Properties

Label 140.6.a.a
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.

\(f(q)\) \(=\) \( q + ( -11 - \beta ) q^{3} -25 q^{5} + 49 q^{7} + ( 130 + 23 \beta ) q^{9} +O(q^{10})\) \( q + ( -11 - \beta ) q^{3} -25 q^{5} + 49 q^{7} + ( 130 + 23 \beta ) q^{9} + ( 441 - 9 \beta ) q^{11} + ( -103 + 21 \beta ) q^{13} + ( 275 + 25 \beta ) q^{15} + ( -1041 + 63 \beta ) q^{17} + ( -310 + 6 \beta ) q^{19} + ( -539 - 49 \beta ) q^{21} + ( -606 - 138 \beta ) q^{23} + 625 q^{25} + ( -4553 - 163 \beta ) q^{27} + ( -729 + 459 \beta ) q^{29} + ( -4516 + 12 \beta ) q^{31} + ( -2583 - 333 \beta ) q^{33} -1225 q^{35} + ( -4318 - 396 \beta ) q^{37} + ( -4159 - 149 \beta ) q^{39} + ( -9108 - 114 \beta ) q^{41} + ( -2194 + 414 \beta ) q^{43} + ( -3250 - 575 \beta ) q^{45} + ( -10419 - 921 \beta ) q^{47} + 2401 q^{49} + ( -4425 + 285 \beta ) q^{51} + ( -12 + 2178 \beta ) q^{53} + ( -11025 + 225 \beta ) q^{55} + ( 1898 + 238 \beta ) q^{57} + ( -14256 - 192 \beta ) q^{59} + ( 5216 + 1962 \beta ) q^{61} + ( 6370 + 1127 \beta ) q^{63} + ( 2575 - 525 \beta ) q^{65} + ( -50452 - 984 \beta ) q^{67} + ( 41442 + 2262 \beta ) q^{69} + ( 46608 - 1584 \beta ) q^{71} + ( -16894 - 2208 \beta ) q^{73} + ( -6875 - 625 \beta ) q^{75} + ( 21609 - 441 \beta ) q^{77} + ( -4525 - 951 \beta ) q^{79} + ( 59569 + 920 \beta ) q^{81} + ( 39276 - 444 \beta ) q^{83} + ( 26025 - 1575 \beta ) q^{85} + ( -107649 - 4779 \beta ) q^{87} + ( 15516 + 2406 \beta ) q^{89} + ( -5047 + 1029 \beta ) q^{91} + ( 46652 + 4372 \beta ) q^{93} + ( 7750 - 150 \beta ) q^{95} + ( -43573 + 1395 \beta ) q^{97} + ( 5166 + 8766 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 23 q^{3} - 50 q^{5} + 98 q^{7} + 283 q^{9} + O(q^{10}) \) \( 2 q - 23 q^{3} - 50 q^{5} + 98 q^{7} + 283 q^{9} + 873 q^{11} - 185 q^{13} + 575 q^{15} - 2019 q^{17} - 614 q^{19} - 1127 q^{21} - 1350 q^{23} + 1250 q^{25} - 9269 q^{27} - 999 q^{29} - 9020 q^{31} - 5499 q^{33} - 2450 q^{35} - 9032 q^{37} - 8467 q^{39} - 18330 q^{41} - 3974 q^{43} - 7075 q^{45} - 21759 q^{47} + 4802 q^{49} - 8565 q^{51} + 2154 q^{53} - 21825 q^{55} + 4034 q^{57} - 28704 q^{59} + 12394 q^{61} + 13867 q^{63} + 4625 q^{65} - 101888 q^{67} + 85146 q^{69} + 91632 q^{71} - 35996 q^{73} - 14375 q^{75} + 42777 q^{77} - 10001 q^{79} + 120058 q^{81} + 78108 q^{83} + 50475 q^{85} - 220077 q^{87} + 33438 q^{89} - 9065 q^{91} + 97676 q^{93} + 15350 q^{95} - 85751 q^{97} + 19098 q^{99} + O(q^{100}) \)

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 −27.3824 0 −25.0000 0 49.0000 0 506.795 0
1.2 0 4.38238 0 −25.0000 0 49.0000 0 −223.795 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.a 2
4.b odd 2 1 560.6.a.p 2
5.b even 2 1 700.6.a.h 2
5.c odd 4 2 700.6.e.e 4
7.b odd 2 1 980.6.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.a 2 1.a even 1 1 trivial
560.6.a.p 2 4.b odd 2 1
700.6.a.h 2 5.b even 2 1
700.6.e.e 4 5.c odd 4 2
980.6.a.g 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 23 T_{3} - 120 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(140))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -120 + 23 T + T^{2} \)
$5$ \( ( 25 + T )^{2} \)
$7$ \( ( -49 + T )^{2} \)
$11$ \( 170100 - 873 T + T^{2} \)
$13$ \( -102686 + 185 T + T^{2} \)
$17$ \( 17910 + 2019 T + T^{2} \)
$19$ \( 85168 + 614 T + T^{2} \)
$23$ \( -4348224 + 1350 T + T^{2} \)
$29$ \( -52894782 + 999 T + T^{2} \)
$31$ \( 20303776 + 9020 T + T^{2} \)
$37$ \( -19162580 + 9032 T + T^{2} \)
$41$ \( 80718984 + 18330 T + T^{2} \)
$43$ \( -39286472 + 3974 T + T^{2} \)
$47$ \( -95605272 + 21759 T + T^{2} \)
$53$ \( -1195434360 - 2154 T + T^{2} \)
$59$ \( 196680960 + 28704 T + T^{2} \)
$61$ \( -932619440 - 12394 T + T^{2} \)
$67$ \( 2351048560 + 101888 T + T^{2} \)
$71$ \( 1466196480 - 91632 T + T^{2} \)
$73$ \( -905857340 + 35996 T + T^{2} \)
$79$ \( -203130152 + 10001 T + T^{2} \)
$83$ \( 1475487360 - 78108 T + T^{2} \)
$89$ \( -1180708920 - 33438 T + T^{2} \)
$97$ \( 1347423694 + 85751 T + T^{2} \)
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