# Properties

 Label 74.2.b.a Level $74$ Weight $2$ Character orbit 74.b Analytic conductor $0.591$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,2,Mod(73,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("74.73");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.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: $$0.590892974957$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 5$$ (v^3 + 6*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$ v^2 + 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ b3 - 6 $$\nu^{3}$$ $$=$$ $$5\beta_{2} - 6\beta_1$$ 5*b2 - 6*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 2.79129i − 1.79129i − 2.79129i 1.79129i
1.00000i −2.79129 −1.00000 3.79129i 2.79129i −2.00000 1.00000i 4.79129 −3.79129
73.2 1.00000i 1.79129 −1.00000 0.791288i 1.79129i −2.00000 1.00000i 0.208712 0.791288
73.3 1.00000i −2.79129 −1.00000 3.79129i 2.79129i −2.00000 1.00000i 4.79129 −3.79129
73.4 1.00000i 1.79129 −1.00000 0.791288i 1.79129i −2.00000 1.00000i 0.208712 0.791288
 $$n$$: 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
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.b.a 4
3.b odd 2 1 666.2.c.b 4
4.b odd 2 1 592.2.g.c 4
5.b even 2 1 1850.2.d.e 4
5.c odd 4 1 1850.2.c.g 4
5.c odd 4 1 1850.2.c.h 4
8.b even 2 1 2368.2.g.j 4
8.d odd 2 1 2368.2.g.h 4
12.b even 2 1 5328.2.h.m 4
37.b even 2 1 inner 74.2.b.a 4
37.d odd 4 1 2738.2.a.h 2
37.d odd 4 1 2738.2.a.k 2
111.d odd 2 1 666.2.c.b 4
148.b odd 2 1 592.2.g.c 4
185.d even 2 1 1850.2.d.e 4
185.h odd 4 1 1850.2.c.g 4
185.h odd 4 1 1850.2.c.h 4
296.e even 2 1 2368.2.g.j 4
296.h odd 2 1 2368.2.g.h 4
444.g even 2 1 5328.2.h.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 1.a even 1 1 trivial
74.2.b.a 4 37.b even 2 1 inner
592.2.g.c 4 4.b odd 2 1
592.2.g.c 4 148.b odd 2 1
666.2.c.b 4 3.b odd 2 1
666.2.c.b 4 111.d odd 2 1
1850.2.c.g 4 5.c odd 4 1
1850.2.c.g 4 185.h odd 4 1
1850.2.c.h 4 5.c odd 4 1
1850.2.c.h 4 185.h odd 4 1
1850.2.d.e 4 5.b even 2 1
1850.2.d.e 4 185.d even 2 1
2368.2.g.h 4 8.d odd 2 1
2368.2.g.h 4 296.h odd 2 1
2368.2.g.j 4 8.b even 2 1
2368.2.g.j 4 296.e even 2 1
2738.2.a.h 2 37.d odd 4 1
2738.2.a.k 2 37.d odd 4 1
5328.2.h.m 4 12.b even 2 1
5328.2.h.m 4 444.g even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + T - 5)^{2}$$
$5$ $$T^{4} + 15T^{2} + 9$$
$7$ $$(T + 2)^{4}$$
$11$ $$(T^{2} - 3 T - 3)^{2}$$
$13$ $$T^{4} + 15T^{2} + 9$$
$17$ $$T^{4} + 60T^{2} + 144$$
$19$ $$T^{4} + 60T^{2} + 144$$
$23$ $$T^{4} + 15T^{2} + 9$$
$29$ $$T^{4} + 15T^{2} + 9$$
$31$ $$T^{4} + 99T^{2} + 2025$$
$37$ $$(T^{2} - 8 T + 37)^{2}$$
$41$ $$(T^{2} - 15 T + 51)^{2}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} + 6 T - 12)^{2}$$
$53$ $$(T^{2} - 6 T - 12)^{2}$$
$59$ $$T^{4} + 60T^{2} + 144$$
$61$ $$T^{4} + 231 T^{2} + 11025$$
$67$ $$(T^{2} + T - 47)^{2}$$
$71$ $$(T^{2} - 84)^{2}$$
$73$ $$(T^{2} - 5 T - 41)^{2}$$
$79$ $$T^{4} + 231 T^{2} + 11025$$
$83$ $$(T^{2} - 12 T - 48)^{2}$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$T^{4} + 204T^{2} + 3600$$