Properties

Label 1425.2.c.i
Level $1425$
Weight $2$
Character orbit 1425.c
Analytic conductor $11.379$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 285)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_1 q^{3} - 5 q^{4} + \beta_{2} q^{6} + ( - \beta_{3} + \beta_1) q^{7} - 3 \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_1 q^{3} - 5 q^{4} + \beta_{2} q^{6} + ( - \beta_{3} + \beta_1) q^{7} - 3 \beta_{3} q^{8} - q^{9} + ( - \beta_{2} + 3) q^{11} + 5 \beta_1 q^{12} + (\beta_{3} - 3 \beta_1) q^{13} + ( - \beta_{2} + 7) q^{14} + 11 q^{16} + 4 \beta_1 q^{17} - \beta_{3} q^{18} + q^{19} + ( - \beta_{2} + 1) q^{21} + (3 \beta_{3} - 7 \beta_1) q^{22} + (2 \beta_{3} + 4 \beta_1) q^{23} - 3 \beta_{2} q^{24} + (3 \beta_{2} - 7) q^{26} + \beta_1 q^{27} + (5 \beta_{3} - 5 \beta_1) q^{28} + ( - 3 \beta_{2} - 1) q^{29} + 6 q^{31} + 5 \beta_{3} q^{32} + (\beta_{3} - 3 \beta_1) q^{33} - 4 \beta_{2} q^{34} + 5 q^{36} + ( - \beta_{3} - \beta_1) q^{37} + \beta_{3} q^{38} + (\beta_{2} - 3) q^{39} + ( - \beta_{2} - 7) q^{41} + (\beta_{3} - 7 \beta_1) q^{42} + (\beta_{3} + 3 \beta_1) q^{43} + (5 \beta_{2} - 15) q^{44} + ( - 4 \beta_{2} - 14) q^{46} + (2 \beta_{3} - 4 \beta_1) q^{47} - 11 \beta_1 q^{48} + (2 \beta_{2} - 1) q^{49} + 4 q^{51} + ( - 5 \beta_{3} + 15 \beta_1) q^{52} + (2 \beta_{3} + 6 \beta_1) q^{53} - \beta_{2} q^{54} + (3 \beta_{2} - 21) q^{56} - \beta_1 q^{57} + ( - \beta_{3} - 21 \beta_1) q^{58} + ( - 2 \beta_{2} - 6) q^{59} + ( - 2 \beta_{2} - 6) q^{61} + 6 \beta_{3} q^{62} + (\beta_{3} - \beta_1) q^{63} - 13 q^{64} + (3 \beta_{2} - 7) q^{66} + (4 \beta_{3} - 4 \beta_1) q^{67} - 20 \beta_1 q^{68} + (2 \beta_{2} + 4) q^{69} + (2 \beta_{2} + 2) q^{71} + 3 \beta_{3} q^{72} + 10 \beta_1 q^{73} + (\beta_{2} + 7) q^{74} - 5 q^{76} + ( - 4 \beta_{3} + 10 \beta_1) q^{77} + ( - 3 \beta_{3} + 7 \beta_1) q^{78} + ( - 4 \beta_{2} + 4) q^{79} + q^{81} + ( - 7 \beta_{3} - 7 \beta_1) q^{82} + 6 \beta_1 q^{83} + (5 \beta_{2} - 5) q^{84} + ( - 3 \beta_{2} - 7) q^{86} + (3 \beta_{3} + \beta_1) q^{87} + ( - 9 \beta_{3} + 21 \beta_1) q^{88} + (3 \beta_{2} + 9) q^{89} + ( - 4 \beta_{2} + 10) q^{91} + ( - 10 \beta_{3} - 20 \beta_1) q^{92} - 6 \beta_1 q^{93} + (4 \beta_{2} - 14) q^{94} + 5 \beta_{2} q^{96} + (3 \beta_{3} - 5 \beta_1) q^{97} + ( - \beta_{3} + 14 \beta_1) q^{98} + (\beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{4} - 4 q^{9} + 12 q^{11} + 28 q^{14} + 44 q^{16} + 4 q^{19} + 4 q^{21} - 28 q^{26} - 4 q^{29} + 24 q^{31} + 20 q^{36} - 12 q^{39} - 28 q^{41} - 60 q^{44} - 56 q^{46} - 4 q^{49} + 16 q^{51} - 84 q^{56} - 24 q^{59} - 24 q^{61} - 52 q^{64} - 28 q^{66} + 16 q^{69} + 8 q^{71} + 28 q^{74} - 20 q^{76} + 16 q^{79} + 4 q^{81} - 20 q^{84} - 28 q^{86} + 36 q^{89} + 40 q^{91} - 56 q^{94} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 3x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
−1.32288 + 0.500000i
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i
2.64575i 1.00000i −5.00000 0 −2.64575 3.64575i 7.93725i −1.00000 0
799.2 2.64575i 1.00000i −5.00000 0 2.64575 1.64575i 7.93725i −1.00000 0
799.3 2.64575i 1.00000i −5.00000 0 2.64575 1.64575i 7.93725i −1.00000 0
799.4 2.64575i 1.00000i −5.00000 0 −2.64575 3.64575i 7.93725i −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 1425.2.c.i 4
5.b even 2 1 inner 1425.2.c.i 4
5.c odd 4 1 285.2.a.d 2
5.c odd 4 1 1425.2.a.p 2
15.e even 4 1 855.2.a.g 2
15.e even 4 1 4275.2.a.u 2
20.e even 4 1 4560.2.a.bo 2
95.g even 4 1 5415.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.d 2 5.c odd 4 1
855.2.a.g 2 15.e even 4 1
1425.2.a.p 2 5.c odd 4 1
1425.2.c.i 4 1.a even 1 1 trivial
1425.2.c.i 4 5.b even 2 1 inner
4275.2.a.u 2 15.e even 4 1
4560.2.a.bo 2 20.e even 4 1
5415.2.a.s 2 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1425, [\chi])\):

\( T_{2}^{2} + 7 \) Copy content Toggle raw display
\( T_{7}^{4} + 16T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 16T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 32T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 88T^{2} + 144 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 62)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 16T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T^{2} + 14 T + 42)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 32T^{2} + 4 \) Copy content Toggle raw display
$47$ \( T^{4} + 88T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{4} + 128T^{2} + 64 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 256T^{2} + 9216 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 176T^{2} + 1444 \) Copy content Toggle raw display
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