# Properties

 Label 6003.2.a.i Level $6003$ Weight $2$ Character orbit 6003.a Self dual yes Analytic conductor $47.934$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6003 = 3^{2} \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6003.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$47.9341963334$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - x^{6} - 9 x^{5} + 10 x^{4} + 19 x^{3} - 20 x^{2} - 5 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2001) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( 1 + \beta_{4} + \beta_{5} ) q^{4} + ( 1 - \beta_{3} + \beta_{6} ) q^{5} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( 1 + \beta_{4} + \beta_{5} ) q^{4} + ( 1 - \beta_{3} + \beta_{6} ) q^{5} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{8} + ( -1 - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{10} + ( 2 + \beta_{5} - \beta_{6} ) q^{11} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{13} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{14} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{16} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{17} + ( -1 - 3 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{19} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{6} ) q^{20} + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{22} + q^{23} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} ) q^{25} + ( 1 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{26} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{28} - q^{29} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{31} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{32} + ( -4 + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{34} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{35} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{37} + ( -1 - 3 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} ) q^{38} + ( -4 + \beta_{1} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{40} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} ) q^{41} + ( 2 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{43} + ( 3 - \beta_{1} + 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{44} + \beta_{3} q^{46} + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{49} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{50} + ( -1 + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{52} + ( -1 - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{53} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{55} + ( 5 + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{56} -\beta_{3} q^{58} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{59} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{61} + ( 3 + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{62} + ( 1 + \beta_{1} - \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{64} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{65} + ( -4 - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{67} + ( 2 \beta_{2} - 5 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{68} + ( 7 - 4 \beta_{1} - \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{70} + ( 3 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{71} + ( -6 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{73} + ( -2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} ) q^{74} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{76} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{77} + ( -6 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{79} + ( -3 - \beta_{1} + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} ) q^{80} + ( 1 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{82} + ( -1 + 2 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{83} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{85} + ( 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{86} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} ) q^{88} + ( 1 - 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{89} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{91} + ( 1 + \beta_{4} + \beta_{5} ) q^{92} + ( 6 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - \beta_{6} ) q^{94} + ( 1 - 2 \beta_{1} - \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{95} + ( -7 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{97} + ( -7 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + q^{2} + 7q^{4} + 5q^{5} - 5q^{7} - 3q^{8} + O(q^{10})$$ $$7q + q^{2} + 7q^{4} + 5q^{5} - 5q^{7} - 3q^{8} - 11q^{10} + 12q^{11} - 13q^{13} + 3q^{14} - 13q^{16} + 12q^{17} - 5q^{19} + 8q^{20} - q^{22} + 7q^{23} - 4q^{25} - 2q^{26} - 21q^{28} - 7q^{29} - 8q^{31} + 5q^{32} - 28q^{34} - 5q^{35} - 24q^{37} + 6q^{38} - 20q^{40} - 9q^{41} - q^{43} + 23q^{44} + q^{46} - 27q^{47} - 14q^{49} - 7q^{50} - 9q^{52} + q^{53} - 11q^{55} + 20q^{56} - q^{58} - 8q^{59} + q^{61} + 3q^{64} - 12q^{65} - 16q^{67} - 15q^{68} + 40q^{70} + 13q^{71} - 23q^{73} + 8q^{74} - 2q^{76} - 13q^{77} - 44q^{79} - 30q^{80} - 10q^{82} - 21q^{83} - 6q^{86} + 21q^{88} + 5q^{89} - 18q^{91} + 7q^{92} + 28q^{94} - 9q^{95} - 55q^{97} - 36q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 9 x^{5} + 10 x^{4} + 19 x^{3} - 20 x^{2} - 5 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{5} - 7 \nu^{4} - 11 \nu^{3} + 14 \nu^{2} + 10 \nu - 7$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 2 \nu^{5} + 7 \nu^{4} + 15 \nu^{3} - 10 \nu^{2} - 26 \nu - 1$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} - 9 \nu^{4} + \nu^{3} + 20 \nu^{2} - 2 \nu - 5$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} - 2 \nu^{5} + 25 \nu^{4} + 9 \nu^{3} - 50 \nu^{2} - 6 \nu + 5$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{6} - 2 \nu^{5} - 25 \nu^{4} + 23 \nu^{3} + 46 \nu^{2} - 46 \nu - 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + 7 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{6} - 10 \beta_{5} - 9 \beta_{4} + 8 \beta_{3} - \beta_{2} + 19 \beta_{1} - 11$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{6} + 44 \beta_{5} + 28 \beta_{4} - 19 \beta_{3} + 34 \beta_{2} - 11 \beta_{1} + 81$$

## 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
 −2.51747 1.40506 0.136094 −1.64802 −0.325238 2.13025 1.81932
−2.35219 0 3.53279 2.63525 0 −4.33765 −3.60540 0 −6.19860
1.2 −1.75752 0 1.08886 2.67591 0 0.0258051 1.60134 0 −4.70295
1.3 −1.17088 0 −0.629030 −0.418864 0 1.98148 3.07829 0 0.490441
1.4 0.865893 0 −1.25023 −1.66575 0 −0.715958 −2.81435 0 −1.44236
1.5 1.49169 0 0.225141 2.94997 0 1.89422 −2.64754 0 4.40045
1.6 1.55072 0 0.404722 0.920116 0 −2.53796 −2.47382 0 1.42684
1.7 2.37229 0 3.62775 −2.09663 0 −1.30993 3.86149 0 −4.97382
 $$n$$: 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
$$3$$ $$-1$$
$$23$$ $$-1$$
$$29$$ $$1$$

## 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 6003.2.a.i 7
3.b odd 2 1 2001.2.a.j 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.j 7 3.b odd 2 1
6003.2.a.i 7 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6003))$$:

 $$T_{2}^{7} - T_{2}^{6} - 10 T_{2}^{5} + 10 T_{2}^{4} + 29 T_{2}^{3} - 29 T_{2}^{2} - 24 T_{2} + 23$$ $$T_{5}^{7} - 5 T_{5}^{6} - 3 T_{5}^{5} + 40 T_{5}^{4} - 15 T_{5}^{3} - 87 T_{5}^{2} + 36 T_{5} + 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$23 - 24 T - 29 T^{2} + 29 T^{3} + 10 T^{4} - 10 T^{5} - T^{6} + T^{7}$$
$3$ $$T^{7}$$
$5$ $$28 + 36 T - 87 T^{2} - 15 T^{3} + 40 T^{4} - 3 T^{5} - 5 T^{6} + T^{7}$$
$7$ $$-1 + 37 T + 68 T^{2} - 3 T^{3} - 38 T^{4} - 5 T^{5} + 5 T^{6} + T^{7}$$
$11$ $$-23 + 90 T + 2 T^{2} - 150 T^{3} + 9 T^{4} + 39 T^{5} - 12 T^{6} + T^{7}$$
$13$ $$-1867 - 4597 T - 4314 T^{2} - 1849 T^{3} - 294 T^{4} + 27 T^{5} + 13 T^{6} + T^{7}$$
$17$ $$-12751 + 14263 T - 3350 T^{2} - 1068 T^{3} + 443 T^{4} - 2 T^{5} - 12 T^{6} + T^{7}$$
$19$ $$-7244 - 3664 T + 3993 T^{2} + 909 T^{3} - 306 T^{4} - 62 T^{5} + 5 T^{6} + T^{7}$$
$23$ $$( -1 + T )^{7}$$
$29$ $$( 1 + T )^{7}$$
$31$ $$-8156 + 6494 T + 7247 T^{2} + 278 T^{3} - 515 T^{4} - 54 T^{5} + 8 T^{6} + T^{7}$$
$37$ $$85172 + 64270 T - 4649 T^{2} - 6432 T^{3} - 483 T^{4} + 146 T^{5} + 24 T^{6} + T^{7}$$
$41$ $$1065508 + 685908 T + 136507 T^{2} + 1335 T^{3} - 2446 T^{4} - 187 T^{5} + 9 T^{6} + T^{7}$$
$43$ $$-15268 + 7782 T + 13761 T^{2} + 2793 T^{3} - 380 T^{4} - 115 T^{5} + T^{6} + T^{7}$$
$47$ $$10031 - 4296 T - 12110 T^{2} - 3545 T^{3} + 326 T^{4} + 226 T^{5} + 27 T^{6} + T^{7}$$
$53$ $$-596 + 1262 T + 955 T^{2} - 525 T^{3} - 506 T^{4} - 105 T^{5} - T^{6} + T^{7}$$
$59$ $$-128 + 128 T + 640 T^{2} + 192 T^{3} - 192 T^{4} - 36 T^{5} + 8 T^{6} + T^{7}$$
$61$ $$-36740 - 16402 T + 7949 T^{2} + 3219 T^{3} - 263 T^{4} - 134 T^{5} - T^{6} + T^{7}$$
$67$ $$124927 + 60419 T - 16308 T^{2} - 14086 T^{3} - 2557 T^{4} - 80 T^{5} + 16 T^{6} + T^{7}$$
$71$ $$16 + 4 T - 367 T^{2} - 479 T^{3} + 201 T^{4} + 24 T^{5} - 13 T^{6} + T^{7}$$
$73$ $$4228 + 55498 T - 11447 T^{2} - 20226 T^{3} - 3953 T^{4} - 59 T^{5} + 23 T^{6} + T^{7}$$
$79$ $$127580 - 162018 T - 78309 T^{2} + 772 T^{3} + 4205 T^{4} + 692 T^{5} + 44 T^{6} + T^{7}$$
$83$ $$212788 + 184452 T + 32991 T^{2} - 7657 T^{3} - 2362 T^{4} - 41 T^{5} + 21 T^{6} + T^{7}$$
$89$ $$18865 - 48069 T + 18732 T^{2} + 11085 T^{3} + 274 T^{4} - 195 T^{5} - 5 T^{6} + T^{7}$$
$97$ $$-1979204 - 998308 T - 62983 T^{2} + 46413 T^{3} + 11671 T^{4} + 1170 T^{5} + 55 T^{6} + T^{7}$$