# 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 $2$

# Related objects

## 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: $$x^{4} - x^{3} + 577x^{2} + 576x + 331776$$ x^4 - x^3 + 577*x^2 + 576*x + 331776 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.

 $$f(q)$$ $$=$$ $$q + ( - 16 \beta_1 + 16) q^{2} + (5 \beta_{2} + 7 \beta_1) q^{3} - 256 \beta_1 q^{4} + ( - 21 \beta_{3} - 21 \beta_{2} - 1365 \beta_1 + 1365) q^{5} + ( - 80 \beta_{3} + 112) q^{6} - 4096 q^{8} + (70 \beta_{3} + 70 \beta_{2} + 37991 \beta_1 - 37991) q^{9}+O(q^{10})$$ q + (-16*b1 + 16) * q^2 + (5*b2 + 7*b1) * q^3 - 256*b1 * q^4 + (-21*b3 - 21*b2 - 1365*b1 + 1365) * q^5 + (-80*b3 + 112) * q^6 - 4096 * q^8 + (70*b3 + 70*b2 + 37991*b1 - 37991) * q^9 $$q + ( - 16 \beta_1 + 16) q^{2} + (5 \beta_{2} + 7 \beta_1) q^{3} - 256 \beta_1 q^{4} + ( - 21 \beta_{3} - 21 \beta_{2} - 1365 \beta_1 + 1365) q^{5} + ( - 80 \beta_{3} + 112) q^{6} - 4096 q^{8} + (70 \beta_{3} + 70 \beta_{2} + 37991 \beta_1 - 37991) q^{9} + ( - 336 \beta_{2} - 21840 \beta_1) q^{10} + (1050 \beta_{2} - 22470 \beta_1) q^{11} + ( - 1280 \beta_{3} - 1280 \beta_{2} - 1792 \beta_1 + 1792) q^{12} + ( - 225 \beta_{3} + 50141) q^{13} + ( - 6972 \beta_{3} + 251580) q^{15} + (65536 \beta_1 - 65536) q^{16} + ( - 1590 \beta_{2} + 435204 \beta_1) q^{17} + (1120 \beta_{2} + 607856 \beta_1) q^{18} + ( - 13455 \beta_{3} - 13455 \beta_{2} + 254387 \beta_1 - 254387) q^{19} + (5376 \beta_{3} - 349440) q^{20} + ( - 16800 \beta_{3} - 359520) q^{22} + ( - 7140 \beta_{3} - 7140 \beta_{2} + 39900 \beta_1 - 39900) q^{23} + ( - 20480 \beta_{2} - 28672 \beta_1) q^{24} + ( - 57330 \beta_{2} - 926605 \beta_1) q^{25} + ( - 3600 \beta_{3} - 3600 \beta_{2} - 802256 \beta_1 + 802256) q^{26} + (92030 \beta_{3} - 934906) q^{27} + ( - 119490 \beta_{3} + 1003164) q^{29} + ( - 111552 \beta_{3} - 111552 \beta_{2} - 4025280 \beta_1 + 4025280) q^{30} + ( - 39330 \beta_{2} - 1094366 \beta_1) q^{31} + 1048576 \beta_1 q^{32} + ( - 105000 \beta_{3} - 105000 \beta_{2} + \cdots - 11943960) q^{33}+ \cdots + (38317650 \beta_{3} + 684240270) q^{99}+O(q^{100})$$ q + (-16*b1 + 16) * q^2 + (5*b2 + 7*b1) * q^3 - 256*b1 * q^4 + (-21*b3 - 21*b2 - 1365*b1 + 1365) * q^5 + (-80*b3 + 112) * q^6 - 4096 * q^8 + (70*b3 + 70*b2 + 37991*b1 - 37991) * q^9 + (-336*b2 - 21840*b1) * q^10 + (1050*b2 - 22470*b1) * q^11 + (-1280*b3 - 1280*b2 - 1792*b1 + 1792) * q^12 + (-225*b3 + 50141) * q^13 + (-6972*b3 + 251580) * q^15 + (65536*b1 - 65536) * q^16 + (-1590*b2 + 435204*b1) * q^17 + (1120*b2 + 607856*b1) * q^18 + (-13455*b3 - 13455*b2 + 254387*b1 - 254387) * q^19 + (5376*b3 - 349440) * q^20 + (-16800*b3 - 359520) * q^22 + (-7140*b3 - 7140*b2 + 39900*b1 - 39900) * q^23 + (-20480*b2 - 28672*b1) * q^24 + (-57330*b2 - 926605*b1) * q^25 + (-3600*b3 - 3600*b2 - 802256*b1 + 802256) * q^26 + (92030*b3 - 934906) * q^27 + (-119490*b3 + 1003164) * q^29 + (-111552*b3 - 111552*b2 - 4025280*b1 + 4025280) * q^30 + (-39330*b2 - 1094366*b1) * q^31 + 1048576*b1 * q^32 + (-105000*b3 - 105000*b2 + 11943960*b1 - 11943960) * q^33 + (25440*b3 + 6963264) * q^34 + (-17920*b3 + 9725696) * q^36 + (143010*b3 + 143010*b2 - 10361788*b1 + 10361788) * q^37 + (-215280*b2 + 4070192*b1) * q^38 + (252280*b2 + 2944112*b1) * q^39 + (86016*b3 + 86016*b2 + 5591040*b1 - 5591040) * q^40 + (-517890*b3 + 9508296) * q^41 + (345870*b3 + 2096858) * q^43 + (-268800*b3 - 268800*b2 + 5752320*b1 - 5752320) * q^44 + (893361*b2 + 55246065*b1) * q^45 + (-114240*b2 + 638400*b1) * q^46 + (200430*b3 + 200430*b2 - 37271262*b1 + 37271262) * q^47 + (327680*b3 - 458752) * q^48 + (917280*b3 - 14825680) * q^50 + (2164890*b3 + 2164890*b2 - 15278322*b1 + 15278322) * q^51 + (-57600*b2 - 12836096*b1) * q^52 + (464520*b2 + 1619874*b1) * q^53 + (1472480*b3 + 1472480*b2 + 14958496*b1 - 14958496) * q^54 + (-961380*b3 + 20153700) * q^55 + (1177750*b3 + 153288166) * q^57 + (-1911840*b3 - 1911840*b2 - 16050624*b1 + 16050624) * q^58 + (1119735*b2 + 66821181*b1) * q^59 + (-1784832*b2 - 64404480*b1) * q^60 + (-866205*b3 - 866205*b2 + 113900843*b1 - 113900843) * q^61 + (629280*b3 - 17509856) * q^62 + 16777216 * q^64 + (-1360086*b3 - 1360086*b2 - 79333590*b1 + 79333590) * q^65 + (-1680000*b2 + 191103360*b1) * q^66 + (1654380*b2 - 166465136*b1) * q^67 + (407040*b3 + 407040*b2 - 111412224*b1 + 111412224) * q^68 + (149520*b3 + 82009200) * q^69 + (-6323940*b3 - 83992860) * q^71 + (-286720*b3 - 286720*b2 - 155611136*b1 + 155611136) * q^72 + (-6043140*b2 + 22342138*b1) * q^73 + (2288160*b2 - 165788608*b1) * q^74 + (-5034335*b3 - 5034335*b2 - 667214485*b1 + 667214485) * q^75 + (3444480*b3 + 65123072) * q^76 + (-4036480*b3 + 47105792) * q^78 + (-2213820*b3 - 2213820*b2 + 134821388*b1 - 134821388) * q^79 + (1376256*b2 + 89456640*b1) * q^80 + (-3940930*b2 - 319413239*b1) * q^81 + (-8286240*b3 - 8286240*b2 - 152132736*b1 + 152132736) * q^82 + (6075435*b3 - 91552881) * q^83 + (-6968934*b3 + 517089510) * q^85 + (5533920*b3 + 5533920*b2 - 33549728*b1 + 33549728) * q^86 + (5852250*b2 + 1384144398*b1) * q^87 + (-4300800*b2 + 92037120*b1) * q^88 + (6785040*b3 + 6785040*b2 + 395828874*b1 - 395828874) * q^89 + (-14293776*b3 + 883937040) * q^90 + (1827840*b3 + 10214400) * q^92 + (-5747140*b3 - 5747140*b2 - 460938812*b1 + 460938812) * q^93 + (3206880*b2 - 596340192*b1) * q^94 + (-13023948*b2 - 304051020*b1) * q^95 + (5242880*b3 + 5242880*b2 + 7340032*b1 - 7340032) * q^96 + (13989690*b3 - 2084740) * q^97 + (38317650*b3 + 684240270) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{2} + 14 q^{3} - 512 q^{4} + 2730 q^{5} + 448 q^{6} - 16384 q^{8} - 75982 q^{9}+O(q^{10})$$ 4 * q + 32 * q^2 + 14 * q^3 - 512 * q^4 + 2730 * q^5 + 448 * q^6 - 16384 * q^8 - 75982 * q^9 $$4 q + 32 q^{2} + 14 q^{3} - 512 q^{4} + 2730 q^{5} + 448 q^{6} - 16384 q^{8} - 75982 q^{9} - 43680 q^{10} - 44940 q^{11} + 3584 q^{12} + 200564 q^{13} + 1006320 q^{15} - 131072 q^{16} + 870408 q^{17} + 1215712 q^{18} - 508774 q^{19} - 1397760 q^{20} - 1438080 q^{22} - 79800 q^{23} - 57344 q^{24} - 1853210 q^{25} + 1604512 q^{26} - 3739624 q^{27} + 4012656 q^{29} + 8050560 q^{30} - 2188732 q^{31} + 2097152 q^{32} - 23887920 q^{33} + 27853056 q^{34} + 38902784 q^{36} + 20723576 q^{37} + 8140384 q^{38} + 5888224 q^{39} - 11182080 q^{40} + 38033184 q^{41} + 8387432 q^{43} - 11504640 q^{44} + 110492130 q^{45} + 1276800 q^{46} + 74542524 q^{47} - 1835008 q^{48} - 59302720 q^{50} + 30556644 q^{51} - 25672192 q^{52} + 3239748 q^{53} - 29916992 q^{54} + 80614800 q^{55} + 613152664 q^{57} + 32101248 q^{58} + 133642362 q^{59} - 128808960 q^{60} - 227801686 q^{61} - 70039424 q^{62} + 67108864 q^{64} + 158667180 q^{65} + 382206720 q^{66} - 332930272 q^{67} + 222824448 q^{68} + 328036800 q^{69} - 335971440 q^{71} + 311222272 q^{72} + 44684276 q^{73} - 331577216 q^{74} + 1334428970 q^{75} + 260492288 q^{76} + 188423168 q^{78} - 269642776 q^{79} + 178913280 q^{80} - 638826478 q^{81} + 304265472 q^{82} - 366211524 q^{83} + 2068358040 q^{85} + 67099456 q^{86} + 2768288796 q^{87} + 184074240 q^{88} - 791657748 q^{89} + 3535748160 q^{90} + 40857600 q^{92} + 921877624 q^{93} - 1192680384 q^{94} - 608102040 q^{95} - 14680064 q^{96} - 8338960 q^{97} + 2736961080 q^{99}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 577x^{2} + 576x + 331776$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 577\nu^{2} - 577\nu + 331776 ) / 332352$$ (-v^3 + 577*v^2 - 577*v + 331776) / 332352 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 577\nu^{2} + 665281\nu - 331776 ) / 332352$$ (v^3 - 577*v^2 + 665281*v - 331776) / 332352 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + 1729 ) / 577$$ (2*v^3 + 1729) / 577
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 1153\beta _1 - 1153 ) / 2$$ (b3 + b2 + 1153*b1 - 1153) / 2 $$\nu^{3}$$ $$=$$ $$( 577\beta_{3} - 1729 ) / 2$$ (577*b3 - 1729) / 2

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{10}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

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