Properties

Label 234.2.b.b
Level $234$
Weight $2$
Character orbit 234.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} -3 i q^{5} -3 i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} -3 i q^{5} -3 i q^{7} -i q^{8} + 3 q^{10} + ( 2 - 3 i ) q^{13} + 3 q^{14} + q^{16} -3 q^{17} + 6 i q^{19} + 3 i q^{20} + 6 q^{23} -4 q^{25} + ( 3 + 2 i ) q^{26} + 3 i q^{28} + i q^{32} -3 i q^{34} -9 q^{35} -3 i q^{37} -6 q^{38} -3 q^{40} - q^{43} + 6 i q^{46} + 3 i q^{47} -2 q^{49} -4 i q^{50} + ( -2 + 3 i ) q^{52} + 6 q^{53} -3 q^{56} -6 i q^{59} -8 q^{61} - q^{64} + ( -9 - 6 i ) q^{65} + 12 i q^{67} + 3 q^{68} -9 i q^{70} + 15 i q^{71} -6 i q^{73} + 3 q^{74} -6 i q^{76} + 10 q^{79} -3 i q^{80} + 6 i q^{83} + 9 i q^{85} -i q^{86} -6 i q^{89} + ( -9 - 6 i ) q^{91} -6 q^{92} -3 q^{94} + 18 q^{95} + 12 i q^{97} -2 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 6q^{10} + 4q^{13} + 6q^{14} + 2q^{16} - 6q^{17} + 12q^{23} - 8q^{25} + 6q^{26} - 18q^{35} - 12q^{38} - 6q^{40} - 2q^{43} - 4q^{49} - 4q^{52} + 12q^{53} - 6q^{56} - 16q^{61} - 2q^{64} - 18q^{65} + 6q^{68} + 6q^{74} + 20q^{79} - 18q^{91} - 12q^{92} - 6q^{94} + 36q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.00000i
1.00000i
1.00000i 0 −1.00000 3.00000i 0 3.00000i 1.00000i 0 3.00000
181.2 1.00000i 0 −1.00000 3.00000i 0 3.00000i 1.00000i 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.2.b.b 2
3.b odd 2 1 26.2.b.a 2
4.b odd 2 1 1872.2.c.f 2
12.b even 2 1 208.2.f.a 2
13.b even 2 1 inner 234.2.b.b 2
13.d odd 4 1 3042.2.a.g 1
13.d odd 4 1 3042.2.a.j 1
15.d odd 2 1 650.2.d.b 2
15.e even 4 1 650.2.c.a 2
15.e even 4 1 650.2.c.d 2
21.c even 2 1 1274.2.d.c 2
21.g even 6 2 1274.2.n.c 4
21.h odd 6 2 1274.2.n.d 4
24.f even 2 1 832.2.f.b 2
24.h odd 2 1 832.2.f.d 2
39.d odd 2 1 26.2.b.a 2
39.f even 4 1 338.2.a.b 1
39.f even 4 1 338.2.a.d 1
39.h odd 6 2 338.2.e.c 4
39.i odd 6 2 338.2.e.c 4
39.k even 12 2 338.2.c.b 2
39.k even 12 2 338.2.c.f 2
52.b odd 2 1 1872.2.c.f 2
156.h even 2 1 208.2.f.a 2
156.l odd 4 1 2704.2.a.j 1
156.l odd 4 1 2704.2.a.k 1
195.e odd 2 1 650.2.d.b 2
195.n even 4 1 8450.2.a.h 1
195.n even 4 1 8450.2.a.u 1
195.s even 4 1 650.2.c.a 2
195.s even 4 1 650.2.c.d 2
273.g even 2 1 1274.2.d.c 2
273.w odd 6 2 1274.2.n.d 4
273.ba even 6 2 1274.2.n.c 4
312.b odd 2 1 832.2.f.d 2
312.h even 2 1 832.2.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 3.b odd 2 1
26.2.b.a 2 39.d odd 2 1
208.2.f.a 2 12.b even 2 1
208.2.f.a 2 156.h even 2 1
234.2.b.b 2 1.a even 1 1 trivial
234.2.b.b 2 13.b even 2 1 inner
338.2.a.b 1 39.f even 4 1
338.2.a.d 1 39.f even 4 1
338.2.c.b 2 39.k even 12 2
338.2.c.f 2 39.k even 12 2
338.2.e.c 4 39.h odd 6 2
338.2.e.c 4 39.i odd 6 2
650.2.c.a 2 15.e even 4 1
650.2.c.a 2 195.s even 4 1
650.2.c.d 2 15.e even 4 1
650.2.c.d 2 195.s even 4 1
650.2.d.b 2 15.d odd 2 1
650.2.d.b 2 195.e odd 2 1
832.2.f.b 2 24.f even 2 1
832.2.f.b 2 312.h even 2 1
832.2.f.d 2 24.h odd 2 1
832.2.f.d 2 312.b odd 2 1
1274.2.d.c 2 21.c even 2 1
1274.2.d.c 2 273.g even 2 1
1274.2.n.c 4 21.g even 6 2
1274.2.n.c 4 273.ba even 6 2
1274.2.n.d 4 21.h odd 6 2
1274.2.n.d 4 273.w odd 6 2
1872.2.c.f 2 4.b odd 2 1
1872.2.c.f 2 52.b odd 2 1
2704.2.a.j 1 156.l odd 4 1
2704.2.a.k 1 156.l odd 4 1
3042.2.a.g 1 13.d odd 4 1
3042.2.a.j 1 13.d odd 4 1
8450.2.a.h 1 195.n even 4 1
8450.2.a.u 1 195.n even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(234, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 + T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 - 4 T + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( 9 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 225 + T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( 36 + T^{2} \)
$97$ \( 144 + T^{2} \)
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