Newform orbit 250.4.a.b

• Label: 250.4.a.b
• Level: $250$
• Weight: $4$
• Character orbit: 250.a
• Self dual: yes
• Analytic conductor: $14.750$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 250 = 2 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	250.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.7504775014\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + (3*b + 4) * q^3 + 4 * q^4 + (-6*b - 8) * q^6 + (-11*b + 7) * q^7 - 8 * q^8 + (33*b - 2) * q^9 + (-18*b + 16) * q^11 + (12*b + 16) * q^12 + (-14*b + 40) * q^13 + (22*b - 14) * q^14 + 16 * q^16 + (46*b + 46) * q^17 + (-66*b + 4) * q^18 + (98*b - 34) * q^19 + (-56*b - 5) * q^21 + (36*b - 32) * q^22 + (13*b + 179) * q^23 + (-24*b - 32) * q^24 + (28*b - 80) * q^26 + (144*b - 17) * q^27 + (-44*b + 28) * q^28 + (-169*b - 18) * q^29 + (-26*b - 100) * q^31 - 32 * q^32 + (-78*b + 10) * q^33 + (-92*b - 92) * q^34 + (132*b - 8) * q^36 + (-24*b + 206) * q^37 + (-196*b + 68) * q^38 + (22*b + 118) * q^39 + (-139*b + 324) * q^41 + (112*b + 10) * q^42 + (-131*b + 196) * q^43 + (-72*b + 64) * q^44 + (-26*b - 358) * q^46 + (-51*b - 18) * q^47 + (48*b + 64) * q^48 + (-33*b - 173) * q^49 + (460*b + 322) * q^51 + (-56*b + 160) * q^52 + (-104*b + 560) * q^53 + (-288*b + 34) * q^54 + (88*b - 56) * q^56 + (584*b + 158) * q^57 + (338*b + 36) * q^58 + (202*b + 344) * q^59 + (-97*b - 17) * q^61 + (52*b + 200) * q^62 + (-110*b - 377) * q^63 + 64 * q^64 + (156*b - 20) * q^66 + (-148*b - 352) * q^67 + (184*b + 184) * q^68 + (628*b + 755) * q^69 + (-882*b + 338) * q^71 + (-264*b + 16) * q^72 + (-334*b - 40) * q^73 + (48*b - 412) * q^74 + (392*b - 136) * q^76 + (-104*b + 310) * q^77 + (-44*b - 236) * q^78 + (-432*b + 626) * q^79 + (66*b + 418) * q^81 + (278*b - 648) * q^82 + (255*b - 877) * q^83 + (-224*b - 20) * q^84 + (262*b - 392) * q^86 + (-1237*b - 579) * q^87 + (144*b - 128) * q^88 + (955*b - 555) * q^89 + (-384*b + 434) * q^91 + (52*b + 716) * q^92 + (-482*b - 478) * q^93 + (102*b + 36) * q^94 + (-96*b - 128) * q^96 + (350*b - 596) * q^97 + (66*b + 346) * q^98 + (-30*b - 626) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 11 * q^3 + 8 * q^4 - 22 * q^6 + 3 * q^7 - 16 * q^8 + 29 * q^9 + 14 * q^11 + 44 * q^12 + 66 * q^13 - 6 * q^14 + 32 * q^16 + 138 * q^17 - 58 * q^18 + 30 * q^19 - 66 * q^21 - 28 * q^22 + 371 * q^23 - 88 * q^24 - 132 * q^26 + 110 * q^27 + 12 * q^28 - 205 * q^29 - 226 * q^31 - 64 * q^32 - 58 * q^33 - 276 * q^34 + 116 * q^36 + 388 * q^37 - 60 * q^38 + 258 * q^39 + 509 * q^41 + 132 * q^42 + 261 * q^43 + 56 * q^44 - 742 * q^46 - 87 * q^47 + 176 * q^48 - 379 * q^49 + 1104 * q^51 + 264 * q^52 + 1016 * q^53 - 220 * q^54 - 24 * q^56 + 900 * q^57 + 410 * q^58 + 890 * q^59 - 131 * q^61 + 452 * q^62 - 864 * q^63 + 128 * q^64 + 116 * q^66 - 852 * q^67 + 552 * q^68 + 2138 * q^69 - 206 * q^71 - 232 * q^72 - 414 * q^73 - 776 * q^74 + 120 * q^76 + 516 * q^77 - 516 * q^78 + 820 * q^79 + 902 * q^81 - 1018 * q^82 - 1499 * q^83 - 264 * q^84 - 522 * q^86 - 2395 * q^87 - 112 * q^88 - 155 * q^89 + 484 * q^91 + 1484 * q^92 - 1438 * q^93 + 174 * q^94 - 352 * q^96 - 842 * q^97 + 758 * q^98 - 1282 * 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

−0.618034
1.61803

−2.00000

2.14590

4.00000

0

−4.29180

13.7984

−8.00000

−22.3951

0

1.2

−2.00000

8.85410

4.00000

0

−17.7082

−10.7984

−8.00000

51.3951

0

Atkin-Lehner signs

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

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	250.4.a.b	✓	2
3.b	odd	2	1		2250.4.a.i		2
4.b	odd	2	1		2000.4.a.a		2
5.b	even	2	1		250.4.a.c	yes	2
5.c	odd	4	2		250.4.b.a		4
15.d	odd	2	1		2250.4.a.d		2
20.d	odd	2	1		2000.4.a.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
250.4.a.b	✓	2	1.a	even	1	1	trivial
250.4.a.c	yes	2	5.b	even	2	1
250.4.b.a		4	5.c	odd	4	2
2000.4.a.a		2	4.b	odd	2	1
2000.4.a.d		2	20.d	odd	2	1
2250.4.a.d		2	15.d	odd	2	1
2250.4.a.i		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 11T_{3} + 19 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(250))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} - 11T + 19 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 3T - 149 \)
$11$	\( T^{2} - 14T - 356 \)