# Properties

 Label 78.6.b.a Level $78$ Weight $6$ Character orbit 78.b Analytic conductor $12.510$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,6,Mod(25,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("78.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 78.b (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: $$12.5099379454$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792$$ x^6 - 2*x^5 + 2*x^4 - 2946*x^3 + 131769*x^2 - 1332936*x + 6741792 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{11}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 9 q^{3} - 16 q^{4} + ( - \beta_{3} + 3 \beta_1) q^{5} + 9 \beta_1 q^{6} + ( - \beta_{5} - 14 \beta_1) q^{7} + 16 \beta_1 q^{8} + 81 q^{9}+O(q^{10})$$ q - b1 * q^2 - 9 * q^3 - 16 * q^4 + (-b3 + 3*b1) * q^5 + 9*b1 * q^6 + (-b5 - 14*b1) * q^7 + 16*b1 * q^8 + 81 * q^9 $$q - \beta_1 q^{2} - 9 q^{3} - 16 q^{4} + ( - \beta_{3} + 3 \beta_1) q^{5} + 9 \beta_1 q^{6} + ( - \beta_{5} - 14 \beta_1) q^{7} + 16 \beta_1 q^{8} + 81 q^{9} + (\beta_{4} + 53) q^{10} + ( - \beta_{5} + 7 \beta_{3} - 11 \beta_1) q^{11} + 144 q^{12} + (3 \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots + 88) q^{13}+ \cdots + ( - 81 \beta_{5} + 567 \beta_{3} - 891 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^2 - 9 * q^3 - 16 * q^4 + (-b3 + 3*b1) * q^5 + 9*b1 * q^6 + (-b5 - 14*b1) * q^7 + 16*b1 * q^8 + 81 * q^9 + (b4 + 53) * q^10 + (-b5 + 7*b3 - 11*b1) * q^11 + 144 * q^12 + (3*b5 + 2*b4 - b3 + b2 - 15*b1 + 88) * q^13 + (2*b2 - 226) * q^14 + (9*b3 - 27*b1) * q^15 + 256 * q^16 + (-2*b4 + 5*b2 - 137) * q^17 - 81*b1 * q^18 + (5*b5 + 34*b3 + 16*b1) * q^19 + (16*b3 - 48*b1) * q^20 + (9*b5 + 126*b1) * q^21 + (-7*b4 + 2*b2 - 213) * q^22 + (4*b4 + 8*b2 - 68) * q^23 - 144*b1 * q^24 + (-14*b4 + 5*b2 + 126) * q^25 + (8*b5 + b4 + 32*b3 - 6*b2 - 79*b1 - 229) * q^26 - 729 * q^27 + (16*b5 + 224*b1) * q^28 + (14*b4 - 5*b2 + 3125) * q^29 + (-9*b4 - 477) * q^30 + (-b5 + 66*b3 - 596*b1) * q^31 - 256*b1 * q^32 + (9*b5 - 63*b3 + 99*b1) * q^33 + (40*b5 - 32*b3 + 122*b1) * q^34 + (28*b4 + 8*b2 + 1012) * q^35 - 1296 * q^36 + (6*b5 + 112*b3 - 612*b1) * q^37 + (-34*b4 - 10*b2 + 96) * q^38 + (-27*b5 - 18*b4 + 9*b3 - 9*b2 + 135*b1 - 792) * q^39 + (-16*b4 - 848) * q^40 + (-54*b5 + b3 + 1521*b1) * q^41 + (-18*b2 + 2034) * q^42 + (10*b4 + 59*b2 - 4017) * q^43 + (16*b5 - 112*b3 + 176*b1) * q^44 + (-81*b3 + 243*b1) * q^45 + (64*b5 + 64*b3 + 80*b1) * q^46 + (-163*b5 + 7*b3 - 311*b1) * q^47 - 2304 * q^48 + (34*b4 + 29*b2 - 508) * q^49 + (40*b5 - 224*b3 - 201*b1) * q^50 + (18*b4 - 45*b2 + 1233) * q^51 + (-48*b5 - 32*b4 + 16*b3 - 16*b2 + 240*b1 - 1408) * q^52 + (132*b4 - 36*b2 - 7122) * q^53 + 729*b1 * q^54 + (102*b4 - 27*b2 + 20733) * q^55 + (-32*b2 + 3616) * q^56 + (-45*b5 - 306*b3 - 144*b1) * q^57 + (-40*b5 + 224*b3 - 3050*b1) * q^58 + (-189*b5 - 263*b3 + 87*b1) * q^59 + (-144*b3 + 432*b1) * q^60 + (-154*b4 - 5*b2 - 483) * q^61 + (-66*b4 + 2*b2 - 9868) * q^62 + (-81*b5 - 1134*b1) * q^63 - 4096 * q^64 + (112*b5 - 38*b4 - 319*b3 - 19*b2 + 5693*b1 - 2855) * q^65 + (63*b4 - 18*b2 + 1917) * q^66 + (121*b5 - 68*b3 + 8858*b1) * q^67 + (32*b4 - 80*b2 + 2192) * q^68 + (-36*b4 - 72*b2 + 612) * q^69 + (64*b5 + 448*b3 - 880*b1) * q^70 + (-27*b5 - 651*b3 - 5937*b1) * q^71 + 1296*b1 * q^72 + (56*b5 + 242*b3 - 5510*b1) * q^73 + (-112*b4 - 12*b2 - 10340) * q^74 + (126*b4 - 45*b2 - 1134) * q^75 + (-80*b5 - 544*b3 - 256*b1) * q^76 + (-162*b4 - 75*b2 - 18975) * q^77 + (-72*b5 - 9*b4 - 288*b3 + 54*b2 + 711*b1 + 2061) * q^78 + (16*b4 + 140*b2 - 28180) * q^79 + (-256*b3 + 768*b1) * q^80 + 6561 * q^81 + (-b4 + 108*b2 + 24223) * q^82 + (-73*b5 + 201*b3 - 3329*b1) * q^83 + (-144*b5 - 2016*b1) * q^84 + (80*b5 + 1034*b3 - 6446*b1) * q^85 + (472*b5 + 160*b3 + 4008*b1) * q^86 + (-126*b4 + 45*b2 - 28125) * q^87 + (112*b4 - 32*b2 + 3408) * q^88 + (194*b5 + 853*b3 - 875*b1) * q^89 + (81*b4 + 4293) * q^90 + (149*b5 - 74*b4 + 1168*b3 + 41*b2 + 5638*b1 + 39397) * q^91 + (-64*b4 - 128*b2 + 1088) * q^92 + (9*b5 - 594*b3 + 5364*b1) * q^93 + (-7*b4 + 326*b2 - 5337) * q^94 + (288*b4 - 210*b2 + 94362) * q^95 + 2304*b1 * q^96 + (556*b5 - 82*b3 - 8722*b1) * q^97 + (232*b5 + 544*b3 + 649*b1) * q^98 + (-81*b5 + 567*b3 - 891*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 54 q^{3} - 96 q^{4} + 486 q^{9}+O(q^{10})$$ 6 * q - 54 * q^3 - 96 * q^4 + 486 * q^9 $$6 q - 54 q^{3} - 96 q^{4} + 486 q^{9} + 320 q^{10} + 864 q^{12} + 530 q^{13} - 1360 q^{14} + 1536 q^{16} - 836 q^{17} - 1296 q^{22} - 416 q^{23} + 718 q^{25} - 1360 q^{26} - 4374 q^{27} + 18788 q^{29} - 2880 q^{30} + 6112 q^{35} - 7776 q^{36} + 528 q^{38} - 4770 q^{39} - 5120 q^{40} + 12240 q^{42} - 24200 q^{43} - 13824 q^{48} - 3038 q^{49} + 7524 q^{51} - 8480 q^{52} - 42396 q^{53} + 124656 q^{55} + 21760 q^{56} - 3196 q^{61} - 59344 q^{62} - 24576 q^{64} - 17168 q^{65} + 11664 q^{66} + 13376 q^{68} + 3744 q^{69} - 62240 q^{74} - 6462 q^{75} - 114024 q^{77} + 12240 q^{78} - 169328 q^{79} + 39366 q^{81} + 145120 q^{82} - 169092 q^{87} + 20736 q^{88} + 25920 q^{90} + 236152 q^{91} + 6656 q^{92} - 32688 q^{94} + 567168 q^{95}+O(q^{100})$$ 6 * q - 54 * q^3 - 96 * q^4 + 486 * q^9 + 320 * q^10 + 864 * q^12 + 530 * q^13 - 1360 * q^14 + 1536 * q^16 - 836 * q^17 - 1296 * q^22 - 416 * q^23 + 718 * q^25 - 1360 * q^26 - 4374 * q^27 + 18788 * q^29 - 2880 * q^30 + 6112 * q^35 - 7776 * q^36 + 528 * q^38 - 4770 * q^39 - 5120 * q^40 + 12240 * q^42 - 24200 * q^43 - 13824 * q^48 - 3038 * q^49 + 7524 * q^51 - 8480 * q^52 - 42396 * q^53 + 124656 * q^55 + 21760 * q^56 - 3196 * q^61 - 59344 * q^62 - 24576 * q^64 - 17168 * q^65 + 11664 * q^66 + 13376 * q^68 + 3744 * q^69 - 62240 * q^74 - 6462 * q^75 - 114024 * q^77 + 12240 * q^78 - 169328 * q^79 + 39366 * q^81 + 145120 * q^82 - 169092 * q^87 + 20736 * q^88 + 25920 * q^90 + 236152 * q^91 + 6656 * q^92 - 32688 * q^94 + 567168 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792$$ :

 $$\beta_{1}$$ $$=$$ $$( 11164\nu^{5} + 33517\nu^{4} + 136007\nu^{3} - 12505719\nu^{2} + 1388809431\nu - 7854226236 ) / 1807933527$$ (11164*v^5 + 33517*v^4 + 136007*v^3 - 12505719*v^2 + 1388809431*v - 7854226236) / 1807933527 $$\beta_{2}$$ $$=$$ $$( -1108\nu^{5} - 131752\nu^{4} - 404420\nu^{3} + 1632084\nu^{2} + 4068576\nu - 9481422087 ) / 35449677$$ (-1108*v^5 - 131752*v^4 - 404420*v^3 + 1632084*v^2 + 4068576*v - 9481422087) / 35449677 $$\beta_{3}$$ $$=$$ $$( 383264 \nu^{5} + 1222625 \nu^{4} - 21601613 \nu^{3} - 1608343917 \nu^{2} + 37929979503 \nu - 221669454492 ) / 1807933527$$ (383264*v^5 + 1222625*v^4 - 21601613*v^3 - 1608343917*v^2 + 37929979503*v - 221669454492) / 1807933527 $$\beta_{4}$$ $$=$$ $$( -8786\nu^{5} - 276878\nu^{4} - 3206890\nu^{3} + 12941778\nu^{2} + 32262192\nu - 10639840401 ) / 35449677$$ (-8786*v^5 - 276878*v^4 - 3206890*v^3 + 12941778*v^2 + 32262192*v - 10639840401) / 35449677 $$\beta_{5}$$ $$=$$ $$( 307633 \nu^{5} + 917590 \nu^{4} + 5937014 \nu^{3} - 1150320390 \nu^{2} + 39082121901 \nu - 220426926804 ) / 1205289018$$ (307633*v^5 + 917590*v^4 + 5937014*v^3 - 1150320390*v^2 + 39082121901*v - 220426926804) / 1205289018
 $$\nu$$ $$=$$ $$( 4\beta_{5} + \beta_{4} - 4\beta_{3} - 2\beta_{2} + 2\beta _1 + 15 ) / 48$$ (4*b5 + b4 - 4*b3 - 2*b2 + 2*b1 + 15) / 48 $$\nu^{2}$$ $$=$$ $$( -8\beta_{5} - \beta_{3} + 365\beta_1 ) / 6$$ (-8*b5 - b3 + 365*b1) / 6 $$\nu^{3}$$ $$=$$ $$( 1388\beta_{5} - 365\beta_{4} - 1460\beta_{3} + 694\beta_{2} - 18386\beta _1 + 71025 ) / 48$$ (1388*b5 - 365*b4 - 1460*b3 + 694*b2 - 18386*b1 + 71025) / 48 $$\nu^{4}$$ $$=$$ $$( 554\beta_{4} - 4393\beta_{2} - 1008681 ) / 12$$ (554*b4 - 4393*b2 - 1008681) / 12 $$\nu^{5}$$ $$=$$ $$( -586204\beta_{5} - 126607\beta_{4} + 506428\beta_{3} + 293102\beta_{2} + 11019394\beta _1 + 43151439 ) / 48$$ (-586204*b5 - 126607*b4 + 506428*b3 + 293102*b2 + 11019394*b1 + 43151439) / 48

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 6.10758 + 6.10758i −15.0768 − 15.0768i 9.96927 + 9.96927i 9.96927 − 9.96927i −15.0768 + 15.0768i 6.10758 − 6.10758i
4.00000i −9.00000 −16.0000 37.1158i 36.0000i 176.407i 64.0000i 81.0000 −148.463
25.2 4.00000i −9.00000 −16.0000 9.73803i 36.0000i 105.184i 64.0000i 81.0000 −38.9521
25.3 4.00000i −9.00000 −16.0000 86.8538i 36.0000i 98.7774i 64.0000i 81.0000 347.415
25.4 4.00000i −9.00000 −16.0000 86.8538i 36.0000i 98.7774i 64.0000i 81.0000 347.415
25.5 4.00000i −9.00000 −16.0000 9.73803i 36.0000i 105.184i 64.0000i 81.0000 −38.9521
25.6 4.00000i −9.00000 −16.0000 37.1158i 36.0000i 176.407i 64.0000i 81.0000 −148.463
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.6.b.a 6
3.b odd 2 1 234.6.b.c 6
4.b odd 2 1 624.6.c.d 6
13.b even 2 1 inner 78.6.b.a 6
13.d odd 4 1 1014.6.a.o 3
13.d odd 4 1 1014.6.a.q 3
39.d odd 2 1 234.6.b.c 6
52.b odd 2 1 624.6.c.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.6.b.a 6 1.a even 1 1 trivial
78.6.b.a 6 13.b even 2 1 inner
234.6.b.c 6 3.b odd 2 1
234.6.b.c 6 39.d odd 2 1
624.6.c.d 6 4.b odd 2 1
624.6.c.d 6 52.b odd 2 1
1014.6.a.o 3 13.d odd 4 1
1014.6.a.q 3 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 9016T_{5}^{4} + 11237904T_{5}^{2} + 985457664$$ acting on $$S_{6}^{\mathrm{new}}(78, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 16)^{3}$$
$3$ $$(T + 9)^{6}$$
$5$ $$T^{6} + 9016 T^{4} + \cdots + 985457664$$
$7$ $$T^{6} + \cdots + 3359273140224$$
$11$ $$T^{6} + \cdots + 871026613185600$$
$13$ $$T^{6} + \cdots + 51\!\cdots\!57$$
$17$ $$(T^{3} + 418 T^{2} + \cdots - 1783218312)^{2}$$
$19$ $$T^{6} + \cdots + 224220196832256$$
$23$ $$(T^{3} + 208 T^{2} + \cdots + 2602266624)^{2}$$
$29$ $$(T^{3} - 9394 T^{2} + \cdots - 2546571960)^{2}$$
$31$ $$T^{6} + \cdots + 18\!\cdots\!00$$
$37$ $$T^{6} + \cdots + 15\!\cdots\!44$$
$41$ $$T^{6} + \cdots + 40\!\cdots\!00$$
$43$ $$(T^{3} + \cdots - 1729964364544)^{2}$$
$47$ $$T^{6} + \cdots + 20\!\cdots\!24$$
$53$ $$(T^{3} + \cdots - 26960052479832)^{2}$$
$59$ $$T^{6} + \cdots + 11\!\cdots\!04$$
$61$ $$(T^{3} + \cdots + 17380383329800)^{2}$$
$67$ $$T^{6} + \cdots + 94\!\cdots\!84$$
$71$ $$T^{6} + \cdots + 22\!\cdots\!00$$
$73$ $$T^{6} + \cdots + 11\!\cdots\!16$$
$79$ $$(T^{3} + \cdots - 35835411156480)^{2}$$
$83$ $$T^{6} + \cdots + 11\!\cdots\!56$$
$89$ $$T^{6} + \cdots + 35\!\cdots\!16$$
$97$ $$T^{6} + \cdots + 30\!\cdots\!64$$