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Properties

Label 89.10.a.b
Level $89$
Weight $10$
Character orbit 89.a
Self dual yes
Analytic conductor $45.838$
Analytic rank $0$
Dimension $36$
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] = [89,10,Mod(1,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 89.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: \(45.8381894186\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 15 q^{2} + 162 q^{3} + 10495 q^{4} + 966 q^{5} + 9759 q^{6} + 26410 q^{7} + 11775 q^{8} + 276410 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 15 q^{2} + 162 q^{3} + 10495 q^{4} + 966 q^{5} + 9759 q^{6} + 26410 q^{7} + 11775 q^{8} + 276410 q^{9} + 65475 q^{10} + 141482 q^{11} + 133527 q^{12} + 217542 q^{13} + 305724 q^{14} + 736836 q^{15} + 3281783 q^{16} - 13450 q^{17} - 74552 q^{18} + 2374934 q^{19} + 489007 q^{20} + 3324392 q^{21} + 2527874 q^{22} + 3869816 q^{23} + 16281709 q^{24} + 18771998 q^{25} - 6195022 q^{26} - 5612058 q^{27} + 347358 q^{28} + 1783858 q^{29} - 34044257 q^{30} + 14456562 q^{31} - 44350765 q^{32} - 9595156 q^{33} - 4602557 q^{34} + 9034040 q^{35} + 75534524 q^{36} + 2074154 q^{37} - 3036099 q^{38} + 72045942 q^{39} + 72740235 q^{40} + 47862096 q^{41} + 105709036 q^{42} + 108738656 q^{43} + 231376478 q^{44} + 54251216 q^{45} + 237627227 q^{46} + 177226686 q^{47} + 300563237 q^{48} + 368390644 q^{49} + 267620962 q^{50} + 297125548 q^{51} + 613926178 q^{52} + 256750550 q^{53} + 945754123 q^{54} + 331243430 q^{55} + 981553052 q^{56} + 377074690 q^{57} + 377892536 q^{58} + 401113052 q^{59} + 1907546283 q^{60} + 289650002 q^{61} + 803459347 q^{62} + 758492436 q^{63} + 1894814207 q^{64} + 380449484 q^{65} + 1557871000 q^{66} + 683028714 q^{67} + 961349631 q^{68} + 491762246 q^{69} + 1504491654 q^{70} + 1002832888 q^{71} + 705971600 q^{72} + 937565554 q^{73} + 2007690592 q^{74} + 580613272 q^{75} + 1773333875 q^{76} + 1164977392 q^{77} + 1690128446 q^{78} + 1941197712 q^{79} + 894979383 q^{80} + 3212128808 q^{81} + 1833150658 q^{82} + 365334864 q^{83} + 3824121972 q^{84} + 1224101222 q^{85} + 135577933 q^{86} + 1785094850 q^{87} + 540764658 q^{88} + 2258720676 q^{89} + 6397770370 q^{90} + 2370736400 q^{91} + 1385750829 q^{92} + 6163875486 q^{93} + 4589849528 q^{94} + 5709294860 q^{95} + 8950995191 q^{96} + 5586363166 q^{97} + 2529336247 q^{98} + 2579880124 q^{99}+O(q^{100}) \) 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 −44.9403 −255.538 1507.63 −2240.65 11484.0 −3038.74 −44743.7 45616.8 100696.
1.2 −43.6815 81.9163 1396.08 2171.91 −3578.23 −27.9067 −38617.9 −12972.7 −94872.2
1.3 −41.8744 83.7852 1241.46 −1001.56 −3508.45 4855.70 −30545.7 −12663.0 41939.9
1.4 −40.5346 −1.36800 1131.05 −53.7432 55.4512 −11281.1 −25092.9 −19681.1 2178.46
1.5 −37.2034 206.336 872.095 1909.44 −7676.40 −5387.73 −13396.8 22891.4 −71037.5
1.6 −31.4352 −161.139 476.174 121.930 5065.46 −537.744 1126.19 6282.90 −3832.89
1.7 −28.7925 146.766 317.006 108.338 −4225.77 11749.0 5614.37 1857.38 −3119.32
1.8 −26.8851 −179.847 210.810 −1975.70 4835.20 10262.7 8097.52 12661.8 53117.1
1.9 −24.8072 7.02111 103.399 765.917 −174.175 −5472.66 10136.3 −19633.7 −19000.3
1.10 −24.5390 −279.426 90.1614 15.4439 6856.82 −120.250 10351.5 58395.8 −378.979
1.11 −22.9184 33.4017 13.2517 −1623.25 −765.513 2512.95 11430.5 −18567.3 37202.3
1.12 −16.3781 256.044 −243.758 −1660.99 −4193.51 10448.2 12377.9 45875.5 27203.8
1.13 −15.7155 148.100 −265.024 −1653.45 −2327.46 −4856.65 12211.3 2250.63 25984.7
1.14 −14.3664 −208.036 −305.607 1549.06 2988.72 9399.58 11746.1 23595.9 −22254.4
1.15 −12.6127 249.655 −352.920 2155.42 −3148.82 1948.72 10909.0 42644.8 −27185.7
1.16 −10.6818 −192.492 −397.898 971.286 2056.17 −1690.89 9719.38 17370.3 −10375.1
1.17 −6.41261 −19.2196 −470.878 278.593 123.248 10880.7 6302.82 −19313.6 −1786.51
1.18 −6.06008 82.4587 −475.275 1256.11 −499.706 −12648.0 5982.96 −12883.6 −7612.15
1.19 5.15842 47.8494 −485.391 2406.85 246.827 3648.49 −5144.96 −17393.4 12415.5
1.20 7.15234 −78.7061 −460.844 −412.208 −562.933 −4083.55 −6958.11 −13488.3 −2948.25
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(89\) \(-1\)

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 89.10.a.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.10.a.b ✓ 36 1.a even 1 1 trivial