Properties

Label 1225.4.a.ba
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $0$
Dimension $4$
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] = [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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.23265040.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 131x^{2} + 3460 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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)
 

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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} + 2) q^{4} + (\beta_{3} + \beta_1) q^{6} + ( - 5 \beta_{2} - 10) q^{8} + (9 \beta_{2} + 43) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} + 2) q^{4} + (\beta_{3} + \beta_1) q^{6} + ( - 5 \beta_{2} - 10) q^{8} + (9 \beta_{2} + 43) q^{9} + ( - 4 \beta_{2} - 23) q^{11} + ( - \beta_{3} + \beta_1) q^{12} + (\beta_{3} - 2 \beta_1) q^{13} + (3 \beta_{2} - 66) q^{16} + (5 \beta_{3} + \beta_1) q^{17} + (34 \beta_{2} + 90) q^{18} + (\beta_{3} + 11 \beta_1) q^{19} + ( - 19 \beta_{2} - 40) q^{22} + (23 \beta_{2} + 125) q^{23} + ( - 5 \beta_{3} - 15 \beta_1) q^{24} + ( - 4 \beta_{3} + 6 \beta_1) q^{26} + (9 \beta_{3} + 25 \beta_1) q^{27} + ( - 57 \beta_{2} - 81) q^{29} + (9 \beta_{3} + 4 \beta_1) q^{31} + ( - 29 \beta_{2} + 110) q^{32} + ( - 4 \beta_{3} - 27 \beta_1) q^{33} + ( - 9 \beta_{3} + 41 \beta_1) q^{34} + ( - 16 \beta_{2} - 4) q^{36} + ( - 55 \beta_{2} + 75) q^{37} + (9 \beta_{3} + 19 \beta_1) q^{38} + (34 \beta_{2} - 120) q^{39} + (4 \beta_{3} + 39 \beta_1) q^{41} + ( - 63 \beta_{2} + 215) q^{43} + (11 \beta_{2} - 6) q^{44} + (102 \beta_{2} + 230) q^{46} + ( - 8 \beta_{3} + 42 \beta_1) q^{47} + (3 \beta_{3} - 63 \beta_1) q^{48} + (269 \beta_{2} + 170) q^{51} + (6 \beta_{3} - 10 \beta_1) q^{52} + (30 \beta_{2} - 330) q^{53} + (7 \beta_{3} + 97 \beta_1) q^{54} + (151 \beta_{2} + 790) q^{57} + ( - 24 \beta_{2} - 570) q^{58} + ( - \beta_{3} - 46 \beta_1) q^{59} + (15 \beta_{3} + 30 \beta_1) q^{61} + ( - 14 \beta_{3} + 76 \beta_1) q^{62} + (115 \beta_{2} + 238) q^{64} + ( - 19 \beta_{3} - 59 \beta_1) q^{66} + ( - 138 \beta_{2} + 135) q^{67} + (19 \beta_{3} - 39 \beta_1) q^{68} + (23 \beta_{3} + 148 \beta_1) q^{69} + ( - 89 \beta_{2} + 383) q^{71} + ( - 260 \beta_{2} - 880) q^{72} + ( - 28 \beta_{3} + 31 \beta_1) q^{73} + (130 \beta_{2} - 550) q^{74} + ( - 7 \beta_{3} + 3 \beta_1) q^{76} + ( - 154 \beta_{2} + 340) q^{78} + ( - 15 \beta_{2} - 119) q^{79} + (450 \beta_{2} + 769) q^{81} + (31 \beta_{3} + 71 \beta_1) q^{82} + ( - 10 \beta_{3} - 41 \beta_1) q^{83} + (278 \beta_{2} - 630) q^{86} + ( - 57 \beta_{3} - 138 \beta_1) q^{87} + (135 \beta_{2} + 430) q^{88} + ( - 38 \beta_{3} + 27 \beta_1) q^{89} + ( - 56 \beta_{2} + 20) q^{92} + (504 \beta_{2} + 460) q^{93} + (58 \beta_{3} - 22 \beta_1) q^{94} + ( - 29 \beta_{3} + 81 \beta_1) q^{96} + ( - 33 \beta_{3} - 56 \beta_1) q^{97} + ( - 343 \beta_{2} - 1349) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 10 q^{4} - 30 q^{8} + 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 10 q^{4} - 30 q^{8} + 154 q^{9} - 84 q^{11} - 270 q^{16} + 292 q^{18} - 122 q^{22} + 454 q^{23} - 210 q^{29} + 498 q^{32} + 16 q^{36} + 410 q^{37} - 548 q^{39} + 986 q^{43} - 46 q^{44} + 716 q^{46} + 142 q^{51} - 1380 q^{53} + 2858 q^{57} - 2232 q^{58} + 722 q^{64} + 816 q^{67} + 1710 q^{71} - 3000 q^{72} - 2460 q^{74} + 1668 q^{78} - 446 q^{79} + 2176 q^{81} - 3076 q^{86} + 1450 q^{88} + 192 q^{92} + 832 q^{93} - 4710 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 131x^{2} + 3460 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 70 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 79\nu ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 9\beta_{2} + 70 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{3} + 79\beta_1 \) 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
−6.05689
6.05689
−9.71154
9.71154
−3.70156 −6.05689 5.70156 0 22.4200 0 8.50781 9.68594 0
1.2 −3.70156 6.05689 5.70156 0 −22.4200 0 8.50781 9.68594 0
1.3 2.70156 −9.71154 −0.701562 0 −26.2363 0 −23.5078 67.3141 0
1.4 2.70156 9.71154 −0.701562 0 26.2363 0 −23.5078 67.3141 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.4.a.ba 4
5.b even 2 1 1225.4.a.bc yes 4
7.b odd 2 1 inner 1225.4.a.ba 4
35.c odd 2 1 1225.4.a.bc yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1225.4.a.ba 4 1.a even 1 1 trivial
1225.4.a.ba 4 7.b odd 2 1 inner
1225.4.a.bc yes 4 5.b even 2 1
1225.4.a.bc yes 4 35.c odd 2 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} - 10 \) Copy content Toggle raw display
\( T_{3}^{4} - 131T_{3}^{2} + 3460 \) Copy content Toggle raw display
\( T_{19}^{4} - 16671T_{19}^{2} + 22144000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T - 10)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 131T^{2} + 3460 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 42 T + 277)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 1656 T^{2} + 13840 \) Copy content Toggle raw display
$17$ \( T^{4} - 27111 T^{2} + \cdots + 158454160 \) Copy content Toggle raw display
$19$ \( T^{4} - 16671 T^{2} + \cdots + 22144000 \) Copy content Toggle raw display
$23$ \( (T^{2} - 227 T + 7460)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 105 T - 30546)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 89036 T^{2} + \cdots + 1894696000 \) Copy content Toggle raw display
$37$ \( (T^{2} - 205 T - 20500)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 212851 T^{2} + \cdots + 2960462500 \) Copy content Toggle raw display
$43$ \( (T^{2} - 493 T + 20080)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 308524 T^{2} + \cdots + 17672296000 \) Copy content Toggle raw display
$53$ \( (T^{2} + 690 T + 109800)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 277176 T^{2} + \cdots + 13427914000 \) Copy content Toggle raw display
$61$ \( T^{4} - 351000 T^{2} + \cdots + 17516250000 \) Copy content Toggle raw display
$67$ \( (T^{2} - 408 T - 153585)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 855 T + 101566)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 996579 T^{2} + \cdots + 25354357540 \) Copy content Toggle raw display
$79$ \( (T^{2} + 223 T + 10126)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 318771 T^{2} + \cdots + 421431460 \) Copy content Toggle raw display
$89$ \( T^{4} - 1685419 T^{2} + \cdots + 207548186500 \) Copy content Toggle raw display
$97$ \( T^{4} - 1546940 T^{2} + \cdots + 427844224000 \) Copy content Toggle raw display
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