# Properties

 Label 5.10.b.a Level $5$ Weight $10$ Character orbit 5.b Analytic conductor $2.575$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.57517918082$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.49740556.1 Defining polynomial: $$x^{4} + 45 x^{2} + 304$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( -342 - \beta_{3} ) q^{4} + ( 285 + 19 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{5} + ( 702 + \beta_{3} ) q^{6} + ( 133 \beta_{1} - 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} + ( 2907 + 18 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( -342 - \beta_{3} ) q^{4} + ( 285 + 19 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{5} + ( 702 + \beta_{3} ) q^{6} + ( 133 \beta_{1} - 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} + ( 2907 + 18 \beta_{3} ) q^{9} + ( -17290 + 1139 \beta_{1} - 8 \beta_{2} - 19 \beta_{3} ) q^{10} + ( 27492 - 20 \beta_{3} ) q^{11} + ( 1044 \beta_{1} + 504 \beta_{2} ) q^{12} + ( -1918 \beta_{1} - 110 \beta_{2} ) q^{13} + ( -106134 - 133 \beta_{3} ) q^{14} + ( -99180 - 987 \beta_{1} - 561 \beta_{2} + 152 \beta_{3} ) q^{15} + ( 407816 + 172 \beta_{3} ) q^{16} + ( -7448 \beta_{1} - 1632 \beta_{2} ) q^{17} + ( 18279 \beta_{1} - 144 \beta_{2} ) q^{18} + ( -159220 + 476 \beta_{3} ) q^{19} + ( -825570 - 23788 \beta_{1} + 3736 \beta_{2} - 627 \beta_{3} ) q^{20} + ( 880992 - 798 \beta_{3} ) q^{21} + ( 10412 \beta_{1} + 160 \beta_{2} ) q^{22} + ( 17005 \beta_{1} + 5199 \beta_{2} ) q^{23} + ( -608760 - 532 \beta_{3} ) q^{24} + ( -334475 + 45410 \beta_{1} - 8270 \beta_{2} + 1140 \beta_{3} ) q^{25} + ( 1654692 + 1918 \beta_{3} ) q^{26} + ( -35226 \beta_{1} + 7362 \beta_{2} ) q^{27} + ( -151620 \beta_{1} - 24024 \beta_{2} ) q^{28} + ( -882930 - 1976 \beta_{3} ) q^{29} + ( 928170 + 30628 \beta_{1} - 1216 \beta_{2} + 987 \beta_{3} ) q^{30} + ( -2646928 - 760 \beta_{3} ) q^{31} + ( 204496 \beta_{1} + 2720 \beta_{2} ) q^{32} + ( -13452 \beta_{1} + 44412 \beta_{2} ) q^{33} + ( 6608656 + 7448 \beta_{3} ) q^{34} + ( 3407460 + 144039 \beta_{1} + 26817 \beta_{2} - 9044 \beta_{3} ) q^{35} + ( -14099994 - 9063 \beta_{3} ) q^{36} + ( -335370 \beta_{1} - 51378 \beta_{2} ) q^{37} + ( 247284 \beta_{1} - 3808 \beta_{2} ) q^{38} + ( 421704 - 4008 \beta_{3} ) q^{39} + ( 10894600 - 777860 \beta_{1} + 920 \beta_{2} + 14060 \beta_{3} ) q^{40} + ( -4197138 + 24890 \beta_{3} ) q^{41} + ( 199500 \beta_{1} + 6384 \beta_{2} ) q^{42} + ( 719131 \beta_{1} - 132643 \beta_{2} ) q^{43} + ( 5159736 - 20652 \beta_{3} ) q^{44} + ( 13934295 + 366453 \beta_{1} - 89991 \beta_{2} + 8037 \beta_{3} ) q^{45} + ( -15312518 - 17005 \beta_{3} ) q^{46} + ( -1012111 \beta_{1} + 214259 \beta_{2} ) q^{47} + ( -528560 \beta_{1} + 262304 \beta_{2} ) q^{48} + ( -11730257 + 27930 \beta_{3} ) q^{49} + ( -37523100 + 639085 \beta_{1} - 9120 \beta_{2} - 45410 \beta_{3} ) q^{50} + ( 21004272 - 38456 \beta_{3} ) q^{51} + ( 2310648 \beta_{1} - 71664 \beta_{2} ) q^{52} + ( -1349666 \beta_{1} - 259794 \beta_{2} ) q^{53} + ( 28963980 + 35226 \beta_{3} ) q^{54} + ( -6726780 + 176548 \beta_{1} + 315044 \beta_{2} + 21792 \beta_{3} ) q^{55} + ( 78794520 + 83524 \beta_{3} ) q^{56} + ( -174932 \beta_{1} - 561916 \beta_{2} ) q^{57} + ( -2570434 \beta_{1} + 15808 \beta_{2} ) q^{58} + ( -115207260 - 52972 \beta_{3} ) q^{59} + ( -76751640 + 1265724 \beta_{1} - 295128 \beta_{2} + 47196 \beta_{3} ) q^{60} + ( 90122642 - 43150 \beta_{3} ) q^{61} + ( -3295968 \beta_{1} + 6080 \beta_{2} ) q^{62} + ( 2297043 \beta_{1} + 591633 \beta_{2} ) q^{63} + ( 33748768 - 116432 \beta_{3} ) q^{64} + ( 45973920 - 2201322 \beta_{1} + 77934 \beta_{2} + 21812 \beta_{3} ) q^{65} + ( 4737384 + 13452 \beta_{3} ) q^{66} + ( 6669647 \beta_{1} + 1448953 \beta_{2} ) q^{67} + ( 9155872 \beta_{1} - 895168 \beta_{2} ) q^{68} + ( -71631216 + 115786 \beta_{3} ) q^{69} + ( -127085490 - 4316116 \beta_{1} + 72352 \beta_{2} - 144039 \beta_{3} ) q^{70} + ( -11902968 - 91200 \beta_{3} ) q^{71} + ( -12480948 \beta_{1} - 1224 \beta_{2} ) q^{72} + ( -1847940 \beta_{1} + 90564 \beta_{2} ) q^{73} + ( 294215436 + 335370 \beta_{3} ) q^{74} + ( 164809800 - 465805 \beta_{1} - 1298915 \beta_{2} - 111720 \beta_{3} ) q^{75} + ( -292122360 - 3572 \beta_{3} ) q^{76} + ( 1533756 \beta_{1} - 2162748 \beta_{2} ) q^{77} + ( -3001128 \beta_{1} + 32064 \beta_{2} ) q^{78} + ( -182010880 - 267976 \beta_{3} ) q^{79} + ( 241460760 + 10722384 \beta_{1} + 1800352 \beta_{2} + 456836 \beta_{3} ) q^{80} + ( -85846959 + 458946 \beta_{3} ) q^{81} + ( 17058922 \beta_{1} - 199120 \beta_{2} ) q^{82} + ( -12737657 \beta_{1} + 1180353 \beta_{2} ) q^{83} + ( 279724536 - 608076 \beta_{3} ) q^{84} + ( 318854960 - 8731336 \beta_{1} + 988192 \beta_{2} - 75544 \beta_{3} ) q^{85} + ( -593976138 - 719131 \beta_{3} ) q^{86} + ( 2270082 \beta_{1} + 788766 \beta_{2} ) q^{87} + ( -7146128 \beta_{1} + 247136 \beta_{2} ) q^{88} + ( -395675190 + 185592 \beta_{3} ) q^{89} + ( -299272230 + 20797893 \beta_{1} - 64296 \beta_{2} - 366453 \beta_{3} ) q^{90} + ( 118330632 + 357504 \beta_{3} ) q^{91} + ( -21128228 \beta_{1} + 2797928 \beta_{2} ) q^{92} + ( 3180448 \beta_{1} - 2003968 \beta_{2} ) q^{93} + ( 831775426 + 1012111 \beta_{3} ) q^{94} + ( 301197900 + 5204860 \beta_{1} - 4032420 \beta_{2} - 23560 \beta_{3} ) q^{95} + ( 99834912 + 256176 \beta_{3} ) q^{96} + ( 14786464 \beta_{1} - 1954216 \beta_{2} ) q^{97} + ( 12121963 \beta_{1} - 223440 \beta_{2} ) q^{98} + ( -182196756 + 436716 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 1368q^{4} + 1140q^{5} + 2808q^{6} + 11628q^{9} + O(q^{10})$$ $$4q - 1368q^{4} + 1140q^{5} + 2808q^{6} + 11628q^{9} - 69160q^{10} + 109968q^{11} - 424536q^{14} - 396720q^{15} + 1631264q^{16} - 636880q^{19} - 3302280q^{20} + 3523968q^{21} - 2435040q^{24} - 1337900q^{25} + 6618768q^{26} - 3531720q^{29} + 3712680q^{30} - 10587712q^{31} + 26434624q^{34} + 13629840q^{35} - 56399976q^{36} + 1686816q^{39} + 43578400q^{40} - 16788552q^{41} + 20638944q^{44} + 55737180q^{45} - 61250072q^{46} - 46921028q^{49} - 150092400q^{50} + 84017088q^{51} + 115855920q^{54} - 26907120q^{55} + 315178080q^{56} - 460829040q^{59} - 307006560q^{60} + 360490568q^{61} + 134995072q^{64} + 183895680q^{65} + 18949536q^{66} - 286524864q^{69} - 508341960q^{70} - 47611872q^{71} + 1176861744q^{74} + 659239200q^{75} - 1168489440q^{76} - 728043520q^{79} + 965843040q^{80} - 343387836q^{81} + 1118898144q^{84} + 1275419840q^{85} - 2375904552q^{86} - 1582700760q^{89} - 1197088920q^{90} + 473322528q^{91} + 3327101704q^{94} + 1204791600q^{95} + 399339648q^{96} - 728787024q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 45 x^{2} + 304$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 37 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} - 7 \nu$$ $$\beta_{3}$$ $$=$$ $$60 \nu^{2} + 1350$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/30$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 1350$$$$)/60$$ $$\nu^{3}$$ $$=$$ $$($$$$-37 \beta_{2} - 14 \beta_{1}$$$$)/30$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 2.87724i − 6.05982i 6.05982i 2.87724i
41.3193i 37.6407i −1195.29 1138.29 810.818i 1555.29 5315.22i 28233.0i 18266.2 −33502.5 47033.3i
4.2 0.843944i 179.263i 511.288 −568.288 1276.78i −151.288 8712.99i 863.597i −12452.2 −1077.53 + 479.603i
4.3 0.843944i 179.263i 511.288 −568.288 + 1276.78i −151.288 8712.99i 863.597i −12452.2 −1077.53 479.603i
4.4 41.3193i 37.6407i −1195.29 1138.29 + 810.818i 1555.29 5315.22i 28233.0i 18266.2 −33502.5 + 47033.3i
 $$n$$: 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 5.10.b.a 4
3.b odd 2 1 45.10.b.b 4
4.b odd 2 1 80.10.c.c 4
5.b even 2 1 inner 5.10.b.a 4
5.c odd 4 2 25.10.a.e 4
15.d odd 2 1 45.10.b.b 4
15.e even 4 2 225.10.a.s 4
20.d odd 2 1 80.10.c.c 4
20.e even 4 2 400.10.a.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.b.a 4 1.a even 1 1 trivial
5.10.b.a 4 5.b even 2 1 inner
25.10.a.e 4 5.c odd 4 2
45.10.b.b 4 3.b odd 2 1
45.10.b.b 4 15.d odd 2 1
80.10.c.c 4 4.b odd 2 1
80.10.c.c 4 20.d odd 2 1
225.10.a.s 4 15.e even 4 2
400.10.a.ba 4 20.e even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(5, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 340 T^{2} - 174912 T^{4} - 89128960 T^{6} + 68719476736 T^{8}$$
$3$ $$1 - 45180 T^{2} + 1049244678 T^{4} - 17503657693020 T^{6} + 150094635296999121 T^{8}$$
$5$ $$1 - 1140 T + 1318750 T^{2} - 2226562500 T^{3} + 3814697265625 T^{4}$$
$7$ $$1 - 57246700 T^{2} + 3508143545353398 T^{4} -$$$$93\!\cdots\!00$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 - 54984 T + 5180465446 T^{2} - 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2}$$
$13$ $$1 - 35613791860 T^{2} +$$$$53\!\cdots\!58$$$$T^{4} -$$$$40\!\cdots\!40$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$17$ $$1 - 285780369220 T^{2} +$$$$48\!\cdots\!18$$$$T^{4} -$$$$40\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!81$$$$T^{8}$$
$19$ $$( 1 + 318440 T + 505756418358 T^{2} + 102756670480744760 T^{3} +$$$$10\!\cdots\!41$$$$T^{4} )^{2}$$
$23$ $$1 - 5779790962540 T^{2} +$$$$14\!\cdots\!38$$$$T^{4} -$$$$18\!\cdots\!60$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 + 1765860 T + 26950935551038 T^{2} + 25617588792948032340 T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$( 1 + 5293856 T + 59464921598526 T^{2} +$$$$13\!\cdots\!76$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$
$37$ $$1 - 231603274936660 T^{2} +$$$$44\!\cdots\!58$$$$T^{4} -$$$$39\!\cdots\!40$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 + 8394276 T + 221313076168966 T^{2} +$$$$27\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 - 614109141147100 T^{2} +$$$$57\!\cdots\!98$$$$T^{4} -$$$$15\!\cdots\!00$$$$T^{6} +$$$$63\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 1368976020813580 T^{2} +$$$$29\!\cdots\!78$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 7684297973864980 T^{2} +$$$$36\!\cdots\!78$$$$T^{4} -$$$$83\!\cdots\!20$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 + 230414520 T + 28555631923987078 T^{2} +$$$$19\!\cdots\!80$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$61$ $$( 1 - 180245284 T + 30154717014478446 T^{2} -$$$$21\!\cdots\!44$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 + 41160407446058180 T^{2} +$$$$18\!\cdots\!18$$$$T^{4} +$$$$30\!\cdots\!20$$$$T^{6} +$$$$54\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 + 23805936 T + 85782754020107086 T^{2} +$$$$10\!\cdots\!16$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$73$ $$1 - 229489314868712740 T^{2} +$$$$20\!\cdots\!38$$$$T^{4} -$$$$79\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 + 364021760 T + 220545463862625438 T^{2} +$$$$43\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} )^{2}$$
$83$ $$1 - 434569632367965820 T^{2} +$$$$10\!\cdots\!18$$$$T^{4} -$$$$15\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 + 791350380 T + 832192702699668118 T^{2} +$$$$27\!\cdots\!20$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$1 - 2561123777205326980 T^{2} +$$$$27\!\cdots\!78$$$$T^{4} -$$$$14\!\cdots\!20$$$$T^{6} +$$$$33\!\cdots\!21$$$$T^{8}$$