Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 2 \beta_{3} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 2 \beta_{3} q^{7} + \beta_{2} q^{9} - 3 q^{11} + (3 \beta_{3} - 3 \beta_1) q^{13} - \beta_1 q^{17} + ( - 3 \beta_{2} - 2) q^{19} + ( - 8 \beta_{2} + 8) q^{21} + (2 \beta_{3} - 2 \beta_1) q^{23} - 2 \beta_{3} q^{27} + \beta_{2} q^{29} - 5 q^{31} - 3 \beta_1 q^{33} - 2 \beta_{3} q^{37} - 12 q^{39} + (2 \beta_{2} - 2) q^{41} + ( - 3 \beta_{3} + 3 \beta_1) q^{47} - 9 q^{49} - 4 \beta_{2} q^{51} + ( - 3 \beta_{3} + 3 \beta_1) q^{53} + ( - 3 \beta_{3} - 2 \beta_1) q^{57} + (\beta_{2} - 1) q^{59} + 7 \beta_{2} q^{61} + ( - 2 \beta_{3} + 2 \beta_1) q^{63} + (7 \beta_{3} - 7 \beta_1) q^{67} - 8 q^{69} + (15 \beta_{2} - 15) q^{71} + 6 \beta_1 q^{73} + 6 \beta_{3} q^{77} + (\beta_{2} - 1) q^{79} + ( - 11 \beta_{2} + 11) q^{81} - 8 \beta_{3} q^{83} + \beta_{3} q^{87} + 17 \beta_{2} q^{89} + 24 \beta_{2} q^{91} - 5 \beta_1 q^{93} - 6 \beta_1 q^{97} - 3 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} - 12 q^{11} - 14 q^{19} + 16 q^{21} + 2 q^{29} - 20 q^{31} - 48 q^{39} - 4 q^{41} - 36 q^{49} - 8 q^{51} - 2 q^{59} + 14 q^{61} - 32 q^{69} - 30 q^{71} - 2 q^{79} + 22 q^{81} + 34 q^{89} + 48 q^{91} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

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

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −1.73205 1.00000i 0 0 0 4.00000i 0 0.500000 + 0.866025i 0
49.2 0 1.73205 + 1.00000i 0 0 0 4.00000i 0 0.500000 + 0.866025i 0
349.1 0 −1.73205 + 1.00000i 0 0 0 4.00000i 0 0.500000 0.866025i 0
349.2 0 1.73205 1.00000i 0 0 0 4.00000i 0 0.500000 0.866025i 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
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.s.b 4
5.b even 2 1 inner 1900.2.s.b 4
5.c odd 4 1 380.2.i.a 2
5.c odd 4 1 1900.2.i.b 2
15.e even 4 1 3420.2.t.b 2
19.c even 3 1 inner 1900.2.s.b 4
20.e even 4 1 1520.2.q.g 2
95.i even 6 1 inner 1900.2.s.b 4
95.l even 12 1 7220.2.a.a 1
95.m odd 12 1 380.2.i.a 2
95.m odd 12 1 1900.2.i.b 2
95.m odd 12 1 7220.2.a.e 1
285.v even 12 1 3420.2.t.b 2
380.v even 12 1 1520.2.q.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.a 2 5.c odd 4 1
380.2.i.a 2 95.m odd 12 1
1520.2.q.g 2 20.e even 4 1
1520.2.q.g 2 380.v even 12 1
1900.2.i.b 2 5.c odd 4 1
1900.2.i.b 2 95.m odd 12 1
1900.2.s.b 4 1.a even 1 1 trivial
1900.2.s.b 4 5.b even 2 1 inner
1900.2.s.b 4 19.c even 3 1 inner
1900.2.s.b 4 95.i even 6 1 inner
3420.2.t.b 2 15.e even 4 1
3420.2.t.b 2 285.v even 12 1
7220.2.a.a 1 95.l even 12 1
7220.2.a.e 1 95.m odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T + 5)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 196 T^{2} + 38416 \) Copy content Toggle raw display
$71$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$79$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 17 T + 289)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
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