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Properties

Label 234.2.t.a
Level $234$
Weight $2$
Character orbit 234.t
Analytic conductor $1.868$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [234,2,Mod(25,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(1.86849940730\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{4} - 16 q^{9} + 2 q^{13} + 8 q^{14} - 14 q^{16} + 16 q^{17} - 8 q^{23} + 14 q^{25} + 8 q^{26} + 18 q^{27} - 16 q^{29} - 8 q^{30} - 68 q^{35} - 8 q^{36} - 8 q^{39} - 10 q^{42} - 4 q^{43} + 10 q^{49} + 58 q^{51} - 2 q^{52} - 120 q^{53} - 8 q^{56} + 28 q^{61} + 68 q^{62} - 28 q^{64} - 8 q^{65} - 24 q^{66} + 8 q^{68} - 92 q^{69} + 16 q^{74} + 32 q^{75} - 24 q^{77} - 22 q^{78} + 28 q^{79} + 8 q^{81} - 48 q^{82} + 68 q^{87} - 50 q^{90} + 8 q^{92}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.866025 − 0.500000i −1.62009 − 0.612619i 0.500000 + 0.866025i 0.548308 − 0.316566i 1.09673 + 1.34059i −2.15366 − 1.24342i − 1.00000i 2.24940 + 1.98500i −0.633132
25.2 −0.866025 − 0.500000i −0.987257 + 1.42314i 0.500000 + 0.866025i 2.53992 − 1.46642i 1.56656 − 0.738845i −1.90832 − 1.10177i − 1.00000i −1.05065 − 2.81001i −2.93284
25.3 −0.866025 − 0.500000i −0.579434 − 1.63225i 0.500000 + 0.866025i −3.32246 + 1.91822i −0.314323 + 1.70329i 2.91802 + 1.68472i − 1.00000i −2.32851 + 1.89157i 3.83645
25.4 −0.866025 − 0.500000i −0.215223 + 1.71863i 0.500000 + 0.866025i −2.40542 + 1.38877i 1.04570 − 1.38076i −0.759011 − 0.438215i − 1.00000i −2.90736 − 0.739776i 2.77754
25.5 −0.866025 − 0.500000i 0.523143 − 1.65116i 0.500000 + 0.866025i 0.419378 − 0.242128i −1.27863 + 1.16837i −4.37722 − 2.52719i − 1.00000i −2.45264 − 1.72758i −0.484256
25.6 −0.866025 − 0.500000i 1.21476 + 1.23465i 0.500000 + 0.866025i 2.73536 − 1.57926i −0.434692 − 1.67662i 1.36392 + 0.787458i − 1.00000i −0.0487053 + 2.99960i −3.15852
25.7 −0.866025 − 0.500000i 1.66410 − 0.480381i 0.500000 + 0.866025i −0.515076 + 0.297379i −1.68134 − 0.416029i 1.45217 + 0.838409i − 1.00000i 2.53847 − 1.59880i 0.594758
25.8 0.866025 + 0.500000i −1.62009 − 0.612619i 0.500000 + 0.866025i −0.548308 + 0.316566i −1.09673 − 1.34059i 2.15366 + 1.24342i 1.00000i 2.24940 + 1.98500i −0.633132
25.9 0.866025 + 0.500000i −0.987257 + 1.42314i 0.500000 + 0.866025i −2.53992 + 1.46642i −1.56656 + 0.738845i 1.90832 + 1.10177i 1.00000i −1.05065 − 2.81001i −2.93284
25.10 0.866025 + 0.500000i −0.579434 − 1.63225i 0.500000 + 0.866025i 3.32246 − 1.91822i 0.314323 − 1.70329i −2.91802 − 1.68472i 1.00000i −2.32851 + 1.89157i 3.83645
25.11 0.866025 + 0.500000i −0.215223 + 1.71863i 0.500000 + 0.866025i 2.40542 − 1.38877i −1.04570 + 1.38076i 0.759011 + 0.438215i 1.00000i −2.90736 − 0.739776i 2.77754
25.12 0.866025 + 0.500000i 0.523143 − 1.65116i 0.500000 + 0.866025i −0.419378 + 0.242128i 1.27863 − 1.16837i 4.37722 + 2.52719i 1.00000i −2.45264 − 1.72758i −0.484256
25.13 0.866025 + 0.500000i 1.21476 + 1.23465i 0.500000 + 0.866025i −2.73536 + 1.57926i 0.434692 + 1.67662i −1.36392 − 0.787458i 1.00000i −0.0487053 + 2.99960i −3.15852
25.14 0.866025 + 0.500000i 1.66410 − 0.480381i 0.500000 + 0.866025i 0.515076 − 0.297379i 1.68134 + 0.416029i −1.45217 − 0.838409i 1.00000i 2.53847 − 1.59880i 0.594758
103.1 −0.866025 + 0.500000i −1.62009 + 0.612619i 0.500000 − 0.866025i 0.548308 + 0.316566i 1.09673 − 1.34059i −2.15366 + 1.24342i 1.00000i 2.24940 − 1.98500i −0.633132
103.2 −0.866025 + 0.500000i −0.987257 − 1.42314i 0.500000 − 0.866025i 2.53992 + 1.46642i 1.56656 + 0.738845i −1.90832 + 1.10177i 1.00000i −1.05065 + 2.81001i −2.93284
103.3 −0.866025 + 0.500000i −0.579434 + 1.63225i 0.500000 − 0.866025i −3.32246 − 1.91822i −0.314323 − 1.70329i 2.91802 − 1.68472i 1.00000i −2.32851 − 1.89157i 3.83645
103.4 −0.866025 + 0.500000i −0.215223 − 1.71863i 0.500000 − 0.866025i −2.40542 − 1.38877i 1.04570 + 1.38076i −0.759011 + 0.438215i 1.00000i −2.90736 + 0.739776i 2.77754
103.5 −0.866025 + 0.500000i 0.523143 + 1.65116i 0.500000 − 0.866025i 0.419378 + 0.242128i −1.27863 − 1.16837i −4.37722 + 2.52719i 1.00000i −2.45264 + 1.72758i −0.484256
103.6 −0.866025 + 0.500000i 1.21476 − 1.23465i 0.500000 − 0.866025i 2.73536 + 1.57926i −0.434692 + 1.67662i 1.36392 − 0.787458i 1.00000i −0.0487053 − 2.99960i −3.15852
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
13.b even 2 1 inner
117.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.2.t.a ✓ 28
3.b odd 2 1 702.2.t.a 28
9.c even 3 1 inner 234.2.t.a ✓ 28
9.c even 3 1 2106.2.b.c 14
9.d odd 6 1 702.2.t.a 28
9.d odd 6 1 2106.2.b.d 14
13.b even 2 1 inner 234.2.t.a ✓ 28
39.d odd 2 1 702.2.t.a 28
117.n odd 6 1 702.2.t.a 28
117.n odd 6 1 2106.2.b.d 14
117.t even 6 1 inner 234.2.t.a ✓ 28
117.t even 6 1 2106.2.b.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.t.a ✓ 28 1.a even 1 1 trivial
234.2.t.a ✓ 28 9.c even 3 1 inner
234.2.t.a ✓ 28 13.b even 2 1 inner
234.2.t.a ✓ 28 117.t even 6 1 inner
702.2.t.a 28 3.b odd 2 1
702.2.t.a 28 9.d odd 6 1
702.2.t.a 28 39.d odd 2 1
702.2.t.a 28 117.n odd 6 1
2106.2.b.c 14 9.c even 3 1
2106.2.b.c 14 117.t even 6 1
2106.2.b.d 14 9.d odd 6 1
2106.2.b.d 14 117.n odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(234, [\chi])\).