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Properties

Label 522.3.f.g
Level $522$
Weight $3$
Character orbit 522.f
Analytic conductor $14.223$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [522,3,Mod(307,522)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(522, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("522.307");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 522.f (of order \(4\), degree \(2\), 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: \(14.2234697996\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} + 8 x^{8} + 96 x^{7} + 1204 x^{6} - 1792 x^{5} + 2144 x^{4} + 13632 x^{3} + 39204 x^{2} + \cdots + 288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + 2 \beta_{3} q^{4} + (\beta_{4} + \beta_{3}) q^{5} + \beta_{2} q^{7} + ( - 2 \beta_{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{2} + 2 \beta_{3} q^{4} + (\beta_{4} + \beta_{3}) q^{5} + \beta_{2} q^{7} + ( - 2 \beta_{3} + 2) q^{8} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{10} + \beta_{6} q^{11} + ( - \beta_{9} + \beta_{8} + \beta_{5}) q^{13} + (\beta_{5} - \beta_{2}) q^{14} - 4 q^{16} + ( - 2 \beta_{8} - \beta_{6} + \cdots - \beta_1) q^{17}+ \cdots + (4 \beta_{8} + 2 \beta_{6} + \beta_{5} + \cdots + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 20 q^{8} + 12 q^{10} - 40 q^{16} + 6 q^{17} - 32 q^{19} - 24 q^{20} + 16 q^{23} - 110 q^{25} - 4 q^{26} + 22 q^{29} - 8 q^{31} + 40 q^{32} + 46 q^{37} + 24 q^{40} - 58 q^{41} + 56 q^{43} - 16 q^{46} + 192 q^{47} - 50 q^{49} + 110 q^{50} + 8 q^{52} - 180 q^{53} + 48 q^{55} - 46 q^{58} - 24 q^{59} + 22 q^{61} - 248 q^{65} - 12 q^{68} - 48 q^{70} - 62 q^{73} - 92 q^{74} + 64 q^{76} - 192 q^{77} - 200 q^{79} + 116 q^{82} + 296 q^{83} - 196 q^{85} + 26 q^{89} - 384 q^{94} + 32 q^{95} + 10 q^{97} + 50 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} + 8 x^{8} + 96 x^{7} + 1204 x^{6} - 1792 x^{5} + 2144 x^{4} + 13632 x^{3} + 39204 x^{2} + \cdots + 288 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 180717 \nu^{9} + 873540 \nu^{8} - 2684924 \nu^{7} - 14018493 \nu^{6} - 207205566 \nu^{5} + \cdots - 6703373544 ) / 5956407672 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15254848 \nu^{9} - 79077552 \nu^{8} + 251396481 \nu^{7} + 1153636722 \nu^{6} + \cdots + 619680794880 ) / 184648637832 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5187263 \nu^{9} - 21110486 \nu^{8} + 43245184 \nu^{7} + 492607400 \nu^{6} + 6217427666 \nu^{5} + \cdots + 12418582896 ) / 11912815344 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5548697 \nu^{9} - 22857566 \nu^{8} + 48615032 \nu^{7} + 520644386 \nu^{6} + 6631838798 \nu^{5} + \cdots + 13912514640 ) / 11912815344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 516596823 \nu^{9} + 2125878906 \nu^{8} - 4532042888 \nu^{7} - 48262554472 \nu^{6} + \cdots - 1286457534144 ) / 369297275664 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 712254965 \nu^{9} - 2804632730 \nu^{8} + 5253798697 \nu^{7} + 69536805390 \nu^{6} + \cdots + 3414130430112 ) / 369297275664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 268341643 \nu^{9} + 1102030518 \nu^{8} - 2266265405 \nu^{7} - 25511020978 \nu^{6} + \cdots - 26649283296 ) / 123099091888 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1253356476 \nu^{9} - 4955997768 \nu^{8} + 9804462461 \nu^{7} + 121107496658 \nu^{6} + \cdots + 6007856872512 ) / 369297275664 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1330900044 \nu^{9} + 5536979354 \nu^{8} - 11457151327 \nu^{7} - 126521926406 \nu^{6} + \cdots - 135890148768 ) / 369297275664 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - 2\beta_{5} + 2\beta_{4} - 18\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 5\beta_{6} + 19\beta_{4} - 36\beta_{3} - 19\beta _1 - 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{9} - 8\beta_{8} + 46\beta_{7} + 46\beta_{6} + 74\beta_{2} - 124\beta _1 - 676 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -72\beta_{9} + 320\beta_{7} + 84\beta_{5} - 845\beta_{4} + 2105\beta_{3} + 84\beta_{2} - 845\beta _1 - 2105 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -432\beta_{9} + 432\beta_{8} + 2122\beta_{7} - 2122\beta_{6} + 2900\beta_{5} - 6716\beta_{4} + 29580\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3898 \beta_{8} - 17330 \beta_{6} + 6768 \beta_{5} - 39478 \beta_{4} + 114384 \beta_{3} - 6768 \beta_{2} + \cdots + 114384 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 21176\beta_{9} + 21176\beta_{8} - 100132\beta_{7} - 100132\beta_{6} - 123044\beta_{2} + 347464\beta _1 + 1371304 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 198528 \beta_{9} - 893456 \beta_{7} - 406488 \beta_{5} + 1890338 \beta_{4} - 5946506 \beta_{3} + \cdots + 5946506 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−3.66257 − 3.66257i
−1.29667 − 1.29667i
−0.0619416 − 0.0619416i
2.05251 + 2.05251i
4.96867 + 4.96867i
4.96867 − 4.96867i
2.05251 − 2.05251i
−0.0619416 + 0.0619416i
−1.29667 + 1.29667i
−3.66257 + 3.66257i
−1.00000 + 1.00000i 0 − 2.00000i − 9.32513i 0 −8.71902 2.00000 + 2.00000i 0 9.32513 + 9.32513i
307.2 −1.00000 + 1.00000i 0 − 2.00000i − 4.59333i 0 10.0165 2.00000 + 2.00000i 0 4.59333 + 4.59333i
307.3 −1.00000 + 1.00000i 0 − 2.00000i − 2.12388i 0 3.35200 2.00000 + 2.00000i 0 2.12388 + 2.12388i
307.4 −1.00000 + 1.00000i 0 − 2.00000i 2.10502i 0 −5.61134 2.00000 + 2.00000i 0 −2.10502 − 2.10502i
307.5 −1.00000 + 1.00000i 0 − 2.00000i 7.93733i 0 0.961839 2.00000 + 2.00000i 0 −7.93733 − 7.93733i
505.1 −1.00000 − 1.00000i 0 2.00000i − 7.93733i 0 0.961839 2.00000 − 2.00000i 0 −7.93733 + 7.93733i
505.2 −1.00000 − 1.00000i 0 2.00000i − 2.10502i 0 −5.61134 2.00000 − 2.00000i 0 −2.10502 + 2.10502i
505.3 −1.00000 − 1.00000i 0 2.00000i 2.12388i 0 3.35200 2.00000 − 2.00000i 0 2.12388 − 2.12388i
505.4 −1.00000 − 1.00000i 0 2.00000i 4.59333i 0 10.0165 2.00000 − 2.00000i 0 4.59333 − 4.59333i
505.5 −1.00000 − 1.00000i 0 2.00000i 9.32513i 0 −8.71902 2.00000 − 2.00000i 0 9.32513 − 9.32513i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.3.f.g ✓ 10
3.b odd 2 1 522.3.f.h yes 10
29.c odd 4 1 inner 522.3.f.g ✓ 10
87.f even 4 1 522.3.f.h yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.3.f.g ✓ 10 1.a even 1 1 trivial
522.3.f.g ✓ 10 29.c odd 4 1 inner
522.3.f.h yes 10 3.b odd 2 1
522.3.f.h yes 10 87.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(522, [\chi])\):

\( T_{5}^{10} + 180T_{5}^{8} + 10192T_{5}^{6} + 196288T_{5}^{4} + 1206336T_{5}^{2} + 2310400 \) Copy content Toggle raw display
\( T_{11}^{10} + 2376 T_{11}^{7} + 78066 T_{11}^{6} + 660312 T_{11}^{5} + 2822688 T_{11}^{4} + \cdots + 4536852768 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 180 T^{8} + \cdots + 2310400 \) Copy content Toggle raw display
$7$ \( (T^{5} - 110 T^{3} + \cdots - 1580)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 4536852768 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 2455598916 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 10686927602 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 12866646528 \) Copy content Toggle raw display
$23$ \( (T^{5} - 8 T^{4} + \cdots + 30656)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 420707233300201 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 13715724075008 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 46\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 64615811072 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 241196699731968 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 36\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{5} + 90 T^{4} + \cdots + 25372888)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + 12 T^{4} + \cdots - 31946608)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 76\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{10} + 7692 T^{8} + \cdots + 90478144 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 95\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{5} - 148 T^{4} + \cdots - 459652096)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 10\!\cdots\!22 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
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