# Properties

 Label 195.3.u.c Level $195$ Weight $3$ Character orbit 195.u Analytic conductor $5.313$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,3,Mod(47,195)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("195.47");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 195.u (of order $$4$$, 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: $$5.31336515503$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{2} + ( - 2 \zeta_{8}^{2} - \zeta_{8} + 2) q^{3} + 4 q^{4} + (3 \zeta_{8}^{3} + 4 \zeta_{8}) q^{5} + (8 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2) q^{6} - 5 q^{7} + (4 \zeta_{8}^{3} - 7 \zeta_{8}^{2} - 4 \zeta_{8}) q^{9}+O(q^{10})$$ q + (2*z^3 - 2*z) * q^2 + (-2*z^2 - z + 2) * q^3 + 4 * q^4 + (3*z^3 + 4*z) * q^5 + (8*z^3 + 2*z^2 + 2) * q^6 - 5 * q^7 + (4*z^3 - 7*z^2 - 4*z) * q^9 $$q + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{2} + ( - 2 \zeta_{8}^{2} - \zeta_{8} + 2) q^{3} + 4 q^{4} + (3 \zeta_{8}^{3} + 4 \zeta_{8}) q^{5} + (8 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2) q^{6} - 5 q^{7} + (4 \zeta_{8}^{3} - 7 \zeta_{8}^{2} - 4 \zeta_{8}) q^{9} + ( - 14 \zeta_{8}^{2} - 2) q^{10} - 13 \zeta_{8} q^{11} + ( - 8 \zeta_{8}^{2} - 4 \zeta_{8} + 8) q^{12} + 13 q^{13} + ( - 10 \zeta_{8}^{3} + 10 \zeta_{8}) q^{14} + ( - 2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 14 \zeta_{8} + 3) q^{15} - 16 q^{16} - 19 \zeta_{8} q^{17} + (14 \zeta_{8}^{3} + 14 \zeta_{8} + 16) q^{18} + (22 \zeta_{8}^{2} - 22) q^{19} + (12 \zeta_{8}^{3} + 16 \zeta_{8}) q^{20} + (10 \zeta_{8}^{2} + 5 \zeta_{8} - 10) q^{21} + (26 \zeta_{8}^{2} + 26) q^{22} - 21 \zeta_{8}^{3} q^{23} + (7 \zeta_{8}^{2} - 24) q^{25} + (26 \zeta_{8}^{3} - 26 \zeta_{8}) q^{26} + (23 \zeta_{8}^{3} - 10 \zeta_{8}^{2} - 10) q^{27} - 20 q^{28} + (27 \zeta_{8}^{3} - 27 \zeta_{8}) q^{29} + (14 \zeta_{8}^{3} - 24 \zeta_{8}^{2} + 2 \zeta_{8} - 32) q^{30} + ( - 13 \zeta_{8}^{2} - 13) q^{31} + ( - 32 \zeta_{8}^{3} + 32 \zeta_{8}) q^{32} + (26 \zeta_{8}^{3} + 13 \zeta_{8}^{2} - 26 \zeta_{8}) q^{33} + (38 \zeta_{8}^{2} + 38) q^{34} + ( - 15 \zeta_{8}^{3} - 20 \zeta_{8}) q^{35} + (16 \zeta_{8}^{3} - 28 \zeta_{8}^{2} - 16 \zeta_{8}) q^{36} - 15 q^{37} - 88 \zeta_{8}^{3} q^{38} + ( - 26 \zeta_{8}^{2} - 13 \zeta_{8} + 26) q^{39} + 35 \zeta_{8}^{3} q^{41} + ( - 40 \zeta_{8}^{3} - 10 \zeta_{8}^{2} - 10) q^{42} + ( - 17 \zeta_{8}^{2} - 17) q^{43} - 52 \zeta_{8} q^{44} + ( - 28 \zeta_{8}^{3} - 28 \zeta_{8}^{2} + 21 \zeta_{8} - 4) q^{45} + (42 \zeta_{8}^{2} - 42) q^{46} + (51 \zeta_{8}^{3} + 51 \zeta_{8}) q^{47} + (32 \zeta_{8}^{2} + 16 \zeta_{8} - 32) q^{48} - 24 q^{49} + ( - 62 \zeta_{8}^{3} + 34 \zeta_{8}) q^{50} + (38 \zeta_{8}^{3} + 19 \zeta_{8}^{2} - 38 \zeta_{8}) q^{51} + 52 q^{52} - 13 \zeta_{8} q^{53} + ( - 46 \zeta_{8}^{2} + 40 \zeta_{8} + 46) q^{54} + ( - 52 \zeta_{8}^{2} + 39) q^{55} + ( - 22 \zeta_{8}^{3} + 88 \zeta_{8}^{2} + 22 \zeta_{8}) q^{57} + 108 q^{58} - 68 \zeta_{8}^{3} q^{59} + ( - 8 \zeta_{8}^{3} - 16 \zeta_{8}^{2} + 56 \zeta_{8} + 12) q^{60} + 67 q^{61} + 52 \zeta_{8} q^{62} + ( - 20 \zeta_{8}^{3} + 35 \zeta_{8}^{2} + 20 \zeta_{8}) q^{63} - 64 q^{64} + (39 \zeta_{8}^{3} + 52 \zeta_{8}) q^{65} + ( - 26 \zeta_{8}^{3} - 26 \zeta_{8} + 104) q^{66} - 100 \zeta_{8}^{2} q^{67} - 76 \zeta_{8} q^{68} + ( - 42 \zeta_{8}^{3} - 42 \zeta_{8} - 21) q^{69} + (70 \zeta_{8}^{2} + 10) q^{70} - 53 \zeta_{8}^{3} q^{71} + 76 \zeta_{8}^{2} q^{73} + ( - 30 \zeta_{8}^{3} + 30 \zeta_{8}) q^{74} + ( - 7 \zeta_{8}^{3} + 62 \zeta_{8}^{2} + 24 \zeta_{8} - 34) q^{75} + (88 \zeta_{8}^{2} - 88) q^{76} + 65 \zeta_{8} q^{77} + (104 \zeta_{8}^{3} + 26 \zeta_{8}^{2} + 26) q^{78} - 89 \zeta_{8}^{2} q^{79} + ( - 48 \zeta_{8}^{3} - 64 \zeta_{8}) q^{80} + (56 \zeta_{8}^{3} + 56 \zeta_{8} - 17) q^{81} + ( - 70 \zeta_{8}^{2} + 70) q^{82} + ( - 66 \zeta_{8}^{3} - 66 \zeta_{8}) q^{83} + (40 \zeta_{8}^{2} + 20 \zeta_{8} - 40) q^{84} + ( - 76 \zeta_{8}^{2} + 57) q^{85} + 68 \zeta_{8} q^{86} + (108 \zeta_{8}^{3} + 27 \zeta_{8}^{2} + 27) q^{87} + 7 \zeta_{8}^{3} q^{89} + (48 \zeta_{8}^{3} + 14 \zeta_{8}^{2} + 64 \zeta_{8} - 98) q^{90} - 65 q^{91} - 84 \zeta_{8}^{3} q^{92} + (13 \zeta_{8}^{3} + 13 \zeta_{8} - 52) q^{93} - 204 \zeta_{8}^{2} q^{94} + (22 \zeta_{8}^{3} - 154 \zeta_{8}) q^{95} + ( - 128 \zeta_{8}^{3} - 32 \zeta_{8}^{2} - 32) q^{96} - 25 \zeta_{8}^{2} q^{97} + ( - 48 \zeta_{8}^{3} + 48 \zeta_{8}) q^{98} + (91 \zeta_{8}^{3} + 52 \zeta_{8}^{2} + 52) q^{99} +O(q^{100})$$ q + (2*z^3 - 2*z) * q^2 + (-2*z^2 - z + 2) * q^3 + 4 * q^4 + (3*z^3 + 4*z) * q^5 + (8*z^3 + 2*z^2 + 2) * q^6 - 5 * q^7 + (4*z^3 - 7*z^2 - 4*z) * q^9 + (-14*z^2 - 2) * q^10 - 13*z * q^11 + (-8*z^2 - 4*z + 8) * q^12 + 13 * q^13 + (-10*z^3 + 10*z) * q^14 + (-2*z^3 - 4*z^2 + 14*z + 3) * q^15 - 16 * q^16 - 19*z * q^17 + (14*z^3 + 14*z + 16) * q^18 + (22*z^2 - 22) * q^19 + (12*z^3 + 16*z) * q^20 + (10*z^2 + 5*z - 10) * q^21 + (26*z^2 + 26) * q^22 - 21*z^3 * q^23 + (7*z^2 - 24) * q^25 + (26*z^3 - 26*z) * q^26 + (23*z^3 - 10*z^2 - 10) * q^27 - 20 * q^28 + (27*z^3 - 27*z) * q^29 + (14*z^3 - 24*z^2 + 2*z - 32) * q^30 + (-13*z^2 - 13) * q^31 + (-32*z^3 + 32*z) * q^32 + (26*z^3 + 13*z^2 - 26*z) * q^33 + (38*z^2 + 38) * q^34 + (-15*z^3 - 20*z) * q^35 + (16*z^3 - 28*z^2 - 16*z) * q^36 - 15 * q^37 - 88*z^3 * q^38 + (-26*z^2 - 13*z + 26) * q^39 + 35*z^3 * q^41 + (-40*z^3 - 10*z^2 - 10) * q^42 + (-17*z^2 - 17) * q^43 - 52*z * q^44 + (-28*z^3 - 28*z^2 + 21*z - 4) * q^45 + (42*z^2 - 42) * q^46 + (51*z^3 + 51*z) * q^47 + (32*z^2 + 16*z - 32) * q^48 - 24 * q^49 + (-62*z^3 + 34*z) * q^50 + (38*z^3 + 19*z^2 - 38*z) * q^51 + 52 * q^52 - 13*z * q^53 + (-46*z^2 + 40*z + 46) * q^54 + (-52*z^2 + 39) * q^55 + (-22*z^3 + 88*z^2 + 22*z) * q^57 + 108 * q^58 - 68*z^3 * q^59 + (-8*z^3 - 16*z^2 + 56*z + 12) * q^60 + 67 * q^61 + 52*z * q^62 + (-20*z^3 + 35*z^2 + 20*z) * q^63 - 64 * q^64 + (39*z^3 + 52*z) * q^65 + (-26*z^3 - 26*z + 104) * q^66 - 100*z^2 * q^67 - 76*z * q^68 + (-42*z^3 - 42*z - 21) * q^69 + (70*z^2 + 10) * q^70 - 53*z^3 * q^71 + 76*z^2 * q^73 + (-30*z^3 + 30*z) * q^74 + (-7*z^3 + 62*z^2 + 24*z - 34) * q^75 + (88*z^2 - 88) * q^76 + 65*z * q^77 + (104*z^3 + 26*z^2 + 26) * q^78 - 89*z^2 * q^79 + (-48*z^3 - 64*z) * q^80 + (56*z^3 + 56*z - 17) * q^81 + (-70*z^2 + 70) * q^82 + (-66*z^3 - 66*z) * q^83 + (40*z^2 + 20*z - 40) * q^84 + (-76*z^2 + 57) * q^85 + 68*z * q^86 + (108*z^3 + 27*z^2 + 27) * q^87 + 7*z^3 * q^89 + (48*z^3 + 14*z^2 + 64*z - 98) * q^90 - 65 * q^91 - 84*z^3 * q^92 + (13*z^3 + 13*z - 52) * q^93 - 204*z^2 * q^94 + (22*z^3 - 154*z) * q^95 + (-128*z^3 - 32*z^2 - 32) * q^96 - 25*z^2 * q^97 + (-48*z^3 + 48*z) * q^98 + (91*z^3 + 52*z^2 + 52) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{3} + 16 q^{4} + 8 q^{6} - 20 q^{7}+O(q^{10})$$ 4 * q + 8 * q^3 + 16 * q^4 + 8 * q^6 - 20 * q^7 $$4 q + 8 q^{3} + 16 q^{4} + 8 q^{6} - 20 q^{7} - 8 q^{10} + 32 q^{12} + 52 q^{13} + 12 q^{15} - 64 q^{16} + 64 q^{18} - 88 q^{19} - 40 q^{21} + 104 q^{22} - 96 q^{25} - 40 q^{27} - 80 q^{28} - 128 q^{30} - 52 q^{31} + 152 q^{34} - 60 q^{37} + 104 q^{39} - 40 q^{42} - 68 q^{43} - 16 q^{45} - 168 q^{46} - 128 q^{48} - 96 q^{49} + 208 q^{52} + 184 q^{54} + 156 q^{55} + 432 q^{58} + 48 q^{60} + 268 q^{61} - 256 q^{64} + 416 q^{66} - 84 q^{69} + 40 q^{70} - 136 q^{75} - 352 q^{76} + 104 q^{78} - 68 q^{81} + 280 q^{82} - 160 q^{84} + 228 q^{85} + 108 q^{87} - 392 q^{90} - 260 q^{91} - 208 q^{93} - 128 q^{96} + 208 q^{99}+O(q^{100})$$ 4 * q + 8 * q^3 + 16 * q^4 + 8 * q^6 - 20 * q^7 - 8 * q^10 + 32 * q^12 + 52 * q^13 + 12 * q^15 - 64 * q^16 + 64 * q^18 - 88 * q^19 - 40 * q^21 + 104 * q^22 - 96 * q^25 - 40 * q^27 - 80 * q^28 - 128 * q^30 - 52 * q^31 + 152 * q^34 - 60 * q^37 + 104 * q^39 - 40 * q^42 - 68 * q^43 - 16 * q^45 - 168 * q^46 - 128 * q^48 - 96 * q^49 + 208 * q^52 + 184 * q^54 + 156 * q^55 + 432 * q^58 + 48 * q^60 + 268 * q^61 - 256 * q^64 + 416 * q^66 - 84 * q^69 + 40 * q^70 - 136 * q^75 - 352 * q^76 + 104 * q^78 - 68 * q^81 + 280 * q^82 - 160 * q^84 + 228 * q^85 + 108 * q^87 - 392 * q^90 - 260 * q^91 - 208 * q^93 - 128 * q^96 + 208 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/195\mathbb{Z}\right)^\times$$.

 $$n$$ $$106$$ $$131$$ $$157$$ $$\chi(n)$$ $$\zeta_{8}^{2}$$ $$-1$$ $$\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
−2.82843 1.29289 2.70711i 4.00000 0.707107 + 4.94975i −3.65685 + 7.65685i −5.00000 0 −5.65685 7.00000i −2.00000 14.0000i
47.2 2.82843 2.70711 1.29289i 4.00000 −0.707107 4.94975i 7.65685 3.65685i −5.00000 0 5.65685 7.00000i −2.00000 14.0000i
83.1 −2.82843 1.29289 + 2.70711i 4.00000 0.707107 4.94975i −3.65685 7.65685i −5.00000 0 −5.65685 + 7.00000i −2.00000 + 14.0000i
83.2 2.82843 2.70711 + 1.29289i 4.00000 −0.707107 + 4.94975i 7.65685 + 3.65685i −5.00000 0 5.65685 + 7.00000i −2.00000 + 14.0000i
 $$n$$: 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
3.b odd 2 1 inner
65.f even 4 1 inner
195.u odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.3.u.c yes 4
3.b odd 2 1 inner 195.3.u.c yes 4
5.c odd 4 1 195.3.j.c 4
13.d odd 4 1 195.3.j.c 4
15.e even 4 1 195.3.j.c 4
39.f even 4 1 195.3.j.c 4
65.f even 4 1 inner 195.3.u.c yes 4
195.u odd 4 1 inner 195.3.u.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.3.j.c 4 5.c odd 4 1
195.3.j.c 4 13.d odd 4 1
195.3.j.c 4 15.e even 4 1
195.3.j.c 4 39.f even 4 1
195.3.u.c yes 4 1.a even 1 1 trivial
195.3.u.c yes 4 3.b odd 2 1 inner
195.3.u.c yes 4 65.f even 4 1 inner
195.3.u.c yes 4 195.u odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 8$$ acting on $$S_{3}^{\mathrm{new}}(195, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 8)^{2}$$
$3$ $$T^{4} - 8 T^{3} + 32 T^{2} - 72 T + 81$$
$5$ $$T^{4} + 48T^{2} + 625$$
$7$ $$(T + 5)^{4}$$
$11$ $$T^{4} + 28561$$
$13$ $$(T - 13)^{4}$$
$17$ $$T^{4} + 130321$$
$19$ $$(T^{2} + 44 T + 968)^{2}$$
$23$ $$T^{4} + 194481$$
$29$ $$(T^{2} - 1458)^{2}$$
$31$ $$(T^{2} + 26 T + 338)^{2}$$
$37$ $$(T + 15)^{4}$$
$41$ $$T^{4} + 1500625$$
$43$ $$(T^{2} + 34 T + 578)^{2}$$
$47$ $$(T^{2} + 5202)^{2}$$
$53$ $$T^{4} + 28561$$
$59$ $$T^{4} + 21381376$$
$61$ $$(T - 67)^{4}$$
$67$ $$(T^{2} + 10000)^{2}$$
$71$ $$T^{4} + 7890481$$
$73$ $$(T^{2} + 5776)^{2}$$
$79$ $$(T^{2} + 7921)^{2}$$
$83$ $$(T^{2} + 8712)^{2}$$
$89$ $$T^{4} + 2401$$
$97$ $$(T^{2} + 625)^{2}$$