Properties

Label 280.6.a.c
Level $280$
Weight $6$
Character orbit 280.a
Self dual yes
Analytic conductor $44.907$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9074695476\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
Defining polynomial: \(x^{2} - x - 27\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{109}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 194 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 194 + 2 \beta ) q^{9} + ( -127 - 18 \beta ) q^{11} + ( -671 - 9 \beta ) q^{13} + ( 25 + 25 \beta ) q^{15} + ( -893 - 29 \beta ) q^{17} + ( 988 - 35 \beta ) q^{19} + ( -49 - 49 \beta ) q^{21} + ( -1056 - 141 \beta ) q^{23} + 625 q^{25} + ( 823 - 47 \beta ) q^{27} + ( -2381 - 92 \beta ) q^{29} + ( 6846 + 33 \beta ) q^{31} + ( -7975 - 145 \beta ) q^{33} -1225 q^{35} + ( 1068 + 381 \beta ) q^{37} + ( -4595 - 680 \beta ) q^{39} + ( -3370 + 193 \beta ) q^{41} + ( -13364 + 97 \beta ) q^{43} + ( 4850 + 50 \beta ) q^{45} + ( -3163 - 541 \beta ) q^{47} + 2401 q^{49} + ( -13537 - 922 \beta ) q^{51} + ( 12312 + 648 \beta ) q^{53} + ( -3175 - 450 \beta ) q^{55} + ( -14272 + 953 \beta ) q^{57} + ( 25668 + 1148 \beta ) q^{59} + ( 2234 + 1095 \beta ) q^{61} + ( -9506 - 98 \beta ) q^{63} + ( -16775 - 225 \beta ) q^{65} + ( -19584 + 1192 \beta ) q^{67} + ( -62532 - 1197 \beta ) q^{69} + ( 20616 + 2504 \beta ) q^{71} + ( -18062 + 2238 \beta ) q^{73} + ( 625 + 625 \beta ) q^{75} + ( 6223 + 882 \beta ) q^{77} + ( -70421 - 362 \beta ) q^{79} + ( -66811 + 290 \beta ) q^{81} + ( -28856 - 2258 \beta ) q^{83} + ( -22325 - 725 \beta ) q^{85} + ( -42493 - 2473 \beta ) q^{87} + ( -10118 - 567 \beta ) q^{89} + ( 32879 + 441 \beta ) q^{91} + ( 21234 + 6879 \beta ) q^{93} + ( 24700 - 875 \beta ) q^{95} + ( -91793 - 2949 \beta ) q^{97} + ( -40334 - 3746 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9} - 254 q^{11} - 1342 q^{13} + 50 q^{15} - 1786 q^{17} + 1976 q^{19} - 98 q^{21} - 2112 q^{23} + 1250 q^{25} + 1646 q^{27} - 4762 q^{29} + 13692 q^{31} - 15950 q^{33} - 2450 q^{35} + 2136 q^{37} - 9190 q^{39} - 6740 q^{41} - 26728 q^{43} + 9700 q^{45} - 6326 q^{47} + 4802 q^{49} - 27074 q^{51} + 24624 q^{53} - 6350 q^{55} - 28544 q^{57} + 51336 q^{59} + 4468 q^{61} - 19012 q^{63} - 33550 q^{65} - 39168 q^{67} - 125064 q^{69} + 41232 q^{71} - 36124 q^{73} + 1250 q^{75} + 12446 q^{77} - 140842 q^{79} - 133622 q^{81} - 57712 q^{83} - 44650 q^{85} - 84986 q^{87} - 20236 q^{89} + 65758 q^{91} + 42468 q^{93} + 49400 q^{95} - 183586 q^{97} - 80668 q^{99} + O(q^{100}) \)

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
−4.72015
5.72015
0 −19.8806 0 25.0000 0 −49.0000 0 152.239 0
1.2 0 21.8806 0 25.0000 0 −49.0000 0 235.761 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 280.6.a.c 2
4.b odd 2 1 560.6.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.c 2 1.a even 1 1 trivial
560.6.a.n 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 T_{3} - 435 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(280))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -435 - 2 T + T^{2} \)
$5$ \( ( -25 + T )^{2} \)
$7$ \( ( 49 + T )^{2} \)
$11$ \( -125135 + 254 T + T^{2} \)
$13$ \( 414925 + 1342 T + T^{2} \)
$17$ \( 430773 + 1786 T + T^{2} \)
$19$ \( 442044 - 1976 T + T^{2} \)
$23$ \( -7552980 + 2112 T + T^{2} \)
$29$ \( 1978857 + 4762 T + T^{2} \)
$31$ \( 46392912 - 13692 T + T^{2} \)
$37$ \( -62149572 - 2136 T + T^{2} \)
$41$ \( -4883664 + 6740 T + T^{2} \)
$43$ \( 174494172 + 26728 T + T^{2} \)
$47$ \( -117604347 + 6326 T + T^{2} \)
$53$ \( -31492800 - 24624 T + T^{2} \)
$59$ \( 84240080 - 51336 T + T^{2} \)
$61$ \( -517784144 - 4468 T + T^{2} \)
$67$ \( -235963648 + 39168 T + T^{2} \)
$71$ \( -2308707520 - 41232 T + T^{2} \)
$73$ \( -1857532940 + 36124 T + T^{2} \)
$79$ \( 4901982057 + 140842 T + T^{2} \)
$83$ \( -1390305168 + 57712 T + T^{2} \)
$89$ \( -37795280 + 20236 T + T^{2} \)
$97$ \( 4634236813 + 183586 T + T^{2} \)
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