Properties

Label 6292.2.a.z
Level $6292$
Weight $2$
Character orbit 6292.a
Self dual yes
Analytic conductor $50.242$
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] = [6292,2,Mod(1,6292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6292, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6292.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6292.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: \(50.2418729518\)
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: \( x^{14} - 3 x^{13} - 31 x^{12} + 97 x^{11} + 339 x^{10} - 1140 x^{9} - 1495 x^{8} + 5888 x^{7} + 1859 x^{6} - 12750 x^{5} + 2100 x^{4} + 9280 x^{3} - 3480 x^{2} - 800 x + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 572)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{11} + 1) q^{5} + (\beta_{8} - \beta_{5}) q^{7} + (\beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{11} + 1) q^{5} + (\beta_{8} - \beta_{5}) q^{7} + (\beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{4} + \beta_{3} + 2) q^{9} - q^{13} + (\beta_{13} + \beta_{12} + \beta_{5} - \beta_{4} + \beta_1 + 1) q^{15} + ( - \beta_{11} - \beta_{7} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{8} - \beta_{4} - \beta_{3} + \beta_{2}) q^{19} + (\beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + \cdots + 1) q^{21}+ \cdots + ( - 2 \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} - 5 \beta_{5} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{3} + 9 q^{5} + 3 q^{7} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{3} + 9 q^{5} + 3 q^{7} + 29 q^{9} - 14 q^{13} + 13 q^{15} - 4 q^{17} - q^{19} - 6 q^{21} + 17 q^{23} + 37 q^{25} - 3 q^{27} + 4 q^{29} + 14 q^{31} + 22 q^{35} + 25 q^{37} - 3 q^{39} - 13 q^{41} - 4 q^{43} + 31 q^{45} + 24 q^{47} + 13 q^{49} + 12 q^{51} + 44 q^{53} - 3 q^{57} + 19 q^{59} - 7 q^{61} + 3 q^{63} - 9 q^{65} + 5 q^{67} + 9 q^{69} + 24 q^{71} + 20 q^{73} + 13 q^{75} + 33 q^{79} + 86 q^{81} - 34 q^{83} + 36 q^{85} + 28 q^{87} + 41 q^{89} - 3 q^{91} + 36 q^{93} - 17 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 31 x^{12} + 97 x^{11} + 339 x^{10} - 1140 x^{9} - 1495 x^{8} + 5888 x^{7} + 1859 x^{6} - 12750 x^{5} + 2100 x^{4} + 9280 x^{3} - 3480 x^{2} - 800 x + 80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5775283 \nu^{13} + 2321514269 \nu^{12} + 3558672231 \nu^{11} - 67918022689 \nu^{10} - 97038374251 \nu^{9} + 719472873164 \nu^{8} + \cdots - 37852890224 ) / 121175686984 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 245328043 \nu^{13} + 197256242 \nu^{12} + 7842640650 \nu^{11} - 7652616356 \nu^{10} - 93504913934 \nu^{9} + 105297707905 \nu^{8} + \cdots + 91651277820 ) / 60587843492 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 637127335 \nu^{13} - 2272404840 \nu^{12} - 21260401790 \nu^{11} + 71427377406 \nu^{10} + 258324698152 \nu^{9} - 812898207663 \nu^{8} + \cdots - 291439579600 ) / 121175686984 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4257495 \nu^{13} - 4589118 \nu^{12} - 135237718 \nu^{11} + 157344602 \nu^{10} + 1578497940 \nu^{9} - 1923112143 \nu^{8} - 8240045685 \nu^{7} + \cdots + 120787552 ) / 579787976 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1127783421 \nu^{13} - 2666917324 \nu^{12} - 36945683090 \nu^{11} + 86732610118 \nu^{10} + 445334526020 \nu^{9} - 1023493623473 \nu^{8} + \cdots - 474742135240 ) / 121175686984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8183367 \nu^{13} - 3255373 \nu^{12} - 255632413 \nu^{11} + 135207135 \nu^{10} + 2930432157 \nu^{9} - 1875090660 \nu^{8} - 14995218209 \nu^{7} + \cdots - 340599600 ) / 579787976 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 868816860 \nu^{13} - 1738407540 \nu^{12} - 28028589771 \nu^{11} + 56074963373 \nu^{10} + 329904865985 \nu^{9} - 653728144131 \nu^{8} + \cdots - 60207035660 ) / 60587843492 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2228654575 \nu^{13} - 8236998381 \nu^{12} - 73864185327 \nu^{11} + 259718289413 \nu^{10} + 894257329331 \nu^{9} + \cdots - 688385818552 ) / 121175686984 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1315256846 \nu^{13} - 666866407 \nu^{12} - 40149095359 \nu^{11} + 27334225413 \nu^{10} + 442212053521 \nu^{9} - 381007894143 \nu^{8} + \cdots - 412162241344 ) / 60587843492 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2728330567 \nu^{13} - 1447117743 \nu^{12} - 86616388515 \nu^{11} + 54918245657 \nu^{10} + 1018812181551 \nu^{9} - 714135651286 \nu^{8} + \cdots - 400106904 ) / 121175686984 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3222881290 \nu^{13} - 457066073 \nu^{12} + 100301580433 \nu^{11} - 1806849319 \nu^{10} - 1153001139495 \nu^{9} + \cdots - 426162526904 ) / 121175686984 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3767681788 \nu^{13} + 1181927833 \nu^{12} + 116432520181 \nu^{11} - 53558914431 \nu^{10} - 1314741426907 \nu^{9} + \cdots + 206569107312 ) / 121175686984 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{4} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} + \beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11 \beta_{13} - 2 \beta_{12} + 9 \beta_{11} + 10 \beta_{10} + \beta_{9} - 10 \beta_{8} + \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 2 \beta_{12} + 4 \beta_{11} - \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - 5 \beta_{7} + 16 \beta_{6} + 6 \beta_{5} - 13 \beta_{4} + 11 \beta_{3} + 4 \beta_{2} + 73 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 113 \beta_{13} - 32 \beta_{12} + 83 \beta_{11} + 98 \beta_{10} + 17 \beta_{9} - 97 \beta_{8} + 22 \beta_{7} - 34 \beta_{6} - 56 \beta_{5} + 117 \beta_{4} + 112 \beta_{3} - \beta_{2} - 8 \beta _1 + 365 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6 \beta_{13} + 42 \beta_{12} + 68 \beta_{11} - 22 \beta_{10} - 40 \beta_{9} + 62 \beta_{8} - 109 \beta_{7} + 204 \beta_{6} + 133 \beta_{5} - 161 \beta_{4} + 98 \beta_{3} + 69 \beta_{2} + 711 \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1135 \beta_{13} - 419 \beta_{12} + 755 \beta_{11} + 967 \beta_{10} + 233 \beta_{9} - 975 \beta_{8} + 352 \beta_{7} - 471 \beta_{6} - 921 \beta_{5} + 1262 \beta_{4} + 1137 \beta_{3} - 44 \beta_{2} - 200 \beta _1 + 3382 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 70 \beta_{13} + 660 \beta_{12} + 875 \beta_{11} - 361 \beta_{10} - 582 \beta_{9} + 929 \beta_{8} - 1682 \beta_{7} + 2421 \beta_{6} + 2144 \beta_{5} - 2011 \beta_{4} + 768 \beta_{3} + 905 \beta_{2} + 7183 \beta _1 - 61 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11368 \beta_{13} - 5161 \beta_{12} + 6731 \beta_{11} + 9665 \beta_{10} + 2953 \beta_{9} - 10150 \beta_{8} + 4958 \beta_{7} - 6026 \beta_{6} - 12712 \beta_{5} + 13823 \beta_{4} + 11635 \beta_{3} + \cdots + 32232 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2606 \beta_{13} + 9292 \beta_{12} + 10245 \beta_{11} - 5270 \beta_{10} - 7546 \beta_{9} + 12433 \beta_{8} - 22776 \beta_{7} + 27893 \beta_{6} + 30492 \beta_{5} - 25117 \beta_{4} + 4978 \beta_{3} + \cdots - 4197 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 114445 \beta_{13} - 61908 \beta_{12} + 58594 \beta_{11} + 97929 \beta_{10} + 35960 \beta_{9} - 108432 \beta_{8} + 65431 \beta_{7} - 74005 \beta_{6} - 162415 \beta_{5} + 153291 \beta_{4} + \cdots + 314208 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 49780 \beta_{13} + 123565 \beta_{12} + 115535 \beta_{11} - 72108 \beta_{10} - 92973 \beta_{9} + 158307 \beta_{8} - 289697 \beta_{7} + 317383 \beta_{6} + 406224 \beta_{5} - 311522 \beta_{4} + \cdots - 89620 \) Copy content Toggle raw display

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
−3.40915
−3.08355
−2.62768
−1.70651
−1.11416
−0.220164
0.0787443
0.784076
0.932094
1.69963
2.40182
3.04872
3.09087
3.12526
0 −3.40915 0 2.99160 0 3.46369 0 8.62230 0
1.2 0 −3.08355 0 −3.63624 0 −1.68288 0 6.50827 0
1.3 0 −2.62768 0 2.18123 0 −1.99183 0 3.90473 0
1.4 0 −1.70651 0 −1.92496 0 −0.214859 0 −0.0878066 0
1.5 0 −1.11416 0 0.822008 0 2.64213 0 −1.75866 0
1.6 0 −0.220164 0 4.37021 0 3.88017 0 −2.95153 0
1.7 0 0.0787443 0 −1.47082 0 −1.85991 0 −2.99380 0
1.8 0 0.784076 0 −2.58682 0 −0.358442 0 −2.38522 0
1.9 0 0.932094 0 3.17488 0 −2.52437 0 −2.13120 0
1.10 0 1.69963 0 −1.37550 0 4.77216 0 −0.111248 0
1.11 0 2.40182 0 2.84757 0 −4.84768 0 2.76875 0
1.12 0 3.04872 0 1.67683 0 2.77040 0 6.29470 0
1.13 0 3.09087 0 4.35475 0 1.15829 0 6.55346 0
1.14 0 3.12526 0 −2.42474 0 −2.20687 0 6.76725 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

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 6292.2.a.z 14
11.b odd 2 1 6292.2.a.y 14
11.c even 5 2 572.2.n.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.n.b 28 11.c even 5 2
6292.2.a.y 14 11.b odd 2 1
6292.2.a.z 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6292))\):

\( T_{3}^{14} - 3 T_{3}^{13} - 31 T_{3}^{12} + 97 T_{3}^{11} + 339 T_{3}^{10} - 1140 T_{3}^{9} - 1495 T_{3}^{8} + 5888 T_{3}^{7} + 1859 T_{3}^{6} - 12750 T_{3}^{5} + 2100 T_{3}^{4} + 9280 T_{3}^{3} - 3480 T_{3}^{2} - 800 T_{3} + 80 \) Copy content Toggle raw display
\( T_{5}^{14} - 9 T_{5}^{13} - 13 T_{5}^{12} + 303 T_{5}^{11} - 269 T_{5}^{10} - 3873 T_{5}^{9} + 6212 T_{5}^{8} + 24482 T_{5}^{7} - 46241 T_{5}^{6} - 82010 T_{5}^{5} + 158492 T_{5}^{4} + 141192 T_{5}^{3} - 247992 T_{5}^{2} + \cdots + 137456 \) Copy content Toggle raw display
\( T_{7}^{14} - 3 T_{7}^{13} - 51 T_{7}^{12} + 132 T_{7}^{11} + 971 T_{7}^{10} - 1854 T_{7}^{9} - 9563 T_{7}^{8} + 10400 T_{7}^{7} + 51024 T_{7}^{6} - 15965 T_{7}^{5} - 130587 T_{7}^{4} - 35979 T_{7}^{3} + 100444 T_{7}^{2} + \cdots + 7051 \) Copy content Toggle raw display
\( T_{17}^{14} + 4 T_{17}^{13} - 135 T_{17}^{12} - 504 T_{17}^{11} + 6531 T_{17}^{10} + 21487 T_{17}^{9} - 140935 T_{17}^{8} - 370673 T_{17}^{7} + 1401049 T_{17}^{6} + 2307128 T_{17}^{5} - 7166788 T_{17}^{4} + \cdots + 1126961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 3 T^{13} - 31 T^{12} + 97 T^{11} + \cdots + 80 \) Copy content Toggle raw display
$5$ \( T^{14} - 9 T^{13} - 13 T^{12} + \cdots + 137456 \) Copy content Toggle raw display
$7$ \( T^{14} - 3 T^{13} - 51 T^{12} + \cdots + 7051 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( (T + 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 4 T^{13} - 135 T^{12} + \cdots + 1126961 \) Copy content Toggle raw display
$19$ \( T^{14} + T^{13} - 159 T^{12} + \cdots + 39883321 \) Copy content Toggle raw display
$23$ \( T^{14} - 17 T^{13} - 6 T^{12} + \cdots - 1658096 \) Copy content Toggle raw display
$29$ \( T^{14} - 4 T^{13} - 180 T^{12} + \cdots + 226845 \) Copy content Toggle raw display
$31$ \( T^{14} - 14 T^{13} - 76 T^{12} + \cdots + 11899525 \) Copy content Toggle raw display
$37$ \( T^{14} - 25 T^{13} + \cdots + 545519696 \) Copy content Toggle raw display
$41$ \( T^{14} + 13 T^{13} + \cdots + 1419085520 \) Copy content Toggle raw display
$43$ \( T^{14} + 4 T^{13} - 115 T^{12} + \cdots + 2542480 \) Copy content Toggle raw display
$47$ \( T^{14} - 24 T^{13} + 11 T^{12} + \cdots + 33644945 \) Copy content Toggle raw display
$53$ \( T^{14} - 44 T^{13} + \cdots + 8309935089 \) Copy content Toggle raw display
$59$ \( T^{14} - 19 T^{13} + \cdots + 2051693385695 \) Copy content Toggle raw display
$61$ \( T^{14} + 7 T^{13} - 382 T^{12} + \cdots - 479049769 \) Copy content Toggle raw display
$67$ \( T^{14} - 5 T^{13} - 329 T^{12} + \cdots - 499497680 \) Copy content Toggle raw display
$71$ \( T^{14} - 24 T^{13} + \cdots - 604146125 \) Copy content Toggle raw display
$73$ \( T^{14} - 20 T^{13} + \cdots - 4338290896 \) Copy content Toggle raw display
$79$ \( T^{14} - 33 T^{13} + \cdots + 5873053744 \) Copy content Toggle raw display
$83$ \( T^{14} + 34 T^{13} + \cdots - 42610610251 \) Copy content Toggle raw display
$89$ \( T^{14} - 41 T^{13} + \cdots + 46139042224 \) Copy content Toggle raw display
$97$ \( T^{14} - 16 T^{13} + \cdots - 97177400144 \) Copy content Toggle raw display
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