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Properties

Label 8022.2.a.w
Level $8022$
Weight $2$
Character orbit 8022.a
Self dual yes
Analytic conductor $64.056$
Analytic rank $0$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 39 x^{11} + 67 x^{10} + 588 x^{9} - 823 x^{8} - 4265 x^{7} + 4419 x^{6} + 14926 x^{5} + \cdots - 984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{12}\) 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 \( q - q^{2} + q^{3} + q^{4} - \beta_1 q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + \beta_1 q^{10} - \beta_{5} q^{11} + q^{12} + ( - \beta_{12} + 1) q^{13} + q^{14} - \beta_1 q^{15} + q^{16} + (\beta_{8} + 1) q^{17} - q^{18} + ( - \beta_{6} - \beta_{2}) q^{19} - \beta_1 q^{20} - q^{21} + \beta_{5} q^{22} + ( - \beta_{10} + \beta_{9} + \cdots + \beta_{3}) q^{23}+ \cdots - \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} - 2 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} - 2 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} + 13 q^{9} + 2 q^{10} - q^{11} + 13 q^{12} + 14 q^{13} + 13 q^{14} - 2 q^{15} + 13 q^{16} + 7 q^{17} - 13 q^{18} - 2 q^{20} - 13 q^{21} + q^{22} + 11 q^{23} - 13 q^{24} + 17 q^{25} - 14 q^{26} + 13 q^{27} - 13 q^{28} + 5 q^{29} + 2 q^{30} - 7 q^{31} - 13 q^{32} - q^{33} - 7 q^{34} + 2 q^{35} + 13 q^{36} + 15 q^{37} + 14 q^{39} + 2 q^{40} - 10 q^{41} + 13 q^{42} + 15 q^{43} - q^{44} - 2 q^{45} - 11 q^{46} - 7 q^{47} + 13 q^{48} + 13 q^{49} - 17 q^{50} + 7 q^{51} + 14 q^{52} + 14 q^{53} - 13 q^{54} + 19 q^{55} + 13 q^{56} - 5 q^{58} - 18 q^{59} - 2 q^{60} + 27 q^{61} + 7 q^{62} - 13 q^{63} + 13 q^{64} + 18 q^{65} + q^{66} + 7 q^{67} + 7 q^{68} + 11 q^{69} - 2 q^{70} - 4 q^{71} - 13 q^{72} + 26 q^{73} - 15 q^{74} + 17 q^{75} + q^{77} - 14 q^{78} + 20 q^{79} - 2 q^{80} + 13 q^{81} + 10 q^{82} + q^{83} - 13 q^{84} + 34 q^{85} - 15 q^{86} + 5 q^{87} + q^{88} - 15 q^{89} + 2 q^{90} - 14 q^{91} + 11 q^{92} - 7 q^{93} + 7 q^{94} + 24 q^{95} - 13 q^{96} + 18 q^{97} - 13 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 39 x^{11} + 67 x^{10} + 588 x^{9} - 823 x^{8} - 4265 x^{7} + 4419 x^{6} + 14926 x^{5} + \cdots - 984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 815360779 \nu^{12} - 2774347109 \nu^{11} + 15438858089 \nu^{10} + 133824114179 \nu^{9} + \cdots + 8061817040013 ) / 2506799845265 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10201546261 \nu^{12} + 56756216984 \nu^{11} + 402919256431 \nu^{10} + \cdots + 2109658591972 ) / 10027199381060 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15975601261 \nu^{12} + 66002861504 \nu^{11} + 516126759411 \nu^{10} + \cdots + 94923743332512 ) / 10027199381060 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1125014597 \nu^{12} - 2040700125 \nu^{11} - 39731854036 \nu^{10} + 65253175552 \nu^{9} + \cdots - 4710957761738 ) / 501359969053 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5778912177 \nu^{12} - 11531211217 \nu^{11} + 224174055562 \nu^{10} + 487969080462 \nu^{9} + \cdots + 7648863907359 ) / 2506799845265 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7570316369 \nu^{12} + 6580424121 \nu^{11} + 267566100459 \nu^{10} - 168708040596 \nu^{9} + \cdots + 6322577194703 ) / 2506799845265 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32194159899 \nu^{12} + 43663595874 \nu^{11} - 1223668410149 \nu^{10} + \cdots - 16751420402088 ) / 10027199381060 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 36445163621 \nu^{12} + 55963442296 \nu^{11} - 1453126690831 \nu^{10} - 2398017470511 \nu^{9} + \cdots - 9593832434452 ) / 10027199381060 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 20200900453 \nu^{12} - 60061658382 \nu^{11} - 725515697533 \nu^{10} + 2033768434187 \nu^{9} + \cdots + 10293529770944 ) / 5013599690530 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23323354941 \nu^{12} + 396387344 \nu^{11} + 928022973631 \nu^{10} + 92773657751 \nu^{9} + \cdots - 29243452320498 ) / 5013599690530 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 48130005127 \nu^{12} - 65641670948 \nu^{11} - 1771870773337 \nu^{10} + \cdots + 33601179974596 ) / 10027199381060 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{3} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 2\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + 10\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + 13 \beta_{11} - \beta_{10} + 13 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - \beta_{6} + \cdots + 69 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} + 3 \beta_{11} - \beta_{10} - 12 \beta_{9} - 32 \beta_{6} - 22 \beta_{5} - 20 \beta_{4} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{12} + 154 \beta_{11} - 16 \beta_{10} + 159 \beta_{9} - 44 \beta_{8} - 54 \beta_{7} + \cdots + 739 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{12} + 76 \beta_{11} - 9 \beta_{10} - 106 \beta_{9} - 18 \beta_{8} + \beta_{7} - 445 \beta_{6} + \cdots + 690 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 140 \beta_{12} + 1796 \beta_{11} - 194 \beta_{10} + 1949 \beta_{9} - 726 \beta_{8} - 777 \beta_{7} + \cdots + 8329 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 32 \beta_{12} + 1376 \beta_{11} + 13 \beta_{10} - 640 \beta_{9} - 566 \beta_{8} - 38 \beta_{7} + \cdots + 10662 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1004 \beta_{12} + 21036 \beta_{11} - 2048 \beta_{10} + 24095 \beta_{9} - 10802 \beta_{8} - 10512 \beta_{7} + \cdots + 97427 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2002 \beta_{12} + 21900 \beta_{11} + 2055 \beta_{10} + 1051 \beta_{9} - 12020 \beta_{8} + \cdots + 157188 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1141 \beta_{12} + 249281 \beta_{11} - 18847 \beta_{10} + 300938 \beta_{9} - 153134 \beta_{8} + \cdots + 1172901 \) 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
3.72176
3.43842
3.09724
2.30480
2.25979
0.465864
0.176422
−0.847733
−0.997264
−2.34123
−2.76832
−3.15631
−3.35344
−1.00000 1.00000 1.00000 −3.72176 −1.00000 −1.00000 −1.00000 1.00000 3.72176
1.2 −1.00000 1.00000 1.00000 −3.43842 −1.00000 −1.00000 −1.00000 1.00000 3.43842
1.3 −1.00000 1.00000 1.00000 −3.09724 −1.00000 −1.00000 −1.00000 1.00000 3.09724
1.4 −1.00000 1.00000 1.00000 −2.30480 −1.00000 −1.00000 −1.00000 1.00000 2.30480
1.5 −1.00000 1.00000 1.00000 −2.25979 −1.00000 −1.00000 −1.00000 1.00000 2.25979
1.6 −1.00000 1.00000 1.00000 −0.465864 −1.00000 −1.00000 −1.00000 1.00000 0.465864
1.7 −1.00000 1.00000 1.00000 −0.176422 −1.00000 −1.00000 −1.00000 1.00000 0.176422
1.8 −1.00000 1.00000 1.00000 0.847733 −1.00000 −1.00000 −1.00000 1.00000 −0.847733
1.9 −1.00000 1.00000 1.00000 0.997264 −1.00000 −1.00000 −1.00000 1.00000 −0.997264
1.10 −1.00000 1.00000 1.00000 2.34123 −1.00000 −1.00000 −1.00000 1.00000 −2.34123
1.11 −1.00000 1.00000 1.00000 2.76832 −1.00000 −1.00000 −1.00000 1.00000 −2.76832
1.12 −1.00000 1.00000 1.00000 3.15631 −1.00000 −1.00000 −1.00000 1.00000 −3.15631
1.13 −1.00000 1.00000 1.00000 3.35344 −1.00000 −1.00000 −1.00000 1.00000 −3.35344
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.w ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8022.2.a.w ✓ 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\( T_{5}^{13} + 2 T_{5}^{12} - 39 T_{5}^{11} - 67 T_{5}^{10} + 588 T_{5}^{9} + 823 T_{5}^{8} - 4265 T_{5}^{7} + \cdots + 984 \) Copy content Toggle raw display
\( T_{11}^{13} + T_{11}^{12} - 86 T_{11}^{11} - 62 T_{11}^{10} + 2683 T_{11}^{9} + 1429 T_{11}^{8} + \cdots + 375808 \) Copy content Toggle raw display
\( T_{13}^{13} - 14 T_{13}^{12} - 16 T_{13}^{11} + 993 T_{13}^{10} - 2357 T_{13}^{9} - 24978 T_{13}^{8} + \cdots - 640192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{13} \) Copy content Toggle raw display
$3$ \( (T - 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + 2 T^{12} + \cdots + 984 \) Copy content Toggle raw display
$7$ \( (T + 1)^{13} \) Copy content Toggle raw display
$11$ \( T^{13} + T^{12} + \cdots + 375808 \) Copy content Toggle raw display
$13$ \( T^{13} - 14 T^{12} + \cdots - 640192 \) Copy content Toggle raw display
$17$ \( T^{13} - 7 T^{12} + \cdots + 938480 \) Copy content Toggle raw display
$19$ \( T^{13} - 154 T^{11} + \cdots - 76307328 \) Copy content Toggle raw display
$23$ \( T^{13} - 11 T^{12} + \cdots + 2632064 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 163938880 \) Copy content Toggle raw display
$31$ \( T^{13} + 7 T^{12} + \cdots + 3200 \) Copy content Toggle raw display
$37$ \( T^{13} - 15 T^{12} + \cdots - 71918592 \) Copy content Toggle raw display
$41$ \( T^{13} + 10 T^{12} + \cdots - 14547392 \) Copy content Toggle raw display
$43$ \( T^{13} - 15 T^{12} + \cdots - 6190592 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 833040512 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 1266825808 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 19501861888 \) Copy content Toggle raw display
$61$ \( T^{13} - 27 T^{12} + \cdots - 88782912 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 110581760 \) Copy content Toggle raw display
$71$ \( T^{13} + 4 T^{12} + \cdots - 168128 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 3215749610 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 6018813376 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 34411204946 \) Copy content Toggle raw display
$89$ \( T^{13} + 15 T^{12} + \cdots + 9504624 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 203692288 \) Copy content Toggle raw display
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