# Properties

 Label 4275.2.a.u Level $4275$ Weight $2$ Character orbit 4275.a Self dual yes Analytic conductor $34.136$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 5 q^{4} + (\beta + 1) q^{7} + 3 \beta q^{8}+O(q^{10})$$ q + b * q^2 + 5 * q^4 + (b + 1) * q^7 + 3*b * q^8 $$q + \beta q^{2} + 5 q^{4} + (\beta + 1) q^{7} + 3 \beta q^{8} + ( - \beta - 3) q^{11} + (\beta + 3) q^{13} + (\beta + 7) q^{14} + 11 q^{16} - 4 q^{17} - q^{19} + ( - 3 \beta - 7) q^{22} + ( - 2 \beta + 4) q^{23} + (3 \beta + 7) q^{26} + (5 \beta + 5) q^{28} + (3 \beta - 1) q^{29} + 6 q^{31} + 5 \beta q^{32} - 4 \beta q^{34} + (\beta - 1) q^{37} - \beta q^{38} + ( - \beta + 7) q^{41} + (\beta - 3) q^{43} + ( - 5 \beta - 15) q^{44} + (4 \beta - 14) q^{46} + (2 \beta + 4) q^{47} + (2 \beta + 1) q^{49} + (5 \beta + 15) q^{52} + ( - 2 \beta + 6) q^{53} + (3 \beta + 21) q^{56} + ( - \beta + 21) q^{58} + (2 \beta - 6) q^{59} + (2 \beta - 6) q^{61} + 6 \beta q^{62} + 13 q^{64} + ( - 4 \beta - 4) q^{67} - 20 q^{68} + (2 \beta - 2) q^{71} - 10 q^{73} + ( - \beta + 7) q^{74} - 5 q^{76} + ( - 4 \beta - 10) q^{77} + ( - 4 \beta - 4) q^{79} + (7 \beta - 7) q^{82} + 6 q^{83} + ( - 3 \beta + 7) q^{86} + ( - 9 \beta - 21) q^{88} + ( - 3 \beta + 9) q^{89} + (4 \beta + 10) q^{91} + ( - 10 \beta + 20) q^{92} + (4 \beta + 14) q^{94} + ( - 3 \beta - 5) q^{97} + (\beta + 14) q^{98} +O(q^{100})$$ q + b * q^2 + 5 * q^4 + (b + 1) * q^7 + 3*b * q^8 + (-b - 3) * q^11 + (b + 3) * q^13 + (b + 7) * q^14 + 11 * q^16 - 4 * q^17 - q^19 + (-3*b - 7) * q^22 + (-2*b + 4) * q^23 + (3*b + 7) * q^26 + (5*b + 5) * q^28 + (3*b - 1) * q^29 + 6 * q^31 + 5*b * q^32 - 4*b * q^34 + (b - 1) * q^37 - b * q^38 + (-b + 7) * q^41 + (b - 3) * q^43 + (-5*b - 15) * q^44 + (4*b - 14) * q^46 + (2*b + 4) * q^47 + (2*b + 1) * q^49 + (5*b + 15) * q^52 + (-2*b + 6) * q^53 + (3*b + 21) * q^56 + (-b + 21) * q^58 + (2*b - 6) * q^59 + (2*b - 6) * q^61 + 6*b * q^62 + 13 * q^64 + (-4*b - 4) * q^67 - 20 * q^68 + (2*b - 2) * q^71 - 10 * q^73 + (-b + 7) * q^74 - 5 * q^76 + (-4*b - 10) * q^77 + (-4*b - 4) * q^79 + (7*b - 7) * q^82 + 6 * q^83 + (-3*b + 7) * q^86 + (-9*b - 21) * q^88 + (-3*b + 9) * q^89 + (4*b + 10) * q^91 + (-10*b + 20) * q^92 + (4*b + 14) * q^94 + (-3*b - 5) * q^97 + (b + 14) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{4} + 2 q^{7}+O(q^{10})$$ 2 * q + 10 * q^4 + 2 * q^7 $$2 q + 10 q^{4} + 2 q^{7} - 6 q^{11} + 6 q^{13} + 14 q^{14} + 22 q^{16} - 8 q^{17} - 2 q^{19} - 14 q^{22} + 8 q^{23} + 14 q^{26} + 10 q^{28} - 2 q^{29} + 12 q^{31} - 2 q^{37} + 14 q^{41} - 6 q^{43} - 30 q^{44} - 28 q^{46} + 8 q^{47} + 2 q^{49} + 30 q^{52} + 12 q^{53} + 42 q^{56} + 42 q^{58} - 12 q^{59} - 12 q^{61} + 26 q^{64} - 8 q^{67} - 40 q^{68} - 4 q^{71} - 20 q^{73} + 14 q^{74} - 10 q^{76} - 20 q^{77} - 8 q^{79} - 14 q^{82} + 12 q^{83} + 14 q^{86} - 42 q^{88} + 18 q^{89} + 20 q^{91} + 40 q^{92} + 28 q^{94} - 10 q^{97} + 28 q^{98}+O(q^{100})$$ 2 * q + 10 * q^4 + 2 * q^7 - 6 * q^11 + 6 * q^13 + 14 * q^14 + 22 * q^16 - 8 * q^17 - 2 * q^19 - 14 * q^22 + 8 * q^23 + 14 * q^26 + 10 * q^28 - 2 * q^29 + 12 * q^31 - 2 * q^37 + 14 * q^41 - 6 * q^43 - 30 * q^44 - 28 * q^46 + 8 * q^47 + 2 * q^49 + 30 * q^52 + 12 * q^53 + 42 * q^56 + 42 * q^58 - 12 * q^59 - 12 * q^61 + 26 * q^64 - 8 * q^67 - 40 * q^68 - 4 * q^71 - 20 * q^73 + 14 * q^74 - 10 * q^76 - 20 * q^77 - 8 * q^79 - 14 * q^82 + 12 * q^83 + 14 * q^86 - 42 * q^88 + 18 * q^89 + 20 * q^91 + 40 * q^92 + 28 * q^94 - 10 * q^97 + 28 * q^98

## 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
 −2.64575 2.64575
−2.64575 0 5.00000 0 0 −1.64575 −7.93725 0 0
1.2 2.64575 0 5.00000 0 0 3.64575 7.93725 0 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$1$$

## 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 4275.2.a.u 2
3.b odd 2 1 1425.2.a.p 2
5.b even 2 1 855.2.a.g 2
15.d odd 2 1 285.2.a.d 2
15.e even 4 2 1425.2.c.i 4
60.h even 2 1 4560.2.a.bo 2
285.b even 2 1 5415.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.d 2 15.d odd 2 1
855.2.a.g 2 5.b even 2 1
1425.2.a.p 2 3.b odd 2 1
1425.2.c.i 4 15.e even 4 2
4275.2.a.u 2 1.a even 1 1 trivial
4560.2.a.bo 2 60.h even 2 1
5415.2.a.s 2 285.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4275))$$:

 $$T_{2}^{2} - 7$$ T2^2 - 7 $$T_{7}^{2} - 2T_{7} - 6$$ T7^2 - 2*T7 - 6 $$T_{11}^{2} + 6T_{11} + 2$$ T11^2 + 6*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 7$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2T - 6$$
$11$ $$T^{2} + 6T + 2$$
$13$ $$T^{2} - 6T + 2$$
$17$ $$(T + 4)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 8T - 12$$
$29$ $$T^{2} + 2T - 62$$
$31$ $$(T - 6)^{2}$$
$37$ $$T^{2} + 2T - 6$$
$41$ $$T^{2} - 14T + 42$$
$43$ $$T^{2} + 6T + 2$$
$47$ $$T^{2} - 8T - 12$$
$53$ $$T^{2} - 12T + 8$$
$59$ $$T^{2} + 12T + 8$$
$61$ $$T^{2} + 12T + 8$$
$67$ $$T^{2} + 8T - 96$$
$71$ $$T^{2} + 4T - 24$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 8T - 96$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 18T + 18$$
$97$ $$T^{2} + 10T - 38$$