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Properties

Label 806.2.bx.a
Level $806$
Weight $2$
Character orbit 806.bx
Analytic conductor $6.436$
Analytic rank $0$
Dimension $304$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [806,2,Mod(173,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([5, 26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.173");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.bx (of order \(30\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(38\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 304 q - 2 q^{3} - 38 q^{4} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 304 q - 2 q^{3} - 38 q^{4} + 40 q^{9} - 30 q^{11} + 2 q^{12} - 12 q^{13} - 12 q^{14} + 38 q^{16} - 32 q^{17} - 24 q^{18} + 62 q^{21} - 6 q^{22} - 6 q^{23} + 160 q^{25} - 24 q^{26} - 2 q^{27} + 4 q^{28} - 18 q^{29} - 50 q^{30} - 4 q^{31} + 84 q^{33} - 4 q^{35} - 320 q^{36} - 30 q^{37} + 38 q^{38} - 68 q^{39} + 36 q^{42} - 56 q^{43} + 6 q^{44} + 4 q^{48} + 94 q^{49} - 68 q^{51} - 6 q^{52} - 30 q^{53} + 18 q^{54} + 24 q^{55} - 4 q^{56} + 102 q^{57} - 6 q^{58} + 48 q^{61} + 58 q^{62} - 288 q^{63} + 76 q^{64} + 80 q^{65} + 24 q^{66} + 34 q^{68} - 6 q^{69} + 30 q^{71} - 24 q^{72} + 24 q^{73} + 56 q^{74} + 28 q^{75} - 10 q^{76} - 2 q^{77} - 30 q^{78} - 92 q^{79} - 62 q^{81} - 48 q^{83} - 18 q^{84} - 24 q^{85} - 78 q^{86} - 36 q^{87} + 28 q^{88} + 42 q^{89} + 24 q^{90} + 6 q^{91} + 8 q^{92} + 136 q^{93} + 52 q^{95} - 98 q^{97} - 12 q^{98} + 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1 −0.994522 + 0.104528i −0.296054 + 2.81677i 0.978148 − 0.207912i 3.61721 − 2.08839i − 2.83229i 3.66260 + 1.19005i −0.951057 + 0.309017i −4.91210 − 1.04410i −3.37909 + 2.45506i
173.2 −0.994522 + 0.104528i −0.270235 + 2.57112i 0.978148 − 0.207912i −0.549868 + 0.317466i − 2.58528i 0.240279 + 0.0780715i −0.951057 + 0.309017i −3.60317 − 0.765877i 0.513671 − 0.373204i
173.3 −0.994522 + 0.104528i −0.265495 + 2.52602i 0.978148 − 0.207912i 0.599874 − 0.346337i − 2.53993i 0.540721 + 0.175691i −0.951057 + 0.309017i −3.37584 − 0.717556i −0.560385 + 0.407144i
173.4 −0.994522 + 0.104528i −0.238239 + 2.26669i 0.978148 − 0.207912i −2.97811 + 1.71941i − 2.27917i −4.55318 − 1.47942i −0.951057 + 0.309017i −2.14667 − 0.456290i 2.78207 − 2.02129i
173.5 −0.994522 + 0.104528i −0.207587 + 1.97506i 0.978148 − 0.207912i 2.69950 − 1.55856i − 1.98594i −3.73793 − 1.21453i −0.951057 + 0.309017i −0.923334 − 0.196261i −2.52180 + 1.83219i
173.6 −0.994522 + 0.104528i −0.188488 + 1.79334i 0.978148 − 0.207912i −2.09200 + 1.20781i − 1.80322i 3.81964 + 1.24108i −0.951057 + 0.309017i −0.246105 − 0.0523112i 1.95428 − 1.41987i
173.7 −0.994522 + 0.104528i −0.102607 + 0.976244i 0.978148 − 0.207912i −2.20650 + 1.27392i − 0.981622i 0.853952 + 0.277466i −0.951057 + 0.309017i 1.99192 + 0.423395i 2.06125 − 1.49759i
173.8 −0.994522 + 0.104528i −0.0494496 + 0.470481i 0.978148 − 0.207912i 0.503173 − 0.290507i − 0.473073i −2.86785 − 0.931821i −0.951057 + 0.309017i 2.71554 + 0.577205i −0.470050 + 0.341511i
173.9 −0.994522 + 0.104528i −0.0120646 + 0.114787i 0.978148 − 0.207912i −0.129303 + 0.0746532i − 0.115419i 0.411567 + 0.133726i −0.951057 + 0.309017i 2.92141 + 0.620965i 0.120791 − 0.0877601i
173.10 −0.994522 + 0.104528i −0.00823831 + 0.0783823i 0.978148 − 0.207912i 0.878541 − 0.507226i − 0.0788140i 2.44297 + 0.793769i −0.951057 + 0.309017i 2.92837 + 0.622444i −0.820709 + 0.596280i
173.11 −0.994522 + 0.104528i 0.00737272 − 0.0701467i 0.978148 − 0.207912i 3.41194 − 1.96988i 0.0705331i −0.580099 − 0.188486i −0.951057 + 0.309017i 2.92958 + 0.622701i −3.18734 + 2.31574i
173.12 −0.994522 + 0.104528i 0.0542574 − 0.516225i 0.978148 − 0.207912i −3.75324 + 2.16693i 0.519069i 2.86869 + 0.932095i −0.951057 + 0.309017i 2.67090 + 0.567717i 3.50617 − 2.54738i
173.13 −0.994522 + 0.104528i 0.120327 − 1.14484i 0.978148 − 0.207912i −0.796643 + 0.459942i 1.15114i −3.18567 − 1.03509i −0.951057 + 0.309017i 1.63827 + 0.348226i 0.744202 − 0.540694i
173.14 −0.994522 + 0.104528i 0.178882 − 1.70195i 0.978148 − 0.207912i 1.62458 − 0.937954i 1.71133i 0.272415 + 0.0885130i −0.951057 + 0.309017i 0.0698008 + 0.0148366i −1.51764 + 1.10263i
173.15 −0.994522 + 0.104528i 0.222724 − 2.11908i 0.978148 − 0.207912i 1.06375 − 0.614157i 2.13075i −4.38516 − 1.42482i −0.951057 + 0.309017i −1.50643 − 0.320202i −0.993726 + 0.721984i
173.16 −0.994522 + 0.104528i 0.265330 − 2.52444i 0.978148 − 0.207912i −1.02070 + 0.589301i 2.53835i 2.55396 + 0.829831i −0.951057 + 0.309017i −3.36797 − 0.715884i 0.953509 − 0.692765i
173.17 −0.994522 + 0.104528i 0.277697 − 2.64211i 0.978148 − 0.207912i 2.96579 − 1.71230i 2.65666i 2.74895 + 0.893189i −0.951057 + 0.309017i −3.96918 − 0.843675i −2.77056 + 2.01293i
173.18 −0.994522 + 0.104528i 0.279669 − 2.66088i 0.978148 − 0.207912i −1.47127 + 0.849439i 2.67553i 2.33714 + 0.759382i −0.951057 + 0.309017i −4.06761 − 0.864597i 1.37442 − 0.998575i
173.19 −0.994522 + 0.104528i 0.336728 − 3.20375i 0.978148 − 0.207912i −2.36673 + 1.36643i 3.22140i −1.39658 − 0.453775i −0.951057 + 0.309017i −7.21618 − 1.53385i 2.21093 − 1.60634i
173.20 0.994522 − 0.104528i −0.314936 + 2.99641i 0.978148 − 0.207912i 3.24638 − 1.87430i 3.01292i −1.23514 − 0.401321i 0.951057 − 0.309017i −5.94487 − 1.26362i 3.03268 − 2.20337i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bt even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.bx.a yes 304
13.e even 6 1 806.2.bo.a ✓ 304
31.g even 15 1 806.2.bo.a ✓ 304
403.bt even 30 1 inner 806.2.bx.a yes 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.bo.a ✓ 304 13.e even 6 1
806.2.bo.a ✓ 304 31.g even 15 1
806.2.bx.a yes 304 1.a even 1 1 trivial
806.2.bx.a yes 304 403.bt even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).