Newform orbit 245.4.a.i

• Label: 245.4.a.i
• Level: $245$
• Weight: $4$
• Character orbit: 245.a
• Self dual: yes
• Analytic conductor: $14.455$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.4554679514\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{11}) \)
Defining polynomial:	\( x^{2} - 11 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 1) * q^2 - 5 * q^3 + (2*b + 4) * q^4 - 5 * q^5 + (-5*b - 5) * q^6 + (-2*b + 18) * q^8 - 2 * q^9 + (-5*b - 5) * q^10 + (-4*b + 33) * q^11 + (-10*b - 20) * q^12 + (-20*b - 5) * q^13 + 25 * q^15 - 36 * q^16 + (20*b - 35) * q^17 + (-2*b - 2) * q^18 + (-20*b - 70) * q^19 + (-10*b - 20) * q^20 + (29*b - 11) * q^22 + (-28*b - 8) * q^23 + (10*b - 90) * q^24 + 25 * q^25 + (-25*b - 225) * q^26 + 145 * q^27 + (-48*b - 129) * q^29 + (25*b + 25) * q^30 + (60*b + 10) * q^31 + (-20*b - 180) * q^32 + (20*b - 165) * q^33 + (-15*b + 185) * q^34 + (-4*b - 8) * q^36 + (44*b + 164) * q^37 + (-90*b - 290) * q^38 + (100*b + 25) * q^39 + (10*b - 90) * q^40 + (100*b - 150) * q^41 + (12*b - 58) * q^43 + (50*b + 44) * q^44 + 10 * q^45 + (-36*b - 316) * q^46 + (40*b + 15) * q^47 + 180 * q^48 + (25*b + 25) * q^50 + (-100*b + 175) * q^51 + (-90*b - 460) * q^52 + (-120*b + 270) * q^53 + (145*b + 145) * q^54 + (20*b - 165) * q^55 + (100*b + 350) * q^57 + (-177*b - 657) * q^58 + (-40*b - 190) * q^59 + (50*b + 100) * q^60 + (60*b - 540) * q^61 + (70*b + 670) * q^62 + (-200*b - 112) * q^64 + (100*b + 25) * q^65 + (-145*b + 55) * q^66 + (-96*b + 234) * q^67 + (10*b + 300) * q^68 + (140*b + 40) * q^69 + (64*b - 528) * q^71 + (4*b - 36) * q^72 + (200*b + 430) * q^73 + (208*b + 648) * q^74 - 125 * q^75 + (-220*b - 720) * q^76 + (125*b + 1125) * q^78 + (348*b + 79) * q^79 + 180 * q^80 - 671 * q^81 + (-50*b + 950) * q^82 + (200*b - 20) * q^83 + (-100*b + 175) * q^85 + (-46*b + 74) * q^86 + (240*b + 645) * q^87 + (-138*b + 682) * q^88 + (-380*b + 120) * q^89 + (10*b + 10) * q^90 + (-128*b - 648) * q^92 + (-300*b - 50) * q^93 + (55*b + 455) * q^94 + (100*b + 350) * q^95 + (100*b + 900) * q^96 + (-180*b - 815) * q^97 + (8*b - 66) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 - 10 * q^3 + 8 * q^4 - 10 * q^5 - 10 * q^6 + 36 * q^8 - 4 * q^9 - 10 * q^10 + 66 * q^11 - 40 * q^12 - 10 * q^13 + 50 * q^15 - 72 * q^16 - 70 * q^17 - 4 * q^18 - 140 * q^19 - 40 * q^20 - 22 * q^22 - 16 * q^23 - 180 * q^24 + 50 * q^25 - 450 * q^26 + 290 * q^27 - 258 * q^29 + 50 * q^30 + 20 * q^31 - 360 * q^32 - 330 * q^33 + 370 * q^34 - 16 * q^36 + 328 * q^37 - 580 * q^38 + 50 * q^39 - 180 * q^40 - 300 * q^41 - 116 * q^43 + 88 * q^44 + 20 * q^45 - 632 * q^46 + 30 * q^47 + 360 * q^48 + 50 * q^50 + 350 * q^51 - 920 * q^52 + 540 * q^53 + 290 * q^54 - 330 * q^55 + 700 * q^57 - 1314 * q^58 - 380 * q^59 + 200 * q^60 - 1080 * q^61 + 1340 * q^62 - 224 * q^64 + 50 * q^65 + 110 * q^66 + 468 * q^67 + 600 * q^68 + 80 * q^69 - 1056 * q^71 - 72 * q^72 + 860 * q^73 + 1296 * q^74 - 250 * q^75 - 1440 * q^76 + 2250 * q^78 + 158 * q^79 + 360 * q^80 - 1342 * q^81 + 1900 * q^82 - 40 * q^83 + 350 * q^85 + 148 * q^86 + 1290 * q^87 + 1364 * q^88 + 240 * q^89 + 20 * q^90 - 1296 * q^92 - 100 * q^93 + 910 * q^94 + 700 * q^95 + 1800 * q^96 - 1630 * q^97 - 132 * 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

−3.31662
3.31662

−2.31662

−5.00000

−2.63325

−5.00000

11.5831

0

24.6332

−2.00000

11.5831

1.2

4.31662

−5.00000

10.6332

−5.00000

−21.5831

0

11.3668

−2.00000

−21.5831

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\( +1 \)
\(7\)	\( -1 \)

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	245.4.a.i	✓	2
3.b	odd	2	1		2205.4.a.x		2
5.b	even	2	1		1225.4.a.q		2
7.b	odd	2	1		245.4.a.j	yes	2
7.c	even	3	2		245.4.e.k		4
7.d	odd	6	2		245.4.e.j		4
21.c	even	2	1		2205.4.a.w		2
35.c	odd	2	1		1225.4.a.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
245.4.a.i	✓	2	1.a	even	1	1	trivial
245.4.a.j	yes	2	7.b	odd	2	1
245.4.e.j		4	7.d	odd	6	2
245.4.e.k		4	7.c	even	3	2
1225.4.a.p		2	35.c	odd	2	1
1225.4.a.q		2	5.b	even	2	1
2205.4.a.w		2	21.c	even	2	1
2205.4.a.x		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{2} - 2T_{2} - 10 \)

\( T_{3} + 5 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T - 10 \)
$3$	\( (T + 5)^{2} \)
$5$	\( (T + 5)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 66T + 913 \)