↑

Properties

Label 197.10.a.b
Level $197$
Weight $10$
Character orbit 197.a
Self dual yes
Analytic conductor $101.462$
Analytic rank $0$
Dimension $76$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,10,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 197.a (trivial)

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: yes
Analytic conductor: \(101.462059724\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 48 q^{2} + 890 q^{3} + 20736 q^{4} + 5171 q^{5} + 2688 q^{6} + 38986 q^{7} + 36507 q^{8} + 518318 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q + 48 q^{2} + 890 q^{3} + 20736 q^{4} + 5171 q^{5} + 2688 q^{6} + 38986 q^{7} + 36507 q^{8} + 518318 q^{9} + 121093 q^{10} + 120464 q^{11} + 415744 q^{12} + 480131 q^{13} + 330849 q^{14} + 544874 q^{15} + 5963776 q^{16} + 942582 q^{17} + 1483945 q^{18} + 3097319 q^{19} + 3237739 q^{20} + 889076 q^{21} + 3791921 q^{22} + 5139200 q^{23} + 1999533 q^{24} + 34080519 q^{25} + 2791454 q^{26} + 24386486 q^{27} + 20891166 q^{28} + 6886818 q^{29} + 14171083 q^{30} + 28002851 q^{31} + 16857332 q^{32} + 30921422 q^{33} + 33506194 q^{34} + 16271736 q^{35} + 151430458 q^{36} + 55950976 q^{37} + 62370882 q^{38} - 11569592 q^{39} + 129854766 q^{40} + 14990859 q^{41} + 82216531 q^{42} + 169867467 q^{43} + 41872434 q^{44} + 205007649 q^{45} + 144032301 q^{46} + 78743342 q^{47} + 156250562 q^{48} + 533861890 q^{49} + 626841163 q^{50} + 477099244 q^{51} + 560784114 q^{52} + 188670216 q^{53} + 525901687 q^{54} + 298497914 q^{55} + 56575048 q^{56} + 213972590 q^{57} + 338315251 q^{58} + 208222151 q^{59} - 615921507 q^{60} - 233556134 q^{61} - 399368105 q^{62} + 329825056 q^{63} + 876517017 q^{64} - 840557006 q^{65} - 2482481592 q^{66} + 1210808414 q^{67} - 1266757099 q^{68} + 327801786 q^{69} - 546384313 q^{70} + 345300221 q^{71} - 1549481681 q^{72} + 1192286460 q^{73} - 1471133595 q^{74} + 761630676 q^{75} - 398699826 q^{76} - 101106252 q^{77} - 2609825943 q^{78} + 955627631 q^{79} + 1059617770 q^{80} + 3387041436 q^{81} + 1062705523 q^{82} + 1538917201 q^{83} + 1394513218 q^{84} + 225481100 q^{85} + 701644810 q^{86} + 1758812842 q^{87} + 3151474875 q^{88} + 855413630 q^{89} + 6070671455 q^{90} + 4652436248 q^{91} + 8082863606 q^{92} + 3462095982 q^{93} + 2660342117 q^{94} + 1036805508 q^{95} + 12370989029 q^{96} + 6393874545 q^{97} + 7510976010 q^{98} + 8731109606 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −44.8173 −222.555 1496.59 1221.65 9974.32 2695.26 −44126.8 29847.7 −54751.2
1.2 −43.4563 176.827 1376.45 −1584.29 −7684.26 6821.55 −37565.9 11584.9 68847.5
1.3 −41.6400 9.57611 1221.89 18.3099 −398.750 11241.8 −29560.0 −19591.3 −762.427
1.4 −40.7455 267.346 1148.19 −514.109 −10893.1 −8392.30 −25921.9 51791.0 20947.6
1.5 −39.6398 96.9325 1059.31 −2160.11 −3842.38 −6397.21 −21695.3 −10287.1 85626.1
1.6 −39.0491 −79.6019 1012.83 −1275.42 3108.38 8453.17 −19557.0 −13346.5 49804.1
1.7 −38.8449 200.986 996.927 −347.642 −7807.27 −7838.83 −18836.9 20712.3 13504.1
1.8 −38.2269 −153.641 949.295 2107.84 5873.22 5202.74 −16716.4 3922.60 −80576.2
1.9 −37.7253 −206.276 911.195 −2075.65 7781.80 −5253.06 −15059.7 22866.6 78304.4
1.10 −36.2963 −223.000 805.418 837.898 8094.08 −7154.81 −10650.0 30046.2 −30412.5
1.11 −35.8261 −60.4243 771.507 1939.07 2164.76 −570.567 −9297.10 −16031.9 −69469.3
1.12 −35.3221 187.873 735.651 2090.93 −6636.07 −77.6611 −7899.83 15613.2 −73855.9
1.13 −34.7070 −78.4883 692.576 −317.779 2724.09 −4384.10 −6267.25 −13522.6 11029.1
1.14 −33.0699 245.298 581.616 1345.23 −8111.96 6878.64 −2302.20 40487.9 −44486.7
1.15 −32.1699 137.841 522.904 −143.135 −4434.35 −2652.42 −350.769 −682.762 4604.64
1.16 −31.2026 46.5765 461.602 1445.56 −1453.31 12293.0 1572.55 −17513.6 −45105.3
1.17 −28.5607 −17.8798 303.711 −2276.73 510.659 4449.75 5948.86 −19363.3 65025.0
1.18 −25.7900 −52.5034 153.122 2513.73 1354.06 −2435.70 9255.45 −16926.4 −64829.0
1.19 −25.7510 108.855 151.115 −962.546 −2803.12 −9096.95 9293.15 −7833.66 24786.5
1.20 −25.4393 83.7395 135.160 −347.412 −2130.28 1070.50 9586.56 −12670.7 8837.93
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.76
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(197\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 197.10.a.b ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.10.a.b ✓ 76 1.a even 1 1 trivial