Properties

Label 29.6.a.b
Level $29$
Weight $6$
Character orbit 29.a
Self dual yes
Analytic conductor $4.651$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.65113077458\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{5} + 4) q^{3} + (\beta_{2} - 3 \beta_1 + 23) q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{5} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{6} + ( - 4 \beta_{6} - 4 \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_1 + 23) q^{7} + (2 \beta_{6} - 6 \beta_{5} + 8 \beta_{4} + \beta_{3} - \beta_{2} - 13 \beta_1 + 140) q^{8} + (7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 139) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{5} + 4) q^{3} + (\beta_{2} - 3 \beta_1 + 23) q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{5} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{6} + ( - 4 \beta_{6} - 4 \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_1 + 23) q^{7} + (2 \beta_{6} - 6 \beta_{5} + 8 \beta_{4} + \beta_{3} - \beta_{2} - 13 \beta_1 + 140) q^{8} + (7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 139) q^{9} + ( - 2 \beta_{6} - 4 \beta_{5} - 10 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + \cdots + 123) q^{10}+ \cdots + (2061 \beta_{6} + 2369 \beta_{5} + 3564 \beta_{4} + 1740 \beta_{3} + \cdots + 43573) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} + 26 q^{3} + 154 q^{4} + 32 q^{5} + 22 q^{6} + 184 q^{7} + 942 q^{8} + 1005 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{2} + 26 q^{3} + 154 q^{4} + 32 q^{5} + 22 q^{6} + 184 q^{7} + 942 q^{8} + 1005 q^{9} + 922 q^{10} + 1106 q^{11} + 214 q^{12} + 408 q^{13} - 2008 q^{14} - 614 q^{15} + 242 q^{16} - 874 q^{17} - 5598 q^{18} + 4288 q^{19} - 6350 q^{20} - 4200 q^{21} - 6114 q^{22} - 4532 q^{23} - 4318 q^{24} + 5527 q^{25} - 19806 q^{26} + 5942 q^{27} - 496 q^{28} + 5887 q^{29} - 16734 q^{30} + 7794 q^{31} + 7898 q^{32} + 34410 q^{33} + 20840 q^{34} + 7088 q^{35} - 572 q^{36} + 5086 q^{37} + 23732 q^{38} + 33394 q^{39} + 22906 q^{40} + 19826 q^{41} - 55440 q^{42} + 19498 q^{43} - 6074 q^{44} + 7854 q^{45} - 12404 q^{46} + 14278 q^{47} - 16406 q^{48} + 38431 q^{49} - 41066 q^{50} + 23892 q^{51} - 34302 q^{52} - 58644 q^{53} - 31194 q^{54} - 25574 q^{55} - 79560 q^{56} - 88540 q^{57} + 3364 q^{58} + 12888 q^{59} - 180822 q^{60} + 102866 q^{61} - 42654 q^{62} - 88632 q^{63} - 10170 q^{64} - 149206 q^{65} + 7710 q^{66} + 102996 q^{67} + 85100 q^{68} - 107244 q^{69} + 349480 q^{70} - 51596 q^{71} + 135568 q^{72} - 17566 q^{73} + 12132 q^{74} + 39356 q^{75} + 360740 q^{76} - 94104 q^{77} + 46386 q^{78} + 212058 q^{79} + 142510 q^{80} - 128285 q^{81} + 201924 q^{82} - 122928 q^{83} - 12328 q^{84} - 109336 q^{85} - 63290 q^{86} + 21866 q^{87} + 136666 q^{88} - 66510 q^{89} + 56084 q^{90} + 194368 q^{91} - 110108 q^{92} - 474274 q^{93} + 438926 q^{94} - 131676 q^{95} - 117018 q^{96} - 118182 q^{97} - 29132 q^{98} + 300668 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 6\nu^{5} - 114\nu^{4} + 618\nu^{3} + 2727\nu^{2} - 10724\nu - 11462 ) / 240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 5\nu^{4} - 119\nu^{3} + 547\nu^{2} + 2890\nu - 11242 ) / 96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 10\nu^{5} - 218\nu^{4} - 1094\nu^{3} + 13231\nu^{2} + 21356\nu - 185766 ) / 960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 2\nu^{5} - 226\nu^{4} - 238\nu^{3} + 14279\nu^{2} + 5916\nu - 208054 ) / 960 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} + 6\beta_{5} - 8\beta_{4} - \beta_{3} + 4\beta_{2} + 71\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -16\beta_{6} + 8\beta_{5} + 2\beta_{3} + 105\beta_{2} - 95\beta _1 + 3846 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -318\beta_{6} + 754\beta_{5} - 856\beta_{4} - 109\beta_{3} + 454\beta_{2} + 5631\beta _1 - 3945 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2496\beta_{6} + 1728\beta_{5} - 192\beta_{4} + 432\beta_{3} + 9495\beta_{2} - 7471\beta _1 + 304316 \) Copy content Toggle raw display

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
9.56883
7.83842
4.90786
3.60554
−4.83960
−8.92709
−9.15396
−8.56883 18.3274 41.4249 −84.3249 −157.045 216.816 −80.7606 92.8952 722.566
1.2 −6.83842 −24.5656 14.7640 −54.3066 167.990 −37.6697 117.867 360.469 371.372
1.3 −3.90786 29.3989 −16.7287 64.0801 −114.887 −138.793 190.425 621.298 −250.416
1.4 −2.60554 −13.2844 −25.2112 69.0035 34.6131 156.573 149.066 −66.5241 −179.792
1.5 5.83960 15.9679 2.10095 31.5616 93.2461 106.304 −174.599 11.9736 184.307
1.6 9.92709 15.4219 66.5471 −58.0818 153.094 −210.388 342.952 −5.16616 −576.583
1.7 10.1540 −15.2661 71.1029 64.0682 −155.012 91.1564 397.049 −9.94539 650.545
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(-1\)

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 29.6.a.b 7
3.b odd 2 1 261.6.a.e 7
4.b odd 2 1 464.6.a.k 7
5.b even 2 1 725.6.a.b 7
29.b even 2 1 841.6.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.b 7 1.a even 1 1 trivial
261.6.a.e 7 3.b odd 2 1
464.6.a.k 7 4.b odd 2 1
725.6.a.b 7 5.b even 2 1
841.6.a.b 7 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 4T_{2}^{6} - 181T_{2}^{5} + 346T_{2}^{4} + 10616T_{2}^{3} + 2416T_{2}^{2} - 186896T_{2} - 351200 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(29))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 4 T^{6} - 181 T^{5} + \cdots - 351200 \) Copy content Toggle raw display
$3$ \( T^{7} - 26 T^{6} + \cdots + 661023756 \) Copy content Toggle raw display
$5$ \( T^{7} - 32 T^{6} + \cdots + 2378174390186 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots + 361848785235968 \) Copy content Toggle raw display
$11$ \( T^{7} - 1106 T^{6} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{7} - 408 T^{6} + \cdots + 10\!\cdots\!34 \) Copy content Toggle raw display
$17$ \( T^{7} + 874 T^{6} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{7} - 4288 T^{6} + \cdots + 15\!\cdots\!92 \) Copy content Toggle raw display
$23$ \( T^{7} + 4532 T^{6} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T - 841)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} - 7794 T^{6} + \cdots + 64\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{7} - 5086 T^{6} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{7} - 19826 T^{6} + \cdots - 56\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{7} - 19498 T^{6} + \cdots - 58\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{7} - 14278 T^{6} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{7} + 58644 T^{6} + \cdots + 84\!\cdots\!94 \) Copy content Toggle raw display
$59$ \( T^{7} - 12888 T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} - 102866 T^{6} + \cdots + 45\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{7} - 102996 T^{6} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{7} + 51596 T^{6} + \cdots - 20\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{7} + 17566 T^{6} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{7} - 212058 T^{6} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + 122928 T^{6} + \cdots - 97\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{7} + 66510 T^{6} + \cdots - 79\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{7} + 118182 T^{6} + \cdots + 62\!\cdots\!52 \) Copy content Toggle raw display
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