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Properties

Label 29.9.c.a
Level $29$
Weight $9$
Character orbit 29.c
Analytic conductor $11.814$
Analytic rank $0$
Dimension $38$
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] = [29,9,Mod(12,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.12");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 29.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: \(11.8139796918\)
Analytic rank: \(0\)
Dimension: \(38\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 8 q^{2} - 2 q^{3} - 4 q^{7} - 3678 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 8 q^{2} - 2 q^{3} - 4 q^{7} - 3678 q^{8} + 15870 q^{10} - 8390 q^{11} + 5022 q^{12} + 12728 q^{14} - 35530 q^{15} - 245912 q^{16} + 76234 q^{17} - 465022 q^{18} - 164488 q^{19} + 4092 q^{20} - 514620 q^{21} + 733568 q^{23} - 482376 q^{24} - 1528354 q^{25} + 1405470 q^{26} - 1279754 q^{27} + 1420996 q^{29} + 3659852 q^{30} + 291062 q^{31} + 1592134 q^{32} - 10223688 q^{36} + 7994574 q^{37} - 309930 q^{39} + 9093994 q^{40} + 9561658 q^{41} - 5181022 q^{43} - 11400526 q^{44} + 9526060 q^{45} - 32561820 q^{46} + 5445034 q^{47} - 17683542 q^{48} + 45106458 q^{49} + 3122002 q^{50} + 56198616 q^{52} - 25331452 q^{53} - 42307016 q^{54} + 3037606 q^{55} - 25661184 q^{56} - 33846328 q^{58} + 27333008 q^{59} + 30592014 q^{60} - 49572406 q^{61} - 121974972 q^{65} + 133443354 q^{66} + 21192980 q^{68} - 28739184 q^{69} - 17976124 q^{70} - 186114192 q^{72} + 45787870 q^{73} + 91919252 q^{74} - 159074460 q^{75} + 243795016 q^{76} + 59851916 q^{77} + 163580208 q^{78} - 99827470 q^{79} - 75138586 q^{81} - 18178416 q^{82} - 177270160 q^{83} - 9436684 q^{84} + 156307616 q^{85} - 20584526 q^{87} + 563433236 q^{88} - 388451858 q^{89} + 306402752 q^{90} + 81569204 q^{94} - 44386260 q^{95} - 121157834 q^{97} - 129908712 q^{98} - 216447384 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −21.4922 − 21.4922i −26.9761 − 26.9761i 667.831i − 209.913i 1159.55i 2370.17 8851.17 − 8851.17i − 5105.58i −4511.49 + 4511.49i
12.2 −18.8438 − 18.8438i 49.8418 + 49.8418i 454.181i 1116.67i − 1878.42i −1871.68 3734.50 − 3734.50i − 1592.59i 21042.4 − 21042.4i
12.3 −16.0506 − 16.0506i −92.3210 − 92.3210i 259.242i − 266.052i 2963.61i −4134.50 52.0309 − 52.0309i 10485.3i −4270.28 + 4270.28i
12.4 −15.3962 − 15.3962i 5.11102 + 5.11102i 218.083i − 625.365i − 157.380i −974.923 −583.770 + 583.770i − 6508.75i −9628.22 + 9628.22i
12.5 −15.2859 − 15.2859i 93.1254 + 93.1254i 211.315i − 424.754i − 2847.00i 996.548 −683.041 + 683.041i 10783.7i −6492.74 + 6492.74i
12.6 −12.8569 − 12.8569i −84.9422 − 84.9422i 74.6002i 733.457i 2184.19i 4371.72 −2332.24 + 2332.24i 7869.35i 9429.98 − 9429.98i
12.7 −6.63336 − 6.63336i −21.3611 − 21.3611i − 167.997i 656.237i 283.392i −2102.40 −2812.53 + 2812.53i − 5648.41i 4353.06 − 4353.06i
12.8 −6.50125 − 6.50125i 25.7599 + 25.7599i − 171.468i 37.7478i − 334.943i 1420.11 −2779.07 + 2779.07i − 5233.85i 245.408 − 245.408i
12.9 −3.03267 − 3.03267i −59.6889 − 59.6889i − 237.606i − 1116.88i 362.033i 2170.16 −1496.94 + 1496.94i 564.525i −3387.14 + 3387.14i
12.10 0.384989 + 0.384989i 93.7530 + 93.7530i − 255.704i 802.752i 72.1877i 2726.98 197.000 − 197.000i 11018.2i −309.050 + 309.050i
12.11 0.492724 + 0.492724i 72.5537 + 72.5537i − 255.514i − 679.404i 71.4979i −4572.34 252.036 − 252.036i 3967.08i 334.759 − 334.759i
12.12 4.53287 + 4.53287i −83.7812 − 83.7812i − 214.906i − 3.13649i − 759.538i −1568.18 2134.56 − 2134.56i 7477.58i 14.2173 − 14.2173i
12.13 8.30345 + 8.30345i −37.9600 − 37.9600i − 118.106i 774.671i − 630.398i −18.6293 3106.37 − 3106.37i − 3679.07i −6432.44 + 6432.44i
12.14 9.93858 + 9.93858i 27.3327 + 27.3327i − 58.4494i − 447.536i 543.297i 2077.42 3125.18 − 3125.18i − 5066.84i 4447.87 − 4447.87i
12.15 13.6258 + 13.6258i 70.4012 + 70.4012i 115.327i 729.288i 1918.55i −3525.11 1916.79 − 1916.79i 3351.66i −9937.16 + 9937.16i
12.16 17.5030 + 17.5030i −100.063 − 100.063i 356.708i 183.024i − 3502.79i 3415.09 −1762.69 + 1762.69i 13464.1i −3203.47 + 3203.47i
12.17 17.9289 + 17.9289i −42.9110 − 42.9110i 386.891i − 832.826i − 1538.69i −3297.00 −2346.74 + 2346.74i − 2878.30i 14931.7 − 14931.7i
12.18 19.3086 + 19.3086i 104.909 + 104.909i 489.642i − 703.899i 4051.27i 1720.21 −4511.30 + 4511.30i 15450.6i 13591.3 − 13591.3i
12.19 20.0740 + 20.0740i 6.21683 + 6.21683i 549.928i 611.920i 249.593i 794.354 −5900.31 + 5900.31i − 6483.70i −12283.7 + 12283.7i
17.1 −21.4922 + 21.4922i −26.9761 + 26.9761i − 667.831i 209.913i − 1159.55i 2370.17 8851.17 + 8851.17i 5105.58i −4511.49 − 4511.49i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.19
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.9.c.a ✓ 38
29.c odd 4 1 inner 29.9.c.a ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.9.c.a ✓ 38 1.a even 1 1 trivial
29.9.c.a ✓ 38 29.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(29, [\chi])\).