# Properties

 Label 1425.2.a.z Level $1425$ Weight $2$ Character orbit 1425.a Self dual yes Analytic conductor $11.379$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.3786822880$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 8 x^{5} + 26 x^{4} + 11 x^{3} - 51 x^{2} + 12 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} + \beta_{1} q^{6} + ( 1 + \beta_{4} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} + \beta_{1} q^{6} + ( 1 + \beta_{4} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{11} + ( 2 + \beta_{2} ) q^{12} + ( 2 - \beta_{1} - \beta_{6} ) q^{13} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{14} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{16} + ( 1 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{17} + \beta_{1} q^{18} + q^{19} + ( 1 + \beta_{4} ) q^{21} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{22} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{24} + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{26} + q^{27} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{28} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{31} + ( 4 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{32} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{33} + ( 1 - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 1 + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + \beta_{1} q^{38} + ( 2 - \beta_{1} - \beta_{6} ) q^{39} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{41} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{42} + ( -\beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{43} + ( -5 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{6} ) q^{44} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{46} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{47} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{48} + ( 2 + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{49} + ( 1 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{51} + ( 5 - 4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{52} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{53} + \beta_{1} q^{54} + ( -5 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{56} + q^{57} + ( -6 - \beta_{1} - 4 \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{58} + ( -3 - \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{59} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{61} + ( 3 - \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{62} + ( 1 + \beta_{4} ) q^{63} + ( 4 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{64} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{66} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{67} + ( -2 + \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{68} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{69} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{73} + ( 1 - \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{74} + ( 2 + \beta_{2} ) q^{76} + ( -6 + 3 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{77} + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{78} + ( 2 - 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{79} + q^{81} + ( -4 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{82} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{83} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{84} + ( -7 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{6} ) q^{86} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} ) q^{87} + ( 2 - 7 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{88} + ( -2 + 3 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( 8 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{92} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{93} + ( -5 + 4 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{94} + ( 4 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{96} + ( 4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{97} + ( -1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{6} ) q^{98} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 3 q^{2} + 7 q^{3} + 11 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 7 q^{9} + O(q^{10})$$ $$7 q + 3 q^{2} + 7 q^{3} + 11 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 7 q^{9} - 4 q^{11} + 11 q^{12} + 8 q^{13} - 4 q^{14} + 19 q^{16} + 4 q^{17} + 3 q^{18} + 7 q^{19} + 8 q^{21} + 12 q^{22} + 10 q^{23} + 9 q^{24} - 20 q^{26} + 7 q^{27} + 14 q^{28} - 6 q^{29} + 4 q^{31} + 31 q^{32} - 4 q^{33} + 2 q^{34} + 11 q^{36} + 14 q^{37} + 3 q^{38} + 8 q^{39} + 2 q^{41} - 4 q^{42} - 2 q^{43} - 32 q^{44} - 4 q^{46} + 30 q^{47} + 19 q^{48} + 17 q^{49} + 4 q^{51} + 18 q^{52} + 3 q^{54} - 22 q^{56} + 7 q^{57} - 40 q^{58} - 18 q^{59} + 12 q^{61} + 18 q^{62} + 8 q^{63} + 11 q^{64} + 12 q^{66} + 18 q^{67} - 12 q^{68} + 10 q^{69} - 18 q^{71} + 9 q^{72} + 10 q^{73} + 6 q^{74} + 11 q^{76} - 18 q^{77} - 20 q^{78} - 4 q^{79} + 7 q^{81} - 16 q^{82} + 18 q^{83} + 14 q^{84} - 46 q^{86} - 6 q^{87} - 18 q^{88} - 8 q^{89} + 12 q^{91} + 34 q^{92} + 4 q^{93} - 20 q^{94} + 31 q^{96} + 20 q^{97} + 5 q^{98} - 4 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 8 x^{5} + 26 x^{4} + 11 x^{3} - 51 x^{2} + 12 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 10 \nu^{3} + 21 \nu - 2$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} - \nu^{5} - 10 \nu^{4} + 8 \nu^{3} + 25 \nu^{2} - 15 \nu - 8$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 16 \nu^{3} + 11 \nu^{2} - 28 \nu + 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 8 \beta_{2} + \beta_{1} + 22$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{4} + 10 \beta_{3} + 10 \beta_{2} + 29 \beta_{1} + 12$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{6} - 8 \beta_{5} + 12 \beta_{4} + 12 \beta_{3} + 57 \beta_{2} + 14 \beta_{1} + 132$$

## 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
 −2.39336 −1.70383 −0.184902 0.498112 1.57229 2.47637 2.73532
−2.39336 1.00000 3.72816 0 −2.39336 4.15221 −4.13612 1.00000 0
1.2 −1.70383 1.00000 0.903045 0 −1.70383 −0.338398 1.86903 1.00000 0
1.3 −0.184902 1.00000 −1.96581 0 −0.184902 −1.90997 0.733287 1.00000 0
1.4 0.498112 1.00000 −1.75188 0 0.498112 4.62756 −1.86886 1.00000 0
1.5 1.57229 1.00000 0.472094 0 1.57229 1.87913 −2.40231 1.00000 0
1.6 2.47637 1.00000 4.13242 0 2.47637 −3.36493 5.28066 1.00000 0
1.7 2.73532 1.00000 5.48197 0 2.73532 2.95440 9.52431 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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 1425.2.a.z 7
3.b odd 2 1 4275.2.a.bv 7
5.b even 2 1 1425.2.a.y 7
5.c odd 4 2 285.2.c.b 14
15.d odd 2 1 4275.2.a.bw 7
15.e even 4 2 855.2.c.g 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.c.b 14 5.c odd 4 2
855.2.c.g 14 15.e even 4 2
1425.2.a.y 7 5.b even 2 1
1425.2.a.z 7 1.a even 1 1 trivial
4275.2.a.bv 7 3.b odd 2 1
4275.2.a.bw 7 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1425))$$:

 $$T_{2}^{7} - 3 T_{2}^{6} - 8 T_{2}^{5} + 26 T_{2}^{4} + 11 T_{2}^{3} - 51 T_{2}^{2} + 12 T_{2} + 4$$ $$T_{7}^{7} - 8 T_{7}^{6} - T_{7}^{5} + 126 T_{7}^{4} - 166 T_{7}^{3} - 418 T_{7}^{2} + 568 T_{7} + 232$$ $$T_{11}^{7} + 4 T_{11}^{6} - 43 T_{11}^{5} - 112 T_{11}^{4} + 680 T_{11}^{3} + 546 T_{11}^{2} - 4016 T_{11} + 3232$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 12 T - 51 T^{2} + 11 T^{3} + 26 T^{4} - 8 T^{5} - 3 T^{6} + T^{7}$$
$3$ $$( -1 + T )^{7}$$
$5$ $$T^{7}$$
$7$ $$232 + 568 T - 418 T^{2} - 166 T^{3} + 126 T^{4} - T^{5} - 8 T^{6} + T^{7}$$
$11$ $$3232 - 4016 T + 546 T^{2} + 680 T^{3} - 112 T^{4} - 43 T^{5} + 4 T^{6} + T^{7}$$
$13$ $$-800 + 3744 T - 2080 T^{2} - 160 T^{3} + 266 T^{4} - 22 T^{5} - 8 T^{6} + T^{7}$$
$17$ $$40000 - 8000 T - 7700 T^{2} + 1740 T^{3} + 348 T^{4} - 83 T^{5} - 4 T^{6} + T^{7}$$
$19$ $$( -1 + T )^{7}$$
$23$ $$-14272 - 19840 T - 5920 T^{2} + 1232 T^{3} + 492 T^{4} - 48 T^{5} - 10 T^{6} + T^{7}$$
$29$ $$20000 - 26240 T + 3760 T^{2} + 2640 T^{3} - 342 T^{4} - 88 T^{5} + 6 T^{6} + T^{7}$$
$31$ $$-13568 - 6016 T + 3888 T^{2} + 2256 T^{3} + 76 T^{4} - 84 T^{5} - 4 T^{6} + T^{7}$$
$37$ $$-21376 + 56768 T - 17656 T^{2} - 2136 T^{3} + 1158 T^{4} - 42 T^{5} - 14 T^{6} + T^{7}$$
$41$ $$128 - 2304 T + 808 T^{2} + 792 T^{3} - 118 T^{4} - 80 T^{5} - 2 T^{6} + T^{7}$$
$43$ $$-12776 - 17784 T + 314 T^{2} + 3630 T^{3} - 188 T^{4} - 125 T^{5} + 2 T^{6} + T^{7}$$
$47$ $$128944 - 161376 T + 62604 T^{2} - 7748 T^{3} - 790 T^{4} + 309 T^{5} - 30 T^{6} + T^{7}$$
$53$ $$-1856 - 2048 T + 384 T^{2} + 752 T^{3} - 12 T^{4} - 64 T^{5} + T^{7}$$
$59$ $$40960 + 81920 T + 12160 T^{2} - 15168 T^{3} - 3368 T^{4} - 100 T^{5} + 18 T^{6} + T^{7}$$
$61$ $$5641360 - 748256 T - 234964 T^{2} + 27128 T^{3} + 3066 T^{4} - 291 T^{5} - 12 T^{6} + T^{7}$$
$67$ $$10240 + 67584 T + 21248 T^{2} - 19136 T^{3} + 3112 T^{4} - 68 T^{5} - 18 T^{6} + T^{7}$$
$71$ $$-10240 + 67584 T - 21248 T^{2} - 19136 T^{3} - 3112 T^{4} - 68 T^{5} + 18 T^{6} + T^{7}$$
$73$ $$-136832 - 117696 T - 12312 T^{2} + 6732 T^{3} + 854 T^{4} - 127 T^{5} - 10 T^{6} + T^{7}$$
$79$ $$-81920 - 225280 T + 34560 T^{2} + 16384 T^{3} - 1296 T^{4} - 328 T^{5} + 4 T^{6} + T^{7}$$
$83$ $$145856 - 222784 T + 127328 T^{2} - 33648 T^{3} + 3924 T^{4} - 84 T^{5} - 18 T^{6} + T^{7}$$
$89$ $$-652000 + 691200 T - 253200 T^{2} + 34320 T^{3} + 82 T^{4} - 328 T^{5} + 8 T^{6} + T^{7}$$
$97$ $$18272 - 46624 T - 36896 T^{2} - 1856 T^{3} + 1778 T^{4} - 18 T^{5} - 20 T^{6} + T^{7}$$