Properties

Label 1850.2.a.bb
Level $1850$
Weight $2$
Character orbit 1850.a
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.48119
2.17009
0.311108
1.00000 −2.67513 1.00000 0 −2.67513 −3.28726 1.00000 4.15633 0
1.2 1.00000 −1.53919 1.00000 0 −1.53919 2.87936 1.00000 −0.630898 0
1.3 1.00000 1.21432 1.00000 0 1.21432 −3.59210 1.00000 −1.52543 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(37\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.a.bb yes 3
5.b even 2 1 1850.2.a.ba 3
5.c odd 4 2 1850.2.b.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.ba 3 5.b even 2 1
1850.2.a.bb yes 3 1.a even 1 1 trivial
1850.2.b.n 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\):

\( T_{3}^{3} + 3T_{3}^{2} - T_{3} - 5 \) Copy content Toggle raw display
\( T_{7}^{3} + 4T_{7}^{2} - 8T_{7} - 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 3T^{2} - T - 5 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 4 T^{2} - 8 T - 34 \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 23 T - 25 \) Copy content Toggle raw display
$13$ \( T^{3} + 12 T^{2} + 44 T + 46 \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 21 T - 29 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 25 T + 79 \) Copy content Toggle raw display
$23$ \( T^{3} + 8T^{2} - 32 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 52 T - 40 \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} - 18 T - 54 \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 21 T^{2} + 131 T + 215 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 128 T - 1076 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 100 T - 632 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 128 T + 1076 \) Copy content Toggle raw display
$61$ \( T^{3} + 8 T^{2} - 44 T - 290 \) Copy content Toggle raw display
$67$ \( T^{3} + 13 T^{2} - 9 T - 67 \) Copy content Toggle raw display
$71$ \( T^{3} + 8 T^{2} - 44 T - 290 \) Copy content Toggle raw display
$73$ \( T^{3} + 31 T^{2} + 315 T + 1049 \) Copy content Toggle raw display
$79$ \( T^{3} - 22 T^{2} + 140 T - 232 \) Copy content Toggle raw display
$83$ \( T^{3} + 7 T^{2} - 227 T - 1819 \) Copy content Toggle raw display
$89$ \( T^{3} + 3 T^{2} - 27 T - 27 \) Copy content Toggle raw display
$97$ \( T^{3} + 6 T^{2} - 4 T - 40 \) Copy content Toggle raw display
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