# Properties

 Label 95.11.d.a Level $95$ Weight $11$ Character orbit 95.d Analytic conductor $60.359$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,11,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, 11, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$11$$ 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: $$60.3589390040$$ 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: $$11$$ 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 $$\beta = \frac{1}{2}(-1 + 11\sqrt{-19})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 1024 q^{4} + (101 \beta + 2026) q^{5} + (354 \beta + 177) q^{7} - 59049 q^{9}+O(q^{10})$$ q - 1024 * q^4 + (101*b + 2026) * q^5 + (354*b + 177) * q^7 - 59049 * q^9 $$q - 1024 q^{4} + (101 \beta + 2026) q^{5} + (354 \beta + 177) q^{7} - 59049 q^{9} - 203523 q^{11} + 1048576 q^{16} + ( - 77546 \beta - 38773) q^{17} - 2476099 q^{19} + ( - 103424 \beta - 2074624) q^{20} + (492304 \beta + 246152) q^{23} + (399051 \beta - 1760899) q^{25} + ( - 362496 \beta - 181248) q^{28} + (699327 \beta - 20199948) q^{35} + 60466176 q^{36} + ( - 8477646 \beta - 4238823) q^{43} + 208407552 q^{44} + ( - 5963949 \beta - 119633274) q^{45} + (1788254 \beta + 894127) q^{47} + 210449878 q^{49} + ( - 20555823 \beta - 412337598) q^{55} + 1606836977 q^{61} + ( - 20903346 \beta - 10451673) q^{63} - 1073741824 q^{64} + (79407104 \beta + 39703552) q^{68} + ( - 112781946 \beta - 56390973) q^{73} + 2535525376 q^{76} + ( - 72047142 \beta - 36023571) q^{77} + (105906176 \beta + 2124414976) q^{80} + 3486784401 q^{81} + ( - 316127296 \beta - 158063648) q^{83} + ( - 153192123 \beta + 4424929852) q^{85} + ( - 504119296 \beta - 252059648) q^{92} + ( - 250085999 \beta - 5016576574) q^{95} + 12017829627 q^{99}+O(q^{100})$$ q - 1024 * q^4 + (101*b + 2026) * q^5 + (354*b + 177) * q^7 - 59049 * q^9 - 203523 * q^11 + 1048576 * q^16 + (-77546*b - 38773) * q^17 - 2476099 * q^19 + (-103424*b - 2074624) * q^20 + (492304*b + 246152) * q^23 + (399051*b - 1760899) * q^25 + (-362496*b - 181248) * q^28 + (699327*b - 20199948) * q^35 + 60466176 * q^36 + (-8477646*b - 4238823) * q^43 + 208407552 * q^44 + (-5963949*b - 119633274) * q^45 + (1788254*b + 894127) * q^47 + 210449878 * q^49 + (-20555823*b - 412337598) * q^55 + 1606836977 * q^61 + (-20903346*b - 10451673) * q^63 - 1073741824 * q^64 + (79407104*b + 39703552) * q^68 + (-112781946*b - 56390973) * q^73 + 2535525376 * q^76 + (-72047142*b - 36023571) * q^77 + (105906176*b + 2124414976) * q^80 + 3486784401 * q^81 + (-316127296*b - 158063648) * q^83 + (-153192123*b + 4424929852) * q^85 + (-504119296*b - 252059648) * q^92 + (-250085999*b - 5016576574) * q^95 + 12017829627 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9}+O(q^{10})$$ 2 * q - 2048 * q^4 + 3951 * q^5 - 118098 * q^9 $$2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9} - 407046 q^{11} + 2097152 q^{16} - 4952198 q^{19} - 4045824 q^{20} - 3920849 q^{25} - 41099223 q^{35} + 120932352 q^{36} + 416815104 q^{44} - 233302599 q^{45} + 420899756 q^{49} - 804119373 q^{55} + 3213673954 q^{61} - 2147483648 q^{64} + 5071050752 q^{76} + 4142923776 q^{80} + 6973568802 q^{81} + 9003051827 q^{85} - 9783067149 q^{95} + 24035659254 q^{99}+O(q^{100})$$ 2 * q - 2048 * q^4 + 3951 * q^5 - 118098 * q^9 - 407046 * q^11 + 2097152 * q^16 - 4952198 * q^19 - 4045824 * q^20 - 3920849 * q^25 - 41099223 * q^35 + 120932352 * q^36 + 416815104 * q^44 - 233302599 * q^45 + 420899756 * q^49 - 804119373 * q^55 + 3213673954 * q^61 - 2147483648 * q^64 + 5071050752 * q^76 + 4142923776 * q^80 + 6973568802 * q^81 + 9003051827 * q^85 - 9783067149 * q^95 + 24035659254 * 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 −1024.00 1975.50 2421.37i 0 8486.78i 0 −59049.0 0
94.2 0 0 −1024.00 1975.50 + 2421.37i 0 8486.78i 0 −59049.0 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 95.11.d.a 2
5.b even 2 1 inner 95.11.d.a 2
19.b odd 2 1 CM 95.11.d.a 2
95.d odd 2 1 inner 95.11.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.11.d.a 2 1.a even 1 1 trivial
95.11.d.a 2 5.b even 2 1 inner
95.11.d.a 2 19.b odd 2 1 CM
95.11.d.a 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 $$S_{11}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3951 T + 9765625$$
$7$ $$T^{2} + 72025371$$
$11$ $$(T + 203523)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 3456191371171$$
$19$ $$(T + 2476099)^{2}$$
$23$ $$T^{2} + \cdots + 139298265532096$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 41\!\cdots\!71$$
$47$ $$T^{2} + 18\!\cdots\!71$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 1606836977)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 73\!\cdots\!71$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 57\!\cdots\!96$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$