Properties

Label 1680.4.a.bo
Level 1680
Weight 4
Character orbit 1680.a
Self dual yes
Analytic conductor 99.123
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 9 q^{9} + ( 8 + 40 \beta ) q^{11} + ( -38 + 4 \beta ) q^{13} + 15 q^{15} + ( -62 + 4 \beta ) q^{17} + ( 48 - 32 \beta ) q^{19} + 21 q^{21} + ( 8 - 68 \beta ) q^{23} + 25 q^{25} + 27 q^{27} + ( 94 + 108 \beta ) q^{29} + ( 60 + 36 \beta ) q^{31} + ( 24 + 120 \beta ) q^{33} + 35 q^{35} + ( -66 + 132 \beta ) q^{37} + ( -114 + 12 \beta ) q^{39} + ( 50 - 160 \beta ) q^{41} + ( 268 + 124 \beta ) q^{43} + 45 q^{45} + ( 464 - 84 \beta ) q^{47} + 49 q^{49} + ( -186 + 12 \beta ) q^{51} + ( 442 - 128 \beta ) q^{53} + ( 40 + 200 \beta ) q^{55} + ( 144 - 96 \beta ) q^{57} + ( -52 - 408 \beta ) q^{59} + ( -234 + 84 \beta ) q^{61} + 63 q^{63} + ( -190 + 20 \beta ) q^{65} + ( 844 + 76 \beta ) q^{67} + ( 24 - 204 \beta ) q^{69} + ( 68 - 300 \beta ) q^{71} + ( 254 - 644 \beta ) q^{73} + 75 q^{75} + ( 56 + 280 \beta ) q^{77} + ( 216 - 464 \beta ) q^{79} + 81 q^{81} + ( 292 - 168 \beta ) q^{83} + ( -310 + 20 \beta ) q^{85} + ( 282 + 324 \beta ) q^{87} + ( -702 - 224 \beta ) q^{89} + ( -266 + 28 \beta ) q^{91} + ( 180 + 108 \beta ) q^{93} + ( 240 - 160 \beta ) q^{95} + ( -594 - 92 \beta ) q^{97} + ( 72 + 360 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + 16q^{11} - 76q^{13} + 30q^{15} - 124q^{17} + 96q^{19} + 42q^{21} + 16q^{23} + 50q^{25} + 54q^{27} + 188q^{29} + 120q^{31} + 48q^{33} + 70q^{35} - 132q^{37} - 228q^{39} + 100q^{41} + 536q^{43} + 90q^{45} + 928q^{47} + 98q^{49} - 372q^{51} + 884q^{53} + 80q^{55} + 288q^{57} - 104q^{59} - 468q^{61} + 126q^{63} - 380q^{65} + 1688q^{67} + 48q^{69} + 136q^{71} + 508q^{73} + 150q^{75} + 112q^{77} + 432q^{79} + 162q^{81} + 584q^{83} - 620q^{85} + 564q^{87} - 1404q^{89} - 532q^{91} + 360q^{93} + 480q^{95} - 1188q^{97} + 144q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
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.bo 2
4.b odd 2 1 105.4.a.e 2
12.b even 2 1 315.4.a.k 2
20.d odd 2 1 525.4.a.l 2
20.e even 4 2 525.4.d.l 4
28.d even 2 1 735.4.a.o 2
60.h even 2 1 1575.4.a.q 2
84.h odd 2 1 2205.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 4.b odd 2 1
315.4.a.k 2 12.b even 2 1
525.4.a.l 2 20.d odd 2 1
525.4.d.l 4 20.e even 4 2
735.4.a.o 2 28.d even 2 1
1575.4.a.q 2 60.h even 2 1
1680.4.a.bo 2 1.a even 1 1 trivial
2205.4.a.bb 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1680))\):

\( T_{11}^{2} - 16 T_{11} - 3136 \)
\( T_{13}^{2} + 76 T_{13} + 1412 \)
\( T_{17}^{2} + 124 T_{17} + 3812 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T )^{2} \)
$5$ \( ( 1 - 5 T )^{2} \)
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 - 16 T - 474 T^{2} - 21296 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 76 T + 5806 T^{2} + 166972 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 124 T + 13638 T^{2} + 609212 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 96 T + 13974 T^{2} - 658464 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 16 T + 15150 T^{2} - 194672 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 188 T + 34286 T^{2} - 4585132 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 120 T + 60590 T^{2} - 3574920 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 132 T + 70814 T^{2} + 6686196 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 100 T + 89142 T^{2} - 6892100 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 536 T + 200086 T^{2} - 42615752 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 928 T + 408830 T^{2} - 96347744 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 884 T + 460350 T^{2} - 131607268 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 104 T + 80534 T^{2} + 21359416 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 468 T + 494606 T^{2} + 106227108 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 1688 T + 1302310 T^{2} - 507687944 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 136 T + 540446 T^{2} - 48675896 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 508 T + 13078 T^{2} - 197620636 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 432 T + 602142 T^{2} - 212992848 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 584 T + 1172390 T^{2} - 333923608 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 + 1404 T + 1802390 T^{2} + 989776476 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 1188 T + 2161254 T^{2} + 1084255524 T^{3} + 832972004929 T^{4} \)
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