Properties

Label 4650.2.d.u
Level $4650$
Weight $2$
Character orbit 4650.d
Analytic conductor $37.130$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.1304369399\)
Analytic rank: \(1\)
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 930)
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.

\(f(q)\) \(=\) \( q + i q^{2} - i q^{3} - q^{4} + q^{6} + 2 i q^{7} - i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - i q^{3} - q^{4} + q^{6} + 2 i q^{7} - i q^{8} - q^{9} + i q^{12} + 4 i q^{13} - 2 q^{14} + q^{16} + 6 i q^{17} - i q^{18} - 8 q^{19} + 2 q^{21} - q^{24} - 4 q^{26} + i q^{27} - 2 i q^{28} + q^{31} + i q^{32} - 6 q^{34} + q^{36} - 4 i q^{37} - 8 i q^{38} + 4 q^{39} - 6 q^{41} + 2 i q^{42} - 8 i q^{43} - 12 i q^{47} - i q^{48} + 3 q^{49} + 6 q^{51} - 4 i q^{52} + 6 i q^{53} - q^{54} + 2 q^{56} + 8 i q^{57} + 6 q^{59} + 2 q^{61} + i q^{62} - 2 i q^{63} - q^{64} + 2 i q^{67} - 6 i q^{68} - 6 q^{71} + i q^{72} - 8 i q^{73} + 4 q^{74} + 8 q^{76} + 4 i q^{78} - 8 q^{79} + q^{81} - 6 i q^{82} - 12 i q^{83} - 2 q^{84} + 8 q^{86} - 8 q^{91} - i q^{93} + 12 q^{94} + q^{96} - 10 i q^{97} + 3 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 16 q^{19} + 4 q^{21} - 2 q^{24} - 8 q^{26} + 2 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 12 q^{41} + 6 q^{49} + 12 q^{51} - 2 q^{54} + 4 q^{56} + 12 q^{59} + 4 q^{61} - 2 q^{64} - 12 q^{71} + 8 q^{74} + 16 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 16 q^{86} - 16 q^{91} + 24 q^{94} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3349.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
3349.2 1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −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 4650.2.d.u 2
5.b even 2 1 inner 4650.2.d.u 2
5.c odd 4 1 930.2.a.n 1
5.c odd 4 1 4650.2.a.d 1
15.e even 4 1 2790.2.a.k 1
20.e even 4 1 7440.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.n 1 5.c odd 4 1
2790.2.a.k 1 15.e even 4 1
4650.2.a.d 1 5.c odd 4 1
4650.2.d.u 2 1.a even 1 1 trivial
4650.2.d.u 2 5.b even 2 1 inner
7440.2.a.b 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4650, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} + 36 \) Copy content Toggle raw display
\( T_{19} + 8 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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