# Properties

 Label 3525.2.a.bd Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $1$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{7} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{8} + q^{9} + ( -3 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{13} + ( -2 + 2 \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{14} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{16} + ( 2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} -\beta_{1} q^{18} + ( -2 + \beta_{1} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{21} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{22} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{23} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{24} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{26} + q^{27} + ( -6 + \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{28} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{29} + ( -2 + \beta_{1} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{31} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{32} + ( -3 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{33} + ( 3 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( 1 - \beta_{2} + 3 \beta_{5} ) q^{37} + ( -6 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} ) q^{38} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{39} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{41} + ( -2 + 2 \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{42} + ( -5 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{44} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{46} - q^{47} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{48} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{49} + ( 2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{51} + ( -3 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{52} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{53} -\beta_{1} q^{54} + ( 2 + 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{56} + ( -2 + \beta_{1} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{57} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{7} ) q^{59} + ( 4 - \beta_{1} - 4 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{61} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{62} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{63} + ( -1 - \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{64} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{66} + ( -5 + 3 \beta_{1} - \beta_{2} + 2 \beta_{6} + 3 \beta_{7} ) q^{67} + ( 3 - 7 \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{68} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{69} + ( 3 - \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{71} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{72} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{73} + ( -7 + 7 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{74} + ( -4 + 8 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{76} + ( -4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{77} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{78} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{79} + q^{81} + ( 7 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{82} + ( -8 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{83} + ( -6 + \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{84} + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{86} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{87} + ( -10 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{88} + ( -1 - \beta_{1} - \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{89} + ( 8 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{91} + ( 7 + 2 \beta_{2} + \beta_{4} + 4 \beta_{5} + 3 \beta_{6} ) q^{92} + ( -2 + \beta_{1} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{93} + \beta_{1} q^{94} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{96} + ( -5 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{97} + ( -1 - 3 \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{98} + ( -3 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} + O(q^{10})$$ $$8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 7q^{12} - 10q^{13} + q^{14} + 5q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 8q^{21} - 10q^{23} - 6q^{24} - 14q^{26} + 8q^{27} - 44q^{28} - 13q^{29} - 10q^{32} - 8q^{33} + 28q^{34} + 7q^{36} - 3q^{37} - 36q^{38} - 10q^{39} - 16q^{41} + q^{42} - 25q^{43} - 17q^{44} - 5q^{46} - 8q^{47} + 5q^{48} + 16q^{49} - 6q^{51} + 17q^{52} - 4q^{53} - 3q^{54} + 37q^{56} - 2q^{57} - 15q^{58} - 8q^{59} + 15q^{61} - 6q^{62} - 8q^{63} - 14q^{64} - 27q^{67} - 14q^{68} - 10q^{69} + 14q^{71} - 6q^{72} - 28q^{73} - 21q^{74} + 6q^{76} - 4q^{77} - 14q^{78} + 7q^{79} + 8q^{81} + 53q^{82} - 60q^{83} - 44q^{84} - 3q^{86} - 13q^{87} - 54q^{88} - 34q^{89} + 23q^{91} + 43q^{92} + 3q^{94} - 10q^{96} - 7q^{97} - 40q^{98} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{7} - 3 \nu^{6} - 7 \nu^{5} + 24 \nu^{4} + 7 \nu^{3} - 46 \nu^{2} + 13 \nu + 9$$ $$\beta_{4}$$ $$=$$ $$\nu^{7} - 4 \nu^{6} - 5 \nu^{5} + 32 \nu^{4} - 7 \nu^{3} - 61 \nu^{2} + 33 \nu + 11$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{7} - 6 \nu^{6} - 13 \nu^{5} + 47 \nu^{4} + 7 \nu^{3} - 87 \nu^{2} + 36 \nu + 14$$ $$\beta_{6}$$ $$=$$ $$3 \nu^{7} - 10 \nu^{6} - 18 \nu^{5} + 79 \nu^{4} + \nu^{3} - 148 \nu^{2} + 64 \nu + 26$$ $$\beta_{7}$$ $$=$$ $$-6 \nu^{7} + 19 \nu^{6} + 38 \nu^{5} - 149 \nu^{4} - 16 \nu^{3} + 275 \nu^{2} - 109 \nu - 45$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{5} - \beta_{4} + 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} + 6 \beta_{2} + \beta_{1} + 13$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 9 \beta_{6} - 6 \beta_{5} - 8 \beta_{4} - \beta_{3} + \beta_{2} + 26 \beta_{1} - 5$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{7} + 20 \beta_{6} + 2 \beta_{5} - 11 \beta_{4} + 7 \beta_{3} + 35 \beta_{2} + 10 \beta_{1} + 65$$ $$\nu^{7}$$ $$=$$ $$13 \beta_{7} + 68 \beta_{6} - 29 \beta_{5} - 58 \beta_{4} - 9 \beta_{3} + 14 \beta_{2} + 140 \beta_{1} - 16$$

## 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
 2.62510 2.19791 1.60965 0.936719 −0.237165 −0.267165 −1.60641 −2.25864
−2.62510 1.00000 4.89115 0 −2.62510 −5.24764 −7.58955 1.00000 0
1.2 −2.19791 1.00000 2.83083 0 −2.19791 1.05086 −1.82609 1.00000 0
1.3 −1.60965 1.00000 0.590961 0 −1.60965 −2.89141 2.26805 1.00000 0
1.4 −0.936719 1.00000 −1.12256 0 −0.936719 3.65526 2.92496 1.00000 0
1.5 0.237165 1.00000 −1.94375 0 0.237165 1.64667 −0.935320 1.00000 0
1.6 0.267165 1.00000 −1.92862 0 0.267165 −0.207232 −1.04959 1.00000 0
1.7 1.60641 1.00000 0.580562 0 1.60641 −2.35394 −2.28020 1.00000 0
1.8 2.25864 1.00000 3.10144 0 2.25864 −3.65257 2.48774 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.bd 8
5.b even 2 1 3525.2.a.be yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.2.a.bd 8 1.a even 1 1 trivial
3525.2.a.be yes 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}^{8} + \cdots$$ $$T_{7}^{8} + \cdots$$ $$T_{11}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 - 13 T + 8 T^{2} + 47 T^{3} + 8 T^{4} - 24 T^{5} - 7 T^{6} + 3 T^{7} + T^{8}$$
$3$ $$( -1 + T )^{8}$$
$5$ $$T^{8}$$
$7$ $$-171 - 723 T + 613 T^{2} + 550 T^{3} - 170 T^{4} - 144 T^{5} - 4 T^{6} + 8 T^{7} + T^{8}$$
$11$ $$-4420 - 2423 T + 3505 T^{2} + 2140 T^{3} - 341 T^{4} - 334 T^{5} - 27 T^{6} + 8 T^{7} + T^{8}$$
$13$ $$6107 - 7887 T + 430 T^{2} + 2142 T^{3} - 167 T^{4} - 236 T^{5} - 2 T^{6} + 10 T^{7} + T^{8}$$
$17$ $$25308 - 23103 T - 9866 T^{2} + 6050 T^{3} + 1396 T^{4} - 377 T^{5} - 70 T^{6} + 6 T^{7} + T^{8}$$
$19$ $$3627 + 19161 T - 14237 T^{2} - 2741 T^{3} + 2513 T^{4} - 33 T^{5} - 93 T^{6} + 2 T^{7} + T^{8}$$
$23$ $$-365714 - 287761 T - 8623 T^{2} + 30413 T^{3} + 2565 T^{4} - 983 T^{5} - 93 T^{6} + 10 T^{7} + T^{8}$$
$29$ $$-1138486 - 530079 T + 112928 T^{2} + 67382 T^{3} + 878 T^{4} - 1812 T^{5} - 108 T^{6} + 13 T^{7} + T^{8}$$
$31$ $$351587 - 412420 T - 150440 T^{2} + 21796 T^{3} + 8661 T^{4} - 287 T^{5} - 168 T^{6} + T^{8}$$
$37$ $$283838 + 95633 T - 149173 T^{2} - 11424 T^{3} + 11209 T^{4} - 128 T^{5} - 194 T^{6} + 3 T^{7} + T^{8}$$
$41$ $$-98438 - 49655 T + 32288 T^{2} + 14196 T^{3} - 2343 T^{4} - 1015 T^{5} - 7 T^{6} + 16 T^{7} + T^{8}$$
$43$ $$22599 + 10449 T - 22356 T^{2} - 16821 T^{3} - 3123 T^{4} + 390 T^{5} + 206 T^{6} + 25 T^{7} + T^{8}$$
$47$ $$( 1 + T )^{8}$$
$53$ $$1104692 - 65167 T - 218058 T^{2} + 14076 T^{3} + 11280 T^{4} - 499 T^{5} - 192 T^{6} + 4 T^{7} + T^{8}$$
$59$ $$3744478 - 6145171 T - 850537 T^{2} + 225639 T^{3} + 28445 T^{4} - 2439 T^{5} - 303 T^{6} + 8 T^{7} + T^{8}$$
$61$ $$24176727 + 5452713 T - 1106768 T^{2} - 233085 T^{3} + 22079 T^{4} + 3300 T^{5} - 234 T^{6} - 15 T^{7} + T^{8}$$
$67$ $$2431981 - 3628639 T + 385043 T^{2} + 185004 T^{3} - 12438 T^{4} - 3749 T^{5} + 28 T^{6} + 27 T^{7} + T^{8}$$
$71$ $$-3420630 + 1496547 T + 472789 T^{2} - 211859 T^{3} + 337 T^{4} + 4411 T^{5} - 251 T^{6} - 14 T^{7} + T^{8}$$
$73$ $$1021770 + 2499183 T + 736567 T^{2} - 40623 T^{3} - 36745 T^{4} - 3574 T^{5} + 100 T^{6} + 28 T^{7} + T^{8}$$
$79$ $$-19475608 + 14632017 T - 856871 T^{2} - 646651 T^{3} + 72996 T^{4} + 4105 T^{5} - 527 T^{6} - 7 T^{7} + T^{8}$$
$83$ $$66012500 - 4940625 T - 12377125 T^{2} - 2683375 T^{3} - 190280 T^{4} + 3498 T^{5} + 1191 T^{6} + 60 T^{7} + T^{8}$$
$89$ $$16076470 + 8648971 T + 688848 T^{2} - 370851 T^{3} - 87773 T^{4} - 5629 T^{5} + 186 T^{6} + 34 T^{7} + T^{8}$$
$97$ $$11093 - 44475 T - 273701 T^{2} + 45082 T^{3} + 18882 T^{4} - 1285 T^{5} - 260 T^{6} + 7 T^{7} + T^{8}$$