↑

Properties

Label 4028.2.a.d
Level $4028$
Weight $2$
Character orbit 4028.a
Self dual yes
Analytic conductor $32.164$
Analytic rank $1$
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: \(1\)
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: \( x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{6} q^{5} + (\beta_{11} - 1) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{6} q^{5} + (\beta_{11} - 1) q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{4} - 1) q^{11} + \beta_{16} q^{13} + ( - \beta_{16} + \beta_{15} - \beta_{14} + \cdots - 1) q^{15}+ \cdots + (\beta_{16} + \beta_{15} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\!\cdots\!25 \nu^{18} + \cdots - 36\!\cdots\!20 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29\!\cdots\!65 \nu^{18} + \cdots + 48\!\cdots\!33 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\!\cdots\!62 \nu^{18} + \cdots - 11\!\cdots\!77 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 59\!\cdots\!83 \nu^{18} + \cdots - 76\!\cdots\!87 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 76\!\cdots\!43 \nu^{18} + \cdots - 42\!\cdots\!91 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!80 \nu^{18} + \cdots - 15\!\cdots\!69 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!78 \nu^{18} + \cdots + 14\!\cdots\!12 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!10 \nu^{18} + \cdots - 22\!\cdots\!98 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27\!\cdots\!49 \nu^{18} + \cdots + 42\!\cdots\!59 ) / 81\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19\!\cdots\!04 \nu^{18} + \cdots - 28\!\cdots\!15 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22\!\cdots\!78 \nu^{18} + \cdots + 28\!\cdots\!58 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 24\!\cdots\!48 \nu^{18} + \cdots + 30\!\cdots\!94 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 29\!\cdots\!44 \nu^{18} + \cdots + 43\!\cdots\!19 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 32\!\cdots\!63 \nu^{18} + \cdots - 42\!\cdots\!33 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 34\!\cdots\!06 \nu^{18} + \cdots - 40\!\cdots\!22 ) / 57\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 36\!\cdots\!24 \nu^{18} + \cdots - 52\!\cdots\!60 ) / 57\!\cdots\!63 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} + \beta_{16} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{9} + \cdots + 7 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + 3 \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{11} - 2 \beta_{10} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{18} - 14 \beta_{17} + 14 \beta_{16} - \beta_{15} - 12 \beta_{14} + 12 \beta_{13} + 14 \beta_{12} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{18} - 13 \beta_{17} + 51 \beta_{16} - 21 \beta_{15} + 15 \beta_{14} + 29 \beta_{13} + \cdots + 220 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 15 \beta_{18} - 160 \beta_{17} + 163 \beta_{16} - 20 \beta_{15} - 123 \beta_{14} + 118 \beta_{13} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 64 \beta_{18} - 143 \beta_{17} + 674 \beta_{16} - 318 \beta_{15} + 175 \beta_{14} + 479 \beta_{13} + \cdots + 1845 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 186 \beta_{18} - 1716 \beta_{17} + 1810 \beta_{16} - 311 \beta_{15} - 1213 \beta_{14} + 1137 \beta_{13} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 963 \beta_{18} - 1529 \beta_{17} + 8115 \beta_{16} - 4215 \beta_{15} + 1882 \beta_{14} + 6487 \beta_{13} + \cdots + 16183 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2222 \beta_{18} - 17931 \beta_{17} + 19781 \beta_{16} - 4373 \beta_{15} - 11876 \beta_{14} + \cdots + 1667 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 12567 \beta_{18} - 16352 \beta_{17} + 93240 \beta_{16} - 51976 \beta_{15} + 19521 \beta_{14} + \cdots + 146604 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 26256 \beta_{18} - 185243 \beta_{17} + 214944 \beta_{16} - 57740 \beta_{15} - 116593 \beta_{14} + \cdots + 29492 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 152326 \beta_{18} - 175877 \beta_{17} + 1042772 \beta_{16} - 612969 \beta_{15} + 198689 \beta_{14} + \cdots + 1360642 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 307369 \beta_{18} - 1905027 \beta_{17} + 2330726 \beta_{16} - 728959 \beta_{15} - 1150881 \beta_{14} + \cdots + 401589 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1768769 \beta_{18} - 1901705 \beta_{17} + 11466798 \beta_{16} - 7018125 \beta_{15} + 2000621 \beta_{14} + \cdots + 12868814 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3559419 \beta_{18} - 19569956 \beta_{17} + 25251168 \beta_{16} - 8902203 \beta_{15} - 11423098 \beta_{14} + \cdots + 4915976 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 19994652 \beta_{18} - 20638398 \beta_{17} + 124696919 \beta_{16} - 78717542 \beta_{15} + \cdots + 123576982 \) 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
3.25955
2.97522
2.44269
2.40459
2.05170
1.96979
1.76613
1.01911
0.383819
−0.0800487
−0.346400
−0.365086
−0.537191
−0.755832
−1.67288
−1.94972
−2.53438
−2.85652
−3.17454
0 −3.25955 0 3.01882 0 1.05463 0 7.62467 0
1.2 0 −2.97522 0 0.840694 0 −4.22061 0 5.85193 0
1.3 0 −2.44269 0 −3.59133 0 0.459217 0 2.96674 0
1.4 0 −2.40459 0 −0.489610 0 4.26722 0 2.78204 0
1.5 0 −2.05170 0 2.67326 0 0.688434 0 1.20949 0
1.6 0 −1.96979 0 −1.29651 0 −5.21755 0 0.880060 0
1.7 0 −1.76613 0 −1.36988 0 0.174249 0 0.119213 0
1.8 0 −1.01911 0 3.39604 0 −2.78127 0 −1.96142 0
1.9 0 −0.383819 0 −4.10226 0 0.287206 0 −2.85268 0
1.10 0 0.0800487 0 0.442813 0 2.60518 0 −2.99359 0
1.11 0 0.346400 0 0.720634 0 −2.22237 0 −2.88001 0
1.12 0 0.365086 0 1.83829 0 2.27559 0 −2.86671 0
1.13 0 0.537191 0 −2.35184 0 −1.00979 0 −2.71143 0
1.14 0 0.755832 0 4.26919 0 −2.09147 0 −2.42872 0
1.15 0 1.67288 0 −0.778334 0 3.24553 0 −0.201474 0
1.16 0 1.94972 0 1.42435 0 −4.01869 0 0.801405 0
1.17 0 2.53438 0 −2.27936 0 1.27718 0 3.42306 0
1.18 0 2.85652 0 −0.0885731 0 −1.03776 0 5.15970 0
1.19 0 3.17454 0 −4.27638 0 −4.73492 0 7.07773 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.d ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4028.2.a.d ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4028))\):

\( T_{3}^{19} + 4 T_{3}^{18} - 30 T_{3}^{17} - 132 T_{3}^{16} + 332 T_{3}^{15} + 1714 T_{3}^{14} + \cdots + 49 \) Copy content Toggle raw display
\( T_{5}^{19} + 2 T_{5}^{18} - 56 T_{5}^{17} - 98 T_{5}^{16} + 1234 T_{5}^{15} + 1831 T_{5}^{14} + \cdots - 1664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} \) Copy content Toggle raw display
$3$ \( T^{19} + 4 T^{18} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{19} + 2 T^{18} + \cdots - 1664 \) Copy content Toggle raw display
$7$ \( T^{19} + 11 T^{18} + \cdots + 9932 \) Copy content Toggle raw display
$11$ \( T^{19} + 11 T^{18} + \cdots + 1670704 \) Copy content Toggle raw display
$13$ \( T^{19} - T^{18} + \cdots + 3319808 \) Copy content Toggle raw display
$17$ \( T^{19} + T^{18} + \cdots + 22680413 \) Copy content Toggle raw display
$19$ \( (T + 1)^{19} \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots - 5404415488 \) Copy content Toggle raw display
$29$ \( T^{19} - 288 T^{17} + \cdots + 400256 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots - 168977759517 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 45012717056 \) Copy content Toggle raw display
$41$ \( T^{19} - T^{18} + \cdots + 267986 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots - 260580334689 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 189806116180 \) Copy content Toggle raw display
$53$ \( (T - 1)^{19} \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots - 15344631283840 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots - 161104986186416 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 65423736896 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 8549930587392 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 108223785080 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 41582941006592 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 8653330539520 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots - 19025578268416 \) Copy content Toggle raw display
show more
show less