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Properties

Label 9.13.d.a
Level $9$
Weight $13$
Character orbit 9.d
Analytic conductor $8.226$
Analytic rank $0$
Dimension $22$
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] = [9,13,Mod(2,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.2");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 9.d (of order \(6\), 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: \(8.22594435549\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 3 q^{2} + 618 q^{3} + 20479 q^{4} - 15987 q^{5} + 67725 q^{6} + 34319 q^{7} - 315126 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 3 q^{2} + 618 q^{3} + 20479 q^{4} - 15987 q^{5} + 67725 q^{6} + 34319 q^{7} - 315126 q^{9} - 8196 q^{10} - 5335311 q^{11} + 2423418 q^{12} - 425041 q^{13} + 12103638 q^{14} + 5132349 q^{15} - 33550337 q^{16} + 177665292 q^{18} + 11877236 q^{19} - 392809506 q^{20} - 143745225 q^{21} + 67326207 q^{22} + 642241437 q^{23} - 539029017 q^{24} + 204473002 q^{25} + 1933854318 q^{27} + 595845116 q^{28} - 1981338411 q^{29} - 3111355998 q^{30} - 994841521 q^{31} + 2176403175 q^{32} - 1798826823 q^{33} + 1745794161 q^{34} - 458986401 q^{36} + 4368665276 q^{37} + 2450754219 q^{38} + 7623183351 q^{39} + 4400823966 q^{40} - 21698696775 q^{41} + 3812969052 q^{42} - 1150355221 q^{43} - 17747793387 q^{45} - 20932186848 q^{46} + 43727781957 q^{47} + 53764264221 q^{48} - 543274020 q^{49} - 78988772733 q^{50} - 33353944236 q^{51} - 14782367488 q^{52} + 31687307163 q^{54} + 63530134986 q^{55} + 166781913810 q^{56} - 20109925464 q^{57} - 37929930258 q^{58} - 68597540607 q^{59} - 322880030370 q^{60} + 39920189231 q^{61} + 322848734319 q^{63} - 152030563718 q^{64} + 213680229117 q^{65} + 81324054090 q^{66} + 83105880059 q^{67} - 315577119015 q^{68} - 817813372995 q^{69} - 89563601604 q^{70} + 1479468022515 q^{72} - 61644834604 q^{73} + 1106310843030 q^{74} + 245546974626 q^{75} - 51092760367 q^{76} - 1702913094723 q^{77} - 2355365683260 q^{78} + 13028834423 q^{79} + 1035862303338 q^{81} + 206299499034 q^{82} + 2772503183157 q^{83} + 2547743632554 q^{84} + 210025544976 q^{85} - 6139247974419 q^{86} - 3758996209563 q^{87} + 10284870963 q^{88} + 2889388695510 q^{90} + 695109549694 q^{91} + 10758288981684 q^{92} + 3442586846799 q^{93} - 659068348404 q^{94} - 7824500713662 q^{95} - 12416772136824 q^{96} + 949029910139 q^{97} + 5474142326565 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −105.404 − 60.8548i −470.679 − 556.688i 5358.61 + 9281.38i −16839.0 + 9722.01i 15734.1 + 87320.0i 88.8404 − 153.876i − 805865.i −88363.1 + 524043.i 2.36652e6
2.2 −82.4480 − 47.6014i −152.574 + 712.855i 2483.79 + 4302.04i 15062.9 − 8696.59i 46512.3 − 51510.7i −42566.0 + 73726.5i − 82976.0i −484883. − 217527.i −1.65588e6
2.3 −64.6760 − 37.3407i 721.758 + 102.503i 740.658 + 1282.86i −13863.9 + 8004.33i −42852.9 − 33580.4i 4312.89 − 7470.15i 195268.i 510427. + 147964.i 1.19555e6
2.4 −50.9664 − 29.4255i 248.904 − 685.192i −316.286 − 547.823i 19006.0 − 10973.1i −32847.8 + 27597.6i 52001.1 − 90068.5i 278281.i −407535. − 341094.i −1.29156e6
2.5 −24.0407 − 13.8799i −728.625 − 23.3851i −1662.70 − 2879.87i −1184.96 + 684.138i 17192.1 + 10675.4i −13455.6 + 23305.8i 206017.i 530347. + 34078.0i 37983.1
2.6 11.8801 + 6.85898i 24.4708 + 728.589i −1953.91 − 3384.27i −12231.6 + 7061.92i −4706.66 + 8823.56i 78889.2 − 136640.i − 109796.i −530243. + 35658.4i −193750.
2.7 24.8632 + 14.3548i 233.332 − 690.650i −1635.88 − 2833.43i −14488.2 + 8364.74i 15715.5 − 13822.4i −84341.3 + 146083.i − 211525.i −422554. − 322301.i −480297.
2.8 36.0381 + 20.8066i 674.632 + 276.248i −1182.17 − 2047.58i 21279.0 − 12285.4i 18564.7 + 23992.3i −34746.6 + 60182.9i − 268836.i 378815. + 372731.i 1.02247e6
2.9 69.7303 + 40.2588i −482.628 − 546.361i 1193.54 + 2067.28i 9375.43 − 5412.91i −11658.0 − 57528.0i 84970.7 − 147174.i − 137598.i −65580.6 + 527379.i 871669.
2.10 83.8736 + 48.4245i −477.099 + 551.196i 2641.86 + 4575.83i 103.730 − 59.8884i −66707.4 + 23127.6i −89255.0 + 154594.i 115029.i −76193.9 − 525951.i 11600.3
2.11 99.6493 + 57.5325i 717.510 − 128.922i 4571.99 + 7918.91i −14212.9 + 8205.84i 78916.5 + 28433.2i 61261.4 − 106108.i 580845.i 498199. − 185005.i −1.88841e6
5.1 −105.404 + 60.8548i −470.679 + 556.688i 5358.61 − 9281.38i −16839.0 − 9722.01i 15734.1 − 87320.0i 88.8404 + 153.876i 805865.i −88363.1 − 524043.i 2.36652e6
5.2 −82.4480 + 47.6014i −152.574 − 712.855i 2483.79 − 4302.04i 15062.9 + 8696.59i 46512.3 + 51510.7i −42566.0 − 73726.5i 82976.0i −484883. + 217527.i −1.65588e6
5.3 −64.6760 + 37.3407i 721.758 − 102.503i 740.658 − 1282.86i −13863.9 − 8004.33i −42852.9 + 33580.4i 4312.89 + 7470.15i − 195268.i 510427. − 147964.i 1.19555e6
5.4 −50.9664 + 29.4255i 248.904 + 685.192i −316.286 + 547.823i 19006.0 + 10973.1i −32847.8 − 27597.6i 52001.1 + 90068.5i − 278281.i −407535. + 341094.i −1.29156e6
5.5 −24.0407 + 13.8799i −728.625 + 23.3851i −1662.70 + 2879.87i −1184.96 − 684.138i 17192.1 − 10675.4i −13455.6 − 23305.8i − 206017.i 530347. − 34078.0i 37983.1
5.6 11.8801 − 6.85898i 24.4708 − 728.589i −1953.91 + 3384.27i −12231.6 − 7061.92i −4706.66 − 8823.56i 78889.2 + 136640.i 109796.i −530243. − 35658.4i −193750.
5.7 24.8632 − 14.3548i 233.332 + 690.650i −1635.88 + 2833.43i −14488.2 − 8364.74i 15715.5 + 13822.4i −84341.3 − 146083.i 211525.i −422554. + 322301.i −480297.
5.8 36.0381 − 20.8066i 674.632 − 276.248i −1182.17 + 2047.58i 21279.0 + 12285.4i 18564.7 − 23992.3i −34746.6 − 60182.9i 268836.i 378815. − 372731.i 1.02247e6
5.9 69.7303 − 40.2588i −482.628 + 546.361i 1193.54 − 2067.28i 9375.43 + 5412.91i −11658.0 + 57528.0i 84970.7 + 147174.i 137598.i −65580.6 − 527379.i 871669.
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.13.d.a ✓ 22
3.b odd 2 1 27.13.d.a 22
9.c even 3 1 27.13.d.a 22
9.c even 3 1 81.13.b.a 22
9.d odd 6 1 inner 9.13.d.a ✓ 22
9.d odd 6 1 81.13.b.a 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.13.d.a ✓ 22 1.a even 1 1 trivial
9.13.d.a ✓ 22 9.d odd 6 1 inner
27.13.d.a 22 3.b odd 2 1
27.13.d.a 22 9.c even 3 1
81.13.b.a 22 9.c even 3 1
81.13.b.a 22 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(9, [\chi])\).