Properties

Label 5.20.b.a
Level $5$
Weight $20$
Character orbit 5.b
Analytic conductor $11.441$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 5.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4408348278\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 726881x^{6} + 160513523376x^{4} + 10607307647230976x^{2} + 32429098232548950016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 5^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{3} - 202593) q^{4} + (\beta_{4} - 2 \beta_{3} + 27 \beta_{2} + 94 \beta_1 + 18375) q^{5} + (\beta_{6} + \beta_{4} - 3 \beta_{3} + 420717) q^{6} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 322 \beta_{2} - 30616 \beta_1) q^{7} + (\beta_{7} + 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 656 \beta_{2} - 92622 \beta_1) q^{8} + (9 \beta_{7} + 12 \beta_{6} + 66 \beta_{4} - 387 \beta_{3} + \cdots + 43169823) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{3} - 202593) q^{4} + (\beta_{4} - 2 \beta_{3} + 27 \beta_{2} + 94 \beta_1 + 18375) q^{5} + (\beta_{6} + \beta_{4} - 3 \beta_{3} + 420717) q^{6} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 322 \beta_{2} - 30616 \beta_1) q^{7} + (\beta_{7} + 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 656 \beta_{2} - 92622 \beta_1) q^{8} + (9 \beta_{7} + 12 \beta_{6} + 66 \beta_{4} - 387 \beta_{3} + \cdots + 43169823) q^{9}+ \cdots + ( - 2123060598 \beta_{7} + \cdots - 20\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1620744 q^{4} + 147000 q^{5} + 3365736 q^{6} + 345358584 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1620744 q^{4} + 147000 q^{5} + 3365736 q^{6} + 345358584 q^{9} - 610691000 q^{10} - 3379575264 q^{11} + 177250591032 q^{14} - 242324628000 q^{15} - 312730276832 q^{16} + 4547188380640 q^{19} - 4180429431000 q^{20} - 2983154334624 q^{21} - 6176642779680 q^{24} + 17715709625000 q^{25} + 15909228128496 q^{26} - 188222300345040 q^{29} + 148501482939000 q^{30} + 72115006686976 q^{31} + 378440221985792 q^{34} - 299115755916000 q^{35} - 964020253238712 q^{36} + 30\!\cdots\!28 q^{39}+ \cdots - 16\!\cdots\!72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 726881x^{6} + 160513523376x^{4} + 10607307647230976x^{2} + 32429098232548950016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 173\nu^{7} + 111026445\nu^{5} + 20083149107568\nu^{3} + 947168012636830720\nu ) / 1505285197258752 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 726881 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24489 \nu^{7} + 1840496 \nu^{6} - 19390777353 \nu^{5} + 1808281798512 \nu^{4} + \cdots + 15\!\cdots\!64 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 254739 \nu^{7} - 7361984 \nu^{6} - 173282696883 \nu^{5} - 7233127194048 \nu^{4} + \cdots - 54\!\cdots\!96 ) / 75\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24489 \nu^{7} - 75460336 \nu^{6} + 19390777353 \nu^{5} - 40236733980912 \nu^{4} + \cdots - 36\!\cdots\!24 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 294733 \nu^{7} + 14723968 \nu^{6} + 233244460461 \nu^{5} + 14466254388096 \nu^{4} + \cdots + 11\!\cdots\!52 ) / 75\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 726881 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 3\beta_{5} + 2\beta_{4} - 2\beta_{3} + 656\beta_{2} - 1141198\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1776\beta_{7} + 444\beta_{6} + 11100\beta_{4} - 377593\beta_{3} - 1776\beta_{2} + 5328\beta _1 + 207328941409 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 319487 \beta_{7} - 1499523 \beta_{5} - 4276994 \beta_{4} + 1819010 \beta_{3} - 792634640 \beta_{2} + 362708638734 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 927045360 \beta_{7} - 436228668 \beta_{6} - 5998500828 \beta_{4} + 137204036265 \beta_{3} + 927045360 \beta_{2} - 2781136080 \beta _1 - 65\!\cdots\!01 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 321125173071 \beta_{7} + 614088211347 \beta_{5} + 2512677114978 \beta_{4} - 935213384418 \beta_{3} + 502145329633296 \beta_{2} - 12\!\cdots\!02 \beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
607.943i
490.322i
337.134i
56.6657i
56.6657i
337.134i
490.322i
607.943i
1215.89i 18754.1i −954092. 4.05746e6 + 1.61570e6i 2.28029e7 1.79878e8i 5.22593e8i 8.10544e8 1.96451e9 4.93341e9i
4.2 980.644i 42067.9i −437374. −2.12371e6 3.81619e6i −4.12537e7 4.16374e7i 8.52313e7i −6.07450e8 −3.74233e9 + 2.08261e9i
4.3 674.268i 35433.1i 69650.8 −4.08230e6 + 1.55186e6i 2.38914e7 1.27830e8i 4.00474e8i −9.32466e7 1.04637e9 + 2.75257e9i
4.4 113.331i 33157.6i 511444. 2.22206e6 + 3.75978e6i −3.75780e6 2.70208e7i 1.17381e8i 6.28322e7 4.26102e8 2.51829e8i
4.5 113.331i 33157.6i 511444. 2.22206e6 3.75978e6i −3.75780e6 2.70208e7i 1.17381e8i 6.28322e7 4.26102e8 + 2.51829e8i
4.6 674.268i 35433.1i 69650.8 −4.08230e6 1.55186e6i 2.38914e7 1.27830e8i 4.00474e8i −9.32466e7 1.04637e9 2.75257e9i
4.7 980.644i 42067.9i −437374. −2.12371e6 + 3.81619e6i −4.12537e7 4.16374e7i 8.52313e7i −6.07450e8 −3.74233e9 2.08261e9i
4.8 1215.89i 18754.1i −954092. 4.05746e6 1.61570e6i 2.28029e7 1.79878e8i 5.22593e8i 8.10544e8 1.96451e9 + 4.93341e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.20.b.a 8
3.b odd 2 1 45.20.b.b 8
4.b odd 2 1 80.20.c.a 8
5.b even 2 1 inner 5.20.b.a 8
5.c odd 4 2 25.20.a.f 8
15.d odd 2 1 45.20.b.b 8
20.d odd 2 1 80.20.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.20.b.a 8 1.a even 1 1 trivial
5.20.b.a 8 5.b even 2 1 inner
25.20.a.f 8 5.c odd 4 2
45.20.b.b 8 3.b odd 2 1
45.20.b.b 8 15.d odd 2 1
80.20.c.a 8 4.b odd 2 1
80.20.c.a 8 20.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{20}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2907524 T^{6} + \cdots + 83\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( T^{8} + 4476366576 T^{6} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{8} - 147000 T^{7} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 66\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 1689787632 T^{3} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} - 2273594190320 T^{3} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + 94111150172520 T^{3} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 36057503343488 T^{3} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 62\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
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