Properties

Label 1.30.a.a
Level $1$
Weight $30$
Character orbit 1.a
Self dual yes
Analytic conductor $5.328$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.32780423830\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{51349}) \)
Defining polynomial: \(x^{2} - x - 12837\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 96\sqrt{51349}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4320 - \beta ) q^{2} + ( -2483820 + 552 \beta ) q^{3} + ( -44976128 - 8640 \beta ) q^{4} + ( -8738894250 - 116000 \beta ) q^{5} + ( -271954378368 + 4868460 \beta ) q^{6} + ( -1510156341400 - 67855536 \beta ) q^{7} + ( 1575148584960 + 544522240 \beta ) q^{8} + ( 81734784761853 - 2742137280 \beta ) q^{9} +O(q^{10})\) \( q +(4320 - \beta) q^{2} +(-2483820 + 552 \beta) q^{3} +(-44976128 - 8640 \beta) q^{4} +(-8738894250 - 116000 \beta) q^{5} +(-271954378368 + 4868460 \beta) q^{6} +(-1510156341400 - 67855536 \beta) q^{7} +(1575148584960 + 544522240 \beta) q^{8} +(81734784761853 - 2742137280 \beta) q^{9} +(17142933384000 + 8237774250 \beta) q^{10} +(-1027690259794308 - 10592091400 \beta) q^{11} +(-2145265138114560 - 3366617856 \beta) q^{12} +(8569685589666110 - 113971542048 \beta) q^{13} +(25587561674029824 + 1217020425880 \beta) q^{14} +(-8596175696253000 - 4535746506000 \beta) q^{15} +(-226734541031604224 + 5415752171520 \beta) q^{16} +(-332397963953874990 + 19090509870144 \beta) q^{17} +(1650762432440880480 - 93580817811453 \beta) q^{18} +(616084722726077540 + 142801363479240 \beta) q^{19} +(867334050906624000 + 80721277168000 \beta) q^{20} +(-13974556738124410848 - 665065363025280 \beta) q^{21} +(572898742456487040 + 981932424946308 \beta) q^{22} +(-9294215252187189960 - 351986937759248 \beta) q^{23} +(\)\(13\!\cdots\!20\)\( - 483013211258880 \beta) q^{24} +(-\)\(10\!\cdots\!25\)\( + 2027423466000000 \beta) q^{25} +(90956066298888877632 - 9062042651313470 \beta) q^{26} +(-\)\(74\!\cdots\!40\)\( + 14044608301937040 \beta) q^{27} +(\)\(34\!\cdots\!60\)\( + 16099630062340608 \beta) q^{28} +(\)\(49\!\cdots\!10\)\( - 63411615083585440 \beta) q^{29} +(\)\(21\!\cdots\!00\)\( - 10998249209667000 \beta) q^{30} +(-\)\(54\!\cdots\!28\)\( + 195576850719316800 \beta) q^{31} +(-\)\(43\!\cdots\!80\)\( - 42207561180512256 \beta) q^{32} +(-\)\(21\!\cdots\!40\)\( - 540976174945310016 \beta) q^{33} +(-\)\(10\!\cdots\!96\)\( + 414868966592897070 \beta) q^{34} +(\)\(16\!\cdots\!00\)\( + 768160488983468000 \beta) q^{35} +(\)\(75\!\cdots\!16\)\( - 582857823043558080 \beta) q^{36} +(\)\(49\!\cdots\!30\)\( - 1363639917490075296 \beta) q^{37} +(-\)\(64\!\cdots\!60\)\( + 817167504239260 \beta) q^{38} +(-\)\(51\!\cdots\!64\)\( + 5013551241065356080 \beta) q^{39} +(-\)\(43\!\cdots\!00\)\( - 4941239507988480000 \beta) q^{40} +(-\)\(53\!\cdots\!38\)\( - 2173657014743715200 \beta) q^{41} +(\)\(25\!\cdots\!60\)\( + 11101474369855201248 \beta) q^{42} +(\)\(25\!\cdots\!00\)\( - 19178453595369039048 \beta) q^{43} +(\)\(89\!\cdots\!24\)\( + 9355635103216920320 \beta) q^{44} +(-\)\(56\!\cdots\!50\)\( + 14482012676527692000 \beta) q^{45} +(\)\(12\!\cdots\!32\)\( + 7773631681067238600 \beta) q^{46} +(-\)\(22\!\cdots\!60\)\( - 16406150082376208416 \beta) q^{47} +(\)\(19\!\cdots\!40\)\( - \)\(13\!\cdots\!48\)\( \beta) q^{48} +(\)\(12\!\cdots\!57\)\( + \)\(20\!\cdots\!00\)\( \beta) q^{49} +(-\)\(14\!\cdots\!00\)\( + \)\(11\!\cdots\!25\)\( \beta) q^{50} +(\)\(58\!\cdots\!92\)\( - \)\(23\!\cdots\!60\)\( \beta) q^{51} +(\)\(80\!\cdots\!00\)\( - 68916084831206960256 \beta) q^{52} +(-\)\(80\!\cdots\!70\)\( - \)\(29\!\cdots\!48\)\( \beta) q^{53} +(-\)\(98\!\cdots\!60\)\( + \)\(80\!\cdots\!40\)\( \beta) q^{54} +(\)\(95\!\cdots\!00\)\( + \)\(21\!\cdots\!00\)\( \beta) q^{55} +(-\)\(19\!\cdots\!60\)\( - \)\(92\!\cdots\!60\)\( \beta) q^{56} +(\)\(35\!\cdots\!20\)\( - 14614115692211094720 \beta) q^{57} +(\)\(32\!\cdots\!60\)\( - \)\(77\!\cdots\!10\)\( \beta) q^{58} +(-\)\(41\!\cdots\!80\)\( + \)\(21\!\cdots\!20\)\( \beta) q^{59} +(\)\(18\!\cdots\!00\)\( + \)\(27\!\cdots\!00\)\( \beta) q^{60} +(-\)\(79\!\cdots\!58\)\( - \)\(24\!\cdots\!00\)\( \beta) q^{61} +(-\)\(94\!\cdots\!60\)\( + \)\(13\!\cdots\!28\)\( \beta) q^{62} +(-\)\(35\!\cdots\!80\)\( - \)\(14\!\cdots\!08\)\( \beta) q^{63} +(\)\(12\!\cdots\!92\)\( + \)\(12\!\cdots\!20\)\( \beta) q^{64} +(-\)\(68\!\cdots\!00\)\( + 1901725065631664000 \beta) q^{65} +(\)\(25\!\cdots\!44\)\( - \)\(21\!\cdots\!80\)\( \beta) q^{66} +(\)\(12\!\cdots\!60\)\( + \)\(53\!\cdots\!44\)\( \beta) q^{67} +(-\)\(63\!\cdots\!20\)\( + \)\(20\!\cdots\!68\)\( \beta) q^{68} +(-\)\(68\!\cdots\!64\)\( - \)\(42\!\cdots\!60\)\( \beta) q^{69} +(-\)\(29\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{70} +(-\)\(94\!\cdots\!68\)\( - \)\(12\!\cdots\!00\)\( \beta) q^{71} +(-\)\(57\!\cdots\!20\)\( + \)\(40\!\cdots\!20\)\( \beta) q^{72} +(\)\(49\!\cdots\!90\)\( + \)\(43\!\cdots\!52\)\( \beta) q^{73} +(\)\(85\!\cdots\!64\)\( - \)\(55\!\cdots\!50\)\( \beta) q^{74} +(\)\(78\!\cdots\!00\)\( - \)\(62\!\cdots\!00\)\( \beta) q^{75} +(-\)\(61\!\cdots\!20\)\( - \)\(11\!\cdots\!20\)\( \beta) q^{76} +(\)\(18\!\cdots\!00\)\( + \)\(85\!\cdots\!88\)\( \beta) q^{77} +(-\)\(25\!\cdots\!00\)\( + \)\(72\!\cdots\!64\)\( \beta) q^{78} +(-\)\(36\!\cdots\!40\)\( - \)\(15\!\cdots\!40\)\( \beta) q^{79} +(\)\(16\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( \beta) q^{80} +(-\)\(80\!\cdots\!79\)\( - \)\(26\!\cdots\!40\)\( \beta) q^{81} +(\)\(79\!\cdots\!40\)\( + \)\(43\!\cdots\!38\)\( \beta) q^{82} +(\)\(11\!\cdots\!20\)\( + \)\(25\!\cdots\!52\)\( \beta) q^{83} +(\)\(33\!\cdots\!44\)\( + \)\(15\!\cdots\!60\)\( \beta) q^{84} +(\)\(18\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( \beta) q^{85} +(\)\(10\!\cdots\!32\)\( - \)\(33\!\cdots\!60\)\( \beta) q^{86} +(-\)\(17\!\cdots\!20\)\( + \)\(43\!\cdots\!20\)\( \beta) q^{87} +(-\)\(43\!\cdots\!80\)\( - \)\(57\!\cdots\!20\)\( \beta) q^{88} +(\)\(29\!\cdots\!30\)\( + \)\(16\!\cdots\!80\)\( \beta) q^{89} +(-\)\(92\!\cdots\!00\)\( + \)\(62\!\cdots\!50\)\( \beta) q^{90} +(-\)\(92\!\cdots\!48\)\( - \)\(40\!\cdots\!60\)\( \beta) q^{91} +(\)\(18\!\cdots\!60\)\( + \)\(96\!\cdots\!44\)\( \beta) q^{92} +(\)\(52\!\cdots\!60\)\( - \)\(78\!\cdots\!56\)\( \beta) q^{93} +(-\)\(20\!\cdots\!56\)\( + \)\(21\!\cdots\!40\)\( \beta) q^{94} +(-\)\(13\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( \beta) q^{95} +(-\)\(12\!\cdots\!08\)\( - \)\(23\!\cdots\!40\)\( \beta) q^{96} +(\)\(54\!\cdots\!90\)\( + \)\(18\!\cdots\!84\)\( \beta) q^{97} +(-\)\(91\!\cdots\!60\)\( - \)\(35\!\cdots\!57\)\( \beta) q^{98} +(-\)\(70\!\cdots\!24\)\( + \)\(19\!\cdots\!40\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8640q^{2} - 4967640q^{3} - 89952256q^{4} - 17477788500q^{5} - 543908756736q^{6} - 3020312682800q^{7} + 3150297169920q^{8} + 163469569523706q^{9} + O(q^{10}) \) \( 2q + 8640q^{2} - 4967640q^{3} - 89952256q^{4} - 17477788500q^{5} - 543908756736q^{6} - 3020312682800q^{7} + 3150297169920q^{8} + 163469569523706q^{9} + 34285866768000q^{10} - 2055380519588616q^{11} - 4290530276229120q^{12} + 17139371179332220q^{13} + 51175123348059648q^{14} - 17192351392506000q^{15} - 453469082063208448q^{16} - 664795927907749980q^{17} + 3301524864881760960q^{18} + 1232169445452155080q^{19} + 1734668101813248000q^{20} - 27949113476248821696q^{21} + 1145797484912974080q^{22} - 18588430504374379920q^{23} + \)\(27\!\cdots\!40\)\(q^{24} - \)\(20\!\cdots\!50\)\(q^{25} + \)\(18\!\cdots\!64\)\(q^{26} - \)\(14\!\cdots\!80\)\(q^{27} + \)\(69\!\cdots\!20\)\(q^{28} + \)\(99\!\cdots\!20\)\(q^{29} + \)\(42\!\cdots\!00\)\(q^{30} - \)\(10\!\cdots\!56\)\(q^{31} - \)\(87\!\cdots\!60\)\(q^{32} - \)\(42\!\cdots\!80\)\(q^{33} - \)\(20\!\cdots\!92\)\(q^{34} + \)\(33\!\cdots\!00\)\(q^{35} + \)\(15\!\cdots\!32\)\(q^{36} + \)\(98\!\cdots\!60\)\(q^{37} - \)\(12\!\cdots\!20\)\(q^{38} - \)\(10\!\cdots\!28\)\(q^{39} - \)\(87\!\cdots\!00\)\(q^{40} - \)\(10\!\cdots\!76\)\(q^{41} + \)\(50\!\cdots\!20\)\(q^{42} + \)\(51\!\cdots\!00\)\(q^{43} + \)\(17\!\cdots\!48\)\(q^{44} - \)\(11\!\cdots\!00\)\(q^{45} + \)\(25\!\cdots\!64\)\(q^{46} - \)\(45\!\cdots\!20\)\(q^{47} + \)\(39\!\cdots\!80\)\(q^{48} + \)\(24\!\cdots\!14\)\(q^{49} - \)\(28\!\cdots\!00\)\(q^{50} + \)\(11\!\cdots\!84\)\(q^{51} + \)\(16\!\cdots\!00\)\(q^{52} - \)\(16\!\cdots\!40\)\(q^{53} - \)\(19\!\cdots\!20\)\(q^{54} + \)\(19\!\cdots\!00\)\(q^{55} - \)\(39\!\cdots\!20\)\(q^{56} + \)\(71\!\cdots\!40\)\(q^{57} + \)\(64\!\cdots\!20\)\(q^{58} - \)\(83\!\cdots\!60\)\(q^{59} + \)\(37\!\cdots\!00\)\(q^{60} - \)\(15\!\cdots\!16\)\(q^{61} - \)\(18\!\cdots\!20\)\(q^{62} - \)\(70\!\cdots\!60\)\(q^{63} + \)\(24\!\cdots\!84\)\(q^{64} - \)\(13\!\cdots\!00\)\(q^{65} + \)\(51\!\cdots\!88\)\(q^{66} + \)\(24\!\cdots\!20\)\(q^{67} - \)\(12\!\cdots\!40\)\(q^{68} - \)\(13\!\cdots\!28\)\(q^{69} - \)\(58\!\cdots\!00\)\(q^{70} - \)\(18\!\cdots\!36\)\(q^{71} - \)\(11\!\cdots\!40\)\(q^{72} + \)\(99\!\cdots\!80\)\(q^{73} + \)\(17\!\cdots\!28\)\(q^{74} + \)\(15\!\cdots\!00\)\(q^{75} - \)\(12\!\cdots\!40\)\(q^{76} + \)\(37\!\cdots\!00\)\(q^{77} - \)\(51\!\cdots\!00\)\(q^{78} - \)\(72\!\cdots\!80\)\(q^{79} + \)\(33\!\cdots\!00\)\(q^{80} - \)\(16\!\cdots\!58\)\(q^{81} + \)\(15\!\cdots\!80\)\(q^{82} + \)\(22\!\cdots\!40\)\(q^{83} + \)\(66\!\cdots\!88\)\(q^{84} + \)\(37\!\cdots\!00\)\(q^{85} + \)\(20\!\cdots\!64\)\(q^{86} - \)\(35\!\cdots\!40\)\(q^{87} - \)\(86\!\cdots\!60\)\(q^{88} + \)\(58\!\cdots\!60\)\(q^{89} - \)\(18\!\cdots\!00\)\(q^{90} - \)\(18\!\cdots\!96\)\(q^{91} + \)\(37\!\cdots\!20\)\(q^{92} + \)\(10\!\cdots\!20\)\(q^{93} - \)\(40\!\cdots\!12\)\(q^{94} - \)\(26\!\cdots\!00\)\(q^{95} - \)\(25\!\cdots\!16\)\(q^{96} + \)\(10\!\cdots\!80\)\(q^{97} - \)\(18\!\cdots\!20\)\(q^{98} - \)\(14\!\cdots\!48\)\(q^{99} + O(q^{100}) \)

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
113.802
−112.802
−17433.9 9.52434e6 −2.32930e8 −1.12623e10 −1.66046e11 −2.98628e12 1.34206e13 2.20826e13 1.96347e14
1.2 26073.9 −1.44920e7 1.42978e8 −6.21544e9 −3.77862e11 −3.40335e10 −1.02703e13 1.41387e14 −1.62061e14
\(n\): 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 1.30.a.a 2
3.b odd 2 1 9.30.a.a 2
4.b odd 2 1 16.30.a.c 2
5.b even 2 1 25.30.a.a 2
5.c odd 4 2 25.30.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.30.a.a 2 1.a even 1 1 trivial
9.30.a.a 2 3.b odd 2 1
16.30.a.c 2 4.b odd 2 1
25.30.a.a 2 5.b even 2 1
25.30.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{30}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -454569984 - 8640 T + T^{2} \)
$3$ \( -138026438541936 + 4967640 T + T^{2} \)
$5$ \( 70000457753579062500 + 17477788500 T + T^{2} \)
$7$ \( \)\(10\!\cdots\!36\)\( + 3020312682800 T + T^{2} \)
$11$ \( \)\(10\!\cdots\!64\)\( + 2055380519588616 T + T^{2} \)
$13$ \( \)\(67\!\cdots\!64\)\( - 17139371179332220 T + T^{2} \)
$17$ \( -\)\(61\!\cdots\!24\)\( + 664795927907749980 T + T^{2} \)
$19$ \( -\)\(92\!\cdots\!00\)\( - 1232169445452155080 T + T^{2} \)
$23$ \( \)\(27\!\cdots\!64\)\( + 18588430504374379920 T + T^{2} \)
$29$ \( -\)\(16\!\cdots\!00\)\( - \)\(99\!\cdots\!20\)\( T + T^{2} \)
$31$ \( -\)\(17\!\cdots\!16\)\( + \)\(10\!\cdots\!56\)\( T + T^{2} \)
$37$ \( \)\(15\!\cdots\!56\)\( - \)\(98\!\cdots\!60\)\( T + T^{2} \)
$41$ \( \)\(61\!\cdots\!44\)\( + \)\(10\!\cdots\!76\)\( T + T^{2} \)
$43$ \( -\)\(10\!\cdots\!36\)\( - \)\(51\!\cdots\!00\)\( T + T^{2} \)
$47$ \( \)\(49\!\cdots\!96\)\( + \)\(45\!\cdots\!20\)\( T + T^{2} \)
$53$ \( \)\(23\!\cdots\!64\)\( + \)\(16\!\cdots\!40\)\( T + T^{2} \)
$59$ \( -\)\(52\!\cdots\!00\)\( + \)\(83\!\cdots\!60\)\( T + T^{2} \)
$61$ \( -\)\(28\!\cdots\!36\)\( + \)\(15\!\cdots\!16\)\( T + T^{2} \)
$67$ \( -\)\(13\!\cdots\!24\)\( - \)\(24\!\cdots\!20\)\( T + T^{2} \)
$71$ \( -\)\(63\!\cdots\!76\)\( + \)\(18\!\cdots\!36\)\( T + T^{2} \)
$73$ \( -\)\(66\!\cdots\!36\)\( - \)\(99\!\cdots\!80\)\( T + T^{2} \)
$79$ \( \)\(12\!\cdots\!00\)\( + \)\(72\!\cdots\!80\)\( T + T^{2} \)
$83$ \( -\)\(29\!\cdots\!36\)\( - \)\(22\!\cdots\!40\)\( T + T^{2} \)
$89$ \( -\)\(39\!\cdots\!00\)\( - \)\(58\!\cdots\!60\)\( T + T^{2} \)
$97$ \( \)\(13\!\cdots\!96\)\( - \)\(10\!\cdots\!80\)\( T + T^{2} \)
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