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Properties

Label 945.2.bt.a
Level $945$
Weight $2$
Character orbit 945.bt
Analytic conductor $7.546$
Analytic rank $0$
Dimension $96$
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] = [945,2,Mod(106,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([10, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bt (of order \(9\), degree \(6\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 3 q^{3} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 3 q^{3} + 9 q^{8} + 3 q^{9} - 3 q^{10} + 9 q^{11} + 60 q^{12} - 6 q^{13} + 6 q^{15} - 12 q^{16} - 72 q^{18} - 6 q^{19} + 9 q^{22} - 6 q^{23} - 60 q^{24} + 54 q^{26} + 36 q^{27} - 78 q^{28} - 18 q^{29} + 3 q^{30} - 51 q^{32} + 39 q^{34} - 48 q^{35} + 42 q^{36} + 54 q^{37} + 105 q^{38} + 45 q^{39} - 9 q^{40} + 45 q^{41} - 3 q^{42} + 9 q^{43} - 48 q^{44} - 12 q^{45} + 24 q^{46} - 6 q^{47} + 6 q^{48} - 6 q^{51} + 3 q^{52} - 36 q^{53} + 90 q^{54} - 24 q^{55} - 9 q^{56} - 30 q^{57} - 27 q^{58} + 12 q^{59} - 15 q^{60} + 24 q^{61} - 27 q^{62} - 3 q^{63} + 15 q^{64} - 12 q^{65} - 27 q^{66} + 30 q^{67} + 69 q^{68} - 42 q^{69} - 42 q^{71} - 177 q^{72} + 6 q^{73} - 51 q^{74} - 24 q^{76} - 15 q^{78} - 48 q^{79} - 42 q^{80} + 99 q^{81} - 36 q^{82} + 30 q^{83} + 12 q^{84} - 21 q^{85} - 75 q^{86} - 18 q^{87} + 87 q^{88} - 75 q^{89} - 9 q^{90} - 18 q^{91} + 123 q^{92} + 24 q^{93} - 3 q^{94} - 21 q^{95} + 24 q^{96} + 102 q^{97} + 3 q^{98} - 75 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −0.465170 + 2.63811i −1.53773 + 0.797118i −4.86387 − 1.77030i −0.766044 + 0.642788i −1.38758 − 4.42749i 0.939693 − 0.342020i 4.25398 − 7.36810i 1.72920 − 2.45150i −1.33940 − 2.31992i
106.2 −0.389135 + 2.20690i −1.42313 − 0.987265i −2.83958 − 1.03352i −0.766044 + 0.642788i 2.73258 − 2.75653i 0.939693 − 0.342020i 1.14492 − 1.98305i 1.05062 + 2.81002i −1.12047 − 1.94071i
106.3 −0.376837 + 2.13715i −0.0790981 + 1.73024i −2.54601 − 0.926671i −0.766044 + 0.642788i −3.66798 − 0.821064i 0.939693 − 0.342020i 0.769747 − 1.33324i −2.98749 − 0.273718i −1.08506 − 1.87938i
106.4 −0.343554 + 1.94839i 0.437318 − 1.67593i −1.79881 − 0.654713i −0.766044 + 0.642788i 3.11513 + 1.42784i 0.939693 − 0.342020i −0.0848237 + 0.146919i −2.61751 − 1.46583i −0.989223 − 1.71339i
106.5 −0.243544 + 1.38121i 1.48888 + 0.885011i 0.0309651 + 0.0112704i −0.766044 + 0.642788i −1.58499 + 1.84091i 0.939693 − 0.342020i −1.42562 + 2.46925i 1.43351 + 2.63535i −0.701257 − 1.21461i
106.6 −0.196671 + 1.11538i −1.15843 − 1.28765i 0.674003 + 0.245317i −0.766044 + 0.642788i 1.66404 − 1.03884i 0.939693 − 0.342020i −1.53876 + 2.66521i −0.316083 + 2.98330i −0.566291 − 0.980844i
106.7 −0.0667532 + 0.378576i 1.08913 + 1.34678i 1.74052 + 0.633498i −0.766044 + 0.642788i −0.582560 + 0.322416i 0.939693 − 0.342020i −0.740429 + 1.28246i −0.627607 + 2.93362i −0.192208 − 0.332914i
106.8 −0.0381502 + 0.216361i −1.72881 − 0.105870i 1.83403 + 0.667532i −0.766044 + 0.642788i 0.0888607 − 0.370008i 0.939693 − 0.342020i −0.434095 + 0.751874i 2.97758 + 0.366058i −0.109849 − 0.190264i
106.9 −0.0134396 + 0.0762199i 1.40028 − 1.01941i 1.87376 + 0.681992i −0.766044 + 0.642788i 0.0588804 + 0.120430i 0.939693 − 0.342020i −0.154560 + 0.267705i 0.921590 − 2.85494i −0.0386979 − 0.0670267i
106.10 0.0980868 − 0.556278i 1.66581 − 0.474436i 1.57956 + 0.574913i −0.766044 + 0.642788i −0.100524 − 0.973187i 0.939693 − 0.342020i 1.03960 − 1.80065i 2.54982 − 1.58064i 0.282430 + 0.489182i
106.11 0.195058 − 1.10623i −0.732344 + 1.56961i 0.693689 + 0.252482i −0.766044 + 0.642788i 1.59350 + 1.11631i 0.939693 − 0.342020i 1.53791 − 2.66373i −1.92734 − 2.29899i 0.561647 + 0.972802i
106.12 0.234962 − 1.33254i −1.29175 − 1.15386i 0.158937 + 0.0578482i −0.766044 + 0.642788i −1.84107 + 1.45019i 0.939693 − 0.342020i 1.46752 − 2.54182i 0.337236 + 2.98098i 0.676547 + 1.17181i
106.13 0.301339 − 1.70898i 0.271092 − 1.71070i −0.950411 − 0.345921i −0.766044 + 0.642788i −2.84186 − 0.978791i 0.939693 − 0.342020i 0.857773 − 1.48571i −2.85302 − 0.927516i 0.867670 + 1.50285i
106.14 0.322199 − 1.82728i 1.73043 − 0.0748678i −1.35575 − 0.493454i −0.766044 + 0.642788i 0.420738 − 3.18610i 0.939693 − 0.342020i 0.516969 − 0.895416i 2.98879 − 0.259107i 0.927734 + 1.60688i
106.15 0.380611 − 2.15855i −1.59962 + 0.664237i −2.63511 − 0.959101i −0.766044 + 0.642788i 0.824957 + 3.70569i 0.939693 − 0.342020i −0.881369 + 1.52658i 2.11758 − 2.12505i 1.09593 + 1.89820i
106.16 0.427350 − 2.42362i 0.201935 + 1.72024i −3.81193 − 1.38743i −0.766044 + 0.642788i 4.25551 + 0.245730i 0.939693 − 0.342020i −2.53062 + 4.38317i −2.91844 + 0.694753i 1.23051 + 2.13130i
211.1 −1.92333 + 1.61386i −1.52351 − 0.823970i 0.747337 − 4.23836i 0.939693 − 0.342020i 4.25998 − 0.873968i −0.173648 − 0.984808i 2.89203 + 5.00914i 1.64215 + 2.51065i −1.25536 + 2.17435i
211.2 −1.84611 + 1.54907i 0.384944 + 1.68873i 0.661207 − 3.74989i 0.939693 − 0.342020i −3.32661 − 2.52128i −0.173648 − 0.984808i 2.17826 + 3.77285i −2.70364 + 1.30013i −1.20496 + 2.08706i
211.3 −1.61713 + 1.35694i 1.70112 + 0.325894i 0.426548 − 2.41908i 0.939693 − 0.342020i −3.19315 + 1.78129i −0.173648 − 0.984808i 0.481727 + 0.834376i 2.78759 + 1.10877i −1.05551 + 1.82819i
211.4 −1.17685 + 0.987492i −0.406656 − 1.68364i 0.0625322 − 0.354638i 0.939693 − 0.342020i 2.14115 + 1.57981i −0.173648 − 0.984808i −1.25965 − 2.18178i −2.66926 + 1.36932i −0.768132 + 1.33044i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bt.a ✓ 96
27.e even 9 1 inner 945.2.bt.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bt.a ✓ 96 1.a even 1 1 trivial
945.2.bt.a ✓ 96 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 7 T_{2}^{93} + 3 T_{2}^{92} + 39 T_{2}^{91} + 611 T_{2}^{90} - 90 T_{2}^{89} + \cdots + 779689 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display