# Properties

 Label 42.4.e.b Level $42$ Weight $4$ Character orbit 42.e Analytic conductor $2.478$ 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] = [42,4,Mod(25,42)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, 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("42.25");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 42.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: $$2.47808022024$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + 3 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 15 \zeta_{6} + 15) q^{5} + 6 q^{6} + (7 \zeta_{6} + 14) q^{7} - 8 q^{8} + (9 \zeta_{6} - 9) q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^2 + 3*z * q^3 - 4*z * q^4 + (-15*z + 15) * q^5 + 6 * q^6 + (7*z + 14) * q^7 - 8 * q^8 + (9*z - 9) * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{2} + 3 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 15 \zeta_{6} + 15) q^{5} + 6 q^{6} + (7 \zeta_{6} + 14) q^{7} - 8 q^{8} + (9 \zeta_{6} - 9) q^{9} - 30 \zeta_{6} q^{10} + 9 \zeta_{6} q^{11} + ( - 12 \zeta_{6} + 12) q^{12} - 88 q^{13} + ( - 28 \zeta_{6} + 42) q^{14} + 45 q^{15} + (16 \zeta_{6} - 16) q^{16} + 84 \zeta_{6} q^{17} + 18 \zeta_{6} q^{18} + (104 \zeta_{6} - 104) q^{19} - 60 q^{20} + (63 \zeta_{6} - 21) q^{21} + 18 q^{22} + ( - 84 \zeta_{6} + 84) q^{23} - 24 \zeta_{6} q^{24} - 100 \zeta_{6} q^{25} + (176 \zeta_{6} - 176) q^{26} - 27 q^{27} + ( - 84 \zeta_{6} + 28) q^{28} + 51 q^{29} + ( - 90 \zeta_{6} + 90) q^{30} - 185 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + (27 \zeta_{6} - 27) q^{33} + 168 q^{34} + ( - 210 \zeta_{6} + 315) q^{35} + 36 q^{36} + (44 \zeta_{6} - 44) q^{37} + 208 \zeta_{6} q^{38} - 264 \zeta_{6} q^{39} + (120 \zeta_{6} - 120) q^{40} - 168 q^{41} + (42 \zeta_{6} + 84) q^{42} + 326 q^{43} + ( - 36 \zeta_{6} + 36) q^{44} + 135 \zeta_{6} q^{45} - 168 \zeta_{6} q^{46} + ( - 138 \zeta_{6} + 138) q^{47} - 48 q^{48} + (245 \zeta_{6} + 147) q^{49} - 200 q^{50} + (252 \zeta_{6} - 252) q^{51} + 352 \zeta_{6} q^{52} - 639 \zeta_{6} q^{53} + (54 \zeta_{6} - 54) q^{54} + 135 q^{55} + ( - 56 \zeta_{6} - 112) q^{56} - 312 q^{57} + ( - 102 \zeta_{6} + 102) q^{58} - 159 \zeta_{6} q^{59} - 180 \zeta_{6} q^{60} + (722 \zeta_{6} - 722) q^{61} - 370 q^{62} + (126 \zeta_{6} - 189) q^{63} + 64 q^{64} + (1320 \zeta_{6} - 1320) q^{65} + 54 \zeta_{6} q^{66} + 166 \zeta_{6} q^{67} + ( - 336 \zeta_{6} + 336) q^{68} + 252 q^{69} + ( - 630 \zeta_{6} + 210) q^{70} + 1086 q^{71} + ( - 72 \zeta_{6} + 72) q^{72} - 218 \zeta_{6} q^{73} + 88 \zeta_{6} q^{74} + ( - 300 \zeta_{6} + 300) q^{75} + 416 q^{76} + (189 \zeta_{6} - 63) q^{77} - 528 q^{78} + ( - 583 \zeta_{6} + 583) q^{79} + 240 \zeta_{6} q^{80} - 81 \zeta_{6} q^{81} + (336 \zeta_{6} - 336) q^{82} - 597 q^{83} + ( - 168 \zeta_{6} + 252) q^{84} + 1260 q^{85} + ( - 652 \zeta_{6} + 652) q^{86} + 153 \zeta_{6} q^{87} - 72 \zeta_{6} q^{88} + ( - 1038 \zeta_{6} + 1038) q^{89} + 270 q^{90} + ( - 616 \zeta_{6} - 1232) q^{91} - 336 q^{92} + ( - 555 \zeta_{6} + 555) q^{93} - 276 \zeta_{6} q^{94} + 1560 \zeta_{6} q^{95} + (96 \zeta_{6} - 96) q^{96} - 169 q^{97} + ( - 294 \zeta_{6} + 784) q^{98} - 81 q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^2 + 3*z * q^3 - 4*z * q^4 + (-15*z + 15) * q^5 + 6 * q^6 + (7*z + 14) * q^7 - 8 * q^8 + (9*z - 9) * q^9 - 30*z * q^10 + 9*z * q^11 + (-12*z + 12) * q^12 - 88 * q^13 + (-28*z + 42) * q^14 + 45 * q^15 + (16*z - 16) * q^16 + 84*z * q^17 + 18*z * q^18 + (104*z - 104) * q^19 - 60 * q^20 + (63*z - 21) * q^21 + 18 * q^22 + (-84*z + 84) * q^23 - 24*z * q^24 - 100*z * q^25 + (176*z - 176) * q^26 - 27 * q^27 + (-84*z + 28) * q^28 + 51 * q^29 + (-90*z + 90) * q^30 - 185*z * q^31 + 32*z * q^32 + (27*z - 27) * q^33 + 168 * q^34 + (-210*z + 315) * q^35 + 36 * q^36 + (44*z - 44) * q^37 + 208*z * q^38 - 264*z * q^39 + (120*z - 120) * q^40 - 168 * q^41 + (42*z + 84) * q^42 + 326 * q^43 + (-36*z + 36) * q^44 + 135*z * q^45 - 168*z * q^46 + (-138*z + 138) * q^47 - 48 * q^48 + (245*z + 147) * q^49 - 200 * q^50 + (252*z - 252) * q^51 + 352*z * q^52 - 639*z * q^53 + (54*z - 54) * q^54 + 135 * q^55 + (-56*z - 112) * q^56 - 312 * q^57 + (-102*z + 102) * q^58 - 159*z * q^59 - 180*z * q^60 + (722*z - 722) * q^61 - 370 * q^62 + (126*z - 189) * q^63 + 64 * q^64 + (1320*z - 1320) * q^65 + 54*z * q^66 + 166*z * q^67 + (-336*z + 336) * q^68 + 252 * q^69 + (-630*z + 210) * q^70 + 1086 * q^71 + (-72*z + 72) * q^72 - 218*z * q^73 + 88*z * q^74 + (-300*z + 300) * q^75 + 416 * q^76 + (189*z - 63) * q^77 - 528 * q^78 + (-583*z + 583) * q^79 + 240*z * q^80 - 81*z * q^81 + (336*z - 336) * q^82 - 597 * q^83 + (-168*z + 252) * q^84 + 1260 * q^85 + (-652*z + 652) * q^86 + 153*z * q^87 - 72*z * q^88 + (-1038*z + 1038) * q^89 + 270 * q^90 + (-616*z - 1232) * q^91 - 336 * q^92 + (-555*z + 555) * q^93 - 276*z * q^94 + 1560*z * q^95 + (96*z - 96) * q^96 - 169 * q^97 + (-294*z + 784) * q^98 - 81 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 3 q^{3} - 4 q^{4} + 15 q^{5} + 12 q^{6} + 35 q^{7} - 16 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 3 * q^3 - 4 * q^4 + 15 * q^5 + 12 * q^6 + 35 * q^7 - 16 * q^8 - 9 * q^9 $$2 q + 2 q^{2} + 3 q^{3} - 4 q^{4} + 15 q^{5} + 12 q^{6} + 35 q^{7} - 16 q^{8} - 9 q^{9} - 30 q^{10} + 9 q^{11} + 12 q^{12} - 176 q^{13} + 56 q^{14} + 90 q^{15} - 16 q^{16} + 84 q^{17} + 18 q^{18} - 104 q^{19} - 120 q^{20} + 21 q^{21} + 36 q^{22} + 84 q^{23} - 24 q^{24} - 100 q^{25} - 176 q^{26} - 54 q^{27} - 28 q^{28} + 102 q^{29} + 90 q^{30} - 185 q^{31} + 32 q^{32} - 27 q^{33} + 336 q^{34} + 420 q^{35} + 72 q^{36} - 44 q^{37} + 208 q^{38} - 264 q^{39} - 120 q^{40} - 336 q^{41} + 210 q^{42} + 652 q^{43} + 36 q^{44} + 135 q^{45} - 168 q^{46} + 138 q^{47} - 96 q^{48} + 539 q^{49} - 400 q^{50} - 252 q^{51} + 352 q^{52} - 639 q^{53} - 54 q^{54} + 270 q^{55} - 280 q^{56} - 624 q^{57} + 102 q^{58} - 159 q^{59} - 180 q^{60} - 722 q^{61} - 740 q^{62} - 252 q^{63} + 128 q^{64} - 1320 q^{65} + 54 q^{66} + 166 q^{67} + 336 q^{68} + 504 q^{69} - 210 q^{70} + 2172 q^{71} + 72 q^{72} - 218 q^{73} + 88 q^{74} + 300 q^{75} + 832 q^{76} + 63 q^{77} - 1056 q^{78} + 583 q^{79} + 240 q^{80} - 81 q^{81} - 336 q^{82} - 1194 q^{83} + 336 q^{84} + 2520 q^{85} + 652 q^{86} + 153 q^{87} - 72 q^{88} + 1038 q^{89} + 540 q^{90} - 3080 q^{91} - 672 q^{92} + 555 q^{93} - 276 q^{94} + 1560 q^{95} - 96 q^{96} - 338 q^{97} + 1274 q^{98} - 162 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 3 * q^3 - 4 * q^4 + 15 * q^5 + 12 * q^6 + 35 * q^7 - 16 * q^8 - 9 * q^9 - 30 * q^10 + 9 * q^11 + 12 * q^12 - 176 * q^13 + 56 * q^14 + 90 * q^15 - 16 * q^16 + 84 * q^17 + 18 * q^18 - 104 * q^19 - 120 * q^20 + 21 * q^21 + 36 * q^22 + 84 * q^23 - 24 * q^24 - 100 * q^25 - 176 * q^26 - 54 * q^27 - 28 * q^28 + 102 * q^29 + 90 * q^30 - 185 * q^31 + 32 * q^32 - 27 * q^33 + 336 * q^34 + 420 * q^35 + 72 * q^36 - 44 * q^37 + 208 * q^38 - 264 * q^39 - 120 * q^40 - 336 * q^41 + 210 * q^42 + 652 * q^43 + 36 * q^44 + 135 * q^45 - 168 * q^46 + 138 * q^47 - 96 * q^48 + 539 * q^49 - 400 * q^50 - 252 * q^51 + 352 * q^52 - 639 * q^53 - 54 * q^54 + 270 * q^55 - 280 * q^56 - 624 * q^57 + 102 * q^58 - 159 * q^59 - 180 * q^60 - 722 * q^61 - 740 * q^62 - 252 * q^63 + 128 * q^64 - 1320 * q^65 + 54 * q^66 + 166 * q^67 + 336 * q^68 + 504 * q^69 - 210 * q^70 + 2172 * q^71 + 72 * q^72 - 218 * q^73 + 88 * q^74 + 300 * q^75 + 832 * q^76 + 63 * q^77 - 1056 * q^78 + 583 * q^79 + 240 * q^80 - 81 * q^81 - 336 * q^82 - 1194 * q^83 + 336 * q^84 + 2520 * q^85 + 652 * q^86 + 153 * q^87 - 72 * q^88 + 1038 * q^89 + 540 * q^90 - 3080 * q^91 - 672 * q^92 + 555 * q^93 - 276 * q^94 + 1560 * q^95 - 96 * q^96 - 338 * q^97 + 1274 * q^98 - 162 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$31$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 + 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 7.50000 + 12.9904i 6.00000 17.5000 6.06218i −8.00000 −4.50000 7.79423i −15.0000 + 25.9808i
37.1 1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 7.50000 12.9904i 6.00000 17.5000 + 6.06218i −8.00000 −4.50000 + 7.79423i −15.0000 25.9808i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.4.e.b 2
3.b odd 2 1 126.4.g.a 2
4.b odd 2 1 336.4.q.d 2
7.b odd 2 1 294.4.e.e 2
7.c even 3 1 inner 42.4.e.b 2
7.c even 3 1 294.4.a.a 1
7.d odd 6 1 294.4.a.g 1
7.d odd 6 1 294.4.e.e 2
21.c even 2 1 882.4.g.l 2
21.g even 6 1 882.4.a.h 1
21.g even 6 1 882.4.g.l 2
21.h odd 6 1 126.4.g.a 2
21.h odd 6 1 882.4.a.r 1
28.f even 6 1 2352.4.a.q 1
28.g odd 6 1 336.4.q.d 2
28.g odd 6 1 2352.4.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 1.a even 1 1 trivial
42.4.e.b 2 7.c even 3 1 inner
126.4.g.a 2 3.b odd 2 1
126.4.g.a 2 21.h odd 6 1
294.4.a.a 1 7.c even 3 1
294.4.a.g 1 7.d odd 6 1
294.4.e.e 2 7.b odd 2 1
294.4.e.e 2 7.d odd 6 1
336.4.q.d 2 4.b odd 2 1
336.4.q.d 2 28.g odd 6 1
882.4.a.h 1 21.g even 6 1
882.4.a.r 1 21.h odd 6 1
882.4.g.l 2 21.c even 2 1
882.4.g.l 2 21.g even 6 1
2352.4.a.q 1 28.f even 6 1
2352.4.a.u 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 15T_{5} + 225$$ acting on $$S_{4}^{\mathrm{new}}(42, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$T^{2} - 15T + 225$$
$7$ $$T^{2} - 35T + 343$$
$11$ $$T^{2} - 9T + 81$$
$13$ $$(T + 88)^{2}$$
$17$ $$T^{2} - 84T + 7056$$
$19$ $$T^{2} + 104T + 10816$$
$23$ $$T^{2} - 84T + 7056$$
$29$ $$(T - 51)^{2}$$
$31$ $$T^{2} + 185T + 34225$$
$37$ $$T^{2} + 44T + 1936$$
$41$ $$(T + 168)^{2}$$
$43$ $$(T - 326)^{2}$$
$47$ $$T^{2} - 138T + 19044$$
$53$ $$T^{2} + 639T + 408321$$
$59$ $$T^{2} + 159T + 25281$$
$61$ $$T^{2} + 722T + 521284$$
$67$ $$T^{2} - 166T + 27556$$
$71$ $$(T - 1086)^{2}$$
$73$ $$T^{2} + 218T + 47524$$
$79$ $$T^{2} - 583T + 339889$$
$83$ $$(T + 597)^{2}$$
$89$ $$T^{2} - 1038 T + 1077444$$
$97$ $$(T + 169)^{2}$$