Properties

Label 1900.4.c.b
Level $1900$
Weight $4$
Character orbit 1900.c
Analytic conductor $112.104$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(112.103629011\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
Defining polynomial: \(x^{4} + 17 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -5 \beta_{1} - \beta_{2} ) q^{7} + ( 12 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -5 \beta_{1} - \beta_{2} ) q^{7} + ( 12 + \beta_{3} ) q^{9} + ( -36 + \beta_{3} ) q^{11} + ( 9 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{17} + 19 q^{19} + ( 73 - 5 \beta_{3} ) q^{21} + ( -29 \beta_{1} + 30 \beta_{2} ) q^{23} + ( 25 \beta_{1} - 2 \beta_{2} ) q^{27} + ( -71 - 13 \beta_{3} ) q^{29} + ( -52 + 16 \beta_{3} ) q^{31} + ( -50 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 34 \beta_{1} + 42 \beta_{2} ) q^{37} + ( -139 + 9 \beta_{3} ) q^{39} + ( -68 - 6 \beta_{3} ) q^{41} + ( -74 \beta_{1} + 43 \beta_{2} ) q^{43} + ( -34 \beta_{1} - 57 \beta_{2} ) q^{47} + ( -26 + 24 \beta_{3} ) q^{49} + ( -51 + 3 \beta_{3} ) q^{51} + ( 89 \beta_{1} - 34 \beta_{2} ) q^{53} + 19 \beta_{1} q^{57} + ( 433 + 7 \beta_{3} ) q^{59} + ( 240 - 35 \beta_{3} ) q^{61} + ( 8 \beta_{1} - 17 \beta_{2} ) q^{63} + ( 95 \beta_{1} + 34 \beta_{2} ) q^{67} + ( 495 - 29 \beta_{3} ) q^{69} + ( -854 - 4 \beta_{3} ) q^{71} + ( 67 \beta_{1} + 131 \beta_{2} ) q^{73} + ( 248 \beta_{1} + 31 \beta_{2} ) q^{77} + ( 638 - 2 \beta_{3} ) q^{79} + ( -55 + 52 \beta_{3} ) q^{81} + ( 66 \beta_{1} - 48 \beta_{2} ) q^{83} + ( 111 \beta_{1} + 26 \beta_{2} ) q^{87} + ( 452 - 16 \beta_{3} ) q^{89} + ( 649 - 47 \beta_{3} ) q^{91} + ( -276 \beta_{1} - 32 \beta_{2} ) q^{93} + ( -158 \beta_{1} + 300 \beta_{2} ) q^{97} + ( -226 - 23 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 50q^{9} + O(q^{10}) \) \( 4q + 50q^{9} - 142q^{11} + 76q^{19} + 282q^{21} - 310q^{29} - 176q^{31} - 538q^{39} - 284q^{41} - 56q^{49} - 198q^{51} + 1746q^{59} + 890q^{61} + 1922q^{69} - 3424q^{71} + 2548q^{79} - 116q^{81} + 1776q^{89} + 2502q^{91} - 950q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 17 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 5 \nu \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{3} - 35 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\( 5 \nu^{2} + 43 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{2} + 3 \beta_{1}\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 43\)\()/5\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} - 7 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
3.37228i
2.37228i
2.37228i
3.37228i
0 5.37228i 0 0 0 26.4891i 0 −1.86141 0
1749.2 0 0.372281i 0 0 0 3.51087i 0 26.8614 0
1749.3 0 0.372281i 0 0 0 3.51087i 0 26.8614 0
1749.4 0 5.37228i 0 0 0 26.4891i 0 −1.86141 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 1900.4.c.b 4
5.b even 2 1 inner 1900.4.c.b 4
5.c odd 4 1 76.4.a.a 2
5.c odd 4 1 1900.4.a.b 2
15.e even 4 1 684.4.a.g 2
20.e even 4 1 304.4.a.f 2
40.i odd 4 1 1216.4.a.o 2
40.k even 4 1 1216.4.a.h 2
95.g even 4 1 1444.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.a 2 5.c odd 4 1
304.4.a.f 2 20.e even 4 1
684.4.a.g 2 15.e even 4 1
1216.4.a.h 2 40.k even 4 1
1216.4.a.o 2 40.i odd 4 1
1444.4.a.d 2 95.g even 4 1
1900.4.a.b 2 5.c odd 4 1
1900.4.c.b 4 1.a even 1 1 trivial
1900.4.c.b 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 29 T_{3}^{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 + 29 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 8649 + 714 T^{2} + T^{4} \)
$11$ \( ( 1054 + 71 T + T^{2} )^{2} \)
$13$ \( 478864 + 2609 T^{2} + T^{4} \)
$17$ \( ( 297 + T^{2} )^{2} \)
$19$ \( ( -19 + T )^{4} \)
$23$ \( 824378944 + 57449 T^{2} + T^{4} \)
$29$ \( ( -28850 + 155 T + T^{2} )^{2} \)
$31$ \( ( -50864 + 88 T + T^{2} )^{2} \)
$37$ \( 1265367184 + 73256 T^{2} + T^{4} \)
$41$ \( ( -2384 + 142 T + T^{2} )^{2} \)
$43$ \( 11433597184 + 237881 T^{2} + T^{4} \)
$47$ \( 2246001664 + 112241 T^{2} + T^{4} \)
$53$ \( 11216504464 + 287441 T^{2} + T^{4} \)
$59$ \( ( 180426 - 873 T + T^{2} )^{2} \)
$61$ \( ( -203150 - 445 T + T^{2} )^{2} \)
$67$ \( 5374062864 + 269409 T^{2} + T^{4} \)
$71$ \( ( 729436 + 1712 T + T^{2} )^{2} \)
$73$ \( 44619380289 + 557634 T^{2} + T^{4} \)
$79$ \( ( 404944 - 1274 T + T^{2} )^{2} \)
$83$ \( 11065356864 + 218484 T^{2} + T^{4} \)
$89$ \( ( 144336 - 888 T + T^{2} )^{2} \)
$97$ \( 2574510102784 + 3713156 T^{2} + T^{4} \)
show more
show less