# Properties

 Label 35.4.e.b Level $35$ Weight $4$ Character orbit 35.e Analytic conductor $2.065$ 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] = [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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{2} + (\beta_{2} + 3 \beta_1 + 1) q^{3} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{4} + 5 \beta_{2} q^{5} + ( - 8 \beta_{3} - 3) q^{6} + (2 \beta_{3} - 5 \beta_{2} - 11 \beta_1 + 3) q^{7} + ( - 13 \beta_{3} + 3) q^{8} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{9}+O(q^{10})$$ q + (b3 - 3*b2 + b1) * q^2 + (b2 + 3*b1 + 1) * q^3 + (-3*b2 + 6*b1 - 3) * q^4 + 5*b2 * q^5 + (-8*b3 - 3) * q^6 + (2*b3 - 5*b2 - 11*b1 + 3) * q^7 + (-13*b3 + 3) * q^8 + (6*b3 - 8*b2 + 6*b1) * q^9 $$q + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{2} + (\beta_{2} + 3 \beta_1 + 1) q^{3} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{4} + 5 \beta_{2} q^{5} + ( - 8 \beta_{3} - 3) q^{6} + (2 \beta_{3} - 5 \beta_{2} - 11 \beta_1 + 3) q^{7} + ( - 13 \beta_{3} + 3) q^{8} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{9} + (15 \beta_{2} - 5 \beta_1 + 15) q^{10} + ( - 14 \beta_{2} - 10 \beta_1 - 14) q^{11} + ( - 3 \beta_{3} + 33 \beta_{2} - 3 \beta_1) q^{12} + ( - 10 \beta_{3} - 18) q^{13} + (42 \beta_{3} - 28 \beta_{2} + \cdots + 7) q^{14}+ \cdots + ( - 4 \beta_{3} + 8) q^{99}+O(q^{100})$$ q + (b3 - 3*b2 + b1) * q^2 + (b2 + 3*b1 + 1) * q^3 + (-3*b2 + 6*b1 - 3) * q^4 + 5*b2 * q^5 + (-8*b3 - 3) * q^6 + (2*b3 - 5*b2 - 11*b1 + 3) * q^7 + (-13*b3 + 3) * q^8 + (6*b3 - 8*b2 + 6*b1) * q^9 + (15*b2 - 5*b1 + 15) * q^10 + (-14*b2 - 10*b1 - 14) * q^11 + (-3*b3 + 33*b2 - 3*b1) * q^12 + (-10*b3 - 18) * q^13 + (42*b3 - 28*b2 + 14*b1 + 7) * q^14 + (15*b3 - 5) * q^15 + (12*b3 - 7*b2 + 12*b1) * q^16 + (38*b2 - 54*b1 + 38) * q^17 + (-36*b2 + 26*b1 - 36) * q^18 + (26*b3 + 80*b2 + 26*b1) * q^19 + (30*b3 + 15) * q^20 + (-26*b3 - 75*b2 - 4*b1 - 4) * q^21 + (16*b3 - 22) * q^22 + (-117*b3 - 11*b2 - 117*b1) * q^23 + (81*b2 + 22*b1 + 81) * q^24 + (-25*b2 - 25) * q^25 + (-48*b3 + 74*b2 - 48*b1) * q^26 + (-99*b3 - 1) * q^27 + (3*b3 - 165*b2 + 57*b1 - 48) * q^28 + (60*b3 - 125) * q^29 + (40*b3 - 15*b2 + 40*b1) * q^30 + (66*b2 + 100*b1 + 66) * q^31 + (-21*b2 + 147*b1 - 21) * q^32 + (-52*b3 - 74*b2 - 52*b1) * q^33 + (200*b3 + 222) * q^34 + (-65*b3 + 40*b2 - 10*b1 + 25) * q^35 + (-66*b3 - 96) * q^36 + (136*b3 - 208*b2 + 136*b1) * q^37 + (188*b2 - 2*b1 + 188) * q^38 + (42*b2 - 44*b1 + 42) * q^39 + (65*b3 + 15*b2 + 65*b1) * q^40 + (-30*b3 - 53) * q^41 + (-70*b3 - 161*b2 - 7*b1 - 217) * q^42 + (-5*b3 - 333) * q^43 + (-54*b3 - 78*b2 - 54*b1) * q^44 + (40*b2 - 30*b1 + 40) * q^45 + (201*b2 - 340*b1 + 201) * q^46 + (-64*b3 - 98*b2 - 64*b1) * q^47 + (-9*b3 - 65) * q^48 + (142*b3 + 275*b2 - 46*b1 + 80) * q^49 + (-25*b3 - 75) * q^50 + (60*b3 - 286*b2 + 60*b1) * q^51 + (174*b2 - 138*b1 + 174) * q^52 + (476*b2 - 142*b1 + 476) * q^53 + (-298*b3 + 201*b2 - 298*b1) * q^54 + (-50*b3 + 70) * q^55 + (-98*b3 - 301*b2 - 98*b1 - 329) * q^56 + (266*b3 - 236) * q^57 + (55*b3 + 255*b2 + 55*b1) * q^58 + (-420*b2 + 266*b1 - 420) * q^59 + (-165*b2 + 15*b1 - 165) * q^60 + (570*b3 + 49*b2 + 570*b1) * q^61 + (-234*b3 - 2) * q^62 + (122*b3 - 88*b2 + 64*b1 + 92) * q^63 + (-366*b3 - 301) * q^64 + (50*b3 - 90*b2 + 50*b1) * q^65 + (-118*b2 - 82*b1 - 118) * q^66 + (643*b2 - 69*b1 + 643) * q^67 + (390*b3 - 762*b2 + 390*b1) * q^68 + (-150*b3 + 713) * q^69 + (-140*b3 + 175*b2 - 210*b1 + 140) * q^70 + (-350*b3 + 532) * q^71 + (-86*b3 + 132*b2 - 86*b1) * q^72 + (86*b2 + 118*b1 + 86) * q^73 + (-896*b2 + 616*b1 - 896) * q^74 + (-75*b3 - 25*b2 - 75*b1) * q^75 + (402*b3 - 72) * q^76 + (204*b3 + 218*b2 + 152*b1 - 72) * q^77 + (174*b3 + 214) * q^78 + (-214*b3 - 620*b2 - 214*b1) * q^79 + (35*b2 - 60*b1 + 35) * q^80 + (377*b2 + 258*b1 + 377) * q^81 + (-143*b3 + 219*b2 - 143*b1) * q^82 + (5*b3 - 953) * q^83 + (-438*b3 + 276*b2 - 90*b1 + 99) * q^84 + (-270*b3 - 190) * q^85 + (-348*b3 + 1009*b2 - 348*b1) * q^86 + (-485*b2 - 435*b1 - 485) * q^87 + (-302*b2 - 212*b1 - 302) * q^88 + (-324*b3 + 325*b2 - 324*b1) * q^89 + (130*b3 + 180) * q^90 + (-116*b3 - 130*b2 + 148*b1 - 314) * q^91 + (285*b3 + 1371) * q^92 + (298*b3 + 666*b2 + 298*b1) * q^93 + (-166*b2 - 94*b1 - 166) * q^94 + (-400*b2 - 130*b1 - 400) * q^95 + (84*b3 + 861*b2 + 84*b1) * q^96 + (500*b3 - 314) * q^97 + (644*b3 + 301*b2 + 231*b1 + 917) * q^98 + (-4*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{2} + 2 q^{3} - 6 q^{4} - 10 q^{5} - 12 q^{6} + 22 q^{7} + 12 q^{8} + 16 q^{9}+O(q^{10})$$ 4 * q + 6 * q^2 + 2 * q^3 - 6 * q^4 - 10 * q^5 - 12 * q^6 + 22 * q^7 + 12 * q^8 + 16 * q^9 $$4 q + 6 q^{2} + 2 q^{3} - 6 q^{4} - 10 q^{5} - 12 q^{6} + 22 q^{7} + 12 q^{8} + 16 q^{9} + 30 q^{10} - 28 q^{11} - 66 q^{12} - 72 q^{13} + 84 q^{14} - 20 q^{15} + 14 q^{16} + 76 q^{17} - 72 q^{18} - 160 q^{19} + 60 q^{20} + 134 q^{21} - 88 q^{22} + 22 q^{23} + 162 q^{24} - 50 q^{25} - 148 q^{26} - 4 q^{27} + 138 q^{28} - 500 q^{29} + 30 q^{30} + 132 q^{31} - 42 q^{32} + 148 q^{33} + 888 q^{34} + 20 q^{35} - 384 q^{36} + 416 q^{37} + 376 q^{38} + 84 q^{39} - 30 q^{40} - 212 q^{41} - 546 q^{42} - 1332 q^{43} + 156 q^{44} + 80 q^{45} + 402 q^{46} + 196 q^{47} - 260 q^{48} - 230 q^{49} - 300 q^{50} + 572 q^{51} + 348 q^{52} + 952 q^{53} - 402 q^{54} + 280 q^{55} - 714 q^{56} - 944 q^{57} - 510 q^{58} - 840 q^{59} - 330 q^{60} - 98 q^{61} - 8 q^{62} + 544 q^{63} - 1204 q^{64} + 180 q^{65} - 236 q^{66} + 1286 q^{67} + 1524 q^{68} + 2852 q^{69} + 210 q^{70} + 2128 q^{71} - 264 q^{72} + 172 q^{73} - 1792 q^{74} + 50 q^{75} - 288 q^{76} - 724 q^{77} + 856 q^{78} + 1240 q^{79} + 70 q^{80} + 754 q^{81} - 438 q^{82} - 3812 q^{83} - 156 q^{84} - 760 q^{85} - 2018 q^{86} - 970 q^{87} - 604 q^{88} - 650 q^{89} + 720 q^{90} - 996 q^{91} + 5484 q^{92} - 1332 q^{93} - 332 q^{94} - 800 q^{95} - 1722 q^{96} - 1256 q^{97} + 3066 q^{98} + 32 q^{99}+O(q^{100})$$ 4 * q + 6 * q^2 + 2 * q^3 - 6 * q^4 - 10 * q^5 - 12 * q^6 + 22 * q^7 + 12 * q^8 + 16 * q^9 + 30 * q^10 - 28 * q^11 - 66 * q^12 - 72 * q^13 + 84 * q^14 - 20 * q^15 + 14 * q^16 + 76 * q^17 - 72 * q^18 - 160 * q^19 + 60 * q^20 + 134 * q^21 - 88 * q^22 + 22 * q^23 + 162 * q^24 - 50 * q^25 - 148 * q^26 - 4 * q^27 + 138 * q^28 - 500 * q^29 + 30 * q^30 + 132 * q^31 - 42 * q^32 + 148 * q^33 + 888 * q^34 + 20 * q^35 - 384 * q^36 + 416 * q^37 + 376 * q^38 + 84 * q^39 - 30 * q^40 - 212 * q^41 - 546 * q^42 - 1332 * q^43 + 156 * q^44 + 80 * q^45 + 402 * q^46 + 196 * q^47 - 260 * q^48 - 230 * q^49 - 300 * q^50 + 572 * q^51 + 348 * q^52 + 952 * q^53 - 402 * q^54 + 280 * q^55 - 714 * q^56 - 944 * q^57 - 510 * q^58 - 840 * q^59 - 330 * q^60 - 98 * q^61 - 8 * q^62 + 544 * q^63 - 1204 * q^64 + 180 * q^65 - 236 * q^66 + 1286 * q^67 + 1524 * q^68 + 2852 * q^69 + 210 * q^70 + 2128 * q^71 - 264 * q^72 + 172 * q^73 - 1792 * q^74 + 50 * q^75 - 288 * q^76 - 724 * q^77 + 856 * q^78 + 1240 * q^79 + 70 * q^80 + 754 * q^81 - 438 * q^82 - 3812 * q^83 - 156 * q^84 - 760 * q^85 - 2018 * q^86 - 970 * q^87 - 604 * q^88 - 650 * q^89 + 720 * q^90 - 996 * q^91 + 5484 * q^92 - 1332 * q^93 - 332 * q^94 - 800 * q^95 - 1722 * q^96 - 1256 * q^97 + 3066 * q^98 + 32 * q^99

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

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

## Character values

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

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
0.792893 + 1.37333i 2.62132 4.54026i 2.74264 4.75039i −2.50000 4.33013i 8.31371 −5.10660 + 17.8023i 21.3848 −0.242641 0.420266i 3.96447 6.86666i
11.2 2.20711 + 3.82282i −1.62132 + 2.80821i −5.74264 + 9.94655i −2.50000 4.33013i −14.3137 16.1066 9.14207i −15.3848 8.24264 + 14.2767i 11.0355 19.1141i
16.1 0.792893 1.37333i 2.62132 + 4.54026i 2.74264 + 4.75039i −2.50000 + 4.33013i 8.31371 −5.10660 17.8023i 21.3848 −0.242641 + 0.420266i 3.96447 + 6.86666i
16.2 2.20711 3.82282i −1.62132 2.80821i −5.74264 9.94655i −2.50000 + 4.33013i −14.3137 16.1066 + 9.14207i −15.3848 8.24264 14.2767i 11.0355 + 19.1141i
 $$n$$: 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.b 4
3.b odd 2 1 315.4.j.c 4
4.b odd 2 1 560.4.q.i 4
5.b even 2 1 175.4.e.c 4
5.c odd 4 2 175.4.k.c 8
7.b odd 2 1 245.4.e.l 4
7.c even 3 1 inner 35.4.e.b 4
7.c even 3 1 245.4.a.g 2
7.d odd 6 1 245.4.a.h 2
7.d odd 6 1 245.4.e.l 4
21.g even 6 1 2205.4.a.bg 2
21.h odd 6 1 315.4.j.c 4
21.h odd 6 1 2205.4.a.bf 2
28.g odd 6 1 560.4.q.i 4
35.i odd 6 1 1225.4.a.v 2
35.j even 6 1 175.4.e.c 4
35.j even 6 1 1225.4.a.x 2
35.l odd 12 2 175.4.k.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b 4 1.a even 1 1 trivial
35.4.e.b 4 7.c even 3 1 inner
175.4.e.c 4 5.b even 2 1
175.4.e.c 4 35.j even 6 1
175.4.k.c 8 5.c odd 4 2
175.4.k.c 8 35.l odd 12 2
245.4.a.g 2 7.c even 3 1
245.4.a.h 2 7.d odd 6 1
245.4.e.l 4 7.b odd 2 1
245.4.e.l 4 7.d odd 6 1
315.4.j.c 4 3.b odd 2 1
315.4.j.c 4 21.h odd 6 1
560.4.q.i 4 4.b odd 2 1
560.4.q.i 4 28.g odd 6 1
1225.4.a.v 2 35.i odd 6 1
1225.4.a.x 2 35.j even 6 1
2205.4.a.bf 2 21.h odd 6 1
2205.4.a.bg 2 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 6T_{2}^{3} + 29T_{2}^{2} - 42T_{2} + 49$$ acting on $$S_{4}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 6 T^{3} + \cdots + 49$$
$3$ $$T^{4} - 2 T^{3} + \cdots + 289$$
$5$ $$(T^{2} + 5 T + 25)^{2}$$
$7$ $$T^{4} - 22 T^{3} + \cdots + 117649$$
$11$ $$T^{4} + 28 T^{3} + \cdots + 16$$
$13$ $$(T^{2} + 36 T + 124)^{2}$$
$17$ $$T^{4} - 76 T^{3} + \cdots + 19254544$$
$19$ $$T^{4} + 160 T^{3} + \cdots + 25482304$$
$23$ $$T^{4} - 22 T^{3} + \cdots + 742944049$$
$29$ $$(T^{2} + 250 T + 8425)^{2}$$
$31$ $$T^{4} - 132 T^{3} + \cdots + 244734736$$
$37$ $$T^{4} - 416 T^{3} + \cdots + 39337984$$
$41$ $$(T^{2} + 106 T + 1009)^{2}$$
$43$ $$(T^{2} + 666 T + 110839)^{2}$$
$47$ $$T^{4} - 196 T^{3} + \cdots + 1993744$$
$53$ $$T^{4} + \cdots + 34688317504$$
$59$ $$T^{4} + \cdots + 1217172544$$
$61$ $$T^{4} + \cdots + 419125465201$$
$67$ $$T^{4} + \cdots + 163157021329$$
$71$ $$(T^{2} - 1064 T + 38024)^{2}$$
$73$ $$T^{4} - 172 T^{3} + \cdots + 418284304$$
$79$ $$T^{4} + \cdots + 85736524864$$
$83$ $$(T^{2} + 1906 T + 908159)^{2}$$
$89$ $$T^{4} + \cdots + 10884122929$$
$97$ $$(T^{2} + 628 T - 401404)^{2}$$