Properties

Label 98.10.c.j
Level $98$
Weight $10$
Character orbit 98.c
Analytic conductor $50.474$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{2305})\)
Defining polynomial: \( x^{4} - x^{3} + 577x^{2} + 576x + 331776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 16 \beta_1 + 16) q^{2} + (5 \beta_{2} + 7 \beta_1) q^{3} - 256 \beta_1 q^{4} + ( - 21 \beta_{3} - 21 \beta_{2} - 1365 \beta_1 + 1365) q^{5} + ( - 80 \beta_{3} + 112) q^{6} - 4096 q^{8} + (70 \beta_{3} + 70 \beta_{2} + 37991 \beta_1 - 37991) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 16 \beta_1 + 16) q^{2} + (5 \beta_{2} + 7 \beta_1) q^{3} - 256 \beta_1 q^{4} + ( - 21 \beta_{3} - 21 \beta_{2} - 1365 \beta_1 + 1365) q^{5} + ( - 80 \beta_{3} + 112) q^{6} - 4096 q^{8} + (70 \beta_{3} + 70 \beta_{2} + 37991 \beta_1 - 37991) q^{9} + ( - 336 \beta_{2} - 21840 \beta_1) q^{10} + (1050 \beta_{2} - 22470 \beta_1) q^{11} + ( - 1280 \beta_{3} - 1280 \beta_{2} - 1792 \beta_1 + 1792) q^{12} + ( - 225 \beta_{3} + 50141) q^{13} + ( - 6972 \beta_{3} + 251580) q^{15} + (65536 \beta_1 - 65536) q^{16} + ( - 1590 \beta_{2} + 435204 \beta_1) q^{17} + (1120 \beta_{2} + 607856 \beta_1) q^{18} + ( - 13455 \beta_{3} - 13455 \beta_{2} + 254387 \beta_1 - 254387) q^{19} + (5376 \beta_{3} - 349440) q^{20} + ( - 16800 \beta_{3} - 359520) q^{22} + ( - 7140 \beta_{3} - 7140 \beta_{2} + 39900 \beta_1 - 39900) q^{23} + ( - 20480 \beta_{2} - 28672 \beta_1) q^{24} + ( - 57330 \beta_{2} - 926605 \beta_1) q^{25} + ( - 3600 \beta_{3} - 3600 \beta_{2} - 802256 \beta_1 + 802256) q^{26} + (92030 \beta_{3} - 934906) q^{27} + ( - 119490 \beta_{3} + 1003164) q^{29} + ( - 111552 \beta_{3} - 111552 \beta_{2} - 4025280 \beta_1 + 4025280) q^{30} + ( - 39330 \beta_{2} - 1094366 \beta_1) q^{31} + 1048576 \beta_1 q^{32} + ( - 105000 \beta_{3} - 105000 \beta_{2} + \cdots - 11943960) q^{33}+ \cdots + (38317650 \beta_{3} + 684240270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 14 q^{3} - 512 q^{4} + 2730 q^{5} + 448 q^{6} - 16384 q^{8} - 75982 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} + 14 q^{3} - 512 q^{4} + 2730 q^{5} + 448 q^{6} - 16384 q^{8} - 75982 q^{9} - 43680 q^{10} - 44940 q^{11} + 3584 q^{12} + 200564 q^{13} + 1006320 q^{15} - 131072 q^{16} + 870408 q^{17} + 1215712 q^{18} - 508774 q^{19} - 1397760 q^{20} - 1438080 q^{22} - 79800 q^{23} - 57344 q^{24} - 1853210 q^{25} + 1604512 q^{26} - 3739624 q^{27} + 4012656 q^{29} + 8050560 q^{30} - 2188732 q^{31} + 2097152 q^{32} - 23887920 q^{33} + 27853056 q^{34} + 38902784 q^{36} + 20723576 q^{37} + 8140384 q^{38} + 5888224 q^{39} - 11182080 q^{40} + 38033184 q^{41} + 8387432 q^{43} - 11504640 q^{44} + 110492130 q^{45} + 1276800 q^{46} + 74542524 q^{47} - 1835008 q^{48} - 59302720 q^{50} + 30556644 q^{51} - 25672192 q^{52} + 3239748 q^{53} - 29916992 q^{54} + 80614800 q^{55} + 613152664 q^{57} + 32101248 q^{58} + 133642362 q^{59} - 128808960 q^{60} - 227801686 q^{61} - 70039424 q^{62} + 67108864 q^{64} + 158667180 q^{65} + 382206720 q^{66} - 332930272 q^{67} + 222824448 q^{68} + 328036800 q^{69} - 335971440 q^{71} + 311222272 q^{72} + 44684276 q^{73} - 331577216 q^{74} + 1334428970 q^{75} + 260492288 q^{76} + 188423168 q^{78} - 269642776 q^{79} + 178913280 q^{80} - 638826478 q^{81} + 304265472 q^{82} - 366211524 q^{83} + 2068358040 q^{85} + 67099456 q^{86} + 2768288796 q^{87} + 184074240 q^{88} - 791657748 q^{89} + 3535748160 q^{90} + 40857600 q^{92} + 921877624 q^{93} - 1192680384 q^{94} - 608102040 q^{95} - 14680064 q^{96} - 8338960 q^{97} + 2736961080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 577x^{2} + 576x + 331776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 577\nu^{2} - 577\nu + 331776 ) / 332352 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 577\nu^{2} + 665281\nu - 331776 ) / 332352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 1729 ) / 577 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 1153\beta _1 - 1153 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 577\beta_{3} - 1729 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−11.7526 + 20.3561i
12.2526 21.2221i
−11.7526 20.3561i
12.2526 + 21.2221i
8.00000 + 13.8564i −116.526 + 201.829i −128.000 + 221.703i 178.391 + 308.982i −3728.83 0 −4096.00 −17315.1 29990.7i −2854.25 + 4943.71i
67.2 8.00000 + 13.8564i 123.526 213.953i −128.000 + 221.703i 1186.61 + 2055.27i 3952.83 0 −4096.00 −20675.9 35811.6i −18985.7 + 32884.3i
79.1 8.00000 13.8564i −116.526 201.829i −128.000 221.703i 178.391 308.982i −3728.83 0 −4096.00 −17315.1 + 29990.7i −2854.25 4943.71i
79.2 8.00000 13.8564i 123.526 + 213.953i −128.000 221.703i 1186.61 2055.27i 3952.83 0 −4096.00 −20675.9 + 35811.6i −18985.7 32884.3i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.10.c.j 4
7.b odd 2 1 98.10.c.h 4
7.c even 3 1 14.10.a.c 2
7.c even 3 1 inner 98.10.c.j 4
7.d odd 6 1 98.10.a.e 2
7.d odd 6 1 98.10.c.h 4
21.h odd 6 1 126.10.a.o 2
28.g odd 6 1 112.10.a.c 2
35.j even 6 1 350.10.a.j 2
35.l odd 12 2 350.10.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.c 2 7.c even 3 1
98.10.a.e 2 7.d odd 6 1
98.10.c.h 4 7.b odd 2 1
98.10.c.h 4 7.d odd 6 1
98.10.c.j 4 1.a even 1 1 trivial
98.10.c.j 4 7.c even 3 1 inner
112.10.a.c 2 28.g odd 6 1
126.10.a.o 2 21.h odd 6 1
350.10.a.j 2 35.j even 6 1
350.10.c.j 4 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 14T_{3}^{3} + 57772T_{3}^{2} + 806064T_{3} + 3314995776 \) acting on \(S_{10}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 16 T + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 14 T^{3} + \cdots + 3314995776 \) Copy content Toggle raw display
$5$ \( T^{4} - 2730 T^{3} + \cdots + 716934758400 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 44940 T^{3} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} - 100282 T + 2397429256)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 870408 T^{3} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{4} + 508774 T^{3} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{4} + 79800 T^{3} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2006328 T - 31904129519604)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2188732 T^{3} + \cdots + 56\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{4} - 20723576 T^{3} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{2} - 19016592 T - 527816477266884)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4193716 T - 271341247682336)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 74542524 T^{3} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} - 3239748 T^{3} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{4} - 133642362 T^{3} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{4} + 227801686 T^{3} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + 332930272 T^{3} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 167985720 T - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 44684276 T^{3} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{4} + 269642776 T^{3} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{2} + 183105762 T - 76\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 791657748 T^{3} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4169480 T - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
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