# Properties

 Label 71.3.b.a Level $71$ Weight $3$ Character orbit 71.b Analytic conductor $1.935$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [71,3,Mod(70,71)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(71, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("71.70");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$71$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 71.b (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: $$1.93460987696$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2836736.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 108x^{2} - 40x + 2825$$ x^4 + 108*x^2 - 40*x + 2825 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_1 - 1) q^{4} + (\beta_1 - 2) q^{5} - q^{6} + \beta_{3} q^{7} + (3 \beta_1 + 1) q^{8} + ( - 2 \beta_1 - 6) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + (b1 - 1) * q^3 + (2*b1 - 1) * q^4 + (b1 - 2) * q^5 - q^6 + b3 * q^7 + (3*b1 + 1) * q^8 + (-2*b1 - 6) * q^9 $$q + ( - \beta_1 - 1) q^{2} + (\beta_1 - 1) q^{3} + (2 \beta_1 - 1) q^{4} + (\beta_1 - 2) q^{5} - q^{6} + \beta_{3} q^{7} + (3 \beta_1 + 1) q^{8} + ( - 2 \beta_1 - 6) q^{9} + \beta_1 q^{10} + \beta_{2} q^{11} + ( - 3 \beta_1 + 5) q^{12} + (\beta_{3} + \beta_{2}) q^{13} + ( - \beta_{3} - \beta_{2}) q^{14} + ( - 3 \beta_1 + 4) q^{15} + ( - 12 \beta_1 - 3) q^{16} - \beta_{2} q^{17} + (8 \beta_1 + 10) q^{18} + (11 \beta_1 + 3) q^{19} + ( - 5 \beta_1 + 6) q^{20} + ( - \beta_{3} + \beta_{2}) q^{21} + ( - 2 \beta_{3} - \beta_{2}) q^{22} + ( - \beta_{3} - 2 \beta_{2}) q^{23} + ( - 2 \beta_1 + 5) q^{24} + ( - 4 \beta_1 - 19) q^{25} + ( - 3 \beta_{3} - 2 \beta_{2}) q^{26} + ( - 13 \beta_1 + 11) q^{27} + ( - \beta_{3} + 2 \beta_{2}) q^{28} + 7 \beta_1 q^{29} + ( - \beta_1 + 2) q^{30} + 3 \beta_{3} q^{31} + (3 \beta_1 + 23) q^{32} + (2 \beta_{3} - \beta_{2}) q^{33} + (2 \beta_{3} + \beta_{2}) q^{34} + ( - 2 \beta_{3} + \beta_{2}) q^{35} + ( - 10 \beta_1 - 2) q^{36} + (37 \beta_1 - 6) q^{37} + ( - 14 \beta_1 - 25) q^{38} + \beta_{3} q^{39} + ( - 5 \beta_1 + 4) q^{40} + (2 \beta_{3} + 3 \beta_{2}) q^{41} - \beta_{3} q^{42} + ( - 19 \beta_1 - 15) q^{43} + (4 \beta_{3} - \beta_{2}) q^{44} + ( - 2 \beta_1 + 8) q^{45} + (5 \beta_{3} + 3 \beta_{2}) q^{46} - 5 \beta_{3} q^{47} + (9 \beta_1 - 21) q^{48} + (20 \beta_1 - 60) q^{49} + (23 \beta_1 + 27) q^{50} + ( - 2 \beta_{3} + \beta_{2}) q^{51} + (3 \beta_{3} + \beta_{2}) q^{52} + (4 \beta_{3} - 2 \beta_{2}) q^{53} + (2 \beta_1 + 15) q^{54} + (2 \beta_{3} - 2 \beta_{2}) q^{55} + (\beta_{3} + 3 \beta_{2}) q^{56} + ( - 8 \beta_1 + 19) q^{57} + ( - 7 \beta_1 - 14) q^{58} - 4 \beta_{3} q^{59} + (11 \beta_1 - 16) q^{60} + (6 \beta_{3} - 2 \beta_{2}) q^{61} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{62} + ( - 6 \beta_{3} - 2 \beta_{2}) q^{63} + (22 \beta_1 - 17) q^{64} - \beta_{2} q^{65} - \beta_{2} q^{66} + ( - 6 \beta_{3} - \beta_{2}) q^{67} + ( - 4 \beta_{3} + \beta_{2}) q^{68} + ( - 3 \beta_{3} + \beta_{2}) q^{69} + \beta_{2} q^{70} + (5 \beta_{3} + 2 \beta_{2} - 20 \beta_1 + 38) q^{71} + ( - 20 \beta_1 - 18) q^{72} + (58 \beta_1 + 17) q^{73} + ( - 31 \beta_1 - 68) q^{74} + ( - 15 \beta_1 + 11) q^{75} + ( - 5 \beta_1 + 41) q^{76} + ( - 109 \beta_1 + 40) q^{77} + ( - \beta_{3} - \beta_{2}) q^{78} + (33 \beta_1 + 48) q^{79} + (21 \beta_1 - 18) q^{80} + (42 \beta_1 + 17) q^{81} + ( - 8 \beta_{3} - 5 \beta_{2}) q^{82} + ( - 24 \beta_1 + 82) q^{83} + (5 \beta_{3} - 3 \beta_{2}) q^{84} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{85} + (34 \beta_1 + 53) q^{86} + ( - 7 \beta_1 + 14) q^{87} + (6 \beta_{3} + \beta_{2}) q^{88} + ( - 22 \beta_1 - 47) q^{89} + ( - 6 \beta_1 - 4) q^{90} + ( - 89 \beta_1 - 69) q^{91} - 7 \beta_{3} q^{92} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{93} + (5 \beta_{3} + 5 \beta_{2}) q^{94} + ( - 19 \beta_1 + 16) q^{95} + (20 \beta_1 - 17) q^{96} + 7 \beta_{2} q^{97} + (40 \beta_1 + 20) q^{98} + ( - 4 \beta_{3} - 6 \beta_{2}) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + (b1 - 1) * q^3 + (2*b1 - 1) * q^4 + (b1 - 2) * q^5 - q^6 + b3 * q^7 + (3*b1 + 1) * q^8 + (-2*b1 - 6) * q^9 + b1 * q^10 + b2 * q^11 + (-3*b1 + 5) * q^12 + (b3 + b2) * q^13 + (-b3 - b2) * q^14 + (-3*b1 + 4) * q^15 + (-12*b1 - 3) * q^16 - b2 * q^17 + (8*b1 + 10) * q^18 + (11*b1 + 3) * q^19 + (-5*b1 + 6) * q^20 + (-b3 + b2) * q^21 + (-2*b3 - b2) * q^22 + (-b3 - 2*b2) * q^23 + (-2*b1 + 5) * q^24 + (-4*b1 - 19) * q^25 + (-3*b3 - 2*b2) * q^26 + (-13*b1 + 11) * q^27 + (-b3 + 2*b2) * q^28 + 7*b1 * q^29 + (-b1 + 2) * q^30 + 3*b3 * q^31 + (3*b1 + 23) * q^32 + (2*b3 - b2) * q^33 + (2*b3 + b2) * q^34 + (-2*b3 + b2) * q^35 + (-10*b1 - 2) * q^36 + (37*b1 - 6) * q^37 + (-14*b1 - 25) * q^38 + b3 * q^39 + (-5*b1 + 4) * q^40 + (2*b3 + 3*b2) * q^41 - b3 * q^42 + (-19*b1 - 15) * q^43 + (4*b3 - b2) * q^44 + (-2*b1 + 8) * q^45 + (5*b3 + 3*b2) * q^46 - 5*b3 * q^47 + (9*b1 - 21) * q^48 + (20*b1 - 60) * q^49 + (23*b1 + 27) * q^50 + (-2*b3 + b2) * q^51 + (3*b3 + b2) * q^52 + (4*b3 - 2*b2) * q^53 + (2*b1 + 15) * q^54 + (2*b3 - 2*b2) * q^55 + (b3 + 3*b2) * q^56 + (-8*b1 + 19) * q^57 + (-7*b1 - 14) * q^58 - 4*b3 * q^59 + (11*b1 - 16) * q^60 + (6*b3 - 2*b2) * q^61 + (-3*b3 - 3*b2) * q^62 + (-6*b3 - 2*b2) * q^63 + (22*b1 - 17) * q^64 - b2 * q^65 - b2 * q^66 + (-6*b3 - b2) * q^67 + (-4*b3 + b2) * q^68 + (-3*b3 + b2) * q^69 + b2 * q^70 + (5*b3 + 2*b2 - 20*b1 + 38) * q^71 + (-20*b1 - 18) * q^72 + (58*b1 + 17) * q^73 + (-31*b1 - 68) * q^74 + (-15*b1 + 11) * q^75 + (-5*b1 + 41) * q^76 + (-109*b1 + 40) * q^77 + (-b3 - b2) * q^78 + (33*b1 + 48) * q^79 + (21*b1 - 18) * q^80 + (42*b1 + 17) * q^81 + (-8*b3 - 5*b2) * q^82 + (-24*b1 + 82) * q^83 + (5*b3 - 3*b2) * q^84 + (-2*b3 + 2*b2) * q^85 + (34*b1 + 53) * q^86 + (-7*b1 + 14) * q^87 + (6*b3 + b2) * q^88 + (-22*b1 - 47) * q^89 + (-6*b1 - 4) * q^90 + (-89*b1 - 69) * q^91 - 7*b3 * q^92 + (-3*b3 + 3*b2) * q^93 + (5*b3 + 5*b2) * q^94 + (-19*b1 + 16) * q^95 + (20*b1 - 17) * q^96 + 7*b2 * q^97 + (40*b1 + 20) * q^98 + (-4*b3 - 6*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 8 q^{5} - 4 q^{6} + 4 q^{8} - 24 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 - 4 * q^3 - 4 * q^4 - 8 * q^5 - 4 * q^6 + 4 * q^8 - 24 * q^9 $$4 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 8 q^{5} - 4 q^{6} + 4 q^{8} - 24 q^{9} + 20 q^{12} + 16 q^{15} - 12 q^{16} + 40 q^{18} + 12 q^{19} + 24 q^{20} + 20 q^{24} - 76 q^{25} + 44 q^{27} + 8 q^{30} + 92 q^{32} - 8 q^{36} - 24 q^{37} - 100 q^{38} + 16 q^{40} - 60 q^{43} + 32 q^{45} - 84 q^{48} - 240 q^{49} + 108 q^{50} + 60 q^{54} + 76 q^{57} - 56 q^{58} - 64 q^{60} - 68 q^{64} + 152 q^{71} - 72 q^{72} + 68 q^{73} - 272 q^{74} + 44 q^{75} + 164 q^{76} + 160 q^{77} + 192 q^{79} - 72 q^{80} + 68 q^{81} + 328 q^{83} + 212 q^{86} + 56 q^{87} - 188 q^{89} - 16 q^{90} - 276 q^{91} + 64 q^{95} - 68 q^{96} + 80 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 - 4 * q^3 - 4 * q^4 - 8 * q^5 - 4 * q^6 + 4 * q^8 - 24 * q^9 + 20 * q^12 + 16 * q^15 - 12 * q^16 + 40 * q^18 + 12 * q^19 + 24 * q^20 + 20 * q^24 - 76 * q^25 + 44 * q^27 + 8 * q^30 + 92 * q^32 - 8 * q^36 - 24 * q^37 - 100 * q^38 + 16 * q^40 - 60 * q^43 + 32 * q^45 - 84 * q^48 - 240 * q^49 + 108 * q^50 + 60 * q^54 + 76 * q^57 - 56 * q^58 - 64 * q^60 - 68 * q^64 + 152 * q^71 - 72 * q^72 + 68 * q^73 - 272 * q^74 + 44 * q^75 + 164 * q^76 + 160 * q^77 + 192 * q^79 - 72 * q^80 + 68 * q^81 + 328 * q^83 + 212 * q^86 + 56 * q^87 - 188 * q^89 - 16 * q^90 - 276 * q^91 + 64 * q^95 - 68 * q^96 + 80 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 108x^{2} - 40x + 2825$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 10\nu^{2} - 53\nu + 570 ) / 155$$ (-v^3 + 10*v^2 - 53*v + 570) / 155 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 10\nu^{2} + 363\nu - 570 ) / 155$$ (v^3 - 10*v^2 + 363*v - 570) / 155 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + 11\nu^{2} + 106\nu + 534 ) / 31$$ (2*v^3 + 11*v^2 + 106*v + 534) / 31
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 10\beta _1 - 54$$ b3 + 10*b1 - 54 $$\nu^{3}$$ $$=$$ $$( 20\beta_{3} - 53\beta_{2} - 163\beta _1 + 60 ) / 2$$ (20*b3 - 53*b2 - 163*b1 + 60) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/71\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
70.1
 0.707107 − 6.35278i 0.707107 + 6.35278i −0.707107 + 8.28505i −0.707107 − 8.28505i
−2.41421 0.414214 1.82843 −0.585786 −1.00000 8.98419i 5.24264 −8.82843 1.41421
70.2 −2.41421 0.414214 1.82843 −0.585786 −1.00000 8.98419i 5.24264 −8.82843 1.41421
70.3 0.414214 −2.41421 −3.82843 −3.41421 −1.00000 11.7168i −3.24264 −3.17157 −1.41421
70.4 0.414214 −2.41421 −3.82843 −3.41421 −1.00000 11.7168i −3.24264 −3.17157 −1.41421
 $$n$$: 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
71.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.3.b.a 4
3.b odd 2 1 639.3.d.a 4
4.b odd 2 1 1136.3.h.a 4
71.b odd 2 1 inner 71.3.b.a 4
213.b even 2 1 639.3.d.a 4
284.c even 2 1 1136.3.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.3.b.a 4 1.a even 1 1 trivial
71.3.b.a 4 71.b odd 2 1 inner
639.3.d.a 4 3.b odd 2 1
639.3.d.a 4 213.b even 2 1
1136.3.h.a 4 4.b odd 2 1
1136.3.h.a 4 284.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} - 1$$ acting on $$S_{3}^{\mathrm{new}}(71, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T - 1)^{2}$$
$3$ $$(T^{2} + 2 T - 1)^{2}$$
$5$ $$(T^{2} + 4 T + 2)^{2}$$
$7$ $$T^{4} + 218 T^{2} + 11081$$
$11$ $$T^{4} + 436 T^{2} + 44324$$
$13$ $$T^{4} + 494 T^{2} + 11081$$
$17$ $$T^{4} + 436 T^{2} + 44324$$
$19$ $$(T^{2} - 6 T - 233)^{2}$$
$23$ $$T^{4} + 1642 T^{2} + 542969$$
$29$ $$(T^{2} - 98)^{2}$$
$31$ $$T^{4} + 1962 T^{2} + 897561$$
$37$ $$(T^{2} + 12 T - 2702)^{2}$$
$41$ $$T^{4} + 3836 T^{2} + \cdots + 2171876$$
$43$ $$(T^{2} + 30 T - 497)^{2}$$
$47$ $$T^{4} + 5450 T^{2} + \cdots + 6925625$$
$53$ $$T^{4} + 6512 T^{2} + 709184$$
$59$ $$T^{4} + 3488 T^{2} + \cdots + 2836736$$
$61$ $$T^{4} + 11512 T^{2} + \cdots + 8687504$$
$67$ $$T^{4} + 7324 T^{2} + \cdots + 12809636$$
$71$ $$T^{4} - 152 T^{3} + \cdots + 25411681$$
$73$ $$(T^{2} - 34 T - 6439)^{2}$$
$79$ $$(T^{2} - 96 T + 126)^{2}$$
$83$ $$(T^{2} - 164 T + 5572)^{2}$$
$89$ $$(T^{2} + 94 T + 1241)^{2}$$
$97$ $$T^{4} + 21364 T^{2} + \cdots + 106421924$$