Properties

Label 1725.2.b.m
Level $1725$
Weight $2$
Character orbit 1725.b
Analytic conductor $13.774$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1725.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7741943487\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 345)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} -\beta_{1} q^{3} -4 q^{4} + \beta_{3} q^{6} -\beta_{1} q^{7} -2 \beta_{2} q^{8} - q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} -\beta_{1} q^{3} -4 q^{4} + \beta_{3} q^{6} -\beta_{1} q^{7} -2 \beta_{2} q^{8} - q^{9} -\beta_{3} q^{11} + 4 \beta_{1} q^{12} + ( -2 \beta_{1} + \beta_{2} ) q^{13} + \beta_{3} q^{14} + 4 q^{16} + ( -3 \beta_{1} + \beta_{2} ) q^{17} -\beta_{2} q^{18} + ( -2 + \beta_{3} ) q^{19} - q^{21} -6 \beta_{1} q^{22} + \beta_{1} q^{23} -2 \beta_{3} q^{24} + ( -6 + 2 \beta_{3} ) q^{26} + \beta_{1} q^{27} + 4 \beta_{1} q^{28} + ( -3 + 3 \beta_{3} ) q^{29} + ( 5 + 2 \beta_{3} ) q^{31} + \beta_{2} q^{33} + ( -6 + 3 \beta_{3} ) q^{34} + 4 q^{36} + ( -\beta_{1} - 2 \beta_{2} ) q^{37} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{38} + ( -2 + \beta_{3} ) q^{39} + ( 3 - \beta_{3} ) q^{41} -\beta_{2} q^{42} -2 \beta_{1} q^{43} + 4 \beta_{3} q^{44} -\beta_{3} q^{46} + ( 6 \beta_{1} - \beta_{2} ) q^{47} -4 \beta_{1} q^{48} + 6 q^{49} + ( -3 + \beta_{3} ) q^{51} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{52} + ( -3 \beta_{1} - \beta_{2} ) q^{53} -\beta_{3} q^{54} -2 \beta_{3} q^{56} + ( 2 \beta_{1} - \beta_{2} ) q^{57} + ( 18 \beta_{1} - 3 \beta_{2} ) q^{58} + ( -3 + 3 \beta_{3} ) q^{59} + ( 8 + 3 \beta_{3} ) q^{61} + ( 12 \beta_{1} + 5 \beta_{2} ) q^{62} + \beta_{1} q^{63} + 8 q^{64} -6 q^{66} -7 \beta_{1} q^{67} + ( 12 \beta_{1} - 4 \beta_{2} ) q^{68} + q^{69} + ( 3 + 3 \beta_{3} ) q^{71} + 2 \beta_{2} q^{72} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 12 + \beta_{3} ) q^{74} + ( 8 - 4 \beta_{3} ) q^{76} + \beta_{2} q^{77} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{78} + 4 q^{79} + q^{81} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{82} + ( -3 \beta_{1} + 5 \beta_{2} ) q^{83} + 4 q^{84} + 2 \beta_{3} q^{86} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{87} + 12 \beta_{1} q^{88} + ( 12 - 2 \beta_{3} ) q^{89} + ( -2 + \beta_{3} ) q^{91} -4 \beta_{1} q^{92} + ( -5 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 6 - 6 \beta_{3} ) q^{94} + ( 8 \beta_{1} + 2 \beta_{2} ) q^{97} + 6 \beta_{2} q^{98} + \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{4} - 4q^{9} + O(q^{10}) \) \( 4q - 16q^{4} - 4q^{9} + 16q^{16} - 8q^{19} - 4q^{21} - 24q^{26} - 12q^{29} + 20q^{31} - 24q^{34} + 16q^{36} - 8q^{39} + 12q^{41} + 24q^{49} - 12q^{51} - 12q^{59} + 32q^{61} + 32q^{64} - 24q^{66} + 4q^{69} + 12q^{71} + 48q^{74} + 32q^{76} + 16q^{79} + 4q^{81} + 16q^{84} + 48q^{89} - 8q^{91} + 24q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1174.1
−1.22474 1.22474i
1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
2.44949i 1.00000i −4.00000 0 −2.44949 1.00000i 4.89898i −1.00000 0
1174.2 2.44949i 1.00000i −4.00000 0 2.44949 1.00000i 4.89898i −1.00000 0
1174.3 2.44949i 1.00000i −4.00000 0 2.44949 1.00000i 4.89898i −1.00000 0
1174.4 2.44949i 1.00000i −4.00000 0 −2.44949 1.00000i 4.89898i −1.00000 0
\(n\): 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 1725.2.b.m 4
5.b even 2 1 inner 1725.2.b.m 4
5.c odd 4 1 345.2.a.i 2
5.c odd 4 1 1725.2.a.y 2
15.e even 4 1 1035.2.a.k 2
15.e even 4 1 5175.2.a.bl 2
20.e even 4 1 5520.2.a.bi 2
115.e even 4 1 7935.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.i 2 5.c odd 4 1
1035.2.a.k 2 15.e even 4 1
1725.2.a.y 2 5.c odd 4 1
1725.2.b.m 4 1.a even 1 1 trivial
1725.2.b.m 4 5.b even 2 1 inner
5175.2.a.bl 2 15.e even 4 1
5520.2.a.bi 2 20.e even 4 1
7935.2.a.t 2 115.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1725, [\chi])\):

\( T_{2}^{2} + 6 \)
\( T_{7}^{2} + 1 \)
\( T_{11}^{2} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 6 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -6 + T^{2} )^{2} \)
$13$ \( 4 + 20 T^{2} + T^{4} \)
$17$ \( 9 + 30 T^{2} + T^{4} \)
$19$ \( ( -2 + 4 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -45 + 6 T + T^{2} )^{2} \)
$31$ \( ( 1 - 10 T + T^{2} )^{2} \)
$37$ \( 529 + 50 T^{2} + T^{4} \)
$41$ \( ( 3 - 6 T + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( 900 + 84 T^{2} + T^{4} \)
$53$ \( 9 + 30 T^{2} + T^{4} \)
$59$ \( ( -45 + 6 T + T^{2} )^{2} \)
$61$ \( ( 10 - 16 T + T^{2} )^{2} \)
$67$ \( ( 49 + T^{2} )^{2} \)
$71$ \( ( -45 - 6 T + T^{2} )^{2} \)
$73$ \( 2500 + 116 T^{2} + T^{4} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( 19881 + 318 T^{2} + T^{4} \)
$89$ \( ( 120 - 24 T + T^{2} )^{2} \)
$97$ \( 1600 + 176 T^{2} + T^{4} \)
show more
show less