# Properties

 Label 35.4.b.a Level $35$ Weight $4$ Character orbit 35.b Analytic conductor $2.065$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [35,4,Mod(29,35)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("35.29");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 35.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: $$2.06506685020$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600$$ x^10 + 55*x^8 + 983*x^6 + 6409*x^4 + 13560*x^2 + 3600 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 7^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 4) q^{4} + ( - \beta_{8} + 1) q^{5} + (\beta_{3} + 1) q^{6} - \beta_{5} q^{7} + (\beta_{9} + \beta_{8} + \cdots - 3 \beta_1) q^{8}+ \cdots + (\beta_{8} + \beta_{7} - \beta_{4} + \cdots - 5) q^{9}+O(q^{10})$$ q + b1 * q^2 - b6 * q^3 + (b2 - 4) * q^4 + (-b8 + 1) * q^5 + (b3 + 1) * q^6 - b5 * q^7 + (b9 + b8 - b7 + b6 + b5 - 3*b1) * q^8 + (b8 + b7 - b4 - b2 - 5) * q^9 $$q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 4) q^{4} + ( - \beta_{8} + 1) q^{5} + (\beta_{3} + 1) q^{6} - \beta_{5} q^{7} + (\beta_{9} + \beta_{8} + \cdots - 3 \beta_1) q^{8}+ \cdots + (44 \beta_{8} + 44 \beta_{7} + \cdots - 536) q^{99}+O(q^{100})$$ q + b1 * q^2 - b6 * q^3 + (b2 - 4) * q^4 + (-b8 + 1) * q^5 + (b3 + 1) * q^6 - b5 * q^7 + (b9 + b8 - b7 + b6 + b5 - 3*b1) * q^8 + (b8 + b7 - b4 - b2 - 5) * q^9 + (-b8 - 2*b7 + b6 + 2*b5 + b4 - b3 + 2*b2 + 2*b1 - 1) * q^10 + (b8 + b7 + b4 - b2 + 8) * q^11 + (-b8 + b7 - 4*b5 - 2*b1) * q^12 + (-2*b9 + b6 + 2*b5 + 6*b1) * q^13 + (-b8 - b7 - b3 + b2 - 5) * q^14 + (-b9 - b6 + 3*b5 - b4 - 2*b3 - 2*b2 + b1 + 2) * q^15 + (4*b8 + 4*b7 - 2*b4 + 2*b3 - 7*b2 + 14) * q^16 + (b8 - b7 + b6 - 8*b5 - 12*b1) * q^17 + (-b9 - 5*b8 + 5*b7 - 5*b6 - b5 - b1) * q^18 + (b8 + b7 + 2*b4 - 4*b3 - 2*b2 + 8) * q^19 + (2*b9 + 3*b8 - 4*b7 - b6 + 8*b5 - b4 + 2*b3 - 2*b2 - 8*b1 - 6) * q^20 + (-2*b8 - 2*b7 + b4 - b2 + 16) * q^21 + (-b9 + b8 - b7 - b6 - 9*b5 + 14*b1) * q^22 + (2*b9 + 2*b8 - 2*b7 + 20*b6 + 10*b5 + 2*b1) * q^23 + (-7*b8 - 7*b7 + 2*b4 + 2*b3 + 6*b2 + 10) * q^24 + (b9 + b8 + 7*b7 + 5*b6 + 5*b5 - 2*b3 + 5*b2 - 13*b1 - 41) * q^25 + (2*b8 + 2*b7 + b3 + 18*b2 - 71) * q^26 + (2*b9 + 7*b8 - 7*b7 - b6 - 10*b5 + 6*b1) * q^27 + (b9 + 3*b8 - 3*b7 - 5*b6 + b5 - 7*b1) * q^28 + (-b8 - b7 - b4 + 12*b3 - 7*b2 + 10) * q^29 + (-2*b9 + 6*b7 - 20*b6 + 10*b5 + 4*b3 + 5*b2 + 21*b1) * q^30 + (-4*b8 - 4*b7 - 2*b4 - 4*b3 - 10*b2 + 20) * q^31 + (b9 - 11*b8 + 11*b7 + 5*b6 - 15*b5 + 23*b1) * q^32 + (2*b9 - 5*b8 + 5*b7 - 21*b6 + 6*b5 - 6*b1) * q^33 + (-5*b8 - 5*b7 - 2*b4 - 7*b3 - 8*b2 + 105) * q^34 + (-b9 - b8 - 2*b7 + 5*b6 + 2*b3 + 5*b2 - 7*b1 - 4) * q^35 + (-8*b8 - 8*b7 + 2*b4 - 6*b3 + 19*b2 - 42) * q^36 + (-2*b9 - 4*b8 + 4*b7 - 20*b6 - 18*b5 - 34*b1) * q^37 + (-2*b9 + 7*b8 - 7*b7 - 32*b6 + 2*b5 + 34*b1) * q^38 + (19*b8 + 19*b7 - b4 + 4*b3 + 3*b2 + 4) * q^39 + (-2*b9 + 4*b8 + 9*b7 + 21*b6 + 12*b5 + b4 + 8*b3 - 28*b2 + 18*b1 + 144) * q^40 + (-14*b8 - 14*b7 - 4*b3 + 12*b2 - 78) * q^41 + (-b9 + 4*b8 - 4*b7 + 5*b6 + 3*b5 + 28*b1) * q^42 + (-4*b9 + 6*b8 - 6*b7 + 20*b6 + 12*b5 - 16*b1) * q^43 + (2*b8 + 2*b7 + 6*b4 - 6*b3 + 18*b2 - 150) * q^44 + (2*b9 + 6*b8 - 3*b7 + 31*b6 + 2*b5 - 4*b4 + 2*b2 + 66*b1 - 53) * q^45 + (16*b8 + 16*b7 - 4*b4 - 6*b3 - 30*b2 + 18) * q^46 + (-2*b9 - 9*b8 + 9*b7 + 21*b6 - 6*b5 - 22*b1) * q^47 + (6*b9 + 9*b8 - 9*b7 + 40*b6 - 14*b5 - 38*b1) * q^48 - 49 * q^49 + (5*b9 - b8 - 7*b7 - 19*b6 - 3*b5 + 6*b4 - 6*b3 - 13*b2 - 78*b1 + 174) * q^50 + (-17*b8 - 17*b7 + 11*b4 - 8*b3 - 3*b2 + 144) * q^51 + (2*b9 + 15*b8 - 15*b7 + 30*b6 + 22*b5 - 156*b1) * q^52 + (2*b9 - 10*b8 + 10*b7 - 28*b6 + 18*b5 - 26*b1) * q^53 + (11*b8 + 11*b7 - 14*b4 + 5*b3 - 26*b2 - 99) * q^54 + (-8*b9 - 13*b8 - 3*b7 - 39*b6 - 28*b5 + 6*b4 - 8*b2 - 24*b1 - 104) * q^55 + (2*b8 + 2*b7 - 6*b4 + 4*b3 - 19*b2 + 64) * q^56 + (6*b9 + 10*b8 - 10*b7 - 18*b6 + 22*b5 + 102*b1) * q^57 + (-7*b9 - 21*b8 + 21*b7 + 89*b6 - 47*b5 + 24*b1) * q^58 + (21*b8 + 21*b7 - 12*b4 + 8*b3 - 20*b2 + 76) * q^59 + (-3*b9 - b8 + 3*b7 + 23*b6 + b5 - 2*b4 + 8*b3 + 21*b2 - 45*b1 - 180) * q^60 + (13*b8 + 13*b7 + 4*b3 + 36*b2 - 114) * q^61 + (-10*b9 - 8*b8 + 8*b7 - 38*b6 + 30*b5 + 108*b1) * q^62 + (-4*b9 - 5*b8 + 5*b7 - 29*b6 - 3*b5) * q^63 + (-16*b8 - 16*b7 + 6*b4 - 26*b3 + 19*b2 - 262) * q^64 + (9*b9 - 6*b8 - 24*b7 - 69*b6 - 43*b5 + b4 - 14*b3 + 32*b2 + 15*b1 + 54) * q^65 + (-9*b8 - 9*b7 + 10*b4 + 17*b3 - 6*b2 + 121) * q^66 + (-4*b9 + 2*b8 - 2*b7 + 18*b6 + 44*b5 + 140*b1) * q^67 + (-8*b9 + 6*b8 - 6*b7 - 50*b6 - 16*b5 + 94*b1) * q^68 + (-16*b8 - 16*b7 + 14*b4 + 12*b3 + 38*b2 + 476) * q^69 + (5*b9 + 6*b8 - 3*b7 + 24*b6 + 3*b5 - b4 - 4*b3 - 2*b2 - 42*b1 + 76) * q^70 + (22*b8 + 22*b7 - 12*b4 + 12*b3 - 10*b2 + 72) * q^71 + (11*b9 - b8 + b7 - 49*b6 + 59*b5 - 147*b1) * q^72 + (10*b9 - 2*b8 + 2*b7 - 48*b6 + 14*b5 + 102*b1) * q^73 + (-30*b8 - 30*b7 + 8*b4 - 6*b3 + 14*b2 + 322) * q^74 + (10*b9 + 2*b8 - 8*b7 + 29*b6 - 42*b5 - 6*b4 - 24*b3 - 2*b2 + 38*b1 + 80) * q^75 + (31*b8 + 31*b7 + 2*b4 + 16*b3 + 2*b2 - 296) * q^76 + (6*b9 - 3*b8 + 3*b7 + 19*b6 + 2*b5 + 42*b1) * q^77 + (3*b9 - 23*b8 + 23*b7 - 5*b6 - 85*b5 - 68*b1) * q^78 + (-19*b8 - 19*b7 - 7*b4 + 4*b3 - 19*b2 - 336) * q^79 + (-12*b9 - 11*b8 + 12*b7 + 17*b6 - 26*b5 - 3*b4 + 2*b3 + 14*b2 + 240*b1 - 238) * q^80 + (-8*b8 - 8*b7 - 4*b4 + 36*b3 - 10*b2 - 27) * q^81 + (12*b9 + 30*b8 - 30*b7 + 8*b6 + 84*b5 - 122*b1) * q^82 + (6*b9 + 9*b8 - 9*b7 + 48*b6 + 50*b5 - 86*b1) * q^83 + (-b8 - b7 + 6*b3 + 8*b2 - 194) * q^84 + (-8*b9 + 3*b8 + b7 + 24*b6 - 32*b5 - 11*b4 + 32*b3 + 13*b2 - 68*b1 + 144) * q^85 + (30*b8 + 30*b7 - 12*b4 + 4*b3 - 24*b2 + 228) * q^86 + (-14*b9 - 47*b8 + 47*b7 - 13*b6 + 14*b5 - 314*b1) * q^87 + (10*b9 + 48*b8 - 48*b7 - 30*b6 - 62*b5 - 144*b1) * q^88 + (-26*b8 - 26*b7 + 30*b4 + 12*b3 + 18*b2 - 138) * q^89 + (2*b9 + 4*b8 + 27*b7 - 9*b6 + 12*b5 - 9*b4 - 20*b3 + 32*b2 - 74*b1 - 796) * q^90 + (8*b8 + 8*b7 + 9*b4 - 16*b3 + 21*b2 + 36) * q^91 + (-14*b9 - 36*b8 + 36*b7 + 42*b6 + 26*b5 + 226*b1) * q^92 + (-4*b9 + 42*b8 - 42*b7 + 70*b6 + 52*b5 + 152*b1) * q^93 + (-33*b8 - 33*b7 + 18*b4 - 45*b3 + 34*b2 + 187) * q^94 + (-7*b9 - 19*b8 - 23*b7 + 7*b6 - 91*b5 + 12*b4 - 18*b3 - 31*b2 - 145*b1 - 130) * q^95 + (-43*b8 - 43*b7 - 2*b4 - 20*b3 - 54*b2 + 468) * q^96 + (12*b9 - 15*b8 + 15*b7 + 29*b6 + 4*b5 + 80*b1) * q^97 - 49*b1 * q^98 + (44*b8 + 44*b7 - 10*b4 - 24*b3 - 62*b2 - 536) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 36 q^{4} + 6 q^{5} + 12 q^{6} - 46 q^{9}+O(q^{10})$$ 10 * q - 36 * q^4 + 6 * q^5 + 12 * q^6 - 46 * q^9 $$10 q - 36 q^{4} + 6 q^{5} + 12 q^{6} - 46 q^{9} - 16 q^{10} + 84 q^{11} - 56 q^{14} + 8 q^{15} + 148 q^{16} + 72 q^{19} - 68 q^{20} + 140 q^{21} + 72 q^{24} - 362 q^{25} - 620 q^{26} + 88 q^{29} + 52 q^{30} + 120 q^{31} + 964 q^{34} - 28 q^{35} - 420 q^{36} + 212 q^{39} + 1396 q^{40} - 852 q^{41} - 1424 q^{44} - 510 q^{45} + 176 q^{46} - 490 q^{49} + 1644 q^{50} + 1276 q^{51} - 996 q^{54} - 1136 q^{55} + 588 q^{56} + 864 q^{59} - 1692 q^{60} - 884 q^{61} - 2724 q^{64} + 520 q^{65} + 1148 q^{66} + 4808 q^{69} + 756 q^{70} + 880 q^{71} + 3024 q^{74} + 720 q^{75} - 2672 q^{76} - 3580 q^{79} - 2316 q^{80} - 302 q^{81} - 1904 q^{84} + 1572 q^{85} + 2432 q^{86} - 1492 q^{89} - 7748 q^{90} + 476 q^{91} + 1652 q^{94} - 1628 q^{95} + 4080 q^{96} - 5304 q^{99}+O(q^{100})$$ 10 * q - 36 * q^4 + 6 * q^5 + 12 * q^6 - 46 * q^9 - 16 * q^10 + 84 * q^11 - 56 * q^14 + 8 * q^15 + 148 * q^16 + 72 * q^19 - 68 * q^20 + 140 * q^21 + 72 * q^24 - 362 * q^25 - 620 * q^26 + 88 * q^29 + 52 * q^30 + 120 * q^31 + 964 * q^34 - 28 * q^35 - 420 * q^36 + 212 * q^39 + 1396 * q^40 - 852 * q^41 - 1424 * q^44 - 510 * q^45 + 176 * q^46 - 490 * q^49 + 1644 * q^50 + 1276 * q^51 - 996 * q^54 - 1136 * q^55 + 588 * q^56 + 864 * q^59 - 1692 * q^60 - 884 * q^61 - 2724 * q^64 + 520 * q^65 + 1148 * q^66 + 4808 * q^69 + 756 * q^70 + 880 * q^71 + 3024 * q^74 + 720 * q^75 - 2672 * q^76 - 3580 * q^79 - 2316 * q^80 - 302 * q^81 - 1904 * q^84 + 1572 * q^85 + 2432 * q^86 - 1492 * q^89 - 7748 * q^90 + 476 * q^91 + 1652 * q^94 - 1628 * q^95 + 4080 * q^96 - 5304 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{9} - 25\nu^{7} + 397\nu^{5} + 11921\nu^{3} + 74220\nu ) / 25200$$ (-v^9 - 25*v^7 + 397*v^5 + 11921*v^3 + 74220*v) / 25200 $$\beta_{2}$$ $$=$$ $$( \nu^{8} + 46\nu^{6} + 611\nu^{4} + 2506\nu^{2} + 4740 ) / 420$$ (v^8 + 46*v^6 + 611*v^4 + 2506*v^2 + 4740) / 420 $$\beta_{3}$$ $$=$$ $$( -5\nu^{8} - 251\nu^{6} - 3895\nu^{4} - 18725\nu^{2} - 11268 ) / 672$$ (-5*v^8 - 251*v^6 - 3895*v^4 - 18725*v^2 - 11268) / 672 $$\beta_{4}$$ $$=$$ $$( -59\nu^{8} - 2945\nu^{6} - 42937\nu^{4} - 159551\nu^{2} - 20940 ) / 3360$$ (-59*v^8 - 2945*v^6 - 42937*v^4 - 159551*v^2 - 20940) / 3360 $$\beta_{5}$$ $$=$$ $$( \nu^{9} + 25\nu^{7} - 397\nu^{5} - 11921\nu^{3} - 49020\nu ) / 3600$$ (v^9 + 25*v^7 - 397*v^5 - 11921*v^3 - 49020*v) / 3600 $$\beta_{6}$$ $$=$$ $$( -29\nu^{9} - 1535\nu^{7} - 25567\nu^{5} - 140561\nu^{3} - 169620\nu ) / 14400$$ (-29*v^9 - 1535*v^7 - 25567*v^5 - 140561*v^3 - 169620*v) / 14400 $$\beta_{7}$$ $$=$$ $$( 89 \nu^{9} + 30 \nu^{8} + 4619 \nu^{7} + 2010 \nu^{6} + 72019 \nu^{5} + 43530 \nu^{4} + \cdots + 585720 ) / 40320$$ (89*v^9 + 30*v^8 + 4619*v^7 + 2010*v^6 + 72019*v^5 + 43530*v^4 + 319109*v^3 + 331590*v^2 + 193380*v + 585720) / 40320 $$\beta_{8}$$ $$=$$ $$( - 89 \nu^{9} + 30 \nu^{8} - 4619 \nu^{7} + 2010 \nu^{6} - 72019 \nu^{5} + 43530 \nu^{4} + \cdots + 585720 ) / 40320$$ (-89*v^9 + 30*v^8 - 4619*v^7 + 2010*v^6 - 72019*v^5 + 43530*v^4 - 319109*v^3 + 331590*v^2 - 193380*v + 585720) / 40320 $$\beta_{9}$$ $$=$$ $$( 227\nu^{9} + 11975\nu^{7} + 199681\nu^{5} + 1155833\nu^{3} + 2178060\nu ) / 25200$$ (227*v^9 + 11975*v^7 + 199681*v^5 + 1155833*v^3 + 2178060*v) / 25200
 $$\nu$$ $$=$$ $$( \beta_{5} + 7\beta_1 ) / 7$$ (b5 + 7*b1) / 7 $$\nu^{2}$$ $$=$$ $$( 2\beta_{8} + 2\beta_{7} + 2\beta_{3} + 5\beta_{2} - 81 ) / 7$$ (2*b8 + 2*b7 + 2*b3 + 5*b2 - 81) / 7 $$\nu^{3}$$ $$=$$ $$( 4\beta_{9} - 2\beta_{8} + 2\beta_{7} + 22\beta_{6} - 21\beta_{5} - 133\beta_1 ) / 7$$ (4*b9 - 2*b8 + 2*b7 + 22*b6 - 21*b5 - 133*b1) / 7 $$\nu^{4}$$ $$=$$ $$( -48\beta_{8} - 48\beta_{7} + 10\beta_{4} - 70\beta_{3} - 115\beta_{2} + 1581 ) / 7$$ (-48*b8 - 48*b7 + 10*b4 - 70*b3 - 115*b2 + 1581) / 7 $$\nu^{5}$$ $$=$$ $$( -102\beta_{9} + 114\beta_{8} - 114\beta_{7} - 694\beta_{6} + 507\beta_{5} + 2933\beta_1 ) / 7$$ (-102*b9 + 114*b8 - 114*b7 - 694*b6 + 507*b5 + 2933*b1) / 7 $$\nu^{6}$$ $$=$$ $$( 1330\beta_{8} + 1330\beta_{7} - 400\beta_{4} + 1986\beta_{3} + 2425\beta_{2} - 35201 ) / 7$$ (1330*b8 + 1330*b7 - 400*b4 + 1986*b3 + 2425*b2 - 35201) / 7 $$\nu^{7}$$ $$=$$ $$( 2268\beta_{9} - 4018\beta_{8} + 4018\beta_{7} + 18438\beta_{6} - 13469\beta_{5} - 68173\beta_1 ) / 7$$ (2268*b9 - 4018*b8 + 4018*b7 + 18438*b6 - 13469*b5 - 68173*b1) / 7 $$\nu^{8}$$ $$=$$ $$( -36864\beta_{8} - 36864\beta_{7} + 12290\beta_{4} - 53598\beta_{3} - 50875\beta_{2} + 823061 ) / 7$$ (-36864*b8 - 36864*b7 + 12290*b4 - 53598*b3 - 50875*b2 + 823061) / 7 $$\nu^{9}$$ $$=$$ $$( -49510\beta_{9} + 121866\beta_{8} - 121866\beta_{7} - 474206\beta_{6} + 361883\beta_{5} + 1626373\beta_1 ) / 7$$ (-49510*b9 + 121866*b8 - 121866*b7 - 474206*b6 + 361883*b5 + 1626373*b1) / 7

## Character values

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

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 4.31366i − 5.04851i − 1.85474i − 2.67516i − 0.555276i 0.555276i 2.67516i 1.85474i 5.04851i 4.31366i
5.31366i 1.93939i −20.2350 −2.32771 10.9353i 10.3052 7.00000i 65.0123i 23.2388 −58.1067 + 12.3686i
29.2 4.04851i 6.52749i −8.39045 6.15172 + 9.33576i −26.4266 7.00000i 1.58074i −15.6081 37.7959 24.9053i
29.3 2.85474i 8.98858i −0.149548 3.91321 + 10.4731i 25.6601 7.00000i 22.4110i −53.7945 29.8981 11.1712i
29.4 1.67516i 2.49396i 5.19383 6.35505 9.19855i 4.17779 7.00000i 22.1018i 20.7802 −15.4091 10.6457i
29.5 1.55528i 4.96149i 5.58112 −11.0923 1.40060i −7.71648 7.00000i 21.1224i 2.38365 −2.17831 + 17.2515i
29.6 1.55528i 4.96149i 5.58112 −11.0923 + 1.40060i −7.71648 7.00000i 21.1224i 2.38365 −2.17831 17.2515i
29.7 1.67516i 2.49396i 5.19383 6.35505 + 9.19855i 4.17779 7.00000i 22.1018i 20.7802 −15.4091 + 10.6457i
29.8 2.85474i 8.98858i −0.149548 3.91321 10.4731i 25.6601 7.00000i 22.4110i −53.7945 29.8981 + 11.1712i
29.9 4.04851i 6.52749i −8.39045 6.15172 9.33576i −26.4266 7.00000i 1.58074i −15.6081 37.7959 + 24.9053i
29.10 5.31366i 1.93939i −20.2350 −2.32771 + 10.9353i 10.3052 7.00000i 65.0123i 23.2388 −58.1067 12.3686i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.b.a 10
3.b odd 2 1 315.4.d.c 10
4.b odd 2 1 560.4.g.f 10
5.b even 2 1 inner 35.4.b.a 10
5.c odd 4 1 175.4.a.i 5
5.c odd 4 1 175.4.a.j 5
7.b odd 2 1 245.4.b.d 10
7.c even 3 2 245.4.j.e 20
7.d odd 6 2 245.4.j.f 20
15.d odd 2 1 315.4.d.c 10
15.e even 4 1 1575.4.a.bn 5
15.e even 4 1 1575.4.a.bq 5
20.d odd 2 1 560.4.g.f 10
35.c odd 2 1 245.4.b.d 10
35.f even 4 1 1225.4.a.be 5
35.f even 4 1 1225.4.a.bh 5
35.i odd 6 2 245.4.j.f 20
35.j even 6 2 245.4.j.e 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.b.a 10 1.a even 1 1 trivial
35.4.b.a 10 5.b even 2 1 inner
175.4.a.i 5 5.c odd 4 1
175.4.a.j 5 5.c odd 4 1
245.4.b.d 10 7.b odd 2 1
245.4.b.d 10 35.c odd 2 1
245.4.j.e 20 7.c even 3 2
245.4.j.e 20 35.j even 6 2
245.4.j.f 20 7.d odd 6 2
245.4.j.f 20 35.i odd 6 2
315.4.d.c 10 3.b odd 2 1
315.4.d.c 10 15.d odd 2 1
560.4.g.f 10 4.b odd 2 1
560.4.g.f 10 20.d odd 2 1
1225.4.a.be 5 35.f even 4 1
1225.4.a.bh 5 35.f even 4 1
1575.4.a.bn 5 15.e even 4 1
1575.4.a.bq 5 15.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 58 T^{8} + \cdots + 25600$$
$3$ $$T^{10} + 158 T^{8} + \cdots + 1982464$$
$5$ $$T^{10} + \cdots + 30517578125$$
$7$ $$(T^{2} + 49)^{5}$$
$11$ $$(T^{5} - 42 T^{4} + \cdots + 11139072)^{2}$$
$13$ $$T^{10} + \cdots + 26\!\cdots\!00$$
$17$ $$T^{10} + \cdots + 12\!\cdots\!64$$
$19$ $$(T^{5} - 36 T^{4} + \cdots + 20133120)^{2}$$
$23$ $$T^{10} + \cdots + 76\!\cdots\!64$$
$29$ $$(T^{5} - 44 T^{4} + \cdots - 126081243400)^{2}$$
$31$ $$(T^{5} - 60 T^{4} + \cdots - 34433580800)^{2}$$
$37$ $$T^{10} + \cdots + 18\!\cdots\!36$$
$41$ $$(T^{5} + 426 T^{4} + \cdots + 109849343072)^{2}$$
$43$ $$T^{10} + \cdots + 24\!\cdots\!00$$
$47$ $$T^{10} + \cdots + 10\!\cdots\!36$$
$53$ $$T^{10} + \cdots + 55\!\cdots\!56$$
$59$ $$(T^{5} + \cdots - 5009454255200)^{2}$$
$61$ $$(T^{5} + 442 T^{4} + \cdots - 650238065792)^{2}$$
$67$ $$T^{10} + \cdots + 25\!\cdots\!24$$
$71$ $$(T^{5} + \cdots - 7767441797120)^{2}$$
$73$ $$T^{10} + \cdots + 88\!\cdots\!64$$
$79$ $$(T^{5} + 1790 T^{4} + \cdots - 201540544400)^{2}$$
$83$ $$T^{10} + \cdots + 78\!\cdots\!96$$
$89$ $$(T^{5} + \cdots - 68855772276480)^{2}$$
$97$ $$T^{10} + \cdots + 15\!\cdots\!64$$