Properties

Label 98.10.a.k
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{4817})\)
Defining polynomial: \( x^{4} - 2x^{3} - 2411x^{2} + 2412x + 1444802 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of 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.

\(f(q)\) \(=\) \( q - 16 q^{2} + (\beta_{3} + 5 \beta_1) q^{3} + 256 q^{4} + ( - 7 \beta_{3} + 104 \beta_1) q^{5} + ( - 16 \beta_{3} - 80 \beta_1) q^{6} - 4096 q^{8} + ( - 9 \beta_{2} + 3978) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} + (\beta_{3} + 5 \beta_1) q^{3} + 256 q^{4} + ( - 7 \beta_{3} + 104 \beta_1) q^{5} + ( - 16 \beta_{3} - 80 \beta_1) q^{6} - 4096 q^{8} + ( - 9 \beta_{2} + 3978) q^{9} + (112 \beta_{3} - 1664 \beta_1) q^{10} + (37 \beta_{2} + 2387) q^{11} + (256 \beta_{3} + 1280 \beta_1) q^{12} + (809 \beta_{3} - 182 \beta_1) q^{13} + ( - 76 \beta_{2} - 104328) q^{15} + 65536 q^{16} + ( - 248 \beta_{3} - 43071 \beta_1) q^{17} + (144 \beta_{2} - 63648) q^{18} + ( - 3679 \beta_{3} - 31639 \beta_1) q^{19} + ( - 1792 \beta_{3} + 26624 \beta_1) q^{20} + ( - 592 \beta_{2} - 38192) q^{22} + ( - 692 \beta_{2} - 709184) q^{23} + ( - 4096 \beta_{3} - 20480 \beta_1) q^{24} + (1505 \beta_{2} + 241536) q^{25} + ( - 12944 \beta_{3} + 2912 \beta_1) q^{26} + ( - 11736 \beta_{3} + 118548 \beta_1) q^{27} + (1087 \beta_{2} + 4167493) q^{29} + (1216 \beta_{2} + 1669248) q^{30} + ( - 41130 \beta_{3} - 100134 \beta_1) q^{31} - 1048576 q^{32} + ( - 13930 \beta_{3} - 798254 \beta_1) q^{33} + (3968 \beta_{3} + 689136 \beta_1) q^{34} + ( - 2304 \beta_{2} + 1018368) q^{36} + ( - 9021 \beta_{2} + 806593) q^{37} + (58864 \beta_{3} + 506224 \beta_1) q^{38} + ( - 3054 \beta_{2} + 17277642) q^{39} + (28672 \beta_{3} - 425984 \beta_1) q^{40} + (56652 \beta_{3} + 1430791 \beta_1) q^{41} + (9147 \beta_{2} - 19120051) q^{43} + (9472 \beta_{2} + 611072) q^{44} + (66969 \beta_{3} - 904500 \beta_1) q^{45} + (11072 \beta_{2} + 11346944) q^{46} + (163054 \beta_{3} - 2664762 \beta_1) q^{47} + (65536 \beta_{3} + 327680 \beta_1) q^{48} + ( - 24080 \beta_{2} - 3864576) q^{50} + (44063 \beta_{2} - 24315399) q^{51} + (207104 \beta_{3} - 46592 \beta_1) q^{52} + ( - 58226 \beta_{2} + 3427028) q^{53} + (187776 \beta_{3} - 1896768 \beta_1) q^{54} + ( - 406504 \beta_{3} + 5667564 \beta_1) q^{55} + (46355 \beta_{2} - 92889423) q^{57} + ( - 17392 \beta_{2} - 66679888) q^{58} + ( - 314395 \beta_{3} + 3105293 \beta_1) q^{59} + ( - 19456 \beta_{2} - 26707968) q^{60} + ( - 340041 \beta_{3} - 6621532 \beta_1) q^{61} + (658080 \beta_{3} + 1602144 \beta_1) q^{62} + 16777216 q^{64} + ( - 91073 \beta_{2} - 128932797) q^{65} + (222880 \beta_{3} + 12772064 \beta_1) q^{66} + (69906 \beta_{2} - 203605194) q^{67} + ( - 63488 \beta_{3} - 11026176 \beta_1) q^{68} + ( - 404012 \beta_{3} + 11606804 \beta_1) q^{69} + ( - 30534 \beta_{2} - 331167186) q^{71} + (36864 \beta_{2} - 16293888) q^{72} + ( - 1351834 \beta_{3} + 24266301 \beta_1) q^{73} + (144336 \beta_{2} - 12905488) q^{74} + ( - 422169 \beta_{3} - 31747305 \beta_1) q^{75} + ( - 941824 \beta_{3} - 8099584 \beta_1) q^{76} + (48864 \beta_{2} - 276442272) q^{78} + ( - 40838 \beta_{2} - 167102794) q^{79} + ( - 458752 \beta_{3} + 6815744 \beta_1) q^{80} + (105543 \beta_{2} - 277826922) q^{81} + ( - 906432 \beta_{3} - 22892656 \beta_1) q^{82} + (1981881 \beta_{3} - 22750843 \beta_1) q^{83} + ( - 273969 \beta_{2} - 414816241) q^{85} + ( - 146352 \beta_{2} + 305920816) q^{86} + (3688126 \beta_{3} - 2964574 \beta_1) q^{87} + ( - 151552 \beta_{2} - 9777152) q^{88} + ( - 1118280 \beta_{3} - 69656107 \beta_1) q^{89} + ( - 1071504 \beta_{3} + 14472000 \beta_1) q^{90} + ( - 177152 \beta_{2} - 181551104) q^{92} + (264654 \beta_{2} - 926644374) q^{93} + ( - 2608864 \beta_{3} + 42636192 \beta_1) q^{94} + (186896 \beta_{2} + 244297172) q^{95} + ( - 1048576 \beta_{3} - 5242880 \beta_1) q^{96} + (79754 \beta_{3} + 89705213 \beta_1) q^{97} + (125703 \beta_{2} - 697895415) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 64 q^{2} + 1024 q^{4} - 16384 q^{8} + 15912 q^{9} + 9548 q^{11} - 417312 q^{15} + 262144 q^{16} - 254592 q^{18} - 152768 q^{22} - 2836736 q^{23} + 966144 q^{25} + 16669972 q^{29} + 6676992 q^{30} - 4194304 q^{32} + 4073472 q^{36} + 3226372 q^{37} + 69110568 q^{39} - 76480204 q^{43} + 2444288 q^{44} + 45387776 q^{46} - 15458304 q^{50} - 97261596 q^{51} + 13708112 q^{53} - 371557692 q^{57} - 266719552 q^{58} - 106831872 q^{60} + 67108864 q^{64} - 515731188 q^{65} - 814420776 q^{67} - 1324668744 q^{71} - 65175552 q^{72} - 51621952 q^{74} - 1105769088 q^{78} - 668411176 q^{79} - 1111307688 q^{81} - 1659264964 q^{85} + 1223683264 q^{86} - 39108608 q^{88} - 726204416 q^{92} - 3706577496 q^{93} + 977188688 q^{95} - 2791581660 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 2411x^{2} + 2412x + 1444802 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 2419\nu - 1210 ) / 687 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} + 14456\nu - 7229 ) / 229 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 1029\nu^{2} - 2240\nu - 1242178 ) / 687 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 6\beta _1 + 21 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 28\beta_{3} + \beta_{2} + 8\beta _1 + 50673 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{3} + 173\beta_{2} - 3096\beta _1 + 10857 ) / 6 \) Copy content Toggle raw display

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
−32.7881
33.7881
36.6165
−35.6165
−16.0000 −191.777 256.000 −33.5898 3068.43 0 −4096.00 17095.5 537.437
1.2 −16.0000 −102.682 256.000 2094.80 1642.91 0 −4096.00 −9139.47 −33516.8
1.3 −16.0000 102.682 256.000 −2094.80 −1642.91 0 −4096.00 −9139.47 33516.8
1.4 −16.0000 191.777 256.000 33.5898 −3068.43 0 −4096.00 17095.5 −537.437
\(n\): 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.k 4
7.b odd 2 1 inner 98.10.a.k 4
7.c even 3 2 98.10.c.m 8
7.d odd 6 2 98.10.c.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.10.a.k 4 1.a even 1 1 trivial
98.10.a.k 4 7.b odd 2 1 inner
98.10.c.m 8 7.c even 3 2
98.10.c.m 8 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 47322T_{3}^{2} + 387774864 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 47322 T^{2} + \cdots + 387774864 \) Copy content Toggle raw display
$5$ \( T^{4} - 4389322 T^{2} + \cdots + 4951092496 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4774 T - 2902464824)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 28441135314 T^{2} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{4} - 364177565476 T^{2} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{4} - 760834631722 T^{2} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T^{2} + 1418368 T - 514307412752)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8334986 T + 14857994423056)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 74580188894856 T^{2} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{2} - 1613186 T - 172221385813328)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 524653229945812 T^{2} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{2} + 38240102 T + 187841499446728)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{2} - 6854056 T - 71\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{2} + 407210388 T + 31\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 662334372 T + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{2} + 334205588 T + 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 89\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 62\!\cdots\!56 \) Copy content Toggle raw display
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