Properties

Label 1775.1.c.a
Level $1775$
Weight $1$
Character orbit 1775.c
Analytic conductor $0.886$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -71
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1775 = 5^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1775.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.885840397424\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 71)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.357911.1
Artin image: $C_4\times D_7$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( -1 + \beta_{2} ) q^{4} + ( -1 + \beta_{4} ) q^{6} + ( \beta_{1} - \beta_{3} ) q^{8} + ( -1 + \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( -1 + \beta_{2} ) q^{4} + ( -1 + \beta_{4} ) q^{6} + ( \beta_{1} - \beta_{3} ) q^{8} + ( -1 + \beta_{4} ) q^{9} -\beta_{5} q^{12} + \beta_{4} q^{16} + ( \beta_{3} - \beta_{5} ) q^{18} + \beta_{4} q^{19} - q^{24} + ( \beta_{1} - \beta_{5} ) q^{27} + ( 1 - \beta_{2} - \beta_{4} ) q^{29} -\beta_{5} q^{32} -\beta_{4} q^{36} -\beta_{1} q^{37} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{38} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{43} + ( \beta_{1} - \beta_{5} ) q^{48} - q^{49} + ( 2 - \beta_{2} - \beta_{4} ) q^{54} + ( \beta_{1} - \beta_{5} ) q^{57} + ( -\beta_{1} + \beta_{5} ) q^{58} + q^{71} + \beta_{1} q^{72} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{73} + ( -2 + \beta_{2} ) q^{74} + ( -1 + \beta_{2} - \beta_{4} ) q^{76} + \beta_{2} q^{79} + ( 1 - \beta_{2} - \beta_{4} ) q^{81} + \beta_{1} q^{83} + ( -1 + \beta_{2} ) q^{86} + ( -\beta_{1} + 2 \beta_{5} ) q^{87} + ( 1 - \beta_{2} - \beta_{4} ) q^{89} + ( 1 - \beta_{2} - \beta_{4} ) q^{96} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + O(q^{10}) \) \( 6 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{16} + 2 q^{19} - 6 q^{24} + 2 q^{29} - 2 q^{36} - 6 q^{49} + 8 q^{54} + 6 q^{71} - 10 q^{74} - 6 q^{76} + 2 q^{79} + 2 q^{81} - 4 q^{86} + 2 q^{89} + 2 q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 5 x^{4} + 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 3 \nu^{2} + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + 4 \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 3 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{3} + 9 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1775\mathbb{Z}\right)^\times\).

\(n\) \(427\) \(1001\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1774.1
1.80194i
1.24698i
0.445042i
0.445042i
1.24698i
1.80194i
1.80194i 0.445042i −2.24698 0 0.801938 0 2.24698i 0.801938 0
1774.2 1.24698i 1.80194i −0.554958 0 −2.24698 0 0.554958i −2.24698 0
1774.3 0.445042i 1.24698i 0.801938 0 −0.554958 0 0.801938i −0.554958 0
1774.4 0.445042i 1.24698i 0.801938 0 −0.554958 0 0.801938i −0.554958 0
1774.5 1.24698i 1.80194i −0.554958 0 −2.24698 0 0.554958i −2.24698 0
1774.6 1.80194i 0.445042i −2.24698 0 0.801938 0 2.24698i 0.801938 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1774.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 CM by \(\Q(\sqrt{-71}) \)
5.b even 2 1 inner
355.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1775.1.c.a 6
5.b even 2 1 inner 1775.1.c.a 6
5.c odd 4 1 71.1.b.a 3
5.c odd 4 1 1775.1.d.b 3
15.e even 4 1 639.1.d.a 3
20.e even 4 1 1136.1.h.a 3
35.f even 4 1 3479.1.d.e 3
35.k even 12 2 3479.1.g.d 6
35.l odd 12 2 3479.1.g.e 6
71.b odd 2 1 CM 1775.1.c.a 6
355.c odd 2 1 inner 1775.1.c.a 6
355.e even 4 1 71.1.b.a 3
355.e even 4 1 1775.1.d.b 3
1065.l odd 4 1 639.1.d.a 3
1420.l odd 4 1 1136.1.h.a 3
2485.j odd 4 1 3479.1.d.e 3
2485.be odd 12 2 3479.1.g.d 6
2485.bg even 12 2 3479.1.g.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.1.b.a 3 5.c odd 4 1
71.1.b.a 3 355.e even 4 1
639.1.d.a 3 15.e even 4 1
639.1.d.a 3 1065.l odd 4 1
1136.1.h.a 3 20.e even 4 1
1136.1.h.a 3 1420.l odd 4 1
1775.1.c.a 6 1.a even 1 1 trivial
1775.1.c.a 6 5.b even 2 1 inner
1775.1.c.a 6 71.b odd 2 1 CM
1775.1.c.a 6 355.c odd 2 1 inner
1775.1.d.b 3 5.c odd 4 1
1775.1.d.b 3 355.e even 4 1
3479.1.d.e 3 35.f even 4 1
3479.1.d.e 3 2485.j odd 4 1
3479.1.g.d 6 35.k even 12 2
3479.1.g.d 6 2485.be odd 12 2
3479.1.g.e 6 35.l odd 12 2
3479.1.g.e 6 2485.bg even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1775, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + 5 T^{4} + T^{6} \)
$3$ \( 1 + 6 T^{2} + 5 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( T^{6} \)
$13$ \( T^{6} \)
$17$ \( T^{6} \)
$19$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$23$ \( T^{6} \)
$29$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$31$ \( T^{6} \)
$37$ \( 1 + 6 T^{2} + 5 T^{4} + T^{6} \)
$41$ \( T^{6} \)
$43$ \( 1 + 6 T^{2} + 5 T^{4} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( T^{6} \)
$61$ \( T^{6} \)
$67$ \( T^{6} \)
$71$ \( ( -1 + T )^{6} \)
$73$ \( 1 + 6 T^{2} + 5 T^{4} + T^{6} \)
$79$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$83$ \( 1 + 6 T^{2} + 5 T^{4} + T^{6} \)
$89$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$97$ \( T^{6} \)
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