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Properties

Label 6422.2.a.bn
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
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: \( x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} + \cdots + 358 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - \beta_{4} q^{5} + \beta_1 q^{6} + \beta_{12} q^{7} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - \beta_{4} q^{5} + \beta_1 q^{6} + \beta_{12} q^{7} + q^{8} + (\beta_{2} + 1) q^{9} - \beta_{4} q^{10} + (\beta_{12} + \beta_{11} + 1) q^{11} + \beta_1 q^{12} + \beta_{12} q^{14} + ( - \beta_{13} + \beta_{11} - \beta_{9} + \cdots + 1) q^{15}+ \cdots + ( - \beta_{13} - \beta_{12} + \beta_{11} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9} - 2 q^{10} + 10 q^{11} + 4 q^{12} + 2 q^{14} - 4 q^{15} + 14 q^{16} + 2 q^{17} + 18 q^{18} + 14 q^{19} - 2 q^{20} - 18 q^{21} + 10 q^{22} + 12 q^{23} + 4 q^{24} + 44 q^{25} + 10 q^{27} + 2 q^{28} + 4 q^{29} - 4 q^{30} + 4 q^{31} + 14 q^{32} + 12 q^{33} + 2 q^{34} + 14 q^{35} + 18 q^{36} + 18 q^{37} + 14 q^{38} - 2 q^{40} + 6 q^{41} - 18 q^{42} + 28 q^{43} + 10 q^{44} + 8 q^{45} + 12 q^{46} - 20 q^{47} + 4 q^{48} + 28 q^{49} + 44 q^{50} + 10 q^{51} + 12 q^{53} + 10 q^{54} - 2 q^{55} + 2 q^{56} + 4 q^{57} + 4 q^{58} + 16 q^{59} - 4 q^{60} + 30 q^{61} + 4 q^{62} + 28 q^{63} + 14 q^{64} + 12 q^{66} + 2 q^{67} + 2 q^{68} - 42 q^{69} + 14 q^{70} + 44 q^{71} + 18 q^{72} - 36 q^{73} + 18 q^{74} + 46 q^{75} + 14 q^{76} + 68 q^{77} + 34 q^{79} - 2 q^{80} - 6 q^{81} + 6 q^{82} + 2 q^{83} - 18 q^{84} - 30 q^{85} + 28 q^{86} + 52 q^{87} + 10 q^{88} + 42 q^{89} + 8 q^{90} + 12 q^{92} - 12 q^{93} - 20 q^{94} - 2 q^{95} + 4 q^{96} + 40 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} + \cdots + 358 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12288 \nu^{13} + 18169 \nu^{12} + 330650 \nu^{11} - 394551 \nu^{10} - 3378153 \nu^{9} + \cdots + 991236 ) / 86614 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29665 \nu^{13} + 68184 \nu^{12} + 769316 \nu^{11} - 1597128 \nu^{10} - 7605118 \nu^{9} + \cdots + 6286514 ) / 86614 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 46556 \nu^{13} + 116698 \nu^{12} + 1206704 \nu^{11} - 2793341 \nu^{10} - 11956519 \nu^{9} + \cdots + 11682236 ) / 86614 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 121238 \nu^{13} + 289609 \nu^{12} + 3135370 \nu^{11} - 6831637 \nu^{10} - 30930578 \nu^{9} + \cdots + 27053054 ) / 86614 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19670 \nu^{13} + 47465 \nu^{12} + 508271 \nu^{11} - 1122043 \nu^{10} - 5009566 \nu^{9} + \cdots + 4454632 ) / 7874 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31215 \nu^{13} + 75531 \nu^{12} + 805617 \nu^{11} - 1783686 \nu^{10} - 7930556 \nu^{9} + \cdots + 7041860 ) / 7874 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 44605 \nu^{13} + 107910 \nu^{12} + 1152434 \nu^{11} - 2551874 \nu^{10} - 11358734 \nu^{9} + \cdots + 10141326 ) / 7874 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 50702 \nu^{13} + 123330 \nu^{12} + 1307720 \nu^{11} - 2914366 \nu^{10} - 12865611 \nu^{9} + \cdots + 11484056 ) / 7874 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 596839 \nu^{13} - 1447234 \nu^{12} - 15406854 \nu^{11} + 34205757 \nu^{10} + 151712350 \nu^{9} + \cdots - 134814398 ) / 86614 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 654970 \nu^{13} + 1595493 \nu^{12} + 16902234 \nu^{11} - 37743422 \nu^{10} - 166397479 \nu^{9} + \cdots + 149804500 ) / 86614 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 850313 \nu^{13} - 2063627 \nu^{12} - 21951921 \nu^{11} + 48790968 \nu^{10} + 216188584 \nu^{9} + \cdots - 192947862 ) / 86614 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 10\beta_{2} + \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{13} - \beta_{12} + 3 \beta_{11} + 13 \beta_{10} - \beta_{9} + 14 \beta_{7} + 3 \beta_{6} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{13} - 5 \beta_{12} + 31 \beta_{11} + 20 \beta_{10} - 3 \beta_{9} + 12 \beta_{8} + 22 \beta_{7} + \cdots + 224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 80 \beta_{13} - 22 \beta_{12} + 64 \beta_{11} + 145 \beta_{10} - 23 \beta_{9} + 4 \beta_{8} + 166 \beta_{7} + \cdots + 223 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 34 \beta_{13} - 100 \beta_{12} + 392 \beta_{11} + 291 \beta_{10} - 66 \beta_{9} + 127 \beta_{8} + \cdots + 2069 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 587 \beta_{13} - 349 \beta_{12} + 971 \beta_{11} + 1582 \beta_{10} - 363 \beta_{9} + 102 \beta_{8} + \cdots + 3241 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 426 \beta_{13} - 1459 \beta_{12} + 4657 \beta_{11} + 3773 \beta_{10} - 1047 \beta_{9} + 1341 \beta_{8} + \cdots + 20367 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4032 \beta_{13} - 4823 \beta_{12} + 12881 \beta_{11} + 17306 \beta_{10} - 4940 \beta_{9} + 1758 \beta_{8} + \cdots + 41712 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4890 \beta_{13} - 18955 \beta_{12} + 53935 \beta_{11} + 46265 \beta_{10} - 14571 \beta_{9} + \cdots + 209179 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 25301 \beta_{13} - 61947 \beta_{12} + 159847 \beta_{11} + 190687 \beta_{10} - 62464 \beta_{9} + \cdots + 507482 \) 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
−2.77993
−2.64536
−2.07634
−2.02205
−1.17869
−0.343703
0.231327
1.30931
1.50159
1.57102
1.58539
2.66506
2.81805
3.36433
1.00000 −2.77993 1.00000 0.764778 −2.77993 −0.951982 1.00000 4.72802 0.764778
1.2 1.00000 −2.64536 1.00000 −3.69209 −2.64536 4.07315 1.00000 3.99793 −3.69209
1.3 1.00000 −2.07634 1.00000 2.52870 −2.07634 3.77437 1.00000 1.31117 2.52870
1.4 1.00000 −2.02205 1.00000 3.58162 −2.02205 0.649114 1.00000 1.08868 3.58162
1.5 1.00000 −1.17869 1.00000 −0.970755 −1.17869 −1.94107 1.00000 −1.61069 −0.970755
1.6 1.00000 −0.343703 1.00000 −2.06376 −0.343703 1.08834 1.00000 −2.88187 −2.06376
1.7 1.00000 0.231327 1.00000 −2.62395 0.231327 −2.76976 1.00000 −2.94649 −2.62395
1.8 1.00000 1.30931 1.00000 −4.16024 1.30931 −4.89391 1.00000 −1.28570 −4.16024
1.9 1.00000 1.50159 1.00000 2.98870 1.50159 3.73146 1.00000 −0.745218 2.98870
1.10 1.00000 1.57102 1.00000 2.81950 1.57102 −4.61884 1.00000 −0.531896 2.81950
1.11 1.00000 1.58539 1.00000 2.09051 1.58539 3.27454 1.00000 −0.486543 2.09051
1.12 1.00000 2.66506 1.00000 −2.40281 2.66506 −0.679986 1.00000 4.10252 −2.40281
1.13 1.00000 2.81805 1.00000 −3.87767 2.81805 2.87057 1.00000 4.94139 −3.87767
1.14 1.00000 3.36433 1.00000 3.01748 3.36433 −1.60598 1.00000 8.31869 3.01748
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(-1\)
\(19\) \(-1\)

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 6422.2.a.bn 14
13.b even 2 1 6422.2.a.bm 14
13.f odd 12 2 494.2.m.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.m.b ✓ 28 13.f odd 12 2
6422.2.a.bm 14 13.b even 2 1
6422.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{14} - 4 T_{3}^{13} - 22 T_{3}^{12} + 98 T_{3}^{11} + 164 T_{3}^{10} - 912 T_{3}^{9} - 374 T_{3}^{8} + \cdots + 358 \) Copy content Toggle raw display
\( T_{5}^{14} + 2 T_{5}^{13} - 55 T_{5}^{12} - 88 T_{5}^{11} + 1245 T_{5}^{10} + 1516 T_{5}^{9} + \cdots - 276992 \) Copy content Toggle raw display
\( T_{7}^{14} - 2 T_{7}^{13} - 61 T_{7}^{12} + 130 T_{7}^{11} + 1357 T_{7}^{10} - 2842 T_{7}^{9} + \cdots - 48128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 4 T^{13} + \cdots + 358 \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots - 276992 \) Copy content Toggle raw display
$7$ \( T^{14} - 2 T^{13} + \cdots - 48128 \) Copy content Toggle raw display
$11$ \( T^{14} - 10 T^{13} + \cdots + 62464 \) Copy content Toggle raw display
$13$ \( T^{14} \) Copy content Toggle raw display
$17$ \( T^{14} - 2 T^{13} + \cdots - 627200 \) Copy content Toggle raw display
$19$ \( (T - 1)^{14} \) Copy content Toggle raw display
$23$ \( T^{14} - 12 T^{13} + \cdots + 931408 \) Copy content Toggle raw display
$29$ \( T^{14} - 4 T^{13} + \cdots + 50879749 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 522788864 \) Copy content Toggle raw display
$37$ \( T^{14} - 18 T^{13} + \cdots - 156608 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 872071616 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 53655128576 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 2200916878 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 13585632587 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 4563741338 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 14608783168 \) Copy content Toggle raw display
$67$ \( T^{14} - 2 T^{13} + \cdots - 27573248 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 8820874624 \) Copy content Toggle raw display
$73$ \( T^{14} + 36 T^{13} + \cdots + 27490129 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 15247371136 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 67167404672 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 7593700864 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 474544504832 \) Copy content Toggle raw display
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