# Properties

 Label 35.8.a.a Level $35$ Weight $8$ Character orbit 35.a Self dual yes Analytic conductor $10.933$ Analytic rank $1$ 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] = [35,8,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, 8, names="a")

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

chi := DirichletCharacter("35.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$10.9334758919$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 8) q^{2} + ( - 6 \beta - 15) q^{3} + (16 \beta - 20) q^{4} + 125 q^{5} + ( - 63 \beta - 384) q^{6} - 343 q^{7} + ( - 20 \beta - 480) q^{8} + (180 \beta - 378) q^{9}+O(q^{10})$$ q + (b + 8) * q^2 + (-6*b - 15) * q^3 + (16*b - 20) * q^4 + 125 * q^5 + (-63*b - 384) * q^6 - 343 * q^7 + (-20*b - 480) * q^8 + (180*b - 378) * q^9 $$q + (\beta + 8) q^{2} + ( - 6 \beta - 15) q^{3} + (16 \beta - 20) q^{4} + 125 q^{5} + ( - 63 \beta - 384) q^{6} - 343 q^{7} + ( - 20 \beta - 480) q^{8} + (180 \beta - 378) q^{9} + (125 \beta + 1000) q^{10} + ( - 380 \beta - 3953) q^{11} + ( - 120 \beta - 3924) q^{12} + ( - 418 \beta - 8909) q^{13} + ( - 343 \beta - 2744) q^{14} + ( - 750 \beta - 1875) q^{15} + ( - 2688 \beta - 2160) q^{16} + (2210 \beta - 1199) q^{17} + (1062 \beta + 4896) q^{18} + (5542 \beta - 1806) q^{19} + (2000 \beta - 2500) q^{20} + (2058 \beta + 5145) q^{21} + ( - 6993 \beta - 48344) q^{22} + (10690 \beta + 6922) q^{23} + (3180 \beta + 12480) q^{24} + 15625 q^{25} + ( - 12253 \beta - 89664) q^{26} + (12690 \beta - 9045) q^{27} + ( - 5488 \beta + 6860) q^{28} + ( - 23772 \beta - 63449) q^{29} + ( - 7875 \beta - 48000) q^{30} + ( - 22554 \beta + 126384) q^{31} + ( - 21104 \beta - 74112) q^{32} + (29418 \beta + 159615) q^{33} + (16481 \beta + 87648) q^{34} - 42875 q^{35} + ( - 9648 \beta + 134280) q^{36} + ( - 43638 \beta - 132930) q^{37} + (42530 \beta + 229400) q^{38} + (59724 \beta + 243987) q^{39} + ( - 2500 \beta - 60000) q^{40} + (37354 \beta - 55960) q^{41} + (21609 \beta + 131712) q^{42} + ( - 24578 \beta + 473786) q^{43} + ( - 55648 \beta - 188460) q^{44} + (22500 \beta - 47250) q^{45} + (92442 \beta + 525736) q^{46} + ( - 56742 \beta + 135637) q^{47} + (53280 \beta + 742032) q^{48} + 117649 q^{49} + (15625 \beta + 125000) q^{50} + ( - 25956 \beta - 565455) q^{51} + ( - 134184 \beta - 116092) q^{52} + ( - 65224 \beta - 633896) q^{53} + (92475 \beta + 486000) q^{54} + ( - 47500 \beta - 494125) q^{55} + (6860 \beta + 164640) q^{56} + ( - 72294 \beta - 1435998) q^{57} + ( - 253625 \beta - 1553560) q^{58} + (170640 \beta - 680060) q^{59} + ( - 15000 \beta - 490500) q^{60} + (11334 \beta - 906840) q^{61} + ( - 54048 \beta + 18696) q^{62} + ( - 61740 \beta + 129654) q^{63} + (101120 \beta - 1244992) q^{64} + ( - 52250 \beta - 1113625) q^{65} + (394959 \beta + 2571312) q^{66} + (506344 \beta - 1094656) q^{67} + ( - 63384 \beta + 1579820) q^{68} + ( - 201882 \beta - 2925990) q^{69} + ( - 42875 \beta - 343000) q^{70} + ( - 222048 \beta - 747464) q^{71} + ( - 78840 \beta + 23040) q^{72} + (212396 \beta + 3584894) q^{73} + ( - 482034 \beta - 2983512) q^{74} + ( - 93750 \beta - 234375) q^{75} + ( - 139736 \beta + 3937688) q^{76} + (130340 \beta + 1355879) q^{77} + (721779 \beta + 4579752) q^{78} + (187504 \beta - 3971487) q^{79} + ( - 336000 \beta - 270000) q^{80} + ( - 529740 \beta - 2387799) q^{81} + (242872 \beta + 1195896) q^{82} + ( - 939444 \beta - 152356) q^{83} + (41160 \beta + 1345932) q^{84} + (276250 \beta - 149875) q^{85} + (277162 \beta + 2708856) q^{86} + (737274 \beta + 7227543) q^{87} + (261460 \beta + 2231840) q^{88} + (247538 \beta - 8971764) q^{89} + (132750 \beta + 612000) q^{90} + (143374 \beta + 3055787) q^{91} + ( - 103048 \beta + 7387320) q^{92} + ( - 419994 \beta + 4058496) q^{93} + ( - 318299 \beta - 1411552) q^{94} + (692750 \beta - 225750) q^{95} + (761232 \beta + 6683136) q^{96} + ( - 680782 \beta + 2129037) q^{97} + (117649 \beta + 941192) q^{98} + ( - 567900 \beta - 1515366) q^{99}+O(q^{100})$$ q + (b + 8) * q^2 + (-6*b - 15) * q^3 + (16*b - 20) * q^4 + 125 * q^5 + (-63*b - 384) * q^6 - 343 * q^7 + (-20*b - 480) * q^8 + (180*b - 378) * q^9 + (125*b + 1000) * q^10 + (-380*b - 3953) * q^11 + (-120*b - 3924) * q^12 + (-418*b - 8909) * q^13 + (-343*b - 2744) * q^14 + (-750*b - 1875) * q^15 + (-2688*b - 2160) * q^16 + (2210*b - 1199) * q^17 + (1062*b + 4896) * q^18 + (5542*b - 1806) * q^19 + (2000*b - 2500) * q^20 + (2058*b + 5145) * q^21 + (-6993*b - 48344) * q^22 + (10690*b + 6922) * q^23 + (3180*b + 12480) * q^24 + 15625 * q^25 + (-12253*b - 89664) * q^26 + (12690*b - 9045) * q^27 + (-5488*b + 6860) * q^28 + (-23772*b - 63449) * q^29 + (-7875*b - 48000) * q^30 + (-22554*b + 126384) * q^31 + (-21104*b - 74112) * q^32 + (29418*b + 159615) * q^33 + (16481*b + 87648) * q^34 - 42875 * q^35 + (-9648*b + 134280) * q^36 + (-43638*b - 132930) * q^37 + (42530*b + 229400) * q^38 + (59724*b + 243987) * q^39 + (-2500*b - 60000) * q^40 + (37354*b - 55960) * q^41 + (21609*b + 131712) * q^42 + (-24578*b + 473786) * q^43 + (-55648*b - 188460) * q^44 + (22500*b - 47250) * q^45 + (92442*b + 525736) * q^46 + (-56742*b + 135637) * q^47 + (53280*b + 742032) * q^48 + 117649 * q^49 + (15625*b + 125000) * q^50 + (-25956*b - 565455) * q^51 + (-134184*b - 116092) * q^52 + (-65224*b - 633896) * q^53 + (92475*b + 486000) * q^54 + (-47500*b - 494125) * q^55 + (6860*b + 164640) * q^56 + (-72294*b - 1435998) * q^57 + (-253625*b - 1553560) * q^58 + (170640*b - 680060) * q^59 + (-15000*b - 490500) * q^60 + (11334*b - 906840) * q^61 + (-54048*b + 18696) * q^62 + (-61740*b + 129654) * q^63 + (101120*b - 1244992) * q^64 + (-52250*b - 1113625) * q^65 + (394959*b + 2571312) * q^66 + (506344*b - 1094656) * q^67 + (-63384*b + 1579820) * q^68 + (-201882*b - 2925990) * q^69 + (-42875*b - 343000) * q^70 + (-222048*b - 747464) * q^71 + (-78840*b + 23040) * q^72 + (212396*b + 3584894) * q^73 + (-482034*b - 2983512) * q^74 + (-93750*b - 234375) * q^75 + (-139736*b + 3937688) * q^76 + (130340*b + 1355879) * q^77 + (721779*b + 4579752) * q^78 + (187504*b - 3971487) * q^79 + (-336000*b - 270000) * q^80 + (-529740*b - 2387799) * q^81 + (242872*b + 1195896) * q^82 + (-939444*b - 152356) * q^83 + (41160*b + 1345932) * q^84 + (276250*b - 149875) * q^85 + (277162*b + 2708856) * q^86 + (737274*b + 7227543) * q^87 + (261460*b + 2231840) * q^88 + (247538*b - 8971764) * q^89 + (132750*b + 612000) * q^90 + (143374*b + 3055787) * q^91 + (-103048*b + 7387320) * q^92 + (-419994*b + 4058496) * q^93 + (-318299*b - 1411552) * q^94 + (692750*b - 225750) * q^95 + (761232*b + 6683136) * q^96 + (-680782*b + 2129037) * q^97 + (117649*b + 941192) * q^98 + (-567900*b - 1515366) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{2} - 30 q^{3} - 40 q^{4} + 250 q^{5} - 768 q^{6} - 686 q^{7} - 960 q^{8} - 756 q^{9}+O(q^{10})$$ 2 * q + 16 * q^2 - 30 * q^3 - 40 * q^4 + 250 * q^5 - 768 * q^6 - 686 * q^7 - 960 * q^8 - 756 * q^9 $$2 q + 16 q^{2} - 30 q^{3} - 40 q^{4} + 250 q^{5} - 768 q^{6} - 686 q^{7} - 960 q^{8} - 756 q^{9} + 2000 q^{10} - 7906 q^{11} - 7848 q^{12} - 17818 q^{13} - 5488 q^{14} - 3750 q^{15} - 4320 q^{16} - 2398 q^{17} + 9792 q^{18} - 3612 q^{19} - 5000 q^{20} + 10290 q^{21} - 96688 q^{22} + 13844 q^{23} + 24960 q^{24} + 31250 q^{25} - 179328 q^{26} - 18090 q^{27} + 13720 q^{28} - 126898 q^{29} - 96000 q^{30} + 252768 q^{31} - 148224 q^{32} + 319230 q^{33} + 175296 q^{34} - 85750 q^{35} + 268560 q^{36} - 265860 q^{37} + 458800 q^{38} + 487974 q^{39} - 120000 q^{40} - 111920 q^{41} + 263424 q^{42} + 947572 q^{43} - 376920 q^{44} - 94500 q^{45} + 1051472 q^{46} + 271274 q^{47} + 1484064 q^{48} + 235298 q^{49} + 250000 q^{50} - 1130910 q^{51} - 232184 q^{52} - 1267792 q^{53} + 972000 q^{54} - 988250 q^{55} + 329280 q^{56} - 2871996 q^{57} - 3107120 q^{58} - 1360120 q^{59} - 981000 q^{60} - 1813680 q^{61} + 37392 q^{62} + 259308 q^{63} - 2489984 q^{64} - 2227250 q^{65} + 5142624 q^{66} - 2189312 q^{67} + 3159640 q^{68} - 5851980 q^{69} - 686000 q^{70} - 1494928 q^{71} + 46080 q^{72} + 7169788 q^{73} - 5967024 q^{74} - 468750 q^{75} + 7875376 q^{76} + 2711758 q^{77} + 9159504 q^{78} - 7942974 q^{79} - 540000 q^{80} - 4775598 q^{81} + 2391792 q^{82} - 304712 q^{83} + 2691864 q^{84} - 299750 q^{85} + 5417712 q^{86} + 14455086 q^{87} + 4463680 q^{88} - 17943528 q^{89} + 1224000 q^{90} + 6111574 q^{91} + 14774640 q^{92} + 8116992 q^{93} - 2823104 q^{94} - 451500 q^{95} + 13366272 q^{96} + 4258074 q^{97} + 1882384 q^{98} - 3030732 q^{99}+O(q^{100})$$ 2 * q + 16 * q^2 - 30 * q^3 - 40 * q^4 + 250 * q^5 - 768 * q^6 - 686 * q^7 - 960 * q^8 - 756 * q^9 + 2000 * q^10 - 7906 * q^11 - 7848 * q^12 - 17818 * q^13 - 5488 * q^14 - 3750 * q^15 - 4320 * q^16 - 2398 * q^17 + 9792 * q^18 - 3612 * q^19 - 5000 * q^20 + 10290 * q^21 - 96688 * q^22 + 13844 * q^23 + 24960 * q^24 + 31250 * q^25 - 179328 * q^26 - 18090 * q^27 + 13720 * q^28 - 126898 * q^29 - 96000 * q^30 + 252768 * q^31 - 148224 * q^32 + 319230 * q^33 + 175296 * q^34 - 85750 * q^35 + 268560 * q^36 - 265860 * q^37 + 458800 * q^38 + 487974 * q^39 - 120000 * q^40 - 111920 * q^41 + 263424 * q^42 + 947572 * q^43 - 376920 * q^44 - 94500 * q^45 + 1051472 * q^46 + 271274 * q^47 + 1484064 * q^48 + 235298 * q^49 + 250000 * q^50 - 1130910 * q^51 - 232184 * q^52 - 1267792 * q^53 + 972000 * q^54 - 988250 * q^55 + 329280 * q^56 - 2871996 * q^57 - 3107120 * q^58 - 1360120 * q^59 - 981000 * q^60 - 1813680 * q^61 + 37392 * q^62 + 259308 * q^63 - 2489984 * q^64 - 2227250 * q^65 + 5142624 * q^66 - 2189312 * q^67 + 3159640 * q^68 - 5851980 * q^69 - 686000 * q^70 - 1494928 * q^71 + 46080 * q^72 + 7169788 * q^73 - 5967024 * q^74 - 468750 * q^75 + 7875376 * q^76 + 2711758 * q^77 + 9159504 * q^78 - 7942974 * q^79 - 540000 * q^80 - 4775598 * q^81 + 2391792 * q^82 - 304712 * q^83 + 2691864 * q^84 - 299750 * q^85 + 5417712 * q^86 + 14455086 * q^87 + 4463680 * q^88 - 17943528 * q^89 + 1224000 * q^90 + 6111574 * q^91 + 14774640 * q^92 + 8116992 * q^93 - 2823104 * q^94 - 451500 * q^95 + 13366272 * q^96 + 4258074 * q^97 + 1882384 * q^98 - 3030732 * 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
 −3.31662 3.31662
1.36675 24.7995 −126.132 125.000 33.8947 −343.000 −347.335 −1571.98 170.844
1.2 14.6332 −54.7995 86.1320 125.000 −801.895 −343.000 −612.665 815.985 1829.16
 $$n$$: 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 35.8.a.a 2
3.b odd 2 1 315.8.a.c 2
4.b odd 2 1 560.8.a.i 2
5.b even 2 1 175.8.a.b 2
5.c odd 4 2 175.8.b.c 4
7.b odd 2 1 245.8.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.a.a 2 1.a even 1 1 trivial
175.8.a.b 2 5.b even 2 1
175.8.b.c 4 5.c odd 4 2
245.8.a.b 2 7.b odd 2 1
315.8.a.c 2 3.b odd 2 1
560.8.a.i 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 16T_{2} + 20$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(35))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 16T + 20$$
$3$ $$T^{2} + 30T - 1359$$
$5$ $$(T - 125)^{2}$$
$7$ $$(T + 343)^{2}$$
$11$ $$T^{2} + 7906 T + 9272609$$
$13$ $$T^{2} + 17818 T + 71682425$$
$17$ $$T^{2} + 2398 T - 213462799$$
$19$ $$T^{2} + \cdots - 1348143980$$
$23$ $$T^{2} + \cdots - 4980234316$$
$29$ $$T^{2} + \cdots - 20838975695$$
$31$ $$T^{2} + \cdots - 6409132848$$
$37$ $$T^{2} + \cdots - 66117717036$$
$41$ $$T^{2} + \cdots - 58262616304$$
$43$ $$T^{2} + \cdots + 197893738100$$
$47$ $$T^{2} + \cdots - 123267405047$$
$53$ $$T^{2} + \cdots + 214640651072$$
$59$ $$T^{2} + \cdots - 818710818800$$
$61$ $$T^{2} + \cdots + 816706565136$$
$67$ $$T^{2} + \cdots - 10082635080448$$
$71$ $$T^{2} + \cdots - 1610731398080$$
$73$ $$T^{2} + \cdots + 10866534315332$$
$79$ $$T^{2} + \cdots + 14225767990465$$
$83$ $$T^{2} + \cdots - 38809208931248$$
$89$ $$T^{2} + \cdots + 77796446568160$$
$97$ $$T^{2} + \cdots - 15859623239687$$