Properties

Label 1856.4.a.be
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 19 x^{6} + 92 x^{4} - 51 x^{2} + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 5 \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 3 + \beta_{2} ) q^{5} -\beta_{4} q^{7} + ( 5 + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 3 + \beta_{2} ) q^{5} -\beta_{4} q^{7} + ( 5 + \beta_{3} ) q^{9} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{11} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{13} + ( -3 \beta_{1} + 2 \beta_{4} + \beta_{6} ) q^{15} + ( -17 + 2 \beta_{2} - 3 \beta_{3} - \beta_{5} ) q^{17} + ( 9 \beta_{1} - 3 \beta_{6} + \beta_{7} ) q^{19} + ( 1 - 9 \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{21} + ( -4 \beta_{1} + 5 \beta_{4} + 2 \beta_{7} ) q^{23} + ( -28 + 7 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{25} + ( 13 \beta_{1} + 3 \beta_{4} - 3 \beta_{6} - 4 \beta_{7} ) q^{27} -29 q^{29} + ( -8 \beta_{1} + 6 \beta_{4} - 6 \beta_{6} - 7 \beta_{7} ) q^{31} + ( -41 - 2 \beta_{2} - 4 \beta_{3} + 6 \beta_{5} ) q^{33} + ( 21 \beta_{1} + \beta_{7} ) q^{35} + ( -42 - 10 \beta_{2} - 10 \beta_{3} - 4 \beta_{5} ) q^{37} + ( -8 \beta_{1} + 6 \beta_{4} - 8 \beta_{6} - 3 \beta_{7} ) q^{39} + ( -93 - 8 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{41} + ( 26 \beta_{1} + 12 \beta_{4} + 5 \beta_{6} + \beta_{7} ) q^{43} + ( 3 - 2 \beta_{2} + 3 \beta_{3} - 6 \beta_{5} ) q^{45} + ( 6 \beta_{1} - 3 \beta_{4} + 7 \beta_{6} + 15 \beta_{7} ) q^{47} + ( -35 - 11 \beta_{2} + 14 \beta_{3} + 5 \beta_{5} ) q^{49} + ( 43 \beta_{1} + 2 \beta_{4} + 8 \beta_{6} + 13 \beta_{7} ) q^{51} + ( 72 - 21 \beta_{2} - 9 \beta_{3} + 14 \beta_{5} ) q^{53} + ( -3 \beta_{1} - 7 \beta_{4} + 12 \beta_{6} + 10 \beta_{7} ) q^{55} + ( -267 - 21 \beta_{2} - 7 \beta_{3} - 9 \beta_{5} ) q^{57} + ( 35 \beta_{1} + 6 \beta_{4} - 11 \beta_{7} ) q^{59} + ( 99 - 27 \beta_{2} + 3 \beta_{3} + 4 \beta_{5} ) q^{61} + ( 12 \beta_{1} - 22 \beta_{4} + 6 \beta_{6} ) q^{63} + ( -218 - 24 \beta_{2} + 17 \beta_{3} - 8 \beta_{5} ) q^{65} + ( 34 \beta_{1} + 12 \beta_{4} - 10 \beta_{6} + 2 \beta_{7} ) q^{67} + ( 105 + 45 \beta_{2} + \beta_{3} - 26 \beta_{5} ) q^{69} + ( 22 \beta_{4} - 20 \beta_{6} - 30 \beta_{7} ) q^{71} + ( -448 - 8 \beta_{3} - 20 \beta_{5} ) q^{73} + ( 47 \beta_{1} + \beta_{4} + 16 \beta_{6} + 7 \beta_{7} ) q^{75} + ( 239 + 3 \beta_{2} + 21 \beta_{3} - 4 \beta_{5} ) q^{77} + ( 21 \beta_{1} - 26 \beta_{4} + 41 \beta_{6} ) q^{79} + ( -488 + 6 \beta_{2} - 15 \beta_{3} - 6 \beta_{5} ) q^{81} + ( 9 \beta_{1} - 14 \beta_{4} - 30 \beta_{6} - 33 \beta_{7} ) q^{83} + ( 137 + 5 \beta_{2} - 3 \beta_{3} + 20 \beta_{5} ) q^{85} + 29 \beta_{1} q^{87} + ( -399 - 82 \beta_{2} + 13 \beta_{3} + 15 \beta_{5} ) q^{89} + ( 27 \beta_{1} - 20 \beta_{4} + 14 \beta_{6} + 11 \beta_{7} ) q^{91} + ( 373 + 12 \beta_{2} + 54 \beta_{3} - 15 \beta_{5} ) q^{93} + ( 111 \beta_{1} - 4 \beta_{4} - 47 \beta_{6} - 33 \beta_{7} ) q^{95} + ( -299 + 8 \beta_{2} + 5 \beta_{3} + 61 \beta_{5} ) q^{97} + ( 56 \beta_{1} - 31 \beta_{4} + \beta_{6} + 10 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{5} + 40 q^{9} + O(q^{10}) \) \( 8 q + 20 q^{5} + 40 q^{9} - 4 q^{13} - 140 q^{17} + 28 q^{21} - 256 q^{25} - 232 q^{29} - 344 q^{33} - 280 q^{37} - 700 q^{41} + 56 q^{45} - 256 q^{49} + 604 q^{53} - 2016 q^{57} + 884 q^{61} - 1616 q^{65} + 764 q^{69} - 3504 q^{73} + 1916 q^{77} - 3904 q^{81} + 996 q^{85} - 2924 q^{89} + 2996 q^{93} - 2668 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 19 x^{6} + 92 x^{4} - 51 x^{2} + 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} + 10 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 10 \)
\(\beta_{3}\)\(=\)\( \nu^{6} - 20 \nu^{4} + 100 \nu^{2} - 32 \)
\(\beta_{4}\)\(=\)\( \nu^{7} - 18 \nu^{5} + 76 \nu^{3} + 5 \nu \)
\(\beta_{5}\)\(=\)\( 2 \nu^{6} - 36 \nu^{4} + 158 \nu^{2} - 42 \)
\(\beta_{6}\)\(=\)\( -2 \nu^{7} + 38 \nu^{5} - 182 \nu^{3} + 90 \nu \)
\(\beta_{7}\)\(=\)\( 5 \nu^{7} - 94 \nu^{5} + 441 \nu^{3} - 165 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{1}\)\()/20\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 10\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{3} + 21 \beta_{2} + 188\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(10 \beta_{7} + 21 \beta_{6} - 8 \beta_{4} - 20 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(5 \beta_{5} - 9 \beta_{3} + 55 \beta_{2} + 472\)
\(\nu^{7}\)\(=\)\((\)\(207 \beta_{7} + 450 \beta_{6} - 131 \beta_{4} - 569 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.68601
0.702141
−3.34163
0.354810
−0.354810
3.34163
−0.702141
−2.68601
0 −7.48152 0 7.42926 0 21.6928 0 28.9731 0
1.2 0 −6.67525 0 −6.01400 0 −26.8310 0 17.5590 0
1.3 0 −3.89793 0 15.3330 0 5.27577 0 −11.8061 0
1.4 0 −3.50343 0 −6.74822 0 −5.06823 0 −14.7260 0
1.5 0 3.50343 0 −6.74822 0 5.06823 0 −14.7260 0
1.6 0 3.89793 0 15.3330 0 −5.27577 0 −11.8061 0
1.7 0 6.67525 0 −6.01400 0 26.8310 0 17.5590 0
1.8 0 7.48152 0 7.42926 0 −21.6928 0 28.9731 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.4.a.be 8
4.b odd 2 1 inner 1856.4.a.be 8
8.b even 2 1 928.4.a.c 8
8.d odd 2 1 928.4.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.c 8 8.b even 2 1
928.4.a.c 8 8.d odd 2 1
1856.4.a.be 8 1.a even 1 1 trivial
1856.4.a.be 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{8} - 128 T_{3}^{6} + 5442 T_{3}^{4} - 87256 T_{3}^{2} + 465125 \)
\( T_{5}^{4} - 10 T_{5}^{3} - 136 T_{5}^{2} + 530 T_{5} + 4623 \)
\( T_{7}^{8} - 1244 T_{7}^{6} + 403200 T_{7}^{4} - 18982336 T_{7}^{2} + 242208000 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 465125 - 87256 T^{2} + 5442 T^{4} - 128 T^{6} + T^{8} \)
$5$ \( ( 4623 + 530 T - 136 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$7$ \( 242208000 - 18982336 T^{2} + 403200 T^{4} - 1244 T^{6} + T^{8} \)
$11$ \( 117236328125 - 1083410000 T^{2} + 2970250 T^{4} - 3048 T^{6} + T^{8} \)
$13$ \( ( -57565 + 51254 T - 2828 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$17$ \( ( 10005088 - 189936 T - 5908 T^{2} + 70 T^{3} + T^{4} )^{2} \)
$19$ \( 864212786208000 - 925631175424 T^{2} + 295992864 T^{4} - 33360 T^{6} + T^{8} \)
$23$ \( 3336667779200000 - 2810979942400 T^{2} + 644884048 T^{4} - 45892 T^{6} + T^{8} \)
$29$ \( ( 29 + T )^{8} \)
$31$ \( 254994533064453125 - 66514196299000 T^{2} + 4946975778 T^{4} - 131168 T^{6} + T^{8} \)
$37$ \( ( -1016954368 - 21516160 T - 94768 T^{2} + 140 T^{3} + T^{4} )^{2} \)
$41$ \( ( -150049680 - 1817816 T + 22600 T^{2} + 350 T^{3} + T^{4} )^{2} \)
$43$ \( 40816899997653125 - 372597257965400 T^{2} + 24428008418 T^{4} - 378560 T^{6} + T^{8} \)
$47$ \( 60179406013916650125 - 3011309734138144 T^{2} + 53468348922 T^{4} - 393176 T^{6} + T^{8} \)
$53$ \( ( -9291290897 + 112049958 T - 299976 T^{2} - 302 T^{3} + T^{4} )^{2} \)
$59$ \( 907188480507008000 - 224971208633344 T^{2} + 16802578128 T^{4} - 390548 T^{6} + T^{8} \)
$61$ \( ( 1059561760 + 19863312 T - 68292 T^{2} - 442 T^{3} + T^{4} )^{2} \)
$67$ \( 10537803207680000000 - 1684578842624000 T^{2} + 68299244032 T^{4} - 502080 T^{6} + T^{8} \)
$71$ \( \)\(57\!\cdots\!00\)\( - 218252583882087424 T^{2} + 1054290301600 T^{4} - 1776416 T^{6} + T^{8} \)
$73$ \( ( -17262411776 - 56713216 T + 681536 T^{2} + 1752 T^{3} + T^{4} )^{2} \)
$79$ \( \)\(16\!\cdots\!25\)\( - 2358006848347465000 T^{2} + 5208747069650 T^{4} - 3985968 T^{6} + T^{8} \)
$83$ \( \)\(33\!\cdots\!00\)\( - 698637672205167616 T^{2} + 3255874242384 T^{4} - 3363092 T^{6} + T^{8} \)
$89$ \( ( -200379381360 - 1193037208 T - 627440 T^{2} + 1462 T^{3} + T^{4} )^{2} \)
$97$ \( ( 3460798083408 - 2798890712 T - 3482912 T^{2} + 1334 T^{3} + T^{4} )^{2} \)
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