Properties

Label 1850.2.b.d
Level $1850$
Weight $2$
Character orbit 1850.b
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
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 370)
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.

\(f(q)\) \(=\) \( q -i q^{2} - q^{4} + i q^{8} + 3 q^{9} +O(q^{10})\) \( q -i q^{2} - q^{4} + i q^{8} + 3 q^{9} -4 q^{11} -2 i q^{13} + q^{16} -2 i q^{17} -3 i q^{18} + 4 q^{19} + 4 i q^{22} -2 q^{26} + 6 q^{29} -4 q^{31} -i q^{32} -2 q^{34} -3 q^{36} -i q^{37} -4 i q^{38} -6 q^{41} -4 i q^{43} + 4 q^{44} -8 i q^{47} + 7 q^{49} + 2 i q^{52} -10 i q^{53} -6 i q^{58} -4 q^{59} + 10 q^{61} + 4 i q^{62} - q^{64} -8 i q^{67} + 2 i q^{68} + 3 i q^{72} -10 i q^{73} - q^{74} -4 q^{76} + 4 q^{79} + 9 q^{81} + 6 i q^{82} -4 q^{86} -4 i q^{88} -2 q^{89} -8 q^{94} + 6 i q^{97} -7 i q^{98} -12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 6q^{9} - 8q^{11} + 2q^{16} + 8q^{19} - 4q^{26} + 12q^{29} - 8q^{31} - 4q^{34} - 6q^{36} - 12q^{41} + 8q^{44} + 14q^{49} - 8q^{59} + 20q^{61} - 2q^{64} - 2q^{74} - 8q^{76} + 8q^{79} + 18q^{81} - 8q^{86} - 4q^{89} - 16q^{94} - 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 3.00000 0
149.2 1.00000i 0 −1.00000 0 0 0 1.00000i 3.00000 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 1850.2.b.d 2
5.b even 2 1 inner 1850.2.b.d 2
5.c odd 4 1 370.2.a.b 1
5.c odd 4 1 1850.2.a.k 1
15.e even 4 1 3330.2.a.w 1
20.e even 4 1 2960.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.b 1 5.c odd 4 1
1850.2.a.k 1 5.c odd 4 1
1850.2.b.d 2 1.a even 1 1 trivial
1850.2.b.d 2 5.b even 2 1 inner
2960.2.a.g 1 20.e even 4 1
3330.2.a.w 1 15.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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3} \)
\( T_{7} \)
\( T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( 36 + T^{2} \)
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