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Properties

Label 197.14.a.b
Level $197$
Weight $14$
Character orbit 197.a
Self dual yes
Analytic conductor $211.245$
Analytic rank $0$
Dimension $109$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(211.244930035\)
Analytic rank: \(0\)
Dimension: \(109\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 109 q + 192 q^{2} + 8018 q^{3} + 471040 q^{4} + 88496 q^{5} + 383232 q^{6} + 1680731 q^{7} + 1820859 q^{8} + 59521391 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 109 q + 192 q^{2} + 8018 q^{3} + 471040 q^{4} + 88496 q^{5} + 383232 q^{6} + 1680731 q^{7} + 1820859 q^{8} + 59521391 q^{9} + 16373653 q^{10} + 21199298 q^{11} + 63225856 q^{12} + 59695238 q^{13} + 37888529 q^{14} + 87246239 q^{15} + 2130706432 q^{16} + 228353715 q^{17} + 400647337 q^{18} + 1139301305 q^{19} + 1109969259 q^{20} + 539982398 q^{21} + 1613315649 q^{22} + 920306804 q^{23} + 5542439613 q^{24} + 31241700999 q^{25} + 1864366110 q^{26} + 17825460755 q^{27} + 20413389070 q^{28} + 7185436621 q^{29} + 2050251883 q^{30} + 28475592572 q^{31} + 8334714660 q^{32} + 19623425846 q^{33} + 37845014194 q^{34} + 25255003636 q^{35} + 287968706746 q^{36} + 71523920490 q^{37} + 67778214914 q^{38} + 44951568463 q^{39} + 169184871486 q^{40} + 69139231052 q^{41} + 58715177635 q^{42} + 247544146139 q^{43} + 63861560722 q^{44} + 257443045479 q^{45} + 160530477869 q^{46} + 308496573061 q^{47} + 412228130018 q^{48} + 1736616239908 q^{49} + 1680360028531 q^{50} + 756579032995 q^{51} + 928015404666 q^{52} + 342783723680 q^{53} - 597894730601 q^{54} + 59276330527 q^{55} - 3822929869144 q^{56} - 562905761941 q^{57} + 62740419347 q^{58} + 827401964151 q^{59} - 2247133283907 q^{60} + 988213134514 q^{61} + 1937380192071 q^{62} + 1788190111357 q^{63} + 11682175668457 q^{64} + 2494670804291 q^{65} + 11819807890512 q^{66} + 8038740399790 q^{67} + 10126245189885 q^{68} + 5225665164579 q^{69} + 11464042631319 q^{70} + 4867145119603 q^{71} + 18133468947055 q^{72} + 9684156738615 q^{73} + 16996786880941 q^{74} + 16718732018262 q^{75} + 21454522032798 q^{76} + 6593100920650 q^{77} + 33749579076633 q^{78} + 7591753073823 q^{79} + 24349241260570 q^{80} + 38778649605417 q^{81} + 25555033184251 q^{82} + 16945724819556 q^{83} + 21855489402730 q^{84} + 15544906794766 q^{85} + 18664144286914 q^{86} + 19049540636401 q^{87} + 17318749473003 q^{88} + 11289674998576 q^{89} + 20983303956671 q^{90} + 47242561944227 q^{91} - 25046698097386 q^{92} - 5411884145985 q^{93} + 18338784709341 q^{94} + 6784117894603 q^{95} - 36827486682955 q^{96} + 45969533477736 q^{97} - 42983409526150 q^{98} + 12084396239183 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 −179.919 949.600 24179.0 −44809.9 −170851. 555727. −2.87636e6 −692583. 8.06216e6
1.2 −177.486 −1858.29 23309.4 5365.93 329822. 583422. −2.68313e6 1.85893e6 −952379.
1.3 −175.365 1521.48 22560.7 48722.0 −266814. 305586. −2.51977e6 720588. −8.54411e6
1.4 −173.852 −154.401 22032.6 10183.6 26843.0 −382540. −2.40622e6 −1.57048e6 −1.77044e6
1.5 −172.035 −63.8588 21404.1 −26311.4 10986.0 −168215. −2.27295e6 −1.59025e6 4.52650e6
1.6 −168.643 977.710 20248.6 23282.0 −164884. −126937. −2.03327e6 −638407. −3.92635e6
1.7 −161.852 −660.311 18003.9 −63584.1 106872. 71804.7 −1.58807e6 −1.15831e6 1.02912e7
1.8 −160.527 1724.90 17577.0 32215.0 −276893. −423232. −1.50655e6 1.38096e6 −5.17138e6
1.9 −158.182 −1675.68 16829.6 7989.80 265063. −477515. −1.36632e6 1.21359e6 −1.26384e6
1.10 −155.970 2509.18 16134.6 −12965.5 −391357. 428722. −1.23881e6 4.70166e6 2.02223e6
1.11 −150.597 −933.812 14487.5 16987.2 140630. 147144. −948089. −722318. −2.55822e6
1.12 −149.393 641.280 14126.2 61634.3 −95802.7 −435746. −886531. −1.18308e6 −9.20772e6
1.13 −145.155 −2025.07 12878.0 −10162.8 293949. 345875. −680200. 2.50658e6 1.47518e6
1.14 −136.248 −2369.27 10371.7 626.226 322809. −79078.9 −296974. 4.01912e6 −85322.4
1.15 −135.924 −2298.95 10283.3 41046.5 312481. 283206. −284251. 3.69083e6 −5.57919e6
1.16 −135.355 −1650.52 10129.0 −19198.7 223406. −557349. −262189. 1.12989e6 2.59865e6
1.17 −134.862 −985.192 9995.81 −46787.8 132865. 482450. −243265. −623719. 6.30990e6
1.18 −133.322 64.3497 9582.75 19563.1 −8579.24 177321. −185418. −1.59018e6 −2.60819e6
1.19 −133.256 1856.77 9565.17 −55057.7 −247426. 87617.4 −182983. 1.85327e6 7.33677e6
1.20 −130.527 1003.60 8845.21 −12039.6 −130997. −78177.1 −85261.9 −587101. 1.57148e6
See next 80 embeddings (of 109 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.109
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 197.14.a.b ✓ 109
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.14.a.b ✓ 109 1.a even 1 1 trivial