Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.65153848837\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} + 3 q^{8} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} + 3 q^{8} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} - 2 q^{13} + \zeta_{6} q^{16} + (\zeta_{6} - 1) q^{17} + (3 \zeta_{6} - 3) q^{18} + 4 \zeta_{6} q^{19} + 2 q^{20} - 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 2 \zeta_{6} q^{26} + 6 q^{29} + (4 \zeta_{6} - 4) q^{31} + ( - 5 \zeta_{6} + 5) q^{32} - q^{34} + 3 q^{36} + 2 \zeta_{6} q^{37} + (4 \zeta_{6} - 4) q^{38} + 6 \zeta_{6} q^{40} - 6 q^{41} + 4 q^{43} + (6 \zeta_{6} - 6) q^{45} + ( - 4 \zeta_{6} + 4) q^{46} + q^{50} + (2 \zeta_{6} - 2) q^{52} + (6 \zeta_{6} - 6) q^{53} + 6 \zeta_{6} q^{58} + ( - 12 \zeta_{6} + 12) q^{59} + 10 \zeta_{6} q^{61} - 4 q^{62} + 7 q^{64} - 4 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{67} + \zeta_{6} q^{68} - 4 q^{71} + 9 \zeta_{6} q^{72} + ( - 6 \zeta_{6} + 6) q^{73} + (2 \zeta_{6} - 2) q^{74} + 4 q^{76} - 12 \zeta_{6} q^{79} + (2 \zeta_{6} - 2) q^{80} + (9 \zeta_{6} - 9) q^{81} - 6 \zeta_{6} q^{82} - 4 q^{83} - 2 q^{85} + 4 \zeta_{6} q^{86} - 10 \zeta_{6} q^{89} - 6 q^{90} - 4 q^{92} + (8 \zeta_{6} - 8) q^{95} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} + 2 q^{5} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} + 2 q^{5} + 6 q^{8} + 3 q^{9} - 2 q^{10} - 4 q^{13} + q^{16} - q^{17} - 3 q^{18} + 4 q^{19} + 4 q^{20} - 4 q^{23} + q^{25} - 2 q^{26} + 12 q^{29} - 4 q^{31} + 5 q^{32} - 2 q^{34} + 6 q^{36} + 2 q^{37} - 4 q^{38} + 6 q^{40} - 12 q^{41} + 8 q^{43} - 6 q^{45} + 4 q^{46} + 2 q^{50} - 2 q^{52} - 6 q^{53} + 6 q^{58} + 12 q^{59} + 10 q^{61} - 8 q^{62} + 14 q^{64} - 4 q^{65} - 4 q^{67} + q^{68} - 8 q^{71} + 9 q^{72} + 6 q^{73} - 2 q^{74} + 8 q^{76} - 12 q^{79} - 2 q^{80} - 9 q^{81} - 6 q^{82} - 8 q^{83} - 4 q^{85} + 4 q^{86} - 10 q^{89} - 12 q^{90} - 8 q^{92} - 8 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(785\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 + 1.73205i 0 0 3.00000 1.50000 + 2.59808i −1.00000 + 1.73205i
324.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 1.73205i 0 0 3.00000 1.50000 2.59808i −1.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
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.b 2
7.b odd 2 1 833.2.e.a 2
7.c even 3 1 17.2.a.a 1
7.c even 3 1 inner 833.2.e.b 2
7.d odd 6 1 833.2.a.a 1
7.d odd 6 1 833.2.e.a 2
21.g even 6 1 7497.2.a.l 1
21.h odd 6 1 153.2.a.c 1
28.g odd 6 1 272.2.a.b 1
35.j even 6 1 425.2.a.d 1
35.l odd 12 2 425.2.b.b 2
56.k odd 6 1 1088.2.a.h 1
56.p even 6 1 1088.2.a.i 1
77.h odd 6 1 2057.2.a.e 1
84.n even 6 1 2448.2.a.o 1
91.r even 6 1 2873.2.a.c 1
105.o odd 6 1 3825.2.a.d 1
119.j even 6 1 289.2.a.a 1
119.n even 12 2 289.2.b.a 2
119.q even 24 4 289.2.c.a 4
119.t odd 48 8 289.2.d.d 8
133.r odd 6 1 6137.2.a.b 1
140.p odd 6 1 6800.2.a.n 1
161.f odd 6 1 8993.2.a.a 1
168.s odd 6 1 9792.2.a.n 1
168.v even 6 1 9792.2.a.i 1
357.q odd 6 1 2601.2.a.g 1
476.o odd 6 1 4624.2.a.d 1
595.z even 6 1 7225.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 7.c even 3 1
153.2.a.c 1 21.h odd 6 1
272.2.a.b 1 28.g odd 6 1
289.2.a.a 1 119.j even 6 1
289.2.b.a 2 119.n even 12 2
289.2.c.a 4 119.q even 24 4
289.2.d.d 8 119.t odd 48 8
425.2.a.d 1 35.j even 6 1
425.2.b.b 2 35.l odd 12 2
833.2.a.a 1 7.d odd 6 1
833.2.e.a 2 7.b odd 2 1
833.2.e.a 2 7.d odd 6 1
833.2.e.b 2 1.a even 1 1 trivial
833.2.e.b 2 7.c even 3 1 inner
1088.2.a.h 1 56.k odd 6 1
1088.2.a.i 1 56.p even 6 1
2057.2.a.e 1 77.h odd 6 1
2448.2.a.o 1 84.n even 6 1
2601.2.a.g 1 357.q odd 6 1
2873.2.a.c 1 91.r even 6 1
3825.2.a.d 1 105.o odd 6 1
4624.2.a.d 1 476.o odd 6 1
6137.2.a.b 1 133.r odd 6 1
6800.2.a.n 1 140.p odd 6 1
7225.2.a.g 1 595.z even 6 1
7497.2.a.l 1 21.g even 6 1
8993.2.a.a 1 161.f odd 6 1
9792.2.a.i 1 168.v even 6 1
9792.2.a.n 1 168.s odd 6 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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