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Properties

Label 630.2.bi.a
Level $630$
Weight $2$
Character orbit 630.bi
Analytic conductor $5.031$
Analytic rank $0$
Dimension $48$
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] = [630,2,Mod(479,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.479");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.bi (of order \(6\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{2} + 48 q^{4} - 3 q^{7} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{2} + 48 q^{4} - 3 q^{7} - 48 q^{8} + 6 q^{11} + 3 q^{14} + 6 q^{15} + 48 q^{16} + 8 q^{21} - 6 q^{22} - 3 q^{23} + 18 q^{25} - 3 q^{28} - 3 q^{29} - 6 q^{30} - 48 q^{32} - 18 q^{35} - 18 q^{39} - 3 q^{41} - 8 q^{42} + 6 q^{44} + 15 q^{45} + 3 q^{46} + 3 q^{49} - 18 q^{50} + 10 q^{51} + 42 q^{55} + 3 q^{56} + 22 q^{57} + 3 q^{58} + 6 q^{60} + 43 q^{63} + 48 q^{64} - 12 q^{65} - 42 q^{69} + 18 q^{70} + 18 q^{73} + 18 q^{75} + 12 q^{77} + 18 q^{78} + 8 q^{81} + 3 q^{82} - 9 q^{83} + 8 q^{84} + 33 q^{85} + 15 q^{87} - 6 q^{88} - 33 q^{89} - 15 q^{90} - 3 q^{92} + 8 q^{93} + 24 q^{97} - 3 q^{98} - 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
479.1 −1.00000 −1.70267 − 0.317671i 1.00000 −2.23224 − 0.130844i 1.70267 + 0.317671i 0.691773 + 2.55371i −1.00000 2.79817 + 1.08178i 2.23224 + 0.130844i
479.2 −1.00000 −1.69918 + 0.335859i 1.00000 1.75304 − 1.38811i 1.69918 − 0.335859i 1.75469 − 1.98017i −1.00000 2.77440 − 1.14137i −1.75304 + 1.38811i
479.3 −1.00000 −1.68632 + 0.395368i 1.00000 0.969500 + 2.01496i 1.68632 − 0.395368i 0.868888 + 2.49901i −1.00000 2.68737 − 1.33344i −0.969500 − 2.01496i
479.4 −1.00000 −1.57419 − 0.722437i 1.00000 0.185127 + 2.22839i 1.57419 + 0.722437i −0.412401 − 2.61341i −1.00000 1.95617 + 2.27451i −0.185127 − 2.22839i
479.5 −1.00000 −1.49245 − 0.878978i 1.00000 0.796362 − 2.08945i 1.49245 + 0.878978i −2.55753 + 0.677517i −1.00000 1.45479 + 2.62366i −0.796362 + 2.08945i
479.6 −1.00000 −1.34174 + 1.09532i 1.00000 −1.79580 + 1.33234i 1.34174 − 1.09532i −2.61687 + 0.389830i −1.00000 0.600540 − 2.93928i 1.79580 − 1.33234i
479.7 −1.00000 −1.13785 − 1.30587i 1.00000 −2.11829 − 0.716121i 1.13785 + 1.30587i 2.41546 − 1.07960i −1.00000 −0.410606 + 2.97177i 2.11829 + 0.716121i
479.8 −1.00000 −0.640430 + 1.60930i 1.00000 −0.945479 − 2.02634i 0.640430 − 1.60930i 2.20085 + 1.46842i −1.00000 −2.17970 − 2.06129i 0.945479 + 2.02634i
479.9 −1.00000 −0.633932 − 1.61187i 1.00000 2.10184 + 0.763064i 0.633932 + 1.61187i −2.20678 + 1.45950i −1.00000 −2.19626 + 2.04363i −2.10184 − 0.763064i
479.10 −1.00000 −0.630353 + 1.61327i 1.00000 1.90522 − 1.17053i 0.630353 − 1.61327i −1.83478 + 1.90619i −1.00000 −2.20531 − 2.03386i −1.90522 + 1.17053i
479.11 −1.00000 −0.284086 + 1.70859i 1.00000 1.93476 + 1.12103i 0.284086 − 1.70859i −1.80444 − 1.93494i −1.00000 −2.83859 − 0.970776i −1.93476 − 1.12103i
479.12 −1.00000 −0.241929 − 1.71507i 1.00000 −1.71718 + 1.43224i 0.241929 + 1.71507i −2.63972 − 0.178501i −1.00000 −2.88294 + 0.829851i 1.71718 − 1.43224i
479.13 −1.00000 −0.193849 − 1.72117i 1.00000 −0.306329 − 2.21499i 0.193849 + 1.72117i 1.57200 − 2.12810i −1.00000 −2.92484 + 0.667295i 0.306329 + 2.21499i
479.14 −1.00000 0.337977 + 1.69876i 1.00000 −2.07176 + 0.841321i −0.337977 − 1.69876i 0.431605 − 2.61031i −1.00000 −2.77154 + 1.14828i 2.07176 − 0.841321i
479.15 −1.00000 0.560822 − 1.63874i 1.00000 −1.37960 + 1.75975i −0.560822 + 1.63874i 1.94702 + 1.79140i −1.00000 −2.37096 − 1.83809i 1.37960 − 1.75975i
479.16 −1.00000 0.767971 − 1.55249i 1.00000 1.47412 − 1.68136i −0.767971 + 1.55249i 1.31170 + 2.29771i −1.00000 −1.82044 − 2.38453i −1.47412 + 1.68136i
479.17 −1.00000 1.04243 + 1.38323i 1.00000 2.22613 − 0.210587i −1.04243 − 1.38323i 2.58635 + 0.557485i −1.00000 −0.826663 + 2.88386i −2.22613 + 0.210587i
479.18 −1.00000 1.26195 − 1.18638i 1.00000 −1.42281 − 1.72500i −1.26195 + 1.18638i −2.33660 − 1.24108i −1.00000 0.185026 − 2.99429i 1.42281 + 1.72500i
479.19 −1.00000 1.31224 + 1.13049i 1.00000 −1.88527 − 1.20239i −1.31224 − 1.13049i −0.994947 + 2.45155i −1.00000 0.443973 + 2.96697i 1.88527 + 1.20239i
479.20 −1.00000 1.48333 + 0.894282i 1.00000 0.990306 − 2.00482i −1.48333 − 0.894282i −2.20215 − 1.46647i −1.00000 1.40052 + 2.65303i −0.990306 + 2.00482i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 479.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
315.bq even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bi.a yes 48
3.b odd 2 1 1890.2.bi.b 48
5.b even 2 1 630.2.bi.b yes 48
7.d odd 6 1 630.2.r.b yes 48
9.c even 3 1 1890.2.r.b 48
9.d odd 6 1 630.2.r.a ✓ 48
15.d odd 2 1 1890.2.bi.a 48
21.g even 6 1 1890.2.r.a 48
35.i odd 6 1 630.2.r.a ✓ 48
45.h odd 6 1 630.2.r.b yes 48
45.j even 6 1 1890.2.r.a 48
63.i even 6 1 630.2.bi.b yes 48
63.t odd 6 1 1890.2.bi.a 48
105.p even 6 1 1890.2.r.b 48
315.q odd 6 1 1890.2.bi.b 48
315.bq even 6 1 inner 630.2.bi.a yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.r.a ✓ 48 9.d odd 6 1
630.2.r.a ✓ 48 35.i odd 6 1
630.2.r.b yes 48 7.d odd 6 1
630.2.r.b yes 48 45.h odd 6 1
630.2.bi.a yes 48 1.a even 1 1 trivial
630.2.bi.a yes 48 315.bq even 6 1 inner
630.2.bi.b yes 48 5.b even 2 1
630.2.bi.b yes 48 63.i even 6 1
1890.2.r.a 48 21.g even 6 1
1890.2.r.a 48 45.j even 6 1
1890.2.r.b 48 9.c even 3 1
1890.2.r.b 48 105.p even 6 1
1890.2.bi.a 48 15.d odd 2 1
1890.2.bi.a 48 63.t odd 6 1
1890.2.bi.b 48 3.b odd 2 1
1890.2.bi.b 48 315.q odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{48} + 177 T_{13}^{46} + 12 T_{13}^{45} + 17958 T_{13}^{44} + 1290 T_{13}^{43} + \cdots + 72\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display