# Properties

 Label 975.2.c.i Level $975$ Weight $2$ Character orbit 975.c Analytic conductor $7.785$ Analytic rank $0$ Dimension $6$ 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 6*x^2 - 8*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} - \beta_{2} q^{3} + (\beta_{4} - 3) q^{4} + \beta_1 q^{6} + \beta_{5} q^{7} + (3 \beta_{3} + 2 \beta_{2}) q^{8} - q^{9}+O(q^{10})$$ q - b3 * q^2 - b2 * q^3 + (b4 - 3) * q^4 + b1 * q^6 + b5 * q^7 + (3*b3 + 2*b2) * q^8 - q^9 $$q - \beta_{3} q^{2} - \beta_{2} q^{3} + (\beta_{4} - 3) q^{4} + \beta_1 q^{6} + \beta_{5} q^{7} + (3 \beta_{3} + 2 \beta_{2}) q^{8} - q^{9} + \beta_{4} q^{11} + (\beta_{5} + 3 \beta_{2}) q^{12} + \beta_{2} q^{13} + ( - 2 \beta_1 + 2) q^{14} + ( - \beta_{4} - 2 \beta_1 + 9) q^{16} + ( - \beta_{5} + 2 \beta_{3}) q^{17} + \beta_{3} q^{18} + ( - 2 \beta_1 - 2) q^{19} - \beta_{4} q^{21} + (2 \beta_{3} + 2 \beta_{2}) q^{22} + (\beta_{5} - 2 \beta_{3} - 2 \beta_{2}) q^{23} + ( - 3 \beta_1 + 2) q^{24} - \beta_1 q^{26} + \beta_{2} q^{27} + ( - 2 \beta_{3} - 10 \beta_{2}) q^{28} - 6 q^{29} + ( - 2 \beta_1 + 2) q^{31} + ( - 2 \beta_{5} - 5 \beta_{3} - 8 \beta_{2}) q^{32} + \beta_{5} q^{33} + ( - 2 \beta_{4} + 2 \beta_1 + 8) q^{34} + ( - \beta_{4} + 3) q^{36} + (\beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{37} + ( - 2 \beta_{5} + 2 \beta_{3} - 10 \beta_{2}) q^{38} + q^{39} + (\beta_{4} - 2 \beta_1) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{42} + 4 \beta_{3} q^{43} + ( - 2 \beta_1 + 10) q^{44} + (2 \beta_{4} - 8) q^{46} + (2 \beta_{3} + 6 \beta_{2}) q^{47} + ( - \beta_{5} - 2 \beta_{3} - 9 \beta_{2}) q^{48} + ( - 3 \beta_{4} + 2 \beta_1 - 3) q^{49} + (\beta_{4} - 2 \beta_1) q^{51} + ( - \beta_{5} - 3 \beta_{2}) q^{52} + (\beta_{5} - 2 \beta_{3} + 4 \beta_{2}) q^{53} - \beta_1 q^{54} + (2 \beta_{4} + 6 \beta_1 - 6) q^{56} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{57} + 6 \beta_{3} q^{58} + (2 \beta_{4} - 2 \beta_1 + 2) q^{59} + ( - 3 \beta_{4} + 2 \beta_1 + 4) q^{61} + ( - 2 \beta_{5} - 2 \beta_{3} - 10 \beta_{2}) q^{62} - \beta_{5} q^{63} + (3 \beta_{4} + 8 \beta_1 - 11) q^{64} + ( - 2 \beta_1 + 2) q^{66} + ( - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{67} + ( - 8 \beta_{3} + 6 \beta_{2}) q^{68} + ( - \beta_{4} + 2 \beta_1 - 2) q^{69} + (\beta_{4} - 4) q^{71} + ( - 3 \beta_{3} - 2 \beta_{2}) q^{72} + (4 \beta_{3} - 2 \beta_{2}) q^{73} + ( - 2 \beta_{4} + 2 \beta_1 + 12) q^{74} + ( - 2 \beta_{4} + 10 \beta_1 + 2) q^{76} + (3 \beta_{5} - 2 \beta_{3} - 10 \beta_{2}) q^{77} - \beta_{3} q^{78} + (\beta_{4} + 2 \beta_1 - 2) q^{79} + q^{81} + ( - 2 \beta_{5} + 2 \beta_{3} - 8 \beta_{2}) q^{82} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2}) q^{83} + (2 \beta_1 - 10) q^{84} + ( - 4 \beta_{4} + 20) q^{86} + 6 \beta_{2} q^{87} + ( - 2 \beta_{5} - 6 \beta_{3} - 6 \beta_{2}) q^{88} + (\beta_{4} - 2 \beta_1 - 4) q^{89} + \beta_{4} q^{91} + (2 \beta_{5} + 8 \beta_{3}) q^{92} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{93} + ( - 2 \beta_{4} - 6 \beta_1 + 10) q^{94} + (2 \beta_{4} + 5 \beta_1 - 8) q^{96} + ( - \beta_{5} + 2 \beta_{3} + 8 \beta_{2}) q^{97} + (2 \beta_{5} - 3 \beta_{3} + 4 \beta_{2}) q^{98} - \beta_{4} q^{99}+O(q^{100})$$ q - b3 * q^2 - b2 * q^3 + (b4 - 3) * q^4 + b1 * q^6 + b5 * q^7 + (3*b3 + 2*b2) * q^8 - q^9 + b4 * q^11 + (b5 + 3*b2) * q^12 + b2 * q^13 + (-2*b1 + 2) * q^14 + (-b4 - 2*b1 + 9) * q^16 + (-b5 + 2*b3) * q^17 + b3 * q^18 + (-2*b1 - 2) * q^19 - b4 * q^21 + (2*b3 + 2*b2) * q^22 + (b5 - 2*b3 - 2*b2) * q^23 + (-3*b1 + 2) * q^24 - b1 * q^26 + b2 * q^27 + (-2*b3 - 10*b2) * q^28 - 6 * q^29 + (-2*b1 + 2) * q^31 + (-2*b5 - 5*b3 - 8*b2) * q^32 + b5 * q^33 + (-2*b4 + 2*b1 + 8) * q^34 + (-b4 + 3) * q^36 + (b5 + 2*b3 - 4*b2) * q^37 + (-2*b5 + 2*b3 - 10*b2) * q^38 + q^39 + (b4 - 2*b1) * q^41 + (-2*b3 - 2*b2) * q^42 + 4*b3 * q^43 + (-2*b1 + 10) * q^44 + (2*b4 - 8) * q^46 + (2*b3 + 6*b2) * q^47 + (-b5 - 2*b3 - 9*b2) * q^48 + (-3*b4 + 2*b1 - 3) * q^49 + (b4 - 2*b1) * q^51 + (-b5 - 3*b2) * q^52 + (b5 - 2*b3 + 4*b2) * q^53 - b1 * q^54 + (2*b4 + 6*b1 - 6) * q^56 + (-2*b3 + 2*b2) * q^57 + 6*b3 * q^58 + (2*b4 - 2*b1 + 2) * q^59 + (-3*b4 + 2*b1 + 4) * q^61 + (-2*b5 - 2*b3 - 10*b2) * q^62 - b5 * q^63 + (3*b4 + 8*b1 - 11) * q^64 + (-2*b1 + 2) * q^66 + (-2*b5 + 2*b3 - 2*b2) * q^67 + (-8*b3 + 6*b2) * q^68 + (-b4 + 2*b1 - 2) * q^69 + (b4 - 4) * q^71 + (-3*b3 - 2*b2) * q^72 + (4*b3 - 2*b2) * q^73 + (-2*b4 + 2*b1 + 12) * q^74 + (-2*b4 + 10*b1 + 2) * q^76 + (3*b5 - 2*b3 - 10*b2) * q^77 - b3 * q^78 + (b4 + 2*b1 - 2) * q^79 + q^81 + (-2*b5 + 2*b3 - 8*b2) * q^82 + (-2*b5 + 2*b3 + 2*b2) * q^83 + (2*b1 - 10) * q^84 + (-4*b4 + 20) * q^86 + 6*b2 * q^87 + (-2*b5 - 6*b3 - 6*b2) * q^88 + (b4 - 2*b1 - 4) * q^89 + b4 * q^91 + (2*b5 + 8*b3) * q^92 + (-2*b3 - 2*b2) * q^93 + (-2*b4 - 6*b1 + 10) * q^94 + (2*b4 + 5*b1 - 8) * q^96 + (-b5 + 2*b3 + 8*b2) * q^97 + (2*b5 - 3*b3 + 4*b2) * q^98 - b4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 16 q^{4} - 6 q^{9}+O(q^{10})$$ 6 * q - 16 * q^4 - 6 * q^9 $$6 q - 16 q^{4} - 6 q^{9} + 2 q^{11} + 12 q^{14} + 52 q^{16} - 12 q^{19} - 2 q^{21} + 12 q^{24} - 36 q^{29} + 12 q^{31} + 44 q^{34} + 16 q^{36} + 6 q^{39} + 2 q^{41} + 60 q^{44} - 44 q^{46} - 24 q^{49} + 2 q^{51} - 32 q^{56} + 16 q^{59} + 18 q^{61} - 60 q^{64} + 12 q^{66} - 14 q^{69} - 22 q^{71} + 68 q^{74} + 8 q^{76} - 10 q^{79} + 6 q^{81} - 60 q^{84} + 112 q^{86} - 22 q^{89} + 2 q^{91} + 56 q^{94} - 44 q^{96} - 2 q^{99}+O(q^{100})$$ 6 * q - 16 * q^4 - 6 * q^9 + 2 * q^11 + 12 * q^14 + 52 * q^16 - 12 * q^19 - 2 * q^21 + 12 * q^24 - 36 * q^29 + 12 * q^31 + 44 * q^34 + 16 * q^36 + 6 * q^39 + 2 * q^41 + 60 * q^44 - 44 * q^46 - 24 * q^49 + 2 * q^51 - 32 * q^56 + 16 * q^59 + 18 * q^61 - 60 * q^64 + 12 * q^66 - 14 * q^69 - 22 * q^71 + 68 * q^74 + 8 * q^76 - 10 * q^79 + 6 * q^81 - 60 * q^84 + 112 * q^86 - 22 * q^89 + 2 * q^91 + 56 * q^94 - 44 * q^96 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2$$ (-v^4 + 2*v^3 - v^2 + 2*v - 2) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4$$ (-v^5 - 3*v^3 + 4*v^2 - 2*v + 8) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4$$ (v^5 - 2*v^4 + 3*v^3 - 6*v^2 + 10*v - 8) / 4 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - \nu^{3} + 5\nu^{2} + 4 ) / 2$$ (-v^5 + v^4 - v^3 + 5*v^2 + 4) / 2 $$\beta_{5}$$ $$=$$ $$( 4\nu^{5} - 3\nu^{4} + 8\nu^{3} - 11\nu^{2} + 8\nu - 20 ) / 2$$ (4*v^5 - 3*v^4 + 8*v^3 - 11*v^2 + 8*v - 20) / 2
 $$\nu$$ $$=$$ $$( \beta_{4} + 2\beta_{3} - \beta _1 + 1 ) / 4$$ (b4 + 2*b3 - b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 2\beta_{4} - \beta_{3} + 3\beta_{2} - 2 ) / 4$$ (b5 + 2*b4 - b3 + 3*b2 - 2) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 5 ) / 4$$ (b4 - 2*b3 - 4*b2 + 3*b1 + 5) / 4 $$\nu^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} + \beta_{3} - 11\beta_{2} - 4\beta _1 + 6 ) / 4$$ (-b5 + 2*b4 + b3 - 11*b2 - 4*b1 + 6) / 4 $$\nu^{5}$$ $$=$$ $$( 4\beta_{5} + 3\beta_{4} - 2\beta_{3} + 8\beta_{2} - 7\beta _1 + 7 ) / 4$$ (4*b5 + 3*b4 - 2*b3 + 8*b2 - 7*b1 + 7) / 4

## Character values

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

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 0.264658 + 1.38923i −0.671462 + 1.24464i 1.40680 + 0.144584i 1.40680 − 0.144584i −0.671462 − 1.24464i 0.264658 − 1.38923i
2.77846i 1.00000i −5.71982 0 −2.77846 2.71982i 10.3354i −1.00000 0
274.2 2.48929i 1.00000i −4.19656 0 2.48929 1.19656i 5.46787i −1.00000 0
274.3 0.289169i 1.00000i 1.91638 0 0.289169 4.91638i 1.13249i −1.00000 0
274.4 0.289169i 1.00000i 1.91638 0 0.289169 4.91638i 1.13249i −1.00000 0
274.5 2.48929i 1.00000i −4.19656 0 2.48929 1.19656i 5.46787i −1.00000 0
274.6 2.77846i 1.00000i −5.71982 0 −2.77846 2.71982i 10.3354i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 274.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.c.i 6
3.b odd 2 1 2925.2.c.w 6
5.b even 2 1 inner 975.2.c.i 6
5.c odd 4 1 195.2.a.e 3
5.c odd 4 1 975.2.a.o 3
15.d odd 2 1 2925.2.c.w 6
15.e even 4 1 585.2.a.n 3
15.e even 4 1 2925.2.a.bh 3
20.e even 4 1 3120.2.a.bj 3
35.f even 4 1 9555.2.a.bq 3
60.l odd 4 1 9360.2.a.dd 3
65.h odd 4 1 2535.2.a.bc 3
195.s even 4 1 7605.2.a.bx 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.e 3 5.c odd 4 1
585.2.a.n 3 15.e even 4 1
975.2.a.o 3 5.c odd 4 1
975.2.c.i 6 1.a even 1 1 trivial
975.2.c.i 6 5.b even 2 1 inner
2535.2.a.bc 3 65.h odd 4 1
2925.2.a.bh 3 15.e even 4 1
2925.2.c.w 6 3.b odd 2 1
2925.2.c.w 6 15.d odd 2 1
3120.2.a.bj 3 20.e even 4 1
7605.2.a.bx 3 195.s even 4 1
9360.2.a.dd 3 60.l odd 4 1
9555.2.a.bq 3 35.f even 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 14T_{2}^{4} + 49T_{2}^{2} + 4$$ T2^6 + 14*T2^4 + 49*T2^2 + 4 $$T_{7}^{6} + 33T_{7}^{4} + 224T_{7}^{2} + 256$$ T7^6 + 33*T7^4 + 224*T7^2 + 256 $$T_{11}^{3} - T_{11}^{2} - 16T_{11} - 16$$ T11^3 - T11^2 - 16*T11 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 14 T^{4} + 49 T^{2} + 4$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 33 T^{4} + 224 T^{2} + \cdots + 256$$
$11$ $$(T^{3} - T^{2} - 16 T - 16)^{2}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 65 T^{4} + 1176 T^{2} + \cdots + 5776$$
$19$ $$(T^{3} + 6 T^{2} - 16 T - 64)^{2}$$
$23$ $$T^{6} + 81 T^{4} + 2048 T^{2} + \cdots + 16384$$
$29$ $$(T + 6)^{6}$$
$31$ $$(T^{3} - 6 T^{2} - 16 T + 32)^{2}$$
$37$ $$T^{6} + 169 T^{4} + 8216 T^{2} + \cdots + 99856$$
$41$ $$(T^{3} - T^{2} - 32 T + 76)^{2}$$
$43$ $$T^{6} + 224 T^{4} + 12544 T^{2} + \cdots + 16384$$
$47$ $$T^{6} + 164 T^{4} + 4096 T^{2} + \cdots + 4096$$
$53$ $$T^{6} + 105 T^{4} + 152 T^{2} + \cdots + 16$$
$59$ $$(T^{3} - 8 T^{2} - 48 T + 128)^{2}$$
$61$ $$(T^{3} - 9 T^{2} - 112 T + 844)^{2}$$
$67$ $$T^{6} + 144 T^{4} + 5120 T^{2} + \cdots + 16384$$
$71$ $$(T^{3} + 11 T^{2} + 24 T - 32)^{2}$$
$73$ $$T^{6} + 236 T^{4} + 14128 T^{2} + \cdots + 118336$$
$79$ $$(T^{3} + 5 T^{2} - 48 T + 64)^{2}$$
$83$ $$T^{6} + 160 T^{4} + 4352 T^{2} + \cdots + 16384$$
$89$ $$(T^{3} + 11 T^{2} + 8 T - 4)^{2}$$
$97$ $$T^{6} + 273 T^{4} + 18776 T^{2} + \cdots + 59536$$