Properties

Label 1850.2.a.u
Level $1850$
Weight $2$
Character orbit 1850.a
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 - \beta ) q^{3} + q^{4} + ( -1 - \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + q^{8} + ( 1 + 3 \beta ) q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 - \beta ) q^{3} + q^{4} + ( -1 - \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + q^{8} + ( 1 + 3 \beta ) q^{9} -\beta q^{11} + ( -1 - \beta ) q^{12} + ( 1 - \beta ) q^{13} + ( -2 + 2 \beta ) q^{14} + q^{16} + 6 q^{17} + ( 1 + 3 \beta ) q^{18} + 2 q^{19} + ( -4 - 2 \beta ) q^{21} -\beta q^{22} + ( 3 - 3 \beta ) q^{23} + ( -1 - \beta ) q^{24} + ( 1 - \beta ) q^{26} + ( -7 - 4 \beta ) q^{27} + ( -2 + 2 \beta ) q^{28} + ( 3 - 3 \beta ) q^{29} + ( 2 - \beta ) q^{31} + q^{32} + ( 3 + 2 \beta ) q^{33} + 6 q^{34} + ( 1 + 3 \beta ) q^{36} - q^{37} + 2 q^{38} + ( 2 + \beta ) q^{39} + ( 3 + 3 \beta ) q^{41} + ( -4 - 2 \beta ) q^{42} + ( 4 - 2 \beta ) q^{43} -\beta q^{44} + ( 3 - 3 \beta ) q^{46} -2 \beta q^{47} + ( -1 - \beta ) q^{48} + ( 9 - 4 \beta ) q^{49} + ( -6 - 6 \beta ) q^{51} + ( 1 - \beta ) q^{52} + 6 q^{53} + ( -7 - 4 \beta ) q^{54} + ( -2 + 2 \beta ) q^{56} + ( -2 - 2 \beta ) q^{57} + ( 3 - 3 \beta ) q^{58} + ( 6 + 2 \beta ) q^{59} + ( -4 + 5 \beta ) q^{61} + ( 2 - \beta ) q^{62} + ( 16 + 2 \beta ) q^{63} + q^{64} + ( 3 + 2 \beta ) q^{66} + ( -8 + 5 \beta ) q^{67} + 6 q^{68} + ( 6 + 3 \beta ) q^{69} + 6 q^{71} + ( 1 + 3 \beta ) q^{72} + ( 10 + \beta ) q^{73} - q^{74} + 2 q^{76} -6 q^{77} + ( 2 + \beta ) q^{78} + ( -7 + 7 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( 3 + 3 \beta ) q^{82} + ( -12 + 4 \beta ) q^{83} + ( -4 - 2 \beta ) q^{84} + ( 4 - 2 \beta ) q^{86} + ( 6 + 3 \beta ) q^{87} -\beta q^{88} -4 \beta q^{89} + ( -8 + 2 \beta ) q^{91} + ( 3 - 3 \beta ) q^{92} + q^{93} -2 \beta q^{94} + ( -1 - \beta ) q^{96} + ( -2 + 8 \beta ) q^{97} + ( 9 - 4 \beta ) q^{98} + ( -9 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{6} - 2q^{7} + 2q^{8} + 5q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{6} - 2q^{7} + 2q^{8} + 5q^{9} - q^{11} - 3q^{12} + q^{13} - 2q^{14} + 2q^{16} + 12q^{17} + 5q^{18} + 4q^{19} - 10q^{21} - q^{22} + 3q^{23} - 3q^{24} + q^{26} - 18q^{27} - 2q^{28} + 3q^{29} + 3q^{31} + 2q^{32} + 8q^{33} + 12q^{34} + 5q^{36} - 2q^{37} + 4q^{38} + 5q^{39} + 9q^{41} - 10q^{42} + 6q^{43} - q^{44} + 3q^{46} - 2q^{47} - 3q^{48} + 14q^{49} - 18q^{51} + q^{52} + 12q^{53} - 18q^{54} - 2q^{56} - 6q^{57} + 3q^{58} + 14q^{59} - 3q^{61} + 3q^{62} + 34q^{63} + 2q^{64} + 8q^{66} - 11q^{67} + 12q^{68} + 15q^{69} + 12q^{71} + 5q^{72} + 21q^{73} - 2q^{74} + 4q^{76} - 12q^{77} + 5q^{78} - 7q^{79} + 38q^{81} + 9q^{82} - 20q^{83} - 10q^{84} + 6q^{86} + 15q^{87} - q^{88} - 4q^{89} - 14q^{91} + 3q^{92} + 2q^{93} - 2q^{94} - 3q^{96} + 4q^{97} + 14q^{98} - 22q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
1.00000 −3.30278 1.00000 0 −3.30278 2.60555 1.00000 7.90833 0
1.2 1.00000 0.302776 1.00000 0 0.302776 −4.60555 1.00000 −2.90833 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.u 2
5.b even 2 1 74.2.a.a 2
5.c odd 4 2 1850.2.b.i 4
15.d odd 2 1 666.2.a.j 2
20.d odd 2 1 592.2.a.f 2
35.c odd 2 1 3626.2.a.a 2
40.e odd 2 1 2368.2.a.ba 2
40.f even 2 1 2368.2.a.s 2
55.d odd 2 1 8954.2.a.p 2
60.h even 2 1 5328.2.a.bf 2
185.d even 2 1 2738.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 5.b even 2 1
592.2.a.f 2 20.d odd 2 1
666.2.a.j 2 15.d odd 2 1
1850.2.a.u 2 1.a even 1 1 trivial
1850.2.b.i 4 5.c odd 4 2
2368.2.a.s 2 40.f even 2 1
2368.2.a.ba 2 40.e odd 2 1
2738.2.a.l 2 185.d even 2 1
3626.2.a.a 2 35.c odd 2 1
5328.2.a.bf 2 60.h even 2 1
8954.2.a.p 2 55.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 3 T_{3} - 1 \)
\( T_{7}^{2} + 2 T_{7} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -12 + 2 T + T^{2} \)
$11$ \( -3 + T + T^{2} \)
$13$ \( -3 - T + T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( -27 - 3 T + T^{2} \)
$29$ \( -27 - 3 T + T^{2} \)
$31$ \( -1 - 3 T + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( -9 - 9 T + T^{2} \)
$43$ \( -4 - 6 T + T^{2} \)
$47$ \( -12 + 2 T + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 36 - 14 T + T^{2} \)
$61$ \( -79 + 3 T + T^{2} \)
$67$ \( -51 + 11 T + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( 107 - 21 T + T^{2} \)
$79$ \( -147 + 7 T + T^{2} \)
$83$ \( 48 + 20 T + T^{2} \)
$89$ \( -48 + 4 T + T^{2} \)
$97$ \( -204 - 4 T + T^{2} \)
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