# Properties

 Label 3525.2.a.z Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $0$ Dimension $7$ 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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - x^{6} - 9 x^{5} + 10 x^{4} + 16 x^{3} - 15 x^{2} - 6 x + 5$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} -\beta_{1} q^{6} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} -\beta_{1} q^{6} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{8} + q^{9} + ( \beta_{1} + \beta_{3} ) q^{11} + ( 1 + \beta_{2} + \beta_{3} ) q^{12} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{13} + ( 1 - 2 \beta_{1} + \beta_{6} ) q^{14} + ( 2 - 2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{16} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{17} -\beta_{1} q^{18} + ( -2 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{19} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{21} + ( -2 - \beta_{1} - \beta_{4} ) q^{22} + ( 1 + \beta_{1} - \beta_{6} ) q^{23} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{24} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{26} + q^{27} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{28} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{31} + ( 5 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{32} + ( \beta_{1} + \beta_{3} ) q^{33} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{34} + ( 1 + \beta_{2} + \beta_{3} ) q^{36} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{37} + ( -1 + 5 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{38} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} ) q^{39} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{41} + ( 1 - 2 \beta_{1} + \beta_{6} ) q^{42} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{43} + ( 3 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{44} + ( -4 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{46} + q^{47} + ( 2 - 2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{48} + ( 1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{49} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{51} + ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{52} + ( 2 + 4 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{53} -\beta_{1} q^{54} + ( 5 - 4 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{56} + ( -2 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{57} + ( 5 + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{58} + ( 1 - 2 \beta_{1} - 4 \beta_{3} - \beta_{4} ) q^{59} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{6} ) q^{61} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{62} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{63} + ( 3 - 5 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{64} + ( -2 - \beta_{1} - \beta_{4} ) q^{66} + ( 2 - 3 \beta_{1} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{67} + ( 2 + 3 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} ) q^{68} + ( 1 + \beta_{1} - \beta_{6} ) q^{69} + ( 3 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{71} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{72} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{73} + ( -4 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{74} + ( -12 + \beta_{1} - 4 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{76} + ( 2 - \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{77} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{78} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{79} + q^{81} + ( -4 - 5 \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{82} + ( 9 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{83} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{84} + ( -3 + 3 \beta_{1} - \beta_{3} + 4 \beta_{4} + 2 \beta_{6} ) q^{86} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{87} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{88} + ( 5 - 5 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{89} + ( -3 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{91} + ( -4 + 5 \beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{92} + ( \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{93} -\beta_{1} q^{94} + ( 5 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{96} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{97} + ( 6 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{98} + ( \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - q^{2} + 7q^{3} + 5q^{4} - q^{6} + 7q^{7} + 6q^{8} + 7q^{9} + O(q^{10})$$ $$7q - q^{2} + 7q^{3} + 5q^{4} - q^{6} + 7q^{7} + 6q^{8} + 7q^{9} + 5q^{12} + 5q^{13} + 7q^{14} + 9q^{16} + 2q^{17} - q^{18} - 13q^{19} + 7q^{21} - 14q^{22} + 6q^{23} + 6q^{24} + 7q^{27} + 30q^{28} + 9q^{29} + 5q^{31} + 26q^{32} - 8q^{34} + 5q^{36} - 5q^{37} - 2q^{38} + 5q^{39} + 18q^{41} + 7q^{42} + 14q^{43} + 17q^{44} - 27q^{46} + 7q^{47} + 9q^{48} + 14q^{49} + 2q^{51} - 3q^{52} + 20q^{53} - q^{54} + 17q^{56} - 13q^{57} + 37q^{58} + 10q^{59} - 8q^{61} - 6q^{62} + 7q^{63} + 18q^{64} - 14q^{66} + 4q^{67} + 10q^{68} + 6q^{69} + 12q^{71} + 6q^{72} + 4q^{73} - 25q^{74} - 66q^{76} + 6q^{77} - 5q^{79} + 7q^{81} - 29q^{82} + 52q^{83} + 30q^{84} - 17q^{86} + 9q^{87} + 26q^{88} + 32q^{89} - 26q^{91} - 17q^{92} + 5q^{93} - q^{94} + 26q^{96} - 12q^{97} + 40q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 9 x^{5} + 10 x^{4} + 16 x^{3} - 15 x^{2} - 6 x + 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{6} + 9 \nu^{4} - \nu^{3} - 17 \nu^{2} - 2 \nu + 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{6} - 9 \nu^{4} + \nu^{3} + 18 \nu^{2} + 2 \nu - 8$$ $$\beta_{4}$$ $$=$$ $$\nu^{6} - 9 \nu^{4} + 2 \nu^{3} + 18 \nu^{2} - 3 \nu - 7$$ $$\beta_{5}$$ $$=$$ $$-\nu^{6} + 10 \nu^{4} - \nu^{3} - 24 \nu^{2} + 10$$ $$\beta_{6}$$ $$=$$ $$-\nu^{6} + \nu^{5} + 9 \nu^{4} - 9 \nu^{3} - 15 \nu^{2} + 8 \nu + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + 7 \beta_{3} + 6 \beta_{2} - 2 \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} + 8 \beta_{4} - 10 \beta_{3} - 3 \beta_{2} + 30 \beta_{1} - 13$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{5} - \beta_{4} + 47 \beta_{3} + 36 \beta_{2} - 25 \beta_{1} + 99$$

## 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
 2.30790 1.85243 0.835360 0.608551 −0.704785 −1.21680 −2.68265
−2.30790 1.00000 3.32640 0 −2.30790 0.860272 −3.06120 1.00000 0
1.2 −1.85243 1.00000 1.43149 0 −1.85243 2.79882 1.05312 1.00000 0
1.3 −0.835360 1.00000 −1.30217 0 −0.835360 1.49880 2.75850 1.00000 0
1.4 −0.608551 1.00000 −1.62967 0 −0.608551 −3.83697 2.20884 1.00000 0
1.5 0.704785 1.00000 −1.50328 0 0.704785 −2.27680 −2.46906 1.00000 0
1.6 1.21680 1.00000 −0.519390 0 1.21680 4.53738 −3.06560 1.00000 0
1.7 2.68265 1.00000 5.19661 0 2.68265 3.41850 8.57540 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$47$$ $$-1$$

## 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.z 7
5.b even 2 1 3525.2.a.ba yes 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.2.a.z 7 1.a even 1 1 trivial
3525.2.a.ba yes 7 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}^{7} + T_{2}^{6} - 9 T_{2}^{5} - 10 T_{2}^{4} + 16 T_{2}^{3} + 15 T_{2}^{2} - 6 T_{2} - 5$$ $$T_{7}^{7} - 7 T_{7}^{6} - 7 T_{7}^{5} + 133 T_{7}^{4} - 181 T_{7}^{3} - 411 T_{7}^{2} + 978 T_{7} - 489$$ $$T_{11}^{7} - 23 T_{11}^{5} + 4 T_{11}^{4} + 133 T_{11}^{3} - 52 T_{11}^{2} - 41 T_{11} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 - 6 T + 15 T^{2} + 16 T^{3} - 10 T^{4} - 9 T^{5} + T^{6} + T^{7}$$
$3$ $$( -1 + T )^{7}$$
$5$ $$T^{7}$$
$7$ $$-489 + 978 T - 411 T^{2} - 181 T^{3} + 133 T^{4} - 7 T^{5} - 7 T^{6} + T^{7}$$
$11$ $$5 - 41 T - 52 T^{2} + 133 T^{3} + 4 T^{4} - 23 T^{5} + T^{7}$$
$13$ $$-365 + 794 T - 192 T^{2} - 352 T^{3} + 213 T^{4} - 27 T^{5} - 5 T^{6} + T^{7}$$
$17$ $$57 - 294 T + 186 T^{2} + 436 T^{3} + 23 T^{4} - 42 T^{5} - 2 T^{6} + T^{7}$$
$19$ $$-219 + 48 T + 1815 T^{2} - 1394 T^{3} - 587 T^{4} - 8 T^{5} + 13 T^{6} + T^{7}$$
$23$ $$909 + 261 T - 861 T^{2} + 43 T^{3} + 161 T^{4} - 23 T^{5} - 6 T^{6} + T^{7}$$
$29$ $$35129 - 13918 T - 26272 T^{2} + 3226 T^{3} + 1020 T^{4} - 116 T^{5} - 9 T^{6} + T^{7}$$
$31$ $$-30829 - 23291 T - 113 T^{2} + 2607 T^{3} + 214 T^{4} - 85 T^{5} - 5 T^{6} + T^{7}$$
$37$ $$-32977 + 4627 T + 10898 T^{2} + 1409 T^{3} - 464 T^{4} - 80 T^{5} + 5 T^{6} + T^{7}$$
$41$ $$-23655 + 13638 T + 5742 T^{2} - 5243 T^{3} + 931 T^{4} + 31 T^{5} - 18 T^{6} + T^{7}$$
$43$ $$1116955 - 80486 T - 84590 T^{2} + 5075 T^{3} + 1928 T^{4} - 122 T^{5} - 14 T^{6} + T^{7}$$
$47$ $$( -1 + T )^{7}$$
$53$ $$294825 + 381558 T - 74040 T^{2} - 16400 T^{3} + 3809 T^{4} - 84 T^{5} - 20 T^{6} + T^{7}$$
$59$ $$3566055 - 339111 T - 171963 T^{2} + 17119 T^{3} + 2443 T^{4} - 247 T^{5} - 10 T^{6} + T^{7}$$
$61$ $$8611 + 83562 T + 61134 T^{2} + 5505 T^{3} - 1402 T^{4} - 162 T^{5} + 8 T^{6} + T^{7}$$
$67$ $$217817 - 126356 T - 13319 T^{2} + 11521 T^{3} + 623 T^{4} - 218 T^{5} - 4 T^{6} + T^{7}$$
$71$ $$24181 + 1763 T - 13461 T^{2} + 719 T^{3} + 837 T^{4} - 63 T^{5} - 12 T^{6} + T^{7}$$
$73$ $$358529 - 135779 T - 31099 T^{2} + 11603 T^{3} + 568 T^{4} - 212 T^{5} - 4 T^{6} + T^{7}$$
$79$ $$-537783 + 76941 T + 114321 T^{2} + 10724 T^{3} - 2279 T^{4} - 287 T^{5} + 5 T^{6} + T^{7}$$
$83$ $$4993687 - 3254513 T + 734573 T^{2} - 51516 T^{3} - 4486 T^{4} + 923 T^{5} - 52 T^{6} + T^{7}$$
$89$ $$-5199685 + 2012222 T - 89973 T^{2} - 43983 T^{3} + 4599 T^{4} + 138 T^{5} - 32 T^{6} + T^{7}$$
$97$ $$-6069845 - 58134 T + 524779 T^{2} + 46493 T^{3} - 4861 T^{4} - 436 T^{5} + 12 T^{6} + T^{7}$$