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Properties

Label 675.2.l.d
Level $675$
Weight $2$
Character orbit 675.l
Analytic conductor $5.390$
Analytic rank $0$
Dimension $30$
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] = [675,2,Mod(76,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([14, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.l (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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 135)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} + 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{3} + 9 q^{8} - 3 q^{9} - 6 q^{11} - 3 q^{12} - 3 q^{13} - 9 q^{14} + 12 q^{16} + 12 q^{17} - 6 q^{18} + 24 q^{19} - 36 q^{21} + 51 q^{22} - 18 q^{23} + 45 q^{24} - 18 q^{26} + 9 q^{27} + 60 q^{28} + 18 q^{29} + 12 q^{31} - 36 q^{32} - 27 q^{33} - 69 q^{34} - 42 q^{36} - 24 q^{37} + 24 q^{38} + 6 q^{39} - 75 q^{41} + 18 q^{42} - 6 q^{43} + 12 q^{44} + 30 q^{46} - 45 q^{47} + 27 q^{48} - 36 q^{49} + 21 q^{51} - 30 q^{52} - 36 q^{53} + 18 q^{54} + 30 q^{56} - 30 q^{57} - 27 q^{58} - 27 q^{59} - 12 q^{61} - 36 q^{62} - 18 q^{63} + 27 q^{64} + 78 q^{66} + 30 q^{67} - 69 q^{68} - 117 q^{69} + 12 q^{71} - 9 q^{72} - 21 q^{73} - 30 q^{76} + 36 q^{77} - 66 q^{78} + 54 q^{79} - 27 q^{81} + 48 q^{82} + 87 q^{83} + 45 q^{84} + 18 q^{86} + 27 q^{87} + 18 q^{88} + 9 q^{89} + 51 q^{91} - 24 q^{92} - 36 q^{93} + 15 q^{94} - 15 q^{96} + 75 q^{97} + 15 q^{98} + 123 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −1.62376 + 1.36249i 0.235856 − 1.71592i 0.432902 − 2.45511i 0 1.95495 + 3.10759i 0.0109747 + 0.0622407i 0.522478 + 0.904959i −2.88874 − 0.809420i 0
76.2 −1.25101 + 1.04972i 0.905836 + 1.47630i 0.115814 − 0.656812i 0 −2.68292 − 0.895989i 0.576430 + 3.26909i −1.08849 − 1.88532i −1.35892 + 2.67457i 0
76.3 −0.186537 + 0.156523i 1.71306 + 0.255766i −0.337000 + 1.91122i 0 −0.359583 + 0.220424i −0.688903 − 3.90696i −0.479795 − 0.831029i 2.86917 + 0.876286i 0
76.4 0.531925 − 0.446338i −0.942993 − 1.45285i −0.263570 + 1.49478i 0 −1.15006 − 0.351912i 0.0506352 + 0.287167i 1.22136 + 2.11545i −1.22153 + 2.74005i 0
76.5 1.76334 − 1.47962i −1.47207 + 0.912694i 0.572799 − 3.24850i 0 −1.24532 + 3.78748i 0.745456 + 4.22769i −1.49463 − 2.58877i 1.33398 − 2.68710i 0
151.1 −1.62376 − 1.36249i 0.235856 + 1.71592i 0.432902 + 2.45511i 0 1.95495 − 3.10759i 0.0109747 − 0.0622407i 0.522478 − 0.904959i −2.88874 + 0.809420i 0
151.2 −1.25101 − 1.04972i 0.905836 − 1.47630i 0.115814 + 0.656812i 0 −2.68292 + 0.895989i 0.576430 − 3.26909i −1.08849 + 1.88532i −1.35892 − 2.67457i 0
151.3 −0.186537 − 0.156523i 1.71306 − 0.255766i −0.337000 − 1.91122i 0 −0.359583 − 0.220424i −0.688903 + 3.90696i −0.479795 + 0.831029i 2.86917 − 0.876286i 0
151.4 0.531925 + 0.446338i −0.942993 + 1.45285i −0.263570 − 1.49478i 0 −1.15006 + 0.351912i 0.0506352 − 0.287167i 1.22136 − 2.11545i −1.22153 − 2.74005i 0
151.5 1.76334 + 1.47962i −1.47207 − 0.912694i 0.572799 + 3.24850i 0 −1.24532 − 3.78748i 0.745456 − 4.22769i −1.49463 + 2.58877i 1.33398 + 2.68710i 0
301.1 −1.73836 − 0.632710i −0.550980 + 1.64208i 1.08947 + 0.914177i 0 1.99676 − 2.50591i 0.872404 − 0.732034i 0.534435 + 0.925669i −2.39284 − 1.80950i 0
301.2 −1.22047 − 0.444217i −1.42893 − 0.978851i −0.239858 − 0.201265i 0 1.30916 + 1.82942i 1.16252 − 0.975467i 1.50214 + 2.60178i 1.08370 + 2.79743i 0
301.3 0.282715 + 0.102900i 0.864728 − 1.50075i −1.46275 − 1.22739i 0 0.398898 − 0.335304i 3.28585 − 2.75715i −0.588102 − 1.01862i −1.50449 − 2.59548i 0
301.4 1.38905 + 0.505573i −1.28409 − 1.16238i 0.141770 + 0.118959i 0 −1.19600 − 2.26380i −2.40771 + 2.02030i −1.34141 − 2.32340i 0.297765 + 2.98519i 0
301.5 2.22676 + 0.810474i 1.72562 + 0.149064i 2.76950 + 2.32388i 0 3.72174 + 1.73050i 0.151115 − 0.126801i 1.91389 + 3.31495i 2.95556 + 0.514456i 0
376.1 −0.451738 + 2.56194i 1.35188 + 1.08278i −4.48006 − 1.63061i 0 −3.38471 + 2.97431i −3.98979 + 1.45216i 3.59987 − 6.23516i 0.655179 + 2.92758i 0
376.2 −0.226731 + 1.28586i −1.31414 − 1.12828i 0.277364 + 0.100952i 0 1.74877 − 1.43398i 0.822529 − 0.299376i −1.49839 + 2.59529i 0.453949 + 2.96546i 0
376.3 0.0302581 − 0.171602i −0.306826 + 1.70466i 1.85085 + 0.673656i 0 0.283239 + 0.104232i −1.63696 + 0.595804i 0.345853 − 0.599035i −2.81172 − 1.04607i 0
376.4 0.130038 − 0.737480i 0.729834 − 1.57078i 1.35242 + 0.492240i 0 −1.06351 − 0.742498i −0.498180 + 0.181323i 1.28774 − 2.23043i −1.93468 − 2.29281i 0
376.5 0.344526 − 1.95390i −1.72679 + 0.134863i −1.81965 − 0.662300i 0 −0.331416 + 3.42045i 1.54362 − 0.561833i 0.0630587 − 0.109221i 2.96362 − 0.465760i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.5
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 675.2.l.d 30
5.b even 2 1 135.2.k.a ✓ 30
5.c odd 4 2 675.2.u.c 60
15.d odd 2 1 405.2.k.a 30
27.e even 9 1 inner 675.2.l.d 30
135.n odd 18 1 405.2.k.a 30
135.n odd 18 1 3645.2.a.g 15
135.p even 18 1 135.2.k.a ✓ 30
135.p even 18 1 3645.2.a.h 15
135.r odd 36 2 675.2.u.c 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.a ✓ 30 5.b even 2 1
135.2.k.a ✓ 30 135.p even 18 1
405.2.k.a 30 15.d odd 2 1
405.2.k.a 30 135.n odd 18 1
675.2.l.d 30 1.a even 1 1 trivial
675.2.l.d 30 27.e even 9 1 inner
675.2.u.c 60 5.c odd 4 2
675.2.u.c 60 135.r odd 36 2
3645.2.a.g 15 135.n odd 18 1
3645.2.a.h 15 135.p even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 7 T_{2}^{27} - 3 T_{2}^{26} + 36 T_{2}^{25} + 169 T_{2}^{24} - 93 T_{2}^{23} - 315 T_{2}^{22} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display