Properties

Label 242.10.a.e
Level $242$
Weight $10$
Character orbit 242.a
Self dual yes
Analytic conductor $124.639$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 242.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(124.638672352\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{889}) \)
Defining polynomial: \( x^{2} - x - 222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 16 q^{2} + ( - 9 \beta - 6) q^{3} + 256 q^{4} + ( - 97 \beta - 212) q^{5} + ( - 144 \beta - 96) q^{6} + (330 \beta + 3580) q^{7} + 4096 q^{8} + (189 \beta - 1665) q^{9}+O(q^{10}) \) 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} + ( - 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} + (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} + (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} + (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} + (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} + (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} + (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} + ( - 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}+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 \( 2 q + 32 q^{2} - 21 q^{3} + 512 q^{4} - 521 q^{5} - 336 q^{6} + 7490 q^{7} + 8192 q^{8} - 3141 q^{9} - 8336 q^{10} - 5376 q^{12} - 150314 q^{13} + 119840 q^{14} + 393519 q^{15} + 131072 q^{16} - 690472 q^{17} - 50256 q^{18} - 511212 q^{19} - 133376 q^{20} - 1398810 q^{21} + 874751 q^{23} - 86016 q^{24} + 411771 q^{25} - 2405024 q^{26} - 309771 q^{27} + 1917440 q^{28} + 2951058 q^{29} + 6296304 q^{30} - 5818705 q^{31} + 2097152 q^{32} - 11047552 q^{34} - 16179590 q^{35} - 804096 q^{36} + 2658905 q^{37} - 8179392 q^{38} - 381948 q^{39} - 2134016 q^{40} - 13427994 q^{41} - 22380960 q^{42} + 17820762 q^{43} - 7330788 q^{45} + 13996016 q^{46} + 56044104 q^{47} - 1376256 q^{48} - 4251114 q^{49} + 6588336 q^{50} - 18401250 q^{51} - 38480384 q^{52} + 96842752 q^{53} - 4956336 q^{54} + 30679040 q^{56} - 141898680 q^{57} + 47216928 q^{58} - 119136183 q^{59} + 100740864 q^{60} + 90424326 q^{61} - 93099280 q^{62} + 15960420 q^{63} + 33554432 q^{64} + 18029712 q^{65} - 295944891 q^{67} - 176760832 q^{68} - 187843215 q^{69} - 258873440 q^{70} - 322953267 q^{71} - 12865536 q^{72} + 255975514 q^{73} + 42542480 q^{74} - 206496864 q^{75} - 130870272 q^{76} - 6111168 q^{78} + 889658 q^{79} - 34144256 q^{80} - 692205750 q^{81} - 214847904 q^{82} + 277699042 q^{83} - 358095360 q^{84} - 96595042 q^{85} + 285132192 q^{86} - 1040096232 q^{87} + 1363672217 q^{89} - 117292608 q^{90} - 491050280 q^{91} + 223936256 q^{92} + 1530132009 q^{93} + 896705664 q^{94} - 1454033872 q^{95} - 22020096 q^{96} + 1398434043 q^{97} - 68017824 q^{98}+O(q^{100}) \) 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 242.10.a.e 2
11.b odd 2 1 22.10.a.d 2
33.d even 2 1 198.10.a.n 2
44.c even 2 1 176.10.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.a.d 2 11.b odd 2 1
176.10.a.e 2 44.c even 2 1
198.10.a.n 2 33.d even 2 1
242.10.a.e 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(242))\):

\( T_{3}^{2} + 21T_{3} - 17892 \) Copy content Toggle raw display
\( T_{7}^{2} - 7490T_{7} - 10178000 \) 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
$11$ \( T^{2} \) 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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