# Properties

 Label 3525.2.a.w Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.414764096.1 Defining polynomial: $$x^{6} - 2 x^{5} - 11 x^{4} + 20 x^{3} + 29 x^{2} - 42 x - 11$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 705) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 11 x^{4} + 20 x^{3} + 29 x^{2} - 42 x - 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} - 8 \nu^{2} + 9$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - 10 \nu^{3} + 21 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} + 8 \beta_{2} + 23$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} + 10 \beta_{3} + 39 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.81413 2.33118 1.52844 −0.231521 −1.77961 −2.66262
−2.81413 −1.00000 5.91931 0 2.81413 3.18049 −11.0294 1.00000 0
1.2 −2.33118 −1.00000 3.43441 0 2.33118 −3.47124 −3.34386 1.00000 0
1.3 −1.52844 −1.00000 0.336137 0 1.52844 −3.11578 2.54312 1.00000 0
1.4 0.231521 −1.00000 −1.94640 0 −0.231521 3.28703 −0.913673 1.00000 0
1.5 1.77961 −1.00000 1.16702 0 −1.77961 −4.15308 −1.48237 1.00000 0
1.6 2.66262 −1.00000 5.08952 0 −2.66262 0.272578 8.22622 1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$47$$ $$-1$$

## 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 3525.2.a.w 6
5.b even 2 1 705.2.a.m 6
15.d odd 2 1 2115.2.a.s 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.m 6 5.b even 2 1
2115.2.a.s 6 15.d odd 2 1
3525.2.a.w 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}^{6} + 2 T_{2}^{5} - 11 T_{2}^{4} - 20 T_{2}^{3} + 29 T_{2}^{2} + 42 T_{2} - 11$$ $$T_{7}^{6} + 4 T_{7}^{5} - 22 T_{7}^{4} - 84 T_{7}^{3} + 133 T_{7}^{2} + 440 T_{7} - 128$$ $$T_{11}^{6} - 4 T_{11}^{5} - 16 T_{11}^{4} + 48 T_{11}^{3} + 68 T_{11}^{2} - 128 T_{11} - 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-11 + 42 T + 29 T^{2} - 20 T^{3} - 11 T^{4} + 2 T^{5} + T^{6}$$
$3$ $$( 1 + T )^{6}$$
$5$ $$T^{6}$$
$7$ $$-128 + 440 T + 133 T^{2} - 84 T^{3} - 22 T^{4} + 4 T^{5} + T^{6}$$
$11$ $$-64 - 128 T + 68 T^{2} + 48 T^{3} - 16 T^{4} - 4 T^{5} + T^{6}$$
$13$ $$1348 + 944 T - 101 T^{2} - 182 T^{3} - 22 T^{4} + 6 T^{5} + T^{6}$$
$17$ $$1348 - 944 T - 101 T^{2} + 182 T^{3} - 22 T^{4} - 6 T^{5} + T^{6}$$
$19$ $$-32 + 148 T + 431 T^{2} + 210 T^{3} - 18 T^{4} - 10 T^{5} + T^{6}$$
$23$ $$-128 - 440 T + 133 T^{2} + 84 T^{3} - 22 T^{4} - 4 T^{5} + T^{6}$$
$29$ $$-18268 - 5988 T + 1737 T^{2} + 452 T^{3} - 70 T^{4} - 8 T^{5} + T^{6}$$
$31$ $$101888 - 53936 T + 3612 T^{2} + 1344 T^{3} - 144 T^{4} - 8 T^{5} + T^{6}$$
$37$ $$16 - 176 T - 44 T^{2} + 144 T^{3} + 84 T^{4} + 16 T^{5} + T^{6}$$
$41$ $$57860 + 14692 T - 6167 T^{2} - 2056 T^{3} - 114 T^{4} + 12 T^{5} + T^{6}$$
$43$ $$T^{6}$$
$47$ $$( -1 + T )^{6}$$
$53$ $$7412 - 8840 T + 523 T^{2} + 746 T^{3} - 82 T^{4} - 10 T^{5} + T^{6}$$
$59$ $$12032 + 3164 T - 2729 T^{2} - 1214 T^{3} - 122 T^{4} + 6 T^{5} + T^{6}$$
$61$ $$-4700 + 5076 T - 1375 T^{2} - 228 T^{3} + 162 T^{4} - 24 T^{5} + T^{6}$$
$67$ $$-25600 - 12384 T + 3468 T^{2} + 1128 T^{3} - 192 T^{4} - 8 T^{5} + T^{6}$$
$71$ $$-524672 - 178148 T + 1231 T^{2} + 3730 T^{3} - 26 T^{4} - 26 T^{5} + T^{6}$$
$73$ $$16720 + 28416 T + 7820 T^{2} - 32 T^{3} - 164 T^{4} - 4 T^{5} + T^{6}$$
$79$ $$627968 - 148672 T - 13868 T^{2} + 4120 T^{3} - 48 T^{4} - 24 T^{5} + T^{6}$$
$83$ $$199936 - 102656 T + 11184 T^{2} + 1392 T^{3} - 232 T^{4} - 4 T^{5} + T^{6}$$
$89$ $$390704 + 48912 T - 21988 T^{2} - 4376 T^{3} - 108 T^{4} + 20 T^{5} + T^{6}$$
$97$ $$-53824 + 7424 T + 5776 T^{2} - 416 T^{3} - 180 T^{4} + T^{6}$$