Properties

Label 2169.4.a.c
Level $2169$
Weight $4$
Character orbit 2169.a
Self dual yes
Analytic conductor $127.975$
Analytic rank $1$
Dimension $29$
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: \(1\)
Dimension: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q - 9 q^{2} + 97 q^{4} - 62 q^{5} - 30 q^{7} - 108 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q - 9 q^{2} + 97 q^{4} - 62 q^{5} - 30 q^{7} - 108 q^{8} + 51 q^{10} - 46 q^{11} + 250 q^{13} - 84 q^{14} + 333 q^{16} - 128 q^{17} + 58 q^{19} - 405 q^{20} - 48 q^{22} - 232 q^{23} + 707 q^{25} - 238 q^{26} - 89 q^{28} - 590 q^{29} - 468 q^{31} - 1068 q^{32} + 287 q^{34} - 474 q^{35} + 842 q^{37} - 160 q^{38} + 434 q^{40} - 814 q^{41} + 20 q^{43} - 150 q^{44} - 37 q^{46} - 1004 q^{47} + 1239 q^{49} - 839 q^{50} + 1928 q^{52} - 2192 q^{53} + 432 q^{55} - 437 q^{56} - 28 q^{58} - 1288 q^{59} + 1502 q^{61} - 3059 q^{62} + 3372 q^{64} - 2312 q^{65} + 358 q^{67} - 4990 q^{68} + 5366 q^{70} - 1938 q^{71} + 3266 q^{73} - 2510 q^{74} + 3591 q^{76} - 5098 q^{77} - 292 q^{79} - 8235 q^{80} + 4511 q^{82} - 4256 q^{83} + 1998 q^{85} - 6860 q^{86} + 5935 q^{88} - 6428 q^{89} - 1650 q^{91} - 6823 q^{92} + 3025 q^{94} - 1802 q^{95} + 5040 q^{97} - 9410 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.53188 0 22.6017 4.41322 0 −24.6542 −80.7747 0 −24.4134
1.2 −5.40987 0 21.2667 −19.5426 0 5.91074 −71.7711 0 105.723
1.3 −5.00185 0 17.0185 −7.18086 0 2.32627 −45.1091 0 35.9176
1.4 −4.92480 0 16.2537 −0.995320 0 35.5796 −40.6479 0 4.90176
1.5 −4.04552 0 8.36624 17.6002 0 −8.19403 −1.48162 0 −71.2021
1.6 −3.42298 0 3.71677 −8.82688 0 −24.4656 14.6614 0 30.2142
1.7 −3.42059 0 3.70041 −14.1333 0 16.0369 14.7071 0 48.3443
1.8 −3.34560 0 3.19303 −17.6990 0 18.2648 16.0822 0 59.2136
1.9 −2.93958 0 0.641136 15.8431 0 −32.0159 21.6320 0 −46.5721
1.10 −2.75926 0 −0.386494 13.3161 0 −12.5857 23.1405 0 −36.7427
1.11 −2.61315 0 −1.17143 6.42911 0 15.9906 23.9664 0 −16.8002
1.12 −1.18901 0 −6.58626 −19.7501 0 −30.4473 17.3432 0 23.4831
1.13 −1.17966 0 −6.60840 5.20729 0 19.7482 17.2329 0 −6.14283
1.14 −1.14345 0 −6.69252 0.465287 0 −21.2063 16.8002 0 −0.532033
1.15 −1.12035 0 −6.74482 −13.1582 0 29.4550 16.5193 0 14.7417
1.16 0.364523 0 −7.86712 −5.79603 0 11.9832 −5.78393 0 −2.11279
1.17 1.16445 0 −6.64406 10.5306 0 9.63072 −17.0522 0 12.2623
1.18 1.28359 0 −6.35240 −0.508995 0 −35.7971 −18.4226 0 −0.653341
1.19 1.68254 0 −5.16907 −2.08567 0 30.8603 −22.1574 0 −3.50922
1.20 1.68847 0 −5.14906 −22.1888 0 −11.1315 −22.2018 0 −37.4651
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
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.c 29
3.b odd 2 1 723.4.a.b 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
723.4.a.b 29 3.b odd 2 1
2169.4.a.c 29 1.a even 1 1 trivial