Properties

Label 83.13.b.a
Level $83$
Weight $13$
Character orbit 83.b
Self dual yes
Analytic conductor $75.861$
Analytic rank $0$
Dimension $1$
CM discriminant -83
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 83.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(75.8614868339\)
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

\(f(q)\) \(=\) \( q - 617 q^{3} + 4096 q^{4} - 231577 q^{7} - 150752 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 617 q^{3} + 4096 q^{4} - 231577 q^{7} - 150752 q^{9} - 3198553 q^{11} - 2527232 q^{12} + 16777216 q^{16} - 30626737 q^{17} + 142883009 q^{21} - 231011422 q^{23} + 244140625 q^{25} + 420913081 q^{27} - 948539392 q^{28} + 991523567 q^{29} - 101624713 q^{31} + 1973507201 q^{33} - 617480192 q^{36} - 5012228257 q^{37} - 9474786718 q^{41} - 13101273088 q^{44} - 10351542272 q^{48} + 39786619728 q^{49} + 18896696729 q^{51} - 48671434393 q^{59} + 86434023647 q^{61} + 34910695904 q^{63} + 68719476736 q^{64} - 125447114752 q^{68} + 142534047374 q^{69} - 150634765625 q^{75} + 740711308081 q^{77} - 179587577345 q^{81} + 326940373369 q^{83} + 585248804864 q^{84} - 611770040839 q^{87} - 946222784512 q^{92} + 62702447921 q^{93} + 482188261856 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/83\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
0
0 −617.000 4096.00 0 0 −231577. 0 −150752. 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
83.b odd 2 1 CM by \(\Q(\sqrt{-83}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 83.13.b.a 1
83.b odd 2 1 CM 83.13.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.13.b.a 1 1.a even 1 1 trivial
83.13.b.a 1 83.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{13}^{\mathrm{new}}(83, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3} + 617 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 617 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 231577 \) Copy content Toggle raw display
$11$ \( T + 3198553 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 30626737 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 231011422 \) Copy content Toggle raw display
$29$ \( T - 991523567 \) Copy content Toggle raw display
$31$ \( T + 101624713 \) Copy content Toggle raw display
$37$ \( T + 5012228257 \) Copy content Toggle raw display
$41$ \( T + 9474786718 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 48671434393 \) Copy content Toggle raw display
$61$ \( T - 86434023647 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 326940373369 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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