# Properties

 Label 975.4.a.l Level $975$ Weight $4$ Character orbit 975.a Self dual yes Analytic conductor $57.527$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 975.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.5268622556$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} - 3 q^{3} + (\beta_{2} + 3) q^{4} + ( - 3 \beta_1 + 3) q^{6} + ( - 6 \beta_1 - 8) q^{7} + ( - 2 \beta_{2} - \beta_1 + 3) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 - 3 * q^3 + (b2 + 3) * q^4 + (-3*b1 + 3) * q^6 + (-6*b1 - 8) * q^7 + (-2*b2 - b1 + 3) * q^8 + 9 * q^9 $$q + (\beta_1 - 1) q^{2} - 3 q^{3} + (\beta_{2} + 3) q^{4} + ( - 3 \beta_1 + 3) q^{6} + ( - 6 \beta_1 - 8) q^{7} + ( - 2 \beta_{2} - \beta_1 + 3) q^{8} + 9 q^{9} + (6 \beta_{2} + 2 \beta_1 - 8) q^{11} + ( - 3 \beta_{2} - 9) q^{12} - 13 q^{13} + ( - 6 \beta_{2} - 14 \beta_1 - 52) q^{14} + ( - 5 \beta_{2} - 6 \beta_1 - 33) q^{16} + (8 \beta_1 + 46) q^{17} + (9 \beta_1 - 9) q^{18} + (16 \beta_{2} - 6 \beta_1 + 28) q^{19} + (18 \beta_1 + 24) q^{21} + ( - 10 \beta_{2} + 18 \beta_1 + 16) q^{22} + (8 \beta_{2} - 32 \beta_1 + 24) q^{23} + (6 \beta_{2} + 3 \beta_1 - 9) q^{24} + ( - 13 \beta_1 + 13) q^{26} - 27 q^{27} + ( - 2 \beta_{2} - 42 \beta_1 - 12) q^{28} + (8 \beta_{2} + 20 \beta_1 - 10) q^{29} + ( - 4 \beta_{2} - 54 \beta_1 + 120) q^{31} + (20 \beta_{2} - 51 \beta_1 - 41) q^{32} + ( - 18 \beta_{2} - 6 \beta_1 + 24) q^{33} + (8 \beta_{2} + 54 \beta_1 + 34) q^{34} + (9 \beta_{2} + 27) q^{36} + (28 \beta_{2} + 48 \beta_1 - 150) q^{37} + ( - 38 \beta_{2} + 86 \beta_1 - 120) q^{38} + 39 q^{39} + (34 \beta_{2} - 4 \beta_1 + 150) q^{41} + (18 \beta_{2} + 42 \beta_1 + 156) q^{42} + ( - 4 \beta_{2} + 60 \beta_1 + 68) q^{43} + ( - 10 \beta_{2} - 22 \beta_1 + 248) q^{44} + ( - 48 \beta_{2} + 24 \beta_1 - 360) q^{46} + (42 \beta_{2} - 54 \beta_1 + 12) q^{47} + (15 \beta_{2} + 18 \beta_1 + 99) q^{48} + (36 \beta_{2} + 168 \beta_1 + 81) q^{49} + ( - 24 \beta_1 - 138) q^{51} + ( - 13 \beta_{2} - 39) q^{52} + (12 \beta_{2} + 108 \beta_1 + 186) q^{53} + ( - 27 \beta_1 + 27) q^{54} + (10 \beta_{2} + 50 \beta_1 + 12) q^{56} + ( - 48 \beta_{2} + 18 \beta_1 - 84) q^{57} + (4 \beta_{2} + 42 \beta_1 + 194) q^{58} + (2 \beta_{2} + 82 \beta_1 - 624) q^{59} + ( - 28 \beta_{2} + 24 \beta_1 + 78) q^{61} + ( - 46 \beta_{2} + 50 \beta_1 - 652) q^{62} + ( - 54 \beta_1 - 72) q^{63} + ( - 51 \beta_{2} + 36 \beta_1 - 245) q^{64} + (30 \beta_{2} - 54 \beta_1 - 48) q^{66} + (76 \beta_{2} - 42 \beta_1 - 36) q^{67} + (38 \beta_{2} + 56 \beta_1 + 122) q^{68} + ( - 24 \beta_{2} + 96 \beta_1 - 72) q^{69} + (14 \beta_{2} - 134 \beta_1 - 276) q^{71} + ( - 18 \beta_{2} - 9 \beta_1 + 27) q^{72} + ( - 12 \beta_{2} + 240 \beta_1 - 2) q^{73} + ( - 8 \beta_{2} + 10 \beta_1 + 574) q^{74} + (34 \beta_{2} - 138 \beta_1 + 832) q^{76} + ( - 24 \beta_{2} - 136 \beta_1 + 16) q^{77} + (39 \beta_1 - 39) q^{78} + ( - 24 \beta_{2} + 48 \beta_1 - 16) q^{79} + 81 q^{81} + ( - 72 \beta_{2} + 282 \beta_1 - 258) q^{82} + (10 \beta_{2} - 30 \beta_1 + 272) q^{83} + (6 \beta_{2} + 126 \beta_1 + 36) q^{84} + (68 \beta_{2} + 112 \beta_1 + 540) q^{86} + ( - 24 \beta_{2} - 60 \beta_1 + 30) q^{87} + (78 \beta_{2} + 42 \beta_1 - 576) q^{88} + (30 \beta_{2} + 116 \beta_1 + 430) q^{89} + (78 \beta_1 + 104) q^{91} + (56 \beta_{2} - 272 \beta_1 + 504) q^{92} + (12 \beta_{2} + 162 \beta_1 - 360) q^{93} + ( - 138 \beta_{2} + 126 \beta_1 - 636) q^{94} + ( - 60 \beta_{2} + 153 \beta_1 + 123) q^{96} + (4 \beta_{2} + 48 \beta_1 - 1098) q^{97} + (96 \beta_{2} + 393 \beta_1 + 1527) q^{98} + (54 \beta_{2} + 18 \beta_1 - 72) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 - 3 * q^3 + (b2 + 3) * q^4 + (-3*b1 + 3) * q^6 + (-6*b1 - 8) * q^7 + (-2*b2 - b1 + 3) * q^8 + 9 * q^9 + (6*b2 + 2*b1 - 8) * q^11 + (-3*b2 - 9) * q^12 - 13 * q^13 + (-6*b2 - 14*b1 - 52) * q^14 + (-5*b2 - 6*b1 - 33) * q^16 + (8*b1 + 46) * q^17 + (9*b1 - 9) * q^18 + (16*b2 - 6*b1 + 28) * q^19 + (18*b1 + 24) * q^21 + (-10*b2 + 18*b1 + 16) * q^22 + (8*b2 - 32*b1 + 24) * q^23 + (6*b2 + 3*b1 - 9) * q^24 + (-13*b1 + 13) * q^26 - 27 * q^27 + (-2*b2 - 42*b1 - 12) * q^28 + (8*b2 + 20*b1 - 10) * q^29 + (-4*b2 - 54*b1 + 120) * q^31 + (20*b2 - 51*b1 - 41) * q^32 + (-18*b2 - 6*b1 + 24) * q^33 + (8*b2 + 54*b1 + 34) * q^34 + (9*b2 + 27) * q^36 + (28*b2 + 48*b1 - 150) * q^37 + (-38*b2 + 86*b1 - 120) * q^38 + 39 * q^39 + (34*b2 - 4*b1 + 150) * q^41 + (18*b2 + 42*b1 + 156) * q^42 + (-4*b2 + 60*b1 + 68) * q^43 + (-10*b2 - 22*b1 + 248) * q^44 + (-48*b2 + 24*b1 - 360) * q^46 + (42*b2 - 54*b1 + 12) * q^47 + (15*b2 + 18*b1 + 99) * q^48 + (36*b2 + 168*b1 + 81) * q^49 + (-24*b1 - 138) * q^51 + (-13*b2 - 39) * q^52 + (12*b2 + 108*b1 + 186) * q^53 + (-27*b1 + 27) * q^54 + (10*b2 + 50*b1 + 12) * q^56 + (-48*b2 + 18*b1 - 84) * q^57 + (4*b2 + 42*b1 + 194) * q^58 + (2*b2 + 82*b1 - 624) * q^59 + (-28*b2 + 24*b1 + 78) * q^61 + (-46*b2 + 50*b1 - 652) * q^62 + (-54*b1 - 72) * q^63 + (-51*b2 + 36*b1 - 245) * q^64 + (30*b2 - 54*b1 - 48) * q^66 + (76*b2 - 42*b1 - 36) * q^67 + (38*b2 + 56*b1 + 122) * q^68 + (-24*b2 + 96*b1 - 72) * q^69 + (14*b2 - 134*b1 - 276) * q^71 + (-18*b2 - 9*b1 + 27) * q^72 + (-12*b2 + 240*b1 - 2) * q^73 + (-8*b2 + 10*b1 + 574) * q^74 + (34*b2 - 138*b1 + 832) * q^76 + (-24*b2 - 136*b1 + 16) * q^77 + (39*b1 - 39) * q^78 + (-24*b2 + 48*b1 - 16) * q^79 + 81 * q^81 + (-72*b2 + 282*b1 - 258) * q^82 + (10*b2 - 30*b1 + 272) * q^83 + (6*b2 + 126*b1 + 36) * q^84 + (68*b2 + 112*b1 + 540) * q^86 + (-24*b2 - 60*b1 + 30) * q^87 + (78*b2 + 42*b1 - 576) * q^88 + (30*b2 + 116*b1 + 430) * q^89 + (78*b1 + 104) * q^91 + (56*b2 - 272*b1 + 504) * q^92 + (12*b2 + 162*b1 - 360) * q^93 + (-138*b2 + 126*b1 - 636) * q^94 + (-60*b2 + 153*b1 + 123) * q^96 + (4*b2 + 48*b1 - 1098) * q^97 + (96*b2 + 393*b1 + 1527) * q^98 + (54*b2 + 18*b1 - 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 9 * q^3 + 10 * q^4 + 6 * q^6 - 30 * q^7 + 6 * q^8 + 27 * q^9 $$3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9} - 16 q^{11} - 30 q^{12} - 39 q^{13} - 176 q^{14} - 110 q^{16} + 146 q^{17} - 18 q^{18} + 94 q^{19} + 90 q^{21} + 56 q^{22} + 48 q^{23} - 18 q^{24} + 26 q^{26} - 81 q^{27} - 80 q^{28} - 2 q^{29} + 302 q^{31} - 154 q^{32} + 48 q^{33} + 164 q^{34} + 90 q^{36} - 374 q^{37} - 312 q^{38} + 117 q^{39} + 480 q^{41} + 528 q^{42} + 260 q^{43} + 712 q^{44} - 1104 q^{46} + 24 q^{47} + 330 q^{48} + 447 q^{49} - 438 q^{51} - 130 q^{52} + 678 q^{53} + 54 q^{54} + 96 q^{56} - 282 q^{57} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 270 q^{63} - 750 q^{64} - 168 q^{66} - 74 q^{67} + 460 q^{68} - 144 q^{69} - 948 q^{71} + 54 q^{72} + 222 q^{73} + 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 78 q^{78} - 24 q^{79} + 243 q^{81} - 564 q^{82} + 796 q^{83} + 240 q^{84} + 1800 q^{86} + 6 q^{87} - 1608 q^{88} + 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 906 q^{93} - 1920 q^{94} + 462 q^{96} - 3242 q^{97} + 5070 q^{98} - 144 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 9 * q^3 + 10 * q^4 + 6 * q^6 - 30 * q^7 + 6 * q^8 + 27 * q^9 - 16 * q^11 - 30 * q^12 - 39 * q^13 - 176 * q^14 - 110 * q^16 + 146 * q^17 - 18 * q^18 + 94 * q^19 + 90 * q^21 + 56 * q^22 + 48 * q^23 - 18 * q^24 + 26 * q^26 - 81 * q^27 - 80 * q^28 - 2 * q^29 + 302 * q^31 - 154 * q^32 + 48 * q^33 + 164 * q^34 + 90 * q^36 - 374 * q^37 - 312 * q^38 + 117 * q^39 + 480 * q^41 + 528 * q^42 + 260 * q^43 + 712 * q^44 - 1104 * q^46 + 24 * q^47 + 330 * q^48 + 447 * q^49 - 438 * q^51 - 130 * q^52 + 678 * q^53 + 54 * q^54 + 96 * q^56 - 282 * q^57 + 628 * q^58 - 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 270 * q^63 - 750 * q^64 - 168 * q^66 - 74 * q^67 + 460 * q^68 - 144 * q^69 - 948 * q^71 + 54 * q^72 + 222 * q^73 + 1724 * q^74 + 2392 * q^76 - 112 * q^77 - 78 * q^78 - 24 * q^79 + 243 * q^81 - 564 * q^82 + 796 * q^83 + 240 * q^84 + 1800 * q^86 + 6 * q^87 - 1608 * q^88 + 1436 * q^89 + 390 * q^91 + 1296 * q^92 - 906 * q^93 - 1920 * q^94 + 462 * q^96 - 3242 * q^97 + 5070 * q^98 - 144 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 16x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 10$$ v^2 - 2*v - 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 10$$ b2 + 2*b1 + 10

## 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
 −3.20905 −0.526440 4.73549
−4.20905 −3.00000 9.71610 0 12.6271 11.2543 −7.22315 9.00000 0
1.2 −1.52644 −3.00000 −5.66998 0 4.57932 −4.84136 20.8664 9.00000 0
1.3 3.73549 −3.00000 5.95388 0 −11.2065 −36.4129 −7.64325 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 975.4.a.l 3
5.b even 2 1 39.4.a.c 3
15.d odd 2 1 117.4.a.f 3
20.d odd 2 1 624.4.a.t 3
35.c odd 2 1 1911.4.a.k 3
40.e odd 2 1 2496.4.a.bp 3
40.f even 2 1 2496.4.a.bl 3
60.h even 2 1 1872.4.a.bk 3
65.d even 2 1 507.4.a.h 3
65.g odd 4 2 507.4.b.g 6
195.e odd 2 1 1521.4.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 5.b even 2 1
117.4.a.f 3 15.d odd 2 1
507.4.a.h 3 65.d even 2 1
507.4.b.g 6 65.g odd 4 2
624.4.a.t 3 20.d odd 2 1
975.4.a.l 3 1.a even 1 1 trivial
1521.4.a.u 3 195.e odd 2 1
1872.4.a.bk 3 60.h even 2 1
1911.4.a.k 3 35.c odd 2 1
2496.4.a.bl 3 40.f even 2 1
2496.4.a.bp 3 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(975))$$:

 $$T_{2}^{3} + 2T_{2}^{2} - 15T_{2} - 24$$ T2^3 + 2*T2^2 - 15*T2 - 24 $$T_{7}^{3} + 30T_{7}^{2} - 288T_{7} - 1984$$ T7^3 + 30*T7^2 - 288*T7 - 1984

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} - 15 T - 24$$
$3$ $$(T + 3)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 30 T^{2} - 288 T - 1984$$
$11$ $$T^{3} + 16 T^{2} - 2256 T + 30336$$
$13$ $$(T + 13)^{3}$$
$17$ $$T^{3} - 146 T^{2} + 6060 T - 71256$$
$19$ $$T^{3} - 94 T^{2} - 14432 T + 779616$$
$23$ $$T^{3} - 48 T^{2} - 20928 T - 534528$$
$29$ $$T^{3} + 2 T^{2} - 10116 T - 199176$$
$31$ $$T^{3} - 302 T^{2} - 17536 T + 7197248$$
$37$ $$T^{3} + 374 T^{2} - 36964 T - 7758104$$
$41$ $$T^{3} - 480 T^{2} + \cdots + 12919824$$
$43$ $$T^{3} - 260 T^{2} - 38096 T + 3663168$$
$47$ $$T^{3} - 24 T^{2} - 168480 T - 18102528$$
$53$ $$T^{3} - 678 T^{2} - 42228 T + 1471608$$
$59$ $$T^{3} + 1788 T^{2} + \cdots + 137423808$$
$61$ $$T^{3} - 230 T^{2} - 44452 T + 6279512$$
$67$ $$T^{3} + 74 T^{2} - 409216 T - 4260896$$
$71$ $$T^{3} + 948 T^{2} + \cdots - 70464384$$
$73$ $$T^{3} - 222 T^{2} + \cdots - 22780552$$
$79$ $$T^{3} + 24 T^{2} - 78336 T + 7757824$$
$83$ $$T^{3} - 796 T^{2} + \cdots - 13963968$$
$89$ $$T^{3} - 1436 T^{2} + \cdots - 30129888$$
$97$ $$T^{3} + 3242 T^{2} + \cdots + 1218481048$$