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

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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: \(\Q(\sqrt{-1}) \)
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:

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

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