# Properties

 Label 175.2.t.a Level $175$ Weight $2$ Character orbit 175.t Analytic conductor $1.397$ Analytic rank $0$ Dimension $144$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,2,Mod(4,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([3, 20]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("175.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.t (of order $$30$$, degree $$8$$, 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: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$18$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 5 q^{2} - 5 q^{3} - 19 q^{4} - 3 q^{5} - 12 q^{6} - 50 q^{8} - 17 q^{9}+O(q^{10})$$ 144 * q - 5 * q^2 - 5 * q^3 - 19 * q^4 - 3 * q^5 - 12 * q^6 - 50 * q^8 - 17 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 5 q^{2} - 5 q^{3} - 19 q^{4} - 3 q^{5} - 12 q^{6} - 50 q^{8} - 17 q^{9} - q^{10} - 5 q^{12} - 20 q^{13} - 18 q^{14} + 12 q^{15} + 5 q^{16} + 5 q^{17} - 11 q^{19} - 24 q^{20} - 9 q^{21} - 60 q^{22} + 25 q^{23} + 50 q^{24} - 11 q^{25} - 60 q^{26} + 40 q^{27} - 24 q^{29} + 53 q^{30} + 15 q^{31} + 20 q^{33} - 20 q^{34} - 14 q^{35} + 16 q^{36} - 5 q^{37} - 20 q^{38} + 13 q^{39} + 7 q^{40} - 62 q^{41} + 40 q^{42} - 15 q^{44} - 41 q^{45} - 27 q^{46} - 5 q^{47} - 38 q^{49} + 54 q^{50} - 8 q^{51} - 130 q^{52} + 25 q^{53} - 29 q^{54} - 20 q^{55} + 32 q^{56} - 65 q^{58} - 39 q^{59} + 79 q^{60} + 7 q^{61} - 20 q^{62} - 45 q^{63} + 34 q^{64} - 13 q^{65} + 11 q^{66} + 25 q^{67} + 74 q^{69} + 85 q^{70} - 46 q^{71} + 60 q^{72} + 35 q^{73} + 6 q^{74} - 107 q^{75} + 180 q^{76} - 5 q^{77} + 10 q^{78} + 9 q^{79} + 88 q^{80} - 59 q^{81} + 90 q^{83} - 51 q^{84} - 6 q^{85} + 11 q^{86} - 5 q^{87} + 140 q^{88} - 42 q^{89} + 4 q^{90} + 22 q^{91} + 10 q^{92} + 5 q^{94} + 13 q^{95} + 53 q^{96} + 120 q^{97} - 180 q^{98} - 44 q^{99}+O(q^{100})$$ 144 * q - 5 * q^2 - 5 * q^3 - 19 * q^4 - 3 * q^5 - 12 * q^6 - 50 * q^8 - 17 * q^9 - q^10 - 5 * q^12 - 20 * q^13 - 18 * q^14 + 12 * q^15 + 5 * q^16 + 5 * q^17 - 11 * q^19 - 24 * q^20 - 9 * q^21 - 60 * q^22 + 25 * q^23 + 50 * q^24 - 11 * q^25 - 60 * q^26 + 40 * q^27 - 24 * q^29 + 53 * q^30 + 15 * q^31 + 20 * q^33 - 20 * q^34 - 14 * q^35 + 16 * q^36 - 5 * q^37 - 20 * q^38 + 13 * q^39 + 7 * q^40 - 62 * q^41 + 40 * q^42 - 15 * q^44 - 41 * q^45 - 27 * q^46 - 5 * q^47 - 38 * q^49 + 54 * q^50 - 8 * q^51 - 130 * q^52 + 25 * q^53 - 29 * q^54 - 20 * q^55 + 32 * q^56 - 65 * q^58 - 39 * q^59 + 79 * q^60 + 7 * q^61 - 20 * q^62 - 45 * q^63 + 34 * q^64 - 13 * q^65 + 11 * q^66 + 25 * q^67 + 74 * q^69 + 85 * q^70 - 46 * q^71 + 60 * q^72 + 35 * q^73 + 6 * q^74 - 107 * q^75 + 180 * q^76 - 5 * q^77 + 10 * q^78 + 9 * q^79 + 88 * q^80 - 59 * q^81 + 90 * q^83 - 51 * q^84 - 6 * q^85 + 11 * q^86 - 5 * q^87 + 140 * q^88 - 42 * q^89 + 4 * q^90 + 22 * q^91 + 10 * q^92 + 5 * q^94 + 13 * q^95 + 53 * q^96 + 120 * q^97 - 180 * q^98 - 44 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −0.561630 2.64226i −0.981775 2.20510i −4.83903 + 2.15447i 2.14180 0.642423i −5.27506 + 3.83256i −2.42948 1.04767i 5.23487 + 7.20518i −1.89120 + 2.10039i −2.90035 5.29838i
4.2 −0.500096 2.35277i 0.932348 + 2.09409i −3.45832 + 1.53974i 0.169836 + 2.22961i 4.46063 3.24084i −0.802852 + 2.52100i 2.52452 + 3.47470i −1.50854 + 1.67540i 5.16081 1.51460i
4.3 −0.482008 2.26767i 0.709985 + 1.59465i −3.08291 + 1.37260i −1.11212 1.93989i 3.27393 2.37865i 1.21596 2.34978i 1.87323 + 2.57828i −0.0314466 + 0.0349249i −3.86299 + 3.45697i
4.4 −0.369436 1.73806i −0.0553691 0.124361i −1.05727 + 0.470728i 2.20183 0.389777i −0.195691 + 0.142178i 1.92211 + 1.81810i −0.880110 1.21137i 1.99499 2.21566i −1.49089 3.68292i
4.5 −0.335784 1.57974i −1.11543 2.50529i −0.555728 + 0.247426i −1.59383 1.56835i −3.58316 + 2.60332i 2.23049 + 1.42300i −1.32111 1.81835i −3.02490 + 3.35950i −1.94240 + 3.04445i
4.6 −0.330614 1.55542i −0.597202 1.34134i −0.482928 + 0.215013i −1.77722 + 1.35701i −1.88890 + 1.37236i −2.59765 0.502201i −1.37525 1.89288i 0.564855 0.627335i 2.69830 + 2.31568i
4.7 −0.175806 0.827100i 0.253333 + 0.568996i 1.17390 0.522655i −0.0289171 2.23588i 0.426079 0.309565i −2.63834 + 0.197839i −1.63270 2.24722i 1.74781 1.94114i −1.84421 + 0.416998i
4.8 −0.147876 0.695704i 0.986806 + 2.21640i 1.36495 0.607717i −1.89164 + 1.19235i 1.39603 1.01428i 2.10313 1.60526i −1.46076 2.01056i −1.93126 + 2.14488i 1.10925 + 1.13970i
4.9 −0.0662594 0.311726i 1.14522 + 2.57220i 1.73431 0.772164i 2.09734 + 0.775343i 0.725939 0.527426i −2.42570 1.05639i −0.730260 1.00512i −3.29729 + 3.66201i 0.102726 0.705170i
4.10 −0.0586647 0.275996i −1.26264 2.83593i 1.75436 0.781091i 1.62491 + 1.53612i −0.708633 + 0.514852i 1.20265 2.35661i −0.650198 0.894920i −4.44086 + 4.93208i 0.328639 0.538584i
4.11 0.0169101 + 0.0795555i −0.239789 0.538575i 1.82105 0.810783i −0.445317 + 2.19128i 0.0387918 0.0281839i 0.791023 + 2.52473i 0.190909 + 0.262763i 1.77483 1.97115i −0.181859 + 0.00162719i
4.12 0.172534 + 0.811707i −0.515214 1.15719i 1.19799 0.533380i −1.94638 1.10073i 0.850407 0.617857i −0.234193 2.63537i 1.61518 + 2.22310i 0.933750 1.03703i 0.557658 1.76980i
4.13 0.208393 + 0.980411i 0.782030 + 1.75647i 0.909313 0.404852i −0.701900 2.12305i −1.55909 + 1.13275i 1.25436 + 2.32950i 1.76471 + 2.42891i −0.466216 + 0.517785i 1.93519 1.13058i
4.14 0.327402 + 1.54030i −0.263147 0.591039i −0.438251 + 0.195122i 2.12141 + 0.706829i 0.824224 0.598834i −2.63114 + 0.277644i 1.40716 + 1.93679i 1.72731 1.91837i −0.394177 + 3.49904i
4.15 0.417054 + 1.96208i 0.502581 + 1.12882i −1.84875 + 0.823117i −0.364398 + 2.20618i −2.00523 + 1.45688i 1.38727 2.25288i −0.0279546 0.0384762i 0.985753 1.09479i −4.48068 + 0.205115i
4.16 0.436418 + 2.05318i −0.881466 1.97981i −2.19801 + 0.978618i 1.11577 1.93779i 3.68022 2.67383i 2.59389 + 0.521300i −0.500948 0.689496i −1.13526 + 1.26083i 4.46559 + 1.44520i
4.17 0.516009 + 2.42763i 1.10824 + 2.48915i −3.80004 + 1.69189i 0.782280 2.09476i −5.47088 + 3.97483i −0.792837 2.52417i −3.15053 4.33633i −2.96028 + 3.28772i 5.48898 + 0.818170i
4.18 0.537983 + 2.53101i −0.0303653 0.0682015i −4.28951 + 1.90981i −2.22433 + 0.228782i 0.156283 0.113546i −0.537537 + 2.59057i −4.09959 5.64260i 2.00366 2.22529i −1.77571 5.50673i
9.1 −1.02647 2.30548i 0.763033 + 0.687038i −2.92336 + 3.24672i 1.22529 + 1.87047i 0.800727 2.46438i 2.48641 0.904292i 5.68568 + 1.84739i −0.203387 1.93510i 3.05462 4.74486i
9.2 −0.885632 1.98916i −1.52446 1.37263i −1.83416 + 2.03704i −0.250174 + 2.22203i −1.38027 + 4.24804i −1.40656 + 2.24089i 1.53472 + 0.498662i 0.126279 + 1.20147i 4.64154 1.47026i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.18 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
25.e even 10 1 inner
175.t even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.t.a 144
5.b even 2 1 875.2.u.a 144
5.c odd 4 2 875.2.q.b 288
7.c even 3 1 inner 175.2.t.a 144
25.d even 5 1 875.2.u.a 144
25.e even 10 1 inner 175.2.t.a 144
25.f odd 20 2 875.2.q.b 288
35.j even 6 1 875.2.u.a 144
35.l odd 12 2 875.2.q.b 288
175.q even 15 1 875.2.u.a 144
175.t even 30 1 inner 175.2.t.a 144
175.w odd 60 2 875.2.q.b 288

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.t.a 144 1.a even 1 1 trivial
175.2.t.a 144 7.c even 3 1 inner
175.2.t.a 144 25.e even 10 1 inner
175.2.t.a 144 175.t even 30 1 inner
875.2.q.b 288 5.c odd 4 2
875.2.q.b 288 25.f odd 20 2
875.2.q.b 288 35.l odd 12 2
875.2.q.b 288 175.w odd 60 2
875.2.u.a 144 5.b even 2 1
875.2.u.a 144 25.d even 5 1
875.2.u.a 144 35.j even 6 1
875.2.u.a 144 175.q even 15 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(175, [\chi])$$.