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Properties

Label 833.2.g.e
Level $833$
Weight $2$
Character orbit 833.g
Analytic conductor $6.652$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [833,2,Mod(344,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.344");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.g (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: \(6.65153848837\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 62x^{12} + 563x^{8} + 910x^{4} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{8}) q^{2} + \beta_{9} q^{3} + ( - \beta_{4} + \beta_{3} - 2) q^{4} - \beta_{15} q^{5} + ( - \beta_{5} - 2 \beta_1) q^{6} + (\beta_{13} + 2 \beta_{11} + \cdots - \beta_{8}) q^{8}+ \cdots + (\beta_{13} + \beta_{11}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_{8}) q^{2} + \beta_{9} q^{3} + ( - \beta_{4} + \beta_{3} - 2) q^{4} - \beta_{15} q^{5} + ( - \beta_{5} - 2 \beta_1) q^{6} + (\beta_{13} + 2 \beta_{11} + \cdots - \beta_{8}) q^{8}+ \cdots + ( - 2 \beta_{13} + \beta_{10} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 28 q^{4} - 12 q^{11} + 20 q^{16} + 52 q^{18} - 4 q^{22} - 8 q^{23} + 12 q^{29} - 20 q^{30} + 32 q^{37} - 28 q^{39} + 12 q^{44} + 72 q^{46} + 56 q^{51} - 36 q^{57} - 28 q^{58} - 36 q^{64} - 8 q^{65} - 96 q^{67} - 24 q^{71} - 160 q^{72} + 88 q^{74} - 116 q^{78} - 36 q^{79} + 40 q^{81} - 52 q^{85} - 20 q^{86} - 48 q^{88} + 120 q^{92} - 84 q^{95} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 62x^{12} + 563x^{8} + 910x^{4} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{13} - 198\nu^{9} - 2350\nu^{5} - 4925\nu ) / 1048 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\nu^{12} + 1057\nu^{8} + 6895\nu^{4} + 3743 ) / 3144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{12} + 1519\nu^{8} + 12029\nu^{4} + 7200 ) / 3144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -27\nu^{13} - 1651\nu^{9} - 13945\nu^{5} - 18518\nu ) / 3144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -27\nu^{12} - 1651\nu^{8} - 13945\nu^{4} - 15374 ) / 3144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26\nu^{13} + 1585\nu^{9} + 12987\nu^{5} + 11287\nu ) / 1572 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\nu^{14} + 2311\nu^{10} + 21953\nu^{6} + 49956\nu^{2} ) / 26724 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\nu^{15} + 2311\nu^{11} + 21953\nu^{7} + 49956\nu^{3} ) / 26724 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -237\nu^{14} - 14201\nu^{10} - 105347\nu^{6} - 68518\nu^{2} ) / 53448 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -405\nu^{14} - 24634\nu^{10} - 198826\nu^{6} - 141923\nu^{2} ) / 53448 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -464\nu^{15} - 28921\nu^{11} - 269103\nu^{7} - 419605\nu^{3} ) / 53448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 627\nu^{14} + 38500\nu^{10} + 330544\nu^{6} + 388211\nu^{2} ) / 53448 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -701\nu^{15} - 43122\nu^{11} - 374450\nu^{7} - 488123\nu^{3} ) / 53448 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 516\nu^{15} + 31567\nu^{11} + 264685\nu^{7} + 265067\nu^{3} ) / 26724 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} - \beta_{11} + 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 2\beta_{14} + 5\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{6} - 9\beta_{4} + 2\beta_{3} - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{7} - 18\beta_{5} + 2\beta_{2} - 32\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 48\beta_{13} + 68\beta_{11} - 22\beta_{10} - 105\beta_{8} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -68\beta_{15} - 138\beta_{14} + 22\beta_{12} - 221\beta_{9} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 337\beta_{6} + 495\beta_{4} - 182\beta_{3} + 731 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 495\beta_{7} + 1014\beta_{5} - 182\beta_{2} + 1563\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2395\beta_{13} - 3567\beta_{11} + 1378\beta_{10} + 5184\beta_{8} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3567\beta_{15} + 7340\beta_{14} - 1378\beta_{12} + 11146\beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -17108\beta_{6} - 25620\beta_{4} + 10096\beta_{3} - 37005 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -25620\beta_{7} - 52824\beta_{5} + 10096\beta_{2} - 79733\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 122461\beta_{13} + 183797\beta_{11} - 73016\beta_{10} - 264819\beta_{8} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -183797\beta_{15} - 379274\beta_{14} + 73016\beta_{12} - 571077\beta_{9} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(52\) \(785\)
\(\chi(n)\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
344.1
0.570362 + 0.570362i
−0.570362 − 0.570362i
1.21285 + 1.21285i
−1.21285 − 1.21285i
0.787063 + 0.787063i
−0.787063 − 0.787063i
1.89320 + 1.89320i
−1.89320 − 1.89320i
1.89320 − 1.89320i
−1.89320 + 1.89320i
0.787063 − 0.787063i
−0.787063 + 0.787063i
1.21285 − 1.21285i
−1.21285 + 1.21285i
0.570362 − 0.570362i
−0.570362 + 0.570362i
− 2.10582i −0.570362 + 0.570362i −2.43447 2.20289 − 2.20289i 1.20108 + 1.20108i 0 0.914908i 2.34937i −4.63888 − 4.63888i
344.2 − 2.10582i 0.570362 − 0.570362i −2.43447 −2.20289 + 2.20289i −1.20108 − 1.20108i 0 0.914908i 2.34937i 4.63888 + 4.63888i
344.3 0.728473i −1.21285 + 1.21285i 1.46933 0.588320 − 0.588320i −0.883529 − 0.883529i 0 2.52731i 0.0579831i 0.428575 + 0.428575i
344.4 0.728473i 1.21285 − 1.21285i 1.46933 −0.588320 + 0.588320i 0.883529 + 0.883529i 0 2.52731i 0.0579831i −0.428575 − 0.428575i
344.5 1.71208i −0.787063 + 0.787063i −0.931205 −2.50697 + 2.50697i −1.34751 − 1.34751i 0 1.82986i 1.76106i −4.29212 − 4.29212i
344.6 1.71208i 0.787063 − 0.787063i −0.931205 2.50697 − 2.50697i 1.34751 + 1.34751i 0 1.82986i 1.76106i 4.29212 + 4.29212i
344.7 2.66527i −1.89320 + 1.89320i −5.10366 1.58628 − 1.58628i −5.04589 − 5.04589i 0 − 8.27208i − 4.16842i 4.22787 + 4.22787i
344.8 2.66527i 1.89320 − 1.89320i −5.10366 −1.58628 + 1.58628i 5.04589 + 5.04589i 0 − 8.27208i − 4.16842i −4.22787 − 4.22787i
540.1 − 2.66527i −1.89320 − 1.89320i −5.10366 1.58628 + 1.58628i −5.04589 + 5.04589i 0 8.27208i 4.16842i 4.22787 − 4.22787i
540.2 − 2.66527i 1.89320 + 1.89320i −5.10366 −1.58628 − 1.58628i 5.04589 − 5.04589i 0 8.27208i 4.16842i −4.22787 + 4.22787i
540.3 − 1.71208i −0.787063 − 0.787063i −0.931205 −2.50697 − 2.50697i −1.34751 + 1.34751i 0 − 1.82986i − 1.76106i −4.29212 + 4.29212i
540.4 − 1.71208i 0.787063 + 0.787063i −0.931205 2.50697 + 2.50697i 1.34751 − 1.34751i 0 − 1.82986i − 1.76106i 4.29212 − 4.29212i
540.5 − 0.728473i −1.21285 − 1.21285i 1.46933 0.588320 + 0.588320i −0.883529 + 0.883529i 0 − 2.52731i − 0.0579831i 0.428575 − 0.428575i
540.6 − 0.728473i 1.21285 + 1.21285i 1.46933 −0.588320 − 0.588320i 0.883529 − 0.883529i 0 − 2.52731i − 0.0579831i −0.428575 + 0.428575i
540.7 2.10582i −0.570362 − 0.570362i −2.43447 2.20289 + 2.20289i 1.20108 − 1.20108i 0 − 0.914908i − 2.34937i −4.63888 + 4.63888i
540.8 2.10582i 0.570362 + 0.570362i −2.43447 −2.20289 − 2.20289i −1.20108 + 1.20108i 0 − 0.914908i − 2.34937i 4.63888 − 4.63888i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 344.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
17.c even 4 1 inner
119.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.g.e ✓ 16
7.b odd 2 1 inner 833.2.g.e ✓ 16
7.c even 3 2 833.2.o.e 32
7.d odd 6 2 833.2.o.e 32
17.c even 4 1 inner 833.2.g.e ✓ 16
119.f odd 4 1 inner 833.2.g.e ✓ 16
119.m odd 12 2 833.2.o.e 32
119.n even 12 2 833.2.o.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
833.2.g.e ✓ 16 1.a even 1 1 trivial
833.2.g.e ✓ 16 7.b odd 2 1 inner
833.2.g.e ✓ 16 17.c even 4 1 inner
833.2.g.e ✓ 16 119.f odd 4 1 inner
833.2.o.e 32 7.c even 3 2
833.2.o.e 32 7.d odd 6 2
833.2.o.e 32 119.m odd 12 2
833.2.o.e 32 119.n even 12 2

Hecke kernels

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

\( T_{2}^{8} + 15T_{2}^{6} + 73T_{2}^{4} + 127T_{2}^{2} + 49 \) Copy content Toggle raw display
\( T_{3}^{16} + 62T_{3}^{12} + 563T_{3}^{8} + 910T_{3}^{4} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 15 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 62 T^{12} + \cdots + 289 \) Copy content Toggle raw display
$5$ \( T^{16} + 278 T^{12} + \cdots + 180625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 6 T^{7} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 94 T^{6} + \cdots + 13328)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( (T^{8} + 124 T^{6} + \cdots + 32912)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 4 T^{7} + \cdots + 4624)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 224783136769 \) Copy content Toggle raw display
$37$ \( (T^{8} - 16 T^{7} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 70556640625 \) Copy content Toggle raw display
$43$ \( (T^{8} + 50 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 118 T^{6} + \cdots + 13328)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 138 T^{6} + \cdots + 17689)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 432 T^{6} + \cdots + 79609232)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 4001448049 \) Copy content Toggle raw display
$67$ \( (T^{4} + 24 T^{3} + \cdots + 139)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 12 T^{7} + \cdots + 559504)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( (T^{8} + 18 T^{7} + \cdots + 4857616)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 106 T^{6} + \cdots + 45968)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 390 T^{6} + \cdots + 3851792)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 2992682023969 \) Copy content Toggle raw display
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