# Properties

 Label 225.2.q.a Level $225$ Weight $2$ Character orbit 225.q Analytic conductor $1.797$ Analytic rank $0$ Dimension $224$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(16,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([20, 6]))

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

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

chi := DirichletCharacter("225.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.q (of order $$15$$, 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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$224$$ Relative dimension: $$28$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$224 q - 3 q^{2} - 8 q^{3} + 23 q^{4} - 8 q^{5} - 10 q^{6} - 8 q^{7} - 20 q^{8} - 8 q^{9}+O(q^{10})$$ 224 * q - 3 * q^2 - 8 * q^3 + 23 * q^4 - 8 * q^5 - 10 * q^6 - 8 * q^7 - 20 * q^8 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$224 q - 3 q^{2} - 8 q^{3} + 23 q^{4} - 8 q^{5} - 10 q^{6} - 8 q^{7} - 20 q^{8} - 8 q^{9} - 20 q^{10} - 11 q^{11} - 4 q^{12} - 3 q^{13} + q^{14} - 48 q^{15} + 23 q^{16} - 24 q^{17} - 12 q^{19} + q^{20} + 15 q^{21} - 11 q^{22} + q^{23} - 30 q^{24} - 16 q^{25} - 136 q^{26} + 7 q^{27} + 4 q^{28} - 15 q^{29} - 24 q^{30} + 3 q^{31} + 12 q^{32} - 5 q^{33} + q^{34} + 14 q^{35} + 38 q^{36} - 24 q^{37} + 55 q^{38} + 20 q^{39} + q^{40} - 19 q^{41} - 38 q^{42} - 8 q^{43} + 4 q^{44} - 38 q^{45} - 20 q^{46} - 10 q^{47} - 25 q^{48} - 72 q^{49} - 3 q^{50} - 26 q^{51} - 25 q^{52} - 12 q^{53} + 53 q^{54} - 20 q^{55} - 60 q^{56} + 38 q^{57} - 23 q^{58} - 30 q^{59} - 33 q^{60} - 3 q^{61} - 44 q^{62} + 46 q^{63} - 44 q^{64} + 51 q^{65} - 134 q^{66} - 12 q^{67} - 156 q^{68} + 4 q^{69} - 16 q^{70} + 42 q^{71} + 74 q^{72} - 12 q^{73} + 90 q^{74} + 67 q^{75} - 8 q^{76} + 31 q^{77} - 92 q^{78} - 15 q^{79} + 298 q^{80} - 104 q^{81} + 8 q^{82} + 59 q^{83} + 115 q^{84} - 11 q^{85} + 9 q^{86} - 59 q^{87} - 23 q^{88} + 106 q^{89} + 107 q^{90} + 30 q^{91} + 11 q^{92} + 32 q^{93} + 25 q^{94} + 7 q^{95} + 35 q^{96} - 21 q^{97} + 146 q^{98} - 20 q^{99}+O(q^{100})$$ 224 * q - 3 * q^2 - 8 * q^3 + 23 * q^4 - 8 * q^5 - 10 * q^6 - 8 * q^7 - 20 * q^8 - 8 * q^9 - 20 * q^10 - 11 * q^11 - 4 * q^12 - 3 * q^13 + q^14 - 48 * q^15 + 23 * q^16 - 24 * q^17 - 12 * q^19 + q^20 + 15 * q^21 - 11 * q^22 + q^23 - 30 * q^24 - 16 * q^25 - 136 * q^26 + 7 * q^27 + 4 * q^28 - 15 * q^29 - 24 * q^30 + 3 * q^31 + 12 * q^32 - 5 * q^33 + q^34 + 14 * q^35 + 38 * q^36 - 24 * q^37 + 55 * q^38 + 20 * q^39 + q^40 - 19 * q^41 - 38 * q^42 - 8 * q^43 + 4 * q^44 - 38 * q^45 - 20 * q^46 - 10 * q^47 - 25 * q^48 - 72 * q^49 - 3 * q^50 - 26 * q^51 - 25 * q^52 - 12 * q^53 + 53 * q^54 - 20 * q^55 - 60 * q^56 + 38 * q^57 - 23 * q^58 - 30 * q^59 - 33 * q^60 - 3 * q^61 - 44 * q^62 + 46 * q^63 - 44 * q^64 + 51 * q^65 - 134 * q^66 - 12 * q^67 - 156 * q^68 + 4 * q^69 - 16 * q^70 + 42 * q^71 + 74 * q^72 - 12 * q^73 + 90 * q^74 + 67 * q^75 - 8 * q^76 + 31 * q^77 - 92 * q^78 - 15 * q^79 + 298 * q^80 - 104 * q^81 + 8 * q^82 + 59 * q^83 + 115 * q^84 - 11 * q^85 + 9 * q^86 - 59 * q^87 - 23 * q^88 + 106 * q^89 + 107 * q^90 + 30 * q^91 + 11 * q^92 + 32 * q^93 + 25 * q^94 + 7 * q^95 + 35 * q^96 - 21 * q^97 + 146 * q^98 - 20 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −2.53861 + 1.13026i 1.12291 1.31874i 3.82877 4.25228i 1.17441 + 1.90283i −1.36012 + 4.61694i −0.665971 + 1.15350i −3.19614 + 9.83670i −0.478132 2.96165i −5.13206 3.50315i
16.2 −2.43895 + 1.08589i −1.70795 0.287957i 3.43105 3.81057i 0.156246 2.23060i 4.47828 1.15233i 0.0677783 0.117395i −2.58030 + 7.94134i 2.83416 + 0.983632i 2.04111 + 5.60999i
16.3 −2.13701 + 0.951457i −0.620400 + 1.61713i 2.32327 2.58025i −1.69271 + 1.46108i −0.212829 4.04610i −0.915602 + 1.58587i −1.06411 + 3.27499i −2.23021 2.00653i 2.22718 4.73287i
16.4 −2.06085 + 0.917550i 1.65499 + 0.510883i 2.06695 2.29558i −2.23029 0.160691i −3.87945 + 0.465685i 2.12428 3.67936i −0.759150 + 2.33642i 2.47800 + 1.69101i 4.74373 1.71524i
16.5 −1.92765 + 0.858246i 1.36960 + 1.06028i 1.64099 1.82251i 0.896552 2.04846i −3.55009 0.868397i −2.48928 + 4.31156i −0.295005 + 0.907932i 0.751612 + 2.90432i 0.0298427 + 4.71818i
16.6 −1.76509 + 0.785867i −1.27173 1.17589i 1.15968 1.28795i −0.197349 + 2.22734i 3.16880 + 1.07614i 1.35187 2.34151i 0.159347 0.490420i 0.234573 + 2.99082i −1.40206 4.08654i
16.7 −1.43805 + 0.640262i −0.230952 1.71658i 0.319797 0.355170i 2.05887 0.872377i 1.43118 + 2.32067i −0.385688 + 0.668032i 0.740392 2.27869i −2.89332 + 0.792896i −2.40222 + 2.57274i
16.8 −1.27611 + 0.568159i −0.563301 + 1.63789i −0.0326201 + 0.0362283i 0.323231 2.21258i −0.211752 2.41017i 2.19012 3.79340i 0.884357 2.72177i −2.36538 1.84525i 0.844622 + 3.00714i
16.9 −1.23977 + 0.551980i 1.19364 1.25508i −0.105919 + 0.117635i −2.19872 + 0.406964i −0.787056 + 2.21487i −1.81169 + 3.13794i 0.905114 2.78565i −0.150451 2.99623i 2.50127 1.71819i
16.10 −1.16983 + 0.520841i 1.71766 0.222825i −0.241041 + 0.267703i 2.12080 + 0.708681i −1.89331 + 1.15529i 1.38994 2.40744i 0.933960 2.87443i 2.90070 0.765476i −2.85007 + 0.275562i
16.11 −0.973390 + 0.433381i −1.71069 0.271186i −0.578592 + 0.642592i −2.03848 0.919011i 1.78270 0.477410i −0.798426 + 1.38291i 0.943229 2.90296i 2.85292 + 0.927831i 2.38252 + 0.0111154i
16.12 −0.808045 + 0.359765i 0.558868 + 1.63941i −0.814755 + 0.904878i 1.09577 + 1.94918i −1.04139 1.12366i −0.458035 + 0.793340i 0.879476 2.70675i −2.37533 + 1.83243i −1.58668 1.18080i
16.13 −0.217479 + 0.0968278i −1.50542 + 0.856567i −1.30034 + 1.44417i 1.93124 1.12708i 0.244458 0.332052i −1.58050 + 2.73751i 0.290090 0.892804i 1.53259 2.57899i −0.310871 + 0.432114i
16.14 −0.0861137 + 0.0383403i 1.36636 1.06445i −1.33232 + 1.47969i −0.0996266 2.23385i −0.0768514 + 0.144050i 0.322971 0.559402i 0.116257 0.357802i 0.733899 2.90885i 0.0942256 + 0.188545i
16.15 0.254613 0.113361i −0.427801 1.67839i −1.28628 + 1.42856i 0.148962 + 2.23110i −0.299187 0.378844i −1.97470 + 3.42027i −0.337813 + 1.03968i −2.63397 + 1.43603i 0.290848 + 0.551181i
16.16 0.309584 0.137836i 1.59476 + 0.675825i −1.26142 + 1.40095i −1.79806 + 1.32928i 0.586865 0.0105903i 0.0614798 0.106486i −0.406855 + 1.25217i 2.08652 + 2.15556i −0.373429 + 0.659360i
16.17 0.322402 0.143543i −0.648399 1.60611i −1.25492 + 1.39373i −1.80765 1.31620i −0.439590 0.424739i 1.84787 3.20061i −0.422642 + 1.30076i −2.15916 + 2.08280i −0.771722 0.164872i
16.18 0.520318 0.231661i −1.62589 0.597061i −1.12120 + 1.24522i 2.21056 + 0.336764i −0.984295 + 0.0659928i 1.58872 2.75174i −0.646919 + 1.99101i 2.28704 + 1.94151i 1.22821 0.336876i
16.19 0.825259 0.367429i −0.453689 + 1.67158i −0.792213 + 0.879842i −1.78097 1.35209i 0.239774 + 1.54618i −1.53265 + 2.65463i −0.888808 + 2.73547i −2.58833 1.51675i −1.96656 0.461443i
16.20 1.00759 0.448609i 1.00870 + 1.40802i −0.524267 + 0.582258i 1.72966 1.41714i 1.64801 + 0.966199i 0.860383 1.49023i −0.948701 + 2.91980i −0.965042 + 2.84054i 1.10705 2.20384i
See next 80 embeddings (of 224 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.d even 5 1 inner
225.q even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.q.a 224
3.b odd 2 1 675.2.r.a 224
9.c even 3 1 inner 225.2.q.a 224
9.d odd 6 1 675.2.r.a 224
25.d even 5 1 inner 225.2.q.a 224
75.j odd 10 1 675.2.r.a 224
225.q even 15 1 inner 225.2.q.a 224
225.t odd 30 1 675.2.r.a 224

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.q.a 224 1.a even 1 1 trivial
225.2.q.a 224 9.c even 3 1 inner
225.2.q.a 224 25.d even 5 1 inner
225.2.q.a 224 225.q even 15 1 inner
675.2.r.a 224 3.b odd 2 1
675.2.r.a 224 9.d odd 6 1
675.2.r.a 224 75.j odd 10 1
675.2.r.a 224 225.t odd 30 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(225, [\chi])$$.