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Properties

Label 89.4.c.a
Level $89$
Weight $4$
Character orbit 89.c
Analytic conductor $5.251$
Analytic rank $0$
Dimension $42$
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] = [89,4,Mod(34,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.34");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 89.c (of order \(4\), degree \(2\), 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: \(5.25116999051\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 8 q^{2} - 8 q^{3} + 148 q^{4} + 10 q^{6} - 2 q^{7} - 84 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 8 q^{2} - 8 q^{3} + 148 q^{4} + 10 q^{6} - 2 q^{7} - 84 q^{8} + 28 q^{11} - 26 q^{12} + 52 q^{13} - 92 q^{14} + 128 q^{15} + 332 q^{16} - 236 q^{22} + 42 q^{23} + 580 q^{24} - 806 q^{25} - 648 q^{26} + 520 q^{27} + 250 q^{28} - 260 q^{29} + 40 q^{30} + 102 q^{31} - 740 q^{32} - 760 q^{33} + 348 q^{35} + 488 q^{37} + 724 q^{38} - 908 q^{39} + 986 q^{41} + 340 q^{43} - 84 q^{44} - 556 q^{45} + 782 q^{46} - 2048 q^{48} - 824 q^{50} + 580 q^{51} + 2424 q^{52} - 3248 q^{54} - 3336 q^{56} + 536 q^{57} - 856 q^{58} + 1560 q^{59} - 992 q^{60} - 356 q^{61} + 1338 q^{62} - 958 q^{63} + 1580 q^{64} + 2212 q^{65} + 3090 q^{66} - 788 q^{67} - 646 q^{70} + 896 q^{73} - 2782 q^{74} - 296 q^{75} + 458 q^{76} - 1488 q^{77} + 1888 q^{78} + 3718 q^{81} - 1374 q^{82} + 3548 q^{83} - 2472 q^{85} - 974 q^{86} + 6028 q^{87} - 6132 q^{88} + 3054 q^{89} + 7768 q^{90} - 2108 q^{91} - 760 q^{92} + 3108 q^{93} - 552 q^{95} + 3690 q^{96} - 480 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −5.34106 −6.52695 − 6.52695i 20.5269 0.0127964i 34.8608 + 34.8608i 11.4409 + 11.4409i −66.9067 58.2021i − 0.0683461i
34.2 −5.15582 3.64079 + 3.64079i 18.5824 − 11.8771i −18.7713 − 18.7713i 3.99104 + 3.99104i −54.5611 − 0.489242i 61.2364i
34.3 −4.68242 0.483732 + 0.483732i 13.9251 7.11281i −2.26504 − 2.26504i −5.50344 − 5.50344i −27.7438 − 26.5320i − 33.3052i
34.4 −3.87010 5.31439 + 5.31439i 6.97770 19.5304i −20.5672 − 20.5672i 2.34350 + 2.34350i 3.95640 29.4855i − 75.5849i
34.5 −3.65169 −2.40290 − 2.40290i 5.33486 9.53882i 8.77464 + 8.77464i 9.22597 + 9.22597i 9.73226 − 15.4522i − 34.8329i
34.6 −3.36411 −3.51363 − 3.51363i 3.31722 − 16.4034i 11.8202 + 11.8202i −7.82574 − 7.82574i 15.7534 − 2.30885i 55.1829i
34.7 −2.42405 4.40251 + 4.40251i −2.12396 − 7.60693i −10.6719 − 10.6719i −22.9201 − 22.9201i 24.5410 11.7642i 18.4396i
34.8 −1.90703 −6.11575 − 6.11575i −4.36322 17.4138i 11.6629 + 11.6629i −24.9333 − 24.9333i 23.5771 47.8048i − 33.2087i
34.9 −1.89558 5.67628 + 5.67628i −4.40676 − 7.51023i −10.7599 − 10.7599i 20.8961 + 20.8961i 23.5181 37.4404i 14.2363i
34.10 −1.23820 −0.766381 − 0.766381i −6.46686 2.02030i 0.948932 + 0.948932i 11.3636 + 11.3636i 17.9129 − 25.8253i − 2.50154i
34.11 0.0150798 −6.25933 − 6.25933i −7.99977 − 8.12166i −0.0943892 − 0.0943892i 12.9083 + 12.9083i −0.241273 51.3585i − 0.122473i
34.12 0.235435 0.886920 + 0.886920i −7.94457 7.39689i 0.208812 + 0.208812i −10.3872 − 10.3872i −3.75392 − 25.4267i 1.74149i
34.13 1.39632 1.34272 + 1.34272i −6.05028 − 21.5200i 1.87487 + 1.87487i 3.07550 + 3.07550i −19.6187 − 23.3942i − 30.0489i
34.14 1.57296 5.62611 + 5.62611i −5.52581 10.6240i 8.84963 + 8.84963i −1.20222 − 1.20222i −21.2755 36.3063i 16.7111i
34.15 2.37961 −3.66894 − 3.66894i −2.33748 21.6553i −8.73063 − 8.73063i 17.0536 + 17.0536i −24.5991 − 0.0777534i 51.5310i
34.16 2.54799 −2.31052 − 2.31052i −1.50772 − 3.47281i −5.88720 − 5.88720i −19.3866 − 19.3866i −24.2256 − 16.3230i − 8.84870i
34.17 3.21090 −5.29416 − 5.29416i 2.30988 − 1.72804i −16.9990 − 16.9990i −5.60858 − 5.60858i −18.2704 29.0562i − 5.54855i
34.18 3.68260 2.90359 + 2.90359i 5.56157 1.03054i 10.6928 + 10.6928i 18.6327 + 18.6327i −8.97976 − 10.1383i 3.79508i
34.19 4.45420 5.26816 + 5.26816i 11.8399 − 9.25982i 23.4654 + 23.4654i −11.2927 − 11.2927i 17.1035 28.5069i − 41.2450i
34.20 5.00045 −3.33233 − 3.33233i 17.0045 − 8.90483i −16.6632 − 16.6632i 7.91556 + 7.91556i 45.0264 − 4.79113i − 44.5282i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.21
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 89.4.c.a ✓ 42
89.c even 4 1 inner 89.4.c.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.4.c.a ✓ 42 1.a even 1 1 trivial
89.4.c.a ✓ 42 89.c even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(89, [\chi])\).