# Properties

 Label 4275.2.a.br Level $4275$ Weight $2$ Character orbit 4275.a Self dual yes Analytic conductor $34.136$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.1360468641$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.66064384.1 Defining polynomial: $$x^{6} - 9x^{4} + 13x^{2} - 1$$ x^6 - 9*x^4 + 13*x^2 - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + ( - \beta_{3} + 1) q^{4} + (\beta_{5} + \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{8}+O(q^{10})$$ q - b4 * q^2 + (-b3 + 1) * q^4 + (b5 + b4) * q^7 + (-b5 - b4 + b1) * q^8 $$q - \beta_{4} q^{2} + ( - \beta_{3} + 1) q^{4} + (\beta_{5} + \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{8} + (\beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} + \beta_1) q^{13} + (2 \beta_{3} - 2) q^{14} + ( - \beta_{3} - \beta_{2} - 1) q^{16} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{17} - q^{19} + (2 \beta_{4} - 4 \beta_1) q^{22} + 2 \beta_1 q^{23} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{26} + (4 \beta_{4} - 2 \beta_1) q^{28} - 6 q^{29} - 2 \beta_{2} q^{31} + (2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{32} + (4 \beta_{3} + 2 \beta_{2}) q^{34} + ( - \beta_{5} - 2 \beta_{4} - \beta_1) q^{37} + \beta_{4} q^{38} + ( - 2 \beta_{2} - 2) q^{41} + ( - \beta_{5} - \beta_{4}) q^{43} + (4 \beta_{3} + 2 \beta_{2} - 2) q^{44} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{46} + ( - \beta_{5} + 3 \beta_{4} - 4 \beta_1) q^{47} + ( - \beta_{3} - \beta_{2} - 1) q^{49} + (\beta_{5} - 2 \beta_{4} + 3 \beta_1) q^{52} + (\beta_{5} + 4 \beta_{4} - \beta_1) q^{53} + (2 \beta_{3} + 2 \beta_{2} - 6) q^{56} + 6 \beta_{4} q^{58} + ( - 2 \beta_{3} - 4) q^{59} + (\beta_{3} + 3 \beta_{2} - 2) q^{61} + (2 \beta_{5} + 6 \beta_1) q^{62} + (3 \beta_{3} - 1) q^{64} + ( - 5 \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{67} + (6 \beta_{4} - 6 \beta_1) q^{68} + (2 \beta_{3} - 8) q^{71} + ( - 3 \beta_{5} + \beta_{4} - 2 \beta_1) q^{73} + ( - 2 \beta_{3} + \beta_{2} + 6) q^{74} + (\beta_{3} - 1) q^{76} + ( - \beta_{5} - 3 \beta_{4} + 4 \beta_1) q^{77} + ( - 2 \beta_{2} - 4) q^{79} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_1) q^{82} + ( - 4 \beta_{4} - 2 \beta_1) q^{83} + ( - 2 \beta_{3} + 2) q^{86} + (2 \beta_{5} + 6 \beta_{4} - 2 \beta_1) q^{88} + (6 \beta_{3} - 2 \beta_{2} - 2) q^{89} + (2 \beta_{2} - 4) q^{91} + ( - 2 \beta_{4} + 4 \beta_1) q^{92} + (6 \beta_{3} + 4 \beta_{2} - 6) q^{94} + ( - \beta_{5} + \beta_1) q^{97} + ( - \beta_{4} + 4 \beta_1) q^{98}+O(q^{100})$$ q - b4 * q^2 + (-b3 + 1) * q^4 + (b5 + b4) * q^7 + (-b5 - b4 + b1) * q^8 + (b3 + b2) * q^11 + (-b5 + b1) * q^13 + (2*b3 - 2) * q^14 + (-b3 - b2 - 1) * q^16 + (b5 + b4 - 2*b1) * q^17 - q^19 + (2*b4 - 4*b1) * q^22 + 2*b1 * q^23 + (-2*b3 - b2 - 2) * q^26 + (4*b4 - 2*b1) * q^28 - 6 * q^29 - 2*b2 * q^31 + (2*b5 + b4 + 2*b1) * q^32 + (4*b3 + 2*b2) * q^34 + (-b5 - 2*b4 - b1) * q^37 + b4 * q^38 + (-2*b2 - 2) * q^41 + (-b5 - b4) * q^43 + (4*b3 + 2*b2 - 2) * q^44 + (-2*b3 - 2*b2 - 2) * q^46 + (-b5 + 3*b4 - 4*b1) * q^47 + (-b3 - b2 - 1) * q^49 + (b5 - 2*b4 + 3*b1) * q^52 + (b5 + 4*b4 - b1) * q^53 + (2*b3 + 2*b2 - 6) * q^56 + 6*b4 * q^58 + (-2*b3 - 4) * q^59 + (b3 + 3*b2 - 2) * q^61 + (2*b5 + 6*b1) * q^62 + (3*b3 - 1) * q^64 + (-5*b5 - 2*b4 - 3*b1) * q^67 + (6*b4 - 6*b1) * q^68 + (2*b3 - 8) * q^71 + (-3*b5 + b4 - 2*b1) * q^73 + (-2*b3 + b2 + 6) * q^74 + (b3 - 1) * q^76 + (-b5 - 3*b4 + 4*b1) * q^77 + (-2*b2 - 4) * q^79 + (2*b5 + 2*b4 + 6*b1) * q^82 + (-4*b4 - 2*b1) * q^83 + (-2*b3 + 2) * q^86 + (2*b5 + 6*b4 - 2*b1) * q^88 + (6*b3 - 2*b2 - 2) * q^89 + (2*b2 - 4) * q^91 + (-2*b4 + 4*b1) * q^92 + (6*b3 + 4*b2 - 6) * q^94 + (-b5 + b1) * q^97 + (-b4 + 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{4}+O(q^{10})$$ 6 * q + 8 * q^4 $$6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74} - 8 q^{76} - 24 q^{79} + 16 q^{86} - 24 q^{89} - 24 q^{91} - 48 q^{94}+O(q^{100})$$ 6 * q + 8 * q^4 - 2 * q^11 - 16 * q^14 - 4 * q^16 - 6 * q^19 - 8 * q^26 - 36 * q^29 - 8 * q^34 - 12 * q^41 - 20 * q^44 - 8 * q^46 - 4 * q^49 - 40 * q^56 - 20 * q^59 - 14 * q^61 - 12 * q^64 - 52 * q^71 + 40 * q^74 - 8 * q^76 - 24 * q^79 + 16 * q^86 - 24 * q^89 - 24 * q^91 - 48 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9x^{4} + 13x^{2} - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 8\nu^{2} + 5 ) / 2$$ (v^4 - 8*v^2 + 5) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 8\nu^{3} + 7\nu ) / 2$$ (v^5 - 8*v^3 + 7*v) / 2 $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 9\nu^{3} - 13\nu$$ -v^5 + 9*v^3 - 13*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + 6\beta_1$$ b5 + 2*b4 + 6*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{3} + 8\beta_{2} + 19$$ 2*b3 + 8*b2 + 19 $$\nu^{5}$$ $$=$$ $$8\beta_{5} + 18\beta_{4} + 41\beta_1$$ 8*b5 + 18*b4 + 41*b1

## 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
 −1.30397 2.68667 0.285442 −0.285442 −2.68667 1.30397
−2.41987 0 3.85577 0 0 3.18676 −4.49073 0 0
1.2 −1.82254 0 1.32164 0 0 1.45033 1.23634 0 0
1.3 −0.906968 0 −1.17741 0 0 −2.59637 2.88181 0 0
1.4 0.906968 0 −1.17741 0 0 2.59637 −2.88181 0 0
1.5 1.82254 0 1.32164 0 0 −1.45033 −1.23634 0 0
1.6 2.41987 0 3.85577 0 0 −3.18676 4.49073 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$1$$

## 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 4275.2.a.br 6
3.b odd 2 1 475.2.a.j 6
5.b even 2 1 inner 4275.2.a.br 6
5.c odd 4 2 855.2.c.d 6
12.b even 2 1 7600.2.a.ck 6
15.d odd 2 1 475.2.a.j 6
15.e even 4 2 95.2.b.b 6
57.d even 2 1 9025.2.a.bx 6
60.h even 2 1 7600.2.a.ck 6
60.l odd 4 2 1520.2.d.h 6
285.b even 2 1 9025.2.a.bx 6
285.j odd 4 2 1805.2.b.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 15.e even 4 2
475.2.a.j 6 3.b odd 2 1
475.2.a.j 6 15.d odd 2 1
855.2.c.d 6 5.c odd 4 2
1520.2.d.h 6 60.l odd 4 2
1805.2.b.e 6 285.j odd 4 2
4275.2.a.br 6 1.a even 1 1 trivial
4275.2.a.br 6 5.b even 2 1 inner
7600.2.a.ck 6 12.b even 2 1
7600.2.a.ck 6 60.h even 2 1
9025.2.a.bx 6 57.d even 2 1
9025.2.a.bx 6 285.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4275))$$:

 $$T_{2}^{6} - 10T_{2}^{4} + 27T_{2}^{2} - 16$$ T2^6 - 10*T2^4 + 27*T2^2 - 16 $$T_{7}^{6} - 19T_{7}^{4} + 104T_{7}^{2} - 144$$ T7^6 - 19*T7^4 + 104*T7^2 - 144 $$T_{11}^{3} + T_{11}^{2} - 16T_{11} - 12$$ T11^3 + T11^2 - 16*T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 10 T^{4} + 27 T^{2} - 16$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 19 T^{4} + 104 T^{2} + \cdots - 144$$
$11$ $$(T^{3} + T^{2} - 16 T - 12)^{2}$$
$13$ $$T^{6} - 28 T^{4} + 236 T^{2} + \cdots - 576$$
$17$ $$T^{6} - 59 T^{4} + 1008 T^{2} + \cdots - 5184$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} - 36 T^{4} + 208 T^{2} + \cdots - 64$$
$29$ $$(T + 6)^{6}$$
$31$ $$(T^{3} - 56 T + 128)^{2}$$
$37$ $$T^{6} - 56 T^{4} + 764 T^{2} + \cdots - 1296$$
$41$ $$(T^{3} + 6 T^{2} - 44 T + 24)^{2}$$
$43$ $$T^{6} - 19 T^{4} + 104 T^{2} + \cdots - 144$$
$47$ $$T^{6} - 187 T^{4} + 7464 T^{2} + \cdots - 85264$$
$53$ $$T^{6} - 156 T^{4} + 2476 T^{2} + \cdots - 64$$
$59$ $$(T^{3} + 10 T^{2} + 8 T - 48)^{2}$$
$61$ $$(T^{3} + 7 T^{2} - 104 T - 776)^{2}$$
$67$ $$T^{6} - 340 T^{4} + 28556 T^{2} + \cdots - 484416$$
$71$ $$(T^{3} + 26 T^{2} + 200 T + 432)^{2}$$
$73$ $$T^{6} - 131 T^{4} + 1616 T^{2} + \cdots - 5184$$
$79$ $$(T^{3} + 12 T^{2} - 8 T - 32)^{2}$$
$83$ $$T^{6} - 228 T^{4} + 11728 T^{2} + \cdots - 141376$$
$89$ $$(T^{3} + 12 T^{2} - 284 T - 3456)^{2}$$
$97$ $$T^{6} - 28 T^{4} + 236 T^{2} + \cdots - 576$$