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Properties

Label 197.2.d.a
Level $197$
Weight $2$
Character orbit 197.d
Analytic conductor $1.573$
Analytic rank $0$
Dimension $90$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,2,Mod(36,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 197.d (of order \(7\), degree \(6\), 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: \(1.57305291982\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(15\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 90 q - 2 q^{2} - 5 q^{3} - 26 q^{4} + q^{5} - 10 q^{6} - 11 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 90 q - 2 q^{2} - 5 q^{3} - 26 q^{4} + q^{5} - 10 q^{6} - 11 q^{7} - 16 q^{9} + 9 q^{10} - 7 q^{11} + 15 q^{12} + 9 q^{13} - 40 q^{14} - 16 q^{15} - 10 q^{16} + 13 q^{17} + 15 q^{18} - 20 q^{19} + 26 q^{20} + 13 q^{21} - 17 q^{22} + 25 q^{23} - 17 q^{24} + 16 q^{25} - 2 q^{26} - 41 q^{27} - 50 q^{28} - 33 q^{29} - 72 q^{30} - 15 q^{31} + 46 q^{32} - 32 q^{33} + 31 q^{34} + 45 q^{35} - 64 q^{36} - 27 q^{37} + 43 q^{38} + 14 q^{39} + 31 q^{40} + 35 q^{41} + 14 q^{42} + 16 q^{43} + 49 q^{44} - 8 q^{45} + 17 q^{46} - 4 q^{47} + 41 q^{48} - 48 q^{49} - 52 q^{50} + 51 q^{51} - 2 q^{52} + 41 q^{53} + 14 q^{54} + 53 q^{55} + 52 q^{56} - 104 q^{57} - 79 q^{58} - 17 q^{59} + 11 q^{60} + 21 q^{61} + 23 q^{62} + 48 q^{63} + 22 q^{64} + 5 q^{65} - 94 q^{66} - 5 q^{67} - 200 q^{68} + 54 q^{69} + 64 q^{70} - 94 q^{71} + 75 q^{72} + 31 q^{73} + 18 q^{74} + 36 q^{75} - 80 q^{76} + 22 q^{77} - 45 q^{78} - 25 q^{79} + 131 q^{80} + 6 q^{81} + 51 q^{82} - 130 q^{83} + 108 q^{84} - 129 q^{85} + 32 q^{86} + 98 q^{87} + 54 q^{88} + 17 q^{89} - 42 q^{90} - 119 q^{91} + 31 q^{92} + 60 q^{93} - 111 q^{94} - 77 q^{95} - 98 q^{96} + 49 q^{97} + 45 q^{98} + 25 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −2.45864 + 1.18402i 0.124979 + 0.0601869i 3.39601 − 4.25846i −0.578930 − 0.725956i −0.378541 −0.626559 − 0.301735i −2.09300 + 9.17002i −1.85847 − 2.33045i 2.28292 + 1.09940i
36.2 −2.08071 + 1.00202i 2.19341 + 1.05629i 2.07835 − 2.60617i 2.34290 + 2.93790i −5.62228 −0.184771 − 0.0889812i −0.685233 + 3.00220i 1.82483 + 2.28827i −7.81874 − 3.76531i
36.3 −1.81302 + 0.873105i −1.68155 − 0.809790i 1.27775 − 1.60225i 1.13035 + 1.41742i 3.75571 −2.93980 − 1.41573i −0.0221004 + 0.0968284i 0.301366 + 0.377901i −3.28690 − 1.58289i
36.4 −1.70417 + 0.820685i 0.577242 + 0.277985i 0.983693 − 1.23351i −1.63883 − 2.05503i −1.21186 1.92605 + 0.927536i 0.177735 − 0.778708i −1.61454 − 2.02456i 4.47938 + 2.15715i
36.5 −0.948843 + 0.456939i −1.49781 − 0.721309i −0.555469 + 0.696537i 0.812989 + 1.01946i 1.75078 −0.337287 − 0.162429i 0.677469 − 2.96818i −0.147309 − 0.184720i −1.23723 − 0.595818i
36.6 −0.593780 + 0.285950i 2.71133 + 1.30571i −0.976172 + 1.22408i −0.811676 − 1.01781i −1.98330 1.33888 + 0.644768i 0.522910 − 2.29102i 3.77595 + 4.73489i 0.773000 + 0.372257i
36.7 −0.448644 + 0.216056i −2.62086 − 1.26214i −1.09238 + 1.36980i −1.50240 − 1.88395i 1.44852 3.49725 + 1.68419i 0.415748 − 1.82151i 3.40543 + 4.27028i 1.08108 + 0.520620i
36.8 −0.120109 + 0.0578412i −0.0851283 − 0.0409956i −1.23590 + 1.54977i −1.69772 − 2.12888i 0.0125959 −3.18358 − 1.53313i 0.118130 − 0.517563i −1.86490 − 2.33851i 0.327048 + 0.157498i
36.9 −0.0351696 + 0.0169368i 1.43946 + 0.693208i −1.24603 + 1.56247i 1.82344 + 2.28652i −0.0623661 −1.43652 − 0.691792i 0.0347315 − 0.152169i −0.278956 − 0.349800i −0.102856 − 0.0495329i
36.10 0.857171 − 0.412792i −2.30589 − 1.11046i −0.682634 + 0.855996i 2.52569 + 3.16712i −2.43493 0.566338 + 0.272734i −0.655194 + 2.87059i 2.21355 + 2.77571i 3.47231 + 1.67218i
36.11 0.887644 − 0.427467i 0.442784 + 0.213234i −0.641795 + 0.804785i 0.198374 + 0.248753i 0.484186 3.85822 + 1.85802i −0.664127 + 2.90973i −1.71988 − 2.15666i 0.282420 + 0.136006i
36.12 1.52637 − 0.735059i −2.33495 − 1.12445i 0.542502 − 0.680276i −1.58350 − 1.98564i −4.39053 −1.94366 − 0.936020i −0.425949 + 1.86621i 2.31713 + 2.90559i −3.87657 − 1.86686i
36.13 1.66827 − 0.803398i 2.04773 + 0.986133i 0.890709 − 1.11691i −1.79031 − 2.24498i 4.20843 0.174909 + 0.0842317i −0.235439 + 1.03153i 1.35026 + 1.69317i −4.79034 − 2.30690i
36.14 1.85821 − 0.894867i 0.813278 + 0.391654i 1.40518 − 1.76204i 1.13765 + 1.42657i 1.86172 −3.23536 − 1.55807i 0.116449 − 0.510195i −1.36244 − 1.70845i 3.39059 + 1.63282i
36.15 2.28193 − 1.09892i −1.44751 − 0.697085i 2.75260 − 3.45165i 0.532937 + 0.668281i −4.06916 1.25930 + 0.606448i 1.36097 − 5.96281i −0.261106 − 0.327417i 1.95051 + 0.939317i
104.1 −2.45864 − 1.18402i 0.124979 − 0.0601869i 3.39601 + 4.25846i −0.578930 + 0.725956i −0.378541 −0.626559 + 0.301735i −2.09300 − 9.17002i −1.85847 + 2.33045i 2.28292 − 1.09940i
104.2 −2.08071 − 1.00202i 2.19341 − 1.05629i 2.07835 + 2.60617i 2.34290 − 2.93790i −5.62228 −0.184771 + 0.0889812i −0.685233 − 3.00220i 1.82483 − 2.28827i −7.81874 + 3.76531i
104.3 −1.81302 − 0.873105i −1.68155 + 0.809790i 1.27775 + 1.60225i 1.13035 − 1.41742i 3.75571 −2.93980 + 1.41573i −0.0221004 − 0.0968284i 0.301366 − 0.377901i −3.28690 + 1.58289i
104.4 −1.70417 − 0.820685i 0.577242 − 0.277985i 0.983693 + 1.23351i −1.63883 + 2.05503i −1.21186 1.92605 − 0.927536i 0.177735 + 0.778708i −1.61454 + 2.02456i 4.47938 − 2.15715i
104.5 −0.948843 − 0.456939i −1.49781 + 0.721309i −0.555469 − 0.696537i 0.812989 − 1.01946i 1.75078 −0.337287 + 0.162429i 0.677469 + 2.96818i −0.147309 + 0.184720i −1.23723 + 0.595818i
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
197.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.2.d.a ✓ 90
197.d even 7 1 inner 197.2.d.a ✓ 90
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.2.d.a ✓ 90 1.a even 1 1 trivial
197.2.d.a ✓ 90 197.d even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(197, [\chi])\).