# Properties

 Label 1425.2.c.i Level $1425$ Weight $2$ Character orbit 1425.c Analytic conductor $11.379$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.3786822880$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3x^{2} + 4$$ x^4 - 3*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 285) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - \beta_1 q^{3} - 5 q^{4} + \beta_{2} q^{6} + ( - \beta_{3} + \beta_1) q^{7} - 3 \beta_{3} q^{8} - q^{9}+O(q^{10})$$ q + b3 * q^2 - b1 * q^3 - 5 * q^4 + b2 * q^6 + (-b3 + b1) * q^7 - 3*b3 * q^8 - q^9 $$q + \beta_{3} q^{2} - \beta_1 q^{3} - 5 q^{4} + \beta_{2} q^{6} + ( - \beta_{3} + \beta_1) q^{7} - 3 \beta_{3} q^{8} - q^{9} + ( - \beta_{2} + 3) q^{11} + 5 \beta_1 q^{12} + (\beta_{3} - 3 \beta_1) q^{13} + ( - \beta_{2} + 7) q^{14} + 11 q^{16} + 4 \beta_1 q^{17} - \beta_{3} q^{18} + q^{19} + ( - \beta_{2} + 1) q^{21} + (3 \beta_{3} - 7 \beta_1) q^{22} + (2 \beta_{3} + 4 \beta_1) q^{23} - 3 \beta_{2} q^{24} + (3 \beta_{2} - 7) q^{26} + \beta_1 q^{27} + (5 \beta_{3} - 5 \beta_1) q^{28} + ( - 3 \beta_{2} - 1) q^{29} + 6 q^{31} + 5 \beta_{3} q^{32} + (\beta_{3} - 3 \beta_1) q^{33} - 4 \beta_{2} q^{34} + 5 q^{36} + ( - \beta_{3} - \beta_1) q^{37} + \beta_{3} q^{38} + (\beta_{2} - 3) q^{39} + ( - \beta_{2} - 7) q^{41} + (\beta_{3} - 7 \beta_1) q^{42} + (\beta_{3} + 3 \beta_1) q^{43} + (5 \beta_{2} - 15) q^{44} + ( - 4 \beta_{2} - 14) q^{46} + (2 \beta_{3} - 4 \beta_1) q^{47} - 11 \beta_1 q^{48} + (2 \beta_{2} - 1) q^{49} + 4 q^{51} + ( - 5 \beta_{3} + 15 \beta_1) q^{52} + (2 \beta_{3} + 6 \beta_1) q^{53} - \beta_{2} q^{54} + (3 \beta_{2} - 21) q^{56} - \beta_1 q^{57} + ( - \beta_{3} - 21 \beta_1) q^{58} + ( - 2 \beta_{2} - 6) q^{59} + ( - 2 \beta_{2} - 6) q^{61} + 6 \beta_{3} q^{62} + (\beta_{3} - \beta_1) q^{63} - 13 q^{64} + (3 \beta_{2} - 7) q^{66} + (4 \beta_{3} - 4 \beta_1) q^{67} - 20 \beta_1 q^{68} + (2 \beta_{2} + 4) q^{69} + (2 \beta_{2} + 2) q^{71} + 3 \beta_{3} q^{72} + 10 \beta_1 q^{73} + (\beta_{2} + 7) q^{74} - 5 q^{76} + ( - 4 \beta_{3} + 10 \beta_1) q^{77} + ( - 3 \beta_{3} + 7 \beta_1) q^{78} + ( - 4 \beta_{2} + 4) q^{79} + q^{81} + ( - 7 \beta_{3} - 7 \beta_1) q^{82} + 6 \beta_1 q^{83} + (5 \beta_{2} - 5) q^{84} + ( - 3 \beta_{2} - 7) q^{86} + (3 \beta_{3} + \beta_1) q^{87} + ( - 9 \beta_{3} + 21 \beta_1) q^{88} + (3 \beta_{2} + 9) q^{89} + ( - 4 \beta_{2} + 10) q^{91} + ( - 10 \beta_{3} - 20 \beta_1) q^{92} - 6 \beta_1 q^{93} + (4 \beta_{2} - 14) q^{94} + 5 \beta_{2} q^{96} + (3 \beta_{3} - 5 \beta_1) q^{97} + ( - \beta_{3} + 14 \beta_1) q^{98} + (\beta_{2} - 3) q^{99}+O(q^{100})$$ q + b3 * q^2 - b1 * q^3 - 5 * q^4 + b2 * q^6 + (-b3 + b1) * q^7 - 3*b3 * q^8 - q^9 + (-b2 + 3) * q^11 + 5*b1 * q^12 + (b3 - 3*b1) * q^13 + (-b2 + 7) * q^14 + 11 * q^16 + 4*b1 * q^17 - b3 * q^18 + q^19 + (-b2 + 1) * q^21 + (3*b3 - 7*b1) * q^22 + (2*b3 + 4*b1) * q^23 - 3*b2 * q^24 + (3*b2 - 7) * q^26 + b1 * q^27 + (5*b3 - 5*b1) * q^28 + (-3*b2 - 1) * q^29 + 6 * q^31 + 5*b3 * q^32 + (b3 - 3*b1) * q^33 - 4*b2 * q^34 + 5 * q^36 + (-b3 - b1) * q^37 + b3 * q^38 + (b2 - 3) * q^39 + (-b2 - 7) * q^41 + (b3 - 7*b1) * q^42 + (b3 + 3*b1) * q^43 + (5*b2 - 15) * q^44 + (-4*b2 - 14) * q^46 + (2*b3 - 4*b1) * q^47 - 11*b1 * q^48 + (2*b2 - 1) * q^49 + 4 * q^51 + (-5*b3 + 15*b1) * q^52 + (2*b3 + 6*b1) * q^53 - b2 * q^54 + (3*b2 - 21) * q^56 - b1 * q^57 + (-b3 - 21*b1) * q^58 + (-2*b2 - 6) * q^59 + (-2*b2 - 6) * q^61 + 6*b3 * q^62 + (b3 - b1) * q^63 - 13 * q^64 + (3*b2 - 7) * q^66 + (4*b3 - 4*b1) * q^67 - 20*b1 * q^68 + (2*b2 + 4) * q^69 + (2*b2 + 2) * q^71 + 3*b3 * q^72 + 10*b1 * q^73 + (b2 + 7) * q^74 - 5 * q^76 + (-4*b3 + 10*b1) * q^77 + (-3*b3 + 7*b1) * q^78 + (-4*b2 + 4) * q^79 + q^81 + (-7*b3 - 7*b1) * q^82 + 6*b1 * q^83 + (5*b2 - 5) * q^84 + (-3*b2 - 7) * q^86 + (3*b3 + b1) * q^87 + (-9*b3 + 21*b1) * q^88 + (3*b2 + 9) * q^89 + (-4*b2 + 10) * q^91 + (-10*b3 - 20*b1) * q^92 - 6*b1 * q^93 + (4*b2 - 14) * q^94 + 5*b2 * q^96 + (3*b3 - 5*b1) * q^97 + (-b3 + 14*b1) * q^98 + (b2 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{4} - 4 q^{9}+O(q^{10})$$ 4 * q - 20 * q^4 - 4 * q^9 $$4 q - 20 q^{4} - 4 q^{9} + 12 q^{11} + 28 q^{14} + 44 q^{16} + 4 q^{19} + 4 q^{21} - 28 q^{26} - 4 q^{29} + 24 q^{31} + 20 q^{36} - 12 q^{39} - 28 q^{41} - 60 q^{44} - 56 q^{46} - 4 q^{49} + 16 q^{51} - 84 q^{56} - 24 q^{59} - 24 q^{61} - 52 q^{64} - 28 q^{66} + 16 q^{69} + 8 q^{71} + 28 q^{74} - 20 q^{76} + 16 q^{79} + 4 q^{81} - 20 q^{84} - 28 q^{86} + 36 q^{89} + 40 q^{91} - 56 q^{94} - 12 q^{99}+O(q^{100})$$ 4 * q - 20 * q^4 - 4 * q^9 + 12 * q^11 + 28 * q^14 + 44 * q^16 + 4 * q^19 + 4 * q^21 - 28 * q^26 - 4 * q^29 + 24 * q^31 + 20 * q^36 - 12 * q^39 - 28 * q^41 - 60 * q^44 - 56 * q^46 - 4 * q^49 + 16 * q^51 - 84 * q^56 - 24 * q^59 - 24 * q^61 - 52 * q^64 - 28 * q^66 + 16 * q^69 + 8 * q^71 + 28 * q^74 - 20 * q^76 + 16 * q^79 + 4 * q^81 - 20 * q^84 - 28 * q^86 + 36 * q^89 + 40 * q^91 - 56 * q^94 - 12 * q^99

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 −1.32288 + 0.500000i 1.32288 − 0.500000i 1.32288 + 0.500000i −1.32288 − 0.500000i
2.64575i 1.00000i −5.00000 0 −2.64575 3.64575i 7.93725i −1.00000 0
799.2 2.64575i 1.00000i −5.00000 0 2.64575 1.64575i 7.93725i −1.00000 0
799.3 2.64575i 1.00000i −5.00000 0 2.64575 1.64575i 7.93725i −1.00000 0
799.4 2.64575i 1.00000i −5.00000 0 −2.64575 3.64575i 7.93725i −1.00000 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
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 1425.2.c.i 4
5.b even 2 1 inner 1425.2.c.i 4
5.c odd 4 1 285.2.a.d 2
5.c odd 4 1 1425.2.a.p 2
15.e even 4 1 855.2.a.g 2
15.e even 4 1 4275.2.a.u 2
20.e even 4 1 4560.2.a.bo 2
95.g even 4 1 5415.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.d 2 5.c odd 4 1
855.2.a.g 2 15.e even 4 1
1425.2.a.p 2 5.c odd 4 1
1425.2.c.i 4 1.a even 1 1 trivial
1425.2.c.i 4 5.b even 2 1 inner
4275.2.a.u 2 15.e even 4 1
4560.2.a.bo 2 20.e even 4 1
5415.2.a.s 2 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1425, [\chi])$$:

 $$T_{2}^{2} + 7$$ T2^2 + 7 $$T_{7}^{4} + 16T_{7}^{2} + 36$$ T7^4 + 16*T7^2 + 36 $$T_{11}^{2} - 6T_{11} + 2$$ T11^2 - 6*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 7)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 16T^{2} + 36$$
$11$ $$(T^{2} - 6 T + 2)^{2}$$
$13$ $$T^{4} + 32T^{2} + 4$$
$17$ $$(T^{2} + 16)^{2}$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 88T^{2} + 144$$
$29$ $$(T^{2} + 2 T - 62)^{2}$$
$31$ $$(T - 6)^{4}$$
$37$ $$T^{4} + 16T^{2} + 36$$
$41$ $$(T^{2} + 14 T + 42)^{2}$$
$43$ $$T^{4} + 32T^{2} + 4$$
$47$ $$T^{4} + 88T^{2} + 144$$
$53$ $$T^{4} + 128T^{2} + 64$$
$59$ $$(T^{2} + 12 T + 8)^{2}$$
$61$ $$(T^{2} + 12 T + 8)^{2}$$
$67$ $$T^{4} + 256T^{2} + 9216$$
$71$ $$(T^{2} - 4 T - 24)^{2}$$
$73$ $$(T^{2} + 100)^{2}$$
$79$ $$(T^{2} - 8 T - 96)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} - 18 T + 18)^{2}$$
$97$ $$T^{4} + 176T^{2} + 1444$$