↑

Properties

Label 630.2.t.c
Level $630$
Weight $2$
Character orbit 630.t
Analytic conductor $5.031$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(311,630)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.311");
 
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.t (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: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{3} + 16 q^{4} + 32 q^{5} + 2 q^{6} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{3} + 16 q^{4} + 32 q^{5} + 2 q^{6} - 2 q^{7} + 6 q^{9} + 4 q^{12} + 2 q^{15} - 16 q^{16} + 6 q^{17} - 4 q^{18} + 16 q^{20} + 16 q^{21} + 4 q^{24} + 32 q^{25} + 12 q^{26} + 8 q^{27} + 2 q^{28} + 6 q^{29} + 2 q^{30} + 18 q^{31} + 16 q^{33} - 2 q^{35} + 2 q^{37} - 18 q^{39} - 6 q^{41} + 6 q^{42} - 28 q^{43} - 6 q^{44} + 6 q^{45} + 24 q^{47} + 2 q^{48} + 32 q^{49} - 26 q^{51} - 36 q^{53} - 32 q^{54} - 6 q^{56} + 18 q^{57} - 30 q^{59} + 4 q^{60} + 54 q^{61} - 94 q^{63} - 32 q^{64} - 44 q^{66} + 4 q^{67} + 12 q^{68} + 28 q^{69} + 4 q^{72} - 30 q^{73} + 2 q^{75} - 6 q^{77} - 22 q^{78} + 4 q^{79} - 16 q^{80} + 26 q^{81} - 24 q^{82} + 6 q^{83} - 4 q^{84} + 6 q^{85} - 8 q^{87} - 12 q^{89} - 4 q^{90} - 66 q^{91} - 18 q^{92} - 32 q^{93} - 42 q^{94} + 2 q^{96} + 96 q^{97} - 24 q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
311.1 −0.866025 − 0.500000i −1.69665 + 0.348398i 0.500000 + 0.866025i 1.00000 1.64354 + 0.546603i −1.74301 − 1.99046i − 1.00000i 2.75724 − 1.18222i −0.866025 − 0.500000i
311.2 −0.866025 − 0.500000i −1.35238 − 1.08216i 0.500000 + 0.866025i 1.00000 0.630115 + 1.61337i −1.80382 + 1.93552i − 1.00000i 0.657860 + 2.92698i −0.866025 − 0.500000i
311.3 −0.866025 − 0.500000i −1.15415 − 1.29148i 0.500000 + 0.866025i 1.00000 0.353784 + 1.69553i 2.60525 − 0.461188i − 1.00000i −0.335862 + 2.98114i −0.866025 − 0.500000i
311.4 −0.866025 − 0.500000i −0.423533 + 1.67947i 0.500000 + 0.866025i 1.00000 1.20653 − 1.24270i 2.20024 − 1.46933i − 1.00000i −2.64124 − 1.42262i −0.866025 − 0.500000i
311.5 −0.866025 − 0.500000i 0.521538 + 1.65167i 0.500000 + 0.866025i 1.00000 0.374168 − 1.69115i −2.64561 + 0.0270445i − 1.00000i −2.45600 + 1.72281i −0.866025 − 0.500000i
311.6 −0.866025 − 0.500000i 1.34284 − 1.09398i 0.500000 + 0.866025i 1.00000 −1.70992 + 0.275993i 1.16109 − 2.37737i − 1.00000i 0.606428 − 2.93807i −0.866025 − 0.500000i
311.7 −0.866025 − 0.500000i 1.61729 − 0.619981i 0.500000 + 0.866025i 1.00000 −1.71060 − 0.271725i −2.63727 − 0.211686i − 1.00000i 2.23125 − 2.00538i −0.866025 − 0.500000i
311.8 −0.866025 − 0.500000i 1.64505 + 0.542044i 0.500000 + 0.866025i 1.00000 −1.15363 − 1.29195i 1.49710 + 2.18144i − 1.00000i 2.41238 + 1.78338i −0.866025 − 0.500000i
311.9 0.866025 + 0.500000i −1.69871 − 0.338184i 0.500000 + 0.866025i 1.00000 −1.30204 − 1.14223i −2.46207 + 0.968625i 1.00000i 2.77126 + 1.14896i 0.866025 + 0.500000i
311.10 0.866025 + 0.500000i −1.55522 + 0.762428i 0.500000 + 0.866025i 1.00000 −1.72807 − 0.117327i 1.13965 − 2.38772i 1.00000i 1.83741 − 2.37148i 0.866025 + 0.500000i
311.11 0.866025 + 0.500000i −0.336991 − 1.69895i 0.500000 + 0.866025i 1.00000 0.557633 − 1.63983i 0.104916 + 2.64367i 1.00000i −2.77287 + 1.14506i 0.866025 + 0.500000i
311.12 0.866025 + 0.500000i −0.261236 − 1.71224i 0.500000 + 0.866025i 1.00000 0.629881 − 1.61346i −2.20349 − 1.46446i 1.00000i −2.86351 + 0.894597i 0.866025 + 0.500000i
311.13 0.866025 + 0.500000i 0.0292487 + 1.73180i 0.500000 + 0.866025i 1.00000 −0.840572 + 1.51441i 2.63859 + 0.194573i 1.00000i −2.99829 + 0.101306i 0.866025 + 0.500000i
311.14 0.866025 + 0.500000i 1.13939 − 1.30453i 0.500000 + 0.866025i 1.00000 1.63900 − 0.560056i 2.26702 + 1.36404i 1.00000i −0.403573 − 2.97273i 0.866025 + 0.500000i
311.15 0.866025 + 0.500000i 1.46653 + 0.921567i 0.500000 + 0.866025i 1.00000 0.809270 + 1.53137i −1.89754 + 1.84373i 1.00000i 1.30143 + 2.70301i 0.866025 + 0.500000i
311.16 0.866025 + 0.500000i 1.71699 − 0.227927i 0.500000 + 0.866025i 1.00000 1.60092 + 0.661104i 0.778946 − 2.52849i 1.00000i 2.89610 − 0.782695i 0.866025 + 0.500000i
551.1 −0.866025 + 0.500000i −1.69665 − 0.348398i 0.500000 − 0.866025i 1.00000 1.64354 − 0.546603i −1.74301 + 1.99046i 1.00000i 2.75724 + 1.18222i −0.866025 + 0.500000i
551.2 −0.866025 + 0.500000i −1.35238 + 1.08216i 0.500000 − 0.866025i 1.00000 0.630115 − 1.61337i −1.80382 − 1.93552i 1.00000i 0.657860 − 2.92698i −0.866025 + 0.500000i
551.3 −0.866025 + 0.500000i −1.15415 + 1.29148i 0.500000 − 0.866025i 1.00000 0.353784 − 1.69553i 2.60525 + 0.461188i 1.00000i −0.335862 − 2.98114i −0.866025 + 0.500000i
551.4 −0.866025 + 0.500000i −0.423533 − 1.67947i 0.500000 − 0.866025i 1.00000 1.20653 + 1.24270i 2.20024 + 1.46933i 1.00000i −2.64124 + 1.42262i −0.866025 + 0.500000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 311.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s 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.t.c ✓ 32
3.b odd 2 1 1890.2.t.c 32
7.d odd 6 1 630.2.bk.c yes 32
9.c even 3 1 1890.2.bk.c 32
9.d odd 6 1 630.2.bk.c yes 32
21.g even 6 1 1890.2.bk.c 32
63.k odd 6 1 1890.2.t.c 32
63.s even 6 1 inner 630.2.t.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.t.c ✓ 32 1.a even 1 1 trivial
630.2.t.c ✓ 32 63.s even 6 1 inner
630.2.bk.c yes 32 7.d odd 6 1
630.2.bk.c yes 32 9.d odd 6 1
1890.2.t.c 32 3.b odd 2 1
1890.2.t.c 32 63.k odd 6 1
1890.2.bk.c 32 9.c even 3 1
1890.2.bk.c 32 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{32} + 186 T_{11}^{30} + 15033 T_{11}^{28} + 694868 T_{11}^{26} + 20357652 T_{11}^{24} + \cdots + 7925984784 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display