Properties

Label 1225.4.a.z
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 32x^{2} - 35x + 120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{3} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + 9) q^{4} + ( - 3 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{6}+ \cdots + ( - 3 \beta_{3} - 6 \beta_1 + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{3} - 1) q^{3} + ( - \beta_{3} + \beta_{2} + 9) q^{4} + ( - 3 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{6}+ \cdots + ( - 253 \beta_{3} - 52 \beta_{2} + \cdots + 672) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26} - 441 q^{27} + 129 q^{29} - 114 q^{31} + 310 q^{32} - 865 q^{33} - 203 q^{34} + 1101 q^{36} - 403 q^{37} + 555 q^{38} + 674 q^{39} - 671 q^{41} + 411 q^{43} + 438 q^{44} - 997 q^{46} - 8 q^{47} - 523 q^{48} - 885 q^{51} + 74 q^{52} - 90 q^{53} + 2777 q^{54} - 233 q^{57} - 673 q^{58} - 1018 q^{59} - 50 q^{61} - 1626 q^{62} - 2421 q^{64} + 3841 q^{66} - 424 q^{67} - 617 q^{68} - 1080 q^{69} + 215 q^{71} - 2940 q^{72} - 1207 q^{73} + 623 q^{74} + 3257 q^{76} - 278 q^{78} - 951 q^{79} + 28 q^{81} + 1695 q^{82} - 3035 q^{83} - 99 q^{86} + 2210 q^{87} - 163 q^{88} - 2819 q^{89} - 3073 q^{92} + 852 q^{93} + 3056 q^{94} + 1345 q^{96} - 1100 q^{97} + 2383 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 32x^{2} - 35x + 120 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 3\nu^{2} - 20\nu + 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu^{2} - 18\nu + 38 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 2\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{3} + 4\beta_{2} + 26\beta _1 + 26 \) 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
−3.84167
−3.53510
1.50478
5.87199
−4.84167 −9.58084 15.4418 0 46.3873 0 −36.0308 64.7926 0
1.2 −4.53510 6.46622 12.5671 0 −29.3249 0 −20.7124 14.8119 0
1.3 0.504784 4.26379 −7.74519 0 2.15229 0 −7.94792 −8.82008 0
1.4 4.87199 −4.14916 15.7363 0 −20.2147 0 37.6910 −9.78444 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.z 4
5.b even 2 1 1225.4.a.bd 4
7.b odd 2 1 175.4.a.g 4
21.c even 2 1 1575.4.a.bl 4
35.c odd 2 1 175.4.a.h yes 4
35.f even 4 2 175.4.b.f 8
105.g even 2 1 1575.4.a.bg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.a.g 4 7.b odd 2 1
175.4.a.h yes 4 35.c odd 2 1
175.4.b.f 8 35.f even 4 2
1225.4.a.z 4 1.a even 1 1 trivial
1225.4.a.bd 4 5.b even 2 1
1575.4.a.bg 4 105.g even 2 1
1575.4.a.bl 4 21.c even 2 1

Hecke kernels

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

\( T_{2}^{4} + 4T_{2}^{3} - 26T_{2}^{2} - 95T_{2} + 54 \) Copy content Toggle raw display
\( T_{3}^{4} + 3T_{3}^{3} - 80T_{3}^{2} - 48T_{3} + 1096 \) Copy content Toggle raw display
\( T_{19}^{4} - 29T_{19}^{3} - 20798T_{19}^{2} + 338180T_{19} + 56837320 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + \cdots + 54 \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots + 1096 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 100 T^{3} + \cdots - 6827031 \) Copy content Toggle raw display
$13$ \( T^{4} - 44 T^{3} + \cdots - 1040448 \) Copy content Toggle raw display
$17$ \( T^{4} + 53 T^{3} + \cdots - 85632 \) Copy content Toggle raw display
$19$ \( T^{4} - 29 T^{3} + \cdots + 56837320 \) Copy content Toggle raw display
$23$ \( T^{4} + 295 T^{3} + \cdots - 138903216 \) Copy content Toggle raw display
$29$ \( T^{4} - 129 T^{3} + \cdots + 18445050 \) Copy content Toggle raw display
$31$ \( T^{4} + 114 T^{3} + \cdots + 44772800 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 1237182776 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 1082974824 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 2143095488 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + \cdots + 466065552 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 24137178144 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 3039063360 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 1189016144 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 1287279571 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 2950374906 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 20054062888 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 178640145850 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 194459351136 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 151094328480 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 802513451424 \) Copy content Toggle raw display
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