Properties

Label 225.2.m.b
Level $225$
Weight $2$
Character orbit 225.m
Analytic conductor $1.797$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,2,Mod(19,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.m (of order \(10\), degree \(4\), 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{10} + \cdots + \beta_1) q^{2}+ \cdots + (2 \beta_{14} - \beta_{13} - \beta_{11} + \cdots + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} - \beta_{10} + \cdots + \beta_1) q^{2}+ \cdots + (3 \beta_{15} + \beta_{13} + 2 \beta_{12} + \cdots - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{4} + 30 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{4} + 30 q^{8} + 6 q^{11} + 12 q^{14} - 10 q^{16} - 10 q^{17} - 2 q^{19} - 20 q^{20} - 30 q^{22} + 20 q^{23} - 10 q^{25} - 12 q^{26} + 30 q^{28} - 16 q^{29} + 6 q^{31} - 36 q^{34} - 10 q^{35} - 10 q^{37} - 30 q^{38} + 10 q^{40} + 14 q^{41} - 26 q^{44} + 16 q^{46} - 40 q^{47} - 20 q^{50} + 40 q^{52} - 10 q^{53} + 10 q^{55} + 10 q^{58} - 12 q^{59} + 10 q^{62} + 8 q^{64} + 70 q^{65} - 40 q^{67} + 30 q^{70} + 8 q^{71} - 20 q^{73} + 52 q^{74} - 32 q^{76} + 40 q^{77} - 20 q^{79} - 10 q^{83} - 20 q^{85} + 36 q^{86} - 40 q^{88} - 18 q^{89} + 26 q^{91} - 10 q^{92} - 38 q^{94} + 40 q^{95} + 40 q^{97} - 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9 \nu^{15} - 11 \nu^{14} + 158 \nu^{13} - 194 \nu^{12} + 1016 \nu^{11} - 1257 \nu^{10} + 2975 \nu^{9} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15 \nu^{15} - 6 \nu^{14} + 263 \nu^{13} - 106 \nu^{12} + 1688 \nu^{11} - 689 \nu^{10} + 4929 \nu^{9} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{15} - 11 \nu^{14} - 158 \nu^{13} - 194 \nu^{12} - 1016 \nu^{11} - 1257 \nu^{10} - 2975 \nu^{9} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15 \nu^{15} - 6 \nu^{14} - 263 \nu^{13} - 106 \nu^{12} - 1688 \nu^{11} - 689 \nu^{10} - 4929 \nu^{9} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -24\nu^{14} - 423\nu^{12} - 2739\nu^{10} - 8128\nu^{8} - 11513\nu^{6} - 7066\nu^{4} - 1247\nu^{2} - 2\nu - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -24\nu^{14} - 423\nu^{12} - 2739\nu^{10} - 8128\nu^{8} - 11513\nu^{6} - 7066\nu^{4} - 1247\nu^{2} + 2\nu - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7 \nu^{15} - 32 \nu^{14} + 124 \nu^{13} - 564 \nu^{12} + 810 \nu^{11} - 3652 \nu^{10} + 2444 \nu^{9} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7 \nu^{15} - 32 \nu^{14} - 124 \nu^{13} - 564 \nu^{12} - 810 \nu^{11} - 3652 \nu^{10} - 2444 \nu^{9} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12 \nu^{15} - 45 \nu^{14} + 208 \nu^{13} - 794 \nu^{12} + 1308 \nu^{11} - 5150 \nu^{10} + 3668 \nu^{9} + \cdots - 17 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10 \nu^{15} + 45 \nu^{14} - 177 \nu^{13} + 794 \nu^{12} - 1154 \nu^{11} + 5150 \nu^{10} - 3465 \nu^{9} + \cdots + 23 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6 \nu^{15} - 45 \nu^{14} + 108 \nu^{13} - 794 \nu^{12} + 724 \nu^{11} - 5150 \nu^{10} + 2282 \nu^{9} + \cdots - 17 ) / 4 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22 \nu^{15} - 29 \nu^{14} - 387 \nu^{13} - 512 \nu^{12} - 2498 \nu^{11} - 3324 \nu^{10} - 7373 \nu^{9} + \cdots - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10 \nu^{15} + 45 \nu^{14} + 177 \nu^{13} + 794 \nu^{12} + 1154 \nu^{11} + 5150 \nu^{10} + 3465 \nu^{9} + \cdots + 23 ) / 4 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29 \nu^{15} + 32 \nu^{14} + 512 \nu^{13} + 564 \nu^{12} + 3325 \nu^{11} + 3652 \nu^{10} + 9921 \nu^{9} + \cdots + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 45 \nu^{15} - 17 \nu^{14} - 794 \nu^{13} - 299 \nu^{12} - 5150 \nu^{11} - 1930 \nu^{10} - 15324 \nu^{9} + \cdots - 8 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{6} - \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{3} - \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{13} - \beta_{12} + \beta_{10} + 2\beta_{8} - 5\beta_{6} + 4\beta_{5} + \beta_{4} + \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} - 9 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} - 5 \beta_{10} - 9 \beta_{9} + 3 \beta_{8} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{15} + 7 \beta_{13} + 8 \beta_{12} - 7 \beta_{10} - 17 \beta_{8} + \beta_{7} + 28 \beta_{6} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19 \beta_{15} + 63 \beta_{13} - 19 \beta_{12} + 62 \beta_{11} + 25 \beta_{10} + 62 \beta_{9} + \cdots - 130 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 57 \beta_{15} - 8 \beta_{14} - 43 \beta_{13} - 57 \beta_{12} + \beta_{11} + 43 \beta_{10} - \beta_{9} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 144 \beta_{15} - 413 \beta_{13} + 144 \beta_{12} - 399 \beta_{11} - 125 \beta_{10} - 399 \beta_{9} + \cdots + 826 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 395 \beta_{15} + 120 \beta_{14} + 259 \beta_{13} + 395 \beta_{12} - 18 \beta_{11} - 259 \beta_{10} + \cdots - 335 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1018 \beta_{15} + 2654 \beta_{13} - 1018 \beta_{12} + 2518 \beta_{11} + 618 \beta_{10} + 2518 \beta_{9} + \cdots - 5209 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2699 \beta_{15} - 1212 \beta_{14} - 1564 \beta_{13} - 2699 \beta_{12} + 200 \beta_{11} + 1564 \beta_{10} + \cdots + 2093 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6987 \beta_{15} - 16954 \beta_{13} + 6987 \beta_{12} - 15819 \beta_{11} - 2980 \beta_{10} + \cdots + 32816 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 18251 \beta_{15} + 10376 \beta_{14} + 9506 \beta_{13} + 18251 \beta_{12} - 1809 \beta_{11} + \cdots - 13063 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 47209 \beta_{15} + 108172 \beta_{13} - 47209 \beta_{12} + 99427 \beta_{11} + 13754 \beta_{10} + \cdots - 207014 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 122384 \beta_{15} - 81360 \beta_{14} - 58148 \beta_{13} - 122384 \beta_{12} + 14675 \beta_{11} + \cdots + 81704 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.53655i
1.35083i
0.536547i
2.35083i
2.53767i
0.0898194i
1.08982i
1.53767i
2.53767i
0.0898194i
1.08982i
1.53767i
1.53655i
1.35083i
0.536547i
2.35083i
−1.46134 + 0.474819i 0 0.292036 0.212177i 2.06122 + 0.866816i 0 1.49550i 1.48030 2.03746i 0 −3.42373 0.288008i
19.2 −1.28472 + 0.417429i 0 −0.141788 + 0.103015i −1.34359 + 1.78739i 0 1.59580i 1.72715 2.37722i 0 0.980025 2.85714i
19.3 0.510286 0.165802i 0 −1.38513 + 1.00636i −2.22820 + 0.187439i 0 2.57318i −1.17071 + 1.61134i 0 −1.10594 + 0.465087i
19.4 2.23577 0.726446i 0 2.85292 2.07277i −0.725498 2.11510i 0 3.48189i 2.10915 2.90300i 0 −3.15856 4.20185i
64.1 −1.49161 2.05302i 0 −1.37197 + 4.22249i −0.227564 + 2.22446i 0 1.04054i 5.88835 1.91324i 0 4.90630 2.85082i
64.2 −0.0527945 0.0726655i 0 0.615541 1.89444i 1.27125 + 1.83954i 0 4.36070i −0.341004 + 0.110799i 0 0.0665563 0.189494i
64.3 0.640580 + 0.881682i 0 0.251013 0.772537i −0.741001 2.10972i 0 3.08724i 2.91489 0.947104i 0 1.38543 2.00477i
64.4 0.903822 + 1.24400i 0 −0.112618 + 0.346603i 1.93338 + 1.12340i 0 1.68601i 2.39187 0.777165i 0 0.349919 + 3.42049i
109.1 −1.49161 + 2.05302i 0 −1.37197 4.22249i −0.227564 2.22446i 0 1.04054i 5.88835 + 1.91324i 0 4.90630 + 2.85082i
109.2 −0.0527945 + 0.0726655i 0 0.615541 + 1.89444i 1.27125 1.83954i 0 4.36070i −0.341004 0.110799i 0 0.0665563 + 0.189494i
109.3 0.640580 0.881682i 0 0.251013 + 0.772537i −0.741001 + 2.10972i 0 3.08724i 2.91489 + 0.947104i 0 1.38543 + 2.00477i
109.4 0.903822 1.24400i 0 −0.112618 0.346603i 1.93338 1.12340i 0 1.68601i 2.39187 + 0.777165i 0 0.349919 3.42049i
154.1 −1.46134 0.474819i 0 0.292036 + 0.212177i 2.06122 0.866816i 0 1.49550i 1.48030 + 2.03746i 0 −3.42373 + 0.288008i
154.2 −1.28472 0.417429i 0 −0.141788 0.103015i −1.34359 1.78739i 0 1.59580i 1.72715 + 2.37722i 0 0.980025 + 2.85714i
154.3 0.510286 + 0.165802i 0 −1.38513 1.00636i −2.22820 0.187439i 0 2.57318i −1.17071 1.61134i 0 −1.10594 0.465087i
154.4 2.23577 + 0.726446i 0 2.85292 + 2.07277i −0.725498 + 2.11510i 0 3.48189i 2.10915 + 2.90300i 0 −3.15856 + 4.20185i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.m.b 16
3.b odd 2 1 75.2.i.a 16
15.d odd 2 1 375.2.i.c 16
15.e even 4 1 375.2.g.d 16
15.e even 4 1 375.2.g.e 16
25.e even 10 1 inner 225.2.m.b 16
25.f odd 20 1 5625.2.a.t 8
25.f odd 20 1 5625.2.a.bd 8
75.h odd 10 1 75.2.i.a 16
75.h odd 10 1 1875.2.b.h 16
75.j odd 10 1 375.2.i.c 16
75.j odd 10 1 1875.2.b.h 16
75.l even 20 1 375.2.g.d 16
75.l even 20 1 375.2.g.e 16
75.l even 20 1 1875.2.a.m 8
75.l even 20 1 1875.2.a.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.i.a 16 3.b odd 2 1
75.2.i.a 16 75.h odd 10 1
225.2.m.b 16 1.a even 1 1 trivial
225.2.m.b 16 25.e even 10 1 inner
375.2.g.d 16 15.e even 4 1
375.2.g.d 16 75.l even 20 1
375.2.g.e 16 15.e even 4 1
375.2.g.e 16 75.l even 20 1
375.2.i.c 16 15.d odd 2 1
375.2.i.c 16 75.j odd 10 1
1875.2.a.m 8 75.l even 20 1
1875.2.a.p 8 75.l even 20 1
1875.2.b.h 16 75.h odd 10 1
1875.2.b.h 16 75.j odd 10 1
5625.2.a.t 8 25.f odd 20 1
5625.2.a.bd 8 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 5 T_{2}^{14} - 10 T_{2}^{13} + 26 T_{2}^{12} + 50 T_{2}^{11} - 95 T_{2}^{10} - 20 T_{2}^{9} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 5 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 56 T^{14} + \cdots + 255025 \) Copy content Toggle raw display
$11$ \( T^{16} - 6 T^{15} + \cdots + 27888961 \) Copy content Toggle raw display
$13$ \( T^{16} - 30 T^{14} + \cdots + 78961 \) Copy content Toggle raw display
$17$ \( T^{16} + 10 T^{15} + \cdots + 53860921 \) Copy content Toggle raw display
$19$ \( T^{16} + 2 T^{15} + \cdots + 6375625 \) Copy content Toggle raw display
$23$ \( T^{16} - 20 T^{15} + \cdots + 4389025 \) Copy content Toggle raw display
$29$ \( T^{16} + 16 T^{15} + \cdots + 156025 \) Copy content Toggle raw display
$31$ \( T^{16} - 6 T^{15} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 8653650625 \) Copy content Toggle raw display
$41$ \( T^{16} - 14 T^{15} + \cdots + 22137025 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 527207521 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 36687479166361 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 40398990025 \) Copy content Toggle raw display
$59$ \( T^{16} + 12 T^{15} + \cdots + 12924025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 275701483356121 \) Copy content Toggle raw display
$67$ \( T^{16} + 40 T^{15} + \cdots + 13980121 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 25529328841 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 757413387025 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 3940125750625 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2356228681 \) Copy content Toggle raw display
$89$ \( T^{16} + 18 T^{15} + \cdots + 25 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 216648961 \) Copy content Toggle raw display
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