# Properties

 Label 3525.2.a.x Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $1$ 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: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} - 15 x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} - q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + \beta_{1} q^{6} + ( -1 - \beta_{5} ) q^{7} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} - q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + \beta_{1} q^{6} + ( -1 - \beta_{5} ) q^{7} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{8} + q^{9} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{11} + ( -\beta_{1} - \beta_{2} ) q^{12} + ( -1 - \beta_{4} + \beta_{6} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{14} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{16} + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{17} -\beta_{1} q^{18} + ( 1 - \beta_{1} - \beta_{3} - \beta_{6} ) q^{19} + ( 1 + \beta_{5} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{22} + ( -\beta_{2} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{23} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{24} + ( 1 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{26} - q^{27} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{28} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{29} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{31} + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{32} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{33} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{34} + ( \beta_{1} + \beta_{2} ) q^{36} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{37} + ( 1 - \beta_{1} + \beta_{4} ) q^{38} + ( 1 + \beta_{4} - \beta_{6} ) q^{39} + ( 2 - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{41} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{42} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{43} + ( -3 + 4 \beta_{1} + 2 \beta_{3} + \beta_{4} + 3 \beta_{6} ) q^{44} + ( 1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} ) q^{46} + q^{47} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{48} + ( -2 + \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{49} + ( 3 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{51} + ( -1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 4 \beta_{6} ) q^{52} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{53} + \beta_{1} q^{54} + ( 3 - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{56} + ( -1 + \beta_{1} + \beta_{3} + \beta_{6} ) q^{57} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{58} + ( 3 + \beta_{2} - \beta_{3} + \beta_{5} ) q^{59} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{61} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{62} + ( -1 - \beta_{5} ) q^{63} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{64} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{66} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{67} + ( -\beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{68} + ( \beta_{2} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{69} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + \beta_{6} ) q^{71} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{72} + ( 2 - 4 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{73} + ( -4 + \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{74} + ( \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{76} + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{77} + ( -1 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{78} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{79} + q^{81} + ( 1 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{82} + ( -6 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{83} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{84} + ( 3 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{86} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{87} + ( -5 + 2 \beta_{1} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{88} + ( \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{89} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{91} + ( -2 + 5 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 7 \beta_{4} + 2 \beta_{6} ) q^{92} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{93} -\beta_{1} q^{94} + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{96} + ( -4 + 5 \beta_{1} + \beta_{3} + \beta_{4} + 4 \beta_{6} ) q^{97} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 2 \beta_{6} ) q^{98} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 3q^{2} - 7q^{3} + 5q^{4} + 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + O(q^{10})$$ $$7q - 3q^{2} - 7q^{3} + 5q^{4} + 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + 4q^{11} - 5q^{12} - 5q^{13} + 5q^{14} + 9q^{16} - 10q^{17} - 3q^{18} + q^{19} + 5q^{21} - 10q^{22} - 10q^{23} + 6q^{24} + 12q^{26} - 7q^{27} - 2q^{28} + 9q^{29} + 3q^{31} - 4q^{33} - 20q^{34} + 5q^{36} - 9q^{37} + 2q^{38} + 5q^{39} + 20q^{41} - 5q^{42} - 16q^{43} - 5q^{44} - q^{46} + 7q^{47} - 9q^{48} - 10q^{49} + 10q^{51} - 21q^{52} + 3q^{54} + 21q^{56} - q^{57} - 19q^{58} + 18q^{59} + 2q^{62} - 5q^{63} - 30q^{64} + 10q^{66} - 8q^{67} - 20q^{68} + 10q^{69} + 14q^{71} - 6q^{72} - 4q^{73} - 17q^{74} + 12q^{76} - 2q^{77} - 12q^{78} - 21q^{79} + 7q^{81} + 7q^{82} - 22q^{83} + 2q^{84} + 35q^{86} - 9q^{87} - 14q^{88} + 2q^{89} - 2q^{91} - 5q^{92} - 3q^{93} - 3q^{94} - 12q^{97} - 30q^{98} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} - 15 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6 \nu^{2} + 4 \nu + 4$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - \nu^{4} - 6 \nu^{3} + 5 \nu^{2} + 4 \nu - 2$$ $$\beta_{5}$$ $$=$$ $$-\nu^{6} + 2 \nu^{5} + 6 \nu^{4} - 11 \nu^{3} - 5 \nu^{2} + 6 \nu + 1$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} - 5 \nu^{4} + 18 \nu^{3} - 15 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + 7 \beta_{1} + 9$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{6} + 7 \beta_{5} + 8 \beta_{4} + \beta_{3} + \beta_{2} + 28 \beta_{1} + 7$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{6} + 8 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 33 \beta_{2} + 44 \beta_{1} + 48$$

## 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
 2.49904 2.29479 1.28015 0.269111 −0.271225 −0.833881 −2.23799
−2.49904 −1.00000 4.24521 0 2.49904 0.534565 −5.61086 1.00000 0
1.2 −2.29479 −1.00000 3.26608 0 2.29479 −4.13749 −2.90540 1.00000 0
1.3 −1.28015 −1.00000 −0.361223 0 1.28015 3.00128 3.02271 1.00000 0
1.4 −0.269111 −1.00000 −1.92758 0 0.269111 −3.07209 1.05695 1.00000 0
1.5 0.271225 −1.00000 −1.92644 0 −0.271225 −0.253445 −1.06495 1.00000 0
1.6 0.833881 −1.00000 −1.30464 0 −0.833881 −1.65674 −2.75568 1.00000 0
1.7 2.23799 −1.00000 3.00859 0 −2.23799 0.583922 2.25722 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.x 7
5.b even 2 1 3525.2.a.bc yes 7

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

 $$T_{2}^{7} + 3 T_{2}^{6} - 5 T_{2}^{5} - 18 T_{2}^{4} + 15 T_{2}^{2} - 1$$ $$T_{7}^{7} + 5 T_{7}^{6} - 7 T_{7}^{5} - 51 T_{7}^{4} - 17 T_{7}^{3} + 53 T_{7}^{2} - 6 T_{7} - 5$$ $$T_{11}^{7} - 4 T_{11}^{6} - 31 T_{11}^{5} + 126 T_{11}^{4} + 161 T_{11}^{3} - 844 T_{11}^{2} + 39 T_{11} + 1137$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 15 T^{2} - 18 T^{4} - 5 T^{5} + 3 T^{6} + T^{7}$$
$3$ $$( 1 + T )^{7}$$
$5$ $$T^{7}$$
$7$ $$-5 - 6 T + 53 T^{2} - 17 T^{3} - 51 T^{4} - 7 T^{5} + 5 T^{6} + T^{7}$$
$11$ $$1137 + 39 T - 844 T^{2} + 161 T^{3} + 126 T^{4} - 31 T^{5} - 4 T^{6} + T^{7}$$
$13$ $$87 + 330 T + 328 T^{2} - 44 T^{3} - 151 T^{4} - 27 T^{5} + 5 T^{6} + T^{7}$$
$17$ $$-39 - 864 T + 688 T^{2} + 212 T^{3} - 197 T^{4} - 10 T^{5} + 10 T^{6} + T^{7}$$
$19$ $$5 + 18 T - 55 T^{2} + 47 T^{4} - 20 T^{5} - T^{6} + T^{7}$$
$23$ $$-27595 + 48243 T + 22587 T^{2} - 87 T^{3} - 991 T^{4} - 75 T^{5} + 10 T^{6} + T^{7}$$
$29$ $$-1123 + 2156 T - 886 T^{2} - 322 T^{3} + 204 T^{4} - 4 T^{5} - 9 T^{6} + T^{7}$$
$31$ $$-6681 - 8217 T - 1175 T^{2} + 1281 T^{3} + 192 T^{4} - 73 T^{5} - 3 T^{6} + T^{7}$$
$37$ $$-6885 + 7587 T + 7722 T^{2} - 975 T^{3} - 1036 T^{4} - 100 T^{5} + 9 T^{6} + T^{7}$$
$41$ $$-16475 + 17164 T + 1152 T^{2} - 3241 T^{3} + 473 T^{4} + 79 T^{5} - 20 T^{6} + T^{7}$$
$43$ $$17715 + 15552 T - 10144 T^{2} - 8263 T^{3} - 1576 T^{4} - 30 T^{5} + 16 T^{6} + T^{7}$$
$47$ $$( -1 + T )^{7}$$
$53$ $$23697 - 146010 T - 472 T^{2} + 10262 T^{3} + 47 T^{4} - 200 T^{5} + T^{7}$$
$59$ $$8247 + 5979 T - 1663 T^{2} - 1283 T^{3} + 231 T^{4} + 71 T^{5} - 18 T^{6} + T^{7}$$
$61$ $$-3265 + 206 T + 4066 T^{2} + 1869 T^{3} - 34 T^{4} - 94 T^{5} + T^{7}$$
$67$ $$-6775 - 694 T + 5849 T^{2} + 1567 T^{3} - 535 T^{4} - 84 T^{5} + 8 T^{6} + T^{7}$$
$71$ $$3460269 + 475179 T - 293419 T^{2} + 4403 T^{3} + 4943 T^{4} - 287 T^{5} - 14 T^{6} + T^{7}$$
$73$ $$203465 - 122491 T - 7785 T^{2} + 8751 T^{3} - 150 T^{4} - 174 T^{5} + 4 T^{6} + T^{7}$$
$79$ $$675901 + 526685 T + 90209 T^{2} - 10200 T^{3} - 3131 T^{4} - 59 T^{5} + 21 T^{6} + T^{7}$$
$83$ $$-2833 - 34557 T + 47091 T^{2} + 6 T^{3} - 2062 T^{4} - 27 T^{5} + 22 T^{6} + T^{7}$$
$89$ $$-8993 - 10062 T + 207 T^{2} + 2247 T^{3} + 99 T^{4} - 92 T^{5} - 2 T^{6} + T^{7}$$
$97$ $$14907 + 127422 T + 102383 T^{2} + 9849 T^{3} - 2313 T^{4} - 208 T^{5} + 12 T^{6} + T^{7}$$