# Properties

 Label 3234.2.a.bk Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.14336.1 Defining polynomial: $$x^{4} - 8x^{2} + 14$$ x^4 - 8*x^2 + 14 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + \beta_{2} q^{5} - q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + b2 * q^5 - q^6 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + \beta_{2} q^{5} - q^{6} - q^{8} + q^{9} - \beta_{2} q^{10} - q^{11} + q^{12} + (\beta_{3} - \beta_1) q^{13} + \beta_{2} q^{15} + q^{16} + ( - 2 \beta_{3} + \beta_{2}) q^{17} - q^{18} + (\beta_{3} - \beta_{2} + \beta_1) q^{19} + \beta_{2} q^{20} + q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{23} - q^{24} + (2 \beta_{3} + 3) q^{25} + ( - \beta_{3} + \beta_1) q^{26} + q^{27} + 2 \beta_{3} q^{29} - \beta_{2} q^{30} + ( - \beta_{3} - \beta_{2} + 4) q^{31} - q^{32} - q^{33} + (2 \beta_{3} - \beta_{2}) q^{34} + q^{36} + ( - 4 \beta_{3} + \beta_1 + 2) q^{37} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{38} + (\beta_{3} - \beta_1) q^{39} - \beta_{2} q^{40} + (2 \beta_{3} - \beta_{2}) q^{41} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{43} - q^{44} + \beta_{2} q^{45} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{46} + ( - \beta_{3} - \beta_{2} + \beta_1 + 4) q^{47} + q^{48} + ( - 2 \beta_{3} - 3) q^{50} + ( - 2 \beta_{3} + \beta_{2}) q^{51} + (\beta_{3} - \beta_1) q^{52} + (4 \beta_{3} + \beta_1 + 2) q^{53} - q^{54} - \beta_{2} q^{55} + (\beta_{3} - \beta_{2} + \beta_1) q^{57} - 2 \beta_{3} q^{58} + (2 \beta_{3} + 8) q^{59} + \beta_{2} q^{60} + ( - \beta_{3} - \beta_1 + 4) q^{61} + (\beta_{3} + \beta_{2} - 4) q^{62} + q^{64} + ( - 8 \beta_{3} + \beta_1 - 4) q^{65} + q^{66} + (2 \beta_{3} + 4 \beta_{2} - \beta_1 + 4) q^{67} + ( - 2 \beta_{3} + \beta_{2}) q^{68} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{69} + (2 \beta_{3} + 2 \beta_1) q^{71} - q^{72} + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1) q^{73} + (4 \beta_{3} - \beta_1 - 2) q^{74} + (2 \beta_{3} + 3) q^{75} + (\beta_{3} - \beta_{2} + \beta_1) q^{76} + ( - \beta_{3} + \beta_1) q^{78} + ( - 6 \beta_{3} + \beta_1 + 4) q^{79} + \beta_{2} q^{80} + q^{81} + ( - 2 \beta_{3} + \beta_{2}) q^{82} + (3 \beta_{3} + \beta_{2}) q^{83} + (2 \beta_{3} - 2 \beta_1 + 8) q^{85} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{86} + 2 \beta_{3} q^{87} + q^{88} + ( - \beta_{3} - 3 \beta_1 + 4) q^{89} - \beta_{2} q^{90} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{92} + ( - \beta_{3} - \beta_{2} + 4) q^{93} + (\beta_{3} + \beta_{2} - \beta_1 - 4) q^{94} + (6 \beta_{3} + \beta_1 - 4) q^{95} - q^{96} + ( - \beta_{3} + 2 \beta_1 - 4) q^{97} - q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + b2 * q^5 - q^6 - q^8 + q^9 - b2 * q^10 - q^11 + q^12 + (b3 - b1) * q^13 + b2 * q^15 + q^16 + (-2*b3 + b2) * q^17 - q^18 + (b3 - b2 + b1) * q^19 + b2 * q^20 + q^22 + (-2*b3 + 2*b2 - b1 - 2) * q^23 - q^24 + (2*b3 + 3) * q^25 + (-b3 + b1) * q^26 + q^27 + 2*b3 * q^29 - b2 * q^30 + (-b3 - b2 + 4) * q^31 - q^32 - q^33 + (2*b3 - b2) * q^34 + q^36 + (-4*b3 + b1 + 2) * q^37 + (-b3 + b2 - b1) * q^38 + (b3 - b1) * q^39 - b2 * q^40 + (2*b3 - b2) * q^41 + (2*b3 + 2*b2 - b1 + 2) * q^43 - q^44 + b2 * q^45 + (2*b3 - 2*b2 + b1 + 2) * q^46 + (-b3 - b2 + b1 + 4) * q^47 + q^48 + (-2*b3 - 3) * q^50 + (-2*b3 + b2) * q^51 + (b3 - b1) * q^52 + (4*b3 + b1 + 2) * q^53 - q^54 - b2 * q^55 + (b3 - b2 + b1) * q^57 - 2*b3 * q^58 + (2*b3 + 8) * q^59 + b2 * q^60 + (-b3 - b1 + 4) * q^61 + (b3 + b2 - 4) * q^62 + q^64 + (-8*b3 + b1 - 4) * q^65 + q^66 + (2*b3 + 4*b2 - b1 + 4) * q^67 + (-2*b3 + b2) * q^68 + (-2*b3 + 2*b2 - b1 - 2) * q^69 + (2*b3 + 2*b1) * q^71 - q^72 + (-4*b3 - b2 + 2*b1) * q^73 + (4*b3 - b1 - 2) * q^74 + (2*b3 + 3) * q^75 + (b3 - b2 + b1) * q^76 + (-b3 + b1) * q^78 + (-6*b3 + b1 + 4) * q^79 + b2 * q^80 + q^81 + (-2*b3 + b2) * q^82 + (3*b3 + b2) * q^83 + (2*b3 - 2*b1 + 8) * q^85 + (-2*b3 - 2*b2 + b1 - 2) * q^86 + 2*b3 * q^87 + q^88 + (-b3 - 3*b1 + 4) * q^89 - b2 * q^90 + (-2*b3 + 2*b2 - b1 - 2) * q^92 + (-b3 - b2 + 4) * q^93 + (b3 + b2 - b1 - 4) * q^94 + (6*b3 + b1 - 4) * q^95 - q^96 + (-b3 + 2*b1 - 4) * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9 $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{11} + 4 q^{12} + 4 q^{16} - 4 q^{18} + 4 q^{22} - 8 q^{23} - 4 q^{24} + 12 q^{25} + 4 q^{27} + 16 q^{31} - 4 q^{32} - 4 q^{33} + 4 q^{36} + 8 q^{37} + 8 q^{43} - 4 q^{44} + 8 q^{46} + 16 q^{47} + 4 q^{48} - 12 q^{50} + 8 q^{53} - 4 q^{54} + 32 q^{59} + 16 q^{61} - 16 q^{62} + 4 q^{64} - 16 q^{65} + 4 q^{66} + 16 q^{67} - 8 q^{69} - 4 q^{72} - 8 q^{74} + 12 q^{75} + 16 q^{79} + 4 q^{81} + 32 q^{85} - 8 q^{86} + 4 q^{88} + 16 q^{89} - 8 q^{92} + 16 q^{93} - 16 q^{94} - 16 q^{95} - 4 q^{96} - 16 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^11 + 4 * q^12 + 4 * q^16 - 4 * q^18 + 4 * q^22 - 8 * q^23 - 4 * q^24 + 12 * q^25 + 4 * q^27 + 16 * q^31 - 4 * q^32 - 4 * q^33 + 4 * q^36 + 8 * q^37 + 8 * q^43 - 4 * q^44 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 12 * q^50 + 8 * q^53 - 4 * q^54 + 32 * q^59 + 16 * q^61 - 16 * q^62 + 4 * q^64 - 16 * q^65 + 4 * q^66 + 16 * q^67 - 8 * q^69 - 4 * q^72 - 8 * q^74 + 12 * q^75 + 16 * q^79 + 4 * q^81 + 32 * q^85 - 8 * q^86 + 4 * q^88 + 16 * q^89 - 8 * q^92 + 16 * q^93 - 16 * q^94 - 16 * q^95 - 4 * q^96 - 16 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 8x^{2} + 14$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v $$\beta_{3}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4$$ b3 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 2\beta_1$$ b2 + 2*b1

## 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
 −2.32685 1.60804 −1.60804 2.32685
−1.00000 1.00000 1.00000 −3.29066 −1.00000 0 −1.00000 1.00000 3.29066
1.2 −1.00000 1.00000 1.00000 −2.27411 −1.00000 0 −1.00000 1.00000 2.27411
1.3 −1.00000 1.00000 1.00000 2.27411 −1.00000 0 −1.00000 1.00000 −2.27411
1.4 −1.00000 1.00000 1.00000 3.29066 −1.00000 0 −1.00000 1.00000 −3.29066
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## 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 3234.2.a.bk yes 4
3.b odd 2 1 9702.2.a.ec 4
7.b odd 2 1 3234.2.a.bj 4
21.c even 2 1 9702.2.a.eb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bj 4 7.b odd 2 1
3234.2.a.bk yes 4 1.a even 1 1 trivial
9702.2.a.eb 4 21.c even 2 1
9702.2.a.ec 4 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5}^{4} - 16T_{5}^{2} + 56$$ T5^4 - 16*T5^2 + 56 $$T_{13}^{4} - 36T_{13}^{2} - 32T_{13} + 164$$ T13^4 - 36*T13^2 - 32*T13 + 164 $$T_{17}^{4} - 32T_{17}^{2} + 32T_{17} - 8$$ T17^4 - 32*T17^2 + 32*T17 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4} - 16T^{2} + 56$$
$7$ $$T^{4}$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} - 36 T^{2} - 32 T + 164$$
$17$ $$T^{4} - 32 T^{2} + 32 T - 8$$
$19$ $$T^{4} - 36 T^{2} + 80 T - 4$$
$23$ $$T^{4} + 8 T^{3} - 56 T^{2} + \cdots - 1168$$
$29$ $$(T^{2} - 8)^{2}$$
$31$ $$T^{4} - 16 T^{3} + 76 T^{2} + \cdots - 100$$
$37$ $$T^{4} - 8 T^{3} - 72 T^{2} + 480 T - 400$$
$41$ $$T^{4} - 32 T^{2} - 32 T - 8$$
$43$ $$T^{4} - 8 T^{3} - 56 T^{2} + \cdots - 1168$$
$47$ $$T^{4} - 16 T^{3} + 60 T^{2} - 48 T - 4$$
$53$ $$T^{4} - 8 T^{3} - 72 T^{2} + 224 T + 112$$
$59$ $$(T^{2} - 16 T + 56)^{2}$$
$61$ $$T^{4} - 16 T^{3} + 60 T^{2} + \cdots - 284$$
$67$ $$T^{4} - 16 T^{3} - 144 T^{2} + \cdots + 3872$$
$71$ $$T^{4} - 144 T^{2} - 256 T + 2624$$
$73$ $$T^{4} - 176 T^{2} - 448 T + 184$$
$79$ $$T^{4} - 16 T^{3} - 80 T^{2} + \cdots - 224$$
$83$ $$T^{4} - 52 T^{2} - 48 T + 92$$
$89$ $$T^{4} - 16 T^{3} - 196 T^{2} + \cdots + 12004$$
$97$ $$T^{4} + 16 T^{3} - 36 T^{2} + \cdots + 1988$$