Properties

Label 345.2.a.i
Level $345$
Weight $2$
Character orbit 345.a
Self dual yes
Analytic conductor $2.755$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + 4 q^{4} - q^{5} + \beta q^{6} - q^{7} + 2 \beta q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} + 4 q^{4} - q^{5} + \beta q^{6} - q^{7} + 2 \beta q^{8} + q^{9} -\beta q^{10} -\beta q^{11} + 4 q^{12} + ( 2 - \beta ) q^{13} -\beta q^{14} - q^{15} + 4 q^{16} + ( -3 + \beta ) q^{17} + \beta q^{18} + ( 2 - \beta ) q^{19} -4 q^{20} - q^{21} -6 q^{22} - q^{23} + 2 \beta q^{24} + q^{25} + ( -6 + 2 \beta ) q^{26} + q^{27} -4 q^{28} + ( 3 - 3 \beta ) q^{29} -\beta q^{30} + ( 5 + 2 \beta ) q^{31} -\beta q^{33} + ( 6 - 3 \beta ) q^{34} + q^{35} + 4 q^{36} + ( -1 - 2 \beta ) q^{37} + ( -6 + 2 \beta ) q^{38} + ( 2 - \beta ) q^{39} -2 \beta q^{40} + ( 3 - \beta ) q^{41} -\beta q^{42} + 2 q^{43} -4 \beta q^{44} - q^{45} -\beta q^{46} + ( 6 - \beta ) q^{47} + 4 q^{48} -6 q^{49} + \beta q^{50} + ( -3 + \beta ) q^{51} + ( 8 - 4 \beta ) q^{52} + ( 3 + \beta ) q^{53} + \beta q^{54} + \beta q^{55} -2 \beta q^{56} + ( 2 - \beta ) q^{57} + ( -18 + 3 \beta ) q^{58} + ( 3 - 3 \beta ) q^{59} -4 q^{60} + ( 8 + 3 \beta ) q^{61} + ( 12 + 5 \beta ) q^{62} - q^{63} -8 q^{64} + ( -2 + \beta ) q^{65} -6 q^{66} -7 q^{67} + ( -12 + 4 \beta ) q^{68} - q^{69} + \beta q^{70} + ( 3 + 3 \beta ) q^{71} + 2 \beta q^{72} + ( 2 + 3 \beta ) q^{73} + ( -12 - \beta ) q^{74} + q^{75} + ( 8 - 4 \beta ) q^{76} + \beta q^{77} + ( -6 + 2 \beta ) q^{78} -4 q^{79} -4 q^{80} + q^{81} + ( -6 + 3 \beta ) q^{82} + ( 3 - 5 \beta ) q^{83} -4 q^{84} + ( 3 - \beta ) q^{85} + 2 \beta q^{86} + ( 3 - 3 \beta ) q^{87} -12 q^{88} + ( -12 + 2 \beta ) q^{89} -\beta q^{90} + ( -2 + \beta ) q^{91} -4 q^{92} + ( 5 + 2 \beta ) q^{93} + ( -6 + 6 \beta ) q^{94} + ( -2 + \beta ) q^{95} + ( 8 + 2 \beta ) q^{97} -6 \beta q^{98} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 8q^{4} - 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 8q^{4} - 2q^{5} - 2q^{7} + 2q^{9} + 8q^{12} + 4q^{13} - 2q^{15} + 8q^{16} - 6q^{17} + 4q^{19} - 8q^{20} - 2q^{21} - 12q^{22} - 2q^{23} + 2q^{25} - 12q^{26} + 2q^{27} - 8q^{28} + 6q^{29} + 10q^{31} + 12q^{34} + 2q^{35} + 8q^{36} - 2q^{37} - 12q^{38} + 4q^{39} + 6q^{41} + 4q^{43} - 2q^{45} + 12q^{47} + 8q^{48} - 12q^{49} - 6q^{51} + 16q^{52} + 6q^{53} + 4q^{57} - 36q^{58} + 6q^{59} - 8q^{60} + 16q^{61} + 24q^{62} - 2q^{63} - 16q^{64} - 4q^{65} - 12q^{66} - 14q^{67} - 24q^{68} - 2q^{69} + 6q^{71} + 4q^{73} - 24q^{74} + 2q^{75} + 16q^{76} - 12q^{78} - 8q^{79} - 8q^{80} + 2q^{81} - 12q^{82} + 6q^{83} - 8q^{84} + 6q^{85} + 6q^{87} - 24q^{88} - 24q^{89} - 4q^{91} - 8q^{92} + 10q^{93} - 12q^{94} - 4q^{95} + 16q^{97} + O(q^{100}) \)

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
−2.44949
2.44949
−2.44949 1.00000 4.00000 −1.00000 −2.44949 −1.00000 −4.89898 1.00000 2.44949
1.2 2.44949 1.00000 4.00000 −1.00000 2.44949 −1.00000 4.89898 1.00000 −2.44949
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

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 345.2.a.i 2
3.b odd 2 1 1035.2.a.k 2
4.b odd 2 1 5520.2.a.bi 2
5.b even 2 1 1725.2.a.y 2
5.c odd 4 2 1725.2.b.m 4
15.d odd 2 1 5175.2.a.bl 2
23.b odd 2 1 7935.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.i 2 1.a even 1 1 trivial
1035.2.a.k 2 3.b odd 2 1
1725.2.a.y 2 5.b even 2 1
1725.2.b.m 4 5.c odd 4 2
5175.2.a.bl 2 15.d odd 2 1
5520.2.a.bi 2 4.b odd 2 1
7935.2.a.t 2 23.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(345))\):

\( T_{2}^{2} - 6 \)
\( T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -6 + T^{2} \)
$13$ \( -2 - 4 T + T^{2} \)
$17$ \( 3 + 6 T + T^{2} \)
$19$ \( -2 - 4 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -45 - 6 T + T^{2} \)
$31$ \( 1 - 10 T + T^{2} \)
$37$ \( -23 + 2 T + T^{2} \)
$41$ \( 3 - 6 T + T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 30 - 12 T + T^{2} \)
$53$ \( 3 - 6 T + T^{2} \)
$59$ \( -45 - 6 T + T^{2} \)
$61$ \( 10 - 16 T + T^{2} \)
$67$ \( ( 7 + T )^{2} \)
$71$ \( -45 - 6 T + T^{2} \)
$73$ \( -50 - 4 T + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -141 - 6 T + T^{2} \)
$89$ \( 120 + 24 T + T^{2} \)
$97$ \( 40 - 16 T + T^{2} \)
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