Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 q^{5} - 2 \zeta_{6} q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 q^{5} - 2 \zeta_{6} q^{7} + q^{8} + ( - 3 \zeta_{6} + 3) q^{10} + ( - 6 \zeta_{6} + 6) q^{11} + ( - \zeta_{6} - 3) q^{13} + 2 q^{14} + (\zeta_{6} - 1) q^{16} - 3 \zeta_{6} q^{17} - 2 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} + 6 \zeta_{6} q^{22} + (6 \zeta_{6} - 6) q^{23} + 4 q^{25} + ( - 3 \zeta_{6} + 4) q^{26} + (2 \zeta_{6} - 2) q^{28} + ( - 3 \zeta_{6} + 3) q^{29} - 4 q^{31} - \zeta_{6} q^{32} + 3 q^{34} + 6 \zeta_{6} q^{35} + ( - 7 \zeta_{6} + 7) q^{37} + 2 q^{38} - 3 q^{40} + (3 \zeta_{6} - 3) q^{41} + 10 \zeta_{6} q^{43} - 6 q^{44} - 6 \zeta_{6} q^{46} - 6 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + (4 \zeta_{6} - 4) q^{50} + (4 \zeta_{6} - 1) q^{52} - 3 q^{53} + (18 \zeta_{6} - 18) q^{55} - 2 \zeta_{6} q^{56} + 3 \zeta_{6} q^{58} + 7 \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + q^{64} + (3 \zeta_{6} + 9) q^{65} + ( - 10 \zeta_{6} + 10) q^{67} + (3 \zeta_{6} - 3) q^{68} - 6 q^{70} + 6 \zeta_{6} q^{71} - 13 q^{73} + 7 \zeta_{6} q^{74} + (2 \zeta_{6} - 2) q^{76} - 12 q^{77} - 4 q^{79} + ( - 3 \zeta_{6} + 3) q^{80} - 3 \zeta_{6} q^{82} + 6 q^{83} + 9 \zeta_{6} q^{85} - 10 q^{86} + ( - 6 \zeta_{6} + 6) q^{88} + ( - 18 \zeta_{6} + 18) q^{89} + (8 \zeta_{6} - 2) q^{91} + 6 q^{92} + ( - 6 \zeta_{6} + 6) q^{94} + 6 \zeta_{6} q^{95} - 14 \zeta_{6} q^{97} + 3 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 6 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - 6 q^{5} - 2 q^{7} + 2 q^{8} + 3 q^{10} + 6 q^{11} - 7 q^{13} + 4 q^{14} - q^{16} - 3 q^{17} - 2 q^{19} + 3 q^{20} + 6 q^{22} - 6 q^{23} + 8 q^{25} + 5 q^{26} - 2 q^{28} + 3 q^{29} - 8 q^{31} - q^{32} + 6 q^{34} + 6 q^{35} + 7 q^{37} + 4 q^{38} - 6 q^{40} - 3 q^{41} + 10 q^{43} - 12 q^{44} - 6 q^{46} - 12 q^{47} + 3 q^{49} - 4 q^{50} + 2 q^{52} - 6 q^{53} - 18 q^{55} - 2 q^{56} + 3 q^{58} + 7 q^{61} + 4 q^{62} + 2 q^{64} + 21 q^{65} + 10 q^{67} - 3 q^{68} - 12 q^{70} + 6 q^{71} - 26 q^{73} + 7 q^{74} - 2 q^{76} - 24 q^{77} - 8 q^{79} + 3 q^{80} - 3 q^{82} + 12 q^{83} + 9 q^{85} - 20 q^{86} + 6 q^{88} + 18 q^{89} + 4 q^{91} + 12 q^{92} + 6 q^{94} + 6 q^{95} - 14 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −3.00000 0 −1.00000 + 1.73205i 1.00000 0 1.50000 + 2.59808i
217.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −3.00000 0 −1.00000 1.73205i 1.00000 0 1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.2.h.a 2
3.b odd 2 1 78.2.e.a 2
4.b odd 2 1 1872.2.t.c 2
12.b even 2 1 624.2.q.g 2
13.c even 3 1 inner 234.2.h.a 2
13.c even 3 1 3042.2.a.i 1
13.e even 6 1 3042.2.a.h 1
13.f odd 12 2 3042.2.b.h 2
15.d odd 2 1 1950.2.i.m 2
15.e even 4 2 1950.2.z.g 4
39.d odd 2 1 1014.2.e.a 2
39.f even 4 2 1014.2.i.b 4
39.h odd 6 1 1014.2.a.f 1
39.h odd 6 1 1014.2.e.a 2
39.i odd 6 1 78.2.e.a 2
39.i odd 6 1 1014.2.a.c 1
39.k even 12 2 1014.2.b.c 2
39.k even 12 2 1014.2.i.b 4
52.j odd 6 1 1872.2.t.c 2
156.p even 6 1 624.2.q.g 2
156.p even 6 1 8112.2.a.m 1
156.r even 6 1 8112.2.a.c 1
195.x odd 6 1 1950.2.i.m 2
195.bl even 12 2 1950.2.z.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 3.b odd 2 1
78.2.e.a 2 39.i odd 6 1
234.2.h.a 2 1.a even 1 1 trivial
234.2.h.a 2 13.c even 3 1 inner
624.2.q.g 2 12.b even 2 1
624.2.q.g 2 156.p even 6 1
1014.2.a.c 1 39.i odd 6 1
1014.2.a.f 1 39.h odd 6 1
1014.2.b.c 2 39.k even 12 2
1014.2.e.a 2 39.d odd 2 1
1014.2.e.a 2 39.h odd 6 1
1014.2.i.b 4 39.f even 4 2
1014.2.i.b 4 39.k even 12 2
1872.2.t.c 2 4.b odd 2 1
1872.2.t.c 2 52.j odd 6 1
1950.2.i.m 2 15.d odd 2 1
1950.2.i.m 2 195.x odd 6 1
1950.2.z.g 4 15.e even 4 2
1950.2.z.g 4 195.bl even 12 2
3042.2.a.h 1 13.e even 6 1
3042.2.a.i 1 13.c even 3 1
3042.2.b.h 2 13.f odd 12 2
8112.2.a.c 1 156.r even 6 1
8112.2.a.m 1 156.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(234, [\chi])\):

\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 13)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
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