Properties

Label 98.10.a.e
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{2305}) \)
Defining polynomial: \( x^{2} - x - 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2305}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 16 q^{2} + (5 \beta + 7) q^{3} + 256 q^{4} + (21 \beta + 1365) q^{5} + ( - 80 \beta - 112) q^{6} - 4096 q^{8} + (70 \beta + 37991) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} + (5 \beta + 7) q^{3} + 256 q^{4} + (21 \beta + 1365) q^{5} + ( - 80 \beta - 112) q^{6} - 4096 q^{8} + (70 \beta + 37991) q^{9} + ( - 336 \beta - 21840) q^{10} + ( - 1050 \beta + 22470) q^{11} + (1280 \beta + 1792) q^{12} + ( - 225 \beta - 50141) q^{13} + (6972 \beta + 251580) q^{15} + 65536 q^{16} + ( - 1590 \beta + 435204) q^{17} + ( - 1120 \beta - 607856) q^{18} + (13455 \beta - 254387) q^{19} + (5376 \beta + 349440) q^{20} + (16800 \beta - 359520) q^{22} + ( - 7140 \beta + 39900) q^{23} + ( - 20480 \beta - 28672) q^{24} + (57330 \beta + 926605) q^{25} + (3600 \beta + 802256) q^{26} + (92030 \beta + 934906) q^{27} + (119490 \beta + 1003164) q^{29} + ( - 111552 \beta - 4025280) q^{30} + ( - 39330 \beta - 1094366) q^{31} - 1048576 q^{32} + (105000 \beta - 11943960) q^{33} + (25440 \beta - 6963264) q^{34} + (17920 \beta + 9725696) q^{36} + (143010 \beta - 10361788) q^{37} + ( - 215280 \beta + 4070192) q^{38} + ( - 252280 \beta - 2944112) q^{39} + ( - 86016 \beta - 5591040) q^{40} + ( - 517890 \beta - 9508296) q^{41} + ( - 345870 \beta + 2096858) q^{43} + ( - 268800 \beta + 5752320) q^{44} + (893361 \beta + 55246065) q^{45} + (114240 \beta - 638400) q^{46} + ( - 200430 \beta + 37271262) q^{47} + (327680 \beta + 458752) q^{48} + ( - 917280 \beta - 14825680) q^{50} + (2164890 \beta - 15278322) q^{51} + ( - 57600 \beta - 12836096) q^{52} + ( - 464520 \beta - 1619874) q^{53} + ( - 1472480 \beta - 14958496) q^{54} + ( - 961380 \beta - 20153700) q^{55} + ( - 1177750 \beta + 153288166) q^{57} + ( - 1911840 \beta - 16050624) q^{58} + (1119735 \beta + 66821181) q^{59} + (1784832 \beta + 64404480) q^{60} + (866205 \beta - 113900843) q^{61} + (629280 \beta + 17509856) q^{62} + 16777216 q^{64} + ( - 1360086 \beta - 79333590) q^{65} + ( - 1680000 \beta + 191103360) q^{66} + ( - 1654380 \beta + 166465136) q^{67} + ( - 407040 \beta + 111412224) q^{68} + (149520 \beta - 82009200) q^{69} + (6323940 \beta - 83992860) q^{71} + ( - 286720 \beta - 155611136) q^{72} + ( - 6043140 \beta + 22342138) q^{73} + ( - 2288160 \beta + 165788608) q^{74} + (5034335 \beta + 667214485) q^{75} + (3444480 \beta - 65123072) q^{76} + (4036480 \beta + 47105792) q^{78} + ( - 2213820 \beta + 134821388) q^{79} + (1376256 \beta + 89456640) q^{80} + (3940930 \beta + 319413239) q^{81} + (8286240 \beta + 152132736) q^{82} + (6075435 \beta + 91552881) q^{83} + (6968934 \beta + 517089510) q^{85} + (5533920 \beta - 33549728) q^{86} + (5852250 \beta + 1384144398) q^{87} + (4300800 \beta - 92037120) q^{88} + ( - 6785040 \beta - 395828874) q^{89} + ( - 14293776 \beta - 883937040) q^{90} + ( - 1827840 \beta + 10214400) q^{92} + ( - 5747140 \beta - 460938812) q^{93} + (3206880 \beta - 596340192) q^{94} + (13023948 \beta + 304051020) q^{95} + ( - 5242880 \beta - 7340032) q^{96} + (13989690 \beta + 2084740) q^{97} + ( - 38317650 \beta + 684240270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} + 14 q^{3} + 512 q^{4} + 2730 q^{5} - 224 q^{6} - 8192 q^{8} + 75982 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{2} + 14 q^{3} + 512 q^{4} + 2730 q^{5} - 224 q^{6} - 8192 q^{8} + 75982 q^{9} - 43680 q^{10} + 44940 q^{11} + 3584 q^{12} - 100282 q^{13} + 503160 q^{15} + 131072 q^{16} + 870408 q^{17} - 1215712 q^{18} - 508774 q^{19} + 698880 q^{20} - 719040 q^{22} + 79800 q^{23} - 57344 q^{24} + 1853210 q^{25} + 1604512 q^{26} + 1869812 q^{27} + 2006328 q^{29} - 8050560 q^{30} - 2188732 q^{31} - 2097152 q^{32} - 23887920 q^{33} - 13926528 q^{34} + 19451392 q^{36} - 20723576 q^{37} + 8140384 q^{38} - 5888224 q^{39} - 11182080 q^{40} - 19016592 q^{41} + 4193716 q^{43} + 11504640 q^{44} + 110492130 q^{45} - 1276800 q^{46} + 74542524 q^{47} + 917504 q^{48} - 29651360 q^{50} - 30556644 q^{51} - 25672192 q^{52} - 3239748 q^{53} - 29916992 q^{54} - 40307400 q^{55} + 306576332 q^{57} - 32101248 q^{58} + 133642362 q^{59} + 128808960 q^{60} - 227801686 q^{61} + 35019712 q^{62} + 33554432 q^{64} - 158667180 q^{65} + 382206720 q^{66} + 332930272 q^{67} + 222824448 q^{68} - 164018400 q^{69} - 167985720 q^{71} - 311222272 q^{72} + 44684276 q^{73} + 331577216 q^{74} + 1334428970 q^{75} - 130246144 q^{76} + 94211584 q^{78} + 269642776 q^{79} + 178913280 q^{80} + 638826478 q^{81} + 304265472 q^{82} + 183105762 q^{83} + 1034179020 q^{85} - 67099456 q^{86} + 2768288796 q^{87} - 184074240 q^{88} - 791657748 q^{89} - 1767874080 q^{90} + 20428800 q^{92} - 921877624 q^{93} - 1192680384 q^{94} + 608102040 q^{95} - 14680064 q^{96} + 4169480 q^{97} + 1368480540 q^{99}+O(q^{100}) \) 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
−23.5052
24.5052
−16.0000 −233.052 256.000 356.781 3728.83 0 −4096.00 34630.3 −5708.50
1.2 −16.0000 247.052 256.000 2373.22 −3952.83 0 −4096.00 41351.7 −37971.5
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 14T_{3} - 57576 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 14T - 57576 \) Copy content Toggle raw display
$5$ \( T^{2} - 2730 T + 846720 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 44940 T - 2036361600 \) Copy content Toggle raw display
$13$ \( T^{2} + 100282 T + 2397429256 \) Copy content Toggle raw display
$17$ \( T^{2} - 870408 T + 183575251116 \) Copy content Toggle raw display
$19$ \( T^{2} + 508774 T - 352577596856 \) Copy content Toggle raw display
$23$ \( T^{2} - 79800 T - 115915968000 \) Copy content Toggle raw display
$29$ \( T^{2} - 2006328 T - 31904129519604 \) Copy content Toggle raw display
$31$ \( T^{2} + 2188732 T - 2367849772544 \) Copy content Toggle raw display
$37$ \( T^{2} + 20723576 T + 60225113026444 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 527816477266884 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 271341247682336 \) Copy content Toggle raw display
$47$ \( T^{2} - 74542524 T + 12\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 494746212296124 \) Copy content Toggle raw display
$59$ \( T^{2} - 133642362 T + 15\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{2} + 227801686 T + 11\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} - 332930272 T + 21\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} + 167985720 T - 85\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} - 44684276 T - 83\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{2} - 269642776 T + 68\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{2} - 183105762 T - 76\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} + 791657748 T + 50\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{2} - 4169480 T - 45\!\cdots\!00 \) Copy content Toggle raw display
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