Properties

Label 3525.2.a.bb
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 9 x^{5} + 6 x^{4} + 20 x^{3} - 9 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 9 x^{5} + 6 x^{4} + 20 x^{3} - 9 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} - \nu^{5} + 7 \nu^{4} + 13 \nu^{3} + \nu^{2} - 24 \nu - 16 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 12 \nu^{4} - 8 \nu^{3} + 34 \nu^{2} + 9 \nu - 19 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{6} - 3 \nu^{5} - 14 \nu^{4} + 19 \nu^{3} + 13 \nu^{2} - 27 \nu + 2 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{6} + 2 \nu^{5} - 19 \nu^{4} - 21 \nu^{3} + 38 \nu^{2} + 28 \nu - 18 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{6} + 2 \nu^{5} + 26 \nu^{4} - 6 \nu^{3} - 52 \nu^{2} - 2 \nu + 22 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 8 \beta_{5} + \beta_{4} - 8 \beta_{3} + 7 \beta_{2} + 9 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(9 \beta_{6} + 12 \beta_{5} + 8 \beta_{4} - 3 \beta_{3} + 10 \beta_{2} + 33 \beta_{1} + 21\)
\(\nu^{6}\)\(=\)\(11 \beta_{6} + 58 \beta_{5} + 12 \beta_{4} - 54 \beta_{3} + 48 \beta_{2} + 72 \beta_{1} + 104\)

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
−2.29205
−1.36884
−0.913267
0.231416
1.10030
1.49858
2.74386
−2.29205 −1.00000 3.25349 0 2.29205 4.47358 −2.87307 1.00000 0
1.2 −1.36884 −1.00000 −0.126278 0 1.36884 −0.111162 2.91053 1.00000 0
1.3 −0.913267 −1.00000 −1.16594 0 0.913267 −2.03699 2.89135 1.00000 0
1.4 0.231416 −1.00000 −1.94645 0 −0.231416 4.71752 −0.913273 1.00000 0
1.5 1.10030 −1.00000 −0.789350 0 −1.10030 1.25231 −3.06911 1.00000 0
1.6 1.49858 −1.00000 0.245756 0 −1.49858 3.73400 −2.62888 1.00000 0
1.7 2.74386 −1.00000 5.52877 0 −2.74386 −1.02926 9.68245 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.bb yes 7
5.b even 2 1 3525.2.a.y 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.2.a.y 7 5.b even 2 1
3525.2.a.bb yes 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2}^{7} - T_{2}^{6} - 9 T_{2}^{5} + 6 T_{2}^{4} + 20 T_{2}^{3} - 9 T_{2}^{2} - 12 T_{2} + 3 \)
\( T_{7}^{7} - 11 T_{7}^{6} + 29 T_{7}^{5} + 45 T_{7}^{4} - 201 T_{7}^{3} - 31 T_{7}^{2} + 206 T_{7} + 23 \)
\( T_{11}^{7} + 8 T_{11}^{6} - 3 T_{11}^{5} - 152 T_{11}^{4} - 365 T_{11}^{3} - 216 T_{11}^{2} + 21 T_{11} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 12 T - 9 T^{2} + 20 T^{3} + 6 T^{4} - 9 T^{5} - T^{6} + T^{7} \)
$3$ \( ( 1 + T )^{7} \)
$5$ \( T^{7} \)
$7$ \( 23 + 206 T - 31 T^{2} - 201 T^{3} + 45 T^{4} + 29 T^{5} - 11 T^{6} + T^{7} \)
$11$ \( 5 + 21 T - 216 T^{2} - 365 T^{3} - 152 T^{4} - 3 T^{5} + 8 T^{6} + T^{7} \)
$13$ \( 2783 - 1934 T - 1080 T^{2} + 510 T^{3} + 139 T^{4} - 39 T^{5} - 5 T^{6} + T^{7} \)
$17$ \( -17567 + 19548 T - 4632 T^{2} - 1272 T^{3} + 587 T^{4} - 30 T^{5} - 10 T^{6} + T^{7} \)
$19$ \( -15 + 714 T + 123 T^{2} - 968 T^{3} + 423 T^{4} - 38 T^{5} - 7 T^{6} + T^{7} \)
$23$ \( 121 + 1265 T - 1565 T^{2} + 9 T^{3} + 375 T^{4} - 77 T^{5} - 4 T^{6} + T^{7} \)
$29$ \( 825 - 2874 T + 2772 T^{2} - 182 T^{3} - 472 T^{4} - 26 T^{5} + 11 T^{6} + T^{7} \)
$31$ \( 43059 - 11781 T - 8979 T^{2} + 2299 T^{3} + 494 T^{4} - 119 T^{5} - 3 T^{6} + T^{7} \)
$37$ \( 30047 - 4321 T - 7498 T^{2} + 609 T^{3} + 516 T^{4} - 40 T^{5} - 11 T^{6} + T^{7} \)
$41$ \( 135321 + 46986 T - 30540 T^{2} - 15547 T^{3} - 2007 T^{4} + 21 T^{5} + 20 T^{6} + T^{7} \)
$43$ \( -2321 + 64040 T - 41712 T^{2} - 3433 T^{3} + 2052 T^{4} - 54 T^{5} - 18 T^{6} + T^{7} \)
$47$ \( ( 1 + T )^{7} \)
$53$ \( 290305 + 39358 T - 50732 T^{2} + 1162 T^{3} + 1883 T^{4} - 140 T^{5} - 12 T^{6} + T^{7} \)
$59$ \( 782875 - 4233 T - 71587 T^{2} + 709 T^{3} + 1987 T^{4} - 61 T^{5} - 18 T^{6} + T^{7} \)
$61$ \( -483289 - 338454 T + 97238 T^{2} + 32193 T^{3} - 1414 T^{4} - 362 T^{5} + 4 T^{6} + T^{7} \)
$67$ \( -272539 + 168586 T + 29675 T^{2} - 25165 T^{3} + 3487 T^{4} - 20 T^{5} - 22 T^{6} + T^{7} \)
$71$ \( 972757 + 903839 T + 212809 T^{2} + 2163 T^{3} - 3789 T^{4} - 235 T^{5} + 14 T^{6} + T^{7} \)
$73$ \( -104159 + 70533 T + 158317 T^{2} - 56101 T^{3} + 5108 T^{4} + 86 T^{5} - 30 T^{6} + T^{7} \)
$79$ \( 57725 - 221935 T + 14413 T^{2} + 16800 T^{3} - 291 T^{4} - 251 T^{5} + T^{6} + T^{7} \)
$83$ \( 896587 - 411837 T - 37633 T^{2} + 51002 T^{3} - 11290 T^{4} + 1129 T^{5} - 54 T^{6} + T^{7} \)
$89$ \( -121325 + 4710 T + 49231 T^{2} + 1827 T^{3} - 1613 T^{4} - 92 T^{5} + 14 T^{6} + T^{7} \)
$97$ \( 564059 + 641482 T - 303841 T^{2} - 21035 T^{3} + 9739 T^{4} - 320 T^{5} - 24 T^{6} + T^{7} \)
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