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 $1$

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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: \(7\)
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:

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

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