Properties

Label 1850.2.b.o
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: 6.0.3182656.1
Defining polynomial: \( x^{6} - 2x^{3} + 25x^{2} - 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{3} q^{3} - q^{4} + \beta_{2} q^{6} - \beta_{5} q^{7} + \beta_{4} q^{8} + (2 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{3} q^{3} - q^{4} + \beta_{2} q^{6} - \beta_{5} q^{7} + \beta_{4} q^{8} + (2 \beta_1 - 3) q^{9} + ( - \beta_{2} + \beta_1 + 4) q^{11} + \beta_{3} q^{12} + 2 \beta_{3} q^{13} - \beta_1 q^{14} + q^{16} + (\beta_{5} + \beta_{3}) q^{17} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{18} - \beta_{2} q^{19} + ( - 2 \beta_{2} + 2) q^{21} + ( - \beta_{5} - 4 \beta_{4} - \beta_{3}) q^{22} + (2 \beta_{5} + 2 \beta_{3}) q^{23} - \beta_{2} q^{24} - 2 \beta_{2} q^{26} + (4 \beta_{4} + 4 \beta_{3}) q^{27} + \beta_{5} q^{28} + (\beta_{2} + \beta_1 + 2) q^{29} - 3 \beta_1 q^{31} - \beta_{4} q^{32} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3}) q^{33} + ( - \beta_{2} + \beta_1) q^{34} + ( - 2 \beta_1 + 3) q^{36} + \beta_{4} q^{37} - \beta_{3} q^{38} + ( - 4 \beta_1 + 12) q^{39} + ( - \beta_{2} + 3 \beta_1 - 2) q^{41} + ( - 2 \beta_{4} - 2 \beta_{3}) q^{42} + ( - \beta_{5} - 4 \beta_{4} + \beta_{3}) q^{43} + (\beta_{2} - \beta_1 - 4) q^{44} + ( - 2 \beta_{2} + 2 \beta_1) q^{46} + (2 \beta_{5} + \beta_{3}) q^{47} - \beta_{3} q^{48} + (\beta_{2} - \beta_1 + 1) q^{49} + (2 \beta_{2} - 2 \beta_1 + 4) q^{51} - 2 \beta_{3} q^{52} + ( - 3 \beta_{5} + 6 \beta_{4} - \beta_{3}) q^{53} + ( - 4 \beta_{2} + 4) q^{54} + \beta_1 q^{56} + (2 \beta_{5} - 6 \beta_{4}) q^{57} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{58} + ( - \beta_{2} + 2 \beta_1 - 4) q^{59} + ( - \beta_{2} - \beta_1 + 2) q^{61} + 3 \beta_{5} q^{62} + (\beta_{5} - 12 \beta_{4} - 2 \beta_{3}) q^{63} - q^{64} + (2 \beta_{2} + 2 \beta_1 - 4) q^{66} + (2 \beta_{5} + \beta_{3}) q^{67} + ( - \beta_{5} - \beta_{3}) q^{68} + (4 \beta_{2} - 4 \beta_1 + 8) q^{69} + (2 \beta_{2} + 2 \beta_1 + 8) q^{71} + (2 \beta_{5} - 3 \beta_{4}) q^{72} + ( - 2 \beta_{5} - 6 \beta_{4} - 2 \beta_{3}) q^{73} + q^{74} + \beta_{2} q^{76} + ( - 5 \beta_{5} - 4 \beta_{4} + \beta_{3}) q^{77} + (4 \beta_{5} - 12 \beta_{4}) q^{78} + ( - \beta_{2} - 2 \beta_1 + 8) q^{79} + ( - 4 \beta_{2} - 2 \beta_1 + 15) q^{81} + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{82} + ( - 2 \beta_{5} + 3 \beta_{3}) q^{83} + (2 \beta_{2} - 2) q^{84} + ( - \beta_{2} - \beta_1 - 4) q^{86} + ( - 2 \beta_{5} + 8 \beta_{4}) q^{87} + (\beta_{5} + 4 \beta_{4} + \beta_{3}) q^{88} - 6 q^{89} + (4 \beta_{2} - 4) q^{91} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{92} + ( - 6 \beta_{4} - 6 \beta_{3}) q^{93} + ( - \beta_{2} + 2 \beta_1) q^{94} + \beta_{2} q^{96} + ( - 3 \beta_{5} + 6 \beta_{4} - \beta_{3}) q^{97} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{98} + (5 \beta_{2} + 7 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56} - 28 q^{59} + 14 q^{61} - 6 q^{64} - 28 q^{66} + 56 q^{69} + 44 q^{71} + 6 q^{74} + 52 q^{79} + 94 q^{81} - 12 q^{84} - 22 q^{86} - 36 q^{89} - 24 q^{91} - 4 q^{94} - 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{3} + 25x^{2} - 10x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 25\nu^{4} + 5\nu^{3} - \nu^{2} + 362 ) / 124 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - \nu^{4} - 25\nu^{3} + 5\nu^{2} + 50 ) / 124 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{5} + \nu^{4} + 25\nu^{3} - 5\nu^{2} + 248\nu - 50 ) / 124 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{5} - 5\nu^{4} - \nu^{3} + 25\nu^{2} - 620\nu + 126 ) / 124 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -75\nu^{5} - 15\nu^{4} - 3\nu^{3} + 199\nu^{2} - 1860\nu + 378 ) / 124 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{4} + 5\beta_{3} - 5\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{2} + 5\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{5} - 16\beta_{4} - 25\beta_{3} - 25\beta_{2} - 2\beta _1 + 16 ) / 2 \) 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
−1.67298 + 1.67298i
0.203364 0.203364i
1.46962 1.46962i
1.46962 + 1.46962i
0.203364 + 0.203364i
−1.67298 1.67298i
1.00000i 3.34596i −1.00000 0 −3.34596 2.59774i 1.00000i −8.19547 0
149.2 1.00000i 0.406728i −1.00000 0 0.406728 2.91729i 1.00000i 2.83457 0
149.3 1.00000i 2.93923i −1.00000 0 2.93923 1.31955i 1.00000i −5.63910 0
149.4 1.00000i 2.93923i −1.00000 0 2.93923 1.31955i 1.00000i −5.63910 0
149.5 1.00000i 0.406728i −1.00000 0 0.406728 2.91729i 1.00000i 2.83457 0
149.6 1.00000i 3.34596i −1.00000 0 −3.34596 2.59774i 1.00000i −8.19547 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.o 6
5.b even 2 1 inner 1850.2.b.o 6
5.c odd 4 1 370.2.a.g 3
5.c odd 4 1 1850.2.a.z 3
15.e even 4 1 3330.2.a.bg 3
20.e even 4 1 2960.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.g 3 5.c odd 4 1
1850.2.a.z 3 5.c odd 4 1
1850.2.b.o 6 1.a even 1 1 trivial
1850.2.b.o 6 5.b even 2 1 inner
2960.2.a.u 3 20.e even 4 1
3330.2.a.bg 3 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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{6} + 20T_{3}^{4} + 100T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{6} + 17T_{7}^{4} + 84T_{7}^{2} + 100 \) Copy content Toggle raw display
\( T_{13}^{6} + 80T_{13}^{4} + 1600T_{13}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 20 T^{4} + 100 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 17 T^{4} + 84 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$11$ \( (T^{3} - 11 T^{2} + 28 T + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 80 T^{4} + 1600 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{6} + 25 T^{4} + 128 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 10 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 100 T^{4} + 2048 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{3} - 5 T^{2} - 16 T + 76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} - 72 T + 270)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} - 40 T - 364)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 89 T^{4} + 2016 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$47$ \( T^{6} + 64 T^{4} + 1124 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$53$ \( T^{6} + 281 T^{4} + 19672 T^{2} + \cdots + 99856 \) Copy content Toggle raw display
$59$ \( (T^{3} + 14 T^{2} + 34 T - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 7 T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 64 T^{4} + 1124 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$71$ \( (T^{3} - 22 T^{2} + 64 T + 640)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 184 T^{4} + 7952 T^{2} + \cdots + 43264 \) Copy content Toggle raw display
$79$ \( (T^{3} - 26 T^{2} + 170 T - 224)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 320 T^{4} + 22308 T^{2} + \cdots + 440896 \) Copy content Toggle raw display
$89$ \( (T + 6)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 281 T^{4} + 19672 T^{2} + \cdots + 99856 \) Copy content Toggle raw display
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