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

Downloads

Learn more

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:

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

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