Properties

Label 225.2.h.d
Level $225$
Weight $2$
Character orbit 225.h
Analytic conductor $1.797$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,2,Mod(46,225)]
 
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, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.h (of order \(5\), 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: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3x^{10} - 2x^{9} + 34x^{8} - 22x^{7} + 236x^{6} - 179x^{5} + 877x^{4} - 409x^{3} + 96x^{2} - 11x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{11} + \beta_{9} - 2 \beta_{8} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{11} + \beta_{9} - 2 \beta_{8} + \cdots - 1) q^{4}+ \cdots + (2 \beta_{11} - 4 \beta_{10} + \cdots + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{4} + 6 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{4} + 6 q^{5} - 12 q^{7} - 9 q^{8} - 9 q^{10} + 4 q^{11} - 2 q^{13} - 6 q^{14} + 16 q^{16} + q^{17} + 7 q^{19} - 26 q^{20} + 13 q^{22} - 19 q^{23} + 4 q^{25} + 56 q^{26} + q^{28} + q^{29} + 13 q^{31} + 32 q^{32} - 25 q^{34} + 10 q^{35} + 8 q^{37} + 22 q^{38} - 28 q^{40} - 8 q^{41} - 4 q^{43} - 33 q^{44} - 22 q^{46} + 13 q^{47} - 28 q^{49} - 81 q^{50} + 44 q^{52} - 44 q^{53} + 9 q^{55} - 45 q^{56} + 41 q^{58} + 22 q^{59} - 8 q^{61} - 41 q^{62} + 49 q^{64} + 38 q^{65} - 6 q^{67} + 100 q^{68} - 45 q^{70} + 21 q^{71} - 16 q^{73} + 44 q^{74} - 52 q^{76} - q^{77} + 10 q^{79} + 99 q^{80} + 26 q^{82} + 10 q^{83} + 23 q^{85} - 56 q^{86} - 16 q^{88} - 57 q^{89} - 7 q^{91} - 3 q^{92} - 23 q^{94} - 21 q^{95} + 4 q^{97} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3x^{10} - 2x^{9} + 34x^{8} - 22x^{7} + 236x^{6} - 179x^{5} + 877x^{4} - 409x^{3} + 96x^{2} - 11x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 78219134 \nu^{11} + 275969449 \nu^{10} + 322545241 \nu^{9} + 688323383 \nu^{8} + \cdots + 1397761481 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 295076203 \nu^{11} - 2737251577 \nu^{10} + 423072977 \nu^{9} - 8708425024 \nu^{8} + \cdots - 16004299508 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 444010163 \nu^{11} - 328887363 \nu^{10} - 1432734827 \nu^{9} - 101573726 \nu^{8} + \cdots - 6770591262 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1397761481 \nu^{11} + 78219134 \nu^{10} - 3917314994 \nu^{9} + 3118068203 \nu^{8} + \cdots - 20040639729 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1818269283 \nu^{11} + 240671723 \nu^{10} + 5444121987 \nu^{9} - 2938441244 \nu^{8} + \cdots - 3685345568 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3685345568 \nu^{11} + 1818269283 \nu^{10} + 11296708427 \nu^{9} - 1926569149 \nu^{8} + \cdots + 24741863047 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6770591262 \nu^{11} + 444010163 \nu^{10} - 19982886423 \nu^{9} + 14973917351 \nu^{8} + \cdots + 51950796787 ) / 30308822750 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5281629612 \nu^{11} + 669745048 \nu^{10} - 15574701453 \nu^{9} + 12663738036 \nu^{8} + \cdots + 81498022997 ) / 15154411375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2874546749 \nu^{11} + 341056225 \nu^{10} + 8742298783 \nu^{9} - 4686864056 \nu^{8} + \cdots - 19681913548 ) / 6061764550 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8838097791 \nu^{11} + 252041914 \nu^{10} - 26000680229 \nu^{9} + 18549046073 \nu^{8} + \cdots + 26628527446 ) / 15154411375 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{7} + 5\beta_{5} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{5} + 6\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{11} - 7 \beta_{10} - 5 \beta_{9} - 10 \beta_{8} + 31 \beta_{7} - 7 \beta_{6} + 10 \beta_{5} + \cdots + 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{11} + \beta_{10} + 11 \beta_{8} - 11 \beta_{7} + 38 \beta_{6} + 9 \beta_{4} - 2 \beta_{3} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37 \beta_{11} + 11 \beta_{10} + 37 \beta_{9} - 116 \beta_{8} - 12 \beta_{6} - 82 \beta_{5} - 12 \beta_{4} + \cdots - 82 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{11} - 26 \beta_{10} - 24 \beta_{9} + 99 \beta_{7} - 12 \beta_{6} + 130 \beta_{5} + 166 \beta_{4} + \cdots + 99 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 142 \beta_{11} + 470 \beta_{10} + 235 \beta_{9} + 673 \beta_{8} - 1294 \beta_{7} + 346 \beta_{6} + \cdots - 673 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 144 \beta_{11} - 255 \beta_{10} - 33 \beta_{9} - 823 \beta_{8} + 1159 \beta_{7} - 1673 \beta_{6} + \cdots - 671 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1428 \beta_{11} + 714 \beta_{10} - 815 \beta_{9} + 4529 \beta_{8} - 4529 \beta_{7} + 2017 \beta_{6} + \cdots + 4085 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1303 \beta_{11} + 934 \beta_{10} + 1303 \beta_{9} - 3124 \beta_{8} - 5243 \beta_{6} - 6568 \beta_{5} + \cdots - 6568 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
1.97423 + 1.43436i
0.199632 + 0.145041i
−2.17386 1.57940i
0.623865 + 1.92006i
0.0437845 + 0.134755i
−0.667650 2.05481i
0.623865 1.92006i
0.0437845 0.134755i
−0.667650 + 2.05481i
1.97423 1.43436i
0.199632 0.145041i
−2.17386 + 1.57940i
−0.754089 + 2.32085i 0 −3.19965 2.32468i −0.824264 2.07860i 0 −3.44028 3.85959 2.80415i 0 5.44569 0.345540i
46.2 −0.0762527 + 0.234682i 0 1.56877 + 1.13978i 2.09387 + 0.784664i 0 −1.24676 −0.786373 + 0.571334i 0 −0.343810 + 0.431561i
46.3 0.830342 2.55553i 0 −4.22323 3.06835i 1.34843 1.78375i 0 1.68704 −7.00026 + 5.08599i 0 −3.43876 4.92706i
91.1 −1.63330 1.18666i 0 0.641469 + 1.97424i 2.07079 0.843702i 0 1.01887 0.0473123 0.145612i 0 −4.38341 1.07931i
91.2 −0.114629 0.0832830i 0 −0.611830 1.88302i −2.14898 0.617963i 0 −0.858311 −0.174259 + 0.536314i 0 0.194870 + 0.249810i
91.3 1.74793 + 1.26995i 0 0.824463 + 2.53744i 0.460159 + 2.18821i 0 −3.16056 −0.446002 + 1.37265i 0 −1.97458 + 4.40921i
136.1 −1.63330 + 1.18666i 0 0.641469 1.97424i 2.07079 + 0.843702i 0 1.01887 0.0473123 + 0.145612i 0 −4.38341 + 1.07931i
136.2 −0.114629 + 0.0832830i 0 −0.611830 + 1.88302i −2.14898 + 0.617963i 0 −0.858311 −0.174259 0.536314i 0 0.194870 0.249810i
136.3 1.74793 1.26995i 0 0.824463 2.53744i 0.460159 2.18821i 0 −3.16056 −0.446002 1.37265i 0 −1.97458 4.40921i
181.1 −0.754089 2.32085i 0 −3.19965 + 2.32468i −0.824264 + 2.07860i 0 −3.44028 3.85959 + 2.80415i 0 5.44569 + 0.345540i
181.2 −0.0762527 0.234682i 0 1.56877 1.13978i 2.09387 0.784664i 0 −1.24676 −0.786373 0.571334i 0 −0.343810 0.431561i
181.3 0.830342 + 2.55553i 0 −4.22323 + 3.06835i 1.34843 + 1.78375i 0 1.68704 −7.00026 5.08599i 0 −3.43876 + 4.92706i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.h.d 12
3.b odd 2 1 75.2.g.c 12
15.d odd 2 1 375.2.g.c 12
15.e even 4 2 375.2.i.d 24
25.d even 5 1 inner 225.2.h.d 12
25.d even 5 1 5625.2.a.p 6
25.e even 10 1 5625.2.a.q 6
75.h odd 10 1 375.2.g.c 12
75.h odd 10 1 1875.2.a.k 6
75.j odd 10 1 75.2.g.c 12
75.j odd 10 1 1875.2.a.j 6
75.l even 20 2 375.2.i.d 24
75.l even 20 2 1875.2.b.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c 12 3.b odd 2 1
75.2.g.c 12 75.j odd 10 1
225.2.h.d 12 1.a even 1 1 trivial
225.2.h.d 12 25.d even 5 1 inner
375.2.g.c 12 15.d odd 2 1
375.2.g.c 12 75.h odd 10 1
375.2.i.d 24 15.e even 4 2
375.2.i.d 24 75.l even 20 2
1875.2.a.j 6 75.j odd 10 1
1875.2.a.k 6 75.h odd 10 1
1875.2.b.f 12 75.l even 20 2
5625.2.a.p 6 25.d even 5 1
5625.2.a.q 6 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 8 T_{2}^{10} + 3 T_{2}^{9} + 34 T_{2}^{8} + 8 T_{2}^{7} + 91 T_{2}^{6} + 96 T_{2}^{5} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 8 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 6 T^{5} + 4 T^{4} + \cdots + 20)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 59536 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 10201 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + \cdots + 21520321 \) Copy content Toggle raw display
$19$ \( T^{12} - 7 T^{11} + \cdots + 144400 \) Copy content Toggle raw display
$23$ \( T^{12} + 19 T^{11} + \cdots + 4080400 \) Copy content Toggle raw display
$29$ \( T^{12} - T^{11} + \cdots + 4431025 \) Copy content Toggle raw display
$31$ \( T^{12} - 13 T^{11} + \cdots + 8410000 \) Copy content Toggle raw display
$37$ \( T^{12} - 8 T^{11} + \cdots + 36300625 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 6831849025 \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} + \cdots - 6284)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 13 T^{11} + \cdots + 5216656 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1189905025 \) Copy content Toggle raw display
$59$ \( T^{12} - 22 T^{11} + \cdots + 15366400 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 28314456361 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 215619856 \) Copy content Toggle raw display
$71$ \( T^{12} - 21 T^{11} + \cdots + 38416 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 493062025 \) Copy content Toggle raw display
$79$ \( T^{12} - 10 T^{11} + \cdots + 64000000 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1683953296 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 142170473025 \) Copy content Toggle raw display
$97$ \( T^{12} - 4 T^{11} + \cdots + 5755201 \) Copy content Toggle raw display
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