Newform orbit 245.4.a.h

• Label: 245.4.a.h
• 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:	$\Q(\sqrt{2})$
Defining polynomial:	$x^{2} - 2$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 35)
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 + (b - 3) * q^2 + (3*b + 1) * q^3 + (-6*b + 3) * q^4 - 5 * q^5 + (-8*b + 3) * q^6 + (13*b + 3) * q^8 + (6*b - 8) * q^9 + (-5*b + 15) * q^10 + (10*b + 14) * q^11 + (3*b - 33) * q^12 + (-10*b + 18) * q^13 + (-15*b - 5) * q^15 + (12*b - 7) * q^16 + (-54*b + 38) * q^17 + (-26*b + 36) * q^18 + (-26*b - 80) * q^19 + (30*b - 15) * q^20 + (-16*b - 22) * q^22 + (-117*b - 11) * q^23 + (22*b + 81) * q^24 + 25 * q^25 + (48*b - 74) * q^26 + (-99*b + 1) * q^27 + (-60*b - 125) * q^29 + (40*b - 15) * q^30 + (100*b + 66) * q^31 + (-147*b + 21) * q^32 + (52*b + 74) * q^33 + (200*b - 222) * q^34 + (66*b - 96) * q^36 + (136*b - 208) * q^37 + (-2*b + 188) * q^38 + (44*b - 42) * q^39 + (-65*b - 15) * q^40 + (-30*b + 53) * q^41 + (5*b - 333) * q^43 + (-54*b - 78) * q^44 + (-30*b + 40) * q^45 + (340*b - 201) * q^46 + (64*b + 98) * q^47 + (-9*b + 65) * q^48 + (25*b - 75) * q^50 + (60*b - 286) * q^51 + (-138*b + 174) * q^52 + (142*b - 476) * q^53 + (298*b - 201) * q^54 + (-50*b - 70) * q^55 + (-266*b - 236) * q^57 + (55*b + 255) * q^58 + (266*b - 420) * q^59 + (-15*b + 165) * q^60 + (-570*b - 49) * q^61 + (-234*b + 2) * q^62 + (366*b - 301) * q^64 + (50*b - 90) * q^65 + (-82*b - 118) * q^66 + (69*b - 643) * q^67 + (-390*b + 762) * q^68 + (-150*b - 713) * q^69 + (350*b + 532) * q^71 + (-86*b + 132) * q^72 + (118*b + 86) * q^73 + (-616*b + 896) * q^74 + (75*b + 25) * q^75 + (402*b + 72) * q^76 + (-174*b + 214) * q^78 + (-214*b - 620) * q^79 + (-60*b + 35) * q^80 + (-258*b - 377) * q^81 + (143*b - 219) * q^82 + (5*b + 953) * q^83 + (270*b - 190) * q^85 + (-348*b + 1009) * q^86 + (-435*b - 485) * q^87 + (212*b + 302) * q^88 + (324*b - 325) * q^89 + (130*b - 180) * q^90 + (-285*b + 1371) * q^92 + (298*b + 666) * q^93 + (-94*b - 166) * q^94 + (130*b + 400) * q^95 + (-84*b - 861) * q^96 + (500*b + 314) * q^97 + (4*b + 8) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 10 * q^5 + 6 * q^6 + 6 * q^8 - 16 * q^9 + 30 * q^10 + 28 * q^11 - 66 * q^12 + 36 * q^13 - 10 * q^15 - 14 * q^16 + 76 * q^17 + 72 * q^18 - 160 * q^19 - 30 * q^20 - 44 * q^22 - 22 * q^23 + 162 * q^24 + 50 * q^25 - 148 * q^26 + 2 * q^27 - 250 * q^29 - 30 * q^30 + 132 * q^31 + 42 * q^32 + 148 * q^33 - 444 * q^34 - 192 * q^36 - 416 * q^37 + 376 * q^38 - 84 * q^39 - 30 * q^40 + 106 * q^41 - 666 * q^43 - 156 * q^44 + 80 * q^45 - 402 * q^46 + 196 * q^47 + 130 * q^48 - 150 * q^50 - 572 * q^51 + 348 * q^52 - 952 * q^53 - 402 * q^54 - 140 * q^55 - 472 * q^57 + 510 * q^58 - 840 * q^59 + 330 * q^60 - 98 * q^61 + 4 * q^62 - 602 * q^64 - 180 * q^65 - 236 * q^66 - 1286 * q^67 + 1524 * q^68 - 1426 * q^69 + 1064 * q^71 + 264 * q^72 + 172 * q^73 + 1792 * q^74 + 50 * q^75 + 144 * q^76 + 428 * q^78 - 1240 * q^79 + 70 * q^80 - 754 * q^81 - 438 * q^82 + 1906 * q^83 - 380 * q^85 + 2018 * q^86 - 970 * q^87 + 604 * q^88 - 650 * q^89 - 360 * q^90 + 2742 * q^92 + 1332 * q^93 - 332 * q^94 + 800 * q^95 - 1722 * q^96 + 628 * q^97 + 16 * 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

−1.41421
1.41421

−4.41421

−3.24264

11.4853

−5.00000

14.3137

0

−15.3848

−16.4853

22.0711

1.2

−1.58579

5.24264

−5.48528

−5.00000

−8.31371

0

21.3848

0.485281

7.92893

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.h		2
3.b	odd	2	1		2205.4.a.bg		2
5.b	even	2	1		1225.4.a.v		2
7.b	odd	2	1		245.4.a.g		2
7.c	even	3	2		245.4.e.l		4
7.d	odd	6	2		35.4.e.b	✓	4
21.c	even	2	1		2205.4.a.bf		2
21.g	even	6	2		315.4.j.c		4
28.f	even	6	2		560.4.q.i		4
35.c	odd	2	1		1225.4.a.x		2
35.i	odd	6	2		175.4.e.c		4
35.k	even	12	4		175.4.k.c		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.e.b	✓	4	7.d	odd	6	2
175.4.e.c		4	35.i	odd	6	2
175.4.k.c		8	35.k	even	12	4
245.4.a.g		2	7.b	odd	2	1
245.4.a.h		2	1.a	even	1	1	trivial
245.4.e.l		4	7.c	even	3	2
315.4.j.c		4	21.g	even	6	2
560.4.q.i		4	28.f	even	6	2
1225.4.a.v		2	5.b	even	2	1
1225.4.a.x		2	35.c	odd	2	1
2205.4.a.bf		2	21.c	even	2	1
2205.4.a.bg		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 $S_{4}^{\mathrm{new}}(\Gamma_0(245))$:

$T_{2}^{2} + 6T_{2} + 7$

$T_{3}^{2} - 2T_{3} - 17$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 6T + 7$
$3$	$T^{2} - 2T - 17$
$5$	$(T + 5)^{2}$
$7$	$T^{2}$
$11$	$T^{2} - 28T - 4$