# Properties

 Label 2925.2.c.w Level $2925$ Weight $2$ Character orbit 2925.c Analytic conductor $23.356$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.3562425912$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

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 $$\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$2251$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2224.1
 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 0 −5.71982 0 0 2.71982i 10.3354i 0 0
2224.2 2.48929i 0 −4.19656 0 0 1.19656i 5.46787i 0 0
2224.3 0.289169i 0 1.91638 0 0 4.91638i 1.13249i 0 0
2224.4 0.289169i 0 1.91638 0 0 4.91638i 1.13249i 0 0
2224.5 2.48929i 0 −4.19656 0 0 1.19656i 5.46787i 0 0
2224.6 2.77846i 0 −5.71982 0 0 2.71982i 10.3354i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2224.6 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 2925.2.c.w 6
3.b odd 2 1 975.2.c.i 6
5.b even 2 1 inner 2925.2.c.w 6
5.c odd 4 1 585.2.a.n 3
5.c odd 4 1 2925.2.a.bh 3
15.d odd 2 1 975.2.c.i 6
15.e even 4 1 195.2.a.e 3
15.e even 4 1 975.2.a.o 3
20.e even 4 1 9360.2.a.dd 3
60.l odd 4 1 3120.2.a.bj 3
65.h odd 4 1 7605.2.a.bx 3
105.k odd 4 1 9555.2.a.bq 3
195.s even 4 1 2535.2.a.bc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.e 3 15.e even 4 1
585.2.a.n 3 5.c odd 4 1
975.2.a.o 3 15.e even 4 1
975.2.c.i 6 3.b odd 2 1
975.2.c.i 6 15.d odd 2 1
2535.2.a.bc 3 195.s even 4 1
2925.2.a.bh 3 5.c odd 4 1
2925.2.c.w 6 1.a even 1 1 trivial
2925.2.c.w 6 5.b even 2 1 inner
3120.2.a.bj 3 60.l odd 4 1
7605.2.a.bx 3 65.h odd 4 1
9360.2.a.dd 3 20.e even 4 1
9555.2.a.bq 3 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2925, [\chi])$$:

 $$T_{2}^{6} + 14T_{2}^{4} + 49T_{2}^{2} + 4$$ 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

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