Newform orbit 70.6.a.g

• Label: 70.6.a.g
• Level: $70$
• Weight: $6$
• Character orbit: 70.a
• Self dual: yes
• Analytic conductor: $11.227$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$70 = 2 \cdot 5 \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	70.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$11.2268673869$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3369})$
Defining polynomial:	$x^{2} - x - 842$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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 $\beta = \frac{1}{2}(1 + \sqrt{3369})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 4 * q^2 + (-b + 2) * q^3 + 16 * q^4 - 25 * q^5 + (-4*b + 8) * q^6 - 49 * q^7 + 64 * q^8 + (-3*b + 603) * q^9 - 100 * q^10 + (3*b + 478) * q^11 + (-16*b + 32) * q^12 + (15*b - 204) * q^13 - 196 * q^14 + (25*b - 50) * q^15 + 256 * q^16 + (9*b - 1120) * q^17 + (-12*b + 2412) * q^18 + (6*b + 1668) * q^19 - 400 * q^20 + (49*b - 98) * q^21 + (12*b + 1912) * q^22 + (150*b - 300) * q^23 + (-64*b + 128) * q^24 + 625 * q^25 + (60*b - 816) * q^26 + (-363*b + 3246) * q^27 - 784 * q^28 + (-171*b - 2172) * q^29 + (100*b - 200) * q^30 + (204*b - 1120) * q^31 + 1024 * q^32 + (-475*b - 1570) * q^33 + (36*b - 4480) * q^34 + 1225 * q^35 + (-48*b + 9648) * q^36 + (-84*b - 922) * q^37 + (24*b + 6672) * q^38 + (219*b - 13038) * q^39 - 1600 * q^40 + (282*b + 9518) * q^41 + (196*b - 392) * q^42 + (126*b - 7136) * q^43 + (48*b + 7648) * q^44 + (75*b - 15075) * q^45 + (600*b - 1200) * q^46 + (-117*b - 9814) * q^47 + (-256*b + 512) * q^48 + 2401 * q^49 + 2500 * q^50 + (1129*b - 9818) * q^51 + (240*b - 3264) * q^52 + (342*b + 13018) * q^53 + (-1452*b + 12984) * q^54 + (-75*b - 11950) * q^55 - 3136 * q^56 + (-1662*b - 1716) * q^57 + (-684*b - 8688) * q^58 + (816*b - 5960) * q^59 + (400*b - 800) * q^60 + (-1194*b + 2374) * q^61 + (816*b - 4480) * q^62 + (147*b - 29547) * q^63 + 4096 * q^64 + (-375*b + 5100) * q^65 + (-1900*b - 6280) * q^66 + (1152*b + 32028) * q^67 + (144*b - 17920) * q^68 + (450*b - 126900) * q^69 + 4900 * q^70 + (-672*b + 8888) * q^71 + (-192*b + 38592) * q^72 + (1992*b - 19366) * q^73 + (-336*b - 3688) * q^74 + (-625*b + 1250) * q^75 + (96*b + 26688) * q^76 + (-147*b - 23422) * q^77 + (876*b - 52152) * q^78 + (873*b + 35686) * q^79 - 6400 * q^80 + (-2880*b + 165609) * q^81 + (1128*b + 38072) * q^82 + (-1524*b + 50600) * q^83 + (784*b - 1568) * q^84 + (-225*b + 28000) * q^85 + (504*b - 28544) * q^86 + (2001*b + 139638) * q^87 + (192*b + 30592) * q^88 + (2922*b + 39766) * q^89 + (300*b - 60300) * q^90 + (-735*b + 9996) * q^91 + (2400*b - 4800) * q^92 + (1324*b - 174008) * q^93 + (-468*b - 39256) * q^94 + (-150*b - 41700) * q^95 + (-1024*b + 2048) * q^96 + (-2979*b - 50824) * q^97 + 9604 * q^98 + (366*b + 280656) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 8 * q^2 + 3 * q^3 + 32 * q^4 - 50 * q^5 + 12 * q^6 - 98 * q^7 + 128 * q^8 + 1203 * q^9 - 200 * q^10 + 959 * q^11 + 48 * q^12 - 393 * q^13 - 392 * q^14 - 75 * q^15 + 512 * q^16 - 2231 * q^17 + 4812 * q^18 + 3342 * q^19 - 800 * q^20 - 147 * q^21 + 3836 * q^22 - 450 * q^23 + 192 * q^24 + 1250 * q^25 - 1572 * q^26 + 6129 * q^27 - 1568 * q^28 - 4515 * q^29 - 300 * q^30 - 2036 * q^31 + 2048 * q^32 - 3615 * q^33 - 8924 * q^34 + 2450 * q^35 + 19248 * q^36 - 1928 * q^37 + 13368 * q^38 - 25857 * q^39 - 3200 * q^40 + 19318 * q^41 - 588 * q^42 - 14146 * q^43 + 15344 * q^44 - 30075 * q^45 - 1800 * q^46 - 19745 * q^47 + 768 * q^48 + 4802 * q^49 + 5000 * q^50 - 18507 * q^51 - 6288 * q^52 + 26378 * q^53 + 24516 * q^54 - 23975 * q^55 - 6272 * q^56 - 5094 * q^57 - 18060 * q^58 - 11104 * q^59 - 1200 * q^60 + 3554 * q^61 - 8144 * q^62 - 58947 * q^63 + 8192 * q^64 + 9825 * q^65 - 14460 * q^66 + 65208 * q^67 - 35696 * q^68 - 253350 * q^69 + 9800 * q^70 + 17104 * q^71 + 76992 * q^72 - 36740 * q^73 - 7712 * q^74 + 1875 * q^75 + 53472 * q^76 - 46991 * q^77 - 103428 * q^78 + 72245 * q^79 - 12800 * q^80 + 328338 * q^81 + 77272 * q^82 + 99676 * q^83 - 2352 * q^84 + 55775 * q^85 - 56584 * q^86 + 281277 * q^87 + 61376 * q^88 + 82454 * q^89 - 120300 * q^90 + 19257 * q^91 - 7200 * q^92 - 346692 * q^93 - 78980 * q^94 - 83550 * q^95 + 3072 * q^96 - 104627 * q^97 + 19208 * q^98 + 561678 * 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

29.5215
−28.5215

4.00000

−27.5215

16.0000

−25.0000

−110.086

−49.0000

64.0000

514.435

−100.000

1.2

4.00000

30.5215

16.0000

−25.0000

122.086

−49.0000

64.0000

688.565

−100.000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$+1$
$7$	$+1$

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	70.6.a.g	✓	2
3.b	odd	2	1		630.6.a.u		2
4.b	odd	2	1		560.6.a.m		2
5.b	even	2	1		350.6.a.q		2
5.c	odd	4	2		350.6.c.j		4
7.b	odd	2	1		490.6.a.v		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.6.a.g	✓	2	1.a	even	1	1	trivial
350.6.a.q		2	5.b	even	2	1
350.6.c.j		4	5.c	odd	4	2
490.6.a.v		2	7.b	odd	2	1
560.6.a.m		2	4.b	odd	2	1
630.6.a.u		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} - 3T_{3} - 840$ acting on $S_{6}^{\mathrm{new}}(\Gamma_0(70))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T - 4)^{2}$
$3$	$T^{2} - 3T - 840$
$5$	$(T + 25)^{2}$
$7$	$(T + 49)^{2}$
$11$	$T^{2} - 959T + 222340$