# Properties

 Label 24.8.f.b Level 24 Weight 8 Character orbit 24.f Analytic conductor 7.497 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 24.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.49724061162$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{6}, \sqrt{-26})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} + \beta_{2} ) q^{2} + ( -9 - 9 \beta_{1} ) q^{3} + ( -80 + 4 \beta_{3} ) q^{4} -20 \beta_{2} q^{5} + ( 468 - 18 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} ) q^{6} -62 \beta_{3} q^{7} + ( -64 \beta_{1} - 288 \beta_{2} ) q^{8} + ( -2025 + 162 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} + \beta_{2} ) q^{2} + ( -9 - 9 \beta_{1} ) q^{3} + ( -80 + 4 \beta_{3} ) q^{4} -20 \beta_{2} q^{5} + ( 468 - 18 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} ) q^{6} -62 \beta_{3} q^{7} + ( -64 \beta_{1} - 288 \beta_{2} ) q^{8} + ( -2025 + 162 \beta_{1} ) q^{9} + ( -480 - 40 \beta_{3} ) q^{10} -350 \beta_{1} q^{11} + ( 720 + 720 \beta_{1} + 936 \beta_{2} - 36 \beta_{3} ) q^{12} -406 \beta_{3} q^{13} + ( -1488 \beta_{1} + 3224 \beta_{2} ) q^{14} + ( 180 \beta_{2} + 180 \beta_{3} ) q^{15} + ( -3584 - 640 \beta_{3} ) q^{16} + 5404 \beta_{1} q^{17} + ( -8424 - 4050 \beta_{1} - 2025 \beta_{2} + 162 \beta_{3} ) q^{18} + 11570 q^{19} + ( -1920 \beta_{1} + 1600 \beta_{2} ) q^{20} + ( -14508 \beta_{2} + 558 \beta_{3} ) q^{21} + ( 18200 - 350 \beta_{3} ) q^{22} -11336 \beta_{2} q^{23} + ( -14976 + 576 \beta_{1} + 2592 \beta_{2} + 2592 \beta_{3} ) q^{24} -68525 q^{25} + ( -9744 \beta_{1} + 21112 \beta_{2} ) q^{26} + ( 56133 + 16767 \beta_{1} ) q^{27} + ( 154752 + 4960 \beta_{3} ) q^{28} + 11180 \beta_{2} q^{29} + ( 4320 + 4320 \beta_{1} - 9360 \beta_{2} + 360 \beta_{3} ) q^{30} + 2870 \beta_{3} q^{31} + ( -22528 \beta_{1} + 29696 \beta_{2} ) q^{32} + ( -81900 + 3150 \beta_{1} ) q^{33} + ( -281008 + 5404 \beta_{3} ) q^{34} + 29760 \beta_{1} q^{35} + ( 162000 - 12960 \beta_{1} - 16848 \beta_{2} - 8100 \beta_{3} ) q^{36} -11322 \beta_{3} q^{37} + ( 23140 \beta_{1} + 11570 \beta_{2} ) q^{38} + ( -95004 \beta_{2} + 3654 \beta_{3} ) q^{39} + ( 138240 + 1280 \beta_{3} ) q^{40} -79160 \beta_{1} q^{41} + ( -348192 + 13392 \beta_{1} - 29016 \beta_{2} - 29016 \beta_{3} ) q^{42} + 495062 q^{43} + ( 28000 \beta_{1} + 36400 \beta_{2} ) q^{44} + ( 40500 \beta_{2} - 3240 \beta_{3} ) q^{45} + ( -272064 - 22672 \beta_{3} ) q^{46} + 231872 \beta_{2} q^{47} + ( 32256 + 32256 \beta_{1} - 149760 \beta_{2} + 5760 \beta_{3} ) q^{48} -1575113 q^{49} + ( -137050 \beta_{1} - 68525 \beta_{2} ) q^{50} + ( 1264536 - 48636 \beta_{1} ) q^{51} + ( 1013376 + 32480 \beta_{3} ) q^{52} + 116844 \beta_{2} q^{53} + ( -871884 + 112266 \beta_{1} + 56133 \beta_{2} + 16767 \beta_{3} ) q^{54} + 7000 \beta_{3} q^{55} + ( 428544 \beta_{1} - 103168 \beta_{2} ) q^{56} + ( -104130 - 104130 \beta_{1} ) q^{57} + ( 268320 + 22360 \beta_{3} ) q^{58} -282506 \beta_{1} q^{59} + ( -449280 + 17280 \beta_{1} - 14400 \beta_{2} - 14400 \beta_{3} ) q^{60} -71510 \beta_{3} q^{61} + ( 68880 \beta_{1} - 149240 \beta_{2} ) q^{62} + ( 261144 \beta_{2} + 125550 \beta_{3} ) q^{63} + ( 1884160 + 36864 \beta_{3} ) q^{64} + 194880 \beta_{1} q^{65} + ( -163800 - 163800 \beta_{1} - 81900 \beta_{2} + 3150 \beta_{3} ) q^{66} + 1400126 q^{67} + ( -432320 \beta_{1} - 562016 \beta_{2} ) q^{68} + ( 102024 \beta_{2} + 102024 \beta_{3} ) q^{69} + ( -1547520 + 29760 \beta_{3} ) q^{70} -728616 \beta_{2} q^{71} + ( 269568 + 129600 \beta_{1} + 583200 \beta_{2} - 46656 \beta_{3} ) q^{72} -2223598 q^{73} + ( -271728 \beta_{1} + 588744 \beta_{2} ) q^{74} + ( 616725 + 616725 \beta_{1} ) q^{75} + ( -925600 + 46280 \beta_{3} ) q^{76} -564200 \beta_{2} q^{77} + ( -2280096 + 87696 \beta_{1} - 190008 \beta_{2} - 190008 \beta_{3} ) q^{78} -223802 \beta_{3} q^{79} + ( 307200 \beta_{1} + 71680 \beta_{2} ) q^{80} + ( 3418281 - 656100 \beta_{1} ) q^{81} + ( 4116320 - 79160 \beta_{3} ) q^{82} + 590962 \beta_{1} q^{83} + ( -1392768 - 1392768 \beta_{1} + 1160640 \beta_{2} - 44640 \beta_{3} ) q^{84} -108080 \beta_{3} q^{85} + ( 990124 \beta_{1} + 495062 \beta_{2} ) q^{86} + ( -100620 \beta_{2} - 100620 \beta_{3} ) q^{87} + ( -582400 + 100800 \beta_{3} ) q^{88} + 1146364 \beta_{1} q^{89} + ( 972000 - 77760 \beta_{1} + 168480 \beta_{2} + 81000 \beta_{3} ) q^{90} -15707328 q^{91} + ( -1088256 \beta_{1} + 906880 \beta_{2} ) q^{92} + ( 671580 \beta_{2} - 25830 \beta_{3} ) q^{93} + ( 5564928 + 463744 \beta_{3} ) q^{94} -231400 \beta_{2} q^{95} + ( -5271552 + 202752 \beta_{1} - 267264 \beta_{2} - 267264 \beta_{3} ) q^{96} + 6867926 q^{97} + ( -3150226 \beta_{1} - 1575113 \beta_{2} ) q^{98} + ( 1474200 + 708750 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 36q^{3} - 320q^{4} + 1872q^{6} - 8100q^{9} + O(q^{10})$$ $$4q - 36q^{3} - 320q^{4} + 1872q^{6} - 8100q^{9} - 1920q^{10} + 2880q^{12} - 14336q^{16} - 33696q^{18} + 46280q^{19} + 72800q^{22} - 59904q^{24} - 274100q^{25} + 224532q^{27} + 619008q^{28} + 17280q^{30} - 327600q^{33} - 1124032q^{34} + 648000q^{36} + 552960q^{40} - 1392768q^{42} + 1980248q^{43} - 1088256q^{46} + 129024q^{48} - 6300452q^{49} + 5058144q^{51} + 4053504q^{52} - 3487536q^{54} - 416520q^{57} + 1073280q^{58} - 1797120q^{60} + 7536640q^{64} - 655200q^{66} + 5600504q^{67} - 6190080q^{70} + 1078272q^{72} - 8894392q^{73} + 2466900q^{75} - 3702400q^{76} - 9120384q^{78} + 13673124q^{81} + 16465280q^{82} - 5571072q^{84} - 2329600q^{88} + 3888000q^{90} - 62829312q^{91} + 22259712q^{94} - 21086208q^{96} + 27471704q^{97} + 5896800q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 18 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 2 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 20$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{2} - 2 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −1.22474 − 2.54951i −1.22474 + 2.54951i 1.22474 − 2.54951i 1.22474 + 2.54951i
−4.89898 10.1980i −9.00000 + 45.8912i −80.0000 + 99.9200i 97.9796 512.091 133.038i 1548.76i 1410.91 + 326.337i −2025.00 826.041i −480.000 999.200i
11.2 −4.89898 + 10.1980i −9.00000 45.8912i −80.0000 99.9200i 97.9796 512.091 + 133.038i 1548.76i 1410.91 326.337i −2025.00 + 826.041i −480.000 + 999.200i
11.3 4.89898 10.1980i −9.00000 + 45.8912i −80.0000 99.9200i −97.9796 423.909 + 316.602i 1548.76i −1410.91 + 326.337i −2025.00 826.041i −480.000 + 999.200i
11.4 4.89898 + 10.1980i −9.00000 45.8912i −80.0000 + 99.9200i −97.9796 423.909 316.602i 1548.76i −1410.91 326.337i −2025.00 + 826.041i −480.000 999.200i
 $$n$$: 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
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.8.f.b 4
3.b odd 2 1 inner 24.8.f.b 4
4.b odd 2 1 96.8.f.b 4
8.b even 2 1 96.8.f.b 4
8.d odd 2 1 inner 24.8.f.b 4
12.b even 2 1 96.8.f.b 4
24.f even 2 1 inner 24.8.f.b 4
24.h odd 2 1 96.8.f.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.f.b 4 1.a even 1 1 trivial
24.8.f.b 4 3.b odd 2 1 inner
24.8.f.b 4 8.d odd 2 1 inner
24.8.f.b 4 24.f even 2 1 inner
96.8.f.b 4 4.b odd 2 1
96.8.f.b 4 8.b even 2 1
96.8.f.b 4 12.b even 2 1
96.8.f.b 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 9600$$ acting on $$S_{8}^{\mathrm{new}}(24, [\chi])$$.