Properties

Label 1850.2.b.l
Level $1850$
Weight $2$
Character orbit 1850.b
Analytic conductor $14.772$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} - q^{4} + ( - \beta_{3} + 1) q^{6} + ( - \beta_{2} - 3 \beta_1) q^{7} + \beta_1 q^{8} + (2 \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} - q^{4} + ( - \beta_{3} + 1) q^{6} + ( - \beta_{2} - 3 \beta_1) q^{7} + \beta_1 q^{8} + (2 \beta_{3} - 1) q^{9} + ( - 2 \beta_{3} - 2) q^{11} + (\beta_{2} - \beta_1) q^{12} + (2 \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{3} - 3) q^{14} + q^{16} + (2 \beta_{2} + 2 \beta_1) q^{17} + ( - 2 \beta_{2} + \beta_1) q^{18} + ( - 3 \beta_{3} - 1) q^{19} - 2 \beta_{3} q^{21} + (2 \beta_{2} + 2 \beta_1) q^{22} + 8 \beta_1 q^{23} + (\beta_{3} - 1) q^{24} + (2 \beta_{3} + 2) q^{26} - 4 \beta_1 q^{27} + (\beta_{2} + 3 \beta_1) q^{28} + ( - 4 \beta_{3} + 2) q^{29} + (\beta_{3} - 1) q^{31} - \beta_1 q^{32} + 4 \beta_1 q^{33} + (2 \beta_{3} + 2) q^{34} + ( - 2 \beta_{3} + 1) q^{36} + \beta_1 q^{37} + (3 \beta_{2} + \beta_1) q^{38} + 4 q^{39} - 2 q^{41} + 2 \beta_{2} q^{42} - 4 \beta_{2} q^{43} + (2 \beta_{3} + 2) q^{44} + 8 q^{46} + ( - \beta_{2} - 3 \beta_1) q^{47} + ( - \beta_{2} + \beta_1) q^{48} + ( - 6 \beta_{3} - 5) q^{49} + 4 q^{51} + ( - 2 \beta_{2} - 2 \beta_1) q^{52} + 6 \beta_1 q^{53} - 4 q^{54} + (\beta_{3} + 3) q^{56} + ( - 2 \beta_{2} + 8 \beta_1) q^{57} + (4 \beta_{2} - 2 \beta_1) q^{58} + (3 \beta_{3} + 5) q^{59} + ( - 4 \beta_{3} + 2) q^{61} + ( - \beta_{2} + \beta_1) q^{62} + ( - 5 \beta_{2} - 3 \beta_1) q^{63} - q^{64} + 4 q^{66} + ( - 5 \beta_{2} + 5 \beta_1) q^{67} + ( - 2 \beta_{2} - 2 \beta_1) q^{68} + (8 \beta_{3} - 8) q^{69} + (4 \beta_{3} - 4) q^{71} + (2 \beta_{2} - \beta_1) q^{72} + (4 \beta_{2} - 6 \beta_1) q^{73} + q^{74} + (3 \beta_{3} + 1) q^{76} + (8 \beta_{2} + 12 \beta_1) q^{77} - 4 \beta_1 q^{78} + ( - \beta_{3} - 7) q^{79} + (2 \beta_{3} + 1) q^{81} + 2 \beta_1 q^{82} + (\beta_{2} + 7 \beta_1) q^{83} + 2 \beta_{3} q^{84} - 4 \beta_{3} q^{86} + ( - 6 \beta_{2} + 14 \beta_1) q^{87} + ( - 2 \beta_{2} - 2 \beta_1) q^{88} + 2 q^{89} + (8 \beta_{3} + 12) q^{91} - 8 \beta_1 q^{92} + (2 \beta_{2} - 4 \beta_1) q^{93} + ( - \beta_{3} - 3) q^{94} + ( - \beta_{3} + 1) q^{96} - 2 \beta_1 q^{97} + (6 \beta_{2} + 5 \beta_1) q^{98} + ( - 2 \beta_{3} - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 8 q^{11} - 12 q^{14} + 4 q^{16} - 4 q^{19} - 4 q^{24} + 8 q^{26} + 8 q^{29} - 4 q^{31} + 8 q^{34} + 4 q^{36} + 16 q^{39} - 8 q^{41} + 8 q^{44} + 32 q^{46} - 20 q^{49} + 16 q^{51} - 16 q^{54} + 12 q^{56} + 20 q^{59} + 8 q^{61} - 4 q^{64} + 16 q^{66} - 32 q^{69} - 16 q^{71} + 4 q^{74} + 4 q^{76} - 28 q^{79} + 4 q^{81} + 8 q^{89} + 48 q^{91} - 12 q^{94} + 4 q^{96} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

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
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
1.00000i 0.732051i −1.00000 0 −0.732051 4.73205i 1.00000i 2.46410 0
149.2 1.00000i 2.73205i −1.00000 0 2.73205 1.26795i 1.00000i −4.46410 0
149.3 1.00000i 2.73205i −1.00000 0 2.73205 1.26795i 1.00000i −4.46410 0
149.4 1.00000i 0.732051i −1.00000 0 −0.732051 4.73205i 1.00000i 2.46410 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.l 4
5.b even 2 1 inner 1850.2.b.l 4
5.c odd 4 1 370.2.a.e 2
5.c odd 4 1 1850.2.a.x 2
15.e even 4 1 3330.2.a.bd 2
20.e even 4 1 2960.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.e 2 5.c odd 4 1
1850.2.a.x 2 5.c odd 4 1
1850.2.b.l 4 1.a even 1 1 trivial
1850.2.b.l 4 5.b even 2 1 inner
2960.2.a.q 2 20.e even 4 1
3330.2.a.bd 2 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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 24T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{13}^{4} + 32T_{13}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 200T^{2} + 2500 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 46)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 104T^{2} + 2116 \) Copy content Toggle raw display
$89$ \( (T - 2)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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