Properties

Label 1805.2.b.h
Level $1805$
Weight $2$
Character orbit 1805.b
Analytic conductor $14.413$
Analytic rank $0$
Dimension $8$
CM discriminant -95
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.280944640000.2
Defining polynomial: \( x^{8} + 16x^{6} + 80x^{4} + 128x^{2} + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{3} q^{5} + (\beta_{6} - \beta_{3}) q^{6} + (\beta_{4} - 2 \beta_1) q^{8} + ( - \beta_{6} - \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{3} q^{5} + (\beta_{6} - \beta_{3}) q^{6} + (\beta_{4} - 2 \beta_1) q^{8} + ( - \beta_{6} - \beta_{2} - 3) q^{9} + (\beta_{5} - \beta_{4}) q^{10} + (\beta_{6} - \beta_{2}) q^{11} + (\beta_{7} - \beta_{5} + \beta_{4}) q^{12} + (\beta_{7} - \beta_1) q^{13} + (\beta_{7} + \beta_1) q^{15} + (3 \beta_{3} - 2 \beta_{2} + 4) q^{16} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_1) q^{18} + (\beta_{6} - 2 \beta_{3} + 2 \beta_{2}) q^{20} + (\beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_1) q^{22} + ( - \beta_{6} + 2 \beta_{3} - 2 \beta_{2} - 1) q^{24} + 5 q^{25} + ( - 2 \beta_{6} - \beta_{2} + 3) q^{26} + ( - \beta_{7} - 3 \beta_{5} + 3 \beta_1) q^{27} + ( - 2 \beta_{6} + \beta_{2} - 5) q^{30} + (3 \beta_{5} - 3 \beta_{4} + 4 \beta_1) q^{32} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{33} + (2 \beta_{6} - 5 \beta_{3} - \beta_{2} - 1) q^{36} + ( - \beta_{5} - 2 \beta_{4}) q^{37} + ( - \beta_{6} - 4 \beta_{3} - 3 \beta_{2}) q^{39} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_1) q^{40} + ( - 2 \beta_{6} - \beta_{3} + 2 \beta_{2} - 9) q^{44} + (\beta_{6} - 3 \beta_{3} - 3 \beta_{2}) q^{45} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 3 \beta_1) q^{48} + 7 q^{49} + 5 \beta_1 q^{50} + (4 \beta_{5} - \beta_{4} + 3 \beta_1) q^{52} + (3 \beta_{5} - 2 \beta_{4}) q^{53} + ( - \beta_{6} + 3 \beta_{3} + 3 \beta_{2} - 11) q^{54} + ( - 3 \beta_{6} - \beta_{2}) q^{55} + (4 \beta_{5} + \beta_{4} - 5 \beta_1) q^{60} + (3 \beta_{6} - \beta_{2}) q^{61} + (3 \beta_{6} - 6 \beta_{3} + 6 \beta_{2} - 8) q^{64} + (3 \beta_{5} + 2 \beta_{4}) q^{65} + (3 \beta_{6} - 7 \beta_{3} + \beta_{2} + 13) q^{66} + ( - \beta_{7} - 5 \beta_1) q^{67} + ( - 5 \beta_{5} + 2 \beta_{4} - \beta_1) q^{72} + ( - \beta_{6} - 5 \beta_{3} + 4 \beta_{2}) q^{74} + 5 \beta_{5} q^{75} + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} + 6 \beta_1) q^{78} + ( - 2 \beta_{6} + 4 \beta_{3} - 4 \beta_{2} + 15) q^{80} + (3 \beta_{6} + 2 \beta_{3} + 3 \beta_{2} + 9) q^{81} + ( - \beta_{5} + \beta_{4} - 9 \beta_1) q^{88} + (\beta_{7} - 5 \beta_{5} + 6 \beta_1) q^{90} + ( - 2 \beta_{6} - 4 \beta_{3} + 3 \beta_{2} - 15) q^{96} + (5 \beta_{5} - 2 \beta_{4}) q^{97} + 7 \beta_1 q^{98} + ( - 3 \beta_{6} + 8 \beta_{3} + 3 \beta_{2} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 24 q^{9} + 32 q^{16} - 8 q^{24} + 40 q^{25} + 24 q^{26} - 40 q^{30} - 8 q^{36} - 72 q^{44} + 56 q^{49} - 88 q^{54} - 64 q^{64} + 104 q^{66} + 120 q^{80} + 72 q^{81} - 120 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 16x^{6} + 80x^{4} + 128x^{2} + 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 8\nu^{2} + 8 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 11\nu^{3} + 26\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 12\nu^{4} + 34\nu^{2} + 8 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 56\nu^{3} + 60\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{3} - 8\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{5} - 11\beta_{4} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{6} - 36\beta_{3} + 62\beta_{2} - 160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{7} - 42\beta_{5} + 98\beta_{4} - 284\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1084.1
2.79913i
2.26640i
1.69217i
0.406045i
0.406045i
1.69217i
2.26640i
2.79913i
2.79913i 1.12228i −5.83513 2.23607 −3.14142 0 10.7350i 1.74048 6.25904i
1084.2 2.26640i 3.11095i −3.13657 −2.23607 7.05066 0 2.57593i −6.67802 5.06783i
1084.3 1.69217i 1.52380i −0.863428 −2.23607 −2.57853 0 1.92327i 0.678024 3.78380i
1084.4 0.406045i 3.27727i 1.83513 2.23607 −1.33072 0 1.55723i −7.74048 0.907944i
1084.5 0.406045i 3.27727i 1.83513 2.23607 −1.33072 0 1.55723i −7.74048 0.907944i
1084.6 1.69217i 1.52380i −0.863428 −2.23607 −2.57853 0 1.92327i 0.678024 3.78380i
1084.7 2.26640i 3.11095i −3.13657 −2.23607 7.05066 0 2.57593i −6.67802 5.06783i
1084.8 2.79913i 1.12228i −5.83513 2.23607 −3.14142 0 10.7350i 1.74048 6.25904i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1084.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.b.h 8
5.b even 2 1 inner 1805.2.b.h 8
5.c odd 4 2 9025.2.a.cb 8
19.b odd 2 1 inner 1805.2.b.h 8
95.d odd 2 1 CM 1805.2.b.h 8
95.g even 4 2 9025.2.a.cb 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.h 8 1.a even 1 1 trivial
1805.2.b.h 8 5.b even 2 1 inner
1805.2.b.h 8 19.b odd 2 1 inner
1805.2.b.h 8 95.d odd 2 1 CM
9025.2.a.cb 8 5.c odd 4 2
9025.2.a.cb 8 95.g even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1805, [\chi])\):

\( T_{2}^{8} + 16T_{2}^{6} + 80T_{2}^{4} + 128T_{2}^{2} + 19 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 16 T^{6} + 80 T^{4} + 128 T^{2} + \cdots + 19 \) Copy content Toggle raw display
$3$ \( T^{8} + 24 T^{6} + 180 T^{4} + \cdots + 304 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 44 T^{2} + 304)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 104 T^{6} + 3380 T^{4} + \cdots + 109744 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 296 T^{6} + 27380 T^{4} + \cdots + 511024 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 424 T^{6} + \cdots + 30935344 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 244 T^{2} + 304)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 536 T^{6} + \cdots + 43666864 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} + 776 T^{6} + \cdots + 116480944 \) Copy content Toggle raw display
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