Properties

Label 8.23.d.a
Level 8
Weight 23
Character orbit 8.d
Self dual yes
Analytic conductor 24.537
Analytic rank 0
Dimension 1
CM discriminant -8
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 23 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5365947873\)
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

\(f(q)\) \(=\) \( q - 2048q^{2} - 199058q^{3} + 4194304q^{4} + 407670784q^{6} - 8589934592q^{8} + 8243027755q^{9} + O(q^{10}) \) \( q - 2048q^{2} - 199058q^{3} + 4194304q^{4} + 407670784q^{6} - 8589934592q^{8} + 8243027755q^{9} - 550486028386q^{11} - 834909765632q^{12} + 17592186044416q^{16} - 41341538741182q^{17} - 16881720842240q^{18} - 86441697072754q^{19} + 1127395386134528q^{22} + 1709895200014336q^{24} + 2384185791015625q^{25} + 4605810344793532q^{27} - 36028797018963968q^{32} + 109578647838460388q^{33} + 84667471341940736q^{34} + 34573764284907520q^{36} + 177032595605000192q^{38} + 291475944231948914q^{41} - 1810527321484332514q^{43} - 2308905750803513344q^{44} - 3501865369629360128q^{48} + 3909821048582988049q^{49} - 4882812500000000000q^{50} + 8229364018742206556q^{51} - 9432699586137153536q^{54} + 17206911335908265732q^{57} + 49867838021809381118q^{59} + 73786976294838206464q^{64} - 224417070773166874624q^{66} + 204466126247920757678q^{67} - 173398981308294627328q^{68} - 70807069255490600960q^{72} + 531355580825828150162q^{73} - 474591255187988281250q^{75} - 362562755799040393216q^{76} - 1175498340952207340651q^{81} - 596942733787031375872q^{82} - 2470066709866614036658q^{83} + 3707959954399912988672q^{86} + 4728638977645595328512q^{88} + 5077570440143736010226q^{89} + 7171820277000929542144q^{96} - 9422684141117979760606q^{97} - 8007313507497959524352q^{98} - 4537671610725515853430q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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
−2048.00 −199058. 4.19430e6 0 4.07671e8 0 −8.58993e9 8.24303e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.23.d.a 1
4.b odd 2 1 32.23.d.a 1
8.b even 2 1 32.23.d.a 1
8.d odd 2 1 CM 8.23.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.23.d.a 1 1.a even 1 1 trivial
8.23.d.a 1 8.d odd 2 1 CM
32.23.d.a 1 4.b odd 2 1
32.23.d.a 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 199058 \) acting on \(S_{23}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2048 T \)
$3$ \( 1 + 199058 T + 31381059609 T^{2} \)
$5$ \( ( 1 - 48828125 T )( 1 + 48828125 T ) \)
$7$ \( ( 1 - 1977326743 T )( 1 + 1977326743 T ) \)
$11$ \( 1 + 550486028386 T + \)\(81\!\cdots\!21\)\( T^{2} \)
$13$ \( ( 1 - 1792160394037 T )( 1 + 1792160394037 T ) \)
$17$ \( 1 + 41341538741182 T + \)\(11\!\cdots\!89\)\( T^{2} \)
$19$ \( 1 + 86441697072754 T + \)\(13\!\cdots\!61\)\( T^{2} \)
$23$ \( ( 1 - 952809757913927 T )( 1 + 952809757913927 T ) \)
$29$ \( ( 1 - 12200509765705829 T )( 1 + 12200509765705829 T ) \)
$31$ \( ( 1 - 25408476896404831 T )( 1 + 25408476896404831 T ) \)
$37$ \( ( 1 - 177917621779460413 T )( 1 + 177917621779460413 T ) \)
$41$ \( 1 - 291475944231948914 T + \)\(30\!\cdots\!81\)\( T^{2} \)
$43$ \( 1 + 1810527321484332514 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$47$ \( ( 1 - 2472159215084012303 T )( 1 + 2472159215084012303 T ) \)
$53$ \( ( 1 - 9269035929372191597 T )( 1 + 9269035929372191597 T ) \)
$59$ \( 1 - 49867838021809381118 T + \)\(90\!\cdots\!81\)\( T^{2} \)
$61$ \( ( 1 - 43513917611435838661 T )( 1 + 43513917611435838661 T ) \)
$67$ \( 1 - \)\(20\!\cdots\!78\)\( T + \)\(14\!\cdots\!89\)\( T^{2} \)
$71$ \( ( 1 - \)\(23\!\cdots\!71\)\( T )( 1 + \)\(23\!\cdots\!71\)\( T ) \)
$73$ \( 1 - \)\(53\!\cdots\!62\)\( T + \)\(98\!\cdots\!29\)\( T^{2} \)
$79$ \( ( 1 - \)\(74\!\cdots\!79\)\( T )( 1 + \)\(74\!\cdots\!79\)\( T ) \)
$83$ \( 1 + \)\(24\!\cdots\!58\)\( T + \)\(16\!\cdots\!89\)\( T^{2} \)
$89$ \( 1 - \)\(50\!\cdots\!26\)\( T + \)\(77\!\cdots\!21\)\( T^{2} \)
$97$ \( 1 + \)\(94\!\cdots\!06\)\( T + \)\(51\!\cdots\!09\)\( T^{2} \)
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