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Properties

Label 3822.2.c.m
Level $3822$
Weight $2$
Character orbit 3822.c
Analytic conductor $30.519$
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] = [3822,2,Mod(883,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3822.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.c (of order \(2\), degree \(1\), 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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 26x^{8} + 245x^{6} + 1000x^{4} + 1644x^{2} + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - q^{3} - q^{4} + \beta_1 q^{5} + \beta_{4} q^{6} + \beta_{4} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - q^{3} - q^{4} + \beta_1 q^{5} + \beta_{4} q^{6} + \beta_{4} q^{8} + q^{9} - \beta_{3} q^{10} + (\beta_{5} - \beta_{4} - \beta_1) q^{11} + q^{12} + (\beta_{8} - \beta_{7}) q^{13} - \beta_1 q^{15} + q^{16} + (\beta_{6} + \beta_{3} - \beta_{2}) q^{17} - \beta_{4} q^{18} + (\beta_{5} - \beta_{4}) q^{19} - \beta_1 q^{20} + ( - \beta_{6} + \beta_{3} - 1) q^{22} + ( - \beta_{6} - \beta_{3} + 2) q^{23} - \beta_{4} q^{24} + \beta_{2} q^{25} + ( - \beta_{9} - \beta_{2}) q^{26} - q^{27} + (\beta_{9} + \beta_{7} + \cdots + \beta_{2}) q^{29}+ \cdots + (\beta_{5} - \beta_{4} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{3} - 10 q^{4} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{3} - 10 q^{4} + 10 q^{9} - 4 q^{10} + 10 q^{12} + 2 q^{13} + 10 q^{16} + 6 q^{17} - 6 q^{22} + 16 q^{23} - 2 q^{25} + 4 q^{26} - 10 q^{27} - 14 q^{29} + 4 q^{30} - 10 q^{36} - 10 q^{38} - 2 q^{39} + 4 q^{40} + 12 q^{43} - 10 q^{48} - 6 q^{51} - 2 q^{52} + 22 q^{53} + 44 q^{55} - 14 q^{61} + 20 q^{62} - 10 q^{64} - 20 q^{65} + 6 q^{66} - 6 q^{68} - 16 q^{69} - 24 q^{74} + 2 q^{75} - 4 q^{78} - 4 q^{79} + 10 q^{81} - 12 q^{82} + 14 q^{87} + 6 q^{88} - 4 q^{90} - 16 q^{92} + 18 q^{94} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 26x^{8} + 245x^{6} + 1000x^{4} + 1644x^{2} + 900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 15\nu^{4} + 62\nu^{2} + 60 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 26\nu^{7} - 230\nu^{5} - 775\nu^{3} - 714\nu ) / 90 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} - 11\nu^{7} + 40\nu^{5} + 650\nu^{3} + 1176\nu ) / 90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 18\nu^{4} + 92\nu^{2} + 102 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 15 \nu^{8} + 41 \nu^{7} + 315 \nu^{6} + 500 \nu^{5} + 2190 \nu^{4} + 2110 \nu^{3} + \cdots + 3960 ) / 180 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 23\nu^{7} + 185\nu^{5} + 589\nu^{3} + 534\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{9} + 15 \nu^{8} - 41 \nu^{7} + 315 \nu^{6} - 500 \nu^{5} + 2190 \nu^{4} - 2110 \nu^{3} + \cdots + 3960 ) / 180 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{7} + \beta_{5} - 2\beta_{4} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 2\beta_{3} - 10\beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{9} + 2\beta_{8} + 11\beta_{7} - 9\beta_{5} + 30\beta_{4} + 66\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{6} + 36\beta_{3} + 88\beta_{2} - 290 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 103\beta_{9} - 36\beta_{8} - 103\beta_{7} + 73\beta_{5} - 356\beta_{4} - 554\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6\beta_{9} + 6\beta_{7} + 169\beta_{6} - 464\beta_{3} - 760\beta_{2} + 2430 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -923\beta_{9} + 476\beta_{8} + 923\beta_{7} - 603\beta_{5} + 3816\beta_{4} + 4710\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times\).

\(n\) \(1471\) \(2549\) \(3433\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
− 2.74967i
− 2.52086i
− 1.08032i
1.32323i
3.02763i
− 3.02763i
− 1.32323i
1.08032i
2.52086i
2.74967i
− 1.00000i −1.00000 −1.00000 − 2.74967i 1.00000i 0 1.00000i 1.00000 −2.74967
883.2 − 1.00000i −1.00000 −1.00000 − 2.52086i 1.00000i 0 1.00000i 1.00000 −2.52086
883.3 − 1.00000i −1.00000 −1.00000 − 1.08032i 1.00000i 0 1.00000i 1.00000 −1.08032
883.4 − 1.00000i −1.00000 −1.00000 1.32323i 1.00000i 0 1.00000i 1.00000 1.32323
883.5 − 1.00000i −1.00000 −1.00000 3.02763i 1.00000i 0 1.00000i 1.00000 3.02763
883.6 1.00000i −1.00000 −1.00000 − 3.02763i − 1.00000i 0 − 1.00000i 1.00000 3.02763
883.7 1.00000i −1.00000 −1.00000 − 1.32323i − 1.00000i 0 − 1.00000i 1.00000 1.32323
883.8 1.00000i −1.00000 −1.00000 1.08032i − 1.00000i 0 − 1.00000i 1.00000 −1.08032
883.9 1.00000i −1.00000 −1.00000 2.52086i − 1.00000i 0 − 1.00000i 1.00000 −2.52086
883.10 1.00000i −1.00000 −1.00000 2.74967i − 1.00000i 0 − 1.00000i 1.00000 −2.74967
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3822.2.c.m 10
7.b odd 2 1 3822.2.c.n 10
7.c even 3 2 546.2.bk.c ✓ 20
13.b even 2 1 inner 3822.2.c.m 10
21.h odd 6 2 1638.2.dm.e 20
91.b odd 2 1 3822.2.c.n 10
91.r even 6 2 546.2.bk.c ✓ 20
273.w odd 6 2 1638.2.dm.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bk.c ✓ 20 7.c even 3 2
546.2.bk.c ✓ 20 91.r even 6 2
1638.2.dm.e 20 21.h odd 6 2
1638.2.dm.e 20 273.w odd 6 2
3822.2.c.m 10 1.a even 1 1 trivial
3822.2.c.m 10 13.b even 2 1 inner
3822.2.c.n 10 7.b odd 2 1
3822.2.c.n 10 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3822, [\chi])\):

\( T_{5}^{10} + 26T_{5}^{8} + 245T_{5}^{6} + 1000T_{5}^{4} + 1644T_{5}^{2} + 900 \) Copy content Toggle raw display
\( T_{11}^{10} + 85T_{11}^{8} + 2374T_{11}^{6} + 22378T_{11}^{4} + 21921T_{11}^{2} + 5625 \) Copy content Toggle raw display
\( T_{17}^{5} - 3T_{17}^{4} - 51T_{17}^{3} + 215T_{17}^{2} - 162T_{17} - 54 \) Copy content Toggle raw display
\( T_{19}^{10} + 67T_{19}^{8} + 1075T_{19}^{6} + 6625T_{19}^{4} + 16944T_{19}^{2} + 14400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 26 T^{8} + \cdots + 900 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 85 T^{8} + \cdots + 5625 \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( (T^{5} - 3 T^{4} - 51 T^{3} + \cdots - 54)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 67 T^{8} + \cdots + 14400 \) Copy content Toggle raw display
$23$ \( (T^{5} - 8 T^{4} + \cdots - 972)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 7 T^{4} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 82 T^{8} + \cdots + 36864 \) Copy content Toggle raw display
$37$ \( T^{10} + 240 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 4671995904 \) Copy content Toggle raw display
$43$ \( (T^{5} - 6 T^{4} + \cdots - 1712)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 199 T^{8} + \cdots + 60516 \) Copy content Toggle raw display
$53$ \( (T^{5} - 11 T^{4} + \cdots - 2529)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 369 T^{8} + \cdots + 51194025 \) Copy content Toggle raw display
$61$ \( (T^{5} + 7 T^{4} + \cdots + 3518)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 463 T^{8} + \cdots + 31337604 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 1913187600 \) Copy content Toggle raw display
$73$ \( T^{10} + 172 T^{8} + \cdots + 12960000 \) Copy content Toggle raw display
$79$ \( (T^{5} + 2 T^{4} + \cdots + 1776)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 482 T^{8} + \cdots + 5817744 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 6074643600 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 15148194084 \) Copy content Toggle raw display
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