Properties

Label 3380.2.a.r
Level $3380$
Weight $2$
Character orbit 3380.a
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} - q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} - q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{9} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{11} + \beta_{1} q^{15} + ( 1 + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{17} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{19} + ( 1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{23} + q^{25} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 3 \beta_{7} + \beta_{8} ) q^{27} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{29} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{31} + ( -3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{33} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{35} + ( -\beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{37} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{41} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{43} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{45} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} ) q^{47} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{7} ) q^{49} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} ) q^{51} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{53} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{55} + ( -4 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{5} - 3 \beta_{7} - \beta_{8} ) q^{57} + ( 2 \beta_{1} + \beta_{2} + 4 \beta_{4} + \beta_{6} + 3 \beta_{7} - \beta_{8} ) q^{59} + ( 3 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{61} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{63} + ( -4 - 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - 6 \beta_{7} - 2 \beta_{8} ) q^{67} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + 6 \beta_{7} + \beta_{8} ) q^{69} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} - 2 \beta_{8} ) q^{71} + ( 4 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{73} -\beta_{1} q^{75} + ( 7 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} ) q^{77} + ( 2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{79} + ( 8 - \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{81} + ( -6 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{83} + ( -1 - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{85} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 8 \beta_{4} + \beta_{5} + 2 \beta_{6} - 6 \beta_{7} ) q^{87} + ( 5 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{8} ) q^{89} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - \beta_{6} - 6 \beta_{7} - 2 \beta_{8} ) q^{93} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{95} + ( 3 + 4 \beta_{1} + \beta_{2} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} ) q^{97} + ( 14 + 2 \beta_{1} - 5 \beta_{2} + 8 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + O(q^{10}) \) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + 7q^{11} + q^{15} + 13q^{17} + 4q^{19} + 3q^{21} + 12q^{23} + 9q^{25} - 4q^{27} + 16q^{29} - 13q^{31} - 34q^{33} + q^{35} - q^{37} + 6q^{41} + q^{43} - 12q^{45} + 2q^{47} + 20q^{49} + 11q^{51} + 30q^{53} - 7q^{55} - 38q^{57} - 15q^{59} + 21q^{61} + 17q^{63} + 7q^{67} + 15q^{69} + 7q^{71} + 28q^{73} - q^{75} + 46q^{77} + 31q^{79} + 41q^{81} - 45q^{83} - 13q^{85} + 28q^{87} + 41q^{89} + 11q^{93} - 4q^{95} - 8q^{97} + 81q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{8} + 42 \nu^{7} - 241 \nu^{6} - 394 \nu^{5} + 1964 \nu^{4} + 214 \nu^{3} - 3215 \nu^{2} + 1620 \nu + 226 \)\()/338\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{8} + 93 \nu^{7} - 256 \nu^{6} - 969 \nu^{5} + 2683 \nu^{4} + 1512 \nu^{3} - 5103 \nu^{2} + 1873 \nu - 79 \)\()/338\)
\(\beta_{4}\)\(=\)\((\)\( -33 \nu^{8} + 27 \nu^{7} + 678 \nu^{6} - 543 \nu^{5} - 4073 \nu^{4} + 3590 \nu^{3} + 6347 \nu^{2} - 6805 \nu + 773 \)\()/338\)
\(\beta_{5}\)\(=\)\((\)\( 84 \nu^{8} - 38 \nu^{7} - 1649 \nu^{6} + 614 \nu^{5} + 9369 \nu^{4} - 3976 \nu^{3} - 13452 \nu^{2} + 8288 \nu - 1645 \)\()/338\)
\(\beta_{6}\)\(=\)\((\)\( 108 \nu^{8} - 73 \nu^{7} - 2096 \nu^{6} + 1055 \nu^{5} + 12070 \nu^{4} - 5450 \nu^{3} - 19082 \nu^{2} + 9473 \nu + 82 \)\()/338\)
\(\beta_{7}\)\(=\)\((\)\( 112 \nu^{8} - 107 \nu^{7} - 2086 \nu^{6} + 1551 \nu^{5} + 11478 \nu^{4} - 7780 \nu^{3} - 16922 \nu^{2} + 13473 \nu - 1292 \)\()/338\)
\(\beta_{8}\)\(=\)\((\)\( -323 \nu^{8} + 295 \nu^{7} + 6206 \nu^{6} - 4731 \nu^{5} - 35175 \nu^{4} + 25992 \nu^{3} + 53223 \nu^{2} - 44995 \nu + 3551 \)\()/338\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{8} - 3 \beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{2} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{8} - 10 \beta_{7} + 8 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 14 \beta_{2} - \beta_{1} + 35\)
\(\nu^{5}\)\(=\)\(-14 \beta_{8} - 38 \beta_{7} - 3 \beta_{6} + 13 \beta_{5} + 28 \beta_{4} + 5 \beta_{3} - 21 \beta_{2} + 53 \beta_{1} + 17\)
\(\nu^{6}\)\(=\)\(-21 \beta_{8} - 106 \beta_{7} - 8 \beta_{6} + 74 \beta_{5} - 18 \beta_{4} + 47 \beta_{3} - 162 \beta_{2} - 7 \beta_{1} + 338\)
\(\nu^{7}\)\(=\)\(-162 \beta_{8} - 418 \beta_{7} - 55 \beta_{6} + 155 \beta_{5} + 330 \beta_{4} + 105 \beta_{3} - 319 \beta_{2} + 428 \beta_{1} + 355\)
\(\nu^{8}\)\(=\)\(-319 \beta_{8} - 1179 \beta_{7} - 160 \beta_{6} + 747 \beta_{5} + 195 \beta_{4} + 599 \beta_{3} - 1817 \beta_{2} + 13 \beta_{1} + 3428\)

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
3.40622
2.52382
1.40883
0.715841
0.161999
0.0545075
−1.65879
−2.79462
−2.81781
0 −3.40622 0 −1.00000 0 −2.33790 0 8.60234 0
1.2 0 −2.52382 0 −1.00000 0 4.54727 0 3.36968 0
1.3 0 −1.40883 0 −1.00000 0 2.17051 0 −1.01520 0
1.4 0 −0.715841 0 −1.00000 0 −3.76323 0 −2.48757 0
1.5 0 −0.161999 0 −1.00000 0 −1.89771 0 −2.97376 0
1.6 0 −0.0545075 0 −1.00000 0 −3.71818 0 −2.99703 0
1.7 0 1.65879 0 −1.00000 0 4.25414 0 −0.248414 0
1.8 0 2.79462 0 −1.00000 0 −1.22908 0 4.80991 0
1.9 0 2.81781 0 −1.00000 0 0.974186 0 4.94004 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

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 3380.2.a.r 9
13.b even 2 1 3380.2.a.s yes 9
13.d odd 4 2 3380.2.f.j 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.2.a.r 9 1.a even 1 1 trivial
3380.2.a.s yes 9 13.b even 2 1
3380.2.f.j 18 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\):

\(T_{3}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{19}^{9} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \)
$3$ \( -1 - 26 T - 149 T^{2} - 153 T^{3} + 87 T^{4} + 106 T^{5} - 16 T^{6} - 19 T^{7} + T^{8} + T^{9} \)
$5$ \( ( 1 + T )^{9} \)
$7$ \( 3121 + 1126 T - 3909 T^{2} - 1877 T^{3} + 951 T^{4} + 518 T^{5} - 60 T^{6} - 41 T^{7} + T^{8} + T^{9} \)
$11$ \( 94016 - 146176 T + 59760 T^{2} + 9440 T^{3} - 9992 T^{4} + 692 T^{5} + 473 T^{6} - 58 T^{7} - 7 T^{8} + T^{9} \)
$13$ \( T^{9} \)
$17$ \( 86528 - 332800 T + 85120 T^{2} + 52000 T^{3} - 17496 T^{4} - 1492 T^{5} + 877 T^{6} - 26 T^{7} - 13 T^{8} + T^{9} \)
$19$ \( 64 - 320 T - 3600 T^{2} - 7568 T^{3} - 3412 T^{4} + 2116 T^{5} + 253 T^{6} - 87 T^{7} - 4 T^{8} + T^{9} \)
$23$ \( 46171 - 29147 T - 35592 T^{2} + 32719 T^{3} - 4363 T^{4} - 2763 T^{5} + 873 T^{6} - 34 T^{7} - 12 T^{8} + T^{9} \)
$29$ \( -96559 + 84569 T + 253348 T^{2} + 53173 T^{3} - 32345 T^{4} - 3127 T^{5} + 1579 T^{6} - 42 T^{7} - 16 T^{8} + T^{9} \)
$31$ \( 105664 + 982272 T - 720592 T^{2} - 30960 T^{3} + 67008 T^{4} + 3144 T^{5} - 1725 T^{6} - 112 T^{7} + 13 T^{8} + T^{9} \)
$37$ \( -18752 + 53568 T + 15792 T^{2} - 97136 T^{3} + 12968 T^{4} + 7832 T^{5} - 321 T^{6} - 169 T^{7} + T^{8} + T^{9} \)
$41$ \( -23863181 + 8113743 T + 2724582 T^{2} - 825051 T^{3} - 96715 T^{4} + 23743 T^{5} + 1311 T^{6} - 264 T^{7} - 6 T^{8} + T^{9} \)
$43$ \( -9484117 + 5928484 T + 1527395 T^{2} - 681337 T^{3} - 41569 T^{4} + 20630 T^{5} + 380 T^{6} - 243 T^{7} - T^{8} + T^{9} \)
$47$ \( -25493819 + 5197257 T + 3326166 T^{2} - 780037 T^{3} - 83501 T^{4} + 23837 T^{5} + 725 T^{6} - 266 T^{7} - 2 T^{8} + T^{9} \)
$53$ \( 118208 + 231296 T - 145232 T^{2} - 70832 T^{3} + 57808 T^{4} - 10400 T^{5} - 325 T^{6} + 287 T^{7} - 30 T^{8} + T^{9} \)
$59$ \( 512 - 5376 T - 11072 T^{2} + 21664 T^{3} + 7440 T^{4} - 10296 T^{5} - 2775 T^{6} - 121 T^{7} + 15 T^{8} + T^{9} \)
$61$ \( 1253057 + 3967046 T + 3193811 T^{2} + 116995 T^{3} - 187917 T^{4} - 4048 T^{5} + 3888 T^{6} - 99 T^{7} - 21 T^{8} + T^{9} \)
$67$ \( -4851601 - 10949069 T + 5222685 T^{2} + 164623 T^{3} - 304407 T^{4} + 25721 T^{5} + 2890 T^{6} - 332 T^{7} - 7 T^{8} + T^{9} \)
$71$ \( -110282432 + 14212672 T + 8691152 T^{2} - 1127472 T^{3} - 228392 T^{4} + 29840 T^{5} + 2321 T^{6} - 310 T^{7} - 7 T^{8} + T^{9} \)
$73$ \( 7046656 - 49346816 T + 12267904 T^{2} + 2750976 T^{3} - 550048 T^{4} - 26112 T^{5} + 7080 T^{6} - 64 T^{7} - 28 T^{8} + T^{9} \)
$79$ \( 342428864 - 98522688 T - 10412416 T^{2} + 5714384 T^{3} - 334396 T^{4} - 61800 T^{5} + 7135 T^{6} + 46 T^{7} - 31 T^{8} + T^{9} \)
$83$ \( 65550407 + 56345154 T + 14665643 T^{2} + 35267 T^{3} - 560869 T^{4} - 79106 T^{5} - 720 T^{6} + 601 T^{7} + 45 T^{8} + T^{9} \)
$89$ \( -54990949 + 69505827 T - 26276881 T^{2} + 2680497 T^{3} + 357987 T^{4} - 86531 T^{5} + 3100 T^{6} + 438 T^{7} - 41 T^{8} + T^{9} \)
$97$ \( -7849472 + 22536192 T + 4079104 T^{2} - 2857376 T^{3} + 62344 T^{4} + 56012 T^{5} - 1859 T^{6} - 401 T^{7} + 8 T^{8} + T^{9} \)
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