Properties

Label 10.4.b.a
Level 10
Weight 4
Character orbit 10.b
Analytic conductor 0.590
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 10.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590019100057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} -2 i q^{3} -4 q^{4} + ( -5 - 10 i ) q^{5} + 4 q^{6} + 26 i q^{7} -8 i q^{8} + 23 q^{9} +O(q^{10})\) \( q + 2 i q^{2} -2 i q^{3} -4 q^{4} + ( -5 - 10 i ) q^{5} + 4 q^{6} + 26 i q^{7} -8 i q^{8} + 23 q^{9} + ( 20 - 10 i ) q^{10} -28 q^{11} + 8 i q^{12} -12 i q^{13} -52 q^{14} + ( -20 + 10 i ) q^{15} + 16 q^{16} -64 i q^{17} + 46 i q^{18} + 60 q^{19} + ( 20 + 40 i ) q^{20} + 52 q^{21} -56 i q^{22} + 58 i q^{23} -16 q^{24} + ( -75 + 100 i ) q^{25} + 24 q^{26} -100 i q^{27} -104 i q^{28} -90 q^{29} + ( -20 - 40 i ) q^{30} -128 q^{31} + 32 i q^{32} + 56 i q^{33} + 128 q^{34} + ( 260 - 130 i ) q^{35} -92 q^{36} + 236 i q^{37} + 120 i q^{38} -24 q^{39} + ( -80 + 40 i ) q^{40} + 242 q^{41} + 104 i q^{42} -362 i q^{43} + 112 q^{44} + ( -115 - 230 i ) q^{45} -116 q^{46} + 226 i q^{47} -32 i q^{48} -333 q^{49} + ( -200 - 150 i ) q^{50} -128 q^{51} + 48 i q^{52} + 108 i q^{53} + 200 q^{54} + ( 140 + 280 i ) q^{55} + 208 q^{56} -120 i q^{57} -180 i q^{58} + 20 q^{59} + ( 80 - 40 i ) q^{60} + 542 q^{61} -256 i q^{62} + 598 i q^{63} -64 q^{64} + ( -120 + 60 i ) q^{65} -112 q^{66} -434 i q^{67} + 256 i q^{68} + 116 q^{69} + ( 260 + 520 i ) q^{70} -1128 q^{71} -184 i q^{72} -632 i q^{73} -472 q^{74} + ( 200 + 150 i ) q^{75} -240 q^{76} -728 i q^{77} -48 i q^{78} + 720 q^{79} + ( -80 - 160 i ) q^{80} + 421 q^{81} + 484 i q^{82} + 478 i q^{83} -208 q^{84} + ( -640 + 320 i ) q^{85} + 724 q^{86} + 180 i q^{87} + 224 i q^{88} + 490 q^{89} + ( 460 - 230 i ) q^{90} + 312 q^{91} -232 i q^{92} + 256 i q^{93} -452 q^{94} + ( -300 - 600 i ) q^{95} + 64 q^{96} + 1456 i q^{97} -666 i q^{98} -644 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} - 10q^{5} + 8q^{6} + 46q^{9} + O(q^{10}) \) \( 2q - 8q^{4} - 10q^{5} + 8q^{6} + 46q^{9} + 40q^{10} - 56q^{11} - 104q^{14} - 40q^{15} + 32q^{16} + 120q^{19} + 40q^{20} + 104q^{21} - 32q^{24} - 150q^{25} + 48q^{26} - 180q^{29} - 40q^{30} - 256q^{31} + 256q^{34} + 520q^{35} - 184q^{36} - 48q^{39} - 160q^{40} + 484q^{41} + 224q^{44} - 230q^{45} - 232q^{46} - 666q^{49} - 400q^{50} - 256q^{51} + 400q^{54} + 280q^{55} + 416q^{56} + 40q^{59} + 160q^{60} + 1084q^{61} - 128q^{64} - 240q^{65} - 224q^{66} + 232q^{69} + 520q^{70} - 2256q^{71} - 944q^{74} + 400q^{75} - 480q^{76} + 1440q^{79} - 160q^{80} + 842q^{81} - 416q^{84} - 1280q^{85} + 1448q^{86} + 980q^{89} + 920q^{90} + 624q^{91} - 904q^{94} - 600q^{95} + 128q^{96} - 1288q^{99} + O(q^{100}) \)

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
2.00000i 2.00000i −4.00000 −5.00000 + 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 + 10.0000i
9.2 2.00000i 2.00000i −4.00000 −5.00000 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 10.0000i
\(n\): 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 10.4.b.a 2
3.b odd 2 1 90.4.c.b 2
4.b odd 2 1 80.4.c.a 2
5.b even 2 1 inner 10.4.b.a 2
5.c odd 4 1 50.4.a.b 1
5.c odd 4 1 50.4.a.d 1
7.b odd 2 1 490.4.c.b 2
8.b even 2 1 320.4.c.d 2
8.d odd 2 1 320.4.c.c 2
12.b even 2 1 720.4.f.f 2
15.d odd 2 1 90.4.c.b 2
15.e even 4 1 450.4.a.j 1
15.e even 4 1 450.4.a.k 1
20.d odd 2 1 80.4.c.a 2
20.e even 4 1 400.4.a.h 1
20.e even 4 1 400.4.a.n 1
35.c odd 2 1 490.4.c.b 2
35.f even 4 1 2450.4.a.o 1
35.f even 4 1 2450.4.a.bb 1
40.e odd 2 1 320.4.c.c 2
40.f even 2 1 320.4.c.d 2
40.i odd 4 1 1600.4.a.u 1
40.i odd 4 1 1600.4.a.bh 1
40.k even 4 1 1600.4.a.t 1
40.k even 4 1 1600.4.a.bg 1
60.h even 2 1 720.4.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 1.a even 1 1 trivial
10.4.b.a 2 5.b even 2 1 inner
50.4.a.b 1 5.c odd 4 1
50.4.a.d 1 5.c odd 4 1
80.4.c.a 2 4.b odd 2 1
80.4.c.a 2 20.d odd 2 1
90.4.c.b 2 3.b odd 2 1
90.4.c.b 2 15.d odd 2 1
320.4.c.c 2 8.d odd 2 1
320.4.c.c 2 40.e odd 2 1
320.4.c.d 2 8.b even 2 1
320.4.c.d 2 40.f even 2 1
400.4.a.h 1 20.e even 4 1
400.4.a.n 1 20.e even 4 1
450.4.a.j 1 15.e even 4 1
450.4.a.k 1 15.e even 4 1
490.4.c.b 2 7.b odd 2 1
490.4.c.b 2 35.c odd 2 1
720.4.f.f 2 12.b even 2 1
720.4.f.f 2 60.h even 2 1
1600.4.a.t 1 40.k even 4 1
1600.4.a.u 1 40.i odd 4 1
1600.4.a.bg 1 40.k even 4 1
1600.4.a.bh 1 40.i odd 4 1
2450.4.a.o 1 35.f even 4 1
2450.4.a.bb 1 35.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(10, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} \)
$3$ \( 1 - 50 T^{2} + 729 T^{4} \)
$5$ \( 1 + 10 T + 125 T^{2} \)
$7$ \( 1 - 10 T^{2} + 117649 T^{4} \)
$11$ \( ( 1 + 28 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 4250 T^{2} + 4826809 T^{4} \)
$17$ \( 1 - 5730 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 - 60 T + 6859 T^{2} )^{2} \)
$23$ \( 1 - 20970 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 + 90 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 128 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 45610 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 - 242 T + 68921 T^{2} )^{2} \)
$43$ \( 1 - 27970 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 - 156570 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 286090 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 - 20 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 - 542 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 413170 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 1128 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 378610 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 720 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 915090 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 - 490 T + 704969 T^{2} )^{2} \)
$97$ \( 1 + 294590 T^{2} + 832972004929 T^{4} \)
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