Properties

Label 1850.2.a.bd
Level $1850$
Weight $2$
Character orbit 1850.a
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1791440.1
Defining polynomial: \( x^{5} - 9x^{3} + 13x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{11} + \beta_1 q^{12} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{13} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{14} + q^{16} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{17} + ( - \beta_{2} - 1) q^{18} + 2 \beta_{3} q^{19} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{21} + (\beta_{4} - \beta_{3} + \beta_1) q^{22} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{23} - \beta_1 q^{24} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{26} + (\beta_{4} - 2 \beta_{3} + 1) q^{27} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{28} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{29} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 5) q^{31} - q^{32} + ( - \beta_{4} - 2 \beta_{2} - 5) q^{33} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{34} + (\beta_{2} + 1) q^{36} - q^{37} - 2 \beta_{3} q^{38} + (\beta_{4} - 2 \beta_{3} + \beta_1 + 5) q^{39} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{41} + (2 \beta_{3} - 2 \beta_1 - 4) q^{42} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{43} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{44} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{46} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{47} + \beta_1 q^{48} + (\beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 + 7) q^{49} + ( - 2 \beta_{3} - 2 \beta_{2} - 4) q^{51} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{52} + (3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{53} + ( - \beta_{4} + 2 \beta_{3} - 1) q^{54} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{56} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{57} + (\beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{58} + ( - 2 \beta_{3} - 2 \beta_{2} + 4) q^{59} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{61} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - 5) q^{62} + (3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 8) q^{63} + q^{64} + (\beta_{4} + 2 \beta_{2} + 5) q^{66} + (\beta_{4} + 2 \beta_{3} + \beta_1 - 3) q^{67} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{68} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{69} + (4 \beta_{4} + 2) q^{71} + ( - \beta_{2} - 1) q^{72} + ( - 3 \beta_{4} + 4 \beta_{3} - \beta_1 - 1) q^{73} + q^{74} + 2 \beta_{3} q^{76} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{77} + ( - \beta_{4} + 2 \beta_{3} - \beta_1 - 5) q^{78} + ( - 2 \beta_{4} + 4 \beta_{3} + \beta_1 + 2) q^{79} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{81} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{82} + (2 \beta_{4} - 4 \beta_1 - 2) q^{83} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{84} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 4) q^{86} + ( - 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 8) q^{87} + (\beta_{4} - \beta_{3} + \beta_1) q^{88} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{89} + ( - 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 6) q^{91} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{92} + (3 \beta_{4} - 5 \beta_{3} - \beta_{2} + 7 \beta_1 + 3) q^{93} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2) q^{94} - \beta_1 q^{96} + (\beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{97} + ( - \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 7) q^{98} + (2 \beta_{3} - 6 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 9x^{3} + 13x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 8 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} + \nu^{3} - 16\nu^{2} - 8\nu + 13 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{3} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 25 \) 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
−2.62545
−1.53175
0.332924
1.09441
2.72987
−1.00000 −2.62545 1.00000 0 2.62545 1.83227 −1.00000 3.89300 0
1.2 −1.00000 −1.53175 1.00000 0 1.53175 −4.67211 −1.00000 −0.653743 0
1.3 −1.00000 0.332924 1.00000 0 −0.332924 −3.51336 −1.00000 −2.88916 0
1.4 −1.00000 1.09441 1.00000 0 −1.09441 3.20984 −1.00000 −1.80226 0
1.5 −1.00000 2.72987 1.00000 0 −2.72987 4.14336 −1.00000 4.45216 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bd 5
5.b even 2 1 1850.2.a.be 5
5.c odd 4 2 370.2.b.d 10
15.e even 4 2 3330.2.d.p 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.d 10 5.c odd 4 2
1850.2.a.bd 5 1.a even 1 1 trivial
1850.2.a.be 5 5.b even 2 1
3330.2.d.p 10 15.e even 4 2

Hecke kernels

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

\( T_{3}^{5} - 9T_{3}^{3} + 13T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{5} - T_{7}^{4} - 32T_{7}^{3} + 44T_{7}^{2} + 240T_{7} - 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 9 T^{3} + 13 T - 4 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - T^{4} - 32 T^{3} + 44 T^{2} + \cdots - 400 \) Copy content Toggle raw display
$11$ \( T^{5} - 3 T^{4} - 23 T^{3} + 71 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{5} - 6 T^{4} - 29 T^{3} + 154 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( T^{5} + 9 T^{4} - 4 T^{3} - 216 T^{2} + \cdots - 368 \) Copy content Toggle raw display
$19$ \( T^{5} - 4 T^{4} - 40 T^{3} + 176 T^{2} + \cdots - 640 \) Copy content Toggle raw display
$23$ \( T^{5} + 6 T^{4} - 25 T^{3} - 82 T^{2} + \cdots - 288 \) Copy content Toggle raw display
$29$ \( T^{5} - 11 T^{4} - 9 T^{3} + 251 T^{2} + \cdots - 335 \) Copy content Toggle raw display
$31$ \( T^{5} - 23 T^{4} + 107 T^{3} + \cdots - 10511 \) Copy content Toggle raw display
$37$ \( (T + 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} + 7 T^{4} - 119 T^{3} + \cdots + 5821 \) Copy content Toggle raw display
$43$ \( T^{5} - 17 T^{4} + 52 T^{3} + \cdots + 3056 \) Copy content Toggle raw display
$47$ \( T^{5} + 12 T^{4} - 44 T^{3} + \cdots + 8000 \) Copy content Toggle raw display
$53$ \( T^{5} - 7 T^{4} - 164 T^{3} + \cdots - 32816 \) Copy content Toggle raw display
$59$ \( T^{5} - 20 T^{4} + 52 T^{3} + \cdots - 3136 \) Copy content Toggle raw display
$61$ \( T^{5} + 9 T^{4} - 81 T^{3} + \cdots - 2323 \) Copy content Toggle raw display
$67$ \( T^{5} + 12 T^{4} - 29 T^{3} + \cdots - 392 \) Copy content Toggle raw display
$71$ \( T^{5} - 6 T^{4} - 184 T^{3} + \cdots - 13792 \) Copy content Toggle raw display
$73$ \( T^{5} - 6 T^{4} - 197 T^{3} + \cdots - 14726 \) Copy content Toggle raw display
$79$ \( T^{5} - 20 T^{4} - 13 T^{3} + \cdots - 8168 \) Copy content Toggle raw display
$83$ \( T^{5} + 12 T^{4} - 120 T^{3} + \cdots - 2752 \) Copy content Toggle raw display
$89$ \( T^{5} - 12 T^{4} - 144 T^{3} + \cdots - 55552 \) Copy content Toggle raw display
$97$ \( T^{5} + 3 T^{4} - 244 T^{3} + \cdots + 1008 \) Copy content Toggle raw display
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