Properties

Label 1805.2.b.f
Level $1805$
Weight $2$
Character orbit 1805.b
Analytic conductor $14.413$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Coefficient field: 6.0.4227136.1
Defining polynomial: \( x^{6} + 6x^{4} + 7x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{3} + ( - \beta_{3} - \beta_{2}) q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{2} + 2) q^{6} + (2 \beta_{5} + \beta_1) q^{7} + ( - 2 \beta_{5} + \beta_{4}) q^{8} + (2 \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{3} + ( - \beta_{3} - \beta_{2}) q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{2} + 2) q^{6} + (2 \beta_{5} + \beta_1) q^{7} + ( - 2 \beta_{5} + \beta_{4}) q^{8} + (2 \beta_{3} - \beta_{2} - 2) q^{9} + (2 \beta_{5} - \beta_{4} + \beta_1 - 1) q^{10} + \beta_{3} q^{11} + (2 \beta_{5} - 2 \beta_{4} + \beta_1) q^{12} + (\beta_{5} - \beta_{4} + 2 \beta_1) q^{13} + ( - 2 \beta_{3} - 2 \beta_{2} - 3) q^{14} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{15} + (\beta_{3} + 2) q^{16} + (\beta_{5} + 2 \beta_{4} + \beta_1) q^{17} + ( - 2 \beta_{4} - \beta_1) q^{18} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{20} + (\beta_{3} + \beta_{2} + 3) q^{21} + (2 \beta_{5} - \beta_{4} + \beta_1) q^{22} + ( - \beta_{4} - \beta_1) q^{23} + ( - 4 \beta_{3} + 1) q^{24} + ( - 2 \beta_{4} + 2 \beta_{2} + 1) q^{25} + ( - 2 \beta_{3} - \beta_{2} - 3) q^{26} + ( - 2 \beta_{5} + 3 \beta_{4} + 3 \beta_1) q^{27} + ( - 7 \beta_{5} + 2 \beta_{4} - 5 \beta_1) q^{28} + ( - 3 \beta_{3} + 3) q^{29} + (\beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1) q^{30} + (\beta_{2} + 5) q^{31} + (\beta_{4} + 3 \beta_1) q^{32} + ( - \beta_{5} + 2 \beta_{4} + \beta_1) q^{33} + (\beta_{3} - \beta_{2} - 2) q^{34} + (4 \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1) q^{35} + (2 \beta_{3} - 2 \beta_{2} - 3) q^{36} + (2 \beta_{4} - 3 \beta_1) q^{37} + (\beta_{3} + 2 \beta_{2}) q^{39} + ( - 5 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1 - 2) q^{40} + ( - 3 \beta_{3} + 2 \beta_{2} + 3) q^{41} + (7 \beta_{5} - \beta_{4} + 5 \beta_1) q^{42} + ( - \beta_{5} + 4 \beta_1) q^{43} + ( - \beta_{3} - 2 \beta_{2} - 3) q^{44} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1 + 6) q^{45} + ( - \beta_{3} + 1) q^{46} + (4 \beta_{5} + 4 \beta_{4} + \beta_1) q^{47} + ( - 3 \beta_{5} - \beta_1) q^{48} + ( - 4 \beta_{3} - 3 \beta_{2} + 1) q^{49} + (5 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{50} + ( - 4 \beta_{3} + \beta_{2} + 8) q^{51} + ( - 7 \beta_{5} - 2 \beta_1) q^{52} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{53} + (5 \beta_{3} + 2 \beta_{2} - 1) q^{54} + ( - \beta_{4} + \beta_{2} + 3) q^{55} + (5 \beta_{3} + 3 \beta_{2} + 6) q^{56} + ( - 3 \beta_{5} + 3 \beta_{4}) q^{58} + ( - \beta_{3} - 4 \beta_{2} - 3) q^{59} + (6 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 6 \beta_1) q^{60} + (2 \beta_{3} + 2 \beta_{2} - 1) q^{61} + (7 \beta_{5} + 6 \beta_1) q^{62} + (\beta_{5} - \beta_{4} - \beta_1) q^{63} + (3 \beta_{3} + 1) q^{64} + (3 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{65} + (3 \beta_{3} + \beta_{2}) q^{66} + (2 \beta_{5} + 4 \beta_1) q^{67} + 3 \beta_{4} q^{68} + (3 \beta_{3} - \beta_{2} - 4) q^{69} + ( - 2 \beta_{5} - 5 \beta_{3} - 4 \beta_{2} - \beta_1 - 6) q^{70} + ( - \beta_{3} + 5 \beta_{2}) q^{71} + ( - 3 \beta_{5} - 6 \beta_{4} - 5 \beta_1) q^{72} + (2 \beta_{5} - \beta_{4} - 5 \beta_1) q^{73} + (2 \beta_{3} + 3) q^{74} + ( - 3 \beta_{5} - \beta_{4} + 4 \beta_{3} - 5 \beta_1 - 6) q^{75} + (4 \beta_{5} - \beta_{4} + 2 \beta_1) q^{77} + (6 \beta_{5} - \beta_{4} + 3 \beta_1) q^{78} + ( - 4 \beta_{2} - 4) q^{79} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{80} + ( - 5 \beta_{3} + 4) q^{81} + (\beta_{5} + 3 \beta_{4} + 2 \beta_1) q^{82} + (5 \beta_{5} - 2 \beta_{4} + 9 \beta_1) q^{83} + ( - 6 \beta_{3} - 5 \beta_{2} - 6) q^{84} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 1) q^{85} + (\beta_{3} + \beta_{2} - 3) q^{86} + ( - 9 \beta_{4} - 6 \beta_1) q^{87} + ( - 5 \beta_{5} - \beta_{4} - 4 \beta_1) q^{88} + (4 \beta_{2} - 6) q^{89} + (2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 2) q^{90} + ( - 3 \beta_{3} - 4 \beta_{2} - 3) q^{91} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{92} + ( - 6 \beta_{5} - 5 \beta_{4} - 7 \beta_1) q^{93} + ( - 4 \beta_{2} - 5) q^{94} + ( - 5 \beta_{3} + 3 \beta_{2} + 6) q^{96} + (5 \beta_{5} + 7 \beta_{4} + \beta_1) q^{97} + ( - 13 \beta_{5} + 4 \beta_{4} - 6 \beta_1) q^{98} + ( - 3 \beta_{3} + \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} + 2 q^{5} + 12 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{4} + 2 q^{5} + 12 q^{6} - 8 q^{9} - 6 q^{10} + 2 q^{11} - 22 q^{14} + 4 q^{15} + 14 q^{16} - 20 q^{20} + 20 q^{21} - 2 q^{24} + 6 q^{25} - 22 q^{26} + 12 q^{29} + 6 q^{30} + 30 q^{31} - 10 q^{34} - 14 q^{36} + 2 q^{39} - 10 q^{40} + 12 q^{41} - 20 q^{44} + 30 q^{45} + 4 q^{46} - 2 q^{49} - 4 q^{50} + 40 q^{51} + 4 q^{54} + 18 q^{55} + 46 q^{56} - 20 q^{59} - 4 q^{60} - 2 q^{61} + 12 q^{64} - 20 q^{65} + 6 q^{66} - 18 q^{69} - 46 q^{70} - 2 q^{71} + 22 q^{74} - 28 q^{75} - 24 q^{79} + 22 q^{80} + 14 q^{81} - 48 q^{84} - 2 q^{85} - 16 q^{86} - 36 q^{89} + 8 q^{90} - 24 q^{91} - 30 q^{94} + 26 q^{96} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 6x^{4} + 7x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} - 5\nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 6\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} - 5\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{5} - 6\beta_{4} + 12\beta_1 \) 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
0.407132i
1.15904i
2.11917i
2.11917i
1.15904i
0.407132i
2.45620i 1.56104i −4.03293 2.19869 0.407132i 3.83424 4.50527i 4.99330i 0.563139 −1.00000 5.40043i
1084.2 0.862781i 3.07914i 1.25561 −1.91223 1.15904i 2.65662 0.566520i 2.80888i −6.48108 −1.00000 + 1.64984i
1084.3 0.471884i 1.04022i 1.77733 0.713538 2.11917i −0.490864 1.17540i 1.78246i 1.91794 −1.00000 0.336707i
1084.4 0.471884i 1.04022i 1.77733 0.713538 + 2.11917i −0.490864 1.17540i 1.78246i 1.91794 −1.00000 + 0.336707i
1084.5 0.862781i 3.07914i 1.25561 −1.91223 + 1.15904i 2.65662 0.566520i 2.80888i −6.48108 −1.00000 1.64984i
1084.6 2.45620i 1.56104i −4.03293 2.19869 + 0.407132i 3.83424 4.50527i 4.99330i 0.563139 −1.00000 + 5.40043i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1084.6
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 1805.2.b.f 6
5.b even 2 1 inner 1805.2.b.f 6
5.c odd 4 2 9025.2.a.bu 6
19.b odd 2 1 1805.2.b.g 6
19.c even 3 2 95.2.i.b 12
57.h odd 6 2 855.2.be.d 12
95.d odd 2 1 1805.2.b.g 6
95.g even 4 2 9025.2.a.bt 6
95.i even 6 2 95.2.i.b 12
95.m odd 12 4 475.2.e.g 12
285.n odd 6 2 855.2.be.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.b 12 19.c even 3 2
95.2.i.b 12 95.i even 6 2
475.2.e.g 12 95.m odd 12 4
855.2.be.d 12 57.h odd 6 2
855.2.be.d 12 285.n odd 6 2
1805.2.b.f 6 1.a even 1 1 trivial
1805.2.b.f 6 5.b even 2 1 inner
1805.2.b.g 6 19.b odd 2 1
1805.2.b.g 6 95.d odd 2 1
9025.2.a.bt 6 95.g even 4 2
9025.2.a.bu 6 5.c odd 4 2

Hecke kernels

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

\( T_{2}^{6} + 7T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{29}^{3} - 6T_{29}^{2} - 27T_{29} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 7 T^{4} + 6 T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 13 T^{4} + 36 T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} - T^{4} + 4 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 22 T^{4} + 35 T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 4 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 31 T^{4} + 239 T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{6} + 35 T^{4} + 198 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{4} + 7 T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{3} - 6 T^{2} - 27 T + 27)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 15 T^{2} + 70 T - 97)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 98 T^{4} + 887 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} - 41 T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 127 T^{4} + 2516 T^{2} + \cdots + 441 \) Copy content Toggle raw display
$47$ \( T^{6} + 214 T^{4} + 13407 T^{2} + \cdots + 218089 \) Copy content Toggle raw display
$53$ \( T^{6} + 99 T^{4} + 2302 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$59$ \( (T^{3} + 10 T^{2} - 55 T - 291)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + T^{2} - 41 T - 113)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 76 T^{4} + 1856 T^{2} + \cdots + 14400 \) Copy content Toggle raw display
$71$ \( (T^{3} + T^{2} - 124 T + 477)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 236 T^{4} + 13331 T^{2} + \cdots + 99225 \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} - 32 T - 448)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 459 T^{4} + 56302 T^{2} + \cdots + 966289 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} + 28 T - 72)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 529 T^{4} + 74192 T^{2} + \cdots + 1238769 \) Copy content Toggle raw display
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