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: \(x^{4} - x^{2} + 1\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(1\)

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 \( T_{3}^{4} - 4 T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

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