Properties

 Label 35.2.a.b Level $35$ Weight $2$ Character orbit 35.a Self dual yes Analytic conductor $0.279$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 35.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$0.279476407074$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( -1 + \beta ) q^{3} + ( 2 + \beta ) q^{4} + q^{5} -4 q^{6} - q^{7} + ( -4 - \beta ) q^{8} + ( 2 - \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{2} + ( -1 + \beta ) q^{3} + ( 2 + \beta ) q^{4} + q^{5} -4 q^{6} - q^{7} + ( -4 - \beta ) q^{8} + ( 2 - \beta ) q^{9} -\beta q^{10} + ( 1 - \beta ) q^{11} + ( 2 + 2 \beta ) q^{12} + ( 3 - \beta ) q^{13} + \beta q^{14} + ( -1 + \beta ) q^{15} + 3 \beta q^{16} + ( -3 + \beta ) q^{17} + ( 4 - \beta ) q^{18} + ( -2 - 2 \beta ) q^{19} + ( 2 + \beta ) q^{20} + ( 1 - \beta ) q^{21} + 4 q^{22} + ( -2 + 2 \beta ) q^{23} -4 \beta q^{24} + q^{25} + ( 4 - 2 \beta ) q^{26} + ( -3 - \beta ) q^{27} + ( -2 - \beta ) q^{28} + ( -1 + 3 \beta ) q^{29} -4 q^{30} + ( -4 - \beta ) q^{32} + ( -5 + \beta ) q^{33} + ( -4 + 2 \beta ) q^{34} - q^{35} -\beta q^{36} + 6 q^{37} + ( 8 + 4 \beta ) q^{38} + ( -7 + 3 \beta ) q^{39} + ( -4 - \beta ) q^{40} + 2 \beta q^{41} + 4 q^{42} + ( 6 - 2 \beta ) q^{43} + ( -2 - 2 \beta ) q^{44} + ( 2 - \beta ) q^{45} -8 q^{46} + ( -1 - 3 \beta ) q^{47} + 12 q^{48} + q^{49} -\beta q^{50} + ( 7 - 3 \beta ) q^{51} + 2 q^{52} -2 \beta q^{53} + ( 4 + 4 \beta ) q^{54} + ( 1 - \beta ) q^{55} + ( 4 + \beta ) q^{56} + ( -6 - 2 \beta ) q^{57} + ( -12 - 2 \beta ) q^{58} -4 q^{59} + ( 2 + 2 \beta ) q^{60} + 6 \beta q^{61} + ( -2 + \beta ) q^{63} + ( 4 - \beta ) q^{64} + ( 3 - \beta ) q^{65} + ( -4 + 4 \beta ) q^{66} + 4 \beta q^{67} -2 q^{68} + ( 10 - 2 \beta ) q^{69} + \beta q^{70} + 8 q^{71} + ( -4 + 3 \beta ) q^{72} + ( -2 - 4 \beta ) q^{73} -6 \beta q^{74} + ( -1 + \beta ) q^{75} + ( -12 - 8 \beta ) q^{76} + ( -1 + \beta ) q^{77} + ( -12 + 4 \beta ) q^{78} + ( -5 + \beta ) q^{79} + 3 \beta q^{80} -7 q^{81} + ( -8 - 2 \beta ) q^{82} + 4 q^{83} + ( -2 - 2 \beta ) q^{84} + ( -3 + \beta ) q^{85} + ( 8 - 4 \beta ) q^{86} + ( 13 - \beta ) q^{87} + 4 \beta q^{88} + ( 4 - 2 \beta ) q^{89} + ( 4 - \beta ) q^{90} + ( -3 + \beta ) q^{91} + ( 4 + 4 \beta ) q^{92} + ( 12 + 4 \beta ) q^{94} + ( -2 - 2 \beta ) q^{95} -4 \beta q^{96} + ( -7 + 5 \beta ) q^{97} -\beta q^{98} + ( 6 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} + 5q^{4} + 2q^{5} - 8q^{6} - 2q^{7} - 9q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} + 5q^{4} + 2q^{5} - 8q^{6} - 2q^{7} - 9q^{8} + 3q^{9} - q^{10} + q^{11} + 6q^{12} + 5q^{13} + q^{14} - q^{15} + 3q^{16} - 5q^{17} + 7q^{18} - 6q^{19} + 5q^{20} + q^{21} + 8q^{22} - 2q^{23} - 4q^{24} + 2q^{25} + 6q^{26} - 7q^{27} - 5q^{28} + q^{29} - 8q^{30} - 9q^{32} - 9q^{33} - 6q^{34} - 2q^{35} - q^{36} + 12q^{37} + 20q^{38} - 11q^{39} - 9q^{40} + 2q^{41} + 8q^{42} + 10q^{43} - 6q^{44} + 3q^{45} - 16q^{46} - 5q^{47} + 24q^{48} + 2q^{49} - q^{50} + 11q^{51} + 4q^{52} - 2q^{53} + 12q^{54} + q^{55} + 9q^{56} - 14q^{57} - 26q^{58} - 8q^{59} + 6q^{60} + 6q^{61} - 3q^{63} + 7q^{64} + 5q^{65} - 4q^{66} + 4q^{67} - 4q^{68} + 18q^{69} + q^{70} + 16q^{71} - 5q^{72} - 8q^{73} - 6q^{74} - q^{75} - 32q^{76} - q^{77} - 20q^{78} - 9q^{79} + 3q^{80} - 14q^{81} - 18q^{82} + 8q^{83} - 6q^{84} - 5q^{85} + 12q^{86} + 25q^{87} + 4q^{88} + 6q^{89} + 7q^{90} - 5q^{91} + 12q^{92} + 28q^{94} - 6q^{95} - 4q^{96} - 9q^{97} - q^{98} + 10q^{99} + O(q^{100})$$

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
 2.56155 −1.56155
−2.56155 1.56155 4.56155 1.00000 −4.00000 −1.00000 −6.56155 −0.561553 −2.56155
1.2 1.56155 −2.56155 0.438447 1.00000 −4.00000 −1.00000 −2.43845 3.56155 1.56155
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$7$$ $$1$$

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 35.2.a.b 2
3.b odd 2 1 315.2.a.e 2
4.b odd 2 1 560.2.a.i 2
5.b even 2 1 175.2.a.f 2
5.c odd 4 2 175.2.b.b 4
7.b odd 2 1 245.2.a.d 2
7.c even 3 2 245.2.e.i 4
7.d odd 6 2 245.2.e.h 4
8.b even 2 1 2240.2.a.bh 2
8.d odd 2 1 2240.2.a.bd 2
11.b odd 2 1 4235.2.a.m 2
12.b even 2 1 5040.2.a.bt 2
13.b even 2 1 5915.2.a.l 2
15.d odd 2 1 1575.2.a.p 2
15.e even 4 2 1575.2.d.e 4
20.d odd 2 1 2800.2.a.bi 2
20.e even 4 2 2800.2.g.t 4
21.c even 2 1 2205.2.a.x 2
28.d even 2 1 3920.2.a.bs 2
35.c odd 2 1 1225.2.a.s 2
35.f even 4 2 1225.2.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 1.a even 1 1 trivial
175.2.a.f 2 5.b even 2 1
175.2.b.b 4 5.c odd 4 2
245.2.a.d 2 7.b odd 2 1
245.2.e.h 4 7.d odd 6 2
245.2.e.i 4 7.c even 3 2
315.2.a.e 2 3.b odd 2 1
560.2.a.i 2 4.b odd 2 1
1225.2.a.s 2 35.c odd 2 1
1225.2.b.f 4 35.f even 4 2
1575.2.a.p 2 15.d odd 2 1
1575.2.d.e 4 15.e even 4 2
2205.2.a.x 2 21.c even 2 1
2240.2.a.bd 2 8.d odd 2 1
2240.2.a.bh 2 8.b even 2 1
2800.2.a.bi 2 20.d odd 2 1
2800.2.g.t 4 20.e even 4 2
3920.2.a.bs 2 28.d even 2 1
4235.2.a.m 2 11.b odd 2 1
5040.2.a.bt 2 12.b even 2 1
5915.2.a.l 2 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(35))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T + T^{2}$$
$3$ $$-4 + T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-4 - T + T^{2}$$
$13$ $$2 - 5 T + T^{2}$$
$17$ $$2 + 5 T + T^{2}$$
$19$ $$-8 + 6 T + T^{2}$$
$23$ $$-16 + 2 T + T^{2}$$
$29$ $$-38 - T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$-16 - 2 T + T^{2}$$
$43$ $$8 - 10 T + T^{2}$$
$47$ $$-32 + 5 T + T^{2}$$
$53$ $$-16 + 2 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$-144 - 6 T + T^{2}$$
$67$ $$-64 - 4 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$-52 + 8 T + T^{2}$$
$79$ $$16 + 9 T + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$-8 - 6 T + T^{2}$$
$97$ $$-86 + 9 T + T^{2}$$
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