Properties

Label 1900.2.c.d
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: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + ( -3 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + ( -3 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} -2 q^{11} + ( -3 \zeta_{8} + 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{13} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{17} + q^{19} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + 6 \zeta_{8}^{2} q^{23} + ( 8 \zeta_{8} - 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{27} + ( -2 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{29} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{31} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{33} + ( -3 \zeta_{8} - 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{37} + ( -10 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{39} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{43} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} + ( -5 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{49} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{51} + ( 5 \zeta_{8} - 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{53} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{57} + ( -8 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{59} + ( -8 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{61} + ( -2 \zeta_{8} - 10 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{67} + ( -12 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{69} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{71} + ( -6 \zeta_{8} - 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{73} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{77} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{79} + ( 23 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{81} + ( -8 \zeta_{8} + 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{83} + ( -10 \zeta_{8} + 8 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{87} + ( -2 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{89} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{91} + ( 8 \zeta_{8} - 12 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{93} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{97} + ( 6 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} - 8q^{11} + 4q^{19} - 8q^{29} - 16q^{31} - 40q^{39} - 8q^{41} - 20q^{49} - 32q^{59} - 32q^{61} - 48q^{69} + 32q^{71} + 16q^{79} + 92q^{81} - 8q^{89} - 32q^{91} + 24q^{99} + O(q^{100}) \)

Character values

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

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

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
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 3.41421i 0 0 0 0.828427i 0 −8.65685 0
1749.2 0 0.585786i 0 0 0 4.82843i 0 2.65685 0
1749.3 0 0.585786i 0 0 0 4.82843i 0 2.65685 0
1749.4 0 3.41421i 0 0 0 0.828427i 0 −8.65685 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.d 4
5.b even 2 1 inner 1900.2.c.d 4
5.c odd 4 1 380.2.a.c 2
5.c odd 4 1 1900.2.a.e 2
15.e even 4 1 3420.2.a.g 2
20.e even 4 1 1520.2.a.o 2
20.e even 4 1 7600.2.a.u 2
40.i odd 4 1 6080.2.a.bl 2
40.k even 4 1 6080.2.a.y 2
95.g even 4 1 7220.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.c 2 5.c odd 4 1
1520.2.a.o 2 20.e even 4 1
1900.2.a.e 2 5.c odd 4 1
1900.2.c.d 4 1.a even 1 1 trivial
1900.2.c.d 4 5.b even 2 1 inner
3420.2.a.g 2 15.e even 4 1
6080.2.a.y 2 40.k even 4 1
6080.2.a.bl 2 40.i odd 4 1
7220.2.a.m 2 95.g even 4 1
7600.2.a.u 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 \(S_{2}^{\mathrm{new}}(1900, [\chi])\):

\( T_{3}^{4} + 12 T_{3}^{2} + 4 \)
\( T_{7}^{4} + 24 T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 + 12 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 + 24 T^{2} + T^{4} \)
$11$ \( ( 2 + T )^{4} \)
$13$ \( 196 + 44 T^{2} + T^{4} \)
$17$ \( 16 + 24 T^{2} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( ( 36 + T^{2} )^{2} \)
$29$ \( ( -68 + 4 T + T^{2} )^{2} \)
$31$ \( ( 8 + 8 T + T^{2} )^{2} \)
$37$ \( 324 + 108 T^{2} + T^{4} \)
$41$ \( ( -28 + 4 T + T^{2} )^{2} \)
$43$ \( 16 + 24 T^{2} + T^{4} \)
$47$ \( 16 + 24 T^{2} + T^{4} \)
$53$ \( 2116 + 108 T^{2} + T^{4} \)
$59$ \( ( 32 + 16 T + T^{2} )^{2} \)
$61$ \( ( 32 + 16 T + T^{2} )^{2} \)
$67$ \( 4 + 12 T^{2} + T^{4} \)
$71$ \( ( 56 - 16 T + T^{2} )^{2} \)
$73$ \( 1296 + 216 T^{2} + T^{4} \)
$79$ \( ( -16 - 8 T + T^{2} )^{2} \)
$83$ \( 15376 + 264 T^{2} + T^{4} \)
$89$ \( ( -68 + 4 T + T^{2} )^{2} \)
$97$ \( 324 + 108 T^{2} + T^{4} \)
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