# Properties

 Label 95.9.d.c Level $95$ Weight $9$ Character orbit 95.d Analytic conductor $38.701$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

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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: no Analytic conductor: $$38.7009679558$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 5$$ x^2 - x + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 $$\beta = \frac{1}{2}(1 + 3\sqrt{-19})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 256 q^{4} + (93 \beta - 191) q^{5} + (730 \beta - 365) q^{7} - 6561 q^{9}+O(q^{10})$$ q - 256 * q^4 + (93*b - 191) * q^5 + (730*b - 365) * q^7 - 6561 * q^9 $$q - 256 q^{4} + (93 \beta - 191) q^{5} + (730 \beta - 365) q^{7} - 6561 q^{9} - 25007 q^{11} + 65536 q^{16} + (24710 \beta - 12355) q^{17} + 130321 q^{19} + ( - 23808 \beta + 48896) q^{20} + (25280 \beta - 12640) q^{23} + ( - 26877 \beta - 335426) q^{25} + ( - 186880 \beta + 93440) q^{28} + ( - 105485 \beta - 2849555) q^{35} + 1679616 q^{36} + ( - 599590 \beta + 299795) q^{43} + 6401792 q^{44} + ( - 610173 \beta + 1253151) q^{45} + (784550 \beta - 392275) q^{47} - 17016674 q^{49} + ( - 2325651 \beta + 4776337) q^{55} + 17661793 q^{61} + ( - 4789530 \beta + 2394765) q^{63} - 16777216 q^{64} + ( - 6325760 \beta + 3162880) q^{68} + (5518150 \beta - 2759075) q^{73} - 33362176 q^{76} + ( - 18255110 \beta + 9127555) q^{77} + (6094848 \beta - 12517376) q^{80} + 43046721 q^{81} + (10901760 \beta - 5450880) q^{83} + ( - 3570595 \beta - 96455485) q^{85} + ( - 6471680 \beta + 3235840) q^{92} + (12119853 \beta - 24891311) q^{95} + 164070927 q^{99}+O(q^{100})$$ q - 256 * q^4 + (93*b - 191) * q^5 + (730*b - 365) * q^7 - 6561 * q^9 - 25007 * q^11 + 65536 * q^16 + (24710*b - 12355) * q^17 + 130321 * q^19 + (-23808*b + 48896) * q^20 + (25280*b - 12640) * q^23 + (-26877*b - 335426) * q^25 + (-186880*b + 93440) * q^28 + (-105485*b - 2849555) * q^35 + 1679616 * q^36 + (-599590*b + 299795) * q^43 + 6401792 * q^44 + (-610173*b + 1253151) * q^45 + (784550*b - 392275) * q^47 - 17016674 * q^49 + (-2325651*b + 4776337) * q^55 + 17661793 * q^61 + (-4789530*b + 2394765) * q^63 - 16777216 * q^64 + (-6325760*b + 3162880) * q^68 + (5518150*b - 2759075) * q^73 - 33362176 * q^76 + (-18255110*b + 9127555) * q^77 + (6094848*b - 12517376) * q^80 + 43046721 * q^81 + (10901760*b - 5450880) * q^83 + (-3570595*b - 96455485) * q^85 + (-6471680*b + 3235840) * q^92 + (12119853*b - 24891311) * q^95 + 164070927 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 512 q^{4} - 289 q^{5} - 13122 q^{9}+O(q^{10})$$ 2 * q - 512 * q^4 - 289 * q^5 - 13122 * q^9 $$2 q - 512 q^{4} - 289 q^{5} - 13122 q^{9} - 50014 q^{11} + 131072 q^{16} + 260642 q^{19} + 73984 q^{20} - 697729 q^{25} - 5804595 q^{35} + 3359232 q^{36} + 12803584 q^{44} + 1896129 q^{45} - 34033348 q^{49} + 7227023 q^{55} + 35323586 q^{61} - 33554432 q^{64} - 66724352 q^{76} - 18939904 q^{80} + 86093442 q^{81} - 196481565 q^{85} - 37662769 q^{95} + 328141854 q^{99}+O(q^{100})$$ 2 * q - 512 * q^4 - 289 * q^5 - 13122 * q^9 - 50014 * q^11 + 131072 * q^16 + 260642 * q^19 + 73984 * q^20 - 697729 * q^25 - 5804595 * q^35 + 3359232 * q^36 + 12803584 * q^44 + 1896129 * q^45 - 34033348 * q^49 + 7227023 * q^55 + 35323586 * q^61 - 33554432 * q^64 - 66724352 * q^76 - 18939904 * q^80 + 86093442 * q^81 - 196481565 * q^85 - 37662769 * q^95 + 328141854 * 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
 0.5 − 2.17945i 0.5 + 2.17945i
0 0 −256.000 −144.500 608.066i 0 4772.99i 0 −6561.00 0
94.2 0 0 −256.000 −144.500 + 608.066i 0 4772.99i 0 −6561.00 0
 $$n$$: 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
5.b even 2 1 inner
95.d 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.c 2
5.b even 2 1 inner 95.9.d.c 2
19.b odd 2 1 CM 95.9.d.c 2
95.d odd 2 1 inner 95.9.d.c 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 289T + 390625$$
$7$ $$T^{2} + 22781475$$
$11$ $$(T + 25007)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 26102470275$$
$19$ $$(T - 130321)^{2}$$
$23$ $$T^{2} + 27320601600$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 15368974186275$$
$47$ $$T^{2} + 26313424531875$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 17661793)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 13\!\cdots\!75$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 50\!\cdots\!00$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$
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