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Properties

Label 29.12.a.b
Level $29$
Weight $12$
Character orbit 29.a
Self dual yes
Analytic conductor $22.282$
Analytic rank $0$
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] = [29,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(22.2819522362\)
Analytic rank: \(0\)
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: \( x^{14} - 23517 x^{12} - 42196 x^{11} + 214206700 x^{10} + 532863376 x^{9} - 951901011680 x^{8} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 34) q^{3} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 1312) q^{4} + ( - \beta_{5} + \beta_{3} + \beta_{2} + \cdots + 698) q^{5}+ \cdots + (9 \beta_{12} - 6 \beta_{11} + \cdots + 98018) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 34) q^{3} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 1312) q^{4} + ( - \beta_{5} + \beta_{3} + \beta_{2} + \cdots + 698) q^{5}+ \cdots + (261501 \beta_{13} + \cdots + 16813511439) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 476 q^{3} + 18362 q^{4} + 9760 q^{5} + 18454 q^{6} + 85024 q^{7} + 126588 q^{8} + 1372146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 476 q^{3} + 18362 q^{4} + 9760 q^{5} + 18454 q^{6} + 85024 q^{7} + 126588 q^{8} + 1372146 q^{9} + 713576 q^{10} + 398020 q^{11} - 4026800 q^{12} + 2272440 q^{13} - 7199712 q^{14} - 4763864 q^{15} + 19015138 q^{16} + 5623508 q^{17} - 204156 q^{18} + 29803300 q^{19} + 65161006 q^{20} + 51227832 q^{21} + 167334266 q^{22} + 52654304 q^{23} + 221514842 q^{24} + 194970462 q^{25} + 373581536 q^{26} + 397348256 q^{27} + 319501772 q^{28} - 287156086 q^{29} + 423014226 q^{30} + 634041348 q^{31} + 1260290884 q^{32} + 1180833420 q^{33} + 1316105060 q^{34} + 1599853768 q^{35} + 3198076132 q^{36} + 488665204 q^{37} + 1892845072 q^{38} + 1972619104 q^{39} + 1826486880 q^{40} + 198215164 q^{41} + 1011384468 q^{42} + 2193188100 q^{43} + 26522720 q^{44} - 1129321956 q^{45} - 1567525268 q^{46} - 4175934476 q^{47} - 15582938120 q^{48} + 1105222462 q^{49} - 6630582612 q^{50} + 3297462720 q^{51} - 4557341374 q^{52} - 13223081840 q^{53} - 8946135054 q^{54} - 2726359424 q^{55} - 27538267872 q^{56} - 24477013312 q^{57} + 352219640 q^{59} - 36042747924 q^{60} - 7658546476 q^{61} - 10024135594 q^{62} - 23037581736 q^{63} + 14721327762 q^{64} + 1152802884 q^{65} - 99505241364 q^{66} + 21781534280 q^{67} - 104178000188 q^{68} - 14601399408 q^{69} - 67948872984 q^{70} - 5573287168 q^{71} - 24062143544 q^{72} + 39661511924 q^{73} + 28506052056 q^{74} + 81845109044 q^{75} + 166950090320 q^{76} + 38773567192 q^{77} + 54249159006 q^{78} + 105565209020 q^{79} + 146242150550 q^{80} + 170581084750 q^{81} + 47345182756 q^{82} + 127846064024 q^{83} + 215311861496 q^{84} + 83883234552 q^{85} - 103162039382 q^{86} - 9763306924 q^{87} + 418253082102 q^{88} + 187826099404 q^{89} + 96335639960 q^{90} + 58390389864 q^{91} - 259645875396 q^{92} + 394641636020 q^{93} + 117694719934 q^{94} + 69935059424 q^{95} + 12533631786 q^{96} + 137285937500 q^{97} - 484896369168 q^{98} + 235419947204 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 23517 x^{12} - 42196 x^{11} + 214206700 x^{10} + 532863376 x^{9} - 951901011680 x^{8} + \cdots + 30\!\cdots\!04 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 65\!\cdots\!65 \nu^{13} + \cdots - 18\!\cdots\!92 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65\!\cdots\!65 \nu^{13} + \cdots + 36\!\cdots\!92 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!87 \nu^{13} + \cdots + 25\!\cdots\!28 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\!\cdots\!27 \nu^{13} + \cdots + 18\!\cdots\!96 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!69 \nu^{13} + \cdots + 67\!\cdots\!80 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 36\!\cdots\!97 \nu^{13} + \cdots - 15\!\cdots\!36 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!69 \nu^{13} + \cdots - 80\!\cdots\!08 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40\!\cdots\!83 \nu^{13} + \cdots + 11\!\cdots\!20 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!83 \nu^{13} + \cdots - 72\!\cdots\!48 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!51 \nu^{13} + \cdots - 84\!\cdots\!60 ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 44\!\cdots\!71 \nu^{13} + \cdots + 32\!\cdots\!24 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!45 \nu^{13} + \cdots + 11\!\cdots\!92 ) / 44\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta _1 + 3360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - 55 \beta_{3} + \cdots + 9049 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 66 \beta_{13} + 117 \beta_{12} - 67 \beta_{11} + 63 \beta_{10} - 92 \beta_{9} + 15 \beta_{8} + \cdots + 17808144 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2552 \beta_{13} - 756 \beta_{12} + 1370 \beta_{11} + 6217 \beta_{10} + 22124 \beta_{9} + \cdots + 164162017 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 788730 \beta_{13} + 1295329 \beta_{12} - 623431 \beta_{11} + 826663 \beta_{10} - 1092372 \beta_{9} + \cdots + 107440043028 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 35663488 \beta_{13} - 2910048 \beta_{12} + 18768470 \beta_{11} + 37096933 \beta_{10} + \cdots + 1603640220849 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7278434650 \beta_{13} + 11269781201 \beta_{12} - 4903315199 \beta_{11} + 7944783867 \beta_{10} + \cdots + 698080000353096 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 366122300872 \beta_{13} + 28779169172 \beta_{12} + 178183947178 \beta_{11} + 247620648481 \beta_{10} + \cdots + 13\!\cdots\!45 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 61606633335418 \beta_{13} + 90523428541985 \beta_{12} - 36795179509175 \beta_{11} + \cdots + 47\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 33\!\cdots\!52 \beta_{13} + 628476013502792 \beta_{12} + \cdots + 10\!\cdots\!05 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 50\!\cdots\!46 \beta_{13} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 28\!\cdots\!04 \beta_{13} + \cdots + 82\!\cdots\!33 \) 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
−85.4920
−67.9786
−62.7554
−61.8148
−55.2092
−26.9644
6.06439
8.22600
20.2914
35.2896
51.1787
73.5071
77.1568
88.5004
−85.4920 −725.551 5260.89 −1144.42 62028.9 2737.98 −274676. 349278. 97838.7
1.2 −67.9786 −249.255 2573.09 12131.4 16944.0 74935.0 −35694.6 −115019. −824677.
1.3 −62.7554 761.773 1890.24 1400.24 −47805.4 77426.1 9900.16 403151. −87872.8
1.4 −61.8148 211.836 1773.06 −11752.2 −13094.6 18695.9 16995.1 −132273. 726462.
1.5 −55.2092 −283.770 1000.05 5132.07 15666.7 −31764.7 57856.2 −96621.5 −283337.
1.6 −26.9644 724.855 −1320.92 −9762.32 −19545.3 −73001.2 90841.0 348267. 263235.
1.7 6.06439 −282.467 −2011.22 −8656.32 −1712.99 −42605.5 −24616.7 −97359.2 −52495.3
1.8 8.22600 −223.329 −1980.33 8469.53 −1837.11 2360.62 −33137.1 −127271. 69670.4
1.9 20.2914 719.778 −1636.26 12035.6 14605.3 8773.61 −74758.8 340933. 244220.
1.10 35.2896 118.146 −802.642 −5887.06 4169.31 66890.4 −100598. −163189. −207752.
1.11 51.1787 −728.829 571.255 −4173.93 −37300.5 −8856.29 −75577.8 354044. −213616.
1.12 73.5071 274.559 3355.30 7553.66 20182.1 46751.0 96095.7 −101764. 555248.
1.13 77.1568 692.210 3905.18 −3176.79 53408.8 −18569.7 143294. 302008. −245111.
1.14 88.5004 −533.955 5784.31 7590.53 −47255.2 −38749.2 330665. 107961. 671765.
\(n\): 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 29.12.a.b ✓ 14
3.b odd 2 1 261.12.a.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.12.a.b ✓ 14 1.a even 1 1 trivial
261.12.a.e 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 23517 T_{2}^{12} - 42196 T_{2}^{11} + 214206700 T_{2}^{10} + 532863376 T_{2}^{9} + \cdots + 30\!\cdots\!04 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(29))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots - 23\!\cdots\!20 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 35\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 23\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T + 20511149)^{14} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 97\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 35\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 63\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 45\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 76\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 31\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 49\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 17\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 62\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 91\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 35\!\cdots\!48 \) Copy content Toggle raw display
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