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Properties

Label 197.2.g.a
Level $197$
Weight $2$
Character orbit 197.g
Analytic conductor $1.573$
Analytic rank $0$
Dimension $630$
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(16,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(98))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 197.g (of order \(49\), degree \(42\), 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: \(630\)
Relative dimension: \(15\) over \(\Q(\zeta_{49})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{49}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 630 q - 42 q^{2} - 42 q^{3} - 28 q^{4} - 42 q^{5} - 21 q^{6} - 28 q^{7} - 28 q^{8} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 630 q - 42 q^{2} - 42 q^{3} - 28 q^{4} - 42 q^{5} - 21 q^{6} - 28 q^{7} - 28 q^{8} - 42 q^{9} - 28 q^{10} - 28 q^{11} - 28 q^{12} - 42 q^{13} - 28 q^{14} - 154 q^{15} - 42 q^{17} - 21 q^{18} - 7 q^{19} - 35 q^{20} - 42 q^{21} - 42 q^{23} + 28 q^{24} - 42 q^{25} - 7 q^{26} + 42 q^{27} - 49 q^{28} - 182 q^{29} + 77 q^{30} - 14 q^{31} - 28 q^{32} + 35 q^{33} - 14 q^{34} - 42 q^{35} + 84 q^{36} + 14 q^{37} - 28 q^{38} - 21 q^{39} + 28 q^{40} - 28 q^{41} - 112 q^{42} - 56 q^{43} - 210 q^{44} + 49 q^{45} - 7 q^{47} + 42 q^{48} + 42 q^{49} + 98 q^{50} - 14 q^{51} + 49 q^{52} - 42 q^{53} + 63 q^{54} - 161 q^{56} - 175 q^{57} - 42 q^{58} + 42 q^{59} + 126 q^{60} - 14 q^{61} + 42 q^{62} + 49 q^{63} + 98 q^{64} + 56 q^{65} + 175 q^{66} + 28 q^{67} + 273 q^{68} - 7 q^{69} - 252 q^{70} - 77 q^{71} - 364 q^{72} - 28 q^{73} + 35 q^{74} + 7 q^{75} + 133 q^{76} + 21 q^{77} + 210 q^{78} + 42 q^{79} - 14 q^{80} - 14 q^{81} - 42 q^{82} + 161 q^{83} - 245 q^{84} - 266 q^{85} - 147 q^{86} - 35 q^{87} + 77 q^{88} + 28 q^{89} + 315 q^{90} + 154 q^{91} + 84 q^{92} - 7 q^{93} + 182 q^{94} + 126 q^{95} + 245 q^{96} + 42 q^{97} - 308 q^{98} - 161 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.48397 − 0.159476i −1.29377 + 1.85472i 4.16109 + 0.536513i −0.977512 + 0.636290i 3.50947 − 4.40074i 5.00407 − 0.321272i −5.36410 − 1.04466i −0.730042 − 1.98376i 2.52958 − 1.42464i
16.2 −2.15065 − 0.138077i −0.639435 + 0.916678i 2.62267 + 0.338156i −0.0368843 + 0.0240090i 1.50178 − 1.88317i −3.79102 + 0.243391i −1.36308 − 0.265461i 0.604674 + 1.64309i 0.0826405 − 0.0465422i
16.3 −2.04022 − 0.130987i 0.485195 − 0.695563i 2.16177 + 0.278730i −1.49439 + 0.972737i −1.08102 + 1.35555i −0.633099 + 0.0406463i −0.360546 − 0.0702164i 0.787701 + 2.14044i 3.17630 − 1.78886i
16.4 −1.70486 − 0.109455i 1.44109 − 2.06591i 0.910971 + 0.117457i 1.97200 − 1.28363i −2.68298 + 3.36435i 1.14000 − 0.0731907i 1.81351 + 0.353180i −1.15516 − 3.13893i −3.50247 + 1.97255i
16.5 −1.15219 − 0.0739728i −1.26737 + 1.81686i −0.661518 − 0.0852934i 3.21973 − 2.09581i 1.59464 − 1.99962i −0.487946 + 0.0313272i 3.02242 + 0.588616i −0.658685 − 1.78986i −3.86476 + 2.17659i
16.6 −0.506206 − 0.0324995i 1.73419 − 2.48609i −1.72839 − 0.222851i −3.12927 + 2.03693i −0.958653 + 1.20211i −4.30048 + 0.276100i 1.86347 + 0.362910i −2.13714 − 5.80729i 1.65025 − 0.929406i
16.7 −0.484111 − 0.0310810i 0.216468 − 0.310323i −1.75018 − 0.225661i −2.33159 + 1.51769i −0.114440 + 0.143503i 2.93627 − 0.188515i 1.79259 + 0.349107i 0.986653 + 2.68106i 1.17592 − 0.662265i
16.8 −0.335970 − 0.0215700i 0.0454721 − 0.0651877i −1.87117 − 0.241261i 0.983419 − 0.640135i −0.0166834 + 0.0209203i 2.21855 − 0.142436i 1.28436 + 0.250129i 1.03391 + 2.80948i −0.344207 + 0.193854i
16.9 0.791569 + 0.0508204i −0.962249 + 1.37946i −1.35958 − 0.175299i −1.15157 + 0.749590i −0.831792 + 1.04303i −2.60996 + 0.167565i −2.62444 − 0.511109i 0.0591181 + 0.160643i −0.949643 + 0.534829i
16.10 1.01041 + 0.0648703i 1.00307 − 1.43798i −0.966867 − 0.124664i 2.59562 − 1.68956i 1.10679 − 1.38788i −2.13216 + 0.136889i −2.95647 − 0.575773i −0.0255388 − 0.0693973i 2.73224 − 1.53877i
16.11 1.37861 + 0.0885098i 1.41762 − 2.03227i −0.0908444 − 0.0117131i −0.970092 + 0.631460i 2.13422 − 2.67623i 2.91441 − 0.187111i −2.83615 − 0.552341i −1.08436 − 2.94655i −1.39327 + 0.784675i
16.12 1.55472 + 0.0998163i −0.748244 + 1.07266i 0.423609 + 0.0546184i 2.57534 − 1.67636i −1.27038 + 1.59300i 2.62646 − 0.168624i −2.40524 − 0.468420i 0.445357 + 1.21018i 4.17126 − 2.34921i
16.13 2.18236 + 0.140112i −1.06546 + 1.52741i 2.75947 + 0.355794i −3.47190 + 2.25995i −2.53922 + 3.18408i 4.33586 − 0.278371i 1.67925 + 0.327034i −0.161699 − 0.439389i −7.89356 + 4.44557i
16.14 2.30859 + 0.148217i 0.781712 − 1.12064i 3.32405 + 0.428589i −1.06628 + 0.694069i 1.97075 − 2.47124i −3.00952 + 0.193218i 3.06898 + 0.597684i 0.391328 + 1.06337i −2.56447 + 1.44428i
16.15 2.56869 + 0.164915i −1.80293 + 2.58464i 4.58739 + 0.591479i 1.88168 − 1.22484i −5.05742 + 6.34180i −4.01439 + 0.257732i 6.63302 + 1.29178i −2.39370 − 6.50444i 5.03544 − 2.83591i
23.1 −0.871876 + 2.36917i 1.75406 − 1.14176i −3.32991 − 2.83479i −0.0953576 − 2.97360i 1.17571 + 5.15114i −1.06898 − 2.90477i 5.22007 − 2.93989i 0.558739 − 1.26220i 7.12811 + 2.36670i
23.2 −0.835546 + 2.27045i 0.441713 − 0.287523i −2.93391 − 2.49767i 0.0569457 + 1.77578i 0.283735 + 1.24313i 0.530984 + 1.44285i 3.90627 − 2.19997i −1.10191 + 2.48923i −4.07939 − 1.35445i
23.3 −0.612843 + 1.66529i −1.31213 + 0.854099i −0.874733 − 0.744671i 0.0995611 + 3.10468i −0.618198 − 2.70850i −0.901602 − 2.44994i −1.31610 + 0.741216i −0.222161 + 0.501866i −5.23122 − 1.73689i
23.4 −0.567149 + 1.54113i −1.88076 + 1.22424i −0.530525 − 0.451643i −0.112394 − 3.50484i −0.820039 − 3.59283i −1.15309 − 3.13332i −1.86479 + 1.05023i 0.824156 − 1.86178i 5.46516 + 1.81456i
23.5 −0.556348 + 1.51178i −0.127734 + 0.0831453i −0.453058 − 0.385694i −0.0999221 − 3.11594i −0.0546329 − 0.239363i 1.75292 + 4.76324i −1.97207 + 1.11065i −1.20495 + 2.72200i 4.76620 + 1.58249i
See next 80 embeddings (of 630 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
197.g even 49 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.2.g.a ✓ 630
197.g even 49 1 inner 197.2.g.a ✓ 630
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.2.g.a ✓ 630 1.a even 1 1 trivial
197.2.g.a ✓ 630 197.g even 49 1 inner

Hecke kernels

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