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Properties

Label 833.2.e.f
Level $833$
Weight $2$
Character orbit 833.e
Analytic conductor $6.652$
Analytic rank $0$
Dimension $8$
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] = [833,2,Mod(18,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.e (of order \(3\), degree \(2\), not 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: \(6.65153848837\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 23x^{4} + x^{3} + 16x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 119)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{7} + \beta_{3} + 1) q^{3} + ( - \beta_{6} - \beta_{2} + \beta_1) q^{4} + (\beta_{4} + \beta_{3} + 1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{2} - 1) q^{6} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - 2) q^{8}+ \cdots + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{7} + \beta_{3} + 1) q^{3} + ( - \beta_{6} - \beta_{2} + \beta_1) q^{4} + (\beta_{4} + \beta_{3} + 1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{2} - 1) q^{6} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - 2) q^{8}+ \cdots + (4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 2 q^{3} - 3 q^{4} + 2 q^{5} - 2 q^{6} - 18 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 2 q^{3} - 3 q^{4} + 2 q^{5} - 2 q^{6} - 18 q^{8} - 6 q^{9} - 2 q^{10} - 2 q^{11} + 6 q^{12} - 16 q^{13} - 32 q^{15} - 5 q^{16} - 4 q^{17} + q^{18} + 10 q^{19} + 6 q^{20} - 44 q^{22} + 6 q^{23} - 26 q^{24} + 2 q^{25} - 4 q^{26} - 4 q^{27} + 4 q^{29} - 5 q^{30} + 12 q^{31} + 14 q^{32} - 2 q^{33} - 2 q^{34} - 40 q^{36} - 6 q^{37} - 22 q^{38} + 18 q^{39} + 17 q^{40} - 24 q^{41} - 24 q^{43} - 22 q^{44} - 2 q^{45} - 24 q^{46} + 2 q^{47} - 30 q^{48} - 4 q^{50} + 2 q^{51} + 26 q^{52} + 26 q^{53} + 23 q^{54} - 8 q^{55} + 48 q^{57} + 22 q^{58} - 4 q^{59} - 17 q^{60} + 12 q^{61} - 32 q^{62} + 34 q^{64} - 26 q^{65} - 20 q^{66} + 12 q^{67} - 3 q^{68} - 24 q^{69} - 28 q^{71} - 6 q^{72} + 20 q^{73} - 18 q^{74} - 30 q^{75} - 16 q^{76} - 36 q^{78} + 14 q^{79} - 10 q^{80} - 8 q^{81} + 17 q^{82} + 56 q^{83} - 4 q^{85} - 27 q^{86} - 2 q^{87} + 22 q^{88} - 10 q^{89} + 54 q^{90} + 8 q^{92} + 20 q^{93} + 22 q^{94} + 14 q^{95} + 18 q^{96} - 52 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} + 3x^{5} + 23x^{4} + x^{3} + 16x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\nu^{7} + 857\nu^{6} - 474\nu^{5} + 4932\nu^{4} + 5726\nu^{3} + 14569\nu^{2} + 4405\nu + 2370 ) / 5817 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -146\nu^{7} - 352\nu^{6} + 792\nu^{5} - 4191\nu^{4} + 1232\nu^{3} - 5984\nu^{2} + 20203\nu - 3960 ) / 5817 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -166\nu^{7} + 556\nu^{6} - 1251\nu^{5} + 1530\nu^{4} - 1946\nu^{3} + 9452\nu^{2} - 1174\nu + 438 ) / 5817 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 197\nu^{7} - 800\nu^{6} + 1800\nu^{5} - 3708\nu^{4} + 2800\nu^{3} - 13600\nu^{2} - 3793\nu - 9000 ) / 5817 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -130\nu^{7} + 85\nu^{6} - 676\nu^{5} - 624\nu^{4} - 3206\nu^{3} - 494\nu^{2} - 312\nu - 498 ) / 1939 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -565\nu^{7} + 444\nu^{6} - 2938\nu^{5} - 2712\nu^{4} - 11249\nu^{3} - 2147\nu^{2} - 1356\nu - 822 ) / 1939 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + 2\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + \beta_{3} - 9\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{7} + 30\beta_{6} - 11\beta_{4} + 6\beta_{3} + 10\beta_{2} - 30\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{7} + 67\beta_{6} + 36\beta_{5} + 36\beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 77\beta_{5} + 88\beta_{4} - 36\beta_{3} + 193\beta _1 + 88 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(785\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−0.844316 − 1.46240i
−0.392238 − 0.679376i
0.434993 + 0.753430i
1.30156 + 2.25437i
−0.844316 + 1.46240i
−0.392238 + 0.679376i
0.434993 − 0.753430i
1.30156 − 2.25437i
−0.844316 − 1.46240i 0.0439772 − 0.0761707i −0.425739 + 0.737401i 0.456023 + 0.789855i −0.148523 0 −1.93943 1.49613 + 2.59138i 0.770055 − 1.33377i
18.2 −0.392238 − 0.679376i 1.51987 − 2.63249i 0.692299 − 1.19910i −1.01987 − 1.76646i −2.38460 0 −2.65514 −3.11999 − 5.40398i −0.800061 + 1.38575i
18.3 0.434993 + 0.753430i −1.28917 + 2.23291i 0.621562 − 1.07658i 1.78917 + 3.09894i −2.24312 0 2.82147 −1.82394 − 3.15915i −1.55656 + 2.69603i
18.4 1.30156 + 2.25437i 0.725330 − 1.25631i −2.38812 + 4.13635i −0.225330 − 0.390283i 3.77625 0 −7.22690 0.447793 + 0.775600i 0.586561 − 1.01595i
324.1 −0.844316 + 1.46240i 0.0439772 + 0.0761707i −0.425739 − 0.737401i 0.456023 − 0.789855i −0.148523 0 −1.93943 1.49613 − 2.59138i 0.770055 + 1.33377i
324.2 −0.392238 + 0.679376i 1.51987 + 2.63249i 0.692299 + 1.19910i −1.01987 + 1.76646i −2.38460 0 −2.65514 −3.11999 + 5.40398i −0.800061 − 1.38575i
324.3 0.434993 − 0.753430i −1.28917 − 2.23291i 0.621562 + 1.07658i 1.78917 − 3.09894i −2.24312 0 2.82147 −1.82394 + 3.15915i −1.55656 − 2.69603i
324.4 1.30156 − 2.25437i 0.725330 + 1.25631i −2.38812 − 4.13635i −0.225330 + 0.390283i 3.77625 0 −7.22690 0.447793 − 0.775600i 0.586561 + 1.01595i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.e.f 8
7.b odd 2 1 833.2.e.e 8
7.c even 3 1 833.2.a.e 4
7.c even 3 1 inner 833.2.e.f 8
7.d odd 6 1 119.2.a.a ✓ 4
7.d odd 6 1 833.2.e.e 8
21.g even 6 1 1071.2.a.k 4
21.h odd 6 1 7497.2.a.bl 4
28.f even 6 1 1904.2.a.s 4
35.i odd 6 1 2975.2.a.k 4
56.j odd 6 1 7616.2.a.bk 4
56.m even 6 1 7616.2.a.bn 4
119.h odd 6 1 2023.2.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.a.a ✓ 4 7.d odd 6 1
833.2.a.e 4 7.c even 3 1
833.2.e.e 8 7.b odd 2 1
833.2.e.e 8 7.d odd 6 1
833.2.e.f 8 1.a even 1 1 trivial
833.2.e.f 8 7.c even 3 1 inner
1071.2.a.k 4 21.g even 6 1
1904.2.a.s 4 28.f even 6 1
2023.2.a.e 4 119.h odd 6 1
2975.2.a.k 4 35.i odd 6 1
7497.2.a.bl 4 21.h odd 6 1
7616.2.a.bk 4 56.j odd 6 1
7616.2.a.bn 4 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):

\( T_{2}^{8} - T_{2}^{7} + 6T_{2}^{6} + 3T_{2}^{5} + 23T_{2}^{4} + T_{2}^{3} + 16T_{2}^{2} + 3T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{8} - 2T_{3}^{7} + 11T_{3}^{6} - 10T_{3}^{5} + 74T_{3}^{4} - 88T_{3}^{3} + 137T_{3}^{2} - 12T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} - 2T_{5}^{7} + 11T_{5}^{6} + 6T_{5}^{5} + 54T_{5}^{4} - 16T_{5}^{3} + 37T_{5}^{2} + 12T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 6 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( (T^{4} + 8 T^{3} + \cdots - 368)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 57600 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} - 20 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 12 T^{7} + \cdots + 840889 \) Copy content Toggle raw display
$37$ \( T^{8} + 6 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 237)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 12 T^{3} + \cdots - 115)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 2 T^{7} + \cdots + 3154176 \) Copy content Toggle raw display
$53$ \( T^{8} - 26 T^{7} + \cdots + 641601 \) Copy content Toggle raw display
$59$ \( T^{8} + 4 T^{7} + \cdots + 589824 \) Copy content Toggle raw display
$61$ \( T^{8} - 12 T^{7} + \cdots + 41615401 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + \cdots + 3798601 \) Copy content Toggle raw display
$71$ \( (T^{4} + 14 T^{3} + \cdots - 3312)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + \cdots + 17161 \) Copy content Toggle raw display
$79$ \( T^{8} - 14 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$83$ \( (T^{4} - 28 T^{3} + \cdots + 1200)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 10 T^{7} + \cdots + 518400 \) Copy content Toggle raw display
$97$ \( (T^{4} + 26 T^{3} + \cdots - 1901)^{2} \) Copy content Toggle raw display
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