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Properties

Label 630.2.r.b
Level $630$
Weight $2$
Character orbit 630.r
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(59,630)]
 
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([5, 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("630.59");
 
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.r (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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{2} - 3 q^{3} - 24 q^{4} - 48 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{2} - 3 q^{3} - 24 q^{4} - 48 q^{8} + 3 q^{9} + 3 q^{12} + 3 q^{14} + 6 q^{15} - 24 q^{16} + 3 q^{18} + 5 q^{21} - 6 q^{22} + 6 q^{23} + 3 q^{24} + 3 q^{28} - 3 q^{29} - 3 q^{30} + 24 q^{32} - 24 q^{33} + 12 q^{35} + 3 q^{41} - 8 q^{42} - 6 q^{44} + 9 q^{45} + 3 q^{46} - 6 q^{49} - 18 q^{50} - 8 q^{51} - 42 q^{55} + 22 q^{57} - 9 q^{60} - 9 q^{61} + 10 q^{63} + 48 q^{64} - 21 q^{65} - 24 q^{66} + 33 q^{67} + 42 q^{69} + 12 q^{70} - 3 q^{72} - 18 q^{73} - 39 q^{75} - 6 q^{77} + 18 q^{78} - 37 q^{81} - 3 q^{82} + 9 q^{83} - 13 q^{84} + 33 q^{85} + 18 q^{87} + 33 q^{89} + 15 q^{90} - 3 q^{92} + 32 q^{93} + 33 q^{95} - 3 q^{96} - 24 q^{97} - 3 q^{98} - 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 0.500000 − 0.866025i −1.71289 + 0.256935i −0.500000 − 0.866025i 0.390088 − 2.20178i −0.633932 + 1.61187i 0.160573 + 2.64087i −1.00000 2.86797 − 0.880202i −1.71175 − 1.43872i
59.2 0.500000 − 0.866025i −1.69984 − 0.332469i −0.500000 − 0.866025i −0.438968 + 2.19256i −1.13785 + 1.30587i 0.272771 − 2.63165i −1.00000 2.77893 + 1.13029i 1.67933 + 1.47644i
59.3 0.500000 − 0.866025i −1.60626 + 0.648019i −0.500000 − 0.866025i −2.09894 + 0.770998i −0.241929 + 1.71507i −1.47445 + 2.19682i −1.00000 2.16014 − 2.08177i −0.381768 + 2.20324i
59.4 0.500000 − 0.866025i −1.58750 + 0.692706i −0.500000 − 0.866025i 1.76507 + 1.37278i −0.193849 + 1.72117i −1.05699 − 2.42544i −1.00000 2.04032 − 2.19934i 2.07140 − 0.842204i
59.5 0.500000 − 0.866025i −1.50744 − 0.853008i −0.500000 − 0.866025i 2.20770 + 0.355056i −1.49245 + 0.878978i −0.692019 + 2.55365i −1.00000 1.54476 + 2.57172i 1.41134 − 1.73440i
59.6 0.500000 − 0.866025i −1.41275 − 1.00207i −0.500000 − 0.866025i −1.83728 − 1.27452i −1.57419 + 0.722437i −2.46948 − 0.949557i −1.00000 0.991700 + 2.83135i −2.02241 + 0.953871i
59.7 0.500000 − 0.866025i −1.13878 + 1.30506i −0.500000 − 0.866025i −2.21378 + 0.314891i 0.560822 + 1.63874i 2.52491 − 0.790468i −1.00000 −0.406351 − 2.97235i −0.834189 + 2.07464i
59.8 0.500000 − 0.866025i −1.12645 − 1.31572i −0.500000 − 0.866025i −1.00280 + 1.99860i −1.70267 + 0.317671i 2.55747 + 0.677764i −1.00000 −0.462238 + 2.96418i 1.22943 + 1.86775i
59.9 0.500000 − 0.866025i −0.960509 + 1.44133i −0.500000 − 0.866025i 2.19316 − 0.435950i 0.767971 + 1.55249i 2.64572 + 0.0128895i −1.00000 −1.15485 − 2.76881i 0.719036 − 2.11731i
59.10 0.500000 − 0.866025i −0.558726 − 1.63946i −0.500000 − 0.866025i 2.07866 − 0.824119i −1.69918 − 0.335859i −0.837534 − 2.50969i −1.00000 −2.37565 + 1.83202i 0.325622 − 2.21223i
59.11 0.500000 − 0.866025i −0.500762 − 1.65808i −0.500000 − 0.866025i −1.26026 − 1.84709i −1.68632 − 0.395368i 2.59865 + 0.497024i −1.00000 −2.49847 + 1.66061i −2.22976 + 0.167870i
59.12 0.500000 − 0.866025i −0.396457 + 1.68607i −0.500000 − 0.866025i 0.782489 + 2.09469i 1.26195 + 1.18638i −2.24311 + 1.40302i −1.00000 −2.68564 − 1.33691i 2.20530 + 0.369687i
59.13 0.500000 − 0.866025i 0.0618143 + 1.73095i −0.500000 − 0.866025i −0.717414 − 2.11786i 1.52995 + 0.811941i 0.647443 + 2.56531i −1.00000 −2.99236 + 0.213994i −2.19282 − 0.437630i
59.14 0.500000 − 0.866025i 0.118189 + 1.72801i −0.500000 − 0.866025i 0.619562 − 2.14852i 1.55560 + 0.761652i −2.06156 − 1.65830i −1.00000 −2.97206 + 0.408466i −1.55089 − 1.61082i
59.15 0.500000 − 0.866025i 0.277706 − 1.70964i −0.500000 − 0.866025i −2.05173 + 0.889037i −1.34174 − 1.09532i −0.970835 + 2.46119i −1.00000 −2.84576 − 0.949556i −0.255938 + 2.22137i
59.16 0.500000 − 0.866025i 1.07348 − 1.35928i −0.500000 − 0.866025i 1.28213 + 1.83198i −0.640430 − 1.60930i 2.37211 − 1.17179i −1.00000 −0.695279 − 2.91832i 2.22760 − 0.194363i
59.17 0.500000 − 0.866025i 1.08196 − 1.35254i −0.500000 − 0.866025i 1.96632 − 1.06470i −0.630353 − 1.61327i 0.733422 + 2.54206i −1.00000 −0.658722 − 2.92679i 0.0611015 − 2.23523i
59.18 0.500000 − 0.866025i 1.11655 + 1.32413i −0.500000 − 0.866025i −1.91482 − 1.15475i 1.70500 − 0.304890i 2.13310 − 1.56522i −1.00000 −0.506654 + 2.95691i −1.95745 + 1.08091i
59.19 0.500000 − 0.866025i 1.13044 + 1.31229i −0.500000 − 0.866025i −1.60670 + 1.55516i 1.70170 − 0.322848i −1.24799 − 2.33292i −1.00000 −0.444199 + 2.96693i 0.543458 + 2.16902i
59.20 0.500000 − 0.866025i 1.33764 − 1.10032i −0.500000 − 0.866025i −0.00346114 − 2.23607i −0.284086 − 1.70859i −2.57793 + 0.595221i −1.00000 0.578578 − 2.94368i −1.93822 − 1.11504i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.24
Significant digits:
Format:

Inner twists

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