Properties

Label 83.13.b.b
Level $83$
Weight $13$
Character orbit 83.b
Self dual yes
Analytic conductor $75.861$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 5\sqrt{249})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 29 \beta + 294) q^{3} + 4096 q^{4} + (915 \beta + 116246) q^{7} + ( - 17893 \beta + 863591) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 29 \beta + 294) q^{3} + 4096 q^{4} + (915 \beta + 116246) q^{7} + ( - 17893 \beta + 863591) q^{9} + (33459 \beta + 1616006) q^{11} + ( - 118784 \beta + 1204224) q^{12} + 16777216 q^{16} + (819195 \beta + 15722966) q^{17} + ( - 3075589 \beta - 7112136) q^{21} - 231011422 q^{23} + 244140625 q^{25} + ( - 15411789 \beta + 905055832) q^{27} + (3747840 \beta + 476143616) q^{28} + (14431227 \beta - 488546170) q^{29} + ( - 38902557 \beta + 31361078) q^{31} + ( - 36056917 \beta - 1034698152) q^{33} + ( - 73289728 \beta + 3537268736) q^{36} + ( - 24141909 \beta + 2494043174) q^{37} - 9474786718 q^{41} + (137048064 \beta + 6619160576) q^{44} + ( - 486539264 \beta + 4932501504) q^{48} + (211892955 \beta + 974567415) q^{49} + ( - 191366029 \beta - 32342803176) q^{51} + (1512658419 \beta + 25092046406) q^{59} + (1231426923 \beta - 42601298362) q^{61} + ( - 1273431818 \beta + 74914019566) q^{63} + 68719476736 q^{64} + (3355422720 \beta + 64401268736) q^{68} + (6699331238 \beta - 67917358068) q^{69} + ( - 7080078125 \beta + 71777343750) q^{75} + (5337505419 \beta + 235491150136) q^{77} + ( - 21715553162 \beta + 502580316813) q^{81} + 326940373369 q^{83} + ( - 12597612544 \beta - 29131309056) q^{84} + (18829125251 \beta - 794827261128) q^{87} - 946222784512 q^{92} + ( - 13474997173 \beta + 1764659139000) q^{93} + (578377798 \beta + 464019221374) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 617 q^{3} + 8192 q^{4} + 231577 q^{7} + 1745075 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 617 q^{3} + 8192 q^{4} + 231577 q^{7} + 1745075 q^{9} + 3198553 q^{11} + 2527232 q^{12} + 33554432 q^{16} + 30626737 q^{17} - 11148683 q^{21} - 462022844 q^{23} + 488281250 q^{25} + 1825523453 q^{27} + 948539392 q^{28} - 991523567 q^{29} + 101624713 q^{31} - 2033339387 q^{33} + 7147827200 q^{36} + 5012228257 q^{37} - 18949573436 q^{41} + 13101273088 q^{44} + 10351542272 q^{48} + 1737241875 q^{49} - 64494240323 q^{51} + 48671434393 q^{59} - 86434023647 q^{61} + 151101470950 q^{63} + 137438953472 q^{64} + 125447114752 q^{68} - 142534047374 q^{69} + 150634765625 q^{75} + 465644794853 q^{77} + 1026876186788 q^{81} + 653880746738 q^{83} - 45665005568 q^{84} - 1608483647507 q^{87} - 1892445569024 q^{92} + 3542793275173 q^{93} + 927460064950 q^{99}+O(q^{100}) \) 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
8.38987
−7.38987
0 −835.531 4096.00 0 0 151885. 0 166671. 0
82.2 0 1452.53 4096.00 0 0 79692.4 0 1.57840e6 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.b 2
83.b odd 2 1 CM 83.13.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.13.b.b 2 1.a even 1 1 trivial
83.13.b.b 2 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
\( T_{3}^{2} - 617T_{3} - 1213634 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 617 T - 1213634 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 231577 T + 12104045326 \) Copy content Toggle raw display
$11$ \( T^{2} - 3198553 T + 815456163646 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 809869692422114 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 231011422)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 991523567 T - 78\!\cdots\!34 \) Copy content Toggle raw display
$31$ \( T^{2} - 101624713 T - 23\!\cdots\!14 \) Copy content Toggle raw display
$37$ \( T^{2} - 5012228257 T + 53\!\cdots\!06 \) Copy content Toggle raw display
$41$ \( (T + 9474786718)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 48671434393 T - 29\!\cdots\!94 \) Copy content Toggle raw display
$61$ \( T^{2} + 86434023647 T - 49\!\cdots\!54 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T - 326940373369)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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