Properties

Label 4028.2.a.f
Level $4028$
Weight $2$
Character orbit 4028.a
Self dual yes
Analytic conductor $32.164$
Analytic rank $0$
Dimension $19$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4028,2,Mod(1,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.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: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 4 x^{18} - 30 x^{17} + 124 x^{16} + 364 x^{15} - 1554 x^{14} - 2310 x^{13} + 10113 x^{12} + 8368 x^{11} - 36567 x^{10} - 18074 x^{9} + 72868 x^{8} + 23819 x^{7} + \cdots + 139 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{7} q^{5} + (\beta_{8} + 1) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{7} q^{5} + (\beta_{8} + 1) q^{7} + (\beta_{2} + 1) q^{9} - \beta_{14} q^{11} + \beta_{16} q^{13} + ( - \beta_{17} - \beta_{14} + \beta_{11} - \beta_{8} + \beta_{7} + \beta_{5} + \beta_1) q^{15} + (\beta_{17} + \beta_{16} - \beta_{15} - \beta_{10}) q^{17} - q^{19} + (\beta_{18} - \beta_{17} + \beta_{11} - \beta_{6} + 1) q^{21} + ( - \beta_{15} + \beta_{13} + \beta_{7} - \beta_{6} + 1) q^{23} + ( - \beta_{16} - \beta_{14} - \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + 2 \beta_1 + 1) q^{25} + ( - \beta_{13} - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{3} + 2 \beta_1 + 1) q^{27} + ( - \beta_{15} + \beta_{13} - \beta_{5}) q^{29} + (\beta_{17} + \beta_{16} + \beta_{14} + \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{31} + ( - \beta_{17} - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{3}) q^{33} + (\beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{3} - \beta_1 + 1) q^{35} + (\beta_{17} + \beta_{15} + 2 \beta_{14} - \beta_{12} - \beta_{11} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \cdots + 1) q^{37}+ \cdots + ( - \beta_{18} - 2 \beta_{17} - \beta_{16} + \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{7} - \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 4 x^{18} - 30 x^{17} + 124 x^{16} + 364 x^{15} - 1554 x^{14} - 2310 x^{13} + 10113 x^{12} + 8368 x^{11} - 36567 x^{10} - 18074 x^{9} + 72868 x^{8} + 23819 x^{7} + \cdots + 139 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!59 \nu^{18} + \cdots + 95\!\cdots\!09 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53\!\cdots\!60 \nu^{18} + \cdots - 70\!\cdots\!42 ) / 26\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80\!\cdots\!18 \nu^{18} + \cdots - 27\!\cdots\!93 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 90\!\cdots\!06 \nu^{18} + \cdots - 10\!\cdots\!39 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!13 \nu^{18} + \cdots + 15\!\cdots\!41 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!71 \nu^{18} + \cdots - 15\!\cdots\!35 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14\!\cdots\!62 \nu^{18} + \cdots - 28\!\cdots\!13 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!64 \nu^{18} + \cdots + 26\!\cdots\!56 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 30\!\cdots\!70 \nu^{18} + \cdots + 11\!\cdots\!00 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 28\!\cdots\!28 \nu^{18} + \cdots + 17\!\cdots\!02 ) / 26\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!13 \nu^{18} + \cdots - 67\!\cdots\!39 ) / 26\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!09 \nu^{18} + \cdots + 18\!\cdots\!86 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 40\!\cdots\!50 \nu^{18} + \cdots + 55\!\cdots\!49 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 48\!\cdots\!29 \nu^{18} + \cdots + 15\!\cdots\!27 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 50\!\cdots\!96 \nu^{18} + \cdots - 17\!\cdots\!96 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 74\!\cdots\!92 \nu^{18} + \cdots - 25\!\cdots\!33 ) / 34\!\cdots\!67 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{3} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} - 3 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{3} + 9 \beta_{2} + 4 \beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{18} - 2 \beta_{16} - 4 \beta_{15} - 12 \beta_{13} - \beta_{12} - 13 \beta_{11} + 13 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} + 10 \beta_{6} + 2 \beta_{5} + \beta_{4} + 13 \beta_{3} + 2 \beta_{2} + 73 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{18} + 5 \beta_{17} + \beta_{16} - 41 \beta_{15} + 20 \beta_{14} + 7 \beta_{13} - 20 \beta_{12} - 21 \beta_{11} + 17 \beta_{10} - 17 \beta_{9} - 9 \beta_{8} + 14 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 33 \beta_{3} + 78 \beta_{2} + 70 \beta _1 + 226 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 41 \beta_{18} + 5 \beta_{17} - 28 \beta_{16} - 74 \beta_{15} + 19 \beta_{14} - 119 \beta_{13} - 31 \beta_{12} - 149 \beta_{11} + 139 \beta_{10} - 53 \beta_{9} - 27 \beta_{8} + 54 \beta_{7} + 87 \beta_{6} + 33 \beta_{5} + 16 \beta_{4} + 151 \beta_{3} + \cdots + 231 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 207 \beta_{18} + 103 \beta_{17} + 24 \beta_{16} - 463 \beta_{15} + 302 \beta_{14} + 16 \beta_{13} - 283 \beta_{12} - 314 \beta_{11} + 219 \beta_{10} - 214 \beta_{9} - 55 \beta_{8} + 160 \beta_{7} + 70 \beta_{6} + 39 \beta_{5} - 77 \beta_{4} + \cdots + 1977 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 599 \beta_{18} + 140 \beta_{17} - 277 \beta_{16} - 1010 \beta_{15} + 472 \beta_{14} - 1147 \beta_{13} - 560 \beta_{12} - 1672 \beta_{11} + 1413 \beta_{10} - 696 \beta_{9} - 249 \beta_{8} + 687 \beta_{7} + 735 \beta_{6} + \cdots + 2842 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2507 \beta_{18} + 1506 \beta_{17} + 386 \beta_{16} - 4994 \beta_{15} + 4028 \beta_{14} - 394 \beta_{13} - 3568 \beta_{12} - 4139 \beta_{11} + 2577 \beta_{10} - 2449 \beta_{9} - 162 \beta_{8} + 1716 \beta_{7} + \cdots + 18218 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7733 \beta_{18} + 2547 \beta_{17} - 2271 \beta_{16} - 12358 \beta_{15} + 8016 \beta_{14} - 11158 \beta_{13} - 8238 \beta_{12} - 18683 \beta_{11} + 14181 \beta_{10} - 8207 \beta_{9} - 1753 \beta_{8} + 7669 \beta_{7} + \cdots + 33065 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 29496 \beta_{18} + 19295 \beta_{17} + 5343 \beta_{16} - 53290 \beta_{15} + 50228 \beta_{14} - 9201 \beta_{13} - 42712 \beta_{12} - 51146 \beta_{11} + 29102 \beta_{10} - 27072 \beta_{9} + 2241 \beta_{8} + \cdots + 174103 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 94228 \beta_{18} + 38386 \beta_{17} - 15205 \beta_{16} - 143737 \beta_{15} + 115928 \beta_{14} - 110631 \beta_{13} - 109371 \beta_{12} - 208329 \beta_{11} + 142341 \beta_{10} - 92230 \beta_{9} + \cdots + 372147 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 341593 \beta_{18} + 231871 \beta_{17} + 68875 \beta_{16} - 568501 \beta_{15} + 602069 \beta_{14} - 141725 \beta_{13} - 497375 \beta_{12} - 608076 \beta_{11} + 321422 \beta_{10} - 295529 \beta_{9} + \cdots + 1709345 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1112893 \beta_{18} + 521542 \beta_{17} - 63831 \beta_{16} - 1627497 \beta_{15} + 1536252 \beta_{14} - 1119294 \beta_{13} - 1368510 \beta_{12} - 2317456 \beta_{11} + 1435963 \beta_{10} + \cdots + 4105878 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 3915869 \beta_{18} + 2691446 \beta_{17} + 854084 \beta_{16} - 6080900 \beta_{15} + 7042028 \beta_{14} - 1893310 \beta_{13} - 5696665 \beta_{12} - 7050409 \beta_{11} + 3503369 \beta_{10} + \cdots + 17133185 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 12898673 \beta_{18} + 6646398 \beta_{17} + 313034 \beta_{16} - 18144121 \beta_{15} + 19290684 \beta_{14} - 11533401 \beta_{13} - 16492848 \beta_{12} - 25708684 \beta_{11} + \cdots + 44764082 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 44557628 \beta_{18} + 30617557 \beta_{17} + 10343134 \beta_{16} - 65241437 \beta_{15} + 81056867 \beta_{14} - 23575127 \beta_{13} - 64539806 \beta_{12} - 80361253 \beta_{11} + \cdots + 174525936 \) 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
−2.84270
−2.63586
−2.55329
−1.93943
−1.53086
−1.37886
−0.828283
−0.414343
−0.284621
0.154590
0.416989
0.667966
1.62054
1.87845
1.95331
2.30353
2.97835
3.13171
3.30281
0 −2.84270 0 0.245225 0 0.879635 0 5.08095 0
1.2 0 −2.63586 0 −2.63773 0 −2.07266 0 3.94775 0
1.3 0 −2.55329 0 3.69573 0 1.87818 0 3.51931 0
1.4 0 −1.93943 0 0.842053 0 −2.29879 0 0.761401 0
1.5 0 −1.53086 0 2.27832 0 5.10455 0 −0.656480 0
1.6 0 −1.37886 0 −2.19516 0 1.41910 0 −1.09875 0
1.7 0 −0.828283 0 0.378111 0 −3.91567 0 −2.31395 0
1.8 0 −0.414343 0 −2.73706 0 4.71163 0 −2.82832 0
1.9 0 −0.284621 0 1.91371 0 0.801952 0 −2.91899 0
1.10 0 0.154590 0 0.882499 0 −1.05782 0 −2.97610 0
1.11 0 0.416989 0 −2.91551 0 3.12812 0 −2.82612 0
1.12 0 0.667966 0 −3.08795 0 −2.50903 0 −2.55382 0
1.13 0 1.62054 0 4.01966 0 2.03129 0 −0.373838 0
1.14 0 1.87845 0 3.30275 0 4.61837 0 0.528558 0
1.15 0 1.95331 0 −3.65783 0 −1.44461 0 0.815417 0
1.16 0 2.30353 0 0.631218 0 −0.267329 0 2.30625 0
1.17 0 2.97835 0 1.17616 0 3.75385 0 5.87055 0
1.18 0 3.13171 0 3.22705 0 −4.22839 0 6.80761 0
1.19 0 3.30281 0 −1.36124 0 2.46762 0 7.90858 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)
\(53\) \(1\)

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 4028.2.a.f 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4028.2.a.f 19 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4028))\):

\( T_{3}^{19} - 4 T_{3}^{18} - 30 T_{3}^{17} + 124 T_{3}^{16} + 364 T_{3}^{15} - 1554 T_{3}^{14} - 2310 T_{3}^{13} + 10113 T_{3}^{12} + 8368 T_{3}^{11} - 36567 T_{3}^{10} - 18074 T_{3}^{9} + 72868 T_{3}^{8} + 23819 T_{3}^{7} + \cdots + 139 \) Copy content Toggle raw display
\( T_{5}^{19} - 4 T_{5}^{18} - 50 T_{5}^{17} + 204 T_{5}^{16} + 1000 T_{5}^{15} - 4277 T_{5}^{14} - 9947 T_{5}^{13} + 47508 T_{5}^{12} + 46799 T_{5}^{11} - 298423 T_{5}^{10} - 36807 T_{5}^{9} + 1030638 T_{5}^{8} + \cdots + 25088 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} \) Copy content Toggle raw display
$3$ \( T^{19} - 4 T^{18} - 30 T^{17} + 124 T^{16} + \cdots + 139 \) Copy content Toggle raw display
$5$ \( T^{19} - 4 T^{18} - 50 T^{17} + \cdots + 25088 \) Copy content Toggle raw display
$7$ \( T^{19} - 13 T^{18} + 3 T^{17} + \cdots - 993916 \) Copy content Toggle raw display
$11$ \( T^{19} - T^{18} - 107 T^{17} + \cdots + 46080 \) Copy content Toggle raw display
$13$ \( T^{19} + T^{18} - 128 T^{17} + \cdots - 20150272 \) Copy content Toggle raw display
$17$ \( T^{19} - 3 T^{18} - 185 T^{17} + \cdots - 150290803 \) Copy content Toggle raw display
$19$ \( (T + 1)^{19} \) Copy content Toggle raw display
$23$ \( T^{19} - 10 T^{18} - 130 T^{17} + \cdots - 1356416 \) Copy content Toggle raw display
$29$ \( T^{19} - 2 T^{18} - 264 T^{17} + \cdots + 232158464 \) Copy content Toggle raw display
$31$ \( T^{19} - 25 T^{18} + \cdots + 192410753 \) Copy content Toggle raw display
$37$ \( T^{19} - 19 T^{18} + \cdots + 5284204160 \) Copy content Toggle raw display
$41$ \( T^{19} + 9 T^{18} + \cdots + 31991364975770 \) Copy content Toggle raw display
$43$ \( T^{19} - 35 T^{18} + \cdots + 63016982467 \) Copy content Toggle raw display
$47$ \( T^{19} - 23 T^{18} + \cdots + 838341337304 \) Copy content Toggle raw display
$53$ \( (T + 1)^{19} \) Copy content Toggle raw display
$59$ \( T^{19} - 16 T^{18} - 216 T^{17} + \cdots - 19167232 \) Copy content Toggle raw display
$61$ \( T^{19} - 21 T^{18} + \cdots + 1208697716736 \) Copy content Toggle raw display
$67$ \( T^{19} - 67 T^{18} + \cdots + 11315577057200 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 128165258278448 \) Copy content Toggle raw display
$73$ \( T^{19} + 20 T^{18} + \cdots - 3399907378688 \) Copy content Toggle raw display
$79$ \( T^{19} - 2 T^{18} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{19} - 38 T^{18} + \cdots - 49\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 192688598935296 \) Copy content Toggle raw display
$97$ \( T^{19} + 21 T^{18} + \cdots - 4096949664256 \) Copy content Toggle raw display
show more
show less