# Properties

 Label 198.10.a.n Level $198$ Weight $10$ Character orbit 198.a Self dual yes Analytic conductor $101.977$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$198 = 2 \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 198.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$101.977095560$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{889})$$ Defining polynomial: $$x^{2} - x - 222$$ x^2 - x - 222 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{889})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + 256 q^{4} + ( - 97 \beta + 309) q^{5} + (330 \beta - 3910) q^{7} + 4096 q^{8}+O(q^{10})$$ q + 16 * q^2 + 256 * q^4 + (-97*b + 309) * q^5 + (330*b - 3910) * q^7 + 4096 * q^8 $$q + 16 q^{2} + 256 q^{4} + ( - 97 \beta + 309) q^{5} + (330 \beta - 3910) q^{7} + 4096 q^{8} + ( - 1552 \beta + 4944) q^{10} - 14641 q^{11} + (490 \beta + 74912) q^{13} + (5280 \beta - 62560) q^{14} + 65536 q^{16} + ( - 6412 \beta - 342030) q^{17} + (36812 \beta + 237200) q^{19} + ( - 24832 \beta + 79104) q^{20} - 234256 q^{22} + (44659 \beta - 459705) q^{23} + ( - 50537 \beta + 231154) q^{25} + (7840 \beta + 1198592) q^{26} + (84480 \beta - 1000960) q^{28} + ( - 252246 \beta + 1601652) q^{29} + (367213 \beta - 3092959) q^{31} + 1048576 q^{32} + ( - 102592 \beta - 5472480) q^{34} + (449230 \beta - 8314410) q^{35} + ( - 403577 \beta + 1531241) q^{37} + (588992 \beta + 3795200) q^{38} + ( - 397312 \beta + 1265664) q^{40} + (228846 \beta - 6828420) q^{41} + (1121834 \beta - 9471298) q^{43} - 3748096 q^{44} + (714544 \beta - 7355280) q^{46} + (2071176 \beta - 29057640) q^{47} + ( - 2471700 \beta - 889707) q^{49} + ( - 808592 \beta + 3698464) q^{50} + (125440 \beta + 19177472) q^{52} + (2579204 \beta - 49710978) q^{53} + (1420177 \beta - 4524069) q^{55} + (1351680 \beta - 16015360) q^{56} + ( - 4035936 \beta + 25626432) q^{58} + ( - 8232723 \beta + 63684453) q^{59} + ( - 11223238 \beta - 39600544) q^{61} + (5875408 \beta - 49487344) q^{62} + 16777216 q^{64} + ( - 7162584 \beta + 12596148) q^{65} + ( - 5360809 \beta - 145292041) q^{67} + ( - 1641472 \beta - 87559680) q^{68} + (7187680 \beta - 133030560) q^{70} + ( - 393879 \beta + 161673573) q^{71} + (1597174 \beta - 128786344) q^{73} + ( - 6457232 \beta + 24499856) q^{74} + (9423872 \beta + 60723200) q^{76} + ( - 4831530 \beta + 57246310) q^{77} + ( - 19867086 \beta + 9488714) q^{79} + ( - 6356992 \beta + 20250624) q^{80} + (3661536 \beta - 109254720) q^{82} + (40242910 \beta + 118728066) q^{83} + (31817566 \beta + 32388738) q^{85} + (17949344 \beta - 151540768) q^{86} - 59969536 q^{88} + ( - 14637263 \beta - 674517477) q^{89} + (22966760 \beta - 257008520) q^{91} + (11432704 \beta - 117684480) q^{92} + (33138816 \beta - 464922240) q^{94} + ( - 15204256 \beta - 719414808) q^{95} + (39733637 \beta + 679350203) q^{97} + ( - 39547200 \beta - 14235312) q^{98}+O(q^{100})$$ q + 16 * q^2 + 256 * q^4 + (-97*b + 309) * q^5 + (330*b - 3910) * q^7 + 4096 * q^8 + (-1552*b + 4944) * q^10 - 14641 * q^11 + (490*b + 74912) * q^13 + (5280*b - 62560) * q^14 + 65536 * q^16 + (-6412*b - 342030) * q^17 + (36812*b + 237200) * q^19 + (-24832*b + 79104) * q^20 - 234256 * q^22 + (44659*b - 459705) * q^23 + (-50537*b + 231154) * q^25 + (7840*b + 1198592) * q^26 + (84480*b - 1000960) * q^28 + (-252246*b + 1601652) * q^29 + (367213*b - 3092959) * q^31 + 1048576 * q^32 + (-102592*b - 5472480) * q^34 + (449230*b - 8314410) * q^35 + (-403577*b + 1531241) * q^37 + (588992*b + 3795200) * q^38 + (-397312*b + 1265664) * q^40 + (228846*b - 6828420) * q^41 + (1121834*b - 9471298) * q^43 - 3748096 * q^44 + (714544*b - 7355280) * q^46 + (2071176*b - 29057640) * q^47 + (-2471700*b - 889707) * q^49 + (-808592*b + 3698464) * q^50 + (125440*b + 19177472) * q^52 + (2579204*b - 49710978) * q^53 + (1420177*b - 4524069) * q^55 + (1351680*b - 16015360) * q^56 + (-4035936*b + 25626432) * q^58 + (-8232723*b + 63684453) * q^59 + (-11223238*b - 39600544) * q^61 + (5875408*b - 49487344) * q^62 + 16777216 * q^64 + (-7162584*b + 12596148) * q^65 + (-5360809*b - 145292041) * q^67 + (-1641472*b - 87559680) * q^68 + (7187680*b - 133030560) * q^70 + (-393879*b + 161673573) * q^71 + (1597174*b - 128786344) * q^73 + (-6457232*b + 24499856) * q^74 + (9423872*b + 60723200) * q^76 + (-4831530*b + 57246310) * q^77 + (-19867086*b + 9488714) * q^79 + (-6356992*b + 20250624) * q^80 + (3661536*b - 109254720) * q^82 + (40242910*b + 118728066) * q^83 + (31817566*b + 32388738) * q^85 + (17949344*b - 151540768) * q^86 - 59969536 * q^88 + (-14637263*b - 674517477) * q^89 + (22966760*b - 257008520) * q^91 + (11432704*b - 117684480) * q^92 + (33138816*b - 464922240) * q^94 + (-15204256*b - 719414808) * q^95 + (39733637*b + 679350203) * q^97 + (-39547200*b - 14235312) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 32 q^{2} + 512 q^{4} + 521 q^{5} - 7490 q^{7} + 8192 q^{8}+O(q^{10})$$ 2 * q + 32 * q^2 + 512 * q^4 + 521 * q^5 - 7490 * q^7 + 8192 * q^8 $$2 q + 32 q^{2} + 512 q^{4} + 521 q^{5} - 7490 q^{7} + 8192 q^{8} + 8336 q^{10} - 29282 q^{11} + 150314 q^{13} - 119840 q^{14} + 131072 q^{16} - 690472 q^{17} + 511212 q^{19} + 133376 q^{20} - 468512 q^{22} - 874751 q^{23} + 411771 q^{25} + 2405024 q^{26} - 1917440 q^{28} + 2951058 q^{29} - 5818705 q^{31} + 2097152 q^{32} - 11047552 q^{34} - 16179590 q^{35} + 2658905 q^{37} + 8179392 q^{38} + 2134016 q^{40} - 13427994 q^{41} - 17820762 q^{43} - 7496192 q^{44} - 13996016 q^{46} - 56044104 q^{47} - 4251114 q^{49} + 6588336 q^{50} + 38480384 q^{52} - 96842752 q^{53} - 7627961 q^{55} - 30679040 q^{56} + 47216928 q^{58} + 119136183 q^{59} - 90424326 q^{61} - 93099280 q^{62} + 33554432 q^{64} + 18029712 q^{65} - 295944891 q^{67} - 176760832 q^{68} - 258873440 q^{70} + 322953267 q^{71} - 255975514 q^{73} + 42542480 q^{74} + 130870272 q^{76} + 109661090 q^{77} - 889658 q^{79} + 34144256 q^{80} - 214847904 q^{82} + 277699042 q^{83} + 96595042 q^{85} - 285132192 q^{86} - 119939072 q^{88} - 1363672217 q^{89} - 491050280 q^{91} - 223936256 q^{92} - 896705664 q^{94} - 1454033872 q^{95} + 1398434043 q^{97} - 68017824 q^{98}+O(q^{100})$$ 2 * q + 32 * q^2 + 512 * q^4 + 521 * q^5 - 7490 * q^7 + 8192 * q^8 + 8336 * q^10 - 29282 * q^11 + 150314 * q^13 - 119840 * q^14 + 131072 * q^16 - 690472 * q^17 + 511212 * q^19 + 133376 * q^20 - 468512 * q^22 - 874751 * q^23 + 411771 * q^25 + 2405024 * q^26 - 1917440 * q^28 + 2951058 * q^29 - 5818705 * q^31 + 2097152 * q^32 - 11047552 * q^34 - 16179590 * q^35 + 2658905 * q^37 + 8179392 * q^38 + 2134016 * q^40 - 13427994 * q^41 - 17820762 * q^43 - 7496192 * q^44 - 13996016 * q^46 - 56044104 * q^47 - 4251114 * q^49 + 6588336 * q^50 + 38480384 * q^52 - 96842752 * q^53 - 7627961 * q^55 - 30679040 * q^56 + 47216928 * q^58 + 119136183 * q^59 - 90424326 * q^61 - 93099280 * q^62 + 33554432 * q^64 + 18029712 * q^65 - 295944891 * q^67 - 176760832 * q^68 - 258873440 * q^70 + 322953267 * q^71 - 255975514 * q^73 + 42542480 * q^74 + 130870272 * q^76 + 109661090 * q^77 - 889658 * q^79 + 34144256 * q^80 - 214847904 * q^82 + 277699042 * q^83 + 96595042 * q^85 - 285132192 * q^86 - 119939072 * q^88 - 1363672217 * q^89 - 491050280 * q^91 - 223936256 * q^92 - 896705664 * q^94 - 1454033872 * q^95 + 1398434043 * q^97 - 68017824 * q^98

## 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
 15.4081 −14.4081
16.0000 0 256.000 −1185.58 0 1174.66 4096.00 0 −18969.3
1.2 16.0000 0 256.000 1706.58 0 −8664.66 4096.00 0 27305.3
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 198.10.a.n 2
3.b odd 2 1 22.10.a.d 2
12.b even 2 1 176.10.a.e 2
33.d even 2 1 242.10.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.a.d 2 3.b odd 2 1
176.10.a.e 2 12.b even 2 1
198.10.a.n 2 1.a even 1 1 trivial
242.10.a.e 2 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 521T_{5} - 2023290$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(198))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 16)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 521 T - 2023290$$
$7$ $$T^{2} + 7490 T - 10178000$$
$11$ $$(T + 14641)^{2}$$
$13$ $$T^{2} - 150314 T + 5595212424$$
$17$ $$T^{2} + 690472 T + 110050366092$$
$19$ $$T^{2} - 511212 T - 235841735968$$
$23$ $$T^{2} + 874751 T - 251963912952$$
$29$ $$T^{2} - 2951058 T - 11964147063840$$
$31$ $$T^{2} + 5818705 T - 21505055373504$$
$37$ $$T^{2} - 2658905 T - 34431390323214$$
$41$ $$T^{2} + 13427994 T + 33438413932128$$
$43$ $$T^{2} + \cdots - 200309296545160$$
$47$ $$T^{2} + \cdots - 168165989315712$$
$53$ $$T^{2} + \cdots + 866157473672220$$
$59$ $$T^{2} - 119136183 T - 11\!\cdots\!48$$
$61$ $$T^{2} + 90424326 T - 25\!\cdots\!60$$
$67$ $$T^{2} + 295944891 T + 15\!\cdots\!68$$
$71$ $$T^{2} - 322953267 T + 26\!\cdots\!60$$
$73$ $$T^{2} + 255975514 T + 15\!\cdots\!08$$
$79$ $$T^{2} + 889658 T - 87\!\cdots\!20$$
$83$ $$T^{2} - 277699042 T - 34\!\cdots\!84$$
$89$ $$T^{2} + 1363672217 T + 41\!\cdots\!62$$
$97$ $$T^{2} - 1398434043 T + 13\!\cdots\!02$$