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Properties

Label 816.2.bl.c
Level $816$
Weight $2$
Character orbit 816.bl
Analytic conductor $6.516$
Analytic rank $0$
Dimension $68$
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] = [816,2,Mod(13,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bl (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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 2 q^{2} + 8 q^{5} + 2 q^{6} + 4 q^{7} - 4 q^{8} - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 2 q^{2} + 8 q^{5} + 2 q^{6} + 4 q^{7} - 4 q^{8} - 68 q^{9} + 14 q^{10} - 8 q^{12} + 4 q^{13} + 20 q^{16} + 8 q^{17} - 2 q^{18} + 4 q^{19} - 8 q^{20} - 4 q^{21} - 18 q^{22} + 28 q^{23} + 16 q^{24} + 92 q^{25} + 6 q^{26} - 12 q^{28} - 2 q^{30} + 12 q^{31} - 28 q^{32} + 18 q^{34} - 14 q^{38} - 4 q^{39} - 8 q^{40} + 20 q^{41} - 8 q^{42} + 12 q^{43} - 4 q^{44} - 8 q^{45} - 26 q^{46} + 16 q^{48} - 28 q^{50} + 12 q^{51} + 16 q^{52} + 4 q^{53} - 2 q^{54} + 12 q^{56} + 4 q^{57} + 4 q^{58} - 28 q^{59} + 8 q^{60} - 16 q^{61} - 76 q^{62} - 4 q^{63} - 12 q^{64} - 8 q^{65} - 10 q^{66} + 20 q^{67} - 32 q^{68} - 28 q^{69} - 24 q^{70} - 12 q^{71} + 4 q^{72} + 44 q^{73} - 56 q^{74} - 68 q^{76} - 32 q^{77} + 6 q^{78} - 20 q^{79} - 20 q^{80} + 68 q^{81} + 22 q^{82} - 4 q^{83} - 16 q^{84} + 8 q^{85} - 22 q^{86} + 32 q^{87} + 52 q^{88} - 14 q^{90} + 56 q^{92} - 12 q^{93} + 28 q^{94} - 28 q^{96} + 20 q^{97} + 14 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.41190 + 0.0809210i 1.00000i 1.98690 − 0.228504i −1.44947 −0.0809210 − 1.41190i 3.47459 + 3.47459i −2.78681 + 0.483407i −1.00000 2.04650 − 0.117292i
13.2 −1.40031 − 0.197822i 1.00000i 1.92173 + 0.554024i −3.74363 0.197822 − 1.40031i −3.52047 − 3.52047i −2.58142 − 1.15597i −1.00000 5.24224 + 0.740572i
13.3 −1.37601 + 0.326492i 1.00000i 1.78681 − 0.898513i 1.56335 −0.326492 − 1.37601i −1.00007 − 1.00007i −2.16530 + 1.81974i −1.00000 −2.15118 + 0.510422i
13.4 −1.37544 − 0.328892i 1.00000i 1.78366 + 0.904741i 3.51113 0.328892 − 1.37544i −0.363009 − 0.363009i −2.15575 − 1.83105i −1.00000 −4.82934 − 1.15478i
13.5 −1.26087 − 0.640479i 1.00000i 1.17957 + 1.61512i 0.405092 0.640479 − 1.26087i −0.219997 − 0.219997i −0.452839 − 2.79194i −1.00000 −0.510767 − 0.259453i
13.6 −1.25706 + 0.647918i 1.00000i 1.16040 − 1.62895i 3.10937 −0.647918 − 1.25706i 0.771262 + 0.771262i −0.403275 + 2.79953i −1.00000 −3.90866 + 2.01462i
13.7 −1.24867 − 0.663952i 1.00000i 1.11834 + 1.65811i −1.75568 0.663952 − 1.24867i 0.904219 + 0.904219i −0.295525 − 2.81295i −1.00000 2.19226 + 1.16569i
13.8 −1.18911 + 0.765525i 1.00000i 0.827942 − 1.82058i −3.96798 −0.765525 − 1.18911i −0.100288 − 0.100288i 0.409191 + 2.79867i −1.00000 4.71834 − 3.03759i
13.9 −0.870719 − 1.11438i 1.00000i −0.483697 + 1.94063i −0.453975 1.11438 − 0.870719i −2.53484 − 2.53484i 2.58377 − 1.15072i −1.00000 0.395284 + 0.505902i
13.10 −0.723621 − 1.21506i 1.00000i −0.952746 + 1.75849i −4.05323 1.21506 − 0.723621i 1.22344 + 1.22344i 2.82610 − 0.114832i −1.00000 2.93300 + 4.92492i
13.11 −0.721944 − 1.21606i 1.00000i −0.957594 + 1.75585i 3.80270 1.21606 − 0.721944i 1.87649 + 1.87649i 2.82655 − 0.103136i −1.00000 −2.74533 − 4.62430i
13.12 −0.579132 + 1.29020i 1.00000i −1.32921 − 1.49439i 3.13507 −1.29020 − 0.579132i −1.35432 − 1.35432i 2.69784 − 0.849496i −1.00000 −1.81562 + 4.04486i
13.13 −0.464771 − 1.33566i 1.00000i −1.56798 + 1.24155i 0.900917 1.33566 − 0.464771i 2.71315 + 2.71315i 2.38704 + 1.51725i −1.00000 −0.418720 − 1.20332i
13.14 −0.392757 + 1.35858i 1.00000i −1.69148 − 1.06718i −1.61413 −1.35858 − 0.392757i −0.473847 − 0.473847i 2.11420 − 1.87888i −1.00000 0.633959 − 2.19292i
13.15 −0.194963 + 1.40071i 1.00000i −1.92398 − 0.546173i −2.56167 −1.40071 − 0.194963i 1.10459 + 1.10459i 1.14014 − 2.58845i −1.00000 0.499430 − 3.58816i
13.16 −0.174697 − 1.40338i 1.00000i −1.93896 + 0.490333i 1.31045 1.40338 − 0.174697i −1.73893 − 1.73893i 1.02686 + 2.63544i −1.00000 −0.228931 − 1.83906i
13.17 0.0421898 + 1.41358i 1.00000i −1.99644 + 0.119278i 0.0529865 −1.41358 + 0.0421898i 1.72535 + 1.72535i −0.252838 − 2.81710i −1.00000 0.00223549 + 0.0749009i
13.18 0.0482765 − 1.41339i 1.00000i −1.99534 − 0.136467i −0.846924 1.41339 + 0.0482765i −1.61860 − 1.61860i −0.289209 + 2.81360i −1.00000 −0.0408865 + 1.19703i
13.19 0.227984 + 1.39572i 1.00000i −1.89605 + 0.636402i 1.87158 −1.39572 + 0.227984i −2.97746 − 2.97746i −1.32050 − 2.50125i −1.00000 0.426689 + 2.61219i
13.20 0.328041 − 1.37564i 1.00000i −1.78478 − 0.902534i −1.77507 1.37564 + 0.328041i 1.80588 + 1.80588i −1.82704 + 2.15915i −1.00000 −0.582296 + 2.44186i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
272.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.bl.c yes 68
16.e even 4 1 816.2.s.c ✓ 68
17.c even 4 1 816.2.s.c ✓ 68
272.j even 4 1 inner 816.2.bl.c yes 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.s.c ✓ 68 16.e even 4 1
816.2.s.c ✓ 68 17.c even 4 1
816.2.bl.c yes 68 1.a even 1 1 trivial
816.2.bl.c yes 68 272.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\):

\( T_{5}^{34} - 4 T_{5}^{33} - 100 T_{5}^{32} + 400 T_{5}^{31} + 4448 T_{5}^{30} - 17792 T_{5}^{29} + \cdots - 6782976 \) Copy content Toggle raw display
\( T_{7}^{68} - 4 T_{7}^{67} + 8 T_{7}^{66} + 1756 T_{7}^{64} - 6944 T_{7}^{63} + 13728 T_{7}^{62} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display