# Properties

 Label 10.5.c.b 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.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 i ) q^{2} + ( 1 - i ) q^{3} + 8 i q^{4} + ( -15 - 20 i ) q^{5} + 4 q^{6} + ( -19 - 19 i ) q^{7} + ( -16 + 16 i ) q^{8} + 79 i q^{9} +O(q^{10})$$ $$q + ( 2 + 2 i ) q^{2} + ( 1 - i ) q^{3} + 8 i q^{4} + ( -15 - 20 i ) q^{5} + 4 q^{6} + ( -19 - 19 i ) q^{7} + ( -16 + 16 i ) q^{8} + 79 i q^{9} + ( 10 - 70 i ) q^{10} + 202 q^{11} + ( 8 + 8 i ) q^{12} + ( -99 + 99 i ) q^{13} -76 i q^{14} + ( -35 - 5 i ) q^{15} -64 q^{16} + ( -239 - 239 i ) q^{17} + ( -158 + 158 i ) q^{18} + 40 i q^{19} + ( 160 - 120 i ) q^{20} -38 q^{21} + ( 404 + 404 i ) q^{22} + ( 541 - 541 i ) q^{23} + 32 i q^{24} + ( -175 + 600 i ) q^{25} -396 q^{26} + ( 160 + 160 i ) q^{27} + ( 152 - 152 i ) q^{28} -200 i q^{29} + ( -60 - 80 i ) q^{30} -758 q^{31} + ( -128 - 128 i ) q^{32} + ( 202 - 202 i ) q^{33} -956 i q^{34} + ( -95 + 665 i ) q^{35} -632 q^{36} + ( 141 + 141 i ) q^{37} + ( -80 + 80 i ) q^{38} + 198 i q^{39} + ( 560 + 80 i ) q^{40} + 1042 q^{41} + ( -76 - 76 i ) q^{42} + ( -759 + 759 i ) q^{43} + 1616 i q^{44} + ( 1580 - 1185 i ) q^{45} + 2164 q^{46} + ( -459 - 459 i ) q^{47} + ( -64 + 64 i ) q^{48} -1679 i q^{49} + ( -1550 + 850 i ) q^{50} -478 q^{51} + ( -792 - 792 i ) q^{52} + ( -1819 + 1819 i ) q^{53} + 640 i q^{54} + ( -3030 - 4040 i ) q^{55} + 608 q^{56} + ( 40 + 40 i ) q^{57} + ( 400 - 400 i ) q^{58} + 4600 i q^{59} + ( 40 - 280 i ) q^{60} + 2082 q^{61} + ( -1516 - 1516 i ) q^{62} + ( 1501 - 1501 i ) q^{63} -512 i q^{64} + ( 3465 + 495 i ) q^{65} + 808 q^{66} + ( 5081 + 5081 i ) q^{67} + ( 1912 - 1912 i ) q^{68} -1082 i q^{69} + ( -1520 + 1140 i ) q^{70} -3478 q^{71} + ( -1264 - 1264 i ) q^{72} + ( -3479 + 3479 i ) q^{73} + 564 i q^{74} + ( 425 + 775 i ) q^{75} -320 q^{76} + ( -3838 - 3838 i ) q^{77} + ( -396 + 396 i ) q^{78} -7680 i q^{79} + ( 960 + 1280 i ) q^{80} -6079 q^{81} + ( 2084 + 2084 i ) q^{82} + ( 6081 - 6081 i ) q^{83} -304 i q^{84} + ( -1195 + 8365 i ) q^{85} -3036 q^{86} + ( -200 - 200 i ) q^{87} + ( -3232 + 3232 i ) q^{88} -5680 i q^{89} + ( 5530 + 790 i ) q^{90} + 3762 q^{91} + ( 4328 + 4328 i ) q^{92} + ( -758 + 758 i ) q^{93} -1836 i q^{94} + ( 800 - 600 i ) q^{95} -256 q^{96} + ( 561 + 561 i ) q^{97} + ( 3358 - 3358 i ) q^{98} + 15958 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} + 2q^{3} - 30q^{5} + 8q^{6} - 38q^{7} - 32q^{8} + O(q^{10})$$ $$2q + 4q^{2} + 2q^{3} - 30q^{5} + 8q^{6} - 38q^{7} - 32q^{8} + 20q^{10} + 404q^{11} + 16q^{12} - 198q^{13} - 70q^{15} - 128q^{16} - 478q^{17} - 316q^{18} + 320q^{20} - 76q^{21} + 808q^{22} + 1082q^{23} - 350q^{25} - 792q^{26} + 320q^{27} + 304q^{28} - 120q^{30} - 1516q^{31} - 256q^{32} + 404q^{33} - 190q^{35} - 1264q^{36} + 282q^{37} - 160q^{38} + 1120q^{40} + 2084q^{41} - 152q^{42} - 1518q^{43} + 3160q^{45} + 4328q^{46} - 918q^{47} - 128q^{48} - 3100q^{50} - 956q^{51} - 1584q^{52} - 3638q^{53} - 6060q^{55} + 1216q^{56} + 80q^{57} + 800q^{58} + 80q^{60} + 4164q^{61} - 3032q^{62} + 3002q^{63} + 6930q^{65} + 1616q^{66} + 10162q^{67} + 3824q^{68} - 3040q^{70} - 6956q^{71} - 2528q^{72} - 6958q^{73} + 850q^{75} - 640q^{76} - 7676q^{77} - 792q^{78} + 1920q^{80} - 12158q^{81} + 4168q^{82} + 12162q^{83} - 2390q^{85} - 6072q^{86} - 400q^{87} - 6464q^{88} + 11060q^{90} + 7524q^{91} + 8656q^{92} - 1516q^{93} + 1600q^{95} - 512q^{96} + 1122q^{97} + 6716q^{98} + O(q^{100})$$

## 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 1.00000 + 1.00000i 8.00000i −15.0000 + 20.0000i 4.00000 −19.0000 + 19.0000i −16.0000 16.0000i 79.0000i 10.0000 + 70.0000i
7.1 2.00000 + 2.00000i 1.00000 1.00000i 8.00000i −15.0000 20.0000i 4.00000 −19.0000 19.0000i −16.0000 + 16.0000i 79.0000i 10.0000 70.0000i
 $$n$$: 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.b 2
3.b odd 2 1 90.5.g.a 2
4.b odd 2 1 80.5.p.c 2
5.b even 2 1 50.5.c.a 2
5.c odd 4 1 inner 10.5.c.b 2
5.c odd 4 1 50.5.c.a 2
8.b even 2 1 320.5.p.d 2
8.d odd 2 1 320.5.p.g 2
15.d odd 2 1 450.5.g.b 2
15.e even 4 1 90.5.g.a 2
15.e even 4 1 450.5.g.b 2
20.d odd 2 1 400.5.p.b 2
20.e even 4 1 80.5.p.c 2
20.e even 4 1 400.5.p.b 2
40.i odd 4 1 320.5.p.d 2
40.k even 4 1 320.5.p.g 2

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

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 8 T^{2}$$
$3$ $$1 - 2 T + 2 T^{2} - 162 T^{3} + 6561 T^{4}$$
$5$ $$1 + 30 T + 625 T^{2}$$
$7$ $$1 + 38 T + 722 T^{2} + 91238 T^{3} + 5764801 T^{4}$$
$11$ $$( 1 - 202 T + 14641 T^{2} )^{2}$$
$13$ $$1 + 198 T + 19602 T^{2} + 5655078 T^{3} + 815730721 T^{4}$$
$17$ $$1 + 478 T + 114242 T^{2} + 39923038 T^{3} + 6975757441 T^{4}$$
$19$ $$1 - 259042 T^{2} + 16983563041 T^{4}$$
$23$ $$1 - 1082 T + 585362 T^{2} - 302787962 T^{3} + 78310985281 T^{4}$$
$29$ $$1 - 1374562 T^{2} + 500246412961 T^{4}$$
$31$ $$( 1 + 758 T + 923521 T^{2} )^{2}$$
$37$ $$1 - 282 T + 39762 T^{2} - 528513402 T^{3} + 3512479453921 T^{4}$$
$41$ $$( 1 - 1042 T + 2825761 T^{2} )^{2}$$
$43$ $$1 + 1518 T + 1152162 T^{2} + 5189739918 T^{3} + 11688200277601 T^{4}$$
$47$ $$1 + 918 T + 421362 T^{2} + 4479547158 T^{3} + 23811286661761 T^{4}$$
$53$ $$1 + 3638 T + 6617522 T^{2} + 28705569878 T^{3} + 62259690411361 T^{4}$$
$59$ $$1 - 3074722 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 - 2082 T + 13845841 T^{2} )^{2}$$
$67$ $$1 - 10162 T + 51633122 T^{2} - 204775691602 T^{3} + 406067677556641 T^{4}$$
$71$ $$( 1 + 3478 T + 25411681 T^{2} )^{2}$$
$73$ $$1 + 6958 T + 24206882 T^{2} + 197594960878 T^{3} + 806460091894081 T^{4}$$
$79$ $$1 - 18917762 T^{2} + 1517108809906561 T^{4}$$
$83$ $$1 - 12162 T + 73957122 T^{2} - 577188100002 T^{3} + 2252292232139041 T^{4}$$
$89$ $$1 - 93222082 T^{2} + 3936588805702081 T^{4}$$
$97$ $$1 - 1122 T + 629442 T^{2} - 99329853282 T^{3} + 7837433594376961 T^{4}$$