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 $1$

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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:

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

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