Properties

Label 98.5.b.a
Level $98$
Weight $5$
Character orbit 98.b
Analytic conductor $10.130$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 98.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1302563822\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta_{2} q^{2} + ( -4 \beta_{1} - 7 \beta_{3} ) q^{3} + 8 q^{4} + ( 5 \beta_{1} + 13 \beta_{3} ) q^{5} + ( 6 \beta_{1} + 22 \beta_{3} ) q^{6} -16 \beta_{2} q^{8} + ( -49 - 89 \beta_{2} ) q^{9} +O(q^{10})\) \( q -2 \beta_{2} q^{2} + ( -4 \beta_{1} - 7 \beta_{3} ) q^{3} + 8 q^{4} + ( 5 \beta_{1} + 13 \beta_{3} ) q^{5} + ( 6 \beta_{1} + 22 \beta_{3} ) q^{6} -16 \beta_{2} q^{8} + ( -49 - 89 \beta_{2} ) q^{9} + ( -16 \beta_{1} - 36 \beta_{3} ) q^{10} + ( 6 - 103 \beta_{2} ) q^{11} + ( -32 \beta_{1} - 56 \beta_{3} ) q^{12} + ( -101 \beta_{1} + 97 \beta_{3} ) q^{13} + ( 222 + 158 \beta_{2} ) q^{15} + 64 q^{16} + ( 282 \beta_{1} + 177 \beta_{3} ) q^{17} + ( 356 + 98 \beta_{2} ) q^{18} + ( -207 \beta_{1} + 10 \beta_{3} ) q^{19} + ( 40 \beta_{1} + 104 \beta_{3} ) q^{20} + ( 412 - 12 \beta_{2} ) q^{22} + ( 26 + 144 \beta_{2} ) q^{23} + ( 48 \beta_{1} + 176 \beta_{3} ) q^{24} + ( 237 - 274 \beta_{2} ) q^{25} + ( -396 \beta_{1} + 8 \beta_{3} ) q^{26} + ( 139 \beta_{1} + 755 \beta_{3} ) q^{27} + ( -352 - 22 \beta_{2} ) q^{29} + ( -632 - 444 \beta_{2} ) q^{30} + ( 698 \beta_{1} - 74 \beta_{3} ) q^{31} -128 \beta_{2} q^{32} + ( 285 \beta_{1} + 1091 \beta_{3} ) q^{33} + ( 210 \beta_{1} - 918 \beta_{3} ) q^{34} + ( -392 - 712 \beta_{2} ) q^{36} + ( -848 - 36 \beta_{2} ) q^{37} + ( -434 \beta_{1} + 394 \beta_{3} ) q^{38} + ( 550 + 764 \beta_{2} ) q^{39} + ( -128 \beta_{1} - 288 \beta_{3} ) q^{40} + ( -251 \beta_{1} + 460 \beta_{3} ) q^{41} + ( -506 + 1175 \beta_{2} ) q^{43} + ( 48 - 824 \beta_{2} ) q^{44} + ( -957 \beta_{1} - 2239 \beta_{3} ) q^{45} + ( -576 - 52 \beta_{2} ) q^{46} + ( 1732 \beta_{1} - 1418 \beta_{3} ) q^{47} + ( -256 \beta_{1} - 448 \beta_{3} ) q^{48} + ( 1096 - 474 \beta_{2} ) q^{50} + ( 4734 + 2793 \beta_{2} ) q^{51} + ( -808 \beta_{1} + 776 \beta_{3} ) q^{52} + ( -4170 - 706 \beta_{2} ) q^{53} + ( -1232 \beta_{1} - 1788 \beta_{3} ) q^{54} + ( -794 \beta_{1} - 1776 \beta_{3} ) q^{55} + ( -1516 - 511 \beta_{2} ) q^{57} + ( 88 + 704 \beta_{2} ) q^{58} + ( -1379 \beta_{1} + 2820 \beta_{3} ) q^{59} + ( 1776 + 1264 \beta_{2} ) q^{60} + ( 1523 \beta_{1} + 365 \beta_{3} ) q^{61} + ( 1544 \beta_{1} - 1248 \beta_{3} ) q^{62} + 512 q^{64} + ( -1512 - 938 \beta_{2} ) q^{65} + ( -1612 \beta_{1} - 2752 \beta_{3} ) q^{66} + ( -5204 - 786 \beta_{2} ) q^{67} + ( 2256 \beta_{1} + 1416 \beta_{3} ) q^{68} + ( -536 \beta_{1} - 1766 \beta_{3} ) q^{69} + ( -496 + 1244 \beta_{2} ) q^{71} + ( 2848 + 784 \beta_{2} ) q^{72} + ( 1385 \beta_{1} - 2736 \beta_{3} ) q^{73} + ( 144 + 1696 \beta_{2} ) q^{74} + ( -126 \beta_{1} + 1355 \beta_{3} ) q^{75} + ( -1656 \beta_{1} + 80 \beta_{3} ) q^{76} + ( -3056 - 1100 \beta_{2} ) q^{78} + ( -7404 - 2270 \beta_{2} ) q^{79} + ( 320 \beta_{1} + 832 \beta_{3} ) q^{80} + ( 7713 + 1513 \beta_{2} ) q^{81} + ( -1422 \beta_{1} - 418 \beta_{3} ) q^{82} + ( 4703 \beta_{1} + 1164 \beta_{3} ) q^{83} + ( -7422 - 5442 \beta_{2} ) q^{85} + ( -4700 + 1012 \beta_{2} ) q^{86} + ( 1474 \beta_{1} + 2706 \beta_{3} ) q^{87} + ( 3296 - 96 \beta_{2} ) q^{88} + ( -3573 \beta_{1} + 3746 \beta_{3} ) q^{89} + ( 2564 \beta_{1} + 6392 \beta_{3} ) q^{90} + ( 208 + 1152 \beta_{2} ) q^{92} + ( 4548 + 1280 \beta_{2} ) q^{93} + ( 6300 \beta_{1} - 628 \beta_{3} ) q^{94} + ( 1810 + 1476 \beta_{2} ) q^{95} + ( 384 \beta_{1} + 1408 \beta_{3} ) q^{96} + ( -5324 \beta_{1} + 141 \beta_{3} ) q^{97} + ( 18040 + 4513 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} - 196q^{9} + O(q^{10}) \) \( 4q + 32q^{4} - 196q^{9} + 24q^{11} + 888q^{15} + 256q^{16} + 1424q^{18} + 1648q^{22} + 104q^{23} + 948q^{25} - 1408q^{29} - 2528q^{30} - 1568q^{36} - 3392q^{37} + 2200q^{39} - 2024q^{43} + 192q^{44} - 2304q^{46} + 4384q^{50} + 18936q^{51} - 16680q^{53} - 6064q^{57} + 352q^{58} + 7104q^{60} + 2048q^{64} - 6048q^{65} - 20816q^{67} - 1984q^{71} + 11392q^{72} + 576q^{74} - 12224q^{78} - 29616q^{79} + 30852q^{81} - 29688q^{85} - 18800q^{86} + 13184q^{88} + 832q^{92} + 18192q^{93} + 7240q^{95} + 72160q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.765367i
0.765367i
1.84776i
1.84776i
−2.82843 15.9958i 8.00000 27.8477i 45.2429i 0 −22.6274 −174.865 78.7652i
97.2 −2.82843 15.9958i 8.00000 27.8477i 45.2429i 0 −22.6274 −174.865 78.7652i
97.3 2.82843 2.03347i 8.00000 0.710974i 5.75152i 0 22.6274 76.8650 2.01094i
97.4 2.82843 2.03347i 8.00000 0.710974i 5.75152i 0 22.6274 76.8650 2.01094i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.5.b.a 4
3.b odd 2 1 882.5.c.c 4
4.b odd 2 1 784.5.c.a 4
7.b odd 2 1 inner 98.5.b.a 4
7.c even 3 2 98.5.d.c 8
7.d odd 6 2 98.5.d.c 8
21.c even 2 1 882.5.c.c 4
28.d even 2 1 784.5.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.5.b.a 4 1.a even 1 1 trivial
98.5.b.a 4 7.b odd 2 1 inner
98.5.d.c 8 7.c even 3 2
98.5.d.c 8 7.d odd 6 2
784.5.c.a 4 4.b odd 2 1
784.5.c.a 4 28.d even 2 1
882.5.c.c 4 3.b odd 2 1
882.5.c.c 4 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 260 T_{3}^{2} + 1058 \) acting on \(S_{5}^{\mathrm{new}}(98, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T^{2} )^{2} \)
$3$ \( 1058 + 260 T^{2} + T^{4} \)
$5$ \( 392 + 776 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -21182 - 12 T + T^{2} )^{2} \)
$13$ \( 707030408 + 78440 T^{2} + T^{4} \)
$17$ \( 43821617058 + 443412 T^{2} + T^{4} \)
$19$ \( 2981309762 + 171796 T^{2} + T^{4} \)
$23$ \( ( -40796 - 52 T + T^{2} )^{2} \)
$29$ \( ( 122936 + 704 T + T^{2} )^{2} \)
$31$ \( 286409447552 + 1970720 T^{2} + T^{4} \)
$37$ \( ( 716512 + 1696 T + T^{2} )^{2} \)
$41$ \( 288069342722 + 1098404 T^{2} + T^{4} \)
$43$ \( ( -2505214 + 1012 T + T^{2} )^{2} \)
$47$ \( 30777535627808 + 20042192 T^{2} + T^{4} \)
$53$ \( ( 16392028 + 8340 T + T^{2} )^{2} \)
$59$ \( 382444812731522 + 39416164 T^{2} + T^{4} \)
$61$ \( 21754848065672 + 9811016 T^{2} + T^{4} \)
$67$ \( ( 25846024 + 10408 T + T^{2} )^{2} \)
$71$ \( ( -2849056 + 992 T + T^{2} )^{2} \)
$73$ \( 345644675616962 + 37615684 T^{2} + T^{4} \)
$79$ \( ( 44513416 + 14808 T + T^{2} )^{2} \)
$83$ \( 2011288822677218 + 93892420 T^{2} + T^{4} \)
$89$ \( 1571934000441218 + 107195380 T^{2} + T^{4} \)
$97$ \( 1439024660341058 + 113459428 T^{2} + T^{4} \)
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