Properties

Label 605.2.a.e
Level $605$
Weight $2$
Character orbit 605.a
Self dual yes
Analytic conductor $4.831$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.83094932229\)
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: yes
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.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + q^{4} - q^{5} -\beta q^{6} -\beta q^{7} -\beta q^{8} -2 q^{9} +O(q^{10})\) \( q + \beta q^{2} - q^{3} + q^{4} - q^{5} -\beta q^{6} -\beta q^{7} -\beta q^{8} -2 q^{9} -\beta q^{10} - q^{12} + 2 \beta q^{13} -3 q^{14} + q^{15} -5 q^{16} -4 \beta q^{17} -2 \beta q^{18} -2 \beta q^{19} - q^{20} + \beta q^{21} + \beta q^{24} + q^{25} + 6 q^{26} + 5 q^{27} -\beta q^{28} + \beta q^{30} -8 q^{31} -3 \beta q^{32} -12 q^{34} + \beta q^{35} -2 q^{36} -8 q^{37} -6 q^{38} -2 \beta q^{39} + \beta q^{40} + 7 \beta q^{41} + 3 q^{42} + 5 \beta q^{43} + 2 q^{45} + 9 q^{47} + 5 q^{48} -4 q^{49} + \beta q^{50} + 4 \beta q^{51} + 2 \beta q^{52} + 6 q^{53} + 5 \beta q^{54} + 3 q^{56} + 2 \beta q^{57} -12 q^{59} + q^{60} + 5 \beta q^{61} -8 \beta q^{62} + 2 \beta q^{63} + q^{64} -2 \beta q^{65} -5 q^{67} -4 \beta q^{68} + 3 q^{70} -12 q^{71} + 2 \beta q^{72} -8 \beta q^{74} - q^{75} -2 \beta q^{76} -6 q^{78} -6 \beta q^{79} + 5 q^{80} + q^{81} + 21 q^{82} + 2 \beta q^{83} + \beta q^{84} + 4 \beta q^{85} + 15 q^{86} + 3 q^{89} + 2 \beta q^{90} -6 q^{91} + 8 q^{93} + 9 \beta q^{94} + 2 \beta q^{95} + 3 \beta q^{96} -10 q^{97} -4 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{4} - 2q^{5} - 4q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{4} - 2q^{5} - 4q^{9} - 2q^{12} - 6q^{14} + 2q^{15} - 10q^{16} - 2q^{20} + 2q^{25} + 12q^{26} + 10q^{27} - 16q^{31} - 24q^{34} - 4q^{36} - 16q^{37} - 12q^{38} + 6q^{42} + 4q^{45} + 18q^{47} + 10q^{48} - 8q^{49} + 12q^{53} + 6q^{56} - 24q^{59} + 2q^{60} + 2q^{64} - 10q^{67} + 6q^{70} - 24q^{71} - 2q^{75} - 12q^{78} + 10q^{80} + 2q^{81} + 42q^{82} + 30q^{86} + 6q^{89} - 12q^{91} + 16q^{93} - 20q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
−1.73205 −1.00000 1.00000 −1.00000 1.73205 1.73205 1.73205 −2.00000 1.73205
1.2 1.73205 −1.00000 1.00000 −1.00000 −1.73205 −1.73205 −1.73205 −2.00000 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.a.e 2
3.b odd 2 1 5445.2.a.u 2
4.b odd 2 1 9680.2.a.bu 2
5.b even 2 1 3025.2.a.l 2
11.b odd 2 1 inner 605.2.a.e 2
11.c even 5 4 605.2.g.i 8
11.d odd 10 4 605.2.g.i 8
33.d even 2 1 5445.2.a.u 2
44.c even 2 1 9680.2.a.bu 2
55.d odd 2 1 3025.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.e 2 1.a even 1 1 trivial
605.2.a.e 2 11.b odd 2 1 inner
605.2.g.i 8 11.c even 5 4
605.2.g.i 8 11.d odd 10 4
3025.2.a.l 2 5.b even 2 1
3025.2.a.l 2 55.d odd 2 1
5445.2.a.u 2 3.b odd 2 1
5445.2.a.u 2 33.d even 2 1
9680.2.a.bu 2 4.b odd 2 1
9680.2.a.bu 2 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2}^{2} - 3 \)
\( T_{3} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -3 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( -48 + T^{2} \)
$19$ \( -12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( -147 + T^{2} \)
$43$ \( -75 + T^{2} \)
$47$ \( ( -9 + T )^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( -75 + T^{2} \)
$67$ \( ( 5 + T )^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( -108 + T^{2} \)
$83$ \( -12 + T^{2} \)
$89$ \( ( -3 + T )^{2} \)
$97$ \( ( 10 + T )^{2} \)
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