Newform orbit 1900.4.a.b

• Label: 1900.4.a.b
• Level: $1900$
• Weight: $4$
• Character orbit: 1900.a
• Self dual: yes
• Analytic conductor: $112.104$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1900.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(112.103629011\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{33}) \)
Defining polynomial:	\( x^{2} - x - 8 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b + 3) * q^3 + (-4*b + 17) * q^7 + (-5*b - 10) * q^9 + (5*b - 38) * q^11 + (-11*b + 23) * q^13 + (6*b - 3) * q^17 - 19 * q^19 + (-25*b + 83) * q^21 + (59*b - 27) * q^23 + (27*b - 71) * q^27 + (65*b + 45) * q^29 + (80*b - 84) * q^31 + (48*b - 154) * q^33 + (-8*b - 186) * q^37 + (-45*b + 157) * q^39 + (-30*b - 56) * q^41 + (117*b - 136) * q^43 + (23*b + 216) * q^47 + (-120*b + 74) * q^49 + (15*b - 57) * q^51 + (-123*b + 199) * q^53 + (19*b - 57) * q^57 + (-35*b - 419) * q^59 + (-175*b + 310) * q^61 + (-25*b - 10) * q^63 + (61*b - 353) * q^67 + (145*b - 553) * q^69 + (-20*b - 846) * q^71 + (64*b + 463) * q^73 + (217*b - 806) * q^77 + (10*b - 642) * q^79 + (260*b - 159) * q^81 + (-114*b + 102) * q^83 + (85*b - 385) * q^87 + (80*b - 484) * q^89 + (-235*b + 743) * q^91 + (244*b - 892) * q^93 + (-458*b - 126) * q^97 + (115*b + 180) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 5 * q^3 + 30 * q^7 - 25 * q^9 - 71 * q^11 + 35 * q^13 - 38 * q^19 + 141 * q^21 + 5 * q^23 - 115 * q^27 + 155 * q^29 - 88 * q^31 - 260 * q^33 - 380 * q^37 + 269 * q^39 - 142 * q^41 - 155 * q^43 + 455 * q^47 + 28 * q^49 - 99 * q^51 + 275 * q^53 - 95 * q^57 - 873 * q^59 + 445 * q^61 - 45 * q^63 - 645 * q^67 - 961 * q^69 - 1712 * q^71 + 990 * q^73 - 1395 * q^77 - 1274 * q^79 - 58 * q^81 + 90 * q^83 - 685 * q^87 - 888 * q^89 + 1251 * q^91 - 1540 * q^93 - 710 * q^97 + 475 * q^99

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

3.37228
−2.37228

0

−0.372281

0

0

0

3.51087

0

−26.8614

0

1.2

0

5.37228

0

0

0

26.4891

0

1.86141

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( +1 \)
\(19\)	\( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1900.4.a.b		2
5.b	even	2	1		76.4.a.a	✓	2
5.c	odd	4	2		1900.4.c.b		4
15.d	odd	2	1		684.4.a.g		2
20.d	odd	2	1		304.4.a.f		2
40.e	odd	2	1		1216.4.a.h		2
40.f	even	2	1		1216.4.a.o		2
95.d	odd	2	1		1444.4.a.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.4.a.a	✓	2	5.b	even	2	1
304.4.a.f		2	20.d	odd	2	1
684.4.a.g		2	15.d	odd	2	1
1216.4.a.h		2	40.e	odd	2	1
1216.4.a.o		2	40.f	even	2	1
1444.4.a.d		2	95.d	odd	2	1
1900.4.a.b		2	1.a	even	1	1	trivial
1900.4.c.b		4	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 5T_{3} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1900))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 5T - 2 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 30T + 93 \)
$11$	\( T^{2} + 71T + 1054 \)