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Properties

Label 833.2.b.e
Level $833$
Weight $2$
Character orbit 833.b
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(50,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("833.50");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.b (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: \(6.65153848837\)
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: \( x^{16} + 24x^{14} + 240x^{12} + 1190x^{10} + 3318x^{8} + 5064x^{6} + 7273x^{4} + 4392x^{2} + 3969 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_1 q^{3} + (\beta_{6} + 1) q^{4} + \beta_{2} q^{5} + \beta_{5} q^{6} + (\beta_{6} - \beta_{4} + 1) q^{8} + ( - \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - \beta_1 q^{3} + (\beta_{6} + 1) q^{4} + \beta_{2} q^{5} + \beta_{5} q^{6} + (\beta_{6} - \beta_{4} + 1) q^{8} + ( - \beta_{4} - 1) q^{9} + (\beta_{3} + \beta_{2} - \beta_1) q^{10} + \beta_{9} q^{11} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{12} + (\beta_{12} + \beta_{10}) q^{13} + ( - 2 \beta_{6} + 2 \beta_{4} + 1) q^{15} + ( - \beta_{7} - \beta_{4} - 1) q^{16} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{17}+ \cdots + (3 \beta_{14} + \beta_{13} + \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 16 q^{9} + 24 q^{15} - 12 q^{16} - 12 q^{18} - 16 q^{25} - 12 q^{30} + 8 q^{32} - 40 q^{36} + 16 q^{43} - 64 q^{50} + 40 q^{51} + 32 q^{53} - 60 q^{60} - 44 q^{64} - 24 q^{67} + 48 q^{72} - 88 q^{81} + 72 q^{85} - 4 q^{86} + 96 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 240x^{12} + 1190x^{10} + 3318x^{8} + 5064x^{6} + 7273x^{4} + 4392x^{2} + 3969 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 14732 \nu^{14} + 1284511 \nu^{12} + 25387883 \nu^{10} + 228180842 \nu^{8} + 1014019573 \nu^{6} + \cdots + 2320363989 ) / 318637071 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 94016 \nu^{14} + 3048151 \nu^{12} + 41477315 \nu^{10} + 297105635 \nu^{8} + 1177066021 \nu^{6} + \cdots + 2435997321 ) / 318637071 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 49730 \nu^{14} + 1343787 \nu^{12} + 15293308 \nu^{10} + 90218987 \nu^{8} + 303335373 \nu^{6} + \cdots + 555476832 ) / 106212357 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 197924 \nu^{14} - 4507528 \nu^{12} - 42045806 \nu^{10} - 186045665 \nu^{8} - 446028880 \nu^{6} + \cdots + 529283079 ) / 318637071 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 275135 \nu^{14} - 6076801 \nu^{12} - 53847449 \nu^{10} - 212465903 \nu^{8} - 404799769 \nu^{6} + \cdots - 79729353 ) / 318637071 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 119188 \nu^{14} + 2857891 \nu^{12} + 28296750 \nu^{10} + 135205555 \nu^{8} + 334876546 \nu^{6} + \cdots + 2882790 ) / 106212357 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 476204 \nu^{14} + 11317561 \nu^{12} + 110846624 \nu^{10} + 522737537 \nu^{8} + 1287612070 \nu^{6} + \cdots + 319643172 ) / 318637071 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 255303 \nu^{15} - 5942276 \nu^{13} - 57157560 \nu^{11} - 266268562 \nu^{9} + \cdots + 17420427 \nu ) / 743486499 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 727511 \nu^{15} + 15911682 \nu^{13} + 137758595 \nu^{11} + 501906300 \nu^{9} + \cdots - 4558733379 \nu ) / 743486499 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 810669 \nu^{15} + 20275861 \nu^{13} + 213205228 \nu^{11} + 1132758837 \nu^{9} + \cdots + 2799088758 \nu ) / 743486499 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2438609 \nu^{15} - 62186671 \nu^{13} - 666461687 \nu^{11} - 3621017729 \nu^{9} + \cdots - 10789800750 \nu ) / 2230459497 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2515250 \nu^{15} + 60320920 \nu^{13} + 600110699 \nu^{11} + 2935490411 \nu^{9} + \cdots + 3604515525 \nu ) / 2230459497 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2811623 \nu^{15} + 66350965 \nu^{13} + 643587944 \nu^{11} + 2990394575 \nu^{9} + \cdots + 11770061508 \nu ) / 2230459497 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4109380 \nu^{15} - 92685620 \nu^{13} - 838020985 \nu^{11} - 3342288901 \nu^{9} + \cdots + 16337921643 \nu ) / 2230459497 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4875289 \nu^{15} + 110512448 \nu^{13} + 1009493665 \nu^{11} + 4141094587 \nu^{9} + \cdots - 11929263930 \nu ) / 2230459497 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} + \beta_{2} - \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{15} - 4\beta_{14} + 2\beta_{13} - 3\beta_{12} - 3\beta_{11} - 3\beta_{10} + \beta_{9} - 10\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{7} + 7\beta_{6} - 4\beta_{5} - 4\beta_{4} - 2\beta_{3} - 10\beta_{2} + 12\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21 \beta_{15} + 13 \beta_{14} - 10 \beta_{13} + 30 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + \cdots + 94 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{7} - 19\beta_{6} + 50\beta_{5} + \beta_{4} + 43\beta_{3} + 91\beta_{2} - 125\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 56 \beta_{15} + 59 \beta_{14} - 15 \beta_{13} - 231 \beta_{12} - 441 \beta_{11} - 462 \beta_{10} + \cdots - 811 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 305\beta_{7} - 305\beta_{6} - 444\beta_{5} + 197\beta_{4} - 456\beta_{3} - 712\beta_{2} + 1060\beta _1 - 627 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2602 \beta_{15} - 1841 \beta_{14} + 1054 \beta_{13} + 1551 \beta_{12} + 3351 \beta_{11} + \cdots + 5833 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6113 \beta_{7} + 6473 \beta_{6} + 2950 \beta_{5} - 3074 \beta_{4} + 3239 \beta_{3} + 4447 \beta_{2} + \cdots + 11067 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 37020 \beta_{15} + 24701 \beta_{14} - 15536 \beta_{13} - 7689 \beta_{12} - 17820 \beta_{11} + \cdots - 30064 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 72758 \beta_{7} - 78101 \beta_{6} - 10844 \beta_{5} + 34091 \beta_{4} - 12550 \beta_{3} - 15436 \beta_{2} + \cdots - 127425 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 381221 \beta_{15} - 249710 \beta_{14} + 161703 \beta_{13} + 2769 \beta_{12} + 14040 \beta_{11} + \cdots + 17965 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 666800 \beta_{7} + 718766 \beta_{6} - 65505 \beta_{5} - 305225 \beta_{4} - 65760 \beta_{3} + \cdots + 1155273 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3101650 \beta_{15} + 2019830 \beta_{14} - 1320061 \beta_{13} + 570138 \beta_{12} + \cdots + 2124260 \beta_{8} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(52\) \(785\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1
−0.707107 + 0.969681i
0.707107 − 0.969681i
0.707107 + 0.969681i
−0.707107 − 0.969681i
0.707107 − 2.00301i
−0.707107 + 2.00301i
−0.707107 − 2.00301i
0.707107 + 2.00301i
0.707107 + 0.762351i
−0.707107 − 0.762351i
−0.707107 + 0.762351i
0.707107 − 0.762351i
−0.707107 + 2.90973i
0.707107 − 2.90973i
0.707107 + 2.90973i
−0.707107 − 2.90973i
−1.94939 − 2.32604i 1.80011 0.954700i 4.53434i 0 0.389667 −2.41044 − 1.86108i
50.2 −1.94939 − 2.32604i 1.80011 0.954700i 4.53434i 0 0.389667 −2.41044 − 1.86108i
50.3 −1.94939 2.32604i 1.80011 − 0.954700i − 4.53434i 0 0.389667 −2.41044 1.86108i
50.4 −1.94939 2.32604i 1.80011 − 0.954700i − 4.53434i 0 0.389667 −2.41044 1.86108i
50.5 −0.563729 − 0.491958i −1.68221 − 2.34072i 0.277331i 0 2.07577 2.75798 1.31953i
50.6 −0.563729 − 0.491958i −1.68221 − 2.34072i 0.277331i 0 2.07577 2.75798 1.31953i
50.7 −0.563729 0.491958i −1.68221 2.34072i − 0.277331i 0 2.07577 2.75798 − 1.31953i
50.8 −0.563729 0.491958i −1.68221 2.34072i − 0.277331i 0 2.07577 2.75798 − 1.31953i
50.9 1.16027 − 2.53482i −0.653767 3.61295i − 2.94109i 0 −3.07909 −3.42533 4.19201i
50.10 1.16027 − 2.53482i −0.653767 3.61295i − 2.94109i 0 −3.07909 −3.42533 4.19201i
50.11 1.16027 2.53482i −0.653767 − 3.61295i 2.94109i 0 −3.07909 −3.42533 − 4.19201i
50.12 1.16027 2.53482i −0.653767 − 3.61295i 2.94109i 0 −3.07909 −3.42533 − 4.19201i
50.13 2.35284 − 1.98046i 3.53587 − 2.13452i − 4.65970i 0 3.61366 −0.922211 − 5.02218i
50.14 2.35284 − 1.98046i 3.53587 − 2.13452i − 4.65970i 0 3.61366 −0.922211 − 5.02218i
50.15 2.35284 1.98046i 3.53587 2.13452i 4.65970i 0 3.61366 −0.922211 5.02218i
50.16 2.35284 1.98046i 3.53587 2.13452i 4.65970i 0 3.61366 −0.922211 5.02218i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.b.e ✓ 16
7.b odd 2 1 inner 833.2.b.e ✓ 16
7.c even 3 2 833.2.j.e 32
7.d odd 6 2 833.2.j.e 32
17.b even 2 1 inner 833.2.b.e ✓ 16
119.d odd 2 1 inner 833.2.b.e ✓ 16
119.h odd 6 2 833.2.j.e 32
119.j even 6 2 833.2.j.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
833.2.b.e ✓ 16 1.a even 1 1 trivial
833.2.b.e ✓ 16 7.b odd 2 1 inner
833.2.b.e ✓ 16 17.b even 2 1 inner
833.2.b.e ✓ 16 119.d odd 2 1 inner
833.2.j.e 32 7.c even 3 2
833.2.j.e 32 7.d odd 6 2
833.2.j.e 32 119.h odd 6 2
833.2.j.e 32 119.j even 6 2

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 3T_{2} + 3 \) Copy content Toggle raw display
\( T_{13}^{8} - 50T_{13}^{6} + 600T_{13}^{4} - 1472T_{13}^{2} + 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} - 5 T^{2} + \cdots + 3)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 16 T^{6} + \cdots + 33)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 24 T^{6} + \cdots + 297)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 56 T^{6} + \cdots + 528)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 50 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( (T^{8} - 62 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 134 T^{6} + \cdots + 4752)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 140 T^{6} + \cdots + 25872)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 156 T^{6} + \cdots + 323433)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 162 T^{6} + \cdots + 384912)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 244 T^{6} + \cdots + 3110217)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} - 5 T^{2} + \cdots + 3)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 360 T^{6} + \cdots + 47279376)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} + \cdots - 369)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 252 T^{6} + \cdots + 1245456)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 156 T^{6} + \cdots + 297)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 6 T^{3} - 15 T^{2} + \cdots + 27)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 302 T^{6} + \cdots + 1267728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 204 T^{6} + \cdots + 2673)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 504 T^{6} + \cdots + 155680272)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 262 T^{6} + \cdots + 3111696)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 496 T^{6} + \cdots + 147768336)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 240 T^{6} + \cdots + 216513)^{2} \) Copy content Toggle raw display
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