# Properties

 Label 35.4.a.b Level $35$ Weight $4$ Character orbit 35.a Self dual yes Analytic conductor $2.065$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [35,4,Mod(1,35)]

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([0, 0]))

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

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

chi := DirichletCharacter("35.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 35.a (trivial)

## 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: yes Analytic conductor: $$2.06506685020$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 4) q^{2} + ( - 4 \beta + 1) q^{3} + (8 \beta + 10) q^{4} - 5 q^{5} + ( - 15 \beta - 4) q^{6} - 7 q^{7} + (34 \beta + 24) q^{8} + ( - 8 \beta + 6) q^{9}+O(q^{10})$$ q + (b + 4) * q^2 + (-4*b + 1) * q^3 + (8*b + 10) * q^4 - 5 * q^5 + (-15*b - 4) * q^6 - 7 * q^7 + (34*b + 24) * q^8 + (-8*b + 6) * q^9 $$q + (\beta + 4) q^{2} + ( - 4 \beta + 1) q^{3} + (8 \beta + 10) q^{4} - 5 q^{5} + ( - 15 \beta - 4) q^{6} - 7 q^{7} + (34 \beta + 24) q^{8} + ( - 8 \beta + 6) q^{9} + ( - 5 \beta - 20) q^{10} + ( - 32 \beta - 7) q^{11} + ( - 32 \beta - 54) q^{12} + (4 \beta + 25) q^{13} + ( - 7 \beta - 28) q^{14} + (20 \beta - 5) q^{15} + (96 \beta + 84) q^{16} + (44 \beta - 25) q^{17} + ( - 26 \beta + 8) q^{18} + (44 \beta + 18) q^{19} + ( - 40 \beta - 50) q^{20} + (28 \beta - 7) q^{21} + ( - 135 \beta - 92) q^{22} + ( - 68 \beta + 122) q^{23} + ( - 62 \beta - 248) q^{24} + 25 q^{25} + (41 \beta + 108) q^{26} + (76 \beta + 43) q^{27} + ( - 56 \beta - 70) q^{28} + (24 \beta - 13) q^{29} + (75 \beta + 20) q^{30} + ( - 180 \beta - 60) q^{31} + (196 \beta + 336) q^{32} + ( - 4 \beta + 249) q^{33} + (151 \beta - 12) q^{34} + 35 q^{35} + ( - 32 \beta - 68) q^{36} + ( - 60 \beta + 282) q^{37} + (194 \beta + 160) q^{38} + ( - 96 \beta - 7) q^{39} + ( - 170 \beta - 120) q^{40} + (124 \beta - 164) q^{41} + (105 \beta + 28) q^{42} + (68 \beta - 130) q^{43} + ( - 376 \beta - 582) q^{44} + (40 \beta - 30) q^{45} + ( - 150 \beta + 352) q^{46} + ( - 132 \beta - 175) q^{47} + ( - 240 \beta - 684) q^{48} + 49 q^{49} + (25 \beta + 100) q^{50} + (144 \beta - 377) q^{51} + (240 \beta + 314) q^{52} + (128 \beta - 28) q^{53} + (347 \beta + 324) q^{54} + (160 \beta + 35) q^{55} + ( - 238 \beta - 168) q^{56} + ( - 28 \beta - 334) q^{57} + (83 \beta - 4) q^{58} - 616 q^{59} + (160 \beta + 270) q^{60} + ( - 108 \beta + 168) q^{61} + ( - 780 \beta - 600) q^{62} + (56 \beta - 42) q^{63} + (352 \beta + 1064) q^{64} + ( - 20 \beta - 125) q^{65} + (233 \beta + 988) q^{66} + ( - 64 \beta - 76) q^{67} + (240 \beta + 454) q^{68} + ( - 556 \beta + 666) q^{69} + (35 \beta + 140) q^{70} - 952 q^{71} + (12 \beta - 400) q^{72} + ( - 344 \beta + 338) q^{73} + (42 \beta + 1008) q^{74} + ( - 100 \beta + 25) q^{75} + (584 \beta + 884) q^{76} + (224 \beta + 49) q^{77} + ( - 391 \beta - 220) q^{78} + (248 \beta + 507) q^{79} + ( - 480 \beta - 420) q^{80} + (120 \beta - 727) q^{81} + (332 \beta - 408) q^{82} + (600 \beta - 188) q^{83} + (224 \beta + 378) q^{84} + ( - 220 \beta + 125) q^{85} + (142 \beta - 384) q^{86} + (76 \beta - 205) q^{87} + ( - 1006 \beta - 2344) q^{88} + (44 \beta - 108) q^{89} + (130 \beta - 40) q^{90} + ( - 28 \beta - 175) q^{91} + (296 \beta + 132) q^{92} + (60 \beta + 1380) q^{93} + ( - 703 \beta - 964) q^{94} + ( - 220 \beta - 90) q^{95} + ( - 1148 \beta - 1232) q^{96} + (220 \beta + 1371) q^{97} + (49 \beta + 196) q^{98} + ( - 136 \beta + 470) q^{99}+O(q^{100})$$ q + (b + 4) * q^2 + (-4*b + 1) * q^3 + (8*b + 10) * q^4 - 5 * q^5 + (-15*b - 4) * q^6 - 7 * q^7 + (34*b + 24) * q^8 + (-8*b + 6) * q^9 + (-5*b - 20) * q^10 + (-32*b - 7) * q^11 + (-32*b - 54) * q^12 + (4*b + 25) * q^13 + (-7*b - 28) * q^14 + (20*b - 5) * q^15 + (96*b + 84) * q^16 + (44*b - 25) * q^17 + (-26*b + 8) * q^18 + (44*b + 18) * q^19 + (-40*b - 50) * q^20 + (28*b - 7) * q^21 + (-135*b - 92) * q^22 + (-68*b + 122) * q^23 + (-62*b - 248) * q^24 + 25 * q^25 + (41*b + 108) * q^26 + (76*b + 43) * q^27 + (-56*b - 70) * q^28 + (24*b - 13) * q^29 + (75*b + 20) * q^30 + (-180*b - 60) * q^31 + (196*b + 336) * q^32 + (-4*b + 249) * q^33 + (151*b - 12) * q^34 + 35 * q^35 + (-32*b - 68) * q^36 + (-60*b + 282) * q^37 + (194*b + 160) * q^38 + (-96*b - 7) * q^39 + (-170*b - 120) * q^40 + (124*b - 164) * q^41 + (105*b + 28) * q^42 + (68*b - 130) * q^43 + (-376*b - 582) * q^44 + (40*b - 30) * q^45 + (-150*b + 352) * q^46 + (-132*b - 175) * q^47 + (-240*b - 684) * q^48 + 49 * q^49 + (25*b + 100) * q^50 + (144*b - 377) * q^51 + (240*b + 314) * q^52 + (128*b - 28) * q^53 + (347*b + 324) * q^54 + (160*b + 35) * q^55 + (-238*b - 168) * q^56 + (-28*b - 334) * q^57 + (83*b - 4) * q^58 - 616 * q^59 + (160*b + 270) * q^60 + (-108*b + 168) * q^61 + (-780*b - 600) * q^62 + (56*b - 42) * q^63 + (352*b + 1064) * q^64 + (-20*b - 125) * q^65 + (233*b + 988) * q^66 + (-64*b - 76) * q^67 + (240*b + 454) * q^68 + (-556*b + 666) * q^69 + (35*b + 140) * q^70 - 952 * q^71 + (12*b - 400) * q^72 + (-344*b + 338) * q^73 + (42*b + 1008) * q^74 + (-100*b + 25) * q^75 + (584*b + 884) * q^76 + (224*b + 49) * q^77 + (-391*b - 220) * q^78 + (248*b + 507) * q^79 + (-480*b - 420) * q^80 + (120*b - 727) * q^81 + (332*b - 408) * q^82 + (600*b - 188) * q^83 + (224*b + 378) * q^84 + (-220*b + 125) * q^85 + (142*b - 384) * q^86 + (76*b - 205) * q^87 + (-1006*b - 2344) * q^88 + (44*b - 108) * q^89 + (130*b - 40) * q^90 + (-28*b - 175) * q^91 + (296*b + 132) * q^92 + (60*b + 1380) * q^93 + (-703*b - 964) * q^94 + (-220*b - 90) * q^95 + (-1148*b - 1232) * q^96 + (220*b + 1371) * q^97 + (49*b + 196) * q^98 + (-136*b + 470) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{2} + 2 q^{3} + 20 q^{4} - 10 q^{5} - 8 q^{6} - 14 q^{7} + 48 q^{8} + 12 q^{9}+O(q^{10})$$ 2 * q + 8 * q^2 + 2 * q^3 + 20 * q^4 - 10 * q^5 - 8 * q^6 - 14 * q^7 + 48 * q^8 + 12 * q^9 $$2 q + 8 q^{2} + 2 q^{3} + 20 q^{4} - 10 q^{5} - 8 q^{6} - 14 q^{7} + 48 q^{8} + 12 q^{9} - 40 q^{10} - 14 q^{11} - 108 q^{12} + 50 q^{13} - 56 q^{14} - 10 q^{15} + 168 q^{16} - 50 q^{17} + 16 q^{18} + 36 q^{19} - 100 q^{20} - 14 q^{21} - 184 q^{22} + 244 q^{23} - 496 q^{24} + 50 q^{25} + 216 q^{26} + 86 q^{27} - 140 q^{28} - 26 q^{29} + 40 q^{30} - 120 q^{31} + 672 q^{32} + 498 q^{33} - 24 q^{34} + 70 q^{35} - 136 q^{36} + 564 q^{37} + 320 q^{38} - 14 q^{39} - 240 q^{40} - 328 q^{41} + 56 q^{42} - 260 q^{43} - 1164 q^{44} - 60 q^{45} + 704 q^{46} - 350 q^{47} - 1368 q^{48} + 98 q^{49} + 200 q^{50} - 754 q^{51} + 628 q^{52} - 56 q^{53} + 648 q^{54} + 70 q^{55} - 336 q^{56} - 668 q^{57} - 8 q^{58} - 1232 q^{59} + 540 q^{60} + 336 q^{61} - 1200 q^{62} - 84 q^{63} + 2128 q^{64} - 250 q^{65} + 1976 q^{66} - 152 q^{67} + 908 q^{68} + 1332 q^{69} + 280 q^{70} - 1904 q^{71} - 800 q^{72} + 676 q^{73} + 2016 q^{74} + 50 q^{75} + 1768 q^{76} + 98 q^{77} - 440 q^{78} + 1014 q^{79} - 840 q^{80} - 1454 q^{81} - 816 q^{82} - 376 q^{83} + 756 q^{84} + 250 q^{85} - 768 q^{86} - 410 q^{87} - 4688 q^{88} - 216 q^{89} - 80 q^{90} - 350 q^{91} + 264 q^{92} + 2760 q^{93} - 1928 q^{94} - 180 q^{95} - 2464 q^{96} + 2742 q^{97} + 392 q^{98} + 940 q^{99}+O(q^{100})$$ 2 * q + 8 * q^2 + 2 * q^3 + 20 * q^4 - 10 * q^5 - 8 * q^6 - 14 * q^7 + 48 * q^8 + 12 * q^9 - 40 * q^10 - 14 * q^11 - 108 * q^12 + 50 * q^13 - 56 * q^14 - 10 * q^15 + 168 * q^16 - 50 * q^17 + 16 * q^18 + 36 * q^19 - 100 * q^20 - 14 * q^21 - 184 * q^22 + 244 * q^23 - 496 * q^24 + 50 * q^25 + 216 * q^26 + 86 * q^27 - 140 * q^28 - 26 * q^29 + 40 * q^30 - 120 * q^31 + 672 * q^32 + 498 * q^33 - 24 * q^34 + 70 * q^35 - 136 * q^36 + 564 * q^37 + 320 * q^38 - 14 * q^39 - 240 * q^40 - 328 * q^41 + 56 * q^42 - 260 * q^43 - 1164 * q^44 - 60 * q^45 + 704 * q^46 - 350 * q^47 - 1368 * q^48 + 98 * q^49 + 200 * q^50 - 754 * q^51 + 628 * q^52 - 56 * q^53 + 648 * q^54 + 70 * q^55 - 336 * q^56 - 668 * q^57 - 8 * q^58 - 1232 * q^59 + 540 * q^60 + 336 * q^61 - 1200 * q^62 - 84 * q^63 + 2128 * q^64 - 250 * q^65 + 1976 * q^66 - 152 * q^67 + 908 * q^68 + 1332 * q^69 + 280 * q^70 - 1904 * q^71 - 800 * q^72 + 676 * q^73 + 2016 * q^74 + 50 * q^75 + 1768 * q^76 + 98 * q^77 - 440 * q^78 + 1014 * q^79 - 840 * q^80 - 1454 * q^81 - 816 * q^82 - 376 * q^83 + 756 * q^84 + 250 * q^85 - 768 * q^86 - 410 * q^87 - 4688 * q^88 - 216 * q^89 - 80 * q^90 - 350 * q^91 + 264 * q^92 + 2760 * q^93 - 1928 * q^94 - 180 * q^95 - 2464 * q^96 + 2742 * q^97 + 392 * q^98 + 940 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
2.58579 6.65685 −1.31371 −5.00000 17.2132 −7.00000 −24.0833 17.3137 −12.9289
1.2 5.41421 −4.65685 21.3137 −5.00000 −25.2132 −7.00000 72.0833 −5.31371 −27.0711
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.a.b 2
3.b odd 2 1 315.4.a.f 2
4.b odd 2 1 560.4.a.r 2
5.b even 2 1 175.4.a.c 2
5.c odd 4 2 175.4.b.c 4
7.b odd 2 1 245.4.a.k 2
7.c even 3 2 245.4.e.h 4
7.d odd 6 2 245.4.e.i 4
8.b even 2 1 2240.4.a.bn 2
8.d odd 2 1 2240.4.a.bo 2
15.d odd 2 1 1575.4.a.z 2
21.c even 2 1 2205.4.a.u 2
35.c odd 2 1 1225.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 1.a even 1 1 trivial
175.4.a.c 2 5.b even 2 1
175.4.b.c 4 5.c odd 4 2
245.4.a.k 2 7.b odd 2 1
245.4.e.h 4 7.c even 3 2
245.4.e.i 4 7.d odd 6 2
315.4.a.f 2 3.b odd 2 1
560.4.a.r 2 4.b odd 2 1
1225.4.a.m 2 35.c odd 2 1
1575.4.a.z 2 15.d odd 2 1
2205.4.a.u 2 21.c even 2 1
2240.4.a.bn 2 8.b even 2 1
2240.4.a.bo 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 8T_{2} + 14$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(35))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 8T + 14$$
$3$ $$T^{2} - 2T - 31$$
$5$ $$(T + 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} + 14T - 1999$$
$13$ $$T^{2} - 50T + 593$$
$17$ $$T^{2} + 50T - 3247$$
$19$ $$T^{2} - 36T - 3548$$
$23$ $$T^{2} - 244T + 5636$$
$29$ $$T^{2} + 26T - 983$$
$31$ $$T^{2} + 120T - 61200$$
$37$ $$T^{2} - 564T + 72324$$
$41$ $$T^{2} + 328T - 3856$$
$43$ $$T^{2} + 260T + 7652$$
$47$ $$T^{2} + 350T - 4223$$
$53$ $$T^{2} + 56T - 31984$$
$59$ $$(T + 616)^{2}$$
$61$ $$T^{2} - 336T + 4896$$
$67$ $$T^{2} + 152T - 2416$$
$71$ $$(T + 952)^{2}$$
$73$ $$T^{2} - 676T - 122428$$
$79$ $$T^{2} - 1014 T + 134041$$
$83$ $$T^{2} + 376T - 684656$$
$89$ $$T^{2} + 216T + 7792$$
$97$ $$T^{2} - 2742 T + 1782841$$