Properties

Label 225.2.w.a
Level $225$
Weight $2$
Character orbit 225.w
Analytic conductor $1.797$
Analytic rank $0$
Dimension $448$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,2,Mod(2,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.w (of order \(60\), degree \(16\), 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: \(448\)
Relative dimension: \(28\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 448 q - 24 q^{2} - 14 q^{3} - 10 q^{4} - 24 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 448 q - 24 q^{2} - 14 q^{3} - 10 q^{4} - 24 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9} - 32 q^{10} - 18 q^{11} - 14 q^{12} - 8 q^{13} - 30 q^{14} - 14 q^{15} - 50 q^{16} - 56 q^{18} - 40 q^{19} - 48 q^{20} - 12 q^{21} - 48 q^{23} + 16 q^{25} - 38 q^{27} - 24 q^{28} - 30 q^{29} - 50 q^{30} - 6 q^{31} - 60 q^{32} - 8 q^{33} - 10 q^{34} + 4 q^{36} - 44 q^{37} - 60 q^{39} - 16 q^{40} - 18 q^{41} + 174 q^{42} - 8 q^{43} - 64 q^{45} - 24 q^{46} - 18 q^{47} - 100 q^{48} + 24 q^{50} - 32 q^{51} + 24 q^{52} - 150 q^{54} - 24 q^{55} - 18 q^{56} - 94 q^{57} - 4 q^{58} + 202 q^{60} - 6 q^{61} - 46 q^{63} - 40 q^{64} - 96 q^{65} + 12 q^{66} - 14 q^{67} + 288 q^{68} + 50 q^{69} - 28 q^{70} + 102 q^{72} - 32 q^{73} + 18 q^{75} - 32 q^{76} + 216 q^{77} + 182 q^{78} - 10 q^{79} - 32 q^{81} - 72 q^{82} + 36 q^{83} + 100 q^{84} - 32 q^{85} - 18 q^{86} + 48 q^{87} - 28 q^{88} + 106 q^{90} - 24 q^{91} + 30 q^{92} + 8 q^{93} - 130 q^{94} + 6 q^{95} - 60 q^{96} - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.08185 1.68585i 0.798183 + 1.53717i 1.07618 + 5.06304i −2.23432 + 0.0883769i 0.929742 4.54577i 0.0946799 0.0253694i 3.86273 7.58103i −1.72581 + 2.45389i 4.80050 + 3.58273i
2.2 −2.00482 1.62347i −0.535601 1.64716i 0.967816 + 4.55321i −0.251379 + 2.22189i −1.60033 + 4.17178i 3.21084 0.860341i 3.10937 6.10249i −2.42626 + 1.76444i 4.11114 4.04638i
2.3 −1.92972 1.56265i 1.72141 0.191662i 0.866095 + 4.07466i 2.21571 0.301067i −3.62134 2.32012i −3.08775 + 0.827360i 2.44137 4.79145i 2.92653 0.659860i −4.74615 2.88141i
2.4 −1.88812 1.52897i −1.53403 0.804204i 0.811429 + 3.81747i −0.154565 2.23072i 1.66684 + 3.86393i −3.12915 + 0.838454i 2.09874 4.11900i 1.70651 + 2.46735i −3.11887 + 4.44820i
2.5 −1.54331 1.24975i 1.24822 1.20081i 0.404110 + 1.90119i −0.779714 2.09572i −3.42709 + 0.293265i 3.96539 1.06252i −0.0507944 + 0.0996896i 0.116102 2.99775i −1.41578 + 4.20878i
2.6 −1.44398 1.16931i 0.219166 + 1.71813i 0.301968 + 1.42065i 2.00481 + 0.990313i 1.69256 2.73722i 0.739070 0.198033i −0.461936 + 0.906601i −2.90393 + 0.753111i −1.73693 3.77425i
2.7 −1.32373 1.07193i −1.37118 + 1.05824i 0.187388 + 0.881593i −2.23256 + 0.125228i 2.94942 + 0.0689920i 0.893133 0.239314i −0.849622 + 1.66748i 0.760261 2.90207i 3.08953 + 2.22738i
2.8 −1.26732 1.02626i 0.979850 1.42825i 0.137078 + 0.644900i −1.08873 + 1.95312i −2.70753 + 0.804473i −3.51917 + 0.942959i −0.992568 + 1.94802i −1.07979 2.79894i 3.38417 1.35792i
2.9 −1.14980 0.931089i −1.68761 0.389856i 0.0392881 + 0.184836i 2.21919 0.274221i 1.57742 + 2.01957i 3.95765 1.06045i −1.21645 + 2.38741i 2.69602 + 1.31585i −2.80695 1.75096i
2.10 −0.870681 0.705063i −0.448846 + 1.67288i −0.154853 0.728525i 0.686598 2.12805i 1.57029 1.14008i −3.87586 + 1.03853i −1.39609 + 2.73998i −2.59708 1.50173i −2.09822 + 1.36875i
2.11 −0.863435 0.699196i 1.54952 + 0.773953i −0.159178 0.748875i −0.594209 + 2.15567i −0.796761 1.75167i 1.93702 0.519023i −1.39497 + 2.73778i 1.80200 + 2.39850i 2.02030 1.44581i
2.12 −0.476183 0.385605i −1.66967 0.460650i −0.337765 1.58906i 0.528402 + 2.17274i 0.617440 + 0.863188i −3.94730 + 1.05768i −1.00826 + 1.97882i 2.57560 + 1.53827i 0.586203 1.23838i
2.13 −0.350664 0.283962i −0.107492 1.72871i −0.373493 1.75714i 0.997580 2.00121i −0.453195 + 0.636721i −0.764485 + 0.204843i −0.777692 + 1.52631i −2.97689 + 0.371647i −0.918083 + 0.418477i
2.14 −0.235350 0.190582i −1.06897 1.36283i −0.396756 1.86659i −2.22710 + 0.200096i −0.00815054 + 0.524468i 2.13576 0.572276i −0.537334 + 1.05458i −0.714618 + 2.91364i 0.562281 + 0.377353i
2.15 −0.123603 0.100092i 1.69052 + 0.376997i −0.410564 1.93155i 1.70202 1.45021i −0.171220 0.215806i −0.310258 + 0.0831334i −0.286998 + 0.563265i 2.71575 + 1.27465i −0.355530 + 0.00889246i
2.16 0.209703 + 0.169814i 0.333716 + 1.69960i −0.400685 1.88507i −1.21183 1.87922i −0.218634 + 0.413080i 4.98040 1.33450i 0.481094 0.944201i −2.77727 + 1.13437i 0.0649950 0.599864i
2.17 0.319818 + 0.258983i 1.43991 0.962634i −0.380612 1.79064i −2.05377 0.884313i 0.709815 + 0.0650446i −2.53527 + 0.679325i 0.715680 1.40460i 1.14667 2.77221i −0.427811 0.814713i
2.18 0.614967 + 0.497991i −1.48472 + 0.891965i −0.285633 1.34380i 2.12267 0.703044i −1.35725 0.190848i 0.864298 0.231588i 1.21204 2.37877i 1.40880 2.64864i 1.65548 + 0.624721i
2.19 0.700361 + 0.567141i 0.0516917 1.73128i −0.246967 1.16189i 1.17286 + 1.90378i 1.01808 1.18320i 1.15542 0.309594i 1.30426 2.55975i −2.99466 0.178986i −0.258288 + 1.99851i
2.20 0.802972 + 0.650234i −1.64915 + 0.529426i −0.193864 0.912056i −1.97227 1.05364i −1.66847 0.647221i −2.82373 + 0.756615i 1.37554 2.69964i 2.43942 1.74621i −0.898562 2.12848i
See next 80 embeddings (of 448 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
25.f odd 20 1 inner
225.w even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.w.a 448
3.b odd 2 1 675.2.bd.a 448
9.c even 3 1 675.2.bd.a 448
9.d odd 6 1 inner 225.2.w.a 448
25.f odd 20 1 inner 225.2.w.a 448
75.l even 20 1 675.2.bd.a 448
225.w even 60 1 inner 225.2.w.a 448
225.x odd 60 1 675.2.bd.a 448
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.w.a 448 1.a even 1 1 trivial
225.2.w.a 448 9.d odd 6 1 inner
225.2.w.a 448 25.f odd 20 1 inner
225.2.w.a 448 225.w even 60 1 inner
675.2.bd.a 448 3.b odd 2 1
675.2.bd.a 448 9.c even 3 1
675.2.bd.a 448 75.l even 20 1
675.2.bd.a 448 225.x odd 60 1

Hecke kernels

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