Properties

Label 2.34.a.b
Level 2
Weight 34
Character orbit 2.a
Self dual yes
Analytic conductor 13.797
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.7965657762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 19957422\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 10560\sqrt{79829689}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 65536 q^{2} + ( 4178244 - \beta ) q^{3} + 4294967296 q^{4} + ( -2666238330 - 3996 \beta ) q^{5} + ( 273825398784 - 65536 \beta ) q^{6} + ( 66359547937928 + 896238 \beta ) q^{7} + 281474976710656 q^{8} + ( 3360493163638413 - 8356488 \beta ) q^{9} +O(q^{10})\) \( q +65536 q^{2} +(4178244 - \beta) q^{3} +4294967296 q^{4} +(-2666238330 - 3996 \beta) q^{5} +(273825398784 - 65536 \beta) q^{6} +(66359547937928 + 896238 \beta) q^{7} +281474976710656 q^{8} +(3360493163638413 - 8356488 \beta) q^{9} +(-174734595194880 - 261881856 \beta) q^{10} +(-79052735711361588 + 2213453277 \beta) q^{11} +(17945421334708224 - 4294967296 \beta) q^{12} +(2491699257749546894 - 13597194492 \beta) q^{13} +(4348939333660049408 + 58735853568 \beta) q^{14} +(35561635450747625880 - 14030024694 \beta) q^{15} +18446744073709551616 q^{16} +(\)\(15\!\cdots\!38\)\( - 833438065608 \beta) q^{17} +(\)\(22\!\cdots\!68\)\( - 547650797568 \beta) q^{18} +(\)\(11\!\cdots\!00\)\( + 13790479962267 \beta) q^{19} +(-11451406430691655680 - 17162689314816 \beta) q^{20} +(-\)\(77\!\cdots\!68\)\( - 62614846891856 \beta) q^{21} +(-\)\(51\!\cdots\!68\)\( + 145060873961472 \beta) q^{22} +(-\)\(35\!\cdots\!76\)\( + 63128735356554 \beta) q^{23} +(\)\(11\!\cdots\!64\)\( - 281474976710656 \beta) q^{24} +(\)\(25\!\cdots\!75\)\( + 21308576733360 \beta) q^{25} +(\)\(16\!\cdots\!84\)\( - 891105738227712 \beta) q^{26} +(\)\(65\!\cdots\!60\)\( + 2163651957070038 \beta) q^{27} +(\)\(28\!\cdots\!88\)\( + 3849312899432448 \beta) q^{28} +(-\)\(12\!\cdots\!10\)\( - 10405860886172556 \beta) q^{29} +(\)\(23\!\cdots\!80\)\( - 919471698345984 \beta) q^{30} +(-\)\(21\!\cdots\!48\)\( + 2188390162942584 \beta) q^{31} +\)\(12\!\cdots\!76\)\( q^{32} +(-\)\(20\!\cdots\!72\)\( + 88301083585267176 \beta) q^{33} +(\)\(99\!\cdots\!68\)\( - 54620197067685888 \beta) q^{34} +(-\)\(32\!\cdots\!40\)\( - 267562337668362828 \beta) q^{35} +(\)\(14\!\cdots\!48\)\( - 35890842669416448 \beta) q^{36} +(\)\(58\!\cdots\!38\)\( + 704679249787856532 \beta) q^{37} +(\)\(73\!\cdots\!00\)\( + 903772894807130112 \beta) q^{38} +(\)\(13\!\cdots\!36\)\( - 2548511654052578942 \beta) q^{39} +(-\)\(75\!\cdots\!80\)\( - 1124774006935781376 \beta) q^{40} +(-\)\(83\!\cdots\!38\)\( + 5060866057679897136 \beta) q^{41} +(-\)\(50\!\cdots\!48\)\( - 4103526605904674816 \beta) q^{42} +(-\)\(18\!\cdots\!76\)\( + 795637191126404277 \beta) q^{43} +(-\)\(33\!\cdots\!48\)\( + 9506709435939028992 \beta) q^{44} +(\)\(28\!\cdots\!10\)\( - 13406250293289313308 \beta) q^{45} +(-\)\(23\!\cdots\!36\)\( + 4137204800327122944 \beta) q^{46} +(\)\(14\!\cdots\!28\)\( - 18059171324272980012 \beta) q^{47} +(\)\(77\!\cdots\!04\)\( - 18446744073709551616 \beta) q^{48} +(\)\(38\!\cdots\!77\)\( + \)\(11\!\cdots\!28\)\( \beta) q^{49} +(\)\(16\!\cdots\!00\)\( + 1396478884797480960 \beta) q^{50} +(\)\(80\!\cdots\!72\)\( - \)\(15\!\cdots\!90\)\( \beta) q^{51} +(\)\(10\!\cdots\!24\)\( - 58399505660491333632 \beta) q^{52} +(-\)\(21\!\cdots\!46\)\( + 12899519303831752788 \beta) q^{53} +(\)\(42\!\cdots\!60\)\( + \)\(14\!\cdots\!68\)\( \beta) q^{54} +(-\)\(78\!\cdots\!60\)\( + \)\(30\!\cdots\!38\)\( \beta) q^{55} +(\)\(18\!\cdots\!68\)\( + \)\(25\!\cdots\!28\)\( \beta) q^{56} +(-\)\(11\!\cdots\!00\)\( - \)\(10\!\cdots\!52\)\( \beta) q^{57} +(-\)\(82\!\cdots\!60\)\( - \)\(68\!\cdots\!16\)\( \beta) q^{58} +(\)\(93\!\cdots\!80\)\( + \)\(15\!\cdots\!73\)\( \beta) q^{59} +(\)\(15\!\cdots\!80\)\( - 60258497222802407424 \beta) q^{60} +(\)\(23\!\cdots\!02\)\( - \)\(42\!\cdots\!92\)\( \beta) q^{61} +(-\)\(14\!\cdots\!28\)\( + \)\(14\!\cdots\!24\)\( \beta) q^{62} +(\)\(15\!\cdots\!64\)\( + \)\(24\!\cdots\!30\)\( \beta) q^{63} +\)\(79\!\cdots\!36\)\( q^{64} +(\)\(47\!\cdots\!80\)\( - \)\(99\!\cdots\!64\)\( \beta) q^{65} +(-\)\(13\!\cdots\!92\)\( + \)\(57\!\cdots\!36\)\( \beta) q^{66} +(-\)\(37\!\cdots\!32\)\( - \)\(98\!\cdots\!21\)\( \beta) q^{67} +(\)\(64\!\cdots\!48\)\( - \)\(35\!\cdots\!68\)\( \beta) q^{68} +(-\)\(71\!\cdots\!44\)\( + \)\(35\!\cdots\!52\)\( \beta) q^{69} +(-\)\(21\!\cdots\!40\)\( - \)\(17\!\cdots\!08\)\( \beta) q^{70} +(\)\(13\!\cdots\!32\)\( - \)\(32\!\cdots\!62\)\( \beta) q^{71} +(\)\(94\!\cdots\!28\)\( - \)\(23\!\cdots\!28\)\( \beta) q^{72} +(-\)\(26\!\cdots\!66\)\( - \)\(70\!\cdots\!72\)\( \beta) q^{73} +(\)\(38\!\cdots\!68\)\( + \)\(46\!\cdots\!52\)\( \beta) q^{74} +(-\)\(82\!\cdots\!00\)\( - \)\(25\!\cdots\!35\)\( \beta) q^{75} +(\)\(48\!\cdots\!00\)\( + \)\(59\!\cdots\!32\)\( \beta) q^{76} +(\)\(12\!\cdots\!36\)\( + \)\(76\!\cdots\!12\)\( \beta) q^{77} +(\)\(86\!\cdots\!96\)\( - \)\(16\!\cdots\!12\)\( \beta) q^{78} +(-\)\(13\!\cdots\!40\)\( - \)\(55\!\cdots\!68\)\( \beta) q^{79} +(-\)\(49\!\cdots\!80\)\( - \)\(73\!\cdots\!36\)\( \beta) q^{80} +(-\)\(37\!\cdots\!59\)\( - \)\(97\!\cdots\!64\)\( \beta) q^{81} +(-\)\(54\!\cdots\!68\)\( + \)\(33\!\cdots\!96\)\( \beta) q^{82} +(-\)\(18\!\cdots\!96\)\( + \)\(35\!\cdots\!35\)\( \beta) q^{83} +(-\)\(33\!\cdots\!28\)\( - \)\(26\!\cdots\!76\)\( \beta) q^{84} +(\)\(29\!\cdots\!60\)\( - \)\(60\!\cdots\!08\)\( \beta) q^{85} +(-\)\(12\!\cdots\!36\)\( + \)\(52\!\cdots\!72\)\( \beta) q^{86} +(\)\(92\!\cdots\!60\)\( + \)\(82\!\cdots\!46\)\( \beta) q^{87} +(-\)\(22\!\cdots\!28\)\( + \)\(62\!\cdots\!12\)\( \beta) q^{88} +(\)\(15\!\cdots\!90\)\( - \)\(50\!\cdots\!40\)\( \beta) q^{89} +(\)\(18\!\cdots\!60\)\( - \)\(87\!\cdots\!88\)\( \beta) q^{90} +(\)\(56\!\cdots\!32\)\( + \)\(13\!\cdots\!96\)\( \beta) q^{91} +(-\)\(15\!\cdots\!96\)\( + \)\(27\!\cdots\!84\)\( \beta) q^{92} +(-\)\(28\!\cdots\!12\)\( + \)\(21\!\cdots\!44\)\( \beta) q^{93} +(\)\(94\!\cdots\!08\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{94} +(-\)\(49\!\cdots\!00\)\( - \)\(45\!\cdots\!10\)\( \beta) q^{95} +(\)\(50\!\cdots\!44\)\( - \)\(12\!\cdots\!76\)\( \beta) q^{96} +(\)\(34\!\cdots\!58\)\( - \)\(24\!\cdots\!20\)\( \beta) q^{97} +(\)\(25\!\cdots\!72\)\( + \)\(77\!\cdots\!08\)\( \beta) q^{98} +(-\)\(43\!\cdots\!44\)\( + \)\(80\!\cdots\!45\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 131072q^{2} + 8356488q^{3} + 8589934592q^{4} - 5332476660q^{5} + 547650797568q^{6} + 132719095875856q^{7} + 562949953421312q^{8} + 6720986327276826q^{9} + O(q^{10}) \) \( 2q + 131072q^{2} + 8356488q^{3} + 8589934592q^{4} - 5332476660q^{5} + 547650797568q^{6} + 132719095875856q^{7} + 562949953421312q^{8} + 6720986327276826q^{9} - 349469190389760q^{10} - 158105471422723176q^{11} + 35890842669416448q^{12} + 4983398515499093788q^{13} + 8697878667320098816q^{14} + 71123270901495251760q^{15} + 36893488147419103232q^{16} + \)\(30\!\cdots\!76\)\(q^{17} + \)\(44\!\cdots\!36\)\(q^{18} + \)\(22\!\cdots\!00\)\(q^{19} - 22902812861383311360q^{20} - \)\(15\!\cdots\!36\)\(q^{21} - \)\(10\!\cdots\!36\)\(q^{22} - \)\(71\!\cdots\!52\)\(q^{23} + \)\(23\!\cdots\!28\)\(q^{24} + \)\(51\!\cdots\!50\)\(q^{25} + \)\(32\!\cdots\!68\)\(q^{26} + \)\(13\!\cdots\!20\)\(q^{27} + \)\(57\!\cdots\!76\)\(q^{28} - \)\(25\!\cdots\!20\)\(q^{29} + \)\(46\!\cdots\!60\)\(q^{30} - \)\(43\!\cdots\!96\)\(q^{31} + \)\(24\!\cdots\!52\)\(q^{32} - \)\(40\!\cdots\!44\)\(q^{33} + \)\(19\!\cdots\!36\)\(q^{34} - \)\(64\!\cdots\!80\)\(q^{35} + \)\(28\!\cdots\!96\)\(q^{36} + \)\(11\!\cdots\!76\)\(q^{37} + \)\(14\!\cdots\!00\)\(q^{38} + \)\(26\!\cdots\!72\)\(q^{39} - \)\(15\!\cdots\!60\)\(q^{40} - \)\(16\!\cdots\!76\)\(q^{41} - \)\(10\!\cdots\!96\)\(q^{42} - \)\(36\!\cdots\!52\)\(q^{43} - \)\(67\!\cdots\!96\)\(q^{44} + \)\(57\!\cdots\!20\)\(q^{45} - \)\(46\!\cdots\!72\)\(q^{46} + \)\(28\!\cdots\!56\)\(q^{47} + \)\(15\!\cdots\!08\)\(q^{48} + \)\(76\!\cdots\!54\)\(q^{49} + \)\(33\!\cdots\!00\)\(q^{50} + \)\(16\!\cdots\!44\)\(q^{51} + \)\(21\!\cdots\!48\)\(q^{52} - \)\(43\!\cdots\!92\)\(q^{53} + \)\(85\!\cdots\!20\)\(q^{54} - \)\(15\!\cdots\!20\)\(q^{55} + \)\(37\!\cdots\!36\)\(q^{56} - \)\(23\!\cdots\!00\)\(q^{57} - \)\(16\!\cdots\!20\)\(q^{58} + \)\(18\!\cdots\!60\)\(q^{59} + \)\(30\!\cdots\!60\)\(q^{60} + \)\(46\!\cdots\!04\)\(q^{61} - \)\(28\!\cdots\!56\)\(q^{62} + \)\(31\!\cdots\!28\)\(q^{63} + \)\(15\!\cdots\!72\)\(q^{64} + \)\(95\!\cdots\!60\)\(q^{65} - \)\(26\!\cdots\!84\)\(q^{66} - \)\(74\!\cdots\!64\)\(q^{67} + \)\(12\!\cdots\!96\)\(q^{68} - \)\(14\!\cdots\!88\)\(q^{69} - \)\(42\!\cdots\!80\)\(q^{70} + \)\(26\!\cdots\!64\)\(q^{71} + \)\(18\!\cdots\!56\)\(q^{72} - \)\(52\!\cdots\!32\)\(q^{73} + \)\(76\!\cdots\!36\)\(q^{74} - \)\(16\!\cdots\!00\)\(q^{75} + \)\(96\!\cdots\!00\)\(q^{76} + \)\(24\!\cdots\!72\)\(q^{77} + \)\(17\!\cdots\!92\)\(q^{78} - \)\(26\!\cdots\!80\)\(q^{79} - \)\(98\!\cdots\!60\)\(q^{80} - \)\(75\!\cdots\!18\)\(q^{81} - \)\(10\!\cdots\!36\)\(q^{82} - \)\(37\!\cdots\!92\)\(q^{83} - \)\(66\!\cdots\!56\)\(q^{84} + \)\(58\!\cdots\!20\)\(q^{85} - \)\(24\!\cdots\!72\)\(q^{86} + \)\(18\!\cdots\!20\)\(q^{87} - \)\(44\!\cdots\!56\)\(q^{88} + \)\(30\!\cdots\!80\)\(q^{89} + \)\(37\!\cdots\!20\)\(q^{90} + \)\(11\!\cdots\!64\)\(q^{91} - \)\(30\!\cdots\!92\)\(q^{92} - \)\(57\!\cdots\!24\)\(q^{93} + \)\(18\!\cdots\!16\)\(q^{94} - \)\(98\!\cdots\!00\)\(q^{95} + \)\(10\!\cdots\!88\)\(q^{96} + \)\(68\!\cdots\!16\)\(q^{97} + \)\(50\!\cdots\!44\)\(q^{98} - \)\(86\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
4467.87
−4466.87
65536.0 −9.01727e7 4.29497e9 −3.79693e11 −5.90956e12 1.50920e14 2.81475e14 2.57205e15 −2.48835e16
1.2 65536.0 9.85292e7 4.29497e9 3.74360e11 6.45721e12 −1.82013e13 2.81475e14 4.14894e15 2.45341e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 2.34.a.b 2
3.b odd 2 1 18.34.a.e 2
4.b odd 2 1 16.34.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.34.a.b 2 1.a even 1 1 trivial
16.34.a.c 2 4.b odd 2 1
18.34.a.e 2 3.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8356488 T_{3} - \)\(88\!\cdots\!64\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 65536 T )^{2} \)
$3$ \( 1 - 8356488 T + 2233482848764182 T^{2} - \)\(46\!\cdots\!24\)\( T^{3} + \)\(30\!\cdots\!29\)\( T^{4} \)
$5$ \( 1 + 5332476660 T + \)\(90\!\cdots\!50\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 132719095875856 T + \)\(12\!\cdots\!98\)\( T^{2} - \)\(10\!\cdots\!92\)\( T^{3} + \)\(59\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + 158105471422723176 T + \)\(90\!\cdots\!06\)\( T^{2} + \)\(36\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!61\)\( T^{4} \)
$13$ \( 1 - 4983398515499093788 T + \)\(16\!\cdots\!42\)\( T^{2} - \)\(28\!\cdots\!64\)\( T^{3} + \)\(33\!\cdots\!09\)\( T^{4} \)
$17$ \( 1 - \)\(30\!\cdots\!76\)\( T + \)\(97\!\cdots\!18\)\( T^{2} - \)\(12\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} \)
$19$ \( 1 - \)\(22\!\cdots\!00\)\( T + \)\(27\!\cdots\!18\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!81\)\( T^{4} \)
$23$ \( 1 + \)\(71\!\cdots\!52\)\( T + \)\(29\!\cdots\!42\)\( T^{2} + \)\(61\!\cdots\!16\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$29$ \( 1 + \)\(25\!\cdots\!20\)\( T + \)\(26\!\cdots\!78\)\( T^{2} + \)\(45\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!21\)\( T^{4} \)
$31$ \( 1 + \)\(43\!\cdots\!96\)\( T + \)\(37\!\cdots\!86\)\( T^{2} + \)\(70\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$37$ \( 1 - \)\(11\!\cdots\!76\)\( T + \)\(10\!\cdots\!38\)\( T^{2} - \)\(65\!\cdots\!72\)\( T^{3} + \)\(31\!\cdots\!09\)\( T^{4} \)
$41$ \( 1 + \)\(16\!\cdots\!76\)\( T + \)\(11\!\cdots\!86\)\( T^{2} + \)\(27\!\cdots\!96\)\( T^{3} + \)\(27\!\cdots\!41\)\( T^{4} \)
$43$ \( 1 + \)\(36\!\cdots\!52\)\( T + \)\(15\!\cdots\!62\)\( T^{2} + \)\(29\!\cdots\!36\)\( T^{3} + \)\(64\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(28\!\cdots\!56\)\( T + \)\(29\!\cdots\!38\)\( T^{2} - \)\(43\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!29\)\( T^{4} \)
$53$ \( 1 + \)\(43\!\cdots\!92\)\( T + \)\(15\!\cdots\!62\)\( T^{2} + \)\(35\!\cdots\!16\)\( T^{3} + \)\(63\!\cdots\!29\)\( T^{4} \)
$59$ \( 1 - \)\(18\!\cdots\!60\)\( T + \)\(42\!\cdots\!58\)\( T^{2} - \)\(51\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!41\)\( T^{4} \)
$61$ \( 1 - \)\(46\!\cdots\!04\)\( T + \)\(21\!\cdots\!66\)\( T^{2} - \)\(38\!\cdots\!24\)\( T^{3} + \)\(67\!\cdots\!61\)\( T^{4} \)
$67$ \( 1 + \)\(74\!\cdots\!64\)\( T + \)\(37\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!68\)\( T^{3} + \)\(33\!\cdots\!69\)\( T^{4} \)
$71$ \( 1 - \)\(26\!\cdots\!64\)\( T + \)\(17\!\cdots\!46\)\( T^{2} - \)\(32\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} \)
$73$ \( 1 + \)\(52\!\cdots\!32\)\( T + \)\(68\!\cdots\!22\)\( T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(95\!\cdots\!89\)\( T^{4} \)
$79$ \( 1 + \)\(26\!\cdots\!80\)\( T + \)\(98\!\cdots\!78\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!21\)\( T^{4} \)
$83$ \( 1 + \)\(37\!\cdots\!92\)\( T + \)\(35\!\cdots\!42\)\( T^{2} + \)\(79\!\cdots\!96\)\( T^{3} + \)\(45\!\cdots\!69\)\( T^{4} \)
$89$ \( 1 - \)\(30\!\cdots\!80\)\( T + \)\(63\!\cdots\!38\)\( T^{2} - \)\(64\!\cdots\!20\)\( T^{3} + \)\(45\!\cdots\!61\)\( T^{4} \)
$97$ \( 1 - \)\(68\!\cdots\!16\)\( T + \)\(68\!\cdots\!18\)\( T^{2} - \)\(24\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
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