Properties

Label 945.2.a.k
Level 945
Weight 2
Character orbit 945.a
Self dual yes
Analytic conductor 7.546
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} + q^{5} - q^{7} + ( 3 + \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} + q^{5} - q^{7} + ( 3 + \beta ) q^{8} + ( 1 + \beta ) q^{10} + ( -1 + \beta ) q^{11} + ( 1 + \beta ) q^{13} + ( -1 - \beta ) q^{14} + 3 q^{16} + ( 2 + 2 \beta ) q^{17} + ( 3 - \beta ) q^{19} + ( 1 + 2 \beta ) q^{20} + q^{22} + ( 2 - 2 \beta ) q^{23} + q^{25} + ( 3 + 2 \beta ) q^{26} + ( -1 - 2 \beta ) q^{28} -4 q^{29} + 6 q^{31} + ( -3 + \beta ) q^{32} + ( 6 + 4 \beta ) q^{34} - q^{35} -6 \beta q^{37} + ( 1 + 2 \beta ) q^{38} + ( 3 + \beta ) q^{40} + ( -5 - 2 \beta ) q^{41} + ( -1 - 4 \beta ) q^{43} + ( 3 - \beta ) q^{44} -2 q^{46} + ( -1 + 6 \beta ) q^{47} + q^{49} + ( 1 + \beta ) q^{50} + ( 5 + 3 \beta ) q^{52} + ( -1 - 7 \beta ) q^{53} + ( -1 + \beta ) q^{55} + ( -3 - \beta ) q^{56} + ( -4 - 4 \beta ) q^{58} + ( 2 - 6 \beta ) q^{59} -6 \beta q^{61} + ( 6 + 6 \beta ) q^{62} + ( -7 - 2 \beta ) q^{64} + ( 1 + \beta ) q^{65} + ( 5 + 2 \beta ) q^{67} + ( 10 + 6 \beta ) q^{68} + ( -1 - \beta ) q^{70} + 2 q^{71} + ( 5 - 5 \beta ) q^{73} + ( -12 - 6 \beta ) q^{74} + ( -1 + 5 \beta ) q^{76} + ( 1 - \beta ) q^{77} + ( 12 + 2 \beta ) q^{79} + 3 q^{80} + ( -9 - 7 \beta ) q^{82} + ( 1 - 6 \beta ) q^{83} + ( 2 + 2 \beta ) q^{85} + ( -9 - 5 \beta ) q^{86} + ( -1 + 2 \beta ) q^{88} + ( -3 - 4 \beta ) q^{89} + ( -1 - \beta ) q^{91} + ( -6 + 2 \beta ) q^{92} + ( 11 + 5 \beta ) q^{94} + ( 3 - \beta ) q^{95} + 10 \beta q^{97} + ( 1 + \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 6q^{8} + 2q^{10} - 2q^{11} + 2q^{13} - 2q^{14} + 6q^{16} + 4q^{17} + 6q^{19} + 2q^{20} + 2q^{22} + 4q^{23} + 2q^{25} + 6q^{26} - 2q^{28} - 8q^{29} + 12q^{31} - 6q^{32} + 12q^{34} - 2q^{35} + 2q^{38} + 6q^{40} - 10q^{41} - 2q^{43} + 6q^{44} - 4q^{46} - 2q^{47} + 2q^{49} + 2q^{50} + 10q^{52} - 2q^{53} - 2q^{55} - 6q^{56} - 8q^{58} + 4q^{59} + 12q^{62} - 14q^{64} + 2q^{65} + 10q^{67} + 20q^{68} - 2q^{70} + 4q^{71} + 10q^{73} - 24q^{74} - 2q^{76} + 2q^{77} + 24q^{79} + 6q^{80} - 18q^{82} + 2q^{83} + 4q^{85} - 18q^{86} - 2q^{88} - 6q^{89} - 2q^{91} - 12q^{92} + 22q^{94} + 6q^{95} + 2q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−0.414214 0 −1.82843 1.00000 0 −1.00000 1.58579 0 −0.414214
1.2 2.41421 0 3.82843 1.00000 0 −1.00000 4.41421 0 2.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 945.2.a.k yes 2
3.b odd 2 1 945.2.a.b 2
5.b even 2 1 4725.2.a.v 2
7.b odd 2 1 6615.2.a.w 2
15.d odd 2 1 4725.2.a.bg 2
21.c even 2 1 6615.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.a.b 2 3.b odd 2 1
945.2.a.k yes 2 1.a even 1 1 trivial
4725.2.a.v 2 5.b even 2 1
4725.2.a.bg 2 15.d odd 2 1
6615.2.a.l 2 21.c even 2 1
6615.2.a.w 2 7.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(945))\):

\( T_{2}^{2} - 2 T_{2} - 1 \)
\( T_{11}^{2} + 2 T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \)
$3$ 1
$5$ \( ( 1 - T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + 2 T + 21 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( 1 - 2 T + 25 T^{2} - 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 4 T + 30 T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( 1 - 6 T + 45 T^{2} - 114 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T + 42 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 6 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 2 T^{2} + 1369 T^{4} \)
$41$ \( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 2 T + 55 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 2 T + 23 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 2 T + 9 T^{2} + 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 4 T + 50 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 50 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 10 T + 151 T^{2} - 670 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 2 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 10 T + 121 T^{2} - 730 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 24 T + 294 T^{2} - 1896 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 2 T + 95 T^{2} - 166 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 6 T + 155 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 6 T^{2} + 9409 T^{4} \)
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