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 $1$

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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: \( x^{2} - x - 222 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + 16 q^{2} + 256 q^{4} + ( - 97 \beta + 309) q^{5} + (330 \beta - 3910) q^{7} + 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 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}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{2} + 512 q^{4} + 521 q^{5} - 7490 q^{7} + 8192 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 32 q^{2} + 512 q^{4} + 521 q^{5} - 7490 q^{7} + 8192 q^{8} + 8336 q^{10} - 29282 q^{11} + 150314 q^{13} - 119840 q^{14} + 131072 q^{16} - 690472 q^{17} + 511212 q^{19} + 133376 q^{20} - 468512 q^{22} - 874751 q^{23} + 411771 q^{25} + 2405024 q^{26} - 1917440 q^{28} + 2951058 q^{29} - 5818705 q^{31} + 2097152 q^{32} - 11047552 q^{34} - 16179590 q^{35} + 2658905 q^{37} + 8179392 q^{38} + 2134016 q^{40} - 13427994 q^{41} - 17820762 q^{43} - 7496192 q^{44} - 13996016 q^{46} - 56044104 q^{47} - 4251114 q^{49} + 6588336 q^{50} + 38480384 q^{52} - 96842752 q^{53} - 7627961 q^{55} - 30679040 q^{56} + 47216928 q^{58} + 119136183 q^{59} - 90424326 q^{61} - 93099280 q^{62} + 33554432 q^{64} + 18029712 q^{65} - 295944891 q^{67} - 176760832 q^{68} - 258873440 q^{70} + 322953267 q^{71} - 255975514 q^{73} + 42542480 q^{74} + 130870272 q^{76} + 109661090 q^{77} - 889658 q^{79} + 34144256 q^{80} - 214847904 q^{82} + 277699042 q^{83} + 96595042 q^{85} - 285132192 q^{86} - 119939072 q^{88} - 1363672217 q^{89} - 491050280 q^{91} - 223936256 q^{92} - 896705664 q^{94} - 1454033872 q^{95} + 1398434043 q^{97} - 68017824 q^{98}+O(q^{100}) \) 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
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:

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 \(S_{10}^{\mathrm{new}}(\Gamma_0(198))\). Copy content Toggle raw display

Hecke characteristic polynomials

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