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Properties

Label 2169.2.d.d
Level $2169$
Weight $2$
Character orbit 2169.d
Analytic conductor $17.320$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2169,2,Mod(1927,2169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2169.1927");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 93 x^{16} + 3580 x^{14} + 73633 x^{12} + 869315 x^{10} + 5859371 x^{8} + 20952638 x^{6} + \cdots + 5917589 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{2} q^{4} + \beta_{13} q^{5} + \beta_1 q^{7} - \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{2} q^{4} + \beta_{13} q^{5} + \beta_1 q^{7} - \beta_{6} q^{8} + (\beta_{13} - \beta_{7} + \cdots - \beta_{5}) q^{10}+ \cdots + (3 \beta_{13} - \beta_{8} + \cdots + 3 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 8 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 8 q^{4} - 4 q^{5} - 6 q^{10} - 12 q^{16} - 12 q^{20} - 14 q^{25} - 4 q^{29} - 20 q^{32} + 4 q^{40} - 22 q^{41} - 12 q^{47} - 60 q^{49} + 52 q^{50} - 6 q^{53} - 18 q^{58} + 4 q^{59} + 8 q^{61} - 40 q^{64} + 12 q^{67} - 2 q^{77} + 18 q^{79} - 22 q^{80} + 10 q^{82} + 36 q^{83} + 36 q^{91} + 20 q^{94} + 6 q^{97} - 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 93 x^{16} + 3580 x^{14} + 73633 x^{12} + 869315 x^{10} + 5859371 x^{8} + 20952638 x^{6} + \cdots + 5917589 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!03 \nu^{16} + \cdots - 53\!\cdots\!46 ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\!\cdots\!06 \nu^{16} + \cdots - 18\!\cdots\!93 ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28\!\cdots\!06 \nu^{17} + \cdots - 18\!\cdots\!93 \nu ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 46\!\cdots\!36 \nu^{16} + \cdots - 58\!\cdots\!05 ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 97\!\cdots\!11 \nu^{16} + \cdots + 20\!\cdots\!24 ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!04 \nu^{16} + \cdots + 10\!\cdots\!75 ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!47 \nu^{16} + \cdots - 77\!\cdots\!24 ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!12 \nu^{17} + \cdots - 17\!\cdots\!11 \nu ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23\!\cdots\!65 \nu^{17} + \cdots - 20\!\cdots\!36 \nu ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23\!\cdots\!65 \nu^{16} + \cdots + 20\!\cdots\!83 ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23\!\cdots\!18 \nu^{17} + \cdots - 11\!\cdots\!51 \nu ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26\!\cdots\!03 \nu^{16} + \cdots + 17\!\cdots\!32 ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!33 \nu^{17} + \cdots - 29\!\cdots\!26 \nu ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 29\!\cdots\!16 \nu^{17} + \cdots - 27\!\cdots\!86 \nu ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 37\!\cdots\!44 \nu^{17} + \cdots - 26\!\cdots\!49 \nu ) / 21\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 79\!\cdots\!84 \nu^{17} + \cdots - 65\!\cdots\!07 \nu ) / 21\!\cdots\!53 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{11} + 2\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{3} + \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{17} + 2\beta_{16} + 3\beta_{15} - \beta_{14} + \beta_{10} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 19 \beta_{13} - 28 \beta_{11} - 42 \beta_{8} + 31 \beta_{7} - 28 \beta_{6} - 14 \beta_{5} + \cdots + 150 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 61 \beta_{17} - 49 \beta_{16} - 85 \beta_{15} + 21 \beta_{14} - 7 \beta_{12} - 47 \beta_{10} + \cdots + 281 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 667 \beta_{13} + 377 \beta_{11} + 854 \beta_{8} - 812 \beta_{7} + 700 \beta_{6} + 118 \beta_{5} + \cdots - 2470 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1521 \beta_{17} + 1064 \beta_{16} + 1961 \beta_{15} - 328 \beta_{14} + 372 \beta_{12} + \cdots - 5186 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 17812 \beta_{13} - 5077 \beta_{11} - 17613 \beta_{8} + 19759 \beta_{7} - 16736 \beta_{6} + \cdots + 43127 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 35425 \beta_{17} - 22461 \beta_{16} - 42214 \beta_{15} + 3766 \beta_{14} - 12970 \beta_{12} + \cdots + 99283 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 434275 \beta_{13} + 65689 \beta_{11} + 368053 \beta_{8} - 463825 \beta_{7} + 392322 \beta_{6} + \cdots - 787744 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 802328 \beta_{17} + 471772 \beta_{16} + 884837 \beta_{15} - 11006 \beta_{14} + 381316 \beta_{12} + \cdots - 1955370 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 10188902 \beta_{13} - 739212 \beta_{11} - 7778591 \beta_{8} + 10684730 \beta_{7} - 9103601 \beta_{6} + \cdots + 14902057 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17967493 \beta_{17} - 9947780 \beta_{16} - 18399099 \beta_{15} - 1149523 \beta_{14} + \cdots + 39393651 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 234788168 \beta_{13} + 4740181 \beta_{11} + 166073637 \beta_{8} - 243610957 \beta_{7} + 209928045 \beta_{6} + \cdots - 289844800 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 400861805 \beta_{17} + 211324093 \beta_{16} + 383037886 \beta_{15} + 51506831 \beta_{14} + \cdots - 808617544 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 5362194600 \beta_{13} + 93160407 \beta_{11} - 3578284053 \beta_{8} + 5522385328 \beta_{7} + \cdots + 5766851926 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 8940478653 \beta_{17} - 4527557722 \beta_{16} - 8020359294 \beta_{15} - 1622733372 \beta_{14} + \cdots + 16861934446 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2169\mathbb{Z}\right)^\times\).

\(n\) \(730\) \(965\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1927.1
− 3.05566i
3.05566i
− 1.17687i
1.17687i
− 4.17304i
4.17304i
− 3.30061i
3.30061i
− 4.24169i
4.24169i
− 0.872558i
0.872558i
− 0.806404i
0.806404i
− 4.75050i
4.75050i
− 3.46393i
3.46393i
−2.42329 0 3.87232 −0.730711 0 − 3.05566i −4.53717 0 1.77072
1927.2 −2.42329 0 3.87232 −0.730711 0 3.05566i −4.53717 0 1.77072
1927.3 −1.71402 0 0.937857 0.783277 0 − 1.17687i 1.82053 0 −1.34255
1927.4 −1.71402 0 0.937857 0.783277 0 1.17687i 1.82053 0 −1.34255
1927.5 −1.24262 0 −0.455890 −1.57905 0 − 4.17304i 3.05174 0 1.96216
1927.6 −1.24262 0 −0.455890 −1.57905 0 4.17304i 3.05174 0 1.96216
1927.7 −0.781140 0 −1.38982 2.95136 0 − 3.30061i 2.64792 0 −2.30543
1927.8 −0.781140 0 −1.38982 2.95136 0 3.30061i 2.64792 0 −2.30543
1927.9 0.0801743 0 −1.99357 −1.66166 0 − 4.24169i −0.320182 0 −0.133222
1927.10 0.0801743 0 −1.99357 −1.66166 0 4.24169i −0.320182 0 −0.133222
1927.11 0.527696 0 −1.72154 −1.78312 0 − 0.872558i −1.96384 0 −0.940947
1927.12 0.527696 0 −1.72154 −1.78312 0 0.872558i −1.96384 0 −0.940947
1927.13 1.29095 0 −0.333457 2.54416 0 − 0.806404i −3.01237 0 3.28438
1927.14 1.29095 0 −0.333457 2.54416 0 0.806404i −3.01237 0 3.28438
1927.15 2.11230 0 2.46180 −3.50370 0 − 4.75050i 0.975459 0 −7.40085
1927.16 2.11230 0 2.46180 −3.50370 0 4.75050i 0.975459 0 −7.40085
1927.17 2.14995 0 2.62230 0.979434 0 − 3.46393i 1.33791 0 2.10574
1927.18 2.14995 0 2.62230 0.979434 0 3.46393i 1.33791 0 2.10574
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1927.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2169.2.d.d 18
3.b odd 2 1 241.2.b.a ✓ 18
241.b even 2 1 inner 2169.2.d.d 18
723.b odd 2 1 241.2.b.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.b.a ✓ 18 3.b odd 2 1
241.2.b.a ✓ 18 723.b odd 2 1
2169.2.d.d 18 1.a even 1 1 trivial
2169.2.d.d 18 241.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 11T_{2}^{7} + 38T_{2}^{5} + 2T_{2}^{4} - 45T_{2}^{3} - 3T_{2}^{2} + 13T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(2169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{9} - 11 T^{7} + 38 T^{5} + \cdots - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( (T^{9} + 2 T^{8} - 17 T^{7} + \cdots + 69)^{2} \) Copy content Toggle raw display
$7$ \( T^{18} + 93 T^{16} + \cdots + 5917589 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 1917298836 \) Copy content Toggle raw display
$13$ \( T^{18} + 161 T^{16} + \cdots + 94681424 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 59175890000 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 15391648989 \) Copy content Toggle raw display
$23$ \( T^{18} + 252 T^{16} + \cdots + 5917589 \) Copy content Toggle raw display
$29$ \( (T^{9} + 2 T^{8} + \cdots - 9565)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 106256228084 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 2130332040000 \) Copy content Toggle raw display
$41$ \( (T^{9} + 11 T^{8} + \cdots + 512081)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 9947467109 \) Copy content Toggle raw display
$47$ \( (T^{9} + 6 T^{8} + \cdots + 3280300)^{2} \) Copy content Toggle raw display
$53$ \( (T^{9} + 3 T^{8} + \cdots - 71875)^{2} \) Copy content Toggle raw display
$59$ \( (T^{9} - 2 T^{8} + \cdots - 21777256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} - 4 T^{8} + \cdots + 8432003)^{2} \) Copy content Toggle raw display
$67$ \( (T^{9} - 6 T^{8} + \cdots - 519348)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 135603735525341 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{9} - 9 T^{8} + \cdots + 825764)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} - 18 T^{8} + \cdots + 9796212)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{9} - 3 T^{8} + \cdots + 3819013)^{2} \) Copy content Toggle raw display
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