Properties

Label 1840.4.a.m
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 68 x^{2} - 111 x + 342\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} -5 q^{5} + ( -2 \beta_{1} - \beta_{2} ) q^{7} + ( 8 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} -5 q^{5} + ( -2 \beta_{1} - \beta_{2} ) q^{7} + ( 8 + \beta_{3} ) q^{9} + ( 10 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -5 + 8 \beta_{1} - \beta_{3} ) q^{13} + ( -5 + 5 \beta_{1} ) q^{15} + ( -6 + 13 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -14 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{19} + ( 74 + \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{21} + 23 q^{23} + 25 q^{25} + ( -35 + \beta_{1} - 3 \beta_{2} ) q^{27} + ( 41 - 14 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{29} + ( -97 - 16 \beta_{1} + 5 \beta_{3} ) q^{31} + ( 22 - 26 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{33} + ( 10 \beta_{1} + 5 \beta_{2} ) q^{35} + ( 116 - 18 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{37} + ( -261 + 15 \beta_{1} + 3 \beta_{2} - 8 \beta_{3} ) q^{39} + ( 121 - \beta_{1} - 10 \beta_{3} ) q^{41} + ( -222 - 24 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{43} + ( -40 - 5 \beta_{3} ) q^{45} + ( 67 - 44 \beta_{1} + 3 \beta_{2} + 11 \beta_{3} ) q^{47} + ( 411 - 49 \beta_{1} + \beta_{2} + 9 \beta_{3} ) q^{49} + ( -458 - 26 \beta_{1} - 7 \beta_{2} - 10 \beta_{3} ) q^{51} + ( 146 - 40 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{53} + ( -50 + 5 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{55} + ( 40 - 23 \beta_{1} - 18 \beta_{2} + 11 \beta_{3} ) q^{57} + ( 20 - 10 \beta_{1} - 14 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 290 + 33 \beta_{1} + 7 \beta_{2} - 19 \beta_{3} ) q^{61} + ( -16 - 115 \beta_{1} - 4 \beta_{2} + 11 \beta_{3} ) q^{63} + ( 25 - 40 \beta_{1} + 5 \beta_{3} ) q^{65} + ( 368 - 28 \beta_{1} + 12 \beta_{3} ) q^{67} + ( 23 - 23 \beta_{1} ) q^{69} + ( -57 - 39 \beta_{1} - 28 \beta_{2} - 4 \beta_{3} ) q^{71} + ( 287 + 44 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{73} + ( 25 - 25 \beta_{1} ) q^{75} + ( -548 - 61 \beta_{1} - 16 \beta_{2} + 17 \beta_{3} ) q^{77} + ( 232 + 72 \beta_{1} + 20 \beta_{2} + 16 \beta_{3} ) q^{79} + ( -267 + 31 \beta_{1} - 12 \beta_{2} - 19 \beta_{3} ) q^{81} + ( 256 + 26 \beta_{1} - 24 \beta_{2} + 6 \beta_{3} ) q^{83} + ( 30 - 65 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{85} + ( 547 + 30 \beta_{1} + 21 \beta_{2} + 5 \beta_{3} ) q^{87} + ( -450 + 30 \beta_{1} - 16 \beta_{2} - 34 \beta_{3} ) q^{89} + ( -576 + 85 \beta_{1} + 25 \beta_{2} - 51 \beta_{3} ) q^{91} + ( 367 + 23 \beta_{1} - 15 \beta_{2} + 16 \beta_{3} ) q^{93} + ( 70 + 10 \beta_{1} + 15 \beta_{2} - 10 \beta_{3} ) q^{95} + ( -504 - 137 \beta_{1} + 31 \beta_{2} + \beta_{3} ) q^{97} + ( 662 + 68 \beta_{1} - 17 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 20q^{5} + q^{7} + 32q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 20q^{5} + q^{7} + 32q^{9} + 39q^{11} - 20q^{13} - 20q^{15} - 23q^{17} - 53q^{19} + 300q^{21} + 92q^{23} + 100q^{25} - 137q^{27} + 161q^{29} - 388q^{31} + 87q^{33} - 5q^{35} + 466q^{37} - 1047q^{39} + 484q^{41} - 894q^{43} - 160q^{45} + 265q^{47} + 1643q^{49} - 1825q^{51} + 576q^{53} - 195q^{55} + 178q^{57} + 94q^{59} + 1153q^{61} - 60q^{63} + 100q^{65} + 1472q^{67} + 92q^{69} - 200q^{71} + 1147q^{73} + 100q^{75} - 2176q^{77} + 908q^{79} - 1056q^{81} + 1048q^{83} + 115q^{85} + 2167q^{87} - 1784q^{89} - 2329q^{91} + 1483q^{93} + 265q^{95} - 2047q^{97} + 2665q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 68 x^{2} - 111 x + 342\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 50 \nu + 18 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 2 \nu - 34 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2 \beta_{1} + 34\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + 3 \beta_{2} + 56 \beta_{1} + 84\)

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
8.73081
1.58997
−3.74869
−6.57209
0 −7.73081 0 −5.00000 0 −23.5622 0 32.7654 0
1.2 0 −0.589969 0 −5.00000 0 18.5077 0 −26.6519 0
1.3 0 4.74869 0 −5.00000 0 −29.3684 0 −4.44993 0
1.4 0 7.57209 0 −5.00000 0 35.4229 0 30.3365 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

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 1840.4.a.m 4
4.b odd 2 1 230.4.a.h 4
12.b even 2 1 2070.4.a.bj 4
20.d odd 2 1 1150.4.a.p 4
20.e even 4 2 1150.4.b.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.h 4 4.b odd 2 1
1150.4.a.p 4 20.d odd 2 1
1150.4.b.n 8 20.e even 4 2
1840.4.a.m 4 1.a even 1 1 trivial
2070.4.a.bj 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{4} - 4 T_{3}^{3} - 62 T_{3}^{2} + 243 T_{3} + 164 \)
\( T_{7}^{4} - T_{7}^{3} - 1507 T_{7}^{2} - 2618 T_{7} + 453664 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 164 + 243 T - 62 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( ( 5 + T )^{4} \)
$7$ \( 453664 - 2618 T - 1507 T^{2} - T^{3} + T^{4} \)
$11$ \( -977536 + 94420 T - 1771 T^{2} - 39 T^{3} + T^{4} \)
$13$ \( 3059002 + 25481 T - 4948 T^{2} + 20 T^{3} + T^{4} \)
$17$ \( 1048184 - 590494 T - 14205 T^{2} + 23 T^{3} + T^{4} \)
$19$ \( -78336 - 340080 T - 15029 T^{2} + 53 T^{3} + T^{4} \)
$23$ \( ( -23 + T )^{4} \)
$29$ \( 1597064 + 1886712 T - 27260 T^{2} - 161 T^{3} + T^{4} \)
$31$ \( -397027152 - 5546709 T + 13076 T^{2} + 388 T^{3} + T^{4} \)
$37$ \( -7921024 - 681856 T + 37920 T^{2} - 466 T^{3} + T^{4} \)
$41$ \( 1506099394 + 17572043 T - 36232 T^{2} - 484 T^{3} + T^{4} \)
$43$ \( -5392023552 - 22823424 T + 166752 T^{2} + 894 T^{3} + T^{4} \)
$47$ \( 1306137888 + 43523988 T - 230390 T^{2} - 265 T^{3} + T^{4} \)
$53$ \( -3236491568 + 79775808 T - 107352 T^{2} - 576 T^{3} + T^{4} \)
$59$ \( 11673423616 + 20596976 T - 243460 T^{2} - 94 T^{3} + T^{4} \)
$61$ \( -42772329400 + 315388730 T - 54057 T^{2} - 1153 T^{3} + T^{4} \)
$67$ \( -10308616192 - 42702336 T + 602032 T^{2} - 1472 T^{3} + T^{4} \)
$71$ \( 274201266224 - 93278335 T - 1067322 T^{2} + 200 T^{3} + T^{4} \)
$73$ \( -4754107544 - 9049904 T + 356120 T^{2} - 1147 T^{3} + T^{4} \)
$79$ \( 145785325568 + 272612224 T - 923664 T^{2} - 908 T^{3} + T^{4} \)
$83$ \( -178673654272 + 675145440 T - 359444 T^{2} - 1048 T^{3} + T^{4} \)
$89$ \( -969417760000 - 2249185600 T - 492084 T^{2} + 1784 T^{3} + T^{4} \)
$97$ \( -658266694276 - 2746911284 T - 716799 T^{2} + 2047 T^{3} + T^{4} \)
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