Properties

Label 225.10.b.e
Level $225$
Weight $10$
Character orbit 225.b
Analytic conductor $115.883$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,10,Mod(199,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.199");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not 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: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} + 496 q^{4} - 3840 \beta q^{7} + 2016 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{2} + 496 q^{4} - 3840 \beta q^{7} + 2016 \beta q^{8} + 86404 q^{11} + 74989 \beta q^{13} + 30720 q^{14} + 237824 q^{16} + 103811 \beta q^{17} - 716284 q^{19} + 172808 \beta q^{22} + 684960 \beta q^{23} - 599912 q^{26} - 1904640 \beta q^{28} - 3194402 q^{29} - 2349000 q^{31} + 1507840 \beta q^{32} - 830488 q^{34} + 9367855 \beta q^{37} - 1432568 \beta q^{38} + 29282630 q^{41} + 758362 \beta q^{43} + 42856384 q^{44} - 5479680 q^{46} - 307876 \beta q^{47} - 18628793 q^{49} + 37194544 \beta q^{52} + 2373715 \beta q^{53} + 30965760 q^{56} - 6388804 \beta q^{58} + 60616076 q^{59} - 126745682 q^{61} - 4698000 \beta q^{62} + 109703168 q^{64} - 55591326 \beta q^{67} + 51490256 \beta q^{68} + 175551608 q^{71} + 30616675 \beta q^{73} - 74942840 q^{74} - 355276864 q^{76} - 331791360 \beta q^{77} - 234431160 q^{79} + 58565260 \beta q^{82} + 59455194 \beta q^{83} - 6066896 q^{86} + 174190464 \beta q^{88} - 316534326 q^{89} + 1151831040 q^{91} + 339740160 \beta q^{92} + 2463008 q^{94} + 121456129 \beta q^{97} - 37257586 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 992 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 992 q^{4} + 172808 q^{11} + 61440 q^{14} + 475648 q^{16} - 1432568 q^{19} - 1199824 q^{26} - 6388804 q^{29} - 4698000 q^{31} - 1660976 q^{34} + 58565260 q^{41} + 85712768 q^{44} - 10959360 q^{46} - 37257586 q^{49} + 61931520 q^{56} + 121232152 q^{59} - 253491364 q^{61} + 219406336 q^{64} + 351103216 q^{71} - 149885680 q^{74} - 710553728 q^{76} - 468862320 q^{79} - 12133792 q^{86} - 633068652 q^{89} + 2303662080 q^{91} + 4926016 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
4.00000i 0 496.000 0 0 7680.00i 4032.00i 0 0
199.2 4.00000i 0 496.000 0 0 7680.00i 4032.00i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.10.b.e 2
3.b odd 2 1 75.10.b.d 2
5.b even 2 1 inner 225.10.b.e 2
5.c odd 4 1 45.10.a.b 1
5.c odd 4 1 225.10.a.c 1
15.d odd 2 1 75.10.b.d 2
15.e even 4 1 15.10.a.a 1
15.e even 4 1 75.10.a.c 1
60.l odd 4 1 240.10.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.a 1 15.e even 4 1
45.10.a.b 1 5.c odd 4 1
75.10.a.c 1 15.e even 4 1
75.10.b.d 2 3.b odd 2 1
75.10.b.d 2 15.d odd 2 1
225.10.a.c 1 5.c odd 4 1
225.10.b.e 2 1.a even 1 1 trivial
225.10.b.e 2 5.b even 2 1 inner
240.10.a.c 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} - 86404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 58982400 \) Copy content Toggle raw display
$11$ \( (T - 86404)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 22493400484 \) Copy content Toggle raw display
$17$ \( T^{2} + 43106894884 \) Copy content Toggle raw display
$19$ \( (T + 716284)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1876680806400 \) Copy content Toggle raw display
$29$ \( (T + 3194402)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2349000)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 351026829204100 \) Copy content Toggle raw display
$41$ \( (T - 29282630)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2300451692176 \) Copy content Toggle raw display
$47$ \( T^{2} + 379150525504 \) Copy content Toggle raw display
$53$ \( T^{2} + 22538091604900 \) Copy content Toggle raw display
$59$ \( (T - 60616076)^{2} \) Copy content Toggle raw display
$61$ \( (T + 126745682)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T - 175551608)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 37\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T + 234431160)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T + 316534326)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 59\!\cdots\!64 \) Copy content Toggle raw display
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