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Properties

Label 630.2.ca.a
Level $630$
Weight $2$
Character orbit 630.ca
Analytic conductor $5.031$
Analytic rank $0$
Dimension $72$
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(113,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.113");
 
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.ca (of order \(12\), degree \(4\), 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: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{3} + 12 q^{11} + 4 q^{12} - 36 q^{14} + 12 q^{15} + 36 q^{16} - 24 q^{17} + 8 q^{18} + 12 q^{20} + 4 q^{21} + 24 q^{23} + 8 q^{24} + 12 q^{25} - 16 q^{27} - 32 q^{30} - 36 q^{33} + 12 q^{34} - 4 q^{36} + 24 q^{37} - 36 q^{38} - 40 q^{39} - 48 q^{41} - 4 q^{42} - 48 q^{43} - 12 q^{45} - 48 q^{47} - 8 q^{48} + 16 q^{51} - 4 q^{54} + 24 q^{55} + 52 q^{57} + 48 q^{58} - 4 q^{60} + 8 q^{66} + 12 q^{67} - 12 q^{68} + 64 q^{69} - 116 q^{75} + 8 q^{78} + 72 q^{79} + 48 q^{82} - 60 q^{83} + 24 q^{85} + 72 q^{86} - 44 q^{87} + 8 q^{90} - 24 q^{91} + 24 q^{92} + 32 q^{93} - 132 q^{94} - 60 q^{95} + 36 q^{97} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 −0.258819 + 0.965926i −1.68178 − 0.414269i −0.866025 − 0.500000i −0.704488 + 2.12219i 0.835429 − 1.51725i 0.965926 + 0.258819i 0.707107 − 0.707107i 2.65676 + 1.39342i −1.86754 − 1.22975i
113.2 −0.258819 + 0.965926i −1.66403 − 0.480632i −0.866025 − 0.500000i 1.26482 − 1.84397i 0.894937 − 1.48293i 0.965926 + 0.258819i 0.707107 − 0.707107i 2.53799 + 1.59957i 1.45378 + 1.69898i
113.3 −0.258819 + 0.965926i −1.48527 + 0.891045i −0.866025 − 0.500000i −1.41907 − 1.72808i −0.476266 − 1.66528i 0.965926 + 0.258819i 0.707107 − 0.707107i 1.41208 − 2.64689i 2.03647 − 0.923456i
113.4 −0.258819 + 0.965926i −0.664059 + 1.59970i −0.866025 − 0.500000i 2.19754 − 0.413301i −1.37332 − 1.05546i 0.965926 + 0.258819i 0.707107 − 0.707107i −2.11805 − 2.12458i −0.169547 + 2.22963i
113.5 −0.258819 + 0.965926i 0.391967 + 1.68712i −0.866025 − 0.500000i −1.20627 − 1.88279i −1.73108 − 0.0580471i 0.965926 + 0.258819i 0.707107 − 0.707107i −2.69272 + 1.32259i 2.13084 − 0.677868i
113.6 −0.258819 + 0.965926i 0.466476 − 1.66805i −0.866025 − 0.500000i −1.96234 + 1.07202i 1.49048 + 0.882305i 0.965926 + 0.258819i 0.707107 − 0.707107i −2.56480 − 1.55621i −0.527600 − 2.17293i
113.7 −0.258819 + 0.965926i 0.925451 − 1.46408i −0.866025 − 0.500000i 1.98898 + 1.02175i 1.17467 + 1.27285i 0.965926 + 0.258819i 0.707107 − 0.707107i −1.28708 − 2.70987i −1.50172 + 1.65676i
113.8 −0.258819 + 0.965926i 1.33038 + 1.10909i −0.866025 − 0.500000i 0.914863 + 2.04035i −1.41563 + 0.997993i 0.965926 + 0.258819i 0.707107 − 0.707107i 0.539818 + 2.95103i −2.20761 + 0.355608i
113.9 −0.258819 + 0.965926i 1.72195 − 0.186782i −0.866025 − 0.500000i −1.94006 + 1.11183i −0.265256 + 1.71162i 0.965926 + 0.258819i 0.707107 − 0.707107i 2.93022 − 0.643259i −0.571826 − 2.16172i
113.10 0.258819 − 0.965926i −1.73003 + 0.0836960i −0.866025 − 0.500000i −1.36521 − 1.77093i −0.366920 + 1.69274i −0.965926 − 0.258819i −0.707107 + 0.707107i 2.98599 − 0.289593i −2.06393 + 0.860339i
113.11 0.258819 − 0.965926i −1.67920 + 0.424585i −0.866025 − 0.500000i −0.399741 + 2.20005i −0.0244928 + 1.73188i −0.965926 − 0.258819i −0.707107 + 0.707107i 2.63946 − 1.42593i 2.02162 + 0.955535i
113.12 0.258819 − 0.965926i −1.02473 − 1.39640i −0.866025 − 0.500000i −1.84758 + 1.25955i −1.61404 + 0.628394i −0.965926 − 0.258819i −0.707107 + 0.707107i −0.899873 + 2.86186i 0.738439 + 2.11062i
113.13 0.258819 − 0.965926i −0.954441 + 1.44535i −0.866025 − 0.500000i 2.09046 + 0.793717i 1.14908 + 1.29600i −0.965926 − 0.258819i −0.707107 + 0.707107i −1.17808 − 2.75901i 1.30772 − 1.81380i
113.14 0.258819 − 0.965926i −0.499696 − 1.65840i −0.866025 − 0.500000i 2.14525 − 0.630806i −1.73123 + 0.0534431i −0.965926 − 0.258819i −0.707107 + 0.707107i −2.50061 + 1.65740i −0.0540811 − 2.23541i
113.15 0.258819 − 0.965926i 0.301698 + 1.70557i −0.866025 − 0.500000i −1.93399 − 1.12236i 1.72554 + 0.150016i −0.965926 − 0.258819i −0.707107 + 0.707107i −2.81796 + 1.02914i −1.58467 + 1.57760i
113.16 0.258819 − 0.965926i 0.602415 − 1.62391i −0.866025 − 0.500000i −0.651455 − 2.13907i −1.41266 − 1.00219i −0.965926 − 0.258819i −0.707107 + 0.707107i −2.27419 − 1.95654i −2.23479 + 0.0756260i
113.17 0.258819 − 0.965926i 1.25668 + 1.19196i −0.866025 − 0.500000i −0.955789 + 2.02150i 1.47659 − 0.905355i −0.965926 − 0.258819i −0.707107 + 0.707107i 0.158472 + 2.99581i 1.70524 + 1.44642i
113.18 0.258819 − 0.965926i 1.65417 − 0.513526i −0.866025 − 0.500000i 2.05203 + 0.888355i −0.0678959 − 1.73072i −0.965926 − 0.258819i −0.707107 + 0.707107i 2.47258 − 1.69892i 1.38919 − 1.75219i
407.1 −0.258819 − 0.965926i −1.68178 + 0.414269i −0.866025 + 0.500000i −0.704488 − 2.12219i 0.835429 + 1.51725i 0.965926 − 0.258819i 0.707107 + 0.707107i 2.65676 − 1.39342i −1.86754 + 1.22975i
407.2 −0.258819 − 0.965926i −1.66403 + 0.480632i −0.866025 + 0.500000i 1.26482 + 1.84397i 0.894937 + 1.48293i 0.965926 − 0.258819i 0.707107 + 0.707107i 2.53799 − 1.59957i 1.45378 − 1.69898i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.ca.a ✓ 72
5.c odd 4 1 630.2.ca.b yes 72
9.d odd 6 1 630.2.ca.b yes 72
45.l even 12 1 inner 630.2.ca.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.ca.a ✓ 72 1.a even 1 1 trivial
630.2.ca.a ✓ 72 45.l even 12 1 inner
630.2.ca.b yes 72 5.c odd 4 1
630.2.ca.b yes 72 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{72} - 144 T_{13}^{70} - 24 T_{13}^{69} + 7071 T_{13}^{68} + 4296 T_{13}^{67} + \cdots + 19\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display