Properties

Label 7098.2.a.cq
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.6148961.1
Defining polynomial: \(x^{6} - 2 x^{5} - 11 x^{4} + 12 x^{3} + 32 x^{2} - 16 x - 29\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + ( -2 + \beta_{5} ) q^{5} - q^{6} + q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} + ( -2 + \beta_{5} ) q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + ( 2 - \beta_{5} ) q^{10} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + q^{12} - q^{14} + ( -2 + \beta_{5} ) q^{15} + q^{16} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{17} - q^{18} + ( \beta_{1} - 2 \beta_{3} ) q^{19} + ( -2 + \beta_{5} ) q^{20} + q^{21} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{22} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{23} - q^{24} + ( 3 + \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{25} + q^{27} + q^{28} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{29} + ( 2 - \beta_{5} ) q^{30} + ( 1 - \beta_{2} - 2 \beta_{3} - 3 \beta_{5} ) q^{31} - q^{32} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( -\beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{34} + ( -2 + \beta_{5} ) q^{35} + q^{36} + ( -2 + \beta_{1} + \beta_{4} ) q^{37} + ( -\beta_{1} + 2 \beta_{3} ) q^{38} + ( 2 - \beta_{5} ) q^{40} + ( -8 + 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{41} - q^{42} + ( 2 + \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{43} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{44} + ( -2 + \beta_{5} ) q^{45} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{46} + ( -4 + 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{47} + q^{48} + q^{49} + ( -3 - \beta_{2} + \beta_{3} + 3 \beta_{5} ) q^{50} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{51} + ( 2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{53} - q^{54} + ( 7 + 3 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{55} - q^{56} + ( \beta_{1} - 2 \beta_{3} ) q^{57} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{58} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{59} + ( -2 + \beta_{5} ) q^{60} + ( 2 - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{61} + ( -1 + \beta_{2} + 2 \beta_{3} + 3 \beta_{5} ) q^{62} + q^{63} + q^{64} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{66} + ( -6 - \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{68} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{69} + ( 2 - \beta_{5} ) q^{70} + ( -\beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{5} ) q^{71} - q^{72} + ( \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{73} + ( 2 - \beta_{1} - \beta_{4} ) q^{74} + ( 3 + \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{75} + ( \beta_{1} - 2 \beta_{3} ) q^{76} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{77} + ( 4 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{79} + ( -2 + \beta_{5} ) q^{80} + q^{81} + ( 8 - 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{82} + ( \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{83} + q^{84} + ( -7 + \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{85} + ( -2 - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{86} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{87} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{88} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{89} + ( 2 - \beta_{5} ) q^{90} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{92} + ( 1 - \beta_{2} - 2 \beta_{3} - 3 \beta_{5} ) q^{93} + ( 4 - 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{94} + ( -2 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{95} - q^{96} + ( \beta_{2} - 6 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{97} - q^{98} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 6q^{3} + 6q^{4} - 9q^{5} - 6q^{6} + 6q^{7} - 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 6q^{3} + 6q^{4} - 9q^{5} - 6q^{6} + 6q^{7} - 6q^{8} + 6q^{9} + 9q^{10} - 10q^{11} + 6q^{12} - 6q^{14} - 9q^{15} + 6q^{16} + 4q^{17} - 6q^{18} - 2q^{19} - 9q^{20} + 6q^{21} + 10q^{22} - 6q^{24} + 9q^{25} + 6q^{27} + 6q^{28} - 4q^{29} + 9q^{30} - 9q^{31} - 6q^{32} - 10q^{33} - 4q^{34} - 9q^{35} + 6q^{36} - 9q^{37} + 2q^{38} + 9q^{40} - 25q^{41} - 6q^{42} + 19q^{43} - 10q^{44} - 9q^{45} - 21q^{47} + 6q^{48} + 6q^{49} - 9q^{50} + 4q^{51} - 4q^{53} - 6q^{54} + 21q^{55} - 6q^{56} - 2q^{57} + 4q^{58} - 20q^{59} - 9q^{60} + 3q^{61} + 9q^{62} + 6q^{63} + 6q^{64} + 10q^{66} - 24q^{67} + 4q^{68} + 9q^{70} - 13q^{71} - 6q^{72} + 9q^{73} + 9q^{74} + 9q^{75} - 2q^{76} - 10q^{77} + 28q^{79} - 9q^{80} + 6q^{81} + 25q^{82} - 15q^{83} + 6q^{84} - 17q^{85} - 19q^{86} - 4q^{87} + 10q^{88} - 11q^{89} + 9q^{90} - 9q^{93} + 21q^{94} - 6q^{96} - 2q^{97} - 6q^{98} - 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 11 x^{4} + 12 x^{3} + 32 x^{2} - 16 x - 29\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 9 \nu^{2} + 7 \nu + 15 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 9 \nu^{3} + 9 \nu^{2} + 13 \nu - 6 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{4} - 3 \nu^{3} - 19 \nu^{2} + 8 \nu + 29 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 12 \nu^{3} - 12 \nu^{2} + 23 \nu + 31 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{3} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + 3 \beta_{4} + \beta_{3} - 4 \beta_{2} + 7 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(-11 \beta_{5} + 15 \beta_{4} + 11 \beta_{3} - 6 \beta_{2} + 16 \beta_{1} + 31\)
\(\nu^{5}\)\(=\)\(-22 \beta_{5} + 48 \beta_{4} + 24 \beta_{3} - 48 \beta_{2} + 73 \beta_{1} + 77\)

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
1.63026
−2.29987
3.58604
−1.33906
1.49793
−1.07530
−1.00000 1.00000 1.00000 −3.93763 −1.00000 1.00000 −1.00000 1.00000 3.93763
1.2 −1.00000 1.00000 1.00000 −3.86789 −1.00000 1.00000 −1.00000 1.00000 3.86789
1.3 −1.00000 1.00000 1.00000 −2.59594 −1.00000 1.00000 −1.00000 1.00000 2.59594
1.4 −1.00000 1.00000 1.00000 −0.404061 −1.00000 1.00000 −1.00000 1.00000 0.404061
1.5 −1.00000 1.00000 1.00000 0.867888 −1.00000 1.00000 −1.00000 1.00000 −0.867888
1.6 −1.00000 1.00000 1.00000 0.937626 −1.00000 1.00000 −1.00000 1.00000 −0.937626
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(1\)

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 7098.2.a.cq 6
13.b even 2 1 7098.2.a.cu yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cq 6 1.a even 1 1 trivial
7098.2.a.cu yes 6 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5}^{6} + 9 T_{5}^{5} + 21 T_{5}^{4} - 9 T_{5}^{3} - 49 T_{5}^{2} + 15 T_{5} + 13 \)
\( T_{11}^{6} + 10 T_{11}^{5} - 9 T_{11}^{4} - 296 T_{11}^{3} - 342 T_{11}^{2} + 1630 T_{11} + 937 \)
\( T_{17}^{6} - 4 T_{17}^{5} - 55 T_{17}^{4} + 256 T_{17}^{3} + 452 T_{17}^{2} - 2486 T_{17} + 337 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( ( -1 + T )^{6} \)
$5$ \( 13 + 15 T - 49 T^{2} - 9 T^{3} + 21 T^{4} + 9 T^{5} + T^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( 937 + 1630 T - 342 T^{2} - 296 T^{3} - 9 T^{4} + 10 T^{5} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( 337 - 2486 T + 452 T^{2} + 256 T^{3} - 55 T^{4} - 4 T^{5} + T^{6} \)
$19$ \( -1 - 54 T + 116 T^{2} - 26 T^{3} - 25 T^{4} + 2 T^{5} + T^{6} \)
$23$ \( 547 - 1442 T + 934 T^{2} + 14 T^{3} - 67 T^{4} + T^{6} \)
$29$ \( 64 + 176 T + 88 T^{2} - 67 T^{3} - 27 T^{4} + 4 T^{5} + T^{6} \)
$31$ \( -5447 + 12287 T + 1201 T^{2} - 845 T^{3} - 95 T^{4} + 9 T^{5} + T^{6} \)
$37$ \( -211 - 601 T - 525 T^{2} - 149 T^{3} + 7 T^{4} + 9 T^{5} + T^{6} \)
$41$ \( -111007 - 93411 T - 25963 T^{2} - 2361 T^{3} + 91 T^{4} + 25 T^{5} + T^{6} \)
$43$ \( 8728 + 6112 T - 9310 T^{2} + 1755 T^{3} - 19 T^{5} + T^{6} \)
$47$ \( -392 - 8232 T - 4214 T^{2} - 385 T^{3} + 105 T^{4} + 21 T^{5} + T^{6} \)
$53$ \( 6728 + 13224 T + 5786 T^{2} - 487 T^{3} - 153 T^{4} + 4 T^{5} + T^{6} \)
$59$ \( 2456 - 15276 T - 14812 T^{2} - 3407 T^{3} - 91 T^{4} + 20 T^{5} + T^{6} \)
$61$ \( 7288 - 8336 T + 2286 T^{2} + 227 T^{3} - 111 T^{4} - 3 T^{5} + T^{6} \)
$67$ \( 87688 - 26100 T - 22750 T^{2} - 3267 T^{3} + 7 T^{4} + 24 T^{5} + T^{6} \)
$71$ \( 203624 + 168468 T + 13574 T^{2} - 3025 T^{3} - 244 T^{4} + 13 T^{5} + T^{6} \)
$73$ \( -14792 + 18276 T - 8344 T^{2} + 1647 T^{3} - 98 T^{4} - 9 T^{5} + T^{6} \)
$79$ \( -169000 + 100100 T - 19720 T^{2} + 791 T^{3} + 215 T^{4} - 28 T^{5} + T^{6} \)
$83$ \( 106952 + 50432 T - 2826 T^{2} - 2679 T^{3} - 162 T^{4} + 15 T^{5} + T^{6} \)
$89$ \( 1163 + 2277 T - 143 T^{2} - 363 T^{3} - 19 T^{4} + 11 T^{5} + T^{6} \)
$97$ \( -569192 + 5356 T + 26936 T^{2} - 149 T^{3} - 343 T^{4} + 2 T^{5} + T^{6} \)
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