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

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

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:

Atkin-Lehner signs

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

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T + 156 \) Copy content Toggle raw display
$5$ \( T - 870 \) Copy content Toggle raw display
$7$ \( T + 952 \) Copy content Toggle raw display
$11$ \( T + 56148 \) Copy content Toggle raw display
$13$ \( T - 178094 \) Copy content Toggle raw display
$17$ \( T + 247662 \) Copy content Toggle raw display
$19$ \( T - 315380 \) Copy content Toggle raw display
$23$ \( T - 204504 \) Copy content Toggle raw display
$29$ \( T + 3840450 \) Copy content Toggle raw display
$31$ \( T + 1309408 \) Copy content Toggle raw display
$37$ \( T - 4307078 \) Copy content Toggle raw display
$41$ \( T - 1512042 \) Copy content Toggle raw display
$43$ \( T - 33670604 \) Copy content Toggle raw display
$47$ \( T + 10581072 \) Copy content Toggle raw display
$53$ \( T - 16616214 \) Copy content Toggle raw display
$59$ \( T - 112235100 \) Copy content Toggle raw display
$61$ \( T + 33197218 \) Copy content Toggle raw display
$67$ \( T + 121372252 \) Copy content Toggle raw display
$71$ \( T + 387172728 \) Copy content Toggle raw display
$73$ \( T - 255240074 \) Copy content Toggle raw display
$79$ \( T - 492101840 \) Copy content Toggle raw display
$83$ \( T + 457420236 \) Copy content Toggle raw display
$89$ \( T + 31809510 \) Copy content Toggle raw display
$97$ \( T + 673532062 \) Copy content Toggle raw display
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