Properties

Label 3822.2.c.e
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{3} - q^{4} + i q^{5} + i q^{6} - i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{3} - q^{4} + i q^{5} + i q^{6} - i q^{8} + q^{9} - q^{10} - i q^{11} - q^{12} + (2 i - 3) q^{13} + i q^{15} + q^{16} - q^{17} + i q^{18} - i q^{19} - i q^{20} + q^{22} + 3 q^{23} - i q^{24} + 4 q^{25} + ( - 3 i - 2) q^{26} + q^{27} + 9 q^{29} - q^{30} - 4 i q^{31} + i q^{32} - i q^{33} - i q^{34} - q^{36} + 9 i q^{37} + q^{38} + (2 i - 3) q^{39} + q^{40} + 8 i q^{41} + 7 q^{43} + i q^{44} + i q^{45} + 3 i q^{46} + 8 i q^{47} + q^{48} + 4 i q^{50} - q^{51} + ( - 2 i + 3) q^{52} - 10 q^{53} + i q^{54} + q^{55} - i q^{57} + 9 i q^{58} - 6 i q^{59} - i q^{60} - 11 q^{61} + 4 q^{62} - q^{64} + ( - 3 i - 2) q^{65} + q^{66} + 12 i q^{67} + q^{68} + 3 q^{69} - 6 i q^{71} - i q^{72} + 11 i q^{73} - 9 q^{74} + 4 q^{75} + i q^{76} + ( - 3 i - 2) q^{78} - 12 q^{79} + i q^{80} + q^{81} - 8 q^{82} + 6 i q^{83} - i q^{85} + 7 i q^{86} + 9 q^{87} - q^{88} + 12 i q^{89} - q^{90} - 3 q^{92} - 4 i q^{93} - 8 q^{94} + q^{95} + i q^{96} - 2 i q^{97} - i q^{99} +O(q^{100}) \) 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 \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 6 q^{13} + 2 q^{16} - 2 q^{17} + 2 q^{22} + 6 q^{23} + 8 q^{25} - 4 q^{26} + 2 q^{27} + 18 q^{29} - 2 q^{30} - 2 q^{36} + 2 q^{38} - 6 q^{39} + 2 q^{40} + 14 q^{43} + 2 q^{48} - 2 q^{51} + 6 q^{52} - 20 q^{53} + 2 q^{55} - 22 q^{61} + 8 q^{62} - 2 q^{64} - 4 q^{65} + 2 q^{66} + 2 q^{68} + 6 q^{69} - 18 q^{74} + 8 q^{75} - 4 q^{78} - 24 q^{79} + 2 q^{81} - 16 q^{82} + 18 q^{87} - 2 q^{88} - 2 q^{90} - 6 q^{92} - 16 q^{94} + 2 q^{95}+O(q^{100}) \) 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 1.00000i 1.00000i 0 1.00000i 1.00000 −1.00000
883.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 −1.00000
\(n\): 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.e 2
7.b odd 2 1 546.2.c.a 2
13.b even 2 1 inner 3822.2.c.e 2
21.c even 2 1 1638.2.c.f 2
28.d even 2 1 4368.2.h.k 2
91.b odd 2 1 546.2.c.a 2
91.i even 4 1 7098.2.a.d 1
91.i even 4 1 7098.2.a.s 1
273.g even 2 1 1638.2.c.f 2
364.h even 2 1 4368.2.h.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.a 2 7.b odd 2 1
546.2.c.a 2 91.b odd 2 1
1638.2.c.f 2 21.c even 2 1
1638.2.c.f 2 273.g even 2 1
3822.2.c.e 2 1.a even 1 1 trivial
3822.2.c.e 2 13.b even 2 1 inner
4368.2.h.k 2 28.d even 2 1
4368.2.h.k 2 364.h even 2 1
7098.2.a.d 1 91.i even 4 1
7098.2.a.s 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 \(S_{2}^{\mathrm{new}}(3822, [\chi])\):

\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 1 \) Copy content Toggle raw display
\( T_{17} + 1 \) 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
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1 \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 81 \) Copy content Toggle raw display
$41$ \( T^{2} + 64 \) Copy content Toggle raw display
$43$ \( (T - 7)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( (T + 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 144 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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