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Properties

Label 684.2.cf.b
Level $684$
Weight $2$
Character orbit 684.cf
Analytic conductor $5.462$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [684,2,Mod(91,684)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(684, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 0, 11])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("684.91"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.cf (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{2} - 3 q^{4} + 3 q^{8} - 6 q^{10} - 6 q^{13} + 9 q^{14} + 21 q^{16} + 18 q^{19} - 30 q^{20} - 12 q^{22} - 18 q^{28} - 12 q^{31} - 33 q^{32} - 15 q^{34} + 84 q^{38} - 87 q^{40} + 12 q^{41} - 18 q^{43}+ \cdots + 105 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.40972 + 0.112689i 0 1.97460 − 0.317720i −1.52856 + 1.28261i 0 −1.51170 − 0.872779i −2.74783 + 0.670411i 0 2.01030 − 1.98037i
91.2 −1.29298 − 0.572888i 0 1.34360 + 1.48147i 2.02034 − 1.69527i 0 −0.752594 − 0.434510i −0.888536 − 2.68524i 0 −3.58346 + 1.03452i
91.3 −1.16563 + 0.800819i 0 0.717378 − 1.86691i 1.07136 − 0.898980i 0 3.16192 + 1.82553i 0.658864 + 2.75062i 0 −0.528890 + 1.90584i
91.4 −0.578232 − 1.29060i 0 −1.33130 + 1.49253i −3.00606 + 2.52238i 0 3.14549 + 1.81605i 2.69606 + 0.855142i 0 4.99358 + 2.42109i
91.5 −0.428246 − 1.34782i 0 −1.63321 + 1.15439i 1.63332 − 1.37052i 0 −1.13591 − 0.655818i 2.25532 + 1.70690i 0 −2.54667 − 1.61449i
91.6 −0.365891 + 1.36606i 0 −1.73225 − 0.999660i −2.56653 + 2.15357i 0 −2.87385 − 1.65922i 1.99941 − 2.00059i 0 −2.00284 − 4.29401i
91.7 0.255165 + 1.39100i 0 −1.86978 + 0.709871i 0.471332 − 0.395495i 0 3.37123 + 1.94638i −1.46454 − 2.41974i 0 0.670402 + 0.554709i
91.8 0.810901 − 1.15864i 0 −0.684878 − 1.87908i 2.11944 − 1.77843i 0 2.80017 + 1.61668i −2.73254 − 0.730223i 0 −0.341888 − 3.89779i
91.9 1.34509 + 0.436740i 0 1.61852 + 1.17491i −1.62636 + 1.36468i 0 1.25845 + 0.726566i 1.66392 + 2.28722i 0 −2.78361 + 1.12532i
91.10 1.38985 − 0.261379i 0 1.86336 − 0.726554i 1.41171 − 1.18456i 0 −4.05173 − 2.33927i 2.39989 − 1.49684i 0 1.65244 − 2.01536i
127.1 −1.39185 + 0.250479i 0 1.87452 − 0.697261i −0.683388 − 3.87569i 0 0.181667 + 0.104885i −2.43441 + 1.44001i 0 1.92196 + 5.22322i
127.2 −1.37128 + 0.345809i 0 1.76083 − 0.948403i 0.540387 + 3.06469i 0 −1.31815 − 0.761035i −2.08663 + 1.90944i 0 −1.80082 − 4.01568i
127.3 −0.947134 − 1.05021i 0 −0.205875 + 1.98938i −0.234529 − 1.33008i 0 1.38264 + 0.798268i 2.28425 − 1.66799i 0 −1.17473 + 1.50607i
127.4 −0.436096 − 1.34530i 0 −1.61964 + 1.17336i 0.711633 + 4.03587i 0 −1.83842 − 1.06142i 2.28483 + 1.66720i 0 5.11910 − 2.71738i
127.5 −0.260406 + 1.39003i 0 −1.86438 − 0.723946i 0.175493 + 0.995269i 0 −2.28954 − 1.32186i 1.49180 − 2.40302i 0 −1.42916 − 0.0152335i
127.6 0.526616 − 1.31251i 0 −1.44535 − 1.38238i 0.0217506 + 0.123354i 0 3.23383 + 1.86705i −2.57552 + 1.16905i 0 0.173357 + 0.0364123i
127.7 0.626457 + 1.26789i 0 −1.21510 + 1.58856i −0.316000 − 1.79212i 0 −0.258690 − 0.149354i −2.77533 − 0.545458i 0 2.07426 − 1.52334i
127.8 1.02960 − 0.969499i 0 0.120142 − 1.99639i −0.162908 − 0.923898i 0 −2.27351 − 1.31261i −1.81180 − 2.17195i 0 −1.06345 − 0.793304i
127.9 1.11501 + 0.869917i 0 0.486490 + 1.93993i 0.379809 + 2.15400i 0 3.95398 + 2.28283i −1.14514 + 2.58625i 0 −1.45031 + 2.73213i
127.10 1.37514 + 0.330144i 0 1.78201 + 0.907986i −0.432248 − 2.45140i 0 −1.95860 − 1.13080i 2.15074 + 1.83693i 0 0.214913 − 3.51371i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.cf.b 60
3.b odd 2 1 228.2.w.b yes 60
4.b odd 2 1 684.2.cf.c 60
12.b even 2 1 228.2.w.a ✓ 60
19.f odd 18 1 684.2.cf.c 60
57.j even 18 1 228.2.w.a ✓ 60
76.k even 18 1 inner 684.2.cf.b 60
228.u odd 18 1 228.2.w.b yes 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.w.a ✓ 60 12.b even 2 1
228.2.w.a ✓ 60 57.j even 18 1
228.2.w.b yes 60 3.b odd 2 1
228.2.w.b yes 60 228.u odd 18 1
684.2.cf.b 60 1.a even 1 1 trivial
684.2.cf.b 60 76.k even 18 1 inner
684.2.cf.c 60 4.b odd 2 1
684.2.cf.c 60 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\):

\( T_{5}^{60} - 12 T_{5}^{57} - 69 T_{5}^{56} - 528 T_{5}^{55} + 4170 T_{5}^{54} + 4236 T_{5}^{53} + \cdots + 2228834013184 \) Copy content Toggle raw display
\( T_{7}^{60} - 120 T_{7}^{58} + 8037 T_{7}^{56} + 234 T_{7}^{55} - 370070 T_{7}^{54} + \cdots + 54\!\cdots\!89 \) Copy content Toggle raw display