# Properties

 Label 175.5.d.e Level $175$ Weight $5$ Character orbit 175.d Analytic conductor $18.090$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,5,Mod(76,175)]

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, 1]))

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

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

chi := DirichletCharacter("175.76");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 175.d (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: $$18.0897435397$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 35) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - \beta_1 q^{3} - 10 q^{4} + \beta_{3} q^{6} + ( - 14 \beta_{2} + 7 \beta_1) q^{7} + 26 \beta_{2} q^{8} + 56 q^{9}+O(q^{10})$$ q - b2 * q^2 - b1 * q^3 - 10 * q^4 + b3 * q^6 + (-14*b2 + 7*b1) * q^7 + 26*b2 * q^8 + 56 * q^9 $$q - \beta_{2} q^{2} - \beta_1 q^{3} - 10 q^{4} + \beta_{3} q^{6} + ( - 14 \beta_{2} + 7 \beta_1) q^{7} + 26 \beta_{2} q^{8} + 56 q^{9} + 89 q^{11} + 10 \beta_1 q^{12} + \beta_1 q^{13} + ( - 7 \beta_{3} + 84) q^{14} + 4 q^{16} - 97 \beta_1 q^{17} - 56 \beta_{2} q^{18} + 18 \beta_{3} q^{19} + (14 \beta_{3} + 175) q^{21} - 89 \beta_{2} q^{22} - 286 \beta_{2} q^{23} - 26 \beta_{3} q^{24} - \beta_{3} q^{26} - 137 \beta_1 q^{27} + (140 \beta_{2} - 70 \beta_1) q^{28} - 191 q^{29} + 86 \beta_{3} q^{31} - 420 \beta_{2} q^{32} - 89 \beta_1 q^{33} + 97 \beta_{3} q^{34} - 560 q^{36} - 666 \beta_{2} q^{37} - 108 \beta_1 q^{38} + 25 q^{39} - 238 \beta_{3} q^{41} + ( - 175 \beta_{2} - 84 \beta_1) q^{42} + 154 \beta_{2} q^{43} - 890 q^{44} + 1716 q^{46} - 439 \beta_1 q^{47} - 4 \beta_1 q^{48} + ( - 196 \beta_{3} - 49) q^{49} - 2425 q^{51} - 10 \beta_1 q^{52} - 648 \beta_{2} q^{53} + 137 \beta_{3} q^{54} + (182 \beta_{3} - 2184) q^{56} + 450 \beta_{2} q^{57} + 191 \beta_{2} q^{58} - 296 \beta_{3} q^{59} - 158 \beta_{3} q^{61} - 516 \beta_1 q^{62} + ( - 784 \beta_{2} + 392 \beta_1) q^{63} + 2456 q^{64} + 89 \beta_{3} q^{66} - 836 \beta_{2} q^{67} + 970 \beta_1 q^{68} + 286 \beta_{3} q^{69} + 4454 q^{71} + 1456 \beta_{2} q^{72} - 1730 \beta_1 q^{73} + 3996 q^{74} - 180 \beta_{3} q^{76} + ( - 1246 \beta_{2} + 623 \beta_1) q^{77} - 25 \beta_{2} q^{78} - 5561 q^{79} + 1111 q^{81} + 1428 \beta_1 q^{82} - 398 \beta_1 q^{83} + ( - 140 \beta_{3} - 1750) q^{84} - 924 q^{86} + 191 \beta_1 q^{87} + 2314 \beta_{2} q^{88} - 66 \beta_{3} q^{89} + ( - 14 \beta_{3} - 175) q^{91} + 2860 \beta_{2} q^{92} + 2150 \beta_{2} q^{93} + 439 \beta_{3} q^{94} + 420 \beta_{3} q^{96} + 1847 \beta_1 q^{97} + (49 \beta_{2} + 1176 \beta_1) q^{98} + 4984 q^{99}+O(q^{100})$$ q - b2 * q^2 - b1 * q^3 - 10 * q^4 + b3 * q^6 + (-14*b2 + 7*b1) * q^7 + 26*b2 * q^8 + 56 * q^9 + 89 * q^11 + 10*b1 * q^12 + b1 * q^13 + (-7*b3 + 84) * q^14 + 4 * q^16 - 97*b1 * q^17 - 56*b2 * q^18 + 18*b3 * q^19 + (14*b3 + 175) * q^21 - 89*b2 * q^22 - 286*b2 * q^23 - 26*b3 * q^24 - b3 * q^26 - 137*b1 * q^27 + (140*b2 - 70*b1) * q^28 - 191 * q^29 + 86*b3 * q^31 - 420*b2 * q^32 - 89*b1 * q^33 + 97*b3 * q^34 - 560 * q^36 - 666*b2 * q^37 - 108*b1 * q^38 + 25 * q^39 - 238*b3 * q^41 + (-175*b2 - 84*b1) * q^42 + 154*b2 * q^43 - 890 * q^44 + 1716 * q^46 - 439*b1 * q^47 - 4*b1 * q^48 + (-196*b3 - 49) * q^49 - 2425 * q^51 - 10*b1 * q^52 - 648*b2 * q^53 + 137*b3 * q^54 + (182*b3 - 2184) * q^56 + 450*b2 * q^57 + 191*b2 * q^58 - 296*b3 * q^59 - 158*b3 * q^61 - 516*b1 * q^62 + (-784*b2 + 392*b1) * q^63 + 2456 * q^64 + 89*b3 * q^66 - 836*b2 * q^67 + 970*b1 * q^68 + 286*b3 * q^69 + 4454 * q^71 + 1456*b2 * q^72 - 1730*b1 * q^73 + 3996 * q^74 - 180*b3 * q^76 + (-1246*b2 + 623*b1) * q^77 - 25*b2 * q^78 - 5561 * q^79 + 1111 * q^81 + 1428*b1 * q^82 - 398*b1 * q^83 + (-140*b3 - 1750) * q^84 - 924 * q^86 + 191*b1 * q^87 + 2314*b2 * q^88 - 66*b3 * q^89 + (-14*b3 - 175) * q^91 + 2860*b2 * q^92 + 2150*b2 * q^93 + 439*b3 * q^94 + 420*b3 * q^96 + 1847*b1 * q^97 + (49*b2 + 1176*b1) * q^98 + 4984 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 40 q^{4} + 224 q^{9}+O(q^{10})$$ 4 * q - 40 * q^4 + 224 * q^9 $$4 q - 40 q^{4} + 224 q^{9} + 356 q^{11} + 336 q^{14} + 16 q^{16} + 700 q^{21} - 764 q^{29} - 2240 q^{36} + 100 q^{39} - 3560 q^{44} + 6864 q^{46} - 196 q^{49} - 9700 q^{51} - 8736 q^{56} + 9824 q^{64} + 17816 q^{71} + 15984 q^{74} - 22244 q^{79} + 4444 q^{81} - 7000 q^{84} - 3696 q^{86} - 700 q^{91} + 19936 q^{99}+O(q^{100})$$ 4 * q - 40 * q^4 + 224 * q^9 + 356 * q^11 + 336 * q^14 + 16 * q^16 + 700 * q^21 - 764 * q^29 - 2240 * q^36 + 100 * q^39 - 3560 * q^44 + 6864 * q^46 - 196 * q^49 - 9700 * q^51 - 8736 * q^56 + 9824 * q^64 + 17816 * q^71 + 15984 * q^74 - 22244 * q^79 + 4444 * q^81 - 7000 * q^84 - 3696 * q^86 - 700 * q^91 + 19936 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 5\nu^{2} ) / 3$$ (5*v^2) / 3 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 3\nu ) / 3$$ (-v^3 + 3*v) / 3 $$\beta_{3}$$ $$=$$ $$( 5\nu^{3} + 15\nu ) / 3$$ (5*v^3 + 15*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + 5\beta_{2} ) / 10$$ (b3 + 5*b2) / 10 $$\nu^{2}$$ $$=$$ $$( 3\beta_1 ) / 5$$ (3*b1) / 5 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} - 15\beta_{2} ) / 10$$ (3*b3 - 15*b2) / 10

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i −1.22474 + 1.22474i
−2.44949 5.00000i −10.0000 0 12.2474i −34.2929 + 35.0000i 63.6867 56.0000 0
76.2 −2.44949 5.00000i −10.0000 0 12.2474i −34.2929 35.0000i 63.6867 56.0000 0
76.3 2.44949 5.00000i −10.0000 0 12.2474i 34.2929 + 35.0000i −63.6867 56.0000 0
76.4 2.44949 5.00000i −10.0000 0 12.2474i 34.2929 35.0000i −63.6867 56.0000 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
7.b odd 2 1 inner
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 175.5.d.e 4
5.b even 2 1 inner 175.5.d.e 4
5.c odd 4 1 35.5.c.c 2
5.c odd 4 1 35.5.c.d yes 2
7.b odd 2 1 inner 175.5.d.e 4
15.e even 4 1 315.5.e.c 2
15.e even 4 1 315.5.e.d 2
20.e even 4 1 560.5.p.d 2
20.e even 4 1 560.5.p.e 2
35.c odd 2 1 inner 175.5.d.e 4
35.f even 4 1 35.5.c.c 2
35.f even 4 1 35.5.c.d yes 2
105.k odd 4 1 315.5.e.c 2
105.k odd 4 1 315.5.e.d 2
140.j odd 4 1 560.5.p.d 2
140.j odd 4 1 560.5.p.e 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 6$$ acting on $$S_{5}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 6)^{2}$$
$3$ $$(T^{2} + 25)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 98 T^{2} + 5764801$$
$11$ $$(T - 89)^{4}$$
$13$ $$(T^{2} + 25)^{2}$$
$17$ $$(T^{2} + 235225)^{2}$$
$19$ $$(T^{2} + 48600)^{2}$$
$23$ $$(T^{2} - 490776)^{2}$$
$29$ $$(T + 191)^{4}$$
$31$ $$(T^{2} + 1109400)^{2}$$
$37$ $$(T^{2} - 2661336)^{2}$$
$41$ $$(T^{2} + 8496600)^{2}$$
$43$ $$(T^{2} - 142296)^{2}$$
$47$ $$(T^{2} + 4818025)^{2}$$
$53$ $$(T^{2} - 2519424)^{2}$$
$59$ $$(T^{2} + 13142400)^{2}$$
$61$ $$(T^{2} + 3744600)^{2}$$
$67$ $$(T^{2} - 4193376)^{2}$$
$71$ $$(T - 4454)^{4}$$
$73$ $$(T^{2} + 74822500)^{2}$$
$79$ $$(T + 5561)^{4}$$
$83$ $$(T^{2} + 3960100)^{2}$$
$89$ $$(T^{2} + 653400)^{2}$$
$97$ $$(T^{2} + 85285225)^{2}$$