Newform orbit 2034.4.a.c

• Label: 2034.4.a.c
• Level: $2034$
• Weight: $4$
• Character orbit: 2034.a
• Self dual: yes
• Analytic conductor: $120.010$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2034.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(120.009884952\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 678)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	q - 2 * q^2 + 4 * q^4 + (5*b + 5) * q^5 + (2*b - 11) * q^7 - 8 * q^8 + (-10*b - 10) * q^10 + (-33*b + 22) * q^11 + (-6*b - 4) * q^13 + (-4*b + 22) * q^14 + 16 * q^16 + (-16*b + 74) * q^17 + (-17*b - 66) * q^19 + (20*b + 20) * q^20 + (66*b - 44) * q^22 + (67*b - 81) * q^23 + (50*b - 50) * q^25 + (12*b + 8) * q^26 + (8*b - 44) * q^28 + (-5*b + 155) * q^29 + (116*b - 126) * q^31 - 32 * q^32 + (32*b - 148) * q^34 + (-45*b - 35) * q^35 + (66*b + 24) * q^37 + (34*b + 132) * q^38 + (-40*b - 40) * q^40 + (264*b + 32) * q^41 + (-134*b + 36) * q^43 + (-132*b + 88) * q^44 + (-134*b + 162) * q^46 + (45*b - 105) * q^47 + (-44*b - 214) * q^49 + (-100*b + 100) * q^50 + (-24*b - 16) * q^52 + (b + 202) * q^53 + (-55*b - 220) * q^55 + (-16*b + 88) * q^56 + (10*b - 310) * q^58 + (-5*b + 281) * q^59 + (-106*b + 87) * q^61 + (-232*b + 252) * q^62 + 64 * q^64 + (-50*b - 80) * q^65 + (-389*b - 306) * q^67 + (-64*b + 296) * q^68 + (90*b + 70) * q^70 + (-233*b + 293) * q^71 + (-161*b + 804) * q^73 + (-132*b - 48) * q^74 + (-68*b - 264) * q^76 + (407*b - 374) * q^77 + (-319*b - 516) * q^79 + (80*b + 80) * q^80 + (-528*b - 64) * q^82 + (-654*b - 278) * q^83 + (290*b + 210) * q^85 + (268*b - 72) * q^86 + (264*b - 176) * q^88 + (-491*b + 345) * q^89 + (58*b + 20) * q^91 + (268*b - 324) * q^92 + (-90*b + 210) * q^94 + (-415*b - 500) * q^95 + (-810*b + 243) * q^97 + (88*b + 428) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 8 * q^4 + 10 * q^5 - 22 * q^7 - 16 * q^8 - 20 * q^10 + 44 * q^11 - 8 * q^13 + 44 * q^14 + 32 * q^16 + 148 * q^17 - 132 * q^19 + 40 * q^20 - 88 * q^22 - 162 * q^23 - 100 * q^25 + 16 * q^26 - 88 * q^28 + 310 * q^29 - 252 * q^31 - 64 * q^32 - 296 * q^34 - 70 * q^35 + 48 * q^37 + 264 * q^38 - 80 * q^40 + 64 * q^41 + 72 * q^43 + 176 * q^44 + 324 * q^46 - 210 * q^47 - 428 * q^49 + 200 * q^50 - 32 * q^52 + 404 * q^53 - 440 * q^55 + 176 * q^56 - 620 * q^58 + 562 * q^59 + 174 * q^61 + 504 * q^62 + 128 * q^64 - 160 * q^65 - 612 * q^67 + 592 * q^68 + 140 * q^70 + 586 * q^71 + 1608 * q^73 - 96 * q^74 - 528 * q^76 - 748 * q^77 - 1032 * q^79 + 160 * q^80 - 128 * q^82 - 556 * q^83 + 420 * q^85 - 144 * q^86 - 352 * q^88 + 690 * q^89 + 40 * q^91 - 648 * q^92 + 420 * q^94 - 1000 * q^95 + 486 * q^97 + 856 * q^98

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

−1.41421
1.41421

−2.00000

0

4.00000

−2.07107

0

−13.8284

−8.00000

0

4.14214

1.2

−2.00000

0

4.00000

12.0711

0

−8.17157

−8.00000

0

−24.1421

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(113\)	\( +1 \)

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	2034.4.a.c		2
3.b	odd	2	1		678.4.a.c	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
678.4.a.c	✓	2	3.b	odd	2	1
2034.4.a.c		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 10T_{5} - 25 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2034))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 10T - 25 \)
$7$	\( T^{2} + 22T + 113 \)
$11$	\( T^{2} - 44T - 1694 \)