Properties

Label 1850.2.b.p
Level $1850$
Weight $2$
Character orbit 1850.b
Analytic conductor $14.772$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.37161216.1
Defining polynomial: \( x^{6} + 15x^{4} + 51x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 15x^{4} + 51x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 8\nu^{2} + 1 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 14\nu^{2} + 31 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 14\nu^{3} + 43\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} + 31\nu^{3} + 110\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 4\beta_{4} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{3} + 14\beta_{2} + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{5} + 62\beta_{4} + 69\beta_1 \) 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
2.27307i
0.140435i
3.13264i
3.13264i
0.140435i
2.27307i
1.00000i 2.27307i −1.00000 0 −2.27307 1.27307i 1.00000i −2.16686 0
149.2 1.00000i 0.140435i −1.00000 0 0.140435 1.14044i 1.00000i 2.98028 0
149.3 1.00000i 3.13264i −1.00000 0 3.13264 4.13264i 1.00000i −6.81342 0
149.4 1.00000i 3.13264i −1.00000 0 3.13264 4.13264i 1.00000i −6.81342 0
149.5 1.00000i 0.140435i −1.00000 0 0.140435 1.14044i 1.00000i 2.98028 0
149.6 1.00000i 2.27307i −1.00000 0 −2.27307 1.27307i 1.00000i −2.16686 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.6
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.p 6
5.b even 2 1 inner 1850.2.b.p 6
5.c odd 4 1 1850.2.a.y 3
5.c odd 4 1 1850.2.a.bc yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.y 3 5.c odd 4 1
1850.2.a.bc yes 3 5.c odd 4 1
1850.2.b.p 6 1.a even 1 1 trivial
1850.2.b.p 6 5.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{6} + 15T_{3}^{4} + 51T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} + 20T_{7}^{4} + 52T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{13}^{6} + 36T_{13}^{4} + 228T_{13}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 15 T^{4} + 51 T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + 52 T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} - 9 T - 51)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 36 T^{4} + 228 T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{6} + 31 T^{4} + 123 T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T^{3} - T^{2} - 25 T - 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 64 T^{4} + 960 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( (T^{3} - 14 T^{2} + 36 T + 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 36 T + 214)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} - 19 T^{2} + 51 T + 399)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 248 T^{4} + 18064 T^{2} + \cdots + 318096 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} + 204 T^{4} + 9648 T^{2} + \cdots + 46656 \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 60 T - 84)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 12 T^{2} - 18 T + 238)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 159 T^{4} + 5523 T^{2} + \cdots + 289 \) Copy content Toggle raw display
$71$ \( (T^{3} - 8 T^{2} - 54 T + 378)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 107 T^{4} + 1195 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$79$ \( (T^{3} + 2 T^{2} - 100 T + 88)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 211 T^{4} + 3231 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$89$ \( (T^{3} + T^{2} - 189 T - 147)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 204 T^{4} + 240 T^{2} + \cdots + 64 \) Copy content Toggle raw display
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