Properties

Label 23.13.b.b
Level $23$
Weight $13$
Character orbit 23.b
Self dual yes
Analytic conductor $21.022$
Analytic rank $0$
Dimension $2$
CM discriminant -23
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 23.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(21.0218577974\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{69}) \)
Defining polynomial: \( x^{2} - x - 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{69})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (21 \beta + 29) q^{2} + ( - 304 \beta + 159) q^{3} + (1659 \beta + 4242) q^{4} + ( - 11861 \beta - 103917) q^{6} + (86016 \beta + 596497) q^{8} + ( - 4256 \beta + 1064912) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (21 \beta + 29) q^{2} + ( - 304 \beta + 159) q^{3} + (1659 \beta + 4242) q^{4} + ( - 11861 \beta - 103917) q^{6} + (86016 \beta + 596497) q^{8} + ( - 4256 \beta + 1064912) q^{9} + ( - 1530123 \beta - 7899234) q^{12} + (960816 \beta + 3761039) q^{13} + (10031973 \beta + 30630893) q^{16} + (22150352 \beta + 29363056) q^{18} + 148035889 q^{23} + ( - 193807408 \beta - 349687665) q^{24} + 244140625 q^{25} + (127022619 \beta + 452081443) q^{26} + ( - 161558064 \beta + 106816897) q^{27} + ( - 242770416 \beta + 243550271) q^{29} + ( - 121268256 \beta - 777878449) q^{31} + (792525867 \beta + 2026458546) q^{32} + (1741574352 \beta + 4397324736) q^{36} + ( - 1282674176 \beta - 4367491887) q^{39} + ( - 1189637376 \beta + 4392959951) q^{41} + (3108753669 \beta + 4293040781) q^{46} + ( - 1320759744 \beta - 9643073281) q^{47} + ( - 10766427557 \beta - 46974924477) q^{48} + 13841287201 q^{49} + (5126953125 \beta + 7080078125) q^{50} + (11909338917 \beta + 43052221086) q^{52} + ( - 5834748363 \beta - 54578538835) q^{54} + ( - 7023965109 \beta - 79606080653) q^{58} - 19874527918 q^{59} + ( - 22398860229 \beta - 65851242413) q^{62} + (41090961408 \beta + 216234894625) q^{64} + ( - 45002910256 \beta + 23537706351) q^{69} + (36090362784 \beta - 112492127329) q^{71} + (88694695264 \beta + 628993383632) q^{72} + (42667271616 \beta - 132842360401) q^{73} + ( - 74218750000 \beta + 38818359375) q^{75} + ( - 155851038427 \beta - 584571945555) q^{78} + ( - 6784604512 \beta + 285978063183) q^{81} + (32770290171 \beta - 297304704653) q^{82} + ( - 38837572064 \beta + 1293362002977) q^{87} + (245591539851 \beta + 627968241138) q^{92} + (254058945616 \beta + 503031673617) q^{93} + ( - 268542526101 \beta - 751160353757) q^{94} + ( - 730959648699 \beta - 3773566771842) q^{96} + (290667031221 \beta + 401397328829) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 79 q^{2} + 14 q^{3} + 10143 q^{4} - 219695 q^{6} + 1279010 q^{8} + 2125568 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 79 q^{2} + 14 q^{3} + 10143 q^{4} - 219695 q^{6} + 1279010 q^{8} + 2125568 q^{9} - 17328591 q^{12} + 8482894 q^{13} + 71293759 q^{16} + 80876464 q^{18} + 296071778 q^{23} - 893182738 q^{24} + 488281250 q^{25} + 1031185505 q^{26} + 52075730 q^{27} + 244330126 q^{29} - 1677025154 q^{31} + 4845442959 q^{32} + 10536223824 q^{36} - 10017657950 q^{39} + 7596282526 q^{41} + 11694835231 q^{46} - 20606906306 q^{47} - 104716276511 q^{48} + 27682574402 q^{49} + 19287109375 q^{50} + 98013781089 q^{52} - 114991826033 q^{54} - 166236126415 q^{58} - 39749055836 q^{59} - 154101345055 q^{62} + 473560750658 q^{64} + 2072502446 q^{69} - 188893891874 q^{71} + 1346681462528 q^{72} - 223017449186 q^{73} + 3417968750 q^{75} - 1324994929537 q^{78} + 565171521854 q^{81} - 561839119135 q^{82} + 2547886433890 q^{87} + 1501528022127 q^{92} + 1260122292850 q^{93} - 1770863233615 q^{94} - 8278093192383 q^{96} + 1093461688879 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−3.65331
4.65331
−47.7196 1269.61 −1818.84 0 −60585.1 0 282254. 1.08046e6 0
22.2 126.720 −1255.61 11961.8 0 −159110. 0 996756. 1.04511e6 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.13.b.b 2
23.b odd 2 1 CM 23.13.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.13.b.b 2 1.a even 1 1 trivial
23.13.b.b 2 23.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 79T_{2} - 6047 \) acting on \(S_{13}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 79T - 6047 \) Copy content Toggle raw display
$3$ \( T^{2} - 14T - 1594127 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8482894 T + 2065235247793 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 148035889)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 244330126 T - 10\!\cdots\!47 \) Copy content Toggle raw display
$31$ \( T^{2} + 1677025154 T + 44\!\cdots\!33 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 7596282526 T - 99\!\cdots\!67 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 20606906306 T + 76\!\cdots\!13 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T + 19874527918)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 188893891874 T - 13\!\cdots\!47 \) Copy content Toggle raw display
$73$ \( T^{2} + 223017449186 T - 18\!\cdots\!67 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) 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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