Properties

Label 1900.3.g.b
Level $1900$
Weight $3$
Character orbit 1900.g
Analytic conductor $51.771$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1900.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.7712502285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{29})\)
Defining polynomial: \(x^{4} + 15 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 76)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} -\beta_{1} q^{7} + 20 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} -\beta_{1} q^{7} + 20 q^{9} + 14 q^{11} + 3 \beta_{3} q^{13} + 23 \beta_{1} q^{17} + ( -10 - 3 \beta_{2} ) q^{19} -\beta_{2} q^{21} + \beta_{1} q^{23} + 11 \beta_{3} q^{27} + 9 \beta_{2} q^{29} -6 \beta_{2} q^{31} + 14 \beta_{3} q^{33} -6 \beta_{3} q^{37} + 87 q^{39} + 6 \beta_{2} q^{41} -68 \beta_{1} q^{43} + 26 \beta_{1} q^{47} + 48 q^{49} + 23 \beta_{2} q^{51} -15 \beta_{3} q^{53} + ( -87 \beta_{1} - 10 \beta_{3} ) q^{57} -3 \beta_{2} q^{59} -40 q^{61} -20 \beta_{1} q^{63} + 3 \beta_{3} q^{67} + \beta_{2} q^{69} -6 \beta_{2} q^{71} + 7 \beta_{1} q^{73} -14 \beta_{1} q^{77} -18 \beta_{2} q^{79} + 139 q^{81} -32 \beta_{1} q^{83} + 261 \beta_{1} q^{87} + 24 \beta_{2} q^{89} -3 \beta_{2} q^{91} -174 \beta_{1} q^{93} + 18 \beta_{3} q^{97} + 280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 80q^{9} + O(q^{10}) \) \( 4q + 80q^{9} + 56q^{11} - 40q^{19} + 348q^{39} + 192q^{49} - 160q^{61} + 556q^{81} + 1120q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 15 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 8 \nu \)\()/7\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 22 \nu \)\()/7\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 15\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{2} + 11 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
3.19258i
3.19258i
2.19258i
2.19258i
0 −5.38516 0 0 0 1.00000i 0 20.0000 0
949.2 0 −5.38516 0 0 0 1.00000i 0 20.0000 0
949.3 0 5.38516 0 0 0 1.00000i 0 20.0000 0
949.4 0 5.38516 0 0 0 1.00000i 0 20.0000 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.b odd 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.3.g.b 4
5.b even 2 1 inner 1900.3.g.b 4
5.c odd 4 1 76.3.c.a 2
5.c odd 4 1 1900.3.e.b 2
15.e even 4 1 684.3.h.c 2
19.b odd 2 1 inner 1900.3.g.b 4
20.e even 4 1 304.3.e.b 2
40.i odd 4 1 1216.3.e.k 2
40.k even 4 1 1216.3.e.l 2
60.l odd 4 1 2736.3.o.i 2
95.d odd 2 1 inner 1900.3.g.b 4
95.g even 4 1 76.3.c.a 2
95.g even 4 1 1900.3.e.b 2
285.j odd 4 1 684.3.h.c 2
380.j odd 4 1 304.3.e.b 2
760.t even 4 1 1216.3.e.k 2
760.y odd 4 1 1216.3.e.l 2
1140.w even 4 1 2736.3.o.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.a 2 5.c odd 4 1
76.3.c.a 2 95.g even 4 1
304.3.e.b 2 20.e even 4 1
304.3.e.b 2 380.j odd 4 1
684.3.h.c 2 15.e even 4 1
684.3.h.c 2 285.j odd 4 1
1216.3.e.k 2 40.i odd 4 1
1216.3.e.k 2 760.t even 4 1
1216.3.e.l 2 40.k even 4 1
1216.3.e.l 2 760.y odd 4 1
1900.3.e.b 2 5.c odd 4 1
1900.3.e.b 2 95.g even 4 1
1900.3.g.b 4 1.a even 1 1 trivial
1900.3.g.b 4 5.b even 2 1 inner
1900.3.g.b 4 19.b odd 2 1 inner
1900.3.g.b 4 95.d odd 2 1 inner
2736.3.o.i 2 60.l odd 4 1
2736.3.o.i 2 1140.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 29 \) acting on \(S_{3}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -29 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -14 + T )^{4} \)
$13$ \( ( -261 + T^{2} )^{2} \)
$17$ \( ( 529 + T^{2} )^{2} \)
$19$ \( ( 361 + 20 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 2349 + T^{2} )^{2} \)
$31$ \( ( 1044 + T^{2} )^{2} \)
$37$ \( ( -1044 + T^{2} )^{2} \)
$41$ \( ( 1044 + T^{2} )^{2} \)
$43$ \( ( 4624 + T^{2} )^{2} \)
$47$ \( ( 676 + T^{2} )^{2} \)
$53$ \( ( -6525 + T^{2} )^{2} \)
$59$ \( ( 261 + T^{2} )^{2} \)
$61$ \( ( 40 + T )^{4} \)
$67$ \( ( -261 + T^{2} )^{2} \)
$71$ \( ( 1044 + T^{2} )^{2} \)
$73$ \( ( 49 + T^{2} )^{2} \)
$79$ \( ( 9396 + T^{2} )^{2} \)
$83$ \( ( 1024 + T^{2} )^{2} \)
$89$ \( ( 16704 + T^{2} )^{2} \)
$97$ \( ( -9396 + T^{2} )^{2} \)
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