Newform orbit 35.4.a.b

• Label: 35.4.a.b
• Level: $35$
• Weight: $4$
• Character orbit: 35.a
• Self dual: yes
• Analytic conductor: $2.065$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	35.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.06506685020\)
Analytic rank:	\(0\)
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:	yes
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 + 4) * q^2 + (-4*b + 1) * q^3 + (8*b + 10) * q^4 - 5 * q^5 + (-15*b - 4) * q^6 - 7 * q^7 + (34*b + 24) * q^8 + (-8*b + 6) * q^9 + (-5*b - 20) * q^10 + (-32*b - 7) * q^11 + (-32*b - 54) * q^12 + (4*b + 25) * q^13 + (-7*b - 28) * q^14 + (20*b - 5) * q^15 + (96*b + 84) * q^16 + (44*b - 25) * q^17 + (-26*b + 8) * q^18 + (44*b + 18) * q^19 + (-40*b - 50) * q^20 + (28*b - 7) * q^21 + (-135*b - 92) * q^22 + (-68*b + 122) * q^23 + (-62*b - 248) * q^24 + 25 * q^25 + (41*b + 108) * q^26 + (76*b + 43) * q^27 + (-56*b - 70) * q^28 + (24*b - 13) * q^29 + (75*b + 20) * q^30 + (-180*b - 60) * q^31 + (196*b + 336) * q^32 + (-4*b + 249) * q^33 + (151*b - 12) * q^34 + 35 * q^35 + (-32*b - 68) * q^36 + (-60*b + 282) * q^37 + (194*b + 160) * q^38 + (-96*b - 7) * q^39 + (-170*b - 120) * q^40 + (124*b - 164) * q^41 + (105*b + 28) * q^42 + (68*b - 130) * q^43 + (-376*b - 582) * q^44 + (40*b - 30) * q^45 + (-150*b + 352) * q^46 + (-132*b - 175) * q^47 + (-240*b - 684) * q^48 + 49 * q^49 + (25*b + 100) * q^50 + (144*b - 377) * q^51 + (240*b + 314) * q^52 + (128*b - 28) * q^53 + (347*b + 324) * q^54 + (160*b + 35) * q^55 + (-238*b - 168) * q^56 + (-28*b - 334) * q^57 + (83*b - 4) * q^58 - 616 * q^59 + (160*b + 270) * q^60 + (-108*b + 168) * q^61 + (-780*b - 600) * q^62 + (56*b - 42) * q^63 + (352*b + 1064) * q^64 + (-20*b - 125) * q^65 + (233*b + 988) * q^66 + (-64*b - 76) * q^67 + (240*b + 454) * q^68 + (-556*b + 666) * q^69 + (35*b + 140) * q^70 - 952 * q^71 + (12*b - 400) * q^72 + (-344*b + 338) * q^73 + (42*b + 1008) * q^74 + (-100*b + 25) * q^75 + (584*b + 884) * q^76 + (224*b + 49) * q^77 + (-391*b - 220) * q^78 + (248*b + 507) * q^79 + (-480*b - 420) * q^80 + (120*b - 727) * q^81 + (332*b - 408) * q^82 + (600*b - 188) * q^83 + (224*b + 378) * q^84 + (-220*b + 125) * q^85 + (142*b - 384) * q^86 + (76*b - 205) * q^87 + (-1006*b - 2344) * q^88 + (44*b - 108) * q^89 + (130*b - 40) * q^90 + (-28*b - 175) * q^91 + (296*b + 132) * q^92 + (60*b + 1380) * q^93 + (-703*b - 964) * q^94 + (-220*b - 90) * q^95 + (-1148*b - 1232) * q^96 + (220*b + 1371) * q^97 + (49*b + 196) * q^98 + (-136*b + 470) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 8 * q^2 + 2 * q^3 + 20 * q^4 - 10 * q^5 - 8 * q^6 - 14 * q^7 + 48 * q^8 + 12 * q^9 - 40 * q^10 - 14 * q^11 - 108 * q^12 + 50 * q^13 - 56 * q^14 - 10 * q^15 + 168 * q^16 - 50 * q^17 + 16 * q^18 + 36 * q^19 - 100 * q^20 - 14 * q^21 - 184 * q^22 + 244 * q^23 - 496 * q^24 + 50 * q^25 + 216 * q^26 + 86 * q^27 - 140 * q^28 - 26 * q^29 + 40 * q^30 - 120 * q^31 + 672 * q^32 + 498 * q^33 - 24 * q^34 + 70 * q^35 - 136 * q^36 + 564 * q^37 + 320 * q^38 - 14 * q^39 - 240 * q^40 - 328 * q^41 + 56 * q^42 - 260 * q^43 - 1164 * q^44 - 60 * q^45 + 704 * q^46 - 350 * q^47 - 1368 * q^48 + 98 * q^49 + 200 * q^50 - 754 * q^51 + 628 * q^52 - 56 * q^53 + 648 * q^54 + 70 * q^55 - 336 * q^56 - 668 * q^57 - 8 * q^58 - 1232 * q^59 + 540 * q^60 + 336 * q^61 - 1200 * q^62 - 84 * q^63 + 2128 * q^64 - 250 * q^65 + 1976 * q^66 - 152 * q^67 + 908 * q^68 + 1332 * q^69 + 280 * q^70 - 1904 * q^71 - 800 * q^72 + 676 * q^73 + 2016 * q^74 + 50 * q^75 + 1768 * q^76 + 98 * q^77 - 440 * q^78 + 1014 * q^79 - 840 * q^80 - 1454 * q^81 - 816 * q^82 - 376 * q^83 + 756 * q^84 + 250 * q^85 - 768 * q^86 - 410 * q^87 - 4688 * q^88 - 216 * q^89 - 80 * q^90 - 350 * q^91 + 264 * q^92 + 2760 * q^93 - 1928 * q^94 - 180 * q^95 - 2464 * q^96 + 2742 * q^97 + 392 * q^98 + 940 * 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

2.58579

6.65685

−1.31371

−5.00000

17.2132

−7.00000

−24.0833

17.3137

−12.9289

1.2

5.41421

−4.65685

21.3137

−5.00000

−25.2132

−7.00000

72.0833

−5.31371

−27.0711

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	35.4.a.b	✓	2
3.b	odd	2	1		315.4.a.f		2
4.b	odd	2	1		560.4.a.r		2
5.b	even	2	1		175.4.a.c		2
5.c	odd	4	2		175.4.b.c		4
7.b	odd	2	1		245.4.a.k		2
7.c	even	3	2		245.4.e.h		4
7.d	odd	6	2		245.4.e.i		4
8.b	even	2	1		2240.4.a.bn		2
8.d	odd	2	1		2240.4.a.bo		2
15.d	odd	2	1		1575.4.a.z		2
21.c	even	2	1		2205.4.a.u		2
35.c	odd	2	1		1225.4.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.4.a.b	✓	2	1.a	even	1	1	trivial
175.4.a.c		2	5.b	even	2	1
175.4.b.c		4	5.c	odd	4	2
245.4.a.k		2	7.b	odd	2	1
245.4.e.h		4	7.c	even	3	2
245.4.e.i		4	7.d	odd	6	2
315.4.a.f		2	3.b	odd	2	1
560.4.a.r		2	4.b	odd	2	1
1225.4.a.m		2	35.c	odd	2	1
1575.4.a.z		2	15.d	odd	2	1
2205.4.a.u		2	21.c	even	2	1
2240.4.a.bn		2	8.b	even	2	1
2240.4.a.bo		2	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 8T_{2} + 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(35))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 8T + 14 \)
$3$	\( T^{2} - 2T - 31 \)
$5$	\( (T + 5)^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} + 14T - 1999 \)