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: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{5} - q^{6} + ( - 2 \beta_{3} + \beta_1) q^{7} + \beta_{3} q^{8} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{5} - q^{6} + ( - 2 \beta_{3} + \beta_1) q^{7} + \beta_{3} q^{8} + ( - \beta_{2} + 1) q^{9} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{10} + (2 \beta_{3} - \beta_1) q^{12} + ( - 3 \beta_{3} + 3 \beta_1) q^{13} + \beta_{2} q^{14} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{15} + (2 \beta_{2} - 1) q^{16} + 2 \beta_{3} q^{17} + ( - \beta_{3} + 3 \beta_1) q^{18} + (3 \beta_{2} - 1) q^{19} + ( - \beta_{3} - \beta_1 + 3) q^{20} + (2 \beta_{2} + 3) q^{21} + ( - \beta_{3} - 3 \beta_1) q^{23} + ( - \beta_{2} - 2) q^{24} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{25} + (3 \beta_{2} - 3) q^{26} + (2 \beta_{3} + \beta_1) q^{27} - 3 \beta_{3} q^{28} - 4 \beta_{2} q^{29} + (\beta_{3} - \beta_{2} - \beta_1) q^{30} + ( - \beta_{2} - 7) q^{31} + (4 \beta_{3} - 5 \beta_1) q^{32} - 2 q^{34} + ( - 3 \beta_{3} - \beta_{2} - 3) q^{35} + (\beta_{2} - 3) q^{36} + (2 \beta_{3} - 4 \beta_1) q^{37} + (3 \beta_{3} - 7 \beta_1) q^{38} + (3 \beta_{2} + 3) q^{39} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{40} + \beta_{2} q^{41} + (2 \beta_{3} - \beta_1) q^{42} + (2 \beta_{3} + 5 \beta_1) q^{43} + (\beta_{2} + 2 \beta_1 - 3) q^{45} + ( - 3 \beta_{2} + 7) q^{46} + (7 \beta_{3} - 4 \beta_1) q^{47} + (3 \beta_{3} - 2 \beta_1) q^{48} + ( - 3 \beta_{2} + 1) q^{49} + ( - 2 \beta_{2} + \beta_1 + 6) q^{50} + ( - 2 \beta_{2} - 4) q^{51} + ( - 3 \beta_{3} - 3 \beta_1) q^{52} + ( - 3 \beta_{3} + 7 \beta_1) q^{53} + (\beta_{2} - 4) q^{54} + (2 \beta_{2} + 3) q^{56} + (5 \beta_{3} - 3 \beta_1) q^{57} + ( - 4 \beta_{3} + 8 \beta_1) q^{58} + ( - \beta_{2} + 3) q^{59} + (3 \beta_{3} + \beta_{2} + 3) q^{60} + ( - \beta_{2} - 10) q^{61} + ( - \beta_{3} - 5 \beta_1) q^{62} + (\beta_{3} + \beta_1) q^{63} + ( - \beta_{2} + 4) q^{64} + ( - 3 \beta_{3} - 3 \beta_1 - 6) q^{65} + ( - \beta_{3} + 8 \beta_1) q^{67} + (4 \beta_{3} - 2 \beta_1) q^{68} + (\beta_{2} + 5) q^{69} + ( - \beta_{3} - \beta_1 + 3) q^{70} + ( - 3 \beta_{2} - 3) q^{71} + ( - \beta_{3} + \beta_1) q^{72} + (2 \beta_{3} + 2 \beta_1) q^{73} + ( - 4 \beta_{2} + 6) q^{74} + (\beta_{3} + 2 \beta_{2} + 6) q^{75} + ( - \beta_{2} + 9) q^{76} + (3 \beta_{3} - 3 \beta_1) q^{78} + ( - 2 \beta_{2} + 2) q^{79} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 6) q^{80} + ( - 5 \beta_{2} - 2) q^{81} + (\beta_{3} - 2 \beta_1) q^{82} + (7 \beta_{3} - 7 \beta_1) q^{83} + (3 \beta_{2} + 6) q^{84} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{85} + (5 \beta_{2} - 12) q^{86} + ( - 8 \beta_{3} + 4 \beta_1) q^{87} + ( - 2 \beta_{2} + 3) q^{89} + (\beta_{3} + 2 \beta_{2} - 5 \beta_1 - 4) q^{90} + ( - 3 \beta_{2} - 9) q^{91} + ( - 5 \beta_{3} + 7 \beta_1) q^{92} + ( - 9 \beta_{3} + \beta_1) q^{93} + ( - 4 \beta_{2} + 1) q^{94} + ( - 2 \beta_{3} - \beta_{2} - 4 \beta_1 + 9) q^{95} + ( - 4 \beta_{2} - 3) q^{96} + ( - 5 \beta_{3} + \beta_1) q^{97} + ( - 3 \beta_{3} + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{6} + 4 q^{9} - 4 q^{10} + 4 q^{15} - 4 q^{16} - 4 q^{19} + 12 q^{20} + 12 q^{21} - 8 q^{24} + 4 q^{25} - 12 q^{26} - 28 q^{31} - 8 q^{34} - 12 q^{35} - 12 q^{36} + 12 q^{39} + 4 q^{40} - 12 q^{45} + 28 q^{46} + 4 q^{49} + 24 q^{50} - 16 q^{51} - 16 q^{54} + 12 q^{56} + 12 q^{59} + 12 q^{60} - 40 q^{61} + 16 q^{64} - 24 q^{65} + 20 q^{69} + 12 q^{70} - 12 q^{71} + 24 q^{74} + 24 q^{75} + 36 q^{76} + 8 q^{79} + 24 q^{80} - 8 q^{81} + 24 q^{84} + 8 q^{85} - 48 q^{86} + 12 q^{89} - 16 q^{90} - 36 q^{91} + 4 q^{94} + 36 q^{95} - 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{4} + 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

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