Properties

Label 1225.4.a.d
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,4,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

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: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 7 q^{3} - 4 q^{4} - 14 q^{6} + 24 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 7 q^{3} - 4 q^{4} - 14 q^{6} + 24 q^{8} + 22 q^{9} - 5 q^{11} - 28 q^{12} - 14 q^{13} - 16 q^{16} - 21 q^{17} - 44 q^{18} - 49 q^{19} + 10 q^{22} + 159 q^{23} + 168 q^{24} + 28 q^{26} - 35 q^{27} + 58 q^{29} - 147 q^{31} - 160 q^{32} - 35 q^{33} + 42 q^{34} - 88 q^{36} - 219 q^{37} + 98 q^{38} - 98 q^{39} - 350 q^{41} + 124 q^{43} + 20 q^{44} - 318 q^{46} + 525 q^{47} - 112 q^{48} - 147 q^{51} + 56 q^{52} - 303 q^{53} + 70 q^{54} - 343 q^{57} - 116 q^{58} + 105 q^{59} + 413 q^{61} + 294 q^{62} + 448 q^{64} + 70 q^{66} - 415 q^{67} + 84 q^{68} + 1113 q^{69} - 432 q^{71} + 528 q^{72} - 1113 q^{73} + 438 q^{74} + 196 q^{76} + 196 q^{78} - 103 q^{79} - 839 q^{81} + 700 q^{82} + 1092 q^{83} - 248 q^{86} + 406 q^{87} - 120 q^{88} + 329 q^{89} - 636 q^{92} - 1029 q^{93} - 1050 q^{94} - 1120 q^{96} - 882 q^{97} - 110 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 7.00000 −4.00000 0 −14.0000 0 24.0000 22.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.4.a.d 1
5.b even 2 1 49.4.a.c 1
7.b odd 2 1 1225.4.a.c 1
7.d odd 6 2 175.4.e.a 2
15.d odd 2 1 441.4.a.e 1
20.d odd 2 1 784.4.a.r 1
35.c odd 2 1 49.4.a.d 1
35.i odd 6 2 7.4.c.a 2
35.j even 6 2 49.4.c.a 2
35.k even 12 4 175.4.k.a 4
105.g even 2 1 441.4.a.d 1
105.o odd 6 2 441.4.e.k 2
105.p even 6 2 63.4.e.b 2
140.c even 2 1 784.4.a.b 1
140.s even 6 2 112.4.i.c 2
280.ba even 6 2 448.4.i.a 2
280.bk odd 6 2 448.4.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 35.i odd 6 2
49.4.a.c 1 5.b even 2 1
49.4.a.d 1 35.c odd 2 1
49.4.c.a 2 35.j even 6 2
63.4.e.b 2 105.p even 6 2
112.4.i.c 2 140.s even 6 2
175.4.e.a 2 7.d odd 6 2
175.4.k.a 4 35.k even 12 4
441.4.a.d 1 105.g even 2 1
441.4.a.e 1 15.d odd 2 1
441.4.e.k 2 105.o odd 6 2
448.4.i.a 2 280.ba even 6 2
448.4.i.f 2 280.bk odd 6 2
784.4.a.b 1 140.c even 2 1
784.4.a.r 1 20.d odd 2 1
1225.4.a.c 1 7.b odd 2 1
1225.4.a.d 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{3} - 7 \) Copy content Toggle raw display
\( T_{19} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 7 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 5 \) Copy content Toggle raw display
$13$ \( T + 14 \) Copy content Toggle raw display
$17$ \( T + 21 \) Copy content Toggle raw display
$19$ \( T + 49 \) Copy content Toggle raw display
$23$ \( T - 159 \) Copy content Toggle raw display
$29$ \( T - 58 \) Copy content Toggle raw display
$31$ \( T + 147 \) Copy content Toggle raw display
$37$ \( T + 219 \) Copy content Toggle raw display
$41$ \( T + 350 \) Copy content Toggle raw display
$43$ \( T - 124 \) Copy content Toggle raw display
$47$ \( T - 525 \) Copy content Toggle raw display
$53$ \( T + 303 \) Copy content Toggle raw display
$59$ \( T - 105 \) Copy content Toggle raw display
$61$ \( T - 413 \) Copy content Toggle raw display
$67$ \( T + 415 \) Copy content Toggle raw display
$71$ \( T + 432 \) Copy content Toggle raw display
$73$ \( T + 1113 \) Copy content Toggle raw display
$79$ \( T + 103 \) Copy content Toggle raw display
$83$ \( T - 1092 \) Copy content Toggle raw display
$89$ \( T - 329 \) Copy content Toggle raw display
$97$ \( T + 882 \) Copy content Toggle raw display
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