Properties

Label 130.2.p.a
Level $130$
Weight $2$
Character orbit 130.p
Analytic conductor $1.038$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.p (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.03805522628\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{11} + \beta_{8} + \beta_{7} + \beta_{5}) q^{3} + \beta_{3} q^{4} + ( - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{5} - \beta_{8} q^{6} + ( - \beta_{7} + \beta_{2} - \beta_1) q^{7} + (\beta_{6} + \beta_{4}) q^{8} + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{11} + \beta_{8} + \beta_{7} + \beta_{5}) q^{3} + \beta_{3} q^{4} + ( - \beta_{11} + \beta_{10} + \beta_{7} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{5} - \beta_{8} q^{6} + ( - \beta_{7} + \beta_{2} - \beta_1) q^{7} + (\beta_{6} + \beta_{4}) q^{8} + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 4) q^{9} + (\beta_{11} + \beta_{7} + \beta_{2}) q^{10} + ( - \beta_{11} + \beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{11} + (\beta_{8} + \beta_{7} + \beta_{5} - \beta_{2} + \beta_1) q^{12} + (2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - 1) q^{13} + ( - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{14} + (2 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \cdots - 1) q^{15}+ \cdots + (3 \beta_{11} - \beta_{10} + 2 \beta_{8} - 7 \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} - 2 \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} - 24 q^{9} + 6 q^{11} - 12 q^{13} - 6 q^{15} - 6 q^{16} + 12 q^{17} + 12 q^{18} + 36 q^{19} - 6 q^{20} - 24 q^{21} + 6 q^{22} + 6 q^{23} - 12 q^{25} + 6 q^{26} - 12 q^{27} + 6 q^{29} + 6 q^{30} - 24 q^{31} - 6 q^{33} - 12 q^{34} - 12 q^{35} - 24 q^{36} - 6 q^{38} + 6 q^{39} + 18 q^{41} + 6 q^{42} + 6 q^{44} + 12 q^{45} - 6 q^{46} + 12 q^{47} - 24 q^{50} - 6 q^{52} + 18 q^{53} - 6 q^{54} - 24 q^{55} + 6 q^{56} + 6 q^{58} + 18 q^{59} + 24 q^{60} + 18 q^{61} + 12 q^{62} + 30 q^{63} - 12 q^{64} + 30 q^{65} - 36 q^{66} + 12 q^{67} - 42 q^{69} + 24 q^{70} + 18 q^{71} + 6 q^{72} + 6 q^{74} - 12 q^{75} + 30 q^{76} + 30 q^{77} + 30 q^{78} - 6 q^{80} + 30 q^{81} - 18 q^{82} - 48 q^{83} - 6 q^{84} - 18 q^{85} + 12 q^{87} + 6 q^{89} - 12 q^{90} - 6 q^{91} - 42 q^{93} + 24 q^{94} + 30 q^{95} - 102 q^{97} - 48 q^{98} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} + 380\nu + 1728 ) / 760 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} - 380\nu + 1728 ) / 760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{11} + 87\nu^{9} + 576\nu^{7} + 1504\nu^{5} + 1542\nu^{3} + 456\nu + 20 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 60 \nu^{11} + 36 \nu^{10} - 1315 \nu^{9} + 808 \nu^{8} - 8820 \nu^{7} + 5634 \nu^{6} - 23280 \nu^{5} + 15336 \nu^{4} - 23250 \nu^{3} + 14368 \nu^{2} - 5040 \nu + 744 ) / 760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{11} + 89 \nu^{10} - 173 \nu^{9} + 1892 \nu^{8} - 154 \nu^{7} + 11886 \nu^{6} + 5194 \nu^{5} + 27654 \nu^{4} + 19402 \nu^{3} + 21292 \nu^{2} + 19216 \nu + 1016 ) / 760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 60 \nu^{11} - 36 \nu^{10} - 1315 \nu^{9} - 808 \nu^{8} - 8820 \nu^{7} - 5634 \nu^{6} - 23280 \nu^{5} - 15336 \nu^{4} - 23250 \nu^{3} - 14368 \nu^{2} - 5040 \nu - 744 ) / 760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11 \nu^{11} + 82 \nu^{10} + 173 \nu^{9} + 1756 \nu^{8} + 154 \nu^{7} + 11218 \nu^{6} - 5194 \nu^{5} + 26952 \nu^{4} - 19402 \nu^{3} + 21116 \nu^{2} - 19596 \nu - 712 ) / 760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 36 \nu^{11} + 125 \nu^{10} - 808 \nu^{9} + 2700 \nu^{8} - 5634 \nu^{7} + 17520 \nu^{6} - 15336 \nu^{5} + 42990 \nu^{4} - 14368 \nu^{3} + 35280 \nu^{2} - 1504 \nu + 240 ) / 760 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} - 134 \nu^{10} + 99 \nu^{9} - 2902 \nu^{8} + 1182 \nu^{7} - 18976 \nu^{6} + 6218 \nu^{5} - 47584 \nu^{4} + 14054 \nu^{3} - 41152 \nu^{2} + 10892 \nu - 1756 ) / 760 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 105 \nu^{11} - 178 \nu^{10} - 2135 \nu^{9} - 3784 \nu^{8} - 11920 \nu^{7} - 23772 \nu^{6} - 18890 \nu^{5} - 55308 \nu^{4} + 9330 \nu^{3} - 42204 \nu^{2} + 28040 \nu - 132 ) / 760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -421\nu^{10} - 9048\nu^{8} - 58144\nu^{6} - 140586\nu^{4} - 112968\nu^{2} + 380\nu - 784 ) / 760 \) Copy content Toggle raw display
\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{11} - 4 \beta_{8} - 2 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} + 10 \beta_{2} - 10 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 13 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} - 27 \beta_{7} + 14 \beta_{6} - 29 \beta_{5} - 16 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 7 \beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{11} - 5 \beta_{10} + 59 \beta_{8} + 24 \beta_{7} - 77 \beta_{6} + 35 \beta_{5} - 72 \beta_{4} - 95 \beta_{3} - 118 \beta_{2} + 110 \beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 168 \beta_{11} - 24 \beta_{10} + 48 \beta_{9} - 24 \beta_{8} + 352 \beta_{7} - 174 \beta_{6} + 376 \beta_{5} + 198 \beta_{4} - 24 \beta_{3} - 56 \beta_{2} + 80 \beta _1 - 292 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 422 \beta_{11} + 96 \beta_{10} - 748 \beta_{8} - 270 \beta_{7} + 1008 \beta_{6} - 478 \beta_{5} + 912 \beta_{4} + 1264 \beta_{3} + 1440 \beta_{2} - 1288 \beta _1 - 680 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2122 \beta_{11} + 270 \beta_{10} - 540 \beta_{9} + 270 \beta_{8} - 4460 \beta_{7} + 2132 \beta_{6} - 4730 \beta_{5} - 2402 \beta_{4} + 270 \beta_{3} + 792 \beta_{2} - 1006 \beta _1 + 3460 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5316 \beta_{11} - 1368 \beta_{10} + 9264 \beta_{8} + 3156 \beta_{7} - 12698 \beta_{6} + 6108 \beta_{5} - 11330 \beta_{4} - 16044 \beta_{3} - 17724 \beta_{2} + 15564 \beta _1 + 8706 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 26474 \beta_{11} - 3156 \beta_{10} + 6312 \beta_{9} - 3156 \beta_{8} + 55718 \beta_{7} - 26196 \beta_{6} + 58874 \beta_{5} + 29352 \beta_{4} - 3156 \beta_{3} - 10290 \beta_{2} + 12642 \beta _1 - 42312 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 66116 \beta_{11} + 17810 \beta_{10} - 114422 \beta_{8} - 38016 \beta_{7} + 158054 \beta_{6} - 76406 \beta_{5} + 140244 \beta_{4} + 200358 \beta_{3} + 218764 \beta_{2} - 190664 \beta _1 - 109084 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-\beta_{4} - \beta_{6}\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.55227i
0.0775341i
1.62980i
3.51623i
1.32083i
2.19540i
1.55227i
0.0775341i
1.62980i
3.51623i
1.32083i
2.19540i
−0.866025 0.500000i −0.776134 2.89657i 0.500000 + 0.866025i 2.12044 0.709753i −0.776134 + 2.89657i −0.638961 1.10671i 1.00000i −5.18966 + 2.99625i −2.19123 0.445554i
7.2 −0.866025 0.500000i −0.0387670 0.144681i 0.500000 + 0.866025i 0.105914 + 2.23356i −0.0387670 + 0.144681i 1.10259 + 1.90974i 1.00000i 2.57865 1.48878i 1.02506 1.98727i
7.3 −0.866025 0.500000i 0.814901 + 3.04125i 0.500000 + 0.866025i −2.22635 + 0.208246i 0.814901 3.04125i 0.402397 + 0.696972i 1.00000i −5.98707 + 3.45663i 2.03220 + 0.932829i
37.1 0.866025 0.500000i −1.75811 0.471085i 0.500000 0.866025i −1.28703 1.82854i −1.75811 + 0.471085i 1.48736 2.57618i 1.00000i 0.270964 + 0.156441i −2.02887 0.940048i
37.2 0.866025 0.500000i 0.660414 + 0.176957i 0.500000 0.866025i 0.483457 + 2.18318i 0.660414 0.176957i 0.189447 0.328132i 1.00000i −2.19324 1.26627i 1.51028 + 1.64896i
37.3 0.866025 0.500000i 1.09770 + 0.294128i 0.500000 0.866025i 0.803571 2.08669i 1.09770 0.294128i −2.54283 + 4.40431i 1.00000i −1.47964 0.854273i −0.347432 2.20891i
93.1 −0.866025 + 0.500000i −0.776134 + 2.89657i 0.500000 0.866025i 2.12044 + 0.709753i −0.776134 2.89657i −0.638961 + 1.10671i 1.00000i −5.18966 2.99625i −2.19123 + 0.445554i
93.2 −0.866025 + 0.500000i −0.0387670 + 0.144681i 0.500000 0.866025i 0.105914 2.23356i −0.0387670 0.144681i 1.10259 1.90974i 1.00000i 2.57865 + 1.48878i 1.02506 + 1.98727i
93.3 −0.866025 + 0.500000i 0.814901 3.04125i 0.500000 0.866025i −2.22635 0.208246i 0.814901 + 3.04125i 0.402397 0.696972i 1.00000i −5.98707 3.45663i 2.03220 0.932829i
123.1 0.866025 + 0.500000i −1.75811 + 0.471085i 0.500000 + 0.866025i −1.28703 + 1.82854i −1.75811 0.471085i 1.48736 + 2.57618i 1.00000i 0.270964 0.156441i −2.02887 + 0.940048i
123.2 0.866025 + 0.500000i 0.660414 0.176957i 0.500000 + 0.866025i 0.483457 2.18318i 0.660414 + 0.176957i 0.189447 + 0.328132i 1.00000i −2.19324 + 1.26627i 1.51028 1.64896i
123.3 0.866025 + 0.500000i 1.09770 0.294128i 0.500000 + 0.866025i 0.803571 + 2.08669i 1.09770 + 0.294128i −2.54283 4.40431i 1.00000i −1.47964 + 0.854273i −0.347432 + 2.20891i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 123.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.p.a 12
5.b even 2 1 650.2.t.e 12
5.c odd 4 1 130.2.s.a yes 12
5.c odd 4 1 650.2.w.e 12
13.f odd 12 1 130.2.s.a yes 12
65.o even 12 1 650.2.t.e 12
65.s odd 12 1 650.2.w.e 12
65.t even 12 1 inner 130.2.p.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.p.a 12 1.a even 1 1 trivial
130.2.p.a 12 65.t even 12 1 inner
130.2.s.a yes 12 5.c odd 4 1
130.2.s.a yes 12 13.f odd 12 1
650.2.t.e 12 5.b even 2 1
650.2.t.e 12 65.o even 12 1
650.2.w.e 12 5.c odd 4 1
650.2.w.e 12 65.s odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 12 T_{3}^{10} + 4 T_{3}^{9} + 24 T_{3}^{8} + 24 T_{3}^{7} - 280 T_{3}^{6} + 48 T_{3}^{5} + 600 T_{3}^{4} - 568 T_{3}^{3} + 144 T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + 12 T^{10} + 4 T^{9} + 24 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{10} + 8 T^{9} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 21 T^{10} - 64 T^{9} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + 9 T^{10} - 28 T^{9} + \cdots + 6889 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + 69 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 12 T^{11} + 30 T^{10} + 32 T^{9} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{12} - 36 T^{11} + 627 T^{10} + \cdots + 221841 \) Copy content Toggle raw display
$23$ \( T^{12} - 6 T^{11} + 66 T^{10} + \cdots + 23078416 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} - 42 T^{10} + \cdots + 8248384 \) Copy content Toggle raw display
$31$ \( T^{12} + 24 T^{11} + 288 T^{10} + \cdots + 26152996 \) Copy content Toggle raw display
$37$ \( T^{12} + 93 T^{10} - 44 T^{9} + \cdots + 769129 \) Copy content Toggle raw display
$41$ \( T^{12} - 18 T^{11} + 126 T^{10} + \cdots + 1690000 \) Copy content Toggle raw display
$43$ \( T^{12} - 36 T^{10} + 200 T^{9} + \cdots + 22886656 \) Copy content Toggle raw display
$47$ \( (T^{6} - 6 T^{5} - 105 T^{4} + 196 T^{3} + \cdots - 6047)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 18 T^{11} + 162 T^{10} + \cdots + 36881329 \) Copy content Toggle raw display
$59$ \( T^{12} - 18 T^{11} - 6 T^{10} + 852 T^{9} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{12} - 18 T^{11} + \cdots + 307721764 \) Copy content Toggle raw display
$67$ \( T^{12} - 12 T^{11} - 12 T^{10} + \cdots + 2304 \) Copy content Toggle raw display
$71$ \( T^{12} - 18 T^{11} + \cdots + 880427584 \) Copy content Toggle raw display
$73$ \( T^{12} + 300 T^{10} + \cdots + 704902500 \) Copy content Toggle raw display
$79$ \( T^{12} + 444 T^{10} + \cdots + 1427177284 \) Copy content Toggle raw display
$83$ \( (T^{6} + 24 T^{5} + 36 T^{4} - 2466 T^{3} + \cdots - 31050)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 6 T^{11} + 159 T^{10} + \cdots + 39828721 \) Copy content Toggle raw display
$97$ \( T^{12} + 102 T^{11} + \cdots + 553217613796 \) Copy content Toggle raw display
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