# Properties

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

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(74,175)]

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([3, 4]))

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

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

chi := DirichletCharacter("175.74");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.k (of order $$6$$, degree $$2$$, 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: $$10.3253342510$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{12} q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} - 6 q^{6} + (7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{7} - 21 \zeta_{12}^{3} q^{8} + (23 \zeta_{12}^{2} - 23) q^{9} +O(q^{10})$$ q + 3*z * q^2 + (2*z^3 - 2*z) * q^3 + z^2 * q^4 - 6 * q^6 + (7*z^3 + 14*z) * q^7 - 21*z^3 * q^8 + (23*z^2 - 23) * q^9 $$q + 3 \zeta_{12} q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} - 6 q^{6} + (7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{7} - 21 \zeta_{12}^{3} q^{8} + (23 \zeta_{12}^{2} - 23) q^{9} + 45 \zeta_{12}^{2} q^{11} - 2 \zeta_{12} q^{12} + 59 \zeta_{12}^{3} q^{13} + (63 \zeta_{12}^{2} - 21) q^{14} + ( - 71 \zeta_{12}^{2} + 71) q^{16} + ( - 54 \zeta_{12}^{3} + 54 \zeta_{12}) q^{17} + (69 \zeta_{12}^{3} - 69 \zeta_{12}) q^{18} + (121 \zeta_{12}^{2} - 121) q^{19} + ( - 14 \zeta_{12}^{2} - 28) q^{21} + 135 \zeta_{12}^{3} q^{22} - 69 \zeta_{12} q^{23} + 42 \zeta_{12}^{2} q^{24} + (177 \zeta_{12}^{2} - 177) q^{26} - 100 \zeta_{12}^{3} q^{27} + (21 \zeta_{12}^{3} - 7 \zeta_{12}) q^{28} + 162 q^{29} + 88 \zeta_{12}^{2} q^{31} + ( - 45 \zeta_{12}^{3} + 45 \zeta_{12}) q^{32} - 90 \zeta_{12} q^{33} + 162 q^{34} - 23 q^{36} - 259 \zeta_{12} q^{37} + (363 \zeta_{12}^{3} - 363 \zeta_{12}) q^{38} - 118 \zeta_{12}^{2} q^{39} + 195 q^{41} + ( - 42 \zeta_{12}^{3} - 84 \zeta_{12}) q^{42} - 286 \zeta_{12}^{3} q^{43} + (45 \zeta_{12}^{2} - 45) q^{44} - 207 \zeta_{12}^{2} q^{46} + 45 \zeta_{12} q^{47} + 142 \zeta_{12}^{3} q^{48} + (392 \zeta_{12}^{2} - 245) q^{49} + (108 \zeta_{12}^{2} - 108) q^{51} + (59 \zeta_{12}^{3} - 59 \zeta_{12}) q^{52} + ( - 597 \zeta_{12}^{3} + 597 \zeta_{12}) q^{53} + ( - 300 \zeta_{12}^{2} + 300) q^{54} + ( - 294 \zeta_{12}^{2} + 441) q^{56} - 242 \zeta_{12}^{3} q^{57} + 486 \zeta_{12} q^{58} - 360 \zeta_{12}^{2} q^{59} + (392 \zeta_{12}^{2} - 392) q^{61} + 264 \zeta_{12}^{3} q^{62} + (322 \zeta_{12}^{3} - 483 \zeta_{12}) q^{63} - 433 q^{64} - 270 \zeta_{12}^{2} q^{66} + ( - 280 \zeta_{12}^{3} + 280 \zeta_{12}) q^{67} + 54 \zeta_{12} q^{68} + 138 q^{69} + 48 q^{71} + 483 \zeta_{12} q^{72} + ( - 668 \zeta_{12}^{3} + 668 \zeta_{12}) q^{73} - 777 \zeta_{12}^{2} q^{74} - 121 q^{76} + (945 \zeta_{12}^{3} - 315 \zeta_{12}) q^{77} - 354 \zeta_{12}^{3} q^{78} + ( - 782 \zeta_{12}^{2} + 782) q^{79} - 421 \zeta_{12}^{2} q^{81} + 585 \zeta_{12} q^{82} + 768 \zeta_{12}^{3} q^{83} + ( - 42 \zeta_{12}^{2} + 14) q^{84} + ( - 858 \zeta_{12}^{2} + 858) q^{86} + (324 \zeta_{12}^{3} - 324 \zeta_{12}) q^{87} + ( - 945 \zeta_{12}^{3} + 945 \zeta_{12}) q^{88} + (1194 \zeta_{12}^{2} - 1194) q^{89} + (826 \zeta_{12}^{2} - 1239) q^{91} - 69 \zeta_{12}^{3} q^{92} - 176 \zeta_{12} q^{93} + 135 \zeta_{12}^{2} q^{94} + (90 \zeta_{12}^{2} - 90) q^{96} - 902 \zeta_{12}^{3} q^{97} + (1176 \zeta_{12}^{3} - 735 \zeta_{12}) q^{98} - 1035 q^{99} +O(q^{100})$$ q + 3*z * q^2 + (2*z^3 - 2*z) * q^3 + z^2 * q^4 - 6 * q^6 + (7*z^3 + 14*z) * q^7 - 21*z^3 * q^8 + (23*z^2 - 23) * q^9 + 45*z^2 * q^11 - 2*z * q^12 + 59*z^3 * q^13 + (63*z^2 - 21) * q^14 + (-71*z^2 + 71) * q^16 + (-54*z^3 + 54*z) * q^17 + (69*z^3 - 69*z) * q^18 + (121*z^2 - 121) * q^19 + (-14*z^2 - 28) * q^21 + 135*z^3 * q^22 - 69*z * q^23 + 42*z^2 * q^24 + (177*z^2 - 177) * q^26 - 100*z^3 * q^27 + (21*z^3 - 7*z) * q^28 + 162 * q^29 + 88*z^2 * q^31 + (-45*z^3 + 45*z) * q^32 - 90*z * q^33 + 162 * q^34 - 23 * q^36 - 259*z * q^37 + (363*z^3 - 363*z) * q^38 - 118*z^2 * q^39 + 195 * q^41 + (-42*z^3 - 84*z) * q^42 - 286*z^3 * q^43 + (45*z^2 - 45) * q^44 - 207*z^2 * q^46 + 45*z * q^47 + 142*z^3 * q^48 + (392*z^2 - 245) * q^49 + (108*z^2 - 108) * q^51 + (59*z^3 - 59*z) * q^52 + (-597*z^3 + 597*z) * q^53 + (-300*z^2 + 300) * q^54 + (-294*z^2 + 441) * q^56 - 242*z^3 * q^57 + 486*z * q^58 - 360*z^2 * q^59 + (392*z^2 - 392) * q^61 + 264*z^3 * q^62 + (322*z^3 - 483*z) * q^63 - 433 * q^64 - 270*z^2 * q^66 + (-280*z^3 + 280*z) * q^67 + 54*z * q^68 + 138 * q^69 + 48 * q^71 + 483*z * q^72 + (-668*z^3 + 668*z) * q^73 - 777*z^2 * q^74 - 121 * q^76 + (945*z^3 - 315*z) * q^77 - 354*z^3 * q^78 + (-782*z^2 + 782) * q^79 - 421*z^2 * q^81 + 585*z * q^82 + 768*z^3 * q^83 + (-42*z^2 + 14) * q^84 + (-858*z^2 + 858) * q^86 + (324*z^3 - 324*z) * q^87 + (-945*z^3 + 945*z) * q^88 + (1194*z^2 - 1194) * q^89 + (826*z^2 - 1239) * q^91 - 69*z^3 * q^92 - 176*z * q^93 + 135*z^2 * q^94 + (90*z^2 - 90) * q^96 - 902*z^3 * q^97 + (1176*z^3 - 735*z) * q^98 - 1035 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 24 q^{6} - 46 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 24 * q^6 - 46 * q^9 $$4 q + 2 q^{4} - 24 q^{6} - 46 q^{9} + 90 q^{11} + 42 q^{14} + 142 q^{16} - 242 q^{19} - 140 q^{21} + 84 q^{24} - 354 q^{26} + 648 q^{29} + 176 q^{31} + 648 q^{34} - 92 q^{36} - 236 q^{39} + 780 q^{41} - 90 q^{44} - 414 q^{46} - 196 q^{49} - 216 q^{51} + 600 q^{54} + 1176 q^{56} - 720 q^{59} - 784 q^{61} - 1732 q^{64} - 540 q^{66} + 552 q^{69} + 192 q^{71} - 1554 q^{74} - 484 q^{76} + 1564 q^{79} - 842 q^{81} - 28 q^{84} + 1716 q^{86} - 2388 q^{89} - 3304 q^{91} + 270 q^{94} - 180 q^{96} - 4140 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 - 24 * q^6 - 46 * q^9 + 90 * q^11 + 42 * q^14 + 142 * q^16 - 242 * q^19 - 140 * q^21 + 84 * q^24 - 354 * q^26 + 648 * q^29 + 176 * q^31 + 648 * q^34 - 92 * q^36 - 236 * q^39 + 780 * q^41 - 90 * q^44 - 414 * q^46 - 196 * q^49 - 216 * q^51 + 600 * q^54 + 1176 * q^56 - 720 * q^59 - 784 * q^61 - 1732 * q^64 - 540 * q^66 + 552 * q^69 + 192 * q^71 - 1554 * q^74 - 484 * q^76 + 1564 * q^79 - 842 * q^81 - 28 * q^84 + 1716 * q^86 - 2388 * q^89 - 3304 * q^91 + 270 * q^94 - 180 * q^96 - 4140 * q^99

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−2.59808 + 1.50000i 1.73205 + 1.00000i 0.500000 0.866025i 0 −6.00000 −12.1244 + 14.0000i 21.0000i −11.5000 19.9186i 0
74.2 2.59808 1.50000i −1.73205 1.00000i 0.500000 0.866025i 0 −6.00000 12.1244 14.0000i 21.0000i −11.5000 19.9186i 0
149.1 −2.59808 1.50000i 1.73205 1.00000i 0.500000 + 0.866025i 0 −6.00000 −12.1244 14.0000i 21.0000i −11.5000 + 19.9186i 0
149.2 2.59808 + 1.50000i −1.73205 + 1.00000i 0.500000 + 0.866025i 0 −6.00000 12.1244 + 14.0000i 21.0000i −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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 9T_{2}^{2} + 81$$ acting on $$S_{4}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 9T^{2} + 81$$
$3$ $$T^{4} - 4T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 98 T^{2} + 117649$$
$11$ $$(T^{2} - 45 T + 2025)^{2}$$
$13$ $$(T^{2} + 3481)^{2}$$
$17$ $$T^{4} - 2916 T^{2} + 8503056$$
$19$ $$(T^{2} + 121 T + 14641)^{2}$$
$23$ $$T^{4} - 4761 T^{2} + 22667121$$
$29$ $$(T - 162)^{4}$$
$31$ $$(T^{2} - 88 T + 7744)^{2}$$
$37$ $$T^{4} + \cdots + 4499860561$$
$41$ $$(T - 195)^{4}$$
$43$ $$(T^{2} + 81796)^{2}$$
$47$ $$T^{4} - 2025 T^{2} + 4100625$$
$53$ $$T^{4} + \cdots + 127027375281$$
$59$ $$(T^{2} + 360 T + 129600)^{2}$$
$61$ $$(T^{2} + 392 T + 153664)^{2}$$
$67$ $$T^{4} + \cdots + 6146560000$$
$71$ $$(T - 48)^{4}$$
$73$ $$T^{4} + \cdots + 199115858176$$
$79$ $$(T^{2} - 782 T + 611524)^{2}$$
$83$ $$(T^{2} + 589824)^{2}$$
$89$ $$(T^{2} + 1194 T + 1425636)^{2}$$
$97$ $$(T^{2} + 813604)^{2}$$
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