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Properties

Label 1444.4.a.h
Level $1444$
Weight $4$
Character orbit 1444.a
Self dual yes
Analytic conductor $85.199$
Analytic rank $0$
Dimension $10$
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] = [1444,4,Mod(1,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, 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("1444.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1444.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: \(85.1987580483\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 203 x^{8} + 374 x^{7} + 13789 x^{6} - 16482 x^{5} - 394948 x^{4} + 124784 x^{3} + \cdots - 6261184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{7} - 2 \beta_{3} + 2) q^{5} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1) q^{7} + (\beta_{9} + 9 \beta_{3} + \beta_{2} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{7} - 2 \beta_{3} + 2) q^{5} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1) q^{7} + (\beta_{9} + 9 \beta_{3} + \beta_{2} + 9) q^{9} + (\beta_{8} + \beta_{7} + \beta_{6} + \cdots + 10) q^{11}+ \cdots + ( - 4 \beta_{9} + 4 \beta_{8} + \cdots + 640) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} + 13 q^{5} - 2 q^{7} + 140 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{3} + 13 q^{5} - 2 q^{7} + 140 q^{9} + 86 q^{11} - 23 q^{13} - 160 q^{15} + 209 q^{17} - 206 q^{21} - 14 q^{23} + 595 q^{25} + 4 q^{27} + 311 q^{29} + 338 q^{31} + 38 q^{33} + 594 q^{35} - 43 q^{37} + 222 q^{39} + 75 q^{41} - 382 q^{43} + 181 q^{45} + 616 q^{47} + 754 q^{49} - 738 q^{51} + 1801 q^{53} - 518 q^{55} - 296 q^{59} + 1755 q^{61} - 1126 q^{63} - 2886 q^{65} + 1454 q^{67} - 1080 q^{69} + 1220 q^{71} + 2205 q^{73} - 3988 q^{75} + 810 q^{77} + 2082 q^{79} + 1614 q^{81} - 1216 q^{83} + 2520 q^{85} - 1018 q^{87} - 281 q^{89} + 1024 q^{91} + 662 q^{93} - 3431 q^{97} + 5670 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 203 x^{8} + 374 x^{7} + 13789 x^{6} - 16482 x^{5} - 394948 x^{4} + 124784 x^{3} + \cdots - 6261184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 461567 \nu^{9} - 36360456 \nu^{8} - 186639133 \nu^{7} + 6047280632 \nu^{6} + \cdots - 20097585656864 ) / 312734355840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1137737 \nu^{9} + 248988 \nu^{8} - 202551571 \nu^{7} + 21149468 \nu^{6} + 10929874389 \nu^{5} + \cdots + 304191664864 ) / 625468711680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1262231 \nu^{9} + 14204520 \nu^{8} - 202182085 \nu^{7} - 2379190552 \nu^{6} + \cdots + 3561790350304 ) / 312734355840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 577709557 \nu^{9} + 1618018372 \nu^{8} - 115024124359 \nu^{7} - 148225791036 \nu^{6} + \cdots - 21\!\cdots\!28 ) / 65361480370560 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1407222919 \nu^{9} + 11272360324 \nu^{8} + 215670930477 \nu^{7} - 1948753620476 \nu^{6} + \cdots + 21\!\cdots\!92 ) / 130722960741120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1583582999 \nu^{9} + 2033071036 \nu^{8} - 248829828957 \nu^{7} - 325688347844 \nu^{6} + \cdots + 892517333990048 ) / 130722960741120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 365284363 \nu^{9} - 1798611356 \nu^{8} - 60762960873 \nu^{7} + 317077472100 \nu^{6} + \cdots - 647312535956960 ) / 21787160123520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11162767 \nu^{9} + 70480020 \nu^{8} + 2196242405 \nu^{7} - 12284906476 \nu^{6} + \cdots + 14940572709472 ) / 625468711680 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 9\beta_{3} + \beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} + 3\beta_{8} - 9\beta_{7} - 3\beta_{6} + 17\beta_{4} - 42\beta_{3} - \beta_{2} + 68\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 106 \beta_{9} - 15 \beta_{8} - 21 \beta_{7} - 6 \beta_{6} - 21 \beta_{5} - 50 \beta_{4} + 1425 \beta_{3} + \cdots + 2026 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 361 \beta_{9} + 258 \beta_{8} - 1107 \beta_{7} - 498 \beta_{6} + 33 \beta_{5} + 2265 \beta_{4} + \cdots - 1145 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10863 \beta_{9} - 2295 \beta_{8} - 2556 \beta_{7} - 939 \beta_{6} - 2400 \beta_{5} - 8579 \beta_{4} + \cdots + 145061 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 47651 \beta_{9} + 22929 \beta_{8} - 113712 \beta_{7} - 59409 \beta_{6} + 6687 \beta_{5} + \cdots - 306975 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1108539 \beta_{9} - 279366 \beta_{8} - 208551 \beta_{7} - 84891 \beta_{6} - 234339 \beta_{5} + \cdots + 12215581 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5774412 \beta_{9} + 2248053 \beta_{8} - 11150373 \beta_{7} - 6300909 \beta_{6} + 965616 \beta_{5} + \cdots - 46079736 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
9.32235
7.31827
6.93954
5.18466
0.979686
−2.00500
−3.66140
−5.50011
−6.23389
−10.3441
0 −9.32235 0 20.9673 0 3.61003 0 59.9062 0
1.2 0 −7.31827 0 14.9641 0 −14.3498 0 26.5571 0
1.3 0 −6.93954 0 −18.9376 0 16.3610 0 21.1572 0
1.4 0 −5.18466 0 −9.85277 0 15.2260 0 −0.119304 0
1.5 0 −0.979686 0 −2.75376 0 −17.9347 0 −26.0402 0
1.6 0 2.00500 0 2.68582 0 −27.2690 0 −22.9800 0
1.7 0 3.66140 0 8.43594 0 18.0742 0 −13.5942 0
1.8 0 5.50011 0 19.7815 0 35.8557 0 3.25119 0
1.9 0 6.23389 0 −11.0164 0 −3.75469 0 11.8614 0
1.10 0 10.3441 0 −11.2742 0 −27.8187 0 80.0005 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 1444.4.a.h ✓ 10
19.b odd 2 1 1444.4.a.i yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1444.4.a.h ✓ 10 1.a even 1 1 trivial
1444.4.a.i yes 10 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 2 T_{3}^{9} - 203 T_{3}^{8} - 374 T_{3}^{7} + 13789 T_{3}^{6} + 16482 T_{3}^{5} + \cdots - 6261184 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1444))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots - 6261184 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 8974190000 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 427210958016 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 461922700458304 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 29\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{10} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 65\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 58\!\cdots\!75 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 12\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 39\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 58\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 17\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 55\!\cdots\!31 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 80\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 32\!\cdots\!11 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 56\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 24\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 10\!\cdots\!71 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 66\!\cdots\!81 \) Copy content Toggle raw display
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