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: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} - 4\beta_1 \) 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
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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{4} + 11T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 28T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{13}^{4} + 7T_{13}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

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