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Properties

Label 869.4.a.d
Level $869$
Weight $4$
Character orbit 869.a
Self dual yes
Analytic conductor $51.273$
Analytic rank $0$
Dimension $54$
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] = [869,4,Mod(1,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, 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("869.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 869.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: \(51.2726597950\)
Analytic rank: \(0\)
Dimension: \(54\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 8 q^{2} + 18 q^{3} + 242 q^{4} + 30 q^{5} + 78 q^{6} + 98 q^{7} + 96 q^{8} + 558 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q + 8 q^{2} + 18 q^{3} + 242 q^{4} + 30 q^{5} + 78 q^{6} + 98 q^{7} + 96 q^{8} + 558 q^{9} + 175 q^{10} + 594 q^{11} + 81 q^{12} + 200 q^{13} + 63 q^{14} + 34 q^{15} + 1246 q^{16} + 446 q^{17} + 195 q^{18} + 1072 q^{19} + 175 q^{20} + 558 q^{21} + 88 q^{22} + 206 q^{23} + 41 q^{24} + 1578 q^{25} + 191 q^{26} + 246 q^{27} + 540 q^{28} + 700 q^{29} - 1327 q^{30} + 1200 q^{31} + 1441 q^{32} + 198 q^{33} + 714 q^{34} + 1066 q^{35} + 1276 q^{36} + 366 q^{37} + 134 q^{38} + 1220 q^{39} + 2406 q^{40} + 1306 q^{41} + 1181 q^{42} + 1752 q^{43} + 2662 q^{44} + 816 q^{45} + 1190 q^{46} + 270 q^{47} + 956 q^{48} + 3750 q^{49} + 1314 q^{50} + 2690 q^{51} + 1321 q^{52} - 8 q^{53} + 1786 q^{54} + 330 q^{55} + 442 q^{56} + 60 q^{57} + 211 q^{58} + 2154 q^{59} - 1758 q^{60} + 3194 q^{61} + 1621 q^{62} + 2354 q^{63} + 8110 q^{64} + 1356 q^{65} + 858 q^{66} + 2244 q^{67} + 4242 q^{68} - 1214 q^{69} + 511 q^{70} + 1140 q^{71} + 2760 q^{72} + 4532 q^{73} + 1620 q^{74} + 1782 q^{75} + 10250 q^{76} + 1078 q^{77} + 2667 q^{78} + 4266 q^{79} + 369 q^{80} + 6602 q^{81} + 2452 q^{82} + 5898 q^{83} + 9090 q^{84} + 658 q^{85} - 785 q^{86} + 526 q^{87} + 1056 q^{88} - 278 q^{89} + 4541 q^{90} + 7798 q^{91} + 1466 q^{92} + 1388 q^{93} + 4666 q^{94} + 2150 q^{95} + 3064 q^{96} + 1356 q^{97} + 3303 q^{98} + 6138 q^{99}+O(q^{100}) \) 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 −5.60920 5.48527 23.4631 −13.4817 −30.7679 −14.8938 −86.7355 3.08816 75.6213
1.2 −5.53825 −1.92471 22.6722 −2.12937 10.6595 26.8333 −81.2585 −23.2955 11.7930
1.3 −4.90370 −9.23045 16.0463 0.433561 45.2634 −22.0505 −39.4567 58.2011 −2.12605
1.4 −4.88219 −5.88543 15.8358 −19.3476 28.7338 25.2150 −38.2560 7.63832 94.4586
1.5 −4.87925 7.49400 15.8071 11.1347 −36.5651 15.3350 −38.0926 29.1600 −54.3290
1.6 −4.81926 −5.13051 15.2252 11.0201 24.7253 −8.05395 −34.8203 −0.677827 −53.1088
1.7 −4.59554 1.26020 13.1190 −4.71669 −5.79128 −17.5996 −23.5244 −25.4119 21.6757
1.8 −4.27358 −2.43415 10.2635 18.5802 10.4026 27.0111 −9.67335 −21.0749 −79.4042
1.9 −4.12652 8.44080 9.02820 −4.91999 −34.8312 −11.9840 −4.24288 44.2472 20.3025
1.10 −4.02047 4.04882 8.16422 7.73716 −16.2782 −24.5258 −0.660230 −10.6071 −31.1070
1.11 −3.95254 7.62099 7.62253 10.4555 −30.1222 36.0848 1.49195 31.0795 −41.3257
1.12 −3.55277 −2.50677 4.62217 −9.33374 8.90598 −5.88532 12.0007 −20.7161 33.1606
1.13 −3.31451 −9.34120 2.98599 −20.0030 30.9615 −16.8759 16.6190 60.2581 66.3000
1.14 −3.30216 −5.59634 2.90425 19.8342 18.4800 −10.7938 16.8270 4.31907 −65.4958
1.15 −2.99942 4.55091 0.996522 −19.6893 −13.6501 20.0024 21.0064 −6.28926 59.0566
1.16 −2.53162 −7.08403 −1.59091 −3.24169 17.9341 −8.26650 24.2805 23.1835 8.20671
1.17 −2.23996 −1.75365 −2.98259 −6.54249 3.92811 19.8087 24.6005 −23.9247 14.6549
1.18 −2.23032 2.00860 −3.02568 −7.35852 −4.47982 −19.5239 24.5908 −22.9655 16.4118
1.19 −1.88454 3.31215 −4.44850 17.6817 −6.24190 4.46789 23.4597 −16.0296 −33.3219
1.20 −1.71370 9.06048 −5.06325 −1.69300 −15.5269 12.5287 22.3864 55.0923 2.90129
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.54
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(79\) \(-1\)

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 869.4.a.d ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.4.a.d ✓ 54 1.a even 1 1 trivial