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Properties

Label 630.2.bi.b
Level $630$
Weight $2$
Character orbit 630.bi
Analytic conductor $5.031$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [630,2,Mod(479,630)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [48,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{2} + 48 q^{4} + 3 q^{7} + 48 q^{8} + 6 q^{11} + 3 q^{14} - 4 q^{15} + 48 q^{16} + 8 q^{21} + 6 q^{22} + 3 q^{23} - 18 q^{25} + 3 q^{28} - 3 q^{29} - 4 q^{30} + 48 q^{32} - 12 q^{35} - 18 q^{39}+ \cdots - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −1.70500 − 0.304890i 1.00000 −1.95745 + 1.08091i −1.70500 − 0.304890i −2.42207 − 1.06471i 1.00000 2.81408 + 1.03968i −1.95745 + 1.08091i
479.2 1.00000 −1.70170 − 0.322848i 1.00000 0.543458 + 2.16902i −1.70170 − 0.322848i −1.39638 + 2.24725i 1.00000 2.79154 + 1.09878i 0.543458 + 2.16902i
479.3 1.00000 −1.55560 + 0.761652i 1.00000 −1.55089 − 1.61082i −1.55560 + 0.761652i −0.405354 + 2.61451i 1.00000 1.83977 − 2.36965i −1.55089 − 1.61082i
479.4 1.00000 −1.52995 + 0.811941i 1.00000 −2.19282 − 0.437630i −1.52995 + 0.811941i 1.89790 − 1.84336i 1.00000 1.68150 − 2.48446i −2.19282 − 0.437630i
479.5 1.00000 −1.48333 − 0.894282i 1.00000 1.24107 − 1.86004i −1.48333 − 0.894282i 2.20215 + 1.46647i 1.00000 1.40052 + 2.65303i 1.24107 − 1.86004i
479.6 1.00000 −1.31224 − 1.13049i 1.00000 1.98394 + 1.03150i −1.31224 − 1.13049i 0.994947 − 2.45155i 1.00000 0.443973 + 2.96697i 1.98394 + 1.03150i
479.7 1.00000 −1.26195 + 1.18638i 1.00000 2.20530 + 0.369687i −1.26195 + 1.18638i 2.33660 + 1.24108i 1.00000 0.185026 − 2.99429i 2.20530 + 0.369687i
479.8 1.00000 −1.04243 − 1.38323i 1.00000 −0.930691 − 2.03318i −1.04243 − 1.38323i −2.58635 − 0.557485i 1.00000 −0.826663 + 2.88386i −0.930691 − 2.03318i
479.9 1.00000 −0.767971 + 1.55249i 1.00000 0.719036 − 2.11731i −0.767971 + 1.55249i −1.31170 − 2.29771i 1.00000 −1.82044 − 2.38453i 0.719036 − 2.11731i
479.10 1.00000 −0.560822 + 1.63874i 1.00000 −0.834189 + 2.07464i −0.560822 + 1.63874i −1.94702 − 1.79140i 1.00000 −2.37096 − 1.83809i −0.834189 + 2.07464i
479.11 1.00000 −0.337977 − 1.69876i 1.00000 0.307273 + 2.21486i −0.337977 − 1.69876i −0.431605 + 2.61031i 1.00000 −2.77154 + 1.14828i 0.307273 + 2.21486i
479.12 1.00000 0.193849 + 1.72117i 1.00000 2.07140 − 0.842204i 0.193849 + 1.72117i −1.57200 + 2.12810i 1.00000 −2.92484 + 0.667295i 2.07140 − 0.842204i
479.13 1.00000 0.241929 + 1.71507i 1.00000 −0.381768 + 2.20324i 0.241929 + 1.71507i 2.63972 + 0.178501i 1.00000 −2.88294 + 0.829851i −0.381768 + 2.20324i
479.14 1.00000 0.284086 − 1.70859i 1.00000 −1.93822 − 1.11504i 0.284086 − 1.70859i 1.80444 + 1.93494i 1.00000 −2.83859 − 0.970776i −1.93822 − 1.11504i
479.15 1.00000 0.630353 − 1.61327i 1.00000 0.0611015 − 2.23523i 0.630353 − 1.61327i 1.83478 − 1.90619i 1.00000 −2.20531 − 2.03386i 0.0611015 − 2.23523i
479.16 1.00000 0.633932 + 1.61187i 1.00000 −1.71175 − 1.43872i 0.633932 + 1.61187i 2.20678 − 1.45950i 1.00000 −2.19626 + 2.04363i −1.71175 − 1.43872i
479.17 1.00000 0.640430 − 1.60930i 1.00000 2.22760 − 0.194363i 0.640430 − 1.60930i −2.20085 − 1.46842i 1.00000 −2.17970 − 2.06129i 2.22760 − 0.194363i
479.18 1.00000 1.13785 + 1.30587i 1.00000 1.67933 + 1.47644i 1.13785 + 1.30587i −2.41546 + 1.07960i 1.00000 −0.410606 + 2.97177i 1.67933 + 1.47644i
479.19 1.00000 1.34174 − 1.09532i 1.00000 −0.255938 + 2.22137i 1.34174 − 1.09532i 2.61687 − 0.389830i 1.00000 0.600540 − 2.93928i −0.255938 + 2.22137i
479.20 1.00000 1.49245 + 0.878978i 1.00000 1.41134 − 1.73440i 1.49245 + 0.878978i 2.55753 − 0.677517i 1.00000 1.45479 + 2.62366i 1.41134 − 1.73440i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 479.24
Significant digits:
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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.b yes 48
3.b odd 2 1 1890.2.bi.a 48
5.b even 2 1 630.2.bi.a yes 48
7.d odd 6 1 630.2.r.a ✓ 48
9.c even 3 1 1890.2.r.a 48
9.d odd 6 1 630.2.r.b yes 48
15.d odd 2 1 1890.2.bi.b 48
21.g even 6 1 1890.2.r.b 48
35.i odd 6 1 630.2.r.b yes 48
45.h odd 6 1 630.2.r.a ✓ 48
45.j even 6 1 1890.2.r.b 48
63.i even 6 1 630.2.bi.a yes 48
63.t odd 6 1 1890.2.bi.b 48
105.p even 6 1 1890.2.r.a 48
315.q odd 6 1 1890.2.bi.a 48
315.bq even 6 1 inner 630.2.bi.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.r.a ✓ 48 7.d odd 6 1
630.2.r.a ✓ 48 45.h odd 6 1
630.2.r.b yes 48 9.d odd 6 1
630.2.r.b yes 48 35.i odd 6 1
630.2.bi.a yes 48 5.b even 2 1
630.2.bi.a yes 48 63.i even 6 1
630.2.bi.b yes 48 1.a even 1 1 trivial
630.2.bi.b yes 48 315.bq even 6 1 inner
1890.2.r.a 48 9.c even 3 1
1890.2.r.a 48 105.p even 6 1
1890.2.r.b 48 21.g even 6 1
1890.2.r.b 48 45.j even 6 1
1890.2.bi.a 48 3.b odd 2 1
1890.2.bi.a 48 315.q odd 6 1
1890.2.bi.b 48 15.d odd 2 1
1890.2.bi.b 48 63.t 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