Properties

Label 1850.2.b.i
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: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} - 4 \beta_{1}\)

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
2.30278i
1.30278i
1.30278i
2.30278i
1.00000i 3.30278i −1.00000 0 −3.30278 2.60555i 1.00000i −7.90833 0
149.2 1.00000i 0.302776i −1.00000 0 0.302776 4.60555i 1.00000i 2.90833 0
149.3 1.00000i 0.302776i −1.00000 0 0.302776 4.60555i 1.00000i 2.90833 0
149.4 1.00000i 3.30278i −1.00000 0 −3.30278 2.60555i 1.00000i −7.90833 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.i 4
5.b even 2 1 inner 1850.2.b.i 4
5.c odd 4 1 74.2.a.a 2
5.c odd 4 1 1850.2.a.u 2
15.e even 4 1 666.2.a.j 2
20.e even 4 1 592.2.a.f 2
35.f even 4 1 3626.2.a.a 2
40.i odd 4 1 2368.2.a.s 2
40.k even 4 1 2368.2.a.ba 2
55.e even 4 1 8954.2.a.p 2
60.l odd 4 1 5328.2.a.bf 2
185.h odd 4 1 2738.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 5.c odd 4 1
592.2.a.f 2 20.e even 4 1
666.2.a.j 2 15.e even 4 1
1850.2.a.u 2 5.c odd 4 1
1850.2.b.i 4 1.a even 1 1 trivial
1850.2.b.i 4 5.b even 2 1 inner
2368.2.a.s 2 40.i odd 4 1
2368.2.a.ba 2 40.k even 4 1
2738.2.a.l 2 185.h odd 4 1
3626.2.a.a 2 35.f even 4 1
5328.2.a.bf 2 60.l odd 4 1
8954.2.a.p 2 55.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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{4} + 11 T_{3}^{2} + 1 \)
\( T_{7}^{4} + 28 T_{7}^{2} + 144 \)
\( T_{13}^{4} + 7 T_{13}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + 11 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 144 + 28 T^{2} + T^{4} \)
$11$ \( ( -3 + T + T^{2} )^{2} \)
$13$ \( 9 + 7 T^{2} + T^{4} \)
$17$ \( ( 36 + T^{2} )^{2} \)
$19$ \( ( 2 + T )^{4} \)
$23$ \( 729 + 63 T^{2} + T^{4} \)
$29$ \( ( -27 + 3 T + T^{2} )^{2} \)
$31$ \( ( -1 - 3 T + T^{2} )^{2} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( -9 - 9 T + T^{2} )^{2} \)
$43$ \( 16 + 44 T^{2} + T^{4} \)
$47$ \( 144 + 28 T^{2} + T^{4} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 36 + 14 T + T^{2} )^{2} \)
$61$ \( ( -79 + 3 T + T^{2} )^{2} \)
$67$ \( 2601 + 223 T^{2} + T^{4} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( 11449 + 227 T^{2} + T^{4} \)
$79$ \( ( -147 - 7 T + T^{2} )^{2} \)
$83$ \( 2304 + 304 T^{2} + T^{4} \)
$89$ \( ( -48 - 4 T + T^{2} )^{2} \)
$97$ \( 41616 + 424 T^{2} + T^{4} \)
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