Properties

Label 4.13.b.a
Level 4
Weight 13
Character orbit 4.b
Self dual yes
Analytic conductor 3.656
Analytic rank 0
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(3.65597526911\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 64q^{2} + 4096q^{4} + 23506q^{5} - 262144q^{8} + 531441q^{9} + O(q^{10}) \) \( q - 64q^{2} + 4096q^{4} + 23506q^{5} - 262144q^{8} + 531441q^{9} - 1504384q^{10} + 6911282q^{13} + 16777216q^{16} - 47295038q^{17} - 34012224q^{18} + 96280576q^{20} + 308391411q^{25} - 442322048q^{26} - 173439758q^{29} - 1073741824q^{32} + 3026882432q^{34} + 2176782336q^{36} - 2050092718q^{37} - 6161956864q^{40} - 2285065118q^{41} + 12492052146q^{45} + 13841287201q^{49} - 19737050304q^{50} + 28308611072q^{52} - 43462597358q^{53} + 11100144512q^{58} - 47844884878q^{61} + 68719476736q^{64} + 162456594692q^{65} - 193720475648q^{68} - 139314069504q^{72} - 119852347678q^{73} + 131205933952q^{74} + 394365239296q^{80} + 282429536481q^{81} + 146244167552q^{82} - 1111717163228q^{85} + 907573615522q^{89} - 799491337344q^{90} + 502341690242q^{97} - 885842380864q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
−64.0000 0 4096.00 23506.0 0 0 −262144. 531441. −1.50438e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.13.b.a 1
3.b odd 2 1 36.13.d.a 1
4.b odd 2 1 CM 4.13.b.a 1
8.b even 2 1 64.13.c.a 1
8.d odd 2 1 64.13.c.a 1
12.b even 2 1 36.13.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.13.b.a 1 1.a even 1 1 trivial
4.13.b.a 1 4.b odd 2 1 CM
36.13.d.a 1 3.b odd 2 1
36.13.d.a 1 12.b even 2 1
64.13.c.a 1 8.b even 2 1
64.13.c.a 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{13}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 64 T \)
$3$ \( ( 1 - 729 T )( 1 + 729 T ) \)
$5$ \( 1 - 23506 T + 244140625 T^{2} \)
$7$ \( ( 1 - 117649 T )( 1 + 117649 T ) \)
$11$ \( ( 1 - 1771561 T )( 1 + 1771561 T ) \)
$13$ \( 1 - 6911282 T + 23298085122481 T^{2} \)
$17$ \( 1 + 47295038 T + 582622237229761 T^{2} \)
$19$ \( ( 1 - 47045881 T )( 1 + 47045881 T ) \)
$23$ \( ( 1 - 148035889 T )( 1 + 148035889 T ) \)
$29$ \( 1 + 173439758 T + 353814783205469041 T^{2} \)
$31$ \( ( 1 - 887503681 T )( 1 + 887503681 T ) \)
$37$ \( 1 + 2050092718 T + 6582952005840035281 T^{2} \)
$41$ \( 1 + 2285065118 T + 22563490300366186081 T^{2} \)
$43$ \( ( 1 - 6321363049 T )( 1 + 6321363049 T ) \)
$47$ \( ( 1 - 10779215329 T )( 1 + 10779215329 T ) \)
$53$ \( 1 + 43462597358 T + \)\(49\!\cdots\!41\)\( T^{2} \)
$59$ \( ( 1 - 42180533641 T )( 1 + 42180533641 T ) \)
$61$ \( 1 + 47844884878 T + \)\(26\!\cdots\!21\)\( T^{2} \)
$67$ \( ( 1 - 90458382169 T )( 1 + 90458382169 T ) \)
$71$ \( ( 1 - 128100283921 T )( 1 + 128100283921 T ) \)
$73$ \( 1 + 119852347678 T + \)\(22\!\cdots\!21\)\( T^{2} \)
$79$ \( ( 1 - 243087455521 T )( 1 + 243087455521 T ) \)
$83$ \( ( 1 - 326940373369 T )( 1 + 326940373369 T ) \)
$89$ \( 1 - 907573615522 T + \)\(24\!\cdots\!21\)\( T^{2} \)
$97$ \( 1 - 502341690242 T + \)\(69\!\cdots\!41\)\( T^{2} \)
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