Properties

Label 89.9.d.a
Level $89$
Weight $9$
Character orbit 89.d
Analytic conductor $36.257$
Analytic rank $0$
Dimension $236$
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] = [89,9,Mod(12,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.12");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 89.d (of order \(8\), degree \(4\), 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: \(36.2566962954\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(59\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 236 q - 8 q^{2} + 220 q^{3} + 29688 q^{4} + 332 q^{5} - 2052 q^{6} - 4 q^{7} - 2056 q^{8} + 14504 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 236 q - 8 q^{2} + 220 q^{3} + 29688 q^{4} + 332 q^{5} - 2052 q^{6} - 4 q^{7} - 2056 q^{8} + 14504 q^{9} - 1028 q^{10} - 97476 q^{12} - 95204 q^{13} + 94204 q^{14} - 4 q^{15} + 3667960 q^{16} - 67456 q^{17} + 120828 q^{18} + 423068 q^{19} + 889080 q^{20} - 1025092 q^{21} + 595868 q^{23} + 353520 q^{24} - 2901512 q^{26} - 1270028 q^{27} + 2301148 q^{28} - 576124 q^{29} + 6928684 q^{30} + 877908 q^{31} - 526344 q^{32} - 2038964 q^{33} - 4691820 q^{34} + 4025468 q^{35} + 8307708 q^{36} + 1546548 q^{37} - 1028 q^{38} - 8 q^{39} + 9711252 q^{40} + 14565884 q^{41} - 9995088 q^{42} + 7940220 q^{43} + 34911848 q^{45} + 11789660 q^{46} - 26481760 q^{47} - 51979188 q^{48} + 85220 q^{49} + 51174744 q^{51} - 73930392 q^{52} + 9747284 q^{53} + 75838728 q^{54} + 47889280 q^{55} - 22593976 q^{56} - 11993252 q^{58} + 97628864 q^{59} - 85858136 q^{60} + 13251468 q^{61} + 92105116 q^{62} + 26029912 q^{63} + 316184792 q^{64} - 156094816 q^{65} - 92328636 q^{66} + 56350304 q^{67} + 83635296 q^{68} - 206335212 q^{69} + 173206640 q^{70} - 82173484 q^{71} - 210835728 q^{72} - 27844820 q^{74} - 29476036 q^{75} + 249929324 q^{76} + 21869340 q^{77} + 242778104 q^{78} - 79268176 q^{79} + 220536952 q^{80} - 193896452 q^{82} + 21717416 q^{83} - 474363188 q^{84} + 154933216 q^{86} + 415677380 q^{89} - 237576720 q^{90} + 571377128 q^{91} + 126840936 q^{92} + 399514672 q^{93} - 651711884 q^{94} + 390077024 q^{95} + 73697656 q^{96} - 456904840 q^{97} - 1341430712 q^{98} + 237646156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −31.0946 −9.36236 22.6027i 710.877 −857.445 857.445i 291.119 + 702.824i 628.442 + 1517.19i −14144.2 4216.10 4216.10i 26662.0 + 26662.0i
12.2 −31.0011 31.3265 + 75.6288i 705.071 51.8392 + 51.8392i −971.157 2344.58i −917.976 2216.19i −13921.7 −99.0429 + 99.0429i −1607.07 1607.07i
12.3 −28.9464 −19.8602 47.9467i 581.896 735.914 + 735.914i 574.881 + 1387.89i 252.151 + 608.747i −9433.52 2734.87 2734.87i −21302.1 21302.1i
12.4 −28.5712 −56.6127 136.675i 560.314 −23.2205 23.2205i 1617.49 + 3904.98i 166.667 + 402.371i −8694.63 −10835.8 + 10835.8i 663.439 + 663.439i
12.5 −27.3135 −25.5602 61.7079i 490.027 −165.674 165.674i 698.140 + 1685.46i −1577.37 3808.10i −6392.10 1484.79 1484.79i 4525.14 + 4525.14i
12.6 −26.9582 8.42763 + 20.3461i 470.746 145.579 + 145.579i −227.194 548.495i 1430.71 + 3454.05i −5789.16 4296.39 4296.39i −3924.54 3924.54i
12.7 −26.3802 46.7395 + 112.839i 439.914 143.497 + 143.497i −1233.00 2976.71i −358.362 865.163i −4851.68 −5908.74 + 5908.74i −3785.47 3785.47i
12.8 −26.2674 26.2029 + 63.2594i 433.974 150.875 + 150.875i −688.281 1661.66i 1068.50 + 2579.60i −4674.91 1324.17 1324.17i −3963.09 3963.09i
12.9 −25.6656 58.9224 + 142.251i 402.721 −721.025 721.025i −1512.28 3650.96i 588.391 + 1420.50i −3765.67 −12124.2 + 12124.2i 18505.5 + 18505.5i
12.10 −23.4949 −24.5114 59.1758i 296.008 −339.543 339.543i 575.892 + 1390.33i 235.030 + 567.412i −939.987 1738.36 1738.36i 7977.52 + 7977.52i
12.11 −20.9644 33.0521 + 79.7947i 183.506 775.587 + 775.587i −692.916 1672.85i −1234.34 2979.97i 1519.80 −635.433 + 635.433i −16259.7 16259.7i
12.12 −20.7331 −27.2807 65.8615i 173.863 410.068 + 410.068i 565.615 + 1365.52i −959.586 2316.64i 1702.97 1045.83 1045.83i −8501.98 8501.98i
12.13 −20.5596 15.9072 + 38.4034i 166.695 −384.710 384.710i −327.045 789.556i −987.510 2384.06i 1836.06 3417.55 3417.55i 7909.46 + 7909.46i
12.14 −18.9016 18.5519 + 44.7883i 101.272 −670.857 670.857i −350.662 846.573i −29.3264 70.8003i 2924.61 2977.51 2977.51i 12680.3 + 12680.3i
12.15 −18.6788 −38.3619 92.6139i 92.8993 −257.482 257.482i 716.557 + 1729.92i 1454.22 + 3510.79i 3046.53 −2466.37 + 2466.37i 4809.47 + 4809.47i
12.16 −18.3903 55.4418 + 133.848i 82.2026 596.658 + 596.658i −1019.59 2461.51i 1012.39 + 2444.13i 3196.18 −10202.3 + 10202.3i −10972.7 10972.7i
12.17 −16.0085 −51.6553 124.707i 0.272565 637.942 + 637.942i 826.924 + 1996.37i 501.601 + 1210.97i 4093.82 −8244.21 + 8244.21i −10212.5 10212.5i
12.18 −13.9550 −1.45574 3.51448i −61.2570 488.794 + 488.794i 20.3150 + 49.0447i 318.064 + 767.876i 4427.33 4629.10 4629.10i −6821.14 6821.14i
12.19 −13.8988 −56.4369 136.251i −62.8227 −785.745 785.745i 784.406 + 1893.72i −1084.96 2619.32i 4431.26 −10739.8 + 10739.8i 10920.9 + 10920.9i
12.20 −12.8757 8.12224 + 19.6088i −90.2159 332.420 + 332.420i −104.580 252.478i 865.626 + 2089.81i 4457.78 4320.79 4320.79i −4280.15 4280.15i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.59
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.d odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 89.9.d.a 236
89.d odd 8 1 inner 89.9.d.a 236
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.9.d.a 236 1.a even 1 1 trivial
89.9.d.a 236 89.d odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(89, [\chi])\).