Properties

 Label 10.6.b.a Level 10 Weight 6 Character orbit 10.b Analytic conductor 1.604 Analytic rank 0 Dimension 2 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$10 = 2 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 10.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$1.60383819813$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 14 i q^{3} -16 q^{4} + ( 55 + 10 i ) q^{5} -56 q^{6} -158 i q^{7} -64 i q^{8} + 47 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} + 14 i q^{3} -16 q^{4} + ( 55 + 10 i ) q^{5} -56 q^{6} -158 i q^{7} -64 i q^{8} + 47 q^{9} + ( -40 + 220 i ) q^{10} -148 q^{11} -224 i q^{12} + 684 i q^{13} + 632 q^{14} + ( -140 + 770 i ) q^{15} + 256 q^{16} -2048 i q^{17} + 188 i q^{18} -2220 q^{19} + ( -880 - 160 i ) q^{20} + 2212 q^{21} -592 i q^{22} -1246 i q^{23} + 896 q^{24} + ( 2925 + 1100 i ) q^{25} -2736 q^{26} + 4060 i q^{27} + 2528 i q^{28} + 270 q^{29} + ( -3080 - 560 i ) q^{30} -2048 q^{31} + 1024 i q^{32} -2072 i q^{33} + 8192 q^{34} + ( 1580 - 8690 i ) q^{35} -752 q^{36} + 4372 i q^{37} -8880 i q^{38} -9576 q^{39} + ( 640 - 3520 i ) q^{40} -2398 q^{41} + 8848 i q^{42} + 2294 i q^{43} + 2368 q^{44} + ( 2585 + 470 i ) q^{45} + 4984 q^{46} + 10682 i q^{47} + 3584 i q^{48} -8157 q^{49} + ( -4400 + 11700 i ) q^{50} + 28672 q^{51} -10944 i q^{52} + 2964 i q^{53} -16240 q^{54} + ( -8140 - 1480 i ) q^{55} -10112 q^{56} -31080 i q^{57} + 1080 i q^{58} + 39740 q^{59} + ( 2240 - 12320 i ) q^{60} -42298 q^{61} -8192 i q^{62} -7426 i q^{63} -4096 q^{64} + ( -6840 + 37620 i ) q^{65} + 8288 q^{66} -32098 i q^{67} + 32768 i q^{68} + 17444 q^{69} + ( 34760 + 6320 i ) q^{70} -4248 q^{71} -3008 i q^{72} + 30104 i q^{73} -17488 q^{74} + ( -15400 + 40950 i ) q^{75} + 35520 q^{76} + 23384 i q^{77} -38304 i q^{78} -35280 q^{79} + ( 14080 + 2560 i ) q^{80} -45419 q^{81} -9592 i q^{82} -27826 i q^{83} -35392 q^{84} + ( 20480 - 112640 i ) q^{85} -9176 q^{86} + 3780 i q^{87} + 9472 i q^{88} + 85210 q^{89} + ( -1880 + 10340 i ) q^{90} + 108072 q^{91} + 19936 i q^{92} -28672 i q^{93} -42728 q^{94} + ( -122100 - 22200 i ) q^{95} -14336 q^{96} + 97232 i q^{97} -32628 i q^{98} -6956 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + 110q^{5} - 112q^{6} + 94q^{9} + O(q^{10})$$ $$2q - 32q^{4} + 110q^{5} - 112q^{6} + 94q^{9} - 80q^{10} - 296q^{11} + 1264q^{14} - 280q^{15} + 512q^{16} - 4440q^{19} - 1760q^{20} + 4424q^{21} + 1792q^{24} + 5850q^{25} - 5472q^{26} + 540q^{29} - 6160q^{30} - 4096q^{31} + 16384q^{34} + 3160q^{35} - 1504q^{36} - 19152q^{39} + 1280q^{40} - 4796q^{41} + 4736q^{44} + 5170q^{45} + 9968q^{46} - 16314q^{49} - 8800q^{50} + 57344q^{51} - 32480q^{54} - 16280q^{55} - 20224q^{56} + 79480q^{59} + 4480q^{60} - 84596q^{61} - 8192q^{64} - 13680q^{65} + 16576q^{66} + 34888q^{69} + 69520q^{70} - 8496q^{71} - 34976q^{74} - 30800q^{75} + 71040q^{76} - 70560q^{79} + 28160q^{80} - 90838q^{81} - 70784q^{84} + 40960q^{85} - 18352q^{86} + 170420q^{89} - 3760q^{90} + 216144q^{91} - 85456q^{94} - 244200q^{95} - 28672q^{96} - 13912q^{99} + O(q^{100})$$

Character values

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

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

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
 − 1.00000i 1.00000i
4.00000i 14.0000i −16.0000 55.0000 10.0000i −56.0000 158.000i 64.0000i 47.0000 −40.0000 220.000i
9.2 4.00000i 14.0000i −16.0000 55.0000 + 10.0000i −56.0000 158.000i 64.0000i 47.0000 −40.0000 + 220.000i
 $$n$$: 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 10.6.b.a 2
3.b odd 2 1 90.6.c.a 2
4.b odd 2 1 80.6.c.c 2
5.b even 2 1 inner 10.6.b.a 2
5.c odd 4 1 50.6.a.c 1
5.c odd 4 1 50.6.a.e 1
8.b even 2 1 320.6.c.b 2
8.d odd 2 1 320.6.c.a 2
12.b even 2 1 720.6.f.a 2
15.d odd 2 1 90.6.c.a 2
15.e even 4 1 450.6.a.c 1
15.e even 4 1 450.6.a.w 1
20.d odd 2 1 80.6.c.c 2
20.e even 4 1 400.6.a.c 1
20.e even 4 1 400.6.a.k 1
40.e odd 2 1 320.6.c.a 2
40.f even 2 1 320.6.c.b 2
60.h even 2 1 720.6.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.b.a 2 1.a even 1 1 trivial
10.6.b.a 2 5.b even 2 1 inner
50.6.a.c 1 5.c odd 4 1
50.6.a.e 1 5.c odd 4 1
80.6.c.c 2 4.b odd 2 1
80.6.c.c 2 20.d odd 2 1
90.6.c.a 2 3.b odd 2 1
90.6.c.a 2 15.d odd 2 1
320.6.c.a 2 8.d odd 2 1
320.6.c.a 2 40.e odd 2 1
320.6.c.b 2 8.b even 2 1
320.6.c.b 2 40.f even 2 1
400.6.a.c 1 20.e even 4 1
400.6.a.k 1 20.e even 4 1
450.6.a.c 1 15.e even 4 1
450.6.a.w 1 15.e even 4 1
720.6.f.a 2 12.b even 2 1
720.6.f.a 2 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(10, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 16 T^{2}$$
$3$ $$1 - 290 T^{2} + 59049 T^{4}$$
$5$ $$1 - 110 T + 3125 T^{2}$$
$7$ $$1 - 8650 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 148 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 274730 T^{2} + 137858491849 T^{4}$$
$17$ $$1 + 1354590 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 2220 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 11320170 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 270 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 2048 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 119573530 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 2398 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 288754450 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 344584890 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 827605690 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 - 39740 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 42298 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 1669968610 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 4248 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3239892370 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 + 35280 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 7103795010 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 85210 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 7720618690 T^{2} + 73742412689492826049 T^{4}$$