Properties

Label 6.28.a.c
Level $6$
Weight $28$
Character orbit 6.a
Self dual yes
Analytic conductor $27.711$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.7113344903\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 8192q^{2} + 1594323q^{3} + 67108864q^{4} + 1220703150q^{5} + 13060694016q^{6} + 96889207016q^{7} + 549755813888q^{8} + 2541865828329q^{9} + O(q^{10}) \) \( q + 8192q^{2} + 1594323q^{3} + 67108864q^{4} + 1220703150q^{5} + 13060694016q^{6} + 96889207016q^{7} + 549755813888q^{8} + 2541865828329q^{9} + 10000000204800q^{10} + 34495064342052q^{11} + 106993205379072q^{12} + 300892562137622q^{13} + 793716383875072q^{14} + 1946195108217450q^{15} + 4503599627370496q^{16} + 11406510312331986q^{17} + 20822964865671168q^{18} + 62694436994411420q^{19} + 81920001677721600q^{20} + 154472691197370168q^{21} + 282583567090089984q^{22} - 894750379460289528q^{23} + 876488338465357824q^{24} - 5960464416503905625q^{25} + 2464911869031399424q^{26} + 4052555153018976267q^{27} + 6502124616704589824q^{28} + 104977877616797619030q^{29} + 15943230326517350400q^{30} + 242604665669598327632q^{31} + 36893488147419103232q^{32} + 54996274467013370796q^{33} + 93442132478623629312q^{34} + 118272960205433300400q^{35} + 170581728179578208256q^{36} + 1615296776273185563326q^{37} + 513592827858218352640q^{38} + 479719932344939919906q^{39} + 671088653743895347200q^{40} - 3845227056141271336998q^{41} + 1265440286288856416256q^{42} - 457279560936215164348q^{43} + 2314924581602017148928q^{44} + 3102863623518569536350q^{45} - 7329795108538691813376q^{46} - 48040279177582596731664q^{47} + 7180192468708211294208q^{48} - 56324843927344976515287q^{49} - 48828124499999994880000q^{50} + 18185661740688068915478q^{51} + 20192558031105224081408q^{52} - 277436637507624408709218q^{53} + 33198531813531453579264q^{54} + 42108233701795553863800q^{55} + 53265404860043999838208q^{56} + 99955182872240998368660q^{57} + 859978773436806095093760q^{58} + 257780373920421815739540q^{59} + 130606942834830134476800q^{60} + 143521812506920157597702q^{61} + 1987417421165349499961344q^{62} + 246279364447864798356264q^{63} + 302231454903657293676544q^{64} + 367300498412965908909300q^{65} + 450529480433773533560832q^{66} - 5989037992702629692139124q^{67} + 765477949264884771323904q^{68} - 1426521109232267181149544q^{69} + 968892090002909596876800q^{70} - 5167007853435876621826728q^{71} + 1397405517247104682033152q^{72} - 12558020215909057855312678q^{73} + 13232511191229936134766592q^{74} - 9502905509913756327766875q^{75} + 4207352445814524744826880q^{76} + 3342199430067316062236832q^{77} + 3929865685769747823869952q^{78} - 60783355718993851022367520q^{79} + 5497558251469990684262400q^{80} + 6461081889226673298932241q^{81} - 31500100043909294792687616q^{82} + 3971159179443692816062812q^{83} + 10366486825278311761969152q^{84} + 13923963068771139155955900q^{85} - 3746034163189474626338816q^{86} + 167368644775645630364766690q^{87} + 18963862172483724484018176q^{88} + 312460635134078620023752010q^{89} + 25418658803864121641779200q^{90} + 29153241742526701439955952q^{91} - 60045681529148963335176192q^{92} + 386790198384351014505233136q^{93} - 393545967022756632425791488q^{94} + 76531296726554552789973000q^{95} + 58820136703657666922151936q^{96} - 548671865528572742983955614q^{97} - 461413121452810047613231104q^{98} + 87681825297072158367591108q^{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
0
8192.00 1.59432e6 6.71089e7 1.22070e9 1.30607e10 9.68892e10 5.49756e11 2.54187e12 1.00000e13
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 6.28.a.c 1
3.b odd 2 1 18.28.a.b 1
4.b odd 2 1 48.28.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.28.a.c 1 1.a even 1 1 trivial
18.28.a.b 1 3.b odd 2 1
48.28.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1220703150 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -8192 + T \)
$3$ \( -1594323 + T \)
$5$ \( -1220703150 + T \)
$7$ \( -96889207016 + T \)
$11$ \( -34495064342052 + T \)
$13$ \( -300892562137622 + T \)
$17$ \( -11406510312331986 + T \)
$19$ \( -62694436994411420 + T \)
$23$ \( 894750379460289528 + T \)
$29$ \( -\)\(10\!\cdots\!30\)\( + T \)
$31$ \( -\)\(24\!\cdots\!32\)\( + T \)
$37$ \( -\)\(16\!\cdots\!26\)\( + T \)
$41$ \( \)\(38\!\cdots\!98\)\( + T \)
$43$ \( \)\(45\!\cdots\!48\)\( + T \)
$47$ \( \)\(48\!\cdots\!64\)\( + T \)
$53$ \( \)\(27\!\cdots\!18\)\( + T \)
$59$ \( -\)\(25\!\cdots\!40\)\( + T \)
$61$ \( -\)\(14\!\cdots\!02\)\( + T \)
$67$ \( \)\(59\!\cdots\!24\)\( + T \)
$71$ \( \)\(51\!\cdots\!28\)\( + T \)
$73$ \( \)\(12\!\cdots\!78\)\( + T \)
$79$ \( \)\(60\!\cdots\!20\)\( + T \)
$83$ \( -\)\(39\!\cdots\!12\)\( + T \)
$89$ \( -\)\(31\!\cdots\!10\)\( + T \)
$97$ \( \)\(54\!\cdots\!14\)\( + T \)
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