Properties

Label 1900.2.c.e
Level $1900$
Weight $2$
Character orbit 1900.c
Analytic conductor $15.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{2} - \beta_1) q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{2} - \beta_1) q^{7} + (\beta_{3} - 1) q^{9} + \beta_{3} q^{11} + \beta_{2} q^{13} + ( - \beta_{2} - \beta_1) q^{17} - q^{19} + ( - \beta_{3} + 2) q^{21} + (\beta_{2} + \beta_1) q^{23} + (2 \beta_{2} - 2 \beta_1) q^{27} + \beta_{3} q^{29} + (\beta_{3} + 2) q^{31} + (4 \beta_{2} - 2 \beta_1) q^{33} + (3 \beta_{2} - 2 \beta_1) q^{37} + ( - \beta_{3} + 4) q^{39} - 6 q^{41} + (\beta_{2} + 3 \beta_1) q^{43} + (5 \beta_{2} - \beta_1) q^{47} + 3 q^{49} + (\beta_{3} - 6) q^{51} + (5 \beta_{2} - 4 \beta_1) q^{53} + \beta_{2} q^{57} + 2 \beta_{3} q^{59} + ( - 3 \beta_{3} + 2) q^{61} + ( - 3 \beta_{2} - \beta_1) q^{63} + (3 \beta_{2} - 2 \beta_1) q^{67} + ( - \beta_{3} + 6) q^{69} + ( - \beta_{3} - 6) q^{71} + (\beta_{2} - 3 \beta_1) q^{73} + ( - 2 \beta_{2} - 2 \beta_1) q^{77} + (2 \beta_{3} + 4) q^{79} + (\beta_{3} + 1) q^{81} + ( - \beta_{2} - \beta_1) q^{83} + (4 \beta_{2} - 2 \beta_1) q^{87} + ( - \beta_{3} + 12) q^{89} + (\beta_{3} - 2) q^{91} + (2 \beta_{2} - 2 \beta_1) q^{93} + ( - 5 \beta_{2} - 4 \beta_1) q^{97} + ( - \beta_{3} + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{19} + 8 q^{21} + 8 q^{31} + 16 q^{39} - 24 q^{41} + 12 q^{49} - 24 q^{51} + 8 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} + 4 q^{81} + 48 q^{89} - 8 q^{91} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 2.73205i 0 0 0 2.00000i 0 −4.46410 0
1749.2 0 0.732051i 0 0 0 2.00000i 0 2.46410 0
1749.3 0 0.732051i 0 0 0 2.00000i 0 2.46410 0
1749.4 0 2.73205i 0 0 0 2.00000i 0 −4.46410 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 1900.2.c.e 4
5.b even 2 1 inner 1900.2.c.e 4
5.c odd 4 1 380.2.a.d 2
5.c odd 4 1 1900.2.a.d 2
15.e even 4 1 3420.2.a.h 2
20.e even 4 1 1520.2.a.l 2
20.e even 4 1 7600.2.a.bf 2
40.i odd 4 1 6080.2.a.z 2
40.k even 4 1 6080.2.a.bj 2
95.g even 4 1 7220.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.d 2 5.c odd 4 1
1520.2.a.l 2 20.e even 4 1
1900.2.a.d 2 5.c odd 4 1
1900.2.c.e 4 1.a even 1 1 trivial
1900.2.c.e 4 5.b even 2 1 inner
3420.2.a.h 2 15.e even 4 1
6080.2.a.z 2 40.i odd 4 1
6080.2.a.bj 2 40.k even 4 1
7220.2.a.h 2 95.g even 4 1
7600.2.a.bf 2 20.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 \(S_{2}^{\mathrm{new}}(1900, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$47$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 6084 \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 104)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 24 T + 132)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 488 T^{2} + 58564 \) Copy content Toggle raw display
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