Properties

Label 22.10.a.d
Level $22$
Weight $10$
Character orbit 22.a
Self dual yes
Analytic conductor $11.331$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.3307883956\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{889}) \)
Defining polynomial: \( x^{2} - x - 222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - 16 q^{2} + ( - 9 \beta - 6) q^{3} + 256 q^{4} + ( - 97 \beta - 212) q^{5} + (144 \beta + 96) q^{6} + ( - 330 \beta - 3580) q^{7} - 4096 q^{8} + (189 \beta - 1665) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} + ( - 9 \beta - 6) q^{3} + 256 q^{4} + ( - 97 \beta - 212) q^{5} + (144 \beta + 96) q^{6} + ( - 330 \beta - 3580) q^{7} - 4096 q^{8} + (189 \beta - 1665) q^{9} + (1552 \beta + 3392) q^{10} + 14641 q^{11} + ( - 2304 \beta - 1536) q^{12} + ( - 490 \beta + 75402) q^{13} + (5280 \beta + 57280) q^{14} + (3363 \beta + 195078) q^{15} + 65536 q^{16} + ( - 6412 \beta + 348442) q^{17} + ( - 3024 \beta + 26640) q^{18} + ( - 36812 \beta + 274012) q^{19} + ( - 24832 \beta - 54272) q^{20} + (37170 \beta + 680820) q^{21} - 234256 q^{22} + (44659 \beta + 415046) q^{23} + (36864 \beta + 24576) q^{24} + (50537 \beta + 180617) q^{25} + (7840 \beta - 1206432) q^{26} + (189297 \beta - 249534) q^{27} + ( - 84480 \beta - 916480) q^{28} + ( - 252246 \beta - 1349406) q^{29} + ( - 53808 \beta - 3121248) q^{30} + ( - 367213 \beta - 2725746) q^{31} - 1048576 q^{32} + ( - 131769 \beta - 87846) q^{33} + (102592 \beta - 5575072) q^{34} + (449230 \beta + 7865180) q^{35} + (48384 \beta - 426240) q^{36} + (403577 \beta + 1127664) q^{37} + (588992 \beta - 4384192) q^{38} + ( - 671268 \beta + 526608) q^{39} + (397312 \beta + 868352) q^{40} + (228846 \beta + 6599574) q^{41} + ( - 594720 \beta - 10893120) q^{42} + ( - 1121834 \beta - 8349464) q^{43} + 3748096 q^{44} + (103104 \beta - 3716946) q^{45} + ( - 714544 \beta - 6640736) q^{46} + (2071176 \beta + 26986464) q^{47} + ( - 589824 \beta - 393216) q^{48} + (2471700 \beta - 3361407) q^{49} + ( - 808592 \beta - 2889872) q^{50} + ( - 3039798 \beta + 10720524) q^{51} + ( - 125440 \beta + 19302912) q^{52} + (2579204 \beta + 47131774) q^{53} + ( - 3028752 \beta + 3992544) q^{54} + ( - 1420177 \beta - 3103892) q^{55} + (1351680 \beta + 14663680) q^{56} + ( - 1913928 \beta + 71906304) q^{57} + (4035936 \beta + 21590496) q^{58} + ( - 8232723 \beta - 55451730) q^{59} + (860928 \beta + 49939968) q^{60} + (11223238 \beta - 50823782) q^{61} + (5875408 \beta + 43611936) q^{62} + ( - 189540 \beta - 7885440) q^{63} + 16777216 q^{64} + ( - 7162584 \beta - 5433564) q^{65} + (2108304 \beta + 1405536) q^{66} + (5360809 \beta - 150652850) q^{67} + ( - 1641472 \beta + 89201152) q^{68} + ( - 4405299 \beta - 91718958) q^{69} + ( - 7187680 \beta - 125842880) q^{70} + ( - 393879 \beta - 161279694) q^{71} + ( - 774144 \beta + 6819840) q^{72} + ( - 1597174 \beta - 127189170) q^{73} + ( - 6457232 \beta - 18042624) q^{74} + ( - 2383608 \beta - 102056628) q^{75} + ( - 9423872 \beta + 70147072) q^{76} + ( - 4831530 \beta - 52414780) q^{77} + (10740288 \beta - 8425728) q^{78} + (19867086 \beta - 10378372) q^{79} + ( - 6356992 \beta - 13893632) q^{80} + ( - 4313736 \beta - 343946007) q^{81} + ( - 3661536 \beta - 105593184) q^{82} + (40242910 \beta - 158970976) q^{83} + (9515520 \beta + 174289920) q^{84} + ( - 31817566 \beta + 64206304) q^{85} + (17949344 \beta + 133591424) q^{86} + (15928344 \beta + 512083944) q^{87} - 59969536 q^{88} + ( - 14637263 \beta + 689154740) q^{89} + ( - 1649664 \beta + 59471136) q^{90} + ( - 22966760 \beta - 234041760) q^{91} + (11432704 \beta + 106251776) q^{92} + (30039909 \beta + 750046050) q^{93} + ( - 33138816 \beta - 431783424) q^{94} + ( - 15204256 \beta + 734619064) q^{95} + (9437184 \beta + 6291456) q^{96} + ( - 39733637 \beta + 719083840) q^{97} + ( - 39547200 \beta + 53782512) q^{98} + (2767149 \beta - 24377265) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} - 21 q^{3} + 512 q^{4} - 521 q^{5} + 336 q^{6} - 7490 q^{7} - 8192 q^{8} - 3141 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{2} - 21 q^{3} + 512 q^{4} - 521 q^{5} + 336 q^{6} - 7490 q^{7} - 8192 q^{8} - 3141 q^{9} + 8336 q^{10} + 29282 q^{11} - 5376 q^{12} + 150314 q^{13} + 119840 q^{14} + 393519 q^{15} + 131072 q^{16} + 690472 q^{17} + 50256 q^{18} + 511212 q^{19} - 133376 q^{20} + 1398810 q^{21} - 468512 q^{22} + 874751 q^{23} + 86016 q^{24} + 411771 q^{25} - 2405024 q^{26} - 309771 q^{27} - 1917440 q^{28} - 2951058 q^{29} - 6296304 q^{30} - 5818705 q^{31} - 2097152 q^{32} - 307461 q^{33} - 11047552 q^{34} + 16179590 q^{35} - 804096 q^{36} + 2658905 q^{37} - 8179392 q^{38} + 381948 q^{39} + 2134016 q^{40} + 13427994 q^{41} - 22380960 q^{42} - 17820762 q^{43} + 7496192 q^{44} - 7330788 q^{45} - 13996016 q^{46} + 56044104 q^{47} - 1376256 q^{48} - 4251114 q^{49} - 6588336 q^{50} + 18401250 q^{51} + 38480384 q^{52} + 96842752 q^{53} + 4956336 q^{54} - 7627961 q^{55} + 30679040 q^{56} + 141898680 q^{57} + 47216928 q^{58} - 119136183 q^{59} + 100740864 q^{60} - 90424326 q^{61} + 93099280 q^{62} - 15960420 q^{63} + 33554432 q^{64} - 18029712 q^{65} + 4919376 q^{66} - 295944891 q^{67} + 176760832 q^{68} - 187843215 q^{69} - 258873440 q^{70} - 322953267 q^{71} + 12865536 q^{72} - 255975514 q^{73} - 42542480 q^{74} - 206496864 q^{75} + 130870272 q^{76} - 109661090 q^{77} - 6111168 q^{78} - 889658 q^{79} - 34144256 q^{80} - 692205750 q^{81} - 214847904 q^{82} - 277699042 q^{83} + 358095360 q^{84} + 96595042 q^{85} + 285132192 q^{86} + 1040096232 q^{87} - 119939072 q^{88} + 1363672217 q^{89} + 117292608 q^{90} - 491050280 q^{91} + 223936256 q^{92} + 1530132009 q^{93} - 896705664 q^{94} + 1454033872 q^{95} + 22020096 q^{96} + 1398434043 q^{97} + 68017824 q^{98} - 45987381 q^{99}+O(q^{100}) \) 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
15.4081
−14.4081
−16.0000 −144.672 256.000 −1706.58 2314.76 −8664.66 −4096.00 1247.12 27305.3
1.2 −16.0000 123.672 256.000 1185.58 −1978.76 1174.66 −4096.00 −4388.12 −18969.3
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)

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 22.10.a.d 2
3.b odd 2 1 198.10.a.n 2
4.b odd 2 1 176.10.a.e 2
11.b odd 2 1 242.10.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.a.d 2 1.a even 1 1 trivial
176.10.a.e 2 4.b odd 2 1
198.10.a.n 2 3.b odd 2 1
242.10.a.e 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 21T_{3} - 17892 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 21T - 17892 \) 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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