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Properties

Label 1444.4.a.j
Level $1444$
Weight $4$
Character orbit 1444.a
Self dual yes
Analytic conductor $85.199$
Analytic rank $1$
Dimension $15$
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: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 270 x^{13} - 73 x^{12} + 27762 x^{11} + 12723 x^{10} - 1362566 x^{9} - 774753 x^{8} + \cdots + 1913127803 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2}\cdot 19 \)
Twist minimal: no (minimal twist has level 76)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{9} + 1) q^{5} - \beta_{11} q^{7} + (\beta_{8} - \beta_{7} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{9} + 1) q^{5} - \beta_{11} q^{7} + (\beta_{8} - \beta_{7} + 9) q^{9} + (\beta_{14} - \beta_{13} - \beta_{12} + \cdots - 3) q^{11}+ \cdots + ( - 14 \beta_{14} - 17 \beta_{13} + \cdots + 170) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 12 q^{5} - 6 q^{7} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 12 q^{5} - 6 q^{7} + 135 q^{9} - 42 q^{11} - 150 q^{13} - 153 q^{17} - 216 q^{21} + 72 q^{23} + 207 q^{25} - 219 q^{27} - 462 q^{29} - 30 q^{31} - 309 q^{33} - 84 q^{35} + 30 q^{37} - 1086 q^{39} - 1368 q^{41} + 345 q^{43} + 882 q^{45} - 1134 q^{47} + 525 q^{49} - 1212 q^{51} - 612 q^{53} + 1536 q^{55} - 2190 q^{59} - 1032 q^{61} - 1770 q^{63} - 1530 q^{65} - 618 q^{67} - 756 q^{69} + 804 q^{71} + 996 q^{73} - 1758 q^{77} - 630 q^{79} + 1947 q^{81} + 2382 q^{83} - 4290 q^{85} + 1110 q^{87} - 4965 q^{89} - 864 q^{91} + 2736 q^{93} - 4410 q^{97} + 2385 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 270 x^{13} - 73 x^{12} + 27762 x^{11} + 12723 x^{10} - 1362566 x^{9} - 774753 x^{8} + \cdots + 1913127803 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38\!\cdots\!21 \nu^{14} + \cdots + 48\!\cdots\!34 ) / 27\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 36\!\cdots\!42 \nu^{14} + \cdots + 21\!\cdots\!51 ) / 27\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 58\!\cdots\!92 \nu^{14} + \cdots + 34\!\cdots\!39 ) / 27\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\!\cdots\!87 \nu^{14} + \cdots + 66\!\cdots\!25 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29\!\cdots\!88 \nu^{14} + \cdots + 18\!\cdots\!27 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38\!\cdots\!79 \nu^{14} + \cdots + 31\!\cdots\!62 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 38\!\cdots\!79 \nu^{14} + \cdots - 60\!\cdots\!50 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55\!\cdots\!31 \nu^{14} + \cdots - 49\!\cdots\!71 ) / 11\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!23 \nu^{14} + \cdots - 27\!\cdots\!09 ) / 27\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!00 \nu^{14} + \cdots - 45\!\cdots\!68 ) / 11\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!30 \nu^{14} + \cdots + 52\!\cdots\!63 ) / 10\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 46\!\cdots\!99 \nu^{14} + \cdots + 65\!\cdots\!07 ) / 34\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20\!\cdots\!54 \nu^{14} + \cdots + 14\!\cdots\!83 ) / 10\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{14} - 3 \beta_{13} + 4 \beta_{12} - 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} - 5 \beta_{10} + 5 \beta_{9} + 87 \beta_{8} + \cdots + 2306 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 223 \beta_{14} - 320 \beta_{13} + 377 \beta_{12} - 201 \beta_{11} - 483 \beta_{10} - 86 \beta_{9} + \cdots + 2331 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1127 \beta_{14} - 400 \beta_{13} + 527 \beta_{12} - 328 \beta_{11} - 1109 \beta_{10} + 579 \beta_{9} + \cdots + 170809 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 22607 \beta_{14} - 29020 \beta_{13} + 32273 \beta_{12} - 18736 \beta_{11} - 36848 \beta_{10} + \cdots + 295920 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 127706 \beta_{14} - 43363 \beta_{13} + 69947 \beta_{12} - 27049 \beta_{11} - 153707 \beta_{10} + \cdots + 13445700 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2209318 \beta_{14} - 2532040 \beta_{13} + 2738475 \beta_{12} - 1685028 \beta_{11} - 2899670 \beta_{10} + \cdots + 34538937 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 13569280 \beta_{14} - 4502326 \beta_{13} + 7810600 \beta_{12} - 2160244 \beta_{11} - 17791069 \beta_{10} + \cdots + 1098327320 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 212278515 \beta_{14} - 218496768 \beta_{13} + 233119024 \beta_{12} - 148194688 \beta_{11} + \cdots + 3791003157 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1400794783 \beta_{14} - 463147772 \beta_{13} + 814995904 \beta_{12} - 184530272 \beta_{11} + \cdots + 92182561636 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20237383495 \beta_{14} - 18825624149 \beta_{13} + 19975875499 \beta_{12} - 12870431849 \beta_{11} + \cdots + 398222188380 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 142142675116 \beta_{14} - 47466233324 \beta_{13} + 82275624424 \beta_{12} - 17260571084 \beta_{11} + \cdots + 7905573692697 \) 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
9.76267
8.83860
6.66222
6.25147
3.34144
2.16097
1.54109
0.636006
−2.09974
−3.51070
−3.62248
−4.76257
−8.14437
−8.21926
−8.83535
0 −9.76267 0 3.39841 0 −9.38413 0 68.3098 0
1.2 0 −8.83860 0 13.5678 0 14.2889 0 51.1208 0
1.3 0 −6.66222 0 12.3991 0 9.15087 0 17.3852 0
1.4 0 −6.25147 0 −17.6888 0 −24.7545 0 12.0809 0
1.5 0 −3.34144 0 −15.5429 0 27.5319 0 −15.8348 0
1.6 0 −2.16097 0 −9.57071 0 0.0933468 0 −22.3302 0
1.7 0 −1.54109 0 −2.13180 0 −26.1233 0 −24.6250 0
1.8 0 −0.636006 0 12.4015 0 23.5447 0 −26.5955 0
1.9 0 2.09974 0 −5.82273 0 −4.74295 0 −22.5911 0
1.10 0 3.51070 0 −0.363401 0 27.1316 0 −14.6750 0
1.11 0 3.62248 0 16.9170 0 8.45166 0 −13.8776 0
1.12 0 4.76257 0 10.8935 0 −32.8605 0 −4.31795 0
1.13 0 8.14437 0 −13.1227 0 8.68030 0 39.3308 0
1.14 0 8.21926 0 −8.81012 0 −0.532378 0 40.5562 0
1.15 0 8.83535 0 15.4758 0 −26.4755 0 51.0635 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.j 15
19.b odd 2 1 1444.4.a.k 15
19.e even 9 2 76.4.i.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.i.a ✓ 30 19.e even 9 2
1444.4.a.j 15 1.a even 1 1 trivial
1444.4.a.k 15 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 270 T_{3}^{13} + 73 T_{3}^{12} + 27762 T_{3}^{11} - 12723 T_{3}^{10} - 1362566 T_{3}^{9} + \cdots - 1913127803 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1444))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 1913127803 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 27747920450664 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 209945114722584 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 21\!\cdots\!11 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 85\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 13\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 24\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 69\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 25\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 31\!\cdots\!29 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 10\!\cdots\!51 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 31\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 13\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 20\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 17\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 48\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 27\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 14\!\cdots\!87 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 11\!\cdots\!59 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 37\!\cdots\!87 \) Copy content Toggle raw display
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