Properties

Label 225.4.b.d
Level $225$
Weight $4$
Character orbit 225.b
Analytic conductor $13.275$
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] = [225,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(13.2754297513\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{2} - q^{4} - 20 i q^{7} + 21 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{2} - q^{4} - 20 i q^{7} + 21 i q^{8} + 24 q^{11} + 74 i q^{13} + 60 q^{14} - 71 q^{16} + 54 i q^{17} + 124 q^{19} + 72 i q^{22} + 120 i q^{23} - 222 q^{26} + 20 i q^{28} - 78 q^{29} + 200 q^{31} - 45 i q^{32} - 162 q^{34} + 70 i q^{37} + 372 i q^{38} - 330 q^{41} + 92 i q^{43} - 24 q^{44} - 360 q^{46} - 24 i q^{47} - 57 q^{49} - 74 i q^{52} - 450 i q^{53} + 420 q^{56} - 234 i q^{58} + 24 q^{59} - 322 q^{61} + 600 i q^{62} - 433 q^{64} + 196 i q^{67} - 54 i q^{68} + 288 q^{71} - 430 i q^{73} - 210 q^{74} - 124 q^{76} - 480 i q^{77} + 520 q^{79} - 990 i q^{82} - 156 i q^{83} - 276 q^{86} + 504 i q^{88} + 1026 q^{89} + 1480 q^{91} - 120 i q^{92} + 72 q^{94} + 286 i q^{97} - 171 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 48 q^{11} + 120 q^{14} - 142 q^{16} + 248 q^{19} - 444 q^{26} - 156 q^{29} + 400 q^{31} - 324 q^{34} - 660 q^{41} - 48 q^{44} - 720 q^{46} - 114 q^{49} + 840 q^{56} + 48 q^{59} - 644 q^{61} - 866 q^{64} + 576 q^{71} - 420 q^{74} - 248 q^{76} + 1040 q^{79} - 552 q^{86} + 2052 q^{89} + 2960 q^{91} + 144 q^{94}+O(q^{100}) \) 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
3.00000i 0 −1.00000 0 0 20.0000i 21.0000i 0 0
199.2 3.00000i 0 −1.00000 0 0 20.0000i 21.0000i 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 225.4.b.d 2
3.b odd 2 1 75.4.b.a 2
5.b even 2 1 inner 225.4.b.d 2
5.c odd 4 1 45.4.a.b 1
5.c odd 4 1 225.4.a.g 1
12.b even 2 1 1200.4.f.m 2
15.d odd 2 1 75.4.b.a 2
15.e even 4 1 15.4.a.b 1
15.e even 4 1 75.4.a.a 1
20.e even 4 1 720.4.a.r 1
35.f even 4 1 2205.4.a.c 1
45.k odd 12 2 405.4.e.k 2
45.l even 12 2 405.4.e.d 2
60.h even 2 1 1200.4.f.m 2
60.l odd 4 1 240.4.a.f 1
60.l odd 4 1 1200.4.a.o 1
105.k odd 4 1 735.4.a.i 1
120.q odd 4 1 960.4.a.l 1
120.w even 4 1 960.4.a.bi 1
165.l odd 4 1 1815.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 15.e even 4 1
45.4.a.b 1 5.c odd 4 1
75.4.a.a 1 15.e even 4 1
75.4.b.a 2 3.b odd 2 1
75.4.b.a 2 15.d odd 2 1
225.4.a.g 1 5.c odd 4 1
225.4.b.d 2 1.a even 1 1 trivial
225.4.b.d 2 5.b even 2 1 inner
240.4.a.f 1 60.l odd 4 1
405.4.e.d 2 45.l even 12 2
405.4.e.k 2 45.k odd 12 2
720.4.a.r 1 20.e even 4 1
735.4.a.i 1 105.k odd 4 1
960.4.a.l 1 120.q odd 4 1
960.4.a.bi 1 120.w even 4 1
1200.4.a.o 1 60.l odd 4 1
1200.4.f.m 2 12.b even 2 1
1200.4.f.m 2 60.h even 2 1
1815.4.a.a 1 165.l odd 4 1
2205.4.a.c 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 400 \) Copy content Toggle raw display
\( T_{11} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 400 \) Copy content Toggle raw display
$11$ \( (T - 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5476 \) Copy content Toggle raw display
$17$ \( T^{2} + 2916 \) Copy content Toggle raw display
$19$ \( (T - 124)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 14400 \) Copy content Toggle raw display
$29$ \( (T + 78)^{2} \) Copy content Toggle raw display
$31$ \( (T - 200)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4900 \) Copy content Toggle raw display
$41$ \( (T + 330)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8464 \) Copy content Toggle raw display
$47$ \( T^{2} + 576 \) Copy content Toggle raw display
$53$ \( T^{2} + 202500 \) Copy content Toggle raw display
$59$ \( (T - 24)^{2} \) Copy content Toggle raw display
$61$ \( (T + 322)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 38416 \) Copy content Toggle raw display
$71$ \( (T - 288)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 184900 \) Copy content Toggle raw display
$79$ \( (T - 520)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 24336 \) Copy content Toggle raw display
$89$ \( (T - 1026)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 81796 \) Copy content Toggle raw display
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