Properties

Label 6900.2.f.e
Level $6900$
Weight $2$
Character orbit 6900.f
Analytic conductor $55.097$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(55.0967773947\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
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.

\(f(q)\) \(=\) \( q + i q^{3} + i q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} + i q^{7} - q^{9} - 4 i q^{13} + 3 i q^{17} + 4 q^{19} - q^{21} - i q^{23} - i q^{27} + 3 q^{29} - 7 q^{31} - 11 i q^{37} + 4 q^{39} - 9 q^{41} - 4 i q^{43} - 6 i q^{47} + 6 q^{49} - 3 q^{51} + 9 i q^{53} + 4 i q^{57} - 3 q^{59} - 10 q^{61} - i q^{63} + 13 i q^{67} + q^{69} + 9 q^{71} - 16 i q^{73} - 8 q^{79} + q^{81} - 15 i q^{83} + 3 i q^{87} + 4 q^{91} - 7 i q^{93} - 2 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 8 q^{19} - 2 q^{21} + 6 q^{29} - 14 q^{31} + 8 q^{39} - 18 q^{41} + 12 q^{49} - 6 q^{51} - 6 q^{59} - 20 q^{61} + 2 q^{69} + 18 q^{71} - 16 q^{79} + 2 q^{81} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1201\) \(3451\) \(4601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6349.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
6349.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6900.2.f.e 2
5.b even 2 1 inner 6900.2.f.e 2
5.c odd 4 1 1380.2.a.c 1
5.c odd 4 1 6900.2.a.b 1
15.e even 4 1 4140.2.a.i 1
20.e even 4 1 5520.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.c 1 5.c odd 4 1
4140.2.a.i 1 15.e even 4 1
5520.2.a.g 1 20.e even 4 1
6900.2.a.b 1 5.c odd 4 1
6900.2.f.e 2 1.a even 1 1 trivial
6900.2.f.e 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6900, [\chi])\):

\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 121 \) Copy content Toggle raw display
$41$ \( (T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 81 \) Copy content Toggle raw display
$59$ \( (T + 3)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 169 \) Copy content Toggle raw display
$71$ \( (T - 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 225 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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