# Properties

 Label 10.5.c.a Level 10 Weight 5 Character orbit 10.c Analytic conductor 1.034 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 10.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.03369963084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 - 2 i ) q^{2} + ( 9 - 9 i ) q^{3} + 8 i q^{4} + ( -15 + 20 i ) q^{5} -36 q^{6} + ( 29 + 29 i ) q^{7} + ( 16 - 16 i ) q^{8} -81 i q^{9} +O(q^{10})$$ $$q + ( -2 - 2 i ) q^{2} + ( 9 - 9 i ) q^{3} + 8 i q^{4} + ( -15 + 20 i ) q^{5} -36 q^{6} + ( 29 + 29 i ) q^{7} + ( 16 - 16 i ) q^{8} -81 i q^{9} + ( 70 - 10 i ) q^{10} -118 q^{11} + ( 72 + 72 i ) q^{12} + ( 69 - 69 i ) q^{13} -116 i q^{14} + ( 45 + 315 i ) q^{15} -64 q^{16} + ( -271 - 271 i ) q^{17} + ( -162 + 162 i ) q^{18} + 280 i q^{19} + ( -160 - 120 i ) q^{20} + 522 q^{21} + ( 236 + 236 i ) q^{22} + ( 269 - 269 i ) q^{23} -288 i q^{24} + ( -175 - 600 i ) q^{25} -276 q^{26} + ( -232 + 232 i ) q^{28} + 680 i q^{29} + ( 540 - 720 i ) q^{30} + 202 q^{31} + ( 128 + 128 i ) q^{32} + ( -1062 + 1062 i ) q^{33} + 1084 i q^{34} + ( -1015 + 145 i ) q^{35} + 648 q^{36} + ( -651 - 651 i ) q^{37} + ( 560 - 560 i ) q^{38} -1242 i q^{39} + ( 80 + 560 i ) q^{40} + 1682 q^{41} + ( -1044 - 1044 i ) q^{42} + ( 1089 - 1089 i ) q^{43} -944 i q^{44} + ( 1620 + 1215 i ) q^{45} -1076 q^{46} + ( 1269 + 1269 i ) q^{47} + ( -576 + 576 i ) q^{48} -719 i q^{49} + ( -850 + 1550 i ) q^{50} -4878 q^{51} + ( 552 + 552 i ) q^{52} + ( -611 + 611 i ) q^{53} + ( 1770 - 2360 i ) q^{55} + 928 q^{56} + ( 2520 + 2520 i ) q^{57} + ( 1360 - 1360 i ) q^{58} + 1160 i q^{59} + ( -2520 + 360 i ) q^{60} -5598 q^{61} + ( -404 - 404 i ) q^{62} + ( 2349 - 2349 i ) q^{63} -512 i q^{64} + ( 345 + 2415 i ) q^{65} + 4248 q^{66} + ( -751 - 751 i ) q^{67} + ( 2168 - 2168 i ) q^{68} -4842 i q^{69} + ( 2320 + 1740 i ) q^{70} + 6442 q^{71} + ( -1296 - 1296 i ) q^{72} + ( -2951 + 2951 i ) q^{73} + 2604 i q^{74} + ( -6975 - 3825 i ) q^{75} -2240 q^{76} + ( -3422 - 3422 i ) q^{77} + ( -2484 + 2484 i ) q^{78} + 10560 i q^{79} + ( 960 - 1280 i ) q^{80} + 6561 q^{81} + ( -3364 - 3364 i ) q^{82} + ( -6231 + 6231 i ) q^{83} + 4176 i q^{84} + ( 9485 - 1355 i ) q^{85} -4356 q^{86} + ( 6120 + 6120 i ) q^{87} + ( -1888 + 1888 i ) q^{88} -14480 i q^{89} + ( -810 - 5670 i ) q^{90} + 4002 q^{91} + ( 2152 + 2152 i ) q^{92} + ( 1818 - 1818 i ) q^{93} -5076 i q^{94} + ( -5600 - 4200 i ) q^{95} + 2304 q^{96} + ( -7311 - 7311 i ) q^{97} + ( -1438 + 1438 i ) q^{98} + 9558 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 18q^{3} - 30q^{5} - 72q^{6} + 58q^{7} + 32q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 18q^{3} - 30q^{5} - 72q^{6} + 58q^{7} + 32q^{8} + 140q^{10} - 236q^{11} + 144q^{12} + 138q^{13} + 90q^{15} - 128q^{16} - 542q^{17} - 324q^{18} - 320q^{20} + 1044q^{21} + 472q^{22} + 538q^{23} - 350q^{25} - 552q^{26} - 464q^{28} + 1080q^{30} + 404q^{31} + 256q^{32} - 2124q^{33} - 2030q^{35} + 1296q^{36} - 1302q^{37} + 1120q^{38} + 160q^{40} + 3364q^{41} - 2088q^{42} + 2178q^{43} + 3240q^{45} - 2152q^{46} + 2538q^{47} - 1152q^{48} - 1700q^{50} - 9756q^{51} + 1104q^{52} - 1222q^{53} + 3540q^{55} + 1856q^{56} + 5040q^{57} + 2720q^{58} - 5040q^{60} - 11196q^{61} - 808q^{62} + 4698q^{63} + 690q^{65} + 8496q^{66} - 1502q^{67} + 4336q^{68} + 4640q^{70} + 12884q^{71} - 2592q^{72} - 5902q^{73} - 13950q^{75} - 4480q^{76} - 6844q^{77} - 4968q^{78} + 1920q^{80} + 13122q^{81} - 6728q^{82} - 12462q^{83} + 18970q^{85} - 8712q^{86} + 12240q^{87} - 3776q^{88} - 1620q^{90} + 8004q^{91} + 4304q^{92} + 3636q^{93} - 11200q^{95} + 4608q^{96} - 14622q^{97} - 2876q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$7$$ $$\chi(n)$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 1.00000i 1.00000i
−2.00000 + 2.00000i 9.00000 + 9.00000i 8.00000i −15.0000 20.0000i −36.0000 29.0000 29.0000i 16.0000 + 16.0000i 81.0000i 70.0000 + 10.0000i
7.1 −2.00000 2.00000i 9.00000 9.00000i 8.00000i −15.0000 + 20.0000i −36.0000 29.0000 + 29.0000i 16.0000 16.0000i 81.0000i 70.0000 10.0000i
 $$n$$: 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.5.c.a 2
3.b odd 2 1 90.5.g.b 2
4.b odd 2 1 80.5.p.b 2
5.b even 2 1 50.5.c.b 2
5.c odd 4 1 inner 10.5.c.a 2
5.c odd 4 1 50.5.c.b 2
8.b even 2 1 320.5.p.b 2
8.d odd 2 1 320.5.p.i 2
15.d odd 2 1 450.5.g.a 2
15.e even 4 1 90.5.g.b 2
15.e even 4 1 450.5.g.a 2
20.d odd 2 1 400.5.p.c 2
20.e even 4 1 80.5.p.b 2
20.e even 4 1 400.5.p.c 2
40.i odd 4 1 320.5.p.b 2
40.k even 4 1 320.5.p.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.5.c.a 2 1.a even 1 1 trivial
10.5.c.a 2 5.c odd 4 1 inner
50.5.c.b 2 5.b even 2 1
50.5.c.b 2 5.c odd 4 1
80.5.p.b 2 4.b odd 2 1
80.5.p.b 2 20.e even 4 1
90.5.g.b 2 3.b odd 2 1
90.5.g.b 2 15.e even 4 1
320.5.p.b 2 8.b even 2 1
320.5.p.b 2 40.i odd 4 1
320.5.p.i 2 8.d odd 2 1
320.5.p.i 2 40.k even 4 1
400.5.p.c 2 20.d odd 2 1
400.5.p.c 2 20.e even 4 1
450.5.g.a 2 15.d odd 2 1
450.5.g.a 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 18 T_{3} + 162$$ acting on $$S_{5}^{\mathrm{new}}(10, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 8 T^{2}$$
$3$ $$( 1 - 9 T )^{2}( 1 + 81 T^{2} )$$
$5$ $$1 + 30 T + 625 T^{2}$$
$7$ $$1 - 58 T + 1682 T^{2} - 139258 T^{3} + 5764801 T^{4}$$
$11$ $$( 1 + 118 T + 14641 T^{2} )^{2}$$
$13$ $$1 - 138 T + 9522 T^{2} - 3941418 T^{3} + 815730721 T^{4}$$
$17$ $$1 + 542 T + 146882 T^{2} + 45268382 T^{3} + 6975757441 T^{4}$$
$19$ $$1 - 182242 T^{2} + 16983563041 T^{4}$$
$23$ $$1 - 538 T + 144722 T^{2} - 150554458 T^{3} + 78310985281 T^{4}$$
$29$ $$1 - 952162 T^{2} + 500246412961 T^{4}$$
$31$ $$( 1 - 202 T + 923521 T^{2} )^{2}$$
$37$ $$1 + 1302 T + 847602 T^{2} + 2440157622 T^{3} + 3512479453921 T^{4}$$
$41$ $$( 1 - 1682 T + 2825761 T^{2} )^{2}$$
$43$ $$1 - 2178 T + 2371842 T^{2} - 7446148578 T^{3} + 11688200277601 T^{4}$$
$47$ $$1 - 2538 T + 3220722 T^{2} - 12384630378 T^{3} + 23811286661761 T^{4}$$
$53$ $$1 + 1222 T + 746642 T^{2} + 9642167782 T^{3} + 62259690411361 T^{4}$$
$59$ $$1 - 22889122 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 + 5598 T + 13845841 T^{2} )^{2}$$
$67$ $$1 + 1502 T + 1128002 T^{2} + 30266983742 T^{3} + 406067677556641 T^{4}$$
$71$ $$( 1 - 6442 T + 25411681 T^{2} )^{2}$$
$73$ $$1 + 5902 T + 17416802 T^{2} + 167606418382 T^{3} + 806460091894081 T^{4}$$
$79$ $$1 + 33613438 T^{2} + 1517108809906561 T^{4}$$
$83$ $$1 + 12462 T + 77650722 T^{2} + 591425596302 T^{3} + 2252292232139041 T^{4}$$
$89$ $$1 + 84185918 T^{2} + 3936588805702081 T^{4}$$
$97$ $$1 + 14622 T + 106901442 T^{2} + 1294475146782 T^{3} + 7837433594376961 T^{4}$$