdots\!00$$$$-$$$$71\!\cdots\!00$$$$T +$$$$15\!\cdots\!00$$$$T^{2} + 51980122560355404800 T^{3} + 101423010754761920 T^{4} + 1256314418240 T^{5} + 520738856 T^{6} + 15224 T^{7} + T^{8}$$
$29$ $$( -39242852020022400 + 9430137809600 T - 442580539 T^{2} - 16262 T^{3} + T^{4} )^{2}$$
$31$ $$42\!\cdots\!21$$$$-$$$$16\!\cdots\!80$$$$T -$$$$92\!\cdots\!06$$$$T^{2} +$$$$43\!\cdots\!80$$$$T^{3} + 2063880899710463727 T^{4} - 70137742150800 T^{5} - 1206039954 T^{6} + 40200 T^{7} + T^{8}$$
$37$ $$25\!\cdots\!00$$$$-$$$$12\!\cdots\!00$$$$T +$$$$21\!\cdots\!00$$$$T^{2} +$$$$27\!\cdots\!00$$$$T^{3} + 10133981768848670525 T^{4} - 93188930684650 T^{5} + 5175060655 T^{6} + 45670 T^{7} + T^{8}$$
$41$ $$53\!\cdots\!16$$$$+$$$$87\!\cdots\!08$$$$T^{2} + 50479416076831845984 T^{4} + 11994986352 T^{6} + T^{8}$$
$43$ $$( -$$$$14\!\cdots\!84$$$$- 2845100102192540 T - 1577056827 T^{2} + 222830 T^{3} + T^{4} )^{2}$$
$47$ $$30\!\cdots\!00$$$$-$$$$12\!\cdots\!00$$$$T +$$$$18\!\cdots\!00$$$$T^{2} -$$$$87\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!20$$$$T^{4} - 1025758292062320 T^{5} - 10085910828 T^{6} + 82884 T^{7} + T^{8}$$
$53$ $$23\!\cdots\!00$$$$-$$$$38\!\cdots\!00$$$$T +$$$$25\!\cdots\!64$$$$T^{2} +$$$$27\!\cdots\!44$$$$T^{3} +$$$$20\!\cdots\!21$$$$T^{4} + 946587441999554 T^{5} + 50746432799 T^{6} + 13034 T^{7} + T^{8}$$
$59$ $$13\!\cdots\!36$$$$+$$$$41\!\cdots\!92$$$$T +$$$$55\!\cdots\!32$$$$T^{2} +$$$$43\!\cdots\!28$$$$T^{3} +$$$$21\!\cdots\!13$$$$T^{4} + 715051889313496398 T^{5} + 1487447487327 T^{6} + 1810362 T^{7} + T^{8}$$
$61$ $$78\!\cdots\!00$$$$-$$$$11\!\cdots\!40$$$$T +$$$$61\!\cdots\!72$$$$T^{2} -$$$$15\!\cdots\!00$$$$T^{3} -$$$$51\!\cdots\!48$$$$T^{4} + 1477092202992000 T^{5} + 47685396912 T^{6} - 392856 T^{7} + T^{8}$$
$67$ $$93\!\cdots\!00$$$$+$$$$14\!\cdots\!40$$$$T +$$$$25\!\cdots\!04$$$$T^{2} +$$$$80\!\cdots\!20$$$$T^{3} +$$$$67\!\cdots\!57$$$$T^{4} - 12184102069585486 T^{5} + 192425428651 T^{6} - 384094 T^{7} + T^{8}$$
$71$ $$( -$$$$25\!\cdots\!12$$$$+ 44439430271275520 T - 187952640388 T^{2} - 112844 T^{3} + T^{4} )^{2}$$
$73$ $$12\!\cdots\!24$$$$-$$$$24\!\cdots\!32$$$$T +$$$$16\!\cdots\!20$$$$T^{2} -$$$$44\!\cdots\!04$$$$T^{3} -$$$$63\!\cdots\!51$$$$T^{4} + 18164126696429538 T^{5} + 291963532599 T^{6} + 903078 T^{7} + T^{8}$$
$79$ $$58\!\cdots\!25$$$$+$$$$18\!\cdots\!80$$$$T +$$$$57\!\cdots\!34$$$$T^{2} +$$$$12\!\cdots\!12$$$$T^{3} +$$$$36\!\cdots\!07$$$$T^{4} + 233175179655084368 T^{5} + 779488395718 T^{6} + 559592 T^{7} + T^{8}$$
$83$ $$36\!\cdots\!24$$$$+$$$$24\!\cdots\!96$$$$T^{2} +$$$$12\!\cdots\!25$$$$T^{4} + 2038066317246 T^{6} + T^{8}$$
$89$ $$13\!\cdots\!44$$$$-$$$$38\!\cdots\!40$$$$T +$$$$51\!\cdots\!48$$$$T^{2} -$$$$40\!\cdots\!80$$$$T^{3} +$$$$20\!\cdots\!44$$$$T^{4} - 673833907778767344 T^{5} + 1425031860636 T^{6} - 1770036 T^{7} + T^{8}$$
$97$ $$22\!\cdots\!00$$$$+$$$$33\!\cdots\!12$$$$T^{2} +$$$$14\!\cdots\!01$$$$T^{4} + 2348711138742 T^{6} + T^{8}$$