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 $2$

Related objects

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: $$x^{2} - x - 17$$ x^2 - x - 17 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.

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

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{13}^{\mathrm{new}}(23, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 79T - 6047$$
$3$ $$T^{2} - 14T - 1594127$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 8482894 T + 2065235247793$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$(T - 148035889)^{2}$$
$29$ $$T^{2} - 244330126 T - 10\!\cdots\!47$$
$31$ $$T^{2} + 1677025154 T + 44\!\cdots\!33$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 7596282526 T - 99\!\cdots\!67$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 20606906306 T + 76\!\cdots\!13$$
$53$ $$T^{2}$$
$59$ $$(T + 19874527918)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2} + 188893891874 T - 13\!\cdots\!47$$
$73$ $$T^{2} + 223017449186 T - 18\!\cdots\!67$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$