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Properties

Label 8022.2.a.y
Level $8022$
Weight $2$
Character orbit 8022.a
Self dual yes
Analytic conductor $64.056$
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] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, 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("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.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: \(64.0559925015\)
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: \( x^{14} - 4 x^{13} - 40 x^{12} + 166 x^{11} + 544 x^{10} - 2476 x^{9} - 2604 x^{8} + 15833 x^{7} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - \beta_1 q^{5} + q^{6} - q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} - \beta_1 q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - \beta_1 q^{10} + (\beta_{6} + 1) q^{11} + q^{12} - \beta_{12} q^{13} - q^{14} - \beta_1 q^{15} + q^{16} + ( - \beta_{11} + 1) q^{17} + q^{18} + (\beta_{4} + 1) q^{19} - \beta_1 q^{20} - q^{21} + (\beta_{6} + 1) q^{22} + ( - \beta_{13} - \beta_{9} - \beta_{8} + \cdots + 2) q^{23}+ \cdots + (\beta_{6} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} + 14 q^{3} + 14 q^{4} - 4 q^{5} + 14 q^{6} - 14 q^{7} + 14 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} + 14 q^{3} + 14 q^{4} - 4 q^{5} + 14 q^{6} - 14 q^{7} + 14 q^{8} + 14 q^{9} - 4 q^{10} + 10 q^{11} + 14 q^{12} - 14 q^{14} - 4 q^{15} + 14 q^{16} + 13 q^{17} + 14 q^{18} + 21 q^{19} - 4 q^{20} - 14 q^{21} + 10 q^{22} + 12 q^{23} + 14 q^{24} + 26 q^{25} + 14 q^{27} - 14 q^{28} + 9 q^{29} - 4 q^{30} + 4 q^{31} + 14 q^{32} + 10 q^{33} + 13 q^{34} + 4 q^{35} + 14 q^{36} + 9 q^{37} + 21 q^{38} - 4 q^{40} + 10 q^{41} - 14 q^{42} + 26 q^{43} + 10 q^{44} - 4 q^{45} + 12 q^{46} + 14 q^{47} + 14 q^{48} + 14 q^{49} + 26 q^{50} + 13 q^{51} + 23 q^{53} + 14 q^{54} + 19 q^{55} - 14 q^{56} + 21 q^{57} + 9 q^{58} + 17 q^{59} - 4 q^{60} + 19 q^{61} + 4 q^{62} - 14 q^{63} + 14 q^{64} + 14 q^{65} + 10 q^{66} + 14 q^{67} + 13 q^{68} + 12 q^{69} + 4 q^{70} + 3 q^{71} + 14 q^{72} + 8 q^{73} + 9 q^{74} + 26 q^{75} + 21 q^{76} - 10 q^{77} + 47 q^{79} - 4 q^{80} + 14 q^{81} + 10 q^{82} + 32 q^{83} - 14 q^{84} + 29 q^{85} + 26 q^{86} + 9 q^{87} + 10 q^{88} + 25 q^{89} - 4 q^{90} + 12 q^{92} + 4 q^{93} + 14 q^{94} + 41 q^{95} + 14 q^{96} + 4 q^{97} + 14 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 40 x^{12} + 166 x^{11} + 544 x^{10} - 2476 x^{9} - 2604 x^{8} + 15833 x^{7} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 368384592388 \nu^{13} - 1483075030140 \nu^{12} - 15009296918997 \nu^{11} + \cdots + 21301144896792 ) / 163937452664119 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6692727436995 \nu^{13} + 22051168540810 \nu^{12} + 283046713040264 \nu^{11} + \cdots + 19\!\cdots\!20 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3653718636678 \nu^{13} + 14130496363929 \nu^{12} + 148126919702262 \nu^{11} + \cdots + 25\!\cdots\!54 ) / 327874905328238 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7674268263627 \nu^{13} + 28706435599820 \nu^{12} + 313306598677332 \nu^{11} + \cdots - 23\!\cdots\!24 ) / 655749810656476 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4077140160858 \nu^{13} - 14424330400033 \nu^{12} - 170379523439682 \nu^{11} + \cdots - 31\!\cdots\!26 ) / 327874905328238 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27197726012453 \nu^{13} - 100717626826736 \nu^{12} + \cdots - 66\!\cdots\!80 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6901299677005 \nu^{13} + 25481538695388 \nu^{12} + 283609893937990 \nu^{11} + \cdots + 16\!\cdots\!88 ) / 327874905328238 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35011646605357 \nu^{13} - 128183320995708 \nu^{12} + \cdots - 15\!\cdots\!04 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 47218928786371 \nu^{13} + 177381490232174 \nu^{12} + \cdots + 15\!\cdots\!84 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55276847995865 \nu^{13} - 201697146137512 \nu^{12} + \cdots - 29\!\cdots\!64 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 72270539621361 \nu^{13} - 265498338537834 \nu^{12} + \cdots - 30\!\cdots\!76 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19721696991254 \nu^{13} + 71949582616243 \nu^{12} + 814141169070996 \nu^{11} + \cdots + 95\!\cdots\!54 ) / 327874905328238 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} + 2\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{10} + 18 \beta_{9} + 20 \beta_{8} + 19 \beta_{7} + \cdots + 66 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 27 \beta_{13} - 2 \beta_{12} + 31 \beta_{11} + 27 \beta_{10} - \beta_{9} + 19 \beta_{8} + 24 \beta_{7} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 69 \beta_{13} - 41 \beta_{12} + 38 \beta_{11} - 45 \beta_{10} + 310 \beta_{9} + 352 \beta_{8} + \cdots + 837 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 538 \beta_{13} - 53 \beta_{12} + 656 \beta_{11} + 560 \beta_{10} - 35 \beta_{9} + 331 \beta_{8} + \cdots + 681 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1278 \beta_{13} - 999 \beta_{12} + 884 \beta_{11} - 758 \beta_{10} + 5256 \beta_{9} + 5961 \beta_{8} + \cdots + 11628 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9719 \beta_{13} - 1152 \beta_{12} + 12219 \beta_{11} + 10534 \beta_{10} - 698 \beta_{9} + 5663 \beta_{8} + \cdots + 14297 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22203 \beta_{13} - 20479 \beta_{12} + 17498 \beta_{11} - 11469 \beta_{10} + 88222 \beta_{9} + \cdots + 172094 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 167974 \beta_{13} - 24081 \beta_{12} + 215145 \beta_{11} + 188683 \beta_{10} - 10797 \beta_{9} + \cdots + 269247 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 376026 \beta_{13} - 388417 \beta_{12} + 322114 \beta_{11} - 163459 \beta_{10} + 1470799 \beta_{9} + \cdots + 2658122 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2836946 \beta_{13} - 492497 \beta_{12} + 3681835 \beta_{11} + 3288710 \beta_{10} - 135401 \beta_{9} + \cdots + 4824795 \) 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
4.10767
3.92544
3.26089
2.59829
1.68942
1.05025
0.365718
0.312189
−0.125328
−0.426756
−2.44918
−3.12326
−3.15707
−4.02827
1.00000 1.00000 1.00000 −4.10767 1.00000 −1.00000 1.00000 1.00000 −4.10767
1.2 1.00000 1.00000 1.00000 −3.92544 1.00000 −1.00000 1.00000 1.00000 −3.92544
1.3 1.00000 1.00000 1.00000 −3.26089 1.00000 −1.00000 1.00000 1.00000 −3.26089
1.4 1.00000 1.00000 1.00000 −2.59829 1.00000 −1.00000 1.00000 1.00000 −2.59829
1.5 1.00000 1.00000 1.00000 −1.68942 1.00000 −1.00000 1.00000 1.00000 −1.68942
1.6 1.00000 1.00000 1.00000 −1.05025 1.00000 −1.00000 1.00000 1.00000 −1.05025
1.7 1.00000 1.00000 1.00000 −0.365718 1.00000 −1.00000 1.00000 1.00000 −0.365718
1.8 1.00000 1.00000 1.00000 −0.312189 1.00000 −1.00000 1.00000 1.00000 −0.312189
1.9 1.00000 1.00000 1.00000 0.125328 1.00000 −1.00000 1.00000 1.00000 0.125328
1.10 1.00000 1.00000 1.00000 0.426756 1.00000 −1.00000 1.00000 1.00000 0.426756
1.11 1.00000 1.00000 1.00000 2.44918 1.00000 −1.00000 1.00000 1.00000 2.44918
1.12 1.00000 1.00000 1.00000 3.12326 1.00000 −1.00000 1.00000 1.00000 3.12326
1.13 1.00000 1.00000 1.00000 3.15707 1.00000 −1.00000 1.00000 1.00000 3.15707
1.14 1.00000 1.00000 1.00000 4.02827 1.00000 −1.00000 1.00000 1.00000 4.02827
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(191\) \(-1\)

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 8022.2.a.y ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8022.2.a.y ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\( T_{5}^{14} + 4 T_{5}^{13} - 40 T_{5}^{12} - 166 T_{5}^{11} + 544 T_{5}^{10} + 2476 T_{5}^{9} + \cdots + 144 \) Copy content Toggle raw display
\( T_{11}^{14} - 10 T_{11}^{13} - 46 T_{11}^{12} + 749 T_{11}^{11} - 627 T_{11}^{10} - 16350 T_{11}^{9} + \cdots + 13056 \) Copy content Toggle raw display
\( T_{13}^{14} - 110 T_{13}^{12} + 41 T_{13}^{11} + 4141 T_{13}^{10} - 3646 T_{13}^{9} - 62555 T_{13}^{8} + \cdots - 95376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{14} \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 4 T^{13} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 10 T^{13} + \cdots + 13056 \) Copy content Toggle raw display
$13$ \( T^{14} - 110 T^{12} + \cdots - 95376 \) Copy content Toggle raw display
$17$ \( T^{14} - 13 T^{13} + \cdots - 2811056 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 149389744 \) Copy content Toggle raw display
$23$ \( T^{14} - 12 T^{13} + \cdots - 2866176 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 392131584 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 887687424 \) Copy content Toggle raw display
$37$ \( T^{14} - 9 T^{13} + \cdots + 80704 \) Copy content Toggle raw display
$41$ \( T^{14} - 10 T^{13} + \cdots + 50788928 \) Copy content Toggle raw display
$43$ \( T^{14} - 26 T^{13} + \cdots - 835584 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 22461468672 \) Copy content Toggle raw display
$53$ \( T^{14} - 23 T^{13} + \cdots + 1131264 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 4068682752 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 928132032 \) Copy content Toggle raw display
$67$ \( T^{14} - 14 T^{13} + \cdots + 238592 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 30151798032 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 350216692 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 151269696576 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 7803529152 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 24229806144 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 143304299264 \) Copy content Toggle raw display
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