Properties

Label 10.10.a.b
Level 10
Weight 10
Character orbit 10.a
Self dual yes
Analytic conductor 5.150
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.15035836164\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 16q^{2} + 46q^{3} + 256q^{4} - 625q^{5} - 736q^{6} - 10318q^{7} - 4096q^{8} - 17567q^{9} + O(q^{10}) \) \( q - 16q^{2} + 46q^{3} + 256q^{4} - 625q^{5} - 736q^{6} - 10318q^{7} - 4096q^{8} - 17567q^{9} + 10000q^{10} - 5568q^{11} + 11776q^{12} + 45986q^{13} + 165088q^{14} - 28750q^{15} + 65536q^{16} - 381318q^{17} + 281072q^{18} + 610460q^{19} - 160000q^{20} - 474628q^{21} + 89088q^{22} - 1447914q^{23} - 188416q^{24} + 390625q^{25} - 735776q^{26} - 1713500q^{27} - 2641408q^{28} + 5385510q^{29} + 460000q^{30} + 3053852q^{31} - 1048576q^{32} - 256128q^{33} + 6101088q^{34} + 6448750q^{35} - 4497152q^{36} + 12889442q^{37} - 9767360q^{38} + 2115356q^{39} + 2560000q^{40} - 33786618q^{41} + 7594048q^{42} - 36886234q^{43} - 1425408q^{44} + 10979375q^{45} + 23166624q^{46} - 44163798q^{47} + 3014656q^{48} + 66107517q^{49} - 6250000q^{50} - 17540628q^{51} + 11772416q^{52} + 29746266q^{53} + 27416000q^{54} + 3480000q^{55} + 42262528q^{56} + 28081160q^{57} - 86168160q^{58} - 65575380q^{59} - 7360000q^{60} + 40183202q^{61} - 48861632q^{62} + 181256306q^{63} + 16777216q^{64} - 28741250q^{65} + 4098048q^{66} - 115706158q^{67} - 97617408q^{68} - 66604044q^{69} - 103180000q^{70} - 231681708q^{71} + 71954432q^{72} + 358691906q^{73} - 206231072q^{74} + 17968750q^{75} + 156277760q^{76} + 57450624q^{77} - 33845696q^{78} - 486017080q^{79} - 40960000q^{80} + 266950261q^{81} + 540585888q^{82} + 251168886q^{83} - 121504768q^{84} + 238323750q^{85} + 590179744q^{86} + 247733460q^{87} + 22806528q^{88} - 526039110q^{89} - 175670000q^{90} - 474483548q^{91} - 370665984q^{92} + 140477192q^{93} + 706620768q^{94} - 381537500q^{95} - 48234496q^{96} - 1075981438q^{97} - 1057720272q^{98} + 97813056q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−16.0000 46.0000 256.000 −625.000 −736.000 −10318.0 −4096.00 −17567.0 10000.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.10.a.b 1
3.b odd 2 1 90.10.a.h 1
4.b odd 2 1 80.10.a.b 1
5.b even 2 1 50.10.a.d 1
5.c odd 4 2 50.10.b.b 2
8.b even 2 1 320.10.a.e 1
8.d odd 2 1 320.10.a.f 1
20.d odd 2 1 400.10.a.h 1
20.e even 4 2 400.10.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.10.a.b 1 1.a even 1 1 trivial
50.10.a.d 1 5.b even 2 1
50.10.b.b 2 5.c odd 4 2
80.10.a.b 1 4.b odd 2 1
90.10.a.h 1 3.b odd 2 1
320.10.a.e 1 8.b even 2 1
320.10.a.f 1 8.d odd 2 1
400.10.a.h 1 20.d odd 2 1
400.10.c.i 2 20.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 46 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16 T \)
$3$ \( 1 - 46 T + 19683 T^{2} \)
$5$ \( 1 + 625 T \)
$7$ \( 1 + 10318 T + 40353607 T^{2} \)
$11$ \( 1 + 5568 T + 2357947691 T^{2} \)
$13$ \( 1 - 45986 T + 10604499373 T^{2} \)
$17$ \( 1 + 381318 T + 118587876497 T^{2} \)
$19$ \( 1 - 610460 T + 322687697779 T^{2} \)
$23$ \( 1 + 1447914 T + 1801152661463 T^{2} \)
$29$ \( 1 - 5385510 T + 14507145975869 T^{2} \)
$31$ \( 1 - 3053852 T + 26439622160671 T^{2} \)
$37$ \( 1 - 12889442 T + 129961739795077 T^{2} \)
$41$ \( 1 + 33786618 T + 327381934393961 T^{2} \)
$43$ \( 1 + 36886234 T + 502592611936843 T^{2} \)
$47$ \( 1 + 44163798 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 29746266 T + 3299763591802133 T^{2} \)
$59$ \( 1 + 65575380 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 40183202 T + 11694146092834141 T^{2} \)
$67$ \( 1 + 115706158 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 231681708 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 358691906 T + 58871586708267913 T^{2} \)
$79$ \( 1 + 486017080 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 251168886 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 526039110 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 1075981438 T + 760231058654565217 T^{2} \)
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