Properties

Label 1850.2.b.m
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: \(\Q(i, \sqrt{33})\)
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) 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)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + 2 \beta_{2} q^{3} - q^{4} + 2 q^{6} + (2 \beta_{2} + \beta_1) q^{7} + \beta_{2} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + 2 \beta_{2} q^{3} - q^{4} + 2 q^{6} + (2 \beta_{2} + \beta_1) q^{7} + \beta_{2} q^{8} - q^{9} + (\beta_{3} - 1) q^{11} - 2 \beta_{2} q^{12} + (2 \beta_{2} + 2 \beta_1) q^{13} + (\beta_{3} + 1) q^{14} + q^{16} + (2 \beta_{2} - \beta_1) q^{17} + \beta_{2} q^{18} + 2 q^{19} + ( - 2 \beta_{3} - 2) q^{21} - \beta_1 q^{22} - 2 \beta_1 q^{23} - 2 q^{24} + 2 \beta_{3} q^{26} + 4 \beta_{2} q^{27} + ( - 2 \beta_{2} - \beta_1) q^{28} + (3 \beta_{3} - 1) q^{29} + ( - \beta_{3} - 5) q^{31} - \beta_{2} q^{32} + 2 \beta_1 q^{33} + ( - \beta_{3} + 3) q^{34} + q^{36} - \beta_{2} q^{37} - 2 \beta_{2} q^{38} - 4 \beta_{3} q^{39} + ( - \beta_{3} + 3) q^{41} + (4 \beta_{2} + 2 \beta_1) q^{42} + (4 \beta_{2} - \beta_1) q^{43} + ( - \beta_{3} + 1) q^{44} + ( - 2 \beta_{3} + 2) q^{46} + (2 \beta_{2} - 2 \beta_1) q^{47} + 2 \beta_{2} q^{48} + ( - 3 \beta_{3} - 2) q^{49} + (2 \beta_{3} - 6) q^{51} + ( - 2 \beta_{2} - 2 \beta_1) q^{52} + ( - 2 \beta_{2} - \beta_1) q^{53} + 4 q^{54} + ( - \beta_{3} - 1) q^{56} + 4 \beta_{2} q^{57} + ( - 2 \beta_{2} - 3 \beta_1) q^{58} + (2 \beta_{3} - 8) q^{59} + (\beta_{3} - 3) q^{61} + (6 \beta_{2} + \beta_1) q^{62} + ( - 2 \beta_{2} - \beta_1) q^{63} - q^{64} + (2 \beta_{3} - 2) q^{66} + (2 \beta_{2} + 2 \beta_1) q^{67} + ( - 2 \beta_{2} + \beta_1) q^{68} + (4 \beta_{3} - 4) q^{69} + (2 \beta_{3} - 2) q^{71} - \beta_{2} q^{72} + (2 \beta_{2} - 2 \beta_1) q^{73} - q^{74} - 2 q^{76} + (8 \beta_{2} + \beta_1) q^{77} + (4 \beta_{2} + 4 \beta_1) q^{78} - 2 \beta_{3} q^{79} - 11 q^{81} + ( - 2 \beta_{2} + \beta_1) q^{82} + (6 \beta_{2} + 2 \beta_1) q^{83} + (2 \beta_{3} + 2) q^{84} + ( - \beta_{3} + 5) q^{86} + (4 \beta_{2} + 6 \beta_1) q^{87} + \beta_1 q^{88} - 10 q^{89} + ( - 4 \beta_{3} - 16) q^{91} + 2 \beta_1 q^{92} + ( - 12 \beta_{2} - 2 \beta_1) q^{93} + ( - 2 \beta_{3} + 4) q^{94} + 2 q^{96} + ( - 10 \beta_{2} - 3 \beta_1) q^{97} + (5 \beta_{2} + 3 \beta_1) q^{98} + ( - \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 2 q^{11} + 6 q^{14} + 4 q^{16} + 8 q^{19} - 12 q^{21} - 8 q^{24} + 4 q^{26} + 2 q^{29} - 22 q^{31} + 10 q^{34} + 4 q^{36} - 8 q^{39} + 10 q^{41} + 2 q^{44} + 4 q^{46} - 14 q^{49} - 20 q^{51} + 16 q^{54} - 6 q^{56} - 28 q^{59} - 10 q^{61} - 4 q^{64} - 4 q^{66} - 8 q^{69} - 4 q^{71} - 4 q^{74} - 8 q^{76} - 4 q^{79} - 44 q^{81} + 12 q^{84} + 18 q^{86} - 40 q^{89} - 72 q^{91} + 12 q^{94} + 8 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 17x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 9\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{2} - 9\beta_1 \) 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
3.37228i
2.37228i
2.37228i
3.37228i
1.00000i 2.00000i −1.00000 0 2.00000 1.37228i 1.00000i −1.00000 0
149.2 1.00000i 2.00000i −1.00000 0 2.00000 4.37228i 1.00000i −1.00000 0
149.3 1.00000i 2.00000i −1.00000 0 2.00000 4.37228i 1.00000i −1.00000 0
149.4 1.00000i 2.00000i −1.00000 0 2.00000 1.37228i 1.00000i −1.00000 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.m 4
5.b even 2 1 inner 1850.2.b.m 4
5.c odd 4 1 370.2.a.f 2
5.c odd 4 1 1850.2.a.q 2
15.e even 4 1 3330.2.a.bb 2
20.e even 4 1 2960.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.f 2 5.c odd 4 1
1850.2.a.q 2 5.c odd 4 1
1850.2.b.m 4 1.a even 1 1 trivial
1850.2.b.m 4 5.b even 2 1 inner
2960.2.a.o 2 20.e even 4 1
3330.2.a.bb 2 15.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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 21T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{13}^{4} + 68T_{13}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 21T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 68T^{2} + 1024 \) Copy content Toggle raw display
$17$ \( T^{4} + 29T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 68T^{2} + 1024 \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 74)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 11 T + 22)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 5 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 57T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{4} + 84T^{2} + 576 \) Copy content Toggle raw display
$53$ \( T^{4} + 21T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T^{2} + 14 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 68T^{2} + 1024 \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 84T^{2} + 576 \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 116T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T + 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 293T^{2} + 4 \) Copy content Toggle raw display
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