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Properties

Label 630.2.t.b
Level $630$
Weight $2$
Character orbit 630.t
Analytic conductor $5.031$
Analytic rank $0$
Dimension $28$
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(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: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{3} + 14 q^{4} - 28 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{3} + 14 q^{4} - 28 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{9} - 4 q^{12} + 6 q^{14} + 2 q^{15} - 14 q^{16} - 6 q^{17} - 4 q^{18} - 6 q^{19} - 14 q^{20} + 6 q^{21} - 6 q^{22} - 4 q^{24} + 28 q^{25} + 12 q^{26} + 28 q^{27} - 8 q^{28} + 8 q^{30} - 12 q^{31} + 26 q^{33} + 4 q^{35} - 6 q^{36} + 4 q^{37} - 12 q^{38} + 54 q^{39} + 18 q^{41} - 32 q^{42} + 28 q^{43} - 6 q^{45} - 18 q^{46} + 30 q^{47} - 2 q^{48} - 14 q^{49} + 34 q^{51} - 42 q^{53} + 14 q^{54} + 6 q^{56} - 30 q^{57} - 12 q^{58} - 24 q^{59} + 4 q^{60} + 24 q^{61} - 12 q^{62} + 44 q^{63} - 28 q^{64} - 10 q^{66} - 40 q^{67} - 12 q^{68} + 8 q^{69} - 6 q^{70} + 4 q^{72} + 6 q^{73} - 2 q^{75} - 6 q^{76} - 24 q^{77} + 14 q^{78} + 2 q^{79} + 14 q^{80} - 46 q^{81} + 24 q^{82} - 18 q^{83} + 18 q^{84} + 6 q^{85} + 104 q^{87} - 12 q^{88} + 6 q^{89} + 4 q^{90} + 66 q^{91} - 30 q^{92} + 40 q^{93} + 42 q^{94} + 6 q^{95} + 4 q^{96} - 72 q^{97} - 24 q^{98} - 42 q^{99}+O(q^{100}) \) 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 −1.71805 − 0.219817i 0.500000 + 0.866025i −1.00000 1.37796 + 1.04939i −2.11261 − 1.59276i − 1.00000i 2.90336 + 0.755312i 0.866025 + 0.500000i
311.2 −0.866025 − 0.500000i −1.18136 − 1.26664i 0.500000 + 0.866025i −1.00000 0.389766 + 1.68763i 2.05946 + 1.66090i − 1.00000i −0.208774 + 2.99273i 0.866025 + 0.500000i
311.3 −0.866025 − 0.500000i −0.790256 + 1.54126i 0.500000 + 0.866025i −1.00000 1.45501 − 0.939646i −1.82075 + 1.91960i − 1.00000i −1.75099 − 2.43599i 0.866025 + 0.500000i
311.4 −0.866025 − 0.500000i 0.404488 − 1.68416i 0.500000 + 0.866025i −1.00000 −1.19238 + 1.25628i 1.07313 + 2.41835i − 1.00000i −2.67278 − 1.36244i 0.866025 + 0.500000i
311.5 −0.866025 − 0.500000i 1.16177 + 1.28464i 0.500000 + 0.866025i −1.00000 −0.363801 − 1.69341i −1.57438 − 2.12634i − 1.00000i −0.300592 + 2.98490i 0.866025 + 0.500000i
311.6 −0.866025 − 0.500000i 1.65157 + 0.521832i 0.500000 + 0.866025i −1.00000 −1.16939 − 1.27771i −2.02958 + 1.69729i − 1.00000i 2.45538 + 1.72369i 0.866025 + 0.500000i
311.7 −0.866025 − 0.500000i 1.70388 − 0.311090i 0.500000 + 0.866025i −1.00000 −1.63115 − 0.582530i 2.53870 − 0.744985i − 1.00000i 2.80645 − 1.06012i 0.866025 + 0.500000i
311.8 0.866025 + 0.500000i −1.51897 + 0.832312i 0.500000 + 0.866025i −1.00000 −1.73162 − 0.0386792i 2.13731 + 1.55945i 1.00000i 1.61451 − 2.52851i −0.866025 − 0.500000i
311.9 0.866025 + 0.500000i −1.28599 + 1.16028i 0.500000 + 0.866025i −1.00000 −1.69383 + 0.361836i −0.555567 − 2.58676i 1.00000i 0.307516 − 2.98420i −0.866025 − 0.500000i
311.10 0.866025 + 0.500000i −1.22158 − 1.22791i 0.500000 + 0.866025i −1.00000 −0.443962 − 1.67419i 2.64030 + 0.169721i 1.00000i −0.0155088 + 2.99996i −0.866025 − 0.500000i
311.11 0.866025 + 0.500000i −0.995677 − 1.41726i 0.500000 + 0.866025i −1.00000 −0.153652 − 1.72522i −2.36047 − 1.19507i 1.00000i −1.01725 + 2.82227i −0.866025 − 0.500000i
311.12 0.866025 + 0.500000i 0.0350998 + 1.73170i 0.500000 + 0.866025i −1.00000 −0.835450 + 1.51724i −1.17997 + 2.36805i 1.00000i −2.99754 + 0.121564i −0.866025 − 0.500000i
311.13 0.866025 + 0.500000i 1.11018 + 1.32948i 0.500000 + 0.866025i −1.00000 0.296702 + 1.70645i −0.142082 − 2.64193i 1.00000i −0.535019 + 2.95191i −0.866025 − 0.500000i
311.14 0.866025 + 0.500000i 1.64488 − 0.542569i 0.500000 + 0.866025i −1.00000 1.69579 + 0.352560i −0.673503 + 2.55859i 1.00000i 2.41124 − 1.78492i −0.866025 − 0.500000i
551.1 −0.866025 + 0.500000i −1.71805 + 0.219817i 0.500000 − 0.866025i −1.00000 1.37796 − 1.04939i −2.11261 + 1.59276i 1.00000i 2.90336 − 0.755312i 0.866025 − 0.500000i
551.2 −0.866025 + 0.500000i −1.18136 + 1.26664i 0.500000 − 0.866025i −1.00000 0.389766 − 1.68763i 2.05946 − 1.66090i 1.00000i −0.208774 − 2.99273i 0.866025 − 0.500000i
551.3 −0.866025 + 0.500000i −0.790256 − 1.54126i 0.500000 − 0.866025i −1.00000 1.45501 + 0.939646i −1.82075 − 1.91960i 1.00000i −1.75099 + 2.43599i 0.866025 − 0.500000i
551.4 −0.866025 + 0.500000i 0.404488 + 1.68416i 0.500000 − 0.866025i −1.00000 −1.19238 − 1.25628i 1.07313 − 2.41835i 1.00000i −2.67278 + 1.36244i 0.866025 − 0.500000i
551.5 −0.866025 + 0.500000i 1.16177 − 1.28464i 0.500000 − 0.866025i −1.00000 −0.363801 + 1.69341i −1.57438 + 2.12634i 1.00000i −0.300592 − 2.98490i 0.866025 − 0.500000i
551.6 −0.866025 + 0.500000i 1.65157 − 0.521832i 0.500000 − 0.866025i −1.00000 −1.16939 + 1.27771i −2.02958 − 1.69729i 1.00000i 2.45538 − 1.72369i 0.866025 − 0.500000i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 311.14
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.b ✓ 28
3.b odd 2 1 1890.2.t.b 28
7.d odd 6 1 630.2.bk.b yes 28
9.c even 3 1 1890.2.bk.b 28
9.d odd 6 1 630.2.bk.b yes 28
21.g even 6 1 1890.2.bk.b 28
63.k odd 6 1 1890.2.t.b 28
63.s even 6 1 inner 630.2.t.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.t.b ✓ 28 1.a even 1 1 trivial
630.2.t.b ✓ 28 63.s even 6 1 inner
630.2.bk.b yes 28 7.d odd 6 1
630.2.bk.b yes 28 9.d odd 6 1
1890.2.t.b 28 3.b odd 2 1
1890.2.t.b 28 63.k odd 6 1
1890.2.bk.b 28 9.c even 3 1
1890.2.bk.b 28 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{28} + 162 T_{11}^{26} + 11301 T_{11}^{24} + 444572 T_{11}^{22} + 10843488 T_{11}^{20} + \cdots + 14197824 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display