# Properties

 Label 975.2.i.k Level $975$ Weight $2$ Character orbit 975.i Analytic conductor $7.785$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + (\beta_{3} + \beta_1) q^{6} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{3} - 4) q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + b2 * q^3 + (b3 + 2*b2 + b1 - 2) * q^4 + (b3 + b1) * q^6 + (b3 + b2 + b1 - 1) * q^7 + (b3 - 4) * q^8 + (b2 - 1) * q^9 $$q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + (\beta_{3} + \beta_1) q^{6} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{3} - 4) q^{8} + (\beta_{2} - 1) q^{9} + 2 \beta_{2} q^{11} + (\beta_{3} - 2) q^{12} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{13} + (2 \beta_{3} - 4) q^{14} - 3 \beta_1 q^{16} + ( - \beta_{3} - \beta_1) q^{17} + \beta_{3} q^{18} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{19} + (\beta_{3} - 1) q^{21} + (2 \beta_{3} + 2 \beta_1) q^{22} + 2 \beta_{2} q^{23} + ( - 4 \beta_{2} - \beta_1) q^{24} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 8) q^{26} - q^{27} + ( - 6 \beta_{2} - 4 \beta_1) q^{28} + ( - 2 \beta_{2} + 3 \beta_1) q^{29} + (\beta_{3} + 1) q^{31} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{32} + (2 \beta_{2} - 2) q^{33} + ( - \beta_{3} + 4) q^{34} + ( - 2 \beta_{2} - \beta_1) q^{36} + (6 \beta_{2} - \beta_1) q^{37} + (2 \beta_{3} + 8) q^{38} + ( - 2 \beta_{3} - \beta_1 - 1) q^{39} - \beta_1 q^{41} + ( - 4 \beta_{2} - 2 \beta_1) q^{42} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{43} + (2 \beta_{3} - 4) q^{44} + (2 \beta_{3} + 2 \beta_1) q^{46} + ( - 4 \beta_{3} - 2) q^{47} + ( - 3 \beta_{3} - 3 \beta_1) q^{48} + (2 \beta_{2} - 3 \beta_1) q^{49} - \beta_{3} q^{51} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{52} + (3 \beta_{3} - 4) q^{53} - \beta_1 q^{54} + ( - 6 \beta_{3} - 8 \beta_{2} - 6 \beta_1 + 8) q^{56} + ( - 2 \beta_{3} - 4) q^{57} + (\beta_{3} + 12 \beta_{2} + \beta_1 - 12) q^{58} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 6) q^{59} + (2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 7) q^{61} - 4 \beta_{2} q^{62} + ( - \beta_{2} - \beta_1) q^{63} + (\beta_{3} + 4) q^{64} + 2 \beta_{3} q^{66} + (3 \beta_{2} - \beta_1) q^{67} + (4 \beta_{2} + 3 \beta_1) q^{68} + (2 \beta_{2} - 2) q^{69} + (14 \beta_{2} - 14) q^{71} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{72} + (2 \beta_{3} + 7) q^{73} + (5 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 4) q^{74} + 2 \beta_1 q^{76} + (2 \beta_{3} - 2) q^{77} + ( - \beta_{3} + 4 \beta_{2} + 4) q^{78} + ( - \beta_{3} + 7) q^{79} - \beta_{2} q^{81} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{82} + ( - 2 \beta_{3} + 4) q^{83} + ( - 4 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 6) q^{84} + ( - 2 \beta_{3} - 4) q^{86} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 2) q^{87} + ( - 8 \beta_{2} - 2 \beta_1) q^{88} + ( - 8 \beta_{2} - 2 \beta_1) q^{89} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 4) q^{91} + (2 \beta_{3} - 4) q^{92} + (\beta_{2} - \beta_1) q^{93} + (16 \beta_{2} + 2 \beta_1) q^{94} + ( - \beta_{3} + 4) q^{96} + ( - \beta_{3} + 7 \beta_{2} - \beta_1 - 7) q^{97} + ( - \beta_{3} - 12 \beta_{2} - \beta_1 + 12) q^{98} - 2 q^{99}+O(q^{100})$$ q + b1 * q^2 + b2 * q^3 + (b3 + 2*b2 + b1 - 2) * q^4 + (b3 + b1) * q^6 + (b3 + b2 + b1 - 1) * q^7 + (b3 - 4) * q^8 + (b2 - 1) * q^9 + 2*b2 * q^11 + (b3 - 2) * q^12 + (-b3 + b2 - 2*b1 - 1) * q^13 + (2*b3 - 4) * q^14 - 3*b1 * q^16 + (-b3 - b1) * q^17 + b3 * q^18 + (-2*b3 + 4*b2 - 2*b1 - 4) * q^19 + (b3 - 1) * q^21 + (2*b3 + 2*b1) * q^22 + 2*b2 * q^23 + (-4*b2 - b1) * q^24 + (-b3 - 4*b2 - b1 + 8) * q^26 - q^27 + (-6*b2 - 4*b1) * q^28 + (-2*b2 + 3*b1) * q^29 + (b3 + 1) * q^31 + (-b3 - 4*b2 - b1 + 4) * q^32 + (2*b2 - 2) * q^33 + (-b3 + 4) * q^34 + (-2*b2 - b1) * q^36 + (6*b2 - b1) * q^37 + (2*b3 + 8) * q^38 + (-2*b3 - b1 - 1) * q^39 - b1 * q^41 + (-4*b2 - 2*b1) * q^42 + (b3 - 3*b2 + b1 + 3) * q^43 + (2*b3 - 4) * q^44 + (2*b3 + 2*b1) * q^46 + (-4*b3 - 2) * q^47 + (-3*b3 - 3*b1) * q^48 + (2*b2 - 3*b1) * q^49 - b3 * q^51 + (-3*b3 + 2*b2 + 2*b1 + 4) * q^52 + (3*b3 - 4) * q^53 - b1 * q^54 + (-6*b3 - 8*b2 - 6*b1 + 8) * q^56 + (-2*b3 - 4) * q^57 + (b3 + 12*b2 + b1 - 12) * q^58 + (-2*b3 - 6*b2 - 2*b1 + 6) * q^59 + (2*b3 + 7*b2 + 2*b1 - 7) * q^61 - 4*b2 * q^62 + (-b2 - b1) * q^63 + (b3 + 4) * q^64 + 2*b3 * q^66 + (3*b2 - b1) * q^67 + (4*b2 + 3*b1) * q^68 + (2*b2 - 2) * q^69 + (14*b2 - 14) * q^71 + (-b3 - 4*b2 - b1 + 4) * q^72 + (2*b3 + 7) * q^73 + (5*b3 - 4*b2 + 5*b1 + 4) * q^74 + 2*b1 * q^76 + (2*b3 - 2) * q^77 + (-b3 + 4*b2 + 4) * q^78 + (-b3 + 7) * q^79 - b2 * q^81 + (-b3 - 4*b2 - b1 + 4) * q^82 + (-2*b3 + 4) * q^83 + (-4*b3 - 6*b2 - 4*b1 + 6) * q^84 + (-2*b3 - 4) * q^86 + (3*b3 - 2*b2 + 3*b1 + 2) * q^87 + (-8*b2 - 2*b1) * q^88 + (-8*b2 - 2*b1) * q^89 + (-2*b3 + 3*b2 + b1 + 4) * q^91 + (2*b3 - 4) * q^92 + (b2 - b1) * q^93 + (16*b2 + 2*b1) * q^94 + (-b3 + 4) * q^96 + (-b3 + 7*b2 - b1 - 7) * q^97 + (-b3 - 12*b2 - b1 + 12) * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 2 q^{3} - 5 q^{4} - q^{6} - 3 q^{7} - 18 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + q^2 + 2 * q^3 - 5 * q^4 - q^6 - 3 * q^7 - 18 * q^8 - 2 * q^9 $$4 q + q^{2} + 2 q^{3} - 5 q^{4} - q^{6} - 3 q^{7} - 18 q^{8} - 2 q^{9} + 4 q^{11} - 10 q^{12} - 2 q^{13} - 20 q^{14} - 3 q^{16} + q^{17} - 2 q^{18} - 6 q^{19} - 6 q^{21} - 2 q^{22} + 4 q^{23} - 9 q^{24} + 25 q^{26} - 4 q^{27} - 16 q^{28} - q^{29} + 2 q^{31} + 9 q^{32} - 4 q^{33} + 18 q^{34} - 5 q^{36} + 11 q^{37} + 28 q^{38} - q^{39} - q^{41} - 10 q^{42} + 5 q^{43} - 20 q^{44} - 2 q^{46} + 3 q^{48} + q^{49} + 2 q^{51} + 28 q^{52} - 22 q^{53} - q^{54} + 22 q^{56} - 12 q^{57} - 25 q^{58} + 14 q^{59} - 16 q^{61} - 8 q^{62} - 3 q^{63} + 14 q^{64} - 4 q^{66} + 5 q^{67} + 11 q^{68} - 4 q^{69} - 28 q^{71} + 9 q^{72} + 24 q^{73} + 3 q^{74} + 2 q^{76} - 12 q^{77} + 26 q^{78} + 30 q^{79} - 2 q^{81} + 9 q^{82} + 20 q^{83} + 16 q^{84} - 12 q^{86} + q^{87} - 18 q^{88} - 18 q^{89} + 27 q^{91} - 20 q^{92} + q^{93} + 34 q^{94} + 18 q^{96} - 13 q^{97} + 25 q^{98} - 8 q^{99}+O(q^{100})$$ 4 * q + q^2 + 2 * q^3 - 5 * q^4 - q^6 - 3 * q^7 - 18 * q^8 - 2 * q^9 + 4 * q^11 - 10 * q^12 - 2 * q^13 - 20 * q^14 - 3 * q^16 + q^17 - 2 * q^18 - 6 * q^19 - 6 * q^21 - 2 * q^22 + 4 * q^23 - 9 * q^24 + 25 * q^26 - 4 * q^27 - 16 * q^28 - q^29 + 2 * q^31 + 9 * q^32 - 4 * q^33 + 18 * q^34 - 5 * q^36 + 11 * q^37 + 28 * q^38 - q^39 - q^41 - 10 * q^42 + 5 * q^43 - 20 * q^44 - 2 * q^46 + 3 * q^48 + q^49 + 2 * q^51 + 28 * q^52 - 22 * q^53 - q^54 + 22 * q^56 - 12 * q^57 - 25 * q^58 + 14 * q^59 - 16 * q^61 - 8 * q^62 - 3 * q^63 + 14 * q^64 - 4 * q^66 + 5 * q^67 + 11 * q^68 - 4 * q^69 - 28 * q^71 + 9 * q^72 + 24 * q^73 + 3 * q^74 + 2 * q^76 - 12 * q^77 + 26 * q^78 + 30 * q^79 - 2 * q^81 + 9 * q^82 + 20 * q^83 + 16 * q^84 - 12 * q^86 + q^87 - 18 * q^88 - 18 * q^89 + 27 * q^91 - 20 * q^92 + q^93 + 34 * q^94 + 18 * q^96 - 13 * q^97 + 25 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 −0.780776 + 1.35234i 1.28078 − 2.21837i −0.780776 − 1.35234i 1.28078 + 2.21837i
−0.780776 + 1.35234i 0.500000 0.866025i −0.219224 0.379706i 0 0.780776 + 1.35234i 0.280776 + 0.486319i −2.43845 −0.500000 0.866025i 0
451.2 1.28078 2.21837i 0.500000 0.866025i −2.28078 3.95042i 0 −1.28078 2.21837i −1.78078 3.08440i −6.56155 −0.500000 0.866025i 0
601.1 −0.780776 1.35234i 0.500000 + 0.866025i −0.219224 + 0.379706i 0 0.780776 1.35234i 0.280776 0.486319i −2.43845 −0.500000 + 0.866025i 0
601.2 1.28078 + 2.21837i 0.500000 + 0.866025i −2.28078 + 3.95042i 0 −1.28078 + 2.21837i −1.78078 + 3.08440i −6.56155 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.i.k 4
5.b even 2 1 39.2.e.b 4
5.c odd 4 2 975.2.bb.i 8
13.c even 3 1 inner 975.2.i.k 4
15.d odd 2 1 117.2.g.c 4
20.d odd 2 1 624.2.q.h 4
60.h even 2 1 1872.2.t.r 4
65.d even 2 1 507.2.e.g 4
65.g odd 4 2 507.2.j.g 8
65.l even 6 1 507.2.a.d 2
65.l even 6 1 507.2.e.g 4
65.n even 6 1 39.2.e.b 4
65.n even 6 1 507.2.a.g 2
65.q odd 12 2 975.2.bb.i 8
65.s odd 12 2 507.2.b.d 4
65.s odd 12 2 507.2.j.g 8
195.x odd 6 1 117.2.g.c 4
195.x odd 6 1 1521.2.a.g 2
195.y odd 6 1 1521.2.a.m 2
195.bh even 12 2 1521.2.b.h 4
260.v odd 6 1 624.2.q.h 4
260.v odd 6 1 8112.2.a.bk 2
260.w odd 6 1 8112.2.a.bo 2
780.br even 6 1 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 5.b even 2 1
39.2.e.b 4 65.n even 6 1
117.2.g.c 4 15.d odd 2 1
117.2.g.c 4 195.x odd 6 1
507.2.a.d 2 65.l even 6 1
507.2.a.g 2 65.n even 6 1
507.2.b.d 4 65.s odd 12 2
507.2.e.g 4 65.d even 2 1
507.2.e.g 4 65.l even 6 1
507.2.j.g 8 65.g odd 4 2
507.2.j.g 8 65.s odd 12 2
624.2.q.h 4 20.d odd 2 1
624.2.q.h 4 260.v odd 6 1
975.2.i.k 4 1.a even 1 1 trivial
975.2.i.k 4 13.c even 3 1 inner
975.2.bb.i 8 5.c odd 4 2
975.2.bb.i 8 65.q odd 12 2
1521.2.a.g 2 195.x odd 6 1
1521.2.a.m 2 195.y odd 6 1
1521.2.b.h 4 195.bh even 12 2
1872.2.t.r 4 60.h even 2 1
1872.2.t.r 4 780.br even 6 1
8112.2.a.bk 2 260.v odd 6 1
8112.2.a.bo 2 260.w odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(975, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16$$ T2^4 - T2^3 + 5*T2^2 + 4*T2 + 16 $$T_{7}^{4} + 3T_{7}^{3} + 11T_{7}^{2} - 6T_{7} + 4$$ T7^4 + 3*T7^3 + 11*T7^2 - 6*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + 5 T^{2} + 4 T + 16$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 3 T^{3} + 11 T^{2} - 6 T + 4$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$(T^{2} + T + 13)^{2}$$
$17$ $$T^{4} - T^{3} + 5 T^{2} + 4 T + 16$$
$19$ $$T^{4} + 6 T^{3} + 44 T^{2} - 48 T + 64$$
$23$ $$(T^{2} - 2 T + 4)^{2}$$
$29$ $$T^{4} + T^{3} + 39 T^{2} - 38 T + 1444$$
$31$ $$(T^{2} - T - 4)^{2}$$
$37$ $$T^{4} - 11 T^{3} + 95 T^{2} + \cdots + 676$$
$41$ $$T^{4} + T^{3} + 5 T^{2} - 4 T + 16$$
$43$ $$T^{4} - 5 T^{3} + 23 T^{2} - 10 T + 4$$
$47$ $$(T^{2} - 68)^{2}$$
$53$ $$(T^{2} + 11 T - 8)^{2}$$
$59$ $$T^{4} - 14 T^{3} + 164 T^{2} + \cdots + 1024$$
$61$ $$T^{4} + 16 T^{3} + 209 T^{2} + \cdots + 2209$$
$67$ $$T^{4} - 5 T^{3} + 23 T^{2} - 10 T + 4$$
$71$ $$(T^{2} + 14 T + 196)^{2}$$
$73$ $$(T^{2} - 12 T + 19)^{2}$$
$79$ $$(T^{2} - 15 T + 52)^{2}$$
$83$ $$(T^{2} - 10 T + 8)^{2}$$
$89$ $$T^{4} + 18 T^{3} + 260 T^{2} + \cdots + 4096$$
$97$ $$T^{4} + 13 T^{3} + 131 T^{2} + \cdots + 1444$$