Properties

Label 22.14.a.b
Level $22$
Weight $14$
Character orbit 22.a
Self dual yes
Analytic conductor $23.591$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.5908043694\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{55441}) \)
Defining polynomial: \( x^{2} - x - 13860 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{55441}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 64 q^{2} + ( - 3 \beta + 813) q^{3} + 4096 q^{4} + (76 \beta + 3833) q^{5} + (192 \beta - 52032) q^{6} + (33 \beta + 318524) q^{7} - 262144 q^{8} + ( - 4878 \beta + 1062522) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 64 q^{2} + ( - 3 \beta + 813) q^{3} + 4096 q^{4} + (76 \beta + 3833) q^{5} + (192 \beta - 52032) q^{6} + (33 \beta + 318524) q^{7} - 262144 q^{8} + ( - 4878 \beta + 1062522) q^{9} + ( - 4864 \beta - 245312) q^{10} + 1771561 q^{11} + ( - 12288 \beta + 3330048) q^{12} + ( - 8279 \beta - 1894334) q^{13} + ( - 2112 \beta - 20385536) q^{14} + (50289 \beta - 47445963) q^{15} + 16777216 q^{16} + ( - 68063 \beta + 18568652) q^{17} + (312192 \beta - 68001408) q^{18} + (227621 \beta - 51230298) q^{19} + (311296 \beta + 15699968) q^{20} + ( - 928743 \beta + 237005376) q^{21} - 113379904 q^{22} + ( - 644389 \beta + 676373521) q^{23} + (786432 \beta - 213123072) q^{24} + (582616 \beta + 74897628) q^{25} + (529856 \beta + 121237376) q^{26} + ( - 2370411 \beta + 2812940163) q^{27} + (135168 \beta + 1304674304) q^{28} + ( - 4578810 \beta + 3712659060) q^{29} + ( - 3218496 \beta + 3036541632) q^{30} + (6082615 \beta + 4081741297) q^{31} - 1073741824 q^{32} + ( - 5314683 \beta + 1440279093) q^{33} + (4356032 \beta - 1188393728) q^{34} + (24334313 \beta + 1777086604) q^{35} + ( - 19980288 \beta + 4352090112) q^{36} + (43794100 \beta - 6036597797) q^{37} + ( - 14567744 \beta + 3278739072) q^{38} + ( - 1047825 \beta + 3967858926) q^{39} + ( - 19922944 \beta - 1004797952) q^{40} + ( - 9606171 \beta + 36896129790) q^{41} + (59439552 \beta - 15168344064) q^{42} + ( - 101899756 \beta + 10225459842) q^{43} + 7256313856 q^{44} + (62054298 \beta - 78141477366) q^{45} + (41240896 \beta - 43287905344) q^{46} + ( - 171862494 \beta + 35653077300) q^{47} + ( - 50331648 \beta + 13639876608) q^{48} + (21022584 \beta + 4810029165) q^{49} + ( - 37287424 \beta - 4793448192) q^{50} + ( - 111041175 \beta + 60378083472) q^{51} + ( - 33910784 \beta - 7759192064) q^{52} + (234108394 \beta + 154788983702) q^{53} + (151706304 \beta - 180028170432) q^{54} + (134638636 \beta + 6790393313) q^{55} + ( - 8650752 \beta - 83499155456) q^{56} + (338746767 \beta - 193084662606) q^{57} + (293043840 \beta - 237610179840) q^{58} + ( - 292525083 \beta - 201901034541) q^{59} + (205983744 \beta - 194338664448) q^{60} + (787404242 \beta - 40609788504) q^{61} + ( - 389287360 \beta - 261231443008) q^{62} + ( - 1518696846 \beta + 302740519392) q^{63} + 68719476736 q^{64} + ( - 175702791 \beta - 146795778078) q^{65} + (340139712 \beta - 92177861952) q^{66} + ( - 605238211 \beta + 114577816551) q^{67} + ( - 278786048 \beta + 76057198592) q^{68} + ( - 2553008820 \beta + 978598519161) q^{69} + ( - 1557396032 \beta - 113733542656) q^{70} + (2199410247 \beta - 580939457289) q^{71} + (1278738432 \beta - 278533767168) q^{72} + (4969469113 \beta + 228018658690) q^{73} + ( - 2802822400 \beta + 386342259008) q^{74} + (248973924 \beta - 326717992308) q^{75} + (932335616 \beta - 209839300608) q^{76} + (58461513 \beta + 564284695964) q^{77} + (67060800 \beta - 253942971264) q^{78} + (1096805448 \beta + 2520544453526) q^{79} + (1275068416 \beta + 64307068928) q^{80} + ( - 2588857038 \beta + 2169932564925) q^{81} + (614794944 \beta - 2361352306560) q^{82} + (3486617858 \beta + 2805453122470) q^{83} + ( - 3804131328 \beta + 970774020096) q^{84} + (1150332073 \beta - 1075964514916) q^{85} + (6521584384 \beta - 654429429888) q^{86} + ( - 14860549710 \beta + 6064637478300) q^{87} - 464404086784 q^{88} + (8358406502 \beta - 1619665313021) q^{89} + ( - 3971475072 \beta + 5001054551424) q^{90} + ( - 2699573218 \beta - 663978320164) q^{91} + ( - 2639417344 \beta + 2770425942016) q^{92} + ( - 7300057896 \beta - 728259424119) q^{93} + (10999199616 \beta - 2281796947200) q^{94} + ( - 3021031355 \beta + 3639973169510) q^{95} + (3221225472 \beta - 872952102912) q^{96} + ( - 2002648840 \beta - 10270683060087) q^{97} + ( - 1345445376 \beta - 307841866560) q^{98} + ( - 8641674558 \beta + 1882322536842) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{2} + 1626 q^{3} + 8192 q^{4} + 7666 q^{5} - 104064 q^{6} + 637048 q^{7} - 524288 q^{8} + 2125044 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 128 q^{2} + 1626 q^{3} + 8192 q^{4} + 7666 q^{5} - 104064 q^{6} + 637048 q^{7} - 524288 q^{8} + 2125044 q^{9} - 490624 q^{10} + 3543122 q^{11} + 6660096 q^{12} - 3788668 q^{13} - 40771072 q^{14} - 94891926 q^{15} + 33554432 q^{16} + 37137304 q^{17} - 136002816 q^{18} - 102460596 q^{19} + 31399936 q^{20} + 474010752 q^{21} - 226759808 q^{22} + 1352747042 q^{23} - 426246144 q^{24} + 149795256 q^{25} + 242474752 q^{26} + 5625880326 q^{27} + 2609348608 q^{28} + 7425318120 q^{29} + 6073083264 q^{30} + 8163482594 q^{31} - 2147483648 q^{32} + 2880558186 q^{33} - 2376787456 q^{34} + 3554173208 q^{35} + 8704180224 q^{36} - 12073195594 q^{37} + 6557478144 q^{38} + 7935717852 q^{39} - 2009595904 q^{40} + 73792259580 q^{41} - 30336688128 q^{42} + 20450919684 q^{43} + 14512627712 q^{44} - 156282954732 q^{45} - 86575810688 q^{46} + 71306154600 q^{47} + 27279753216 q^{48} + 9620058330 q^{49} - 9586896384 q^{50} + 120756166944 q^{51} - 15518384128 q^{52} + 309577967404 q^{53} - 360056340864 q^{54} + 13580786626 q^{55} - 166998310912 q^{56} - 386169325212 q^{57} - 475220359680 q^{58} - 403802069082 q^{59} - 388677328896 q^{60} - 81219577008 q^{61} - 522462886016 q^{62} + 605481038784 q^{63} + 137438953472 q^{64} - 293591556156 q^{65} - 184355723904 q^{66} + 229155633102 q^{67} + 152114397184 q^{68} + 1957197038322 q^{69} - 227467085312 q^{70} - 1161878914578 q^{71} - 557067534336 q^{72} + 456037317380 q^{73} + 772684518016 q^{74} - 653435984616 q^{75} - 419678601216 q^{76} + 1128569391928 q^{77} - 507885942528 q^{78} + 5041088907052 q^{79} + 128614137856 q^{80} + 4339865129850 q^{81} - 4722704613120 q^{82} + 5610906244940 q^{83} + 1941548040192 q^{84} - 2151929029832 q^{85} - 1308858859776 q^{86} + 12129274956600 q^{87} - 928808173568 q^{88} - 3239330626042 q^{89} + 10002109102848 q^{90} - 1327956640328 q^{91} + 5540851884032 q^{92} - 1456518848238 q^{93} - 4563593894400 q^{94} + 7279946339020 q^{95} - 1745904205824 q^{96} - 20541366120174 q^{97} - 615683733120 q^{98} + 3764645073684 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
118.230
−117.230
−64.0000 −599.755 4096.00 39622.8 38384.3 334064. −262144. −1.23462e6 −2.53586e6
1.2 −64.0000 2225.75 4096.00 −31956.8 −142448. 302984. −262144. 3.35966e6 2.04523e6
\(n\): 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 22.14.a.b 2
3.b odd 2 1 198.14.a.d 2
4.b odd 2 1 176.14.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.14.a.b 2 1.a even 1 1 trivial
176.14.a.a 2 4.b odd 2 1
198.14.a.d 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 1626T_{3} - 1334907 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 1626 T - 1334907 \) Copy content Toggle raw display
$5$ \( T^{2} - 7666 T - 1266216975 \) Copy content Toggle raw display
$7$ \( T^{2} - 637048 T + 101216037580 \) Copy content Toggle raw display
$11$ \( (T - 1771561)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3788668 T - 11611611523968 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 682542853036212 \) Copy content Toggle raw display
$19$ \( T^{2} + 102460596 T - 88\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} - 1352747042 T + 36\!\cdots\!97 \) Copy content Toggle raw display
$29$ \( T^{2} - 7425318120 T + 91\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} - 8163482594 T + 84\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{2} + 12073195594 T - 38\!\cdots\!91 \) Copy content Toggle raw display
$41$ \( T^{2} - 73792259580 T + 13\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{2} - 20450919684 T - 21\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{2} - 71306154600 T - 52\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} - 309577967404 T + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{2} + 403802069082 T + 21\!\cdots\!85 \) Copy content Toggle raw display
$61$ \( T^{2} + 81219577008 T - 13\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{2} - 229155633102 T - 68\!\cdots\!43 \) Copy content Toggle raw display
$71$ \( T^{2} + 1161878914578 T - 73\!\cdots\!55 \) Copy content Toggle raw display
$73$ \( T^{2} - 456037317380 T - 54\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{2} - 5041088907052 T + 60\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{2} - 5610906244940 T + 51\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{2} + 3239330626042 T - 12\!\cdots\!15 \) Copy content Toggle raw display
$97$ \( T^{2} + 20541366120174 T + 10\!\cdots\!69 \) Copy content Toggle raw display
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