Properties

Label 22.14.a.c
Level $22$
Weight $14$
Character orbit 22.a
Self dual yes
Analytic conductor $23.591$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{45769}) \)
Defining polynomial: \( x^{2} - x - 11442 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = 2\sqrt{45769}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 64 q^{2} + ( - \beta - 463) q^{3} + 4096 q^{4} + ( - 48 \beta + 1457) q^{5} + ( - 64 \beta - 29632) q^{6} + (1155 \beta - 85280) q^{7} + 262144 q^{8} + (926 \beta - 1196878) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 64 q^{2} + ( - \beta - 463) q^{3} + 4096 q^{4} + ( - 48 \beta + 1457) q^{5} + ( - 64 \beta - 29632) q^{6} + (1155 \beta - 85280) q^{7} + 262144 q^{8} + (926 \beta - 1196878) q^{9} + ( - 3072 \beta + 93248) q^{10} + 1771561 q^{11} + ( - 4096 \beta - 1896448) q^{12} + ( - 27285 \beta - 3474458) q^{13} + (73920 \beta - 5457920) q^{14} + (20767 \beta + 8113057) q^{15} + 16777216 q^{16} + (26787 \beta - 140072144) q^{17} + (59264 \beta - 76600192) q^{18} + ( - 453753 \beta - 174615894) q^{19} + ( - 196608 \beta + 5967872) q^{20} + ( - 449485 \beta - 171968140) q^{21} + 113379904 q^{22} + (2138661 \beta - 75639147) q^{23} + ( - 262144 \beta - 121372672) q^{24} + ( - 139872 \beta - 796773172) q^{25} + ( - 1746240 \beta - 222365312) q^{26} + (2362463 \beta + 1122797687) q^{27} + (4730880 \beta - 349306880) q^{28} + ( - 6597054 \beta - 385810964) q^{29} + (1329088 \beta + 519235648) q^{30} + ( - 4337487 \beta - 4313241035) q^{31} + 1073741824 q^{32} + ( - 1771561 \beta - 820232743) q^{33} + (1714368 \beta - 8964617216) q^{34} + (5776275 \beta - 10273986400) q^{35} + (3792896 \beta - 4902412288) q^{36} + (16483008 \beta - 13166949245) q^{37} + ( - 29040192 \beta - 11175417216) q^{38} + (16107413 \beta + 6603902714) q^{39} + ( - 12582912 \beta + 381943808) q^{40} + ( - 101787201 \beta - 1780495918) q^{41} + ( - 28767040 \beta - 11005960960) q^{42} + (159822924 \beta - 8164997966) q^{43} + 7256313856 q^{44} + (58799326 \beta - 9881213294) q^{45} + (136874304 \beta - 4840905408) q^{46} + ( - 71686146 \beta + 4756675852) q^{47} + ( - 16777216 \beta - 7767851008) q^{48} + ( - 196996800 \beta + 154611628893) q^{49} + ( - 8951808 \beta - 50993483008) q^{50} + (127669763 \beta + 59949345860) q^{51} + ( - 111759360 \beta - 14231379968) q^{52} + ( - 351052074 \beta + 145681949326) q^{53} + (151197632 \beta + 71859051968) q^{54} + ( - 85034928 \beta + 2581164377) q^{55} + (302776320 \beta - 22355640320) q^{56} + (384703533 \beta + 163918443150) q^{57} + ( - 422211456 \beta - 24691901696) q^{58} + (19558743 \beta + 356009505591) q^{59} + (85061632 \beta + 33231081472) q^{60} + (1248591462 \beta + 69459291472) q^{61} + ( - 277599168 \beta - 276047426240) q^{62} + ( - 1461363370 \beta + 297875030120) q^{63} + 68719476736 q^{64} + (127019739 \beta + 234708690374) q^{65} + ( - 113379904 \beta - 52494895552) q^{66} + ( - 357146649 \beta + 456299597787) q^{67} + (109719552 \beta - 573735501824) q^{68} + ( - 914560896 \beta - 356516576175) q^{69} + (369681600 \beta - 657535129600) q^{70} + (1042775529 \beta - 780424171861) q^{71} + (242745344 \beta - 313754386432) q^{72} + (1766329659 \beta + 203691208998) q^{73} + (1054912512 \beta - 842684751680) q^{74} + (861533908 \beta + 394513184908) q^{75} + ( - 1858572288 \beta - 715226701824) q^{76} + (2046152955 \beta - 151078722080) q^{77} + (1030874432 \beta + 422649773696) q^{78} + ( - 2887412976 \beta - 24051338234) q^{79} + ( - 805306368 \beta + 24444403712) q^{80} + ( - 3692961154 \beta + 955844518325) q^{81} + ( - 6514380864 \beta - 113951738752) q^{82} + ( - 3544871850 \beta - 3156910275506) q^{83} + ( - 1841090560 \beta - 704381501440) q^{84} + (6762491571 \beta - 439479840784) q^{85} + (10228667136 \beta - 522559869824) q^{86} + (3440246966 \beta + 1386392734436) q^{87} + 464404086784 q^{88} + ( - 12836101782 \beta + 488846162731) q^{89} + (3763156864 \beta - 632397650816) q^{90} + ( - 1686134190 \beta - 5473187324060) q^{91} + (8759955456 \beta - 309817946112) q^{92} + (6321497516 \beta + 2791120369217) q^{93} + ( - 4587913344 \beta + 304427254528) q^{94} + (7720444791 \beta + 3733006285386) q^{95} + ( - 1073741824 \beta - 497142464512) q^{96} + (12244042032 \beta - 512488651751) q^{97} + ( - 12607795200 \beta + 9895144249152) q^{98} + (1640465486 \beta - 2120342386558) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 128 q^{2} - 926 q^{3} + 8192 q^{4} + 2914 q^{5} - 59264 q^{6} - 170560 q^{7} + 524288 q^{8} - 2393756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 128 q^{2} - 926 q^{3} + 8192 q^{4} + 2914 q^{5} - 59264 q^{6} - 170560 q^{7} + 524288 q^{8} - 2393756 q^{9} + 186496 q^{10} + 3543122 q^{11} - 3792896 q^{12} - 6948916 q^{13} - 10915840 q^{14} + 16226114 q^{15} + 33554432 q^{16} - 280144288 q^{17} - 153200384 q^{18} - 349231788 q^{19} + 11935744 q^{20} - 343936280 q^{21} + 226759808 q^{22} - 151278294 q^{23} - 242745344 q^{24} - 1593546344 q^{25} - 444730624 q^{26} + 2245595374 q^{27} - 698613760 q^{28} - 771621928 q^{29} + 1038471296 q^{30} - 8626482070 q^{31} + 2147483648 q^{32} - 1640465486 q^{33} - 17929234432 q^{34} - 20547972800 q^{35} - 9804824576 q^{36} - 26333898490 q^{37} - 22350834432 q^{38} + 13207805428 q^{39} + 763887616 q^{40} - 3560991836 q^{41} - 22011921920 q^{42} - 16329995932 q^{43} + 14512627712 q^{44} - 19762426588 q^{45} - 9681810816 q^{46} + 9513351704 q^{47} - 15535702016 q^{48} + 309223257786 q^{49} - 101986966016 q^{50} + 119898691720 q^{51} - 28462759936 q^{52} + 291363898652 q^{53} + 143718103936 q^{54} + 5162328754 q^{55} - 44711280640 q^{56} + 327836886300 q^{57} - 49383803392 q^{58} + 712019011182 q^{59} + 66462162944 q^{60} + 138918582944 q^{61} - 552094852480 q^{62} + 595750060240 q^{63} + 137438953472 q^{64} + 469417380748 q^{65} - 104989791104 q^{66} + 912599195574 q^{67} - 1147471003648 q^{68} - 713033152350 q^{69} - 1315070259200 q^{70} - 1560848343722 q^{71} - 627508772864 q^{72} + 407382417996 q^{73} - 1685369503360 q^{74} + 789026369816 q^{75} - 1430453403648 q^{76} - 302157444160 q^{77} + 845299547392 q^{78} - 48102676468 q^{79} + 48888807424 q^{80} + 1911689036650 q^{81} - 227903477504 q^{82} - 6313820551012 q^{83} - 1408763002880 q^{84} - 878959681568 q^{85} - 1045119739648 q^{86} + 2772785468872 q^{87} + 928808173568 q^{88} + 977692325462 q^{89} - 1264795301632 q^{90} - 10946374648120 q^{91} - 619635892224 q^{92} + 5582240738434 q^{93} + 608854509056 q^{94} + 7466012570772 q^{95} - 994284929024 q^{96} - 1024977303502 q^{97} + 19790288498304 q^{98} - 4240684773116 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
107.468
−106.468
64.0000 −890.874 4096.00 −19080.9 −57015.9 408914. 262144. −800667. −1.22118e6
1.2 64.0000 −35.1262 4096.00 21994.9 −2248.08 −579474. 262144. −1.59309e6 1.40768e6
\(n\): 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.c 2
3.b odd 2 1 198.14.a.a 2
4.b odd 2 1 176.14.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.14.a.c 2 1.a even 1 1 trivial
176.14.a.c 2 4.b odd 2 1
198.14.a.a 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 926T_{3} + 31293 \) 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
$3$ \( T^{2} + 926T + 31293 \) Copy content Toggle raw display
$5$ \( T^{2} - 2914 T - 419684255 \) Copy content Toggle raw display
$7$ \( T^{2} + 170560 T - 236955282500 \) Copy content Toggle raw display
$11$ \( (T - 1771561)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 124222955594336 \) Copy content Toggle raw display
$17$ \( T^{2} + 280144288 T + 19\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{2} + 349231788 T - 72\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( T^{2} + 151278294 T - 83\!\cdots\!87 \) Copy content Toggle raw display
$29$ \( T^{2} + 771621928 T - 78\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{2} + 8626482070 T + 15\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{2} + 26333898490 T + 12\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{2} + 3560991836 T - 18\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{2} + 16329995932 T - 46\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{2} - 9513351704 T - 91\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{2} - 291363898652 T - 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{2} - 712019011182 T + 12\!\cdots\!57 \) Copy content Toggle raw display
$61$ \( T^{2} - 138918582944 T - 28\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{2} - 912599195574 T + 18\!\cdots\!93 \) Copy content Toggle raw display
$71$ \( T^{2} + 1560848343722 T + 40\!\cdots\!05 \) Copy content Toggle raw display
$73$ \( T^{2} - 407382417996 T - 52\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{2} + 48102676468 T - 15\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{2} + 6313820551012 T + 76\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{2} - 977692325462 T - 29\!\cdots\!63 \) Copy content Toggle raw display
$97$ \( T^{2} + 1024977303502 T - 27\!\cdots\!23 \) Copy content Toggle raw display
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