Properties

Label 225.6.b.k
Level $225$
Weight $6$
Character orbit 225.b
Analytic conductor $36.086$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.199");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{145})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 73x^{2} + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 45)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} - 11) q^{4} + (\beta_{2} - 19 \beta_1) q^{7} + (15 \beta_{2} + 6 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} - 11) q^{4} + (\beta_{2} - 19 \beta_1) q^{7} + (15 \beta_{2} + 6 \beta_1) q^{8} + (4 \beta_{3} + 398) q^{11} + (14 \beta_{2} - 66 \beta_1) q^{13} + (18 \beta_{3} - 834) q^{14} + (11 \beta_{3} - 349) q^{16} + ( - 105 \beta_{2} - 73 \beta_1) q^{17} + ( - 64 \beta_{3} + 788) q^{19} + (60 \beta_{2} - 290 \beta_1) q^{22} + (60 \beta_{2} - 444 \beta_1) q^{23} + (52 \beta_{3} - 3076) q^{26} + (302 \beta_{2} + 712 \beta_1) q^{28} + (256 \beta_{3} - 778) q^{29} + ( - 88 \beta_{3} - 2868) q^{31} + (645 \beta_{2} + 838 \beta_1) q^{32} + (178 \beta_{3} - 1354) q^{34} + (268 \beta_{2} + 1708 \beta_1) q^{37} + ( - 960 \beta_{2} - 2516 \beta_1) q^{38} + (272 \beta_{3} + 64) q^{41} + (1604 \beta_{2} + 324 \beta_1) q^{43} + (358 \beta_{3} - 754) q^{44} + (384 \beta_{3} - 20112) q^{46} + ( - 900 \beta_{2} - 1084 \beta_1) q^{47} + (320 \beta_{3} + 547) q^{49} + (1228 \beta_{2} + 2368 \beta_1) q^{52} + (2625 \beta_{2} + 2017 \beta_1) q^{53} + ( - 438 \beta_{3} - 1206) q^{56} + (3840 \beta_{2} + 7690 \beta_1) q^{58} + ( - 196 \beta_{3} - 31502) q^{59} + ( - 1088 \beta_{3} - 24014) q^{61} + ( - 1320 \beta_{2} + 492 \beta_1) q^{62} + ( - 1131 \beta_{3} + 13901) q^{64} + (1726 \beta_{2} - 6394 \beta_1) q^{67} + ( - 690 \beta_{2} + 3824 \beta_1) q^{68} + (296 \beta_{3} + 32452) q^{71} + (6047 \beta_{2} + 1407 \beta_1) q^{73} + ( - 1976 \beta_{3} + 68888) q^{74} + (1428 \beta_{3} - 66652) q^{76} + (1650 \beta_{2} - 5550 \beta_1) q^{77} + ( - 536 \beta_{3} + 23412) q^{79} + (4080 \beta_{2} + 7280 \beta_1) q^{82} + ( - 5550 \beta_{2} + 5178 \beta_1) q^{83} + ( - 1928 \beta_{3} - 13336) q^{86} + (7290 \beta_{2} + 1140 \beta_1) q^{88} + ( - 1680 \beta_{3} + 44340) q^{89} + (880 \beta_{3} - 60840) q^{91} + (7680 \beta_{2} + 16272 \beta_1) q^{92} + (1984 \beta_{3} - 31312) q^{94} + ( - 1229 \beta_{2} + 1651 \beta_1) q^{97} + (4800 \beta_{2} + 8093 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 42 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 42 q^{4} + 1600 q^{11} - 3300 q^{14} - 1374 q^{16} + 3024 q^{19} - 12200 q^{26} - 2600 q^{29} - 11648 q^{31} - 5060 q^{34} + 800 q^{41} - 2300 q^{44} - 79680 q^{46} + 2828 q^{49} - 5700 q^{56} - 126400 q^{59} - 98232 q^{61} + 53342 q^{64} + 130400 q^{71} + 271600 q^{74} - 263752 q^{76} + 92576 q^{79} - 57200 q^{86} + 174000 q^{89} - 241600 q^{91} - 121280 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 73x^{2} + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 19\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 83\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\nu^{2} + 183 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 4\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 183 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -19\beta_{2} + 166\beta_1 ) / 5 \) 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
6.52080i
5.52080i
5.52080i
6.52080i
8.52080i 0 −40.6040 0 0 160.416i 73.3128i 0 0
199.2 3.52080i 0 19.6040 0 0 80.4159i 181.687i 0 0
199.3 3.52080i 0 19.6040 0 0 80.4159i 181.687i 0 0
199.4 8.52080i 0 −40.6040 0 0 160.416i 73.3128i 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 225.6.b.k 4
3.b odd 2 1 225.6.b.j 4
5.b even 2 1 inner 225.6.b.k 4
5.c odd 4 1 45.6.a.f yes 2
5.c odd 4 1 225.6.a.k 2
15.d odd 2 1 225.6.b.j 4
15.e even 4 1 45.6.a.d 2
15.e even 4 1 225.6.a.r 2
20.e even 4 1 720.6.a.be 2
60.l odd 4 1 720.6.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.6.a.d 2 15.e even 4 1
45.6.a.f yes 2 5.c odd 4 1
225.6.a.k 2 5.c odd 4 1
225.6.a.r 2 15.e even 4 1
225.6.b.j 4 3.b odd 2 1
225.6.b.j 4 15.d odd 2 1
225.6.b.k 4 1.a even 1 1 trivial
225.6.b.k 4 5.b even 2 1 inner
720.6.a.y 2 60.l odd 4 1
720.6.a.be 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{4} + 85T_{2}^{2} + 900 \) Copy content Toggle raw display
\( T_{7}^{4} + 32200T_{7}^{2} + 166410000 \) Copy content Toggle raw display
\( T_{11}^{2} - 800T_{11} + 145500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 85T^{2} + 900 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 32200 T^{2} + 166410000 \) Copy content Toggle raw display
$11$ \( (T^{2} - 800 T + 145500)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 52166560000 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 816818288400 \) Copy content Toggle raw display
$19$ \( (T^{2} - 1512 T - 3140464)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 76956405350400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1300 T - 58969500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5824 T + 1461744)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} - 400 T - 67008000)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{2} + 63200 T + 963745500)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 49116 T - 469672636)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} - 65200 T + 983358000)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} - 46288 T + 275282736)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{2} - 87000 T - 665550000)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
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