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Properties

Label 45.3.k.a
Level $45$
Weight $3$
Character orbit 45.k
Analytic conductor $1.226$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,3,Mod(7,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([8, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 45.k (of order \(12\), degree \(4\), 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: \(1.22616118962\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 6 q^{3} - 2 q^{5} - 24 q^{6} - 2 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 6 q^{3} - 2 q^{5} - 24 q^{6} - 2 q^{7} - 24 q^{8} - 8 q^{10} + 8 q^{11} - 30 q^{12} - 2 q^{13} - 30 q^{15} + 28 q^{16} + 28 q^{17} + 48 q^{18} - 114 q^{20} + 12 q^{21} + 14 q^{22} + 82 q^{23} - 8 q^{25} - 112 q^{26} - 198 q^{27} - 88 q^{28} + 162 q^{30} - 4 q^{31} - 14 q^{32} + 96 q^{33} + 352 q^{35} + 264 q^{36} - 92 q^{37} + 330 q^{38} + 30 q^{40} - 28 q^{41} + 498 q^{42} - 2 q^{43} - 72 q^{45} - 136 q^{46} + 64 q^{47} - 510 q^{48} - 458 q^{50} - 396 q^{51} - 74 q^{52} - 608 q^{53} + 224 q^{55} - 192 q^{56} - 114 q^{57} + 30 q^{58} - 798 q^{60} + 92 q^{61} - 100 q^{62} + 24 q^{63} - 326 q^{65} + 588 q^{66} - 80 q^{67} + 626 q^{68} - 102 q^{70} + 248 q^{71} - 162 q^{72} - 8 q^{73} + 810 q^{75} - 96 q^{76} + 338 q^{77} + 1062 q^{78} + 1444 q^{80} + 204 q^{81} + 104 q^{82} + 370 q^{83} - 98 q^{85} - 328 q^{86} + 534 q^{87} - 210 q^{88} + 462 q^{90} + 152 q^{91} + 388 q^{92} - 1062 q^{93} - 360 q^{95} - 876 q^{96} + 292 q^{97} - 1876 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.30223 + 0.884830i −2.58106 − 1.52909i 6.65771 − 3.84383i 3.76250 + 3.29296i 9.87625 + 2.76562i 6.43786 − 1.72502i −8.91454 + 8.91454i 4.32375 + 7.89336i −15.3384 − 7.54494i
7.2 −3.22423 + 0.863931i 1.08621 + 2.79645i 6.18520 − 3.57103i −4.88999 + 1.04305i −5.91814 − 8.07800i −8.79970 + 2.35787i −7.41620 + 7.41620i −6.64028 + 6.07508i 14.8654 − 7.58766i
7.3 −2.38711 + 0.639624i 2.13208 − 2.11050i 1.82507 − 1.05371i −1.83989 − 4.64917i −3.73959 + 6.40173i 10.3622 − 2.77655i 3.30727 − 3.30727i 0.0915623 − 8.99953i 7.36573 + 9.92126i
7.4 −0.891829 + 0.238965i −2.93652 − 0.613889i −2.72585 + 1.57377i −1.31764 − 4.82326i 2.76557 − 0.154241i −11.0534 + 2.96175i 4.66637 − 4.66637i 8.24628 + 3.60539i 2.32770 + 3.98665i
7.5 −0.725667 + 0.194442i 2.68329 + 1.34163i −2.97532 + 1.71780i 4.81907 + 1.33289i −2.20804 − 0.451834i −1.79727 + 0.481578i 3.94998 − 3.94998i 5.40005 + 7.19996i −3.75621 − 0.0302083i
7.6 −0.183646 + 0.0492077i −1.82902 + 2.37796i −3.43280 + 1.98193i −1.61810 + 4.73094i 0.218877 − 0.526704i 8.70492 − 2.33248i 1.07064 − 1.07064i −2.30940 − 8.69866i 0.0643587 − 0.948439i
7.7 1.62192 − 0.434591i −0.485971 − 2.96038i −1.02236 + 0.590260i 4.97136 − 0.534392i −2.07476 − 4.59028i 2.65078 − 0.710273i −6.15096 + 6.15096i −8.52766 + 2.87732i 7.83089 − 3.02725i
7.8 1.84813 − 0.495206i 2.82294 − 1.01538i −0.293735 + 0.169588i −4.92689 + 0.851907i 4.71435 − 3.27449i −3.01652 + 0.808273i −5.87059 + 5.87059i 6.93801 − 5.73271i −8.68368 + 4.01426i
7.9 2.40598 − 0.644680i −0.325974 + 2.98224i 1.90902 − 1.10217i 0.360208 − 4.98701i 1.13830 + 7.38535i −0.0658171 + 0.0176356i −3.16269 + 3.16269i −8.78748 − 1.94426i −2.34837 − 12.2309i
7.10 3.47266 − 0.930497i −2.93201 − 0.635068i 7.72947 − 4.46261i −2.41870 + 4.37606i −10.7728 + 0.522853i −4.78910 + 1.28323i 12.5207 − 12.5207i 8.19338 + 3.72405i −4.32742 + 17.4472i
13.1 −3.30223 − 0.884830i −2.58106 + 1.52909i 6.65771 + 3.84383i 3.76250 − 3.29296i 9.87625 − 2.76562i 6.43786 + 1.72502i −8.91454 − 8.91454i 4.32375 − 7.89336i −15.3384 + 7.54494i
13.2 −3.22423 − 0.863931i 1.08621 − 2.79645i 6.18520 + 3.57103i −4.88999 − 1.04305i −5.91814 + 8.07800i −8.79970 − 2.35787i −7.41620 − 7.41620i −6.64028 − 6.07508i 14.8654 + 7.58766i
13.3 −2.38711 − 0.639624i 2.13208 + 2.11050i 1.82507 + 1.05371i −1.83989 + 4.64917i −3.73959 − 6.40173i 10.3622 + 2.77655i 3.30727 + 3.30727i 0.0915623 + 8.99953i 7.36573 − 9.92126i
13.4 −0.891829 − 0.238965i −2.93652 + 0.613889i −2.72585 − 1.57377i −1.31764 + 4.82326i 2.76557 + 0.154241i −11.0534 − 2.96175i 4.66637 + 4.66637i 8.24628 − 3.60539i 2.32770 − 3.98665i
13.5 −0.725667 − 0.194442i 2.68329 − 1.34163i −2.97532 − 1.71780i 4.81907 − 1.33289i −2.20804 + 0.451834i −1.79727 − 0.481578i 3.94998 + 3.94998i 5.40005 − 7.19996i −3.75621 + 0.0302083i
13.6 −0.183646 − 0.0492077i −1.82902 − 2.37796i −3.43280 − 1.98193i −1.61810 − 4.73094i 0.218877 + 0.526704i 8.70492 + 2.33248i 1.07064 + 1.07064i −2.30940 + 8.69866i 0.0643587 + 0.948439i
13.7 1.62192 + 0.434591i −0.485971 + 2.96038i −1.02236 − 0.590260i 4.97136 + 0.534392i −2.07476 + 4.59028i 2.65078 + 0.710273i −6.15096 − 6.15096i −8.52766 − 2.87732i 7.83089 + 3.02725i
13.8 1.84813 + 0.495206i 2.82294 + 1.01538i −0.293735 − 0.169588i −4.92689 − 0.851907i 4.71435 + 3.27449i −3.01652 − 0.808273i −5.87059 − 5.87059i 6.93801 + 5.73271i −8.68368 − 4.01426i
13.9 2.40598 + 0.644680i −0.325974 − 2.98224i 1.90902 + 1.10217i 0.360208 + 4.98701i 1.13830 − 7.38535i −0.0658171 − 0.0176356i −3.16269 − 3.16269i −8.78748 + 1.94426i −2.34837 + 12.2309i
13.10 3.47266 + 0.930497i −2.93201 + 0.635068i 7.72947 + 4.46261i −2.41870 − 4.37606i −10.7728 − 0.522853i −4.78910 − 1.28323i 12.5207 + 12.5207i 8.19338 − 3.72405i −4.32742 − 17.4472i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.c even 3 1 inner
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.3.k.a ✓ 40
3.b odd 2 1 135.3.l.a 40
5.b even 2 1 225.3.o.b 40
5.c odd 4 1 inner 45.3.k.a ✓ 40
5.c odd 4 1 225.3.o.b 40
9.c even 3 1 inner 45.3.k.a ✓ 40
9.c even 3 1 405.3.g.h 20
9.d odd 6 1 135.3.l.a 40
9.d odd 6 1 405.3.g.g 20
15.e even 4 1 135.3.l.a 40
45.j even 6 1 225.3.o.b 40
45.k odd 12 1 inner 45.3.k.a ✓ 40
45.k odd 12 1 225.3.o.b 40
45.k odd 12 1 405.3.g.h 20
45.l even 12 1 135.3.l.a 40
45.l even 12 1 405.3.g.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.k.a ✓ 40 1.a even 1 1 trivial
45.3.k.a ✓ 40 5.c odd 4 1 inner
45.3.k.a ✓ 40 9.c even 3 1 inner
45.3.k.a ✓ 40 45.k odd 12 1 inner
135.3.l.a 40 3.b odd 2 1
135.3.l.a 40 9.d odd 6 1
135.3.l.a 40 15.e even 4 1
135.3.l.a 40 45.l even 12 1
225.3.o.b 40 5.b even 2 1
225.3.o.b 40 5.c odd 4 1
225.3.o.b 40 45.j even 6 1
225.3.o.b 40 45.k odd 12 1
405.3.g.g 20 9.d odd 6 1
405.3.g.g 20 45.l even 12 1
405.3.g.h 20 9.c even 3 1
405.3.g.h 20 45.k odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(45, [\chi])\).