Properties

Label 5.26.a.a
Level $5$
Weight $26$
Character orbit 5.a
Self dual yes
Analytic conductor $19.800$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7998389976\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 1769856x^{2} + 106836475x + 628040620025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 150) q^{2} + (\beta_{2} + 17 \beta_1 - 199650) q^{3} + (\beta_{3} - 4 \beta_{2} - 1018 \beta_1 + 23103472) q^{4} - 244140625 q^{5} + ( - 324 \beta_{3} - 1904 \beta_{2} + 301082 \beta_1 - 991293708) q^{6} + (600 \beta_{3} - 16251 \beta_{2} - 518011 \beta_1 - 12234526750) q^{7} + (3800 \beta_{3} - 134432 \beta_{2} - 1131120 \beta_1 + 56108371200) q^{8} + ( - 8208 \beta_{3} - 520068 \beta_{2} + \cdots + 521891451873) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 150) q^{2} + (\beta_{2} + 17 \beta_1 - 199650) q^{3} + (\beta_{3} - 4 \beta_{2} - 1018 \beta_1 + 23103472) q^{4} - 244140625 q^{5} + ( - 324 \beta_{3} - 1904 \beta_{2} + 301082 \beta_1 - 991293708) q^{6} + (600 \beta_{3} - 16251 \beta_{2} - 518011 \beta_1 - 12234526750) q^{7} + (3800 \beta_{3} - 134432 \beta_{2} - 1131120 \beta_1 + 56108371200) q^{8} + ( - 8208 \beta_{3} - 520068 \beta_{2} + \cdots + 521891451873) q^{9}+ \cdots + (70\!\cdots\!04 \beta_{3} + \cdots - 48\!\cdots\!04) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 600 q^{2} - 798600 q^{3} + 92413888 q^{4} - 976562500 q^{5} - 3965174832 q^{6} - 48938107000 q^{7} + 224433484800 q^{8} + 2087565807492 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 600 q^{2} - 798600 q^{3} + 92413888 q^{4} - 976562500 q^{5} - 3965174832 q^{6} - 48938107000 q^{7} + 224433484800 q^{8} + 2087565807492 q^{9} - 146484375000 q^{10} - 23641453790592 q^{11} - 42051883516800 q^{12} + 109063914225800 q^{13} + 109980501036336 q^{14} + 194970703125000 q^{15} - 28\!\cdots\!76 q^{16}+ \cdots - 19\!\cdots\!16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 1769856x^{2} + 106836475x + 628040620025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 8\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{3} + 3344\nu^{2} - 8182184\nu - 2326754810 ) / 1863 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32\nu^{3} + 132608\nu^{2} - 22087280\nu - 114821444708 ) / 1863 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 4\beta_{2} - 714\beta _1 + 56635408 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -209\beta_{3} + 8288\beta_{2} + 4240318\beta _1 - 2521598848 ) / 32 \) 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
1046.70
785.715
−642.094
−1189.32
−8221.63 987550. 3.40407e7 −2.44141e8 −8.11927e9 −1.93681e10 −3.99805e9 1.27967e11 2.00723e12
1.2 −6133.72 −1.60154e6 4.06809e6 −2.44141e8 9.82339e9 −2.17553e9 1.80861e11 1.71764e12 1.49749e12
1.3 5288.75 887362. −5.58351e6 −2.44141e8 4.69304e9 −4.61897e10 −2.06991e11 −5.98771e10 −1.29120e12
1.4 9666.59 −1.07197e6 5.98886e7 −2.44141e8 −1.03623e10 1.87952e10 2.54562e11 3.01839e11 −2.36001e12
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

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 5.26.a.a 4
3.b odd 2 1 45.26.a.c 4
5.b even 2 1 25.26.a.b 4
5.c odd 4 2 25.26.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.26.a.a 4 1.a even 1 1 trivial
25.26.a.b 4 5.b even 2 1
25.26.b.b 8 5.c odd 4 2
45.26.a.c 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 600T_{2}^{3} - 113135808T_{2}^{2} - 20279449600T_{2} + 2578152318959616 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 600 T^{3} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{4} + 798600 T^{3} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( (T + 244140625)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 48938107000 T^{3} + \cdots - 36\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{4} + 23641453790592 T^{3} + \cdots - 12\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{4} - 109063914225800 T^{3} + \cdots - 27\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{4} + 337440158866200 T^{3} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 75\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 26\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 88\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 89\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 89\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 18\!\cdots\!64 \) Copy content Toggle raw display
show more
show less