Properties

Label 2.10.a.a
Level 2
Weight 10
Character orbit 2.a
Self dual yes
Analytic conductor 1.030
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(1.03007167233\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 16q^{2} - 156q^{3} + 256q^{4} + 870q^{5} - 2496q^{6} - 952q^{7} + 4096q^{8} + 4653q^{9} + O(q^{10}) \) \( q + 16q^{2} - 156q^{3} + 256q^{4} + 870q^{5} - 2496q^{6} - 952q^{7} + 4096q^{8} + 4653q^{9} + 13920q^{10} - 56148q^{11} - 39936q^{12} + 178094q^{13} - 15232q^{14} - 135720q^{15} + 65536q^{16} - 247662q^{17} + 74448q^{18} + 315380q^{19} + 222720q^{20} + 148512q^{21} - 898368q^{22} + 204504q^{23} - 638976q^{24} - 1196225q^{25} + 2849504q^{26} + 2344680q^{27} - 243712q^{28} - 3840450q^{29} - 2171520q^{30} - 1309408q^{31} + 1048576q^{32} + 8759088q^{33} - 3962592q^{34} - 828240q^{35} + 1191168q^{36} + 4307078q^{37} + 5046080q^{38} - 27782664q^{39} + 3563520q^{40} + 1512042q^{41} + 2376192q^{42} + 33670604q^{43} - 14373888q^{44} + 4048110q^{45} + 3272064q^{46} - 10581072q^{47} - 10223616q^{48} - 39447303q^{49} - 19139600q^{50} + 38635272q^{51} + 45592064q^{52} + 16616214q^{53} + 37514880q^{54} - 48848760q^{55} - 3899392q^{56} - 49199280q^{57} - 61447200q^{58} + 112235100q^{59} - 34744320q^{60} - 33197218q^{61} - 20950528q^{62} - 4429656q^{63} + 16777216q^{64} + 154941780q^{65} + 140145408q^{66} - 121372252q^{67} - 63401472q^{68} - 31902624q^{69} - 13251840q^{70} - 387172728q^{71} + 19058688q^{72} + 255240074q^{73} + 68913248q^{74} + 186611100q^{75} + 80737280q^{76} + 53452896q^{77} - 444522624q^{78} + 492101840q^{79} + 57016320q^{80} - 457355079q^{81} + 24192672q^{82} - 457420236q^{83} + 38019072q^{84} - 215465940q^{85} + 538729664q^{86} + 599110200q^{87} - 229982208q^{88} - 31809510q^{89} + 64769760q^{90} - 169545488q^{91} + 52353024q^{92} + 204267648q^{93} - 169297152q^{94} + 274380600q^{95} - 163577856q^{96} - 673532062q^{97} - 631156848q^{98} - 261256644q^{99} + O(q^{100}) \)

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 −156.000 256.000 870.000 −2496.00 −952.000 4096.00 4653.00 13920.0
\(n\): 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 2.10.a.a 1
3.b odd 2 1 18.10.a.a 1
4.b odd 2 1 16.10.a.d 1
5.b even 2 1 50.10.a.c 1
5.c odd 4 2 50.10.b.a 2
7.b odd 2 1 98.10.a.c 1
7.c even 3 2 98.10.c.c 2
7.d odd 6 2 98.10.c.b 2
8.b even 2 1 64.10.a.h 1
8.d odd 2 1 64.10.a.b 1
9.c even 3 2 162.10.c.b 2
9.d odd 6 2 162.10.c.i 2
11.b odd 2 1 242.10.a.a 1
12.b even 2 1 144.10.a.d 1
13.b even 2 1 338.10.a.a 1
16.e even 4 2 256.10.b.g 2
16.f odd 4 2 256.10.b.e 2
20.d odd 2 1 400.10.a.b 1
20.e even 4 2 400.10.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.10.a.a 1 1.a even 1 1 trivial
16.10.a.d 1 4.b odd 2 1
18.10.a.a 1 3.b odd 2 1
50.10.a.c 1 5.b even 2 1
50.10.b.a 2 5.c odd 4 2
64.10.a.b 1 8.d odd 2 1
64.10.a.h 1 8.b even 2 1
98.10.a.c 1 7.b odd 2 1
98.10.c.b 2 7.d odd 6 2
98.10.c.c 2 7.c even 3 2
144.10.a.d 1 12.b even 2 1
162.10.c.b 2 9.c even 3 2
162.10.c.i 2 9.d odd 6 2
242.10.a.a 1 11.b odd 2 1
256.10.b.e 2 16.f odd 4 2
256.10.b.g 2 16.e even 4 2
338.10.a.a 1 13.b even 2 1
400.10.a.b 1 20.d odd 2 1
400.10.c.d 2 20.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 16 T \)
$3$ \( 1 + 156 T + 19683 T^{2} \)
$5$ \( 1 - 870 T + 1953125 T^{2} \)
$7$ \( 1 + 952 T + 40353607 T^{2} \)
$11$ \( 1 + 56148 T + 2357947691 T^{2} \)
$13$ \( 1 - 178094 T + 10604499373 T^{2} \)
$17$ \( 1 + 247662 T + 118587876497 T^{2} \)
$19$ \( 1 - 315380 T + 322687697779 T^{2} \)
$23$ \( 1 - 204504 T + 1801152661463 T^{2} \)
$29$ \( 1 + 3840450 T + 14507145975869 T^{2} \)
$31$ \( 1 + 1309408 T + 26439622160671 T^{2} \)
$37$ \( 1 - 4307078 T + 129961739795077 T^{2} \)
$41$ \( 1 - 1512042 T + 327381934393961 T^{2} \)
$43$ \( 1 - 33670604 T + 502592611936843 T^{2} \)
$47$ \( 1 + 10581072 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 16616214 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 112235100 T + 8662995818654939 T^{2} \)
$61$ \( 1 + 33197218 T + 11694146092834141 T^{2} \)
$67$ \( 1 + 121372252 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 387172728 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 255240074 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 492101840 T + 119851595982618319 T^{2} \)
$83$ \( 1 + 457420236 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 31809510 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 673532062 T + 760231058654565217 T^{2} \)
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