# Properties

 Label 175.6.b.a Level $175$ Weight $6$ Character orbit 175.b Analytic conductor $28.067$ 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] = [175,6,Mod(99,175)]

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([1, 0]))

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

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

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not 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: $$28.0671684673$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 10 i q^{2} - 14 i q^{3} - 68 q^{4} + 140 q^{6} + 49 i q^{7} - 360 i q^{8} + 47 q^{9} +O(q^{10})$$ q + 10*i * q^2 - 14*i * q^3 - 68 * q^4 + 140 * q^6 + 49*i * q^7 - 360*i * q^8 + 47 * q^9 $$q + 10 i q^{2} - 14 i q^{3} - 68 q^{4} + 140 q^{6} + 49 i q^{7} - 360 i q^{8} + 47 q^{9} + 232 q^{11} + 952 i q^{12} - 140 i q^{13} - 490 q^{14} + 1424 q^{16} + 1722 i q^{17} + 470 i q^{18} + 98 q^{19} + 686 q^{21} + 2320 i q^{22} + 1824 i q^{23} - 5040 q^{24} + 1400 q^{26} - 4060 i q^{27} - 3332 i q^{28} - 3418 q^{29} - 7644 q^{31} + 2720 i q^{32} - 3248 i q^{33} - 17220 q^{34} - 3196 q^{36} + 10398 i q^{37} + 980 i q^{38} - 1960 q^{39} - 17962 q^{41} + 6860 i q^{42} + 10880 i q^{43} - 15776 q^{44} - 18240 q^{46} - 9324 i q^{47} - 19936 i q^{48} - 2401 q^{49} + 24108 q^{51} + 9520 i q^{52} + 2262 i q^{53} + 40600 q^{54} + 17640 q^{56} - 1372 i q^{57} - 34180 i q^{58} + 2730 q^{59} + 25648 q^{61} - 76440 i q^{62} + 2303 i q^{63} + 18368 q^{64} + 32480 q^{66} + 48404 i q^{67} - 117096 i q^{68} + 25536 q^{69} - 58560 q^{71} - 16920 i q^{72} + 68082 i q^{73} - 103980 q^{74} - 6664 q^{76} + 11368 i q^{77} - 19600 i q^{78} - 31784 q^{79} - 45419 q^{81} - 179620 i q^{82} - 20538 i q^{83} - 46648 q^{84} - 108800 q^{86} + 47852 i q^{87} - 83520 i q^{88} + 50582 q^{89} + 6860 q^{91} - 124032 i q^{92} + 107016 i q^{93} + 93240 q^{94} + 38080 q^{96} + 58506 i q^{97} - 24010 i q^{98} + 10904 q^{99} +O(q^{100})$$ q + 10*i * q^2 - 14*i * q^3 - 68 * q^4 + 140 * q^6 + 49*i * q^7 - 360*i * q^8 + 47 * q^9 + 232 * q^11 + 952*i * q^12 - 140*i * q^13 - 490 * q^14 + 1424 * q^16 + 1722*i * q^17 + 470*i * q^18 + 98 * q^19 + 686 * q^21 + 2320*i * q^22 + 1824*i * q^23 - 5040 * q^24 + 1400 * q^26 - 4060*i * q^27 - 3332*i * q^28 - 3418 * q^29 - 7644 * q^31 + 2720*i * q^32 - 3248*i * q^33 - 17220 * q^34 - 3196 * q^36 + 10398*i * q^37 + 980*i * q^38 - 1960 * q^39 - 17962 * q^41 + 6860*i * q^42 + 10880*i * q^43 - 15776 * q^44 - 18240 * q^46 - 9324*i * q^47 - 19936*i * q^48 - 2401 * q^49 + 24108 * q^51 + 9520*i * q^52 + 2262*i * q^53 + 40600 * q^54 + 17640 * q^56 - 1372*i * q^57 - 34180*i * q^58 + 2730 * q^59 + 25648 * q^61 - 76440*i * q^62 + 2303*i * q^63 + 18368 * q^64 + 32480 * q^66 + 48404*i * q^67 - 117096*i * q^68 + 25536 * q^69 - 58560 * q^71 - 16920*i * q^72 + 68082*i * q^73 - 103980 * q^74 - 6664 * q^76 + 11368*i * q^77 - 19600*i * q^78 - 31784 * q^79 - 45419 * q^81 - 179620*i * q^82 - 20538*i * q^83 - 46648 * q^84 - 108800 * q^86 + 47852*i * q^87 - 83520*i * q^88 + 50582 * q^89 + 6860 * q^91 - 124032*i * q^92 + 107016*i * q^93 + 93240 * q^94 + 38080 * q^96 + 58506*i * q^97 - 24010*i * q^98 + 10904 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 136 q^{4} + 280 q^{6} + 94 q^{9}+O(q^{10})$$ 2 * q - 136 * q^4 + 280 * q^6 + 94 * q^9 $$2 q - 136 q^{4} + 280 q^{6} + 94 q^{9} + 464 q^{11} - 980 q^{14} + 2848 q^{16} + 196 q^{19} + 1372 q^{21} - 10080 q^{24} + 2800 q^{26} - 6836 q^{29} - 15288 q^{31} - 34440 q^{34} - 6392 q^{36} - 3920 q^{39} - 35924 q^{41} - 31552 q^{44} - 36480 q^{46} - 4802 q^{49} + 48216 q^{51} + 81200 q^{54} + 35280 q^{56} + 5460 q^{59} + 51296 q^{61} + 36736 q^{64} + 64960 q^{66} + 51072 q^{69} - 117120 q^{71} - 207960 q^{74} - 13328 q^{76} - 63568 q^{79} - 90838 q^{81} - 93296 q^{84} - 217600 q^{86} + 101164 q^{89} + 13720 q^{91} + 186480 q^{94} + 76160 q^{96} + 21808 q^{99}+O(q^{100})$$ 2 * q - 136 * q^4 + 280 * q^6 + 94 * q^9 + 464 * q^11 - 980 * q^14 + 2848 * q^16 + 196 * q^19 + 1372 * q^21 - 10080 * q^24 + 2800 * q^26 - 6836 * q^29 - 15288 * q^31 - 34440 * q^34 - 6392 * q^36 - 3920 * q^39 - 35924 * q^41 - 31552 * q^44 - 36480 * q^46 - 4802 * q^49 + 48216 * q^51 + 81200 * q^54 + 35280 * q^56 + 5460 * q^59 + 51296 * q^61 + 36736 * q^64 + 64960 * q^66 + 51072 * q^69 - 117120 * q^71 - 207960 * q^74 - 13328 * q^76 - 63568 * q^79 - 90838 * q^81 - 93296 * q^84 - 217600 * q^86 + 101164 * q^89 + 13720 * q^91 + 186480 * q^94 + 76160 * q^96 + 21808 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
10.0000i 14.0000i −68.0000 0 140.000 49.0000i 360.000i 47.0000 0
99.2 10.0000i 14.0000i −68.0000 0 140.000 49.0000i 360.000i 47.0000 0
 $$n$$: 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
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 175.6.b.a 2
5.b even 2 1 inner 175.6.b.a 2
5.c odd 4 1 7.6.a.a 1
5.c odd 4 1 175.6.a.b 1
15.e even 4 1 63.6.a.e 1
20.e even 4 1 112.6.a.g 1
35.f even 4 1 49.6.a.a 1
35.k even 12 2 49.6.c.b 2
35.l odd 12 2 49.6.c.c 2
40.i odd 4 1 448.6.a.m 1
40.k even 4 1 448.6.a.c 1
55.e even 4 1 847.6.a.b 1
60.l odd 4 1 1008.6.a.y 1
105.k odd 4 1 441.6.a.k 1
140.j odd 4 1 784.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 5.c odd 4 1
49.6.a.a 1 35.f even 4 1
49.6.c.b 2 35.k even 12 2
49.6.c.c 2 35.l odd 12 2
63.6.a.e 1 15.e even 4 1
112.6.a.g 1 20.e even 4 1
175.6.a.b 1 5.c odd 4 1
175.6.b.a 2 1.a even 1 1 trivial
175.6.b.a 2 5.b even 2 1 inner
441.6.a.k 1 105.k odd 4 1
448.6.a.c 1 40.k even 4 1
448.6.a.m 1 40.i odd 4 1
784.6.a.c 1 140.j odd 4 1
847.6.a.b 1 55.e even 4 1
1008.6.a.y 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 100$$ acting on $$S_{6}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 100$$
$3$ $$T^{2} + 196$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T - 232)^{2}$$
$13$ $$T^{2} + 19600$$
$17$ $$T^{2} + 2965284$$
$19$ $$(T - 98)^{2}$$
$23$ $$T^{2} + 3326976$$
$29$ $$(T + 3418)^{2}$$
$31$ $$(T + 7644)^{2}$$
$37$ $$T^{2} + 108118404$$
$41$ $$(T + 17962)^{2}$$
$43$ $$T^{2} + 118374400$$
$47$ $$T^{2} + 86936976$$
$53$ $$T^{2} + 5116644$$
$59$ $$(T - 2730)^{2}$$
$61$ $$(T - 25648)^{2}$$
$67$ $$T^{2} + 2342947216$$
$71$ $$(T + 58560)^{2}$$
$73$ $$T^{2} + 4635158724$$
$79$ $$(T + 31784)^{2}$$
$83$ $$T^{2} + 421809444$$
$89$ $$(T - 50582)^{2}$$
$97$ $$T^{2} + 3422952036$$