# Properties

 Label 234.2.a.c Level $234$ Weight $2$ Character orbit 234.a Self dual yes Analytic conductor $1.868$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 234.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 2 q^{5} + 4 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - 2 * q^5 + 4 * q^7 + q^8 $$q + q^{2} + q^{4} - 2 q^{5} + 4 q^{7} + q^{8} - 2 q^{10} + 4 q^{11} + q^{13} + 4 q^{14} + q^{16} - 2 q^{17} - 8 q^{19} - 2 q^{20} + 4 q^{22} - q^{25} + q^{26} + 4 q^{28} - 6 q^{29} - 4 q^{31} + q^{32} - 2 q^{34} - 8 q^{35} - 2 q^{37} - 8 q^{38} - 2 q^{40} + 10 q^{41} + 4 q^{43} + 4 q^{44} - 8 q^{47} + 9 q^{49} - q^{50} + q^{52} + 10 q^{53} - 8 q^{55} + 4 q^{56} - 6 q^{58} - 4 q^{59} - 2 q^{61} - 4 q^{62} + q^{64} - 2 q^{65} - 16 q^{67} - 2 q^{68} - 8 q^{70} + 8 q^{71} + 2 q^{73} - 2 q^{74} - 8 q^{76} + 16 q^{77} + 8 q^{79} - 2 q^{80} + 10 q^{82} - 12 q^{83} + 4 q^{85} + 4 q^{86} + 4 q^{88} - 14 q^{89} + 4 q^{91} - 8 q^{94} + 16 q^{95} + 10 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 + q^4 - 2 * q^5 + 4 * q^7 + q^8 - 2 * q^10 + 4 * q^11 + q^13 + 4 * q^14 + q^16 - 2 * q^17 - 8 * q^19 - 2 * q^20 + 4 * q^22 - q^25 + q^26 + 4 * q^28 - 6 * q^29 - 4 * q^31 + q^32 - 2 * q^34 - 8 * q^35 - 2 * q^37 - 8 * q^38 - 2 * q^40 + 10 * q^41 + 4 * q^43 + 4 * q^44 - 8 * q^47 + 9 * q^49 - q^50 + q^52 + 10 * q^53 - 8 * q^55 + 4 * q^56 - 6 * q^58 - 4 * q^59 - 2 * q^61 - 4 * q^62 + q^64 - 2 * q^65 - 16 * q^67 - 2 * q^68 - 8 * q^70 + 8 * q^71 + 2 * q^73 - 2 * q^74 - 8 * q^76 + 16 * q^77 + 8 * q^79 - 2 * q^80 + 10 * q^82 - 12 * q^83 + 4 * q^85 + 4 * q^86 + 4 * q^88 - 14 * q^89 + 4 * q^91 - 8 * q^94 + 16 * q^95 + 10 * q^97 + 9 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 1.00000 −2.00000 0 4.00000 1.00000 0 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.2.a.c 1
3.b odd 2 1 78.2.a.a 1
4.b odd 2 1 1872.2.a.c 1
5.b even 2 1 5850.2.a.d 1
5.c odd 4 2 5850.2.e.bb 2
8.b even 2 1 7488.2.a.bz 1
8.d odd 2 1 7488.2.a.bk 1
9.c even 3 2 2106.2.e.j 2
9.d odd 6 2 2106.2.e.q 2
12.b even 2 1 624.2.a.h 1
13.b even 2 1 3042.2.a.f 1
13.d odd 4 2 3042.2.b.g 2
15.d odd 2 1 1950.2.a.w 1
15.e even 4 2 1950.2.e.i 2
21.c even 2 1 3822.2.a.j 1
24.f even 2 1 2496.2.a.b 1
24.h odd 2 1 2496.2.a.t 1
33.d even 2 1 9438.2.a.t 1
39.d odd 2 1 1014.2.a.d 1
39.f even 4 2 1014.2.b.b 2
39.h odd 6 2 1014.2.e.c 2
39.i odd 6 2 1014.2.e.f 2
39.k even 12 4 1014.2.i.d 4
156.h even 2 1 8112.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 3.b odd 2 1
234.2.a.c 1 1.a even 1 1 trivial
624.2.a.h 1 12.b even 2 1
1014.2.a.d 1 39.d odd 2 1
1014.2.b.b 2 39.f even 4 2
1014.2.e.c 2 39.h odd 6 2
1014.2.e.f 2 39.i odd 6 2
1014.2.i.d 4 39.k even 12 4
1872.2.a.c 1 4.b odd 2 1
1950.2.a.w 1 15.d odd 2 1
1950.2.e.i 2 15.e even 4 2
2106.2.e.j 2 9.c even 3 2
2106.2.e.q 2 9.d odd 6 2
2496.2.a.b 1 24.f even 2 1
2496.2.a.t 1 24.h odd 2 1
3042.2.a.f 1 13.b even 2 1
3042.2.b.g 2 13.d odd 4 2
3822.2.a.j 1 21.c even 2 1
5850.2.a.d 1 5.b even 2 1
5850.2.e.bb 2 5.c odd 4 2
7488.2.a.bk 1 8.d odd 2 1
7488.2.a.bz 1 8.b even 2 1
8112.2.a.v 1 156.h even 2 1
9438.2.a.t 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(234))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T - 4$$
$11$ $$T - 4$$
$13$ $$T - 1$$
$17$ $$T + 2$$
$19$ $$T + 8$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T - 10$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T + 2$$
$67$ $$T + 16$$
$71$ $$T - 8$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T + 14$$
$97$ $$T - 10$$