# Properties

 Label 6034.2.a.h Level 6034 Weight 2 Character orbit 6034.a Self dual yes Analytic conductor 48.182 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$6034 = 2 \cdot 7 \cdot 431$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6034.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.1817325796$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + ( 1 - 2 \beta ) q^{5} - q^{6} - q^{7} + q^{8} -2 q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + ( 1 - 2 \beta ) q^{5} - q^{6} - q^{7} + q^{8} -2 q^{9} + ( 1 - 2 \beta ) q^{10} + ( 1 + 2 \beta ) q^{11} - q^{12} + ( 2 - 2 \beta ) q^{13} - q^{14} + ( -1 + 2 \beta ) q^{15} + q^{16} + ( -4 + 2 \beta ) q^{17} -2 q^{18} + 3 q^{19} + ( 1 - 2 \beta ) q^{20} + q^{21} + ( 1 + 2 \beta ) q^{22} + ( -3 - 2 \beta ) q^{23} - q^{24} + ( 2 - 2 \beta ) q^{26} + 5 q^{27} - q^{28} + ( -3 + 4 \beta ) q^{29} + ( -1 + 2 \beta ) q^{30} + ( -6 + 6 \beta ) q^{31} + q^{32} + ( -1 - 2 \beta ) q^{33} + ( -4 + 2 \beta ) q^{34} + ( -1 + 2 \beta ) q^{35} -2 q^{36} + ( -2 + 8 \beta ) q^{37} + 3 q^{38} + ( -2 + 2 \beta ) q^{39} + ( 1 - 2 \beta ) q^{40} + ( 2 - 4 \beta ) q^{41} + q^{42} + ( -6 + 8 \beta ) q^{43} + ( 1 + 2 \beta ) q^{44} + ( -2 + 4 \beta ) q^{45} + ( -3 - 2 \beta ) q^{46} + ( 4 - 4 \beta ) q^{47} - q^{48} + q^{49} + ( 4 - 2 \beta ) q^{51} + ( 2 - 2 \beta ) q^{52} + 3 q^{53} + 5 q^{54} + ( -3 - 4 \beta ) q^{55} - q^{56} -3 q^{57} + ( -3 + 4 \beta ) q^{58} -3 q^{59} + ( -1 + 2 \beta ) q^{60} + ( -6 - 4 \beta ) q^{61} + ( -6 + 6 \beta ) q^{62} + 2 q^{63} + q^{64} + ( 6 - 2 \beta ) q^{65} + ( -1 - 2 \beta ) q^{66} + ( 6 - 6 \beta ) q^{67} + ( -4 + 2 \beta ) q^{68} + ( 3 + 2 \beta ) q^{69} + ( -1 + 2 \beta ) q^{70} -6 \beta q^{71} -2 q^{72} + ( -6 + 2 \beta ) q^{73} + ( -2 + 8 \beta ) q^{74} + 3 q^{76} + ( -1 - 2 \beta ) q^{77} + ( -2 + 2 \beta ) q^{78} -12 q^{79} + ( 1 - 2 \beta ) q^{80} + q^{81} + ( 2 - 4 \beta ) q^{82} + ( -10 - 2 \beta ) q^{83} + q^{84} + ( -8 + 6 \beta ) q^{85} + ( -6 + 8 \beta ) q^{86} + ( 3 - 4 \beta ) q^{87} + ( 1 + 2 \beta ) q^{88} -4 \beta q^{89} + ( -2 + 4 \beta ) q^{90} + ( -2 + 2 \beta ) q^{91} + ( -3 - 2 \beta ) q^{92} + ( 6 - 6 \beta ) q^{93} + ( 4 - 4 \beta ) q^{94} + ( 3 - 6 \beta ) q^{95} - q^{96} + ( -5 + 6 \beta ) q^{97} + q^{98} + ( -2 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 2q^{7} + 2q^{8} - 4q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 2q^{7} + 2q^{8} - 4q^{9} + 4q^{11} - 2q^{12} + 2q^{13} - 2q^{14} + 2q^{16} - 6q^{17} - 4q^{18} + 6q^{19} + 2q^{21} + 4q^{22} - 8q^{23} - 2q^{24} + 2q^{26} + 10q^{27} - 2q^{28} - 2q^{29} - 6q^{31} + 2q^{32} - 4q^{33} - 6q^{34} - 4q^{36} + 4q^{37} + 6q^{38} - 2q^{39} + 2q^{42} - 4q^{43} + 4q^{44} - 8q^{46} + 4q^{47} - 2q^{48} + 2q^{49} + 6q^{51} + 2q^{52} + 6q^{53} + 10q^{54} - 10q^{55} - 2q^{56} - 6q^{57} - 2q^{58} - 6q^{59} - 16q^{61} - 6q^{62} + 4q^{63} + 2q^{64} + 10q^{65} - 4q^{66} + 6q^{67} - 6q^{68} + 8q^{69} - 6q^{71} - 4q^{72} - 10q^{73} + 4q^{74} + 6q^{76} - 4q^{77} - 2q^{78} - 24q^{79} + 2q^{81} - 22q^{83} + 2q^{84} - 10q^{85} - 4q^{86} + 2q^{87} + 4q^{88} - 4q^{89} - 2q^{91} - 8q^{92} + 6q^{93} + 4q^{94} - 2q^{96} - 4q^{97} + 2q^{98} - 8q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
1.00000 −1.00000 1.00000 −2.23607 −1.00000 −1.00000 1.00000 −2.00000 −2.23607
1.2 1.00000 −1.00000 1.00000 2.23607 −1.00000 −1.00000 1.00000 −2.00000 2.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 6034.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.h 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$431$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6034))$$:

 $$T_{3} + 1$$ $$T_{5}^{2} - 5$$ $$T_{11}^{2} - 4 T_{11} - 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$( 1 + T + 3 T^{2} )^{2}$$
$5$ $$1 + 5 T^{2} + 25 T^{4}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4}$$
$17$ $$1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 3 T + 19 T^{2} )^{2}$$
$23$ $$1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4}$$
$29$ $$1 + 2 T + 39 T^{2} + 58 T^{3} + 841 T^{4}$$
$31$ $$1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$1 - 4 T - 2 T^{2} - 148 T^{3} + 1369 T^{4}$$
$41$ $$1 + 62 T^{2} + 1681 T^{4}$$
$43$ $$1 + 4 T + 10 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 - 4 T + 78 T^{2} - 188 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 3 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 3 T + 59 T^{2} )^{2}$$
$61$ $$1 + 16 T + 166 T^{2} + 976 T^{3} + 3721 T^{4}$$
$67$ $$1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4}$$
$71$ $$1 + 6 T + 106 T^{2} + 426 T^{3} + 5041 T^{4}$$
$73$ $$1 + 10 T + 166 T^{2} + 730 T^{3} + 5329 T^{4}$$
$79$ $$( 1 + 12 T + 79 T^{2} )^{2}$$
$83$ $$1 + 22 T + 282 T^{2} + 1826 T^{3} + 6889 T^{4}$$
$89$ $$1 + 4 T + 162 T^{2} + 356 T^{3} + 7921 T^{4}$$
$97$ $$1 + 4 T + 153 T^{2} + 388 T^{3} + 9409 T^{4}$$
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