Properties

Label 3.67.b.a
Level 3
Weight 67
Character orbit 3.b
Self dual yes
Analytic conductor 82.760
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 67 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(82.7604085389\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 5559060566555523q^{3} + 73786976294838206464q^{4} - 15459460129734036008401737214q^{7} + 30903154382632612361920641803529q^{9} + O(q^{10}) \) \( q - 5559060566555523q^{3} + 73786976294838206464q^{4} - 15459460129734036008401737214q^{7} + 30903154382632612361920641803529q^{9} - 410186270246002225336426103593500672q^{12} - 10993813372461732057454331126658365206q^{13} + 5444517870735015415413993718908291383296q^{16} - 832798423753884127074595015476794447985382q^{19} + 85940075187441809311980382917737461386332922q^{21} + 13552527156068805425093160010874271392822265625q^{25} - 171792506910670443678820376588540424234035840667q^{27} - 1140706818123681698887574716220886699357604151296q^{28} - 20829684874384381866590946759750099977154079183918q^{31} + 2280250319867037997421842330085227917956272625811456q^{36} - 7417939125211054386881016651493135320203925642789094q^{37} + 61115274394922801210762477289866071761175701610332738q^{39} - 1245239512199512679501766979615043759328112882443972886q^{43} - 30266404599109864532334369015971799232027475459658743808q^{48} + 179226643608680348145022620544381817220795446800681800147q^{49} - 811200246703709101178369640233561344627077870999341891584q^{52} + 4629576877379813618912743062882067950665990502774395364786q^{57} + 76718687326389894248157038524144813488758607942856930706762q^{61} - 477746083061324508954274974277993621793321305799473675828206q^{63} + 401734511064747568885490523085290650630550748445698208825344q^{64} + 236443545730387202454481566607273858427413476877399945116026q^{67} + 56436561604451561325982235028839350532357796943958932674439634q^{73} - 75339319290474964364683063688943320812541060149669647216796875q^{75} - 61449677551906471596875542797057717992763026840834966969909248q^{76} + 624527582495370862537524794357610311427789739490612827187896178q^{79} + 955004950796825236893190701774414011919935138974343129836853841q^{81} + 6341258290632381916746324700733293965031930411594725724476407808q^{84} + 169958419505309028205531820178646770327376511268675550415652976084q^{91} + 115793479798968249791323839952406459314371397286968580708075679114q^{93} + 244251200682546708355692627372351299246029736144700040416696428546q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0
0 −5.55906e15 7.37870e19 0 0 −1.54595e28 0 3.09032e31 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.67.b.a 1
3.b odd 2 1 CM 3.67.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.67.b.a 1 1.a even 1 1 trivial
3.67.b.a 1 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{67}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8589934592 T )( 1 + 8589934592 T ) \)
$3$ \( 1 + 5559060566555523 T \)
$5$ \( ( 1 - \)\(11\!\cdots\!25\)\( T )( 1 + \)\(11\!\cdots\!25\)\( T ) \)
$7$ \( 1 + \)\(15\!\cdots\!14\)\( T + \)\(59\!\cdots\!49\)\( T^{2} \)
$11$ \( ( 1 - \)\(23\!\cdots\!31\)\( T )( 1 + \)\(23\!\cdots\!31\)\( T ) \)
$13$ \( 1 + \)\(10\!\cdots\!06\)\( T + \)\(33\!\cdots\!09\)\( T^{2} \)
$17$ \( ( 1 - \)\(40\!\cdots\!37\)\( T )( 1 + \)\(40\!\cdots\!37\)\( T ) \)
$19$ \( 1 + \)\(83\!\cdots\!82\)\( T + \)\(24\!\cdots\!81\)\( T^{2} \)
$23$ \( ( 1 - \)\(86\!\cdots\!83\)\( T )( 1 + \)\(86\!\cdots\!83\)\( T ) \)
$29$ \( ( 1 - \)\(18\!\cdots\!89\)\( T )( 1 + \)\(18\!\cdots\!89\)\( T ) \)
$31$ \( 1 + \)\(20\!\cdots\!18\)\( T + \)\(26\!\cdots\!81\)\( T^{2} \)
$37$ \( 1 + \)\(74\!\cdots\!94\)\( T + \)\(31\!\cdots\!09\)\( T^{2} \)
$41$ \( ( 1 - \)\(16\!\cdots\!21\)\( T )( 1 + \)\(16\!\cdots\!21\)\( T ) \)
$43$ \( 1 + \)\(12\!\cdots\!86\)\( T + \)\(64\!\cdots\!49\)\( T^{2} \)
$47$ \( ( 1 - \)\(15\!\cdots\!27\)\( T )( 1 + \)\(15\!\cdots\!27\)\( T ) \)
$53$ \( ( 1 - \)\(79\!\cdots\!73\)\( T )( 1 + \)\(79\!\cdots\!73\)\( T ) \)
$59$ \( ( 1 - \)\(27\!\cdots\!79\)\( T )( 1 + \)\(27\!\cdots\!79\)\( T ) \)
$61$ \( 1 - \)\(76\!\cdots\!62\)\( T + \)\(67\!\cdots\!61\)\( T^{2} \)
$67$ \( 1 - \)\(23\!\cdots\!26\)\( T + \)\(33\!\cdots\!69\)\( T^{2} \)
$71$ \( ( 1 - \)\(12\!\cdots\!11\)\( T )( 1 + \)\(12\!\cdots\!11\)\( T ) \)
$73$ \( 1 - \)\(56\!\cdots\!34\)\( T + \)\(95\!\cdots\!89\)\( T^{2} \)
$79$ \( 1 - \)\(62\!\cdots\!78\)\( T + \)\(17\!\cdots\!21\)\( T^{2} \)
$83$ \( ( 1 - \)\(21\!\cdots\!63\)\( T )( 1 + \)\(21\!\cdots\!63\)\( T ) \)
$89$ \( ( 1 - \)\(21\!\cdots\!69\)\( T )( 1 + \)\(21\!\cdots\!69\)\( T ) \)
$97$ \( 1 - \)\(24\!\cdots\!46\)\( T + \)\(13\!\cdots\!29\)\( T^{2} \)
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