# 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 $1$

# Related objects

## 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: $$x^{2} - x - 576$$ x^2 - x - 576 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{10}^{\mathrm{new}}(\Gamma_0(98))$$.

## Hecke characteristic polynomials

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