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Properties

Label 833.2.e.j
Level $833$
Weight $2$
Character orbit 833.e
Analytic conductor $6.652$
Analytic rank $0$
Dimension $14$
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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} + 17 x^{12} - 34 x^{11} + 150 x^{10} - 279 x^{9} + 758 x^{8} - 926 x^{7} + 1840 x^{6} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} - \beta_{13} q^{3} + ( - \beta_{10} - \beta_{4} + \beta_{3} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{10} + \beta_{5} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} - \beta_{13} q^{3} + ( - \beta_{10} - \beta_{4} + \beta_{3} + \cdots - 1) q^{4}+ \cdots + ( - 2 \beta_{12} - 2 \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{2} + q^{3} - 11 q^{4} + 2 q^{5} - 10 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{2} + q^{3} - 11 q^{4} + 2 q^{5} - 10 q^{6} - 8 q^{9} - 10 q^{10} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 16 q^{15} - 23 q^{16} + 7 q^{17} - 15 q^{18} - 5 q^{19} - 54 q^{20} + 18 q^{22} - 22 q^{23} - 7 q^{24} - 7 q^{25} - 10 q^{26} - 20 q^{27} + 2 q^{29} - 15 q^{30} + 2 q^{31} - 8 q^{32} - 27 q^{33} - 6 q^{34} - 36 q^{36} - 19 q^{37} - 35 q^{38} - 2 q^{39} - 25 q^{40} - 24 q^{41} + 38 q^{43} + 2 q^{44} + 9 q^{45} - 7 q^{47} - 60 q^{48} - 44 q^{50} - q^{51} - 7 q^{52} + 8 q^{53} + 11 q^{54} + 18 q^{55} - 6 q^{57} - 15 q^{58} + 5 q^{59} + q^{60} - 4 q^{61} - 34 q^{62} + 80 q^{64} - 2 q^{65} - 16 q^{66} - 24 q^{67} + 11 q^{68} - 32 q^{69} - 2 q^{71} + 19 q^{73} + 18 q^{74} - 2 q^{75} + 26 q^{76} - 90 q^{78} - 15 q^{79} - 9 q^{80} + 29 q^{81} + 41 q^{82} + 4 q^{83} + 4 q^{85} + 21 q^{86} + 7 q^{87} - 62 q^{88} - 32 q^{89} - 98 q^{90} + 16 q^{92} + q^{93} - 15 q^{94} - 12 q^{95} + 18 q^{96} + 46 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} + 17 x^{12} - 34 x^{11} + 150 x^{10} - 279 x^{9} + 758 x^{8} - 926 x^{7} + 1840 x^{6} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 199906140083 \nu^{13} + 469206844927 \nu^{12} - 3004451441292 \nu^{11} + \cdots + 29406783323570 ) / 148309106292801 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2792097226232 \nu^{13} + 8411207980771 \nu^{12} - 36284852048484 \nu^{11} + \cdots - 70466935458025 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5881356664714 \nu^{13} - 16644539293727 \nu^{12} + 97637029075503 \nu^{11} + \cdots - 984138852425080 ) / 741545531464005 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1192926934119 \nu^{13} + 2610072241415 \nu^{12} - 17527678935804 \nu^{11} + \cdots - 151814698786083 ) / 98872737528534 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3716255419655 \nu^{13} + 9096019543750 \nu^{12} - 57341853007500 \nu^{11} + \cdots - 884032607393329 ) / 296618212585602 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1239189977293 \nu^{13} + 2660241511287 \nu^{12} - 18023575216020 \nu^{11} + \cdots + 161076267013849 ) / 98872737528534 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12771604787171 \nu^{13} - 29042075247991 \nu^{12} + 190929012173292 \nu^{11} + \cdots + 15\!\cdots\!59 ) / 593236425171204 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12958107274442 \nu^{13} + 39709465020316 \nu^{12} - 222529256804469 \nu^{11} + \cdots + 22\!\cdots\!65 ) / 494363687642670 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39385352967736 \nu^{13} + 112217973144323 \nu^{12} - 644713096711182 \nu^{11} + \cdots + 14\!\cdots\!75 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 48395584515827 \nu^{13} + 164461976293366 \nu^{12} - 868957573535889 \nu^{11} + \cdots + 85\!\cdots\!15 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3364574061058 \nu^{13} + 7831710698991 \nu^{12} - 50662403331000 \nu^{11} + \cdots - 408133316602400 ) / 98872737528534 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 137749466751956 \nu^{13} + 393580174427563 \nu^{12} + \cdots + 50\!\cdots\!75 ) / 29\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{7} - \beta_{5} + 3\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{6} - \beta_{5} - 7\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} - 9\beta_{10} - 17\beta_{4} + 6\beta_{3} - 7\beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{13} + 9 \beta_{11} - 10 \beta_{10} - 11 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 10 \beta_{5} + \cdots - 39 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{12} + 24\beta_{8} - 38\beta_{7} + 3\beta_{6} + 71\beta_{5} + 48\beta_{2} + 112 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 30 \beta_{13} - 68 \beta_{12} - 68 \beta_{11} + 84 \beta_{10} + 96 \beta_{9} + 96 \beta_{6} + \cdots - 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 222 \beta_{13} + 12 \beta_{11} + 540 \beta_{10} + 46 \beta_{9} - 222 \beta_{8} + 253 \beta_{7} + \cdots + 350 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 494\beta_{12} - 314\beta_{8} - 202\beta_{7} - 774\beta_{6} - 671\beta_{5} - 2408\beta_{2} + 326 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1862 \beta_{13} - 103 \beta_{12} - 103 \beta_{11} - 4055 \beta_{10} - 491 \beta_{9} - 491 \beta_{6} + \cdots - 5612 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2844 \beta_{13} + 3564 \beta_{11} - 5257 \beta_{10} - 6020 \beta_{9} + 2844 \beta_{8} + \cdots - 13852 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 763\beta_{12} + 14884\beta_{8} - 12159\beta_{7} + 4537\beta_{6} + 30300\beta_{5} + 18131\beta_{2} + 40664 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 23958 \beta_{13} - 25763 \beta_{12} - 25763 \beta_{11} + 40809 \beta_{10} + 45947 \beta_{9} + \cdots - 14115 \) Copy content Toggle raw display

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

Character values

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

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

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
1.37084 − 2.37437i
1.10660 − 1.91669i
0.782162 − 1.35474i
0.276007 − 0.478058i
0.110464 − 0.191329i
−0.805430 + 1.39505i
−1.34065 + 2.32207i
1.37084 + 2.37437i
1.10660 + 1.91669i
0.782162 + 1.35474i
0.276007 + 0.478058i
0.110464 + 0.191329i
−0.805430 − 1.39505i
−1.34065 − 2.32207i
−1.37084 − 2.37437i 0.638645 − 1.10617i −2.75842 + 4.77772i −0.296687 − 0.513876i −3.50193 0 9.64207 0.684264 + 1.18518i −0.813421 + 1.40889i
18.2 −1.10660 − 1.91669i −1.08975 + 1.88751i −1.44914 + 2.50999i 0.217658 + 0.376994i 4.82369 0 1.98810 −0.875118 − 1.51575i 0.481721 − 0.834366i
18.3 −0.782162 − 1.35474i 1.50806 − 2.61203i −0.223554 + 0.387208i 1.48987 + 2.58053i −4.71818 0 −2.42922 −3.04848 − 5.28012i 2.33064 − 4.03679i
18.4 −0.276007 − 0.478058i −0.898729 + 1.55664i 0.847640 − 1.46816i −2.00822 − 3.47834i 0.992222 0 −2.03985 −0.115427 − 0.199925i −1.10857 + 1.92009i
18.5 −0.110464 − 0.191329i 1.16018 − 2.00948i 0.975595 − 1.68978i −0.894042 − 1.54853i −0.512630 0 −0.872928 −1.19202 − 2.06463i −0.197519 + 0.342112i
18.6 0.805430 + 1.39505i −1.07693 + 1.86529i −0.297435 + 0.515173i 0.922414 + 1.59767i −3.46956 0 2.26347 −0.819549 − 1.41950i −1.48588 + 2.57362i
18.7 1.34065 + 2.32207i 0.258529 − 0.447786i −2.59468 + 4.49412i 1.56901 + 2.71760i 1.38639 0 −8.55164 1.36633 + 2.36654i −4.20697 + 7.28669i
324.1 −1.37084 + 2.37437i 0.638645 + 1.10617i −2.75842 − 4.77772i −0.296687 + 0.513876i −3.50193 0 9.64207 0.684264 − 1.18518i −0.813421 − 1.40889i
324.2 −1.10660 + 1.91669i −1.08975 − 1.88751i −1.44914 − 2.50999i 0.217658 − 0.376994i 4.82369 0 1.98810 −0.875118 + 1.51575i 0.481721 + 0.834366i
324.3 −0.782162 + 1.35474i 1.50806 + 2.61203i −0.223554 − 0.387208i 1.48987 − 2.58053i −4.71818 0 −2.42922 −3.04848 + 5.28012i 2.33064 + 4.03679i
324.4 −0.276007 + 0.478058i −0.898729 − 1.55664i 0.847640 + 1.46816i −2.00822 + 3.47834i 0.992222 0 −2.03985 −0.115427 + 0.199925i −1.10857 − 1.92009i
324.5 −0.110464 + 0.191329i 1.16018 + 2.00948i 0.975595 + 1.68978i −0.894042 + 1.54853i −0.512630 0 −0.872928 −1.19202 + 2.06463i −0.197519 − 0.342112i
324.6 0.805430 − 1.39505i −1.07693 − 1.86529i −0.297435 − 0.515173i 0.922414 − 1.59767i −3.46956 0 2.26347 −0.819549 + 1.41950i −1.48588 − 2.57362i
324.7 1.34065 − 2.32207i 0.258529 + 0.447786i −2.59468 − 4.49412i 1.56901 − 2.71760i 1.38639 0 −8.55164 1.36633 − 2.36654i −4.20697 − 7.28669i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.7
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.j 14
7.b odd 2 1 119.2.e.b ✓ 14
7.c even 3 1 833.2.a.h 7
7.c even 3 1 inner 833.2.e.j 14
7.d odd 6 1 119.2.e.b ✓ 14
7.d odd 6 1 833.2.a.i 7
21.c even 2 1 1071.2.i.i 14
21.g even 6 1 1071.2.i.i 14
21.g even 6 1 7497.2.a.bz 7
21.h odd 6 1 7497.2.a.ca 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.e.b ✓ 14 7.b odd 2 1
119.2.e.b ✓ 14 7.d odd 6 1
833.2.a.h 7 7.c even 3 1
833.2.a.i 7 7.d odd 6 1
833.2.e.j 14 1.a even 1 1 trivial
833.2.e.j 14 7.c even 3 1 inner
1071.2.i.i 14 21.c even 2 1
1071.2.i.i 14 21.g even 6 1
7497.2.a.bz 7 21.g even 6 1
7497.2.a.ca 7 21.h odd 6 1

Hecke kernels

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

\( T_{2}^{14} + 3 T_{2}^{13} + 17 T_{2}^{12} + 34 T_{2}^{11} + 150 T_{2}^{10} + 279 T_{2}^{9} + 758 T_{2}^{8} + \cdots + 25 \) Copy content Toggle raw display
\( T_{3}^{14} - T_{3}^{13} + 15 T_{3}^{12} - 4 T_{3}^{11} + 144 T_{3}^{10} - 31 T_{3}^{9} + 752 T_{3}^{8} + \cdots + 1521 \) Copy content Toggle raw display
\( T_{5}^{14} - 2 T_{5}^{13} + 23 T_{5}^{12} - 48 T_{5}^{11} + 387 T_{5}^{10} - 705 T_{5}^{9} + 2669 T_{5}^{8} + \cdots + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 3 T^{13} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{14} - T^{13} + \cdots + 1521 \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 4 T^{13} + \cdots + 12321 \) Copy content Toggle raw display
$13$ \( (T^{7} - 2 T^{6} + \cdots + 2083)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$19$ \( T^{14} + 5 T^{13} + \cdots + 135424 \) Copy content Toggle raw display
$23$ \( T^{14} + 22 T^{13} + \cdots + 16000000 \) Copy content Toggle raw display
$29$ \( (T^{7} - T^{6} - 45 T^{5} + \cdots + 480)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 5851026064 \) Copy content Toggle raw display
$37$ \( T^{14} + 19 T^{13} + \cdots + 102400 \) Copy content Toggle raw display
$41$ \( (T^{7} + 12 T^{6} + \cdots + 72480)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} - 19 T^{6} + \cdots - 142928)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + 7 T^{13} + \cdots + 19927296 \) Copy content Toggle raw display
$53$ \( T^{14} - 8 T^{13} + \cdots + 9641025 \) Copy content Toggle raw display
$59$ \( T^{14} - 5 T^{13} + \cdots + 389376 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 1023690674176 \) Copy content Toggle raw display
$67$ \( T^{14} + 24 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( (T^{7} + T^{6} + \cdots - 22955)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} - 19 T^{13} + \cdots + 6718464 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 129679932321 \) Copy content Toggle raw display
$83$ \( (T^{7} - 2 T^{6} + \cdots - 6592)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 36419776614400 \) Copy content Toggle raw display
$97$ \( (T^{7} - 23 T^{6} + \cdots - 2716304)^{2} \) Copy content Toggle raw display
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