Properties

Label 98.10.a.d
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{43}) \)
Defining polynomial: \(x^{2} - 43\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{43}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -16 q^{2} + \beta q^{3} + 256 q^{4} + 35 \beta q^{5} -16 \beta q^{6} -4096 q^{8} -16931 q^{9} +O(q^{10})\) \( q -16 q^{2} + \beta q^{3} + 256 q^{4} + 35 \beta q^{5} -16 \beta q^{6} -4096 q^{8} -16931 q^{9} -560 \beta q^{10} + 9380 q^{11} + 256 \beta q^{12} + 3413 \beta q^{13} + 96320 q^{15} + 65536 q^{16} + 1866 \beta q^{17} + 270896 q^{18} + 10727 \beta q^{19} + 8960 \beta q^{20} -150080 q^{22} -978936 q^{23} -4096 \beta q^{24} + 1418075 q^{25} -54608 \beta q^{26} -36614 \beta q^{27} -4317214 q^{29} -1541120 q^{30} + 152014 \beta q^{31} -1048576 q^{32} + 9380 \beta q^{33} -29856 \beta q^{34} -4334336 q^{36} + 2714394 q^{37} -171632 \beta q^{38} + 9392576 q^{39} -143360 \beta q^{40} + 171606 \beta q^{41} -34755692 q^{43} + 2401280 q^{44} -592585 \beta q^{45} + 15662976 q^{46} + 804002 \beta q^{47} + 65536 \beta q^{48} -22689200 q^{50} + 5135232 q^{51} + 873728 \beta q^{52} + 68067926 q^{53} + 585824 \beta q^{54} + 328300 \beta q^{55} + 29520704 q^{57} + 69075424 q^{58} -652537 \beta q^{59} + 24657920 q^{60} -3161153 \beta q^{61} -2432224 \beta q^{62} + 16777216 q^{64} + 328740160 q^{65} -150080 \beta q^{66} + 242944420 q^{67} + 477696 \beta q^{68} -978936 \beta q^{69} -94292464 q^{71} + 69349376 q^{72} + 2541664 \beta q^{73} -43430304 q^{74} + 1418075 \beta q^{75} + 2746112 \beta q^{76} -150281216 q^{78} + 677625160 q^{79} + 2293760 \beta q^{80} + 232491145 q^{81} -2745696 \beta q^{82} + 8412459 \beta q^{83} + 179733120 q^{85} + 556091072 q^{86} -4317214 \beta q^{87} -38420480 q^{88} + 10584880 \beta q^{89} + 9481360 \beta q^{90} -250607616 q^{92} + 418342528 q^{93} -12864032 \beta q^{94} + 1033224640 q^{95} -1048576 \beta q^{96} -24189050 \beta q^{97} -158812780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} + 512 q^{4} - 8192 q^{8} - 33862 q^{9} + O(q^{10}) \) \( 2 q - 32 q^{2} + 512 q^{4} - 8192 q^{8} - 33862 q^{9} + 18760 q^{11} + 192640 q^{15} + 131072 q^{16} + 541792 q^{18} - 300160 q^{22} - 1957872 q^{23} + 2836150 q^{25} - 8634428 q^{29} - 3082240 q^{30} - 2097152 q^{32} - 8668672 q^{36} + 5428788 q^{37} + 18785152 q^{39} - 69511384 q^{43} + 4802560 q^{44} + 31325952 q^{46} - 45378400 q^{50} + 10270464 q^{51} + 136135852 q^{53} + 59041408 q^{57} + 138150848 q^{58} + 49315840 q^{60} + 33554432 q^{64} + 657480320 q^{65} + 485888840 q^{67} - 188584928 q^{71} + 138698752 q^{72} - 86860608 q^{74} - 300562432 q^{78} + 1355250320 q^{79} + 464982290 q^{81} + 359466240 q^{85} + 1112182144 q^{86} - 76840960 q^{88} - 501215232 q^{92} + 836685056 q^{93} + 2066449280 q^{95} - 317625560 q^{99} + O(q^{100}) \)

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
−6.55744
6.55744
−16.0000 −52.4595 256.000 −1836.08 839.352 0 −4096.00 −16931.0 29377.3
1.2 −16.0000 52.4595 256.000 1836.08 −839.352 0 −4096.00 −16931.0 −29377.3
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 98.10.a.d 2
7.b odd 2 1 inner 98.10.a.d 2
7.c even 3 2 98.10.c.i 4
7.d odd 6 2 98.10.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.10.a.d 2 1.a even 1 1 trivial
98.10.a.d 2 7.b odd 2 1 inner
98.10.c.i 4 7.c even 3 2
98.10.c.i 4 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2752 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 + T )^{2} \)
$3$ \( -2752 + T^{2} \)
$5$ \( -3371200 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -9380 + T )^{2} \)
$13$ \( -32056861888 + T^{2} \)
$17$ \( -9582342912 + T^{2} \)
$19$ \( -316668591808 + T^{2} \)
$23$ \( ( 978936 + T )^{2} \)
$29$ \( ( 4317214 + T )^{2} \)
$31$ \( -63593921051392 + T^{2} \)
$37$ \( ( -2714394 + T )^{2} \)
$41$ \( -81042600137472 + T^{2} \)
$43$ \( ( 34755692 + T )^{2} \)
$47$ \( -1778945682443008 + T^{2} \)
$53$ \( ( -68067926 + T )^{2} \)
$59$ \( -1171814084087488 + T^{2} \)
$61$ \( -27500428572453568 + T^{2} \)
$67$ \( ( -242944420 + T )^{2} \)
$71$ \( ( 94292464 + T )^{2} \)
$73$ \( -17778073806241792 + T^{2} \)
$79$ \( ( -677625160 + T )^{2} \)
$83$ \( -194757571606226112 + T^{2} \)
$89$ \( -308333212058828800 + T^{2} \)
$97$ \( -1610223105011680000 + T^{2} \)
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