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Properties

Label 847.2.ba.b
Level $847$
Weight $2$
Character orbit 847.ba
Analytic conductor $6.763$
Analytic rank $0$
Dimension $3360$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(6,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([55, 89]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.ba (of order \(110\), degree \(40\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(3360\)
Relative dimension: \(84\) over \(\Q(\zeta_{110})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{110}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3360 q - 78 q^{2} - 158 q^{4} - 34 q^{7} - 98 q^{8} + 814 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3360 q - 78 q^{2} - 158 q^{4} - 34 q^{7} - 98 q^{8} + 814 q^{9} - 68 q^{11} - 34 q^{14} - 176 q^{15} - 2 q^{16} - 90 q^{18} - 77 q^{21} - 174 q^{22} - 190 q^{23} - 138 q^{25} - 34 q^{28} - 98 q^{29} - 114 q^{30} - 88 q^{32} - 84 q^{35} + 14 q^{36} - 150 q^{37} - 184 q^{39} - 121 q^{42} - 88 q^{43} - 210 q^{44} - 78 q^{46} - 2 q^{49} - 28 q^{50} - 232 q^{51} - 48 q^{53} + 61 q^{56} - 64 q^{57} - 2 q^{58} - 262 q^{60} - 35 q^{63} - 234 q^{64} - 88 q^{65} - 80 q^{67} - 56 q^{70} - 48 q^{71} - 8 q^{72} - 178 q^{74} + 86 q^{77} - 198 q^{78} - 50 q^{79} - 682 q^{81} - 7 q^{84} - 278 q^{85} - 94 q^{86} - 222 q^{88} - 55 q^{91} - 572 q^{92} - 56 q^{93} - 116 q^{95} - 209 q^{98} - 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −2.18366 + 1.49315i −1.85628 + 2.55495i 1.81326 − 4.65732i 0.742004 + 1.75579i 0.238565 − 8.35085i 2.64575 + 0.00242856i 1.79620 + 7.72427i −2.15495 − 6.63226i −4.24194 − 2.72613i
6.2 −2.18366 + 1.49315i 1.85628 − 2.55495i 1.81326 − 4.65732i −0.742004 − 1.75579i −0.238565 + 8.35085i −1.36834 − 2.26443i 1.79620 + 7.72427i −2.15495 − 6.63226i 4.24194 + 2.72613i
6.3 −2.13940 + 1.46288i −0.786627 + 1.08270i 1.71138 − 4.39566i −0.133575 − 0.316076i 0.0990463 − 3.46707i −2.29275 + 1.32034i 1.59496 + 6.85887i 0.373595 + 1.14981i 0.748152 + 0.480808i
6.4 −2.13940 + 1.46288i 0.786627 − 1.08270i 1.71138 − 4.39566i 0.133575 + 0.316076i −0.0990463 + 3.46707i 0.0532928 + 2.64521i 1.59496 + 6.85887i 0.373595 + 1.14981i −0.748152 − 0.480808i
6.5 −2.07686 + 1.42012i −1.96587 + 2.70579i 1.57098 − 4.03503i −1.15298 − 2.72827i 0.240292 − 8.41132i −2.31158 − 1.28709i 1.32781 + 5.71001i −2.52961 − 7.78534i 6.26904 + 4.02887i
6.6 −2.07686 + 1.42012i 1.96587 − 2.70579i 1.57098 − 4.03503i 1.15298 + 2.72827i −0.240292 + 8.41132i 2.29589 + 1.31487i 1.32781 + 5.71001i −2.52961 − 7.78534i −6.26904 − 4.02887i
6.7 −2.01014 + 1.37450i −0.0974592 + 0.134141i 1.42581 − 3.66217i −1.14344 − 2.70571i 0.0115300 − 0.403601i −2.64573 + 0.0112936i 1.06446 + 4.57754i 0.918555 + 2.82702i 6.01750 + 3.86721i
6.8 −2.01014 + 1.37450i 0.0974592 − 0.134141i 1.42581 − 3.66217i 1.14344 + 2.70571i −0.0115300 + 0.403601i 1.35658 + 2.27150i 1.06446 + 4.57754i 0.918555 + 2.82702i −6.01750 − 3.86721i
6.9 −1.93467 + 1.32289i −0.779141 + 1.07240i 1.26728 − 3.25497i −1.14595 − 2.71164i 0.0887154 − 3.10544i 2.50966 − 0.837626i 0.792530 + 3.40815i 0.384080 + 1.18208i 5.80423 + 3.73015i
6.10 −1.93467 + 1.32289i 0.779141 − 1.07240i 1.26728 − 3.25497i 1.14595 + 2.71164i −0.0887154 + 3.10544i −0.578680 − 2.58169i 0.792530 + 3.40815i 0.384080 + 1.18208i −5.80423 − 3.73015i
6.11 −1.91634 + 1.31036i −1.01137 + 1.39203i 1.22970 − 3.15845i 1.50386 + 3.55857i 0.114066 − 3.99285i −2.16882 + 1.51533i 0.730562 + 3.14166i 0.0121755 + 0.0374724i −7.54491 − 4.84882i
6.12 −1.91634 + 1.31036i 1.01137 − 1.39203i 1.22970 − 3.15845i −1.50386 − 3.55857i −0.114066 + 3.99285i −0.177683 + 2.63978i 0.730562 + 3.14166i 0.0121755 + 0.0374724i 7.54491 + 4.84882i
6.13 −1.69790 + 1.16099i −1.30442 + 1.79538i 0.809333 − 2.07875i −0.624978 − 1.47888i 0.130348 − 4.56279i 1.43686 + 2.22158i 0.107500 + 0.462288i −0.594828 − 1.83069i 2.77811 + 1.78538i
6.14 −1.69790 + 1.16099i 1.30442 − 1.79538i 0.809333 − 2.07875i 0.624978 + 1.47888i −0.130348 + 4.56279i −2.64444 − 0.0832360i 0.107500 + 0.462288i −0.594828 − 1.83069i −2.77811 − 1.78538i
6.15 −1.59549 + 1.09097i −1.39552 + 1.92077i 0.629760 − 1.61753i 1.20552 + 2.85261i 0.131041 − 4.58704i 1.81317 − 1.92677i −0.115663 − 0.497390i −0.814831 − 2.50779i −5.03550 − 3.23612i
6.16 −1.59549 + 1.09097i 1.39552 − 1.92077i 0.629760 − 1.61753i −1.20552 − 2.85261i −0.131041 + 4.58704i 0.713670 − 2.54768i −0.115663 − 0.497390i −0.814831 − 2.50779i 5.03550 + 3.23612i
6.17 −1.51558 + 1.03633i −0.173946 + 0.239416i 0.497392 − 1.27754i −0.878361 − 2.07845i 0.0155156 − 0.543118i −0.345045 − 2.62316i −0.261591 − 1.12493i 0.899988 + 2.76988i 3.48518 + 2.23979i
6.18 −1.51558 + 1.03633i 0.173946 − 0.239416i 0.497392 − 1.27754i 0.878361 + 2.07845i −0.0155156 + 0.543118i 2.42452 − 1.05911i −0.261591 − 1.12493i 0.899988 + 2.76988i −3.48518 − 2.23979i
6.19 −1.44240 + 0.986287i −1.40489 + 1.93367i 0.382138 − 0.981513i −0.171087 − 0.404841i 0.119263 − 4.17474i −0.790810 + 2.52480i −0.374686 − 1.61127i −0.838295 − 2.58001i 0.646066 + 0.415201i
6.20 −1.44240 + 0.986287i 1.40489 − 1.93367i 0.382138 − 0.981513i 0.171087 + 0.404841i −0.119263 + 4.17474i −1.75374 + 1.98101i −0.374686 − 1.61127i −0.838295 − 2.58001i −0.646066 − 0.415201i
See next 80 embeddings (of 3360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.84
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
121.h odd 110 1 inner
847.ba even 110 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.ba.b ✓ 3360
7.b odd 2 1 inner 847.2.ba.b ✓ 3360
121.h odd 110 1 inner 847.2.ba.b ✓ 3360
847.ba even 110 1 inner 847.2.ba.b ✓ 3360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.ba.b ✓ 3360 1.a even 1 1 trivial
847.2.ba.b ✓ 3360 7.b odd 2 1 inner
847.2.ba.b ✓ 3360 121.h odd 110 1 inner
847.2.ba.b ✓ 3360 847.ba even 110 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{1680} + 39 T_{2}^{1679} + 842 T_{2}^{1678} + 13088 T_{2}^{1677} + 162441 T_{2}^{1676} + \cdots + 81\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display