# Properties

 Label 96.9.g.b Level $96$ Weight $9$ Character orbit 96.g Analytic conductor $39.108$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,9,Mod(31,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("96.31");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 96.g (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: $$39.1083465659$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3468738816.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484$$ x^8 - 2*x^7 + 2*x^6 + 18*x^5 + 77*x^4 + 8*x^2 + 88*x + 484 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{40}\cdot 3^{12}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + (\beta_{4} + 70) q^{5} + ( - 7 \beta_{3} - 7 \beta_1) q^{7} - 2187 q^{9}+O(q^{10})$$ q + b2 * q^3 + (b4 + 70) * q^5 + (-7*b3 - 7*b1) * q^7 - 2187 * q^9 $$q + \beta_{2} q^{3} + (\beta_{4} + 70) q^{5} + ( - 7 \beta_{3} - 7 \beta_1) q^{7} - 2187 q^{9} + ( - 2 \beta_{5} - 18 \beta_{3} + 68 \beta_{2} - 135 \beta_1) q^{11} + ( - 4 \beta_{7} - 19 \beta_{6} + 7 \beta_{4} - 6782) q^{13} + ( - 9 \beta_{5} - 63 \beta_{3} + 94 \beta_{2} - 279 \beta_1) q^{15} + (7 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 54522) q^{17} + ( - 8 \beta_{5} + 350 \beta_{3} - 316 \beta_{2} - 1067 \beta_1) q^{19} + (14 \beta_{7} - 63 \beta_{6} - 98 \beta_{4} + 6804) q^{21} + (88 \beta_{5} - 206 \beta_{3} + 1656 \beta_{2} + 302 \beta_1) q^{23} + ( - 66 \beta_{7} + 418 \beta_{6} - 182 \beta_{4} + 307059) q^{25} - 2187 \beta_{2} q^{27} + ( - 78 \beta_{7} - 62 \beta_{6} + 637 \beta_{4} + 24494) q^{29} + ( - 100 \beta_{5} + 1057 \beta_{3} + 6456 \beta_{2} + 6259 \beta_1) q^{31} + ( - 61 \beta_{7} - 36 \beta_{6} - 680 \beta_{4} - 133164) q^{33} + (294 \beta_{5} - 3486 \beta_{3} + 35224 \beta_{2} - 12061 \beta_1) q^{35} + ( - 256 \beta_{7} - 509 \beta_{6} - 1393 \beta_{4} + \cdots - 877926) q^{37}+ \cdots + (4374 \beta_{5} + 39366 \beta_{3} - 148716 \beta_{2} + 295245 \beta_1) q^{99}+O(q^{100})$$ q + b2 * q^3 + (b4 + 70) * q^5 + (-7*b3 - 7*b1) * q^7 - 2187 * q^9 + (-2*b5 - 18*b3 + 68*b2 - 135*b1) * q^11 + (-4*b7 - 19*b6 + 7*b4 - 6782) * q^13 + (-9*b5 - 63*b3 + 94*b2 - 279*b1) * q^15 + (7*b7 - 2*b6 + 2*b4 + 54522) * q^17 + (-8*b5 + 350*b3 - 316*b2 - 1067*b1) * q^19 + (14*b7 - 63*b6 - 98*b4 + 6804) * q^21 + (88*b5 - 206*b3 + 1656*b2 + 302*b1) * q^23 + (-66*b7 + 418*b6 - 182*b4 + 307059) * q^25 - 2187*b2 * q^27 + (-78*b7 - 62*b6 + 637*b4 + 24494) * q^29 + (-100*b5 + 1057*b3 + 6456*b2 + 6259*b1) * q^31 + (-61*b7 - 36*b6 - 680*b4 - 133164) * q^33 + (294*b5 - 3486*b3 + 35224*b2 - 12061*b1) * q^35 + (-256*b7 - 509*b6 - 1393*b4 - 877926) * q^37 + (-477*b5 + 1278*b3 - 7562*b2 + 7812*b1) * q^39 + (-427*b7 - 1506*b6 + 930*b4 - 1504390) * q^41 + (-748*b5 + 2226*b3 + 9884*b2 - 3473*b1) * q^43 + (-2187*b4 - 153090) * q^45 + (212*b5 + 382*b3 + 86480*b2 + 47694*b1) * q^47 + (-1764*b7 - 1470*b6 - 4606*b4 - 1200255) * q^49 + (72*b5 + 990*b3 + 54114*b2 - 11619*b1) * q^51 + (-2578*b7 + 7962*b6 - 27*b4 + 3417774) * q^53 + (2620*b5 - 19544*b3 + 224648*b2 - 4470*b1) * q^55 + (-2037*b7 + 3654*b6 + 1290*b4 + 343116) * q^57 + (2508*b5 + 18856*b3 + 174684*b2 + 11480*b1) * q^59 + (-76*b7 - 1817*b6 - 6561*b4 + 6350410) * q^61 + (15309*b3 + 15309*b1) * q^63 + (-5049*b7 - 5918*b6 - 27922*b4 + 4256236) * q^65 + (-5924*b5 + 3368*b3 + 220340*b2 - 126528*b1) * q^67 + (40*b7 - 7398*b6 + 6668*b4 - 3335904) * q^69 + (-11188*b5 + 16970*b3 + 75352*b2 - 210458*b1) * q^71 + (-11772*b7 + 6862*b6 - 3746*b4 + 7442706) * q^73 + (7974*b5 - 45756*b3 + 330939*b2 + 86814*b1) * q^75 + (574*b7 - 22624*b6 - 19432*b4 - 22311184) * q^77 + (12556*b5 + 349*b3 + 756776*b2 - 300283*b1) * q^79 + 4782969 * q^81 + (-7614*b5 - 102050*b3 + 414124*b2 + 212329*b1) * q^83 + (528*b7 + 5456*b6 + 50702*b4 + 3678588) * q^85 + (-8253*b5 - 42705*b3 + 39806*b2 - 40059*b1) * q^87 + (-17354*b7 + 34236*b6 - 44196*b4 - 21349374) * q^89 + (2240*b5 + 141162*b3 - 625408*b2 + 689479*b1) * q^91 + (4088*b7 + 15813*b6 + 15502*b4 - 15243876) * q^93 + (13232*b5 + 108288*b3 - 733560*b2 + 822032*b1) * q^95 + (-11910*b7 + 3104*b6 - 65576*b4 + 34080898) * q^97 + (4374*b5 + 39366*b3 - 148716*b2 + 295245*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 560 q^{5} - 17496 q^{9}+O(q^{10})$$ 8 * q + 560 * q^5 - 17496 * q^9 $$8 q + 560 q^{5} - 17496 q^{9} - 54256 q^{13} + 436176 q^{17} + 54432 q^{21} + 2456472 q^{25} + 195952 q^{29} - 1065312 q^{33} - 7023408 q^{37} - 12035120 q^{41} - 1224720 q^{45} - 9602040 q^{49} + 27342192 q^{53} + 2744928 q^{57} + 50803280 q^{61} + 34049888 q^{65} - 26687232 q^{69} + 59541648 q^{73} - 178489472 q^{77} + 38263752 q^{81} + 29428704 q^{85} - 170794992 q^{89} - 121951008 q^{93} + 272647184 q^{97}+O(q^{100})$$ 8 * q + 560 * q^5 - 17496 * q^9 - 54256 * q^13 + 436176 * q^17 + 54432 * q^21 + 2456472 * q^25 + 195952 * q^29 - 1065312 * q^33 - 7023408 * q^37 - 12035120 * q^41 - 1224720 * q^45 - 9602040 * q^49 + 27342192 * q^53 + 2744928 * q^57 + 50803280 * q^61 + 34049888 * q^65 - 26687232 * q^69 + 59541648 * q^73 - 178489472 * q^77 + 38263752 * q^81 + 29428704 * q^85 - 170794992 * q^89 - 121951008 * q^93 + 272647184 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484$$ :

 $$\beta_{1}$$ $$=$$ $$( - 16096 \nu^{7} + 120192 \nu^{6} - 193760 \nu^{5} - 125344 \nu^{4} + 178112 \nu^{3} + 9196000 \nu^{2} - 480768 \nu - 770176 ) / 348777$$ (-16096*v^7 + 120192*v^6 - 193760*v^5 - 125344*v^4 + 178112*v^3 + 9196000*v^2 - 480768*v - 770176) / 348777 $$\beta_{2}$$ $$=$$ $$( 225\nu^{7} - 846\nu^{6} + 1836\nu^{5} + 2466\nu^{4} + 7425\nu^{3} - 7920\nu^{2} + 26550\nu + 14256 ) / 572$$ (225*v^7 - 846*v^6 + 1836*v^5 + 2466*v^4 + 7425*v^3 - 7920*v^2 + 26550*v + 14256) / 572 $$\beta_{3}$$ $$=$$ $$( - 272869 \nu^{7} + 1208730 \nu^{6} - 2417762 \nu^{5} - 2834248 \nu^{4} - 6832507 \nu^{3} + 24871726 \nu^{2} - 27854202 \nu - 16524376 ) / 116259$$ (-272869*v^7 + 1208730*v^6 - 2417762*v^5 - 2834248*v^4 - 6832507*v^3 + 24871726*v^2 - 27854202*v - 16524376) / 116259 $$\beta_{4}$$ $$=$$ $$( 37279 \nu^{7} - 57072 \nu^{6} - 78202 \nu^{5} + 1067344 \nu^{4} + 2474125 \nu^{3} - 149116 \nu^{2} - 6430182 \nu + 6174496 ) / 10569$$ (37279*v^7 - 57072*v^6 - 78202*v^5 + 1067344*v^4 + 2474125*v^3 - 149116*v^2 - 6430182*v + 6174496) / 10569 $$\beta_{5}$$ $$=$$ $$( - 2413799 \nu^{7} + 15867858 \nu^{6} - 33579772 \nu^{5} - 15096278 \nu^{4} + 78107953 \nu^{3} + 408021944 \nu^{2} + 30698358 \nu - 97405616 ) / 348777$$ (-2413799*v^7 + 15867858*v^6 - 33579772*v^5 - 15096278*v^4 + 78107953*v^3 + 408021944*v^2 + 30698358*v - 97405616) / 348777 $$\beta_{6}$$ $$=$$ $$( 74545 \nu^{7} - 94800 \nu^{6} - 407590 \nu^{5} + 3516016 \nu^{4} + 2531875 \nu^{3} - 298180 \nu^{2} - 8181690 \nu + 43030240 ) / 10569$$ (74545*v^7 - 94800*v^6 - 407590*v^5 + 3516016*v^4 + 2531875*v^3 - 298180*v^2 - 8181690*v + 43030240) / 10569 $$\beta_{7}$$ $$=$$ $$( 122002 \nu^{7} - 155424 \nu^{6} - 663532 \nu^{5} + 5323360 \nu^{4} + 4177750 \nu^{3} - 488008 \nu^{2} - 13456212 \nu + 46326976 ) / 10569$$ (122002*v^7 - 155424*v^6 - 663532*v^5 + 5323360*v^4 + 4177750*v^3 - 488008*v^2 - 13456212*v + 46326976) / 10569
 $$\nu$$ $$=$$ $$( 13\beta_{7} - 12\beta_{6} + 12\beta_{5} - 34\beta_{4} - 96\beta_{3} - 208\beta_{2} + 129\beta _1 + 6912 ) / 27648$$ (13*b7 - 12*b6 + 12*b5 - 34*b4 - 96*b3 - 208*b2 + 129*b1 + 6912) / 27648 $$\nu^{2}$$ $$=$$ $$( -144\beta_{3} - 704\beta_{2} + 1323\beta_1 ) / 13824$$ (-144*b3 - 704*b2 + 1323*b1) / 13824 $$\nu^{3}$$ $$=$$ $$( - 161 \beta_{7} + 204 \beta_{6} + 60 \beta_{5} + 122 \beta_{4} - 1056 \beta_{3} - 4880 \beta_{2} + 3345 \beta _1 - 200448 ) / 27648$$ (-161*b7 + 204*b6 + 60*b5 + 122*b4 - 1056*b3 - 4880*b2 + 3345*b1 - 200448) / 27648 $$\nu^{4}$$ $$=$$ $$( -355\beta_{7} + 576\beta_{6} + 10\beta_{4} - 794880 ) / 13824$$ (-355*b7 + 576*b6 + 10*b4 - 794880) / 13824 $$\nu^{5}$$ $$=$$ $$( - 235 \beta_{7} + 332 \beta_{6} - 52 \beta_{5} + 70 \beta_{4} + 1536 \beta_{3} + 7856 \beta_{2} - 6049 \beta _1 - 367872 ) / 3072$$ (-235*b7 + 332*b6 - 52*b5 + 70*b4 + 1536*b3 + 7856*b2 - 6049*b1 - 367872) / 3072 $$\nu^{6}$$ $$=$$ $$( -744\beta_{5} + 35472\beta_{3} + 179744\beta_{2} - 160413\beta_1 ) / 13824$$ (-744*b5 + 35472*b3 + 179744*b2 - 160413*b1) / 13824 $$\nu^{7}$$ $$=$$ $$( 28819 \beta_{7} - 42324 \beta_{6} - 5172 \beta_{5} - 5374 \beta_{4} + 189984 \beta_{3} + 981040 \beta_{2} - 794535 \beta _1 + 48487680 ) / 27648$$ (28819*b7 - 42324*b6 - 5172*b5 - 5374*b4 + 189984*b3 + 981040*b2 - 794535*b1 + 48487680) / 27648

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 1.10829 − 1.10829i 2.64070 + 2.64070i −1.47432 + 1.47432i −1.27467 − 1.27467i 1.10829 + 1.10829i 2.64070 − 2.64070i −1.47432 − 1.47432i −1.27467 + 1.27467i
0 46.7654i 0 −1159.45 0 1312.73i 0 −2187.00 0
31.2 0 46.7654i 0 −149.264 0 3207.68i 0 −2187.00 0
31.3 0 46.7654i 0 509.633 0 42.2502i 0 −2187.00 0
31.4 0 46.7654i 0 1079.08 0 3980.70i 0 −2187.00 0
31.5 0 46.7654i 0 −1159.45 0 1312.73i 0 −2187.00 0
31.6 0 46.7654i 0 −149.264 0 3207.68i 0 −2187.00 0
31.7 0 46.7654i 0 509.633 0 42.2502i 0 −2187.00 0
31.8 0 46.7654i 0 1079.08 0 3980.70i 0 −2187.00 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.9.g.b 8
3.b odd 2 1 288.9.g.c 8
4.b odd 2 1 inner 96.9.g.b 8
8.b even 2 1 192.9.g.d 8
8.d odd 2 1 192.9.g.d 8
12.b even 2 1 288.9.g.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.g.b 8 1.a even 1 1 trivial
96.9.g.b 8 4.b odd 2 1 inner
192.9.g.d 8 8.b even 2 1
192.9.g.d 8 8.d odd 2 1
288.9.g.c 8 3.b odd 2 1
288.9.g.c 8 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 280T_{5}^{3} - 1356168T_{5}^{2} + 444757280T_{5} + 95173421200$$ acting on $$S_{9}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 2187)^{4}$$
$5$ $$(T^{4} - 280 T^{3} + \cdots + 95173421200)^{2}$$
$7$ $$T^{8} + 27860224 T^{6} + \cdots + 50\!\cdots\!84$$
$11$ $$T^{8} + 649524160 T^{6} + \cdots + 89\!\cdots\!56$$
$13$ $$(T^{4} + 27128 T^{3} + \cdots + 78\!\cdots\!96)^{2}$$
$17$ $$(T^{4} - 218088 T^{3} + \cdots + 69\!\cdots\!44)^{2}$$
$19$ $$T^{8} + 83516612544 T^{6} + \cdots + 23\!\cdots\!64$$
$23$ $$T^{8} + 249703618816 T^{6} + \cdots + 21\!\cdots\!64$$
$29$ $$(T^{4} - 97976 T^{3} + \cdots + 81\!\cdots\!44)^{2}$$
$31$ $$T^{8} + 1943611464448 T^{6} + \cdots + 24\!\cdots\!96$$
$37$ $$(T^{4} + 3511704 T^{3} + \cdots + 36\!\cdots\!28)^{2}$$
$41$ $$(T^{4} + 6017560 T^{3} + \cdots - 53\!\cdots\!16)^{2}$$
$43$ $$T^{8} + 18578691622336 T^{6} + \cdots + 10\!\cdots\!00$$
$47$ $$T^{8} + 104203981431808 T^{6} + \cdots + 23\!\cdots\!76$$
$53$ $$(T^{4} - 13671096 T^{3} + \cdots - 44\!\cdots\!24)^{2}$$
$59$ $$T^{8} + 685106367582400 T^{6} + \cdots + 31\!\cdots\!04$$
$61$ $$(T^{4} - 25401640 T^{3} + \cdots - 27\!\cdots\!32)^{2}$$
$67$ $$T^{8} + \cdots + 22\!\cdots\!56$$
$71$ $$T^{8} + \cdots + 33\!\cdots\!76$$
$73$ $$(T^{4} - 29770824 T^{3} + \cdots - 16\!\cdots\!68)^{2}$$
$79$ $$T^{8} + \cdots + 31\!\cdots\!84$$
$83$ $$T^{8} + \cdots + 15\!\cdots\!24$$
$89$ $$(T^{4} + 85397496 T^{3} + \cdots - 47\!\cdots\!32)^{2}$$
$97$ $$(T^{4} - 136323592 T^{3} + \cdots - 88\!\cdots\!36)^{2}$$