Properties

Label 4.34.a.a
Level 4
Weight 34
Character orbit 4.a
Self dual yes
Analytic conductor 27.593
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5931315524\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 65185566 x - 173679864984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{3}\cdot 7\cdot 11\cdot 29 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 30830596 + \beta_{1} ) q^{3} + ( -17960227962 - 1535 \beta_{1} + \beta_{2} ) q^{5} + ( 1513669971464 - 122346 \beta_{1} + 180 \beta_{2} ) q^{7} + ( 2010788144563341 + 75248670 \beta_{1} - 10530 \beta_{2} ) q^{9} +O(q^{10})\) \( q +(30830596 + \beta_{1}) q^{3} +(-17960227962 - 1535 \beta_{1} + \beta_{2}) q^{5} +(1513669971464 - 122346 \beta_{1} + 180 \beta_{2}) q^{7} +(2010788144563341 + 75248670 \beta_{1} - 10530 \beta_{2}) q^{9} +(75872552434149900 + 2215839115 \beta_{1} + 256840 \beta_{2}) q^{11} +(90990147405786254 + 6383188809 \beta_{1} - 3668535 \beta_{2}) q^{13} +(-10714157827080368616 - 382917087630 \beta_{1} + 33406668 \beta_{2}) q^{15} +(31012396103938905618 + 1069301430158 \beta_{1} - 184208050 \beta_{2}) q^{17} +(\)\(44\!\cdots\!76\)\( + 11850406238205 \beta_{1} + 320516280 \beta_{2}) q^{19} +(-\)\(76\!\cdots\!84\)\( - 57340224604180 \beta_{1} + 4392064620 \beta_{2}) q^{21} +(\)\(38\!\cdots\!32\)\( - 94999163347438 \beta_{1} - 49295676820 \beta_{2}) q^{23} +(\)\(86\!\cdots\!07\)\( + 615284668657260 \beta_{1} + 277757168364 \beta_{2}) q^{25} +(\)\(38\!\cdots\!08\)\( + 2919170542770738 \beta_{1} - 973938527640 \beta_{2}) q^{27} +(\)\(96\!\cdots\!26\)\( - 17291424904695515 \beta_{1} + 1814914414885 \beta_{2}) q^{29} +(\)\(52\!\cdots\!76\)\( + 3377924000980200 \beta_{1} + 1180388959200 \beta_{2}) q^{31} +(\)\(17\!\cdots\!60\)\( + 98072137402840890 \beta_{1} - 18904063453830 \beta_{2}) q^{33} +(\)\(34\!\cdots\!16\)\( + 51799596297276380 \beta_{1} + 55139397898832 \beta_{2}) q^{35} +(\)\(11\!\cdots\!14\)\( - 1038005378158598451 \beta_{1} - 52265853646515 \beta_{2}) q^{37} +(\)\(45\!\cdots\!56\)\( + 1463249000741125850 \beta_{1} - 130471960050900 \beta_{2}) q^{39} +(-\)\(22\!\cdots\!54\)\( - 641632766910683060 \beta_{1} + 448924689742540 \beta_{2}) q^{41} +(-\)\(51\!\cdots\!40\)\( + 9557870528390708475 \beta_{1} - 210366843981840 \beta_{2}) q^{43} +(-\)\(27\!\cdots\!62\)\( - 29103707165110673535 \beta_{1} - 950908515500799 \beta_{2}) q^{45} +(-\)\(24\!\cdots\!92\)\( + 13537930853481011508 \beta_{1} - 576631108849560 \beta_{2}) q^{47} +(-\)\(15\!\cdots\!03\)\( + 496168262546838840 \beta_{1} + 10601270675344440 \beta_{2}) q^{49} +(\)\(80\!\cdots\!12\)\( + \)\(13\!\cdots\!10\)\( \beta_{1} - 14436065202263640 \beta_{2}) q^{51} +(\)\(18\!\cdots\!78\)\( - 27337706038779017987 \beta_{1} - 40829066390340995 \beta_{2}) q^{53} +(\)\(24\!\cdots\!60\)\( - \)\(84\!\cdots\!50\)\( \beta_{1} + 158548384447532820 \beta_{2}) q^{55} +(\)\(92\!\cdots\!16\)\( + \)\(87\!\cdots\!06\)\( \beta_{1} - 119258077651337610 \beta_{2}) q^{57} +(\)\(67\!\cdots\!92\)\( + \)\(66\!\cdots\!35\)\( \beta_{1} - 355142845670518240 \beta_{2}) q^{59} +(\)\(19\!\cdots\!22\)\( + \)\(11\!\cdots\!45\)\( \beta_{1} + 864805591092629745 \beta_{2}) q^{61} +(-\)\(41\!\cdots\!56\)\( - \)\(39\!\cdots\!06\)\( \beta_{1} - 321105448391693580 \beta_{2}) q^{63} +(-\)\(75\!\cdots\!04\)\( - \)\(25\!\cdots\!20\)\( \beta_{1} - 991922296965526108 \beta_{2}) q^{65} +(-\)\(96\!\cdots\!76\)\( + \)\(61\!\cdots\!09\)\( \beta_{1} + 745104731828763960 \beta_{2}) q^{67} +(-\)\(50\!\cdots\!72\)\( + \)\(14\!\cdots\!80\)\( \beta_{1} + 150330017751397380 \beta_{2}) q^{69} +(\)\(56\!\cdots\!48\)\( + \)\(17\!\cdots\!70\)\( \beta_{1} + 4210979720475567620 \beta_{2}) q^{71} +(\)\(43\!\cdots\!34\)\( - \)\(54\!\cdots\!86\)\( \beta_{1} - 9892260507865032810 \beta_{2}) q^{73} +(\)\(67\!\cdots\!76\)\( + \)\(30\!\cdots\!55\)\( \beta_{1} - 1689547931514628848 \beta_{2}) q^{75} +(\)\(69\!\cdots\!40\)\( - \)\(13\!\cdots\!40\)\( \beta_{1} + 25628357491802757380 \beta_{2}) q^{77} +(\)\(14\!\cdots\!68\)\( + \)\(14\!\cdots\!60\)\( \beta_{1} - 24410580389385198840 \beta_{2}) q^{79} +(\)\(20\!\cdots\!09\)\( + \)\(38\!\cdots\!30\)\( \beta_{1} + 11004304993611004530 \beta_{2}) q^{81} +(-\)\(23\!\cdots\!88\)\( - \)\(34\!\cdots\!63\)\( \beta_{1} - 30337551274260013440 \beta_{2}) q^{83} +(-\)\(45\!\cdots\!28\)\( - \)\(41\!\cdots\!90\)\( \beta_{1} - 21434939812222425306 \beta_{2}) q^{85} +(-\)\(84\!\cdots\!64\)\( - \)\(34\!\cdots\!14\)\( \beta_{1} + \)\(21\!\cdots\!80\)\( \beta_{2}) q^{87} +(-\)\(64\!\cdots\!66\)\( + \)\(68\!\cdots\!30\)\( \beta_{1} - \)\(17\!\cdots\!70\)\( \beta_{2}) q^{89} +(-\)\(12\!\cdots\!36\)\( - \)\(23\!\cdots\!40\)\( \beta_{1} - \)\(19\!\cdots\!40\)\( \beta_{2}) q^{91} +(\)\(18\!\cdots\!96\)\( + \)\(50\!\cdots\!76\)\( \beta_{1} - 15215953620938720400 \beta_{2}) q^{93} +(-\)\(68\!\cdots\!92\)\( - \)\(46\!\cdots\!10\)\( \beta_{1} + \)\(56\!\cdots\!16\)\( \beta_{2}) q^{95} +(\)\(18\!\cdots\!74\)\( - \)\(10\!\cdots\!46\)\( \beta_{1} + \)\(87\!\cdots\!90\)\( \beta_{2}) q^{97} +(\)\(75\!\cdots\!00\)\( + \)\(14\!\cdots\!15\)\( \beta_{1} - \)\(27\!\cdots\!60\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 92491788q^{3} - 53880683886q^{5} + 4541009914392q^{7} + 6032364433690023q^{9} + O(q^{10}) \) \( 3q + 92491788q^{3} - 53880683886q^{5} + 4541009914392q^{7} + 6032364433690023q^{9} + 227617657302449700q^{11} + 272970442217358762q^{13} - 32142473481241105848q^{15} + 93037188311816716854q^{17} + \)\(13\!\cdots\!28\)\(q^{19} - \)\(22\!\cdots\!52\)\(q^{21} + \)\(11\!\cdots\!96\)\(q^{23} + \)\(25\!\cdots\!21\)\(q^{25} + \)\(11\!\cdots\!24\)\(q^{27} + \)\(28\!\cdots\!78\)\(q^{29} + \)\(15\!\cdots\!28\)\(q^{31} + \)\(51\!\cdots\!80\)\(q^{33} + \)\(10\!\cdots\!48\)\(q^{35} + \)\(34\!\cdots\!42\)\(q^{37} + \)\(13\!\cdots\!68\)\(q^{39} - \)\(67\!\cdots\!62\)\(q^{41} - \)\(15\!\cdots\!20\)\(q^{43} - \)\(82\!\cdots\!86\)\(q^{45} - \)\(72\!\cdots\!76\)\(q^{47} - \)\(47\!\cdots\!09\)\(q^{49} + \)\(24\!\cdots\!36\)\(q^{51} + \)\(54\!\cdots\!34\)\(q^{53} + \)\(72\!\cdots\!80\)\(q^{55} + \)\(27\!\cdots\!48\)\(q^{57} + \)\(20\!\cdots\!76\)\(q^{59} + \)\(59\!\cdots\!66\)\(q^{61} - \)\(12\!\cdots\!68\)\(q^{63} - \)\(22\!\cdots\!12\)\(q^{65} - \)\(29\!\cdots\!28\)\(q^{67} - \)\(15\!\cdots\!16\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} + \)\(13\!\cdots\!02\)\(q^{73} + \)\(20\!\cdots\!28\)\(q^{75} + \)\(20\!\cdots\!20\)\(q^{77} + \)\(43\!\cdots\!04\)\(q^{79} + \)\(60\!\cdots\!27\)\(q^{81} - \)\(71\!\cdots\!64\)\(q^{83} - \)\(13\!\cdots\!84\)\(q^{85} - \)\(25\!\cdots\!92\)\(q^{87} - \)\(19\!\cdots\!98\)\(q^{89} - \)\(38\!\cdots\!08\)\(q^{91} + \)\(55\!\cdots\!88\)\(q^{93} - \)\(20\!\cdots\!76\)\(q^{95} + \)\(56\!\cdots\!22\)\(q^{97} + \)\(22\!\cdots\!00\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 65185566 x - 173679864984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -128 \nu^{2} + 542720 \nu + 5562320768 \)\()/25\)
\(\beta_{2}\)\(=\)\((\)\( 343168 \nu^{2} + 6776698880 \nu - 14915325889408 \)\()/125\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{2} + 2681 \beta_{1} + 109756416\)\()/ 329269248 \)
\(\nu^{2}\)\(=\)\((\)\(1325 \beta_{2} - 3308935 \beta_{1} + 894316769246208\)\()/20579328\)

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
−6032.20
9172.25
−3139.05
0 −6.39317e7 0 −2.18954e11 0 −4.92542e13 0 −1.47179e15 0
1.2 0 2.16957e7 0 6.04966e11 0 1.12234e14 0 −5.08836e15 0
1.3 0 1.34728e8 0 −4.39893e11 0 −5.84388e13 0 1.25925e16 0
\(n\): 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 4.34.a.a 3
4.b odd 2 1 16.34.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.34.a.a 3 1.a even 1 1 trivial
16.34.a.d 3 4.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{34}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 92491788 T + 9599774056706745 T^{2} - \)\(84\!\cdots\!36\)\( T^{3} + \)\(53\!\cdots\!35\)\( T^{4} - \)\(28\!\cdots\!52\)\( T^{5} + \)\(17\!\cdots\!67\)\( T^{6} \)
$5$ \( 1 + 53880683886 T + \)\(46\!\cdots\!75\)\( T^{2} - \)\(45\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!75\)\( T^{4} + \)\(73\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 4541009914392 T + \)\(13\!\cdots\!97\)\( T^{2} - \)\(39\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!79\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(46\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 - 227617657302449700 T + \)\(19\!\cdots\!93\)\( T^{2} + \)\(71\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!83\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} \)
$13$ \( 1 - 272970442217358762 T + \)\(13\!\cdots\!95\)\( T^{2} - \)\(28\!\cdots\!04\)\( T^{3} + \)\(75\!\cdots\!35\)\( T^{4} - \)\(90\!\cdots\!58\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \)
$17$ \( 1 - 93037188311816716854 T + \)\(10\!\cdots\!75\)\( T^{2} - \)\(76\!\cdots\!52\)\( T^{3} + \)\(41\!\cdots\!75\)\( T^{4} - \)\(15\!\cdots\!26\)\( T^{5} + \)\(65\!\cdots\!53\)\( T^{6} \)
$19$ \( 1 - \)\(13\!\cdots\!28\)\( T + \)\(39\!\cdots\!05\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(61\!\cdots\!95\)\( T^{4} - \)\(33\!\cdots\!68\)\( T^{5} + \)\(39\!\cdots\!79\)\( T^{6} \)
$23$ \( 1 - \)\(11\!\cdots\!96\)\( T + \)\(18\!\cdots\!73\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!59\)\( T^{4} - \)\(87\!\cdots\!44\)\( T^{5} + \)\(64\!\cdots\!87\)\( T^{6} \)
$29$ \( 1 - \)\(28\!\cdots\!78\)\( T + \)\(43\!\cdots\!95\)\( T^{2} - \)\(47\!\cdots\!60\)\( T^{3} + \)\(79\!\cdots\!55\)\( T^{4} - \)\(95\!\cdots\!38\)\( T^{5} + \)\(59\!\cdots\!69\)\( T^{6} \)
$31$ \( 1 - \)\(15\!\cdots\!28\)\( T + \)\(13\!\cdots\!01\)\( T^{2} - \)\(66\!\cdots\!72\)\( T^{3} + \)\(21\!\cdots\!91\)\( T^{4} - \)\(42\!\cdots\!68\)\( T^{5} + \)\(44\!\cdots\!71\)\( T^{6} \)
$37$ \( 1 - \)\(34\!\cdots\!42\)\( T + \)\(58\!\cdots\!87\)\( T^{2} - \)\(50\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!39\)\( T^{4} - \)\(11\!\cdots\!78\)\( T^{5} + \)\(17\!\cdots\!73\)\( T^{6} \)
$41$ \( 1 + \)\(67\!\cdots\!62\)\( T + \)\(59\!\cdots\!11\)\( T^{2} + \)\(21\!\cdots\!68\)\( T^{3} + \)\(98\!\cdots\!31\)\( T^{4} + \)\(18\!\cdots\!42\)\( T^{5} + \)\(46\!\cdots\!61\)\( T^{6} \)
$43$ \( 1 + \)\(15\!\cdots\!20\)\( T + \)\(22\!\cdots\!29\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!47\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{5} + \)\(51\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + \)\(72\!\cdots\!76\)\( T + \)\(60\!\cdots\!05\)\( T^{2} + \)\(22\!\cdots\!28\)\( T^{3} + \)\(91\!\cdots\!35\)\( T^{4} + \)\(16\!\cdots\!04\)\( T^{5} + \)\(34\!\cdots\!83\)\( T^{6} \)
$53$ \( 1 - \)\(54\!\cdots\!34\)\( T + \)\(28\!\cdots\!83\)\( T^{2} - \)\(80\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!59\)\( T^{4} - \)\(34\!\cdots\!86\)\( T^{5} + \)\(50\!\cdots\!17\)\( T^{6} \)
$59$ \( 1 - \)\(20\!\cdots\!76\)\( T + \)\(56\!\cdots\!29\)\( T^{2} - \)\(65\!\cdots\!96\)\( T^{3} + \)\(15\!\cdots\!91\)\( T^{4} - \)\(15\!\cdots\!16\)\( T^{5} + \)\(20\!\cdots\!39\)\( T^{6} \)
$61$ \( 1 - \)\(59\!\cdots\!66\)\( T + \)\(26\!\cdots\!95\)\( T^{2} - \)\(28\!\cdots\!40\)\( T^{3} + \)\(21\!\cdots\!95\)\( T^{4} - \)\(40\!\cdots\!26\)\( T^{5} + \)\(55\!\cdots\!41\)\( T^{6} \)
$67$ \( 1 + \)\(29\!\cdots\!28\)\( T + \)\(81\!\cdots\!37\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!19\)\( T^{4} + \)\(96\!\cdots\!32\)\( T^{5} + \)\(60\!\cdots\!03\)\( T^{6} \)
$71$ \( 1 - \)\(16\!\cdots\!44\)\( T + \)\(30\!\cdots\!45\)\( T^{2} - \)\(32\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!95\)\( T^{4} - \)\(25\!\cdots\!24\)\( T^{5} + \)\(18\!\cdots\!31\)\( T^{6} \)
$73$ \( 1 - \)\(13\!\cdots\!02\)\( T + \)\(36\!\cdots\!75\)\( T^{2} - \)\(19\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!75\)\( T^{4} - \)\(12\!\cdots\!78\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \)
$79$ \( 1 - \)\(43\!\cdots\!04\)\( T + \)\(15\!\cdots\!89\)\( T^{2} - \)\(35\!\cdots\!44\)\( T^{3} + \)\(62\!\cdots\!71\)\( T^{4} - \)\(75\!\cdots\!84\)\( T^{5} + \)\(73\!\cdots\!19\)\( T^{6} \)
$83$ \( 1 + \)\(71\!\cdots\!64\)\( T + \)\(66\!\cdots\!93\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!59\)\( T^{4} + \)\(32\!\cdots\!16\)\( T^{5} + \)\(97\!\cdots\!47\)\( T^{6} \)
$89$ \( 1 + \)\(19\!\cdots\!98\)\( T + \)\(63\!\cdots\!75\)\( T^{2} + \)\(76\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!75\)\( T^{4} + \)\(88\!\cdots\!78\)\( T^{5} + \)\(97\!\cdots\!09\)\( T^{6} \)
$97$ \( 1 - \)\(56\!\cdots\!22\)\( T - \)\(50\!\cdots\!73\)\( T^{2} + \)\(35\!\cdots\!00\)\( T^{3} - \)\(18\!\cdots\!21\)\( T^{4} - \)\(75\!\cdots\!38\)\( T^{5} + \)\(49\!\cdots\!33\)\( T^{6} \)
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