# Properties

 Label 96.4.d.a Level $96$ Weight $4$ Character orbit 96.d Analytic conductor $5.664$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,4,Mod(49,96)]

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([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("96.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 96.d (of order $$2$$, degree $$1$$, not 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: $$5.66418336055$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8248384.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$$ x^6 + x^4 - 12*x^3 + 4*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 24) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_1 - 5) q^{7} - 9 q^{9}+O(q^{10})$$ q + b2 * q^3 + (b3 - b2) * q^5 + (b1 - 5) * q^7 - 9 * q^9 $$q + \beta_{2} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_1 - 5) q^{7} - 9 q^{9} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{11} + ( - \beta_{4} + 3 \beta_{3} + 6 \beta_{2}) q^{13} + ( - \beta_{5} + 10) q^{15} + ( - 2 \beta_{5} + 2 \beta_1 + 8) q^{17} + (2 \beta_{4} + 6 \beta_{3} + 4 \beta_{2}) q^{19} + ( - 3 \beta_{4} - 5 \beta_{2}) q^{21} + ( - 2 \beta_1 - 54) q^{23} + (4 \beta_{5} + 4 \beta_1 - 19) q^{25} - 9 \beta_{2} q^{27} + (2 \beta_{4} - 5 \beta_{3} + 19 \beta_{2}) q^{29} + (4 \beta_{5} + 3 \beta_1 + 105) q^{31} + ( - 2 \beta_{5} - 6 \beta_1 + 2) q^{33} + ( - 2 \beta_{4} - 14 \beta_{3} - 12 \beta_{2}) q^{35} + (11 \beta_{4} - 3 \beta_{3} - 32 \beta_{2}) q^{37} + ( - 3 \beta_{5} - 3 \beta_1 - 51) q^{39} + ( - 2 \beta_{5} - 14 \beta_1 + 44) q^{41} + (10 \beta_{4} - 18 \beta_{3} + 28 \beta_{2}) q^{43} + ( - 9 \beta_{3} + 9 \beta_{2}) q^{45} + ( - 4 \beta_{5} - 6 \beta_1 + 70) q^{47} + (8 \beta_{5} + 109) q^{49} + ( - 6 \beta_{4} - 18 \beta_{3} + 6 \beta_{2}) q^{51} + ( - 10 \beta_{4} - 11 \beta_{3} - 123 \beta_{2}) q^{53} + (12 \beta_{5} + 4 \beta_1 - 172) q^{55} + ( - 6 \beta_{5} + 6 \beta_1 - 30) q^{57} + (8 \beta_{4} + 8 \beta_{3} - 20 \beta_{2}) q^{59} + (9 \beta_{4} + 3 \beta_{3} + 172 \beta_{2}) q^{61} + ( - 9 \beta_1 + 45) q^{63} + (6 \beta_{5} + 10 \beta_1 - 294) q^{65} + (48 \beta_{3} - 68 \beta_{2}) q^{67} + (6 \beta_{4} - 54 \beta_{2}) q^{69} + ( - 12 \beta_{5} + 6 \beta_1 + 282) q^{71} + ( - 8 \beta_{5} + 16 \beta_1 + 154) q^{73} + ( - 12 \beta_{4} + 36 \beta_{3} - 15 \beta_{2}) q^{75} + ( - 20 \beta_{4} + 20 \beta_{3} + 256 \beta_{2}) q^{77} + ( - 4 \beta_{5} + 7 \beta_1 + 5) q^{79} + 81 q^{81} + (2 \beta_{4} + 14 \beta_{3} + 128 \beta_{2}) q^{83} + ( - 28 \beta_{4} + 42 \beta_{3} - 302 \beta_{2}) q^{85} + (5 \beta_{5} + 6 \beta_1 - 176) q^{87} + ( - 4 \beta_{5} + 4 \beta_1 - 38) q^{89} + ( - 38 \beta_{4} - 18 \beta_{3} + 64 \beta_{2}) q^{91} + ( - 9 \beta_{4} + 36 \beta_{3} + 109 \beta_{2}) q^{93} + (8 \beta_{5} + 28 \beta_1 - 860) q^{95} + ( - 12 \beta_{5} - 4 \beta_1 - 406) q^{97} + (18 \beta_{4} - 18 \beta_{3}) q^{99}+O(q^{100})$$ q + b2 * q^3 + (b3 - b2) * q^5 + (b1 - 5) * q^7 - 9 * q^9 + (-2*b4 + 2*b3) * q^11 + (-b4 + 3*b3 + 6*b2) * q^13 + (-b5 + 10) * q^15 + (-2*b5 + 2*b1 + 8) * q^17 + (2*b4 + 6*b3 + 4*b2) * q^19 + (-3*b4 - 5*b2) * q^21 + (-2*b1 - 54) * q^23 + (4*b5 + 4*b1 - 19) * q^25 - 9*b2 * q^27 + (2*b4 - 5*b3 + 19*b2) * q^29 + (4*b5 + 3*b1 + 105) * q^31 + (-2*b5 - 6*b1 + 2) * q^33 + (-2*b4 - 14*b3 - 12*b2) * q^35 + (11*b4 - 3*b3 - 32*b2) * q^37 + (-3*b5 - 3*b1 - 51) * q^39 + (-2*b5 - 14*b1 + 44) * q^41 + (10*b4 - 18*b3 + 28*b2) * q^43 + (-9*b3 + 9*b2) * q^45 + (-4*b5 - 6*b1 + 70) * q^47 + (8*b5 + 109) * q^49 + (-6*b4 - 18*b3 + 6*b2) * q^51 + (-10*b4 - 11*b3 - 123*b2) * q^53 + (12*b5 + 4*b1 - 172) * q^55 + (-6*b5 + 6*b1 - 30) * q^57 + (8*b4 + 8*b3 - 20*b2) * q^59 + (9*b4 + 3*b3 + 172*b2) * q^61 + (-9*b1 + 45) * q^63 + (6*b5 + 10*b1 - 294) * q^65 + (48*b3 - 68*b2) * q^67 + (6*b4 - 54*b2) * q^69 + (-12*b5 + 6*b1 + 282) * q^71 + (-8*b5 + 16*b1 + 154) * q^73 + (-12*b4 + 36*b3 - 15*b2) * q^75 + (-20*b4 + 20*b3 + 256*b2) * q^77 + (-4*b5 + 7*b1 + 5) * q^79 + 81 * q^81 + (2*b4 + 14*b3 + 128*b2) * q^83 + (-28*b4 + 42*b3 - 302*b2) * q^85 + (5*b5 + 6*b1 - 176) * q^87 + (-4*b5 + 4*b1 - 38) * q^89 + (-38*b4 - 18*b3 + 64*b2) * q^91 + (-9*b4 + 36*b3 + 109*b2) * q^93 + (8*b5 + 28*b1 - 860) * q^95 + (-12*b5 - 4*b1 - 406) * q^97 + (18*b4 - 18*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 28 q^{7} - 54 q^{9}+O(q^{10})$$ 6 * q - 28 * q^7 - 54 * q^9 $$6 q - 28 q^{7} - 54 q^{9} + 60 q^{15} + 52 q^{17} - 328 q^{23} - 106 q^{25} + 636 q^{31} - 312 q^{39} + 236 q^{41} + 408 q^{47} + 654 q^{49} - 1024 q^{55} - 168 q^{57} + 252 q^{63} - 1744 q^{65} + 1704 q^{71} + 956 q^{73} + 44 q^{79} + 486 q^{81} - 1044 q^{87} - 220 q^{89} - 5104 q^{95} - 2444 q^{97}+O(q^{100})$$ 6 * q - 28 * q^7 - 54 * q^9 + 60 * q^15 + 52 * q^17 - 328 * q^23 - 106 * q^25 + 636 * q^31 - 312 * q^39 + 236 * q^41 + 408 * q^47 + 654 * q^49 - 1024 * q^55 - 168 * q^57 + 252 * q^63 - 1744 * q^65 + 1704 * q^71 + 956 * q^73 + 44 * q^79 + 486 * q^81 - 1044 * q^87 - 220 * q^89 - 5104 * q^95 - 2444 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{4} + 3\nu^{2} + 12\nu - 1$$ -v^4 + 3*v^2 + 12*v - 1 $$\beta_{2}$$ $$=$$ $$( -3\nu^{5} - 6\nu^{4} + 9\nu^{3} + 6\nu^{2} + 24\nu - 96 ) / 32$$ (-3*v^5 - 6*v^4 + 9*v^3 + 6*v^2 + 24*v - 96) / 32 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} + 6\nu^{4} - 17\nu^{3} + 90\nu^{2} - 184\nu + 96 ) / 32$$ (-5*v^5 + 6*v^4 - 17*v^3 + 90*v^2 - 184*v + 96) / 32 $$\beta_{4}$$ $$=$$ $$( -5\nu^{5} + 6\nu^{4} + 111\nu^{3} + 90\nu^{2} - 56\nu - 672 ) / 32$$ (-5*v^5 + 6*v^4 + 111*v^3 + 90*v^2 - 56*v - 672) / 32 $$\beta_{5}$$ $$=$$ $$( -3\nu^{5} + 6\nu^{4} - 3\nu^{3} + 18\nu^{2} - 36\nu + 8 ) / 2$$ (-3*v^5 + 6*v^4 - 3*v^3 + 18*v^2 - 36*v + 8) / 2
 $$\nu$$ $$=$$ $$( \beta_{5} - 6\beta_{3} - 6\beta_{2} + 3\beta _1 - 1 ) / 48$$ (b5 - 6*b3 - 6*b2 + 3*b1 - 1) / 48 $$\nu^{2}$$ $$=$$ $$( 3\beta_{4} + 9\beta_{3} - 20\beta_{2} + 6\beta _1 - 18 ) / 48$$ (3*b4 + 9*b3 - 20*b2 + 6*b1 - 18) / 48 $$\nu^{3}$$ $$=$$ $$( -\beta_{5} + 12\beta_{4} - 6\beta_{3} + 6\beta_{2} - 3\beta _1 + 289 ) / 48$$ (-b5 + 12*b4 - 6*b3 + 6*b2 - 3*b1 + 289) / 48 $$\nu^{4}$$ $$=$$ $$( 4\beta_{5} + 3\beta_{4} - 15\beta_{3} - 44\beta_{2} + 2\beta _1 - 38 ) / 16$$ (4*b5 + 3*b4 - 15*b3 - 44*b2 + 2*b1 - 38) / 16 $$\nu^{5}$$ $$=$$ $$( -19\beta_{5} + 24\beta_{4} + 42\beta_{3} - 318\beta_{2} + 15\beta _1 - 485 ) / 48$$ (-19*b5 + 24*b4 + 42*b3 - 318*b2 + 15*b1 - 485) / 48

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.641412 + 1.89436i 1.88322 + 0.673417i −1.24181 − 1.56777i −1.24181 + 1.56777i 1.88322 − 0.673417i −0.641412 − 1.89436i
0 3.00000i 0 9.15486i 0 −27.4175 0 −9.00000 0
49.2 0 3.00000i 0 0.612661i 0 22.7441 0 −9.00000 0
49.3 0 3.00000i 0 18.5422i 0 −9.32669 0 −9.00000 0
49.4 0 3.00000i 0 18.5422i 0 −9.32669 0 −9.00000 0
49.5 0 3.00000i 0 0.612661i 0 22.7441 0 −9.00000 0
49.6 0 3.00000i 0 9.15486i 0 −27.4175 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.4.d.a 6
3.b odd 2 1 288.4.d.d 6
4.b odd 2 1 24.4.d.a 6
8.b even 2 1 inner 96.4.d.a 6
8.d odd 2 1 24.4.d.a 6
12.b even 2 1 72.4.d.d 6
16.e even 4 1 768.4.a.q 3
16.e even 4 1 768.4.a.t 3
16.f odd 4 1 768.4.a.r 3
16.f odd 4 1 768.4.a.s 3
24.f even 2 1 72.4.d.d 6
24.h odd 2 1 288.4.d.d 6
48.i odd 4 1 2304.4.a.bu 3
48.i odd 4 1 2304.4.a.bw 3
48.k even 4 1 2304.4.a.bt 3
48.k even 4 1 2304.4.a.bv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 4.b odd 2 1
24.4.d.a 6 8.d odd 2 1
72.4.d.d 6 12.b even 2 1
72.4.d.d 6 24.f even 2 1
96.4.d.a 6 1.a even 1 1 trivial
96.4.d.a 6 8.b even 2 1 inner
288.4.d.d 6 3.b odd 2 1
288.4.d.d 6 24.h odd 2 1
768.4.a.q 3 16.e even 4 1
768.4.a.r 3 16.f odd 4 1
768.4.a.s 3 16.f odd 4 1
768.4.a.t 3 16.e even 4 1
2304.4.a.bt 3 48.k even 4 1
2304.4.a.bu 3 48.i odd 4 1
2304.4.a.bv 3 48.k even 4 1
2304.4.a.bw 3 48.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(96, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} + 9)^{3}$$
$5$ $$T^{6} + 428 T^{4} + 28976 T^{2} + \cdots + 10816$$
$7$ $$(T^{3} + 14 T^{2} - 580 T - 5816)^{2}$$
$11$ $$T^{6} + 5632 T^{4} + \cdots + 2415919104$$
$13$ $$T^{6} + 4912 T^{4} + \cdots + 3121680384$$
$17$ $$(T^{3} - 26 T^{2} - 11124 T + 477576)^{2}$$
$19$ $$T^{6} + 22960 T^{4} + \cdots + 75488661504$$
$23$ $$(T^{3} + 164 T^{2} + 6384 T + 45504)^{2}$$
$29$ $$T^{6} + 22348 T^{4} + \cdots + 3766031424$$
$31$ $$(T^{3} - 318 T^{2} + 4476 T + 3749624)^{2}$$
$37$ $$T^{6} + 179776 T^{4} + \cdots + 6879707136$$
$41$ $$(T^{3} - 118 T^{2} - 117300 T + 19985976)^{2}$$
$43$ $$T^{6} + 229552 T^{4} + \cdots + 73984219582464$$
$47$ $$(T^{3} - 204 T^{2} - 27792 T + 1964736)^{2}$$
$53$ $$T^{6} + \cdots + 427051482970176$$
$59$ $$T^{6} + 138416 T^{4} + \cdots + 72651484205056$$
$61$ $$T^{6} + 902016 T^{4} + \cdots + 10\!\cdots\!56$$
$67$ $$T^{6} + 1054512 T^{4} + \cdots + 10\!\cdots\!84$$
$71$ $$(T^{3} - 852 T^{2} - 66960 T + 85084992)^{2}$$
$73$ $$(T^{3} - 478 T^{2} - 255956 T + 120833304)^{2}$$
$79$ $$(T^{3} - 22 T^{2} - 71524 T + 7902616)^{2}$$
$83$ $$T^{6} + 520448 T^{4} + \cdots + 14\!\cdots\!96$$
$89$ $$(T^{3} + 110 T^{2} - 41364 T + 1423656)^{2}$$
$97$ $$(T^{3} + 1222 T^{2} + 251660 T - 74802424)^{2}$$
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