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Properties

Label 2169.4.a.f
Level $2169$
Weight $4$
Character orbit 2169.a
Self dual yes
Analytic conductor $127.975$
Analytic rank $0$
Dimension $35$
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] = [2169,4,Mod(1,2169)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2169.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2169.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: \(127.975142802\)
Analytic rank: \(0\)
Dimension: \(35\)
Twist minimal: no (minimal twist has level 723)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 35 q - 9 q^{2} + 173 q^{4} - 66 q^{5} + 110 q^{7} - 108 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q - 9 q^{2} + 173 q^{4} - 66 q^{5} + 110 q^{7} - 108 q^{8} + 111 q^{10} - 36 q^{11} + 432 q^{13} - 120 q^{14} + 949 q^{16} - 272 q^{17} + 424 q^{19} - 469 q^{20} + 404 q^{22} - 144 q^{23} + 1369 q^{25} - 544 q^{26} + 1023 q^{28} - 510 q^{29} + 1552 q^{31} - 1118 q^{32} + 935 q^{34} - 542 q^{35} + 1900 q^{37} - 690 q^{38} + 946 q^{40} - 582 q^{41} + 1462 q^{43} - 568 q^{44} + 1103 q^{46} - 1592 q^{47} + 3137 q^{49} - 1839 q^{50} + 3904 q^{52} - 1568 q^{53} + 2072 q^{55} - 1241 q^{56} + 1200 q^{58} - 1144 q^{59} + 1730 q^{61} + 1763 q^{62} + 2724 q^{64} + 48 q^{65} + 2422 q^{67} + 2472 q^{68} - 2858 q^{70} + 846 q^{71} + 3458 q^{73} + 2192 q^{74} + 3373 q^{76} - 2398 q^{77} + 2684 q^{79} + 1399 q^{80} + 447 q^{82} + 2260 q^{83} + 4958 q^{85} + 4258 q^{86} + 251 q^{88} - 3084 q^{89} + 3186 q^{91} + 329 q^{92} + 2733 q^{94} + 950 q^{95} + 4916 q^{97} + 5024 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.38545 0 21.0031 −7.22190 0 16.9338 −70.0274 0 38.8932
1.2 −5.31520 0 20.2514 4.14139 0 −6.43170 −65.1184 0 −22.0123
1.3 −5.30659 0 20.1599 18.4879 0 27.9877 −64.5279 0 −98.1078
1.4 −5.28377 0 19.9182 −14.3643 0 0.953059 −62.9729 0 75.8974
1.5 −4.97881 0 16.7886 −8.67277 0 −9.77632 −43.7568 0 43.1801
1.6 −4.61556 0 13.3034 −18.9324 0 −22.5896 −24.4783 0 87.3839
1.7 −4.36903 0 11.0884 −12.6312 0 25.0684 −13.4934 0 55.1861
1.8 −4.29858 0 10.4778 2.32031 0 27.5818 −10.6511 0 −9.97403
1.9 −3.92027 0 7.36855 13.3755 0 −16.2022 2.47546 0 −52.4358
1.10 −3.23131 0 2.44139 −21.9660 0 −0.0149568 17.9616 0 70.9789
1.11 −2.94100 0 0.649490 −5.07234 0 26.7993 21.6179 0 14.9178
1.12 −2.41627 0 −2.16165 −2.81723 0 3.47106 24.5533 0 6.80719
1.13 −2.24313 0 −2.96835 19.4522 0 0.718192 24.6035 0 −43.6338
1.14 −1.75070 0 −4.93504 11.8443 0 −29.4586 22.6454 0 −20.7359
1.15 −1.41379 0 −6.00120 −12.3467 0 −26.1311 19.7948 0 17.4556
1.16 −0.896315 0 −7.19662 13.7222 0 35.0755 13.6210 0 −12.2994
1.17 −0.654407 0 −7.57175 −15.0511 0 0.978677 10.1903 0 9.84955
1.18 −0.599969 0 −7.64004 −21.7839 0 29.0635 9.38353 0 13.0697
1.19 −0.560169 0 −7.68621 9.69072 0 −28.3724 8.78693 0 −5.42844
1.20 0.110671 0 −7.98775 −8.19248 0 13.3555 −1.76938 0 −0.906671
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(241\) \(-1\)

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 2169.4.a.f 35
3.b odd 2 1 723.4.a.d ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
723.4.a.d ✓ 35 3.b odd 2 1
2169.4.a.f 35 1.a even 1 1 trivial