# Properties

 Label 175.6.a.b Level $175$ Weight $6$ Character orbit 175.a Self dual yes Analytic conductor $28.067$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,6,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("175.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$28.0671684673$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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)

 $$f(q)$$ $$=$$ $$q + 10 q^{2} + 14 q^{3} + 68 q^{4} + 140 q^{6} + 49 q^{7} + 360 q^{8} - 47 q^{9}+O(q^{10})$$ q + 10 * q^2 + 14 * q^3 + 68 * q^4 + 140 * q^6 + 49 * q^7 + 360 * q^8 - 47 * q^9 $$q + 10 q^{2} + 14 q^{3} + 68 q^{4} + 140 q^{6} + 49 q^{7} + 360 q^{8} - 47 q^{9} + 232 q^{11} + 952 q^{12} + 140 q^{13} + 490 q^{14} + 1424 q^{16} + 1722 q^{17} - 470 q^{18} - 98 q^{19} + 686 q^{21} + 2320 q^{22} - 1824 q^{23} + 5040 q^{24} + 1400 q^{26} - 4060 q^{27} + 3332 q^{28} + 3418 q^{29} - 7644 q^{31} + 2720 q^{32} + 3248 q^{33} + 17220 q^{34} - 3196 q^{36} + 10398 q^{37} - 980 q^{38} + 1960 q^{39} - 17962 q^{41} + 6860 q^{42} - 10880 q^{43} + 15776 q^{44} - 18240 q^{46} - 9324 q^{47} + 19936 q^{48} + 2401 q^{49} + 24108 q^{51} + 9520 q^{52} - 2262 q^{53} - 40600 q^{54} + 17640 q^{56} - 1372 q^{57} + 34180 q^{58} - 2730 q^{59} + 25648 q^{61} - 76440 q^{62} - 2303 q^{63} - 18368 q^{64} + 32480 q^{66} + 48404 q^{67} + 117096 q^{68} - 25536 q^{69} - 58560 q^{71} - 16920 q^{72} - 68082 q^{73} + 103980 q^{74} - 6664 q^{76} + 11368 q^{77} + 19600 q^{78} + 31784 q^{79} - 45419 q^{81} - 179620 q^{82} + 20538 q^{83} + 46648 q^{84} - 108800 q^{86} + 47852 q^{87} + 83520 q^{88} - 50582 q^{89} + 6860 q^{91} - 124032 q^{92} - 107016 q^{93} - 93240 q^{94} + 38080 q^{96} + 58506 q^{97} + 24010 q^{98} - 10904 q^{99}+O(q^{100})$$ q + 10 * q^2 + 14 * q^3 + 68 * q^4 + 140 * q^6 + 49 * q^7 + 360 * q^8 - 47 * q^9 + 232 * q^11 + 952 * q^12 + 140 * q^13 + 490 * q^14 + 1424 * q^16 + 1722 * q^17 - 470 * q^18 - 98 * q^19 + 686 * q^21 + 2320 * q^22 - 1824 * q^23 + 5040 * q^24 + 1400 * q^26 - 4060 * q^27 + 3332 * q^28 + 3418 * q^29 - 7644 * q^31 + 2720 * q^32 + 3248 * q^33 + 17220 * q^34 - 3196 * q^36 + 10398 * q^37 - 980 * q^38 + 1960 * q^39 - 17962 * q^41 + 6860 * q^42 - 10880 * q^43 + 15776 * q^44 - 18240 * q^46 - 9324 * q^47 + 19936 * q^48 + 2401 * q^49 + 24108 * q^51 + 9520 * q^52 - 2262 * q^53 - 40600 * q^54 + 17640 * q^56 - 1372 * q^57 + 34180 * q^58 - 2730 * q^59 + 25648 * q^61 - 76440 * q^62 - 2303 * q^63 - 18368 * q^64 + 32480 * q^66 + 48404 * q^67 + 117096 * q^68 - 25536 * q^69 - 58560 * q^71 - 16920 * q^72 - 68082 * q^73 + 103980 * q^74 - 6664 * q^76 + 11368 * q^77 + 19600 * q^78 + 31784 * q^79 - 45419 * q^81 - 179620 * q^82 + 20538 * q^83 + 46648 * q^84 - 108800 * q^86 + 47852 * q^87 + 83520 * q^88 - 50582 * q^89 + 6860 * q^91 - 124032 * q^92 - 107016 * q^93 - 93240 * q^94 + 38080 * q^96 + 58506 * q^97 + 24010 * q^98 - 10904 * 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
 0
10.0000 14.0000 68.0000 0 140.000 49.0000 360.000 −47.0000 0
 $$n$$: 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 175.6.a.b 1
5.b even 2 1 7.6.a.a 1
5.c odd 4 2 175.6.b.a 2
15.d odd 2 1 63.6.a.e 1
20.d odd 2 1 112.6.a.g 1
35.c odd 2 1 49.6.a.a 1
35.i odd 6 2 49.6.c.b 2
35.j even 6 2 49.6.c.c 2
40.e odd 2 1 448.6.a.c 1
40.f even 2 1 448.6.a.m 1
55.d odd 2 1 847.6.a.b 1
60.h even 2 1 1008.6.a.y 1
105.g even 2 1 441.6.a.k 1
140.c even 2 1 784.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 5.b even 2 1
49.6.a.a 1 35.c odd 2 1
49.6.c.b 2 35.i odd 6 2
49.6.c.c 2 35.j even 6 2
63.6.a.e 1 15.d odd 2 1
112.6.a.g 1 20.d odd 2 1
175.6.a.b 1 1.a even 1 1 trivial
175.6.b.a 2 5.c odd 4 2
441.6.a.k 1 105.g even 2 1
448.6.a.c 1 40.e odd 2 1
448.6.a.m 1 40.f even 2 1
784.6.a.c 1 140.c even 2 1
847.6.a.b 1 55.d odd 2 1
1008.6.a.y 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 10$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 10$$
$3$ $$T - 14$$
$5$ $$T$$
$7$ $$T - 49$$
$11$ $$T - 232$$
$13$ $$T - 140$$
$17$ $$T - 1722$$
$19$ $$T + 98$$
$23$ $$T + 1824$$
$29$ $$T - 3418$$
$31$ $$T + 7644$$
$37$ $$T - 10398$$
$41$ $$T + 17962$$
$43$ $$T + 10880$$
$47$ $$T + 9324$$
$53$ $$T + 2262$$
$59$ $$T + 2730$$
$61$ $$T - 25648$$
$67$ $$T - 48404$$
$71$ $$T + 58560$$
$73$ $$T + 68082$$
$79$ $$T - 31784$$
$83$ $$T - 20538$$
$89$ $$T + 50582$$
$97$ $$T - 58506$$