Properties

Label 3525.2.a.w
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.414764096.1
Defining polynomial: \(x^{6} - 2 x^{5} - 11 x^{4} + 20 x^{3} + 29 x^{2} - 42 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 705)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 11 x^{4} + 20 x^{3} + 29 x^{2} - 42 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 8 \nu^{2} + 9 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 10 \nu^{3} + 21 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 8 \beta_{2} + 23\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + 10 \beta_{3} + 39 \beta_{1}\)

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
2.81413
2.33118
1.52844
−0.231521
−1.77961
−2.66262
−2.81413 −1.00000 5.91931 0 2.81413 3.18049 −11.0294 1.00000 0
1.2 −2.33118 −1.00000 3.43441 0 2.33118 −3.47124 −3.34386 1.00000 0
1.3 −1.52844 −1.00000 0.336137 0 1.52844 −3.11578 2.54312 1.00000 0
1.4 0.231521 −1.00000 −1.94640 0 −0.231521 3.28703 −0.913673 1.00000 0
1.5 1.77961 −1.00000 1.16702 0 −1.77961 −4.15308 −1.48237 1.00000 0
1.6 2.66262 −1.00000 5.08952 0 −2.66262 0.272578 8.22622 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.w 6
5.b even 2 1 705.2.a.m 6
15.d odd 2 1 2115.2.a.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.m 6 5.b even 2 1
2115.2.a.s 6 15.d odd 2 1
3525.2.a.w 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2}^{6} + 2 T_{2}^{5} - 11 T_{2}^{4} - 20 T_{2}^{3} + 29 T_{2}^{2} + 42 T_{2} - 11 \)
\( T_{7}^{6} + 4 T_{7}^{5} - 22 T_{7}^{4} - 84 T_{7}^{3} + 133 T_{7}^{2} + 440 T_{7} - 128 \)
\( T_{11}^{6} - 4 T_{11}^{5} - 16 T_{11}^{4} + 48 T_{11}^{3} + 68 T_{11}^{2} - 128 T_{11} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -11 + 42 T + 29 T^{2} - 20 T^{3} - 11 T^{4} + 2 T^{5} + T^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( T^{6} \)
$7$ \( -128 + 440 T + 133 T^{2} - 84 T^{3} - 22 T^{4} + 4 T^{5} + T^{6} \)
$11$ \( -64 - 128 T + 68 T^{2} + 48 T^{3} - 16 T^{4} - 4 T^{5} + T^{6} \)
$13$ \( 1348 + 944 T - 101 T^{2} - 182 T^{3} - 22 T^{4} + 6 T^{5} + T^{6} \)
$17$ \( 1348 - 944 T - 101 T^{2} + 182 T^{3} - 22 T^{4} - 6 T^{5} + T^{6} \)
$19$ \( -32 + 148 T + 431 T^{2} + 210 T^{3} - 18 T^{4} - 10 T^{5} + T^{6} \)
$23$ \( -128 - 440 T + 133 T^{2} + 84 T^{3} - 22 T^{4} - 4 T^{5} + T^{6} \)
$29$ \( -18268 - 5988 T + 1737 T^{2} + 452 T^{3} - 70 T^{4} - 8 T^{5} + T^{6} \)
$31$ \( 101888 - 53936 T + 3612 T^{2} + 1344 T^{3} - 144 T^{4} - 8 T^{5} + T^{6} \)
$37$ \( 16 - 176 T - 44 T^{2} + 144 T^{3} + 84 T^{4} + 16 T^{5} + T^{6} \)
$41$ \( 57860 + 14692 T - 6167 T^{2} - 2056 T^{3} - 114 T^{4} + 12 T^{5} + T^{6} \)
$43$ \( T^{6} \)
$47$ \( ( -1 + T )^{6} \)
$53$ \( 7412 - 8840 T + 523 T^{2} + 746 T^{3} - 82 T^{4} - 10 T^{5} + T^{6} \)
$59$ \( 12032 + 3164 T - 2729 T^{2} - 1214 T^{3} - 122 T^{4} + 6 T^{5} + T^{6} \)
$61$ \( -4700 + 5076 T - 1375 T^{2} - 228 T^{3} + 162 T^{4} - 24 T^{5} + T^{6} \)
$67$ \( -25600 - 12384 T + 3468 T^{2} + 1128 T^{3} - 192 T^{4} - 8 T^{5} + T^{6} \)
$71$ \( -524672 - 178148 T + 1231 T^{2} + 3730 T^{3} - 26 T^{4} - 26 T^{5} + T^{6} \)
$73$ \( 16720 + 28416 T + 7820 T^{2} - 32 T^{3} - 164 T^{4} - 4 T^{5} + T^{6} \)
$79$ \( 627968 - 148672 T - 13868 T^{2} + 4120 T^{3} - 48 T^{4} - 24 T^{5} + T^{6} \)
$83$ \( 199936 - 102656 T + 11184 T^{2} + 1392 T^{3} - 232 T^{4} - 4 T^{5} + T^{6} \)
$89$ \( 390704 + 48912 T - 21988 T^{2} - 4376 T^{3} - 108 T^{4} + 20 T^{5} + T^{6} \)
$97$ \( -53824 + 7424 T + 5776 T^{2} - 416 T^{3} - 180 T^{4} + T^{6} \)
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