Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
Defining polynomial: \(x^{2} - 22\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 116)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 5 + \beta ) q^{3} -15 q^{5} + 2 \beta q^{7} + ( 20 + 10 \beta ) q^{9} +O(q^{10})\) \( q + ( 5 + \beta ) q^{3} -15 q^{5} + 2 \beta q^{7} + ( 20 + 10 \beta ) q^{9} + ( -23 - 5 \beta ) q^{11} + ( 5 + 10 \beta ) q^{13} + ( -75 - 15 \beta ) q^{15} + ( 70 - 14 \beta ) q^{17} + ( -30 - 20 \beta ) q^{19} + ( 44 + 10 \beta ) q^{21} + ( -30 - 12 \beta ) q^{23} + 100 q^{25} + ( 185 + 43 \beta ) q^{27} -29 q^{29} + ( 29 - 65 \beta ) q^{31} + ( -225 - 48 \beta ) q^{33} -30 \beta q^{35} + ( 60 + 8 \beta ) q^{37} + ( 245 + 55 \beta ) q^{39} + ( 140 + 50 \beta ) q^{41} + ( -75 - 77 \beta ) q^{43} + ( -300 - 150 \beta ) q^{45} + ( 275 - 37 \beta ) q^{47} -255 q^{49} + 42 q^{51} + ( -245 + 34 \beta ) q^{53} + ( 345 + 75 \beta ) q^{55} + ( -590 - 130 \beta ) q^{57} + ( -638 - 20 \beta ) q^{59} + ( -198 - 70 \beta ) q^{61} + ( 440 + 40 \beta ) q^{63} + ( -75 - 150 \beta ) q^{65} + ( -40 + 108 \beta ) q^{67} + ( -414 - 90 \beta ) q^{69} + ( 862 + 10 \beta ) q^{71} + ( 100 + 32 \beta ) q^{73} + ( 500 + 100 \beta ) q^{75} + ( -220 - 46 \beta ) q^{77} + ( -827 - 15 \beta ) q^{79} + ( 1331 + 130 \beta ) q^{81} + ( -750 + 60 \beta ) q^{83} + ( -1050 + 210 \beta ) q^{85} + ( -145 - 29 \beta ) q^{87} + ( -108 + 90 \beta ) q^{89} + ( 440 + 10 \beta ) q^{91} + ( -1285 - 296 \beta ) q^{93} + ( 450 + 300 \beta ) q^{95} + ( 940 - 82 \beta ) q^{97} + ( -1560 - 330 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{3} - 30 q^{5} + 40 q^{9} + O(q^{10}) \) \( 2 q + 10 q^{3} - 30 q^{5} + 40 q^{9} - 46 q^{11} + 10 q^{13} - 150 q^{15} + 140 q^{17} - 60 q^{19} + 88 q^{21} - 60 q^{23} + 200 q^{25} + 370 q^{27} - 58 q^{29} + 58 q^{31} - 450 q^{33} + 120 q^{37} + 490 q^{39} + 280 q^{41} - 150 q^{43} - 600 q^{45} + 550 q^{47} - 510 q^{49} + 84 q^{51} - 490 q^{53} + 690 q^{55} - 1180 q^{57} - 1276 q^{59} - 396 q^{61} + 880 q^{63} - 150 q^{65} - 80 q^{67} - 828 q^{69} + 1724 q^{71} + 200 q^{73} + 1000 q^{75} - 440 q^{77} - 1654 q^{79} + 2662 q^{81} - 1500 q^{83} - 2100 q^{85} - 290 q^{87} - 216 q^{89} + 880 q^{91} - 2570 q^{93} + 900 q^{95} + 1880 q^{97} - 3120 q^{99} + O(q^{100}) \)

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
−4.69042
4.69042
0 0.309584 0 −15.0000 0 −9.38083 0 −26.9042 0
1.2 0 9.69042 0 −15.0000 0 9.38083 0 66.9042 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1856.4.a.m 2
4.b odd 2 1 1856.4.a.g 2
8.b even 2 1 464.4.a.c 2
8.d odd 2 1 116.4.a.b 2
24.f even 2 1 1044.4.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.4.a.b 2 8.d odd 2 1
464.4.a.c 2 8.b even 2 1
1044.4.a.b 2 24.f even 2 1
1856.4.a.g 2 4.b odd 2 1
1856.4.a.m 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{2} - 10 T_{3} + 3 \)
\( T_{5} + 15 \)
\( T_{7}^{2} - 88 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 - 10 T + T^{2} \)
$5$ \( ( 15 + T )^{2} \)
$7$ \( -88 + T^{2} \)
$11$ \( -21 + 46 T + T^{2} \)
$13$ \( -2175 - 10 T + T^{2} \)
$17$ \( 588 - 140 T + T^{2} \)
$19$ \( -7900 + 60 T + T^{2} \)
$23$ \( -2268 + 60 T + T^{2} \)
$29$ \( ( 29 + T )^{2} \)
$31$ \( -92109 - 58 T + T^{2} \)
$37$ \( 2192 - 120 T + T^{2} \)
$41$ \( -35400 - 280 T + T^{2} \)
$43$ \( -124813 + 150 T + T^{2} \)
$47$ \( 45507 - 550 T + T^{2} \)
$53$ \( 34593 + 490 T + T^{2} \)
$59$ \( 398244 + 1276 T + T^{2} \)
$61$ \( -68596 + 396 T + T^{2} \)
$67$ \( -255008 + 80 T + T^{2} \)
$71$ \( 740844 - 1724 T + T^{2} \)
$73$ \( -12528 - 200 T + T^{2} \)
$79$ \( 678979 + 1654 T + T^{2} \)
$83$ \( 483300 + 1500 T + T^{2} \)
$89$ \( -166536 + 216 T + T^{2} \)
$97$ \( 735672 - 1880 T + T^{2} \)
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