↑

Properties

Label 2009.4.a.g
Level $2009$
Weight $4$
Character orbit 2009.a
Self dual yes
Analytic conductor $118.535$
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] = [2009,4,Mod(1,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, 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("2009.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2009.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: \(118.534837202\)
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: \( x^{19} - 4 x^{18} - 112 x^{17} + 431 x^{16} + 5147 x^{15} - 18874 x^{14} - 125634 x^{13} + \cdots - 4783968 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 287)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} - 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{11} - 1) q^{5} + ( - \beta_{4} - \beta_1 - 1) q^{6} + (\beta_{3} + 5 \beta_1 + 2) q^{8} + ( - \beta_{11} + \beta_{10} + \cdots + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} - 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{11} - 1) q^{5} + ( - \beta_{4} - \beta_1 - 1) q^{6} + (\beta_{3} + 5 \beta_1 + 2) q^{8} + ( - \beta_{11} + \beta_{10} + \cdots + 11) q^{9}+ \cdots + ( - 3 \beta_{18} - 2 \beta_{17} + \cdots + 88) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 4 q^{2} - 13 q^{3} + 88 q^{4} - 26 q^{5} - 30 q^{6} + 51 q^{8} + 206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 4 q^{2} - 13 q^{3} + 88 q^{4} - 26 q^{5} - 30 q^{6} + 51 q^{8} + 206 q^{9} - 108 q^{10} - 4 q^{11} - 52 q^{12} - 239 q^{13} - 144 q^{15} + 484 q^{16} - 117 q^{17} - 193 q^{18} - 323 q^{19} - 519 q^{20} + 105 q^{22} + 121 q^{23} + 127 q^{24} + 989 q^{25} - 134 q^{26} - 328 q^{27} + 72 q^{29} + 311 q^{30} - 872 q^{31} + 422 q^{32} - 746 q^{33} - 639 q^{34} + 552 q^{36} + 419 q^{37} - 106 q^{38} + 420 q^{39} - 1463 q^{40} - 779 q^{41} + 1231 q^{43} - 183 q^{44} - 1638 q^{45} - 630 q^{46} + 163 q^{47} - 1694 q^{48} + 791 q^{50} - 1020 q^{51} - 2737 q^{52} - 308 q^{53} - 932 q^{54} - 1668 q^{55} + 3346 q^{57} + 495 q^{58} - 902 q^{59} - 1080 q^{60} - 1390 q^{61} - 946 q^{62} + 2239 q^{64} - 1500 q^{65} - 3336 q^{66} + 86 q^{67} - 849 q^{68} - 1459 q^{69} + 784 q^{71} - 2013 q^{72} - 4270 q^{73} - 2009 q^{74} + 1841 q^{75} - 155 q^{76} - 1633 q^{78} - 392 q^{79} - 1160 q^{80} + 451 q^{81} - 164 q^{82} - 112 q^{83} + 2740 q^{85} - 6521 q^{86} - 1868 q^{87} - 1540 q^{88} - 687 q^{89} + 7589 q^{90} - 3552 q^{92} - 3908 q^{93} + 1705 q^{94} - 3636 q^{95} + 7802 q^{96} - 577 q^{97} + 1040 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 4 x^{18} - 112 x^{17} + 431 x^{16} + 5147 x^{15} - 18874 x^{14} - 125634 x^{13} + \cdots - 4783968 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 21\nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79\!\cdots\!21 \nu^{18} + \cdots + 41\!\cdots\!44 ) / 10\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!55 \nu^{18} + \cdots + 16\!\cdots\!56 ) / 21\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 86\!\cdots\!67 \nu^{18} + \cdots + 58\!\cdots\!24 ) / 35\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\!\cdots\!35 \nu^{18} + \cdots + 10\!\cdots\!68 ) / 52\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50\!\cdots\!31 \nu^{18} + \cdots + 33\!\cdots\!20 ) / 10\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32\!\cdots\!61 \nu^{18} + \cdots + 26\!\cdots\!80 ) / 52\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40\!\cdots\!35 \nu^{18} + \cdots + 33\!\cdots\!96 ) / 63\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!61 \nu^{18} + \cdots + 10\!\cdots\!20 ) / 21\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 20\!\cdots\!25 \nu^{18} + \cdots + 13\!\cdots\!84 ) / 20\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!95 \nu^{18} + \cdots - 92\!\cdots\!32 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15\!\cdots\!25 \nu^{18} + \cdots + 10\!\cdots\!92 ) / 56\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 16\!\cdots\!61 \nu^{18} + \cdots - 11\!\cdots\!56 ) / 58\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 60\!\cdots\!75 \nu^{18} + \cdots + 43\!\cdots\!12 ) / 18\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 46\!\cdots\!32 \nu^{18} + \cdots + 30\!\cdots\!52 ) / 13\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 46\!\cdots\!15 \nu^{18} + \cdots + 32\!\cdots\!12 ) / 10\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 21\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{15} - 2 \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} + \beta_{6} + 3 \beta_{5} + \cdots + 276 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{18} - \beta_{17} + \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} - 4 \beta_{12} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{18} + 45 \beta_{17} + 4 \beta_{16} - 32 \beta_{15} - 77 \beta_{14} + 52 \beta_{13} - 9 \beta_{12} + \cdots + 6706 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 75 \beta_{18} - 32 \beta_{17} + 65 \beta_{16} - 50 \beta_{15} + 46 \beta_{14} + 73 \beta_{13} + \cdots + 1486 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 56 \beta_{18} + 1619 \beta_{17} + 199 \beta_{16} - 802 \beta_{15} - 2385 \beta_{14} + 1873 \beta_{13} + \cdots + 174756 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3514 \beta_{18} - 782 \beta_{17} + 2927 \beta_{16} - 1918 \beta_{15} + 1617 \beta_{14} + 3465 \beta_{13} + \cdots + 10754 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2789 \beta_{18} + 53649 \beta_{17} + 6880 \beta_{16} - 18073 \beta_{15} - 69324 \beta_{14} + \cdots + 4731975 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 136978 \beta_{18} - 18920 \beta_{17} + 111737 \beta_{16} - 67725 \beta_{15} + 52951 \beta_{14} + \cdots - 699962 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 130517 \beta_{18} + 1706025 \beta_{17} + 202346 \beta_{16} - 368939 \beta_{15} - 1971132 \beta_{14} + \cdots + 131032547 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4865652 \beta_{18} - 522444 \beta_{17} + 3899892 \beta_{16} - 2302499 \beta_{15} + 1705535 \beta_{14} + \cdots - 49536523 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5649187 \beta_{18} + 52983637 \beta_{17} + 5349244 \beta_{16} - 6403262 \beta_{15} - 55674099 \beta_{14} + \cdots + 3681242672 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 163827610 \beta_{18} - 17166271 \beta_{17} + 129109682 \beta_{16} - 76454676 \beta_{15} + \cdots - 2268955627 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 227723351 \beta_{18} + 1621549651 \beta_{17} + 128050895 \beta_{16} - 69553627 \beta_{15} + \cdots + 104498613119 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 5334928245 \beta_{18} - 631904628 \beta_{17} + 4136141141 \beta_{16} - 2494860297 \beta_{15} + \cdots - 89605874763 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 8673586080 \beta_{18} + 49155634762 \beta_{17} + 2687051846 \beta_{16} + 1106680653 \beta_{15} + \cdots + 2990516237201 \) 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
−5.53337
−5.02056
−3.82132
−3.43575
−3.32075
−3.17602
−2.59782
−0.484766
0.138695
0.697028
0.715263
0.948961
2.09684
3.25987
3.63775
4.38524
4.95189
5.25811
5.30071
−5.53337 −6.88370 22.6182 −1.88501 38.0901 0 −80.8880 20.3854 10.4305
1.2 −5.02056 5.21517 17.2061 3.80949 −26.1831 0 −46.2196 0.197954 −19.1258
1.3 −3.82132 9.69544 6.60248 −17.6188 −37.0494 0 5.34035 67.0016 67.3270
1.4 −3.43575 −3.06482 3.80437 20.7461 10.5299 0 14.4151 −17.6069 −71.2785
1.5 −3.32075 4.17440 3.02740 8.80981 −13.8622 0 16.5128 −9.57439 −29.2552
1.6 −3.17602 −8.48387 2.08707 −11.5871 26.9449 0 18.7795 44.9760 36.8010
1.7 −2.59782 −1.99766 −1.25134 −7.25464 5.18956 0 24.0333 −23.0094 18.8462
1.8 −0.484766 6.99783 −7.76500 −16.1477 −3.39231 0 7.64234 21.9696 7.82787
1.9 0.138695 −10.3059 −7.98076 12.9025 −1.42938 0 −2.21645 79.2122 1.78951
1.10 0.697028 1.53792 −7.51415 −0.772366 1.07197 0 −10.8138 −24.6348 −0.538361
1.11 0.715263 −1.93213 −7.48840 22.2651 −1.38198 0 −11.0783 −23.2669 15.9254
1.12 0.948961 −5.37861 −7.09947 −19.3158 −5.10409 0 −14.3288 1.92948 −18.3300
1.13 2.09684 7.20675 −3.60327 16.1298 15.1114 0 −24.3302 24.9373 33.8216
1.14 3.25987 −8.46195 2.62673 −5.57919 −27.5848 0 −17.5161 44.6046 −18.1874
1.15 3.63775 5.37879 5.23323 0.401776 19.5667 0 −10.0648 1.93139 1.46156
1.16 4.38524 −2.55101 11.2303 −20.1177 −11.1868 0 14.1657 −20.4924 −88.2210
1.17 4.95189 −7.22769 16.5213 −0.511871 −35.7908 0 42.1964 25.2395 −2.53473
1.18 5.25811 −3.01212 19.6478 7.00206 −15.8381 0 61.2453 −17.9271 36.8176
1.19 5.30071 6.09319 20.0975 −17.2764 32.2982 0 64.1253 10.1270 −91.5772
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(41\) \(1\)

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 2009.4.a.g 19
7.b odd 2 1 287.4.a.e ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.4.a.e ✓ 19 7.b odd 2 1
2009.4.a.g 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{19} - 4 T_{2}^{18} - 112 T_{2}^{17} + 431 T_{2}^{16} + 5147 T_{2}^{15} - 18874 T_{2}^{14} + \cdots - 4783968 \) Copy content Toggle raw display
\( T_{3}^{19} + 13 T_{3}^{18} - 275 T_{3}^{17} - 3964 T_{3}^{16} + 28686 T_{3}^{15} + 492183 T_{3}^{14} + \cdots + 9655701376416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} - 4 T^{18} + \cdots - 4783968 \) Copy content Toggle raw display
$3$ \( T^{19} + \cdots + 9655701376416 \) Copy content Toggle raw display
$5$ \( T^{19} + \cdots + 60\!\cdots\!28 \) Copy content Toggle raw display
$7$ \( T^{19} \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots - 56\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots + 73\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots - 12\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots - 21\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T + 41)^{19} \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 53\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots - 37\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 84\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 45\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots + 52\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots + 60\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots - 21\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 27\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots + 22\!\cdots\!12 \) Copy content Toggle raw display
show more
show less