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Properties

Label 69.9.d.a
Level $69$
Weight $9$
Character orbit 69.d
Analytic conductor $28.109$
Analytic rank $0$
Dimension $32$
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] = [69,9,Mod(22,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.22");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 69.d (of order \(2\), degree \(1\), 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: \(28.1091240942\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{2} + 4944 q^{4} + 2268 q^{6} - 5388 q^{8} + 69984 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{2} + 4944 q^{4} + 2268 q^{6} - 5388 q^{8} + 69984 q^{9} - 38696 q^{13} + 791848 q^{16} + 26244 q^{18} - 767892 q^{23} + 655452 q^{24} - 2738824 q^{25} + 584136 q^{26} - 1303416 q^{29} + 1506832 q^{31} - 2618436 q^{32} + 11918352 q^{35} + 10812528 q^{36} - 2576448 q^{39} - 13351248 q^{41} - 2948048 q^{46} - 24811944 q^{47} - 21832416 q^{48} - 35733688 q^{49} + 43152348 q^{50} - 43838232 q^{52} + 4960116 q^{54} + 7466016 q^{55} - 2091112 q^{58} - 71697648 q^{59} + 73314648 q^{62} + 260117312 q^{64} + 5409180 q^{69} - 139970040 q^{70} + 13940688 q^{71} - 11783556 q^{72} + 49015072 q^{73} - 10099080 q^{75} + 62260632 q^{77} + 995328 q^{78} + 153055008 q^{81} - 279133000 q^{82} + 275584584 q^{85} + 11511072 q^{87} - 248107476 q^{92} - 222420168 q^{93} - 310157320 q^{94} - 443162688 q^{95} + 537882444 q^{96} - 693231852 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −31.5501 −46.7654 739.411 − 1078.83i 1475.45 3052.63i −15251.7 2187.00 34037.1i
22.2 −31.5501 −46.7654 739.411 1078.83i 1475.45 − 3052.63i −15251.7 2187.00 − 34037.1i
22.3 −28.1835 46.7654 538.309 − 383.804i −1318.01 − 1554.75i −7956.45 2187.00 10816.9i
22.4 −28.1835 46.7654 538.309 383.804i −1318.01 1554.75i −7956.45 2187.00 − 10816.9i
22.5 −23.2092 −46.7654 282.666 − 197.732i 1085.39 − 1785.72i −618.888 2187.00 4589.19i
22.6 −23.2092 −46.7654 282.666 197.732i 1085.39 1785.72i −618.888 2187.00 − 4589.19i
22.7 −21.7075 46.7654 215.216 − 988.761i −1015.16 2326.76i 885.320 2187.00 21463.5i
22.8 −21.7075 46.7654 215.216 988.761i −1015.16 − 2326.76i 885.320 2187.00 − 21463.5i
22.9 −14.0862 −46.7654 −57.5797 − 871.246i 658.745 819.788i 4417.14 2187.00 12272.5i
22.10 −14.0862 −46.7654 −57.5797 871.246i 658.745 − 819.788i 4417.14 2187.00 − 12272.5i
22.11 −13.9187 46.7654 −62.2690 − 558.538i −650.915 − 3444.18i 4429.90 2187.00 7774.14i
22.12 −13.9187 46.7654 −62.2690 558.538i −650.915 3444.18i 4429.90 2187.00 − 7774.14i
22.13 −6.96130 −46.7654 −207.540 − 175.957i 325.548 3704.58i 3226.84 2187.00 1224.89i
22.14 −6.96130 −46.7654 −207.540 175.957i 325.548 − 3704.58i 3226.84 2187.00 − 1224.89i
22.15 −1.25560 46.7654 −254.423 − 242.953i −58.7184 1140.24i 640.885 2187.00 305.051i
22.16 −1.25560 46.7654 −254.423 242.953i −58.7184 − 1140.24i 640.885 2187.00 − 305.051i
22.17 2.59284 −46.7654 −249.277 − 1021.64i −121.255 − 2633.25i −1310.10 2187.00 − 2648.94i
22.18 2.59284 −46.7654 −249.277 1021.64i −121.255 2633.25i −1310.10 2187.00 2648.94i
22.19 9.97644 46.7654 −156.471 − 1114.92i 466.552 3273.66i −4114.99 2187.00 − 11123.0i
22.20 9.97644 46.7654 −156.471 1114.92i 466.552 − 3273.66i −4114.99 2187.00 11123.0i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.9.d.a ✓ 32
3.b odd 2 1 207.9.d.c 32
23.b odd 2 1 inner 69.9.d.a ✓ 32
69.c even 2 1 207.9.d.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.9.d.a ✓ 32 1.a even 1 1 trivial
69.9.d.a ✓ 32 23.b odd 2 1 inner
207.9.d.c 32 3.b odd 2 1
207.9.d.c 32 69.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(69, [\chi])\).