# Properties

 Label 60.7.g.a Level $60$ Weight $7$ Character orbit 60.g Analytic conductor $13.803$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.8032450172$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} - 202 x^{6} + 620 x^{5} + 12167 x^{4} - 25372 x^{3} - 177926 x^{2} + 190716 x + 977814$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}\cdot 3^{7}\cdot 5^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -3 - \beta_{1} ) q^{3} + \beta_{2} q^{5} + ( -71 - \beta_{4} + \beta_{5} ) q^{7} + ( 187 + 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -3 - \beta_{1} ) q^{3} + \beta_{2} q^{5} + ( -71 - \beta_{4} + \beta_{5} ) q^{7} + ( 187 + 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{9} + ( -11 - 22 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{11} + ( -805 - 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{13} + ( 250 - 3 \beta_{2} - 5 \beta_{3} ) q^{15} + ( 44 + 88 \beta_{1} - 33 \beta_{2} - 12 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 7 \beta_{6} + 5 \beta_{7} ) q^{17} + ( -1897 + 10 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 12 \beta_{5} - 8 \beta_{6} + 13 \beta_{7} ) q^{19} + ( -63 + 70 \beta_{1} + 6 \beta_{2} - 9 \beta_{3} - 21 \beta_{5} + 3 \beta_{6} + 27 \beta_{7} ) q^{21} + ( -12 - 24 \beta_{1} - 65 \beta_{2} - 35 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} - 33 \beta_{6} - 14 \beta_{7} ) q^{23} -3125 q^{25} + ( -2480 - 259 \beta_{1} + 131 \beta_{2} - 30 \beta_{3} - 16 \beta_{4} + 30 \beta_{5} - 17 \beta_{6} + 65 \beta_{7} ) q^{27} + ( -123 - 246 \beta_{1} - 128 \beta_{2} - 77 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 18 \beta_{6} + 67 \beta_{7} ) q^{29} + ( 4313 + 22 \beta_{1} + 8 \beta_{2} + 17 \beta_{3} - 94 \beta_{4} + 60 \beta_{5} - 26 \beta_{6} + 43 \beta_{7} ) q^{31} + ( -13857 + 24 \beta_{1} + 162 \beta_{2} - 9 \beta_{3} - 6 \beta_{4} - 48 \beta_{5} - 36 \beta_{6} - 87 \beta_{7} ) q^{33} + ( 30 + 60 \beta_{1} - 113 \beta_{2} + 5 \beta_{3} - 10 \beta_{4} - 10 \beta_{5} - 45 \beta_{6} - 70 \beta_{7} ) q^{35} + ( 12577 + 28 \beta_{1} + 11 \beta_{2} + 23 \beta_{3} + 55 \beta_{4} - 101 \beta_{5} - 35 \beta_{6} + 58 \beta_{7} ) q^{37} + ( 5211 + 670 \beta_{1} + 330 \beta_{2} + 27 \beta_{3} - 72 \beta_{4} + 78 \beta_{5} + 30 \beta_{6} - 45 \beta_{7} ) q^{39} + ( 131 + 262 \beta_{1} - 460 \beta_{2} + 5 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} + 196 \beta_{6} + 211 \beta_{7} ) q^{41} + ( -20566 - 1006 \beta_{1} + 76 \beta_{2} - 41 \beta_{3} + 103 \beta_{4} - 21 \beta_{5} + 158 \beta_{6} - 199 \beta_{7} ) q^{43} + ( -745 - 290 \beta_{1} + 154 \beta_{2} - 15 \beta_{3} + 115 \beta_{4} - 60 \beta_{5} - 70 \beta_{6} - 20 \beta_{7} ) q^{45} + ( 62 + 124 \beta_{1} - 845 \beta_{2} + 295 \beta_{3} + 48 \beta_{4} + 48 \beta_{5} + 103 \beta_{6} - 96 \beta_{7} ) q^{47} + ( 26286 + 1802 \beta_{1} - 230 \beta_{2} - 83 \beta_{3} + 148 \beta_{4} + 18 \beta_{5} - 64 \beta_{6} - 19 \beta_{7} ) q^{49} + ( 53508 - 762 \beta_{1} + 1971 \beta_{2} + 222 \beta_{3} + 339 \beta_{4} - 159 \beta_{5} + 153 \beta_{6} + 15 \beta_{7} ) q^{51} + ( 606 + 1212 \beta_{1} - 1444 \beta_{2} + 254 \beta_{3} - 70 \beta_{4} - 70 \beta_{5} - 138 \beta_{6} - 532 \beta_{7} ) q^{53} + ( -3795 + 860 \beta_{1} - 125 \beta_{2} - 65 \beta_{3} - 35 \beta_{4} + 165 \beta_{5} + 5 \beta_{6} - 70 \beta_{7} ) q^{55} + ( -3189 + 1180 \beta_{1} + 2037 \beta_{2} + 117 \beta_{3} - 405 \beta_{4} + 363 \beta_{5} + 249 \beta_{6} - 108 \beta_{7} ) q^{57} + ( -1233 - 2466 \beta_{1} - 2705 \beta_{2} + 505 \beta_{3} + 235 \beta_{4} + 235 \beta_{5} - 93 \beta_{6} - 128 \beta_{7} ) q^{59} + ( -18293 - 1802 \beta_{1} + 194 \beta_{2} + 23 \beta_{3} - 464 \beta_{4} + 418 \beta_{5} + 148 \beta_{6} - 125 \beta_{7} ) q^{61} + ( 46400 - 1274 \beta_{1} + 4021 \beta_{2} + 297 \beta_{3} - 92 \beta_{4} + 144 \beta_{5} + 455 \beta_{6} - 416 \beta_{7} ) q^{63} + ( -755 - 1510 \beta_{1} - 697 \beta_{2} - 105 \beta_{3} + 85 \beta_{4} + 85 \beta_{5} - 55 \beta_{6} + 220 \beta_{7} ) q^{65} + ( 50370 - 1042 \beta_{1} + 22 \beta_{2} - 137 \beta_{3} + 413 \beta_{4} - 139 \beta_{5} + 296 \beta_{6} - 433 \beta_{7} ) q^{67} + ( -32508 - 1500 \beta_{1} + 3330 \beta_{2} - 366 \beta_{3} + 456 \beta_{4} + 237 \beta_{5} - 873 \beta_{6} + 618 \beta_{7} ) q^{69} + ( 1306 + 2612 \beta_{1} - 4088 \beta_{2} + 628 \beta_{3} - 106 \beta_{4} - 106 \beta_{5} - 64 \beta_{6} - 904 \beta_{7} ) q^{71} + ( -48844 - 8820 \beta_{1} + 1188 \beta_{2} + 510 \beta_{3} - 324 \beta_{4} - 696 \beta_{5} + 168 \beta_{6} + 342 \beta_{7} ) q^{73} + ( 9375 + 3125 \beta_{1} ) q^{75} + ( -696 - 1392 \beta_{1} - 4772 \beta_{2} - 1388 \beta_{3} - 116 \beta_{4} - 116 \beta_{5} - 120 \beta_{6} + 1036 \beta_{7} ) q^{77} + ( 96767 + 16386 \beta_{1} - 1914 \beta_{2} - 459 \beta_{3} + 570 \beta_{4} + 348 \beta_{5} - 996 \beta_{6} + 537 \beta_{7} ) q^{79} + ( -148628 + 1898 \beta_{1} + 3980 \beta_{2} - 651 \beta_{3} - 514 \beta_{4} - 888 \beta_{5} - 14 \beta_{6} - 433 \beta_{7} ) q^{81} + ( 5382 + 10764 \beta_{1} - 3776 \beta_{2} + 211 \beta_{3} - 383 \beta_{4} - 383 \beta_{5} + 2490 \beta_{6} + 1513 \beta_{7} ) q^{83} + ( 107130 + 7510 \beta_{1} - 775 \beta_{2} - 40 \beta_{3} - 85 \beta_{4} + 165 \beta_{5} - 695 \beta_{6} + 655 \beta_{7} ) q^{85} + ( -190917 - 1158 \beta_{1} + 6777 \beta_{2} + 579 \beta_{3} + 519 \beta_{4} - 915 \beta_{5} - 675 \beta_{6} - 1182 \beta_{7} ) q^{87} + ( -4614 - 9228 \beta_{1} - 11040 \beta_{2} - 2850 \beta_{3} - 60 \beta_{4} - 60 \beta_{5} - 2184 \beta_{6} + 546 \beta_{7} ) q^{89} + ( -255924 - 13116 \beta_{1} + 1488 \beta_{2} + 294 \beta_{3} + 854 \beta_{4} - 1442 \beta_{5} + 900 \beta_{6} - 606 \beta_{7} ) q^{91} + ( -62139 - 6836 \beta_{1} + 7383 \beta_{2} - 513 \beta_{3} - 1359 \beta_{4} - 897 \beta_{5} + 1113 \beta_{6} + 2340 \beta_{7} ) q^{93} + ( -4375 - 8750 \beta_{1} - 1474 \beta_{2} - 375 \beta_{3} + 500 \beta_{4} + 500 \beta_{5} - 500 \beta_{6} + 875 \beta_{7} ) q^{95} + ( -168786 - 5088 \beta_{1} + 984 \beta_{2} + 792 \beta_{3} - 1244 \beta_{4} - 340 \beta_{5} - 600 \beta_{6} + 1392 \beta_{7} ) q^{97} + ( 336369 + 11166 \beta_{1} + 969 \beta_{2} - 1827 \beta_{3} + 591 \beta_{4} + 1539 \beta_{5} - 1491 \beta_{6} + 1482 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 20q^{3} - 560q^{7} + 1492q^{9} + O(q^{10})$$ $$8q - 20q^{3} - 560q^{7} + 1492q^{9} - 6440q^{13} + 2000q^{15} - 15272q^{19} - 868q^{21} - 25000q^{25} - 18620q^{27} + 35032q^{31} - 111120q^{33} + 99880q^{37} + 39608q^{39} - 161000q^{43} - 5500q^{45} + 202560q^{49} + 429120q^{51} - 33000q^{55} - 27160q^{57} - 135608q^{61} + 377240q^{63} + 404920q^{67} - 254940q^{69} - 356960q^{73} + 62500q^{75} + 707704q^{79} - 1198112q^{81} + 828000q^{85} - 1528440q^{87} - 2004112q^{91} - 467920q^{93} - 1326320q^{97} + 2650080q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 202 x^{6} + 620 x^{5} + 12167 x^{4} - 25372 x^{3} - 177926 x^{2} + 190716 x + 977814$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$23087 \nu^{7} + 111002 \nu^{6} - 5186194 \nu^{5} - 22636834 \nu^{4} + 334471142 \nu^{3} + 1309113875 \nu^{2} - 5239001274 \nu - 11257731684$$$$)/ 281404449$$ $$\beta_{2}$$ $$=$$ $$($$$$200 \nu^{7} - 700 \nu^{6} - 40750 \nu^{5} + 103625 \nu^{4} + 2282600 \nu^{3} - 3527875 \nu^{2} - 16480050 \nu + 8831475$$$$)/437643$$ $$\beta_{3}$$ $$=$$ $$($$$$157940 \nu^{7} - 1552345 \nu^{6} - 29586835 \nu^{5} + 339001955 \nu^{4} + 1585530710 \nu^{3} - 19979889265 \nu^{2} - 16503039600 \nu + 153005351670$$$$)/ 281404449$$ $$\beta_{4}$$ $$=$$ $$($$$$260806 \nu^{7} - 2593154 \nu^{6} - 41476847 \nu^{5} + 462577573 \nu^{4} + 1454099926 \nu^{3} - 24393919850 \nu^{2} + 15960017778 \nu + 240704956512$$$$)/ 281404449$$ $$\beta_{5}$$ $$=$$ $$($$$$273517 \nu^{7} - 3245480 \nu^{6} - 41161250 \nu^{5} + 558849883 \nu^{4} + 1242783286 \nu^{3} - 25876704038 \nu^{2} + 18783592692 \nu + 149114143197$$$$)/ 281404449$$ $$\beta_{6}$$ $$=$$ $$($$$$113413 \nu^{7} + 39797 \nu^{6} - 20577944 \nu^{5} - 34593850 \nu^{4} + 986989690 \nu^{3} + 3243325376 \nu^{2} - 2200376928 \nu - 37919990457$$$$)/93801483$$ $$\beta_{7}$$ $$=$$ $$($$$$152056 \nu^{7} - 721301 \nu^{6} - 28663523 \nu^{5} + 103151545 \nu^{4} + 1599227110 \nu^{3} - 3735736193 \nu^{2} - 17874824016 \nu + 18159439041$$$$)/93801483$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 61 \beta_{6} - 62 \beta_{5} - 17 \beta_{4} + 44 \beta_{3} + 140 \beta_{2} - 1073 \beta_{1} + 2186$$$$)/5400$$ $$\nu^{2}$$ $$=$$ $$($$$$-47 \beta_{7} + 103 \beta_{6} + 214 \beta_{5} - 371 \beta_{4} + 146 \beta_{3} + 152 \beta_{2} - 779 \beta_{1} + 141068$$$$)/2700$$ $$\nu^{3}$$ $$=$$ $$($$$$2803 \beta_{7} + 4663 \beta_{6} - 5171 \beta_{5} - 2831 \beta_{4} + 4910 \beta_{3} + 3593 \beta_{2} - 84464 \beta_{1} + 382838$$$$)/5400$$ $$\nu^{4}$$ $$=$$ $$($$$$-3934 \beta_{7} + 7211 \beta_{6} + 9953 \beta_{5} - 19567 \beta_{4} + 13177 \beta_{3} + 5269 \beta_{2} - 44908 \beta_{1} + 6072211$$$$)/1350$$ $$\nu^{5}$$ $$=$$ $$($$$$164846 \beta_{7} + 267296 \beta_{6} - 207097 \beta_{5} - 169072 \beta_{4} + 280096 \beta_{3} - 164915 \beta_{2} - 3830413 \beta_{1} + 28297906$$$$)/2700$$ $$\nu^{6}$$ $$=$$ $$($$$$-547093 \beta_{7} + 1009802 \beta_{6} + 786311 \beta_{5} - 1894204 \beta_{4} + 1812889 \beta_{3} - 16061 \beta_{2} - 4301971 \beta_{1} + 560236357$$$$)/1350$$ $$\nu^{7}$$ $$=$$ $$($$$$17051159 \beta_{7} + 31778429 \beta_{6} - 16276903 \beta_{5} - 18521053 \beta_{4} + 32474356 \beta_{3} - 45319931 \beta_{2} - 339979522 \beta_{1} + 3669549064$$$$)/2700$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 −2.67163 − 0.707107i −2.67163 + 0.707107i −9.26938 − 0.707107i −9.26938 + 0.707107i 3.67163 + 0.707107i 3.67163 − 0.707107i 10.2694 + 0.707107i 10.2694 − 0.707107i
0 −24.1542 12.0655i 0 55.9017i 0 −436.815 0 437.848 + 582.864i 0
41.2 0 −24.1542 + 12.0655i 0 55.9017i 0 −436.815 0 437.848 582.864i 0
41.3 0 −22.1779 15.3993i 0 55.9017i 0 437.053 0 254.720 + 683.051i 0
41.4 0 −22.1779 + 15.3993i 0 55.9017i 0 437.053 0 254.720 683.051i 0
41.5 0 11.2485 24.5453i 0 55.9017i 0 −414.697 0 −475.944 552.194i 0
41.6 0 11.2485 + 24.5453i 0 55.9017i 0 −414.697 0 −475.944 + 552.194i 0
41.7 0 25.0836 9.99060i 0 55.9017i 0 134.460 0 529.376 501.201i 0
41.8 0 25.0836 + 9.99060i 0 55.9017i 0 134.460 0 529.376 + 501.201i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 41.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.g.a 8
3.b odd 2 1 inner 60.7.g.a 8
4.b odd 2 1 240.7.l.c 8
5.b even 2 1 300.7.g.h 8
5.c odd 4 2 300.7.b.e 16
12.b even 2 1 240.7.l.c 8
15.d odd 2 1 300.7.g.h 8
15.e even 4 2 300.7.b.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.g.a 8 1.a even 1 1 trivial
60.7.g.a 8 3.b odd 2 1 inner
240.7.l.c 8 4.b odd 2 1
240.7.l.c 8 12.b even 2 1
300.7.b.e 16 5.c odd 4 2
300.7.b.e 16 15.e even 4 2
300.7.g.h 8 5.b even 2 1
300.7.g.h 8 15.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$282429536481 + 7748409780 T - 290166786 T^{2} - 5380020 T^{3} + 559386 T^{4} - 7380 T^{5} - 546 T^{6} + 20 T^{7} + T^{8}$$
$5$ $$( 3125 + T^{2} )^{4}$$
$7$ $$( 10645216936 - 53487320 T - 246738 T^{2} + 280 T^{3} + T^{4} )^{2}$$
$11$ $$22\!\cdots\!00$$$$+ 5861305881531360000 T^{2} + 10809709880400 T^{4} + 6152760 T^{6} + T^{8}$$
$13$ $$( -261697247744 - 3344852480 T + 252132 T^{2} + 3220 T^{3} + T^{4} )^{2}$$
$17$ $$26\!\cdots\!00$$$$+$$$$90\!\cdots\!00$$$$T^{2} + 7295360344322400 T^{4} + 161822160 T^{6} + T^{8}$$
$19$ $$( 1136545793559136 - 455354684384 T - 78241764 T^{2} + 7636 T^{3} + T^{4} )^{2}$$
$23$ $$17\!\cdots\!00$$$$+$$$$59\!\cdots\!00$$$$T^{2} + 143743273758558900 T^{4} + 715419540 T^{6} + T^{8}$$
$29$ $$75\!\cdots\!00$$$$+$$$$42\!\cdots\!00$$$$T^{2} + 787104956899904400 T^{4} + 2283274440 T^{6} + T^{8}$$
$31$ $$( 43347363402701056 + 54665955569344 T - 2254268604 T^{2} - 17516 T^{3} + T^{4} )^{2}$$
$37$ $$( -159628149302495744 + 76493244951040 T - 2637555132 T^{2} - 49940 T^{3} + T^{4} )^{2}$$
$41$ $$64\!\cdots\!00$$$$+$$$$36\!\cdots\!00$$$$T^{2} +$$$$17\!\cdots\!00$$$$T^{4} + 24416952840 T^{6} + T^{8}$$
$43$ $$( -26033578463732758784 - 1458179882816000 T - 15204873558 T^{2} + 80500 T^{3} + T^{4} )^{2}$$
$47$ $$88\!\cdots\!00$$$$+$$$$34\!\cdots\!00$$$$T^{2} +$$$$25\!\cdots\!00$$$$T^{4} + 35727961140 T^{6} + T^{8}$$
$53$ $$23\!\cdots\!00$$$$+$$$$49\!\cdots\!00$$$$T^{2} +$$$$34\!\cdots\!00$$$$T^{4} + 97591865040 T^{6} + T^{8}$$
$59$ $$11\!\cdots\!00$$$$+$$$$33\!\cdots\!00$$$$T^{2} +$$$$18\!\cdots\!00$$$$T^{4} + 265901508360 T^{6} + T^{8}$$
$61$ $$($$$$48\!\cdots\!76$$$$+ 711714612123904 T - 62487698844 T^{2} + 67804 T^{3} + T^{4} )^{2}$$
$67$ $$( 93721981619142833536 - 4426242213673760 T - 84864962838 T^{2} - 202460 T^{3} + T^{4} )^{2}$$
$71$ $$89\!\cdots\!00$$$$+$$$$40\!\cdots\!00$$$$T^{2} +$$$$62\!\cdots\!00$$$$T^{4} + 415954038960 T^{6} + T^{8}$$
$73$ $$($$$$23\!\cdots\!56$$$$- 48317504323887680 T - 358394569032 T^{2} + 178480 T^{3} + T^{4} )^{2}$$
$79$ $$($$$$63\!\cdots\!36$$$$+ 130563026559349312 T - 567974089236 T^{2} - 353852 T^{3} + T^{4} )^{2}$$
$83$ $$61\!\cdots\!00$$$$+$$$$85\!\cdots\!00$$$$T^{2} +$$$$25\!\cdots\!00$$$$T^{4} + 2736050734260 T^{6} + T^{8}$$
$89$ $$98\!\cdots\!00$$$$+$$$$40\!\cdots\!00$$$$T^{2} +$$$$62\!\cdots\!00$$$$T^{4} + 4108796641440 T^{6} + T^{8}$$
$97$ $$($$$$19\!\cdots\!76$$$$- 388062618322727840 T - 1095298435848 T^{2} + 663160 T^{3} + T^{4} )^{2}$$