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: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{3} - q^{4} + i q^{5} + i q^{6} -i q^{8} + q^{9} +O(q^{10})\) \( q + i q^{2} + q^{3} - q^{4} + i q^{5} + i q^{6} -i q^{8} + q^{9} - q^{10} -i q^{11} - q^{12} + ( -3 + 2 i ) q^{13} + i q^{15} + q^{16} - q^{17} + i q^{18} -i q^{19} -i q^{20} + q^{22} + 3 q^{23} -i q^{24} + 4 q^{25} + ( -2 - 3 i ) q^{26} + q^{27} + 9 q^{29} - q^{30} -4 i q^{31} + i q^{32} -i q^{33} -i q^{34} - q^{36} + 9 i q^{37} + q^{38} + ( -3 + 2 i ) q^{39} + q^{40} + 8 i q^{41} + 7 q^{43} + i q^{44} + i q^{45} + 3 i q^{46} + 8 i q^{47} + q^{48} + 4 i q^{50} - q^{51} + ( 3 - 2 i ) q^{52} -10 q^{53} + i q^{54} + q^{55} -i q^{57} + 9 i q^{58} -6 i q^{59} -i q^{60} -11 q^{61} + 4 q^{62} - q^{64} + ( -2 - 3 i ) q^{65} + q^{66} + 12 i q^{67} + q^{68} + 3 q^{69} -6 i q^{71} -i q^{72} + 11 i q^{73} -9 q^{74} + 4 q^{75} + i q^{76} + ( -2 - 3 i ) q^{78} -12 q^{79} + i q^{80} + q^{81} -8 q^{82} + 6 i q^{83} -i q^{85} + 7 i q^{86} + 9 q^{87} - q^{88} + 12 i q^{89} - q^{90} -3 q^{92} -4 i q^{93} -8 q^{94} + q^{95} + i q^{96} -2 i q^{97} -i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 2q^{12} - 6q^{13} + 2q^{16} - 2q^{17} + 2q^{22} + 6q^{23} + 8q^{25} - 4q^{26} + 2q^{27} + 18q^{29} - 2q^{30} - 2q^{36} + 2q^{38} - 6q^{39} + 2q^{40} + 14q^{43} + 2q^{48} - 2q^{51} + 6q^{52} - 20q^{53} + 2q^{55} - 22q^{61} + 8q^{62} - 2q^{64} - 4q^{65} + 2q^{66} + 2q^{68} + 6q^{69} - 18q^{74} + 8q^{75} - 4q^{78} - 24q^{79} + 2q^{81} - 16q^{82} + 18q^{87} - 2q^{88} - 2q^{90} - 6q^{92} - 16q^{94} + 2q^{95} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(3822, [\chi])\):

\( T_{5}^{2} + 1 \)
\( T_{11}^{2} + 1 \)
\( T_{17} + 1 \)
\( T_{19}^{2} + 1 \)

Hecke characteristic polynomials

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