Properties

Label 1617.2.a.o
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + (\beta + 3) q^{4} - 3 q^{5} - \beta q^{6} + ( - 2 \beta - 5) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + (\beta + 3) q^{4} - 3 q^{5} - \beta q^{6} + ( - 2 \beta - 5) q^{8} + q^{9} + 3 \beta q^{10} - q^{11} + (\beta + 3) q^{12} - q^{13} - 3 q^{15} + (5 \beta + 4) q^{16} + (2 \beta - 4) q^{17} - \beta q^{18} + ( - 2 \beta + 3) q^{19} + ( - 3 \beta - 9) q^{20} + \beta q^{22} + (2 \beta - 2) q^{23} + ( - 2 \beta - 5) q^{24} + 4 q^{25} + \beta q^{26} + q^{27} + (4 \beta - 1) q^{29} + 3 \beta q^{30} + 2 \beta q^{31} + ( - 5 \beta - 15) q^{32} - q^{33} + (2 \beta - 10) q^{34} + (\beta + 3) q^{36} + q^{37} + ( - \beta + 10) q^{38} - q^{39} + (6 \beta + 15) q^{40} + ( - 4 \beta + 4) q^{41} + ( - 2 \beta - 2) q^{43} + ( - \beta - 3) q^{44} - 3 q^{45} - 10 q^{46} + ( - 2 \beta - 5) q^{47} + (5 \beta + 4) q^{48} - 4 \beta q^{50} + (2 \beta - 4) q^{51} + ( - \beta - 3) q^{52} + (2 \beta - 6) q^{53} - \beta q^{54} + 3 q^{55} + ( - 2 \beta + 3) q^{57} + ( - 3 \beta - 20) q^{58} + (2 \beta - 1) q^{59} + ( - 3 \beta - 9) q^{60} - 10 q^{61} + ( - 2 \beta - 10) q^{62} + (10 \beta + 17) q^{64} + 3 q^{65} + \beta q^{66} + ( - 2 \beta + 5) q^{67} + (4 \beta - 2) q^{68} + (2 \beta - 2) q^{69} + ( - 4 \beta + 4) q^{71} + ( - 2 \beta - 5) q^{72} - 7 q^{73} - \beta q^{74} + 4 q^{75} + ( - 5 \beta - 1) q^{76} + \beta q^{78} - 4 \beta q^{79} + ( - 15 \beta - 12) q^{80} + q^{81} + 20 q^{82} + ( - 2 \beta + 8) q^{83} + ( - 6 \beta + 12) q^{85} + (4 \beta + 10) q^{86} + (4 \beta - 1) q^{87} + (2 \beta + 5) q^{88} + (4 \beta - 2) q^{89} + 3 \beta q^{90} + (6 \beta + 4) q^{92} + 2 \beta q^{93} + (7 \beta + 10) q^{94} + (6 \beta - 9) q^{95} + ( - 5 \beta - 15) q^{96} + (2 \beta + 6) q^{97} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} + 7 q^{4} - 6 q^{5} - q^{6} - 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} + 7 q^{4} - 6 q^{5} - q^{6} - 12 q^{8} + 2 q^{9} + 3 q^{10} - 2 q^{11} + 7 q^{12} - 2 q^{13} - 6 q^{15} + 13 q^{16} - 6 q^{17} - q^{18} + 4 q^{19} - 21 q^{20} + q^{22} - 2 q^{23} - 12 q^{24} + 8 q^{25} + q^{26} + 2 q^{27} + 2 q^{29} + 3 q^{30} + 2 q^{31} - 35 q^{32} - 2 q^{33} - 18 q^{34} + 7 q^{36} + 2 q^{37} + 19 q^{38} - 2 q^{39} + 36 q^{40} + 4 q^{41} - 6 q^{43} - 7 q^{44} - 6 q^{45} - 20 q^{46} - 12 q^{47} + 13 q^{48} - 4 q^{50} - 6 q^{51} - 7 q^{52} - 10 q^{53} - q^{54} + 6 q^{55} + 4 q^{57} - 43 q^{58} - 21 q^{60} - 20 q^{61} - 22 q^{62} + 44 q^{64} + 6 q^{65} + q^{66} + 8 q^{67} - 2 q^{69} + 4 q^{71} - 12 q^{72} - 14 q^{73} - q^{74} + 8 q^{75} - 7 q^{76} + q^{78} - 4 q^{79} - 39 q^{80} + 2 q^{81} + 40 q^{82} + 14 q^{83} + 18 q^{85} + 24 q^{86} + 2 q^{87} + 12 q^{88} + 3 q^{90} + 14 q^{92} + 2 q^{93} + 27 q^{94} - 12 q^{95} - 35 q^{96} + 14 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
2.79129
−1.79129
−2.79129 1.00000 5.79129 −3.00000 −2.79129 0 −10.5826 1.00000 8.37386
1.2 1.79129 1.00000 1.20871 −3.00000 1.79129 0 −1.41742 1.00000 −5.37386
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 1617.2.a.o 2
3.b odd 2 1 4851.2.a.ba 2
7.b odd 2 1 231.2.a.b 2
21.c even 2 1 693.2.a.j 2
28.d even 2 1 3696.2.a.bl 2
35.c odd 2 1 5775.2.a.bn 2
77.b even 2 1 2541.2.a.z 2
231.h odd 2 1 7623.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 7.b odd 2 1
693.2.a.j 2 21.c even 2 1
1617.2.a.o 2 1.a even 1 1 trivial
2541.2.a.z 2 77.b even 2 1
3696.2.a.bl 2 28.d even 2 1
4851.2.a.ba 2 3.b odd 2 1
5775.2.a.bn 2 35.c odd 2 1
7623.2.a.bf 2 231.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{2} + T_{2} - 5 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 17 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 20 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 83 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 20 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 80 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 21 \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 5 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 80 \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 80 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 28 \) Copy content Toggle raw display
$89$ \( T^{2} - 84 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 28 \) Copy content Toggle raw display
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