# Properties

 Label 95.9.d.d Level $95$ Weight $9$ Character orbit 95.d Self dual yes Analytic conductor $38.701$ Analytic rank $0$ Dimension $2$ CM discriminant -95 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,9,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, 9, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$9$$ 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: $$38.7009679558$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{95})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 95$$ x^2 - 95 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 $$\beta = \sqrt{95}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta q^{2} - 8 \beta q^{3} + 599 q^{4} + 625 q^{5} - 2280 q^{6} + 1029 \beta q^{8} - 481 q^{9} +O(q^{10})$$ q + 3*b * q^2 - 8*b * q^3 + 599 * q^4 + 625 * q^5 - 2280 * q^6 + 1029*b * q^8 - 481 * q^9 $$q + 3 \beta q^{2} - 8 \beta q^{3} + 599 q^{4} + 625 q^{5} - 2280 q^{6} + 1029 \beta q^{8} - 481 q^{9} + 1875 \beta q^{10} + 25438 q^{11} - 4792 \beta q^{12} - 2280 \beta q^{13} - 5000 \beta q^{15} + 139921 q^{16} - 1443 \beta q^{18} + 130321 q^{19} + 374375 q^{20} + 76314 \beta q^{22} - 782040 q^{24} + 390625 q^{25} - 649800 q^{26} + 56336 \beta q^{27} - 1425000 q^{30} + 156339 \beta q^{32} - 203504 \beta q^{33} - 288119 q^{36} + 193848 \beta q^{37} + 390963 \beta q^{38} + 1732800 q^{39} + 643125 \beta q^{40} + 15237362 q^{44} - 300625 q^{45} - 1119368 \beta q^{48} + 5764801 q^{49} + 1171875 \beta q^{50} - 1365720 \beta q^{52} + 451704 \beta q^{53} + 16055760 q^{54} + 15898750 q^{55} - 1042568 \beta q^{57} - 2995000 \beta q^{60} - 23259362 q^{61} + 8736839 q^{64} - 1425000 \beta q^{65} - 57998640 q^{66} - 4120488 \beta q^{67} - 494949 \beta q^{72} + 55246680 q^{74} - 3125000 \beta q^{75} + 78062279 q^{76} + 5198400 \beta q^{78} + 87450625 q^{80} - 39659519 q^{81} + 26175702 \beta q^{88} - 901875 \beta q^{90} + 81450625 q^{95} - 118817640 q^{96} - 17069544 \beta q^{97} + 17294403 \beta q^{98} - 12235678 q^{99} +O(q^{100})$$ q + 3*b * q^2 - 8*b * q^3 + 599 * q^4 + 625 * q^5 - 2280 * q^6 + 1029*b * q^8 - 481 * q^9 + 1875*b * q^10 + 25438 * q^11 - 4792*b * q^12 - 2280*b * q^13 - 5000*b * q^15 + 139921 * q^16 - 1443*b * q^18 + 130321 * q^19 + 374375 * q^20 + 76314*b * q^22 - 782040 * q^24 + 390625 * q^25 - 649800 * q^26 + 56336*b * q^27 - 1425000 * q^30 + 156339*b * q^32 - 203504*b * q^33 - 288119 * q^36 + 193848*b * q^37 + 390963*b * q^38 + 1732800 * q^39 + 643125*b * q^40 + 15237362 * q^44 - 300625 * q^45 - 1119368*b * q^48 + 5764801 * q^49 + 1171875*b * q^50 - 1365720*b * q^52 + 451704*b * q^53 + 16055760 * q^54 + 15898750 * q^55 - 1042568*b * q^57 - 2995000*b * q^60 - 23259362 * q^61 + 8736839 * q^64 - 1425000*b * q^65 - 57998640 * q^66 - 4120488*b * q^67 - 494949*b * q^72 + 55246680 * q^74 - 3125000*b * q^75 + 78062279 * q^76 + 5198400*b * q^78 + 87450625 * q^80 - 39659519 * q^81 + 26175702*b * q^88 - 901875*b * q^90 + 81450625 * q^95 - 118817640 * q^96 - 17069544*b * q^97 + 17294403*b * q^98 - 12235678 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 1198 q^{4} + 1250 q^{5} - 4560 q^{6} - 962 q^{9}+O(q^{10})$$ 2 * q + 1198 * q^4 + 1250 * q^5 - 4560 * q^6 - 962 * q^9 $$2 q + 1198 q^{4} + 1250 q^{5} - 4560 q^{6} - 962 q^{9} + 50876 q^{11} + 279842 q^{16} + 260642 q^{19} + 748750 q^{20} - 1564080 q^{24} + 781250 q^{25} - 1299600 q^{26} - 2850000 q^{30} - 576238 q^{36} + 3465600 q^{39} + 30474724 q^{44} - 601250 q^{45} + 11529602 q^{49} + 32111520 q^{54} + 31797500 q^{55} - 46518724 q^{61} + 17473678 q^{64} - 115997280 q^{66} + 110493360 q^{74} + 156124558 q^{76} + 174901250 q^{80} - 79319038 q^{81} + 162901250 q^{95} - 237635280 q^{96} - 24471356 q^{99}+O(q^{100})$$ 2 * q + 1198 * q^4 + 1250 * q^5 - 4560 * q^6 - 962 * q^9 + 50876 * q^11 + 279842 * q^16 + 260642 * q^19 + 748750 * q^20 - 1564080 * q^24 + 781250 * q^25 - 1299600 * q^26 - 2850000 * q^30 - 576238 * q^36 + 3465600 * q^39 + 30474724 * q^44 - 601250 * q^45 + 11529602 * q^49 + 32111520 * q^54 + 31797500 * q^55 - 46518724 * q^61 + 17473678 * q^64 - 115997280 * q^66 + 110493360 * q^74 + 156124558 * q^76 + 174901250 * q^80 - 79319038 * q^81 + 162901250 * q^95 - 237635280 * q^96 - 24471356 * q^99

## 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
 −9.74679 9.74679
−29.2404 77.9744 599.000 625.000 −2280.00 0 −10029.5 −481.000 −18275.2
94.2 29.2404 −77.9744 599.000 625.000 −2280.00 0 10029.5 −481.000 18275.2
 $$n$$: 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 95.9.d.d 2
5.b even 2 1 inner 95.9.d.d 2
19.b odd 2 1 inner 95.9.d.d 2
95.d odd 2 1 CM 95.9.d.d 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 855$$ acting on $$S_{9}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 855$$
$3$ $$T^{2} - 6080$$
$5$ $$(T - 625)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 25438)^{2}$$
$13$ $$T^{2} - 493848000$$
$17$ $$T^{2}$$
$19$ $$(T - 130321)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 3569819474880$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 19383467843520$$
$59$ $$T^{2}$$
$61$ $$(T + 23259362)^{2}$$
$67$ $$T^{2} - 16\!\cdots\!80$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 27\!\cdots\!20$$