Newform orbit 115.4.a.f

• Label: 115.4.a.f
• Level: $115$
• Weight: $4$
• Character orbit: 115.a
• Self dual: yes
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$115 = 5 \cdot 23$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	115.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$6.78521965066$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
Defining polynomial:	$x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400$
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 a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-b1 + 1) * q^2 + b6 * q^3 + (b2 + 5) * q^4 + 5 * q^5 + (b7 + b5 + b3 - b1 + 1) * q^6 + (-b7 + b6 - b5 - b1 + 1) * q^7 + (b6 - b5 + b4 - b3 - 4*b1 + 1) * q^8 + (-b7 - b6 + b5 - b4 - b3 - b2 + 20) * q^9 + (-5*b1 + 5) * q^10 + (b7 + b6 - 3*b4 - b2 - b1 + 5) * q^11 + (b7 + 3*b6 + 2*b4 - 3*b3 - b2 - 2*b1 + 7) * q^12 + (-b7 - 3*b6 - b5 + b4 + b3 - 3*b2 - 4*b1 + 5) * q^13 + (-2*b7 - 8*b6 + 3*b3 + 2*b2 - b1 + 20) * q^14 + 5*b6 * q^15 + (2*b7 - 5*b6 + b5 + 2*b4 + 2*b3 + 3*b1 + 13) * q^16 + (-2*b7 + b6 - 2*b5 + 5*b4 + b3 + b2 + 11*b1 + 6) * q^17 + (-8*b6 + 2*b5 - 7*b4 + 4*b3 - 3*b2 - 8*b1 + 13) * q^18 + (4*b7 - b6 - b5 - 3*b3 + 2*b2 + 5*b1 + 20) * q^19 + (5*b2 + 25) * q^20 + (b7 - 2*b6 + 5*b5 + 7*b4 - b3 - 5*b2 + 20*b1 + 37) * q^21 + (3*b7 + 5*b6 + 2*b5 - 13*b4 - 4*b3 - b2 + b1 + 19) * q^22 + 23 * q^23 + (3*b7 - 4*b6 - b5 + b4 - 2*b3 + 8*b2 + 12*b1 + 17) * q^24 + 25 * q^25 + (-8*b7 - 8*b6 - 2*b5 + 3*b4 + 2*b3 + 5*b2 + 21*b1 + 42) * q^26 + (-2*b7 + 20*b6 - b5 - 2*b4 + 7*b3 + 4*b2 + 27*b1 - 70) * q^27 + (-10*b7 - 9*b6 - 5*b5 + 8*b4 - 7*b3 - 4*b2 - 21*b1 + 26) * q^28 + (-2*b7 + 2*b6 - 6*b5 - 5*b4 - 5*b3 + 11*b2 + 9*b1 + 18) * q^29 + (5*b7 + 5*b5 + 5*b3 - 5*b1 + 5) * q^30 + (4*b7 + 17*b6 - 3*b5 - 3*b4 - 8*b3 - 5*b2 + 16*b1 - 10) * q^31 + (-4*b7 + 12*b6 + 2*b5 + 4*b4 - 2*b3 - 3*b2 + 16*b1 - 47) * q^32 + (10*b7 + 5*b6 + 9*b5 + b3 - 6*b2 - 5*b1 + 2) * q^33 + (-7*b7 - 12*b6 - 3*b5 + 23*b4 + 8*b3 - 5*b2 - 27*b1 - 119) * q^34 + (-5*b7 + 5*b6 - 5*b5 - 5*b1 + 5) * q^35 + (-6*b7 + 21*b6 - 15*b5 - 15*b4 - 6*b3 + b2 + 18*b1 - 67) * q^36 + (5*b7 + 2*b6 + 6*b5 - 14*b4 + 7*b3 - 4*b2 + 6*b1 - 43) * q^37 + (12*b7 + 15*b6 - b5 - 4*b4 - 13*b3 + 4*b2 - 37*b1 - 32) * q^38 + (2*b7 - 2*b6 + 5*b5 + 4*b4 + 17*b3 - 2*b2 + 7*b1 - 130) * q^39 + (5*b6 - 5*b5 + 5*b4 - 5*b3 - 20*b1 + 5) * q^40 + (-b7 - 23*b6 + 4*b5 - 6*b4 + 9*b3 - 4*b2 + 36*b1 + 71) * q^41 + (7*b7 + 8*b6 + 9*b5 + 21*b4 + 13*b3 - 21*b2 - 14*b1 - 265) * q^42 + (4*b7 - 2*b6 + 14*b5 + 10*b4 + 10*b3 - 6*b2 - 48*b1 - 24) * q^43 + (13*b7 + b6 + 12*b5 - 37*b4 - 10*b3 - 3*b2 - 3*b1 - 19) * q^44 + (-5*b7 - 5*b6 + 5*b5 - 5*b4 - 5*b3 - 5*b2 + 100) * q^45 + (-23*b1 + 23) * q^46 + (3*b7 - 10*b6 - 2*b5 + 11*b4 - 4*b3 + 21*b2 + 35*b1 + 3) * q^47 + (-3*b7 - 6*b6 - 11*b5 - 8*b4 + 5*b3 + 4*b2 - 65*b1 - 155) * q^48 + (-4*b7 - 43*b6 + 4*b5 - 7*b4 - 11*b3 + b2 - 11*b1 + 111) * q^49 + (-25*b1 + 25) * q^50 + (-24*b7 - 7*b6 - 15*b5 + 8*b4 - 15*b3 - 2*b2 + 23*b1 + 104) * q^51 + (-22*b7 - 17*b6 - 9*b5 + 13*b4 + 2*b3 - 56*b1 - 226) * q^52 + (7*b7 - 36*b6 + 10*b5 + 14*b4 + 3*b3 + 20*b2 - 42*b1 + 191) * q^53 + (b7 + 11*b6 + 8*b5 + 10*b4 + 12*b3 - 36*b2 - 18*b1 - 347) * q^54 + (5*b7 + 5*b6 - 15*b4 - 5*b2 - 5*b1 + 25) * q^55 + (-4*b7 - 31*b6 - 3*b5 + 14*b4 + 11*b3 + 20*b2 + 59*b1 + 112) * q^56 + (9*b7 + 18*b6 - 25*b5 - 5*b4 - 27*b3 + 31*b2 - 14*b1 - 163) * q^57 + (2*b7 - 28*b6 - 10*b5 - 19*b4 - 9*b3 + b2 - 100*b1 - 2) * q^58 + (-11*b7 + 14*b6 - 22*b5 + 32*b4 - 21*b3 - 10*b2 + 16*b1 - 93) * q^59 + (5*b7 + 15*b6 + 10*b4 - 15*b3 - 5*b2 - 10*b1 + 35) * q^60 + (5*b7 + b6 - 2*b5 + 9*b4 - 6*b3 + 27*b2 + 17*b1 + 171) * q^61 + (38*b7 - 11*b6 + 27*b5 - 33*b4 + 12*b3 - b2 + 21*b1 - 188) * q^62 + (-30*b7 + 66*b6 - 27*b5 - 51*b4 + 8*b3 + 43*b2 + 4*b1 - 48) * q^63 + (-6*b7 + 11*b6 + 11*b5 - 7*b4 + 19*b3 - 18*b2 + 26*b1 - 357) * q^64 + (-5*b7 - 15*b6 - 5*b5 + 5*b4 + 5*b3 - 15*b2 - 20*b1 + 25) * q^65 + (32*b7 + 75*b6 + 19*b5 - 4*b4 - 11*b3 - 4*b2 + 37*b1 - 12) * q^66 + (-19*b7 - 34*b6 + 8*b5 + 18*b4 + 13*b3 - 4*b2 - 32*b1 - 181) * q^67 + (-29*b7 - 37*b6 - 2*b5 + 63*b4 + 18*b3 + 33*b2 + 93*b1 + 101) * q^68 + 23*b6 * q^69 + (-10*b7 - 40*b6 + 15*b3 + 10*b2 - 5*b1 + 100) * q^70 + (21*b7 - 45*b6 + 24*b5 - 4*b4 + 31*b3 - 26*b2 - 34*b1 + 207) * q^71 + (6*b7 - 19*b6 - 5*b5 - 15*b4 - 18*b3 + 21*b2 + 88*b1 - 251) * q^72 + (27*b7 + 8*b6 + 16*b5 - 7*b4 + 38*b3 + 7*b2 + 7*b1 + 13) * q^73 + (4*b7 + 59*b6 + 5*b5 - 46*b4 - 24*b3 - 34*b2 + 50*b1 - 136) * q^74 + 25*b6 * q^75 + (32*b7 + 43*b6 + 31*b5 - 38*b4 + 7*b3 + 44*b2 - b1 + 284) * q^76 + (-19*b7 - 66*b6 - 5*b5 - 25*b4 + 3*b3 - 33*b2 + 22*b1 + 53) * q^77 + (-27*b7 + 71*b6 - 12*b5 + 48*b4 - 14*b3 - 28*b2 + 108*b1 - 259) * q^78 + (22*b7 - 18*b6 + 10*b5 - 16*b4 - 6*b3 - 16*b2 - 26*b1 + 102) * q^79 + (10*b7 - 25*b6 + 5*b5 + 10*b4 + 10*b3 + 15*b1 + 65) * q^80 + (-11*b7 - 98*b6 + 5*b5 + 37*b4 - 17*b3 - 59*b2 - 30*b1 + 570) * q^81 + (-39*b7 + 25*b6 - 24*b5 - 10*b4 - 27*b3 - 60*b2 - 69*b1 - 407) * q^82 + (9*b7 + 76*b6 + 8*b5 + 12*b4 - 23*b3 - 38*b2 - 26*b1 + 207) * q^83 + (-3*b7 + 94*b6 - 15*b5 + 33*b4 + 33*b3 + 51*b2 + 234*b1 - 563) * q^84 + (-10*b7 + 5*b6 - 10*b5 + 25*b4 + 5*b3 + 5*b2 + 55*b1 + 30) * q^85 + (76*b6 + 8*b5 + 54*b4 + 6*b3 + 24*b2 + 106*b1 + 432) * q^86 + (29*b7 + 28*b6 - 2*b5 + 61*b4 - 68*b3 - 5*b2 + 133*b1 - 123) * q^87 + (35*b7 + 29*b6 + 10*b5 - 67*b4 - 18*b3 - 27*b2 + 53*b1 - 145) * q^88 + (-10*b7 + 46*b6 - 8*b5 - 18*b4 + 14*b3 - 26*b2 - 14*b1 + 172) * q^89 + (-40*b6 + 10*b5 - 35*b4 + 20*b3 - 15*b2 - 40*b1 + 65) * q^90 + (13*b7 - 9*b6 - 8*b5 - 39*b4 + 24*b3 + 27*b2 - 81*b1 + 341) * q^91 + (23*b2 + 115) * q^92 + (-12*b7 + 32*b6 - 27*b5 - 26*b4 - 51*b3 + 44*b2 + 97*b1 + 416) * q^93 + (2*b7 + 23*b6 - 29*b5 + 57*b4 - 27*b3 - 13*b2 - 172*b1 - 358) * q^94 + (20*b7 - 5*b6 - 5*b5 - 15*b3 + 10*b2 + 25*b1 + 100) * q^95 + (-57*b7 + 11*b6 - 18*b5 - 26*b4 - 9*b3 + 7*b2 + 84*b1 + 573) * q^96 + (9*b7 - 91*b6 - 26*b5 + 7*b4 + 48*b3 + 29*b2 + 43*b1 - 129) * q^97 + (-25*b7 - 48*b6 - 29*b5 - 49*b4 - 24*b3 + 3*b2 - 34*b1 + 202) * q^98 + (-40*b7 - 29*b6 - 15*b5 - 10*b4 + 27*b3 + 60*b2 - 183*b1 + 184) * q^99
$\operatorname{Tr}(f)(q)$	$=$	8 * q + 6 * q^2 + 42 * q^4 + 40 * q^5 + 11 * q^7 + 3 * q^8 + 158 * q^9 + 30 * q^10 + 41 * q^11 + 48 * q^12 + 28 * q^13 + 161 * q^14 + 98 * q^16 + 71 * q^17 + 84 * q^18 + 177 * q^19 + 210 * q^20 + 292 * q^21 + 171 * q^22 + 184 * q^23 + 177 * q^24 + 200 * q^25 + 396 * q^26 - 495 * q^27 + 179 * q^28 + 225 * q^29 - 36 * q^31 - 360 * q^32 - 53 * q^33 - 1051 * q^34 + 55 * q^35 - 396 * q^36 - 348 * q^37 - 309 * q^38 - 1077 * q^39 + 15 * q^40 + 620 * q^41 - 2288 * q^42 - 390 * q^43 - 141 * q^44 + 790 * q^45 + 138 * q^46 + 123 * q^47 - 1304 * q^48 + 881 * q^49 + 150 * q^50 + 957 * q^51 - 1890 * q^52 + 1406 * q^53 - 2949 * q^54 + 205 * q^55 + 1031 * q^56 - 1142 * q^57 - 129 * q^58 - 676 * q^59 + 240 * q^60 + 1447 * q^61 - 1556 * q^62 - 58 * q^63 - 2883 * q^64 + 140 * q^65 - 119 * q^66 - 1582 * q^67 + 953 * q^68 + 805 * q^70 + 1396 * q^71 - 1728 * q^72 + 17 * q^73 - 964 * q^74 + 2271 * q^76 + 488 * q^77 - 1919 * q^78 + 708 * q^79 + 490 * q^80 + 4316 * q^81 - 3332 * q^82 + 1486 * q^83 - 3970 * q^84 + 355 * q^85 + 3570 * q^86 - 803 * q^87 - 1031 * q^88 + 1360 * q^89 + 420 * q^90 + 2693 * q^91 + 966 * q^92 + 3833 * q^93 - 3207 * q^94 + 885 * q^95 + 4956 * q^96 - 855 * q^97 + 1817 * q^98 + 1319 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400$ :

$\beta_{1}$	$=$	$\nu$
$\beta_{2}$	$=$	$\nu^{2} - 2\nu - 12$
$\beta_{3}$	$=$	$( -21\nu^{7} + 372\nu^{6} - 611\nu^{5} - 10501\nu^{4} + 19940\nu^{3} + 88331\nu^{2} - 77178\nu - 104240 ) / 3680$
$\beta_{4}$	$=$	$( -43\nu^{7} + 236\nu^{6} + 1027\nu^{5} - 3803\nu^{4} - 10340\nu^{3} + 373\nu^{2} + 31226\nu + 17520 ) / 3680$
$\beta_{5}$	$=$	$( 57\nu^{7} - 484\nu^{6} - 1233\nu^{5} + 15097\nu^{4} + 6860\nu^{3} - 135927\nu^{2} - 25774\nu + 211440 ) / 3680$
$\beta_{6}$	$=$	$( 79\nu^{7} - 348\nu^{6} - 2871\nu^{5} + 8399\nu^{4} + 33460\nu^{3} - 36929\nu^{2} - 71618\nu + 30800 ) / 3680$
$\beta_{7}$	$=$	$( 233\nu^{7} - 1236\nu^{6} - 6977\nu^{5} + 29593\nu^{4} + 63260\nu^{3} - 146183\nu^{2} - 44766\nu + 109520 ) / 3680$

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

6.03341
4.85207
1.76920
0.954235
−1.16661
−3.05234
−3.57019
−3.81977

−5.03341

−4.21852

17.3352

5.00000

21.2335

−19.9088

−46.9881

−9.20409

−25.1671

1.2

−3.85207

8.45088

6.83843

5.00000

−32.5534

16.9190

4.47444

44.4174

−19.2603

1.3

−0.769202

−0.0181662

−7.40833

5.00000

0.0139735

10.0918

11.8521

−26.9997

−3.84601

1.4

0.0457645

−10.2193

−7.99791

5.00000

−0.467681

−33.8594

−0.732137

77.4338

0.228823

1.5

2.16661

8.59146

−3.30580

5.00000

18.6143

9.86011

−24.4953

46.8131

10.8331

1.6

4.05234

−8.94300

8.42146

5.00000

−36.2401

32.7645

1.70790

52.9772

20.2617

1.7

4.57019

4.96142

12.8867

5.00000

22.6746

−18.4259

22.3330

−2.38431

22.8510

1.8

4.81977

1.39521

15.2302

5.00000

6.72460

13.5587

34.8481

−25.0534

24.0989

Atkin-Lehner signs

$p$	Sign
$5$	$-1$
$23$	$-1$

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	115.4.a.f	✓	8
3.b	odd	2	1		1035.4.a.r		8
4.b	odd	2	1		1840.4.a.v		8
5.b	even	2	1		575.4.a.n		8
5.c	odd	4	2		575.4.b.k		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.4.a.f	✓	8	1.a	even	1	1	trivial
575.4.a.n		8	5.b	even	2	1
575.4.b.k		16	5.c	odd	4	2
1035.4.a.r		8	3.b	odd	2	1
1840.4.a.v		8	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{8} - 6T_{2}^{7} - 35T_{2}^{6} + 249T_{2}^{5} + 170T_{2}^{4} - 2565T_{2}^{3} + 1916T_{2}^{2} + 2802T_{2} - 132$ acting on $S_{4}^{\mathrm{new}}(\Gamma_0(115))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 - 6*T^7 - 35*T^6 + 249*T^5 + 170*T^4 - 2565*T^3 + 1916*T^2 + 2802*T - 132
$3$	T^8 - 187*T^6 + 165*T^5 + 10290*T^4 - 15441*T^3 - 137140*T^2 + 191280*T + 3520
$5$	$(T - 5)^{8}$
$7$	T^8 - 11*T^7 - 1752*T^6 + 22539*T^5 + 783793*T^4 - 12717810*T^3 - 76746972*T^2 + 2135252960*T - 9289584128
$11$	T^8 - 41*T^7 - 4775*T^6 + 224992*T^5 + 2597924*T^4 - 152142824*T^3 - 522343552*T^2 + 21630255072*T - 11345758080