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Properties

Label 3822.2.c.o
Level $3822$
Weight $2$
Character orbit 3822.c
Analytic conductor $30.519$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18 x^{14} - 24 x^{13} + 172 x^{12} + 212 x^{11} + 148 x^{10} + 2780 x^{9} + 3533 x^{8} + \cdots + 19984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - q^{3} - q^{4} - \beta_{4} q^{5} - \beta_{7} q^{6} - \beta_{7} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - q^{3} - q^{4} - \beta_{4} q^{5} - \beta_{7} q^{6} - \beta_{7} q^{8} + q^{9} + ( - \beta_{2} + 1) q^{10} + ( - \beta_{14} - \beta_{13} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{14} - \beta_{13} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} - 16 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{3} - 16 q^{4} + 16 q^{9} + 8 q^{10} + 16 q^{12} + 16 q^{16} + 8 q^{22} + 8 q^{23} - 48 q^{25} - 4 q^{26} - 16 q^{27} + 8 q^{29} - 8 q^{30} - 16 q^{36} + 32 q^{38} - 8 q^{40} - 24 q^{43} - 16 q^{48} - 48 q^{55} + 32 q^{61} + 16 q^{62} - 16 q^{64} + 36 q^{65} - 8 q^{66} - 8 q^{69} + 24 q^{74} + 48 q^{75} + 4 q^{78} - 8 q^{79} + 16 q^{81} + 24 q^{82} - 8 q^{87} - 8 q^{88} + 8 q^{90} - 8 q^{92} - 24 q^{94} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 18 x^{14} - 24 x^{13} + 172 x^{12} + 212 x^{11} + 148 x^{10} + 2780 x^{9} + 3533 x^{8} + \cdots + 19984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 22\!\cdots\!01 \nu^{15} + \cdots + 59\!\cdots\!72 ) / 40\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!39 \nu^{15} + \cdots + 14\!\cdots\!12 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 59\!\cdots\!07 \nu^{15} + \cdots + 25\!\cdots\!80 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 59\!\cdots\!07 \nu^{15} + \cdots + 25\!\cdots\!80 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65\!\cdots\!88 \nu^{15} + \cdots + 32\!\cdots\!72 ) / 86\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!33 \nu^{15} + \cdots + 10\!\cdots\!32 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!79 \nu^{15} + \cdots + 54\!\cdots\!68 ) / 60\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 68\!\cdots\!47 \nu^{15} + \cdots - 33\!\cdots\!08 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 71\!\cdots\!73 \nu^{15} + \cdots - 31\!\cdots\!88 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72\!\cdots\!97 \nu^{15} + \cdots + 26\!\cdots\!24 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 548998223148784 \nu^{15} - 495395574026484 \nu^{14} + \cdots + 15\!\cdots\!48 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 78\!\cdots\!83 \nu^{15} + \cdots - 45\!\cdots\!24 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!67 \nu^{15} + \cdots + 33\!\cdots\!48 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!87 \nu^{15} + \cdots - 68\!\cdots\!96 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 25\!\cdots\!11 \nu^{15} + \cdots + 43\!\cdots\!64 ) / 17\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} - \beta_{12} + 5 \beta_{11} - 3 \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{15} - 21 \beta_{14} - 8 \beta_{13} + 2 \beta_{12} + 15 \beta_{11} - 16 \beta_{10} + \cdots + 25 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 41 \beta_{15} - 15 \beta_{14} - 6 \beta_{13} - 7 \beta_{12} + 92 \beta_{11} - 61 \beta_{10} + \cdots - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 263 \beta_{15} - 265 \beta_{14} - 168 \beta_{13} - 82 \beta_{12} + 717 \beta_{11} - 454 \beta_{10} + \cdots + 307 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 592 \beta_{15} - 523 \beta_{14} - 268 \beta_{13} + 57 \beta_{12} + 895 \beta_{11} - 838 \beta_{10} + \cdots - 906 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3963 \beta_{15} - 2037 \beta_{14} - 2708 \beta_{13} - 2048 \beta_{12} + 12847 \beta_{11} + \cdots - 6035 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5607 \beta_{15} - 6679 \beta_{14} - 3644 \beta_{13} + 1397 \beta_{12} + 6178 \beta_{11} + \cdots - 18357 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 23851 \beta_{15} - 9641 \beta_{14} - 31640 \beta_{13} - 13942 \beta_{12} + 81981 \beta_{11} + \cdots - 230557 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5436 \beta_{15} + 6041 \beta_{14} - 22136 \beta_{13} + 2643 \beta_{12} - 17947 \beta_{11} + \cdots - 331646 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 528143 \beta_{15} + 278349 \beta_{14} - 55956 \beta_{13} + 184092 \beta_{12} - 1450775 \beta_{11} + \cdots - 3662557 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1771279 \beta_{15} + 1811607 \beta_{14} + 400312 \beta_{13} - 10173 \beta_{12} - 3193426 \beta_{11} + \cdots - 5183759 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 20797071 \beta_{15} + 13853741 \beta_{14} + 7396136 \beta_{13} + 5472654 \beta_{12} - 52503833 \beta_{11} + \cdots - 29145959 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 47805324 \beta_{15} + 41160251 \beta_{14} + 18653696 \beta_{13} + 5358937 \beta_{12} - 101246465 \beta_{11} + \cdots - 41417966 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 410480521 \beta_{15} + 328264379 \beta_{14} + 204094324 \beta_{13} + 66785612 \beta_{12} + \cdots + 150987917 ) / 4 \) 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
0.634457 − 1.92544i
0.670375 − 1.83475i
1.12708 + 0.0252048i
0.379526 + 0.125183i
−2.57586 + 0.660733i
3.91146 + 1.28776i
−1.29888 + 1.59336i
−2.84817 + 2.06794i
−2.84817 − 2.06794i
−1.29888 − 1.59336i
3.91146 − 1.28776i
−2.57586 − 0.660733i
0.379526 − 0.125183i
1.12708 − 0.0252048i
0.670375 + 1.83475i
0.634457 + 1.92544i
− 1.00000i −1.00000 −1.00000 − 3.85087i 1.00000i 0 1.00000i 1.00000 −3.85087
883.2 − 1.00000i −1.00000 −1.00000 − 3.66950i 1.00000i 0 1.00000i 1.00000 −3.66950
883.3 − 1.00000i −1.00000 −1.00000 0.0504097i 1.00000i 0 1.00000i 1.00000 0.0504097
883.4 − 1.00000i −1.00000 −1.00000 0.250366i 1.00000i 0 1.00000i 1.00000 0.250366
883.5 − 1.00000i −1.00000 −1.00000 1.32147i 1.00000i 0 1.00000i 1.00000 1.32147
883.6 − 1.00000i −1.00000 −1.00000 2.57552i 1.00000i 0 1.00000i 1.00000 2.57552
883.7 − 1.00000i −1.00000 −1.00000 3.18672i 1.00000i 0 1.00000i 1.00000 3.18672
883.8 − 1.00000i −1.00000 −1.00000 4.13588i 1.00000i 0 1.00000i 1.00000 4.13588
883.9 1.00000i −1.00000 −1.00000 − 4.13588i − 1.00000i 0 − 1.00000i 1.00000 4.13588
883.10 1.00000i −1.00000 −1.00000 − 3.18672i − 1.00000i 0 − 1.00000i 1.00000 3.18672
883.11 1.00000i −1.00000 −1.00000 − 2.57552i − 1.00000i 0 − 1.00000i 1.00000 2.57552
883.12 1.00000i −1.00000 −1.00000 − 1.32147i − 1.00000i 0 − 1.00000i 1.00000 1.32147
883.13 1.00000i −1.00000 −1.00000 − 0.250366i − 1.00000i 0 − 1.00000i 1.00000 0.250366
883.14 1.00000i −1.00000 −1.00000 − 0.0504097i − 1.00000i 0 − 1.00000i 1.00000 0.0504097
883.15 1.00000i −1.00000 −1.00000 3.66950i − 1.00000i 0 − 1.00000i 1.00000 −3.66950
883.16 1.00000i −1.00000 −1.00000 3.85087i − 1.00000i 0 − 1.00000i 1.00000 −3.85087
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.16
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.o ✓ 16
7.b odd 2 1 3822.2.c.p yes 16
13.b even 2 1 inner 3822.2.c.o ✓ 16
91.b odd 2 1 3822.2.c.p yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3822.2.c.o ✓ 16 1.a even 1 1 trivial
3822.2.c.o ✓ 16 13.b even 2 1 inner
3822.2.c.p yes 16 7.b odd 2 1
3822.2.c.p yes 16 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 \(S_{2}^{\mathrm{new}}(3822, [\chi])\):

\( T_{5}^{16} + 64 T_{5}^{14} + 1626 T_{5}^{12} + 20700 T_{5}^{10} + 136089 T_{5}^{8} + 419436 T_{5}^{6} + \cdots + 64 \) Copy content Toggle raw display
\( T_{11}^{16} + 124 T_{11}^{14} + 5862 T_{11}^{12} + 134316 T_{11}^{10} + 1593857 T_{11}^{8} + \cdots + 802816 \) Copy content Toggle raw display
\( T_{17}^{8} - 68T_{17}^{6} - 52T_{17}^{5} + 917T_{17}^{4} - 388T_{17}^{3} - 3278T_{17}^{2} + 4632T_{17} - 1736 \) Copy content Toggle raw display
\( T_{19}^{16} + 188 T_{19}^{14} + 14278 T_{19}^{12} + 557732 T_{19}^{10} + 11769841 T_{19}^{8} + \cdots + 200704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 64 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 124 T^{14} + \cdots + 802816 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( (T^{8} - 68 T^{6} + \cdots - 1736)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 188 T^{14} + \cdots + 200704 \) Copy content Toggle raw display
$23$ \( (T^{8} - 4 T^{7} + \cdots - 12544)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 4 T^{7} + \cdots + 315616)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1073741824 \) Copy content Toggle raw display
$37$ \( T^{16} + 264 T^{14} + \cdots + 17909824 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 83239174144 \) Copy content Toggle raw display
$43$ \( (T^{8} + 12 T^{7} + \cdots - 2944)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 89405784064 \) Copy content Toggle raw display
$53$ \( (T^{8} - 234 T^{6} + \cdots - 253952)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 13687128064 \) Copy content Toggle raw display
$61$ \( (T^{8} - 16 T^{7} + \cdots + 35312)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 10993097261056 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1331826786304 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 3316838464 \) Copy content Toggle raw display
$79$ \( (T^{8} + 4 T^{7} + \cdots - 3874304)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 50780104032256 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 49939734784 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 731590841860096 \) Copy content Toggle raw display
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