Properties

 Label 15.10.b.a Level 15 Weight 10 Character orbit 15.b Analytic conductor 7.726 Analytic rank 0 Dimension 8 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 15.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.72553754246$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 939 x^{6} + 217699 x^{4} + 14559561 x^{2} + 31136400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 3^{12}\cdot 5^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + \beta_{4} q^{3} + ( -149 - \beta_{1} ) q^{4} + ( -86 - 2 \beta_{1} + 13 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{5} + ( 61 - \beta_{1} + \beta_{2} ) q^{6} + ( 76 \beta_{3} + 28 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{7} + ( -176 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 13 \beta_{7} ) q^{8} -6561 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + \beta_{4} q^{3} + ( -149 - \beta_{1} ) q^{4} + ( -86 - 2 \beta_{1} + 13 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{5} + ( 61 - \beta_{1} + \beta_{2} ) q^{6} + ( 76 \beta_{3} + 28 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{7} + ( -176 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 13 \beta_{7} ) q^{8} -6561 q^{9} + ( 8371 + 37 \beta_{1} + \beta_{2} + 434 \beta_{3} - 48 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} + 64 \beta_{7} ) q^{10} + ( -9030 + 80 \beta_{1} + 2 \beta_{2} - 23 \beta_{5} + 23 \beta_{6} ) q^{11} + ( 162 \beta_{3} - 132 \beta_{4} + 81 \beta_{7} ) q^{12} + ( -262 \beta_{3} - 200 \beta_{4} + 21 \beta_{5} + 21 \beta_{6} + 214 \beta_{7} ) q^{13} + ( 51962 + 206 \beta_{1} + 42 \beta_{2} - 64 \beta_{5} + 64 \beta_{6} ) q^{14} + ( 10061 - 68 \beta_{1} - 13 \beta_{2} + 486 \beta_{3} - 63 \beta_{4} + 81 \beta_{5} + 162 \beta_{7} ) q^{15} + ( -188155 - 219 \beta_{1} - 38 \beta_{2} - 12 \beta_{5} + 12 \beta_{6} ) q^{16} + ( -1342 \beta_{3} + 1812 \beta_{4} + 147 \beta_{5} + 147 \beta_{6} - 978 \beta_{7} ) q^{17} + 6561 \beta_{3} q^{18} + ( 106784 - 1176 \beta_{1} + 200 \beta_{2} + 84 \beta_{5} - 84 \beta_{6} ) q^{19} + ( 259690 + 290 \beta_{1} - 176 \beta_{2} - 9222 \beta_{3} - 1100 \beta_{4} - 88 \beta_{5} - 88 \beta_{6} - 13 \beta_{7} ) q^{20} + ( -198954 - 252 \beta_{1} - 72 \beta_{2} + 243 \beta_{5} - 243 \beta_{6} ) q^{21} + ( -7054 \beta_{3} - 16 \beta_{4} - 696 \beta_{5} - 696 \beta_{6} + 844 \beta_{7} ) q^{22} + ( -112 \beta_{3} - 3676 \beta_{4} - 324 \beta_{5} - 324 \beta_{6} - 2472 \beta_{7} ) q^{23} + ( 155291 + 835 \beta_{1} + 218 \beta_{2} + 324 \beta_{5} - 324 \beta_{6} ) q^{24} + ( 212075 - 700 \beta_{1} - 740 \beta_{2} + 12920 \beta_{3} + 9700 \beta_{4} - 495 \beta_{5} - 345 \beta_{6} - 1520 \beta_{7} ) q^{25} + ( -110478 + 4190 \beta_{1} - 586 \beta_{2} + 520 \beta_{5} - 520 \beta_{6} ) q^{26} -6561 \beta_{4} q^{27} + ( -53388 \beta_{3} - 8776 \beta_{4} - 480 \beta_{5} - 480 \beta_{6} + 2994 \beta_{7} ) q^{28} + ( -7720 - 3108 \beta_{1} + 2104 \beta_{2} + 1399 \beta_{5} - 1399 \beta_{6} ) q^{29} + ( 384390 + 5490 \beta_{1} - 306 \beta_{2} + 4293 \beta_{3} + 7400 \beta_{4} + 972 \beta_{5} + 972 \beta_{6} - 3078 \beta_{7} ) q^{30} + ( 60680 - 10296 \beta_{1} - 916 \beta_{2} - 2310 \beta_{5} + 2310 \beta_{6} ) q^{31} + ( 137004 \beta_{3} + 8900 \beta_{4} + 2884 \beta_{5} + 2884 \beta_{6} + 7831 \beta_{7} ) q^{32} + ( 7290 \beta_{3} - 11052 \beta_{4} + 1863 \beta_{5} + 1863 \beta_{6} - 6642 \beta_{7} ) q^{33} + ( -1017286 - 5578 \beta_{1} + 4062 \beta_{2} - 6264 \beta_{5} + 6264 \beta_{6} ) q^{34} + ( -4542742 - 5064 \beta_{1} - 2558 \beta_{2} + 33420 \beta_{3} + 32716 \beta_{4} + 1771 \beta_{5} - 787 \beta_{6} + 2604 \beta_{7} ) q^{35} + ( 977589 + 6561 \beta_{1} ) q^{36} + ( -295062 \beta_{3} + 54080 \beta_{4} - 2955 \beta_{5} - 2955 \beta_{6} + 8742 \beta_{7} ) q^{37} + ( 124696 \beta_{3} - 146976 \beta_{4} + 5248 \beta_{5} + 5248 \beta_{6} + 22504 \beta_{7} ) q^{38} + ( 1616670 + 16560 \beta_{1} + 774 \beta_{2} + 1701 \beta_{5} - 1701 \beta_{6} ) q^{39} + ( -1917691 + 4493 \beta_{1} - 738 \beta_{2} - 98048 \beta_{3} + 91048 \beta_{4} - 5244 \beta_{5} + 4332 \beta_{6} + 17030 \beta_{7} ) q^{40} + ( 11656530 + 41464 \beta_{1} + 208 \beta_{2} + 3770 \beta_{5} - 3770 \beta_{6} ) q^{41} + ( 256122 \beta_{3} + 43776 \beta_{4} + 5184 \beta_{5} + 5184 \beta_{6} - 20088 \beta_{7} ) q^{42} + ( 220600 \beta_{3} - 208604 \beta_{4} - 12408 \beta_{5} - 12408 \beta_{6} - 9640 \beta_{7} ) q^{43} + ( -9317162 - 3826 \beta_{1} - 2072 \beta_{2} + 2736 \beta_{5} - 2736 \beta_{6} ) q^{44} + ( 564246 + 13122 \beta_{1} - 85293 \beta_{3} + 13122 \beta_{4} - 6561 \beta_{6} + 6561 \beta_{7} ) q^{45} + ( -1197548 - 51252 \beta_{1} + 620 \beta_{2} - 4704 \beta_{5} + 4704 \beta_{6} ) q^{46} + ( -110224 \beta_{3} + 138116 \beta_{4} - 12914 \beta_{5} - 12914 \beta_{6} - 64424 \beta_{7} ) q^{47} + ( -203796 \beta_{3} - 181732 \beta_{4} + 972 \beta_{5} + 972 \beta_{6} + 20817 \beta_{7} ) q^{48} + ( 6433491 - 35480 \beta_{1} - 16792 \beta_{2} + 16842 \beta_{5} - 16842 \beta_{6} ) q^{49} + ( 8486580 - 45940 \beta_{1} + 11900 \beta_{2} - 111715 \beta_{3} + 446560 \beta_{4} + 5200 \beta_{5} + 3920 \beta_{6} - 35520 \beta_{7} ) q^{50} + ( -13498778 - 79192 \beta_{1} - 26 \beta_{2} + 11907 \beta_{5} - 11907 \beta_{6} ) q^{51} + ( -801252 \beta_{3} + 355384 \beta_{4} + 4656 \beta_{5} + 4656 \beta_{6} - 24270 \beta_{7} ) q^{52} + ( -1041082 \beta_{3} - 632324 \beta_{4} - 5145 \beta_{5} - 5145 \beta_{6} - 133614 \beta_{7} ) q^{53} + ( -400221 + 6561 \beta_{1} - 6561 \beta_{2} ) q^{54} + ( 8058812 + 6664 \beta_{1} + 12992 \beta_{2} + 1505788 \beta_{3} + 491544 \beta_{4} + 5871 \beta_{5} - 21801 \beta_{6} - 7732 \beta_{7} ) q^{55} + ( -8495754 + 66182 \beta_{1} + 5780 \beta_{2} - 13112 \beta_{5} + 13112 \beta_{6} ) q^{56} + ( 1443096 \beta_{3} + 113024 \beta_{4} - 6804 \beta_{5} - 6804 \beta_{6} + 79056 \beta_{7} ) q^{57} + ( 737408 \beta_{3} - 1352672 \beta_{4} + 26400 \beta_{5} + 26400 \beta_{6} + 77152 \beta_{7} ) q^{58} + ( 29536702 + 50928 \beta_{1} - 35914 \beta_{2} - 32641 \beta_{5} + 32641 \beta_{6} ) q^{59} + ( 7895499 - 9897 \beta_{1} + 8844 \beta_{2} - 1173204 \beta_{3} + 268488 \beta_{4} - 7128 \beta_{5} + 7128 \beta_{6} - 9234 \beta_{7} ) q^{60} + ( -44686366 + 93600 \beta_{1} + 26192 \beta_{2} + 31008 \beta_{5} - 31008 \beta_{6} ) q^{61} + ( 1711576 \beta_{3} + 328704 \beta_{4} + 7888 \beta_{5} + 7888 \beta_{6} + 247120 \beta_{7} ) q^{62} + ( -498636 \beta_{3} - 183708 \beta_{4} - 19683 \beta_{5} - 19683 \beta_{6} + 26244 \beta_{7} ) q^{63} + ( -1607525 + 319659 \beta_{1} - 20450 \beta_{2} - 20964 \beta_{5} + 20964 \beta_{6} ) q^{64} + ( 2471570 - 17620 \beta_{1} + 9816 \beta_{2} - 2717328 \beta_{3} + 969920 \beta_{4} - 24367 \beta_{5} + 31123 \beta_{6} + 72168 \beta_{7} ) q^{65} + ( 2870874 + 62406 \beta_{1} + 5958 \beta_{2} - 56376 \beta_{5} + 56376 \beta_{6} ) q^{66} + ( -4872160 \beta_{3} + 100708 \beta_{4} + 51762 \beta_{5} + 51762 \beta_{6} + 78784 \beta_{7} ) q^{67} + ( 1193772 \beta_{3} - 1818728 \beta_{4} - 18896 \beta_{5} - 18896 \beta_{6} + 324058 \beta_{7} ) q^{68} + ( 20917852 - 194104 \beta_{1} - 6128 \beta_{2} - 26244 \beta_{5} + 26244 \beta_{6} ) q^{69} + ( 25096588 + 68996 \beta_{1} + 28492 \beta_{2} + 5464130 \beta_{3} + 1515376 \beta_{4} + 53496 \beta_{5} + 48408 \beta_{6} - 205316 \beta_{7} ) q^{70} + ( -19531716 - 381456 \beta_{1} + 35124 \beta_{2} - 31494 \beta_{5} + 31494 \beta_{6} ) q^{71} + ( 1154736 \beta_{3} + 131220 \beta_{4} - 26244 \beta_{5} - 26244 \beta_{6} - 85293 \beta_{7} ) q^{72} + ( 5456652 \beta_{3} - 1747832 \beta_{4} + 123870 \beta_{5} + 123870 \beta_{6} - 349500 \beta_{7} ) q^{73} + ( -190271014 - 442346 \beta_{1} + 30686 \beta_{2} + 82248 \beta_{5} - 82248 \beta_{6} ) q^{74} + ( -66062340 - 105480 \beta_{1} - 17640 \beta_{2} - 4597560 \beta_{3} + 280045 \beta_{4} - 27945 \beta_{5} + 40095 \beta_{6} + 116640 \beta_{7} ) q^{75} + ( 137279328 + 329472 \beta_{1} - 79088 \beta_{2} + 49056 \beta_{5} - 49056 \beta_{6} ) q^{76} + ( 8973804 \beta_{3} + 2248096 \beta_{4} + 23154 \beta_{5} + 23154 \beta_{6} - 316860 \beta_{7} ) q^{77} + ( -4597074 \beta_{3} - 124560 \beta_{4} - 42120 \beta_{5} - 42120 \beta_{6} - 291924 \beta_{7} ) q^{78} + ( 108038736 + 43112 \beta_{1} + 67740 \beta_{2} + 118458 \beta_{5} - 118458 \beta_{6} ) q^{79} + ( 78777142 + 148854 \beta_{1} - 34036 \beta_{2} - 3919708 \beta_{3} - 89436 \beta_{4} - 61268 \beta_{5} - 212100 \beta_{6} + 248639 \beta_{7} ) q^{80} + 43046721 q^{81} + ( -19206770 \beta_{3} + 759808 \beta_{4} - 106368 \beta_{5} - 106368 \beta_{6} - 811408 \beta_{7} ) q^{82} + ( -10152952 \beta_{3} - 1552220 \beta_{4} - 121680 \beta_{5} - 121680 \beta_{6} + 195000 \beta_{7} ) q^{83} + ( 66062250 + 185058 \beta_{1} + 57456 \beta_{2} - 38880 \beta_{5} + 38880 \beta_{6} ) q^{84} + ( -277899398 + 33724 \beta_{1} - 196376 \beta_{2} + 14078432 \beta_{3} - 1104516 \beta_{4} + 45987 \beta_{5} - 65355 \beta_{6} + 305704 \beta_{7} ) q^{85} + ( 124926452 - 564676 \beta_{1} - 214140 \beta_{2} + 159968 \beta_{5} - 159968 \beta_{6} ) q^{86} + ( 13513716 \beta_{3} - 88218 \beta_{4} - 113319 \beta_{5} - 113319 \beta_{6} + 81324 \beta_{7} ) q^{87} + ( 6488656 \beta_{3} + 1251992 \beta_{4} - 288984 \beta_{5} - 288984 \beta_{6} + 133034 \beta_{7} ) q^{88} + ( 44358538 + 901712 \beta_{1} + 189328 \beta_{2} - 177624 \beta_{5} + 177624 \beta_{6} ) q^{89} + ( -54922131 - 242757 \beta_{1} - 6561 \beta_{2} - 2847474 \beta_{3} + 314928 \beta_{4} + 78732 \beta_{5} - 78732 \beta_{6} - 419904 \beta_{7} ) q^{90} + ( 26662012 + 568208 \beta_{1} + 107908 \beta_{2} - 80454 \beta_{5} + 80454 \beta_{6} ) q^{91} + ( 10502040 \beta_{3} - 3369296 \beta_{4} - 38624 \beta_{5} - 38624 \beta_{6} - 199724 \beta_{7} ) q^{92} + ( -3445092 \beta_{3} + 256340 \beta_{4} + 187110 \beta_{5} + 187110 \beta_{6} + 908172 \beta_{7} ) q^{93} + ( -89738056 - 1989832 \beta_{1} + 241136 \beta_{2} - 51072 \beta_{5} + 51072 \beta_{6} ) q^{94} + ( 212010816 + 986032 \beta_{1} - 229952 \beta_{2} - 8742832 \beta_{3} - 1329248 \beta_{4} + 120524 \beta_{5} + 383548 \beta_{6} - 780600 \beta_{7} ) q^{95} + ( -59447067 + 744117 \beta_{1} - 109806 \beta_{2} + 233604 \beta_{5} - 233604 \beta_{6} ) q^{96} + ( -32681192 \beta_{3} - 1940200 \beta_{4} + 226608 \beta_{5} + 226608 \beta_{6} + 1758824 \beta_{7} ) q^{97} + ( 472541 \beta_{3} + 10038080 \beta_{4} + 478560 \beta_{5} + 478560 \beta_{6} - 1960608 \beta_{7} ) q^{98} + ( 59245830 - 524880 \beta_{1} - 13122 \beta_{2} + 150903 \beta_{5} - 150903 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 1194q^{4} - 690q^{5} + 486q^{6} - 52488q^{9} + O(q^{10})$$ $$8q - 1194q^{4} - 690q^{5} + 486q^{6} - 52488q^{9} + 67090q^{10} - 71988q^{11} + 416364q^{14} + 80190q^{15} - 1505630q^{16} + 851584q^{19} + 2078100q^{20} - 1593108q^{21} + 1242702q^{24} + 1695500q^{25} - 877524q^{26} - 73572q^{29} + 3086100q^{30} + 474088q^{31} - 8124388q^{34} - 36357180q^{35} + 7833834q^{36} + 12959676q^{39} - 15313390q^{40} + 93320088q^{41} - 74555892q^{44} + 4527090q^{45} - 9664072q^{46} + 51329600q^{49} + 67798200q^{50} - 108196236q^{51} - 3188646q^{54} + 64428480q^{55} - 67781220q^{56} + 236526036q^{59} + 63172710q^{60} - 357427760q^{61} - 12137026q^{64} + 19848300q^{65} + 23317308q^{66} + 167059584q^{69} + 200900520q^{70} - 156890664q^{71} - 1523381796q^{74} - 528573600q^{75} + 1098697344q^{76} + 863922280q^{79} + 630213180q^{80} + 344373768q^{81} + 529023636q^{84} - 2223350420q^{85} + 997642392q^{86} + 357382224q^{89} - 440177490q^{90} + 214754328q^{91} - 721679824q^{94} + 1698584640q^{95} - 475022718q^{96} + 472313268q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 939 x^{6} + 217699 x^{4} + 14559561 x^{2} + 31136400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$201 \nu^{6} + 179562 \nu^{4} + 32600241 \nu^{2} + 600254096$$$$)/2360192$$ $$\beta_{2}$$ $$=$$ $$($$$$1023 \nu^{6} + 834630 \nu^{4} + 129381687 \nu^{2} + 3318414960$$$$)/1180096$$ $$\beta_{3}$$ $$=$$ $$($$$$20285 \nu^{7} + 18191922 \nu^{5} + 3715480325 \nu^{3} + 205760824848 \nu$$$$)/ 9877403520$$ $$\beta_{4}$$ $$=$$ $$($$$$597 \nu^{7} + 559746 \nu^{5} + 132904173 \nu^{3} + 10255365504 \nu$$$$)/ 182914880$$ $$\beta_{5}$$ $$=$$ $$($$$$31883 \nu^{7} + 652860 \nu^{6} + 40544322 \nu^{5} + 486481140 \nu^{4} + 15087510287 \nu^{3} + 42767050680 \nu^{2} + 1410309996408 \nu - 2075307082560$$$$)/ 2469350880$$ $$\beta_{6}$$ $$=$$ $$($$$$31883 \nu^{7} - 652860 \nu^{6} + 40544322 \nu^{5} - 486481140 \nu^{4} + 15087510287 \nu^{3} - 42767050680 \nu^{2} + 1410309996408 \nu + 2075307082560$$$$)/ 2469350880$$ $$\beta_{7}$$ $$=$$ $$($$$$-7429 \nu^{7} - 6671874 \nu^{5} - 1369843021 \nu^{3} - 67422946896 \nu$$$$)/ 318625920$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$9 \beta_{7} + 2 \beta_{4} + 99 \beta_{3}$$$$)/270$$ $$\nu^{2}$$ $$=$$ $$($$$$6 \beta_{6} - 6 \beta_{5} + 7 \beta_{2} - 34 \beta_{1} - 21122$$$$)/90$$ $$\nu^{3}$$ $$=$$ $$($$$$-421 \beta_{7} - 16 \beta_{6} - 16 \beta_{5} + 1070 \beta_{4} - 6279 \beta_{3}$$$$)/30$$ $$\nu^{4}$$ $$=$$ $$($$$$-2766 \beta_{6} + 2766 \beta_{5} - 4567 \beta_{2} + 29314 \beta_{1} + 10036322$$$$)/90$$ $$\nu^{5}$$ $$=$$ $$($$$$704463 \beta_{7} + 46440 \beta_{6} + 46440 \beta_{5} - 2544506 \beta_{4} + 11457813 \beta_{3}$$$$)/90$$ $$\nu^{6}$$ $$=$$ $$($$$$1497846 \beta_{6} - 1497846 \beta_{5} + 2944567 \beta_{2} - 19616194 \beta_{1} - 5808868802$$$$)/90$$ $$\nu^{7}$$ $$=$$ $$($$$$-143622821 \beta_{7} - 10952096 \beta_{6} - 10952096 \beta_{5} + 562412110 \beta_{4} - 2372069319 \beta_{3}$$$$)/30$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 25.1000i 1.48693i 13.6993i − 10.9137i 10.9137i − 13.6993i − 1.48693i − 25.1000i
33.3847i 81.0000i −602.537 −1324.50 + 445.892i −2704.16 176.118i 3022.54i −6561.00 14886.0 + 44218.1i
4.2 29.7516i 81.0000i −373.156 343.445 + 1354.68i 2409.88 10707.3i 4130.82i −6561.00 40304.0 10218.0i
4.3 20.9703i 81.0000i 72.2477 −743.946 1183.08i 1698.59 3575.78i 12251.8i −6561.00 −24809.4 + 15600.8i
4.4 14.3372i 81.0000i 306.446 1380.00 + 220.717i −1161.31 2878.61i 11734.2i −6561.00 3164.45 19785.3i
4.5 14.3372i 81.0000i 306.446 1380.00 220.717i −1161.31 2878.61i 11734.2i −6561.00 3164.45 + 19785.3i
4.6 20.9703i 81.0000i 72.2477 −743.946 + 1183.08i 1698.59 3575.78i 12251.8i −6561.00 −24809.4 15600.8i
4.7 29.7516i 81.0000i −373.156 343.445 1354.68i 2409.88 10707.3i 4130.82i −6561.00 40304.0 + 10218.0i
4.8 33.3847i 81.0000i −602.537 −1324.50 445.892i −2704.16 176.118i 3022.54i −6561.00 14886.0 44218.1i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.10.b.a 8
3.b odd 2 1 45.10.b.c 8
4.b odd 2 1 240.10.f.c 8
5.b even 2 1 inner 15.10.b.a 8
5.c odd 4 1 75.10.a.i 4
5.c odd 4 1 75.10.a.l 4
15.d odd 2 1 45.10.b.c 8
15.e even 4 1 225.10.a.q 4
15.e even 4 1 225.10.a.u 4
20.d odd 2 1 240.10.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.b.a 8 1.a even 1 1 trivial
15.10.b.a 8 5.b even 2 1 inner
45.10.b.c 8 3.b odd 2 1
45.10.b.c 8 15.d odd 2 1
75.10.a.i 4 5.c odd 4 1
75.10.a.l 4 5.c odd 4 1
225.10.a.q 4 15.e even 4 1
225.10.a.u 4 15.e even 4 1
240.10.f.c 8 4.b odd 2 1
240.10.f.c 8 20.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(15, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1451 T^{2} + 1581940 T^{4} - 1146579392 T^{6} + 685942325248 T^{8} - 300568908136448 T^{10} + 108710089027747840 T^{12} - 26138892237258358784 T^{14} +$$$$47\!\cdots\!96$$$$T^{16}$$
$3$ $$( 1 + 6561 T^{2} )^{4}$$
$5$ $$1 + 690 T - 609700 T^{2} - 1699931250 T^{3} - 2538785156250 T^{4} - 3320178222656250 T^{5} - 2325820922851562500 T^{6} +$$$$51\!\cdots\!50$$$$T^{7} +$$$$14\!\cdots\!25$$$$T^{8}$$
$7$ $$1 - 187079228 T^{2} + 15253712953314292 T^{4} -$$$$75\!\cdots\!16$$$$T^{6} +$$$$30\!\cdots\!14$$$$T^{8} -$$$$12\!\cdots\!84$$$$T^{10} +$$$$40\!\cdots\!92$$$$T^{12} -$$$$80\!\cdots\!72$$$$T^{14} +$$$$70\!\cdots\!01$$$$T^{16}$$
$11$ $$( 1 + 35994 T + 5523704672 T^{2} + 186668393765490 T^{3} + 15626722625814770286 T^{4} +$$$$44\!\cdots\!90$$$$T^{5} +$$$$30\!\cdots\!32$$$$T^{6} +$$$$47\!\cdots\!74$$$$T^{7} +$$$$30\!\cdots\!61$$$$T^{8} )^{2}$$
$13$ $$1 - 57204710684 T^{2} +$$$$16\!\cdots\!00$$$$T^{4} -$$$$29\!\cdots\!48$$$$T^{6} +$$$$37\!\cdots\!18$$$$T^{8} -$$$$33\!\cdots\!92$$$$T^{10} +$$$$20\!\cdots\!00$$$$T^{12} -$$$$81\!\cdots\!76$$$$T^{14} +$$$$15\!\cdots\!81$$$$T^{16}$$
$17$ $$1 - 118674378380 T^{2} +$$$$17\!\cdots\!36$$$$T^{4} -$$$$19\!\cdots\!60$$$$T^{6} +$$$$18\!\cdots\!86$$$$T^{8} -$$$$27\!\cdots\!40$$$$T^{10} +$$$$34\!\cdots\!16$$$$T^{12} -$$$$33\!\cdots\!20$$$$T^{14} +$$$$39\!\cdots\!61$$$$T^{16}$$
$19$ $$( 1 - 425792 T + 640639588492 T^{2} - 117316391220409664 T^{3} +$$$$17\!\cdots\!54$$$$T^{4} -$$$$37\!\cdots\!56$$$$T^{5} +$$$$66\!\cdots\!72$$$$T^{6} -$$$$14\!\cdots\!88$$$$T^{7} +$$$$10\!\cdots\!81$$$$T^{8} )^{2}$$
$23$ $$1 - 10272938895992 T^{2} +$$$$48\!\cdots\!32$$$$T^{4} -$$$$14\!\cdots\!84$$$$T^{6} +$$$$30\!\cdots\!94$$$$T^{8} -$$$$46\!\cdots\!96$$$$T^{10} +$$$$51\!\cdots\!52$$$$T^{12} -$$$$35\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!21$$$$T^{16}$$
$29$ $$( 1 + 36786 T + 11698955610980 T^{2} - 38232367405598309058 T^{3} +$$$$21\!\cdots\!18$$$$T^{4} -$$$$55\!\cdots\!02$$$$T^{5} +$$$$24\!\cdots\!80$$$$T^{6} +$$$$11\!\cdots\!74$$$$T^{7} +$$$$44\!\cdots\!21$$$$T^{8} )^{2}$$
$31$ $$( 1 - 237044 T + 50199397014268 T^{2} -$$$$19\!\cdots\!72$$$$T^{3} +$$$$11\!\cdots\!74$$$$T^{4} -$$$$51\!\cdots\!12$$$$T^{5} +$$$$35\!\cdots\!88$$$$T^{6} -$$$$43\!\cdots\!84$$$$T^{7} +$$$$48\!\cdots\!81$$$$T^{8} )^{2}$$
$37$ $$1 - 624655700715068 T^{2} +$$$$20\!\cdots\!12$$$$T^{4} -$$$$45\!\cdots\!76$$$$T^{6} +$$$$70\!\cdots\!14$$$$T^{8} -$$$$77\!\cdots\!04$$$$T^{10} +$$$$59\!\cdots\!92$$$$T^{12} -$$$$30\!\cdots\!52$$$$T^{14} +$$$$81\!\cdots\!81$$$$T^{16}$$
$41$ $$( 1 - 46660044 T + 1629598509234068 T^{2} -$$$$39\!\cdots\!32$$$$T^{3} +$$$$78\!\cdots\!54$$$$T^{4} -$$$$12\!\cdots\!52$$$$T^{5} +$$$$17\!\cdots\!28$$$$T^{6} -$$$$16\!\cdots\!64$$$$T^{7} +$$$$11\!\cdots\!41$$$$T^{8} )^{2}$$
$43$ $$1 - 1288760154444440 T^{2} +$$$$10\!\cdots\!96$$$$T^{4} -$$$$57\!\cdots\!80$$$$T^{6} +$$$$31\!\cdots\!06$$$$T^{8} -$$$$14\!\cdots\!20$$$$T^{10} +$$$$65\!\cdots\!96$$$$T^{12} -$$$$20\!\cdots\!60$$$$T^{14} +$$$$40\!\cdots\!01$$$$T^{16}$$
$47$ $$1 - 4836473602067240 T^{2} +$$$$12\!\cdots\!56$$$$T^{4} -$$$$21\!\cdots\!80$$$$T^{6} +$$$$28\!\cdots\!26$$$$T^{8} -$$$$27\!\cdots\!20$$$$T^{10} +$$$$19\!\cdots\!76$$$$T^{12} -$$$$95\!\cdots\!60$$$$T^{14} +$$$$24\!\cdots\!41$$$$T^{16}$$
$53$ $$1 - 4341506689340012 T^{2} +$$$$22\!\cdots\!72$$$$T^{4} -$$$$59\!\cdots\!84$$$$T^{6} +$$$$28\!\cdots\!74$$$$T^{8} -$$$$64\!\cdots\!76$$$$T^{10} +$$$$26\!\cdots\!12$$$$T^{12} -$$$$56\!\cdots\!28$$$$T^{14} +$$$$14\!\cdots\!41$$$$T^{16}$$
$59$ $$( 1 - 118263018 T + 24386862278408432 T^{2} -$$$$17\!\cdots\!86$$$$T^{3} +$$$$26\!\cdots\!54$$$$T^{4} -$$$$15\!\cdots\!54$$$$T^{5} +$$$$18\!\cdots\!72$$$$T^{6} -$$$$76\!\cdots\!42$$$$T^{7} +$$$$56\!\cdots\!41$$$$T^{8} )^{2}$$
$61$ $$( 1 + 178713880 T + 47272829372304076 T^{2} +$$$$59\!\cdots\!20$$$$T^{3} +$$$$82\!\cdots\!06$$$$T^{4} +$$$$69\!\cdots\!20$$$$T^{5} +$$$$64\!\cdots\!56$$$$T^{6} +$$$$28\!\cdots\!80$$$$T^{7} +$$$$18\!\cdots\!61$$$$T^{8} )^{2}$$
$67$ $$1 - 128915336297443400 T^{2} +$$$$88\!\cdots\!36$$$$T^{4} -$$$$39\!\cdots\!00$$$$T^{6} +$$$$12\!\cdots\!86$$$$T^{8} -$$$$29\!\cdots\!00$$$$T^{10} +$$$$48\!\cdots\!16$$$$T^{12} -$$$$52\!\cdots\!00$$$$T^{14} +$$$$30\!\cdots\!61$$$$T^{16}$$
$71$ $$( 1 + 78445332 T + 132742682733898508 T^{2} +$$$$14\!\cdots\!24$$$$T^{3} +$$$$78\!\cdots\!70$$$$T^{4} +$$$$65\!\cdots\!44$$$$T^{5} +$$$$27\!\cdots\!88$$$$T^{6} +$$$$75\!\cdots\!12$$$$T^{7} +$$$$44\!\cdots\!21$$$$T^{8} )^{2}$$
$73$ $$1 - 112790784848235992 T^{2} +$$$$16\!\cdots\!32$$$$T^{4} -$$$$11\!\cdots\!84$$$$T^{6} +$$$$89\!\cdots\!94$$$$T^{8} -$$$$39\!\cdots\!96$$$$T^{10} +$$$$19\!\cdots\!52$$$$T^{12} -$$$$46\!\cdots\!28$$$$T^{14} +$$$$14\!\cdots\!21$$$$T^{16}$$
$79$ $$( 1 - 431961140 T + 452072920533003676 T^{2} -$$$$14\!\cdots\!80$$$$T^{3} +$$$$79\!\cdots\!66$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{5} +$$$$64\!\cdots\!36$$$$T^{6} -$$$$74\!\cdots\!60$$$$T^{7} +$$$$20\!\cdots\!21$$$$T^{8} )^{2}$$
$83$ $$1 - 981210392397464024 T^{2} +$$$$42\!\cdots\!20$$$$T^{4} -$$$$11\!\cdots\!28$$$$T^{6} +$$$$22\!\cdots\!98$$$$T^{8} -$$$$38\!\cdots\!52$$$$T^{10} +$$$$51\!\cdots\!20$$$$T^{12} -$$$$41\!\cdots\!96$$$$T^{14} +$$$$14\!\cdots\!61$$$$T^{16}$$
$89$ $$( 1 - 178691112 T + 708605882924008892 T^{2} -$$$$39\!\cdots\!44$$$$T^{3} +$$$$23\!\cdots\!94$$$$T^{4} -$$$$13\!\cdots\!96$$$$T^{5} +$$$$86\!\cdots\!52$$$$T^{6} -$$$$76\!\cdots\!48$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8} )^{2}$$
$97$ $$1 - 1313769052010852360 T^{2} +$$$$18\!\cdots\!56$$$$T^{4} -$$$$20\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!26$$$$T^{8} -$$$$11\!\cdots\!80$$$$T^{10} +$$$$62\!\cdots\!76$$$$T^{12} -$$$$25\!\cdots\!40$$$$T^{14} +$$$$11\!\cdots\!41$$$$T^{16}$$