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Properties

Label 13.12.e.a
Level $13$
Weight $12$
Character orbit 13.e
Analytic conductor $9.988$
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] = [13,12,Mod(4,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.4");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 13.e (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: \(9.98846134727\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9} - 572989 q^{10} + 941556 q^{11} - 954740 q^{12} - 1999101 q^{13} - 4389420 q^{14} - 1912146 q^{15} - 2217465 q^{16} + 1266933 q^{17} - 26218578 q^{19} + 73151169 q^{20} + 1338614 q^{22} - 47071890 q^{23} + 176666646 q^{24} - 102279200 q^{25} + 86807799 q^{26} - 316653808 q^{27} - 127105884 q^{28} + 113029131 q^{29} - 161927730 q^{30} + 750398445 q^{32} + 588949482 q^{33} - 300427704 q^{35} - 173622325 q^{36} - 570835935 q^{37} + 925613832 q^{38} + 2656614142 q^{39} - 4665525182 q^{40} - 3125495451 q^{41} + 312136374 q^{42} - 687379108 q^{43} + 291815697 q^{45} + 1973786088 q^{46} - 4495108546 q^{48} - 2113923323 q^{49} + 10799589288 q^{50} + 18922873548 q^{51} + 5816536466 q^{52} - 11409380982 q^{53} - 7220008734 q^{54} + 2659826936 q^{55} - 5191105920 q^{56} - 17750605839 q^{58} + 12280821000 q^{59} - 388343883 q^{61} - 4489966530 q^{62} + 12000810648 q^{63} - 27417445590 q^{64} - 7047855789 q^{65} + 29852276436 q^{66} - 24954230718 q^{67} + 18618567747 q^{68} - 17332878054 q^{69} + 7855228734 q^{71} - 2312207943 q^{72} + 14656656585 q^{74} + 34533247192 q^{75} - 91364144550 q^{76} + 1417426884 q^{77} + 51502665942 q^{78} - 15237937464 q^{79} + 187084997403 q^{80} - 66245225351 q^{81} + 38170027049 q^{82} - 221463259788 q^{84} - 144387435243 q^{85} + 133939610130 q^{87} + 191677651072 q^{88} - 22557105234 q^{89} - 28770934134 q^{90} - 26918267220 q^{91} - 31256021460 q^{92} + 169636911984 q^{93} + 114241933898 q^{94} - 72874928466 q^{95} - 318460072098 q^{97} - 223225841301 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −74.2981 − 42.8960i −148.593 + 257.371i 2656.14 + 4600.57i − 10410.3i 22080.4 − 12748.1i 18278.8 − 10553.3i − 280049.i 44413.7 + 76926.7i −446559. + 773464.i
4.2 −57.1912 − 33.0193i 212.159 − 367.469i 1156.55 + 2003.21i 3712.50i −24267.2 + 14010.7i −35175.6 + 20308.7i − 17507.1i −1448.99 − 2509.72i 122584. − 212322.i
4.3 −47.1587 − 27.2271i −219.942 + 380.951i 458.627 + 794.365i 8334.97i 20744.3 − 11976.8i 19275.1 − 11128.5i 61573.8i −8175.47 − 14160.3i 226937. − 393066.i
4.4 −28.4687 − 16.4364i 286.164 − 495.650i −483.689 − 837.774i − 6962.98i −16293.4 + 9407.01i 68090.0 − 39311.8i 99124.0i −75206.1 − 130261.i −114446. + 198227.i
4.5 −18.1006 − 10.4504i −200.570 + 347.397i −805.579 − 1395.30i − 5339.78i 7260.86 − 4192.06i −15062.6 + 8696.38i 76479.2i 8117.03 + 14059.1i −55802.7 + 96653.1i
4.6 8.35978 + 4.82652i 155.366 − 269.102i −977.409 − 1692.92i − 71.9990i 2597.66 − 1499.76i −53916.5 + 31128.7i − 38639.4i 40296.2 + 69795.0i 347.505 − 601.896i
4.7 15.7596 + 9.09880i 80.7581 − 139.877i −858.424 − 1486.83i 13894.4i 2545.43 − 1469.60i 55086.8 − 31804.4i − 68511.2i 75529.7 + 130821.i −126422. + 218969.i
4.8 35.3488 + 20.4086i −366.998 + 635.659i −190.977 − 330.782i 3052.06i −25945.9 + 14979.8i −6894.91 + 3980.78i − 99184.0i −180802. − 313158.i −62288.3 + 107887.i
4.9 42.3913 + 24.4746i −27.6563 + 47.9021i 174.016 + 301.405i − 10452.4i −2344.78 + 1353.76i 39866.5 − 23016.9i − 83212.2i 87043.8 + 150764.i 255820. − 443093.i
4.10 53.9750 + 31.1625i 401.709 − 695.781i 918.197 + 1590.36i − 673.285i 43364.5 − 25036.5i −9262.47 + 5347.69i − 13188.3i −234167. − 405589.i 20981.2 − 36340.5i
4.11 67.8828 + 39.1921i −51.3966 + 89.0216i 2048.05 + 3547.32i 2746.61i −6977.89 + 4028.69i −16037.2 + 9259.07i 160539.i 83290.3 + 144263.i −107645. + 186447.i
10.1 −74.2981 + 42.8960i −148.593 − 257.371i 2656.14 − 4600.57i 10410.3i 22080.4 + 12748.1i 18278.8 + 10553.3i 280049.i 44413.7 − 76926.7i −446559. − 773464.i
10.2 −57.1912 + 33.0193i 212.159 + 367.469i 1156.55 − 2003.21i − 3712.50i −24267.2 − 14010.7i −35175.6 − 20308.7i 17507.1i −1448.99 + 2509.72i 122584. + 212322.i
10.3 −47.1587 + 27.2271i −219.942 − 380.951i 458.627 − 794.365i − 8334.97i 20744.3 + 11976.8i 19275.1 + 11128.5i − 61573.8i −8175.47 + 14160.3i 226937. + 393066.i
10.4 −28.4687 + 16.4364i 286.164 + 495.650i −483.689 + 837.774i 6962.98i −16293.4 − 9407.01i 68090.0 + 39311.8i − 99124.0i −75206.1 + 130261.i −114446. − 198227.i
10.5 −18.1006 + 10.4504i −200.570 − 347.397i −805.579 + 1395.30i 5339.78i 7260.86 + 4192.06i −15062.6 − 8696.38i − 76479.2i 8117.03 − 14059.1i −55802.7 − 96653.1i
10.6 8.35978 − 4.82652i 155.366 + 269.102i −977.409 + 1692.92i 71.9990i 2597.66 + 1499.76i −53916.5 − 31128.7i 38639.4i 40296.2 − 69795.0i 347.505 + 601.896i
10.7 15.7596 − 9.09880i 80.7581 + 139.877i −858.424 + 1486.83i − 13894.4i 2545.43 + 1469.60i 55086.8 + 31804.4i 68511.2i 75529.7 − 130821.i −126422. − 218969.i
10.8 35.3488 − 20.4086i −366.998 − 635.659i −190.977 + 330.782i − 3052.06i −25945.9 − 14979.8i −6894.91 − 3980.78i 99184.0i −180802. + 313158.i −62288.3 − 107887.i
10.9 42.3913 − 24.4746i −27.6563 − 47.9021i 174.016 − 301.405i 10452.4i −2344.78 − 1353.76i 39866.5 + 23016.9i 83212.2i 87043.8 − 150764.i 255820. + 443093.i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.12.e.a ✓ 22
13.e even 6 1 inner 13.12.e.a ✓ 22
13.f odd 12 2 169.12.a.g 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.12.e.a ✓ 22 1.a even 1 1 trivial
13.12.e.a ✓ 22 13.e even 6 1 inner
169.12.a.g 22 13.f odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(13, [\chi])\).