↑

Properties

Label 522.2.k.i
Level $522$
Weight $2$
Character orbit 522.k
Analytic conductor $4.168$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [522,2,Mod(181,522)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(522, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("522.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.k (of order \(7\), degree \(6\), minimal)

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: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 53x^{8} + 603x^{6} + 4405x^{4} + 20045x^{2} + 44521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + \beta_{4} q^{4} + ( - \beta_{9} - \beta_1) q^{5} + (\beta_{10} - \beta_{9} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{9} - \beta_{7} + \beta_{5} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + \beta_{4} q^{4} + ( - \beta_{9} - \beta_1) q^{5} + (\beta_{10} - \beta_{9} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - 2 \beta_{11} + 4 \beta_{9} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 8 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 8 q^{7} + 2 q^{8} - 2 q^{10} + 18 q^{11} - 24 q^{13} + 6 q^{14} - 2 q^{16} - 4 q^{17} - 16 q^{19} + 2 q^{20} + 10 q^{22} - 16 q^{23} + 18 q^{25} - 4 q^{26} + 8 q^{28} + 14 q^{29} + 36 q^{31} + 2 q^{32} + 4 q^{34} + 14 q^{35} + 24 q^{37} - 12 q^{38} - 2 q^{40} - 40 q^{41} - 12 q^{43} + 4 q^{44} - 12 q^{46} - 16 q^{47} - 34 q^{49} + 10 q^{50} + 18 q^{52} - 12 q^{53} - 2 q^{55} + 6 q^{56} - 16 q^{59} - 18 q^{61} + 6 q^{62} - 2 q^{64} - 14 q^{65} + 30 q^{67} - 18 q^{68} - 28 q^{70} - 12 q^{71} - 14 q^{73} - 24 q^{74} + 12 q^{76} + 36 q^{77} - 16 q^{79} + 2 q^{80} - 2 q^{82} - 28 q^{83} - 28 q^{85} + 12 q^{86} + 24 q^{88} + 64 q^{89} - 38 q^{91} - 16 q^{92} - 26 q^{94} - 34 q^{95} + 26 q^{97} + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 53x^{8} + 603x^{6} + 4405x^{4} + 20045x^{2} + 44521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3677051 \nu^{10} + 398521378 \nu^{8} - 6946905647 \nu^{6} + 83487408428 \nu^{4} + \cdots - 100723436227 ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3677051 \nu^{11} + 398521378 \nu^{9} - 6946905647 \nu^{7} + 83487408428 \nu^{5} + \cdots - 100723436227 \nu ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9032458 \nu^{10} - 7460388 \nu^{8} + 1215197417 \nu^{6} - 11870715831 \nu^{4} + \cdots - 319869227322 ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 123109921 \nu^{10} - 1913160148 \nu^{8} + 20746444620 \nu^{6} - 34396729264 \nu^{4} + \cdots + 249877384477 ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 123109921 \nu^{11} - 1913160148 \nu^{9} + 20746444620 \nu^{7} - 34396729264 \nu^{5} + \cdots + 249877384477 \nu ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 184338432 \nu^{10} - 777279118 \nu^{8} + 9627307015 \nu^{6} + 121099769372 \nu^{4} + \cdots + 4120277217769 ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 184338432 \nu^{11} + 777279118 \nu^{9} - 9627307015 \nu^{7} - 121099769372 \nu^{5} + \cdots - 4120277217769 \nu ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 329308827 \nu^{10} + 3310167727 \nu^{8} - 27626968940 \nu^{6} - 111673289406 \nu^{4} + \cdots - 1630212364450 ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 329308827 \nu^{11} + 3310167727 \nu^{9} - 27626968940 \nu^{7} - 111673289406 \nu^{5} + \cdots - 1630212364450 \nu ) / 2243841320183 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1808080 \nu^{11} + 10269319 \nu^{9} - 72476938 \nu^{7} - 1322466843 \nu^{5} + \cdots - 16950900971 \nu ) / 10634319053 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{5} + 8\beta_{4} + 3\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{11} - 8\beta_{10} - 7\beta_{8} - 7\beta_{6} + 3\beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 18\beta_{9} + 47\beta_{5} + 47\beta_{4} + 77\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 47\beta_{11} - 29\beta_{10} - 47\beta_{8} + 77\beta_{3} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 346\beta_{9} + 243\beta_{7} + 572\beta_{5} + 346\beta_{2} - 243 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 346\beta_{10} - 243\beta_{8} + 572\beta_{6} + 346\beta_{3} - 243\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2065\beta_{9} + 3949\beta_{7} - 6914\beta_{4} - 3949\beta_{2} - 6914 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -6914\beta_{11} + 8979\beta_{10} + 2965\beta_{8} + 6914\beta_{6} - 3949\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -21188\beta_{9} + 21188\beta_{7} - 65805\beta_{5} - 100472\beta_{4} - 100472\beta_{2} - 65805 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -100472\beta_{11} + 79284\beta_{10} + 79284\beta_{8} + 34667\beta_{6} - 100472\beta_{3} + 34667\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.436116 + 1.91075i
−0.436116 − 1.91075i
0.436116 − 1.91075i
−0.436116 + 1.91075i
1.37492 + 1.72410i
−1.37492 − 1.72410i
3.02810 − 1.45826i
−3.02810 + 1.45826i
3.02810 + 1.45826i
−3.02810 − 1.45826i
1.37492 − 1.72410i
−1.37492 + 1.72410i
0.222521 + 0.974928i 0 −0.900969 + 0.433884i −0.213595 − 0.935823i 0 2.92444 + 1.40833i −0.623490 − 0.781831i 0 0.864830 − 0.416480i
181.2 0.222521 + 0.974928i 0 −0.900969 + 0.433884i 0.658637 + 2.88568i 0 −1.47939 − 0.712439i −0.623490 − 0.781831i 0 −2.66677 + 1.28425i
199.1 0.222521 − 0.974928i 0 −0.900969 − 0.433884i −0.213595 + 0.935823i 0 2.92444 − 1.40833i −0.623490 + 0.781831i 0 0.864830 + 0.416480i
199.2 0.222521 − 0.974928i 0 −0.900969 − 0.433884i 0.658637 − 2.88568i 0 −1.47939 + 0.712439i −0.623490 + 0.781831i 0 −2.66677 − 1.28425i
343.1 −0.623490 − 0.781831i 0 −0.222521 + 0.974928i −1.99841 − 2.50593i 0 0.760729 + 3.33297i 0.900969 − 0.433884i 0 −0.713226 + 3.12485i
343.2 −0.623490 − 0.781831i 0 −0.222521 + 0.974928i 0.751434 + 0.942269i 0 −1.00771 − 4.41506i 0.900969 − 0.433884i 0 0.268184 − 1.17499i
397.1 0.900969 − 0.433884i 0 0.623490 − 0.781831i −2.12713 + 1.02437i 0 2.33356 + 2.92619i 0.222521 − 0.974928i 0 −1.47202 + 1.84585i
397.2 0.900969 − 0.433884i 0 0.623490 − 0.781831i 3.92907 − 1.89214i 0 0.468379 + 0.587329i 0.222521 − 0.974928i 0 2.71900 − 3.40952i
451.1 0.900969 + 0.433884i 0 0.623490 + 0.781831i −2.12713 − 1.02437i 0 2.33356 − 2.92619i 0.222521 + 0.974928i 0 −1.47202 − 1.84585i
451.2 0.900969 + 0.433884i 0 0.623490 + 0.781831i 3.92907 + 1.89214i 0 0.468379 − 0.587329i 0.222521 + 0.974928i 0 2.71900 + 3.40952i
487.1 −0.623490 + 0.781831i 0 −0.222521 − 0.974928i −1.99841 + 2.50593i 0 0.760729 − 3.33297i 0.900969 + 0.433884i 0 −0.713226 − 3.12485i
487.2 −0.623490 + 0.781831i 0 −0.222521 − 0.974928i 0.751434 − 0.942269i 0 −1.00771 + 4.41506i 0.900969 + 0.433884i 0 0.268184 + 1.17499i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.k.i yes 12
3.b odd 2 1 522.2.k.f ✓ 12
29.d even 7 1 inner 522.2.k.i yes 12
87.j odd 14 1 522.2.k.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.k.f ✓ 12 3.b odd 2 1
522.2.k.f ✓ 12 87.j odd 14 1
522.2.k.i yes 12 1.a even 1 1 trivial
522.2.k.i yes 12 29.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 2 T_{5}^{11} - 2 T_{5}^{10} - 12 T_{5}^{9} + 106 T_{5}^{8} + 154 T_{5}^{7} + 1331 T_{5}^{6} + \cdots + 12769 \) acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{11} + \cdots + 12769 \) Copy content Toggle raw display
$7$ \( T^{12} - 8 T^{11} + \cdots + 53824 \) Copy content Toggle raw display
$11$ \( T^{12} - 18 T^{11} + \cdots + 2286144 \) Copy content Toggle raw display
$13$ \( T^{12} + 24 T^{11} + \cdots + 877969 \) Copy content Toggle raw display
$17$ \( (T^{6} + 2 T^{5} + \cdots - 503)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 16 T^{11} + \cdots + 3136 \) Copy content Toggle raw display
$23$ \( T^{12} + 16 T^{11} + \cdots + 53824 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} - 36 T^{11} + \cdots + 40043584 \) Copy content Toggle raw display
$37$ \( T^{12} - 24 T^{11} + \cdots + 63001 \) Copy content Toggle raw display
$41$ \( (T^{6} + 20 T^{5} + \cdots - 287)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16906240576 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1987733056 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 33946957009 \) Copy content Toggle raw display
$59$ \( (T^{6} + 8 T^{5} + \cdots + 26552)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 339996721 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 370793536 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 18887554624 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 21473678521 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 73345638976 \) Copy content Toggle raw display
$83$ \( T^{12} + 28 T^{11} + \cdots + 3182656 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 124117403809 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 124970041 \) Copy content Toggle raw display
show more
show less