# Properties

 Label 3234.2.a.w Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + b * q^5 + q^6 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{8} + q^{9} - \beta q^{10} + q^{11} - q^{12} - 4 q^{13} - \beta q^{15} + q^{16} - 3 q^{17} - q^{18} + 2 \beta q^{19} + \beta q^{20} - q^{22} - \beta q^{23} + q^{24} + 2 q^{25} + 4 q^{26} - q^{27} + 2 q^{29} + \beta q^{30} - 4 q^{31} - q^{32} - q^{33} + 3 q^{34} + q^{36} + (2 \beta - 4) q^{37} - 2 \beta q^{38} + 4 q^{39} - \beta q^{40} + 9 q^{41} + (2 \beta + 4) q^{43} + q^{44} + \beta q^{45} + \beta q^{46} + ( - 3 \beta + 4) q^{47} - q^{48} - 2 q^{50} + 3 q^{51} - 4 q^{52} + 4 q^{53} + q^{54} + \beta q^{55} - 2 \beta q^{57} - 2 q^{58} + (4 \beta - 4) q^{59} - \beta q^{60} + (3 \beta + 4) q^{61} + 4 q^{62} + q^{64} - 4 \beta q^{65} + q^{66} + (4 \beta - 3) q^{67} - 3 q^{68} + \beta q^{69} + (2 \beta - 8) q^{71} - q^{72} + ( - 2 \beta - 10) q^{73} + ( - 2 \beta + 4) q^{74} - 2 q^{75} + 2 \beta q^{76} - 4 q^{78} + (\beta + 8) q^{79} + \beta q^{80} + q^{81} - 9 q^{82} + (4 \beta + 5) q^{83} - 3 \beta q^{85} + ( - 2 \beta - 4) q^{86} - 2 q^{87} - q^{88} + ( - 2 \beta + 8) q^{89} - \beta q^{90} - \beta q^{92} + 4 q^{93} + (3 \beta - 4) q^{94} + 14 q^{95} + q^{96} + ( - 4 \beta - 1) q^{97} + q^{99} +O(q^{100})$$ q - q^2 - q^3 + q^4 + b * q^5 + q^6 - q^8 + q^9 - b * q^10 + q^11 - q^12 - 4 * q^13 - b * q^15 + q^16 - 3 * q^17 - q^18 + 2*b * q^19 + b * q^20 - q^22 - b * q^23 + q^24 + 2 * q^25 + 4 * q^26 - q^27 + 2 * q^29 + b * q^30 - 4 * q^31 - q^32 - q^33 + 3 * q^34 + q^36 + (2*b - 4) * q^37 - 2*b * q^38 + 4 * q^39 - b * q^40 + 9 * q^41 + (2*b + 4) * q^43 + q^44 + b * q^45 + b * q^46 + (-3*b + 4) * q^47 - q^48 - 2 * q^50 + 3 * q^51 - 4 * q^52 + 4 * q^53 + q^54 + b * q^55 - 2*b * q^57 - 2 * q^58 + (4*b - 4) * q^59 - b * q^60 + (3*b + 4) * q^61 + 4 * q^62 + q^64 - 4*b * q^65 + q^66 + (4*b - 3) * q^67 - 3 * q^68 + b * q^69 + (2*b - 8) * q^71 - q^72 + (-2*b - 10) * q^73 + (-2*b + 4) * q^74 - 2 * q^75 + 2*b * q^76 - 4 * q^78 + (b + 8) * q^79 + b * q^80 + q^81 - 9 * q^82 + (4*b + 5) * q^83 - 3*b * q^85 + (-2*b - 4) * q^86 - 2 * q^87 - q^88 + (-2*b + 8) * q^89 - b * q^90 - b * q^92 + 4 * q^93 + (3*b - 4) * q^94 + 14 * q^95 + q^96 + (-4*b - 1) * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 8 q^{13} + 2 q^{16} - 6 q^{17} - 2 q^{18} - 2 q^{22} + 2 q^{24} + 4 q^{25} + 8 q^{26} - 2 q^{27} + 4 q^{29} - 8 q^{31} - 2 q^{32} - 2 q^{33} + 6 q^{34} + 2 q^{36} - 8 q^{37} + 8 q^{39} + 18 q^{41} + 8 q^{43} + 2 q^{44} + 8 q^{47} - 2 q^{48} - 4 q^{50} + 6 q^{51} - 8 q^{52} + 8 q^{53} + 2 q^{54} - 4 q^{58} - 8 q^{59} + 8 q^{61} + 8 q^{62} + 2 q^{64} + 2 q^{66} - 6 q^{67} - 6 q^{68} - 16 q^{71} - 2 q^{72} - 20 q^{73} + 8 q^{74} - 4 q^{75} - 8 q^{78} + 16 q^{79} + 2 q^{81} - 18 q^{82} + 10 q^{83} - 8 q^{86} - 4 q^{87} - 2 q^{88} + 16 q^{89} + 8 q^{93} - 8 q^{94} + 28 q^{95} + 2 q^{96} - 2 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 - 8 * q^13 + 2 * q^16 - 6 * q^17 - 2 * q^18 - 2 * q^22 + 2 * q^24 + 4 * q^25 + 8 * q^26 - 2 * q^27 + 4 * q^29 - 8 * q^31 - 2 * q^32 - 2 * q^33 + 6 * q^34 + 2 * q^36 - 8 * q^37 + 8 * q^39 + 18 * q^41 + 8 * q^43 + 2 * q^44 + 8 * q^47 - 2 * q^48 - 4 * q^50 + 6 * q^51 - 8 * q^52 + 8 * q^53 + 2 * q^54 - 4 * q^58 - 8 * q^59 + 8 * q^61 + 8 * q^62 + 2 * q^64 + 2 * q^66 - 6 * q^67 - 6 * q^68 - 16 * q^71 - 2 * q^72 - 20 * q^73 + 8 * q^74 - 4 * q^75 - 8 * q^78 + 16 * q^79 + 2 * q^81 - 18 * q^82 + 10 * q^83 - 8 * q^86 - 4 * q^87 - 2 * q^88 + 16 * q^89 + 8 * q^93 - 8 * q^94 + 28 * q^95 + 2 * q^96 - 2 * q^97 + 2 * q^99

## 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
 −2.64575 2.64575
−1.00000 −1.00000 1.00000 −2.64575 1.00000 0 −1.00000 1.00000 2.64575
1.2 −1.00000 −1.00000 1.00000 2.64575 1.00000 0 −1.00000 1.00000 −2.64575
 $$n$$: 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.w 2
3.b odd 2 1 9702.2.a.db 2
7.b odd 2 1 3234.2.a.ba 2
7.c even 3 2 462.2.i.e 4
21.c even 2 1 9702.2.a.dm 2
21.h odd 6 2 1386.2.k.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.e 4 7.c even 3 2
1386.2.k.r 4 21.h odd 6 2
3234.2.a.w 2 1.a even 1 1 trivial
3234.2.a.ba 2 7.b odd 2 1
9702.2.a.db 2 3.b odd 2 1
9702.2.a.dm 2 21.c even 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 7$$ T5^2 - 7 $$T_{13} + 4$$ T13 + 4 $$T_{17} + 3$$ T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 7$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T + 4)^{2}$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} - 28$$
$23$ $$T^{2} - 7$$
$29$ $$(T - 2)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 8T - 12$$
$41$ $$(T - 9)^{2}$$
$43$ $$T^{2} - 8T - 12$$
$47$ $$T^{2} - 8T - 47$$
$53$ $$(T - 4)^{2}$$
$59$ $$T^{2} + 8T - 96$$
$61$ $$T^{2} - 8T - 47$$
$67$ $$T^{2} + 6T - 103$$
$71$ $$T^{2} + 16T + 36$$
$73$ $$T^{2} + 20T + 72$$
$79$ $$T^{2} - 16T + 57$$
$83$ $$T^{2} - 10T - 87$$
$89$ $$T^{2} - 16T + 36$$
$97$ $$T^{2} + 2T - 111$$