# Properties

 Label 35.4.e.a Level $35$ Weight $4$ Character orbit 35.e Analytic conductor $2.065$ 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] = [35,4,Mod(11,35)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, 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("35.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 35.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: $$2.06506685020$$ 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.

 $$f(q)$$ $$=$$ $$q - 3 \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + (\zeta_{6} - 1) q^{4} - 5 \zeta_{6} q^{5} - 6 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} - 21 q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})$$ q - 3*z * q^2 + (-2*z + 2) * q^3 + (z - 1) * q^4 - 5*z * q^5 - 6 * q^6 + (-14*z - 7) * q^7 - 21 * q^8 + 23*z * q^9 $$q - 3 \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + (\zeta_{6} - 1) q^{4} - 5 \zeta_{6} q^{5} - 6 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} - 21 q^{8} + 23 \zeta_{6} q^{9} + (15 \zeta_{6} - 15) q^{10} + ( - 45 \zeta_{6} + 45) q^{11} + 2 \zeta_{6} q^{12} + 59 q^{13} + (63 \zeta_{6} - 42) q^{14} - 10 q^{15} + 71 \zeta_{6} q^{16} + ( - 54 \zeta_{6} + 54) q^{17} + ( - 69 \zeta_{6} + 69) q^{18} + 121 \zeta_{6} q^{19} + 5 q^{20} + (14 \zeta_{6} - 42) q^{21} - 135 q^{22} - 69 \zeta_{6} q^{23} + (42 \zeta_{6} - 42) q^{24} + (25 \zeta_{6} - 25) q^{25} - 177 \zeta_{6} q^{26} + 100 q^{27} + ( - 7 \zeta_{6} + 21) q^{28} - 162 q^{29} + 30 \zeta_{6} q^{30} + ( - 88 \zeta_{6} + 88) q^{31} + ( - 45 \zeta_{6} + 45) q^{32} - 90 \zeta_{6} q^{33} - 162 q^{34} + (105 \zeta_{6} - 70) q^{35} - 23 q^{36} + 259 \zeta_{6} q^{37} + ( - 363 \zeta_{6} + 363) q^{38} + ( - 118 \zeta_{6} + 118) q^{39} + 105 \zeta_{6} q^{40} + 195 q^{41} + (84 \zeta_{6} + 42) q^{42} - 286 q^{43} + 45 \zeta_{6} q^{44} + ( - 115 \zeta_{6} + 115) q^{45} + (207 \zeta_{6} - 207) q^{46} - 45 \zeta_{6} q^{47} + 142 q^{48} + (392 \zeta_{6} - 147) q^{49} + 75 q^{50} - 108 \zeta_{6} q^{51} + (59 \zeta_{6} - 59) q^{52} + (597 \zeta_{6} - 597) q^{53} - 300 \zeta_{6} q^{54} - 225 q^{55} + (294 \zeta_{6} + 147) q^{56} + 242 q^{57} + 486 \zeta_{6} q^{58} + ( - 360 \zeta_{6} + 360) q^{59} + ( - 10 \zeta_{6} + 10) q^{60} - 392 \zeta_{6} q^{61} - 264 q^{62} + ( - 483 \zeta_{6} + 322) q^{63} + 433 q^{64} - 295 \zeta_{6} q^{65} + (270 \zeta_{6} - 270) q^{66} + ( - 280 \zeta_{6} + 280) q^{67} + 54 \zeta_{6} q^{68} - 138 q^{69} + ( - 105 \zeta_{6} + 315) q^{70} + 48 q^{71} - 483 \zeta_{6} q^{72} + (668 \zeta_{6} - 668) q^{73} + ( - 777 \zeta_{6} + 777) q^{74} + 50 \zeta_{6} q^{75} - 121 q^{76} + (315 \zeta_{6} - 945) q^{77} - 354 q^{78} - 782 \zeta_{6} q^{79} + ( - 355 \zeta_{6} + 355) q^{80} + (421 \zeta_{6} - 421) q^{81} - 585 \zeta_{6} q^{82} + 768 q^{83} + ( - 42 \zeta_{6} + 28) q^{84} - 270 q^{85} + 858 \zeta_{6} q^{86} + (324 \zeta_{6} - 324) q^{87} + (945 \zeta_{6} - 945) q^{88} + 1194 \zeta_{6} q^{89} - 345 q^{90} + ( - 826 \zeta_{6} - 413) q^{91} + 69 q^{92} - 176 \zeta_{6} q^{93} + (135 \zeta_{6} - 135) q^{94} + ( - 605 \zeta_{6} + 605) q^{95} - 90 \zeta_{6} q^{96} + 902 q^{97} + ( - 735 \zeta_{6} + 1176) q^{98} + 1035 q^{99} +O(q^{100})$$ q - 3*z * q^2 + (-2*z + 2) * q^3 + (z - 1) * q^4 - 5*z * q^5 - 6 * q^6 + (-14*z - 7) * q^7 - 21 * q^8 + 23*z * q^9 + (15*z - 15) * q^10 + (-45*z + 45) * q^11 + 2*z * q^12 + 59 * q^13 + (63*z - 42) * q^14 - 10 * q^15 + 71*z * q^16 + (-54*z + 54) * q^17 + (-69*z + 69) * q^18 + 121*z * q^19 + 5 * q^20 + (14*z - 42) * q^21 - 135 * q^22 - 69*z * q^23 + (42*z - 42) * q^24 + (25*z - 25) * q^25 - 177*z * q^26 + 100 * q^27 + (-7*z + 21) * q^28 - 162 * q^29 + 30*z * q^30 + (-88*z + 88) * q^31 + (-45*z + 45) * q^32 - 90*z * q^33 - 162 * q^34 + (105*z - 70) * q^35 - 23 * q^36 + 259*z * q^37 + (-363*z + 363) * q^38 + (-118*z + 118) * q^39 + 105*z * q^40 + 195 * q^41 + (84*z + 42) * q^42 - 286 * q^43 + 45*z * q^44 + (-115*z + 115) * q^45 + (207*z - 207) * q^46 - 45*z * q^47 + 142 * q^48 + (392*z - 147) * q^49 + 75 * q^50 - 108*z * q^51 + (59*z - 59) * q^52 + (597*z - 597) * q^53 - 300*z * q^54 - 225 * q^55 + (294*z + 147) * q^56 + 242 * q^57 + 486*z * q^58 + (-360*z + 360) * q^59 + (-10*z + 10) * q^60 - 392*z * q^61 - 264 * q^62 + (-483*z + 322) * q^63 + 433 * q^64 - 295*z * q^65 + (270*z - 270) * q^66 + (-280*z + 280) * q^67 + 54*z * q^68 - 138 * q^69 + (-105*z + 315) * q^70 + 48 * q^71 - 483*z * q^72 + (668*z - 668) * q^73 + (-777*z + 777) * q^74 + 50*z * q^75 - 121 * q^76 + (315*z - 945) * q^77 - 354 * q^78 - 782*z * q^79 + (-355*z + 355) * q^80 + (421*z - 421) * q^81 - 585*z * q^82 + 768 * q^83 + (-42*z + 28) * q^84 - 270 * q^85 + 858*z * q^86 + (324*z - 324) * q^87 + (945*z - 945) * q^88 + 1194*z * q^89 - 345 * q^90 + (-826*z - 413) * q^91 + 69 * q^92 - 176*z * q^93 + (135*z - 135) * q^94 + (-605*z + 605) * q^95 - 90*z * q^96 + 902 * q^97 + (-735*z + 1176) * q^98 + 1035 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 2 q^{3} - q^{4} - 5 q^{5} - 12 q^{6} - 28 q^{7} - 42 q^{8} + 23 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 2 * q^3 - q^4 - 5 * q^5 - 12 * q^6 - 28 * q^7 - 42 * q^8 + 23 * q^9 $$2 q - 3 q^{2} + 2 q^{3} - q^{4} - 5 q^{5} - 12 q^{6} - 28 q^{7} - 42 q^{8} + 23 q^{9} - 15 q^{10} + 45 q^{11} + 2 q^{12} + 118 q^{13} - 21 q^{14} - 20 q^{15} + 71 q^{16} + 54 q^{17} + 69 q^{18} + 121 q^{19} + 10 q^{20} - 70 q^{21} - 270 q^{22} - 69 q^{23} - 42 q^{24} - 25 q^{25} - 177 q^{26} + 200 q^{27} + 35 q^{28} - 324 q^{29} + 30 q^{30} + 88 q^{31} + 45 q^{32} - 90 q^{33} - 324 q^{34} - 35 q^{35} - 46 q^{36} + 259 q^{37} + 363 q^{38} + 118 q^{39} + 105 q^{40} + 390 q^{41} + 168 q^{42} - 572 q^{43} + 45 q^{44} + 115 q^{45} - 207 q^{46} - 45 q^{47} + 284 q^{48} + 98 q^{49} + 150 q^{50} - 108 q^{51} - 59 q^{52} - 597 q^{53} - 300 q^{54} - 450 q^{55} + 588 q^{56} + 484 q^{57} + 486 q^{58} + 360 q^{59} + 10 q^{60} - 392 q^{61} - 528 q^{62} + 161 q^{63} + 866 q^{64} - 295 q^{65} - 270 q^{66} + 280 q^{67} + 54 q^{68} - 276 q^{69} + 525 q^{70} + 96 q^{71} - 483 q^{72} - 668 q^{73} + 777 q^{74} + 50 q^{75} - 242 q^{76} - 1575 q^{77} - 708 q^{78} - 782 q^{79} + 355 q^{80} - 421 q^{81} - 585 q^{82} + 1536 q^{83} + 14 q^{84} - 540 q^{85} + 858 q^{86} - 324 q^{87} - 945 q^{88} + 1194 q^{89} - 690 q^{90} - 1652 q^{91} + 138 q^{92} - 176 q^{93} - 135 q^{94} + 605 q^{95} - 90 q^{96} + 1804 q^{97} + 1617 q^{98} + 2070 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 2 * q^3 - q^4 - 5 * q^5 - 12 * q^6 - 28 * q^7 - 42 * q^8 + 23 * q^9 - 15 * q^10 + 45 * q^11 + 2 * q^12 + 118 * q^13 - 21 * q^14 - 20 * q^15 + 71 * q^16 + 54 * q^17 + 69 * q^18 + 121 * q^19 + 10 * q^20 - 70 * q^21 - 270 * q^22 - 69 * q^23 - 42 * q^24 - 25 * q^25 - 177 * q^26 + 200 * q^27 + 35 * q^28 - 324 * q^29 + 30 * q^30 + 88 * q^31 + 45 * q^32 - 90 * q^33 - 324 * q^34 - 35 * q^35 - 46 * q^36 + 259 * q^37 + 363 * q^38 + 118 * q^39 + 105 * q^40 + 390 * q^41 + 168 * q^42 - 572 * q^43 + 45 * q^44 + 115 * q^45 - 207 * q^46 - 45 * q^47 + 284 * q^48 + 98 * q^49 + 150 * q^50 - 108 * q^51 - 59 * q^52 - 597 * q^53 - 300 * q^54 - 450 * q^55 + 588 * q^56 + 484 * q^57 + 486 * q^58 + 360 * q^59 + 10 * q^60 - 392 * q^61 - 528 * q^62 + 161 * q^63 + 866 * q^64 - 295 * q^65 - 270 * q^66 + 280 * q^67 + 54 * q^68 - 276 * q^69 + 525 * q^70 + 96 * q^71 - 483 * q^72 - 668 * q^73 + 777 * q^74 + 50 * q^75 - 242 * q^76 - 1575 * q^77 - 708 * q^78 - 782 * q^79 + 355 * q^80 - 421 * q^81 - 585 * q^82 + 1536 * q^83 + 14 * q^84 - 540 * q^85 + 858 * q^86 - 324 * q^87 - 945 * q^88 + 1194 * q^89 - 690 * q^90 - 1652 * q^91 + 138 * q^92 - 176 * q^93 - 135 * q^94 + 605 * q^95 - 90 * q^96 + 1804 * q^97 + 1617 * q^98 + 2070 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 2.59808i 1.00000 1.73205i −0.500000 + 0.866025i −2.50000 4.33013i −6.00000 −14.0000 12.1244i −21.0000 11.5000 + 19.9186i −7.50000 + 12.9904i
16.1 −1.50000 + 2.59808i 1.00000 + 1.73205i −0.500000 0.866025i −2.50000 + 4.33013i −6.00000 −14.0000 + 12.1244i −21.0000 11.5000 19.9186i −7.50000 12.9904i
 $$n$$: 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 35.4.e.a 2
3.b odd 2 1 315.4.j.b 2
4.b odd 2 1 560.4.q.b 2
5.b even 2 1 175.4.e.b 2
5.c odd 4 2 175.4.k.b 4
7.b odd 2 1 245.4.e.a 2
7.c even 3 1 inner 35.4.e.a 2
7.c even 3 1 245.4.a.e 1
7.d odd 6 1 245.4.a.f 1
7.d odd 6 1 245.4.e.a 2
21.g even 6 1 2205.4.a.g 1
21.h odd 6 1 315.4.j.b 2
21.h odd 6 1 2205.4.a.e 1
28.g odd 6 1 560.4.q.b 2
35.i odd 6 1 1225.4.a.a 1
35.j even 6 1 175.4.e.b 2
35.j even 6 1 1225.4.a.b 1
35.l odd 12 2 175.4.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 1.a even 1 1 trivial
35.4.e.a 2 7.c even 3 1 inner
175.4.e.b 2 5.b even 2 1
175.4.e.b 2 35.j even 6 1
175.4.k.b 4 5.c odd 4 2
175.4.k.b 4 35.l odd 12 2
245.4.a.e 1 7.c even 3 1
245.4.a.f 1 7.d odd 6 1
245.4.e.a 2 7.b odd 2 1
245.4.e.a 2 7.d odd 6 1
315.4.j.b 2 3.b odd 2 1
315.4.j.b 2 21.h odd 6 1
560.4.q.b 2 4.b odd 2 1
560.4.q.b 2 28.g odd 6 1
1225.4.a.a 1 35.i odd 6 1
1225.4.a.b 1 35.j even 6 1
2205.4.a.e 1 21.h odd 6 1
2205.4.a.g 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 9$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} + 28T + 343$$
$11$ $$T^{2} - 45T + 2025$$
$13$ $$(T - 59)^{2}$$
$17$ $$T^{2} - 54T + 2916$$
$19$ $$T^{2} - 121T + 14641$$
$23$ $$T^{2} + 69T + 4761$$
$29$ $$(T + 162)^{2}$$
$31$ $$T^{2} - 88T + 7744$$
$37$ $$T^{2} - 259T + 67081$$
$41$ $$(T - 195)^{2}$$
$43$ $$(T + 286)^{2}$$
$47$ $$T^{2} + 45T + 2025$$
$53$ $$T^{2} + 597T + 356409$$
$59$ $$T^{2} - 360T + 129600$$
$61$ $$T^{2} + 392T + 153664$$
$67$ $$T^{2} - 280T + 78400$$
$71$ $$(T - 48)^{2}$$
$73$ $$T^{2} + 668T + 446224$$
$79$ $$T^{2} + 782T + 611524$$
$83$ $$(T - 768)^{2}$$
$89$ $$T^{2} - 1194 T + 1425636$$
$97$ $$(T - 902)^{2}$$