Properties

Label 1850.2.b.k
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 2 \) 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
−1.22474 1.22474i
1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
1.00000i 1.44949i −1.00000 0 −1.44949 2.44949i 1.00000i 0.898979 0
149.2 1.00000i 3.44949i −1.00000 0 3.44949 2.44949i 1.00000i −8.89898 0
149.3 1.00000i 3.44949i −1.00000 0 3.44949 2.44949i 1.00000i −8.89898 0
149.4 1.00000i 1.44949i −1.00000 0 −1.44949 2.44949i 1.00000i 0.898979 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.k 4
5.b even 2 1 inner 1850.2.b.k 4
5.c odd 4 1 1850.2.a.r 2
5.c odd 4 1 1850.2.a.w yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.r 2 5.c odd 4 1
1850.2.a.w yes 2 5.c odd 4 1
1850.2.b.k 4 1.a even 1 1 trivial
1850.2.b.k 4 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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{4} + 14T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{2} + 6 \) Copy content Toggle raw display
\( T_{13}^{4} + 20T_{13}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T - 5)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T - 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 120T^{2} + 144 \) Copy content Toggle raw display
$47$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T - 2)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 110T^{2} + 1849 \) Copy content Toggle raw display
$71$ \( (T^{2} + 20 T + 94)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 210T^{2} + 7569 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 92)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14 T - 5)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
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