# Properties

 Label 35.5.c.c Level $35$ Weight $5$ Character orbit 35.c Analytic conductor $3.618$ 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] = [35,5,Mod(34,35)]

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, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("35.34");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 35.c (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: $$3.61794870793$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 6$$ x^2 + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{-6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 5 q^{3} + 10 q^{4} + (10 \beta - 5) q^{5} - 5 \beta q^{6} + (14 \beta - 35) q^{7} + 26 \beta q^{8} - 56 q^{9} +O(q^{10})$$ q + b * q^2 - 5 * q^3 + 10 * q^4 + (10*b - 5) * q^5 - 5*b * q^6 + (14*b - 35) * q^7 + 26*b * q^8 - 56 * q^9 $$q + \beta q^{2} - 5 q^{3} + 10 q^{4} + (10 \beta - 5) q^{5} - 5 \beta q^{6} + (14 \beta - 35) q^{7} + 26 \beta q^{8} - 56 q^{9} + ( - 5 \beta - 60) q^{10} + 89 q^{11} - 50 q^{12} + 5 q^{13} + ( - 35 \beta - 84) q^{14} + ( - 50 \beta + 25) q^{15} + 4 q^{16} + 485 q^{17} - 56 \beta q^{18} + 90 \beta q^{19} + (100 \beta - 50) q^{20} + ( - 70 \beta + 175) q^{21} + 89 \beta q^{22} - 286 \beta q^{23} - 130 \beta q^{24} + ( - 100 \beta - 575) q^{25} + 5 \beta q^{26} + 685 q^{27} + (140 \beta - 350) q^{28} + 191 q^{29} + (25 \beta + 300) q^{30} - 430 \beta q^{31} + 420 \beta q^{32} - 445 q^{33} + 485 \beta q^{34} + ( - 420 \beta - 665) q^{35} - 560 q^{36} + 666 \beta q^{37} - 540 q^{38} - 25 q^{39} + ( - 130 \beta - 1560) q^{40} + 1190 \beta q^{41} + (175 \beta + 420) q^{42} + 154 \beta q^{43} + 890 q^{44} + ( - 560 \beta + 280) q^{45} + 1716 q^{46} + 2195 q^{47} - 20 q^{48} + ( - 980 \beta + 49) q^{49} + ( - 575 \beta + 600) q^{50} - 2425 q^{51} + 50 q^{52} - 648 \beta q^{53} + 685 \beta q^{54} + (890 \beta - 445) q^{55} + ( - 910 \beta - 2184) q^{56} - 450 \beta q^{57} + 191 \beta q^{58} - 1480 \beta q^{59} + ( - 500 \beta + 250) q^{60} + 790 \beta q^{61} + 2580 q^{62} + ( - 784 \beta + 1960) q^{63} - 2456 q^{64} + (50 \beta - 25) q^{65} - 445 \beta q^{66} + 836 \beta q^{67} + 4850 q^{68} + 1430 \beta q^{69} + ( - 665 \beta + 2520) q^{70} + 4454 q^{71} - 1456 \beta q^{72} - 8650 q^{73} - 3996 q^{74} + (500 \beta + 2875) q^{75} + 900 \beta q^{76} + (1246 \beta - 3115) q^{77} - 25 \beta q^{78} + 5561 q^{79} + (40 \beta - 20) q^{80} + 1111 q^{81} - 7140 q^{82} - 1990 q^{83} + ( - 700 \beta + 1750) q^{84} + (4850 \beta - 2425) q^{85} - 924 q^{86} - 955 q^{87} + 2314 \beta q^{88} - 330 \beta q^{89} + (280 \beta + 3360) q^{90} + (70 \beta - 175) q^{91} - 2860 \beta q^{92} + 2150 \beta q^{93} + 2195 \beta q^{94} + ( - 450 \beta - 5400) q^{95} - 2100 \beta q^{96} - 9235 q^{97} + (49 \beta + 5880) q^{98} - 4984 q^{99} +O(q^{100})$$ q + b * q^2 - 5 * q^3 + 10 * q^4 + (10*b - 5) * q^5 - 5*b * q^6 + (14*b - 35) * q^7 + 26*b * q^8 - 56 * q^9 + (-5*b - 60) * q^10 + 89 * q^11 - 50 * q^12 + 5 * q^13 + (-35*b - 84) * q^14 + (-50*b + 25) * q^15 + 4 * q^16 + 485 * q^17 - 56*b * q^18 + 90*b * q^19 + (100*b - 50) * q^20 + (-70*b + 175) * q^21 + 89*b * q^22 - 286*b * q^23 - 130*b * q^24 + (-100*b - 575) * q^25 + 5*b * q^26 + 685 * q^27 + (140*b - 350) * q^28 + 191 * q^29 + (25*b + 300) * q^30 - 430*b * q^31 + 420*b * q^32 - 445 * q^33 + 485*b * q^34 + (-420*b - 665) * q^35 - 560 * q^36 + 666*b * q^37 - 540 * q^38 - 25 * q^39 + (-130*b - 1560) * q^40 + 1190*b * q^41 + (175*b + 420) * q^42 + 154*b * q^43 + 890 * q^44 + (-560*b + 280) * q^45 + 1716 * q^46 + 2195 * q^47 - 20 * q^48 + (-980*b + 49) * q^49 + (-575*b + 600) * q^50 - 2425 * q^51 + 50 * q^52 - 648*b * q^53 + 685*b * q^54 + (890*b - 445) * q^55 + (-910*b - 2184) * q^56 - 450*b * q^57 + 191*b * q^58 - 1480*b * q^59 + (-500*b + 250) * q^60 + 790*b * q^61 + 2580 * q^62 + (-784*b + 1960) * q^63 - 2456 * q^64 + (50*b - 25) * q^65 - 445*b * q^66 + 836*b * q^67 + 4850 * q^68 + 1430*b * q^69 + (-665*b + 2520) * q^70 + 4454 * q^71 - 1456*b * q^72 - 8650 * q^73 - 3996 * q^74 + (500*b + 2875) * q^75 + 900*b * q^76 + (1246*b - 3115) * q^77 - 25*b * q^78 + 5561 * q^79 + (40*b - 20) * q^80 + 1111 * q^81 - 7140 * q^82 - 1990 * q^83 + (-700*b + 1750) * q^84 + (4850*b - 2425) * q^85 - 924 * q^86 - 955 * q^87 + 2314*b * q^88 - 330*b * q^89 + (280*b + 3360) * q^90 + (70*b - 175) * q^91 - 2860*b * q^92 + 2150*b * q^93 + 2195*b * q^94 + (-450*b - 5400) * q^95 - 2100*b * q^96 - 9235 * q^97 + (49*b + 5880) * q^98 - 4984 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{3} + 20 q^{4} - 10 q^{5} - 70 q^{7} - 112 q^{9}+O(q^{10})$$ 2 * q - 10 * q^3 + 20 * q^4 - 10 * q^5 - 70 * q^7 - 112 * q^9 $$2 q - 10 q^{3} + 20 q^{4} - 10 q^{5} - 70 q^{7} - 112 q^{9} - 120 q^{10} + 178 q^{11} - 100 q^{12} + 10 q^{13} - 168 q^{14} + 50 q^{15} + 8 q^{16} + 970 q^{17} - 100 q^{20} + 350 q^{21} - 1150 q^{25} + 1370 q^{27} - 700 q^{28} + 382 q^{29} + 600 q^{30} - 890 q^{33} - 1330 q^{35} - 1120 q^{36} - 1080 q^{38} - 50 q^{39} - 3120 q^{40} + 840 q^{42} + 1780 q^{44} + 560 q^{45} + 3432 q^{46} + 4390 q^{47} - 40 q^{48} + 98 q^{49} + 1200 q^{50} - 4850 q^{51} + 100 q^{52} - 890 q^{55} - 4368 q^{56} + 500 q^{60} + 5160 q^{62} + 3920 q^{63} - 4912 q^{64} - 50 q^{65} + 9700 q^{68} + 5040 q^{70} + 8908 q^{71} - 17300 q^{73} - 7992 q^{74} + 5750 q^{75} - 6230 q^{77} + 11122 q^{79} - 40 q^{80} + 2222 q^{81} - 14280 q^{82} - 3980 q^{83} + 3500 q^{84} - 4850 q^{85} - 1848 q^{86} - 1910 q^{87} + 6720 q^{90} - 350 q^{91} - 10800 q^{95} - 18470 q^{97} + 11760 q^{98} - 9968 q^{99}+O(q^{100})$$ 2 * q - 10 * q^3 + 20 * q^4 - 10 * q^5 - 70 * q^7 - 112 * q^9 - 120 * q^10 + 178 * q^11 - 100 * q^12 + 10 * q^13 - 168 * q^14 + 50 * q^15 + 8 * q^16 + 970 * q^17 - 100 * q^20 + 350 * q^21 - 1150 * q^25 + 1370 * q^27 - 700 * q^28 + 382 * q^29 + 600 * q^30 - 890 * q^33 - 1330 * q^35 - 1120 * q^36 - 1080 * q^38 - 50 * q^39 - 3120 * q^40 + 840 * q^42 + 1780 * q^44 + 560 * q^45 + 3432 * q^46 + 4390 * q^47 - 40 * q^48 + 98 * q^49 + 1200 * q^50 - 4850 * q^51 + 100 * q^52 - 890 * q^55 - 4368 * q^56 + 500 * q^60 + 5160 * q^62 + 3920 * q^63 - 4912 * q^64 - 50 * q^65 + 9700 * q^68 + 5040 * q^70 + 8908 * q^71 - 17300 * q^73 - 7992 * q^74 + 5750 * q^75 - 6230 * q^77 + 11122 * q^79 - 40 * q^80 + 2222 * q^81 - 14280 * q^82 - 3980 * q^83 + 3500 * q^84 - 4850 * q^85 - 1848 * q^86 - 1910 * q^87 + 6720 * q^90 - 350 * q^91 - 10800 * q^95 - 18470 * q^97 + 11760 * q^98 - 9968 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 − 2.44949i 2.44949i
2.44949i −5.00000 10.0000 −5.00000 24.4949i 12.2474i −35.0000 34.2929i 63.6867i −56.0000 −60.0000 + 12.2474i
34.2 2.44949i −5.00000 10.0000 −5.00000 + 24.4949i 12.2474i −35.0000 + 34.2929i 63.6867i −56.0000 −60.0000 12.2474i
 $$n$$: 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
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.5.c.c 2
3.b odd 2 1 315.5.e.d 2
4.b odd 2 1 560.5.p.e 2
5.b even 2 1 35.5.c.d yes 2
5.c odd 4 2 175.5.d.e 4
7.b odd 2 1 35.5.c.d yes 2
15.d odd 2 1 315.5.e.c 2
20.d odd 2 1 560.5.p.d 2
21.c even 2 1 315.5.e.c 2
28.d even 2 1 560.5.p.d 2
35.c odd 2 1 inner 35.5.c.c 2
35.f even 4 2 175.5.d.e 4
105.g even 2 1 315.5.e.d 2
140.c even 2 1 560.5.p.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.c 2 1.a even 1 1 trivial
35.5.c.c 2 35.c odd 2 1 inner
35.5.c.d yes 2 5.b even 2 1
35.5.c.d yes 2 7.b odd 2 1
175.5.d.e 4 5.c odd 4 2
175.5.d.e 4 35.f even 4 2
315.5.e.c 2 15.d odd 2 1
315.5.e.c 2 21.c even 2 1
315.5.e.d 2 3.b odd 2 1
315.5.e.d 2 105.g even 2 1
560.5.p.d 2 20.d odd 2 1
560.5.p.d 2 28.d even 2 1
560.5.p.e 2 4.b odd 2 1
560.5.p.e 2 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(35, [\chi])$$:

 $$T_{2}^{2} + 6$$ T2^2 + 6 $$T_{3} + 5$$ T3 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 6$$
$3$ $$(T + 5)^{2}$$
$5$ $$T^{2} + 10T + 625$$
$7$ $$T^{2} + 70T + 2401$$
$11$ $$(T - 89)^{2}$$
$13$ $$(T - 5)^{2}$$
$17$ $$(T - 485)^{2}$$
$19$ $$T^{2} + 48600$$
$23$ $$T^{2} + 490776$$
$29$ $$(T - 191)^{2}$$
$31$ $$T^{2} + 1109400$$
$37$ $$T^{2} + 2661336$$
$41$ $$T^{2} + 8496600$$
$43$ $$T^{2} + 142296$$
$47$ $$(T - 2195)^{2}$$
$53$ $$T^{2} + 2519424$$
$59$ $$T^{2} + 13142400$$
$61$ $$T^{2} + 3744600$$
$67$ $$T^{2} + 4193376$$
$71$ $$(T - 4454)^{2}$$
$73$ $$(T + 8650)^{2}$$
$79$ $$(T - 5561)^{2}$$
$83$ $$(T + 1990)^{2}$$
$89$ $$T^{2} + 653400$$
$97$ $$(T + 9235)^{2}$$