Properties

Label 1680.4.a.y
Level 1680
Weight 4
Character orbit 1680.a
Self dual yes
Analytic conductor 99.123
Analytic rank 1
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1680.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(99.1232088096\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{41}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} -5 q^{5} -7 q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} -5 q^{5} -7 q^{7} + 9 q^{9} + ( -31 - \beta ) q^{11} + ( -3 + 5 \beta ) q^{13} + 15 q^{15} + ( 20 + 6 \beta ) q^{17} + ( 61 - \beta ) q^{19} + 21 q^{21} + ( -8 - 24 \beta ) q^{23} + 25 q^{25} -27 q^{27} + ( 176 - 6 \beta ) q^{29} + ( -33 - 19 \beta ) q^{31} + ( 93 + 3 \beta ) q^{33} + 35 q^{35} + ( -94 - 40 \beta ) q^{37} + ( 9 - 15 \beta ) q^{39} + ( 8 + 54 \beta ) q^{41} + ( 198 + 50 \beta ) q^{43} -45 q^{45} + ( 94 + 70 \beta ) q^{47} + 49 q^{49} + ( -60 - 18 \beta ) q^{51} + ( 491 - 29 \beta ) q^{53} + ( 155 + 5 \beta ) q^{55} + ( -183 + 3 \beta ) q^{57} + ( -258 + 38 \beta ) q^{59} + ( -440 + 42 \beta ) q^{61} -63 q^{63} + ( 15 - 25 \beta ) q^{65} + ( 178 - 114 \beta ) q^{67} + ( 24 + 72 \beta ) q^{69} + ( -155 + 43 \beta ) q^{71} + ( 163 + 19 \beta ) q^{73} -75 q^{75} + ( 217 + 7 \beta ) q^{77} + ( -916 - 4 \beta ) q^{79} + 81 q^{81} + ( 340 - 112 \beta ) q^{83} + ( -100 - 30 \beta ) q^{85} + ( -528 + 18 \beta ) q^{87} + ( 398 - 168 \beta ) q^{89} + ( 21 - 35 \beta ) q^{91} + ( 99 + 57 \beta ) q^{93} + ( -305 + 5 \beta ) q^{95} + ( -335 - 139 \beta ) q^{97} + ( -279 - 9 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 10q^{5} - 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 10q^{5} - 14q^{7} + 18q^{9} - 62q^{11} - 6q^{13} + 30q^{15} + 40q^{17} + 122q^{19} + 42q^{21} - 16q^{23} + 50q^{25} - 54q^{27} + 352q^{29} - 66q^{31} + 186q^{33} + 70q^{35} - 188q^{37} + 18q^{39} + 16q^{41} + 396q^{43} - 90q^{45} + 188q^{47} + 98q^{49} - 120q^{51} + 982q^{53} + 310q^{55} - 366q^{57} - 516q^{59} - 880q^{61} - 126q^{63} + 30q^{65} + 356q^{67} + 48q^{69} - 310q^{71} + 326q^{73} - 150q^{75} + 434q^{77} - 1832q^{79} + 162q^{81} + 680q^{83} - 200q^{85} - 1056q^{87} + 796q^{89} + 42q^{91} + 198q^{93} - 610q^{95} - 670q^{97} - 558q^{99} + O(q^{100}) \)

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
3.70156
−2.70156
0 −3.00000 0 −5.00000 0 −7.00000 0 9.00000 0
1.2 0 −3.00000 0 −5.00000 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1680.4.a.y 2
4.b odd 2 1 105.4.a.g 2
12.b even 2 1 315.4.a.g 2
20.d odd 2 1 525.4.a.i 2
20.e even 4 2 525.4.d.j 4
28.d even 2 1 735.4.a.q 2
60.h even 2 1 1575.4.a.y 2
84.h odd 2 1 2205.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 4.b odd 2 1
315.4.a.g 2 12.b even 2 1
525.4.a.i 2 20.d odd 2 1
525.4.d.j 4 20.e even 4 2
735.4.a.q 2 28.d even 2 1
1575.4.a.y 2 60.h even 2 1
1680.4.a.y 2 1.a even 1 1 trivial
2205.4.a.v 2 84.h odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1680))\):

\( T_{11}^{2} + 62 T_{11} + 920 \)
\( T_{13}^{2} + 6 T_{13} - 1016 \)
\( T_{17}^{2} - 40 T_{17} - 1076 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 3 T )^{2} \)
$5$ \( ( 1 + 5 T )^{2} \)
$7$ \( ( 1 + 7 T )^{2} \)
$11$ \( 1 + 62 T + 3582 T^{2} + 82522 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 6 T + 3378 T^{2} + 13182 T^{3} + 4826809 T^{4} \)
$17$ \( 1 - 40 T + 8750 T^{2} - 196520 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 122 T + 17398 T^{2} - 836798 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 16 T + 782 T^{2} + 194672 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 352 T + 78278 T^{2} - 8584928 T^{3} + 594823321 T^{4} \)
$31$ \( 1 + 66 T + 45870 T^{2} + 1966206 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 188 T + 44542 T^{2} + 9522764 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 16 T + 18350 T^{2} - 1102736 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 396 T + 95718 T^{2} - 31484772 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 188 T + 15582 T^{2} - 19518724 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 982 T + 504354 T^{2} - 146197214 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 516 T + 418118 T^{2} + 105975564 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 880 T + 575238 T^{2} + 199743280 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 356 T + 100374 T^{2} - 107071628 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 + 310 T + 664038 T^{2} + 110952410 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 326 T + 789802 T^{2} - 126819542 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 1832 T + 1824478 T^{2} + 903247448 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 680 T + 744870 T^{2} - 388815160 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 796 T + 411158 T^{2} - 561155324 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 670 T + 1145410 T^{2} + 611490910 T^{3} + 832972004929 T^{4} \)
show more
show less