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Properties

Label 8214.2.a.bj
Level $8214$
Weight $2$
Character orbit 8214.a
Self dual yes
Analytic conductor $65.589$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8214,2,Mod(1,8214)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8214, 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("8214.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8214 = 2 \cdot 3 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8214.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: \(65.5891202203\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 42 x^{10} - 2 x^{9} + 630 x^{8} + 60 x^{7} - 4148 x^{6} - 594 x^{5} + 12267 x^{4} + 2502 x^{3} + \cdots - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 222)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} + (\beta_{6} + \beta_{4}) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} + (\beta_{6} + \beta_{4}) q^{7} + q^{8} + q^{9} + \beta_1 q^{10} + ( - \beta_{11} + \beta_{9} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - \beta_{11} + \beta_{9} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 12 q^{3} + 12 q^{4} - 12 q^{6} - 6 q^{7} + 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} - 12 q^{3} + 12 q^{4} - 12 q^{6} - 6 q^{7} + 12 q^{8} + 12 q^{9} - 6 q^{11} - 12 q^{12} - 12 q^{13} - 6 q^{14} + 12 q^{16} + 12 q^{18} - 18 q^{19} + 6 q^{21} - 6 q^{22} - 12 q^{23} - 12 q^{24} + 24 q^{25} - 12 q^{26} - 12 q^{27} - 6 q^{28} - 18 q^{29} - 24 q^{31} + 12 q^{32} + 6 q^{33} + 12 q^{35} + 12 q^{36} - 18 q^{38} + 12 q^{39} - 6 q^{41} + 6 q^{42} - 18 q^{43} - 6 q^{44} - 12 q^{46} + 6 q^{47} - 12 q^{48} + 12 q^{49} + 24 q^{50} - 12 q^{52} + 24 q^{53} - 12 q^{54} - 6 q^{55} - 6 q^{56} + 18 q^{57} - 18 q^{58} - 24 q^{59} - 24 q^{62} - 6 q^{63} + 12 q^{64} - 42 q^{65} + 6 q^{66} - 36 q^{67} + 12 q^{69} + 12 q^{70} - 42 q^{71} + 12 q^{72} - 6 q^{73} - 24 q^{75} - 18 q^{76} - 30 q^{77} + 12 q^{78} - 42 q^{79} + 12 q^{81} - 6 q^{82} + 12 q^{83} + 6 q^{84} - 36 q^{85} - 18 q^{86} + 18 q^{87} - 6 q^{88} - 18 q^{89} - 24 q^{91} - 12 q^{92} + 24 q^{93} + 6 q^{94} - 60 q^{95} - 12 q^{96} - 48 q^{97} + 12 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 42 x^{10} - 2 x^{9} + 630 x^{8} + 60 x^{7} - 4148 x^{6} - 594 x^{5} + 12267 x^{4} + 2502 x^{3} + \cdots - 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 62 \nu^{11} + 4521 \nu^{10} + 9343 \nu^{9} - 175449 \nu^{8} - 308676 \nu^{7} + 2290080 \nu^{6} + \cdots + 3666861 ) / 1990122 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14235 \nu^{11} + 5677868 \nu^{10} - 45802 \nu^{9} - 213022180 \nu^{8} - 25597664 \nu^{7} + \cdots + 439704180 ) / 220903542 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 75023 \nu^{11} - 2234635 \nu^{10} - 1041752 \nu^{9} + 84132447 \nu^{8} - 28163208 \nu^{7} + \cdots + 741508038 ) / 220903542 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 269936 \nu^{11} + 1448956 \nu^{10} + 11083203 \nu^{9} - 53286662 \nu^{8} - 161132306 \nu^{7} + \cdots - 635635314 ) / 220903542 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 28612 \nu^{11} + 85547 \nu^{10} + 1063377 \nu^{9} - 3102259 \nu^{8} - 12962041 \nu^{7} + \cdots + 4543542 ) / 12994326 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33357 \nu^{11} + 122626 \nu^{10} - 1319280 \nu^{9} - 4617954 \nu^{8} + 17664448 \nu^{7} + \cdots - 20404557 ) / 12994326 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 594547 \nu^{11} + 1279317 \nu^{10} - 21882101 \nu^{9} - 49355548 \nu^{8} + 256629001 \nu^{7} + \cdots + 446858289 ) / 220903542 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 615920 \nu^{11} + 1258200 \nu^{10} + 23781476 \nu^{9} - 45851739 \nu^{8} - 312417868 \nu^{7} + \cdots + 302740740 ) / 220903542 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 656515 \nu^{11} + 4848904 \nu^{10} - 23347905 \nu^{9} - 183262187 \nu^{8} + 243748534 \nu^{7} + \cdots + 725815080 ) / 220903542 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1543745 \nu^{11} + 412703 \nu^{10} - 54631356 \nu^{9} - 18677146 \nu^{8} + 591291485 \nu^{7} + \cdots + 1059559641 ) / 220903542 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{9} - 3\beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{4} - \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 2\beta_{10} + \beta_{9} + 3\beta_{7} + \beta_{6} + \beta_{5} - 4\beta_{3} + 6\beta_{2} + 13\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 19 \beta_{11} - 6 \beta_{10} + 39 \beta_{9} - 44 \beta_{8} + 18 \beta_{7} - 29 \beta_{6} - 6 \beta_{5} + \cdots + 66 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{11} + 39 \beta_{10} + 8 \beta_{9} - 15 \beta_{8} + 55 \beta_{7} + 31 \beta_{6} + 15 \beta_{5} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 323 \beta_{11} - 160 \beta_{10} + 670 \beta_{9} - 669 \beta_{8} + 327 \beta_{7} - 446 \beta_{6} + \cdots + 895 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 137 \beta_{11} + 636 \beta_{10} - 87 \beta_{9} - 434 \beta_{8} + 813 \beta_{7} + 757 \beta_{6} + \cdots + 189 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5380 \beta_{11} - 3372 \beta_{10} + 11207 \beta_{9} - 10425 \beta_{8} + 5923 \beta_{7} - 6941 \beta_{6} + \cdots + 13206 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 691 \beta_{11} + 9977 \beta_{10} - 4787 \beta_{9} - 9459 \beta_{8} + 11370 \beta_{7} + 16108 \beta_{6} + \cdots + 2092 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 89284 \beta_{11} - 65070 \beta_{10} + 185961 \beta_{9} - 164813 \beta_{8} + 105549 \beta_{7} + \cdots + 201453 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 11575 \beta_{11} + 154983 \beta_{10} - 125095 \beta_{9} - 185751 \beta_{8} + 157510 \beta_{7} + \cdots + 19134 \) 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
−4.05055
−3.61987
−2.20276
−2.13652
−1.52497
−0.293772
−0.00711703
1.78922
2.08778
2.17316
3.70326
4.08215
1.00000 −1.00000 1.00000 −4.05055 −1.00000 −3.87963 1.00000 1.00000 −4.05055
1.2 1.00000 −1.00000 1.00000 −3.61987 −1.00000 −3.14591 1.00000 1.00000 −3.61987
1.3 1.00000 −1.00000 1.00000 −2.20276 −1.00000 2.77882 1.00000 1.00000 −2.20276
1.4 1.00000 −1.00000 1.00000 −2.13652 −1.00000 2.79824 1.00000 1.00000 −2.13652
1.5 1.00000 −1.00000 1.00000 −1.52497 −1.00000 −3.78315 1.00000 1.00000 −1.52497
1.6 1.00000 −1.00000 1.00000 −0.293772 −1.00000 −2.64876 1.00000 1.00000 −0.293772
1.7 1.00000 −1.00000 1.00000 −0.00711703 −1.00000 3.57693 1.00000 1.00000 −0.00711703
1.8 1.00000 −1.00000 1.00000 1.78922 −1.00000 −1.97596 1.00000 1.00000 1.78922
1.9 1.00000 −1.00000 1.00000 2.08778 −1.00000 2.04673 1.00000 1.00000 2.08778
1.10 1.00000 −1.00000 1.00000 2.17316 −1.00000 2.13284 1.00000 1.00000 2.17316
1.11 1.00000 −1.00000 1.00000 3.70326 −1.00000 −2.70142 1.00000 1.00000 3.70326
1.12 1.00000 −1.00000 1.00000 4.08215 −1.00000 −1.19872 1.00000 1.00000 4.08215
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(37\) \(-1\)

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 8214.2.a.bj 12
37.b even 2 1 8214.2.a.bh 12
37.i odd 36 2 222.2.n.a ✓ 24
111.q even 36 2 666.2.bj.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.n.a ✓ 24 37.i odd 36 2
666.2.bj.e 24 111.q even 36 2
8214.2.a.bh 12 37.b even 2 1
8214.2.a.bj 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8214))\):

\( T_{5}^{12} - 42 T_{5}^{10} - 2 T_{5}^{9} + 630 T_{5}^{8} + 60 T_{5}^{7} - 4148 T_{5}^{6} - 594 T_{5}^{5} + \cdots - 27 \) Copy content Toggle raw display
\( T_{7}^{12} + 6 T_{7}^{11} - 30 T_{7}^{10} - 224 T_{7}^{9} + 267 T_{7}^{8} + 3204 T_{7}^{7} + \cdots - 95016 \) Copy content Toggle raw display
\( T_{13}^{12} + 12 T_{13}^{11} - 12 T_{13}^{10} - 598 T_{13}^{9} - 1302 T_{13}^{8} + 8406 T_{13}^{7} + \cdots - 4107 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 42 T^{10} + \cdots - 27 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots - 95016 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 124416 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots - 4107 \) Copy content Toggle raw display
$17$ \( T^{12} - 87 T^{10} + \cdots + 5913 \) Copy content Toggle raw display
$19$ \( T^{12} + 18 T^{11} + \cdots - 49221864 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 85752 \) Copy content Toggle raw display
$29$ \( T^{12} + 18 T^{11} + \cdots - 13859667 \) Copy content Toggle raw display
$31$ \( T^{12} + 24 T^{11} + \cdots + 261816 \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + 6 T^{11} + \cdots + 14001984 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 1034807808 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 178453368 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 126592227 \) Copy content Toggle raw display
$59$ \( T^{12} + 24 T^{11} + \cdots - 6112584 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 3106715121 \) Copy content Toggle raw display
$67$ \( T^{12} + 36 T^{11} + \cdots + 97950232 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 739072512 \) Copy content Toggle raw display
$73$ \( T^{12} + 6 T^{11} + \cdots + 19945792 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 9891329496 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 248798952 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 8376244263 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 947511651 \) Copy content Toggle raw display
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