Properties

Label 140.6.a.b
Level $140$
Weight $6$
Character orbit 140.a
Self dual yes
Analytic conductor $22.454$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 140.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.4537347738\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1009}) \)
Defining polynomial: \(x^{2} - x - 252\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{1009})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -8 - \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 73 + 17 \beta ) q^{9} +O(q^{10})\) \( q + ( -8 - \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 73 + 17 \beta ) q^{9} + ( -88 + 9 \beta ) q^{11} + ( 202 + 41 \beta ) q^{13} + ( -200 - 25 \beta ) q^{15} + ( 38 + 23 \beta ) q^{17} + ( -644 - 46 \beta ) q^{19} + ( 392 + 49 \beta ) q^{21} + ( -176 + 62 \beta ) q^{23} + 625 q^{25} + ( -2924 + 17 \beta ) q^{27} + ( -3350 - 259 \beta ) q^{29} + ( -5184 - 112 \beta ) q^{31} + ( -1564 + 7 \beta ) q^{33} -1225 q^{35} + ( -906 + 224 \beta ) q^{37} + ( -11948 - 571 \beta ) q^{39} + ( -11982 - 146 \beta ) q^{41} + ( -9820 - 766 \beta ) q^{43} + ( 1825 + 425 \beta ) q^{45} + ( -60 + 739 \beta ) q^{47} + 2401 q^{49} + ( -6100 - 245 \beta ) q^{51} + ( -10786 - 222 \beta ) q^{53} + ( -2200 + 225 \beta ) q^{55} + ( 16744 + 1058 \beta ) q^{57} + ( 492 + 992 \beta ) q^{59} + ( 10598 + 1418 \beta ) q^{61} + ( -3577 - 833 \beta ) q^{63} + ( 5050 + 1025 \beta ) q^{65} + ( 1716 + 636 \beta ) q^{67} + ( -14216 - 382 \beta ) q^{69} + ( -22616 - 1296 \beta ) q^{71} + ( 12922 + 1252 \beta ) q^{73} + ( -5000 - 625 \beta ) q^{75} + ( 4312 - 441 \beta ) q^{77} + ( -5276 - 1929 \beta ) q^{79} + ( 1369 - 1360 \beta ) q^{81} + ( -48012 - 4064 \beta ) q^{83} + ( 950 + 575 \beta ) q^{85} + ( 92068 + 5681 \beta ) q^{87} + ( 19282 - 4766 \beta ) q^{89} + ( -9898 - 2009 \beta ) q^{91} + ( 69696 + 6192 \beta ) q^{93} + ( -16100 - 1150 \beta ) q^{95} + ( 70598 - 2205 \beta ) q^{97} + ( 32132 - 686 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 17 q^{3} + 50 q^{5} - 98 q^{7} + 163 q^{9} + O(q^{10}) \) \( 2 q - 17 q^{3} + 50 q^{5} - 98 q^{7} + 163 q^{9} - 167 q^{11} + 445 q^{13} - 425 q^{15} + 99 q^{17} - 1334 q^{19} + 833 q^{21} - 290 q^{23} + 1250 q^{25} - 5831 q^{27} - 6959 q^{29} - 10480 q^{31} - 3121 q^{33} - 2450 q^{35} - 1588 q^{37} - 24467 q^{39} - 24110 q^{41} - 20406 q^{43} + 4075 q^{45} + 619 q^{47} + 4802 q^{49} - 12445 q^{51} - 21794 q^{53} - 4175 q^{55} + 34546 q^{57} + 1976 q^{59} + 22614 q^{61} - 7987 q^{63} + 11125 q^{65} + 4068 q^{67} - 28814 q^{69} - 46528 q^{71} + 27096 q^{73} - 10625 q^{75} + 8183 q^{77} - 12481 q^{79} + 1378 q^{81} - 100088 q^{83} + 2475 q^{85} + 189817 q^{87} + 33798 q^{89} - 21805 q^{91} + 145584 q^{93} - 33350 q^{95} + 138991 q^{97} + 63578 q^{99} + O(q^{100}) \)

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
16.3824
−15.3824
0 −24.3824 0 25.0000 0 −49.0000 0 351.500 0
1.2 0 7.38238 0 25.0000 0 −49.0000 0 −188.500 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 140.6.a.b 2
4.b odd 2 1 560.6.a.o 2
5.b even 2 1 700.6.a.g 2
5.c odd 4 2 700.6.e.f 4
7.b odd 2 1 980.6.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.b 2 1.a even 1 1 trivial
560.6.a.o 2 4.b odd 2 1
700.6.a.g 2 5.b even 2 1
700.6.e.f 4 5.c odd 4 2
980.6.a.f 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 17 T_{3} - 180 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(140))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -180 + 17 T + T^{2} \)
$5$ \( ( -25 + T )^{2} \)
$7$ \( ( 49 + T )^{2} \)
$11$ \( -13460 + 167 T + T^{2} \)
$13$ \( -374526 - 445 T + T^{2} \)
$17$ \( -130990 - 99 T + T^{2} \)
$19$ \( -88872 + 1334 T + T^{2} \)
$23$ \( -948624 + 290 T + T^{2} \)
$29$ \( -4814262 + 6959 T + T^{2} \)
$31$ \( 24293376 + 10480 T + T^{2} \)
$37$ \( -12026460 + 1588 T + T^{2} \)
$41$ \( 139946064 + 24110 T + T^{2} \)
$43$ \( -43907992 + 20406 T + T^{2} \)
$47$ \( -137663232 - 619 T + T^{2} \)
$53$ \( 106312720 + 21794 T + T^{2} \)
$59$ \( -247254000 - 1976 T + T^{2} \)
$61$ \( -379356880 - 22614 T + T^{2} \)
$67$ \( -97896960 - 4068 T + T^{2} \)
$71$ \( 117530560 + 46528 T + T^{2} \)
$73$ \( -211854580 - 27096 T + T^{2} \)
$79$ \( -899688752 + 12481 T + T^{2} \)
$83$ \( -1661783280 + 100088 T + T^{2} \)
$89$ \( -5444221000 - 33798 T + T^{2} \)
$97$ \( 3603178714 - 138991 T + T^{2} \)
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