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Properties

Label 29.14.a.a
Level $29$
Weight $14$
Character orbit 29.a
Self dual yes
Analytic conductor $31.097$
Analytic rank $1$
Dimension $14$
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] = [29,14,Mod(1,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, 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("29.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 29.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: \(31.0969693961\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 77733 x^{12} - 493192 x^{11} + 2246976740 x^{10} + 33435934528 x^{9} - 29924537865600 x^{8} + \cdots + 93\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{24}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - \beta_1 - 156) q^{3} + ( - \beta_{3} + \beta_{2} + \cdots + 2912) q^{4}+ \cdots + ( - 4 \beta_{13} - \beta_{7} + \cdots + 545548) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - \beta_1 - 156) q^{3} + ( - \beta_{3} + \beta_{2} + \cdots + 2912) q^{4}+ \cdots + (6754789 \beta_{13} + \cdots - 3056381816118) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2188 q^{3} + 40778 q^{4} - 67808 q^{5} - 137306 q^{6} - 669728 q^{7} + 1479576 q^{8} + 7638914 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2188 q^{3} + 40778 q^{4} - 67808 q^{5} - 137306 q^{6} - 669728 q^{7} + 1479576 q^{8} + 7638914 q^{9} - 7503188 q^{10} - 16280180 q^{11} - 46898620 q^{12} - 37136680 q^{13} + 51642432 q^{14} + 49714712 q^{15} + 215723298 q^{16} + 79868564 q^{17} + 318018808 q^{18} - 465401780 q^{19} - 124045178 q^{20} + 78869912 q^{21} - 745450982 q^{22} - 1312479712 q^{23} - 4059224862 q^{24} + 2526740014 q^{25} - 6711277324 q^{26} - 8637726112 q^{27} - 8915008596 q^{28} - 8327526494 q^{29} - 22780170094 q^{30} - 11616204772 q^{31} - 23975950328 q^{32} + 4311019660 q^{33} - 35106028460 q^{34} - 27626054504 q^{35} - 30951028108 q^{36} - 30092534588 q^{37} - 79178540456 q^{38} - 55162483424 q^{39} - 130313334660 q^{40} - 126315491876 q^{41} - 316043733436 q^{42} - 207175772980 q^{43} - 306614754460 q^{44} - 521061430804 q^{45} - 170789610388 q^{46} - 224846938388 q^{47} - 347699531340 q^{48} - 897550882 q^{49} - 536101370280 q^{50} - 466442281920 q^{51} - 471947782662 q^{52} + 243182624080 q^{53} - 420584623766 q^{54} + 212372999168 q^{55} - 138607894272 q^{56} + 352184690624 q^{57} - 386322820120 q^{59} + 819803661600 q^{60} + 60552116484 q^{61} + 1075411063358 q^{62} + 1535344683368 q^{63} - 265778722622 q^{64} + 1944817664196 q^{65} + 1389299047216 q^{66} - 1297283291880 q^{67} + 1205232072244 q^{68} + 2508831703888 q^{69} + 2422232973864 q^{70} + 2109294816512 q^{71} + 9208499294808 q^{72} + 4150692120468 q^{73} + 6798848907024 q^{74} + 2704659106652 q^{75} + 5313845829480 q^{76} + 7477663023736 q^{77} + 15568809877422 q^{78} - 4591599097500 q^{79} + 8812400207918 q^{80} + 10430714802350 q^{81} + 6521746738748 q^{82} - 2836206320632 q^{83} - 3009811841016 q^{84} + 7297205928408 q^{85} + 14037854265898 q^{86} + 1301473426348 q^{87} - 11829684362466 q^{88} + 193234513516 q^{89} + 13809729852752 q^{90} - 18617850475976 q^{91} + 14965878068292 q^{92} - 17788504573980 q^{93} - 45421230915274 q^{94} + 6170458778912 q^{95} - 24238978244334 q^{96} - 17955927241380 q^{97} - 7438858047936 q^{98} - 42793606257044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 77733 x^{12} - 493192 x^{11} + 2246976740 x^{10} + 33435934528 x^{9} - 29924537865600 x^{8} + \cdots + 93\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64\!\cdots\!43 \nu^{13} + \cdots + 91\!\cdots\!08 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 64\!\cdots\!43 \nu^{13} + \cdots + 13\!\cdots\!48 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47\!\cdots\!19 \nu^{13} + \cdots + 93\!\cdots\!64 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!27 \nu^{13} + \cdots - 42\!\cdots\!24 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60\!\cdots\!79 \nu^{13} + \cdots + 11\!\cdots\!28 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 52\!\cdots\!71 \nu^{13} + \cdots + 97\!\cdots\!28 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!19 \nu^{13} + \cdots - 46\!\cdots\!80 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!69 \nu^{13} + \cdots - 53\!\cdots\!92 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\!\cdots\!29 \nu^{13} + \cdots + 47\!\cdots\!04 ) / 95\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 53\!\cdots\!35 \nu^{13} + \cdots - 98\!\cdots\!12 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 90\!\cdots\!42 \nu^{13} + \cdots - 16\!\cdots\!04 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!31 \nu^{13} + \cdots - 28\!\cdots\!52 ) / 23\!\cdots\!60 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 10\beta _1 + 11104 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + 3\beta_{6} - 3\beta_{5} + 7\beta_{4} - 119\beta_{3} - 18\beta_{2} + 19924\beta _1 + 105656 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 90 \beta_{13} + 12 \beta_{12} + 30 \beta_{11} + 10 \beta_{10} + 127 \beta_{9} + 59 \beta_{8} + \cdots + 221187964 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 29323 \beta_{13} + 2056 \beta_{12} + 720 \beta_{11} - 2856 \beta_{10} + 946 \beta_{9} + \cdots + 1749304134 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3281066 \beta_{13} + 222604 \beta_{12} + 1129270 \beta_{11} + 395730 \beta_{10} + \cdots + 5119572935484 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 758411459 \beta_{13} + 81087368 \beta_{12} + 42593152 \beta_{11} - 94423160 \beta_{10} + \cdots + 21272455749886 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 93710813138 \beta_{13} + 1961194444 \beta_{12} + 31184889238 \beta_{11} + 8507793970 \beta_{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 19119494181011 \beta_{13} + 2465004657384 \beta_{12} + 1402953811824 \beta_{11} - 2597016738472 \beta_{10} + \cdots + 13\!\cdots\!06 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 24\!\cdots\!74 \beta_{13} - 37369814008308 \beta_{12} + 780567216245910 \beta_{11} + \cdots + 31\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 47\!\cdots\!75 \beta_{13} + \cdots - 37\!\cdots\!30 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 63\!\cdots\!14 \beta_{13} + \cdots + 78\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12\!\cdots\!47 \beta_{13} + \cdots - 22\!\cdots\!94 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−160.442
−159.092
−98.5312
−77.2894
−51.3126
−49.2799
−39.8225
−7.22662
30.5207
66.9312
105.083
122.192
159.058
159.211
−160.442 654.040 17549.6 40600.4 −104936. 93010.6 −1.50135e6 −1.16655e6 −6.51400e6
1.2 −159.092 −1139.67 17118.4 −4982.53 181313. −444048. −1.42012e6 −295475. 792683.
1.3 −98.5312 633.130 1516.39 −31179.7 −62383.0 281091. 657755. −1.19347e6 3.07218e6
1.4 −77.2894 −2360.04 −2218.34 −69032.4 182406. −285354. 804610. 3.97548e6 5.33548e6
1.5 −51.3126 −196.391 −5559.01 66089.8 10077.4 −535523. 705601. −1.55575e6 −3.39124e6
1.6 −49.2799 2079.99 −5763.49 20688.1 −102502. −212806. 687725. 2.73203e6 −1.01951e6
1.7 −39.8225 −1193.00 −6606.17 −14351.7 47508.1 58441.2 589301. −171082. 571520.
1.8 −7.22662 1912.34 −8139.78 −52589.0 −13819.8 419173. 118024. 2.06272e6 380041.
1.9 30.5207 −2038.84 −7260.49 17678.5 −62226.7 300810. −471621. 2.56253e6 539561.
1.10 66.9312 710.140 −3712.22 30189.9 47530.5 −111854. −796763. −1.09002e6 2.02065e6
1.11 105.083 33.6242 2850.53 761.249 3533.35 152509. −561300. −1.59319e6 79994.7
1.12 122.192 1616.33 6738.90 −55103.7 197503. −340514. −177557. 1.01820e6 −6.73324e6
1.13 159.058 −2267.80 17107.5 −12756.3 −360712. 320365. 1.41808e6 3.54858e6 −2.02900e6
1.14 159.211 −631.860 17156.2 −3820.67 −100599. −365030. 1.42719e6 −1.19508e6 −608293.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)

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 29.14.a.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.14.a.a ✓ 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 77733 T_{2}^{12} - 493192 T_{2}^{11} + 2246976740 T_{2}^{10} + 33435934528 T_{2}^{9} + \cdots + 93\!\cdots\!36 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(29))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 93\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 18\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 78\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 59\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 38\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T + 594823321)^{14} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 37\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 33\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 43\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 22\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 31\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 41\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 19\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 36\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
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