# Properties

 Label 64.8.a.j Level $64$ Weight $8$ Character orbit 64.a Self dual yes Analytic conductor $19.993$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.a (trivial)

## 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: yes Analytic conductor: $$19.9926416310$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 10$$ x^2 - 10 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 32) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = 16\sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 8) q^{3} + (8 \beta + 90) q^{5} + (14 \beta + 624) q^{7} + (16 \beta + 437) q^{9}+O(q^{10})$$ q + (b + 8) * q^3 + (8*b + 90) * q^5 + (14*b + 624) * q^7 + (16*b + 437) * q^9 $$q + (\beta + 8) q^{3} + (8 \beta + 90) q^{5} + (14 \beta + 624) q^{7} + (16 \beta + 437) q^{9} + (75 \beta - 4520) q^{11} + ( - 120 \beta + 1250) q^{13} + (154 \beta + 21200) q^{15} + ( - 240 \beta + 8610) q^{17} + ( - 155 \beta - 37080) q^{19} + (736 \beta + 40832) q^{21} + ( - 1302 \beta + 9552) q^{23} + (1440 \beta + 93815) q^{25} + ( - 1622 \beta + 26960) q^{27} + ( - 2200 \beta + 122514) q^{29} + (2200 \beta - 125760) q^{31} + ( - 3920 \beta + 155840) q^{33} + (6252 \beta + 342880) q^{35} + (2280 \beta + 265370) q^{37} + (290 \beta - 297200) q^{39} + (1696 \beta + 363450) q^{41} + ( - 6109 \beta - 22248) q^{43} + (4936 \beta + 367010) q^{45} + (6228 \beta + 247456) q^{47} + (17472 \beta + 67593) q^{49} + (6690 \beta - 545520) q^{51} + ( - 18200 \beta - 659670) q^{53} + ( - 29410 \beta + 1129200) q^{55} + ( - 38320 \beta - 693440) q^{57} + ( - 27945 \beta - 1309000) q^{59} + (19272 \beta - 418350) q^{61} + (16102 \beta + 846128) q^{63} + ( - 800 \beta - 2345100) q^{65} + (1505 \beta - 1187064) q^{67} + ( - 864 \beta - 3256704) q^{69} + (22110 \beta - 1418000) q^{71} + (79440 \beta - 85590) q^{73} + (105335 \beta + 4436920) q^{75} + ( - 16480 \beta - 132480) q^{77} + (1900 \beta - 1249440) q^{79} + ( - 21008 \beta - 4892359) q^{81} + ( - 67803 \beta + 4765992) q^{83} + (47280 \beta - 4140300) q^{85} + (104914 \beta - 4651888) q^{87} + ( - 103600 \beta + 3659034) q^{89} + ( - 57380 \beta - 3520800) q^{91} + ( - 108160 \beta + 4625920) q^{93} + ( - 310590 \beta - 6511600) q^{95} + ( - 127920 \beta - 4658030) q^{97} + ( - 39545 \beta + 1096760) q^{99}+O(q^{100})$$ q + (b + 8) * q^3 + (8*b + 90) * q^5 + (14*b + 624) * q^7 + (16*b + 437) * q^9 + (75*b - 4520) * q^11 + (-120*b + 1250) * q^13 + (154*b + 21200) * q^15 + (-240*b + 8610) * q^17 + (-155*b - 37080) * q^19 + (736*b + 40832) * q^21 + (-1302*b + 9552) * q^23 + (1440*b + 93815) * q^25 + (-1622*b + 26960) * q^27 + (-2200*b + 122514) * q^29 + (2200*b - 125760) * q^31 + (-3920*b + 155840) * q^33 + (6252*b + 342880) * q^35 + (2280*b + 265370) * q^37 + (290*b - 297200) * q^39 + (1696*b + 363450) * q^41 + (-6109*b - 22248) * q^43 + (4936*b + 367010) * q^45 + (6228*b + 247456) * q^47 + (17472*b + 67593) * q^49 + (6690*b - 545520) * q^51 + (-18200*b - 659670) * q^53 + (-29410*b + 1129200) * q^55 + (-38320*b - 693440) * q^57 + (-27945*b - 1309000) * q^59 + (19272*b - 418350) * q^61 + (16102*b + 846128) * q^63 + (-800*b - 2345100) * q^65 + (1505*b - 1187064) * q^67 + (-864*b - 3256704) * q^69 + (22110*b - 1418000) * q^71 + (79440*b - 85590) * q^73 + (105335*b + 4436920) * q^75 + (-16480*b - 132480) * q^77 + (1900*b - 1249440) * q^79 + (-21008*b - 4892359) * q^81 + (-67803*b + 4765992) * q^83 + (47280*b - 4140300) * q^85 + (104914*b - 4651888) * q^87 + (-103600*b + 3659034) * q^89 + (-57380*b - 3520800) * q^91 + (-108160*b + 4625920) * q^93 + (-310590*b - 6511600) * q^95 + (-127920*b - 4658030) * q^97 + (-39545*b + 1096760) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{3} + 180 q^{5} + 1248 q^{7} + 874 q^{9}+O(q^{10})$$ 2 * q + 16 * q^3 + 180 * q^5 + 1248 * q^7 + 874 * q^9 $$2 q + 16 q^{3} + 180 q^{5} + 1248 q^{7} + 874 q^{9} - 9040 q^{11} + 2500 q^{13} + 42400 q^{15} + 17220 q^{17} - 74160 q^{19} + 81664 q^{21} + 19104 q^{23} + 187630 q^{25} + 53920 q^{27} + 245028 q^{29} - 251520 q^{31} + 311680 q^{33} + 685760 q^{35} + 530740 q^{37} - 594400 q^{39} + 726900 q^{41} - 44496 q^{43} + 734020 q^{45} + 494912 q^{47} + 135186 q^{49} - 1091040 q^{51} - 1319340 q^{53} + 2258400 q^{55} - 1386880 q^{57} - 2618000 q^{59} - 836700 q^{61} + 1692256 q^{63} - 4690200 q^{65} - 2374128 q^{67} - 6513408 q^{69} - 2836000 q^{71} - 171180 q^{73} + 8873840 q^{75} - 264960 q^{77} - 2498880 q^{79} - 9784718 q^{81} + 9531984 q^{83} - 8280600 q^{85} - 9303776 q^{87} + 7318068 q^{89} - 7041600 q^{91} + 9251840 q^{93} - 13023200 q^{95} - 9316060 q^{97} + 2193520 q^{99}+O(q^{100})$$ 2 * q + 16 * q^3 + 180 * q^5 + 1248 * q^7 + 874 * q^9 - 9040 * q^11 + 2500 * q^13 + 42400 * q^15 + 17220 * q^17 - 74160 * q^19 + 81664 * q^21 + 19104 * q^23 + 187630 * q^25 + 53920 * q^27 + 245028 * q^29 - 251520 * q^31 + 311680 * q^33 + 685760 * q^35 + 530740 * q^37 - 594400 * q^39 + 726900 * q^41 - 44496 * q^43 + 734020 * q^45 + 494912 * q^47 + 135186 * q^49 - 1091040 * q^51 - 1319340 * q^53 + 2258400 * q^55 - 1386880 * q^57 - 2618000 * q^59 - 836700 * q^61 + 1692256 * q^63 - 4690200 * q^65 - 2374128 * q^67 - 6513408 * q^69 - 2836000 * q^71 - 171180 * q^73 + 8873840 * q^75 - 264960 * q^77 - 2498880 * q^79 - 9784718 * q^81 + 9531984 * q^83 - 8280600 * q^85 - 9303776 * q^87 + 7318068 * q^89 - 7041600 * q^91 + 9251840 * q^93 - 13023200 * q^95 - 9316060 * q^97 + 2193520 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.16228 3.16228
0 −42.5964 0 −314.772 0 −84.3502 0 −372.543 0
1.2 0 58.5964 0 494.772 0 1332.35 0 1246.54 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.8.a.j 2
3.b odd 2 1 576.8.a.bf 2
4.b odd 2 1 64.8.a.h 2
8.b even 2 1 32.8.a.b 2
8.d odd 2 1 32.8.a.d yes 2
12.b even 2 1 576.8.a.be 2
16.e even 4 2 256.8.b.h 4
16.f odd 4 2 256.8.b.j 4
24.f even 2 1 288.8.a.n 2
24.h odd 2 1 288.8.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.8.a.b 2 8.b even 2 1
32.8.a.d yes 2 8.d odd 2 1
64.8.a.h 2 4.b odd 2 1
64.8.a.j 2 1.a even 1 1 trivial
256.8.b.h 4 16.e even 4 2
256.8.b.j 4 16.f odd 4 2
288.8.a.n 2 24.f even 2 1
288.8.a.o 2 24.h odd 2 1
576.8.a.be 2 12.b even 2 1
576.8.a.bf 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 16T_{3} - 2496$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 16T - 2496$$
$5$ $$T^{2} - 180T - 155740$$
$7$ $$T^{2} - 1248 T - 112384$$
$11$ $$T^{2} + 9040 T + 6030400$$
$13$ $$T^{2} - 2500 T - 35301500$$
$17$ $$T^{2} - 17220 T - 73323900$$
$19$ $$T^{2} + 74160 T + 1313422400$$
$23$ $$T^{2} - 19104 T - 4248481536$$
$29$ $$T^{2} - 245028 T + 2619280196$$
$31$ $$T^{2} + 251520 T + 3425177600$$
$37$ $$T^{2} - 530740 T + 57113332900$$
$41$ $$T^{2} - 726900 T + 124732277540$$
$43$ $$T^{2} + 44496 T - 95043921856$$
$47$ $$T^{2} - 494912 T - 38062767104$$
$53$ $$T^{2} + 1319340 T - 412809891100$$
$59$ $$T^{2} + 2618000 T - 285681944000$$
$61$ $$T^{2} + 836700 T - 775792836540$$
$67$ $$T^{2} + 2374128 T + 1403322476096$$
$71$ $$T^{2} + 2836000 T + 759262624000$$
$73$ $$T^{2} + 171180 T - 16148101167900$$
$79$ $$T^{2} + 2498880 T + 1551858713600$$
$83$ $$T^{2} - 9531984 T + 10945727913024$$
$89$ $$T^{2} - 7318068 T - 14087847786844$$
$97$ $$T^{2} + 9316060 T - 20193384103100$$