Properties

Label 605.2.b.d
Level $605$
Weight $2$
Character orbit 605.b
Analytic conductor $4.831$
Analytic rank $0$
Dimension $4$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} - q^{6} + ( \beta_{1} - 2 \beta_{3} ) q^{7} + \beta_{3} q^{8} + ( 1 - \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} - q^{6} + ( \beta_{1} - 2 \beta_{3} ) q^{7} + \beta_{3} q^{8} + ( 1 - \beta_{2} ) q^{9} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{10} + ( -\beta_{1} + 2 \beta_{3} ) q^{12} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{13} + \beta_{2} q^{14} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{15} + ( -1 + 2 \beta_{2} ) q^{16} + 2 \beta_{3} q^{17} + ( 3 \beta_{1} - \beta_{3} ) q^{18} + ( -1 + 3 \beta_{2} ) q^{19} + ( 3 - \beta_{1} - \beta_{3} ) q^{20} + ( 3 + 2 \beta_{2} ) q^{21} + ( -3 \beta_{1} - \beta_{3} ) q^{23} + ( -2 - \beta_{2} ) q^{24} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{25} + ( -3 + 3 \beta_{2} ) q^{26} + ( \beta_{1} + 2 \beta_{3} ) q^{27} -3 \beta_{3} q^{28} -4 \beta_{2} q^{29} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{30} + ( -7 - \beta_{2} ) q^{31} + ( -5 \beta_{1} + 4 \beta_{3} ) q^{32} -2 q^{34} + ( -3 - \beta_{2} - 3 \beta_{3} ) q^{35} + ( -3 + \beta_{2} ) q^{36} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{37} + ( -7 \beta_{1} + 3 \beta_{3} ) q^{38} + ( 3 + 3 \beta_{2} ) q^{39} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{40} + \beta_{2} q^{41} + ( -\beta_{1} + 2 \beta_{3} ) q^{42} + ( 5 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{45} + ( 7 - 3 \beta_{2} ) q^{46} + ( -4 \beta_{1} + 7 \beta_{3} ) q^{47} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{48} + ( 1 - 3 \beta_{2} ) q^{49} + ( 6 + \beta_{1} - 2 \beta_{2} ) q^{50} + ( -4 - 2 \beta_{2} ) q^{51} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{52} + ( 7 \beta_{1} - 3 \beta_{3} ) q^{53} + ( -4 + \beta_{2} ) q^{54} + ( 3 + 2 \beta_{2} ) q^{56} + ( -3 \beta_{1} + 5 \beta_{3} ) q^{57} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{58} + ( 3 - \beta_{2} ) q^{59} + ( 3 + \beta_{2} + 3 \beta_{3} ) q^{60} + ( -10 - \beta_{2} ) q^{61} + ( -5 \beta_{1} - \beta_{3} ) q^{62} + ( \beta_{1} + \beta_{3} ) q^{63} + ( 4 - \beta_{2} ) q^{64} + ( -6 - 3 \beta_{1} - 3 \beta_{3} ) q^{65} + ( 8 \beta_{1} - \beta_{3} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{68} + ( 5 + \beta_{2} ) q^{69} + ( 3 - \beta_{1} - \beta_{3} ) q^{70} + ( -3 - 3 \beta_{2} ) q^{71} + ( \beta_{1} - \beta_{3} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( 6 - 4 \beta_{2} ) q^{74} + ( 6 + 2 \beta_{2} + \beta_{3} ) q^{75} + ( 9 - \beta_{2} ) q^{76} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{78} + ( 2 - 2 \beta_{2} ) q^{79} + ( 6 - 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{80} + ( -2 - 5 \beta_{2} ) q^{81} + ( -2 \beta_{1} + \beta_{3} ) q^{82} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{83} + ( 6 + 3 \beta_{2} ) q^{84} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -12 + 5 \beta_{2} ) q^{86} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{87} + ( 3 - 2 \beta_{2} ) q^{89} + ( -4 - 5 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{90} + ( -9 - 3 \beta_{2} ) q^{91} + ( 7 \beta_{1} - 5 \beta_{3} ) q^{92} + ( \beta_{1} - 9 \beta_{3} ) q^{93} + ( 1 - 4 \beta_{2} ) q^{94} + ( 9 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{95} + ( -3 - 4 \beta_{2} ) q^{96} + ( \beta_{1} - 5 \beta_{3} ) q^{97} + ( 7 \beta_{1} - 3 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{6} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{6} + 4q^{9} - 4q^{10} + 4q^{15} - 4q^{16} - 4q^{19} + 12q^{20} + 12q^{21} - 8q^{24} + 4q^{25} - 12q^{26} - 28q^{31} - 8q^{34} - 12q^{35} - 12q^{36} + 12q^{39} + 4q^{40} - 12q^{45} + 28q^{46} + 4q^{49} + 24q^{50} - 16q^{51} - 16q^{54} + 12q^{56} + 12q^{59} + 12q^{60} - 40q^{61} + 16q^{64} - 24q^{65} + 20q^{69} + 12q^{70} - 12q^{71} + 24q^{74} + 24q^{75} + 36q^{76} + 8q^{79} + 24q^{80} - 8q^{81} + 24q^{84} + 8q^{85} - 48q^{86} + 12q^{89} - 16q^{90} - 36q^{91} + 4q^{94} + 36q^{95} - 12q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 4 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
1.93185i
0.517638i
0.517638i
1.93185i
1.93185i 0.517638i −1.73205 −1.73205 1.41421i −1.00000 0.896575i 0.517638i 2.73205 −2.73205 + 3.34607i
364.2 0.517638i 1.93185i 1.73205 1.73205 + 1.41421i −1.00000 3.34607i 1.93185i −0.732051 0.732051 0.896575i
364.3 0.517638i 1.93185i 1.73205 1.73205 1.41421i −1.00000 3.34607i 1.93185i −0.732051 0.732051 + 0.896575i
364.4 1.93185i 0.517638i −1.73205 −1.73205 + 1.41421i −1.00000 0.896575i 0.517638i 2.73205 −2.73205 3.34607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.b.d 4
5.b even 2 1 inner 605.2.b.d 4
5.c odd 4 2 3025.2.a.y 4
11.b odd 2 1 605.2.b.e yes 4
11.c even 5 4 605.2.j.f 16
11.d odd 10 4 605.2.j.e 16
55.d odd 2 1 605.2.b.e yes 4
55.e even 4 2 3025.2.a.z 4
55.h odd 10 4 605.2.j.e 16
55.j even 10 4 605.2.j.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.d 4 1.a even 1 1 trivial
605.2.b.d 4 5.b even 2 1 inner
605.2.b.e yes 4 11.b odd 2 1
605.2.b.e yes 4 55.d odd 2 1
605.2.j.e 16 11.d odd 10 4
605.2.j.e 16 55.h odd 10 4
605.2.j.f 16 11.c even 5 4
605.2.j.f 16 55.j even 10 4
3025.2.a.y 4 5.c odd 4 2
3025.2.a.z 4 55.e even 4 2

Hecke kernels

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

\( T_{2}^{4} + 4 T_{2}^{2} + 1 \)
\( T_{19}^{2} + 2 T_{19} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} + T^{4} \)
$3$ \( 1 + 4 T^{2} + T^{4} \)
$5$ \( 25 - 2 T^{2} + T^{4} \)
$7$ \( 9 + 12 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 18 + T^{2} )^{2} \)
$17$ \( 16 + 16 T^{2} + T^{4} \)
$19$ \( ( -26 + 2 T + T^{2} )^{2} \)
$23$ \( 484 + 52 T^{2} + T^{4} \)
$29$ \( ( -48 + T^{2} )^{2} \)
$31$ \( ( 46 + 14 T + T^{2} )^{2} \)
$37$ \( 144 + 48 T^{2} + T^{4} \)
$41$ \( ( -3 + T^{2} )^{2} \)
$43$ \( 4761 + 156 T^{2} + T^{4} \)
$47$ \( 2209 + 148 T^{2} + T^{4} \)
$53$ \( 676 + 148 T^{2} + T^{4} \)
$59$ \( ( 6 - 6 T + T^{2} )^{2} \)
$61$ \( ( 97 + 20 T + T^{2} )^{2} \)
$67$ \( 1089 + 228 T^{2} + T^{4} \)
$71$ \( ( -18 + 6 T + T^{2} )^{2} \)
$73$ \( ( 24 + T^{2} )^{2} \)
$79$ \( ( -8 - 4 T + T^{2} )^{2} \)
$83$ \( ( 98 + T^{2} )^{2} \)
$89$ \( ( -3 - 6 T + T^{2} )^{2} \)
$97$ \( 36 + 84 T^{2} + T^{4} \)
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