# Properties

 Label 175.4.e.b Level $175$ Weight $4$ Character orbit 175.e Analytic conductor $10.325$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(51,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("175.51");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.e (of order $$3$$, degree $$2$$, 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: $$10.3253342510$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - 6 q^{6} + (14 \zeta_{6} + 7) q^{7} + 21 q^{8} + 23 \zeta_{6} q^{9}+O(q^{10})$$ q + 3*z * q^2 + (2*z - 2) * q^3 + (z - 1) * q^4 - 6 * q^6 + (14*z + 7) * q^7 + 21 * q^8 + 23*z * q^9 $$q + 3 \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} - 6 q^{6} + (14 \zeta_{6} + 7) q^{7} + 21 q^{8} + 23 \zeta_{6} q^{9} + ( - 45 \zeta_{6} + 45) q^{11} - 2 \zeta_{6} q^{12} - 59 q^{13} + (63 \zeta_{6} - 42) q^{14} + 71 \zeta_{6} q^{16} + (54 \zeta_{6} - 54) q^{17} + (69 \zeta_{6} - 69) q^{18} + 121 \zeta_{6} q^{19} + (14 \zeta_{6} - 42) q^{21} + 135 q^{22} + 69 \zeta_{6} q^{23} + (42 \zeta_{6} - 42) q^{24} - 177 \zeta_{6} q^{26} - 100 q^{27} + (7 \zeta_{6} - 21) q^{28} - 162 q^{29} + ( - 88 \zeta_{6} + 88) q^{31} + (45 \zeta_{6} - 45) q^{32} + 90 \zeta_{6} q^{33} - 162 q^{34} - 23 q^{36} - 259 \zeta_{6} q^{37} + (363 \zeta_{6} - 363) q^{38} + ( - 118 \zeta_{6} + 118) q^{39} + 195 q^{41} + ( - 84 \zeta_{6} - 42) q^{42} + 286 q^{43} + 45 \zeta_{6} q^{44} + (207 \zeta_{6} - 207) q^{46} + 45 \zeta_{6} q^{47} - 142 q^{48} + (392 \zeta_{6} - 147) q^{49} - 108 \zeta_{6} q^{51} + ( - 59 \zeta_{6} + 59) q^{52} + ( - 597 \zeta_{6} + 597) q^{53} - 300 \zeta_{6} q^{54} + (294 \zeta_{6} + 147) q^{56} - 242 q^{57} - 486 \zeta_{6} q^{58} + ( - 360 \zeta_{6} + 360) q^{59} - 392 \zeta_{6} q^{61} + 264 q^{62} + (483 \zeta_{6} - 322) q^{63} + 433 q^{64} + (270 \zeta_{6} - 270) q^{66} + (280 \zeta_{6} - 280) q^{67} - 54 \zeta_{6} q^{68} - 138 q^{69} + 48 q^{71} + 483 \zeta_{6} q^{72} + ( - 668 \zeta_{6} + 668) q^{73} + ( - 777 \zeta_{6} + 777) q^{74} - 121 q^{76} + ( - 315 \zeta_{6} + 945) q^{77} + 354 q^{78} - 782 \zeta_{6} q^{79} + (421 \zeta_{6} - 421) q^{81} + 585 \zeta_{6} q^{82} - 768 q^{83} + ( - 42 \zeta_{6} + 28) q^{84} + 858 \zeta_{6} q^{86} + ( - 324 \zeta_{6} + 324) q^{87} + ( - 945 \zeta_{6} + 945) q^{88} + 1194 \zeta_{6} q^{89} + ( - 826 \zeta_{6} - 413) q^{91} - 69 q^{92} + 176 \zeta_{6} q^{93} + (135 \zeta_{6} - 135) q^{94} - 90 \zeta_{6} q^{96} - 902 q^{97} + (735 \zeta_{6} - 1176) q^{98} + 1035 q^{99} +O(q^{100})$$ q + 3*z * q^2 + (2*z - 2) * q^3 + (z - 1) * q^4 - 6 * q^6 + (14*z + 7) * q^7 + 21 * q^8 + 23*z * q^9 + (-45*z + 45) * q^11 - 2*z * q^12 - 59 * q^13 + (63*z - 42) * q^14 + 71*z * q^16 + (54*z - 54) * q^17 + (69*z - 69) * q^18 + 121*z * q^19 + (14*z - 42) * q^21 + 135 * q^22 + 69*z * q^23 + (42*z - 42) * q^24 - 177*z * q^26 - 100 * q^27 + (7*z - 21) * q^28 - 162 * q^29 + (-88*z + 88) * q^31 + (45*z - 45) * q^32 + 90*z * q^33 - 162 * q^34 - 23 * q^36 - 259*z * q^37 + (363*z - 363) * q^38 + (-118*z + 118) * q^39 + 195 * q^41 + (-84*z - 42) * q^42 + 286 * q^43 + 45*z * q^44 + (207*z - 207) * q^46 + 45*z * q^47 - 142 * q^48 + (392*z - 147) * q^49 - 108*z * q^51 + (-59*z + 59) * q^52 + (-597*z + 597) * q^53 - 300*z * q^54 + (294*z + 147) * q^56 - 242 * q^57 - 486*z * q^58 + (-360*z + 360) * q^59 - 392*z * q^61 + 264 * q^62 + (483*z - 322) * q^63 + 433 * q^64 + (270*z - 270) * q^66 + (280*z - 280) * q^67 - 54*z * q^68 - 138 * q^69 + 48 * q^71 + 483*z * q^72 + (-668*z + 668) * q^73 + (-777*z + 777) * q^74 - 121 * q^76 + (-315*z + 945) * q^77 + 354 * q^78 - 782*z * q^79 + (421*z - 421) * q^81 + 585*z * q^82 - 768 * q^83 + (-42*z + 28) * q^84 + 858*z * q^86 + (-324*z + 324) * q^87 + (-945*z + 945) * q^88 + 1194*z * q^89 + (-826*z - 413) * q^91 - 69 * q^92 + 176*z * q^93 + (135*z - 135) * q^94 - 90*z * q^96 - 902 * q^97 + (735*z - 1176) * q^98 + 1035 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 2 q^{3} - q^{4} - 12 q^{6} + 28 q^{7} + 42 q^{8} + 23 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 - 2 * q^3 - q^4 - 12 * q^6 + 28 * q^7 + 42 * q^8 + 23 * q^9 $$2 q + 3 q^{2} - 2 q^{3} - q^{4} - 12 q^{6} + 28 q^{7} + 42 q^{8} + 23 q^{9} + 45 q^{11} - 2 q^{12} - 118 q^{13} - 21 q^{14} + 71 q^{16} - 54 q^{17} - 69 q^{18} + 121 q^{19} - 70 q^{21} + 270 q^{22} + 69 q^{23} - 42 q^{24} - 177 q^{26} - 200 q^{27} - 35 q^{28} - 324 q^{29} + 88 q^{31} - 45 q^{32} + 90 q^{33} - 324 q^{34} - 46 q^{36} - 259 q^{37} - 363 q^{38} + 118 q^{39} + 390 q^{41} - 168 q^{42} + 572 q^{43} + 45 q^{44} - 207 q^{46} + 45 q^{47} - 284 q^{48} + 98 q^{49} - 108 q^{51} + 59 q^{52} + 597 q^{53} - 300 q^{54} + 588 q^{56} - 484 q^{57} - 486 q^{58} + 360 q^{59} - 392 q^{61} + 528 q^{62} - 161 q^{63} + 866 q^{64} - 270 q^{66} - 280 q^{67} - 54 q^{68} - 276 q^{69} + 96 q^{71} + 483 q^{72} + 668 q^{73} + 777 q^{74} - 242 q^{76} + 1575 q^{77} + 708 q^{78} - 782 q^{79} - 421 q^{81} + 585 q^{82} - 1536 q^{83} + 14 q^{84} + 858 q^{86} + 324 q^{87} + 945 q^{88} + 1194 q^{89} - 1652 q^{91} - 138 q^{92} + 176 q^{93} - 135 q^{94} - 90 q^{96} - 1804 q^{97} - 1617 q^{98} + 2070 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 - 2 * q^3 - q^4 - 12 * q^6 + 28 * q^7 + 42 * q^8 + 23 * q^9 + 45 * q^11 - 2 * q^12 - 118 * q^13 - 21 * q^14 + 71 * q^16 - 54 * q^17 - 69 * q^18 + 121 * q^19 - 70 * q^21 + 270 * q^22 + 69 * q^23 - 42 * q^24 - 177 * q^26 - 200 * q^27 - 35 * q^28 - 324 * q^29 + 88 * q^31 - 45 * q^32 + 90 * q^33 - 324 * q^34 - 46 * q^36 - 259 * q^37 - 363 * q^38 + 118 * q^39 + 390 * q^41 - 168 * q^42 + 572 * q^43 + 45 * q^44 - 207 * q^46 + 45 * q^47 - 284 * q^48 + 98 * q^49 - 108 * q^51 + 59 * q^52 + 597 * q^53 - 300 * q^54 + 588 * q^56 - 484 * q^57 - 486 * q^58 + 360 * q^59 - 392 * q^61 + 528 * q^62 - 161 * q^63 + 866 * q^64 - 270 * q^66 - 280 * q^67 - 54 * q^68 - 276 * q^69 + 96 * q^71 + 483 * q^72 + 668 * q^73 + 777 * q^74 - 242 * q^76 + 1575 * q^77 + 708 * q^78 - 782 * q^79 - 421 * q^81 + 585 * q^82 - 1536 * q^83 + 14 * q^84 + 858 * q^86 + 324 * q^87 + 945 * q^88 + 1194 * q^89 - 1652 * q^91 - 138 * q^92 + 176 * q^93 - 135 * q^94 - 90 * q^96 - 1804 * q^97 - 1617 * q^98 + 2070 * q^99

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 2.59808i −1.00000 1.73205i −0.500000 0.866025i 0 −6.00000 14.0000 12.1244i 21.0000 11.5000 19.9186i 0
151.1 1.50000 + 2.59808i −1.00000 + 1.73205i −0.500000 + 0.866025i 0 −6.00000 14.0000 + 12.1244i 21.0000 11.5000 + 19.9186i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.e.b 2
5.b even 2 1 35.4.e.a 2
5.c odd 4 2 175.4.k.b 4
7.c even 3 1 inner 175.4.e.b 2
7.c even 3 1 1225.4.a.b 1
7.d odd 6 1 1225.4.a.a 1
15.d odd 2 1 315.4.j.b 2
20.d odd 2 1 560.4.q.b 2
35.c odd 2 1 245.4.e.a 2
35.i odd 6 1 245.4.a.f 1
35.i odd 6 1 245.4.e.a 2
35.j even 6 1 35.4.e.a 2
35.j even 6 1 245.4.a.e 1
35.l odd 12 2 175.4.k.b 4
105.o odd 6 1 315.4.j.b 2
105.o odd 6 1 2205.4.a.e 1
105.p even 6 1 2205.4.a.g 1
140.p odd 6 1 560.4.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 5.b even 2 1
35.4.e.a 2 35.j even 6 1
175.4.e.b 2 1.a even 1 1 trivial
175.4.e.b 2 7.c even 3 1 inner
175.4.k.b 4 5.c odd 4 2
175.4.k.b 4 35.l odd 12 2
245.4.a.e 1 35.j even 6 1
245.4.a.f 1 35.i odd 6 1
245.4.e.a 2 35.c odd 2 1
245.4.e.a 2 35.i odd 6 1
315.4.j.b 2 15.d odd 2 1
315.4.j.b 2 105.o odd 6 1
560.4.q.b 2 20.d odd 2 1
560.4.q.b 2 140.p odd 6 1
1225.4.a.a 1 7.d odd 6 1
1225.4.a.b 1 7.c even 3 1
2205.4.a.e 1 105.o odd 6 1
2205.4.a.g 1 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 9$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 28T + 343$$
$11$ $$T^{2} - 45T + 2025$$
$13$ $$(T + 59)^{2}$$
$17$ $$T^{2} + 54T + 2916$$
$19$ $$T^{2} - 121T + 14641$$
$23$ $$T^{2} - 69T + 4761$$
$29$ $$(T + 162)^{2}$$
$31$ $$T^{2} - 88T + 7744$$
$37$ $$T^{2} + 259T + 67081$$
$41$ $$(T - 195)^{2}$$
$43$ $$(T - 286)^{2}$$
$47$ $$T^{2} - 45T + 2025$$
$53$ $$T^{2} - 597T + 356409$$
$59$ $$T^{2} - 360T + 129600$$
$61$ $$T^{2} + 392T + 153664$$
$67$ $$T^{2} + 280T + 78400$$
$71$ $$(T - 48)^{2}$$
$73$ $$T^{2} - 668T + 446224$$
$79$ $$T^{2} + 782T + 611524$$
$83$ $$(T + 768)^{2}$$
$89$ $$T^{2} - 1194 T + 1425636$$
$97$ $$(T + 902)^{2}$$