# Properties

 Label 95.5.d.d Level $95$ Weight $5$ Character orbit 95.d Self dual yes Analytic conductor $9.820$ Analytic rank $0$ Dimension $4$ CM discriminant -95 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,5,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 95.d (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: yes Analytic conductor: $$9.82014649297$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{10}, \sqrt{38})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 24x^{2} + 49$$ x^4 - 24*x^2 + 49 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + ( - 3 \beta_{2} + 4 \beta_1) q^{3} + (3 \beta_{3} + 16) q^{4} - 25 q^{5} + (5 \beta_{3} + 23) q^{6} + (6 \beta_{2} + 45 \beta_1) q^{8} + ( - 8 \beta_{3} + 81) q^{9}+O(q^{10})$$ q + (b2 + b1) * q^2 + (-3*b2 + 4*b1) * q^3 + (3*b3 + 16) * q^4 - 25 * q^5 + (5*b3 + 23) * q^6 + (6*b2 + 45*b1) * q^8 + (-8*b3 + 81) * q^9 $$q + (\beta_{2} + \beta_1) q^{2} + ( - 3 \beta_{2} + 4 \beta_1) q^{3} + (3 \beta_{3} + 16) q^{4} - 25 q^{5} + (5 \beta_{3} + 23) q^{6} + (6 \beta_{2} + 45 \beta_1) q^{8} + ( - 8 \beta_{3} + 81) q^{9} + ( - 25 \beta_{2} - 25 \beta_1) q^{10} - 24 \beta_{3} q^{11} + (81 \beta_{2} + 34 \beta_1) q^{12} + ( - 53 \beta_{2} + 76 \beta_1) q^{13} + (75 \beta_{2} - 100 \beta_1) q^{15} + (48 \beta_{3} + 599) q^{16} + (65 \beta_{2} - 39 \beta_1) q^{18} - 361 q^{19} + ( - 75 \beta_{3} - 400) q^{20} + ( - 48 \beta_{2} - 360 \beta_1) q^{22} + (69 \beta_{3} + 1425) q^{24} + 625 q^{25} + (99 \beta_{3} + 497) q^{26} + ( - 344 \beta_{2} + 80 \beta_1) q^{27} + ( - 125 \beta_{3} - 575) q^{30} + (599 \beta_{2} + 599 \beta_1) q^{32} + ( - 1032 \beta_{2} + 240 \beta_1) q^{33} + (115 \beta_{3} - 984) q^{36} + (509 \beta_{2} + 164 \beta_1) q^{37} + ( - 361 \beta_{2} - 361 \beta_1) q^{38} + ( - 136 \beta_{3} + 3038) q^{39} + ( - 150 \beta_{2} - 1125 \beta_1) q^{40} + ( - 384 \beta_{3} - 6840) q^{44} + (200 \beta_{3} - 2025) q^{45} + (267 \beta_{2} + 1916 \beta_1) q^{48} + 2401 q^{49} + (625 \beta_{2} + 625 \beta_1) q^{50} + (1543 \beta_{2} + 766 \beta_1) q^{52} + (461 \beta_{2} - 1276 \beta_1) q^{53} + ( - 184 \beta_{3} - 3800) q^{54} + 600 \beta_{3} q^{55} + (1083 \beta_{2} - 1444 \beta_1) q^{57} + ( - 2025 \beta_{2} - 850 \beta_1) q^{60} - 216 \beta_{3} q^{61} + (1029 \beta_{3} + 9584) q^{64} + (1325 \beta_{2} - 1900 \beta_1) q^{65} + ( - 552 \beta_{3} - 11400) q^{66} + ( - 2117 \beta_{2} + 1516 \beta_1) q^{67} + ( - 1794 \beta_{2} + 1365 \beta_1) q^{72} + (837 \beta_{3} + 10423) q^{74} + ( - 1875 \beta_{2} + 2500 \beta_1) q^{75} + ( - 1083 \beta_{3} - 5776) q^{76} + (2766 \beta_{2} + 998 \beta_1) q^{78} + ( - 1200 \beta_{3} - 14975) q^{80} + ( - 648 \beta_{3} - 481) q^{81} + ( - 6840 \beta_{2} - 6840 \beta_1) q^{88} + ( - 1625 \beta_{2} + 975 \beta_1) q^{90} + 9025 q^{95} + (2995 \beta_{3} + 13777) q^{96} + (4685 \beta_{2} - 2476 \beta_1) q^{97} + (2401 \beta_{2} + 2401 \beta_1) q^{98} + ( - 1944 \beta_{3} + 18240) q^{99}+O(q^{100})$$ q + (b2 + b1) * q^2 + (-3*b2 + 4*b1) * q^3 + (3*b3 + 16) * q^4 - 25 * q^5 + (5*b3 + 23) * q^6 + (6*b2 + 45*b1) * q^8 + (-8*b3 + 81) * q^9 + (-25*b2 - 25*b1) * q^10 - 24*b3 * q^11 + (81*b2 + 34*b1) * q^12 + (-53*b2 + 76*b1) * q^13 + (75*b2 - 100*b1) * q^15 + (48*b3 + 599) * q^16 + (65*b2 - 39*b1) * q^18 - 361 * q^19 + (-75*b3 - 400) * q^20 + (-48*b2 - 360*b1) * q^22 + (69*b3 + 1425) * q^24 + 625 * q^25 + (99*b3 + 497) * q^26 + (-344*b2 + 80*b1) * q^27 + (-125*b3 - 575) * q^30 + (599*b2 + 599*b1) * q^32 + (-1032*b2 + 240*b1) * q^33 + (115*b3 - 984) * q^36 + (509*b2 + 164*b1) * q^37 + (-361*b2 - 361*b1) * q^38 + (-136*b3 + 3038) * q^39 + (-150*b2 - 1125*b1) * q^40 + (-384*b3 - 6840) * q^44 + (200*b3 - 2025) * q^45 + (267*b2 + 1916*b1) * q^48 + 2401 * q^49 + (625*b2 + 625*b1) * q^50 + (1543*b2 + 766*b1) * q^52 + (461*b2 - 1276*b1) * q^53 + (-184*b3 - 3800) * q^54 + 600*b3 * q^55 + (1083*b2 - 1444*b1) * q^57 + (-2025*b2 - 850*b1) * q^60 - 216*b3 * q^61 + (1029*b3 + 9584) * q^64 + (1325*b2 - 1900*b1) * q^65 + (-552*b3 - 11400) * q^66 + (-2117*b2 + 1516*b1) * q^67 + (-1794*b2 + 1365*b1) * q^72 + (837*b3 + 10423) * q^74 + (-1875*b2 + 2500*b1) * q^75 + (-1083*b3 - 5776) * q^76 + (2766*b2 + 998*b1) * q^78 + (-1200*b3 - 14975) * q^80 + (-648*b3 - 481) * q^81 + (-6840*b2 - 6840*b1) * q^88 + (-1625*b2 + 975*b1) * q^90 + 9025 * q^95 + (2995*b3 + 13777) * q^96 + (4685*b2 - 2476*b1) * q^97 + (2401*b2 + 2401*b1) * q^98 + (-1944*b3 + 18240) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 64 q^{4} - 100 q^{5} + 92 q^{6} + 324 q^{9}+O(q^{10})$$ 4 * q + 64 * q^4 - 100 * q^5 + 92 * q^6 + 324 * q^9 $$4 q + 64 q^{4} - 100 q^{5} + 92 q^{6} + 324 q^{9} + 2396 q^{16} - 1444 q^{19} - 1600 q^{20} + 5700 q^{24} + 2500 q^{25} + 1988 q^{26} - 2300 q^{30} - 3936 q^{36} + 12152 q^{39} - 27360 q^{44} - 8100 q^{45} + 9604 q^{49} - 15200 q^{54} + 38336 q^{64} - 45600 q^{66} + 41692 q^{74} - 23104 q^{76} - 59900 q^{80} - 1924 q^{81} + 36100 q^{95} + 55108 q^{96} + 72960 q^{99}+O(q^{100})$$ 4 * q + 64 * q^4 - 100 * q^5 + 92 * q^6 + 324 * q^9 + 2396 * q^16 - 1444 * q^19 - 1600 * q^20 + 5700 * q^24 + 2500 * q^25 + 1988 * q^26 - 2300 * q^30 - 3936 * q^36 + 12152 * q^39 - 27360 * q^44 - 8100 * q^45 + 9604 * q^49 - 15200 * q^54 + 38336 * q^64 - 45600 * q^66 + 41692 * q^74 - 23104 * q^76 - 59900 * q^80 - 1924 * q^81 + 36100 * q^95 + 55108 * q^96 + 72960 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 24x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 17\nu ) / 7$$ (v^3 - 17*v) / 7 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 12$$ v^2 - 12
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 12$$ b3 + 12 $$\nu^{3}$$ $$=$$ $$7\beta_{2} + 17\beta_1$$ 7*b2 + 17*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1
 −4.66335 1.50107 −1.50107 4.66335
−7.82562 −9.16655 45.2404 −25.0000 71.7340 0 −228.824 3.02565 195.641
94.2 −1.66121 15.4911 −13.2404 −25.0000 −25.7340 0 48.5744 158.974 41.5302
94.3 1.66121 −15.4911 −13.2404 −25.0000 −25.7340 0 −48.5744 158.974 −41.5302
94.4 7.82562 9.16655 45.2404 −25.0000 71.7340 0 228.824 3.02565 −195.641
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.5.d.d 4
5.b even 2 1 inner 95.5.d.d 4
19.b odd 2 1 inner 95.5.d.d 4
95.d odd 2 1 CM 95.5.d.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.5.d.d 4 1.a even 1 1 trivial
95.5.d.d 4 5.b even 2 1 inner
95.5.d.d 4 19.b odd 2 1 inner
95.5.d.d 4 95.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 64T_{2}^{2} + 169$$ acting on $$S_{5}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 64T^{2} + 169$$
$3$ $$T^{4} - 324 T^{2} + 20164$$
$5$ $$(T + 25)^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 54720)^{2}$$
$13$ $$T^{4} - 114244 T^{2} + \cdots + 2769074884$$
$17$ $$T^{4}$$
$19$ $$(T + 361)^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4} - 7496644 T^{2} + \cdots + 10480098340804$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + \cdots + 229655293801924$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 4432320)^{2}$$
$67$ $$T^{4} - 80604484 T^{2} + \cdots + 11320681202884$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} - 354117124 T^{2} + \cdots + 36\!\cdots\!24$$