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Properties

Label 89.12.a.b
Level $89$
Weight $12$
Character orbit 89.a
Self dual yes
Analytic conductor $68.383$
Analytic rank $0$
Dimension $43$
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] = [89,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(68.3825430698\)
Analytic rank: \(0\)
Dimension: \(43\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 43 q + 87 q^{2} + 243 q^{3} + 52671 q^{4} + 8781 q^{5} + 35103 q^{6} + 201620 q^{7} + 110079 q^{8} + 2866430 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 43 q + 87 q^{2} + 243 q^{3} + 52671 q^{4} + 8781 q^{5} + 35103 q^{6} + 201620 q^{7} + 110079 q^{8} + 2866430 q^{9} + 561299 q^{10} + 1121126 q^{11} + 1413543 q^{12} + 4088846 q^{13} + 1470724 q^{14} + 14362689 q^{15} + 64144887 q^{16} + 4312525 q^{17} + 23443408 q^{18} + 22364319 q^{19} + 43366583 q^{20} + 49267028 q^{21} + 3259354 q^{22} + 23147687 q^{23} - 6310001 q^{24} + 613136872 q^{25} + 457201950 q^{26} + 425201265 q^{27} + 801317316 q^{28} + 324076592 q^{29} + 765939253 q^{30} + 1003007335 q^{31} + 283946439 q^{32} - 201145642 q^{33} - 24457441 q^{34} - 96857428 q^{35} + 767628836 q^{36} - 205471934 q^{37} - 4568246209 q^{38} + 1787725734 q^{39} - 2065186801 q^{40} + 80676768 q^{41} - 8908050584 q^{42} + 929052253 q^{43} - 1313677574 q^{44} + 1475328374 q^{45} + 587575763 q^{46} - 776244932 q^{47} - 7098440065 q^{48} + 17209789683 q^{49} + 1226099158 q^{50} + 2288048881 q^{51} + 5458547710 q^{52} + 1093115807 q^{53} + 17136494485 q^{54} + 19308156506 q^{55} + 20182549412 q^{56} + 8674427563 q^{57} + 26080299900 q^{58} + 13695889932 q^{59} + 85111910901 q^{60} + 14311084002 q^{61} + 39415263015 q^{62} + 56791243680 q^{63} + 93438775375 q^{64} + 54873718786 q^{65} + 164438289730 q^{66} + 71075520052 q^{67} + 72379716995 q^{68} + 79182102923 q^{69} + 129409107808 q^{70} + 88356964806 q^{71} + 198981401108 q^{72} + 48074684977 q^{73} + 142967135330 q^{74} + 107398365700 q^{75} + 253487363775 q^{76} + 77474089672 q^{77} + 221627964182 q^{78} + 194629252826 q^{79} + 270205139703 q^{80} + 278343939275 q^{81} + 351071507348 q^{82} + 59914185996 q^{83} + 360067715496 q^{84} + 225391259295 q^{85} + 435693342077 q^{86} + 289461792296 q^{87} + 277857359754 q^{88} - 240114556307 q^{89} + 265245384034 q^{90} + 53768886520 q^{91} + 775578899139 q^{92} - 317570061261 q^{93} + 277948948404 q^{94} + 245798141501 q^{95} + 641947432223 q^{96} + 281030701719 q^{97} + 483600997507 q^{98} + 844748131084 q^{99}+O(q^{100}) \) 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 −89.5080 −111.027 5963.68 −7809.58 9937.78 60241.9 −350485. −164820. 699020.
1.2 −87.8496 −165.069 5669.56 8096.01 14501.3 −18700.6 −318153. −149899. −711232.
1.3 −80.4431 −590.978 4423.09 5597.75 47540.1 −79097.3 −191059. 172108. −450300.
1.4 −77.2183 235.796 3914.66 10633.2 −18207.8 18876.3 −144140. −121547. −821080.
1.5 −77.1160 740.661 3898.88 9053.61 −57116.8 78944.8 −142733. 371431. −698179.
1.6 −75.4512 −400.333 3644.89 −6899.02 30205.6 19192.2 −120487. −16880.5 520540.
1.7 −69.9404 552.807 2843.67 −13842.1 −38663.6 18650.5 −55649.2 128449. 968125.
1.8 −64.8149 −71.1676 2152.97 7513.35 4612.71 −21160.1 −6803.36 −172082. −486977.
1.9 −62.6251 230.632 1873.90 264.992 −14443.3 −46645.7 10902.8 −123956. −16595.1
1.10 −51.2393 795.808 577.468 −2733.91 −40776.7 3513.43 75349.1 456164. 140084.
1.11 −50.2071 −698.187 472.754 −12347.2 35053.9 43407.9 79088.5 310318. 619916.
1.12 −47.8985 416.692 246.269 −7700.20 −19958.9 41884.6 86300.3 −3514.88 368828.
1.13 −47.2746 −232.813 186.889 −6433.52 11006.2 −9220.91 87983.3 −122945. 304142.
1.14 −43.6393 −690.968 −143.614 −615.772 30153.3 −54138.5 95640.4 300290. 26871.9
1.15 −35.6689 312.136 −775.732 11795.0 −11133.5 34982.4 100719. −79718.3 −420715.
1.16 −28.4542 −724.154 −1238.36 −463.864 20605.2 25255.6 93510.8 347252. 13198.9
1.17 −20.3056 −34.8204 −1635.68 −2981.09 707.048 15868.5 74799.3 −175935. 60532.7
1.18 −15.9417 −375.538 −1793.86 3912.50 5986.70 59100.1 61245.7 −36118.2 −62371.7
1.19 −14.6220 620.956 −1834.20 −9072.92 −9079.59 −81147.1 56765.3 208439. 132664.
1.20 −5.26703 −81.4560 −2020.26 8602.03 429.032 −67087.9 21427.6 −170512. −45307.1
See all 43 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.43
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 89.12.a.b ✓ 43
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.12.a.b ✓ 43 1.a even 1 1 trivial