# Properties

 Label 95.4.e.a Level $95$ Weight $4$ Character orbit 95.e Analytic conductor $5.605$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,4,Mod(11,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("95.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 95.e (of order $$3$$, degree $$2$$, 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: $$5.60518145055$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + ( - 5 \zeta_{6} + 5) q^{3} + 7 \zeta_{6} q^{4} + (5 \zeta_{6} - 5) q^{5} - 5 \zeta_{6} q^{6} + 22 q^{7} + 15 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + (-5*z + 5) * q^3 + 7*z * q^4 + (5*z - 5) * q^5 - 5*z * q^6 + 22 * q^7 + 15 * q^8 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + ( - 5 \zeta_{6} + 5) q^{3} + 7 \zeta_{6} q^{4} + (5 \zeta_{6} - 5) q^{5} - 5 \zeta_{6} q^{6} + 22 q^{7} + 15 q^{8} + 2 \zeta_{6} q^{9} + 5 \zeta_{6} q^{10} + 9 q^{11} + 35 q^{12} - 54 \zeta_{6} q^{13} + ( - 22 \zeta_{6} + 22) q^{14} + 25 \zeta_{6} q^{15} + (41 \zeta_{6} - 41) q^{16} + ( - 54 \zeta_{6} + 54) q^{17} + 2 q^{18} + ( - 57 \zeta_{6} - 38) q^{19} - 35 q^{20} + ( - 110 \zeta_{6} + 110) q^{21} + ( - 9 \zeta_{6} + 9) q^{22} + 92 \zeta_{6} q^{23} + ( - 75 \zeta_{6} + 75) q^{24} - 25 \zeta_{6} q^{25} - 54 q^{26} + 145 q^{27} + 154 \zeta_{6} q^{28} + 134 \zeta_{6} q^{29} + 25 q^{30} - 252 q^{31} + 161 \zeta_{6} q^{32} + ( - 45 \zeta_{6} + 45) q^{33} - 54 \zeta_{6} q^{34} + (110 \zeta_{6} - 110) q^{35} + (14 \zeta_{6} - 14) q^{36} - 236 q^{37} + (38 \zeta_{6} - 95) q^{38} - 270 q^{39} + (75 \zeta_{6} - 75) q^{40} + ( - 243 \zeta_{6} + 243) q^{41} - 110 \zeta_{6} q^{42} + (496 \zeta_{6} - 496) q^{43} + 63 \zeta_{6} q^{44} - 10 q^{45} + 92 q^{46} - 502 \zeta_{6} q^{47} + 205 \zeta_{6} q^{48} + 141 q^{49} - 25 q^{50} - 270 \zeta_{6} q^{51} + ( - 378 \zeta_{6} + 378) q^{52} - 62 \zeta_{6} q^{53} + ( - 145 \zeta_{6} + 145) q^{54} + (45 \zeta_{6} - 45) q^{55} + 330 q^{56} + (190 \zeta_{6} - 475) q^{57} + 134 q^{58} + (681 \zeta_{6} - 681) q^{59} + (175 \zeta_{6} - 175) q^{60} + 142 \zeta_{6} q^{61} + (252 \zeta_{6} - 252) q^{62} + 44 \zeta_{6} q^{63} - 167 q^{64} + 270 q^{65} - 45 \zeta_{6} q^{66} - 55 \zeta_{6} q^{67} + 378 q^{68} + 460 q^{69} + 110 \zeta_{6} q^{70} + ( - 974 \zeta_{6} + 974) q^{71} + 30 \zeta_{6} q^{72} + (695 \zeta_{6} - 695) q^{73} + (236 \zeta_{6} - 236) q^{74} - 125 q^{75} + ( - 665 \zeta_{6} + 399) q^{76} + 198 q^{77} + (270 \zeta_{6} - 270) q^{78} + ( - 736 \zeta_{6} + 736) q^{79} - 205 \zeta_{6} q^{80} + ( - 671 \zeta_{6} + 671) q^{81} - 243 \zeta_{6} q^{82} - 63 q^{83} + 770 q^{84} + 270 \zeta_{6} q^{85} + 496 \zeta_{6} q^{86} + 670 q^{87} + 135 q^{88} - 726 \zeta_{6} q^{89} + (10 \zeta_{6} - 10) q^{90} - 1188 \zeta_{6} q^{91} + (644 \zeta_{6} - 644) q^{92} + (1260 \zeta_{6} - 1260) q^{93} - 502 q^{94} + ( - 190 \zeta_{6} + 475) q^{95} + 805 q^{96} + ( - 1167 \zeta_{6} + 1167) q^{97} + ( - 141 \zeta_{6} + 141) q^{98} + 18 \zeta_{6} q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (-5*z + 5) * q^3 + 7*z * q^4 + (5*z - 5) * q^5 - 5*z * q^6 + 22 * q^7 + 15 * q^8 + 2*z * q^9 + 5*z * q^10 + 9 * q^11 + 35 * q^12 - 54*z * q^13 + (-22*z + 22) * q^14 + 25*z * q^15 + (41*z - 41) * q^16 + (-54*z + 54) * q^17 + 2 * q^18 + (-57*z - 38) * q^19 - 35 * q^20 + (-110*z + 110) * q^21 + (-9*z + 9) * q^22 + 92*z * q^23 + (-75*z + 75) * q^24 - 25*z * q^25 - 54 * q^26 + 145 * q^27 + 154*z * q^28 + 134*z * q^29 + 25 * q^30 - 252 * q^31 + 161*z * q^32 + (-45*z + 45) * q^33 - 54*z * q^34 + (110*z - 110) * q^35 + (14*z - 14) * q^36 - 236 * q^37 + (38*z - 95) * q^38 - 270 * q^39 + (75*z - 75) * q^40 + (-243*z + 243) * q^41 - 110*z * q^42 + (496*z - 496) * q^43 + 63*z * q^44 - 10 * q^45 + 92 * q^46 - 502*z * q^47 + 205*z * q^48 + 141 * q^49 - 25 * q^50 - 270*z * q^51 + (-378*z + 378) * q^52 - 62*z * q^53 + (-145*z + 145) * q^54 + (45*z - 45) * q^55 + 330 * q^56 + (190*z - 475) * q^57 + 134 * q^58 + (681*z - 681) * q^59 + (175*z - 175) * q^60 + 142*z * q^61 + (252*z - 252) * q^62 + 44*z * q^63 - 167 * q^64 + 270 * q^65 - 45*z * q^66 - 55*z * q^67 + 378 * q^68 + 460 * q^69 + 110*z * q^70 + (-974*z + 974) * q^71 + 30*z * q^72 + (695*z - 695) * q^73 + (236*z - 236) * q^74 - 125 * q^75 + (-665*z + 399) * q^76 + 198 * q^77 + (270*z - 270) * q^78 + (-736*z + 736) * q^79 - 205*z * q^80 + (-671*z + 671) * q^81 - 243*z * q^82 - 63 * q^83 + 770 * q^84 + 270*z * q^85 + 496*z * q^86 + 670 * q^87 + 135 * q^88 - 726*z * q^89 + (10*z - 10) * q^90 - 1188*z * q^91 + (644*z - 644) * q^92 + (1260*z - 1260) * q^93 - 502 * q^94 + (-190*z + 475) * q^95 + 805 * q^96 + (-1167*z + 1167) * q^97 + (-141*z + 141) * q^98 + 18*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 5 q^{3} + 7 q^{4} - 5 q^{5} - 5 q^{6} + 44 q^{7} + 30 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 + 5 * q^3 + 7 * q^4 - 5 * q^5 - 5 * q^6 + 44 * q^7 + 30 * q^8 + 2 * q^9 $$2 q + q^{2} + 5 q^{3} + 7 q^{4} - 5 q^{5} - 5 q^{6} + 44 q^{7} + 30 q^{8} + 2 q^{9} + 5 q^{10} + 18 q^{11} + 70 q^{12} - 54 q^{13} + 22 q^{14} + 25 q^{15} - 41 q^{16} + 54 q^{17} + 4 q^{18} - 133 q^{19} - 70 q^{20} + 110 q^{21} + 9 q^{22} + 92 q^{23} + 75 q^{24} - 25 q^{25} - 108 q^{26} + 290 q^{27} + 154 q^{28} + 134 q^{29} + 50 q^{30} - 504 q^{31} + 161 q^{32} + 45 q^{33} - 54 q^{34} - 110 q^{35} - 14 q^{36} - 472 q^{37} - 152 q^{38} - 540 q^{39} - 75 q^{40} + 243 q^{41} - 110 q^{42} - 496 q^{43} + 63 q^{44} - 20 q^{45} + 184 q^{46} - 502 q^{47} + 205 q^{48} + 282 q^{49} - 50 q^{50} - 270 q^{51} + 378 q^{52} - 62 q^{53} + 145 q^{54} - 45 q^{55} + 660 q^{56} - 760 q^{57} + 268 q^{58} - 681 q^{59} - 175 q^{60} + 142 q^{61} - 252 q^{62} + 44 q^{63} - 334 q^{64} + 540 q^{65} - 45 q^{66} - 55 q^{67} + 756 q^{68} + 920 q^{69} + 110 q^{70} + 974 q^{71} + 30 q^{72} - 695 q^{73} - 236 q^{74} - 250 q^{75} + 133 q^{76} + 396 q^{77} - 270 q^{78} + 736 q^{79} - 205 q^{80} + 671 q^{81} - 243 q^{82} - 126 q^{83} + 1540 q^{84} + 270 q^{85} + 496 q^{86} + 1340 q^{87} + 270 q^{88} - 726 q^{89} - 10 q^{90} - 1188 q^{91} - 644 q^{92} - 1260 q^{93} - 1004 q^{94} + 760 q^{95} + 1610 q^{96} + 1167 q^{97} + 141 q^{98} + 18 q^{99}+O(q^{100})$$ 2 * q + q^2 + 5 * q^3 + 7 * q^4 - 5 * q^5 - 5 * q^6 + 44 * q^7 + 30 * q^8 + 2 * q^9 + 5 * q^10 + 18 * q^11 + 70 * q^12 - 54 * q^13 + 22 * q^14 + 25 * q^15 - 41 * q^16 + 54 * q^17 + 4 * q^18 - 133 * q^19 - 70 * q^20 + 110 * q^21 + 9 * q^22 + 92 * q^23 + 75 * q^24 - 25 * q^25 - 108 * q^26 + 290 * q^27 + 154 * q^28 + 134 * q^29 + 50 * q^30 - 504 * q^31 + 161 * q^32 + 45 * q^33 - 54 * q^34 - 110 * q^35 - 14 * q^36 - 472 * q^37 - 152 * q^38 - 540 * q^39 - 75 * q^40 + 243 * q^41 - 110 * q^42 - 496 * q^43 + 63 * q^44 - 20 * q^45 + 184 * q^46 - 502 * q^47 + 205 * q^48 + 282 * q^49 - 50 * q^50 - 270 * q^51 + 378 * q^52 - 62 * q^53 + 145 * q^54 - 45 * q^55 + 660 * q^56 - 760 * q^57 + 268 * q^58 - 681 * q^59 - 175 * q^60 + 142 * q^61 - 252 * q^62 + 44 * q^63 - 334 * q^64 + 540 * q^65 - 45 * q^66 - 55 * q^67 + 756 * q^68 + 920 * q^69 + 110 * q^70 + 974 * q^71 + 30 * q^72 - 695 * q^73 - 236 * q^74 - 250 * q^75 + 133 * q^76 + 396 * q^77 - 270 * q^78 + 736 * q^79 - 205 * q^80 + 671 * q^81 - 243 * q^82 - 126 * q^83 + 1540 * q^84 + 270 * q^85 + 496 * q^86 + 1340 * q^87 + 270 * q^88 - 726 * q^89 - 10 * q^90 - 1188 * q^91 - 644 * q^92 - 1260 * q^93 - 1004 * q^94 + 760 * q^95 + 1610 * q^96 + 1167 * q^97 + 141 * q^98 + 18 * q^99

## Character values

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

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 2.50000 4.33013i 3.50000 + 6.06218i −2.50000 + 4.33013i −2.50000 4.33013i 22.0000 15.0000 1.00000 + 1.73205i 2.50000 + 4.33013i
26.1 0.500000 + 0.866025i 2.50000 + 4.33013i 3.50000 6.06218i −2.50000 4.33013i −2.50000 + 4.33013i 22.0000 15.0000 1.00000 1.73205i 2.50000 4.33013i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.4.e.a 2
19.c even 3 1 inner 95.4.e.a 2
19.c even 3 1 1805.4.a.e 1
19.d odd 6 1 1805.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.4.e.a 2 1.a even 1 1 trivial
95.4.e.a 2 19.c even 3 1 inner
1805.4.a.e 1 19.c even 3 1
1805.4.a.g 1 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 5T + 25$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$(T - 22)^{2}$$
$11$ $$(T - 9)^{2}$$
$13$ $$T^{2} + 54T + 2916$$
$17$ $$T^{2} - 54T + 2916$$
$19$ $$T^{2} + 133T + 6859$$
$23$ $$T^{2} - 92T + 8464$$
$29$ $$T^{2} - 134T + 17956$$
$31$ $$(T + 252)^{2}$$
$37$ $$(T + 236)^{2}$$
$41$ $$T^{2} - 243T + 59049$$
$43$ $$T^{2} + 496T + 246016$$
$47$ $$T^{2} + 502T + 252004$$
$53$ $$T^{2} + 62T + 3844$$
$59$ $$T^{2} + 681T + 463761$$
$61$ $$T^{2} - 142T + 20164$$
$67$ $$T^{2} + 55T + 3025$$
$71$ $$T^{2} - 974T + 948676$$
$73$ $$T^{2} + 695T + 483025$$
$79$ $$T^{2} - 736T + 541696$$
$83$ $$(T + 63)^{2}$$
$89$ $$T^{2} + 726T + 527076$$
$97$ $$T^{2} - 1167 T + 1361889$$