Properties

Label 3822.2.c.a
Level $3822$
Weight $2$
Character orbit 3822.c
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times\).

\(n\) \(1471\) \(2549\) \(3433\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 3.00000i 1.00000i 0 1.00000i 1.00000 −3.00000
883.2 1.00000i −1.00000 −1.00000 3.00000i 1.00000i 0 1.00000i 1.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3822.2.c.a 2
7.b odd 2 1 546.2.c.d 2
13.b even 2 1 inner 3822.2.c.a 2
21.c even 2 1 1638.2.c.g 2
28.d even 2 1 4368.2.h.b 2
91.b odd 2 1 546.2.c.d 2
91.i even 4 1 7098.2.a.p 1
91.i even 4 1 7098.2.a.x 1
273.g even 2 1 1638.2.c.g 2
364.h even 2 1 4368.2.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.d 2 7.b odd 2 1
546.2.c.d 2 91.b odd 2 1
1638.2.c.g 2 21.c even 2 1
1638.2.c.g 2 273.g even 2 1
3822.2.c.a 2 1.a even 1 1 trivial
3822.2.c.a 2 13.b even 2 1 inner
4368.2.h.b 2 28.d even 2 1
4368.2.h.b 2 364.h even 2 1
7098.2.a.p 1 91.i even 4 1
7098.2.a.x 1 91.i even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3822, [\chi])\):

\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 25 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display
\( T_{19}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 25 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T - 14)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 196 \) Copy content Toggle raw display
$61$ \( (T - 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 100 \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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