Properties

Label 735.2.a.h
Level $735$
Weight $2$
Character orbit 735.a
Self dual yes
Analytic conductor $5.869$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.86900454856\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 2 - 2 \beta ) q^{4} - q^{5} + ( -1 + \beta ) q^{6} + ( -6 + 2 \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 2 - 2 \beta ) q^{4} - q^{5} + ( -1 + \beta ) q^{6} + ( -6 + 2 \beta ) q^{8} + q^{9} + ( 1 - \beta ) q^{10} + ( -1 - \beta ) q^{11} + ( 2 - 2 \beta ) q^{12} + ( -4 - \beta ) q^{13} - q^{15} + ( 8 - 4 \beta ) q^{16} + ( -5 - \beta ) q^{17} + ( -1 + \beta ) q^{18} + ( -1 + 2 \beta ) q^{19} + ( -2 + 2 \beta ) q^{20} -2 q^{22} + ( -3 + \beta ) q^{23} + ( -6 + 2 \beta ) q^{24} + q^{25} + ( 1 - 3 \beta ) q^{26} + q^{27} + ( 1 + 3 \beta ) q^{29} + ( 1 - \beta ) q^{30} + ( -3 - 2 \beta ) q^{31} + ( -8 + 8 \beta ) q^{32} + ( -1 - \beta ) q^{33} + ( 2 - 4 \beta ) q^{34} + ( 2 - 2 \beta ) q^{36} + ( 2 + 3 \beta ) q^{37} + ( 7 - 3 \beta ) q^{38} + ( -4 - \beta ) q^{39} + ( 6 - 2 \beta ) q^{40} + ( -1 - \beta ) q^{41} + ( -2 - 3 \beta ) q^{43} + 4 q^{44} - q^{45} + ( 6 - 4 \beta ) q^{46} -2 q^{47} + ( 8 - 4 \beta ) q^{48} + ( -1 + \beta ) q^{50} + ( -5 - \beta ) q^{51} + ( -2 + 6 \beta ) q^{52} + ( 2 - 6 \beta ) q^{53} + ( -1 + \beta ) q^{54} + ( 1 + \beta ) q^{55} + ( -1 + 2 \beta ) q^{57} + ( 8 - 2 \beta ) q^{58} + ( -5 - 3 \beta ) q^{59} + ( -2 + 2 \beta ) q^{60} -4 q^{61} + ( -3 - \beta ) q^{62} + ( 16 - 8 \beta ) q^{64} + ( 4 + \beta ) q^{65} -2 q^{66} + ( -6 + 5 \beta ) q^{67} + ( -4 + 8 \beta ) q^{68} + ( -3 + \beta ) q^{69} + ( 1 - 3 \beta ) q^{71} + ( -6 + 2 \beta ) q^{72} + ( -4 + 5 \beta ) q^{73} + ( 7 - \beta ) q^{74} + q^{75} + ( -14 + 6 \beta ) q^{76} + ( 1 - 3 \beta ) q^{78} + ( 3 + 6 \beta ) q^{79} + ( -8 + 4 \beta ) q^{80} + q^{81} -2 q^{82} + ( -3 + 7 \beta ) q^{83} + ( 5 + \beta ) q^{85} + ( -7 + \beta ) q^{86} + ( 1 + 3 \beta ) q^{87} + 4 \beta q^{88} + ( -3 + 7 \beta ) q^{89} + ( 1 - \beta ) q^{90} + ( -12 + 8 \beta ) q^{92} + ( -3 - 2 \beta ) q^{93} + ( 2 - 2 \beta ) q^{94} + ( 1 - 2 \beta ) q^{95} + ( -8 + 8 \beta ) q^{96} + ( -8 + 4 \beta ) q^{97} + ( -1 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} - 12q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} - 12q^{8} + 2q^{9} + 2q^{10} - 2q^{11} + 4q^{12} - 8q^{13} - 2q^{15} + 16q^{16} - 10q^{17} - 2q^{18} - 2q^{19} - 4q^{20} - 4q^{22} - 6q^{23} - 12q^{24} + 2q^{25} + 2q^{26} + 2q^{27} + 2q^{29} + 2q^{30} - 6q^{31} - 16q^{32} - 2q^{33} + 4q^{34} + 4q^{36} + 4q^{37} + 14q^{38} - 8q^{39} + 12q^{40} - 2q^{41} - 4q^{43} + 8q^{44} - 2q^{45} + 12q^{46} - 4q^{47} + 16q^{48} - 2q^{50} - 10q^{51} - 4q^{52} + 4q^{53} - 2q^{54} + 2q^{55} - 2q^{57} + 16q^{58} - 10q^{59} - 4q^{60} - 8q^{61} - 6q^{62} + 32q^{64} + 8q^{65} - 4q^{66} - 12q^{67} - 8q^{68} - 6q^{69} + 2q^{71} - 12q^{72} - 8q^{73} + 14q^{74} + 2q^{75} - 28q^{76} + 2q^{78} + 6q^{79} - 16q^{80} + 2q^{81} - 4q^{82} - 6q^{83} + 10q^{85} - 14q^{86} + 2q^{87} - 6q^{89} + 2q^{90} - 24q^{92} - 6q^{93} + 4q^{94} + 2q^{95} - 16q^{96} - 16q^{97} - 2q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
−2.73205 1.00000 5.46410 −1.00000 −2.73205 0 −9.46410 1.00000 2.73205
1.2 0.732051 1.00000 −1.46410 −1.00000 0.732051 0 −2.53590 1.00000 −0.732051
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 735.2.a.h 2
3.b odd 2 1 2205.2.a.ba 2
5.b even 2 1 3675.2.a.be 2
7.b odd 2 1 735.2.a.g 2
7.c even 3 2 735.2.i.l 4
7.d odd 6 2 105.2.i.d 4
21.c even 2 1 2205.2.a.z 2
21.g even 6 2 315.2.j.c 4
28.f even 6 2 1680.2.bg.o 4
35.c odd 2 1 3675.2.a.bg 2
35.i odd 6 2 525.2.i.f 4
35.k even 12 2 525.2.r.a 4
35.k even 12 2 525.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 7.d odd 6 2
315.2.j.c 4 21.g even 6 2
525.2.i.f 4 35.i odd 6 2
525.2.r.a 4 35.k even 12 2
525.2.r.f 4 35.k even 12 2
735.2.a.g 2 7.b odd 2 1
735.2.a.h 2 1.a even 1 1 trivial
735.2.i.l 4 7.c even 3 2
1680.2.bg.o 4 28.f even 6 2
2205.2.a.z 2 21.c even 2 1
2205.2.a.ba 2 3.b odd 2 1
3675.2.a.be 2 5.b even 2 1
3675.2.a.bg 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2}^{2} + 2 T_{2} - 2 \)
\( T_{13}^{2} + 8 T_{13} + 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + 2 T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -2 + 2 T + T^{2} \)
$13$ \( 13 + 8 T + T^{2} \)
$17$ \( 22 + 10 T + T^{2} \)
$19$ \( -11 + 2 T + T^{2} \)
$23$ \( 6 + 6 T + T^{2} \)
$29$ \( -26 - 2 T + T^{2} \)
$31$ \( -3 + 6 T + T^{2} \)
$37$ \( -23 - 4 T + T^{2} \)
$41$ \( -2 + 2 T + T^{2} \)
$43$ \( -23 + 4 T + T^{2} \)
$47$ \( ( 2 + T )^{2} \)
$53$ \( -104 - 4 T + T^{2} \)
$59$ \( -2 + 10 T + T^{2} \)
$61$ \( ( 4 + T )^{2} \)
$67$ \( -39 + 12 T + T^{2} \)
$71$ \( -26 - 2 T + T^{2} \)
$73$ \( -59 + 8 T + T^{2} \)
$79$ \( -99 - 6 T + T^{2} \)
$83$ \( -138 + 6 T + T^{2} \)
$89$ \( -138 + 6 T + T^{2} \)
$97$ \( 16 + 16 T + T^{2} \)
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