# Properties

 Label 60.7.b.a Level $60$ Weight $7$ Character orbit 60.b Analytic conductor $13.803$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 60.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.8032450172$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 1880 x^{10} + 1266870 x^{8} + 399545800 x^{6} + 62009694600 x^{4} + 4432082624000 x^{2} + 109931031040000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{22}\cdot 3^{8}\cdot 5^{9}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{3} q^{5} + ( -\beta_{1} + \beta_{5} ) q^{7} + ( 59 + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{3} q^{5} + ( -\beta_{1} + \beta_{5} ) q^{7} + ( 59 + \beta_{6} ) q^{9} -\beta_{11} q^{11} + ( 5 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{10} ) q^{13} + ( 207 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{15} + ( -3 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} ) q^{17} + ( -14 + \beta_{1} + \beta_{3} - 4 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{19} + ( 446 + 2 \beta_{1} + 10 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - 7 \beta_{11} ) q^{21} + ( 20 \beta_{1} - 3 \beta_{2} + 14 \beta_{3} + 4 \beta_{4} - 3 \beta_{8} + 3 \beta_{9} ) q^{23} + ( 1553 - 22 \beta_{1} + \beta_{2} - \beta_{3} + 13 \beta_{5} + 10 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{25} + ( 49 \beta_{1} + 14 \beta_{2} - 29 \beta_{3} - 5 \beta_{4} - 13 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 7 \beta_{10} ) q^{27} + ( -12 - 6 \beta_{1} - 42 \beta_{3} + 9 \beta_{4} + 30 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 9 \beta_{11} ) q^{29} + ( 3402 - \beta_{1} - \beta_{3} + 16 \beta_{6} - \beta_{7} + 5 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{31} + ( -57 \beta_{1} - 11 \beta_{2} - 82 \beta_{3} - 16 \beta_{4} + 79 \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{33} + ( 12 + 183 \beta_{1} - 21 \beta_{2} + 7 \beta_{3} + 20 \beta_{4} - 30 \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - 24 \beta_{11} ) q^{35} + ( 255 \beta_{1} + 35 \beta_{2} + 53 \beta_{5} + 7 \beta_{10} ) q^{37} + ( -3780 + 42 \beta_{1} + 174 \beta_{3} - 33 \beta_{4} + 9 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 15 \beta_{11} ) q^{39} + ( 12 - 49 \beta_{1} - 233 \beta_{3} + 46 \beta_{4} - 30 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 43 \beta_{11} ) q^{41} + ( -56 \beta_{1} - 33 \beta_{2} - 208 \beta_{5} ) q^{43} + ( 3782 + 242 \beta_{1} - 71 \beta_{2} - 10 \beta_{3} - 70 \beta_{4} - 143 \beta_{5} - 5 \beta_{6} - 9 \beta_{7} + 8 \beta_{8} - 11 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} ) q^{45} + ( -1530 \beta_{1} + 147 \beta_{2} + 384 \beta_{3} + 74 \beta_{4} + 7 \beta_{8} - 7 \beta_{9} ) q^{47} + ( -20231 + 7 \beta_{1} + 7 \beta_{3} - 100 \beta_{6} + 7 \beta_{7} - 29 \beta_{8} - 22 \beta_{9} - 22 \beta_{10} + 7 \beta_{11} ) q^{49} + ( -3390 + 97 \beta_{1} + 521 \beta_{3} - 106 \beta_{4} - 12 \beta_{6} - 9 \beta_{7} - \beta_{8} - 10 \beta_{9} - 10 \beta_{10} - 2 \beta_{11} ) q^{51} + ( 793 \beta_{1} - 84 \beta_{2} + 643 \beta_{3} + 115 \beta_{4} + 34 \beta_{8} - 34 \beta_{9} ) q^{53} + ( -2032 - 1987 \beta_{1} - 229 \beta_{2} - 6 \beta_{3} + 273 \beta_{5} + 30 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} - 9 \beta_{9} + 22 \beta_{10} - 6 \beta_{11} ) q^{55} + ( -342 \beta_{1} + 327 \beta_{2} - 867 \beta_{3} - 183 \beta_{4} - 195 \beta_{5} + 24 \beta_{8} - 24 \beta_{9} + 57 \beta_{10} ) q^{57} + ( -24 - 176 \beta_{1} - 904 \beta_{3} + 182 \beta_{4} + 60 \beta_{6} + 6 \beta_{7} - 6 \beta_{8} + 99 \beta_{11} ) q^{59} + ( -8286 - 5 \beta_{1} - 5 \beta_{3} - 52 \beta_{6} - 5 \beta_{7} - 41 \beta_{8} - 46 \beta_{9} - 46 \beta_{10} - 5 \beta_{11} ) q^{61} + ( -73 \beta_{1} - 209 \beta_{2} - 1099 \beta_{3} - 223 \beta_{4} + 430 \beta_{5} + 8 \beta_{8} - 8 \beta_{9} + \beta_{10} ) q^{63} + ( 12 + 3738 \beta_{1} - 396 \beta_{2} - 38 \beta_{3} + 265 \beta_{4} - 30 \beta_{6} - 3 \beta_{7} + 26 \beta_{8} - 23 \beta_{9} + 151 \beta_{11} ) q^{65} + ( 6344 \beta_{1} + 821 \beta_{2} + 24 \beta_{5} - 50 \beta_{10} ) q^{67} + ( 9036 + 290 \beta_{1} + 1558 \beta_{3} - 317 \beta_{4} - 9 \beta_{6} - 27 \beta_{7} - 8 \beta_{8} - 35 \beta_{9} - 35 \beta_{10} - 16 \beta_{11} ) q^{69} + ( -284 \beta_{1} - 1420 \beta_{3} + 284 \beta_{4} - 254 \beta_{11} ) q^{71} + ( -3634 \beta_{1} - 538 \beta_{2} - 366 \beta_{5} + 76 \beta_{10} ) q^{73} + ( 10552 + 1747 \beta_{1} - 741 \beta_{2} + 41 \beta_{3} - 325 \beta_{4} - 273 \beta_{5} - 10 \beta_{6} + 36 \beta_{7} - 22 \beta_{8} + 44 \beta_{9} - 37 \beta_{10} - 59 \beta_{11} ) q^{75} + ( -14676 \beta_{1} + 1428 \beta_{2} + 2028 \beta_{3} + 404 \beta_{4} + 4 \beta_{8} - 4 \beta_{9} ) q^{77} + ( 52170 - 21 \beta_{1} - 21 \beta_{3} + 348 \beta_{6} - 21 \beta_{7} + 111 \beta_{8} + 90 \beta_{9} + 90 \beta_{10} - 21 \beta_{11} ) q^{79} + ( -66527 + 483 \beta_{1} + 2235 \beta_{3} - 438 \beta_{4} - 16 \beta_{6} + 45 \beta_{7} - 15 \beta_{8} + 30 \beta_{9} + 30 \beta_{10} + 213 \beta_{11} ) q^{81} + ( 6856 \beta_{1} - 759 \beta_{2} + 1966 \beta_{3} + 450 \beta_{4} - 142 \beta_{8} + 142 \beta_{9} ) q^{83} + ( -61006 - 9466 \beta_{1} - 1197 \beta_{2} + 27 \beta_{3} - 311 \beta_{5} - 180 \beta_{6} + 27 \beta_{7} - 9 \beta_{8} + 18 \beta_{9} - 39 \beta_{10} + 27 \beta_{11} ) q^{85} + ( -1467 \beta_{1} + 2343 \beta_{2} - 1812 \beta_{3} - 318 \beta_{4} + 507 \beta_{5} - 111 \beta_{8} + 111 \beta_{9} - 132 \beta_{10} ) q^{87} + ( 288 - 532 \beta_{1} - 2372 \beta_{3} + 460 \beta_{4} - 720 \beta_{6} - 72 \beta_{7} + 72 \beta_{8} - 318 \beta_{11} ) q^{89} + ( 166400 + 32 \beta_{1} + 32 \beta_{3} + 136 \beta_{6} + 32 \beta_{7} + 164 \beta_{8} + 196 \beta_{9} + 196 \beta_{10} + 32 \beta_{11} ) q^{91} + ( 3058 \beta_{1} - 1155 \beta_{2} - 2517 \beta_{3} - 501 \beta_{4} - 741 \beta_{5} - 6 \beta_{8} + 6 \beta_{9} - 75 \beta_{10} ) q^{93} + ( -300 + 20545 \beta_{1} - 2100 \beta_{2} + 111 \beta_{3} + 330 \beta_{4} + 750 \beta_{6} + 75 \beta_{7} - 175 \beta_{8} + 100 \beta_{9} - 325 \beta_{11} ) q^{95} + ( 27276 \beta_{1} + 3400 \beta_{2} + 196 \beta_{5} + 68 \beta_{10} ) q^{97} + ( 136096 + 228 \beta_{1} + 1212 \beta_{3} - 246 \beta_{4} + 68 \beta_{6} - 18 \beta_{7} + 168 \beta_{8} + 150 \beta_{9} + 150 \beta_{10} - 393 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 712q^{9} + O(q^{10})$$ $$12q + 712q^{9} + 2480q^{15} - 192q^{19} + 5348q^{21} + 18660q^{25} + 40848q^{31} - 45312q^{39} + 45340q^{45} - 242940q^{49} - 40720q^{51} - 24240q^{55} - 99312q^{61} + 108460q^{69} + 126640q^{75} + 626544q^{79} - 798268q^{81} - 732720q^{85} + 1996032q^{91} + 1632080q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 1880 x^{10} + 1266870 x^{8} + 399545800 x^{6} + 62009694600 x^{4} + 4432082624000 x^{2} + 109931031040000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2161174211789 \nu^{11} + 480484922547160 \nu^{10} - 3184303169286640 \nu^{9} + 766542184953294600 \nu^{8} - 1297601919861485230 \nu^{7} + 394800526498394495200 \nu^{6} - 102680523661566568600 \nu^{5} + 85715934913900011034000 \nu^{4} + 24679618512876342676600 \nu^{3} + 7909049931361256107236000 \nu^{2} + 2273205106370932270032000 \nu + 238344569813409910695520000$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$-2161174211789 \nu^{11} - 384387938037728 \nu^{10} - 3184303169286640 \nu^{9} - 613233747962635680 \nu^{8} - 1297601919861485230 \nu^{7} - 315840421198715596160 \nu^{6} - 102680523661566568600 \nu^{5} - 68572747931120008827200 \nu^{4} + 24679618512876342676600 \nu^{3} - 6327239945089004885788800 \nu^{2} + 2273205106370932270032000 \nu - 190675655850727928556416000$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$14002169117329 \nu^{11} - 782772258266396 \nu^{10} + 23031666619364800 \nu^{9} - 1141139858880097760 \nu^{8} + 12411881740750125830 \nu^{7} - 490174618297762769320 \nu^{6} + 2813510140949802987800 \nu^{5} - 77702288100895866594400 \nu^{4} + 272810676523576274991400 \nu^{3} - 4274179677195675252053600 \nu^{2} + 11190527948968977356944000 \nu - 48219832208291779317152000$$$$)/$$$$26\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-89134437642148 \nu^{11} - 4531962922254605 \nu^{10} - 146336143747994980 \nu^{9} - 6557217479285640050 \nu^{8} - 78547462319584400360 \nu^{7} - 2767490969487221436850 \nu^{6} - 17661448773682443600200 \nu^{5} - 421352349445174157939500 \nu^{4} - 1686557014387694461688800 \nu^{3} - 20781835533952028244908000 \nu^{2} - 68235895851277909278376000 \nu - 122615523941766187710560000$$$$)/$$$$33\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$27240080124843 \nu^{11} + 96096984509432 \nu^{10} + 47429962122598080 \nu^{9} + 153308436990658920 \nu^{8} + 28052555955197219010 \nu^{7} + 78960105299678899040 \nu^{6} + 7203044892847178392200 \nu^{5} + 17143186982780002206800 \nu^{4} + 800386696448907098935800 \nu^{3} + 1581809986272251221447200 \nu^{2} + 29213744136700452449616000 \nu + 47668913962681982139104000$$$$)/$$$$89\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$881775295611 \nu^{11} - 15699416590600 \nu^{10} + 1182517185355900 \nu^{9} - 26668473966991600 \nu^{8} + 416749291051518170 \nu^{7} - 15045650595148350000 \nu^{6} + 42750008345038431200 \nu^{5} - 3539031638755691652000 \nu^{4} + 1606538739258979670600 \nu^{3} - 331321523682455423480000 \nu^{2} + 196113358764323204176000 \nu - 9024566708955550716128000$$$$)/$$$$19\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$62913551899733 \nu^{11} + 7088792759002196 \nu^{10} + 74769096794570630 \nu^{9} + 11996558150956738160 \nu^{8} + 18458285237961948010 \nu^{7} + 6733948652627128027720 \nu^{6} + 654192334995046443700 \nu^{5} + 1574046063689129398710400 \nu^{4} + 424288383436038731358800 \nu^{3} + 144657369969017936072897600 \nu^{2} + 84180774999801132086336000 \nu + 3698258478339303494518256000$$$$)/$$$$66\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$494627258408677 \nu^{11} + 7744368545303355 \nu^{10} + 932830647249693670 \nu^{9} + 13128154976838029550 \nu^{8} + 609085850997861033890 \nu^{7} + 7399378108506328798350 \nu^{6} + 168750654784886128685300 \nu^{5} + 1732559832651384068884500 \nu^{4} + 18848305539731890937651200 \nu^{3} + 152346284344880422876848000 \nu^{2} + 632124018703190564213464000 \nu + 3041059798899173836641840000$$$$)/$$$$33\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$526390919884316 \nu^{11} - 4487268979257375 \nu^{10} + 985633359044175710 \nu^{9} - 7050745183199751750 \nu^{8} + 638169152469944120620 \nu^{7} - 3421815603544074237750 \nu^{6} + 175630409351768286301900 \nu^{5} - 605889497538782164102500 \nu^{4} + 19567351658810146139144600 \nu^{3} - 33215843337055182799440000 \nu^{2} + 663510146235169406010872000 \nu - 948691444751058853270320000$$$$)/$$$$33\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-110435887939899 \nu^{11} - 120121230636790 \nu^{10} - 193869369603471240 \nu^{9} - 191635546238323650 \nu^{8} - 115919459073381576930 \nu^{7} - 98700131624598623800 \nu^{6} - 29927484751307722812600 \nu^{5} - 21428983728475002758500 \nu^{4} - 3247492755298865421929400 \nu^{3} - 1977262482840314026809000 \nu^{2} - 113607772809337622972088000 \nu - 59586142453352477673880000$$$$)/$$$$27\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$24017870888963 \nu^{11} + 37642739533404200 \nu^{9} + 18736813838662692610 \nu^{7} + 3844083953784745264600 \nu^{5} + 323206330838086289139800 \nu^{3} + 8107191565790634863408000 \nu$$$$)/$$$$58\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{11} + \beta_{8} - \beta_{7} - 10 \beta_{6} - 90 \beta_{5} - 57 \beta_{4} + 284 \beta_{3} + 70 \beta_{2} + 706 \beta_{1} + 4$$$$)/3600$$ $$\nu^{2}$$ $$=$$ $$($$$$-16 \beta_{11} - 21 \beta_{10} - 54 \beta_{9} + 28 \beta_{8} - 16 \beta_{7} + 86 \beta_{6} + 42 \beta_{4} + 260 \beta_{3} - 116 \beta_{2} + 1168 \beta_{1} - 225632$$$$)/720$$ $$\nu^{3}$$ $$=$$ $$($$$$-1948 \beta_{11} - 1665 \beta_{10} - 569 \beta_{8} + 569 \beta_{7} + 5690 \beta_{6} + 9900 \beta_{5} + 12021 \beta_{4} - 59536 \beta_{3} - 26650 \beta_{2} - 241212 \beta_{1} - 2276$$$$)/1440$$ $$\nu^{4}$$ $$=$$ $$($$$$3650 \beta_{11} + 2715 \beta_{10} + 11481 \beta_{9} - 9701 \beta_{8} + 3650 \beta_{7} - 23770 \beta_{6} - 8226 \beta_{4} - 55012 \beta_{3} + 23472 \beta_{2} - 240376 \beta_{1} + 30027100$$$$)/180$$ $$\nu^{5}$$ $$=$$ $$($$$$421258 \beta_{11} + 461625 \beta_{10} + 122204 \beta_{8} - 122204 \beta_{7} - 1222040 \beta_{6} - 408450 \beta_{5} - 2299488 \beta_{4} + 11375236 \beta_{3} + 5528100 \beta_{2} + 48657034 \beta_{1} + 488816$$$$)/360$$ $$\nu^{6}$$ $$=$$ $$($$$$-1260784 \beta_{11} - 578199 \beta_{10} - 3682554 \beta_{9} + 3786940 \beta_{8} - 1260784 \beta_{7} + 8929874 \beta_{6} + 2702478 \beta_{4} + 18460316 \beta_{3} - 7968220 \beta_{2} + 81927648 \beta_{1} - 8380472768$$$$)/72$$ $$\nu^{7}$$ $$=$$ $$($$$$-140785604 \beta_{11} - 170671515 \beta_{10} - 41417707 \beta_{8} + 41417707 \beta_{7} + 414177070 \beta_{6} - 80505420 \beta_{5} + 765470271 \beta_{4} - 3785933648 \beta_{3} - 1826039790 \beta_{2} - 15934551524 \beta_{1} - 165670828$$$$)/144$$ $$\nu^{8}$$ $$=$$ $$($$$$89987290 \beta_{11} + 30408210 \beta_{10} + 251957571 \beta_{9} - 281128441 \beta_{8} + 89987290 \beta_{7} - 659081900 \beta_{6} - 188815056 \beta_{4} - 1297186712 \beta_{3} + 568646412 \beta_{2} - 5850760496 \beta_{1} + 548126853980$$$$)/6$$ $$\nu^{9}$$ $$=$$ $$($$$$29579633162 \beta_{11} + 37371854700 \beta_{10} + 8793634981 \beta_{8} - 8793634981 \beta_{7} - 87936349810 \beta_{6} + 35230688190 \beta_{5} - 161881371597 \beta_{4} + 800613223004 \beta_{3} + 381797620630 \beta_{2} + 3321725432266 \beta_{1} + 35174539924$$$$)/36$$ $$\nu^{10}$$ $$=$$ $$($$$$-460452788480 \beta_{11} - 133110669705 \beta_{10} - 1264027192830 \beta_{9} + 1458258641900 \beta_{8} - 460452788480 \beta_{7} + 3417400968430 \beta_{6} + 956011717650 \beta_{4} + 6581438846020 \beta_{3} - 2913358610500 \beta_{2} + 29978954645120 \beta_{1} - 2707224311836960$$$$)/36$$ $$\nu^{11}$$ $$=$$ $$($$$$-49974254127980 \beta_{11} - 64228670133525 \beta_{10} - 14944476215965 \beta_{8} + 14944476215965 \beta_{7} + 149444762159650 \beta_{6} - 72009824818500 \beta_{5} + 274751399804745 \beta_{4} - 1358812522807760 \beta_{3} - 643999910605250 \beta_{2} - 5596711064146380 \beta_{1} - 59777904863860$$$$)/72$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 18.2420i 18.2420i 11.1131i − 11.1131i − 29.1428i 29.1428i 13.3561i − 13.3561i − 7.15535i 7.15535i − 18.5699i 18.5699i
0 −24.4963 11.3548i 0 −124.726 8.26448i 0 577.271i 0 471.136 + 556.302i 0
29.2 0 −24.4963 + 11.3548i 0 −124.726 + 8.26448i 0 577.271i 0 471.136 556.302i 0
29.3 0 −22.8848 14.3278i 0 93.9389 82.4651i 0 279.744i 0 318.429 + 655.777i 0
29.4 0 −22.8848 + 14.3278i 0 93.9389 + 82.4651i 0 279.744i 0 318.429 655.777i 0
29.5 0 −7.66273 25.8898i 0 37.2664 + 119.316i 0 46.7246i 0 −611.565 + 396.774i 0
29.6 0 −7.66273 + 25.8898i 0 37.2664 119.316i 0 46.7246i 0 −611.565 396.774i 0
29.7 0 7.66273 25.8898i 0 −37.2664 119.316i 0 46.7246i 0 −611.565 396.774i 0
29.8 0 7.66273 + 25.8898i 0 −37.2664 + 119.316i 0 46.7246i 0 −611.565 + 396.774i 0
29.9 0 22.8848 14.3278i 0 −93.9389 + 82.4651i 0 279.744i 0 318.429 655.777i 0
29.10 0 22.8848 + 14.3278i 0 −93.9389 82.4651i 0 279.744i 0 318.429 + 655.777i 0
29.11 0 24.4963 11.3548i 0 124.726 + 8.26448i 0 577.271i 0 471.136 556.302i 0
29.12 0 24.4963 + 11.3548i 0 124.726 8.26448i 0 577.271i 0 471.136 + 556.302i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.7.b.a 12
3.b odd 2 1 inner 60.7.b.a 12
4.b odd 2 1 240.7.c.e 12
5.b even 2 1 inner 60.7.b.a 12
5.c odd 4 2 300.7.g.i 12
12.b even 2 1 240.7.c.e 12
15.d odd 2 1 inner 60.7.b.a 12
15.e even 4 2 300.7.g.i 12
20.d odd 2 1 240.7.c.e 12
60.h even 2 1 240.7.c.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.b.a 12 1.a even 1 1 trivial
60.7.b.a 12 3.b odd 2 1 inner
60.7.b.a 12 5.b even 2 1 inner
60.7.b.a 12 15.d odd 2 1 inner
240.7.c.e 12 4.b odd 2 1
240.7.c.e 12 12.b even 2 1
240.7.c.e 12 20.d odd 2 1
240.7.c.e 12 60.h even 2 1
300.7.g.i 12 5.c odd 4 2
300.7.g.i 12 15.e even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$150094635296999121 - 100544914987236 T^{2} + 139734439335 T^{4} + 355606200 T^{6} + 262935 T^{8} - 356 T^{10} + T^{12}$$
$5$ $$14\!\cdots\!25$$$$-$$$$55\!\cdots\!50$$$$T^{2} - 10288238525390625 T^{4} - 1333635937500 T^{6} - 42140625 T^{8} - 9330 T^{10} + T^{12}$$
$7$ $$( 56934147702784 + 26976849408 T^{2} + 413682 T^{4} + T^{6} )^{2}$$
$11$ $$( 463787739345920000 + 3200405184000 T^{2} + 4656720 T^{4} + T^{6} )^{2}$$
$13$ $$($$$$15\!\cdots\!04$$$$+ 90569687597568 T^{2} + 16949952 T^{4} + T^{6} )^{2}$$
$17$ $$( -$$$$37\!\cdots\!00$$$$+ 337924580874000 T^{2} - 44984520 T^{4} + T^{6} )^{2}$$
$19$ $$( 543444769296 - 128591532 T + 48 T^{2} + T^{3} )^{4}$$
$23$ $$( -$$$$37\!\cdots\!00$$$$+ 37535620594644000 T^{2} - 417908730 T^{4} + T^{6} )^{2}$$
$29$ $$($$$$64\!\cdots\!00$$$$+ 2049299400838464000 T^{2} + 2831129280 T^{4} + T^{6} )^{2}$$
$31$ $$( -5778576379264 - 828201852 T - 10212 T^{2} + T^{3} )^{4}$$
$37$ $$($$$$17\!\cdots\!96$$$$+ 1425670256955283968 T^{2} + 2388481248 T^{4} + T^{6} )^{2}$$
$41$ $$($$$$62\!\cdots\!00$$$$+ 40341176509563264000 T^{2} + 15214549380 T^{4} + T^{6} )^{2}$$
$43$ $$($$$$35\!\cdots\!64$$$$+ 68025096365278693248 T^{2} + 17921583762 T^{4} + T^{6} )^{2}$$
$47$ $$( -$$$$10\!\cdots\!00$$$$+ 67293495198670164000 T^{2} - 27594267930 T^{4} + T^{6} )^{2}$$
$53$ $$( -$$$$55\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T^{2} - 91610745480 T^{4} + T^{6} )^{2}$$
$59$ $$($$$$56\!\cdots\!00$$$$+$$$$47\!\cdots\!00$$$$T^{2} + 122334999120 T^{4} + T^{6} )^{2}$$
$61$ $$( 2163159473987776 - 58827359772 T + 24828 T^{2} + T^{3} )^{4}$$
$67$ $$($$$$42\!\cdots\!84$$$$+$$$$18\!\cdots\!08$$$$T^{2} + 247387997682 T^{4} + T^{6} )^{2}$$
$71$ $$($$$$80\!\cdots\!00$$$$+$$$$56\!\cdots\!00$$$$T^{2} + 479400318720 T^{4} + T^{6} )^{2}$$
$73$ $$($$$$41\!\cdots\!96$$$$+$$$$13\!\cdots\!68$$$$T^{2} + 219153404448 T^{4} + T^{6} )^{2}$$
$79$ $$( -74225002555506528 - 413760141468 T - 156636 T^{2} + T^{3} )^{4}$$
$83$ $$( -$$$$18\!\cdots\!00$$$$+$$$$46\!\cdots\!00$$$$T^{2} - 1440801120570 T^{4} + T^{6} )^{2}$$
$89$ $$($$$$35\!\cdots\!00$$$$+$$$$17\!\cdots\!00$$$$T^{2} + 2643942113280 T^{4} + T^{6} )^{2}$$
$97$ $$($$$$40\!\cdots\!24$$$$+$$$$25\!\cdots\!28$$$$T^{2} + 3890595351072 T^{4} + T^{6} )^{2}$$