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 disc. -3
Inner twists 2

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Newspace parameters

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

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: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes

Hecke kernels

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