# Properties

 Label 24.9.e.a Level $24$ Weight $9$ Character orbit 24.e Analytic conductor $9.777$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,9,Mod(17,24)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("24.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 24.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$9.77708664147$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633$$ x^8 - 4*x^7 - 78*x^6 + 144*x^5 + 2079*x^4 + 936*x^3 - 658*x^2 + 2884*x + 30633 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{42}\cdot 3^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 7) q^{3} + (\beta_{4} - \beta_{2}) q^{5} + (\beta_1 + 198) q^{7} + ( - 2 \beta_{4} - \beta_{3} + \cdots + 41) q^{9}+O(q^{10})$$ q + (b2 + 7) * q^3 + (b4 - b2) * q^5 + (b1 + 198) * q^7 + (-2*b4 - b3 + 8*b2 - 2*b1 + 41) * q^9 $$q + (\beta_{2} + 7) q^{3} + (\beta_{4} - \beta_{2}) q^{5} + (\beta_1 + 198) q^{7} + ( - 2 \beta_{4} - \beta_{3} + \cdots + 41) q^{9}+ \cdots + ( - 31716 \beta_{7} + 17025 \beta_{6} + \cdots - 46212688) q^{99}+O(q^{100})$$ q + (b2 + 7) * q^3 + (b4 - b2) * q^5 + (b1 + 198) * q^7 + (-2*b4 - b3 + 8*b2 - 2*b1 + 41) * q^9 + (b6 + b5 - 7*b4 - 2*b3 - b2) * q^11 + (-b7 - 2*b6 - b5 + b4 - 4*b3 - 52*b2 + 8*b1 + 3154) * q^13 + (4*b7 + 6*b6 + b5 + 45*b4 - 6*b3 - 22*b2 + 13*b1 + 4792) * q^15 + (-7*b7 - 10*b6 + 5*b5 + 81*b4 - 10*b3 + 270*b2) * q^17 + (-4*b7 + 25*b6 - 4*b5 + 4*b4 - 16*b3 + 683*b2 - 10*b1 + 19742) * q^19 + (39*b7 - 42*b6 + 3*b5 - 138*b4 - 6*b3 + 231*b2 + 36*b1 + 3810) * q^21 + (-68*b7 + 66*b6 - 112*b4 - 1806*b2) * q^23 + (5*b7 - 98*b6 + 5*b5 - 5*b4 + 20*b3 - 2656*b2 - 28*b1 - 72463) * q^25 + (180*b7 + 141*b6 - 9*b5 + 95*b4 + 34*b3 + 196*b2 - 130*b1 - 34505) * q^27 + (-294*b7 - 212*b6 - 38*b5 - 203*b4 + 76*b3 + 4617*b2) * q^29 + (36*b7 + 222*b6 + 36*b5 - 36*b4 + 144*b3 + 5922*b2 + 127*b1 + 100694) * q^31 + (559*b7 - 270*b6 - 29*b5 + 597*b4 + 99*b3 + 182*b2 - 230*b1 + 12856) * q^33 + (-792*b7 + 308*b6 - 55*b5 + 353*b4 + 110*b3 - 11298*b2) * q^35 + (9*b7 - 390*b6 + 9*b5 - 9*b4 + 36*b3 - 10548*b2 - 304*b1 - 498126) * q^37 + (1320*b7 + 312*b6 + 33*b5 - 2355*b4 + 42*b3 + 4516*b2 + 450*b1 - 287138) * q^39 + (-1596*b7 - 160*b6 + 56*b5 - 1498*b4 - 112*b3 + 3690*b2) * q^41 + (-100*b7 + 253*b6 - 100*b5 + 100*b4 - 400*b3 + 7031*b2 - 950*b1 + 870334) * q^43 + (2358*b7 - 228*b6 + 90*b5 + 877*b4 - 250*b3 + 4061*b2 + 724*b1 + 1083824) * q^45 + (-2600*b7 - 96*b6 + 210*b5 + 3106*b4 - 420*b3 - 1724*b2) * q^47 + (-77*b7 + 770*b6 - 77*b5 + 77*b4 - 308*b3 + 20944*b2 + 2220*b1 - 735565) * q^49 + (3224*b7 - 654*b6 - 85*b5 + 7227*b4 - 606*b3 + 1528*b2 - 340*b1 - 1948768) * q^51 + (-3416*b7 + 576*b6 + 240*b5 + 4565*b4 - 480*b3 - 22389*b2) * q^53 + (-12*b7 - 1578*b6 - 12*b5 + 12*b4 - 48*b3 - 42582*b2 + 2902*b1 + 3387664) * q^55 + (3642*b7 + 3012*b6 + 6*b5 - 5406*b4 - 471*b3 + 18458*b2 - 1494*b1 + 4594586) * q^57 + (-3568*b7 - 2361*b6 + 12*b5 - 14868*b4 - 24*b3 + 71707*b2) * q^59 + (-11*b7 + 1250*b6 - 11*b5 + 11*b4 - 44*b3 + 33772*b2 - 4640*b1 - 6447950) * q^61 + (2808*b7 - 3024*b6 + 216*b5 - 7584*b4 + 744*b3 + 17052*b2 - 1617*b1 - 8719914) * q^63 + (-2182*b7 + 5268*b6 - 690*b5 + 6720*b4 + 1380*b3 - 166430*b2) * q^65 + (488*b7 - 3365*b6 + 488*b5 - 488*b4 + 1952*b3 - 91831*b2 - 392*b1 + 7275086) * q^67 + (-788*b7 - 672*b6 - 656*b5 - 3108*b4 + 2454*b3 - 12472*b2 + 3196*b1 + 11778256) * q^69 + (2020*b7 - 3102*b6 - 1182*b5 + 18306*b4 + 2364*b3 + 47030*b2) * q^71 + (524*b7 + 6832*b6 + 524*b5 - 524*b4 + 2096*b3 + 183416*b2 - 1656*b1 - 14606846) * q^73 + (-6132*b7 - 2064*b6 - 129*b5 + 15951*b4 + 2634*b3 - 61381*b2 + 4014*b1 - 17897737) * q^75 + (8336*b7 - 5936*b6 - 440*b5 - 29166*b4 + 880*b3 + 197750*b2) * q^77 + (-516*b7 + 1086*b6 - 516*b5 + 516*b4 - 2064*b3 + 30354*b2 - 17329*b1 + 21556822) * q^79 + (-13329*b7 + 6978*b6 + 1359*b5 + 19873*b4 - 418*b3 - 70564*b2 - 4472*b1 + 24337505) * q^81 + (16904*b7 + 3815*b6 + 965*b5 - 17283*b4 - 1930*b3 - 33579*b2) * q^83 + (-132*b7 - 16680*b6 - 132*b5 + 132*b4 - 528*b3 - 450096*b2 + 18880*b1 - 33041728) * q^85 + (-24588*b7 + 7446*b6 - 1719*b5 - 60243*b4 - 5646*b3 + 38046*b2 - 2049*b1 - 30356376) * q^87 + (28165*b7 + 6406*b6 + 2341*b5 - 1609*b4 - 4682*b3 - 70544*b2) * q^89 + (-364*b7 + 9436*b6 - 364*b5 + 364*b4 - 1456*b3 + 255500*b2 + 30166*b1 + 47766060) * q^91 + (-31335*b7 - 28662*b6 + 57*b5 + 19410*b4 - 8052*b3 + 41057*b2 - 9360*b1 + 39183794) * q^93 + (32852*b7 + 14142*b6 + 2520*b5 + 94024*b4 - 5040*b3 - 362274*b2) * q^95 + (-2121*b7 + 8754*b6 - 2121*b5 + 2121*b4 - 8484*b3 + 240600*b2 - 13852*b1 - 42165838) * q^97 + (-31716*b7 + 17025*b6 + 5058*b5 + 49954*b4 - 3940*b3 + 34877*b2 - 3470*b1 - 46212688) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 56 q^{3} + 1584 q^{7} + 328 q^{9}+O(q^{10})$$ 8 * q + 56 * q^3 + 1584 * q^7 + 328 * q^9 $$8 q + 56 q^{3} + 1584 q^{7} + 328 q^{9} + 25232 q^{13} + 38336 q^{15} + 157936 q^{19} + 30480 q^{21} - 579704 q^{25} - 276040 q^{27} + 805552 q^{31} + 102848 q^{33} - 3985008 q^{37} - 2297104 q^{39} + 6962672 q^{43} + 8670592 q^{45} - 5884520 q^{49} - 15590144 q^{51} + 27101312 q^{55} + 36756688 q^{57} - 51583600 q^{61} - 69759312 q^{63} + 58200688 q^{67} + 94226048 q^{69} - 116854768 q^{73} - 143181896 q^{75} + 172454576 q^{79} + 194700040 q^{81} - 264333824 q^{85} - 242851008 q^{87} + 382128480 q^{91} + 313470352 q^{93} - 337326704 q^{97} - 369701504 q^{99}+O(q^{100})$$ 8 * q + 56 * q^3 + 1584 * q^7 + 328 * q^9 + 25232 * q^13 + 38336 * q^15 + 157936 * q^19 + 30480 * q^21 - 579704 * q^25 - 276040 * q^27 + 805552 * q^31 + 102848 * q^33 - 3985008 * q^37 - 2297104 * q^39 + 6962672 * q^43 + 8670592 * q^45 - 5884520 * q^49 - 15590144 * q^51 + 27101312 * q^55 + 36756688 * q^57 - 51583600 * q^61 - 69759312 * q^63 + 58200688 * q^67 + 94226048 * q^69 - 116854768 * q^73 - 143181896 * q^75 + 172454576 * q^79 + 194700040 * q^81 - 264333824 * q^85 - 242851008 * q^87 + 382128480 * q^91 + 313470352 * q^93 - 337326704 * q^97 - 369701504 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 78x^{6} + 144x^{5} + 2079x^{4} + 936x^{3} - 658x^{2} + 2884x + 30633$$ :

 $$\beta_{1}$$ $$=$$ $$( - 14134 \nu^{7} - 759906 \nu^{6} + 8175192 \nu^{5} + 32119854 \nu^{4} - 356737260 \nu^{3} + \cdots - 10631678772 ) / 6908733$$ (-14134*v^7 - 759906*v^6 + 8175192*v^5 + 32119854*v^4 - 356737260*v^3 - 138768366*v^2 + 1513490626*v - 10631678772) / 6908733 $$\beta_{2}$$ $$=$$ $$( 17611 \nu^{7} - 68427 \nu^{6} - 1422267 \nu^{5} + 2154414 \nu^{4} + 41574963 \nu^{3} + \cdots + 10880913 ) / 4605822$$ (17611*v^7 - 68427*v^6 - 1422267*v^5 + 2154414*v^4 + 41574963*v^3 + 25987890*v^2 - 113453212*v + 10880913) / 4605822 $$\beta_{3}$$ $$=$$ $$( 123374 \nu^{7} + 7690446 \nu^{6} - 57670500 \nu^{5} - 575178978 \nu^{4} + 2325486552 \nu^{3} + \cdots - 59720251260 ) / 20726199$$ (123374*v^7 + 7690446*v^6 - 57670500*v^5 - 575178978*v^4 + 2325486552*v^3 + 18189038634*v^2 - 477576806*v - 59720251260) / 20726199 $$\beta_{4}$$ $$=$$ $$( 1598449 \nu^{7} - 8139321 \nu^{6} - 118604085 \nu^{5} + 385353606 \nu^{4} + 2846090037 \nu^{3} + \cdots + 7308583167 ) / 41452398$$ (1598449*v^7 - 8139321*v^6 - 118604085*v^5 + 385353606*v^4 + 2846090037*v^3 - 2275859670*v^2 + 6015844520*v + 7308583167) / 41452398 $$\beta_{5}$$ $$=$$ $$( 2094373 \nu^{7} - 84180717 \nu^{6} + 131294271 \nu^{5} + 5971966230 \nu^{4} - 4358693679 \nu^{3} + \cdots - 134348701677 ) / 41452398$$ (2094373*v^7 - 84180717*v^6 + 131294271*v^5 + 5971966230*v^4 - 4358693679*v^3 - 143382703398*v^2 - 74094140896*v - 134348701677) / 41452398 $$\beta_{6}$$ $$=$$ $$( 90977 \nu^{7} - 306225 \nu^{6} - 8157405 \nu^{5} + 15590670 \nu^{4} + 246555357 \nu^{3} + \cdots - 77404449 ) / 1535274$$ (90977*v^7 - 306225*v^6 - 8157405*v^5 + 15590670*v^4 + 246555357*v^3 - 22273518*v^2 - 1519133552*v - 77404449) / 1535274 $$\beta_{7}$$ $$=$$ $$( 1492189 \nu^{7} - 6796125 \nu^{6} - 103901037 \nu^{5} + 240389346 \nu^{4} + 2483495733 \nu^{3} + \cdots + 6097404135 ) / 20726199$$ (1492189*v^7 - 6796125*v^6 - 103901037*v^5 + 240389346*v^4 + 2483495733*v^3 + 694439742*v^2 + 10473206396*v + 6097404135) / 20726199
 $$\nu$$ $$=$$ $$( 31\beta_{7} - 16\beta_{6} + 4\beta_{5} - 4\beta_{4} + 16\beta_{3} - 386\beta_{2} - 24\beta _1 + 13824 ) / 27648$$ (31*b7 - 16*b6 + 4*b5 - 4*b4 + 16*b3 - 386*b2 - 24*b1 + 13824) / 27648 $$\nu^{2}$$ $$=$$ $$( 11\beta_{7} + 14\beta_{6} + 2\beta_{5} - 8\beta_{4} + 14\beta_{3} - 382\beta_{2} + 18\beta _1 + 74304 ) / 3456$$ (11*b7 + 14*b6 + 2*b5 - 8*b4 + 14*b3 - 382*b2 + 18*b1 + 74304) / 3456 $$\nu^{3}$$ $$=$$ $$( 719 \beta_{7} + 136 \beta_{6} + 80 \beta_{5} - 1160 \beta_{4} + 248 \beta_{3} - 5506 \beta_{2} + \cdots + 654336 ) / 9216$$ (719*b7 + 136*b6 + 80*b5 - 1160*b4 + 248*b3 - 5506*b2 - 216*b1 + 654336) / 9216 $$\nu^{4}$$ $$=$$ $$( 3059 \beta_{7} + 2672 \beta_{6} + 278 \beta_{5} - 2822 \beta_{4} + 1352 \beta_{3} - 75922 \beta_{2} + \cdots + 5871744 ) / 6912$$ (3059*b7 + 2672*b6 + 278*b5 - 2822*b4 + 1352*b3 - 75922*b2 + 756*b1 + 5871744) / 6912 $$\nu^{5}$$ $$=$$ $$( 41012 \beta_{7} + 16690 \beta_{6} + 2645 \beta_{5} - 64655 \beta_{4} + 7790 \beta_{3} + \cdots + 29137536 ) / 6912$$ (41012*b7 + 16690*b6 + 2645*b5 - 64655*b4 + 7790*b3 - 426976*b2 - 1992*b1 + 29137536) / 6912 $$\nu^{6}$$ $$=$$ $$( 184959 \beta_{7} + 135704 \beta_{6} + 7824 \beta_{5} - 219672 \beta_{4} + 35016 \beta_{3} + \cdots + 130065408 ) / 4608$$ (184959*b7 + 135704*b6 + 7824*b5 - 219672*b4 + 35016*b3 - 3526722*b2 + 2808*b1 + 130065408) / 4608 $$\nu^{7}$$ $$=$$ $$( 11041321 \beta_{7} + 6016160 \beta_{6} + 336388 \beta_{5} - 16342420 \beta_{4} + 852640 \beta_{3} + \cdots + 4132200960 ) / 27648$$ (11041321*b7 + 6016160*b6 + 336388*b5 - 16342420*b4 + 852640*b3 - 134748782*b2 + 214680*b1 + 4132200960) / 27648

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.26427 − 1.41421i 1.26427 + 1.41421i 7.73966 + 1.41421i 7.73966 − 1.41421i −5.49843 − 1.41421i −5.49843 + 1.41421i −1.50551 − 1.41421i −1.50551 + 1.41421i
0 −76.4245 26.8384i 0 295.589i 0 −843.136 0 5120.40 + 4102.23i 0
17.2 0 −76.4245 + 26.8384i 0 295.589i 0 −843.136 0 5120.40 4102.23i 0
17.3 0 −4.95410 80.8484i 0 404.296i 0 −262.245 0 −6511.91 + 801.062i 0
17.4 0 −4.95410 + 80.8484i 0 404.296i 0 −262.245 0 −6511.91 801.062i 0
17.5 0 28.6420 75.7670i 0 868.404i 0 3909.88 0 −4920.27 4340.23i 0
17.6 0 28.6420 + 75.7670i 0 868.404i 0 3909.88 0 −4920.27 + 4340.23i 0
17.7 0 80.7366 6.52720i 0 920.542i 0 −2012.50 0 6475.79 1053.97i 0
17.8 0 80.7366 + 6.52720i 0 920.542i 0 −2012.50 0 6475.79 + 1053.97i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.9.e.a 8
3.b odd 2 1 inner 24.9.e.a 8
4.b odd 2 1 48.9.e.e 8
8.b even 2 1 192.9.e.i 8
8.d odd 2 1 192.9.e.j 8
12.b even 2 1 48.9.e.e 8
24.f even 2 1 192.9.e.j 8
24.h odd 2 1 192.9.e.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.9.e.a 8 1.a even 1 1 trivial
24.9.e.a 8 3.b odd 2 1 inner
48.9.e.e 8 4.b odd 2 1
48.9.e.e 8 12.b even 2 1
192.9.e.i 8 8.b even 2 1
192.9.e.i 8 24.h odd 2 1
192.9.e.j 8 8.d odd 2 1
192.9.e.j 8 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(24, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + \cdots + 18\!\cdots\!41$$
$5$ $$T^{8} + \cdots + 91\!\cdots\!00$$
$7$ $$(T^{4} + \cdots - 1739819024496)^{2}$$
$11$ $$T^{8} + \cdots + 54\!\cdots\!24$$
$13$ $$(T^{4} + \cdots + 50\!\cdots\!00)^{2}$$
$17$ $$T^{8} + \cdots + 68\!\cdots\!36$$
$19$ $$(T^{4} + \cdots + 14\!\cdots\!56)^{2}$$
$23$ $$T^{8} + \cdots + 10\!\cdots\!56$$
$29$ $$T^{8} + \cdots + 20\!\cdots\!76$$
$31$ $$(T^{4} + \cdots - 24\!\cdots\!88)^{2}$$
$37$ $$(T^{4} + \cdots + 13\!\cdots\!76)^{2}$$
$41$ $$T^{8} + \cdots + 71\!\cdots\!76$$
$43$ $$(T^{4} + \cdots + 40\!\cdots\!44)^{2}$$
$47$ $$T^{8} + \cdots + 14\!\cdots\!56$$
$53$ $$T^{8} + \cdots + 12\!\cdots\!44$$
$59$ $$T^{8} + \cdots + 36\!\cdots\!44$$
$61$ $$(T^{4} + \cdots - 77\!\cdots\!96)^{2}$$
$67$ $$(T^{4} + \cdots + 23\!\cdots\!36)^{2}$$
$71$ $$T^{8} + \cdots + 11\!\cdots\!96$$
$73$ $$(T^{4} + \cdots + 58\!\cdots\!04)^{2}$$
$79$ $$(T^{4} + \cdots - 61\!\cdots\!00)^{2}$$
$83$ $$T^{8} + \cdots + 98\!\cdots\!96$$
$89$ $$T^{8} + \cdots + 25\!\cdots\!36$$
$97$ $$(T^{4} + \cdots - 90\!\cdots\!04)^{2}$$