# Properties

 Label 21.7.d.a Level $21$ Weight $7$ Character orbit 21.d Analytic conductor $4.831$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 21.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83113575602$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 212 x^{6} - 1123 x^{5} + 44168 x^{4} - 138697 x^{3} + 660109 x^{2} + 680340 x + 1040400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}\cdot 3^{7}\cdot 7^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{4} q^{3} + ( 44 + \beta_{1} - \beta_{2} ) q^{4} + ( -2 \beta_{4} - \beta_{6} ) q^{5} + ( -\beta_{3} - \beta_{4} ) q^{6} + ( -14 \beta_{1} - \beta_{3} + 4 \beta_{4} + \beta_{7} ) q^{7} + ( -198 - 63 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} ) q^{8} -243 q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{4} q^{3} + ( 44 + \beta_{1} - \beta_{2} ) q^{4} + ( -2 \beta_{4} - \beta_{6} ) q^{5} + ( -\beta_{3} - \beta_{4} ) q^{6} + ( -14 \beta_{1} - \beta_{3} + 4 \beta_{4} + \beta_{7} ) q^{7} + ( -198 - 63 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} ) q^{8} -243 q^{9} + ( -10 \beta_{3} + 48 \beta_{4} - 5 \beta_{5} - 6 \beta_{6} - 5 \beta_{7} ) q^{10} + ( -857 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{5} + 3 \beta_{7} ) q^{11} + ( 7 \beta_{3} - 43 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} ) q^{12} + ( 56 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} ) q^{13} + ( 1526 + 49 \beta_{1} - 7 \beta_{2} + 10 \beta_{3} - 138 \beta_{4} + 7 \beta_{5} + 7 \beta_{6} + 11 \beta_{7} ) q^{14} + ( -597 - 114 \beta_{1} + 3 \beta_{2} + 6 \beta_{5} - 6 \beta_{7} ) q^{15} + ( 3600 + 137 \beta_{1} - 15 \beta_{2} + 5 \beta_{5} - 5 \beta_{7} ) q^{16} + ( -28 \beta_{3} + 50 \beta_{4} - 18 \beta_{5} + 3 \beta_{6} - 18 \beta_{7} ) q^{17} + ( -243 + 243 \beta_{1} ) q^{18} + ( 30 \beta_{3} + 84 \beta_{4} + 21 \beta_{5} + 36 \beta_{6} + 21 \beta_{7} ) q^{19} + ( 100 \beta_{3} - 224 \beta_{4} + 18 \beta_{6} ) q^{20} + ( 924 + 336 \beta_{1} + 21 \beta_{2} - 25 \beta_{3} + 9 \beta_{4} - 21 \beta_{6} - 3 \beta_{7} ) q^{21} + ( -690 + 1222 \beta_{1} + 50 \beta_{2} - 15 \beta_{5} + 15 \beta_{7} ) q^{22} + ( -3255 - 642 \beta_{1} + 109 \beta_{2} - 19 \beta_{5} + 19 \beta_{7} ) q^{23} + ( -35 \beta_{3} + 181 \beta_{4} + 9 \beta_{5} + 45 \beta_{6} + 9 \beta_{7} ) q^{24} + ( -9727 - 424 \beta_{1} - 22 \beta_{2} + 31 \beta_{5} - 31 \beta_{7} ) q^{25} + ( 24 \beta_{3} + 644 \beta_{4} - 54 \beta_{5} - 68 \beta_{6} - 54 \beta_{7} ) q^{26} + 243 \beta_{4} q^{27} + ( -4102 - 1211 \beta_{1} + 63 \beta_{2} - 134 \beta_{3} - 332 \beta_{4} + 28 \beta_{5} + 84 \beta_{6} - 34 \beta_{7} ) q^{28} + ( 7386 - 1176 \beta_{1} - 236 \beta_{2} - 34 \beta_{5} + 34 \beta_{7} ) q^{29} + ( 11478 + 816 \beta_{1} - 192 \beta_{2} + 21 \beta_{5} - 21 \beta_{7} ) q^{30} + ( -124 \beta_{3} - 968 \beta_{4} + 98 \beta_{5} - 180 \beta_{6} + 98 \beta_{7} ) q^{31} + ( 268 - 1267 \beta_{1} + 101 \beta_{2} - 29 \beta_{5} + 29 \beta_{7} ) q^{32} + ( -82 \beta_{3} + 908 \beta_{4} - 27 \beta_{5} - 135 \beta_{6} - 27 \beta_{7} ) q^{33} + ( 370 \beta_{3} - 1772 \beta_{4} + 75 \beta_{5} - 42 \beta_{6} + 75 \beta_{7} ) q^{34} + ( 13237 + 434 \beta_{1} - 343 \beta_{2} - 290 \beta_{3} - 772 \beta_{4} - 161 \beta_{5} - 196 \beta_{6} - 109 \beta_{7} ) q^{35} + ( -10692 - 243 \beta_{1} + 243 \beta_{2} ) q^{36} + ( -4926 - 808 \beta_{1} - 330 \beta_{2} - 31 \beta_{5} + 31 \beta_{7} ) q^{37} + ( 48 \beta_{3} - 48 \beta_{4} + 126 \beta_{5} + 294 \beta_{6} + 126 \beta_{7} ) q^{38} + ( 13326 - 1428 \beta_{1} - 150 \beta_{2} + 24 \beta_{5} - 24 \beta_{7} ) q^{39} + ( -240 \beta_{3} + 6404 \beta_{4} - 190 \beta_{5} + 192 \beta_{6} - 190 \beta_{7} ) q^{40} + ( 100 \beta_{3} - 2014 \beta_{4} - 90 \beta_{5} - 49 \beta_{6} - 90 \beta_{7} ) q^{41} + ( -33621 + 42 \beta_{1} + 357 \beta_{2} + 47 \beta_{3} - 1483 \beta_{4} + 21 \beta_{5} - 63 \beta_{6} + 51 \beta_{7} ) q^{42} + ( -4952 - 4508 \beta_{1} + 332 \beta_{2} + 153 \beta_{5} - 153 \beta_{7} ) q^{43} + ( -72206 + 2798 \beta_{1} + 1340 \beta_{2} + 82 \beta_{5} - 82 \beta_{7} ) q^{44} + ( 486 \beta_{4} + 243 \beta_{6} ) q^{45} + ( 74242 + 14166 \beta_{1} - 158 \beta_{2} - 185 \beta_{5} + 185 \beta_{7} ) q^{46} + ( 508 \beta_{3} + 4968 \beta_{4} + 378 \beta_{5} + 196 \beta_{6} + 378 \beta_{7} ) q^{47} + ( 337 \beta_{3} - 3675 \beta_{4} + 105 \beta_{5} + 255 \beta_{6} + 105 \beta_{7} ) q^{48} + ( -11907 + 784 \beta_{1} + 490 \beta_{2} + 336 \beta_{3} - 4676 \beta_{4} - 147 \beta_{5} + 294 \beta_{6} + 203 \beta_{7} ) q^{49} + ( 32343 + 7491 \beta_{1} - 902 \beta_{2} + 146 \beta_{5} - 146 \beta_{7} ) q^{50} + ( 14211 + 3798 \beta_{1} - 765 \beta_{2} - 72 \beta_{5} + 72 \beta_{7} ) q^{51} + ( 124 \beta_{3} + 6892 \beta_{4} - 552 \beta_{5} - 528 \beta_{6} - 552 \beta_{7} ) q^{52} + ( 27126 - 6624 \beta_{1} + 508 \beta_{2} + 176 \beta_{5} - 176 \beta_{7} ) q^{53} + ( 243 \beta_{3} + 243 \beta_{4} ) q^{54} + ( -1150 \beta_{3} - 6960 \beta_{4} - 365 \beta_{5} - 60 \beta_{6} - 365 \beta_{7} ) q^{55} + ( 30548 + 7189 \beta_{1} - 1071 \beta_{2} + 784 \beta_{3} - 9688 \beta_{4} + 819 \beta_{5} + 434 \beta_{6} + 441 \beta_{7} ) q^{56} + ( 22392 + 720 \beta_{1} + 774 \beta_{2} - 153 \beta_{5} + 153 \beta_{7} ) q^{57} + ( 118366 - 22722 \beta_{1} - 1172 \beta_{2} + 100 \beta_{5} - 100 \beta_{7} ) q^{58} + ( -712 \beta_{3} + 9468 \beta_{4} - 468 \beta_{5} - 580 \beta_{6} - 468 \beta_{7} ) q^{59} + ( -52434 - 22248 \beta_{1} - 54 \beta_{2} - 108 \beta_{5} + 108 \beta_{7} ) q^{60} + ( 1184 \beta_{3} + 7476 \beta_{4} + 748 \beta_{5} - 132 \beta_{6} + 748 \beta_{7} ) q^{61} + ( -1808 \beta_{3} - 12168 \beta_{4} + 432 \beta_{5} + 76 \beta_{6} + 432 \beta_{7} ) q^{62} + ( 3402 \beta_{1} + 243 \beta_{3} - 972 \beta_{4} - 243 \beta_{7} ) q^{63} + ( -85768 + 2897 \beta_{1} + 301 \beta_{2} - 537 \beta_{5} + 537 \beta_{7} ) q^{64} + ( -101708 + 12824 \beta_{1} + 752 \beta_{2} - 86 \beta_{5} + 86 \beta_{7} ) q^{65} + ( 592 \beta_{3} + 910 \beta_{4} - 345 \beta_{5} - 780 \beta_{6} - 345 \beta_{7} ) q^{66} + ( -265694 + 19096 \beta_{1} - 26 \beta_{2} + 305 \beta_{5} - 305 \beta_{7} ) q^{67} + ( -3576 \beta_{3} + 30948 \beta_{4} - 828 \beta_{5} - 954 \beta_{6} - 828 \beta_{7} ) q^{68} + ( -1714 \beta_{3} + 3488 \beta_{4} - 711 \beta_{5} - 1125 \beta_{6} - 711 \beta_{7} ) q^{69} + ( -56098 - 40796 \beta_{1} + 112 \beta_{2} + 520 \beta_{3} - 11992 \beta_{4} + 189 \beta_{5} - 1386 \beta_{6} - 289 \beta_{7} ) q^{70} + ( -64557 - 6606 \beta_{1} - 761 \beta_{2} - 211 \beta_{5} + 211 \beta_{7} ) q^{71} + ( 48114 + 15309 \beta_{1} + 243 \beta_{2} - 243 \beta_{5} + 243 \beta_{7} ) q^{72} + ( 1784 \beta_{3} - 4996 \beta_{4} + 1164 \beta_{5} + 2886 \beta_{6} + 1164 \beta_{7} ) q^{73} + ( 59836 - 18886 \beta_{1} - 1034 \beta_{2} + 206 \beta_{5} - 206 \beta_{7} ) q^{74} + ( 390 \beta_{3} + 9191 \beta_{4} + 225 \beta_{5} + 1368 \beta_{6} + 225 \beta_{7} ) q^{75} + ( -504 \beta_{3} - 24108 \beta_{4} + 594 \beta_{5} + 324 \beta_{6} + 594 \beta_{7} ) q^{76} + ( 193718 + 10528 \beta_{1} + 1904 \beta_{2} + 1156 \beta_{3} - 3294 \beta_{4} - 1148 \beta_{5} + 7 \beta_{6} - 904 \beta_{7} ) q^{77} + ( 154128 - 23628 \beta_{1} - 2064 \beta_{2} + 246 \beta_{5} - 246 \beta_{7} ) q^{78} + ( -17594 - 18656 \beta_{1} + 2542 \beta_{2} - 157 \beta_{5} + 157 \beta_{7} ) q^{79} + ( 4220 \beta_{3} - 620 \beta_{4} + 1260 \beta_{5} - 800 \beta_{6} + 1260 \beta_{7} ) q^{80} + 59049 q^{81} + ( -2846 \beta_{3} + 17516 \beta_{4} - 1385 \beta_{5} - 1314 \beta_{6} - 1385 \beta_{7} ) q^{82} + ( 1268 \beta_{3} - 21116 \beta_{4} + 126 \beta_{5} + 580 \beta_{6} + 126 \beta_{7} ) q^{83} + ( -71064 + 41580 \beta_{1} - 378 \beta_{2} - 907 \beta_{3} + 3481 \beta_{4} - 798 \beta_{5} + 1113 \beta_{6} + 228 \beta_{7} ) q^{84} + ( 484560 + 66600 \beta_{1} - 630 \beta_{2} + 675 \beta_{5} - 675 \beta_{7} ) q^{85} + ( 492408 + 34434 \beta_{1} - 5986 \beta_{2} + 280 \beta_{5} - 280 \beta_{7} ) q^{86} + ( -508 \beta_{3} - 6538 \beta_{4} + 1314 \beta_{5} - 720 \beta_{6} + 1314 \beta_{7} ) q^{87} + ( -236884 + 93498 \beta_{1} + 1130 \beta_{2} - 52 \beta_{5} + 52 \beta_{7} ) q^{88} + ( 5868 \beta_{3} - 4890 \beta_{4} - 270 \beta_{5} + 3261 \beta_{6} - 270 \beta_{7} ) q^{89} + ( 2430 \beta_{3} - 11664 \beta_{4} + 1215 \beta_{5} + 1458 \beta_{6} + 1215 \beta_{7} ) q^{90} + ( 228886 - 36484 \beta_{1} - 3430 \beta_{2} - 8 \beta_{3} + 1236 \beta_{4} - 210 \beta_{5} - 588 \beta_{6} - 734 \beta_{7} ) q^{91} + ( -1234058 - 66514 \beta_{1} + 9464 \beta_{2} + 634 \beta_{5} - 634 \beta_{7} ) q^{92} + ( -264612 + 27840 \beta_{1} + 4656 \beta_{2} + 1374 \beta_{5} - 1374 \beta_{7} ) q^{93} + ( 960 \beta_{3} + 21800 \beta_{4} + 200 \beta_{5} + 2676 \beta_{6} + 200 \beta_{7} ) q^{94} + ( 506094 - 51252 \beta_{1} + 894 \beta_{2} - 2352 \beta_{5} + 2352 \beta_{7} ) q^{95} + ( -2511 \beta_{3} + 153 \beta_{4} - 693 \beta_{5} - 1521 \beta_{6} - 693 \beta_{7} ) q^{96} + ( 152 \beta_{3} + 46116 \beta_{4} + 124 \beta_{5} + 1974 \beta_{6} + 124 \beta_{7} ) q^{97} + ( -51205 + 57379 \beta_{1} + 4214 \beta_{2} - 5124 \beta_{3} + 14224 \beta_{4} - 1568 \beta_{5} + 980 \beta_{6} + 812 \beta_{7} ) q^{98} + ( 208251 - 486 \beta_{1} - 729 \beta_{2} + 729 \beta_{5} - 729 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 10q^{2} + 346q^{4} + 28q^{7} - 1462q^{8} - 1944q^{9} + O(q^{10})$$ $$8q + 10q^{2} + 346q^{4} + 28q^{7} - 1462q^{8} - 1944q^{9} - 6848q^{11} + 12082q^{14} - 4536q^{15} + 28466q^{16} - 2430q^{18} + 6804q^{21} - 7764q^{22} - 24320q^{23} - 77056q^{25} - 30142q^{28} + 60496q^{29} + 89424q^{30} + 5082q^{32} + 103656q^{35} - 84078q^{36} - 39112q^{37} + 108864q^{39} - 267624q^{42} - 29272q^{43} - 577884q^{44} + 564972q^{46} - 94864q^{49} + 240154q^{50} + 103032q^{51} + 232288q^{53} + 225722q^{56} + 180792q^{57} + 987684q^{58} - 375192q^{60} - 6804q^{63} - 690734q^{64} - 836304q^{65} - 2163848q^{67} - 366744q^{70} - 506288q^{71} + 355266q^{72} + 512324q^{74} + 1536304q^{77} + 1272024q^{78} - 93272q^{79} + 472392q^{81} - 653184q^{84} + 3740760q^{85} + 3846452q^{86} - 2077548q^{88} + 1890336q^{91} - 9701580q^{92} - 2153952q^{93} + 4154832q^{95} - 507542q^{98} + 1664064q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 212 x^{6} - 1123 x^{5} + 44168 x^{4} - 138697 x^{3} + 660109 x^{2} + 680340 x + 1040400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$9082495 \nu^{7} + 20968969 \nu^{6} + 1852872025 \nu^{5} - 4069000805 \nu^{4} + 387692440864 \nu^{3} + 57699627205 \nu^{2} + 74069253300 \nu + 6424259322060$$$$)/ 6112279547031$$ $$\beta_{2}$$ $$=$$ $$($$$$57298949 \nu^{7} - 9710008 \nu^{6} + 11689257155 \nu^{5} - 25670200711 \nu^{4} + 2480491758812 \nu^{3} + 364010989991 \nu^{2} + 467282433660 \nu + 663837261700497$$$$)/ 6112279547031$$ $$\beta_{3}$$ $$=$$ $$($$$$27247485 \nu^{7} + 62906907 \nu^{6} + 5558616075 \nu^{5} - 12207002415 \nu^{4} + 1163077322592 \nu^{3} + 173098881615 \nu^{2} + 36895885042086 \nu + 19272777966180$$$$)/ 2037426515677$$ $$\beta_{4}$$ $$=$$ $$($$$$-6298293453 \nu^{7} + 15562438353 \nu^{6} - 1313849863656 \nu^{5} + 8962913013219 \nu^{4} - 282333406053204 \nu^{3} + 1269000696732021 \nu^{2} - 4098706573217277 \nu - 1092167760462210$$$$)/ 346362507665090$$ $$\beta_{5}$$ $$=$$ $$($$$$-15106603126 \nu^{7} + 163811250526 \nu^{6} - 3388053356392 \nu^{5} + 46949412040243 \nu^{4} - 857197973159188 \nu^{3} + 8225104922622397 \nu^{2} - 33706047073110594 \nu + 31447186628724765$$$$)/ 222661612070415$$ $$\beta_{6}$$ $$=$$ $$($$$$236085965056 \nu^{7} - 195554084551 \nu^{6} + 52131292245397 \nu^{5} - 254393550332128 \nu^{4} + 10753186644989518 \nu^{3} - 32709038594896462 \nu^{2} + 214551508656106419 \nu + 84662516979660870$$$$)/ 1558631284492905$$ $$\beta_{7}$$ $$=$$ $$($$$$-76111842031 \nu^{7} + 12621498841 \nu^{6} - 15833411217367 \nu^{5} + 74280048194038 \nu^{4} - 3236061928204333 \nu^{3} + 7837548470087002 \nu^{2} - 34203554915733294 \nu - 53574117242368005$$$$)/ 222661612070415$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - 9 \beta_{1}$$$$)/18$$ $$\nu^{2}$$ $$=$$ $$($$$$-6 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + 106 \beta_{4} - 9 \beta_{3} + 9 \beta_{2} - 27 \beta_{1} - 963$$$$)/18$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{5} - 2 \beta_{2} + 197 \beta_{1} + 392$$ $$\nu^{4}$$ $$=$$ $$($$$$1266 \beta_{7} + 669 \beta_{6} + 1248 \beta_{5} - 20974 \beta_{4} + 2379 \beta_{3} + 1881 \beta_{2} - 9927 \beta_{1} - 190485$$$$)/18$$ $$\nu^{5}$$ $$=$$ $$($$$$-6552 \beta_{7} + 6264 \beta_{6} - 2736 \beta_{5} + 128220 \beta_{4} - 42061 \beta_{3} + 7920 \beta_{2} - 373005 \beta_{1} - 1196244$$$$)/18$$ $$\nu^{6}$$ $$=$$ $$244 \beta_{7} - 244 \beta_{5} - 43533 \beta_{2} + 319627 \beta_{1} + 4485223$$ $$\nu^{7}$$ $$=$$ $$($$$$1163448 \beta_{7} - 959112 \beta_{6} + 1933866 \beta_{5} - 36227340 \beta_{4} + 9695467 \beta_{3} + 2515590 \beta_{2} - 80640261 \beta_{1} - 335495412$$$$)/18$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −7.66250 − 13.2718i −7.66250 + 13.2718i −0.564972 − 0.978561i −0.564972 + 0.978561i 2.28617 + 3.95977i 2.28617 − 3.95977i 6.44130 + 11.1567i 6.44130 − 11.1567i
−14.3250 15.5885i 141.206 175.511i 223.305i −202.362 + 276.945i −1105.97 −243.000 2514.19i
13.2 −14.3250 15.5885i 141.206 175.511i 223.305i −202.362 276.945i −1105.97 −243.000 2514.19i
13.3 −0.129945 15.5885i −63.9831 126.447i 2.02564i −213.284 + 268.624i 16.6307 −243.000 16.4311i
13.4 −0.129945 15.5885i −63.9831 126.447i 2.02564i −213.284 268.624i 16.6307 −243.000 16.4311i
13.5 5.57235 15.5885i −32.9490 205.672i 86.8643i 342.970 + 4.53729i −540.233 −243.000 1146.08i
13.6 5.57235 15.5885i −32.9490 205.672i 86.8643i 342.970 4.53729i −540.233 −243.000 1146.08i
13.7 13.8826 15.5885i 128.727 109.244i 216.408i 86.6767 331.868i 898.572 −243.000 1516.59i
13.8 13.8826 15.5885i 128.727 109.244i 216.408i 86.6767 + 331.868i 898.572 −243.000 1516.59i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.8 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 21.7.d.a 8
3.b odd 2 1 63.7.d.f 8
4.b odd 2 1 336.7.f.a 8
7.b odd 2 1 inner 21.7.d.a 8
7.c even 3 1 147.7.f.b 8
7.c even 3 1 147.7.f.c 8
7.d odd 6 1 147.7.f.b 8
7.d odd 6 1 147.7.f.c 8
21.c even 2 1 63.7.d.f 8
28.d even 2 1 336.7.f.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.d.a 8 1.a even 1 1 trivial
21.7.d.a 8 7.b odd 2 1 inner
63.7.d.f 8 3.b odd 2 1
63.7.d.f 8 21.c even 2 1
147.7.f.b 8 7.c even 3 1
147.7.f.b 8 7.d odd 6 1
147.7.f.c 8 7.c even 3 1
147.7.f.c 8 7.d odd 6 1
336.7.f.a 8 4.b odd 2 1
336.7.f.a 8 28.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 144 + 1082 T - 202 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$3$ $$( 243 + T^{2} )^{4}$$
$5$ $$248637860496000000 + 50334222960000 T^{2} + 3535164900 T^{4} + 101028 T^{6} + T^{8}$$
$7$ $$19\!\cdots\!01$$$$- 45595580741492572 T + 661945719100624 T^{2} - 6986500687124 T^{3} + 4099289726 T^{4} - 59384276 T^{5} + 47824 T^{6} - 28 T^{7} + T^{8}$$
$11$ $$( -3045338564760 - 3608107144 T + 1862438 T^{2} + 3424 T^{3} + T^{4} )^{2}$$
$13$ $$32\!\cdots\!24$$$$+ 156777390691559424 T^{2} + 2310387045696 T^{4} + 10860816 T^{6} + T^{8}$$
$17$ $$11\!\cdots\!00$$$$+$$$$12\!\cdots\!68$$$$T^{2} + 3251319447148452 T^{4} + 117194148 T^{6} + T^{8}$$
$19$ $$94\!\cdots\!84$$$$+$$$$47\!\cdots\!60$$$$T^{2} + 5328681864565392 T^{4} + 152146296 T^{6} + T^{8}$$
$23$ $$( 1471912296313032 - 3449298419896 T - 365711962 T^{2} + 12160 T^{3} + T^{4} )^{2}$$
$29$ $$( -181031780063400624 + 30531450886304 T - 1012022104 T^{2} - 30248 T^{3} + T^{4} )^{2}$$
$31$ $$55\!\cdots\!96$$$$+$$$$27\!\cdots\!08$$$$T^{2} + 25124309320567488768 T^{4} + 8559348960 T^{6} + T^{8}$$
$37$ $$( 141186162311041472 + 2747896697056 T - 1865323500 T^{2} + 19556 T^{3} + T^{4} )^{2}$$
$41$ $$13\!\cdots\!00$$$$+$$$$33\!\cdots\!92$$$$T^{2} + 27918880472230013412 T^{4} + 9054088644 T^{6} + T^{8}$$
$43$ $$( 20499573010124873696 - 7651077476192 T - 11412638436 T^{2} + 14636 T^{3} + T^{4} )^{2}$$
$47$ $$30\!\cdots\!44$$$$+$$$$96\!\cdots\!44$$$$T^{2} +$$$$87\!\cdots\!16$$$$T^{4} + 58642480416 T^{6} + T^{8}$$
$53$ $$( 6687489114740863440 + 1606292747973248 T - 15541117528 T^{2} - 116144 T^{3} + T^{4} )^{2}$$
$59$ $$76\!\cdots\!44$$$$+$$$$27\!\cdots\!76$$$$T^{2} +$$$$55\!\cdots\!92$$$$T^{4} + 143141376000 T^{6} + T^{8}$$
$61$ $$21\!\cdots\!00$$$$+$$$$41\!\cdots\!92$$$$T^{2} +$$$$17\!\cdots\!12$$$$T^{4} + 254327829888 T^{6} + T^{8}$$
$67$ $$( -$$$$64\!\cdots\!72$$$$+ 22209650987981152 T + 337043996052 T^{2} + 1081924 T^{3} + T^{4} )^{2}$$
$71$ $$( 36669339944726089608 - 1432702776161848 T - 3137327482 T^{2} + 253144 T^{3} + T^{4} )^{2}$$
$73$ $$36\!\cdots\!04$$$$+$$$$43\!\cdots\!28$$$$T^{2} +$$$$13\!\cdots\!76$$$$T^{4} + 708435882000 T^{6} + T^{8}$$
$79$ $$($$$$16\!\cdots\!52$$$$+ 19369457184455680 T - 202514723292 T^{2} + 46636 T^{3} + T^{4} )^{2}$$
$83$ $$36\!\cdots\!64$$$$+$$$$32\!\cdots\!84$$$$T^{2} +$$$$76\!\cdots\!24$$$$T^{4} + 523700180256 T^{6} + T^{8}$$
$89$ $$28\!\cdots\!44$$$$+$$$$16\!\cdots\!96$$$$T^{2} +$$$$33\!\cdots\!64$$$$T^{4} + 2998700175204 T^{6} + T^{8}$$
$97$ $$39\!\cdots\!44$$$$+$$$$48\!\cdots\!40$$$$T^{2} +$$$$19\!\cdots\!68$$$$T^{4} + 2469548167824 T^{6} + T^{8}$$