# Properties

 Label 10.10.b.a Level 10 Weight 10 Character orbit 10.b Analytic conductor 5.150 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.15035836164$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{319})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{10}\cdot 5^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -5 \beta_{1} - \beta_{3} ) q^{3} -256 q^{4} + ( -645 + 55 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{5} + ( -1216 - 2 \beta_{1} - 4 \beta_{2} ) q^{6} + ( 170 \beta_{1} - 43 \beta_{3} ) q^{7} + 256 \beta_{1} q^{8} + ( -17993 - 19 \beta_{1} - 38 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -5 \beta_{1} - \beta_{3} ) q^{3} -256 q^{4} + ( -645 + 55 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{5} + ( -1216 - 2 \beta_{1} - 4 \beta_{2} ) q^{6} + ( 170 \beta_{1} - 43 \beta_{3} ) q^{7} + 256 \beta_{1} q^{8} + ( -17993 - 19 \beta_{1} - 38 \beta_{2} ) q^{9} + ( 13760 + 635 \beta_{1} + 12 \beta_{2} - 64 \beta_{3} ) q^{10} + ( -25908 + 34 \beta_{1} + 68 \beta_{2} ) q^{11} + ( 1280 \beta_{1} + 256 \beta_{3} ) q^{12} + ( -5799 \beta_{1} - 222 \beta_{3} ) q^{13} + ( 46272 - 86 \beta_{1} - 172 \beta_{2} ) q^{14} + ( 161060 - 4690 \beta_{1} + 272 \beta_{2} + 341 \beta_{3} ) q^{15} + 65536 q^{16} + ( -2866 \beta_{1} + 168 \beta_{3} ) q^{17} + ( 18601 \beta_{1} + 2432 \beta_{3} ) q^{18} + ( 79380 + 66 \beta_{1} + 132 \beta_{2} ) q^{19} + ( 165120 - 14080 \beta_{1} - 256 \beta_{2} - 768 \beta_{3} ) q^{20} + ( -1151908 - 47 \beta_{1} - 94 \beta_{2} ) q^{21} + ( 24820 \beta_{1} - 4352 \beta_{3} ) q^{22} + ( -33140 \beta_{1} - 8931 \beta_{3} ) q^{23} + ( 311296 + 512 \beta_{1} + 1024 \beta_{2} ) q^{24} + ( -100275 - 22025 \beta_{1} - 2580 \beta_{2} + 3010 \beta_{3} ) q^{25} + ( -1470336 - 444 \beta_{1} - 888 \beta_{2} ) q^{26} + ( 297488 \beta_{1} + 9862 \beta_{3} ) q^{27} + ( -43520 \beta_{1} + 11008 \beta_{3} ) q^{28} + ( 1667070 + 1692 \beta_{1} + 3384 \beta_{2} ) q^{29} + ( -1257280 - 164730 \beta_{1} + 1364 \beta_{2} - 17408 \beta_{3} ) q^{30} + ( 4811872 - 1452 \beta_{1} - 2904 \beta_{2} ) q^{31} -65536 \beta_{1} q^{32} + ( -417928 \beta_{1} + 5236 \beta_{3} ) q^{33} + ( -744448 + 336 \beta_{1} + 672 \beta_{2} ) q^{34} + ( 1627980 - 446145 \beta_{1} + 7076 \beta_{2} + 39303 \beta_{3} ) q^{35} + ( 4606208 + 4864 \beta_{1} + 9728 \beta_{2} ) q^{36} + ( 1048169 \beta_{1} - 19882 \beta_{3} ) q^{37} + ( -81492 \beta_{1} - 8448 \beta_{3} ) q^{38} + ( -14065896 - 13596 \beta_{1} - 27192 \beta_{2} ) q^{39} + ( -3522560 - 162560 \beta_{1} - 3072 \beta_{2} + 16384 \beta_{3} ) q^{40} + ( 5693562 + 6457 \beta_{1} + 12914 \beta_{2} ) q^{41} + ( 1153412 \beta_{1} + 6016 \beta_{3} ) q^{42} + ( 325357 \beta_{1} - 16133 \beta_{3} ) q^{43} + ( 6632448 - 8704 \beta_{1} - 17408 \beta_{2} ) q^{44} + ( -7789715 - 1919190 \beta_{1} + 6517 \beta_{2} - 184699 \beta_{3} ) q^{45} + ( -7912256 - 17862 \beta_{1} - 35724 \beta_{2} ) q^{46} + ( 1425730 \beta_{1} + 7617 \beta_{3} ) q^{47} + ( -327680 \beta_{1} - 65536 \beta_{3} ) q^{48} + ( -26993157 + 31089 \beta_{1} + 62178 \beta_{2} ) q^{49} + ( -5500800 + 147575 \beta_{1} + 12040 \beta_{2} + 165120 \beta_{3} ) q^{50} + ( 1823072 - 4220 \beta_{1} - 8440 \beta_{2} ) q^{51} + ( 1484544 \beta_{1} + 56832 \beta_{3} ) q^{52} + ( -3405969 \beta_{1} + 221118 \beta_{3} ) q^{53} + ( 75525760 + 19724 \beta_{1} + 39448 \beta_{2} ) q^{54} + ( 51417860 + 238510 \beta_{1} - 69768 \beta_{2} + 156196 \beta_{3} ) q^{55} + ( -11845632 + 22016 \beta_{1} + 44032 \beta_{2} ) q^{56} + ( -1459632 \beta_{1} - 119508 \beta_{3} ) q^{57} + ( -1721214 \beta_{1} - 216576 \beta_{3} ) q^{58} + ( 63665340 + 19654 \beta_{1} + 39308 \beta_{2} ) q^{59} + ( -41231360 + 1200640 \beta_{1} - 69632 \beta_{2} - 87296 \beta_{3} ) q^{60} + ( -71645458 - 14559 \beta_{1} - 29118 \beta_{2} ) q^{61} + ( -4765408 \beta_{1} + 185856 \beta_{3} ) q^{62} + ( 9862444 \beta_{1} + 334115 \beta_{3} ) q^{63} -16777216 q^{64} + ( 100275960 + 1936335 \beta_{1} + 116652 \beta_{2} - 224394 \beta_{3} ) q^{65} + ( -107324672 + 10472 \beta_{1} + 20944 \beta_{2} ) q^{66} + ( -1120619 \beta_{1} - 193169 \beta_{3} ) q^{67} + ( 733696 \beta_{1} - 43008 \beta_{3} ) q^{68} + ( -322482116 - 146659 \beta_{1} - 293318 \beta_{2} ) q^{69} + ( -117634240 - 1662590 \beta_{1} + 157212 \beta_{2} - 452864 \beta_{3} ) q^{70} + ( 200717832 - 44904 \beta_{1} - 89808 \beta_{2} ) q^{71} + ( -4761856 \beta_{1} - 622592 \beta_{3} ) q^{72} + ( 945916 \beta_{1} - 1788932 \beta_{3} ) q^{73} + ( 269603712 - 39764 \beta_{1} - 79528 \beta_{2} ) q^{74} + ( 69890200 + 21258575 \beta_{1} - 28760 \beta_{2} + 884595 \beta_{3} ) q^{75} + ( -20321280 - 16896 \beta_{1} - 33792 \beta_{2} ) q^{76} + ( -27526604 \beta_{1} + 1900668 \beta_{3} ) q^{77} + ( 14500968 \beta_{1} + 1740288 \beta_{3} ) q^{78} + ( -258292080 - 34884 \beta_{1} - 69768 \beta_{2} ) q^{79} + ( -42270720 + 3604480 \beta_{1} + 65536 \beta_{2} + 196608 \beta_{3} ) q^{80} + ( 319188941 + 309757 \beta_{1} + 619514 \beta_{2} ) q^{81} + ( -5900186 \beta_{1} - 826496 \beta_{3} ) q^{82} + ( -9424211 \beta_{1} + 1263015 \beta_{3} ) q^{83} + ( 294888448 + 12032 \beta_{1} + 24064 \beta_{2} ) q^{84} + ( 23936480 + 3141230 \beta_{1} - 1224 \beta_{2} - 294472 \beta_{3} ) q^{85} + ( 84323904 - 32266 \beta_{1} - 64532 \beta_{2} ) q^{86} + ( -35579934 \beta_{1} - 2695806 \beta_{3} ) q^{87} + ( -6353920 \beta_{1} + 1114112 \beta_{3} ) q^{88} + ( -69089190 + 568580 \beta_{1} + 1137160 \beta_{2} ) q^{89} + ( -480326080 + 7316045 \beta_{1} - 738796 \beta_{2} - 417088 \beta_{3} ) q^{90} + ( -38754168 - 413688 \beta_{1} - 827376 \beta_{2} ) q^{91} + ( 8483840 \beta_{1} + 2286336 \beta_{3} ) q^{92} + ( -679256 \beta_{1} - 3929056 \beta_{3} ) q^{93} + ( 364499392 + 15234 \beta_{1} + 30468 \beta_{2} ) q^{94} + ( 16172700 + 7594950 \beta_{1} - 5760 \beta_{2} + 692220 \beta_{3} ) q^{95} + ( -79691776 - 131072 \beta_{1} - 262144 \beta_{2} ) q^{96} + ( 66096682 \beta_{1} + 3138400 \beta_{3} ) q^{97} + ( 25998309 \beta_{1} - 3979392 \beta_{3} ) q^{98} + ( -852710956 - 119510 \beta_{1} - 239020 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 1024q^{4} - 2580q^{5} - 4864q^{6} - 71972q^{9} + O(q^{10})$$ $$4q - 1024q^{4} - 2580q^{5} - 4864q^{6} - 71972q^{9} + 55040q^{10} - 103632q^{11} + 185088q^{14} + 644240q^{15} + 262144q^{16} + 317520q^{19} + 660480q^{20} - 4607632q^{21} + 1245184q^{24} - 401100q^{25} - 5881344q^{26} + 6668280q^{29} - 5029120q^{30} + 19247488q^{31} - 2977792q^{34} + 6511920q^{35} + 18424832q^{36} - 56263584q^{39} - 14090240q^{40} + 22774248q^{41} + 26529792q^{44} - 31158860q^{45} - 31649024q^{46} - 107972628q^{49} - 22003200q^{50} + 7292288q^{51} + 302103040q^{54} + 205671440q^{55} - 47382528q^{56} + 254661360q^{59} - 164925440q^{60} - 286581832q^{61} - 67108864q^{64} + 401103840q^{65} - 429298688q^{66} - 1289928464q^{69} - 470536960q^{70} + 802871328q^{71} + 1078414848q^{74} + 279560800q^{75} - 81285120q^{76} - 1033168320q^{79} - 169082880q^{80} + 1276755764q^{81} + 1179553792q^{84} + 95745920q^{85} + 337295616q^{86} - 276356760q^{89} - 1921304320q^{90} - 155016672q^{91} + 1457997568q^{94} + 64690800q^{95} - 318767104q^{96} - 3410843824q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 159 x^{2} + 6400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 79 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{3} + 637 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 400 \nu^{2} + 79 \nu - 31800$$$$)/20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1}$$$$)/80$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{3} + \beta_{1} + 6360$$$$)/80$$ $$\nu^{3}$$ $$=$$ $$($$$$79 \beta_{2} + 637 \beta_{1}$$$$)/80$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 8.93029 + 0.500000i −8.93029 + 0.500000i −8.93029 − 0.500000i 8.93029 − 0.500000i
16.0000i 254.606i −256.000 69.4228 + 1395.82i −4073.69 4788.05i 4096.00i −45141.1 22333.1 1110.77i
9.2 16.0000i 102.606i −256.000 −1359.42 + 324.183i 1641.69 10572.0i 4096.00i 9155.07 5186.93 + 21750.8i
9.3 16.0000i 102.606i −256.000 −1359.42 324.183i 1641.69 10572.0i 4096.00i 9155.07 5186.93 21750.8i
9.4 16.0000i 254.606i −256.000 69.4228 1395.82i −4073.69 4788.05i 4096.00i −45141.1 22333.1 + 1110.77i
 $$n$$: e.g. 2-40 or 990-1000 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 10.10.b.a 4
3.b odd 2 1 90.10.c.b 4
4.b odd 2 1 80.10.c.a 4
5.b even 2 1 inner 10.10.b.a 4
5.c odd 4 1 50.10.a.h 2
5.c odd 4 1 50.10.a.i 2
15.d odd 2 1 90.10.c.b 4
20.d odd 2 1 80.10.c.a 4
20.e even 4 1 400.10.a.m 2
20.e even 4 1 400.10.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.10.b.a 4 1.a even 1 1 trivial
10.10.b.a 4 5.b even 2 1 inner
50.10.a.h 2 5.c odd 4 1
50.10.a.i 2 5.c odd 4 1
80.10.c.a 4 4.b odd 2 1
80.10.c.a 4 20.d odd 2 1
90.10.c.b 4 3.b odd 2 1
90.10.c.b 4 15.d odd 2 1
400.10.a.m 2 20.e even 4 1
400.10.a.s 2 20.e even 4 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 256 T^{2} )^{2}$$
$3$ $$1 - 3380 T^{2} + 40679478 T^{4} - 1309481252820 T^{6} + 150094635296999121 T^{8}$$
$5$ $$1 + 2580 T + 3528750 T^{2} + 5039062500 T^{3} + 3814697265625 T^{4}$$
$7$ $$1 - 26720900 T^{2} + 1462069499709798 T^{4} -$$$$43\!\cdots\!00$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 51816 T + 3027030246 T^{2} + 122179417556856 T^{3} + 5559917313492231481 T^{4} )^{2}$$
$13$ $$1 - 22383928660 T^{2} +$$$$29\!\cdots\!58$$$$T^{4} -$$$$25\!\cdots\!40$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$17$ $$1 - 468221105220 T^{2} +$$$$82\!\cdots\!18$$$$T^{4} -$$$$65\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!81$$$$T^{8}$$
$19$ $$( 1 - 158760 T + 642783370358 T^{2} - 51229898899394040 T^{3} +$$$$10\!\cdots\!41$$$$T^{4} )^{2}$$
$23$ $$1 - 1626654345540 T^{2} +$$$$46\!\cdots\!38$$$$T^{4} -$$$$52\!\cdots\!60$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 3334140 T + 25948591194238 T^{2} - 48368855683983867660 T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$( 1 - 9623744 T + 71729043019326 T^{2} -$$$$25\!\cdots\!24$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$
$37$ $$1 + 73233430078540 T^{2} +$$$$20\!\cdots\!58$$$$T^{4} +$$$$12\!\cdots\!60$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 - 11387124 T + 602060396517366 T^{2} -$$$$37\!\cdots\!64$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 - 1938214042750100 T^{2} +$$$$14\!\cdots\!98$$$$T^{4} -$$$$48\!\cdots\!00$$$$T^{6} +$$$$63\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 3434853059674980 T^{2} +$$$$54\!\cdots\!78$$$$T^{4} -$$$$43\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 3945786642536180 T^{2} +$$$$65\!\cdots\!78$$$$T^{4} -$$$$42\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 - 127330680 T + 20590638486439878 T^{2} -$$$$11\!\cdots\!20$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$61$ $$( 1 + 143290916 T + 28088617153288446 T^{2} +$$$$16\!\cdots\!56$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 - 105856746688500020 T^{2} +$$$$42\!\cdots\!18$$$$T^{4} -$$$$78\!\cdots\!80$$$$T^{6} +$$$$54\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 401435664 T + 127868030128292686 T^{2} -$$$$18\!\cdots\!84$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$73$ $$1 - 30314907095525540 T^{2} +$$$$69\!\cdots\!38$$$$T^{4} -$$$$10\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 + 516584160 T + 303933580876193438 T^{2} +$$$$61\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} )^{2}$$
$83$ $$1 - 597414961848210420 T^{2} +$$$$15\!\cdots\!18$$$$T^{4} -$$$$20\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 + 138178380 T + 45471108987586518 T^{2} +$$$$48\!\cdots\!20$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$1 - 228500609440802180 T^{2} -$$$$20\!\cdots\!22$$$$T^{4} -$$$$13\!\cdots\!20$$$$T^{6} +$$$$33\!\cdots\!21$$$$T^{8}$$