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Properties

Label 197.14.a.a
Level $197$
Weight $14$
Character orbit 197.a
Self dual yes
Analytic conductor $211.245$
Analytic rank $1$
Dimension $104$
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: \(1\)
Dimension: \(104\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 128 q^{2} - 8020 q^{3} + 409600 q^{4} - 99004 q^{5} - 83328 q^{6} - 2084037 q^{7} - 2111301 q^{8} + 51549776 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 128 q^{2} - 8020 q^{3} + 409600 q^{4} - 99004 q^{5} - 83328 q^{6} - 2084037 q^{7} - 2111301 q^{8} + 51549776 q^{9} - 9626347 q^{10} - 10688800 q^{11} - 68157440 q^{12} - 94762650 q^{13} - 52465903 q^{14} - 186128761 q^{15} + 1543503872 q^{16} - 109572251 q^{17} - 449658263 q^{18} - 1495268031 q^{19} - 1194030741 q^{20} - 832275538 q^{21} - 2695120703 q^{22} - 2632554532 q^{23} - 190649667 q^{24} + 22696779124 q^{25} - 2460454754 q^{26} - 16267542277 q^{27} - 18137835250 q^{28} + 47556769 q^{29} - 9613748117 q^{30} - 24774628288 q^{31} - 29246249180 q^{32} - 42367036666 q^{33} - 39395206606 q^{34} - 26216433864 q^{35} + 190013501626 q^{36} - 82419664050 q^{37} - 34593622142 q^{38} - 39498281801 q^{39} - 150303128514 q^{40} - 25862853768 q^{41} - 138889965149 q^{42} - 258164897781 q^{43} - 197365738094 q^{44} - 240782892021 q^{45} - 218441397971 q^{46} - 208905762731 q^{47} - 664063830814 q^{48} + 1113758315863 q^{49} - 1516068087607 q^{50} - 1019253294393 q^{51} - 1162941441840 q^{52} + 161684855900 q^{53} + 1679137315823 q^{54} - 557952233701 q^{55} + 2842561845328 q^{56} + 801593765429 q^{57} - 7249760433 q^{58} - 775487110641 q^{59} - 57598844627 q^{60} - 36786715662 q^{61} + 681529132643 q^{62} - 4349115033663 q^{63} + 4600420238797 q^{64} - 2531819073161 q^{65} - 3958922748734 q^{66} - 7314072405766 q^{67} - 10297353950393 q^{68} - 2089200206275 q^{69} - 14268943913713 q^{70} - 6055724651085 q^{71} - 26860198563805 q^{72} - 7572811533391 q^{73} - 11175265675817 q^{74} - 24431657592434 q^{75} - 26425925198106 q^{76} - 9735327037686 q^{77} - 35310017230907 q^{78} - 8981210762721 q^{79} - 25913753436330 q^{80} + 15437375349812 q^{81} - 19721330628227 q^{82} - 22209909714532 q^{83} - 18837805278768 q^{84} - 5905005171430 q^{85} - 8913876797772 q^{86} - 7341562344401 q^{87} - 35154329886441 q^{88} - 4646484034146 q^{89} + 9564902968095 q^{90} - 22752431457047 q^{91} - 13601121953458 q^{92} - 9615114240293 q^{93} + 15352521272967 q^{94} + 5258551767043 q^{95} + 87277848810881 q^{96} - 42298840804040 q^{97} + 27000102375354 q^{98} - 8666459567773 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 −178.085 −2212.53 23522.4 69269.3 394020. −342836. −2.73012e6 3.30098e6 −1.23359e7
1.2 −174.990 1982.02 22429.6 −59494.1 −346834. −613520. −2.49144e6 2.33409e6 1.04109e7
1.3 −169.954 −1349.68 20692.5 −8715.05 229384. −22887.7 −2.12452e6 227318. 1.48116e6
1.4 −168.777 −2402.96 20293.8 −64762.9 405565. −252510. −2.04252e6 4.17988e6 1.09305e7
1.5 −167.056 2054.37 19715.7 12824.5 −343195. 34898.6 −1.92511e6 2.62611e6 −2.14242e6
1.6 −164.231 −1747.83 18780.0 −30963.6 287048. 120554. −1.73888e6 1.46057e6 5.08520e6
1.7 −164.173 −6.28213 18760.6 49948.6 1031.35 199919. −1.73508e6 −1.59428e6 −8.20019e6
1.8 −162.329 −746.765 18158.7 42494.0 121222. 350398. −1.61789e6 −1.03666e6 −6.89802e6
1.9 −156.355 1756.50 16255.0 −30943.3 −274637. 65618.9 −1.26069e6 1.49095e6 4.83814e6
1.10 −156.182 145.398 16200.7 −42219.2 −22708.4 140720. −1.25081e6 −1.57318e6 6.59387e6
1.11 −154.569 1022.51 15699.6 −29720.6 −158048. 28477.1 −1.16045e6 −548801. 4.59389e6
1.12 −147.901 534.557 13682.8 −7800.19 −79061.7 515509. −812098. −1.30857e6 1.15366e6
1.13 −146.034 −203.966 13133.8 −50263.7 29785.9 −418854. −721672. −1.55272e6 7.34019e6
1.14 −145.508 −1012.29 12980.6 58530.7 147296. 112683. −696787. −569599. −8.51670e6
1.15 −140.225 −1671.46 11470.9 31759.0 234379. −135007. −459789. 1.19945e6 −4.45339e6
1.16 −138.688 508.810 11042.3 10487.4 −70565.7 498775. −395298. −1.33544e6 −1.45448e6
1.17 −136.815 1868.60 10526.4 67606.5 −255652. 369826. −319377. 1.89733e6 −9.24960e6
1.18 −135.981 2210.13 10298.9 24727.3 −300537. −333758. −286501. 3.29036e6 −3.36245e6
1.19 −130.362 −354.130 8802.19 43213.6 46165.1 −431996. −79545.1 −1.46891e6 −5.63341e6
1.20 −130.183 784.591 8755.73 −22486.1 −102141. −529213. −73387.7 −978739. 2.92731e6
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.104
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.a ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
197.14.a.a ✓ 104 1.a even 1 1 trivial