Properties

Label 2169.4.a.e
Level $2169$
Weight $4$
Character orbit 2169.a
Self dual yes
Analytic conductor $127.975$
Analytic rank $1$
Dimension $33$
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] = [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: \(33\)
Twist minimal: no (minimal twist has level 241)
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.

\(\operatorname{Tr}(f)(q) = \) \( 33 q - 9 q^{2} + 161 q^{4} - 50 q^{5} + 58 q^{7} - 111 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q - 9 q^{2} + 161 q^{4} - 50 q^{5} + 58 q^{7} - 111 q^{8} + q^{10} - 274 q^{11} + 24 q^{13} - 220 q^{14} + 789 q^{16} - 140 q^{17} + 164 q^{19} - 630 q^{20} + 270 q^{22} - 980 q^{23} + 1121 q^{25} - 484 q^{26} + 433 q^{28} - 772 q^{29} + 804 q^{31} - 947 q^{32} + 211 q^{34} - 558 q^{35} + 342 q^{37} - 1092 q^{38} + 564 q^{40} - 978 q^{41} + 230 q^{43} - 2788 q^{44} + 1103 q^{46} - 2312 q^{47} + 2501 q^{49} - 1787 q^{50} - 57 q^{52} - 1352 q^{53} + 1058 q^{55} - 2367 q^{56} + 2237 q^{58} - 3546 q^{59} + 156 q^{61} - 1349 q^{62} + 47 q^{64} - 900 q^{65} - 236 q^{67} + 3439 q^{68} - 7312 q^{70} - 5966 q^{71} - 2020 q^{73} - 488 q^{74} - 7552 q^{76} + 8 q^{77} - 198 q^{79} + 261 q^{80} - 4465 q^{82} - 502 q^{83} - 4778 q^{85} + 185 q^{86} - 9372 q^{88} + 192 q^{89} + 1896 q^{91} - 3214 q^{92} - 14222 q^{94} - 5610 q^{95} - 846 q^{97} + 3726 q^{98}+O(q^{100}) \) 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 −5.50714 0 22.3286 4.56136 0 16.8108 −78.9099 0 −25.1200
1.2 −5.34167 0 20.5335 −15.6482 0 −35.3555 −66.9498 0 83.5875
1.3 −5.14541 0 18.4753 −9.13578 0 11.4940 −53.8997 0 47.0074
1.4 −5.06398 0 17.6439 7.53582 0 −4.42337 −48.8363 0 −38.1612
1.5 −4.68737 0 13.9714 20.5680 0 17.2543 −27.9902 0 −96.4100
1.6 −4.50393 0 12.2854 −16.5922 0 27.9465 −19.3010 0 74.7302
1.7 −4.28480 0 10.3595 −5.61150 0 −3.11768 −10.1099 0 24.0441
1.8 −4.22357 0 9.83857 −18.6441 0 −0.526916 −7.76535 0 78.7445
1.9 −4.21161 0 9.73762 10.4748 0 24.1777 −7.31816 0 −44.1159
1.10 −2.89955 0 0.407364 −15.7141 0 −3.82589 22.0152 0 45.5639
1.11 −2.52337 0 −1.63263 2.94898 0 −33.1605 24.3066 0 −7.44136
1.12 −2.45382 0 −1.97876 9.95086 0 24.2976 24.4861 0 −24.4176
1.13 −1.89779 0 −4.39839 −10.1327 0 24.3749 23.5296 0 19.2298
1.14 −1.71404 0 −5.06206 2.30722 0 0.735682 22.3889 0 −3.95467
1.15 −0.975987 0 −7.04745 19.7613 0 −7.10811 14.6861 0 −19.2868
1.16 −0.664281 0 −7.55873 −16.8228 0 18.5044 10.3354 0 11.1751
1.17 −0.270698 0 −7.92672 11.6718 0 13.4994 4.31133 0 −3.15954
1.18 −0.0980206 0 −7.99039 13.5070 0 −15.0904 1.56739 0 −1.32396
1.19 0.139152 0 −7.98064 −14.6080 0 −24.3125 −2.22374 0 −2.03273
1.20 0.788730 0 −7.37790 5.19898 0 −27.1369 −12.1290 0 4.10059
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
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.e 33
3.b odd 2 1 241.4.a.b 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.4.a.b 33 3.b odd 2 1
2169.4.a.e 33 1.a even 1 1 trivial